phase behaviour of polymer blends

Preface

The observation that blending existing polymers can produce new and im-proved materials with the beneficial features of each component has focusedattention on understanding the phase behavior of polymer blends because theresulting composites may form homogeneous phases or may be phase sep-arated with a particular desirable morphology. While Flory–Huggins theorycorrectly predicts that liquid-state polymers generally do not tend to mix, itspredictive ability for designing these new materials are rather limited. Thus,in order to rectify this deficiency in the theory, several approaches have beenused to develop more ambitious theories, while experimental efforts have beendevoted to devising various strategies for improving mixing including, for in-stance, the use of flow, as described in the article by Clarke in this volume. Thus,progress in elucidating the factors affecting and promoting blend miscibilityhas benefited from strong interactions between experimental and theoreticalgroups, as emphasized by the articles herein.

The subject of the phase behavior of polymer blends is quite vast and muchtoo large to fit in a single volume of Advances in Polymer Science, so thepresent volume is the first in a planned series. The lead article in this vol-ume by Schwahn describes his small angle neutron scattering experimentsfor a variety of polymer blend systems, including homopolymer and copoly-mer blends, as well as the fascinating mixtures of diblock copolymers withhomopolymers and the technologically important influences of pressure andadditives on blend phase behavior. Schwahn’s article focuses on the importantinfluences of critical fluctuations on the phase behavior and emphasizes theneed for analyzing experimental data accordingly in order to provide meaning-ful comparisons between experiments and theories of polymer blends, whichare generally of the mean field variety. The second article by Freed and Du-dowicz is devoted to a description of several applications of the lattice clustertheory (LCT) in the simplified high molecular weight, incompressible limitto demonstrate how monomer molecular structure strongly affects the phasebehavior of polymer blends. Examples of LCT predictions that can not be ob-tained from Flory–Huggins theory involve, for instance, the possibility of lowercritical solution temperature phase behavior for incompressible binary blendsand microphase ordering of diblock copolymers upon heating. When appliedto copolymer systems, LCT provides a significant extension of Flory–Huggins

X Preface

type theories for copolymers by incorporating the important contributionsfrom the temperature-independent portion of the Flory effective interactionparameter. Several direct comparisons of the LCT to experiments serves toillustrate the utility of this simple approach to interpret measurements andto elucidate the physics governing blend miscibility. The last article by Clarkedescribes both theoretical methods and experimental observations of the in-fluence of shear flow on the properties of polymer blends. Comparisons aremade between the predictions of two conflicting theories: a quasiequilibriumtheory in which a shear-dependent contribution is appended to the free en-ergy, and a theory based on the modification of the equations of motion by thepresence of the shear flow. Clarke discusses several of the challenges towardsfurther theoretical progress.

Chicago, June 2005 Karl. F. Freed

Contents

Critical to Mean Field Crossover in Polymer BlendsD. Schwahn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Influence of Monomer Molecular Structureon the Miscibility of Polymer BlendsK. F. Freed · J. Dudowicz . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Effect of Shear Flow on Polymer BlendsN. Clarke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Author Index Volumes 101–183 . . . . . . . . . . . . . . . . . . . . . . 175

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Adv Polym Sci (2005) 183: 1–61DOI 10.1007/b135882© Springer-Verlag Berlin Heidelberg 2005Published online: 26 July 2005

Critical to Mean Field Crossover in Polymer Blends

Dietmar Schwahn

Institut für Festkörperforschung, Forschungszentrum Jülich GmbH, 52425 Jülich,[email protected]

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Small Angle Neutron Scattering . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Experimental Design of a SANS Diffractometer . . . . . . . . . . . . . . . 92.2 Scattering Cross Section of an Ideal Polymer Mixture . . . . . . . . . . . . 112.3 Neutron Scattering Contrast . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Thermodynamic Model for Polymer Blends . . . . . . . . . . . . . . . . . 153.1 Flory–Huggins Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Random Phase Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Determination of a Polymer Blend Phase Diagram by SANS . . . . . . . . 18

4 Mean Field to 3D-Ising Crossover in Polymer Blends . . . . . . . . . . . . 214.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Binary Homopolymer Blends . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Binary Blends of Statistical Copolymers . . . . . . . . . . . . . . . . . . . . 274.3.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.4 Polymer Blends in an External Pressure Field . . . . . . . . . . . . . . . . . 324.4.1 SANS Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4.2 Clausius–Clapeyron Equation . . . . . . . . . . . . . . . . . . . . . . . . . 354.5 Binary Blends with Small Additions of a Non-Selective Solvent . . . . . . . 384.5.1 Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.5.2 Susceptibility and Correlation Length . . . . . . . . . . . . . . . . . . . . . 394.5.3 Ginzburg Number and Critical Amplitudes . . . . . . . . . . . . . . . . . . 39

5 Crossover to the Renormalized 3D-Ising Critical Behavior . . . . . . . . . 425.1 Hidden Variables – Fisher Renormalization . . . . . . . . . . . . . . . . . . 435.2 SANS Results on Blends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.3 Renormalized 3D-Ising Critical Behavior

in Blends and Blend-Solvent Systems . . . . . . . . . . . . . . . . . . . . . 45

6 Crossover to Isotropic Lifshitz Critical Behaviorin (A/B) Polymer Blend/(A-B) Diblock Copolymer Mixtures . . . . . . . . 46

6.1 Phase Diagramof a (A/B) Polymer Blend/(A-B) Diblock Copolymer Mixture . . . . . . . 47

6.2 Structure Factor within Mean Field Approximation . . . . . . . . . . . . . 49

2 D. Schwahn

6.3 Effect of Thermal Composition Fluctuations . . . . . . . . . . . . . . . . . 516.4 SANS Results from below the Lifshitz Line . . . . . . . . . . . . . . . . . . 52

7 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Abstract Crossover phenomena due to thermal composition fluctuations play a multifar-ious role in polymer blends. This is demonstrated in this article by describing resultsfrom small angle scattering experiments in particular with neutrons. Scattering methodsare a direct tool to measure the strength and correlation length of composition fluctua-tions. We will review the effects of thermal fluctuations in binary (A/B) polymer blendsunder various conditions of external temperatures and pressures and additives, such asnon-selective solvents and (A-B) diblock copolymers, and will give an interpretation withthe corresponding crossover theories. General conclusions are that the effects from ther-mal composition fluctuations have to be more seriously considered in polymer blendsand that the more sophisticated crossover theories are needed for a precise determinationof the Flory–Huggins interaction parameter and the phase boundaries. In addition, wediscuss observations of crossover to other universality classes such as the 3D-Ising case,namely the transition to the renormalized Ising case when the composition of a thirdcomponent starts to fluctuate and to the isotropic Lifshitz critical behavior when an (A-B)diblock copolymer is added.

Keywords Polymer blends · Crossover by thermal composition fluctuations ·3D-Ising critical behavior · Renormalized Ising critical behavior ·Lifshitz critical behavior · Ginzburg criterion · Flory–Huggins parameter ·External fields of temperature and pressure ·Additives of non-selective solvent and diblock copolymers

1Introduction

Polymer blends represent materials for a large class of industrial productsand are the subject of intensive academic research. Blending of chemicallydifferent polymers is an important tool in industrial production for tailoringproducts with optimized material properties [1]. On the other hand polymermixtures are model systems in statistical physics for studying fundamentalaspects of equilibrium and non-equilibrium properties such as phase dia-grams, thermal composition fluctuations far and near the critical point ofa second order phase transition, conformational properties of the chains, thekinetics of phase transitions as well as the detailed dynamics of diffusionprocesses. There are strong activities in this field from both theory and ex-periment as can be realized from numerous review articles [2–5].

In this article we will focus on the behavior of thermal composition fluctu-ations in binary (A/B) homopolymer blends (A, B denote the monomers) indifferent external fields such as temperature and pressure, in their pure state,

Critical to Mean Field Crossover in Polymer Blends 3

as well as with small additions of a non-selective solvent and symmetric (A-B)diblock copolymers. The external field conditions as well as additives canstrongly influence the degree of thermal fluctuations and lead to crossoverphenomena between different universality classes of mean field, 3D-Ising,renormalized 3D-Ising, and isotropic Lifshitz critical behavior. Each univer-sality class is characterized by a unique set of critical exponents describingthe divergence of several thermodynamic parameters in the near vicinity ofa second order phase transition. Two of those prominent parameters, namely,the correlation length of thermal composition fluctuations and the corres-ponding susceptibility are determined in scattering experiments. Generally,scattering methods are a sensitive tool to measure spatial inhomogeneities,such as thermal composition fluctuations in blends, and neutrons have theparticularly important advantages of strong scattering contrast and deep pen-etration into materials [6, 7]. So, it is quite natural that most of the relevantstudies in this field have been performed with neutron small angle scattering(SANS) techniques.

Thermal composition fluctuations are always present in multi-componentsystems. If the interaction energy between same components is stronger, clus-tering of (A/A) and (B/B) monomers is preferred, and phase separation willoccur at low temperatures. A schematic phase diagram of a symmetric bi-nary polymer blend is depicted in Fig. 1. It represents an “upper criticalsolution temperature” (UCST) system; at high temperatures the blend is ho-mogeneously mixed, while at low temperatures the system decomposes intomacroscopically large domains being rich in A or B polymers. Such a pro-

Fig. 1 Schematic phase diagram of binary polymer melt of equal molar volume VA = VB.At high temperature both polymers are miscible; at low temperature below the binodalthe sample separates into two macroscopic large domains with compositions determinedby the binodal. The spinodal separates metastable and unstable regions which determinethe process of phase separation. The disordered regime is separated by the Ginzburgnumber Gi domain into regions of small and large degrees of thermal composition fluc-tuations

4 D. Schwahn

cess can be easily followed by eye as the sample becomes turbid from thestrong scattering of light by the µm large domains. The binodal is the bor-der line between the one and two-phase regimes, while the spinodal separatesthe meta-stable and unstable parts of the two-phase regime. The binodal andspinodal meet at the critical point which also represents the highest tem-perature of immiscibility in monodisperse polymer systems [8]. In symmetricblends with equal molar volumes VA = VB theory predicts a critical compo-sition of ΦC = 50% volume fraction. The one-phase regime is homogeneouson a macroscopic length scale and usually appears transparent to the eye. Onsmaller length scales, however, the composition is heterogeneous due to ther-mal fluctuations. The occurrence of thermal fluctuations is a dynamic processas they are created with a given probability and decay afterwards with a re-laxation rate determined by the interdiffusion constant [3, 4]. In static elasticneutron scattering experiments one measures the equilibrium mean squaredeviation 〈δΦ2〉 from the average composition Φ given by the volume frac-tion of one of the components. The range of those fluctuations is determinedby the correlation length ξ which sensitively depends on external parameterssuch as temperature and pressure: ξ is of the order of interatomic distances athigh temperatures when entropy is dominating; it is large at low temperaturesand becomes infinite at the critical point. The critical point represents a par-ticular position of the phase boundary where a continuous (second order)phase transition is observed.

According to the fluctuation–dissipation theorem the equilibrium meansquare deviation of thermal composition fluctuations 〈δΦ2〉 is related to thefirst derivative of the order parameter with respect to the chemical poten-tial [3, 4]. If the order parameter is defined as the composition Φ of thecomponent “A” then the conjugate field is represented by the difference ofthe chemical potentials ∆µ(= µA – µB); so the degree of thermal fluctuationsis related to ∂∆µΦ(∂∆µ ≡ ∂/∂∆µ) which is equivalent to 1/∂2

Φ∆G, where∆G represents the Gibbs free energy of mixing with its natural parameterstemperature T, pressure P, and composition Φ. This shows how the ther-modynamic parameters can be determined from measurements of thermalfluctuations.

The border line between strong and weak thermal fluctuations is esti-mated by the Ginzburg number Gi := (1 – TC/TX), which is evaluated from theGinzburg criterion [3, 4] and which represents a reduced temperature TX be-low which deviations from the mean field approach are observed (see hatchedarea in Fig. 1). Fluctuations are considered to be weak as long as the fluctu-ation modes superimpose linearly. In this weak limit the fluctuation modescan be described within the Gaussian approximation, and mean field theoryis a good approximation. Near the critical point thermal fluctuations becomestrong and lead to visible non-linear effects. In this range more sophisticatedtheories as the 3D-Ising model and crossover theories are needed [9, 10].


Fluctuation effects are usually neglected in theoretical descriptions ofpolymer blends which are therefore described within the mean field the-oretical approach of the Flory–Huggins (FH) theory [2–4]. In this modelthe Gibbs free energy of mixing ∆G is depicted by a combinatorial en-tropy of mixing being inversely proportional to the molar volume V andby the FH parameter Γ = Γh/T – Γσ being described by enthalpic and non-combinatorial entropic terms Γh and Γσ , respectively. The general neglect ofthermal fluctuations in polymer blends might be explained with the originalestimate of the Ginzburg criterion by deGennes on the basis of incompress-ible FH theory. This relationship represents a universal criterion accordingto Gi ∝ 1/N (N degree of polymerization) and proposes an extremely smallGinzburg number [3]. Since in low molecular liquids a Gi ∼= 10–2 is the ex-pected value, a roughly N times smaller Gi is estimated in polymer blendsindicating a small region with strong thermal fluctuations [10]. Meanwhile,it has been shown that the Ginzburg criterion for polymer blends is non-universal with appreciably larger Ginzburg numbers and therefore larger re-gions where thermal fluctuations are significant. Theoretical considerations,computer simulations, and scattering experiments show that the compress-ibility is dependent on the non-combinatorial entropy Γσ which has a stronginfluence on the Ginzburg criterion and thereby on the degree of thermalcomposition fluctuations. The effect of thermal fluctuations will be the mainemphasis of this article.

The influence of several chain parameters on Γσ and therefore on Gi hasbeen identified. These are polymer asymmetry, different monomer struc-tures, chain stiffness, compressibility, and chain end-effects. The effects ofcompressibility and monomer structure have been discussed by Lifshitzet al. [11] and Dudowicz et al. [12] on basis of the Lattice Cluster The-ory (LCT), respectively. Considering these effects a one to several orders ofmagnitude increase of Gi is found. The present theoretical ideas behind an ex-tended Ginzburg criterion have in common the consideration of correlations(or non-random mixing) on length scales between the coil and monomer size.These shorter length scale fluctuations lead to an extended mean field de-scription and are comprised into a non-combinatorial entropic term Γσ of theFH parameter. It will be shown later (Eq. 18) in the context of the crossovertheories for the susceptibility and correlation length that Γσ strongly influ-ences the Ginzburg number and thereby the strength of thermal fluctuationsand the domain over which they must be considered in the analysis of experi-mental data.

Another theoretical approach including the effects of thermal fluctua-tions has been derived from field-theoretical methods (see recent review byFredrickson et al. [13]). Wang [14] quite recently gave a compact overviewand a derivation from a systematic renormalization procedure of an effectiveFH parameter and Ginzburg criterion. Starting from a “bare” FH parameteras originally derived by Flory, which only considers the microscopic enthalpic

6 D. Schwahn

interaction, Wang formulated an “effective” FH parameter comprising the ef-fects of local correlations of wavelengths between monomer and chain sizes.The last mentioned parameter is determined from SANS experiments withina mean field approximation, is comparable with the FH parameter derivedfrom the LC-theory, and also includes molecular conformational asymmetryeffects as discussed in [15]. In addition, an “apparent” FH parameter is deter-mined which also includes contributions from long wavelength fluctuationsthat become relevant near the critical point. Fluctuations lead to a reduced ef-fective FH parameter, to a decrease of the critical temperature, and to a largerGi. In the limit of large degree of polymerization, a 1/N scaling of Gi is pre-dicted.

Monte Carlo simulations very early demonstrated the effect of ther-mal composition fluctuations in low molecular blends. Studies by Saribanet al. [16] exclusively found Ising critical behavior in blends of molar volumeup to about 16 000 cm3/mol and no indications of a crossover to mean fieldbehavior. Such a mean field crossover was later detected by Deutsch et al. [17]in blends with an order of magnitude larger chains. These results and thetechniques of Monte Carlo simulations have been extensively reviewed byBinder in [4].

These theoretical considerations and Monte Carlo simulations were ac-companied by scattering experiments, which consistently show that the rangeof relevant thermal fluctuations and thereby Gi is at least of the order ofmagnitude larger than originally estimated from simple incompressible FHtheory [18–26]. An important development in the description of thermalfluctuations is the derivation of proper analytic crossover functions for thesusceptibility and correlation length which also can be relatively easily han-dled by the experimentalist [10, 27–30]. These equations allow an analysis ofthe susceptibility and correlation length over the whole experimental rangeincluding mean field and Ising behavior and therefore lead to much moreprecisely determined thermodynamic parameters. A Ginzburg criterion isquite naturally derived from these crossover functions and is determinedby the ratio of the mean field and Ising critical amplitudes for the suscep-tibility and/or correlation length. The so determined Ginzburg number isa non-universal function, which sensitively depends on the degree of poly-merization and on the FH parameter non-combinatorial entropy of mixingΓσ [31, 32]. An increase of Γσ leads to a strong increase of Gi, and only in thelimit of negligible Γσ is a universal Ginzburg criterion with the proposed 1/Nscaling behavior obtained consistent with earlier predictions for incompress-ible polymer blends.

High external pressure fields can lead to an appreciable change of theFH interaction parameter, the phase boundaries, and the Ginzburg number.Pressure usually induces an increase of the phase transition temperature,a decrease of the Ginzburg parameter and of the enthalpic (Γh) and abso-lute value of the entropic (Γσ ) terms of the FH parameter. An increase of


the phase boundary with pressure is expected as it is related to a decreaseof the free volume and a corresponding decrease of the entropic Γσ [33–41].In LCST systems (phase transition occurs at high temperatures) such as thePS/PVME (polystyrene/poly(vinyl methyl ether) blend Γh is constant within1 and 120 MPa while Γσ is negative and increases with pressure [34]. Ina few blends, however, an “abnormal” pressure induced decrease of the phaseboundary and a corresponding increase of the Γh and Γσ FH terms are ob-served [42, 43]. The shift of the phase boundary with pressure is describedby the Clausius–Clapeyron equation in terms of the FH parameter and theGinzburg number [40]. In some blends one actually finds a shift of the criticaltemperature dominated by the Ginzburg number according to ∂PGi [40]. Theexploitation of the Clausius–Clapeyron equation also allows a check of con-sistency with respect to the underlying theory, such as the dependence of theGinzburg criterion on the entropic FH term Γσ . In this respect, the study ofpressure induced changes of thermal fluctuations may lead to relevant insightinto the mechanisms governing the polymer blend thermodynamic proper-ties and their interrelations.

Mixing a binary polymer blend with a small amount of a third com-ponent usually leads to strong changes of the phase behavior and in somecases even to a crossover to a different universality class. The addition ofa non-selective solvent generally leads to improved compatibilization and toa larger Ginzburg number. Furthermore, very near the critical temperature,a crossover from Ising to the renormalized Ising behavior is predicted forthose blend-solvent systems. Such crossover phenomena have been inten-sively explored by the group of Nose [44, 45] with light scattering. They foundthis type of crossover very pronounced after adding a selective solvent butonly rather weak in a few special cases after adding a non-selective solvent.These observations are consistent with the SANS studies discussed in thisarticle.

A much more complex situation is achieved if an (A-B) diblock copoly-mer is mixed with a binary (A/B) homopolymer blend. Generally the diblockcopolymer leads to a better compatibilization of the homopolymers similar tothe action of surfactant molecules in oil-water mixtures. A complex phase di-agram is obtained that depends upon the diblock content: Several disorderedand ordered phases appear at, respectively, high and low temperatures. Atlow and high diblock content a two-phase region of macroscopically large do-mains and an ordered phase known from diblock copolymer melts emerge,respectively, and intermediate diblock content leads to droplet and bicon-tinuous microemulsion phases [46–52]. Bicontinuous microemulsion andordered phases at high diblock copolymer concentrations are characterizedby a periodicity length Λ which in scattering experiments becomes visibleas a peak at Q∗ = 2π/Λ. Within the disordered phases several crossover phe-nomena are observed, namely, from mean field to 3D-Ising and to isotropicLifshitz critical behavior, as well as to the Brasovskii type of pure diblock

8 D. Schwahn

copolymers [53]. One reason for this complex behavior is that homopoly-mer mixtures and diblock copolymers belong to different universality classes.Diblock copolymers shows much stronger thermal fluctuations which evenchange the character of the disorder-order phase transition from second tofirst order and lead to a broader critical range with a weaker N proportion-ality of Gi according to Gi ∝ 1/

√N [54].

Mean field theory predicts that the critical lines of “blend like” and “di-block like” behavior meet at the isotropic critical Lifshitz point and theLifshitz line (LL) which is defined when Q∗ becomes zero. The isotropic crit-ical Lifshitz point represents a new universality class [54–56]. Under specialconditions even a tricritical Lifshitz point is predicted [55]. In this article wewill discuss in some detail SANS experiments on a mixture of a critical bi-nary (A/B) polymer blend with different concentrations of a symmetric (A-B)diblock copolymer of roughly five times larger molar volume. Under suchconditions an isotropic critical Lifshitz point is predicted [55].

Near the critical Lifshitz point the phase behavior and the correspondingphase diagram are strongly influenced by thermal composition fluctuations.The strength of these fluctuations can be understood from a decrease of thesurface energy acting as a restoring force for thermal fluctuations. A furtherpeculiarity is, that a critical Lifshitz point can only exist within mean field ap-proximation. It is “destroyed” by thermal fluctuations because the Ginzburgcriterion and therefore the stabilization of the miscible phase is of differentstrengths on both sides of the Lifshitz line. A further observation is a changeof the LL concentration with temperature near the two-phase regime [48, 49];recent renormalization group calculations explain these observations as dueto thermal fluctuations [57]. In the range near the LL one also observes a mi-croemulsion phase not predicted by the corresponding existing mean fieldtheories [47].

From these introductory remarks one already gets the impression aboutthe prominent and diversified role of thermal composition fluctuations on theproperties of polymer blends, which will be detailed in the following sectionsmainly from an experimental point of view.

2Small Angle Neutron Scattering

A separate section is devoted to describing small angle neutron scattering(SANS) techniques as this experimental method plays a prominent role inthe investigation of polymer blends. Neutrons are a particularly appropriateprobe for explorations of polymer properties because of the good scatteringcontrast conditions and their weak interactions with material in general [6, 7].The last mentioned condition relies on the fact that neutrons have no charge


and thus allow the exploration of several mm thick samples, making SANSa non-destructive tool. The above mentioned scattering contrast is propor-tional to the difference of the coherent scattering length density of the poly-mer components and is mainly determined from the neutron interaction withthe atomic nuclei. This interaction can lead to appreciably different scatteringlengths for isotopes as most prominently found for hydrogen and deuterium.Consequently, a mixture of deuterated and protonated polymer chains showsa strong scattering contrast and thereby makes neutrons a sensitive tool. Be-yond that the contrast variation with H/D content of selected polymers offersdetailed insight into single chain properties in complex systems as the abovementioned (A/B/A-B) polymer blend-diblock copolymer mixtures.

2.1Experimental Design of a SANS Diffractometer

A schematic lay-out and photography of the SANS experiments at the re-search reactor FRJ2 in Jülich are given in Fig. 2a and b. Neutrons are releasedin nuclear reactors by fission reactions of the uranium isotope 235 or areevaporated in spallation sources from a heavy metal target after being bom-barded with high energy protons. For example about 20 neutrons are releasedfrom a lead nucleus per 1 GeV proton [7]. After being released neutrons aremoderated to lower kinetic energies. For SANS and other neutron scatter-ing methods as spin-echo and back scattering spectrometers a special “coldsource” moderator of liquid hydrogen or deuterium is installed near the reac-tor core which delivers more than one order of magnitude larger intensity oflong wavelength neutrons between 7 and 15 A. Neutrons of this wavelengthapproach a smaller momentum transfer Q, as will be defined in Eq. 1, andavoid double Bragg scattering in crystalline materials. These “cold” neutronsare guided to the instrument through neutron guides that operate by totalreflection. Neutron guides consist of evacuated glass channels with good sur-face quality and are coated with an element having a large coherent scatteringlength such as the Nickel 58 isotope.

The neutrons entering the instrument first pass a velocity selector, thena collimator with two apertures, a sample, and are finally detected in a pos-ition sensitive detector if scattered into a given angular interval. The selectorand collimator, respectively, determine the wavelength with a relative distri-bution of typically ∆λ/λ ∼= 0.10 and the divergence of the neutron beam. Inorder to keep multiple scattering low, the sample should scatter only part(ca. 10%) of the incident neutrons (intensity I0) whereas the non-scatteredneutrons (intensity I∗

0 ) are absorbed in a beam stopper in front of the detec-tor. The distances of sample to detector as well as of the two apertures canbe changed typically from 1 to 20 m allowing the measurement of scatteringangles from roughly 0.1 to 20 and to the adjustment of an optimized re-lationship between maximum intensity and sufficiently good resolution. In

10 D. Schwahn

Fig. 2 Top: Schematic lay-out of a pin hole small angle neutron diffractometer (SANS).Bottom: Photography of one of the two SANS instruments at the research reactor FRJ-2at Jülich. Vacuum sample chamber and detector tank are visible. Both Jülich SANS in-struments have been in operation since 1987 and have up until now been seen worldwideto the most powerful instruments after the two SANS instruments at the ILL in Grenoble(France)

order to gain optimized intensity conditions, additional neutron guides canbe inserted into the neutron beam axis so that neutrons are always trans-ported through neutron guides until the first aperture. The scattered intensityis usually given as a function of the momentum transfer Q whose absolutevalue Q is determined according to

Q = (4π/λ) sin(Θ/2) (1)

Q is inversely proportional to the neutron wave length λ and is proportionalto the scattering angle Θ. The typical wavelengths between 5 and 10 A lead toa Q range of 10–3 < Q[A–1] < 0.3 and correspond in real space to a resolutionrange between 10 < R[A] < 103. The Q range can be extended to smaller Q byother SANS techniques using a focusing mirror optic or perfect Silicon singlecrystals as monochromator and analysor [58].

The neutron intensity ID(Q) scattered by the momentum transfer Q is de-scribed in Eq. 2; it is proportional to the incident intensity I0, the samplethickness D, the sample transmission T = I∗

0 /I0, and the space angle ∆ΩD of


a detector element,

ID(Q) = I0DTdΣ

dΩ(Q)∆ΩD (2)

The transmission describes the attenuation of the neutrons non-scattered bythe sample, and the macroscopic cross section dΣ/dΩ in units of cm–1 is thequantity to be determined. The calibration of the macroscopic cross sectiondΣ/dΩ in absolute units is usually performed by an additional measurementwith a sample of known dΣ/dΩ or by measuring the direct beam in bothcases with the same collimator setting [59, 60].

2.2Scattering Cross Section of an Ideal Polymer Mixture

Blending two polymers with zero energy of mixing (FH parameter χ = 0)leads to an ideal mixture. Such ideal mixtures do not show any phase behaviorand can be realized within good approximation by blending protonated anddeuterated polymers of the same species of not too large molar volumes V. Inorder to determine the structure factor S(Q) of such a melt, we consider thechains on a so-called “Flory” lattice. All lattice points are of the same size andare occupied by the monomers of the two A/B polymers. The basic equationfor the macroscopic cross section in SANS within first Born approximation isgiven according to [6, 7]

dΣ/dΩ(Q) = (1/VS)∣∣∣

∑

bi exp(iQri)∣∣∣

2(3)

The scattering amplitude is determined from the sum of the coherent scatter-ing length, bi of the monomer at position i over all lattice points multipliedwith the corresponding phase factor exp(iQri). The square of the scatteringamplitude divided by the sample volume VS gives dΣ/dΩ(Q). From now onthe momentum transfer will be regarded as a scalar as we always will discussisotropically scattering samples.

To proceed, we define a variable σi with the following meaning,

σi =

1 bi = bA

0 bi = bB(4)

which determines the occupation of the Flory lattice by the monomers ofthe polymers A and B as represented by their coherent scattering length.The scattering length at position i is then given as bi = σi(bA – bB) + bB = σi ·∆b + bB and the cross section becomes

dΣ

dΩ

(

Q)

=∆b2

VS

N∑

ij

〈σiσj exp(iQrij)〉 = K ·S(Q) (5)

12 D. Schwahn

With the contrast factor K = ∆ρ2/NA determined from the difference of thecoherent scattering length density ρA,B =

∑bj/Ω of both monomers and

Avogadro’s number NA· (bj is the coherent scattering length of atom j andΩ the volume of both a lattice site and a monomer). The definition of Eq. 4implies an incompressible melt as no free volume has been considered. Thestructure factor S(Q) represents an average over all chain conformations asindicated by the brackets in Eq. 5; it is defined in units of a molar volume[cm3/mol] and is identical to the partial structure factor SAA(Q) determinedby intra- and intermolecular correlation between the monomers of poly-mer A. A separation of SAA(Q) into intra- P(Q) and intermolecular W(Q)interference gives

S(Q) = SAA(Q) = ΦVP [P(

Q)

+ Φ nAW(

Q)

] (6)

with the polymer molar volume VP and the volume fraction Φ of poly-mer A [6]. The structure factor can be simplified to

S(

Q)

= Φ(

1 – Φ)

VP PDebye(

Q)

(7)

which means that S(Q) is determined from the polymer form factor P(Q)alone. This can be understood from the Babinet principle and more easilyfrom the following argument. When Φ = 1, one has a pure melt of A poly-mers which in the incompressible limit does not scatter, e.g. S(Q) = 0 in Eq. 6.Therefore P

(

Q)

= – nAW(Q) must hold. Moreover, the Debye form factorPDebye(Q) = 2[x – 1 + exp(– x)]/x2 with x = R2

gQ2 (Rg radius of gyration) de-scribes the conformation of polymers in melts to a good approximation as theexcluded volume interaction is screened out by other chains and the confor-mation becomes that of a freely jointed chain [3, 4].

The structure factor in Eq. 7 has a maximum at Q = 0 whose magnitudeis determined by the polymer molar volume VP and the composition Φ ofthe A/B components. The shape of S(Q) is determined by the radius of gyra-tion Rg . In analyzing experimental data one often prefers approximate forms,namely P –1

Debye∼= 1 + (Rg Q)2/3 (Zimm approximation) and PDebye ∝ 1/(Q ·

Rg)2 that are valid at Q < 1/Rg and Q > 1/Rg, respectively. For Q larger than1/Rg one realizes a scaling law with the so-called fractal dimension D = 2. Anexperimental verification is demonstrated in Fig. 3 showing a 50% isotopicmixture of protonated and deuterated polystyrene [61]. The double logarith-mic presentation of the data shows a – 2 power law over a large Q intervalwhich corresponds to the expected dimension being equivalent to the expo-nent ν = 1/2 of the freely jointed chain.

Two more comments have to be given: (1) We neglected the monomerstructure (Pm(Q) = 1) as its size is too small for the available Q range ofSANS. (2) In dΣ/dΩ of Eq. 5 a constant contribution from all monomers,namely bi = bB was neglected as their integration over the sample volumegives a δ(Q) like contribution to the scattering. This scattering from an order


Fig. 3 Structure factor S(Q) versus momentum transfer Q of a 48% isotopic mixture ofpolystyrene in double logarithmic representation. The Q–2 power law at large Q showsthat the polymer conformation in melts is that of Gaussian freely jointed chains

of cm large sample would be visible in a Q range of the order of 10–8A–1. Thesmallest available Q with neutrons is of the order of 10–5A–1 by the doublecrystal diffractometer technique [58]. This means that this δ(Q) contributioncan safely be ignored.

2.3Neutron Scattering Contrast

The contrast factor K = (ρA – ρB)2/NA = ∆ρ2/NA in Eq. 5 describes the in-teraction between the neutrons and the atoms within the sample which mustbe sufficiently large for a strong scattering signal. The coherent scatteringlengths had to be determined experimentally, and their values can be foundin several publications, e.g., in ref. [62]. In polymer research the very dif-ferent scattering lengths of deuterium and hydrogen, being bH = – 3.739 andbD = 6.671 in units of 10–13 cm, are of special importance. This leads to strongdifferences of the scattering length densities of deuterated and protonatedpolymers as listed in Table 1. Mixing two (A/B) polymers their ∆ρ2 canbe nearly zero when both components are deuterated or protonated as forPB/PVE or d-PB/d-PI, and ∆ρ2 is usually very large if only one of the com-ponents is deuterated. A further advantage of choosing deuterated polymer isthe much lower “background” from incoherent scattering in comparison withthe protonated ones.

Another possibility is to adjust the coherent scattering lengths of twochemically different polymers in order to make them indistinguishable forSANS studies. In this way individual polymer chains in more complex mix-tures can be explored. An example may be polystyrene PS and polybutadiene

14 D. Schwahn

Tabl

e1

Coh

eren

tsc

atte

ring

leng

thof

seve

ralp

olym

ers

and

the

corr

espo

ndin

gco

ntra

stfa

ctor

betw

een

them

∆ρ

2 [1020

cm–4

]→

Syst

emρ

PBd-

PBPV

EdP

VE

PIdP

IPS

dPS

PVM

E[1

010cm

–2]

PB0.

416

040

.30

38.7

0.02

43.2

136

.80.

004

dPB

6.76

040

.30.

014

42.1

0.05

328

.50.

0841

PVE

0.41

4–

038

.80.

021

43.2

136

.80.

004

dPV

E6.

64–

–0

40.6

0.12

227

.20.

026

39.6

PI0.

27–

––

031

1.32

38.6

0.00

6dP

I6.

99–

––

–0

310.

2644

PS1.

42–

––

––

025

.61.

14dP

S6.

48–

––

––

–0

37.6

PVM

E0.

35–

––

––

––

0C

6H4C

l 22.

36–

––

––

––

––


PB which are indistinguishable for neutrons if a fully deuterated PS is mixedwith a partially deuterated PB consisting statistically of 4.4% protonated and95.6% deuterated monomers.

3Thermodynamic Model for Polymer Blends

An “ideal” mixture of two chemically distinct polymers represents a modelwhich is not fulfilled in reality. Even mixtures of polymers which are distin-guished only by their degree of deuteration or their microstructure such asthe vinyl content in polybutadiene (PB) show a small but finite energy of mix-ing which can easily be measured and also can lead to phase separation. Thereason is that the conformational entropy of mixing is inversely proportionalto the polymer molar volumes and can therefore be made sufficiently smalland properly adjusted to be comparable to the enthalpy of mixing.

3.1Flory–Huggins Theory

A schematic phase diagram of a symmetrical binary mixture is shown inFig. 1 in a temperature versus composition representation. A symmetric poly-mer blend is characterized by two polymer components of the same molarvolume and, therefore, with a 50% critical composition. Within mean fieldapproximation the binodal and spinodal phase boundaries of a binary (A/B)incompressible polymer mixture are described by the Gibbs free energy ofmixing ∆G according to

∆G/RT =Φ

VAln Φ +

(1 – Φ)VB

ln(1 – Φ) + Φ(1 – Φ)χ (8)

as was originally formulated by Flory and Huggins [2]. The first two termsdescribe the combinatorial entropy of mixing, while the last term representsa free energy of mixing involving the FH parameter χ. Originally, χ was de-fined being proportional to an enthalpic term χh according to χ = χh/T; laterthis parameter had to be extended to a free energy of mixing χ = χh/T – χσ

by introducing an empirical non-combinatorial entropic term χσ . The neces-sary extension of the FH parameter to a free energy of mixing was notedby Flory [63] and later mainly forced by SANS experiments. ∆G in Eq. 8 isnormalized by the gas constant R and the absolute temperature T. The combi-natorial entropic part is inversely proportional to the polymer molar volumesVA and VB. The FH parameter depends on molar volume due to chain-endeffects that introduce an additional 1/V term and that can be considered asa segmental quantity in the limit of large molar volume [4, 64]. This all meansthat the phase boundaries of many polymer blends can easily be shifted into

16 D. Schwahn

an experimentally accessible temperature range by a proper choice of the mo-lar volume.

3.2Random Phase Approximation

The structure factor S(Q) of an incompressible non-ideal mixture of two poly-mers with an enthalpy of mixing χ has been derived by deGennes [3] withinthe random phase approximation (RPA) according to

S –1 (

Q)

=1

ΦVA PADebye

(

Q) +

1(

1 – Φ)

VB PBDebye

(

Q) – 2χ (9)

The RPA is a mean field approximation that neglects contributions from ther-mal composition fluctuations and that assumes the chain conformations to beunperturbed Gaussian chains. The last assumption becomes visible from theDebye form factor in the first two terms, which for VA = VB are in accordancewith Eq. 7, while the third term involves the FH interaction parameter.

The inverse structure factor at Q = 0 is a susceptibility and is related to theGibbs free energy of mixing ∆G according to

S –1(0) = ∂2Φ

(

∆G/RT)

(10)

Equation 10 is a general result based on the fluctuation-dissipation theo-rem [3, 4, 9]. So the susceptibility can independently be evaluated from ∆Gwhich in the case of the mean field FH theory leads to a similar result as theRPA,

S –1(0) = 2 [χS – Γ ] = (2Γh/TS)(1/TS – 1/T)

1/TS= C –1

MF τ (11)

with the critical amplitude CMF = TS/2Γh, the reduced temperature τ =[1/TS – 1/T]/1/TS, and the FH parameter at the spinodal 2χS = [ΦVA]–1 +[(1 – Φ)VB]–1 which at the critical point becomes χC = 2/V for V = VA = VBand ΦC = 0.5. The susceptibilities S(0) in Eq. 11 and those derived from theRPA (Eq. 8) are identical only if the FH parameter is not dependent on com-position. We therefore redefine the FH parameter in the context of SANSexperiments according to 2Γ = ∂2

Φ [Φ(1 – Φ)χ] as Eq. 10 is the more basicequation. This leads to an effective FH parameter Γ = Γh/T – Γσ with theenthalpic and entropic terms Γh and Γσ , respectively, while ΓC ≡ χC = 2/Vremains the same.

For small Q (< 1/Rg) the structure factor in Eq. 9 can be expanded inZimm approximation according to

S –1(Q) = S –1(0) + L2Q2 (12)

with L2 = [R2g,A/ΦVA + R2

g,B/(1 – Φ)VB]/3 determined by the ratios R2g/V for

the two components and describing the square of the chain statistical segment


Fig. 4 Structure factor of binary blend dPB/PS in Zimm representation at different tem-peratures (top) and pressures (middle), and polymer concentrations (bottom), with theother parameters constant. From the fitted straight lines the susceptibility and correlationlength is evaluated. The increase of scattering is caused by stronger thermal compositionfluctuations when approaching the critical point

length. An experimental verification of Eq. 12 is shown in Fig. 4 for a blend ofdeuterated polybutadiene and protonated polystyrene of about 2000 cm3/molmolar volume and critical composition described by sample 2 in Table 2.The structure factors are depicted in a Zimm representation of S –1(Q) ver-sus Q2. All scattering curves follow a straight line in the given Q range inaccordance with Eq. 12. The scattering is shown as a function of three sepa-rate parameters. The top figures show S(Q) at four temperatures; decreasingthe temperature leads to stronger scattering because thermal fluctuations in-crease when approaching the critical temperature. The middle figures showthe sample for different pressures; at higher pressures the scattering becomesstronger because the critical temperature is shifting to higher values andtherefore is approaching the sample temperature of 29.5 C. The lower fig-ures show the blend at constant external conditions, however with differentconcentrations of a non-selective solvent as specified by the total polymer

18 D. Schwahn

content; lower scattering is observed for larger solvent content as the solventleads to a larger miscibility and therefore to a lower critical temperature.

Equation 12 can also be considered as an Ornstein–Zernicke equationdescribing the degree of thermal composition fluctuations of correlationlength ξ . The correlation length ξ is evaluated from

√L2 ·S(0) and becomes

infinite at the critical point as described by the scaling law ξ = ξMF0 τ–0.5. The

thermodynamic information derived from the susceptibility S(Q = 0) repre-sents a property averaged over a macroscopically large volume due to theinverse relationship between reciprocal (Q) space and real (r) spaces.

3.3Determination of a Polymer Blend Phase Diagram by SANS

The inverse susceptibility in Eq. 11 is proportional to the reduced tempera-ture τ and is proportional to (1/TS – 1/T) with TS the spinodal temperaturethat becomes the critical temperature TC in the case of the critical concen-tration. This means that the inverse susceptibility S–1(0) is positive in theone-phase regime and becomes zero at the spinodal as well as at the criticaltemperature. This result is consistent with the Gibbs conditions of stabilitywhich according to [9]

∂2Φ

(

∆G/RT)

⎧

⎪⎨

⎪⎩

> 0 Stable= 0 Spinodal< 0 Unstable

(13)

determines the thermodynamic conditions of stability as well as the spinodalborder line and the critical point. In this respect the susceptibility in Eq. 10 isa measure of the stability conditions.

An experimental example is given in Fig. 5 where S–1(0) is plotted versus1/T, and in Fig. 5b the corresponding phase diagrams of polystyrene (PS) anddeuterated polybutadiene (dPB) with different vinyl contents are depicted(Samples 2–4 in Table 2). S–1(0) was measured in a sample with off-criticalcomposition; within the homogeneous regime S–1(0) follows a straight linedependence on τ in agreement with Eq. 11 and extrapolates to S–1(0) = 0 atthe spinodal temperature TS. Slightly above TS the scattering shows strong de-viations from the straight line; the neutron intensity abruptly decreases onfurther lowering the temperature. This behavior is understood from phasedecomposition below the binodal line. Below the binodal the precipitateddomains grow very fast to sizes of the order of µm length as can easily bevisualized from the turbidity of the system. Scattering from the shape ofthese domains occurs in a Q range that is too small for conventional SANSdiffractometers. The observed scattering below TB therefore measures ther-mal composition fluctuations within the domains whose strength decreaseswith diminishing temperature as the distance from the critical temperature


Fig. 5 Left: Inverse susceptibility S–1(0) versus inverse temperature for an off-criticalblend. The extrapolated straight line gives the spinodal temperature, while the bend yieldsthe binodal temperature. Below the binodal fluctuations within the domains are visi-ble which decrease with decreasing temperature. Right: Phase diagram of three dPB/PSblends of about 2000 molar volume and with different vinyl content of the dPB com-ponent. The different miscibilities and positions of the critical point show that the FHparameter depends on the vinyl content

becomes larger. The critical scattering from fluctuations above and below thephase boundary has been extensively studied in ref. [65]. The so determinedbinodal and spinodal have been plotted in Fig. 5b. The solid and verticaldashed lines represent a guide for the eye and the critical composition, re-spectively. This type of experiment becomes necessary in order to determinethe critical composition in blends for later analysis of crossover experiments.

Some interesting phenomena can be learned already from the phaseboundaries in Fig. 5b. The three mixtures of PS/dPB of about 2000 cm3/molwere mixed with PB components of different vinyl content (Samples 2–4 inTable 2) [37]. With increasing vinyl content the compatibility is increasedby about 25 K, and the critical composition is shifted to about a 5% largerPB volume fraction. For the blend with the statistical dPB(1,2;1,4) copoly-mer, the critical temperature does not represent the highest temperature forphase decomposition as usually expected (see Fig. 1). This observation can beinterpreted as arising from a statistical distribution of vinyl contents for in-dividual, relatively short PB chains. Therefore, here we have the situation ofa multi-component blend because the FH parameter changes with the vinylcontent as already became clear from the different positions of the phaseboundaries. Those blends are so-called “quasi binary” blends when as inthis example the PB copolymer chains are considered as a single compon-ent. Similar effects are known from polymer polydispersity where the entropyof mixing is proportional to 1/V as has been considered by Koningsveld [8](see also [19]) who in particular shows that the shape of the binodal is sensi-tively affected by polydispersity.

20 D. Schwahn

Tabl

e2

Poly

mer

blen

ds.T

hevi

nylc

onte

ntof

PB(1

,4),

PB(1

,2;1

,4),

and

PB(1

,2)

is7,

54,a

nd96

%,r

espe

ctiv

ely

No.

Syst

emV

AN

AV

BN

BΦ

A〈V

〉〈N

〉[c

m3/

mol

][c

m3/

mol

][c

m3/

mol

]

1dP

S/PS

870

000

9060

950

000

9900

0.48

910

000

9100

2dP

B(1

,4)/

PS22

0036

.518

0018

.80.

4222

1528

.23

dPB

(1,2

;1,4

)/PS

2000

33.2

0.50

519

0023

.94

dPB

(1,2

)/PS

2100

34.9

0.5

1940

24.4

5dP

S/PV

ME

218

000

2270

192

000

3390

0.19

212

500

2422

6dP

B(1

,4)/

PB(1

,2;1

,4)

2800

046

3.6

2060

034

10.

523

700

392

7dP

B(1

,4)/

PB(1

,4)

2750

045

5.3

0.5

2770

045

8.6

8dP

B(1

,2)/

PB(1

,2;1

,4)

2050

033

9.4

2060

034

10.

520

600

341

9PP

MS/

dPS

3960

33.3

4320

43.6

0.5

4130

37.8

10PE

E/PD

MS/

PEE-

PDM

S20

1030

.522

7029

.20.

516

1200

016

811

dPB

(1,4

)/PS

/dP

B(1

,4)-

PS27

2045

2180

220.

4215

400

209.

2


4Mean Field to 3D-Ising Crossover in Polymer Blends

Thermal fluctuations can be described within the Gaussian approximation atsufficiently high temperatures above the critical temperature. For these sit-uations, the system fulfills the conditions of mean field approximation [9].On the other hand, thermal composition fluctuations become strong nearthe critical temperature, leading to non-linear effects which asymptoticallyclose to the critical temperature imply that the system obeys the universal-ity class of 3D-Ising critical behavior. Thermal fluctuations are described bythe Ginzburg–Landau Hamiltonian which is written as a functional of thespatially varying order parameter Φ(x),

H =∫

ddx

c2 [∇Φ(x)]2 + rΦ2(x) + uΦ4(x)

(14)

The order parameter Φ(x) is represented by the volume fraction Φ of one ofthe two polymer components (see Fig. 1). The first term represents the surfaceenergy and leads to the correlation length of thermal fluctuations; the secondterm corresponds to the susceptibility according to r–1 ≡ 2S(0), and the thirdterm describes non-linear effects from the fluctuation fields Φ(x) [9]. Whenthe Gaussian approximation is valid, the last term is negligible.

Linear and non-linear effects of thermal composition fluctuations becomevisible in a scattering experiment. Within the mean field and Ising regimesthe susceptibility S(0) and the correlation length ξ are described by simplescaling laws as functions of the reduced temperature τ according to Cτ–γ

(ξ0τ–ν ) with the critical amplitudes C(ξ0) and the critical exponents γ (ν). The

critical exponents γ (ν) are known to be equal to γ = 1 and 1.239 ± 0.003 andν = 0.5 and 0.634 ± 0.001 in the mean field and Ising cases, respectively [66].The mean field case has already been discussed in the context with Eq. 11.

Mean field and Ising behavior are observed in the limits where the sys-tem is asymptotically far and close to the critical temperature, respectively.A further theoretical approach by so-called crossover functions is neededin order to describe the susceptibility and correlation length over the fullmiscible range of the blend and thereby to bridge both asymptotic limits.Such crossover functions and the corresponding SANS results from polymerblends under various conditions will be discussed in the following sections.

4.1Theoretical Background

To a large extent, the formalism of crossover functions has been developedby Sengers and coworkers [10] in recent years. It has been shown that thecrossover from mean field to Ising behavior could be non-universal andrather complex. In the asymptotic critical Ising type regime the correla-

22 D. Schwahn

tion length ξ is the only relevant length, while in the crossover regime anadditional microscopic length could become relevant and enforce a non-universal and non-homogeneous crossover behavior [29]. Such a non-universal crossover behavior has been observed in dilute polymer solutionsin which the size of the polymer was quite naturally interpreted as the mi-croscopic length scale [67]. In most polymer blends so far, the variation ofS(0) can be described by a universal and continuous crossover function that issolely determined by a crossover temperature. Only in a few cases does the ra-dius of gyrations of the chains seem to appear as a microscopic length that isvisible in the crossover behavior [68]. This means that a relevant microscopiclength scale usually does not appear in polymer blends and can therefore beneglected. A corresponding crossover formalism for the susceptibility andcorrelation length has been derived from a more general formalism as

S –1(0) = (1/CMF) τ Y(γ–1)/∆

[

1 +u∗ν∆

(1

2∆+

Y1 – Y

)–1]

(15)

ξ = ξMF0 τ–1/2Y–(2ν–1)/2∆ . (16)

Upon taking the cut off wave number Λ → ∞ which corresponds to a van-ishing microscopic cut-off length [10, 30]. The susceptibility and correlationlength are functions of the reduced temperature τ and the crossover func-tion Y. The critical exponents γ and ν are those of the 3D-Ising modeland ∆ = 0.54 ± 0.03 is a universal correction-to-scaling exponent [66]. Thecrossover function Y is either given as an implicit function of the inversecorrelation length κ = ξMF

0 /ξ or of the reduced temperature τ according to1 – Y = (uΛ/κ) Yν/∆ or (1 – Y)2Y–1/∆ = u2Λ2/τ , respectively. A vanishingmicroscopic length (Λ → ∞) also means that the reduced coupling con-stant u → 0 because of its definition u = u/u∗Λ. The parameters u andu∗ ∼= 0.472 represent the coupling constant in the Ginzburg–Landau Hamil-tonian in Eq. 14 and the RG fixed-point coupling constant, respectively. Forthe mean field approach, using u2Λ2/τ 1 and Y = 1 gives the scaling lawsS(0) = CMFτ

–1 and ξ = ξMF0 τ–1/2 with the corresponding mean field critical

amplitudes CMF and ξMF0 in conformity with the expression in Eq. 11. Within

the asymptotic regime of 3D-Ising behavior one finds τ → 0 and Y → 0 andtherefore Y = (u2Λ2/τ)–∆. From Eqs. 15 and 16 and using the proper Y onegets S(0) = C+τ–γ and ξ = ξ0τ

–ν gives the asymptotic scaling laws of the 3D-Ising model. The ratios of the Ising and mean field critical amplitudes aregiven as

(C+/CMF)1/(γ–1) = u2Λ2 (

ξ0/ξMF0

)2/(2ν–1)= u2Λ2 (17)

In the present case of a zero microscopic length (Λ → ∞) the crossoverbetween the two universality classes is determined by a single parameter,namely (uΛ)2 which is proportional to the Ginzburg number Gi and therebyto the crossover temperature TX (see Fig. 1). In conformity with our earlier


work [31] we define the Ginzburg number as Gi = 0.069 u2Λ2 and finally ob-tain in Eq. 18

Gi = 0.069 (C+/CMF)1/(γ–1) = 0.069(

ξ0/ξMF0

)2/(2ν–1)(18)

the Ginzburg number represented in terms of the ratio of the Ising and meanfield critical amplitudes and a large exponent 1/(γ – 1) [2/(2ν – 1)] of about 4as determined from the Ising critical exponents. Quite generally, Gi can beidentified with the reduced temperature at which the difference between theIsing approximation and the crossover function reaches 10%.

Another form of crossover function for the susceptibility

τ =(

1 + 2.333S(0)∆/γ)(γ–1)/∆ ×

[

S –1(0) +(

1 + 2.333S(0)∆/γ)–γ/∆

]

(19)

has been derived by Belyakov and Kiselev from a renormalization groupapproach and an ε-expansion [27]. In this expression the rescaled reducedtemperature τ = τ/Gi is an explicit function of the rescaled susceptibilityS(0) = S(0) Gi/CMF, the critical exponents γ and ∆ are from the 3D-Isingmodel. So, the parameters Gi, CMF, and TC are the parameters of the sus-ceptibility that are fit to the experimental SANS data. Most of the SANSsusceptibility data so far were analyzed with Eq. 19 as it was the only availableand proper crossover function at the time when the experiments were per-formed. But in any case both crossover functions describe a similar situationas in both cases a microscopic length scale was neglected.

The expressions for the critical amplitudes are given for the mean fieldcase in Eq. 11. For the Ising case they are expressed in terms of the molecu-lar volumes for symmetric blends according to C+ ∝ V(2–γ ) and ξ0 ∝ V(1–ν) asderived by Binder [4, 16]. These expressions lead to the representations

Gi ∝[

V(2–γ )(2/V + Γσ )]1/(γ–1)

Gi ∝[

V(2–2ν)(2/V + Γσ )]1/(2ν–1)

(20)

and in case of a zero entropic term Γσ to the universal scaling law Gi ∝ 1/V.The last expression was originally derived by deGennes assuming an incom-pressible polymer melt [3]. It becomes apparent from the large exponents1/(γ – 1) ∼= 4.23 and 1/(2ν – 1) ∼= 3.73 that a positive finite entropic term Γσ

strongly enhances Gi and thereby the effect of thermal composition fluctua-tions near the critical temperature.

The above expressions for Gi are valid for UCST systems showing phasedecomposition at low temperatures. These systems are characterized by(Γh and Γσ ) > 0 and |Γσ | < ΓC if Γσ < 0 as otherwise no miscibility isachieved. There are several blends, showing phase decomposition at hightemperatures, and which are so-called LCST systems. The FH parameter ofsuch blends have to fulfill the following conditions, namely (Γσ and Γσ ) < 0and |Γσ | > ΓC = 2/V, in order to show phase transitions at a finite criticaltemperature. The corresponding critical amplitude for the mean field suscep-tibility becomes C –1

MF = 2[

–(

2/V + Γσ

)]

because of the necessary redefinition

24 D. Schwahn

of the reduced temperature according to τ ∝ (1/T – 1/TC). For a positive crit-ical amplitude and thereby positive susceptibility the above conditions fora finite TC have to be fulfilled, and the corresponding Ginzburg criterion forS(0) is expressed as

Gi ∝[

V(2–γ )(– 2/V + |Γσ |)]1/(γ–1)

(20a)

This means that in polymer blends quite large values of Gi are possible andalso implies a general need for a more sophisticated analysis of the scatter-ing data using Eqs. 15,16 or 19. This also means that the FH parameter has tobe evaluated from the mean field critical amplitudes CMF in Eq. 11 and fromthe “mean field” critical temperature TMF

C which is approximately related tothe “real” critical temperature TC according to TMF

C = TC(1 + Gi) [10, 28]. TCis lower than TMF

C by several degrees Kelvin because of the stabilization effectof thermal fluctuations.

The dependence of Gi in Eq. 20 on the entropic term Γσ leads to a connec-tion with other theoretical approaches to Gi for polymer blends. Accordingto equation of states theories a finite entropic term Γσ arises from the sam-ple compressibility [33] which also becomes apparent from the Lattice ClusterTheory (LCT) [12]. It has to be mentioned here that the Ginzburg criterionderived on the basis of mean field parameters by Bates et al. [20] and Hairet al. [21] behave inversely proportional to the non-combinatorial entropyterm Γσ [see also Eq. (4.2) in Dudowicz et al. [12]]. So there still seems toexist some confusion with respect to the effect of Γσ on Gi.

4.2Binary Homopolymer Blends

A first experimental example exhibiting a crossover from mean field to Isingcritical behavior is depicted in Fig. 6 showing the inverse susceptibility andthe square of the inverse correlation length versus inverse temperature fora blend of high molar volume deuterated polystyrene (dPS) and poly(vinylmethylether) (PVME) with the critical composition (Sample 5 in Table 2).The susceptibility and correlation length presented were analyzed with theasymptotic crossover function of Eqs. 19 and 16, respectively, and the bestfits are shown by the solid lines; the agreement between experiments and the-ory is excellent. The experimental conditions for determining the correlationlength corresponds to those of the susceptibility giving the same Ginzburgnumber Gi = (4.5±0.5)10–5 and roughly a 0.5 K shift of the critical tempera-ture from the mean field estimate. The correlation length shows qualitativelythe same behavior as S(0) and leads to an additional critical amplitude pa-rameter. From the fit of parameters, the critical temperature, the Ginzburgnumber and the mean field and 3D-Ising critical amplitudes were derivedfrom which the FH parameter were evaluated. The positive slope of S–1(0) in


Fig. 6 Inverse susceptibility S –1(0) and inverse square of the correlation length versus in-verse temperature for the critical LCST high molar volume blend dPS/PVME. The solidlines represent a fit of the corresponding crossover functions. The mean field approxima-tion is visible in the insets, the Ising behavior by the dashed lines

Fig. 6 is related to a negative enthalpic Γh (Eq. 11). In the case of a negative Γh,the PS and PVME heterocontacts are preferred. Only in the case of a negativeentropy of mixing (Γσ < 0) that is larger than the always positive combinato-rial entropy of mixing does a positive driving force for phase separation exist.The critical temperature is defined to occur at S–1(0) = 0 which corresponds toan infinite correlation length. The critical condition of S–1(0) = 0 is very wellapproached, indicating that the experiments were performed at the criticalcomposition.

Figure 7 depicts the susceptibility of two more conventional UCST blendsof critical concentration and about 2000 cm3/mol molar volume. The phase

Fig. 7 S –1(0) versus inverse temperature for dPB(1,4)/PS and dPB(1,4;1,2)/PS blends. Thesolid lines represent a fit of the crossover function. Corresponding asymptotic scalinglaws of mean field and 3D-Ising behavior are depicted as dashed lines. A lower criticaltemperature is observed for the dPB(1,4) blend in accord with the phase diagram in Fig. 5

26 D. Schwahn

diagram of two dPB/PS blends with 7 and 54% vinyl contents for the PB com-ponent (Table 2) are shown in Fig. 5b. The dashed lines give the asymptoticscaling laws corresponding to mean field and Ising behaviors. These blendsshow a larger crossover range of the order of 13 K in comparison with therange of 1 K for the high molecular dPS/PVME blend.

The phenomenon of crossover behavior in polymer blends was first ob-served in this large molar volume dPS/PVME blend [18, 19] and subse-quently observed by other groups [20–26]. At that time the observation of thecrossover to the Ising case in the high molecular PS/PVME blend was surpris-ing because based on the “incompressible” Ginzburg criterion, a temperaturerange of roughly 0.05 K was expected.

Figure 8 shows a collection of Ginzburg numbers from various polymerblends versus the degree of polymerization [31]. A strong variation of Gi overfour orders of magnitude is obtained for samples ranging over two ordersof magnitude in N. The points lie within an area described by N–1 and N–2

scaling laws, the first of which was proposed by deGennes [3] and the sec-ond by SANS experiments. A remarkable observation is related to the orderof magnitude larger Gi that is found for blends composed of shorter chainsbetween N = 10 and 100 as compared with the Gi ∼= 10–2 of low molecular li-quids. These observations in the context of Eq. 20 clearly demonstrate thatthe entropic term Γσ and thereby the compressibility plays an important rolein polymer mixtures and is responsible for the non-universal behavior of thevariation of Gi with N. These observations also must be considered with theunderstanding that the degree of polymerization of ordinary UCST polymer

Fig. 8 Ginzburg number versus degree of polymerization from different polymer blendsas determined from scattering experiments. A universal N–1 scaling behavior has beenpredicted for incompressible polymer blends. Polymer packing effects enter through thenon-combinatorial entropy and have a strong influence that leads to an increase of Gi


blends with at least partial miscibility are usually not larger than of the orderof N = 100.

4.3Binary Blends of Statistical Copolymers

Statistical copolymers often show the interesting effect of better compatibil-ity; comparing polymers of similar length, one may find a “C” homopolymermiscible with an “A-B” statistical copolymer even though the “A, B, and C”homopolymers are not miscible with each other. This phenomenon has beeninterpreted by the interplay of the intermolecular interactions between theA-C and B-C monomers and the intramolecular interaction between the A-Bmonomers reducing the total enthalpy of mixing [69, 70]. While this explana-tion seems to be sufficient to interpret an enhanced compatibility determinedfrom cloud point experiments, it completely failed to correctly describe re-cent SANS experiments on binary blends of PS and PB with different vinylcontents [37]. For a better understanding one needs a full analysis of theFH parameter including the enthalpic and non-combinatorial part of a freeenergy of mixing which cannot separately be determined from cloud pointmeasurements alone.

In this paragraph we will discuss three blends of dPB(1,2)x-dPB(1,4)(1–x)/

PB(1,2)y-PB(1,4)(1–y) which consist of the same PB statistical copolymer withdifferent vinyl content x and y of the deuterated and protonated compon-ent (Sample 6–8 in Table 2). These systems represent the simplest realizationof copolymer blends of the general type Ax-B(1–x)/Cy-D(1–y) with x and ythe volume fraction of the A and C monomers, respectively. The A and Bmonomers were deuterated in order to have good neutron scattering contrast.Symmetric mixtures with chains of similar molar volume between 2 to 28k(k ≡ 103 cm3/mol) and with x and y of 0.07, 0.54, 0.91 were recently publishedin ref. [64]. Figure 9 depicts the susceptibility versus inverse temperature1/T of the larger molar volume blends, namely dPB(1,2)/PB(1,2;1,4) anddPB(1,4)/PB(1,2;1,4) together with that of the isotopic mixture of PB(1,4).The scattering from the PB(1,4) isotopic mixture is rather weak as expected,while the other two samples give very different results in spite of their simi-lar molar volumes and symmetries with respect to their microstructure. Atfirst glance, this strong difference is very surprising as the classical theoriesof copolymers predict for |x – y| ∼= 0.5 the same FH parameter and thereforethe same degree of scattering. The blend containing dPB(1,2) and PB(1,2;1,4)shows large scattering with S –1(0) = 0 at the critical temperature of TC = 69 Cand, furthermore, a crossover behavior according to Eq. 15 and Eq. 19 as de-picted by the solid line in Fig. 9. A Ginzburg number of Gi = (1.9 ± 0.8)10–3

was evaluated and is depicted by the symbol in Fig. 8. The susceptibility ofthe other two blends in Fig. 9 were fitted by straight lines in accordance withthe mean field approximation of Eq. 11.

28 D. Schwahn

Fig. 9 Inverse susceptibility S –1(0) versus inverse temperature for three dPB/PB statis-tical copolymer blends with different vinyl contents and molar volumes as indicated.All blends are of 50% composition. A remarkably strong scattering is found for thedPB(1,2)/PB(1,2;1,4) sample that is described by a crossover function in contrast to theweak scattering from dPB(1,4)/PB(1,2;1,4). The overall decrease of the interaction energyon deuteration explains this observation

A first observation can be established, namely, that blends with deuter-ated PB(1,2) show a much stronger tendency towards incompatibility thanblends with deuterated PB(1,4). This observation is supported by the findingthat a corresponding mixture of dPB(1,2) and PB(1,4) was not miscible below200 C.

4.3.1Theory

Quite recently, Dudowicz and Freed formulated an expression for the FH pa-rameter for statistical copolymers within their “Lattice Cluster Theory forPedestrians” (see Eq. 13 in [71] or Eq. 1 in [72]). This expression leads to theclassical theory of

Γ ≡Γh(AC)xy + Γh(BC)(1 – x)y + Γh(AD)x(1 – y) (21)

+ Γh(BD)(1 – x)(1 – y) – Γh(AB)x(1 – x) – Γh(CD)y(1 – y)

only if Γσ is neglected and the monomers are taken to occupy single latticesites [69, 70]. In the case of a positive intramolecular Γh(AB) and Γh(CD), theoverall enthalpic term in Eq. 21 is reduced with a resultant better compatibil-ity between the polymers in the blend.

The enthalpy of the deuterated dPB(1,2)xdPB(1,4)(1–x)and protonatedPB(1,2)yPB(1,4)(1–y)are determined by the nearest neighbor attractive vander Waals energies εCC, εDD, and εCD, respectively, describing the interac-


tion between the protonated 1,2–1,2, 1,4–1,4, and 1,2–1,4 pairs of monomers.Hence, the enthalpic term, representing the intramolecular interactions suchas between the C and D monomers, is determined by the exchange en-ergy ε = εCC + εDD – 2 εCD in the form with Γh(CD) = (2z/Ω) ε, with zand Ω representing the number of nearest neighbors (theoretical numberis z = 6) and the monomer volume (Ω = 60.4 cm3/mol for PB), respec-tively. A simplification is achieved if the interaction energy of deuteratedmonomers is described by the polarizability model of Bates et al. [73], whichleads to εAA = γ 2εCC, εBB = γ 2εDD, εAB = γ 2εCD, εAC = γεCC, εBD = γεDD,εAD = εBC = γεCD. The value of γ is slightly less than one because ofthe smaller interaction between deuterated monomers in comparison withthe corresponding protonated ones. In the case of symmetric blends withΦC = 0.5, one gets

Γh = (2z/Ω)[

α2 εCC + β2 εDD – 2αβ εCD]

(22)

With the coefficients α = (y – γx) and β = [γ (1 – x) – (1 – y)] (see Eqs. 1 and11 in [71, 72]). In the case of a fully protonated blend with γ = 1, one getsΓh = (2z/Ω)ε [x – y]2, and for the PB(1,4)/PB(1,2) homopolymer blend thisbecomes Γh = (2z/Ω)ε. For blends with one component deuterated, one getsfor (x = 0; y = 1) and (x = 1; y = 0), respectively, Γh = (2z/Ω) (εCC + γ 2εDD –2γεCD) and Γh = (2z/Ω) (γ 2εCC + εDD – 2γεCD), e.g., in the case of differentvan der Waals energies εDD and εCC, deuteration leads to a different Γh.

Lattice Cluster Theory gives in addition an entropic term which for the PBcopolymer blends reads

Γσ =– (x – y)2/(4z2Ω) (23)

This negative non-combinatorial entropy of mixing Γσ is related to the struc-ture of the monomers, e.g., depending on the number of lattice sides permonomer that are occupied by tri- and tetrafunctional united atomic units ofthe (1,2) and (1,4) monomers; compressibility and chain-end effects were notconsidered [71, 72]. A more detailed description of this theory is found in thereview article by Freed and Dudowicz of the same volume.

4.3.2Experimental Results

The FH parameters Γh and Γσ for the PB copolymer blends are depicted inFig. 10 versus the vinyl content of the protonated PB component. The solidand open triangles and spheres represent blends with fixed vinyl content ofx = 0.07, 0.16, and 0.91 of the deuterated PB component, respectively. The FHparameters were corrected for chain-end effects which are rather strong andare not shown here (see Fig. 6 in [64]).

The blend with x = 0.91 (dPB(1,2)) and y = 0.54 vinyl contents yields a rela-tively large enthalpic Γh as already evident from the data in Fig. 9; on the

30 D. Schwahn

Fig. 10 Enthalpic and entropic FH parameters for dPB/PB statistical copolymer blendsin the limit of infinite molar volume as a function of the vinyl content of the deuteratedand protonated chains. The enthalpic terms represented in terms of the van der Waalsinteraction energy and a polarization parameter for deuteration as depicted by the the-oretical solid lines. The entropic terms are represented in terms of an expression for thenon-combinatorial entropy from LCT calculations with a slightly smaller coordinationnumber

other hand, a relatively small Γh is observed for the blends with dPB(1,4), fora limited interval of the vinyl content of the protonated component Γh evenbecomes negative. The entropic Γσ is negative for the dPB(1,4) blend, whilea slightly positive value has been found in the case of the dPB(1,2) blend.

The solid lines in Fig. 10 represent the LCT calculations using Eqs. 21and 22, while the dashed and dotted lines are a guide for the eye, the latterline showing the isotope effect. The corresponding fit parameters are givenin Table 3. Excellent agreement between experiment and theory is achieved,thus explaining the physical reason for the observed quite strong differencesdue to microstructure and deuteration. The enthalpic Γh represented by thesolid symbols is fully described by a single set of van der Waals energy pa-rameters εij and a single deuteration polarizability factor γ . To make these fitsunique, a fixed value of εCC was taken from PVT measurements of a dPB(1,2)melt [39].

The enthalpic part of the FH parameters appears very much influencedby the isotope effect, e.g., whether the PB(1,2) or PB(1,4) is deuterated ornot. If one considers a PB copolymer mixture of only protonated compo-nents, using the van der Waals interaction parameters in Table 3 and setting


Table 3 Parameters from fit of copolymer theory

z = 5.7εCC = 211.6 K∗)

εDD = 250.18 KεCD = 230.56 K

γ = 0.988

∗) This number was taken from Frielinghaus et al. as a fixed parameter [39]. εCC, εDD,and εCD, respectively, describe the interaction between the protonated 1,2–1,2,1,4–1,4,and 1,2–1,4 pairs of monomers

γ = 1, one gets Γh = 0.131(x – y)2 [Γh is always given in units of K mol/cm3],which is symmetric in x and y, always positive, and zero only for x = y, i.e.,when the microstructure in both components is the same. A maximum valuefor Γh = 0.13 is evaluated in the case of

∣∣x – y

∣∣ = 1, which corresponds to

the PB(1,4)/PB(1,2) blend. If one component is fully deuterated, as describedin Eq. 22, one gets very different values for Γh, namely, 0.044 and 0.228 fordPB(1,4)/PB(1,2) and dPB(1,2)/PB(1,4), respectively (see also Fig. 10). So,depending on whether PB(1,4) or PB(1,2) is the deuterated component a re-spective 70% decrease or increase of Γh is observed. The smaller Γh of thedPB(1,4)/PB(1,2) blend is caused by the 18% larger interaction energy be-tween the (1,4) monomer units (εDD) in comparison with those between the(1,2) units (εCC). Hence, a 1.2% change in ε upon deuteration leads to a rela-tively large change of the enthalpic FH parameter, and, furthermore, in somecases to a transition from an UCST to LCST phase behavior.

The physical basis of the behavior of Γh in Fig. 10 becomes more transpar-ent if a simplified version of Eq. 22, namely Eq. 24, is used, which is

Γh = (2z/Ω)[

βε0.5DD – αε0.5

CC]2

(24)

based on the assumption of dispersive (van der Waals) interactions fulfill-ing the relationship εCD =

√εCCεDD. In our case this relationship is fulfilled

within 0.2%. The form of Eq. 24 is similar to the relation for FH parame-ter Γh ∝ [β∆DD – α∆CC]2 in terms of the solubility parameters for the (1,4)(∆DD) and (1,2) (∆CC) monomers. The asymmetry effect of deuteration canbe more easily understood on the basis of Eq. 24, and the correspondingpresentation in Fig. 11. The (1,4) monomers are more stronger bound thanthe (1,2) monomers as they show a stronger attractive energy and therebya larger solubility parameter. This difference of solubility parameters is ofmicrostructural origin. A general trend is reported in the literature, namely,the solubility parameter decreases with increasing content of the branchedvinyl content [74]. As demonstrated by the dashed and dotted lines in Fig. 11,deuteration of the (1,4) and (1,2) polymers leads to a decrease and increase

32 D. Schwahn

Fig. 11 The square root of the van der Waals interaction energies are proportional to thesolubility parameters and are depicted for the (1,4) and (1,2) PB monomers as well astheir decrease caused by deuteration. In the case of dispersion forces, the enthalpic termis determined by their difference (vertical lines); so deuteration of PB(1,2) and PB(1,4)leads to a larger and a smaller Γh, respectively. The different compatibility of PB/PSblends in Fig. 5 can also be concluded from the data in this figure and the larger solubilityparameter of PS

of Γh, respectively. Similar effects were found in blends of saturated polybu-tadiene polymers [74].

The expression for entropic Γσ in Eq. 23 describes the data with thex = 0.07 samples and the results by Jinnai et al. [75] very well assuming z = 5.7nearest neighbors. Because of obvious limitations inherent in any latticemodel, the parameter z is allowed to vary freely to compensate for deficien-cies of the model. The closeness of z = 5.7 to the theoretical z = 6 is, however,quite convincing. According to the theory, there seems to be no isotope ef-fect on Γσ . Deviations from theory were found for the dPB(1,2)/ PB(1,2;1,4)blend, which shows a positive Γσ that cannot be described without possiblyincluding chain semi flexibility effects.

4.4Polymer Blends in an External Pressure Field

Besides temperature and polymer composition, pressure is a third parame-ter needed to completely determine the thermodynamic equilibrium state ofa binary mixture. So far, only a few systematic SANS studies exist for poly-mer blends in external pressure fields [34–41]. Those experiments were alsoperformed in our laboratory for which a temperature-pressure cell was de-veloped for in-situ investigations allowing pressure and temperature fieldsbetween 0.1 ≤ P(MPa) ≤ 200 and – 20 ≤ T(C) ≤ 200, respectively. A tempera-ture control better than 0.01 K allowed also precise exploration of thermalcomposition fluctuations near the critical point [34].

The SANS data near the critical point were analyzed with the correspond-ing crossover functions (Eqs. 15–19) and the effect of pressure was mainly


interpreted in terms of the FH parameter. A theory of compressible blendsincluding the effects of pressure has been incorporated in equation of statestheories by Patterson and Robard [33] and included in the LCT by Tang andFreed [76]. A comparison between LCT theory and SANS data for the pres-sure dependence has been presented in a recent publication by Frielinghauset al. [39].

4.4.1SANS Results

As an example Fig. 12 presents the inverse susceptibility of a dPB(1,4)/PSblend (Sample 2 in Table 2) versus the inverse temperature measured in fourpressure fields between 0.1 and 150 MPa. The solid lines represent a fit withthe crossover function in Eq. 19 and show good agreement. The parame-ters derived from S(0), namely, the pressure dependent phase boundaries, theGinzburg number, and the FH parameter will be discussed in the followingfigures.

The spinodal (critical) and binodal temperatures of the three samples aredepicted in Fig. 13 versus pressure. In all samples an increase of the phaseboundaries with pressure is observed. Such a behavior is expected because ofthe reduced free volume. The shape of the phase boundary is linear for theblends with the dPB(1,4) and dPB(1,4;1,2) copolymers and is more parabolicfor the blend with dPB(1,2). The dPB(1,2) sample was at the critical com-position while the other two samples are slightly off critical compositionas seen from the deviation between the spinodals and binodals. The worstcompatibility is observed for the dPB(1,2)/PS sample, the best one for thedPB(1,4)/PS, and for the copolymer sample it lies in between.

Fig. 12 S –1(0) versus inverse temperature of a dPB(1,4)/PS blend in different pressurefields. Pressure leads to reduced polymer compatibility, and the solid line represents a fitof the crossover function

34 D. Schwahn

Fig. 13 Binodal and spinodal temperature versus pressure for the three dPB/PS blendsas indicated in the inset. The phase boundaries of the PB(1,2) blend increase witha parabolic shape, while the variation is linear for the other two blends

The Ginzburg number of the PPMS/PS and PB/PS blends in Fig. 14 al-ways decreases with pressure but shows characteristic differences [36, 37].The PPMS/PS blend is characterized by a relatively large Gi at ambient pres-sures and a rather fast decrease at larger pressures. On the other hand, Gi forthe PB/PS blends rather strongly depends on the vinyl content of the polybu-tadiene chains, which, of course, is due to the corresponding entropic Γσ

contributions (see Eq. 20). The largest Gi appear for the blend with dPB(1,2),which is constant up to 100 MP, followed by a rather strong decrease. Inthe dPB(1,4) blend, pressure induces a linear decrease of Gi. The lowest Gi

Fig. 14 Ginzburg number versus pressure of the three dPB/PS and PPMS/dPS blends. Inall cases pressure reduces the Ginzburg number and thereby reduces the range of strongthermal fluctuations around TC


Fig. 15 Enthalpic and entropic parts of FH parameters for the three dPB/PS blends versuspressure. In all cases a decrease with pressure is found. The lowest numbers are found forthe PB copolymer mixtures which saturates above 100 MPa

is observed for the copolymer blend for which Gi = (1.7 ± 0.5)10–3 becomesconstant above ambient pressure. The last number appears very near the esti-mated Gi for incompressible polymer blends (see Fig. 8).

It is found from the analysis that the effect of pressure on thermal fluctu-ations is mainly determined by the variation of the mean field parameters;at least for PB/PS the Ising critical amplitude C+ (defined above in Eq. 17) isindependent of pressure within an accuracy of 3% (Fig. 9 in ref. [37]).

The FH parameters Γh and Γσ derived from the mean field critical ampli-tude CMF in Eq. 11 are depicted in Fig. 15. In all cases Γh and Γσ decrease withpressure. The enthalpic term is largest for the PB(1,2)/PS blend. This can beunderstood from the solubility parameters of the monomers; the PS chainshave a larger solubility parameter than both PB monomers as can be seenfrom Table 5 in ref. [77]), while the solubility parameter of PB(1,2) is smallerthan PB(1,4) as discussed in context with Fig. 11. Therefore, mixing PB(1,2)and PS leads to a larger difference in solubility parameters and thereby toa larger enthalpic term. The Γh of the copolymer PB(1,2;1,4) is lowest. But thelow entropic term counters a stronger enhanced compatibility from Γh.

4.4.2Clausius–Clapeyron Equation

Within a mean field approximation the pressure induced changes of the bin-odal and critical temperature are determined by the corresponding changes

36 D. Schwahn

of the FH parameter as described by the Clausius–Clapeyron equation.

∂PTC = TC∆V/∆H (25)

The change of the phase transition (critical) temperature TC with pressuredepends on the sample volume (∆V) and enthalpy (∆H) of mixing [78].The sample volume change is derived from the Gibbs free energy of mix-ing ∆G(T, P, Φ) = ∆H – T∆S according to ∆V = ∂P∆G and to the enthalpyand entropy terms within FH theory, respectively, as ∆H/R = Φ(1 – Φ)χh and∆S/R =– (Φ/VA) ln Φ + [(1 – Φ)/VB] ln(1 – Φ) + Φ(1 – Φ)χσ .

At the critical temperature the interpretation of the thermodynamic pa-rameters ∆V and ∆H is not correctly represented within FH theory becausethermal fluctuations lead to a renormalization of the mean field critical tem-perature TMF

C which in first order is determined as TC = TMFC (1 – Gi) [10].

Thermal fluctuations yield a slightly enhanced miscibility whose degree is de-termined by Gi. This is reasonable as thermal fluctuations also enhance theentropy. At elevated pressure this stabilization effect is reduced as is visiblefrom Gi in Fig. 14.

Also including contributions from thermal composition fluctuations theClausius–Clapeyron equation for critical polymer blends can be expressedas [40]

∂PTC = TC(1) [∂PΓh – TC(1)∂PΓσ ] /Γh(1) – [TC(1)/(1 – Gi(1))] ∂PGi (26)

The first term is determined from the effective FH parameter Γ , while thesecond term describes the stabilization effect of the thermal fluctuations. Theparameters TC(1), Gi(1), Γ (1) represent the corresponding values at the am-bient pressure field of 0.1 MPa.

Figure 16 compares ∂PTC as experimentally determined from the data inFig. 13 and as calculated with Eq. 26 from the Ginzburg numbers and FH pa-rameters in Figs. 14 and 15 for all vinyl contents. The full dots show ∆PTC(≡ ∆TC/∆P) directly derived from the phase boundaries while the open sym-bols represent the evaluated numbers. The two triangular symbols show thechanges caused separately from the FH parameter and the Ginzburg number,while the open dots represent the sum of both effects. For the two sampleswith smaller vinyl contents the effect from the FH parameter is about twiceas large as from the Ginzburg number Gi, while for the dPB(1,2)/PS samplea negative ∆PTC is obtained from Γ (P) which is more than compensated bythe Gi(P) contributions, giving an overall positive ∆PTC. A comparison of the“total” ∆PTC from the Clausius–Clapeyron equation with the experimentalone shows good agreement.

The effect of pressure on thermal fluctuations in PB/PS always seems tolead to a unique behavior, namely, to an increase of the critical temperature.But the situation is more complex if one considers the contribution from thepressure dependence of the FH parameter Γ (P) alone and as already evidentfrom the application of the Clausius–Clapeyron equation. The enthalpic term


Fig. 16 Pressure induced changes of the spinodal temperature as obtained from experi-ment (Fig. 13) and from an extended Clausius–Clapeyron equation with the enthalpic andentropic terms in Fig. 15 and the Ginzburg numbers in Fig. 14. Both values are depictedas solid and open points which show good agreement and demonstrate consistency

is always positive, and both FH parameter terms Γh and Γσ decrease withpressure. Therefore, one expects at low temperature a negative change of thecritical temperature with pressure when Γh becomes the leading term. Fromthe data in Fig. 15 one evaluates the characteristic temperatures of 14 C,60 C, and 132 C for the dPB(1,4), dPB(1,2;1,4), and dPB(1,2) samples, re-spectively, below which a negative ∆PTB is expected. So, because the first twosamples have critical temperatures larger than 14 and 60 C, a positive shift of∆PTC is evaluated in agreement with experimental findings. For the dPB(1,2)sample, however, the critical temperature of 101 C is smaller than the cal-culated 132 C, and therefore a negative change of the critical temperaturewith pressure is predicted from the contribution of Γ (P) alone (see discussionof Fig. 16).

The good agreement of ∆PTC from experiment and the Clausius–Clapeyron equation represents a consistency check. This finding shows thatthe underlying theory of the crossover function in Eqs. 15–19 and the subse-quent analyses lead to reasonable values of the FH parameter and Ginzburgnumber. Furthermore, it is a clear demonstration that pressure inducedchanges of the phase boundaries are generally influenced by two pressuredependences, namely, those of the interaction parameter and the degree ofthermal composition fluctuations.

While the pressure induced increase of the two-phase boundary seemsto be the normal case, there exist a few polymer blends which showan enhanced miscibility with pressure. Such a behavior was observed inPEP/PDMS (poly(ethylenepropylene)/polydimethylsiloxane) and PDMS/

PEMS (polyethyl-methylsiloxane) blends [42, 43]. In both blends an increaseof the enthalpic and entropic terms of the FH parameter with pressure, i.e., to

38 D. Schwahn

∂PΓh > 0 and ∂PΓσ > 0, is observed while the total FH parameter is decreas-ing (∂PΓ < 0) as the entropic term is dominant at the transition temperature.For those blends one expects a positive ∆PTC at sufficiently low tempera-ture when the enthalpic term becomes dominating. The microscopic originof such behavior is still unknown.

4.5Binary Blends with Small Additions of a Non-Selective Solvent

The addition of a non-selective solvent generally leads to a larger compati-bility of the binary blend and an increase of the Ginzburg number. Pressureexperiments visualize an interchange of solvent molecules with free volume.After a short presentation of the scattering law, a series of experiments isdiscussed.

4.5.1Structure Factor

The scattering cross-section of a binary polymer blend with small addition ofa non-selective solvent is given as

dΣ/dΩ(Q) = KABΦPS(Q) + KPSSPP(Q) (27)

The first term is determined by the polymer structure factor given in Eq. 9and is proportional to the contrast factor KAB of the two polymers and thetotal polymer concentration ΦP. The second term contains the polymer-polymer partial structure factor SPP(Q) describing the overall polymer dens-ity fluctuations which would become zero for an incompressible polymer meltwith ΦP = 1. The scattering contrasts KAB = (ρA – ρB)2/NA and KPS = (ρ –ρS)2/NA are determined by the coherent scattering length densities of thepolymers and solvent molecules, the average ρ = (ΦAρA + ΦBρB)/ΦP, andAvogadro’s number NA (see given values in Table 1). As we are interested ininvestigating thermal composition fluctuations, S(Q) has to become the dom-inating scattering contribution. The relevant contrast KAB can be made anorder of magnitude larger than KPS by, e.g., mixing deuterated PB with pro-tonated PS and solvent molecule.

Within the FH model the susceptibility and correlation length are, re-spectively, given as S –1(0) = 2(ΓC – ΦPΓ ) and ξ–2 = [2(ΓC – ΦPΓ )]/R2

gΓC.In the “dilution approximation” of blend–solvent systems, Γ is replaced byΦPΓ [79]. Such a replacement delivers the mean field critical amplitudes andGinzburg criterion in terms of the FH parameters according to

CMF+ = 1/(2 |ΓC + ΦPΓσ |) = TMF

C /2(ΦPΓh) (28)

ξMF0 =

√

2R2gΓCCMF/3 = Rg/

√

3(1 + ΦPΓσ/ΓC) (29)


and

Gi ∝[

V(2–γ )(2/V + ΦPΓσ )]1/(γ–1) ∝

[

V(2–2ν)(2/V + ΦPΓσ )]1/(2ν–1)

(30)

4.5.2Susceptibility and Correlation Length

The susceptibility and correlation length of several dPB(1,4)/PS blends (sam-ple 2 in Table 2) mixed with the non-selective solvent ortho-dichloro-benzene(oDCB) between zero and 16% volume fraction are depicted in Fig. 17 versusinverse temperature. At the maximum solvent content the polymer miscibil-ity has been appreciably improved as evident by the strong decrease of thecritical temperature from about 70 to 30 C [32]. The susceptibility and corre-lation length show the typical deviation from linear mean field behavior, andthe results of the fit with the crossover functions of Eqs. 19 and 16 are shownby the solid lines; the agreement between experiment and theory is alwaysgood. In nearly all samples the critical condition can well be approached.

Fig. 17 Susceptibility and correlation length versus inverse temperature of a PB(1,4)/PSblend with different concentrations of a non-selective solvent. The experimental pointsare fitted by the crossover function. Improved miscibility is observed with increasedsolvent content

4.5.3Ginzburg Number and Critical Amplitudes

The effect of solvent on the Ginzburg number is shown in Fig. 18 for the twodPB/PS blends that differ in the vinyl content of the PB component (Sam-ple 2 and 3 in Table 2). Gi is plotted versus the total polymer concentrationon a double logarithmic scale. A strong increase of Gi with solvent content isobserved; at 80% polymer concentration a 5 to 10 times larger Gi is observedand over this concentration range the Ginzburg number approximately fol-

40 D. Schwahn

Fig. 18 Ginzburg number versus total polymer concentration in double logarithmic rep-resentation. As indicated by the dashed lines a much stronger increase of Gi with solventcontent is observed and is described by an exponent between 8 and 11 instead of 2 as ispredicted from theory

lows a scaling behavior with a large exponent between – 8.6 and – 10.8. Thisobservation is in strong contrast to the mean field theory prediction of a – 2power law behavior [80]. Similar experiments by Nose et al. [44, 45] with lightscattering also show deviations from this – 2 power law; in Fig. 7 of ref. [44],they present the Ginzburg parameter from two samples with larger solventconcentrations of about 50 and 80% and find a power law with an exponentof about – 4. An interesting observation is the stronger increase of Gi from thePB(1,2;1,4) blend. The overall Gi is lower and seems to approach the Gi fromthe other blend at about 70% polymer concentration.

Figure 19 depicts the critical amplitudes of the susceptibility and correla-tion length in the mean field and Ising approximations versus the polymerconcentration. While the amplitudes C+show a slight linear decrease by about20%, the mean field CMF show an increase; more quantitatively, in doublelogarithmic representation one evaluates exponents from the slope which are(1.1±0.1) and – (1±0.25) for PB(1,4)/PS, while (0.35±0.5) and – (2.4±0.5)for PB(1,2;1,4)/PS within the mean field and Ising regimes, respectively.

The mean field critical amplitudes of the correlation length ξ of bothblends are the same within experimental uncertainty and increase by about10% for a 80% polymer content. The “Ising” amplitudes are smaller thantheir mean field numbers, and, instead, they decrease with polymer con-tent by about 20%. In addition, the amplitudes are different in both blends;PB(1,4)/PS shows slightly more than 10% larger values than PB(1,2;1,4)/PS.The mean field amplitude ξ0 is proportional to the polymer radius of gyrationand the square root of (1 + ΦPΓσ/ΓC) (Eq. 29); the polymer parameters arevery similar, (Table 2) so the result of equal amplitudes is not surprising. Onthe other hand, the Ising critical amplitudes seem to depend more sensitively


Fig. 19 Critical amplitudes C of the susceptibility and ξ0 of the correlation length ver-sus total polymer concentration. The “Ising” C slightly decreases by about 10%, while themean field amplitude shows a slightly stronger relative increase of 20%. The critical am-plitudes of the correlation length in the lower figure exhibits an increase of the mean fieldamplitudes and a decrease of the “Ising” amplitudes with polymer concentration

on the microstructure of the PB polymer component and may be affected bythe smaller segment length of the PB with higher vinyl content; a 62% vinylcontent PB shows a 9.3% smaller segment length than a 7% one [38].

In Fig. 20 the FH parameters of the blend-solvent systems have been de-picted versus the solvent content. In the upper figure the enthalpic Γh of bothblends are presented for ambient pressure, while in the two lower figures theentropic Γσ is depicted at ambient and 200 MPa pressure fields. At ambientconditions one observes for the PB(1,4) blend a linear increase of about 50%for Γh and Γσ while for the (1,2;1,4) copolymer blends yield an oscillatorybehavior with relative maximum and minimum values at about 3 and 12%solvent concentration, respectively. The difference of Γσ at the two pressurefields may give information about the free volume. One can reasonably as-sume that practically no free volume is “surviving” at 200 MPa. We proceedwith this analysis by applying a simplified ansatz Γσ = Γ 0

σ + ΦS/ΩS + ΦV/ΩVwith Ωi and Φi, respectively, the molar volume and volume fraction of thesolvent molecules and the units of free volume [36]. The straight line for Γσ at200 MPa describes the function Γσ = Γ 0

σ + ΦS/ΩS as the contribution fromthe solvent molecules, while its difference from the ambient pressure value

42 D. Schwahn

Fig. 20 FH parameter of two PB/PS blends (same as in Fig. 19) versus solvent concen-tration and at ambient and 200 MPa pressure fields. The Γσ at the two pressure fieldscan be used to estimate the volume fraction of cavities and their compensation by thenon-selective solvent molecules

is described by the ΦV/ΩV term. According to this ansatz a 10 to 15% freevolume content is evaluated for the solvent free sample, and this content con-tinuously decreases with solvent content. Above 10% solvent content no freevolume was observed for the copolymer blend, while a constant fraction ofabout 8% is found in the PB(1,4) sample. These experiments clearly show aninterrelation between free volume and non-selective solvent.

5Crossover to the Renormalized 3D-Ising Critical Behavior

In some polymer blends an additional crossover from 3D-Ising to the so-called renormalized 3D-Ising behavior is observed near the critical point.Such a crossover is depicted in the upper and lower Fig. 21 for the sus-ceptibility and correlation length of two PB/PS blends (Samples 2 and 4 inTable 2) [32, 81]. The characteristics of such a crossover are larger criticalexponents in comparison to the Ising case and a shift of the critical tempera-ture. This type of crossover behavior has been systematically studied by thegroup of Nose in polymer blend-solvent systems [45]. In those systems the


Fig. 21 Crossover of susceptibility and correlation length from Ising to the renormalizedIsing behavior near the critical point as observed in the dPB(1,2)/PS and dPB(1,4)/PSblends. The crossover is much more pronounced in the PB(1,2) blend than in the PB(1,4)blend. A selectivity of the cavities in the PB(1,2) blend might be an explanation of thelarger effect

renormalized Ising behavior is interpreted based on the appearance of thetotal polymer density fluctuations near the critical point. So, in order to inter-pret the observations of the renormalized Ising behavior in “pure” polymerblends, one might think of the stronger polymer density fluctuations and thecavities or excess free volume distribution as a third component.

5.1Hidden Variables – Fisher Renormalization

Binary polymer blend–solvent or blend-cavity samples can be considered asquasi binary mixtures as long as the total polymer concentration ΦP = ΦA +ΦB is constant and thermal fluctuations are only considered for the orderparameter represented by the composition of one component. Under suchconditions, the system obeys Ising critical behavior asymptotically close tothe critical temperature TC as discussed in the former sections. If the Gibbsfree energy of mixing ∆G is considered as the relevant thermodynamic po-tential, the equilibrium state is determined by the order parameter Φ, thetemperature and the pressure fields. In polymer-solvent systems one has inaddition the solvent volume fraction ΦS and its conjugate chemical potentialµS = ∂ΦS∆G

∣∣T,P,Φ (polymer and solvent concentrations give ΦP + ΦS = 1). In

44 D. Schwahn

the so-called isomorphism approach discussed by Anisimov et al. µS and ΦSare considered as “hidden” variables [82]. The scaling form of the ∆G “fluctu-ation” part is described as a function of two relevant parameter fields, namely,the temperature and the order parameter, which are both a function of pres-sure and the solvent chemical potential [9, 82]. Generally, one observes Isingcritical exponents when the pressure and solvent chemical potential are con-stant during a so-called isochoric approach to TC. Such an approach, however,is experimentally not possible. An approach to TC can only be realized withconstant solvent concentration, and this leads to a further crossover with therenormalized critical exponents determined as

γ ∗ = γ/(1 – α) ν∗ = ν/(1 – α) (31)

In terms of the critical exponent α = 0.11 of the heat capacity [83].For polymeric blends mixed with an non-selective solvent the transition

temperature T∗∗ or the temperature range ∆T∗∗ = (T∗∗ – TC) to a renormal-ized Ising behavior was estimated by Broseta et al. [80]; on the basis of meanfield parameters they found the scaling law ∆T∗∗ ∼= N–13.6Φ–17. This expres-sion leads to an extremely small temperature interval around TC, which is notvisible in our experiments, and suggests the behavior as being that of a quasibinary blend characterized by Ising universality near TC.

5.2SANS Results on Blends

The blend-solvent systems, whose susceptibility and correlation length aredepicted in Fig. 17, show a crossover from mean field to Ising critical behav-ior and are well described by the crossover functions (Eqs. 15 to 19), and nofurther transition to Fisher’s renormalized Ising behavior is observed. Thismeans that all blend-solvent samples can be considered as quasi-binary blendsystems in accord with the former discussion. The observation of a crossoverto a Fisher renormalized Ising behavior in the two solvent free dPB/PSblends as depicted in Fig. 21 is therefore surprising. The susceptibility of thedPB(1,2)/PS blend is plotted showing the crossover at T∗∗ = 124 C with thecritical exponents γ ∗ = 1.39 and 2ν∗ = 1.43 in agreement with the theoretic-ally expected number. In this blend the crossover occurs over a large tempera-ture range of about 18 K and leads to an about 12 K larger range of miscibility.The crossover observed for the correlation length of the dPB(1,4)/PS blend asdepicted in the lower figure looks qualitatively different and a slightly largerexponent than expected from theory was determined. The crossover startsat about 2 K above TC and leads to a slight enhancement of the critical tem-perature as indicated by the dotted line. It has to be mentioned that in orderto observe a renormalized crossover, the experiments have to be performedvery slowly over several hours in small temperature steps; so in a second andfaster experimental run no renormalized crossover was observed for the same


Fig. 22 The coefficient L2 of the Q2 term of the structure factor versus correlation lengthin double logarithmic presentation. According to theory the slope gives the Fisher ex-ponent η = 0.034 which is found a factor of two larger than predicted. The crossovertransition at T∗∗ has no effect on η as the critical exponents γ and ν are renormalizedin a same manner

dPB(1,2)/PS sample, and the phase transition occurred at a critical tempera-ture as extrapolated from the γ = 1.24 scaling law. This means that densityfluctuations in polymer melts must have large relaxation times of the order ofhours.

An additional proof of the crossover from Ising to renormalized Isingcritical behavior might be the functional dependence of the coefficient L2of the Q2 term in Eq. 12 versus the correlation length ξ . L2 is proportionalto the chain statistical segment length and follows a scaling law accordingto ξη=(2–γ/ν) in both regimes of Ising and renormalized Ising universalityclasses [24]. In Fig. 22 L2 of the d-PB(1,2)/PS sample has been depicted ver-sus the correlation length in double logarithmic presentation. The data followquite well a single line with a slope of η = (0.07±0.01) and no transition is ob-served at the crossover temperature T∗∗ which is indicated by the dashed line.The Fisher exponent is about twice as large as the predicted one.

5.3Renormalized 3D-Ising Critical Behaviorin Blends and Blend-Solvent Systems

Crossover to the renormalized Ising critical behavior was observed in twosymmetric (ΦA

∼= ΦB) PB/PS polymer blends without any solvent but notin the corresponding blends mixed with the non-selective solvent oDCP upto a concentration of 20 vol %. It might be instructive to compare the ob-servations in both blends with results from polymer blend-solution mix-tures obtained by the group of Nose [44, 45]. In their experimental workno crossover was found in symmetric blend-solvent samples in accord with

46 D. Schwahn

our SANS results. A transition to the renormalized Ising case was observedin two asymmetric blends with ΦA = ΦB and mixed with a selective andnon-selective solvent as shown in Figs. 22 and 23 of ref. [44]. Both samplesshow a characteristically different crossover behavior which looks very muchlike those observed in Fig. 21 for the pure blends. The blend with the selec-tive solvent shows qualitatively the same behavior as dPB(1,2)/PS, while thecrossover of the non-selective solvent-blend sample is small and similar tothat in the PB(1,4)/PS sample. This observation seems plausible as a selectivesolvent is stronger correlated with polymer composition fluctuations. Fromthese findings one might argue that cavities or free volume represent a thirdcomponent in polymer blend systems, which in some cases may lead to theobserved renormalized critical behavior. Furthermore, the observation of dif-ferent renormalized crossover characteristics in the upper and lower parts ofFig. 21 might give a hint of a distinct selectivity of the cavities with respect tothe PB(1,4) and PB(1,2) monomers. So, the absence of a renormalized Isingcrossover in the PB/PS/solvent samples might be understood from a compen-sation of the cavities by the non-selective solvent as discussed in context withthe pressure experiments in Fig. 20.

6Crossover to Isotropic Lifshitz Critical Behaviorin (A/B) Polymer Blend/(A-B) Diblock Copolymer Mixtures

Mixing two partially incompatible (A/B) homopolymers with small amountsof an (A-B) diblock copolymer leads to an improved miscibility in a simi-lar way as amphipathic molecules do in oil-water mixtures [for conveniencewe will abbreviate those systems as (A;B)]. At larger diblock content, mi-croemulsion and ordered phases are formed. A reason of this rather complexphase behavior is that the two constituents belong to different universal-ity classes. Diblock copolymers are of Brasovskii type [53]; they decomposeinto spatially modulated phases of mesoscopic length scale determined bythe polymer size, while homopolymer blends are of 3D-Ising type and de-compose into macroscopically large domains (see discussions in context withFig. 1). Within mean field approximation these two phases meet at the Lif-shitz line and the corresponding critical point. An important parameter is theratio of the homopolymer and copolymer molar volumes α =

√VAVB/VA–B.

We will limit our discussions to systems with α ≈ 0.17; in those systems oneexpects no tricritical Lifshitz point and no three-phase liquid region [55].

A first systematic study of such systems with SANS was performed onthe relatively large molar mass symmetric polyolefins PE and PEP and thecorresponding diblock copolymer PE-PEP, PE being polyethylene and PEPbeing poly(ethylene propylene) [46]. A mean field Lifshitz-like behavior was


observed near the predicted isotropic Lifshitz critical point with the criticalexponents γ = 1 and ν = 0.25 of the susceptibility and correlation length, andthe structure factor following the characteristic mean field Lifshitz behavioraccording to S(Q) ∝ Q–4. Thermal composition fluctuations seem not to berelevant here from the observation of the mean field critical exponents. Onthe other hand, fluctuations seem to exercise an influence as the Lifshitz crit-ical point is not observed and instead a one-phase channel of a polymericbicontinuous microemulsion phase appears [46], which could be clearly iden-tified by transmission electron microscopy (TEM) [47]. Equivalent one-phase“channels” were also observed in other systems [48–50]. In a quite recentstudy on (dPB;PS), which will be discussed in more detail below, the mi-croemulsion channel is separated be another Lifshitz line into a droplet andbicontinuous microemulsion phases [52, 84].

As this article is devoted to critical crossover phenomena in polymerblends, we limit our considerations to homopolymer blends of critical com-position mixed with diblock concentrations below the Lifshitz line.

6.1Phase Diagram of a (A/B) Polymer Blend/(A-B) Diblock Copolymer Mixture

The temperature-copolymer concentration plane at fixed ambient pressureof a (dPB;PS) blend phase diagram (sample 11 in Table 2) below Φ ∼= 0.1diblock content is depicted in Fig. 23. The homopolymer dPB/PS blend ofcritical composition was mixed with a symmetric dPB-PS diblock copoly-mer of about six times larger molar volume giving α =

√VAVB/VA–B

∼= 0.16.This phase diagram is quite typical for those systems. It is divided into sev-eral sections. At high temperature one has two homogeneous (disordered)phases which are separated by the Lifshitz line (LL). At concentrations be-low the LL at Φ < ΦLL the structure factor S(Q) looks similar to that ofbinary blends with its maximum at Q = 0. Above the Lifshitz line Φ > ΦLLone observes the characteristics of diblock copolymer melts with the max-imum S(Q∗) at a finite Q = Q∗. In all cases S(Q∗) represents a susceptibilitywith Q∗ = 0 and Q∗ = 0 below and above ΦLL, respectively. Within mean fieldapproximation the Lifshitz line is estimated at a constant concentration ac-cording to ΦLL = 2α2/(1 + 2α2) and should here be observed at ΦLL = 0.048.The experimental Lifshitz line, however, shows characteristic deviations; it isfound at slightly larger diblock concentrations and shows a pronounced de-pendence on temperature near the two-phase boundary. At high temperaturesthe Lifshitz line approaches a constant value of about 0.06; at around 70 Cit displays its largest value of 0.084, and approaches a value of 0.07 at lowtemperatures. A similar observation has been reported in ref. [48, 49] andwill later be explained as due to the effects of thermal composition fluctua-tions [57].

48 D. Schwahn

Fig. 23 Phase diagram in the temperature-diblock copolymer plane for the (dPB;PS) mix-ture below the Lifshitz line separating blend like from diblock-like phase behavior. Thefull dots and the solid line represent the critical points of a two-phase region. The hatchedarea indicates a crossover from Ising to isotropic Lifshitz critical behavior, and a dou-ble critical point DCP is at 7% diblock concentration. The Lifshitz line separates at highand low temperatures the disordered phases and droplet and bicontinuous microemul-sion phases (µE). Its non-monotonic shape near the DCP is caused by the strong thermalfluctuations

At diblock concentrations above ΦLL structures of bicontinuous mi-croemulsion (µE) and of lamella ordered phases were identified [46]. Whilethe lamella phase was predicted by mean field theory, the observation ofa microemulsion phase was unexpected. Polymer microemulsion phases arepresently the subject of active research both from theory [85] and experi-ment [52] and, in addition, seem to generate interest for industrial applicationbecause of its homogeneous structure on the µm length scale.

Below the Lifshitz line (Φ < ΦLL), a line of critical points has been deter-mined (full dots in Fig. 23) separating the homogeneous from the two-phaseregion. This phase boundary of critical temperatures was determined fromthe SANS susceptibility in a similar way to that described in the context ofFig. 5. These blends always fulfill the conditions of criticality as binodal andspinodal curves were observed at the same temperature. The solid line in thephase diagram is a guide for the eye and represents the critical points whichdecrease with increasing diblock concentration and terminate at a co-calleddouble critical point (DCP) at about Φ = 0.07. In the range between Φ = 0.06and 0.07, both an upper and lower critical solution phase boundary is ob-served. Below the LCST boundary a droplet microemulsion phase is formedwhich at higher diblock content above LL transforms into a bicontinuous mi-croemulsion phase.

Two critical universality classes are observed along the line of criticalpoints and are separated by the dashed area at Φ = 0.048. Below Φ < 0.048


a mean field to Ising crossover is present, while above Φ > 0.048 a mean fieldto isotropic Lifshitz critical behavior crossover is observed. Finally, aboveΦ > 0.06 the effect of the double critical point becomes visible. Quite gener-ally, the Lifshitz critical point is a multiple critical point connecting the phaseboundaries of macro-phase separation and of disorder-order transition. Sucha critical point, however, is only realized within a mean field approximation.Because thermal fluctuations are relevant, such a Lifshitz critical point can-not exist. This is because of the different strength of fluctuations above andbelow ΦLL, expressed by the Ginzburg criteria for blends in Eq. 20 and fordiblock copolymers in ref. [56].

6.2Structure Factor within Mean Field Approximation

Here we have the situation that one of the (A/B) monomers of the homopoly-mer and of the copolymer was always deuterated with the same relativeamount of deuterium. Under such conditions the structure factor S(Q) mea-sures thermal composition fluctuations with respect to the total monomerfraction, which corresponds to a scalar order parameter represented by thelocal concentration Φ = Φ(x) of the A or B monomers. The basic thermody-namic features of those systems near their binodal line are well described bythe common Landau expansion of the free energy according to

H =∫

ddx

c2(∇Φ)2 + c4(∇2Φ)2 + rΦ2 + uΦ4 + u6Φ6 (32)

representing a functional of the order parameter [9, 54, 55]. This Hamilto-nian H represents an extended form of Eq. 14 that includes higher expansionparameters in the gradient energy and in the order parameter. A principal ef-fect of the dissolved diblock copolymers is the reduction of the surface energywhich according to the Hamiltonian is described by a smaller parameter c2.This parameter is positive at low copolymer concentration, becomes zero atthe Lifshitz line ΦLL, and is negative for larger copolymer contents.

The Hamiltonian accounts for thermal composition fluctuations in thedisordered regime. For positive c2-values the structure factor S(Q) has thecharacteristic behavior of polymer blends: S(Q) is maximum at Q = 0, and thesusceptibility r–1 is correspondingly given by this S(Q = 0) value: r–1 = 2S(0).For negative c2-values the structure factor S(Q) has the basic characteristicsof block copolymer melts, i.e., the maximum value of S(Q) appears at a fi-nite Q-value, Q = Q∗. The susceptibility is then given by the structure factorat this Q∗-value. Mean field theory for symmetric copolymers predicts thatS(Q∗) will diverge at a critical point, and beyond that the system will order ona mesoscopic length scale through microphase separation. The Lifshitz criti-cal point is determined by two conditions, namely c2 = 0 and S–1(0) = 0. Withc2

∼= 0 in the vicinity of the Lifshitz point, the fourth order term of the gra-

50 D. Schwahn

dient energy, c4, becomes a leading term in the free energy, giving rise to thecharacteristic Lifshitz S(Q) ∝ Q–4 behavior of the structure factor. A Lifshitztricritical point would be realized in the case of the three conditions, namelyS–1(0) = 0, c2

∼= 0, and u = 0 and u6 and c4 all positive.The structure factor for the polymer blend-diblock copolymer system is

described within the random phase approximation by [54–56]

S–1(Q) = F(Q)/V – 2Γ (33)

where Γ is the FH parameter and F(Q) the inverse form factor. F(Q) canbe calculated in terms of the partial structure factors SAA, SBB and SABof all A and B monomers and describes the correlation between the A/Bmonomers,

F(Q)/V =SAA(Q) + SBB(Q) + 2SAB(Q)

SAA(Q)SBB(Q) – S2AB(Q)

(34)

For the ternary systems discussed in Fig. 23, F(Q) can be simplified to

F(Q)/V = 2/[SAA(Q) – SAB(Q)] (35)

for A-B diblock copolymers with molar volume V and homopolymers ofequal molar volume VA = VB and equal partial structure factors SAA = SBB.Assuming a polymer conformation of unperturbed Gaussian chains, F(Q) canbe written in terms of the Debye-function (Eq. 7) as

F(x) = 4/[

(1 – Φ)αPDebye(1, xα) – ΦPDebye(1, x) + 4ΦPDebye(0.5, x)]

(36)

where x = R2g Q2, Rg being the radius of gyration of the diblock copolymer;

and α the ratio of the molar volumes defined above. From the minimum ofF(x) one gets both the FH parameter ΓS at the spinodal and critical point, andthe corresponding characteristic Q = Q∗ value [54–56]. The structure factorin Eq. 33 can be expanded into powers of Q2 according to

S–1(Q) = S–1(0) + L2Q2 + L4Q4 + ... (37)

with coefficients determined from the parameters of the Hamiltonian(Eq. 32). The first term is equal to the susceptibility, S–1(0) = 2(ΓS – Γ ) forconcentrations less than the Lifshitz value. The coefficients L2 and L4 are pro-portional, respectively, to c2 and c4, and can be determined in terms of thepolymer parameters Rg, V, α, and the diblock concentration Φ. At the Lifshitzconcentration (c2 = L2 = 0), a characteristic mean field behavior S –1(Q) ∝ Q4

appears from this equation. A realization of such a transition from S(Q) ∝ Q–2

to ∝ Q–4 is demonstrated in Fig. 24 showing S(Q) from the (PEE;PDMS) blendin a Zimm representation for three copolymer concentrations below the Lif-shitz line [49]. The solid lines represent fits of Eq. 37 from which the threeparameters, namely, the susceptibility S(0) and the coefficients L2 and L4 areobtained. At Φ = 4.3%, S(Q) is well described by the Ornstein–Zernike ap-proximation with L4 = 0. For larger Φ the Q4 term in S(Q) becomes stronger


Fig. 24 Structure factor in Zimm representation for the PEE;PDMS blend with differentdiblock concentrations. The contribution of the Q2 term decreases with diblock contentbecause of the reduced strength of the surface energy. Instead the next higher expansionterm Q4 becomes relevant and dominates

as demonstrated for the Φ = 6% and 8.3% samples. In addition, S(0) is dimin-ished with increasing Φ according to the increased stabilization of the samplemiscibility caused by the diblock copolymers.

6.3Effect of Thermal Composition Fluctuations

Near the Lifshitz line thermal composition fluctuations are expected to be-come strong over a larger temperature range because of the reduced surfaceenergy (c2 ∝ L2 = 0), leading to a lower threshold force for thermal fluctua-tions. On a more abstract level this effect can also be interpreted in terms ofa larger upper critical dimension DU = 8 beyond which thermal fluctuationsbecome irrelevant, and Gi is twice as large as for ordinary binary polymerblends [86].

Near the critical temperature the structure function can be written in thefollowing form [54]

S–1(Q) = S–1(0)[1 + (Qξ)2 + Kp–2(Qξ)4] (38)

with the susceptibility S(0), the correlation length ξ , and the pre-factor Kp–2.The susceptibility follows the well known scaling law S–1(0) = C–1

+ τγ with thereduced temperature τ and the critical amplitude C+ (see Eq. 11), the cor-relation length is determined by ξ =

√S(0)L2, and the pre-factor varies as

Kp–2 = L4/(L22 ·S(0)) in terms of the susceptibility and the coefficients L2 and

52 D. Schwahn

L4 defined in Eq. 37. The parameter p represents a scaling field amplitudedetermined from the square gradient term of the Hamiltonian (Eq. 32) ac-

cording to p = c2/

√

4c4∣∣S–1(0)

∣∣. It is a measure of deviation from the Lifshitz

critical point; it becomes infinite at the ordinary critical point (S –1(0) = 0 andc2 > 0) and remains finite at the Lifshitz critical point when S –1(0) = 0 andc2 = 0 [54].

At the Lifshitz line and the corresponding critical temperature, the corre-lation length ξ looses its meaning as L2 = 0. ξ has to be redefined from thecoefficient of the then dominate Q4 term in Eq. 38. At smaller copolymer con-tents, the Q4 term in the structure factor in Eqs. 37 and 38 becomes negligible,and ξ and p follow, respectively, the scaling laws ξ0 t–νand p2 ∝ ξ(2+η)/L4.Because of L2 ∝ ξη the latter emerges with the Fisher exponent η = 2 – γ/ν

evaluated from the critical exponents γ and ν of the susceptibility and cor-relation length, respectively [34]. ξ and p become accordingly both infinite atthe critical temperature.

6.4SANS Results from below the Lifshitz Line

Figure 25 displays the effects of thermal composition fluctuations on the in-verse susceptibility S(0) for a (PEE;PDMS) mixture (sample 10 in Table 2)versus 1/T for different diblock concentrations below the Lifshitz line [48].The critical temperatures determined from S–1(0) = 0 decrease with increas-ing diblock content in a similar way as shown for the (PB;PS) blend (Fig. 23).The Φ = 4.3% sample behaves as a pure blend: At high temperatures S–1(0)

Fig. 25 Susceptibility versus inverse temperature of the PEE;PDMS blend at differentdiblock content. The increase of the curvature is clearly visible at higher diblock content


shows the mean field and near the critical point the 3D-Ising characteristicswith a crossover regime in between as demonstrated by the fitted solid lineof the crossover function (Eq. 19). The dashed line in the figure representsthe asymptotic 3D-Ising scaling law with the critical exponent γ = 1.24 as cal-culated from the fit parameters of the crossover function. The susceptibilityof the 0.06 to 0.083 diblock concentration samples shows an increased curva-ture which can be described by the scaling law S–1(0) = C–1τγ with the criticalexponent γ larger than 1.24 as depicted by the solid lines.

The increase of the γ -exponent indicates a crossover to the universal-ity class of the isotropic Lifshitz case. In contrast to the Ising case of the4.3% sample (dashed line), an appreciably larger asymptotic critical rangeis observed as the corresponding asymptotic scaling law describes the sus-ceptibility over the whole experimental temperature range. This observationmeans a strong increase of the Ginzburg number as qualitatively expectedfrom the corresponding Ginzburg criterion Gi ∝ 1/V0.4 which shows a muchweaker dependence on polymer size than polymer blends (Gi ∝ 1/V) [54].A similar analysis was also performed for the correlation length according toξ = ξ0τ

–ν with the critical exponent ν as published in [49].In two samples with concentrations 6.7% (not shown in Fig. 25 for clarity)

and 8.3% an additional crossover was observed very near the critical tem-perature in which there is further stabilization of the homogeneous phase,e.g. a lower critical temperature. In this range the susceptibility was analyzedby the same scaling law and is demonstrated for the 8.3% sample in the insetof Fig. 25. Such a crossover was not observed in the 7.4% sample. The lat-ter experiment was performed in larger temperature steps as visualized bythe positions of experimental points. Therefore, this sample spent less timein the region of strong thermal fluctuations. This means that two binodal andspinodal lines exist in the range of 0.06 ≤ Φ ≤ ΦLL; phase separation occurseither at the lower or upper boundary, depending on whether a crossoverto the “new” universality class could occur or not. Such a crossover, how-ever, is only observable if the experiments are performed sufficiently slowly.We abbreviate this new universality class as the renormalized Lifshitz criticalbehavior. An explanation of this crossover might be related to rearrange-ments of the diblock copolymer as caused by strong thermal compositionfluctuations near TC that would further stabilize the sample against phasedecomposition. Such an effect was indeed observed by simulation studies onspinodal decomposition; within the early to intermediate time regime of spin-odal phase decomposition the block copolymers became accumulated at theinterface of the domains [87]. So, this universality class seems quite analogousto the renormalized 3D-Ising critical behavior as discussed in Sect. 5.

The SANS data from the (PEE;PDMS) mixture between 9 and 11% diblockconcentration (not shown here) [49] and the (PB;PS) mixture between 6 and8% [52] exhibit at low and high temperatures the characteristics of diblockcopolymers, i.e. S(Q) shows a maximum at the finite Q∗, while at intermediate

54 D. Schwahn

temperatures the behavior corresponds to those of homopolymer blends, i.e.S(Q) has a maximum at Q = 0. This means that the Lifshitz line was crossedtwice as its concentration changes with temperature. This is shown in thephase diagram of the (PB;PS) mixture (Fig. 23); the diblock concentrationof the Lifshitz line changes with temperature near the two phase region andis similar to that of the (PEE;PDMS) mixture. The apparent non-monotonicshape of the Lifshitz line could be explained quite recently as due to thermalcomposition fluctuations. Renormalization group calculations by Kudlay andStepanow [57] show that at high and low temperatures ΦLL approaches theconstant mean field value when thermal fluctuations become negligible, andΦLL changes with temperature when fluctuations become strong.

In the (PB;PS) mixture a crossover from Ising to isotropic critical Lif-shitz behavior was observed at about 4.8% diblock concentration as indicatedby the dashed area in the phase diagram of Fig. 23. On the other hand,a crossover to a “renormalized” Lifshitz critical behavior was not observed inthis system. The susceptibility critical exponent γ of both systems has beendepicted in Fig. 26 versus diblock concentration. A crossover of the exponentγ at about 4.8% and 6.2% from the Ising 1.24 to larger values is visible forthe (PB;PS) and (PEE;PDMS), respectively. A constant γ = (1.62 ± 0.01) and

Fig. 26 Critical exponent of the susceptibility γ versus diblock content of two homopoly-mer blend-diblock copolymer melts. The crossover to the renormalized Lifshitz criticalbehavior with γ = 1.62 is clearly visible. Differences of the two blends are remarkable:In PEE;PDMS a plateau of γ = 1.62 and a further crossover to a “renormalized” Lifshitzbehavior is found. In PB;PS, on the other hand a double critical point with the Lifshitzcritical exponent γ = 1.62 is observed


renormalized γ = (2.42 ± 0.01) is obtained for (PEE;PDMS). In (PB;PS) thesituation is more complex because of the occurrence of a double critical pointat about 7% diblock concentration.

A double critical point (DCP) represents an endpoint of an UCST and LCSTline of critical points. The LCST critical line is observed over a limited diblockconcentration of about 1%. This means that the homogeneous phases aboveand below the two-phase regime must represent different phases, as the com-patibility is achieved by, respectively, the entropic and enthalpic term of theGibbs free energy of mixing; the SANS experiments show that the homoge-neous phase at lower temperature is represented by a droplet microemulsionphase which must represent a more ordered state than the other two phasesat higher temperature.

Double critical points in general and the observed one in the (PB;PS)system are characterized by a strong increase of γ if one applies the scal-ing law of S–1(0) = C–1τγ with the “conventional” reduced temperature field.If one, however, applies a modified reduced temperature field according toτ = (1 – TUC/T)(1 – TLC/T) then the inverse susceptibility delivers a criti-cal exponent γ of the corresponding universality class, which here is of theisotropic Lifshitz class [88]. At the DCP one has TC = TUC = TLC and thereforeS–1(0) = C–1τ2γ . The solid lines in the lower Fig. 26 describe such a behav-ior for the isotropic Lifshitz case. The critical exponents shown by the fulltriangles together with the critical temperatures TUC and TLC given as fullpoints in the phase diagram (Fig. 23) indeed lead to a crossover to the Lifshitzcritical exponent γ = 1.62 a number which was formerly determined for the(PEE;PDMS) system. Up to now the Lifshitz critical exponents in Fig. 26 couldexclusively be determined from the SANS experiments discussed in this art-icle. A theoretical determination was not yet possible because the theory lacksa small expansion parameter caused by the large upper critical dimension(DU = 8) [86].

7Summary and Outlook

In this article we considered thermal composition fluctuations in binary poly-mer blends under various conditions. Samples of critical composition werestudied in temperature and pressure fields, with small additions of a non-selective solvent, or in mixtures with a symmetric diblock copolymer with thesame monomers as the homopolymers. Blends of critical composition werechosen in order to follow the fluctuations up to the critical point which rep-resents the stability limit of miscibility. The strength of thermal fluctuationsis estimated by the Ginzburg criterion which in the incompressible limit fol-lows the universal scaling law 1/V and predicts that binary polymer blends

56 D. Schwahn

should in principle be described by the FH mean field theory. Recent theoret-ical considerations, however, show that compressibility or packing effects ofthe polymer chains lead to much larger Ginzburg numbers and therefore toa stronger effect of thermal composition fluctuations. These considerationsare supported by Monte Carlo simulations and scattering experiments.

Most of the data presented here were obtained from small angle neutronscattering (SANS) experiments. The main reason is the strong scattering con-trast between the chain components of deuterated and protonated monomersand the low background. The structure factor S(Q) measures the degreeof thermal composition fluctuations and delivers the susceptibility S(Q = 0)which is linked to the corresponding thermodynamic potential ∆G(T, P, Φ)by the fluctuation-dissipation theorem. S(Q = 0) follows a scaling law in termsof the reduced temperature in the asymptotic mean field and Ising regimesand becomes infinite at the critical point.

Temperature and pressure fields determine the equilibrium state of a bi-nary polymer blend of fixed composition. Some arguments in the literaturestate that temperatures and pressures have similar effects on S(0), so thatS(0) can be transformed into a universal curve. Such an argument might betrue within mean field approximation. However, as the Ginzburg number isnon-universal and is a function of pressure, both external fields influencethe equilibrium state in different manners. Pressure reduces Gi as the pack-ing of the polymers is increased. In most cases, increased pressure leads toa lower miscibility and to a decrease of the enthalpic and entropic termsof the FH parameter. This means that the temperature of the phase bound-ary is sufficiently large that the entropic part is dominant. In a few polymerblends, however, an abnormal pressure induced increase of the miscibility isobserved. In those blends, in contrast to most of the other blends, an increaseof enthalpic and entropic terms is found, again the entropic one the dominateterm. The microscopic origin of such opposite behavior of the FH parameteris not yet understood.

Polymer blends mixed with small amounts of a non-selective solvent wereexplored in both external fields. The Ginzburg number shows a remarkablystrong increase with solvent content; a power law of the total polymer con-tent with exponents between – 9 and – 11 is observed. This result is in strongcontrast to mean field predictions proposing an exponent of minus two. Thisagain is a strong hint that the Ginzburg criterion in Eq. 20 is strongly affectedby the non-combinatorial entropy of mixing Γσ , e.g., a larger Γσ leads toa huge increase of Gi because of the large exponents in the Ginzburg criterionof Eq. 20. A further observation is that the excess free volume is compensatedby the solvent molecules.

The rare case of a crossover to the Fisher renormalized Ising behavior wasfound in the two PB(1,4)/PS and PB(1,2)/PS blends near TC (see Figs. 22–23),but these two cases look quite different. Similar observations were made inasymmetric (ΦA = ΦB) polymer blend-solvent systems. A large effect was


only found if a highly selective solvent was chosen. The dPB(1,4)/PS re-sults shows a crossover very near TC similarly to their non-selective solventcase, while the crossover in dPB(1,2)/PS qualitatively agrees with the be-havior expected when their system is mixed with a highly selective solvent.This coincidence is a hint that cavities of excess free volume, mathemat-ically representing a third component, lead to the observed renormalizedcritical behavior. One may furthermore speculate about a possible selectiv-ity of voids in the dPB(1,2)/PS blend. No renormalized Ising behavior wasobserved in the PB/PS/solvent samples which can be understood from com-pensation of the selectivity of the cavities by the non-selective solvent (seeFig. 20).

A mixture of a symmetrical A-B diblock copolymer and a critical binaryA,B homopolymer blend leads to a complex phase diagram representativeof the characteristic phase behavior of binary blends and diblock copoly-mers, namely, phase decomposition into macroscopically large domains andlamella ordered phases on a length scale determined by the diblock size, re-spectively. This polymer system displays several similarities with mixturesof amphipathic molecules in oil-water. Here we discussed two systems con-taining a symmetric diblock copolymer with roughly five times larger molarvolume than the homopolymers. In these systems an isotropic Lifshitz criti-cal point and Lifshitz line is predicted in the transition range of both orderedphases. An isotropic Lifshitz critical point, however, is not observed as it is de-stroyed by the different strengths of the fluctuations in homopolymer blendsand diblock copolymers. Instead a droplet and bicontinuous microemulsionstructure appears in this range. Below the characteristic Lifshitz border linethermal composition fluctuations show a crossover from 3D-Ising to isotropicLifshitz critical universality. In the (PEE;PDMS) blend a further crossover toa renormalized Lifshitz behavior is observed, while in the (dPB;PS) systema double critical point (DCP) is found.

From these findings it becomes clear that the mean field approximationcan only be a poor approximation in those three component polymer mix-tures. In particular, critical fluctuations become strong near the isotropicLifshitz critical point since the stabilizing force from the surface energy be-comes small. There are indeed several observations which originate fromstrong thermal fluctuations such as the microemulsion phase, the destruc-tion of the isotropic Lifshitz critical point, and the bending of the Lifshitzline near the two-phase region. The criticality is quantitatively character-ized by the susceptibility critical exponent γ which shows a crossover fromIsing to isotropic Lifshitz critical behavior (Fig. 26). There are a few dis-tinctions between both systems studied. In the (PEE;PDMS) mixture theLifshitz critical exponent of γ = 1.62 is observed over a plateau of about 2%diblock concentration. In the (dPB;PS) blend, γ depends on the defined re-duced temperature field. Use of the conventional reduced temperature leadsto a 2γ = 3.24 at the DCP, while use of the modified reduced temperature

58 D. Schwahn

field, namely τ = (1 – TUC/T)(1 – TLC/T), which is defined by the UCST andLCST critical temperatures TUC and TLC, leads to γ = 1.62 at the DCP, whichis the same as for the other blend (Fig. 26). The disordered phases aboveand below the two-phase region must be different as the driving force forphase decomposition is of enthalpic and entropic origins, respectively; thedisordered phase at lower temperature is represented by a droplet microemul-sion phase which must be the more ordered state in comparison with theother two phases.

A principle message from this article should be that thermal composi-tion fluctuations play an essential and complex role in polymer blends. Thiswas demonstrated by the crossover phenomena between mean field to 3D-Ising, renormalized 3D-Ising, isotropic Lifshitz, and renormalized isotropicLifshitz critical behavior. The observation of such crossover phenomena maygive hints to microstructural effects such as density fluctuations for the renor-malized 3D-Ising case. The range of relevant thermal fluctuations in the Isingcase is sensitively determined by the entropic term of the FH parameter. So,in order to accurately determine the FH parameter from scattering experi-ments the susceptibility must be analyzed in terms of a crossover functiondescribing the transition from mean field to 3D-Ising behavior.

The FH parameter is a phenomenological parameter describing the phasebehavior of polymer blends and is determined from the structure factorat Q = 0, thereby representing an average over the whole sample volume.Information about the microstructural origin of the FH parameter can beachieved from exploration of the blends under different conditions of molarvolume, chain microstructure, additives, pressure and temperature fields. Inthis respect, the investigation of simple model systems is most promising asdemonstrated by the SANS studies of the dPB/PB copolymer mixtures withdifferent vinyl contents and molar volumes (Figs. 9 and 10). In this blendseveral microstructural contributions to the FH parameter could be sepa-rated, such as the chain-end effect, the polarization effect due to deuteration,and finally the enthalpic and entropic terms of the FH parameter could befully analyzed in terms of the van der Waals interaction energy and a non-combinatorial entropic expression from the lattice cluster theory calculations,respectively (Fig. 10). Binary blends of chemically different chains quite natu-rally lead to a more complex situation which cannot be analyzed so easily. Butsome microstructural aspects can be visualized for the blend-solvent systemsin large pressure fields showing an interrelation between cavities and solventmolecules.

The presented experimental studies and corresponding theoretical back-ground on fluctuation phenomena in polymer blends have shown a broadfield of exploration. Many of the observed phenomena are not yet understoodand could only be explained qualitatively. Hence, this article will hopefullyhelp to stimulate further experimental and theoretical interest in this field.


Acknowledgements I would like to sincerely thank Dr. Henrich Frielinghaus and Dr. Vi-taliy Pipich for their cooperative work over the recent years and for critically reading themanuscript. Finally, I would like to express my thanks to Prof. Karl F. Freed for carefullyreading the manuscript and for his several comments.

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Influence of Monomer Molecular Structureon the Miscibility of Polymer Blends

Karl F. Freed () · Jacek Dudowicz

James Franck Institute and Department of Chemistry, The University of Chicago,Chicago, IL 60637, [email protected]

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641.1 Critical Review of Flory–Huggins Theory:

Motivation for Developing Improved Analytical Theory . . . . . . . . . . . 651.2 Strategies for Developing Improved Analytical Theories . . . . . . . . . . . 68

2 Theoretical Basis for the Lattice Cluster Theory . . . . . . . . . . . . . . . 692.1 Brief Description of Basic Theoretical Principles . . . . . . . . . . . . . . . 702.2 Tests of the LCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3 Simplifying High Molecular Weight,Incompressible Limit of the LCT (the SLCT) . . . . . . . . . . . . . . . . . 74

3.1 Description of Monomer Structures in the LCT . . . . . . . . . . . . . . . . 743.2 Basic Thermodynamic Properties . . . . . . . . . . . . . . . . . . . . . . . 75

4 Classes of Blend MiscibilityAssociated with Structural Asymmetries . . . . . . . . . . . . . . . . . . . 78

4.1 Basic Critical Properties of Binary Blends in the SLCT . . . . . . . . . . . . 804.2 Essential Characteristics

of the Four Classes of Binary Blend Miscibility . . . . . . . . . . . . . . . . 824.3 Experimental Support for New Classes of Polymer Blend Miscibility . . . . 85

5 Application of the SLCT to LCST Polyolefin Blends . . . . . . . . . . . . . 86

6 Introduction of Chain Stiffness into the SLCT . . . . . . . . . . . . . . . . 88

7 Application of the SLCT to UCST Polyolefin Blends . . . . . . . . . . . . . 917.1 Analysis of the PP/hhPP blend . . . . . . . . . . . . . . . . . . . . . . . . . 93

8 Copolymers Blends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.1 The BLCT: General Assumptions . . . . . . . . . . . . . . . . . . . . . . . . 968.1.1 The BLCT Free Energy Expression . . . . . . . . . . . . . . . . . . . . . . . 978.1.2 The SANS χSANS Parameter in the BLCT . . . . . . . . . . . . . . . . . . . 988.1.3 Special Limits of AxB1–x/CyD1–y Copolymer Blends . . . . . . . . . . . . . 998.2 Application of the BLCT

to Norbornene/Ethylene Copolymer Blends . . . . . . . . . . . . . . . . . 1018.2.1 Influence of Stiffness and Steric Interactions . . . . . . . . . . . . . . . . . 1038.3 Isotopic Mixtures of Saturated Polybutadienes (sPB) . . . . . . . . . . . . . 1058.3.1 The χSANS Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.3.2 Comparison with Experimental Data . . . . . . . . . . . . . . . . . . . . . 108

64 K.F. Freed · J. Dudowicz

8.4 Unsaturated Isotopic Polybutadiene Blends . . . . . . . . . . . . . . . . . . 1118.4.1 The χSANS Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1128.4.2 Comparison with Experimental Data . . . . . . . . . . . . . . . . . . . . . 113

9 Order–Disorder Transitionin Polystyrene-b-Poly(n-Alkyl Methacrylate) Diblock Copolymers . . . . . 115

9.1 Unified Modeling of the n-alkyl Methacrylate Systems . . . . . . . . . . . . 115

10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12010.1 Improved Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Abstract Polymer blends are formulated by mixing polymers with different chemicalstructures to create new material with beneficial properties of the individual components.While Flory–Huggins (FH) theory explains some basic trends in blend miscibility, thetheory completely neglects the dissimilarity in monomer structures that is central to thefabrication of real blends. We systematically investigate the influence of monomer struc-ture on blend miscibility using the lattice cluster theory (LCT) generalization of the FHmodel in the limit of incompressible, high molecular weight blends where analytical cal-culations are tractable. The well-known miscibility pattern predicted by FH theory isrecovered only for a limited range of monomer size and shape asymmetries, but addi-tional contributions to the LCT entropy and internal energy of mixing for polymers withdissimilarly shaped monomers leads to three additional blend miscibility classes whosebehavior is quite different from the prediction of classical FH theory. Several illustrativeapplications of the LCT provide new molecular-scale interpretations for many nontrivialphenomena occurring in polymer systems. The applications also illustrate the predictiveability of the theory and its usefulness in analyzing thermodynamic data for a wide var-iety of polymer mixtures, ranging from binary homopolymer blends to various types ofcopolymer systems.

Keywords Polymer blends · Miscibility · Statistical copolymers ·Nonrandom mixing effects · Monomer structural asymmetry

1Introduction

Despite the fact that the production of polymers (including the preparationof monomers) accounts for roughly half of the chemical industry in the US,polymer science represents a minuscule portion of the education of chem-istry students here. Nevertheless, the fields of polymer physics and polymerchemistry afford great scientific and technological interest. Polymer systemssustain a wide variety of phases, and materials fabricated from the differentphases may exhibit enormously varying properties. A number of challeng-ing conceptual questions concern phase separation in polymer blends, phasetransitions in liquid crystals and gels, and the glass transition in polymersystems. For instance, many commercially important materials are alloys

Influence of Monomer Molecular Structure on the Miscibility of Polymer Blends 65

of polymers having different chemical and physical characteristics, and thestability and state of dispersion of polymer blends are often crucial in applica-tions. Thus, it is both a scientific challenge and of technological importance toelucidate the molecular features promoting the mixing of polymers and gov-erning the compositions of the coexisting phases when polymers do not mix.This challenge is heightened by the recent development [1, 2] of metallocenecatalysts for polyolefin production, which has revolutionized the tailoring ofpolyolefins into thermoplastic elastomers and other improved materials. Thisprogress in polyolefin synthesis creates a demand for a structure-propertytheory to guide the syntheses. The utility of theory for designing novel ad-vanced materials depends, to a large extent, on its ability to describe theessential molecular factors determining the system’s thermodynamics andconsequently its phase behavior.

The overwhelming majority of thermodynamic treatments for multicom-ponent polymer systems are based on Flory–Huggins (FH) [3–7] theory andvarious embellishments thereof [8–16]. While these theories are extremelyuseful in correlating data from measurements of different properties and inexplaining many facets of the miscibility of polymer blends, the theories suf-fer from several significant deficiencies that limit their predictive abilitiesand, in particular, their utility for tackling many technologically importantissues [17]. The limitations of prior theories are discussed in Sect. 1.1 to em-phasize the scientific challenges and opportunities open to theories that aredevoid of these deficiencies and to accentuate the motivation for the devel-opment of the lattice cluster theory (LCT) [17–22], which is the subject ofthe present review. Several illustrative applications of the LCT provide newmolecular-scale interpretations for many enigmatic phenomena occurring inpolymer systems. The applications further demonstrate the predictive poten-tial of the theory and its usefulness in analyzing thermodynamic data fora wide variety of systems, ranging from binary homopolymer blends to var-ious types of copolymer mixtures.

1.1Critical Review of Flory–Huggins Theory:Motivation for Developing Improved Analytical Theory

FH theory [3–7] was originally derived from a simple lattice model thatimposes a discrete spatial representation of the conformations accessible toindividual polymer chains and, per force, to collections of many intertwinedpolymers in the liquid phase. The polymers are taken to have random con-figurations, which arise from placing individual monomers at single latticesites, subject to strict constraints of single occupancy for any lattice site. Thesame single occupancy constraint applies to solvent molecules when theyare also present. The exact counting of configurations for a dense system offlexible polymers is beyond feasibility, so FH theory [3–7] employs a very


simple mean-field approximation that effectively ignores the details of thepolymer chain connectivity and, therefore, that cannot distinguish betweenlinear, star, branch, comb, etc. polymer architectures.

The basic physical content of FH theory [3–7] is embodied in its treat-ment for the entropy and enthalpy of mixing. In this regard, polymers behavequite differently from small molecules. The large mixing entropy in mixturesof small molecules is generally responsible for their miscibility in the liquidphase because the enthalpy of mixing is frequently unfavorable. The entropyof mixing for small molecule fluids is proportional to the familiar

∑

i niln(ni)term (called the combinatorial entropy), where ni denotes the number ofmolecules of species i in the system. As extended to large irregular shapedobjects such as polymers, FH theory replaces the

∑

i niln(ni) term by the ex-pression

∑

i niln(ϕi), where ϕi is the volume fraction for polymer species i.The large number of monomers present in individual polymer chains im-plies that a given mass of a polymer blend contains a rather small number ofmolecules compared, for instance, to the same mass of a monomer mixture.Consequently, the entropy of mixing is small, and the entropic force drivingpolymer miscibility is weak.

FH theory [3–7] employs a random mixing (regular solution) approxima-tion for the energy of mixing. More specifically, if εij is the van der Waals at-traction energy between monomers of species i and j that are placed at neigh-boring lattice sites, and if the exchange energy is defined as ε = εii + εjj – 2εij,the dimensionless energy of mixing for an incompressible binary mixture isequal to the product of volume fractions ϕiϕj and the effective Flory interac-tion parameter,

χ = zε/2kT , (1)

where kT is the thermal energy and z is the number of lattice sites that arenearest neighbors to a given lattice site. For interactions dominated by disper-sion forces, the χ parameter is expected to be positive, so the occurrence ofa positive χ and a small entropy of mixing explains why polymers of differ-ent species generally do not mix in the liquid phase. However, some polymerspecies do mix well. Thus, many technologically important materials are fab-ricated from mixtures of different polymers, and the resultant compositematerials often exhibit a synergistic combination of beneficial characteristicsof the individual blended components. For instance, blending a soft rubberypolymer and a brittle tough polymer may yield a strong, elastic substance.Consequently, a greater understanding of the factors governing blend misci-bility is of technological interest, and the present review illustrates how theseinsights into polymer blend miscibility can emerge from theories that de-scribe how the χ parameter depends on the detailed monomer molecularstructure and the intermolecular interactions.

The FH expression for χ in Eq. 1 provides little further insight into thephysical features that control polymer miscibility in view of the fact that typ-


ical values of χ for miscible polymers are only of the order of a few Kelvin,while typical values for the individual εij are hundreds of Kelvin, leading toa low likelihood for computing the χ parameter from first principles. Con-sequently, the χ parameter is traditionally treated in a phenomenologicalfashion. Nevertheless, the theory has been immensely useful in correlatingdata from different types of measurements, including those associated withthe miscibility of polymer blends. Hence, studying the behavior of χ fora set of blends formed from a homologous series of polymers should assistin identifying the factors governing the thermodynamic behavior of poly-mer systems. Indeed, a large body of empirical data for a series of polyolefinblends exhibits glaring discrepancies from the predictions of FH theory.

Inspection of Eq. 1 indicates that χ is predicted by FH theory [3–7] asinversely proportional to temperature and as independent of polymer com-position (the ϕi), molecular weights, and pressure. However, χ is often foundempirically to depend on the blend composition, sometimes to vary with mo-lecular weights, and to be a function of pressure. The pressure dependence ofχ has been observed in several experiments [23, 24] subsequent to our predic-tions [21] that χ is not insensitive to pressure, and the experiments show thatthe theory correctly estimates [25, 26] the magnitude of this technologicallyimportant dependence. Moreover, as reviewed by Flory [12] over 30 years ago,the temperature dependence of χ generally exhibits the form χ = A + B/T,with the presence of a temperature-independent portion A, called the “en-tropic” part of χ, standing in sharp contrast to the predictions of FH the-ory [3–7], which rigorously omits the A term. These strong contradictionsbetween theory and experiment logically imply (as noted by Flory [12]) thatFH theory is wrong concerning the predicted behavior of χ (i.e., the theoryis grossly inadequate to explain general observations for χ). Thus, treating χ

as a phenomenological parameter and allowing this empirical χ to displaythe observed variations (with temperature, composition, pressure, etc.) tech-nically represents pure empiricism (as Flory recognized) and not consistentuse of FH theory, per se. While this empiricism is perfectly reasonable for ana-lyzing experimental data, the apparent successes of this empiricism cannotovershadow the lack of theoretical underpinnings that are necessary for moreambitious predictive purposes than just correlating data for a specific system.All mention of FH theory below refers to the original definition in Eq. 1 andnot to the phenomenological treatment of χ that effectively is a tautology.

The above-mentioned limitations to FH theory [3–7] are merely the tipof the iceberg when questions are raised concerning the molecular design ofnew materials. Within a lattice model, FH theory specifies, for instance, thatmonomers of different polymer species and solvent molecules (if present) alloccupy single-lattice sites, i.e., have the same shapes and volumes. Realistsmay argue that actual differences in monomer volumes are maintained bythe theory if different numbers of monomers of different species are takento reside at a given lattice site. However, this assumption still grossly ignores


significant details of monomer structure, such as disparate monomer mo-lecular sizes and shapes and, consequently, the influences of these structuraldifferences on the thermodynamics of polymer systems. Thus, a fundamentalquestion posed in formulating an analytical molecular-based theory for thestatistical thermodynamics of liquid-phase polymers is associated with rec-ognizing the degree to which the distinct chemical structures of the individ-ual monomers are relevant. This question represents a substantial departurefrom the traditional strategy used in theoretical polymer physics. The lattertheoretical treatments focus on universal polymer properties that arise intrin-sically from polymer chain connectivity and that are consequently indepen-dent of the detailed chemical nature of the system, apart from a few empiricalparameters, such as the χ parameter discussed above. Our molecular-basedtheory, instead, relates these empirical parameters, and particularly the χ

parameter, to specific chemical characteristics of individual monomers andsolvent molecules and to the thermodynamic state of the system.

1.2Strategies for Developing Improved Analytical Theories

A considerable body of recent theoretical analyses for polyolefinblends [27–32] attempts to relate the χ parameter to differences (δ1 – δ2)2

in solubility parameters, where (δi)2 is proportional to the cohesive energyof the pure component i (i.e., to the difference between the internal energiesof the pure liquid and vapor). This cohesive energy is often approximatedby the more accessible internal pressure (∂U/∂V)T and, therefore, does notcontain contributions from the athermal packing entropy that is responsiblefor the appearance of an important component of the entropic χ parameterfor many polymer systems. Hence, when the empirical χ parameter containsa significant temperature-independent portion χs, solubility parameter the-ory can at most only include χs as a theoretically unjustified, purely empiricalquantity, just as in traditional applications of FH theory. Moreover, manypolymer solubility parameters are determined from thermodynamic data forpolymer solutions where the entropic component of χs may be dominant [12].Therefore, this type of empiricism contradicts the fundamental assumption ofsolubility parameter theory that χ is of energetic origin. Thus, a major con-ceptual extension of solubility parameter theory would involve incorporatingthe entropic χ in a theoretically correct fashion and relating both the entropicand energetic contributions of the χ parameter to the monomer chemicalstructures and their interactions.

A number of analytical off-lattice approaches are based on the extensionof various simple one-component analytical models (such as Flory–Hugginstheory, Flory’s equation of state model, or Guggenheim theory) to multicom-ponent systems by the use of some ad hoc combining rules [5–8, 15, 16]. Theseapproaches likewise neglect the influence of chain connectivity and monomer


structure on local nonrandom mixing effects whose many ramifications wewish to understand. While improving the predictive abilities of analytical the-ories, it is also desirable simultaneously to develop the numerically intensiveoff-lattice integral equation methods that are capable of describing the prop-erties of polymer systems that contain structured monomers, of course, at theexpense of heavy numerical computations. (See, for example, recent numeri-cal studies of polyolefin melts using PRISM theory [33–35].)

Another strategy for developing molecular-based theories consists in gen-erating systematic corrections to FH theory that include details of monomerstructure and describe the influence of short range correlations [17–22].While FH theory may be derived without invoking a lattice model, it istraditionally regarded as a lattice model approach because of its originalformulation. Since the traditional FH lattice model treats the monomers asstructureless units, a first step in this strategy of systematically improvingFH theory involves replacing the FH model with a modified (extended) lat-tice model in which monomers are represented as structured entities. Thus,the monomers are permitted to occupy several neighboring lattice sites tomimic the sizes, shapes, and structures of the actual molecules. Althoughchemists and chemical engineers often view lattice models as providingan oversimplified representation of reality, physicists find these models tobe extremely attractive since their use introduces enormous mathematicaltractability while still preserving the essential physics of the system. Becausethe simple mean-field approximations inherent to FH theory [3–7] do notdistinguish between different monomer structures, we have developed [18–22] a vastly superior, approximate analytical solution to the lattice model ofdense polymer systems. The resulting approach is called the lattice clustertheory to emphasize similarities with the Mayer cluster theory of nonidealgases [36].

2Theoretical Basis for the Lattice Cluster Theory

The lattice cluster theory (LCT) [18–22] is based on an extended lattice modelwhere monomers are endowed with molecular structures, as illustrated inFig. 1a and b, which depict united atom group structures for the monomersof several polyolefins and portions of a few polyolefin chains, respectively.Individual CHn (n = 0–3) groups are taken as occupying single lattice sites,and the bonds between united atom groups correspond to the C – C bondsin the actual molecules. The terminal segments of the side groups in Fig. 1aand b are methyl (CH3) groups, while the backbone united atom groups rep-resent CHn units, with n = 2, 1, or 0. These united atom models have longbeen applied in off-lattice theories and in computer simulations of linear and


Fig. 1 (a) United atom group models for monomers of poly(ethylene) (PE), poly(pro-pylene) (PP), poly(ethyl ethylene) (PEE), poly(hexene-1) (PH1), poly(ethylene propylene)(PEP), Poly(isobutylene) (PIB), and head-to-head poly(propylene) (hhPP). Circles des-ignate CHn groups, solid lines represent C – C bonds inside the monomer, and dottedlines indicate the C – C bonds linking the monomer to its neighbors along the chain.(b) United atom models of PP, hhPP, and PEP homopolymer chains constructed by link-ing monomers of (a)

branched alkanes [37–40], perhaps the simplest of all homologous molecularsystems.

FH theory [3–7] neglects all correlations that are present in the systemdue to bonding and excluded volume constraints and to the van der Waalsinteractions. The LCT, on the other hand, describes the thermodynamic con-sequences of the existence of short-range correlations that arise from packingconstraints and the different monomer–monomer interactions. Because onlyshort-range correlations are included, the LCT technically is also a mean-field theory. It is clear that the inclusion of these correlations in a theory ofpolymer systems requires developing fundamentally new theoretical conceptsand counting methods. The goal of the LCT is to determine the influenceof monomer structure on the thermodynamic properties of specific poly-mer systems. The basic physical underpinnings of the lattice cluster theoryare sketched in the remainder of this section, and the interested reader mayconsult the original literature for the lengthy, complicated details of the the-ory [18–22]. Those more interested in applications may skip ahead to Sect. 3and the following sections, which describe several predictions of a simpli-fied version of the LCT. In these sections, we analyze basic phenomena whoseexplanation lies outside the scope of simple FH theory.

2.1Brief Description of Basic Theoretical Principles

The partition function for the extended lattice model of polymer systemsis evaluated within the LCT by introducing approximations for the bondingconstraints and for the excluded volume and van der Waals interactions. Forsimplicity, we assume for now that all chains are fully flexible and focus firston the description of the partition function Zpack for a single athermal limit


polymer chain on a hypercubic lattice (a cubic lattice in d dimensions, soz = 2d). Generally, the chain connectivity constraint (i.e., the condition thatsuccessive monomers are linked to each other) is represented in Zpack bya product of the Boltzmann factors

bi = exp[– Ubond(ri–1 – ri)/kT] , (2)

where Ubond(ri–1 – ri) is the potential energy for the chemical bond betweenthe united atom units, labeled i – 1 and i, in the polymer and kT is the abso-lute temperature in energy units. Each of these Boltzmann factors bi simplifiesenormously in the lattice model to the condition that the vector ri–1 – ri con-necting units i – 1 and i is one of the z possible lattice vectors aβ(β = 1, ..., z)between nearest-neighbor sites on the lattice, where z is the number of near-est neighbors to any lattice site. The above simplification may be applied toeach bonding interaction as

exp[– Ubond(ri–1 – ri)/kT] →z

∑

βi–1=1

δ(ri–1 – ri – aβi–1 ) , (3)

where δ is the Kronecker delta function,

δ(i, j) = 1 if i = j ,

= 0 otherwise ,

and the sum over β appears because the bond may lie along any of z possibledirections.

The approximate treatment of the bonding constraints in Eq. 3 may bemotivated by recourse to the Flory theorem [3, 4], which states that in poly-mer melts it is impossible to discern whether a pair of nonbonded nearest-neighbor united atom groups belongs to different polymer chains or todistant portions of the same polymer molecule. Thus, in the lattice model de-scription of polymer systems, the excluded volume prohibition of multipleoccupancy of a site is more important than the consequences of long-rangechain connectivity. Based on the Flory theorem, we introduce the zeroth-order mean-field average

A = 〈δ(ri–1 – ri – aβi–1 )〉to reexpress the δ on the right-hand side of Eq. 3 exactly as

δ(ri–1 – ri – aβi–1 ) ≡ zA(1 + Xi–1,i) , (4)

with the bond correlation factor Xi–1,i given by

Xi–1,i = (A)–1[δ(ri–1 – ri – aβi–1 ) – A] . (5)

The overall chain connectivity Boltzmann factor for a single polymer chainwith specified positions ri of the united atom groups is the product (over all


N bonds in the chain) of individual terms in Eq. 3 and can be written usingEqs. 4 and 5 as

N∏

i=1

exp[– Ubond(ri–1 – ri)/kT] =N

∏

i=1

z∑

βi–1

A(1 + Xi–1,i) . (6)

The athermal limit partition function Zpack for packing a system of nmonodisperse purely flexible linear chains is computed by constructing theproduct of factors from the right-hand side of Eq. 6 for each polymer chainin the system and then by summing the resultant expression over all possiblepositions ri of the united atom units, subject to the strict excluded volumeconstraints that no two segments may occupy the same lattice site. Thus, theathermal packing partition function Zpack takes the form

Zpack =1

2nn!

∑

r01 =r1

2 =... =rNn

n∏

c=1

N∏

i=1

z∑

βci–1=1

[A(1 + Xci–1,i)] , (7)

where the index c = 1, ..., n labels the chains, i = 1, ..., N labels the bonds be-tween monomers i and i – 1 within a chain, and the factors of 2–n and 1/n!account, respectively, for the indistinguishability of the two chain ends and ofthe chains themselves.

The product in Eq. 7 generates a cluster expansion for the partition func-tion Zpack that bears some similarity to the Mayer cluster expansion [36],which is used in the theoretical description of nonideal gases. The averagequantity A is defined such that the leading zeroth-order contribution to Zpackrecovers the Flory combinatorial entropy [the

∑

s ns ln(ϕs) term, where ϕs isthe volume fraction of species s], while the Xi–1,i terms represent nonrandommixing corrections, including a temperature-independent contribution to theχ parameter that arises from the correlations introduced by the chain connec-tivity and excluded volume constraints. The free energy Fpack for a mixtureof athermal, flexible chains is obtained from Eq. 7 by applying the standardrelation

Fpack =– kT ln Zpack

and has the form of a series in powers of 1/z.Equations 3–7 have also been extended to describe nonathermal chains

interacting with nearest-neighbor van der Waals energies εcd, where thesubscripts c and d label the polymer species and all united atom groups ofa given monomer are assumed, for simplicity, to interact with a commoninteraction energy. (Sect. 9 briefly sketches a straightforward generalizationof the LCT to cases in which different interaction energies are ascribed tochemically distinct groups within the monomers.) The resultant free energy Femerges as a double expansion in 1/z and εcd/kT. The athermal and orderεcd contributions have been evaluated through order 1/z2, while the order


(εcd)2 terms have been determined so far through order 1/z. The truncationof the series in εcd/kT implies that the LCT applies only in the range ofhigher relative temperatures. A significant benefit of the LCT lies in the alge-braic nature of the expressions for thermodynamic properties that apply forall compositions, molecular weights, interaction energies εij, monomer struc-tures, and temperatures, with the only constraint that the εij/T not be toolarge.

The LCT excess free energy for a binary homopolymer blend in the in-compressible (infinite pressure) limit depends on the interaction energiesonly through the single dimensionless exchange energy ε/kT = (ε11 + ε22 –2ε12)/kT. Since ε/kT for polymer blends is typically of order 10–2 (or less)per united atom group, the LCT free energy series in ε/kT may safely betruncated at order (ε)2 when invoking the simplifying incompressible limitdiscussed in Sect. 3.

2.2Tests of the LCT

The accuracy of the LCT has been tested against Monte Carlo (MC) simula-tions of various lattice model polymer systems. Recent simulations by Butaet al. [41] generate MC data for several thermodynamic properties [chemicalpotential, entropy, internal energy, isothermal compressibility, and specificheat] of the linear chain polymer/solvent system for a wide range of tempera-tures and for polymer volume fractions up to 0.8. These simulations demon-strate that the LCT provides an excellent estimation of all thermodynamicproperties in the mean-field regions, with the exception of the specific heatat lower temperatures. Errors are, of course, evident for dilute and semidi-lute systems where non-mean-field effects become important. These tests ofthe LCT have been performed without the use of any adjustable parame-ters and, therefore, enable distinguishing whether differences between theoryand experiment, if any, are due to deficiencies in approximating the partitionfunction for the lattice model system or to inherent deficiencies associatedwith the lattice model itself. Indeed, an earlier comparative analysis [42]quickly indicated the need for introducing monomer structures to explainand describe the presence of the entropic contribution to the χ parame-ter that is rigorously absent in FH theory. Subsequent MC simulations formelts [43] and blends [44] of polymers with structured monomers imply thatthe LCT still exhibits some deficiencies in determining the variations of ther-modynamic properties of various polymer systems with specific monomerstructures. The elimination of these deficiencies requires the inclusion in thefree energy expression of higher-order contributions beyond those calculatedso far (Sect. 2.1). The evaluation of these higher-order contributions is, how-ever, nontrivial.


3Simplifying High Molecular Weight,Incompressible Limit of the LCT (the SLCT)

The theoretical description of polymer thermodynamics at a fixed pres-sure requires treating the system as compressible. The system’s compress-ibility is introduced into the extended lattice models by allowing for thepresence of empty lattice sites, called voids. The voids do not representa species of particles in the thermodynamic sense, nor do they have in-teractions; they are merely a simple bookkeeping device for keeping trackof the excess free volume. When computations are performed for constantpressure systems, the excess free volume fraction is determined numericallyfrom the equation of state, derived as usual by defining the pressure asthe derivative of the free energy with respect to total volume. The LCT ex-pressions for the free energy of polymer melts, blends, and solutions arelengthy because local correlations are treated in a systematic, perturbativefashion [20, 21]. Nevertheless, the fact that the algebraic, analytical expres-sions apply for all compositions, monomer structures, molecular weights,temperatures, and interaction energies εij (provided the ratios εij/kT aresufficiently small) is a significant benefit of the LCT. The analytical tractabil-ity of the LCT contrasts with the numerical complexity of other theories,such as PRISM theory [45], which involves the use of special numericaltechniques to solve a set of integral equations separately for each systemstudied (i.e., for each temperature, composition, etc.). Moreover, in certainlimits the analytical expressions of the LCT become enormously compact andnumerically tractable. For example, the lengthy LCT expression for the freeenergy of binary polymer blends reduces in the incompressible, high molecu-lar weight limit to a simple, physically transparent form that provides greatinsights on the molecular scale into the behavior of a wide variety of poly-mer phenomena and systems. This particular version of the LCT is calledthe simplified lattice cluster theory (SLCT) [46–48] and is described belowafter a discussion of how the monomer structures are specified within thetheory.

3.1Description of Monomer Structures in the LCT

The dependence on monomer molecular structures enters the LCT free en-ergy through a series of geometrical indices [18, 20] N(α)

i that enumerate thenumber of distinct sets of α sequential bonds in a single chain of species i.The only geometrical coefficients, appearing in the SLCT (and the LCT) ex-pression for the free energy of mixing for a homopolymer blend, are the


ratios [49]

ri ≡ N(2)i

Miand pi ≡ N(3)

i

Mi,

where Mi is the number of united atom units in a polymer chain of species i.The quantity ri may be represented more conveniently (using Euler rela-tions [50]) in terms of the respective numbers s(tri)

i and s(tetra)i of tri- and

tetrafunctional united atom groups in a single monomer of species i as

ri ≡ N(2)i

Mi= 1 +

s(tri)i

si+ 3

s(tetra)i

si, (8)

where si designates the number of united atom units in a monomer ofspecies i and where the large Mi limit has been invoked [51, 52].

The technical details associated with the application of Eq. 8 are explainedbelow using the first three polyolefin structures depicted in Fig. 1b as ex-amples. None of these three polyolefins contains tetrafunctional groups, sos(tetra)

i /si = 0 for all of them. Because only one of the three united atomgroups of the PP monomer (sPP = 3) is trifunctional, s(tri)

PP /sPP = 1/3 andrPP = 1 + 1/3 = 4/3. The hhPP monomer contains six united atom groups, twoof which are trifunctional, and, therefore, rhhPP = 1 + 2/6, which equals rPP.Similar counting for a PEP monomer yields rPEP = 6/5.

While the evaluation of pi can be simplified using Euler relations for cer-tain classes of monomer structures, these Euler relations [50] do not apply,for example, to the three polyolefin chains depicted in Fig. 1b because theyhave short side chains. (Of the monomer structures in Fig. 1a, the Euler rela-tions for pi are valid only for PEE and PH1.) Therefore, the geometrical indexpi ≡ N(3)

i /Mi = n(3)i /si is evaluated by directly enumerating all sets of three se-

quential bonds (n(3)i ) that traverse a monomer of species i. [46] summarizes

the details of these calculations and tabulates values of pi for several monomerstructures, so we pass now to a consideration of how thermodynamic proper-ties depend on ri and pi in the high molecular weight, incompressible systemlimit of the LCT.

3.2Basic Thermodynamic Properties

The specific free energy of mixing for a binary polymer blend emerges fromthe SLCT as [46]

∆Fmix

NlkT=

ϕ

M1ln ϕ +

1 – ϕ

M2ln(1 – ϕ) (9a)

+ ϕ(1 – ϕ)[

(r1 – r2)2

z2 +ε

kT

(z – 2

2–

1z

[

p1(1 – ϕ) + p2ϕ])]

,


where Nl is the total number of united atom groups in the system, the site oc-cupancy index Mi denotes the number of united atom groups in a single chainof species i, the blend composition variable ϕ ≡ ϕ1 = 1 – ϕ2 is chosen as thevolume fraction for species 1, and ε = ε11 + ε22 – 2ε12 is the blend exchangeenergy. The corresponding small angle neutron scattering (SANS) interactionparameter χ ′

SANS is defined in terms of the free energy of mixing ∆Fmix ofEq. 9a by

2χ ′SANS =

[

1M1ϕ

+1

M2(1 – ϕ)–

∂2[∆Fmix/NlkT]∂ϕ2

∣∣∣∣T,Nl

]

(9b)

and, therefore, is expressed per united atom group. The prime in χ ′SANS is

used to distinguish between the theoretical LCT united atom group-unitedatom group parameter χ ′

SANS and the experimental monomer-monomer in-teraction parameter χSANS. These two parameters are related to each otherby

χSANS = Cχ ′SANS , (9c)

where the conversion factor C depends on the definition of the normalizingvolume v0 used to calculate χSANS from the experimental zero-angle scatter-ing intensities. If v0 is chosen as the geometric mean of the monomer volumesof the two blend components [i.e., if v0 = (v1v2)1/2], C is given by

C = (s1s2)1/2 , (9d)

while an alternative choice [53] of v0 = [ϕv1 + (1 – ϕ)v2] implies the scaling

C =s1s2

s1(1 – ϕ) + s2ϕ. (9e)

When s1 = s2 = s (i.e., v1 = v2), Eqs. 9d and 9e coincide with each other.Evaluating the derivative of the free energy in Eq. 9b converts χSANS into

the rather compact expression

χSANS

C=

( r1 – r2

z

)2+

( ε

kT

)(z – 2

2+

1z

p1[1 – 3(1 – ϕ)] + p2[1 – 3ϕ])

+ O(( ε

kT

)2)

, (10)

with the scaling factor given by Eq. 9d or 9e. The first term on the right-hand side of Eq. 10 is the athermal limit entropic portion of χSANS, whichdepends on the chemical structures of the two blend components. The ap-pearance of this temperature-independent portion in Eq. 10 becomes moreunderstandable by referring to Eq. 7 and to the observation that Zpack ofEq. 7 may be expressed as the sum of the leading term (without any Xc

i–1,ifactors), which reproduces the FH combinatorial entropy SFH

comb, and the re-mainder that accounts for corrections to SFH

comb. These corrections are present


Fig. 2 Three possible overlapping configurations for two pairs of bonds on a two-dimensional square lattice

because the calculation of SFHcomb by Flory does not omit numerous classes

of unphysical configurations in which excluded volume interactions are vio-lated. Figure 2 displays examples of some disallowed configurations for pairsof two-bond sequences, configurations whose proper treatment contributes tothe temperature-independent term in Eq. 10.

The two remaining terms in χSANS of Eq. 10 are of energetic origin andcontain both monomer-structure-dependent and independent contributions.The leading energetic contribution of zε/2kT is merely the Flory–Hugginsinteraction term that grossly overestimates the number of nearest-neighborheterocontacts [54, 55]. The replacement of the factor of z in the FH approxi-mation zε/2kT by z – 2 is consistent with the arguments of Guggenheim [56,57] that each interior unit in a linear chain is linked by chemical bonds totwo nearest-neighbor units. Consequently, these two neighboring sites areunavailable for occupancy by units belonging to the other species, therebyreducing the probability of heterocontact interactions. The second term oforder ε/kT in Eq. 10 is a correction to the number of heterocontacts due tothe packing constraints imposed by the monomer molecular structures. Thislast term is an explicit function of composition and the geometrical indicespi. The composition-dependent portion of this term provides a correction tothe random mixing approximation (in the strict probabilistic sense [58]). The(ε/kT)2 contributions in Eq. 10 are generally negligible compared to the ε/kTin this equation. (As is well known, several definitions of theoretical χSANS pa-rameters are used in the literature, and generally these definitions differ whenχSANS depends on composition.)

The appearance of the counting indices pi in the coefficient of ε in Eq. 10represents a generalization of the Guggenheim surface fraction concept tostructured monomers. This important observation is rendered more evidentby considering the representative sequences of three consecutive bonds evenon a linear chain that are depicted in Fig. 3. For any of the configurationsin Fig. 2a–c, all neighboring sites to the four bonded united atom units canbe occupied by a united atom unit of another polymer species. When thesethree bonds occur in a U-shaped conformation as in (3d), the two end units inthe three-bond sequence are nearest neighbors to each other. Consequently,one less neighboring lattice site is available for possible heterocontact inter-actions. These considerations indicate that the previously vague concept ofsurface fractions [56, 57] is automatically quantified and extended to struc-


Fig. 3 Some planar configurations of three sequential bonds. All available neighboringsites to the bonded united atom groups in the configurations (a)–(c) may be occupied byunited atom units of another species, while the end units in configuration (d) are near-est neighbors to each other, which effectively decreases the probability of heterocontactinteractions with units of another species

tured monomer systems by the SLCT. Moreover, the properly determinedsurface fractions must be composition dependent, and Eq. 10, indeed, ex-hibits this dependence. The determination of the statistical distribution ofcontacts becomes more tedious for compressible systems because neighbor-ing sites may be vacant or may be occupied by united atom units of eitherspecies. (The SLCT expression for the free energy of polymer solutions issimilarly more complicated than Eq. 9a.)

4Classes of Blend MiscibilityAssociated with Structural Asymmetries

Early LCT computations [22, 59] indicated that changes in monomer molecu-lar structure alone could produce large alterations in the χ parameter and,hence, in the miscibility of polymer blends. Subsequent small angle neu-tron scattering (SANS) experiments for binary polyolefin blends by Graessley,Lhose, and their coworkers [27–32, 60] confirm these LCT predictions. In par-ticular, these experiments reveal the existence of a wide range of miscibilitypatterns that contrast with the naive view of polyolefins as a homologous fam-ily of chemically similar compounds that should therefore be mutually misci-ble. The present and next three sections describe some SLCT calculations forbinary homopolymer blends. As illustrated below, monomer structural asym-metry is found to affect profoundly not only the blend critical temperatureand composition, but also the chain swelling and the scale and intensity ofcomposition fluctuations [47, 48]. The SLCT predicts [47, 48] the existence offour qualitatively different categories of blend miscibility, only one of whichcan be explained by FH theory. Before characterizing these three additionalcategories, we note that retaining all contributions involving the excess freevolume and all 1/M corrections in the LCT free energy expression leads to aneven wider range of types of (constant pressure) phase diagrams.


For notational simplicity and for greater transparency of analysis, theχSANS parameter of Eq. 10 is rewritten in the more compact form

χSANS

C= a +

b + cϕT

, (11a)

where

a =(r1 – r2

z

)2, (11b)

b =ε

k

[z – 2

2+

– 2p1 + p2

z

]

, (11c)

and

c =ε

k3(p1 – p2)

z. (11d)

Equation 11a emphasizes the presence of an entropic component a anda composition-dependent energetic term, proportional to c, that both van-ish when the two blend components have identical monomer structures (andhence r1 = r2 and p1 = p2). The sign of b determines whether the phase sepa-ration may be of upper critical solution temperature (UCST) or lower criticalsolution temperature (LCST) types. The Flory formula for χ in Eq. 1 admitsonly of UCST phase diagrams or complete miscibility, but the SLCT pre-dicts [47, 48] the possibility of an LCST phase separation when the entropicterm a is sufficiently large and the exchange energy ε is negative. Noticethat this mechanism for the occurrence of LCST phase behavior for binaryblends departs from the customary explanation [9, 61] for LCST phase di-agrams in polymer solutions: When the temperature of a polymer solutionincreases, the densities of the pure polymer and pure solvent become increas-ingly disparate, producing an entropic penalty toward mixing and, therefore,an LCST phase diagram. The LCST of polymer solutions generally appearsat a temperature about 70–90% of the critical temperature of the solvent.Hence, the solvent is near the boiling point, while the pure polymer is still ei-ther solid or liquid. On the other hand, the components of binary polyolefinblends, for instance, have very similar densities and coefficients of thermalexpansion. Consequently, an LCST phase behavior in these systems cannotbe governed by the same mechanism that is applicable to polymer solutions.Rather, the competition between a positive entropic portion of the χ pa-rameter and a negative energetic part of χ appears to drive the LCST phasebehavior of these polyolefin blends. Recall that the SLCT assumes the blend tobe incompressible. Hence, the appearance of LCST phase behavior for binaryblends does not automatically imply that it arises because of compressibilityeffects.


4.1Basic Critical Properties of Binary Blends in the SLCT

The critical composition ϕc and the critical temperature Tc are obtained asthe solutions to the pair of equations

(∂2∆Fmix)∂ϕ2

∣∣∣∣T,V

= 0 (12a)

and

(∂3∆Fmix)∂ϕ3

∣∣∣∣T,V

= 0 . (12b)

The condition of Eq. 12b in conjunction with Eq. 9a produces [47] a quadraticequation for ϕc:

– 2ac(1 – ϕc)ϕ2c (13)

+ [2c(λ – 1)ϕ3c – b(λ – 1) – c(4λ – 1)ϕ2

c + 2(c – b)λϕc + b]/(Mλ) = 0 ,

where λ = M2/M1 is a measure of the polymer size asymmetry. The parame-ter λ replaces the polymerization index ratio λN = N2/N1 of FH theory, whereNi denotes the polymerization index for species i. Specializing the constraintof Eq. 12a to T = Tc and ϕ = ϕc implies that the critical temperature Tc de-pends on ϕc through

Tc =2(b + cϕc)

1Mϕc

+ 1Mλ(1–ϕc) – 2a

. (14)

Since |c/b| is typically small, the largest contribution to the shift of Tc fromits FH value arises according to Eq. 14 from a nonzero value of parameter a.An increase of a generally leads to decreased blend miscibility.

The correlation length amplitude ξ is another characteristic propertyof polymer blends that is strongly influenced by monomer size and shapeanisotropy. Within a mean-field approximation, such as the SLCT, whichis valid [62–64] sufficiently far from the critical point, the correlationlength amplitude ξ0 is independent of temperature and is defined as ξ ≡ξ0(|T – Tc|/T)1/2, where ξ designates the static correlation length associatedwith composition fluctuations [65]. Since a mean-field sum rule exhibits ξ2

as proportional to the structure factor S(0) in the long wavelength limit,ξ0 controls both the amplitude [S(0)] and the length scale (ξ) of compositionfluctuations. The correlation length amplitude at the critical composition ϕcis related to the SLCT quantities b and c of Eqs. 11c and 11d by

ξ0 =[

d0Tc

2|b + c|]1/2

, (15)


where the square gradient coefficient d0 can be determined within the ran-dom phase approximation [62] (RPA) in terms of the critical composition ϕcand the Kuhn lengths a1 and a2 as

d0 =1

18

[a2

1

s1ϕc+

a22

s2(1 – ϕc)

]

, (15a)

with s1 and s2 denoting the numbers of united atom groups in singlemonomers of blend components 1 and 2, respectively.

The monomer shape and size asymmetry also affects the ranges of T overwhich mean-field and Ising-type critical behaviors are observed. These tem-perature ranges are expressed in terms of the Ginzburg number Gi, whichprovides a rough estimate of the magnitude of the reduced temperatureτ ≡ (T – Tc)/T at which the crossover from mean-field to Ising-type behav-ior occurs. We have introduced [65] a more refined criterion for specifyingthe three different regimes. Mean-field theory holds for τ ≥ 10 Gi, while theIsing critical behavior corresponds to τ ≤ Gi/10. The range Gi/10 < τ < 10 Gidescribes a crossover domain with τ ≈ Gi in the middle of this range. For anincompressible blend, Gi is given by

Gi =v2

cell

[

M–1ϕ–3c + (Mλ)–1(1 – ϕc)–3

]2

64π2∣∣(Mϕc)–1 + (Mλ)–1(1 – ϕc)–1 – 2ad3

0

∣∣

, (15b)

where v2cell is the volume associated with a single united atom group and a

and d0 are defined by Eqs. 11b and 15a, respectively. The article by Schwahnin this volume discusses the details of the crossover analysis required to ex-tract from neutron scattering data quantities suitable for comparison withmean-field theories. Schwahn’s review also demonstrates the importance ofstrong thermal composition fluctuations in modifying blend thermodynamicproperties in the vicinity of the critical point.

In analogy to the description of polymer solutions where the thetatemperature Tθ is normally identified as an essential reference tempera-ture [4, 62, 66], Tθ can also be defined for dilute polymer blends. Since eitherof the two components may be the dilute species, there are two osmotic virialexpansions and two theta temperatures whose SLCT expressions are [47]

T(1)θ =

2(b + c)M1

1 – 2aM1(16a)

and

T(2)θ =

2bM2

1 – 2aM2. (16b)

The reduced temperature gap δTθ between the theta and critical tempera-tures, δT(i)

θ ≡ (T(i)θ – Tc)/Tc, is another useful property for characterizing


blends. When M1 = M2, both theta temperatures coincide T(1)θ ≈ T(2)

θ ≡ Tθ ,and δT(1)

θ ≈ δT(2)θ since |c/b| is generally small [47, 48].

Based on an analysis of the critical parameters (ϕc, Tc, ξ0, Gi, δT(1)θ , δT(2)

θ )determined from Eqs. 13–15, binary polymer blends may be classified intofour distinct classes of critical behavior.

4.2Essential Characteristics of the Four Classes of Binary Blend Miscibility

This subsection provides a brief description of the four classes of binaryblend miscibility patterns that are predicted by the SLCT. Eight potentialtypes of binary blend phase behavior can be derived form the SLCT becauseb may be positive or negative and because a and c may each be either zeroor nonzero. However, only four blend categories are classified since the re-maining four types of blends are either completely miscible or completelyimmiscible systems. (Note that Eq. 11b implies that a is nonnegative in theSLCT. Classes of critical behavior for a < 0 are not discussed here as they donot emerge from the SLCT.)

Class I corresponds to blends specified by a = 0, b > 0, and either c = 0 orc = 0. Blends in this class exhibit UCST phase diagrams that are very similarto those predicted by FH theory. The basic quantities scale with M ≡ M1 as

ϕc ≈ ϕ(I)c ≡ λ1/2

1 + λ1/2 ,

Tc ∼ M ,

ξ0 ∼ M1/2 ,

Gi ∼ M–1 ,

δT(1)θ ∼ 1 + 2λ1/2

λ, δT(2)

θ ∼ 1 + 2λ1/2 , (17.I)

where λ = M2/M1 and ϕ(I)c is a generalization of the FH theory critical com-

position ϕ(FH)c = λ

1/2N /[1 + λ

1/2N ] to structured monomer blends.

The three other classes of blends display qualitative departures from theFH type behavior of class I. Class II is defined by a = 0, b > 0, and either c = 0or c = 0 and is also characterized by a UCST phase diagram, but a nonlineardependence of Tc on M. The critical properties vary with M as

ϕc ≈ ϕ(I)c ,

Tc ∼ M1 – 2alM

,

ξ0 ∼ M/[1 – 2alM]1/2 ,


Gi ∼ M–1

1 – 2alM,

δT(1)θ ∼ 1 + 2λ1/2

λ(1 – 2aM), δT(2)

θ ∼ 1 + 2λ1/2

(1 – 2aM), (17.II)

where l ≡ λ/[1 + λ1/2]2. The occurrence of the class II type miscibility patternis limited to M < M∗ = 1/2al because the expression for Tc diverges for M∗.

Class III and IV type blends both have LCST miscibility patterns. Class IIIcorresponds to a = 0, b < 0, and c = 0, and the predicted Tc for this class nolonger scales as a power of M but instead approaches the constant |b|/a asM → ∞:

ϕc = ϕ(I)c ,

Tc = |b|/a ,

ξ0 = constant ,

Gi ∼ M–2 ,

δT(1)θ ∼ M–1 , δT(2)

θ ∼ M–1 . (17.III)

Class IV (a = 0, b < 0, and c = 0) exhibits the strongest departures from thepredictions of FH theory since the dependence of both Tc and ϕc on M devi-ates significantly from those emerging from FH theory. The critical propertiesof class IV blends approach different limits depending on the sign of c. Thescaling laws in the M → ∞ limit for c < 0 are

ϕc ∼ M–1/2 ,

Tc = |b|/a ,

ξ0 ∼ M1/4 ,

Gi ∼ M–1/2 ,

δT(1)θ = c/b ,

δT(2)θ ∼ M–1/2 . (17.IVa)

The sign of c is reversed upon interchanging the labels of the two compo-nents, but the scaling laws for c > 0 are not symmetric to those in Eq. 17.IVa,as evident from the following:

ϕc = 1 – O(

M–1/2) ,

Tc = |b – c|/a ,

ξ0 ∼ M1/4 ,

Gi ∼ M–1/2 ,

δT(1)θ ∼ M–1/2 ,

δT(2)θ = c/|b – c| . (17.IVb)


Figure 4a–e illustrates the dependence of the critical properties for somesymmetric polyolefin blends (M1 = M2 = M; λ = 1) specified in the figurecaption. The linear scaling of Tc with M in class I is one hallmark of FH the-ory [3–7]; blends of class II exhibit a stronger than linear dependence of Tcon M; while the critical temperatures for those in classes III and IV approachconstants as M → ∞. The critical composition ϕc is insensitive to M for sym-metric blends of classes I–III but for class IV decreases toward zero (or unity)with the scaling ϕc ∼ M–1/2. Hence, the FH estimate for ϕc can be grosslyin error for LCST blends. The correlation length amplitude ξ0 for blends ofclass II increases more rapidly with M than for class I (ξ0 ∼ M1/2), but ξ0 forblends of classes III and IV is significantly smaller than the chain radius of gy-ration, Rg ∼ M1/2. An insensitivity of ξ0 to M occurs for class III blends whereξ0 is comparable to the statistical segment length rather than to Rg. The thetapoint shift δTθ is also a useful indicator of the blend miscibility class. The

Fig. 4 The critical temperature Tc (a) critical composition ϕc (b) correlation length am-plitude ξ0 (c) reduced theta temperature δTθ = (Tθ – Tc)/Tc (d) and the reduced Ginzburgnumber Gi/Gi(ref)

i (e) for symmetric blends (λ = 1, M1 = M2 = M) as a function of thenumber of united atom groups M in a single chain. (Gi(ref)

i = 0.01 is a typical value ofGi for mixtures of small molecules.) Classes II and IV are represented by PH1/PEP andPIB/PEP blends, respectively. The exchange energy ε for the PH1/PEP and PIB/PEP sys-tems are taken as ε = 0.01 K and ε =– 1 K, respectively. The example for class I blends(a = c = 0, b > 0) is constructed assuming the same b as is chosen for class II, while theexample for class III blends (a = 0, c = 0, b < 0) is generated by taking a and b equal tothose for the PH1/PEP mixture


large values of δTθ ≡ δT(1)θ ≈ δT(2)

θ for blends of classes I and II suggest thatthe theta point is experimentally inaccessible. (Most polymers with Tc nearroom temperature would become thermally unstable at these Tθ .) In contrast,Tθ is predicted to lie close to Tc for blends of classes III and IV.

The scaling of Gi for FH type (class I) blends [62–64, 67], Gi ∼ M–1, indi-cates a strong decrease of the width of the critical region with increasing M.However, the magnitude of Gi can be larger for class II blends that exhibita shallow minimum in Gi as a function of M (Fig. 4e). A typical magnitude forthe minimum value of Gi for class II blends is about 0.001, which is small rela-tive to typical values for Gi of small-molecule mixtures (Gi ≈ 0.01) [68, 69].The Gi for class III blends decreases even more rapidly than Gi for class I,while the dependence of Gi on M for class IV (Gi ∼ M–1/2) is weaker thanfor class I, resembling the Gi scaling for UCST polymer solutions [70]. Ap-parently, the examples of class II–IV blends indicate the presence of largedepartures from the Gi ∼ M–1 scaling of FH theory, but there is neverthelessa general tendency for Gi to become small for large M.

4.3Experimental Support for New Classes of Polymer Blend Miscibility

A linear scaling of Tc with M has been confirmed by SANS experiments [71]for symmetric (N1 = N2) isotopic polyolefin blends (where monomer molecu-lar structures are almost identical) and by Monte Carlo simulations [72, 73]for symmetric linear chain polymer blends. Several experiments, on the otherhand, indicate that this FH pattern of blend miscibility is not general. Perhapsthe best documented example of a gross departure from FH type phase be-havior is provided by the polystyrene/poly(vinyl methyl ether) (PS/PVME)blend whose Tc is found to be nearly insensitive to M (Tc = 145±5 C), whileϕc is observed [74, 75] to be highly asymmetric even for samples with iden-tical N1 and N2 [74, 75]. Based on our fits [17] to the scattering data thatyield b < 0, a > 0, and c = 0, PS/PVME systems are classified as class IV blendsfor which the SLCT predicts that Tc approaches the ratio |b|/a as M → ∞,a prediction that is in accord with the observed insensitivity of Tc to vari-ations in M. In addition, the SLCT scaling relation ϕc ∼ M–1/2 is consistentwith the finding of Han and coworkers [74, 75] that increasing M by a fac-tor of 3 reduces ϕc by roughly a factor of 2. Other SLCT predictions that thecorrelation length amplitude ξ0 for PS/PVME blends weakly depends on Mand that Tc and Tθ are almost equal also agree with the existing experimentaldata [74–77].

A similar insensitivity of Tc to M has been observed [60] for binary blendsof poly(isobutylene) (PIB) with several other polyolefins, systems for whichit is difficult to imagine that any specific interactions could be responsible forthe observed dramatic departures from the predictions of FH theory. (Note


that our explanation for the unusual critical behavior of PS/PVME blendslikewise does not resort to invoking the presence of specific interactions.)The LCST phase separation in binary blends of PIB with other polyolefinsarises according to the SLCT from the competition between a negative en-ergetic portion of the χSANS parameter and a sufficiently positive entropicportion a of χSANS. The large values of a for the PIB/polyolefin blends stemfrom a rather large partial entropic parameter rPIB = 7/4, which, in turn,arises because of the presence of the tetrafunctional carbon atom in eachPIB monomer (Eq. 8). A negative exchange energy ε (implying a negative b)may occur because 50% of the PIB united atom groups are CH3 groups thathave larger attractive interactions (i.e., Lennard–Jones potential interactionparameters) than the CH2, CH, and C united atom groups [37–39]. This effectproduces a larger self-interaction energy ε11 = εPIB–PIB (relative to ε22) anda larger heterocontact interaction energy ε12 with other polyolefins, leadingto a negative ε = ε11 + ε22 – 2ε12.

5Application of the SLCT to LCST Polyolefin Blends

LCST polyolefin blends are quite unusual systems because the dispersivevan der Waals nature of the interactions between polyolefin species fa-vors UCST phase separation, as found in the majority of binary poly-olefin mixtures. LCST phase diagrams have been observed [60] in binaryblends of poly(isobutylene) (PIB) with, for instance, poly(propylene) (PP),head-to-head poly(propylene) (hhPP), saturated polybutadienes (sPB), orpoly(ethylene propylene) (PEP). All these LCST systems exhibit large nega-tive values of the χSANS parameter at lower temperatures and positive χSANSat higher temperatures [60]. Not surprisingly, this LCST behavior cannot beexplained by solubility parameter theory, which has been applied with appar-ent success to a number of UCST polyolefin blends by Graessley et al. [27–32].Thus, the thermodynamic description of these LCST blends of PIB presentsa significant challenge to any polymer theory concerned with modeling themolecular factors controlling blend miscibility. Consequently, these blends ofPIB with other polyolefins are analyzed here using the high pressure, highmolecular weight limit of the LCT.

The incompressibility assumption implies that the SLCT for binary blendscontains only one adjustable parameter, the exchange energy ε. All remain-ing quantities of the SLCT (i.e., rα and pα, α = 1, 2) are determined fromthe united atom group monomer structures (Fig. 1a) by elementary count-ing. Figure 5 compares the calculated and experimental χSANS parametersfor poly(isobutylene)/head-to-head polypropylene (PIB/DhhPP) blends asa function of the inverse temperature. Squares denote the experimental data


Fig. 5 Comparison of calculated and experimental interaction parameter χSANS forPIB/DhhPP blends (ϕPIB = 0.475) as a function of inverse temperature. Squares are theexperimental data of Krishnamoorti et al. [60], while the solid line is a least-squares fit ofthe SLCT χSANS for the exchange energy ε =– 1.16 K

of Krishnamoorti et al. [60] for ϕPIB = 0.475, while the solid line representsa least-squares fit of the SLCT Eq. 11a to the data for χSANS. The least-squaresfit yields ε =– 1.16 K, guaranteeing that the theory reproduces the slope ofχSANS vs. 1/T, but the remarkable feature of Fig. 5 is the excellent theoreticalrepresentation for the entropic portion of χSANS.

Given the empirical value for ε, the SLCT relation in Eq. 10 can then beused to generate predictions for the composition dependence of the χSANSparameter, which to our knowledge has not yet been studied. This composi-tion dependence is delineated for four different temperatures in Fig. 6. Thesquares correspond to the experimental data [60] that are available only forthe single composition ϕPIB = 0.475. The convex parabolic variation of χ(ϕ)for PIB blends in Fig. 6 becomes less pronounced at higher temperatures, al-most disappearing for T ≥ 440 K.

Figure 7 further illustrates the influence of monomer molecular structureson the miscibility of PIB blends by presenting computed spinodal curvesfor three binary blends of PIB with other polyolefins. The homologous na-ture of the second component provides a motivation for using the samePIB/DhhPP exchange energy ε =– 1.16 K in the SLCT calculations for all threesystems. The computed phase diagrams (in Fig. 7) accord with the experi-mental observations that the PIB/DhhPP blend is a strongly miscible system,whereas mixtures of PIB with either poly(ethyl propylene) (PEP) or satu-rated 1,2-polybutadiene (H1,2PB) are phase separated at room temperatures.The computed critical temperature Tc for the PIB/H1,2PB blend is higher byat least 20 K than the Tc deduced from the visual observation [60] of cloudpoints for this system, a small difference considering that the same exchangeenergy ε has been assumed for PIB/H1,2PB and PIB/DhhPP mixtures. Ex-periments for a 50–50 mixture of PIB and DPEP reveal [60] that this systemis phase separated at 298 K, whereas the computed spinodal temperature is260 K (Fig. 7), consistent with this observation.


Fig. 6 SLCT predictions for the composition dependence of the interaction parameterχSANS for PIB/DhhPP blends (ϕPIB = 0.475) at several temperatures. Crosses represent ex-perimental data of Krishnamoorti et al. [60]. The exchange energy ε =– 1.16 K is takenfrom the fit in Fig. 5

Fig. 7 SLCT spinodal curves for PIB/DhhPP, PIB/H1,2PB, and PIB/DPEP blends that arecomputed by assuming the common exchange energy ε =– 1.16 K for all three systems.The molecular weights of the blend components correspond to the experimental samplesstudied by Krishnamoorti et al. [60]

6Introduction of Chain Stiffness into the SLCT

The model employed by the SLCT so far treats each polymer as a com-pletely flexible entity, subject only to the excluded volume constraints againstmultiple occupancy of a lattice site by any two united atom groups. How-ever, bond angle constraints and steric interactions of, e.g., hydrogen atomsand side groups lead to local chain stiffness and the semiflexibility of poly-mer molecules. For example, torsional motions about single C – C backbonebonds in many olefins are limited to a trans and a pair of gauche conforma-tions with differing energies and, therefore, temperature-dependent probabil-


ities of occurrence. Following Flory [78], these different conformational statesare modeled on a simple cubic lattice by defining the trans conformation asthe collinear configuration of two consecutive bonds, i.e., the two bonds liealong the same direction, and by identifying the gauche conformation withsituations in which the two consecutive bonds reside along orthogonal direc-tions [79]. The energy of the trans configuration is taken as zero, while thatof the gauche configuration is E(i)

b , where the superscript i denotes the chainspecies.

This definition of trans and gauche configurations is quite clear for lin-ear chains of polyethylene, but requires modification when the monomerscontain side groups (Fig. 1). For example, the methyl side groups inpoly(propylene) must lie in a direction orthogonal to at least one of thebackbone bonds that emanate from a backbone trifunctional CH group.Thus, there is no physical sense in assigning a gauche energy penalty tothis conformation. Hence, only the backbone bonds in poly(propylene) canhave the bending energies E(i)

b in the LCT, and, consequently, the maximumnumber of gauche conformations (ignoring excluded volume constraints)in a poly(propylene) chain is Nb – 2, where Nb is the number of back-bone C – C bonds. Further complexity arises for polymer species in whicheach side group contains two or more united atom units because a pair ofsuccessive side group bonds may have either a trans or gauche configura-tion. Consequently, the maximum number of gauche conformations (againignoring excluded volume constraints) in a poly(ethyl ethylene) chain isNb – 2 + (1/2)Nb, where the extra (1/2)Nb contribution stems from the pres-ence of the side groups. Although it is possible to assign different gaucheenergies E(i)

b to pairs of backbone or side group bonds [49], a single energyE(i)

b is assumed for each species in order to minimize the number of adjustableparameters. The bending energies E(i)

b are determined by fitting calculatedthermodynamic properties to experimental data. Because of the inadequacyof a cubic lattice to represent the actual numbers and geometrical structuresof the trans and gauche configurations in real olefins, it is best not to deter-mine the Flory model bending energies E(i)

b from fits to structural data suchas the average chain radii of gyration. The Flory trans–gauche model [78],with one trans and (z – 2) gauche conformations for a bond pair, thus emergesas a simple method for including the thermodynamic consequences of localchain stiffness.

The introduction of chain semiflexibility into the SLCT leads to the re-placement [46, 49, 80, 81] of the geometrical parameters ri and pi in Eqs. 9a,10, and 11b–11d by polynomials in the bending energy statistical factor

gi =z

z – 1 + exp[

E(i)b /kT

] , (18)


where z is the lattice coordination number, which equals 6 for a simple cubiclattice. The resultant polynomials have the forms

ri → r(semi)i (gi) = α

(i)0 + α

(i)1 gi , (19a)

pi → p(semi)i (gi) = γ

(i)0 + γ

(i)1 gi + γ

(i)2 g2

i , (19b)

where the coefficients α(i)0 , α

(i)1 , γ

(i)0 , γ

(i)1 , and γ

(i)2 are partitions of the SLCT

parameters ri and pi,

α(i)0 + α

(i)1 = ri , (19c)

γ(i)0 + γ

(i)1 + γ

(i)2 = pi . (19d)

The fully flexible chain limit is recovered from Eqs. 19a and 19b by settingE(i)

b = 0 or, equivalently, gi = 1:

r(semi)i (gi = 1) = α

(i)0 + α

(i)1 = ri ,

p(semi)i (gi = 1) = γ

(i)0 + γ

(i)1 + γ

(i)2 = pi .

Due to the existence of Euler relations for N(i)2 , the coefficients α

(i)0 and α

(i)1

can simply be related [46] to the numbers s(tri)i and s(tetra)

i of tri- and tetra-functional united atom groups in a single monomer of species i as

α(i)0 = 2

[

s(tri)i

si+ 2

s(tetra)i

si

]

and

α(i)1 = 1 –

s(tri)i

si– 3

s(tetra)i

si.

The absence of a Euler relation for N(i)3 for arbitrary monomer structures ren-

ders the calculation of γ(i)0 , γ (i)

1 , and γ(i)2 much more involved. The coefficients

γ(i)0 , γ (i)

1 , and γ(i)2 are obtained by counting the numbers n(i)

j of configurationsfor three successive bonds that traverse a monomer of species i and that havej = 0-, 1-, or 2-bond pairs belonging to identical subchains. More specifically,

γ(i)j =

n(i)j

si, j = 0, 1, 2 .

The concept of subchains is explained in several prior papers [49, 80, 81],which may be consulted for more details. Briefly, the chain backbone is thefirst subchain. Other subchains are formed from sequences of side groupbonds to which gauche conformations may be assigned.

Because the introduction of chain semiflexibility imparts a temperaturedependence to r(semi)

i and p(semi)i , the free energy of mixing ∆Fmix and the

χSANS parameter vary with temperature in a more complicated fashion than


these quantities vary in the completely flexible chain SLCT. Equations 13 and14 for the critical temperature and composition of a binary, incompressibleblend are still maintained in the semiflexible chain extension of the SLCT, butnow a, b, and c are functions of temperature. Consequently, the pair of equa-tions for Tc and ϕc must be solved numerically. While it is expected that thesemiflexible chain SLCT admits of a wider range of miscibility patterns thanthose discussed in Sect. 4.2, a detailed analysis has yet to be made.

It is straightforward to reformulate the SLCT in order to account for rigid-ity within any of the components of the blend because a pair of rigid bondscorresponds to a situation in which the two semiflexible bonds have an in-finite bending energy E(i)

b or, equivalently, a vanishing gi. Thus, the rigidbond constraint implies that these rigid bond configurations are effectivelyexcluded from the set of configurations contributing to the geometrical pa-rameters ri and pi or r(semi)

i and p(semi)i . Section 8.2.1 describes the evaluation

of these geometrical parameters for norbornene-co-ethylene binary blendswhere the rigidity of the norbornene monomers is modeled by taking the sidegroup bonds in the structure of Fig. 10b as completely rigid.

7Application of the SLCT to UCST Polyolefin Blends

The majority of binary polyolefin blends exhibit upper critical solution tem-perature (UCST) phase diagrams. The earlier common view that these UCSTpolyolefin mixtures are not of great scientific interest has been dispelled dueto recent experiments by Graessley et al. [31] that reveal quite different misci-bilities of poly(propylene) (PP) and head-to-head poly(propylene) (hhPP) inbinary blends with poly(ethylene propylene) (PEP). The critical temperaturesdiffer by more than 100 K for blends with polymerization indices in the hun-dreds, as illustrated by the LCT spinodal curves [82] in Fig. 8 that have beencomputed for binary mixtures of either PP or hhPP with PEP. This huge dif-ference in miscibility arises merely from the swapping of alternate methyl sidegroups along the poly(propylene) chain.

While the spinodal curves [82] depicted in Fig. 8 refer to LCT calculationsfor compressible blends of semiflexible polymers at a pressure of 1 atm, simi-lar behavior emerges [83] from the SLCT where the pressure is infinite. Thus,we analyze the molecular features contributing to this remarkable differencein miscibilities within the framework of the SLCT because of its analyticaltractability. In order to elucidate the contributing features to the observedmiscibility difference, it is convenient to assume first that both blend compo-nents are completely flexible. Then, the partial entropic structural parametersrPP and rhhPP are identical (Sect. 3.1), so that all of the calculated miscibilitydifferences between PP/PEP and hhPP/PEP blends emerge from the differ-


Fig. 8 LCT spinodal curves for PP/PEP and hhPP/PEP blends at P = 1 atm. The siteoccupancy indices MPP = 1560 and MPEP = 4275 used in the calculations correspond toone of the blends studied by Graessley et al. [31], and MhhPP = 1560 is selected to en-sure the same molecular weights for the PP and hhPP components. Both blends areassumed to have the same interaction energies εij in order to illustrate the sole in-fluence of monomer structure and chain semiflexibility on the blend miscibilities. Theself-interaction energies ε11 = (1/2)(εPP–PP + εhhPP–hhPP) = 205.40 K and ε22 = εPEP–PEP =207.49 K are obtained from our earlier fits to PVT data for the pure melts. The bendingenergies E(PP)

b = 219 K, E(hhPP)b = 277 K, and E(PEP)

b = 460 K are taken form [80], while theheterocontact energy is an adjustable parameter

ent geometrical indices pPP and phhPP. Inspection of Fig. 1b indicates that theunited atom structure for hhPP generates one three-bond run not present forthe PP structure, namely, the three-bond run that comprises the bonds be-tween two adjacent methyl side groups and the chain backbone along with thejoining backbone bond. Thus, use of Eq. 10 implies that hhPP presents a lowersurface fraction for interaction with PEP than does PP, i.e., hhPP has a lowertotal probability of heterocontact interactions with PEP than PP, leading toa higher compatibility of hhPP/PEP blends compared to PP/PEP mixtures.(Because molecular weights have enormous influence on the critical tempera-tures, the calculated spinodal curves in Fig. 8 have been determined assumingcommon molecular weights for PP and hhPP and the same molecular weightfor PEP in both blends.)

Additional contributions to the disparate miscibilities of hhPP/PEP andPP/PEP blends arise from the obvious difference in chain stiffness betweenPP and hhPP that is induced by the presence of methyl side groups on adja-cent backbone carbons in hhPP. As explained below, lifting the assumptionthat the polymer chains are fully flexible provides additional factors favor-ing the miscibility of PP/PEP blends with respect to PP blends. Clearly, thepresence of methyl groups attached to adjacent backbone carbons in hhPPleads to greater steric interactions in hhPP than in PP. Thus, a larger trans↔ gauche energy difference E(i)

b must be ascribed to hhPP, which in turnimplies the presence of a nonzero difference in the entropic χSANS between


hhPP/PEP and PP/PEP blends. This stiffness-induced entropic χSANS favorsthe miscibility of the hhPP/PEP blends. In other words, the greater stiffnessof hhPP enables improved packing of hhPP with PEP than of PP with PEP. Inaddition, the difference in chain stiffness affects the magnitudes of the geo-metric indices p(semi)

PP and p(semi)hhPP that influence the energetic portion of χSANS.

Again, the greater stiffness of hhPP reduces the probability of heterocontactswith PEP as compared to PP. In summary, the greater miscibility of hhPP/PEPblends over PP/PEP blends stems from the synergistic combination of threefactors: an inherently lower surface fraction for hhPP, a stiffness-induced im-proved packing for hhPP, and a stiffness-driven diminished probability forhhPP–PEP heterocontact interactions [82].

7.1Analysis of the PP/hhPP blend

Binary blends of PP with hhPP represent another interesting system froma molecular standpoint. Fitting the experimental χSANS for a symmetric(ϕ = 0.5) PP/DhhPP mixture [31] to the common empirical form χ = A + B/Tyields [46] a negative A =– 4.46×10–3 and a large B = 2.38 K (per monomer).The value of B is typical for fairly immiscible UCST blends, in sharp contrastto intuitive expectations that the almost chemical identity of PP and hhPPshould lead to a nearly vanishing exchange energy, i.e., a very small B. More-over, the rather large, negative A appears inconsistent with the SLCT Eq. 9,which defines the entropic portion of χSANS as a nonnegative quantity. Be-cause differences in steric interactions are anticipated to be crucial factorsdriving the system’s thermodynamics for mixtures of chemically very similarcomponents, both PP and hhPP are treated within the SLCT as semiflexiblepolymers.

The parameters r(semi)i and p(semi)

i for PP and hhPP are evaluated as

r(semi)PP =

23

+23

gPP , (20a)

gPP =z

z – 1 + exp[

E(PP)b /kT

] ,

r(semi)hhPP =

23

+23

ghhP , (20b)

ghhPP =z

z – 1 + exp[

E(hhPP)b kT

] ,

p(semi)PP =

23

+23

gPP +23

g2PP , (20c)

p(semi)hhPP =

16

+23

ghhPP +23

g2hhPP , (20d)


and the corresponding SLCT expression for χSANS is obtained from Eq. 10 byreplacing ri and pi by r(semi)

i and p(semi)i , respectively:

χSANS

[sPPshhPP]1/2 =

(

r(semi)PP – r(semi)

hhPP

z

)2

(21)

+( ε

kT

)(z – 2

2+

1z

p(semi)PP [1 – 3(1 – ϕ)] + p(semi)

hhPP [1 – 3ϕ])

+ O(( ε

kT

)2)

,

where sPP = 3 and shhPP = 6 are the numbers of united atom groups in individ-ual monomers of PP and hhPP, respectively.

An examination of Eqs. 20a–20c reveals that the term(

r(semi)PP – r(semi)

hhPP

)2

in Eq. 21 is temperature dependent, so that the observed [31] large negativetemperature-independent portion of χSANS can only arise from an interplaybetween the ε/kT factor and the p(semi)

PP and p(semi)hhPP parameters that are tem-

perature dependent through the bending energy factors gPP and ghhPP. Whilethe SLCT for blends of semiflexible polymers contains three adjustable pa-rameters (the exchange energy ε and the two bending energies E(1)

b and E(2)b ),

calculations for various systems suggest that χSANS effectively depends onthe difference ∆Eb = E(1)

b – E(2)b in the bending energies for the two compo-

nents, thereby effectively diminishing the number of adjustable parameters totwo [46].

We find [46] a wide range of parameter pairs that produce good agreementwith the experimental data [31], and Fig. 9 illustrates two examples of these

Fig. 9 Two fits to the SANS data of Graessley et al. [31] for the hhPP/PP blend withMhhPP = 1920, MPP = 780, and ϕ1 = ϕ2 = 0.5. The solid line represents one fit obtainedfor ε = 0.000415 K and ∆Eb = E(hhPP)

b – E(PP)b = 294 K, while the dashed line indicates a fit

generated for ε =– 0.0166 K and ∆Eb = 290 K


fits for ε =– 0.0166 K and ε = + 0.000415 K and for ∆Eb ≡ E(hhPP)b – E(PP)

b =290 K and ∆Eb = 294 K, respectively. The minuscule values of the exchangeenergy ε accord with our expectation that PP and hhPP are almost identicalfrom an interaction energy standpoint. The greater steric interactions be-tween the neighboring side groups in hhPP are likewise anticipated to renderhhPP stiffer than PP, in agreement with our fits that yield E(hhPP)

b > E(PP)b .

8Copolymers Blends

Statistical copolymers are of considerable technological and scientific im-portance. The earliest theoretical treatments [84–87] of these systems werebased on a direct extension [88, 89] of Flory–Huggins theory to randomcopolymers. While the resultant random copolymer FH theory has beenvery successful in explaining the enhanced miscibilities [84–86] observedin many systems containing random copolymers, the theory exhibits someserious limitations. First of all, the theory is insensitive to monomer se-quence and to the detailed chemical structure of the individual monomers.Some attempts [90–92] to improve the random copolymer FH theory intro-duce a set of phenomenological interaction parameters χijk,lmn (or χij,kl) thatexplicitly depend on the monomer sequence. However, these phenomenolog-ical models are required to invoke additional (often arbitrary) assumptionsto reduce the huge number of resulting χijk,lmn parameters to a manage-able few. Thus, there is a great need for developing analytically simple,more realistic, improved theories with a minimal number of adjustable pa-rameters. These improved, yet simple, theories should provide an adequatetool for probing general physical trends. An example of this type of sim-ple, molecular-based theory is given by the SLCT approach [46–48] forbinary homopolymer blends. A similar, but somewhat more simplified, ver-sion of the LCT has also been developed for random copolymer systems.The current section reviews this theory and describes particular applica-tions to blends of random copolymer polyolefins and to diblock copolymersystems.

In addition to the immense technological importance of polyolefins andto the enormous progress [1, 2] in their synthesis that enables the control oftheir chemical structures, binary blends of polyolefins represent a class ofweakly interacting systems. Our earlier LCT predictions [22] that slight mod-ifications of chemical structure can significantly alter the properties of blendshave been confirmed by the observation of considerably different miscibilitiesin a variety of polyolefin mixtures [27–32, 60], for example, a large body ofexperimental data for polyolefin blends [31, 60], the majority of which haveone or both of the components as random copolymers. These data indicate


that there is a strong sensitivity of the thermodynamic properties of binarypolymer blends to the monomer molecular structures of the blend compo-nents. An understanding of this relationship can only be achieved withina molecular-based theory for statistical copolymers systems, which are quiteprevalent in technological applications.

The development of the equivalent of the homopolymer blend SLCT (re-viewed in Sect. 3) for blends containing random copolymers has been ex-tremely lengthy and algebraically rather complicated [93]. This complexityarises not only from the random nature of the copolymers (i.e., from thestatistical positioning of the different monomer species along the polymerchain) but also from the fact that local correlations, which are responsible fornonrandom mixing effects, must be averaged over distances that are largerthan a few bond lengths in order to provide an adequate description of thedependence of the system’s free energy on the monomer sequence [94]. Con-sequently, the SLCT free energy expressions for copolymer blends does notexhibit [93] the analytical simplicity of the SLCT Eq. 9a for homopolymerblends. Hence, additional assumptions are introduced for copolymer systemsbeyond the neglect of excess free volume and 1/M contributions. The result-ing theory is termed the basic lattice cluster theory (BLCT). We now describethe main physical content of the BLCT and its relation to previous theories ofcopolymer systems.

8.1The BLCT: General Assumptions

The BLCT is a molecular generalization of random copolymer Flory–Huggins(FH) theory [88] and is based on two significant improvements. The firstimprovement lies in the description of the polymer–polymer interactions interms of the united atom groups that are used to represent the monomerstructures, an approach that is chemically more realistic than the tradi-tional FH model that is based on monomer–monomer interactions betweenunited monomer groups. The second modification consists in the add-ition of entropic contributions [52] to the interaction parameters χij. Thesetemperature-independent portions of χij are completely determined withinthe infinite pressure, fully flexible, long chain limit by the monomer mo-lecular structures and the copolymer compositions [95]. The entropic contri-butions to the χij are evaluated simply by counting the numbers of varioustypes of united atom groups (e.g., tri- and tetrafunctional groups) in themonomer structures of the two copolymer blend components and by aver-aging these numbers over the monomer distributions in the copolymers (seebelow) [52, 95].


8.1.1The BLCT Free Energy Expression

The specific Helmholtz free energy of mixing f for a binary incompressibleAxB1–x/CyD1–y copolymer blend emerges from the BLCT in the apparentlyfamiliar form [95]

fkT

=ϕ1

M1ln ϕ1 +

ϕ2

M2ln ϕ2 + ϕ2

1χ11 + ϕ22χ22 + ϕ1ϕ2χ12 , (22)

where Mi is the number of united atom groups in a single chain of blendcomponent i and ϕi is its volume fraction. The assumption of blend incom-pressibility implies that ϕ1 + ϕ2 = 1. The chain occupancy indices M1 and M2are simply related to the average numbers nα of α-type monomers in individ-ual copolymer chains by

M1 = nAsA + nBsB , M2 = nCsC + nDsD . (22a)

The interaction parameters χij of Eq. 22 are given as

χ11 =–z2

[εAA

kTm2

A +εBB

kTm2

B + 2εAB

kTmAmB

]

–r2

1

z2 , (22b)

χ22 =–z2

[εCC

kTm2

C +εDD

kTm2

D + 2εCD

kTmCmD

]

–r2

2

z2 , (22c)

and

χ12 =–z2

[εAC

kTmAmC +

εAD

kTmAmD +

εBC

kTmBmC +

εBD

kTmBmD

]

–2r1r2

z2 ,

(22d)

where the εαβ(α, β ≡ A, B, C, D) are monomer averaged nearest-neighbor vander Waals attractive energies between united atom groups of species α and β

and z is the lattice coordination number (taken as six for a cubic lattice). Theratios mA = nAsA/M1, mB = nBsB/M1, mC = nCsC/M2, and mD = nDsD/M2 canconveniently be expressed in terms of the monomer site occupancy numberssα and the copolymer compositions x = nA/(nA + nB) and y = nC/(nC + nD)as

mA = 1 – mB =xsA

xsA + (1 – x)sB(23a)

and

mC = 1 – mD =ysC

ysC + (1 – y)sD. (23b)

The entropic structural parameters r1 and r2 in Eqs. 22b–22d are the tempera-ture independent portions of χij that arise from chains of blend components 1


and 2, respectively. The r1 and r2 are weighted averages [95] of the corres-ponding parameters rα for the individual pure species α

r1 = rAxsA

s1+ rB

(1 – x)sB

s1(23c)

and

r2 = rCysC

s2+ rD

(1 – y)sD

s2, (23d)

where

s1 = xsA + (1 – x)sB (23e)

and

s2 = ysC + (1 – y)sD (23f)

denote the average monomer site occupancy indices for the AxB1–x andCyD1–y copolymer chains, respectively. The quantities rα (α ≡ A, B, C, D)

are computed by counting the numbers of tri-(

s(tri)α

)

and tetrafunctional(

s(tetra)α

)

units in single monomers of species α:

rα =sα + s(tri)

α + 3s(tetra)α

sα. (23g)

The scaling factor of z/2 in Eqs. 22b–22d is the same as in standard FH theoryand formally should be replaced within the BLCT by (z – 2)/2. Introducingthis correction is unnecessary in the BLCT since it merely alters the values ofthe εαβ that are generally treated as adjustable parameters. The correction,however, becomes relevant when energetic contributions to χij are retainedbeyond the leading energy terms that are included in Eqs. 22b–22d.

8.1.2The SANS χSANS Parameter in the BLCT

The small angle neutron scattering (SANS) monomer–monomer interactionparameter χSANS is defined in terms of the derivative of the specific freeenergy f as

χSANS = C12

[

1M1ϕ1

+1

M2ϕ2–

∂2( f /kT)

∂ϕ21

∣∣∣∣V,T

]

. (24a)

The factor of C equals (s1s2)1/2 or s1s2/[s1(1 – ϕ) + s2ϕ] (as explained in Eq. 9)converts the SLCT united atom group–united atom group interaction param-eter into the monomer–monomer interaction parameter χSANS. After some


lengthy algebra, Eq. 20a can be transformed into [95]

χSANS = C |r1 – r2|2

z2

+C

s1s2

[

χACxysAsC + χBC(1 – x)ysBsC (24b)

+ χADx(1 – y)sAsD + χBD(1 – x)(1 – y)sBsD

– χABx(1 – x)sAsBs2/s1 – χCDy(1 – y)sCsDs1/s2

]

,

where the binary interaction parameters are defined as

χαβ =z

2kT[εαα + εββ – 2εαβ] , α = β ∈ A, B, C, D . (24c)

If the temperature-independent contribution to χSANS in Eq. 24b is ignoredand each monomer is treated as an entity occupying a single lattice site (i.e.,sA = sB = sC = sD = 1), Eq. 24b reduces to the well-known random copolymerFH expression [88]

χFH = χACxy + χBC(1 – x)y + χADx(1 – y) + χBD(1 – x)(1 – y)

– χABx(1 – x) – χCDy(1 – y) . (25)

An additional dependence on the copolymer compositions x and y beyondthat predicted by simple random copolymer FH theory appears in Eq. 24bthrough the front conversion factor C/s1s2 as well as through factors of s1/s2and s2/s1. The main novel feature of Eq. 24b, however, lies in the presenceof the temperature-independent portion of χSANS. This term represents theinfluence of monomer structural asymmetry on the nonrandom chain pack-ing and coincides with the leading contribution of the LCT for binary blendsin the incompressible, high molecular weight, athermal, fully flexible chainlimit.

As illustrated in the next subsections, the BLCT has successfully been ap-plied to a wide variety of copolymer blends whose thermodynamics cannotbe described by random copolymer FH theory. While both theories share theinability to distinguish between random, diblock, or alternating copolymersof the same composition, the sensitivity of computed thermodynamic prop-erties to the detailed monomer molecular structure of the blend componentsmakes the BLCT a much more powerful theoretical tool than random copoly-mer FH theory. The relative simplicity of Eq. 24b and the readily computedentropic factors r1 and r2 render the BLCT useful for interpreting thermody-namic data and for generating new predictions.

8.1.3Special Limits of AxB1–x/CyD1–y Copolymer Blends

Equation 24b can be used to evaluate χSANS for special cases of AxB1–x/CyD1–ysystems. For instance, the interaction parameter χSANS for a binary blendAxB1–x/AyB1–y of two copolymers with the same monomer species but differ-


ent compositions emerges directly from Eq. 24b by replacing the subscripts Cby A and D by B:


z2

+C

s1s2χAB(x – y)2(sAsB)2 , (26a)

where

s1 = xsA + (1 – x)sB , (26b)

s2 = ysA + (1 – y)sB , (26c)

r1 = rAxsA

s1+ rB

(1 – x)sB

s1, (26d)

and

r2 = rAysA

s2+ rB

(1 – y)sB

s2, (26e)

with rα(α = A, B) denoting the pure species α entropic parameters givenby Eq. 23g. When the monomer site occupancy indices are set to unity(sA = sB = 1), the temperature-dependent portion of χSANS from Eq. 26a re-duces to the well-known FH random copolymer theory form [88]

χFH = χAB(x – y)2. (27)

The χSANS parameter for a mixture of a pure homopolymer, say A, witha copolymer CyD1–y likewise follows from Eq. 24b by introducing the substi-tutions B ≡ A and x = 1. Thus, we find


z2 +1s2

[

χACysC + χAD(1 – y)sD – χCDy(1 – y)sCsD/s2]

(28a)

with s1 = sA,

r1 =[

sA + s(tri)A + s(tetra)

A

]

/sA , (28b)

and r2 given by Eq. 23d. The energetic portion of χSANS in Eq. 28a reduces tothe well-known FH random copolymer theory expression [89]

χFH = χACy + χAD(1 – y) – χCDy(1 – y) (28c)

for an A/CyD1–y system when monomers of species A, C, and D all are as-sumed to occupy identical volumes.

Equation 24b can similarly be used to generate the χSANS parameter fora binary homopolymer blend, say A/C, by replacing subscript B with A andD with C. The resulting expression,

χSANS = C |rA – rC|2

z2 +z

2kT(εAA + εCC – 2εAC)

, (28d)


is a simplified form of the SLCT Eq. 10. A comparison of Eqs. 28d and 10indicates that the SLCT reduces to the BLCT when the monomer-structure-dependent correction to the interaction energy term is neglected in the SLCTEq. 9a. As mentioned earlier, the factor of z/2 in Eq. 28d should formally bereplaced by (z – 2)/2, but this difference is irrelevant when the interactionenergies are treated as adjustable parameters.

8.2Application of the BLCT to Norbornene/Ethylene Copolymer Blends

The observation by Delfolie et al. [96] that the miscibility of norbornene-co-ethylene (NxE1–x/NyE1–y) binary copolymer blends improves significantlywhen both components are rich in norbornene, i.e., when x > 1/2 and y > 1/2,stands in sharp contrast to the predictions (Eq. 27) of random copolymerFH theory [88], which implies that the blends are miscible when the com-position difference |x – y| is less than a critical constant value. The generaltheory of Sect. 8.1 that has been used to explain this observation is de-scribed elsewhere in some detail [96]. Here we summarize the main resultsof this theoretical analysis to illustrate some computational aspects of theBLCT.

Due to restrictions imposed by the geometry of a simple cubic lattice anddue to the inability [18, 20] of the current LCT to treat polymer systems whosemonomers contain closed rings, the cyclic olefin norbornene monomer ismodeled by an approximate compact structure (Fig. 10) that is composed ofseven united atom groups (sA = 7). The ethylene monomer is represented bya two-bead dimer with two CH2 groups, i.e., sB = 2. Thus, this simple modelincorporates the actual size disparity between these two types of monomers,a feature that influences the magnitude of the entropic component χs ofthe interaction parameter χSANS, and it also affects the dependence of theenergetic portion of χSANS on the compositions x and y of the two blendcomponents.

Fig. 10 United atom group model (a) of the actual norbornene monomer and (b) the sim-plified norbornene structure used in the BLCT calculations. The dashed lines representthe bonds linking the monomer to its neighbors along the chain


The average monomer site occupancy indices s1 and s2 of Eqs. 26b and 26cfor the two blend components are

s1 = 7x + 2(1 – x) (29a)

and

s2 = 7y + 2(1 – y) . (29b)

Because each norbornene monomer in our model has three trifunctionalunited atom CHn groups and because both norbornene and ethylene unitshave no tetrafunctional carbons, rA = (7 + 3)/7 and rB = 2/2. Hence, the par-tial entropic structure factors r1 and r2 of Eqs. 26d and 26e take the form

r1 =(7 + 3)x + 2(1 – x)

7x + 2(1 – x)(30a)

and

r2 =(7 + 3)y + 2(1 – y)

7y + 2(1 – y). (30b)

Since SANS experiments have not been performed for the norbornene-co-ethylene mixtures, the conversion factor C for transforming the BLCT χSANSparameter to a monomer–monomer interaction is taken as (s1s2)1/2, whichis the more commonly used form. Thus, the entropic portion χs of χSANSbecomes

χs = (s1s2)1/2 |r1 – r2|2

z2

(31a)

= [(2 + 5x)(2 + 5y)]1/2 1z2

[6(x – y)

(2 + 5x)(2 + 5y)

]2

. (31b)

According to Eq. 26a, the temperature-dependent portion of χSANS is given by

χh = (s1s2)1/2χAB(x – y)2(sAsB)2 , (32a)

with

χAB =z

2kT[εAA + εBB – 2εAB]

=z

2kT[εN-N + εE-E – 2εE-N] ≡ zε

2kT, (32b)

where ε is the binary E/N homopolymer blend exchange interaction energy.Combining Eqs. 29a, 29b, 32a, and 32b produces the final expression for χh as

χh = [(2 + 5x)(2 + 5y)]1/2[

14(x – y)(2 + 5x)(2 + 5y)

]2 zε2kT

. (32c)

The total interaction parameter χSANS is the sum of the entropic and ener-getic contributions from Eqs. 31b and 32c. Figure 11 depicts the computedχSANS parameter as a function of the composition x of component 1 (NxE1–x).


Fig. 11 Interaction parameter χSANS for ethylene-norbornene copolymer blends com-puted from Eqs. 31b and 32c as a function of the composition of component 1 (NxE1–x).Different curves correspond to various compositions y of component 2 (NyE1–y). Thedimensionless exchange energy ε is chosen as ε/kT = 10–4, and z = 6. Note that the com-puted χSANS is independent of blend composition within the BLCT

The curves correspond to different compositions y of component 2 (NyE1–y).When both x and y are large, χSANS indeed becomes small, suggesting a bet-ter miscibility of NxE1–x and NyE1–y random copolymers, in accord with theexperimental observation of Delfolie et al. [96].

8.2.1Influence of Stiffness and Steric Interactions

Equations 31b and 32c have been derived assuming that all bonds in therandom copolymer chains are completely flexible. However, the actual nor-bornene monomers are rather rigid objects whose steric interactions areconsiderable. As mentioned in Sect. 6, both internal monomer rigidity andsteric interactions influence the partial structural parameters ri. The rigid-ity of norbornene monomers (Fig. 10) is introduced into our model by takingthe two pairs of side group bonds as completely rigid. The parameter ri foran AxB1–x random copolymer chain whose monomers of type A and B haven(A)

2,stiff and n(B)2,stiff stiff pairs of bonds, respectively, is given by the general for-

mula [96]

ri =

[

sA + s(tri)A + 3s(tetra)

A – n(A)2,stiff

]

x

sAx + sB(1 – x)

+

[

sB + s(tri)B + 3s(tetra)

B – n(B)2,stiff

]

(1 – x)

sAx + sB(1 – x), i = 1, 2 , (33a)

which is a direct extension of Eq. 23g to copolymer chains composed ofmonomers with some rigid bonds. Notice that the presence of rigid bonds inthe side groups of an α-type monomer always causes a diminution of rα for


that monomer. Specializing Eq. 33a to the NxE1–x/NyE1–y species (i.e., settingn(A)

2,stiff = 2 and n(B)2,stiff = 0) yields r1 and r2 as

r1 =(7 + 3 – 2)x + 2(1 – x)

7x + 2(1 – x)(33b)

and

r2 =(7 + 3 – 2)x + 2(1 – y)

7y + 2(1 – y), (33c)

which are smaller than the corresponding ri parameters of Eqs. 30a and 30bfor fully flexible chains. Consequently, the corresponding entropic portion χsof χSANS decreases compared to χs of Eq. 31b, thereby leading to improvedmiscibility of norbornene-ethylene copolymer blends. (Introducing rigidityof the norbornene monomers does not affect the calculated temperature de-pendent portion χh of χSANS in the BLCT, so that Eq. 32c remains valid.)

The bulkiness of the actual norbornene ring implies the presence of thestrong steric interactions between bonded norbornene units. These steric in-teractions have been modeled [96] by treating the bonds connecting adjacentnorbornene monomers as semiflexible. The structural parameter r(semi)

i forAxB1–x random copolymer chains with nAA semiflexible bonds between suc-cessive monomers of species A emerges from the BLCT as [96]

r(semi)i =

[

sA + s(tri)A + 3s(tetra)

A – n(A)2,stiff

]

x

sAx + sB(1 – x)

+

[

sB + s(tri)B + 3s(tetra)

B – 2n(B)2,stiff

]

(1 – x)

sAx + sB(1 – x)

+2(gA – 1)nAA/(nA + nB)

sAx + sB(1 – x), (34)

where

gA =z

z – 1 + exp(

E(A)b /kT

) (34a)

and E(A)b is the trans–gauche conformational energy for a pair of bonds

containing a bond between successive norbornene monomers. Equation 34becomes identical to Eq. 33a as it must when gA = 1 (i.e., when all backbonebonds are assumed to be fully flexible). Note that the quantity nAA is sequencedependent: nAA vanishes for a perfectly alternating (AB)n copolymer, equals(nA – 1) for AnBm diblock copolymer, and reduces to nA(nA – 1)/(nA + nB)for a statistically random copolymer composed of nA and nB monomers oftypes A and B, respectively. The presence of the stiffness factor gA in Eq. 34implies that χs now depends on temperature, so the term entropic χ is no


longer appropriate. Because gA < 1 by definition, an increase in the stiffnessof the bonds connecting norbornene monomers is predicted to improve themiscibility of norbornene-co-ethylene blends. A more detailed theoretical an-alysis [96] indicates that the entropic χs exerts a dominant role in influencingthe miscibilities of the norbornene-co-ethylene blends. A very good repre-sentation of the experimental data is obtained when either the rigidity of thenorbornene side group or the semiflexibility of the bond between adjacentnorbornene monomers is described within the BLCT [96].

8.3Isotopic Mixtures of Saturated Polybutadienes (sPB)

The simplest AxB1–x/CyD1–y blends, perhaps, are those in which the AxB1–xand CyD1–y components differ only in the percentage of hydrogen and deu-terium atoms [and, of course, in copolymer composition and blend com-positions (ϕ1, ϕ2)]. An example of these systems is provided by mixturesof hydrogenated and deuterated saturated poly(butadiene) (sPB) polymers.Both blend components are random copolymers because of the presence ofvarying degrees of 1,2 and 1,4 monomers. These weakly interacting isotopicsPB systems have been extensively investigated by SANS techniques [28] andby a nuclear reaction analysis [97]. The same blend samples are used in bothexperiments and cover a wide range of copolymer compositions (x, y).

Scheffold et al. [97] claim general agreement in the overall values of χ

between these two very different measurements, but some systematic dis-crepancies exist. In particular, fits of the SANS and nuclear reaction analysisdata to the form χSANS = χs + χ ′

h/T yield rather different χs and χ ′h for the

three binary blends probed by both methods. (See Table 1 for three exam-ples.) While the temperature-independent portion χs of χSANS, as determinedby Scheffold el al. [97], is generally rather small (on the order of 10–5) withboth positive and negative signs, the SANS values of χs are always nega-tive and large (– 10–3 < χs <– 10–4). Also, significant differences (up to 250%)appear in the temperature dependence of the χ parameters (i.e., in the co-efficients χ ′

h) obtained by these two experimental methods. The disparitiesin χs and in χ ′

h between the data sets (Table 1) represent one source of un-certainty. Hence, we should not attempt to obtain better agreement betweentheory and experiment than the magnitude of these discrepancies and/orthe stated experimental errors. Another possible source of uncertainty arisesfrom the method of preparation of the samples. In order to have matchedpairs of hydrogenated and deuterated sPB chains for the determination ofradii of gyration, etc., the same unsaturated perhydro PB samples are theneither hydrogenated or deuterated. Unfortunately, the deuteration process isnot simple, and a certain degree of H/D scrambling occurs. Thus, the deu-terium content varies in unknown fashion among the individual monomers ofa single chain and also within the united atom CHn groups of each monomer.


Table 1 Examples of disparate values of the interaction parameter χSANS = χs + χ ′h/T

for saturated PB blends as obtained from SANS [28] and nuclear reaction analysis [97]experiments for identical blend samples

H52/D66 H66/D52 H88/D78χs ×104 χ ′

h[K] χs ×104 χ ′h[K] χs ×104 χ ′

h[K]

Graessley – 13.6 1.12 – 6.57 0.656 – 8.17 0.732et al.

Scheffold 4.56 0.428 – 1.22 0.458 – 9.28×10–3 0.469et al.

Since the SANS scattering intensity is generally described as depending, inpart, on differences in the interaction energies εαβ of the hydrogenated anddeuterated species, the random partial deuterium substitution leads to anadditional interaction randomness that cannot presently be treated with thetheory developed here. (The random partial deuteration presumably likewiseaffects the nuclear reaction analysis data.)

The BLCT is applied to isotopic sPB blends by fitting the adjustable pa-rameters of the theory (Sect. 8.3.2) to the two separate data sets generatedfrom the two experimental techniques. First, we describe the fits to the SANSdata of Graessley et al. [28]. For notational specificity, let component 1 desig-nate an AxB1–x random copolymer with a fraction x of hydrogenated 1,2 unitsand a fraction (1 – x) of hydrogenated 1,4 units (A ≡ H1,2 and B ≡ H1,4),whereas component 2 denotes a statistical copolymer CyD1–y of partiallydeuterated 1,2 and 1,4 PB monomers (C ≡ D1,2 and D ≡ D1,4). The unitedatom monomer structures for both 1,2 and 1,4 monomers are depictedin Fig. 12. The eight adjustable interaction energies that determine χ ′

h forAxB1–x/CyD1–y binary systems (Eqs. 24b and 24c) can be reduced to a fewfor the isotopic sPB mixtures by introducing simple assumptions describedbelow. The monomer structures for 1,2 and 1,4 sPB in Fig. 12 indicate thatsA = sB = sC = sD = 4, s(tri)

A = s(tri)C = 1, s(tri)

B = s(tri)D = 0, and s(tetra)

A = s(tetra)C =

s(tetra)B = s(tetra)

D = 0, and, consequently, the average monomer site indices si

Fig. 12 United atom group models for 1,2 and 1,4 poly(butadiene) (PB) monomers. Themodels do not distinguish between saturated and unsaturated PB or between cis andtrans 1,4 units


(i = 1, 2) in the copolymer occupancy indices coincide with those correspond-ing to the individual monomers of either species:

s1 = 4x + 4(1 – x) = 4

and

s2 = 4y + 4(1 – y) = 4 .

The partial entropic structural parameters r1 and r2 of Eqs. 23c and 23d nowsimplify to

r1 =(4 + 1)x + 4(1 – x)

4x + 4(1 – x)=

x + 44

(35a)

and

r2 =(4 + 1)y + 4(1 – y)

4y + 4(1 – y)=

y + 44

. (35b)

In order to reduce the number of adjustable parameters to the bare min-imum, all heterocontact interaction energies εαβ(α = β) are assumed tosatisfy the Berthelot combining rule [98], i.e.,

εαβ = (εααεββ)1/2 , α, β = A, B, C, D , α = β . (36a)

The second simplifying assumption involves the use of the polarizabilitymodel of Bates et al. [99], which postulates a simple scaling relation betweenthe interaction energies for two hydrogenated and two deuterated monomergroups:

εCC = γ 2εAA (36b)

and

εDD = γ 2εBB , (36c)

where the scaling factor γ is less than unity to reflect the weaker attractionbetween two partially deuterated species. Again, to minimize the number ofparameters, a common value of γ is assigned for both 1,2 and 1,4 monomers.This assumption also ignores the fact that the experimental samples havevarying degrees of partial deuteration and should therefore be analyzedwithin the BLCT by allowing the γ factors (and scattering lengths) to varywith the monomers’ deuterium content. As already mentioned, a theoreticaltreatment of variable deuteration poses serious technical difficulties.


8.3.1The χSANS Parameter

The small angle neutron scattering χSANS parameter of Eq. 24b can be con-verted by using Eqs. 35a–36c to the rather compact expression

χSANS =(x – y)2

4z2 +2zkT

(1 – γ )[

yε1/2AA + (1 – y)ε1/2

BB

]

(37)

+ (x – y)[(

ε1/2AA – ε

1/2BB

)]2,

which contains only three adjustable parameters: two self-interaction van derWaals energies εAA and εBB for the 1,2 and 1,4 perhydrogenated species, re-spectively, and the polarizability scaling factor γ . The first term in braces inEq. 37 dominates over the second, but the hydrogen/deuterium swap effect islost without this second term, a feature consistent with the nature of the swapphenomenon that arises from very small differences between the two isotopicspecies. Note that the energetic portion of the χSANS from Eq. 37 exhibitsa structure similar to the form

χδ =v0

2kT(δ1 – δ2)2 (38)

that establishes the basis of solubility parameter solution theory and thathas been used by Graessley, Lohse, and coworkers [27–32] to describe theirexperimental data for χSANS in terms of the pure component solubility pa-rameters δ1 and δ2. Because the squares of the solubility parameters are bydefinition proportional to the cohesive energy densities of the pure melts, thesolubility parameters rigorously do not contain contributions from the ather-mal packing entropy that are embodied in the entropic portion of χ in theBLCT. Thus, the use of Eq. 38 to interpret experimental χSANS data representstheoretically unjustified empiricism when the χSANS parameter has a signifi-cant temperature-independent component of pure entropic origin. The lackof a separate entropic portion χs in solubility parameter theory is a seri-ous deficiency of that approach. An improved version of solubility parametertheory could be formulated, for instance, by merging Eq. 38 with the SLCTEq. 11b.

8.3.2Comparison with Experimental Data

The three parameters in Eq. 37 are fit to experimental χSANS(T) data for a se-ries of ten isotopic sPB random copolymer blends that differ in the composi-tions x and y of the two blend components. The values of the fitting param-eters are, however, not unique, and many sets of εAA, εBB, and γ reproducethe data with similar accuracy and standard deviations. Among the manysets, we choose the following: εAA ≡ ε1,2–1,2 = 238.3 K, εBB ≡ ε1,4–1,4 = 240.8 K,


and γ = 0.9901. The value of γ agrees with the polarizability model pre-diction [99] of αC–D/αC–H = 0.98–0.99. Moreover, the higher self-interactionenergy εαα for the less branched 1,4 polybutadiene species (ε1,4–1,4 > ε1,2–1,2)is consistent with our previous fits [80] of the LCT to equation of state (i.e.,PVT) data for polyolefin melts as well as with the analysis of Schweizer andcoworkers [100–102].

Figure 13 illustrates the comparison between the experimental χSANS(T)data for a pair of systems that are labeled by Graessley et al. [28] as H38/D25and H25/D38 and the BLCT χSANS(T) as calculated from Eq. 37 for the setof εAA, εBB, εAB, and γ specified above. The notation H38/D25, for example,denotes the isotopic blend of hydrogenated (H) and deuterated (D) polybuta-diene random copolymers, and the numbers indicate the fractions of 1,2 unitsin each of the blend components. The experimental points [28] are designatedin Fig. 13 by circles and triangles, whereas the lines represent the theoreticalfits. Figure 13 demonstrates that the theory reproduces the overall values ofthe χ parameters within the experimental error bars of ±1.5×10–4 [28].

Note that in order to describe the χSANS(T) data for ten binary sPB blends,we employ only three adjustable parameters, while the analysis of Graess-ley et al. [28–30] requires a separate set of temperature-dependent solubilityparameters for each individual copolymer species (i.e., 20 sets of solubilityparameters in all). Despite the huge contraction of the δ1(T) and δ2(T) valuesinto just the three parameters (εAA, εBB, and γ ) of the BLCT, the slopes χ ′

hof χSANS(T) vs. 1/T, however, differ considerably between theory and ex-periment. Some decrease in this discrepancy can be obtained by introducingdifferent chain stiffness for the two blend components or by allowing γ todiffer slightly for the 1,2 and 1,4 monomer units, but given the experimental

Fig. 13 Interaction parameter χSANS as a function of inverse temperature 1/T for sym-metric (ϕ1 = ϕ2 = 0.5) H25/D38 and H38/D25 isotopic saturated poly(butadiene) blends.Solid lines are the BLCT fits to a collective set of experimental data [28] for ten Hx/Dyand Hy/Dx mixtures. Circles and triangles indicate data for the H25/D38 and H38/D25mixtures, respectively. The experimental error bars for χSANS are taken as the upper limitsfor the stated accuracy (±1.5×10–4) of the SANS measurements [28]


error bars, there appears to be no point in adding extra parameters to achievethese improvements.

An important feature of the BLCT Eq. 37 is the proper reproduction ofthe deuterium swap effect, i.e., a larger χ is always obtained when the morebranched polyolefin component is deuterated. This deuterium swap effect hasbeen found in numerous experiments [28, 97, 103, 104] and follows as a dir-ect consequence of Eq. 37 provided that the self-interaction energy εαα ishigher for the less branched polyolefin species. It should be possible to ra-tionalize the latter constraint on the interaction energies from the modelsemployed for off-lattice simulations of alkane systems. These continually re-fined models use considerably larger Lennard–Jones interaction parametersfor CH3 groups than for CHn (n = 0, 1, 2) units.

One feature contributing to the differences between the experimental andtheoretical slopes χ ′

h in Fig. 13 is the presence of a rather huge negativetemperature-independent portion χs of χSANS for all the binary isotopic sPBblends studied by Graessley et al. [28] (for instance, χs =– 0.0035 and – 0.0022for the H08/D25 and H25/D08 blends, respectively). In contrast, the com-puted entropic part χs of χSANS in Eq. 37 is always positive

χs =(x – y)2

4z2

and small. [A slightly negative χs can, however, be predicted by the SLCTwhen chain semiflexibility is included (Sect. 7) or by the LCT that treats thesystem as compressible.] On the other hand, both the entropic and energeticportions of χSANS obtained by Graessley et al. [28] for some systems are quitedifferent from those found by Scheffold et al. [97] (Table 1). Therefore, it isalso relevant to describe the neutron reaction analysis data with the BLCT.

Fits to the nuclear reaction analysis data [97] for eight binary sPB blendshave been performed using the interaction energies εAA = 238.3 K and εBB =240.8 K obtained from fits to the SANS data [28] and by allowing γ to be theonly adjustable parameter (although allowing the energies to be adjustableparameters could, in principle, lead to improved fits). As shown in Table 2,theoretical and experimental slopes χ ′

h agree well, with the discrepancy ex-ceeding 20% only for the H52/D66 sample. All other samples exhibit quitesmall values of χs that are presumably smaller than the experimental errorbars. If we assume that the SANS error bars of ±1.5×10–4 also representthe accuracy of the nuclear reaction analysis data [97], there is good agree-ment between the theoretical and experimental [97] χSANS parameters overa wide temperature range (between 400 and 500 K) for half of the samplesconsidered, but further refinement of the current theoretical model (such asintroducing chain semiflexibility or differing γ for the 1,2 and 1,4 monomers)would undoubtedly improve the agreement.


Table 2 SANS interaction parameter χSANS = χs + χ ′h/T of saturated poly(butadiene)

blends as determined from the nuclear reaction analysis [97] experiments and from theBLCT fits to the data

χs ×104 χs ×104 χ ′h[K] χ ′

h[K]Blend Experiment Theory Experiment Theory

H52/D66 3.87 1.36 0.3633 0.518H66/D52 – 1.20 1.36 0.459 0.412H66/D75 0.27 0.56 0.410 0.497H75/D86 0.8 0.84 0.543 0.505H78/D88 – 0.95 0.69 0.583 0.501H86/D75 – 0.70 0.84 0.488 0.422H86/D94 0 0.25 0.647 0.492H88/D78 – 0.009 0.69 0.469 0.425

8.4Unsaturated Isotopic Polybutadiene Blends

Blends of protonated (HPB) and perdeuterated (DPB) unsaturated polybu-tadienes have also been the subject of extensive experimental investiga-tions [53, 87, 106, 107]. Both blend components are typical random copoly-mers because of the presence of varying degrees of randomly placed 1,2 and1,4 PB addition units along the polymer chain.

Early studies by Bates et al. [106, 107] and by Sakurai et al. [87] reveal thatthese systems exhibit an upper critical solution temperature (UCST), i.e., un-dergo phase separation upon cooling. Subsequent investigations have focusedon the combined influence of isotope effects and the blend microstructureon the miscibility patterns of these random copolymer binary mixtures. Inparticular, a series of systematic SANS experiments by Jinnai et al. [53]demonstrate that the UCST phase behavior that had previously been observedfor these systems [87, 106, 107] remarkably converts to a lower critical so-lution temperature (LCST) phase separation with an increase in the vinylcontent of the HPB component when the vinyl content of the DPB compon-ent remains fixed. This phenomenon cannot be explained by the traditionalextension of FH theory to random copolymers since this theory is derivedunder the assumption that the individual χαβ are of purely energetic origin.Thus, the FH random copolymer theory [88] is capable, at most, of predictingthe conversion of a UCST phase separation system into a completly misciblesystem.


8.4.1The χSANS Parameter

The BLCT is applied [108] to these isotopic PB blends in order to elucidatethe molecular factors responsible for the transition from UCST to LCST be-havior that occurs when the blend microstructure is changed [53]. The unitedatom model for the 1,2 and 1,4 PB species does not distinguish between cisand trans 1,4 units, nor between head-to-head and head-to-tail placements ofsuccessive 1,4 units, but this deficiency of the model is not considered by usas very serious since these fine details probably exert a secondary influenceon the blend compatibility. The BLCT computations are performed assumingthe presence of only three independent microscopic van der Waals energiesεAA, εBB, and εAB as representing the interactions between two 1,2, two 1,4,and a pair of 1,2 and 1,4 united atom CHn groups, respectively. The remainingseven energy parameters are determined by using the simple scaling relationsin the spirit of the polarizability model of Bates et al. [99]:

εCC = γ 2εAA , εDD = γ 2εBB , εCD = γ 2εAB , εAC = γεAA , (39)

εBD = γεBB , εBC = εAD = γεAB .

Given these assumptions, the SANS parameter χSANS of Eq. 24b takes theform

χSANS =(x – y)2

4z2 +2zkT

εAA(γy – x)2 + εBB[γ (1 – y) – (1 – x)]2 (40)

– 2εAB(x – γy)[γ (1 – y) – (1 – x)]

,

where following Jinnai et al. C has been replaced by s1s2/[s1(1 – ϕ) + s2ϕ] andwhere x and y are the compositions of 1,2 units in the HPB and DPB compo-nents, respectively.

If both blend components are hydrogenated or deuterated polybutadienes(i.e., if γ = 1), Eq. 40 simplifies to

χSANS =(x – y)2

4z2 +2zkT

(x – y)2[εAA + εBB – 2εAB] . (41)

Equation 41 is a special case of Eq. 26a for a binary AxB1–x/AyB1–y mixtureof two copolymers that emerges when the monomer of both species A and Bhave identical site occupancy indices (sA = sB = 4). Thus, the SANS interac-tion parameter of the PB AxB1–x/AyB1–y blends is proportional to the squareof the difference in the copolymer compositions x and y [χSANS ∼ (x – y)2].Since, as noted by Schwahn and coworkers [109], the exchange energyεAA + εBB – 2εAB is positive for the PB AxB1–x/AyB1–y blends (with any xand y), only UCST phase behavior is possible for these mixtures of two per-hydro polybutadiene copolymers. The remarkable conversion from UCSTto LCST phase diagrams that has been observed by Jinnai et al. [53] as|x – y| is varied arises within the BLCT from a change of the sign of χh with


|x – y|. While both the BLCT and random copolymer FH theory [88] admitof a variation of the sign of χh with the microstructure of AxB1–x/CyD1–yblends, only the BLCT can explain the occurrence of LCST phase behav-ior because the BLCT provides an estimate of the temperature-independentportion of χSANS that is absent in FH theory. The example of HPB/DPB iso-topic blends demonstrates that the very small energetic difference betweenthe H/D isotopes is sufficient to induce this remarkable change in phasebehavior.

8.4.2Comparison with Experimental Data

The four parameters of Eq. 40 are fit [108] to the experimental data of Jinnaiet al. [53] for five isotopic PB random copolymer blend samples with vary-ing compositions x of the protonated PB species and a constant composition yof the deuterated polybutadiene. Table 3 summarizes the theoretical and ex-perimental values of both χs and χ ′

h and indicates that the theory reproducessemiquantitatively the SANS data [53] in the sense that it precisely describesthe variation in the nature of the phase transition (upper vs. lower criticalsolution temperature phase diagrams) with the microstructure of the iso-topic PB blends. The type of phase behavior is correctly predicted [108]by the BLCT for all five PB blends considered. Within the BLCT, an LCSTphase diagram for the isotopic PB blends stems from the competition be-tween a negative energetic portion χ ′

h and a sufficiently positive entropic partχs of χSANS. This competition is more noticeable by considering the spinodalcondition that has the following form in the BLCT:

1M1ϕ

+1

M2(1 – ϕ)– 2

[

χs +χ ′

h

T

]

= 0 , (42)

where ϕ is the blend composition and M1 and M2 are the numbers of unitedatom groups in the individual chains of species 1 and 2, respectively. When χ ′

his negative, a sufficiently large χs > 0 leads to LCST phase behavior.

Except for one blend sample (where both x and y are low and very similarto each other), the experimental and theoretical values of χ ′

h accord reason-ably well. Good agreement for the temperature-independent portion χs ofχSANS is confined, however, to the two samples with the largest composi-tional difference ∆ = |y – x| between the two blend components and with thelargest experimental χs. The experimental values of χs are negative for theblend samples for which ∆ = 0.01 and 0.23, but the flexible chain BLCT pre-dicts χs as always positive. This limitation precludes obtaining a perfect fit tothe data [53]. However, as discussed above, the more general LCT, which ap-plies to compressible polymer systems with arbitrary molecular weights andchain semiflexibility, can admit a small negative χs of a magnitude compara-ble to those observed for the two samples. When both χs and χ ′

h are negative,


the system must be treated as compressible and possibly the polymers de-scribed as semiflexible molecules in order to explain an LCST phase diagram.Another neglected feature that may affect χs is the presence of cis-trans ran-domness of the 1,4 butadiene units, a feature not easily modeled by a simpleanalytical approach.

The values of the fitting parameters again are not unique. Many sets ofεAA, εBB, and εAB, and γ lead to similar results as illustrated by the set inTable 3 that are obtained for εAA ≡ ε1,2–1,2 = 260.1 K, εBB ≡ ε1,4–1,4 = 271.3 K,εAB ≡ ε1,2–1,4 = 265.7 K, and γ = 0.987. The higher interaction energies for theunsaturated PB blends than the εαβ determined for saturated PB mixtures(Sect. 8.3.2) are consistent with the presence of double bonds in the unsatu-rated species. Moreover, the differences between ε1,2–1,2, ε1,4–1,4, and ε1,2–1,4are indeed small as expected for weakly interacting systems. The value ofheterocontact interaction energy ε1,2–1,4 departs from that resulting from theBerthelot geometric combining rule [98] by about 0.02%. Note that forcingε1,2–1,4 to equal the geometrical mean (ε1,2–1,2ε1,4–1,4)1/2 automatically leadswithin the BLCT to a positive χ ′

h for arbitrary microstructures of the isotopicPB blends and, therefore, always to the prediction of a UCST phase behav-ior [108].

The review article by Schwahn in this volume describes more recent testsof the BLCT for isotopic PB blends by Schwahn and coworkers [109]. Thesetests have been applied to data sets roughly three times larger than the dataset of Jinnai et al. that is analyzed by us. Moreover, the analysis by Schwahnand coworkers was performed using only three adjustable parameters sinceone of the self-interaction energies εii was determined from fits of the LCT topure component equation of state (PVT) data. Excellent agreement is foundbetween theory and experimental for both the entropic χs and energetic χ ′

hportions of the χSANS parameter.

Table 3 SANS interaction parameter χSANS = χs + χ ′h/T of isotopic DPB(x)/HPB(y) unsat-

urated poly(butadiene) blendsa

Blend sample χs ×104 χs ×104 χ ′h[K] χ ′

h[K]Experiment Theory Experiment Theory Experiment

DPB(16)/HPB(17) – 4.38 1.36×10–2 0.352 0.784DPB(16)/HPB(39) – 5.19 3.80 0.234 0.214DPB(16)/HPB(52) 3.57 9.30 – 0.144 – 0.128DPB(16)/HPB(65) 14.6 16.9 – 0.556 – 0.457DPB(16)/HPB(80) 30.3 29.2 – 0.814 – 0.859

a Numbers in parentheses are rounded off values for percentage of 1,2 units in the blendcomponents


9Order–Disorder Transitionin Polystyrene-b-Poly(n-Alkyl Methacrylate) Diblock Copolymers

Since our LCT prediction [110] of the possible existence of a lower order–disorder transition (LDOT) in diblock copolymers, interest in this phe-nomenon has increased significantly due to experiments of Russell andcoworkers [111, 112] that confirm our predictions and due to experiments ofRuzette et al. [113] that reveal the existence of different types of phase behav-ior in a homologous family of diblock copolymers formed from styrene anda series of n-alkyl methacrylates. The extremely well-studied system of PS-b-PMMA [poly(styrene)-b-poly(methyl methacrylate)] is known to undergoan upper order–disorder transition (UODT); the short side chain n-alkylmethacrylate copolymers (n = 2, 3, and 4) are found to order upon heating(LDOT systems); and diblock copolymers with longer n-alkyl group (n = 6, 8,12) are observed to order on cooling. Quite remarkably, the block copolymer,consisting of styrene as one block and the other block a 53–47 weight per-cent random copolymer of methyl (n = 1) and lauryl (n = 12) methacrylates[PS-b-P(MMA0.74-co-LMA0.26], orders on heating, despite the fact that bothPS-b-PMMA and PS-b-PLMA copolymers individually exhibit classical upperorder–disorder transitions! These striking results clearly indicate that the mi-crophase separation in these systems is subtly, but decisively, affected by themonomer molecular structures and the detailed molecular interactions.

9.1Unified Modeling of the n-alkyl Methacrylate Systems

This striking variation in the nature of the order–disorder transition with thelength of the alkyl side group in the n-alkyl methacrylate (nAMA) block com-ponent can only be explained by a theory that distinguishes not only betweenthe chemical structures of the styrene and n-alkyl methacrylate monomers,but also between different n-alkyl methacrylate molecules. Flory–Hugginstheory is of no use in these regards since it treats monomers as structurelessentities whose only differences are reflected by ascribing different monomer–monomer interaction parameters χij to the individual components in thesystem. The BLCT provides a more realistic molecular description by spec-ifying united atom type structures for individual monomers, by explicitlycomputing (with no adjustable parameters) the entropic portion of χ fromthe numbers of tri- and tetrafunctional united atom groups in these monomerstructures, and by basing the computation of the temperature-independentportion of χ on the more microscopically faithful interactions between unitedatom groups. However, in order to minimize the number of adjustable pa-rameters in the BLCT, all united atom groups of a monomer have heretofore


been assumed to interact with a common, monomer averaged energy. Whilethe approximation of monomer averaged interaction energies may suffice asa zeroth-order model for some chemically fairly homogeneous systems likepolyolefins, the monomer averaged interaction models are clearly inadequatefor describing the observed [113] striking variation in the ordering behaviorof styrene-b-n-alkyl methacrylate diblock copolymers with the length of thealkyl chain. Thus, the BLCT has been extended [114] to treat binary blendsin which each pair of chemically distinct united atom groups is permitted tointeract with different interaction energies [49].

For simplicity, we invoke the traditional assumption that the SANS inter-action parameters for A/B binary blends and A-b-B diblock copolymer meltsare identical. The BLCT Eq. 28d for χSANS of an A/B binary homopolymerblend,

χSANS = χs + χ ′h/T (43)

= C |r1 – r2|2

z2

+ C[εAA + εBB – 2εAB]z – 2kT

,

is still valid when a monomer of polymer species A has sA1 , sA2 , ..., sAk unitedatom groups of type A1, A2, ..., Ak, respectively (where sA1 + sA2 + ... + sAk =sA), and a monomer of polymer species B has sB1 , sB2 , ..., sBl united atomgroups of type B1, B2, ..., Bk, respectively (sB1 + sB2 + ... + sBl = sB). However,the effective attractive van der Waals energies εAA, εBB, and εAB are now ex-plicit functions of the specific group site occupancy indices sAi and sBi and thespecific interaction energies εAiBj

εAA =k

∑

i=1

εAiAi

(sAi

sA

)2

+ 2k

∑

i=1

k∑

j>i

εAiAj

sAi sAj

s2A

, (44a)

εBB =l

∑

i=1

εBiBi

(sBi

sB

)2

+ 2l

∑

i=1

l∑

j>i

εBiBj

sBi sBj

s2B

, (44b)

εAB =k

∑

i=1

l∑

j=1

εAiBj

sAi sBj

sAsB. (44c)

The specific interaction energies εαiβj in Eqs. 44a–44c are the adjustableparameters of the theory, but, as shown below for the PS-b-PnAMA sys-tems, not all are independent. The factor C in Eq. 43 again ensures that χ

is normalized as a traditional monomer–monomer interaction parameter.Equations 43 and 44 can readily be extended to systems in which one com-ponent is a copolymer, and the details are given in [114].

The knowledge of both portions of χSANS = χs + χ ′h/T for the A-b-B di-

block copolymer system enables evaluating the order–disorder transitiontemperature that is generally determined from the vanishing of the inverse


structure factor 1/S(q∗), where q∗ denotes the wavevector for which S(q) hasa maximum. A modified version of Leibler theory [115] relates χ and S(q∗)through the expression

1S(q∗)

=F(x1, x2, f )

N– 2χSANS , (45)

where N is the polymerization index of the diblock and where the function Fis generalized [116] to treat unequal Kuhn lengths for the two segment types,

F(x1, x2, f ) =f 2g(x1) + 2f (1 – f )h(x2) + (1 – f )2g(x2)

f 2(1 – f )2[g(x1)g(x2) – h2(x1)h2(x2)], (46)

with

g(xi) = 2x–2i [xi + exp(– xi) – 1] , i = 1, 2 ,

h(xi) = 2x–1i [1 – exp(– xi)] , i = 1, 2 ,

and

xi ≡ (q∗Ri)2 , i = 1, 2 .

In the above equations, the Ri = (Nia2i /6)1/2(i = 1, 2) are the radii of gyration

of the blocks, the ai are the Kuhn lengths, f ≡ f1 is the volume fraction ofblock 1, and N1 + N2 = N. The volume fraction f may be determined from themole fraction y ≡ y1 = N1/N and from the monomer molar volumes v1 and v2through the standard relation

f =yv1

yv1 + (1 – y)v2.

The order–disorder transition temperature TODT simply follows from Eqs. 45and 46 as

TODT =χ ′

hN

(1/2)F(x1, x2, f ) – χsN. (47)

Equation 47 differs from the standard Leibler formula [115] by the presenceof the extra term χsN and the more general formulation of the function F asgiven by Eq. 46.

The expression for the interaction parameter χSANS in Eq. 43 is special-ized below for PS-b-poly(n-alkyl methacrylate) diblock copolymer melts. Themonomer structures for both the styrene (sA ≡ sS = 8) and n-alkyl methacry-late (sB ≡ sn-AMA = 6 + n) monomers are depicted in Fig. 14. Stiffness in thestyrene molecule is introduced into our model by treating the two pairsof styrene side group bonds (Fig. 14) as completely rigid (n(S)

2,stiff = 2). Inorder to minimize the number of adjustable parameters, we distinguishonly three types of united atom units in all these diblock copolymer sys-tems considered: alkyl CHn groups (n = 0, 1, 2, or 3), aromatic CHn groups


Fig. 14 United atom group structures for the styrene (S) and n-alkyl methacrylate(nAMA) monomers

(n = 0 or 1), and oxygen atoms. This assumption formally leads to the pres-ence of six different interaction energies εAlk-Alk, εAr-Ar, εO-O, εAlk-Ar, εAlk-O,and εAr-O, but only three combinations of these energies are found [114] tobe independent parameters in the BLCT for the systems studied. No inter-action is ascribed to the united atom group pairs involving the tetrafunc-tional carbon of the n-alkyl methacrylate monomer due to the obvious sterichindrances.

The entropic and energetic portions of the effective interaction parame-ter χSANS for the PS-b-P(n-alkyl methacrylate) melt can be expressed finallyas [114]

χs = [8(6 + n)]1/2 1z2

(98

–10 + n6 + n

)2

(48a)

and

χ ′h = [8(6 + n)]1/2

[z – 2

2k1

16(6 + n)2

]

9(2 + n)2ε1 + 12(2 + n)ε2 + 16ε3

,

(48b)

where the three independent combinations of interaction energies,

ε1 ≡ εAlk-Alk + εAr-Ar – 2εAlk-Ar ,

ε2 ≡ 6(εAr-Ar – εAlk-Ar) + 4(εAlk-O – εAr-O) ,

ε3 ≡ εO-O + 9εAlk-Ar – 12εAr-O ,

are the only three adjustable parameters of the theory. The correspond-ing expressions χs and χ ′

h for the random copolymer PS-b-P(XxY1–x) di-block copolymers, with X and Y denoting, respectively, n-alkyl methacrylate


monomers with n = p and n′ = q united atom groups, take the form [114]

χs = 8[6 + q + x(p – q)]1/2 1z2

[98

–10 + q + x(p – q)6 + q + x(p – q))

]2

, (49a)

χ ′h = 8[6 + q + x(p – q)]1/2 z – 2

2k1

16[6 + q + x(p – q)]2 (49b)

×

9[2 + q + x(p – q)]2ε1 + 12[2 + q + x(p – q)]ε2 + 16ε3

.

Setting x = 0 (or x = 1) in Eqs. 49a and 49b recovers Eqs. 48a and 48b, re-spectively. When Eqs. 49a and 49b are specialized to p = 1 and q = 12, theresulting expressions correspond to the PS-b-P(MMAx-co-LMA1–x) diblockcopolymers studied by Ruzette et al. [113].

Within the BLCT, χs is positive, while χ ′h can formally be of either sign, de-

pending on the signs of the three effective interaction energies ε1, ε2, and ε3(and on the composition x when one of the blocks is a random copolymer).We have found [114] a plethora of sets εi for which the enthalpic portionχ ′

h of the interaction parameter χ is positive for PS-PMMA (n = 1), negativefor PS-PEMA, PS-PPMA, and PS-PBMA diblock copolymers (n = 2, 3, and 4,respectively), positive again for PS-PHMA, PS-POMA, and PS-PLMA melts(n = 6, 8, and 12, respectively), and, finally, negative for the block copolymerof styrene and a 53–47 weight percent random copolymer of methyl (n = 1)and lauryl methacrylates (n = 12) [i.e. for PS-b-P(MMA0.74-co-LMA0.26)]. Sys-tems with a positive χ ′

h order on cooling, while systems with a negative χ ′h

order on heating. The existence of numerous acceptable sets of energy param-eters satisfies our primary goal of elucidating a molecular mechanism thatexplains the observed variation of the order–disorder transition type with thelength of the alkyl side group in the n-alkyl methacrylates. This remarkablebehavior is explained in terms of different interactions between chemicallydifferent united atom groups. The interpretation of this behavior does notrequire considering finite compressibility effects nor resorting to a sophisti-cated theory, although finite compressibility, nonrandom mixing effects, etc.must affect the experimental χSANS parameters, and, hence, the observedODT transition temperatures.

Table 4 summarizes the χ ′h that are determined from Eqs. 48b and 49b for

a typical set of εi (with the constraints ε1 > 0, ε2 < 0, and ε3 > 0), alongwith the calculated entropic portions χs and the order–disorder transitiontemperatures TODT for the eight PS-b-PnAMA systems studied experimen-tally [113]. The computed values of TODT for certain diblock copolymer meltsturn out to be very sensitive to the value of χs that is substituted into thedenominator of Eq. 47. A small variation in χs for these systems leads toa huge change in the computed TODT. Consequently, instead of a single valuefor TODT, Table 4 presents ranges of TODT that are obtained by ascribingan uncertainty of ±20% to the computed χs. Taking into account the obvi-ous simplification inherent in the use of a lattice model and, in particular,


Table 4 Order–disorder transition temperature of polystyrene-b-poly(n-alkyl methacry-late) diblock copolymers and the polystyrene-b-poly(MMA0.74 -co-LMA0.26) copolymersystem as determined from SANS experiments [113] and from BLCT calculationsa

Theory ExperimentSample χs ×102 χ ′

h[K] TODT[K] TODT[K]

28 K PS-b-PMMA 4.14 2.14 303 to ∞ < 37350 K PS-b-PEMA 3.12 – 3.11 214–1501 > 463110 K PS-b-PPMA 2.41 – 3.75 220–505 47375 K PS-b-PBMA 1.88 – 1.67 0 to ∞ 47327 K PS-b-PHMA 1.18 6.78 163–184 < 37323 K PS-b-POMA 0.759 17.8 327–346 < 37319 K PS-b-PLMA 0.315 41.7 446–452 40880 K PS-b-(MMA0.74 1.95 – 2.12 248 to ∞ 441-co-LMA0.26)

aThe adjustable parameters of the theory are chosen as ε1 = 12.45 K, ε2 =– 91.33 K, andε3 = 149.45 K and are assumed to be independent of the deuterium content of the PSblocks

the modeling of monomer structures within the geometric constraints ofa cubic lattice (z = 6), this choice of uncertainty is quite reasonable. Given theerror bars for the computed χs, the agreement between theoretical and ex-perimental transition temperatures TODT appear satisfactory for all but thePS-b-PHMA system where the computed TODT is probably too low.

10Discussion

The lattice cluster theory (LCT) has been developed [12–22] to rectify sev-eral glaring deficiencies in the highly useful Flory–Huggins (FH) theory [3–7]for the thermodynamics of polymer systems. More specifically, the LCT de-scribes the variation of thermodynamic properties of polymer systems withthe monomer molecular structures of the constituents. This description isbased on the systematic computation of the nonrandom mixing correctionsto FH that arise from the presence of local correlations, which are present onlength scales smaller than average monomer sizes and which are completelyneglected in FH theory. The present review focuses on two simplifying lim-its of the LCT that clearly illustrate the manifold physical ramifications of thismolecular theory.

The simplified lattice cluster theory (SLCT) [46–48] emerges from the LCTby neglecting contributions in the noncombinatorial portion of the specificfree energy of mixing that depend on the excess free volume fraction and on


the molecular weights. Thus, the SLCT coincides with the LCT in the highmolecular weight, high pressure limit. The binary blend SLCT differs fromthe corresponding FH theory in two respects. Firstly, the SLCT predicts thepresence of a temperature-independent entropic contribution χs to the effect-ive interaction parameter χSANS that stems from the structural asymmetrybetween the two components of the blend and that is absent in FH theory. Im-portantly, the evaluation of χs in the SLCT proceeds by the trivial countingof the numbers of different types of united atom groups in single monomersof the two blend components. Comparisons with experiment indicate that thecomputed values of χs are quite good so long as χs is not too small or is nega-tive. (Indeed, negative values of χs are rare for homopolymer blends.) Thesecond relevant feature distinguishing the SLCT from FH theory is the de-pendence of the energetic portion χ ′

h of the SLCT χSANS on the monomerstructures of the two blend components and on the blend composition. Whilea composition-dependent term in χ ′

h is introduced into several phenomeno-logical treatments of polymer blends in a physically vague or ad hoc fashion,this term appears in the SLCT as a direct consequence of monomer structuralasymmetry. Both χs and the composition-dependent portion of χ ′

h vanishidentically when the two blend species have identical structures. These twobasic improvements beyond FH theory are responsible for the ability of theSLCT to predict several types of critical phase behavior whose explanationlies beyond the scope of FH theory [47, 48].

The SLCT has been applied to several binary homopolymer polyolefinblends that have been widely studied experimentally and that display behav-iors that cannot be rationalized with FH theory. These applications illustratehow the molecular formulation of the SLCT aids in understanding nontriv-ial miscibility phenomena in these systems. For instance, the SLCT explainswhy hhPP is far more miscible with PEP than is PP and why blends of PIBwith several other polyolefins tend to form lower critical solution temperature(LCST) miscibility patterns. Further comparisons of the SLCT with experi-ment will undoubtedly reveal limitations of the theory that result from the useof a lattice model, of oversimplistic united atom models, of monomer aver-aged interaction energies, etc., as well as limitations due to the fact that theSLCT only contains the leading monomer-structure-dependent contributionto the free energy. Thus, some significant improvements should emerge fromthe computation of higher order contributions to the LCT free energy [117].

When applied to systems containing statistical copolymers, the SLCT losesits enormous analytical and physical simplicity due to the greater complex-ity of these systems compared to binary homopolymer blends. The lack ofmathematical simplicity in describing copolymer blends arises, in part, fromthe dependence of the free energy on the monomer sequence. Therefore,a further approximation is introduced into the SLCT to generate a theoreticalapproach that is simple and easy to use but is devoid of a serious deficiency ofthe extensions of FH theory to random copolymer systems, namely, the neg-


lect of the temperature-independent portion χs of χSANS. The resulting theoryis termed the basic lattice cluster theory (BLCT) to emphasize its relation tothe LCT. Like the SLCT, the BLCT applies to binary blends in the high molecu-lar weight, high pressure limit, but, in contrast to the SLCT, the BLCT ignoresthe monomer-structure-dependent contribution to the interaction term χ ′

h.Thus, the neglect of this term is the only difference between the two theories,but this omitted term implies that the BLCT χSANS is independent of compo-sition. Applications of the BLCT to several copolymer systems demonstratethat χs plays an essential role in explaining a wide range of thermodynamicbehavior.

10.1Improved Theories

The SLCT and the BLCT are suitable theoretical tools to illustrate and predictthe influences of monomer molecular structure and nonrandom mixing ef-fects on thermodynamic properties of many polymer systems, but the moregeneral LCT should be used when the dual assumptions of high molecularweights and/or incompressibility are no longer applicable. For instance, thepressure dependence of χSANS, which is discussed in the article by Schwahnin this volume, can be investigated theoretically with the LCT. Although theLCT expressions for the free energy of mixing ∆Fmix and the χSANS param-eter for a compressible system are far lengthier than those of the SLCT, theLCT corrections to ∆Fmix of FH theory are simple polynomials in the vol-ume fractions of the constituents of the system [20, 21]. The volume fractionfor the excess free volume is obtained numerically from the equation of statethat is derived from the LCT free energy expression [25]. The LCT predictsa wider range of possible phase behaviors than the SLCT, such as ones withclosed-loop phase diagrams or with miscibility windows [59, 118]. The de-scription of compressible systems, however, necessitates the introduction ofadditional adjustable parameters into the theory and, consequently, dependson the availability of more extensive experimental data. For example, theequivalent of SLCT Eq. 9a for compressible binary blends contains three inde-pendent monomer averaged interaction energies ε11, ε22, and ε12 rather thanthe single exchange energy ε that appears in the SLCT Eq. 9a. While the self-interaction energies ε11 and ε22 are usually determined from fits to data forpure melts, the heterocontact energy ε12 can only be determined from fitsto experimental data for the binary blend. Prior papers explain the technicaldetails of these fits [25, 26].

As mentioned earlier, the SLCT has been derived for AxB1–x/CyD1–ycopolymer systems [93], but the theory has not yet been applied to analyzeexperimental data. In contrast to the BLCT, the SLCT predicts that the ener-getic portion χ ′

h of the χSANS parameter for copolymer blends depends onthe blend composition and is sensitive to the monomer sequence distribu-


tion, i.e., differs for blends of alternating, diblock, and random copolymers.The SLCT expression for ∆Fmix of AxB1–x/CyD1–y systems has a form similarto Eq. 9a, but ∆Fmix for the copolymer system also contains a monomer-sequence-dependent contribution.

The remarkable ordering behavior in the series of PS-b-PnAMA diblockcopolymer melts has been explained by lifting the assumption of monomeraveraged interaction energies and by ascribing different interaction energiesto chemically different united atom groups within a monomer. However, thisgeneralization leads inevitably to an increase in the number of adjustable pa-rameters in the theory. While it would be highly desirable to accumulate a setof united-atom-group-dependent interaction energies that could be used fora wide range of different systems, this worthwhile goal requires considerableanalysis and the collection of a rather large body of experimental data.

The LCT and its more simplified descendants, the SLCT and BLCT, containonly the leading monomer-structure-dependent contributions from the clus-ter expansion. While this leading contribution has already provided a wealthof molecular-scale understanding concerning the miscibility and thermody-namic properties of binary polymer blends, the computation of the next ordercontributions [117] should improve the predictive powers of the theory to en-able describing, for instance, the tacticity and sequence dependence of ∆Fmixin copolymer systems since the inclusion of these latter effects requires treat-ing local correlations on the longer-length scales that have been considered sofar.

Recently, we have extended [81] the Gibbs–DiMarzio theory of the glasstransition in systems of semiflexible, interacting polymers composed ofstructured monomers. This extension is based on a reformulation of the LCTthat enables the computation of the configurational entropy Sc that is a ba-sic quantity in many theoretical treatments of the glass transition. This newtheoretical approach provides a valuable tool for determining how monomerstructure affects glass transition temperatures of polymer melts (J. Dudowicz,K.F. Freed, J.F. Douglas unpubl. results) by [119].

Lattice models have also been very popular in theoretical studies of pro-teins and, in particular, protein folding. Each amino acid is generally treatedin these lattice models as a united residue entity occupying a single latticesite in the spirit of the original Flory–Huggins lattice model for polymers.The striking differences in miscibilities observed for the chemically very simi-lar PP/PEP and hhPP/PEP blends engender some skepticism toward the useof these popular protein folding lattice models. Given the presence of spe-cific hydrogen bonding interactions in proteins, it is questionable whetherextended lattice models in which individual amino acids can occupy severalneighboring lattice sites can offer as much insights as found for binary blendsof synthetic polymers.


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Effect of Shear Flow on Polymer Blends

Nigel Clarke

Department of Chemistry, University of Durham, via Cintia, Durham DH1 3LE, [email protected]

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

2 Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3 Theoretical Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333.1 The Thermodynamic Approach . . . . . . . . . . . . . . . . . . . . . . . . 1353.2 Dynamic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.2.1 The Cahn–Hilliard Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.2.2 The Doi-Onuki Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.2.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.2.4 The Asymmetric Shear Response of Polymer Blends . . . . . . . . . . . . . 1513.2.5 Non-linear Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583.2.6 Model Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613.2.7 Temperature Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1643.3 Scattering Under Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1673.4 Numerical Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Abstract An overview of the various theoretical approaches to understanding the effectof shear flow on polymer blends is presented. Two differing models have emerged; oneproposes the addition of a shear-dependent term directly into the free energy, whereas theother considers how shear modifies the equation of motion for concentration fluctuations.As discussed in detail, despite these differences, the models share much in common, andsome degree of convergence between the two approaches has recently emerged. Althoughthe focus is on modelling, a brief review of the varied range of experimental observations,highlighting the challenges that face theoreticians, is also given.

1Introduction

The consequences of flow on the structure of fluid mixtures have receivedconsiderable attention. Interests in the subject stem not only from its indus-trial relevance – many processing technologies involve the flow of mixtures,but also from its importance from a fundamental point of view – under-standing and predicting behaviour provides a challenge and a test for non-equilibrium thermodynamics. A significant proportion of industrial mixtures

128 N. Clarke

can be considered as either polymer solutions or blends, the latter in particu-lar will form the basis for this review. Since the effects of shear on polymersolutions have been the subject of at least two extensive reviews, we will focushere attention on polymer blends, and the unique behaviour associated withthem.

Firstly, it is useful to define the specific nature of flow that is of relevance.The two principal types of flow are shear and extensional, both of which playan important role during processing. Much attention has been devoted to theformer since it is more straightforward to establish a steady-state shear flowthan a steady-state extensional flow, and, in general, the response of single-component polymeric liquids to shear flow is better understood than theirresponse to extensional flow. The three most common geometries for study-ing polymer systems subjected to flow are concentric cylinders, cone andplate, and parallel plates (see Fig. 1). The shear flow is characterised by theshear rate, γ , such that if we denote x as the flow direction, the velocity isgiven by v = γyx, where y denotes the shear gradient direction. Concentriccylinders are useful for low viscosity solutions, whereas the cone and plateand parallel plate set-ups are utilised for more viscous solutions and blends.In the cone and plate geometry the shear rate is constant across the radius,whereas in the parallel plate geometry the shear rate increases linearly withdistance from the centre of the plates.

Interest in the response of polymer blends subjected to shear ranges fromshear-induced homogenisation of incompatible blends to shear -inducedphase separation of miscible blends. Shear has also been used to deform ex-isting microstructures in immiscible polymer mixtures. However, this reviewarticle is mostly concerned with partially miscible polymer blends; i.e., blendswhose miscibility changes with temperature. Such blends are categorised asexhibiting either lower critical solution temperature (LCST) or upper criti-cal solution temperature (UCST) behaviour. The former phase separate uponheating, whilst the latter phase separate upon cooling. The phase separation

Fig. 1 Geometries used for applying shear flows. (a) Concentric cylinders (Couette flow)– either the inner or outer cylinder is rotated, (b) Cone and plate, and (c) parallel plates

Effect of Shear Flow on Polymer Blends 129

temperature, often referred to as the cloud point (CP), since transparent mis-cible mixtures become cloudy when they start to phase separate, dependsupon the chemical nature of the two components, the molecular weights andpolydispersity of each and the composition.

2Experimental Studies

It is useful to first discuss some of the unusual observations that continueto motivate interest in this subject. Perhaps the simplest and most import-ant question to address in partially miscible polymer blends is whether shearaffects the phase boundary. In particular, it is desirable to know whethershearing increases or decreases the range over which a blend is compatible.A number of experimental techniques have been utilised to help answer thisquestion. An early study by Mazich and Carr [1] inferred shifts in the phaseboundary of polystyrene/polyvinylmethylether (PS/PVME). This is an LCSTblend in quiescent conditions, and is ideal for shear flow studies, since thephase boundary is generally well above the glass transition temperature of theblend. The location of the phase boundary was assumed to be indicated by anabrupt change in the temperature dependence of the viscosity. For two dif-ferent mixtures of PS/PVME (see blends and 1 and 2 in Table 1 for details),it was found that the boundary was shifted to higher temperatures by 2–7 K,across a range of compositions, indicating shear-induced mixing. Rather thanusing the shear rate as the fixed parameter, they chose to fix the first normalstress, arguing that this leads to constant stretching of the chain molecules atdifferent temperatures. In many instances it is more convenient to use shearrate as the control parameter; in such a case it is necessary to interpret resultsin view of the changing viscosity with temperature. In other words, at a fixedshear rate the shear-induced deformation of individual chains will be stronglytemperature dependent.

Katsaros et al. [7, 8] studied PS/PVME blends and determined shifts inthe phase boundary using optical clarity. The disadvantage of this techniqueis that although the shear may prevent large-scale morphologies from de-veloping, the blend may remain phase separated at a much smaller lengthscale than that of the wavelength of light, so that it appears transparent. Moredirect evidence was gathered by the same group using fluorescence stud-ies. Anthracene labelled polystyrene (PS∗) was added in trace quantities toa PS/PVME blend. The PS∗ fluorescence is quenched by the presence of theether group in PVME, so that molecular level mixing or de-mixing can bedetermined by the degree of fluorescence. In the preliminary work of Maniet al. [2] (see blend 3 in Table 1 for details), blends were held at 3 K abovetheir cloud points, so that the mixtures were phase separated. They were

130 N. Clarke

Table 1 Details of some of the partially miscible blends used for studying the effect ofshear on phase separation

Blend Components Molecular weight/kg mol–1 Reference

1 Polystyrene 93 [1]Poly(vinylmethylether) 102




5 poly(ethylene-co-vinyl acetate) 100 [4]solution-chlorinated polyethylene 332

6 Poly(butyl acrylate) 75.5 [4]Solution-chlorinated polyethylene 236

7 Polystyrene 302 [4]Poly(vinylmethylether) 69.7



10 Poly (styrene-acrlyonitrile) 190 [11]Poly ε-caprolactone 95

then sheared for different amounts of time and rapidly quenched below theglass transition temperature of the blend to freeze-in the shear-induced struc-tures. Above a critical shear rate it was found that the blends became mixed;however a reduction of the critical shear rate with shearing time was alsoobserved. This phenomenon highlights the importance of time-dependent ef-fects: even if it is in some sense thermodynamically preferable to remix, thesystem has to evolve from one state to another. It is therefore essential to en-sure that a steady state has been achieved. In such an experiment, it is also tobe expected that the time the blend is held above the CP prior to shearing willalso play an important role: the longer the blend has been allowed to phaseseparate, the larger the phase separated structures will be. At the very least itis to be expected that the size of the microstructures will affect the time takenfor a steady state to be reached.

Larbi et al. [3] performed in-situ fluorescence studies (see blend 4 inTable 1 for details), using a modified rheometer. Samples were heated slowlyat constant shear rates, and again phase separation was suppressed to highertemperatures. For a range of different compositions the shift, ∆T, in the de-mixing temperature with respect to the quiescent phase boundary, Ts, was


well described by the empirical relation,

∆T/Ts = (0.015±0.002)γ 0.59±0.04 (1)

Further fluorescence studies by Mani et al. [9] on the same blend discov-ered that de-mixing could be induced at as much as 40 K below the quiescentboundary at higher shear rates.

Hindawi et al. [4] utilised small-angle light scattering (SALS) to probeshear-induced changes in blends of poly(ethylene-co-vinyl acetate)/solu-tion-chlorinated polyethylene (EVA/SCPE), poly(butyl acrylate)/SCPE (PBA/

SCPE) and PS/PVME (see blends 5–7 in Table 1 for details). The advantageof SALS over fluorescence and rheological techniques is that it is particu-larly sensitive to small fluctuations in local concentration. It has been widelyused to study phase separation in the quiescent state, since it allows not onlyphase boundaries to be determined but also the dynamics of the phase sepa-ration process to be followed. When a homogeneous blend is quenched intothe unstable region of its phase diagram it phase separates by the mechan-ism of spinodal decomposition. Ever present fluctuations in the concentrationstart to grow rather than decay, and a dominant length-scale emerges. Thislength scale is, in many cases, comparable to the wavelength of light, and sothe blend, rather than being transparent, becomes turbid due to scattering.Since the extent of scattering increases with time, SALS provides a quanti-tative method for probing phase separation. One disadvantage of the SALStechnique is that it is not sensitive to phase separation at sub-micron length-scales, and so it is necessary to be aware that a lack of scattering does notalways imply a lack of phase separation. In reference [4] the blends weresheared at a constant rate using a plate-plate shear cell and then slowly heated(both EVA/SCPE and PBA/SCPE exhibit quiescent LCST phase behaviour);the onset of phase separation was indicated by an increase in the light scat-tered from the sample at small scattering angles. A remarkable variety ofbehaviour was observed in the blends; in EVA/SCPE an increase in scatter-ing occurred at lower temperatures, with respect to the quiescent boundary,at low shear rates, indicating an enhancement of the two-phase region, butat higher shear rates the behaviour was reversed with shear apparently sup-pressing phase separation. The maximum change in the phase boundary,– 5 C, occurred at a shear rate of ∼1.5 s–1, This behaviour was observedfor a wide range of compositions. Similar trends were also observed in thePBA/SCPE and PS/PVME blends, although the maximum shifts, at a shearrate of ∼3–4 s–1, were as high as – 10 and – 30 C, respectively. It has beensuggested that the changes in the boundary may be an artefact due to vis-cous heating of the sample. It was estimated that such heating would atmost change the sample temperature by ∼1 C, which is too small to accountfor the observed shifts. In order to provide further evidence for the shear-induced de-mixing at low shear rates and shear-induced mixing at highershear rates, sheared blends were quenched below the glass transition tem-

132 N. Clarke

perature (Tg and heat flow scans were performed using differential scanningcalorimetry). Phase separated blends show two Tgs, which for a blend withthe two phases composed of pure A and pure B, will correspond to the glasstransition temperatures of the two pure components. Thus the values of themeasured glass transition temperatures also provide information on the de-gree of phase separation. The results from the SALS were indeed confirmed,suggesting that it was not the case that phase separation was occurring overdifferent length scales inaccessible to light scattering. Fernandez et al. [5] fol-lowed this work with studies of PS/PVME blends and found that at a constantshear rate as the temperature was increased phase separation occurred asmuch as 15 C below the quiescent phase boundary, depending on the shearrate. As the temperature was further increased remixing was observed and fi-nally de-mixing at even higher temperatures. This behaviour, shown in Fig. 2,was interpreted as being the result of an island of miscibility in the shear rate– temperature “phase diagram”.

Madbouly et al. [10] carried out a systematic study on blends of polystyrene-acrlyonitrile (SAN) and poly methyl methacrylate (PMMA) withvarying PMMA molecular weights ranging from 7 to 396 kg mol–1. Signifi-cantly, when compared to other results, only shear-induced mixing was ob-served for all samples. Unsurprisingly, the shear effect was found to be com-position and molecular weight dependent. As the shear range was increased,the shift in the cloud point first increased monotonically with shear rateand then become almost constant regardless of the applied shear rate values.A maximum elevation in the normalized shift of the cloud points was found

Fig. 2 Normalised shift in the cloud point curve as a function of shear rate for blends ofpolystyrene and poly(vinyl methyl ether). The blend labelled PS1/PVME (squares) andPS2/PVME (circles) correspond to blend 8 and 9 respectively in Table 1. In each case,two cloud points were observed as the blends were simultaneously sheared and heated.Reproduced with permission from reference [5]


when the viscosity ratio is close to unity; a result which is difficult to under-stand in the framework of the current theories discussed below that suggestthat shifts in phase boundaries should be greater when the viscosity mis-match is greater.

More recently [11], for a blend of SAN and poly ε-caprolactone (see blend10 in Table 1 for details), no shear-induced de-mixing was observed. However,only one measurement at one shear rate and one temperature within the one-phase region (10 C below the phase boundary) was reported, and since shiftsare often by smaller amounts this result is far from conclusive.

The dynamics of shear-induced phase separation during the early stagesof the process have been studied by Gerard et al. [12–14] immediately fol-lowing the imposition of shear using SALS. It was found that the growth rateof concentration fluctuations was strongly dependent on the shear rate, andqualitative agreement with some of the theoretical predictions to be discussedbelow was found.

3Theoretical Descriptions

In order to understand how shear flow affects the phase behaviour of poly-mer blends, it is useful to consider first how homogeneous polymeric systemsrespond to an imposed shear flow. This is a subject that has received con-siderable attention, and many of the details are unnecessary for this review;however, a basic understanding of polymer rheology [15] is essential, and sowe present a brief overview of the most relevant ideas. When a polymer so-lution or blend (or indeed any viscoelastic medium) is subjected to a steadyshear, its response can be characterised by the viscosity,

η =σxy

γ(2)

where σxy is the shear stress, and the first normal stress coefficient,

Ψ1 =N1

γ 2 =σxx – σyy

γ 2 (3)

where N1 is the first normal stress difference. Many polymeric systems alsoexhibit second normal stress differences, N2 = σyy – σzz, but these are usuallysmall compared to N1, and their consequences have so far been neglected inmost theories related to polymer blends. It is well known that polymers haveunusual properties in the liquid state, the clearest example being the observa-tion that the zero shear viscosity, η0, the viscosity in the limit γ → 0, scaleswith molecular weight as η0 ∝ M3.4 for polymers above a critical molecu-lar weight. At low shear rates polymer solutions and blends exhibit so-calledNewtonian behaviour, that is the both the viscosity and the first normal stress

134 N. Clarke

coefficient are independent of the shear rate. At higher shear rates, shear thin-ning is observed.

The behaviour of short polymers in the melt is reasonably well describedby the Rouse model, originally proposed to explain the dynamics of dilutepolymer solutions, in which the motion of a polymer is represented by theBrownian motion of beads connected by Hookian springs with no other inter-actions. However, from Table 1, it is clear that in order for shear to affect thestability of blends, the components generally have reasonably high molecularweights. The foundation for modern theories of the flow behaviour of suchpolymers was proposed by deGennes [16, 17], who, by analogy with the the-ory of rubber elasticity for crosslinked polymers, proposed the tube model.This was then explored rigorously by Doi and Edwards [18–21], and has beensuccessfully applied to many aspects of the dynamics of polymers. The tubemodel is an elegant attempt to overcome the mathematical difficulty of de-scribing the consequences of the interaction of all the surrounding polymerson the motion of any given polymer. The effect of the neighbours is mod-elled by a tube with a given diameter around a polymer, and it is assumedthat the only motion is along the axis of the tube – chains cannot move later-ally through tubes. This motion has been called reptation, and the chains aredescribed as entangled. The diameter of the tube is the average distance be-tween effective crosslinks. These “crosslinks”, or entanglements, only preventthe chains from moving through each other, not sliding around each other.An important property of an entangled chain is its relaxation function, thisdescribes the fraction of the chain that at time t still resides within the tubesurrounding the chain at some earlier time t′. To a reasonable approximationthe relaxation function is single exponential with a time constant, τR, knownas the reptation time. The tube model predicts that τR scales with molecularweight as M3. Following a small deformation the stress is assumed to be pro-portional to the number of initial entanglements that have not relaxed, witha proportionality constant, the plateau modulus G0, given by the theory ofrubber elasticity for the modulus of a crosslinked rubber, with the molecularweight between crosslinks replaced by the molecular weight between entan-glements. Hence, if a polymeric system is subjected to a small step strain thestress relaxes as,

G(t – t′) = G0exp

–t – t′

τR

(4)

During steady state shearing, stress bearing entanglements are continuallycreated and relaxed, and the stress tensor is given by,

σ =

t∫

–∞

∂G(t – t′)∂t

C –1(t, t′)dt′ (5)


where C –1 is known as the Finger tensor, which for shear flow is given by,

C –1(t, t′) =

⎛

⎝

1 + γ 2(

t – t′)2

γ(

t – t′)

0γ(

t – t′)

1 00 0 1

⎞

⎠ (6)

This model, often referred to as the upper convective Maxwell model, isweakly non-linear in that it predicts a first normal stress, but no shear thin-ning effects, i.e, the shear stress increases linearly with shear rate so that theviscosity is independent of shear rate. Combining Eqs. 2, 4, 5 and 6, we seethat the tube model predicts the viscosity to be,

η = G0τR ∝ M3 (7)

and the first normal stress coefficient to be,

Ψ1 = 2G0τ2Rγ 2 (8)

The molecular weight dependence of the viscosity arises because the plateaumodulus is independent of chain length. The discrepancy between the pre-dicted and observed scaling of the viscosity has been the subject of intensiveresearch. One of the principal assumptions of the original tube model is thatthe entanglements that form the surrounding tube are fixed. Since these en-tanglements arise from interactions with other chains which are themselvescontinuously moving, much effort has been put into developing models inwhich not only a polymer reptates from a tube but also the tube itself changeswith time. A key feature of the chains in a binary polymer blend is how thedynamics depends on the local concentration. We will consider one of theconceptually simpler approaches to this problem in a later section.

3.1The Thermodynamic Approach

Since shear induces apparent shifts in the phase boundary of blends, it is nat-ural to attempt an explanation based on equilibrium thermodynamics. Horstand Wolf [22] extended a phenomenological model devised for polymer solu-tions proposed by Wolf [23]. The affect of shear was accounted for by addinga shear-dependent term, Es, to the Gibbs free energy of mixing, Fmix,

Fγ = Fmix + Es (9)

The usual Flory-Huggins form was adopted for the zero-shear term,

Fmix

kBT=

φA

NAln φA +

φB

NBln φB + χφAφB (10)

where φA and φB are the volume fractions of A and B which have degrees ofpolymerisation NA and NB, respectively. The Flory-Huggins parameter, χ, is

136 N. Clarke

a measure of interactions between components. For convenience incompress-ibility, such that φA + φB = 1, is often assumed.

The stored energy term was assumed to be equal to the first normal stressdifference, and to be given by the general expression,

Es = Ψ1γ2 = (xAVA + xBVB)

⟨

J0e

⟩ 〈η〉2 γ 2 |〈η〉 γ |–2d∗(11)

where xA,B and VA,B are the mole fractions and molar volumes of the poly-mers A and B, so that the first term in brackets represents the average molarvolume. J0

e , the steady-state shear compliance, is measured in experiments inwhich a steady shear stress is applied and the resultant shear rate measured,as opposed to determining the stress required to maintain a given shear rate.In general, J0

e is related to the plateau modulus by,

J0e ≈ 1/G0 (12)

The angular brackets denote that those parameters of Eq. 11 are measured forthe homogeneous state. Finally, d∗ is given by,

d∗ =–∂ ln 〈η〉∂ ln γ

(13)

The physical basis of Eq. 11 is that the energy of the chains is increased dur-ing shear, due to the entropy of a stretched chain being less than that of freechain. Since chains with different chemical structures and molecular weightswill respond differently to shear, the energy changes will depend on the con-centration.

Equation 11 can be broken down into two factors. In the Newtonian regimethe first normal stress for a polymer melt can be rewritten using Eqs. 8 and 12as,

N1 = Ψ1γ2 = 2G0τ

2Rγ 2 = 2J0

e η2γ 2 (14)

It is important to note that the latter expression is completely general andapplies independently of any model. However, as already noted, polymer sys-tems become non-Newtonian at a critical shear rate, and shear-thinning isobserved. An approximate relation gives the shear rate at which this occurs as,

γ τR ≈ 1 (15)

so that as the relaxation time, and hence viscosity, increases the range of shearrates at which Newtonian flow occurs decreases. At higher shear rates a powerlaw relationship between N1 and γ is found, hence Horst and Wolf proposedthe phenomenological relationship leading to Eq. 13. In order to proceed,it is necessary to know how d∗ varies with shear rate. An early version ofthe tube model for non-linear flows proposed by Graessley [24] was utilised.Shear thinning can be explained within the reptation framework by consid-ering that at higher shear rates the number of entanglements experienced bya chain decreases because the contact time between two molecules decreases


below the time required for an entanglement to form. Consequently, the vis-cosity and the first normal stress decrease as the number of entanglementsdecrease. Just as importantly it is necessary to consider how J0

e and η dependon the concentration of the components. Within the context of the originaltube model the blending law is straightforward; since the entanglements arefixed, each polymer behaves independently of its surroundings, and the vis-cosity and first normal stress are simply the weighted sum, by the weightfractions of the two components, wA and wB, of the respective pure com-ponent values. However, Horst and Wolf made use of a different blendingrule suggested by Schuch [25]. Rather than the stress relaxation functions be-ing additive, it was assumed that the viscosity and compliance of a blend isgiven by that which would be appropriate for a melt with a molecular weightequivalent to the weight average molecular weight of the blend. This leads to,

〈η0〉1/3.4 = wAη1/3.40A + wBη

1/3.40B (16)

and,⟨

J0e

⟩

=(

wAη4.4/3.40A J0

eA + wBη4.4/3.40B J0

eB

)

/η4.4/3.40 (17)

One of the key features of this and other theories is that there exists an asym-metry in the molecular response of each component to shear. Horst and Wolfdetermined shear dependent phase diagrams by applying the usual laws ofthermodynamics to the total free energy. In reference [22] they determinedshifts in the phase boundary using model parameters. They predicted that atlow shear rates, shear-induced mixing should occur; followed by an inversionat higher shear rates to shear-induced de-mixing, followed by a further in-version at even higher shear rates to shear-induced mixing again. That twoinversions occur is due to the fact that there are two different shear ratesat which non-Newtonian flow behaviour occurs. The shear rate of the firstinversion is determined by the more viscous component, whilst the second in-version is due to the onset of shear-thinning in the less viscous component.This argument is backed-up by the behaviour of polymer solutions [23], inwhich only the first inversion occurs, and there is only one critical shear rate.These comments only explain the existence of complex behaviour. However,the form of the mixing rules used makes it difficult to understand quali-tatively the underlying physical mechanisms beyond the explanation givenabove.

It is worth highlighting the necessity of a non-linear mixing rule such asgiven by Eqs. 16 and 17. This is most easily done by considering the conse-quences of utilising a linear mixing rule, as would be predicted by the originaltube model; in which the blend rheology is linearly dependent on the volumefraction, e.g.,

ηBlend = φAηA + φBηB ; ΨBlend = φAΨA + φBΨB (18)

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This would result in no predicted perturbation due to shear. This is clear ifwe consider the thermodynamic rule for coexistence between two phases thatthe chemical potential be equal in each phase. For the approach of Horst andWolf, the contribution to the chemical potential from the stored energy, givenby,

µshearA = ES +

(

1 – φA) ∂ES

∂φA(19)

is simply a constant, since ES would be linearly dependent on φA and willtherefore not affect the coexistence conditions. Whether a more complex, andphysically realistic, mixing rule leads to shear-induced mixing or de-mixingis more easily argued by considering the spinodal curve rather than the coex-istence curve. The spinodal separates the regions in which the homogeneousstate is metastable and unstable, and is related to the free energy by,

∂2F∂φ2 = 0 (20)

If ∂2F/∂φ2 < 0 the mixed state of the blend is unstable, so if the contribu-tion that the stored energy makes to the spinodal is such that the curvatureis negative than shear-induced de-mixing will result, whereas if the curvatureis positive then shear-induced mixing occurs.

Fernandez et al. [5] applied the HW model, to explain the observed ex-istence of an immiscibility gap, with limited success. As a consequence ofthis Horst and Wolf [26, 27] later modified their model to include importanttemperature dependencies of the rheological parameters. This lead to somecomplex predictions, most notably the possibility of closed loop miscibilitygaps, as can be seen in Fig. 3. At certain shear rates, an LCST blend whichis miscible at low temperatures, become immiscible far below the quiescentphase boundary as heated, but then become miscible again at a tempera-ture still below quiescent phase boundary, finally becoming immiscible atsome temperature above the quiescent phase boundary, which is in qualitativeagreement with the findings of reference [5].

There has been a steady output modifying and further applying the the-ory. Horst and Wolf [28] calculated the influence on blends exhibiting UCSTbehaviour. Their predictions were similar to those for an LCST blend. Asthe shear rate is increased the blend experiences shear-induced mixing, thende-mixing, followed by the effect becoming negligible at higher shear rates.Miscibility gaps at certain shear rates, at temperatures above the quiescentphase boundary were also found. Krause and Wolf [29, 30] extended themodel to include ternary blends of immiscible A and B polymers dissolvedin a common solvent, and showed that their predictions compared favourablywith experimental observations. Phase diagrams of sheared blends of ho-mopolymer A and a random A-B copolymer [31] and ternary blends ofhomopolymer A, homopolymer B and a copolymeric compatibiliser have also


Fig. 3 Phase diagram for a model blend with components having molecular weights of75 kg mol–1 and 200 kg mol–1, calculated according to theory of Horst and Wolf. The ex-istence of closed miscibility gaps above a critical shear rate is apparent. Reproduced withpermission from reference [27]

been calculated [32, 33]. Sun et al. [34] have recently predicted the combinedeffects of shear and pressure changes on miscibility within the framework ofthe HW model.

Chopra et al. [35] have attempted to develop a microscopic basis for themethodology adopted phenomenologically by HW. They considered how flowaffects the configurational choices, and thus the entropy, of a sheared poly-mer chain. It was proposed that the additional contribution to the free energycould be determined within the framework of the Flory-Huggins lattice the-ory, by weighting the probability that a monomer would occupy a given latticespace with respect to the previous monomer (on the same chain) in the direc-tion of the flow. At higher shear rates, lattice spaces in the direction of flowbecome more heavily weighted. However a link between polymer dynamics,either for single chains or entangled chains was not established, and for calcu-lation purposes, the authors resorted to utilising the same phenomenologicalideas as HW. We also note that Jou et al. have discussed in detail for poly-mer solutions a justification based on extended irreversible thermodynamics(EIT). However, since the ideas have already been the subject of extensive re-views [36, 37], no further details will be included here. We note however thatthe EIT concept has recently been adapted for polymer blends [38], but sincethis work shares much in common with the dynamic approach introduced inthe next few sections, further discussion will be deferred.

140 N. Clarke

3.2Dynamic Approach

One of the principle drawbacks of the thermodynamic approach is that, todate, the possibility of a variation of the flow field has not been allowed for.Consider a phase separated mixture that forms two layers. If the viscosity ofone layer is much greater than that of the other, it is easy to imagine that whensheared, the viscous layer may resist motion whilst the other layer will flowat a compensatory faster shear rate. In partially miscible systems, it is com-mon to consider the effects of shear by considering the behaviour of a localfluctuation in concentration. In the quiescent state within the one-phase re-gion, such fluctuations continually appear, but will then decay. Once a jumpinto the two phase region takes place some of these fluctuations become un-stable and grow. The concentration dependent temperature at which this firstoccurs defines the spinodal curve. In a flowing system, the existence of localfluctuations, and hence gradients, in concentration clearly results in gradi-ents in the stress and the possibility of local deviations from the mean of theflow field itself. This in turn results in an additional factor that affects thestability of concentration fluctuations and hence the requirement for phaseseparation.

3.2.1The Cahn–Hilliard Theory

It is essential to first briefly review the model for the dynamics of con-centration fluctuations in the absence of shear, first derived by Cahn andHilliard [39] for binary alloys, later modified by Cook [40] to include ther-mal noise, and more recently adapted by deGennes [41] and Pincus [42] fordescribing phase separation in polymer blends.

Since the total concentration of each polymer is conserved, then the conti-nuity equation must be obeyed,

∂φA(r, t)∂t

+ ∇ ·[

vA(r, t)φA(r, t)]

= 0 (21)

where vA(r, t) is the flux or the net flow of polymer out of a small volume atposition r and time t. Equation 21 is exact, but in order to make progress a re-lation between flux and concentration is required, which results in a closedexpression. The simplest such relation is known as Fick’s law in which the fluxis proportional to the spatial gradient of the concentration, the physical rea-soning of which is that at equilibrium the concentration is constant. Howeverthis is only true for non-interacting systems, which would not phase sepa-rate anyway. When interactions are present, it is not the concentration but thechemical potential of each component which is constant at equilibrium; hence


a generalised version of Fick’s law may be written,

JA(r, t) =–M

kBT∇µ(r, t) (22)

where M is a mobility and µ is the chemical potential difference defined by,

µ(

r, t)

= µA(

r, t)

– µB(

r, t)

(23)

The chemical potential difference is related to the free energy. Using the FloryHuggins theory modified to include energetic costs for gradients in concen-tration, the free energy is given by,

F =∫

dr(

fmix(r, t) + fgradient(r, t))

(24)

=∫

drφA

NAln φA +

φB

NBln φB + χφAφB + κ

(∇φA)2

where χ = a + b/T is the temperature dependent Flory - Huggins interactionparameter and κ is the interfacial tension. Since F is a functional integral,∇µ in Eq. 22 must be replaced by a functional derivative, which physicallycorresponds to the total change in the free energy due to a small change incomposition at r′,

µ(r′, t) =δF

δφA(

r′) ≡ limε→0

F[

φA + εδ(

r – r′)] – F [φA]

ε(25)

=∂fmix(r′, t)∂φA(r′, t)

– 2κ∇2φA(r′, t)

where δ(

r – r′) is the Kroneker delta function. Substitution of Eq. 25 intoEq. 22 results in a non-linear partial differential equation that cannot besolved analytically without further approximation. There are two approachesto making progress; one is to solve the equation of motion numerically, of-ten using the finite difference method, the other is to linearise with respectto φA. The former approach is particularly valuable for exploring structuraldevelopment, but the latter as we will see below can be used to determine sta-bility limits and predict scattering patterns, both of which can be accessedexperimentally. Linearisation involves expanding the concentration as a Tay-lor series, and retaining only terms proportional to δφA, so that the chemicalpotential can be expanded as,

µ(r, t) ≈ constant + 2[(

χ – χS)

– κ∇2] δφA(r, t) + · · · (26)

where χS is the value of the interaction parameter on the quiescent spinodal,

χS =12

(1

NAφA+

1NBφB

)

(27)

142 N. Clarke

Assuming that the mobility is not spatially dependent, the resulting linearisedequation of motion,

∂φA(r, t)∂t

– 2M∇2[(

χ – χS)

– κ∇2]

δφA(r, t) = 0 (28)

can be cast in a form that is straightforward for analytical manipulation byuse of the Fourier transform,

f (q) =1

(2π)3/2

∫

V

drf (r) exp– iq · r (29)

The result is a first-order linear differential equation,

∂δφA(

q, t)

∂t=– 2q2M

[(

χS – χ)

+ κq2]

δφA(q, t) (30)

We have effectively decomposed the real space fluctuations into an uncoupledspectrum of fluctuations, each with wavevector, q. Equation 30 has the simplesolution,

δφA(

q, t)

= δφA(

q, t0)

exp

– 2q2M[(

χS – χ)

+ κq2] t

(31)

Hence spontaneous random fluctuations decay exponentially with a well-defined wavevector dependence. Equation 31 can also be used to help usunderstand the early stages of phase separation, since following a jump intothe two-phase region, χS < χ, and for small enough values of q the decayrate becomes negative and fluctuations will therefore grow exponentially. Byusing the dependence of the interfacial tension derived by deGennes [41],κ = b2/36φ(– 1φ), the well-known predictions of the Cahn–Hilliard theory fora binary monodisperse blend undergoing phase separation, follow. Firstly,there is a fastest growing wavevector, q2

m = (χ – χS)/2κ, which from Eq. 32can be seen to grow at a rate R(qm) = M(χS – χ)2/2. It is this which givesrise to the characteristic co-continuous morphology associated with spinodaldecomposition. Secondly, for wavevectors above a critical value, qc =

√2qm,

the rate becomes negative and such fluctuations decay. Concentration fluctu-ations can be probed by scattering techniques, which measure the structurefactor,

S(q, t) =⟨|δφA(q, t)|2⟩ (32)

In the one-phase region of the phase diagram, the Cahn–Hilliard theory pre-dicts that S(q, t) → 0 as t → ∞, which is incorrect since fluctuations arecontinually appearing and decaying due to thermal motion, so that the struc-ture factor has a finite value at all q. To account for this Cook [40] addeda random noise term, θ(r, t), which, in order to ensure the correct equilibriumbehaviour, has the following properties,⟨

θ(

r, t)⟩

= 0 ;⟨

θ(

r, t)

θ(

r′, t′)⟩

= 2M∇2δ(

r – r′) δ(

t – t′)

(33)


or, after Fourier transformation,⟨

θ(

q, t)⟩

= 0;⟨

θ(

q, t)

θ(

q, t′)⟩

= 2Mq2δ(

t – t′)

(34)

The addition of a noise term to Eq. 30 means that a solution is only possibleif it is converted into an equation of motion for the structure factor, using thedefinition of Eq. 32, and the average properties of the noise term, Eq. 34, areapplied. The result is an evolution equation for the structure factor,

∂S(

q, t)

∂t=– 4q2M

[(

χS – χ)

+ κq2] S(q, t) + 2Mq2 (35)

which in the steady state, ∂S(q, t)/∂t = 0, predicts the correct form of the staticstructure factor, S –1(q) = 2

(

χS – χ + κq2)

. This latter expression can be de-rived using the random phase approximation (RPA) [17], which has provenextremely successful for predicting equilibrium scattering from polymer mix-tures in the one-phase region. The RPA relates the collective structure factorto the structure factor of individual non-interacting chains and the Flory-Huggins interaction parameter, and is based on the idea that interactions actas a perturbation to ideal behaviour.

The simplest extension of the Cahn–Hilliard–Cook model to include shearis to assume that the only consequence is that concentration fluctuations areconvected with the macroscopic flow field, as suggested by Lai and Fuller [43],adopting a model proposed by Ohta et al. [44]. With such an assumption,stress, and hence flow rate, gradients are neglected. This is only likely to bea physically realistic assumption if the viscoelastic properties of the two com-ponents are identical. The equation of motion is then a simple modificationof Eq. 28,

∂δφA(r, t)∂t

– γyx∇δφA(r, t) (36)

= 2M∇2[(

χ – χS)

– κ∇2]

δφA(r, t) + θ(

r, t)

which, after applying the Fourier transform, becomes,

∂δφA(q, t)∂t

– γqx∂δφA(q, t)

∂qx,y(37)

=– 2Mq2 (χS – χ + κq2) δφA(q, t) + θ(

q, t)

The structure factor then evolves according to,

∂S(

q, t)

∂t– γqx

∂S(

q, t)

∂qy(38)

=– 4q2M[(

χS – χ)

+ κq2] S(q, t) + 2Mq2

144 N. Clarke

The second term on the left hand side of Eqs. 36 and 38 represents the con-vective effect of shear. The steady state solution to Eq. 38, is,

2[

q2 (χS – χ + κq2) – γqx∂

∂qy

]

S(q, t) = q2 (39)

Since this is still a differential equation, albeit in the variable qy, a simple in-terpretation is not straightforward. Hence it is necessary to consider the timedependence of Eq. 38, which can be found using the method of characteris-tics. Through a transformation of variables, a first-order partial differentialequation is converted into a first-order ordinary differential equation, whichcan then be solved using standard techniques. Due to the usefulness of thismethod for solving equations describing coupled shear flow and concen-tration fluctuation dynamics, it is worth briefly outlining the ideas. If weintroduce a variable, t′, such that,

dtdt′

= 1 ;dqy

dt′=– γqx (40)

and make use of the relation,

dSdt′

=dtdt′

∂S∂t

+dqy

dt′∂S∂qy

(41)

then Eq. 38 can be rewritten as,

dSdt

=– 4q2(t)M[(

χS – χ)

+ κq2(t)]

S + 2Mq2(t) (42)

where we have dropped the ′ notation for convenience, and q = (qx, qy, qz)evolves according to the solution of Eq. 40,

q(t) = q0 – γ tq0xqy (43)

The subscript 0 denotes the values of the wavevector at t = 0. The methodof characteristics is not only valuable from a mathematical viewpoint; it alsoallows insight into the physical implications of the model. For example, animportant consequence of Eq. 43 is that that any fluctuation with a non-zero qx component, will eventually be convected to large values of q. Aswe have already discussed, during the early stages of phase separation, onlyfluctuations with a wavevector less than a critical value grow. Hence, even-tually all fluctuations with qx = 0 will be convected to a regime in whichthey decay. Furthermore, it is clear that if a fluctuation appears with q0x = 0,then the right-hand side of Eq. 42 becomes independent of time, and thestructure factor evolves in exactly the same manner as if no shear wereapplied.

It is worth noting that several years prior to the approach adopted Lai andFuller [43], Pistoor and Binder [45] had already attempted to include molecu-lar effects of shear within the Cahn–Hilliard theory. They utilised the randomphase approximation which offers an alternative formulism for determining


the expansion of the chemical potential, in terms of local deviations from themean concentration, which as noted in the previous section is an essential in-gredient of the Cahn–Hilliard theory. In reference [45], the structure factorof individual non-interacting chains under the influence of a shear flow wasfirst calculated assuming that the chains deformed affinely, i.e., with the flow.As a consequence the chemical potential becomes explicitly, rather than onlyimplicitly due to convection, dependent on the shear rate. Their equation ofmotion for the structure factor took a modified form of Eq. 42,

dSdt

= 2Mq2(t) (44)

– 4q2(t)M[(

χS – χ)

+ κ

q2(t) +2π2

15γ τqxqy(t) +

4π4

315

(

γ τqx)2]

S

where τ is a relaxation time related to the relaxation times and compositionsof each of the components. Since only non-interacting chains were consid-ered the model is only applicable to un-entangled chains that obey Rousedynamics. A further consequence of this assumption is that since the chainstretching is irrespective of the environment there is no additional drivingforce for phase separation, and Eq. 44 only results in modified predictions forthe structure factor rather than stability.

From the above discussion, it is clear that convection with or without anenvironment independent response of chains to shear is not enough to shiftthe conditions for stability of concentration fluctuations in the one-phase re-gion. Its usefulness as a theory lies in its ability to predict the anisotropicq dependence of the structure factor in the one-phase region, which can bemeasured using scattering techniques. In order to predict apparent shifts inphase boundaries an extra ingredient is required – stress and flow rate varia-tions within the blend.

The principal works in which the effects of stress gradients on the dynam-ics of such fluctuations in polymer solutions have been considered are byHelfand and Fredrickson [46], Onuki [47, 48], and Milner [49]. In all refer-ences the physical ideas are very similar, with variations only in the details ofthe theory developed.

Doi and Onuki [50] (DO), extended the models to polymer blends in whichboth components are entangled. The key aspect to address is how to in-corporate stress into the equation of motion for concentration fluctuations.Effectively, by determining the conditions for force balance, it was shownthat the stress enters the equation of motion at the same level as the chem-ical potential. Such an approach enabled the development of a frameworkthat coupled the dynamics of concentration fluctuations to the flow fields andstress gradients; however, only the simplest form of constitutive relation forthe stress was treated. In entangled polymer solutions, the tube model pre-dicts that the relaxation of an imposed stress is well described by a singleexponential decay, with the characteristic time-scale being that required for

146 N. Clarke

a polymer to reptate entirely out of its original tube [21]. Hence DO assumedthat the dynamics of stress relaxation were well described by a single relax-ation time.

A significant feature of the rheological behaviour of polymer blends isthat, in general, stress relaxation cannot be adequately described using a sin-gle relaxation time. Essentially simultaneous publications by Sun et al. [51](SBJ) and Clarke and McLeish. [52] (CM) attempted to address this short-coming. SBJ proposed a model in which the shear modulus of the blend wasallowed to be concentration dependent, whilst CM proposed the use of a spe-cific mixing rule, which also included finite relaxation times for each of thecomponents. In both models stress was incorporated into the modified Cahn–Hilliard equation of motion for concentration fluctuations derived by DO.The CM model has the advantage of being able to quantifiably predict thatthe effect of shear on the behaviour of the stability of concentration fluctua-tions in polymer blends. These changes lead to shifts in the spinodal curve,as defined dynamically rather than thermodynamically, of polymer blends.Dynamically the point at which small fluctuations first become unstable andgrow defines the spinodal. This of course coincides with the thermodynamicdefinition in the quiescent state, but in the presence of shear enables us toconsider apparent shifts in the spinodal without invoking unjustified thermo-dynamic concepts. Significantly, the dynamic equations also enable fluctua-tions in the flow field to be incorporated and determined. Since the flow itselfis anisotropic it is to be expected that variations in the flow field will also varywith direction. As we will show below, it is simplest to understand, from theviewpoint of stability, the behaviour in the shear gradient and vorticity direc-tions. In steady state shear the only stresses that act in these directions are thenormal forces; hence, the existence of normal forces is the key to explainingexperimental observations. This is perhaps the major similarity between thedynamic and the thermodynamic approach; however, the dynamic approachresults in a greater degree of subtlety. More precisely, it is gradients in the nor-mal stress, due to gradients in the concentration, which alter the conditionsfor the growth of small concentration fluctuations from that of the quiescentblend. A second important difference is that the dynamic approach considersenhancement and stability of concentration fluctuations rather than coexis-tence of phases in a de-mixed blend. In this respect it can be argued thatthe thermodynamic approach has greater potential for predicting experimen-tally useful and technologically essential parameters; however there remaina number of questions with regards to the fundamental physical basis of theconcept.

In polymer solutions, both the longest relaxation time associated with thepolymer and the plateau modulus scale with concentration, although the ex-act scaling law has yet to be established. The consequent increase in the firstnormal stress with concentration leads to the prediction that fluctuationsin the shear gradient direction become unstable at temperatures within the


quiescent one phase region, whereas in the vorticity direction, fluctuationsremain stable for a range of temperatures within the quiescent two-phase re-gion. In entangled polymer blends, the normal stress is dependent on therelative concentration of each component, with a relationship that is morecomplex than for solutions.

3.2.2The Doi-Onuki Model

In order to understand the essential physical phenomena that occur whenblends are sheared, DO utilized Rayleighs variational principle, whichamounts to minimizing the energy dissipation function with respect to theaverage velocities of the two components. There are three contributions tothe energy dissipation: friction between polymers, rate of change of the freeenergy due to concentration fluctuations and rate of change of the free en-ergy due to stress. The friction was determined using the two-fluid model ofBrochard [53]. In the tube model each polymer, with a degree of polymeri-sation, N, moves through a fixed network with a center of mass velocity, v,which can be related to the curvilinear velocity of a chain, w, within its tube.When a polymer segment has diffused a distance L, the contour length ofthe tube, the center of mass of the chain has only diffused a distance R, theend-to-end distance of the chain. Hence,

v =RL

w (45)

The tube comprises N/Ne entanglement segments, where Ne is the aver-age degree of polymerisation between entanglements, such that the mean-squared end-to-end distance is given by,

⟨

R2⟩ =NNe

⟨

L2⟩ (46)

Hence the center of mass velocity of a chain through the network is,

v2 =Ne

Nw2 (47)

To account for the fact that the polymers are not moving in a fixed network,Brochard introduced the concept of the tube velocity, vT, which is the velocityof a hypothetical moving network, and is common to all polymers. Conse-quently, the chains now move with a velocity given by,

v =RL

w + vT (48)

148 N. Clarke

The energy dissipated during this process is proportional to the square of thecurvilinear velocity, and can be written as,

Wfriction =∫

dr[

φAζ0Aw2A + φBζ0Bw2

B

]

(49)

=∫

dr[

φANA

NeAζ0A

(

vA – vT)2 + φB

NB

NeBζ0B

(

vB – vT)2]

where ζ0i is the monomeric friction coefficient. For simplicity it has, to date,always been assumed that is equal for each component; i.e., ζ0A = ζ0A = ζ0.In the second line, the energy dissipation is written in terms of the averagevelocities rather than the curvilinear velocities, since these are the naturalvariables when considering diffusion processes. It is clearly necessary to self-consistently eliminate the molecular field, vT. DO proposed that this can doneby minimizing Eq. 49 with respect to the tube velocity, δW/δvT = 0, whichamounts to balancing the forces on the network. Hence,

vT =ζAvA + ζBvB

ζA + ζB(50)

whereζi is the frictional drag of each polymer, given by,

ζi = φiNi

Neiζ0 (51)

The appearance and decay of concentration fluctuations results in energy dis-sipation, which can be determined by the rate of change of the correspondingfree energy fluctuations,

Wmix = F =∫

dr∂F∂φA

φA(r, t) ≡∫

drµAφA(r, t) (52)

The last equality serves to define µA, a quantity that is similar, but not iden-tical to the chemical potential. After substitution of the continuity equationinto Eq. 52, it is found that,

Wmix =–∫

drµA∇ · (vAφA)

(53)

The stress dissipation is incorporated by determining the resultant changein the free energy when a small instantaneous deformation r → r + u(r) isimposed,

Wel =∂δFel

∂t=

∂

∂t

∫

drσαβ∂uα

∂rβ

=∫

drσαβ∂vTα

∂rβ

≡∫

drσ : ∇vT (54)

This formalism is similar to the thermodynamic approach, and is based onthe same principle that strain in polymeric systems results in a change in thefree energy, which can be determined by calculating the work done. An im-portant difference is that all components of the stress, and not just the firstnormal stress, are incorporated. DO also proposed that the stress arises from


the motion of the tube, and thus is coupled to the tube velocity rather thanthe macroscopic flow velocity. An interesting consequence of this assump-tion is that stresses arise even when no macroscopic flow is imposed. Thisso called viscoelastic effect has been explored in some detail and a compre-hensive review by Tanaka can be found in reference [54]. An closely relatedimplication is an effective isotropic stress at non zero q. This results in un-usual effects, such as a q dependent kinetic coefficient in the early stages ofspinodal decomposition [55, 56]. When determining stability limits, it is onlyeffects that occur in the limit of small q that are important. Hence, it is onlynecessary to consider the effects of the macroscopic velocity.

Equations 49, 53 and 54 are combined to give the total dissipation func-tion. The minimization needs to be carried out with the additional constraintof incompressibility,

∇·v = ∇ · (φAvA + φBvB)

= 0 (55)

Hence, the function to be minimized with respect to each of the average poly-mer velocities is,

Wfriction + Wmix + Wel +∫

drp(r, t)∇ · (φAvA + φBvB)

(56)

where p(r, t) is a Lagrange multiplier. The minimization results in expressionsfor the total forces acting on component A,

ζAζB

ζA + ζB(vA – vB) + φA∇µA + φA∇p –

ζA

ζA + ζB∇ ·σ (n) = 0 (57)

and those acting on component B,

ζAζB

ζB + ζA(vB – vA) + φB∇p –

ζB

ζA + ζB∇ ·σ (n) = 0 (58)

The nomenclature σ (n) has been introduced to clarify that it represents thetotal network stress. By eliminating the Lagrange multiplier from Eqs. 57 and58, and inserting the resultant expression into the continuity equation, DO ar-rived at an equation of motion for concentration fluctuations that includedthe effects of stress,

∂φA(r, t)∂t

=– ∇ · (φ2AvA + φAφBvB

)

+ ∇(φ2Aφ2

B/ζ)[∇µA – α∇ ·σ (n)] (59)

where,

ζ =ζAζB

ζB + ζA(60)

and,

α =(NA/NeA)ζA

0 – (NB/NeB)ζB0

(NA/NeA)φAζA0 + (NB/NeB)(1 – φA)ζB

0(61)

The importance of this prefactor will be discussed later.

150 N. Clarke

An important simplification in terms of enabling analytical progress is toassume that the stresses are given by expressions appropriate to the macro-scopically imposed flow field, and that the stress at any point, r, in the systemdepends only on the local concentration. That is, when the concentrationchanges, it is assumed that the stress instantaneously changes; the dynamicevolution of the stress is assumed to be much faster than that for the concen-tration fluctuations. This is the so-called adiabatic approximation [49].

Linearisation and Fourier transformation of Eq. 59 results in the requiredequation of motion, for a fluctuation, in q space, δφA(q, t),

∂δφA(q, t)∂t

– γqx∂δφA(q, t)

∂qy= (62)

– M[

2q2(χS – χ + κq2) –α

kBT

∑

i,j

qiqj∂

∂φAσ

(n)ij

]

δφA(q, t)

Without stress gradients, i.e., for a blend whose components have identicalresponses to shear, Eq. 62 reduces to Eq. 36, as would be expected. The nat-ural progression from the model of Lai and Fuller [43] is evident if Eq. 62 iscompared with Eq. 37.

3.2.3Stability Analysis

In order to determine the conditions for fluctuations to grow or decay, atthe level of approximations discussed above, it is only necessary to considerchanges in the condition for the growth or decay of fluctuations in the qx = 0plane, and it is sufficient to focus our attention on the qy and qz directions.In these directions, the convective term of Eq. 62 vanishes, and an effectivediffusion coefficient, Deff may be defined using,

∂δφA(qy,z, t)

∂t=– q2

y,zDeffδφA(qy,z, t) (63)

which is no more than the Cahn–Hilliard [39] equation for the dynamicsof concentration fluctuations with a modified diffusion coefficient which de-pends on gradients in the stress as well as gradients in the chemical potential.If thermal noise is neglected, Eq. 63 has the simple solution,

δφA(qy,z, t) = δφA(qy,z, t = 0) exp

– q2y,zDefft

(64)

An apparent spinodal can be defined by the condition Deff = 0 for q → 0. IfDeff < 0, fluctuations become unstable and grow, rather than decay.

As a consequence of the approximations made, Eq. 63 is a linearised equa-tion in δφA(q, t); hence, the model is not able to predict the behaviour assecond and higher-order fluctuation effects become important once the fluc-tuations have grown beyond a certain magnitude.


3.2.4The Asymmetric Shear Response of Polymer Blends

A key assumption of DO is that, although there are two components, there isonly one network stress. The contribution of each component to this stressshould however be determined from some mixing rule which relates blenddynamics to that of the pure components.

An alternative approach to including the complexity of blend rheologywas considered by CM. At low enough shear rates; i.e., γ τ < 1 (where τ isthe longest molecular relaxation time in the blend), a general description ofthe relation between stress and strain is provided by the upper convectiveMaxwell model, Eq. 5. From the viewpoint of constructing a model for poly-mer blends, an important and useful consequence of this model, is that theoverall stress factors into a term due to the molecular dynamics and a termdue to the flow field. In reference [52], it was proposed that the molecular re-sponse of the blend, as determined by the stress relaxation function, G(t – t′),can be most simply modeled using the double reptation concept introducedby des Cloiseaux [57] and independently by Tsenoglou [58]. Although eachapproached the problem from slightly different physical viewpoints, the con-sequences are similar. A discussion of the differences between the two ap-proaches and the good agreement between the model and experiment canbe found in reference [59]. A word of caution is necessary at this point;the model has been developed and verified for blends of chemically iden-tical polymers with different molecular weights, whereas in references [52]and [60], it was assumed that it is also applicable to blends of chemicallydissimilar polymers. This would only be true if the monomeric friction coef-ficients for each component do not depend on concentration.

Physically, double reptation accounts, in a simple way, for the fact thatpolymers do not reptate in fixed tubes, as assumed in the original tube model.The surrounding polymers, which form the tube, also relax, and so the con-straints which form the tube decay with a characteristic time-scale. In otherwords, stress relaxation depends not only the dynamics of each individualpolymer, but also on the dynamics of the surrounding polymers. It shouldbe noted that more sophisticated and therefore complex rheological modelsfor polymer mixtures have been proposed; however, a particular appeal ofthis model is the simple relation between the stress relaxation function of theblend and the Doi-Edwards stress relaxation function for polymers in a fixednetwork,

G(t) =[

φA GAFA(t)1/2 + φB GBFB(t)1/2]2

(65)

where Gi is the pure component plateau modulus of component i. Fi(t) is therelaxation function, which, in the case of a homogeneous blend, physicallycorresponds to the probability of tube survival for component i in a matrix

152 N. Clarke

of identical relaxing obstacles. As was shown by Doi and Edwards [21], theprobability of tube survival in a matrix of fixed obstacles is almost singleexponential; hence, the dynamics are dominated by a single timescale, theterminal relaxation time. Consequently, for the double reptation model wehave [59],

G(t) =[

φA

GAexp(– t/τA)1/2 + φB

GBexp(– t/τB)1/2

]2(66)

τi represents some relaxation time, of component i, which in terms of thetube model, is related to the idealised Doi-Edwards relaxation time for com-ponent i in a matrix of fixed obstacles, τDE, by, τi = (1/2)τDE. Hence, in thedouble reptation model, the effect of constraint release is to half the relaxationtime (if single exponential decay is assumed), from that predicted for a poly-mer in a fixed matrix. In the heterogeneous blends considered here, the τi arethe tube survival times for chains of species i in an idealised environment,in which the chemical heterogeneity matches that of the blend, but all chainsshare the same relaxation time. That is, double reptation accounts for mu-tual effects in topological stress relaxation, but not for direct effects of localcomposition on the monomeric friction factors. The parameters of the dou-ble reptation model should be treated as phenomenological, to be determinedfrom independent linear rheology experiments in the one phase region (seefor example reference [61]).

In the steady state, Eqs. 11 and 14 clearly lead to expressions for the shearstress, σxy, and the first normal stress, N1, which are functions of concentra-tion,

σxy = ηγ (67)

= γ

[

φ2AGAτA + 4φA(1 – φA)(GAGB)1/2

( τAτB

τA + τB

)

+ φ2BGBτB

]

and,

N1 = ψ1γ2 (68)

= 2γ 2[

φ2AGAτ2

A + 8φA(1 – φA)(GAGB)1/2( τAτB

τA + τB

)2+ φB

2GBτB2]

The Maxwell model results in a second normal stress, N2 = σyy – σzz, equal tozero. In determining the contribution of the stress field to the effective diffu-sion coefficient, we also need to consider how the shear rate is perturbed bythe presence of concentration fluctuations. If the relative velocities are elimi-nated from Eqs. 57 and 58, then an important constraint on the stress tensorcan be found,

∇ ×∇σ (n) = 0 (69)


Consequently, in the velocity gradient direction the shear stress is con-strained to be constant, so that,

δσxy(qy) =∂σxy

∂φA

∣∣∣∣γ

δφA +∂σxy

∂γ

∣∣∣∣φA

δφA = 0 (70)

Hence, the variation of the shear rate in the velocity gradient direction withrespect to the concentration is,

∂γ

∂φA

∣∣∣∣σxy

=–∂σxy

∂φA

∣∣∣∣γ

/∂σxy

∂γ

∣∣∣∣φA

(71)

as was first proposed by Onuki [48] for polymer solutions. By contrast, inthe vorticity direction the shear stress may vary, and no perturbation of theshear field is required to satisfy Eq. 18. It is these differences between the be-haviour in two directions that gives an indication of the importance of usinga dynamic approach.

From Eqs. 62 and 63, the effective diffusion coefficients in the qy and qzdirections are given by,

Deff(qi) = 2M(χc – χ + κq2i ) –

MkBT

∂σii

∂φA(72)

≡ 2M[

χc – χ + κq2i + ∆χc(qi)

] (

i = y, z)

which serves to define ∆χc(qy) and ∆χc(qz). If ∆χc > 0 fluctuations are sup-pressed, whereas if ∆χc < 0 fluctuations are enhanced. In the latter case, themagnitude of ∆χc may be sufficient for Deff to become negative so that fluc-tuations become unstable and grow.

Hence, the effective diffusion coefficient depends on the choice of σyy andσzz. Rheological constitutive equations are only enable the stress tensor to bedetermined to within an isotropic constant. DO argued briefly that, in thetube model, if the length of a polymer is equal to its equilibrium value, sothat stress arises from orientation, Tr σ = 0. For example, if the diagonal com-ponents of the solution to the constitutive equation for the stress tensor aregiven by σ ′

xx = N1, σ ′yy = N2, σ ′

zz = 0, then the appropriate components of thestress tensor should be written, σ ′

xx = N1 + c, σ ′yy = N2 + c, σ ′

zz = c, so that,

σxx =23

N1 –13

N2 ; σyy =23

N2 –13

N1 ; σzz =–13

N1 (73)

By comparison, HF, considered the enhancement of fluctuations in polymersolutions using a constitutive equation of the form,

σ = 2ηD – ψ1D∇ + 4ψ2D2 (74)

where,

D =12

(

∇v + ∇vT)

(75)

154 N. Clarke

and,

D∇ ≡ ∂

∂tσ + v · (∇D

)

–(∇v

)T ·D – D · (∇v)

(76)

ψ2 is the second normal stress coefficient, which is often taken to be zerofor polymer solutions and blends. Equation 74 is known as a second-orderfluid [62], and it leads to,

σxy = ηγ , σxx = ψ1γ2, σyy = ψ2γ

2, σzz = 0 (77)

In considering the consequences of such stresses on sheared polymer so-lutions HF did not assume that Tr σ = 0. Clearly, whether this condition isimposed will have significant consequences for the predicted effects of shearflow on blends, as noted in reference [38]. In reference [60], it was assumedthat the argument put forward by DO was applicable and hence determinedthe shift in the stability conditions was determined using Eq. 73, which leadsto,

∆χc(qy) =2α

3kBTγ 2GAτ2

A (78)

×[

φA + 4(1 – 2φA)√

G′(

τ ′1 + τ ′

)2

– (1 – φA)G′τ ′2

–[

φ2Aτ2

A + 8φA(1 – φA)√

G′(

τ ′

1 + τ ′

)2

+ φ2BG′τ2

B

]

×φA + 2(1 – 2φA)√

G′( τ ′1+τ ′

)

– (1 – φA)G′τ ′

φ2AτA + 4φA(1 – φA)

√G′( τ ′

1+τ ′)

+ φ2BG′τ ′

]

and

∆χc(qz) =2α

3kBTγ 2GAτ2

A (79)

×

φA + 4(1 – 2φA)√

G′(

τ ′

1 + τ ′

)2

– (1 – φA)G′τ ′

These expressions show that the important dimensionless parameters in theproblem are the relative modulae, G′ = GB/GA, and the relative relaxationtimes, τ ′ = τB/τA. If, on the other hand, Eq. 77 is used then there is no shiftin the stability condition in the vorticity direction; i.e., ∆χc(qz) = 0 whilst inthe flow-gradient direction,

∆χc(qy) =2α

kBT

[

γ 2 ∂ψ2

∂φA+ 2ψ2

∂γ

∂φA

]

(80)

which is also zero for the weakly non-linear Maxwell model, since ψ2 = 0.That experimental observations show that the stability condition does de-


pend on shear rate in the vorticity direction provides some support for theargument of DO leading to the use of Eq. 73.

In reference [51], SBJ explore a phenomenological version of the DO the-ory. However, their starting point is less microscopic than DO. Firstly, ratherthan proposing the existence of a self-consistent field, such as the tube vel-ocity, and hence determining the friction using the ideas that lead to Eq. 45to Eq. 49, they assume, that the dissipation due to friction is given by,

Wfriction =∫

drφAζ0(

vA – vB)2 (81)

A second crucial difference, which again arises from the lack of a tube vel-ocity, is that they assume that there is a stress associated with each compon-ent, rather than a network stress which is common to both, which couples tothe individual velocities rather than the tube velocity. The resultant contri-bution to the total dissipation due to deformation can be written (cf. Eq. 54)as,

Wel =∫

dr [σA : ∇vA + σB : ∇vB] (82)

It was assumed that the individual stresses are related to the total stress by,

σ i = φiGi(

φi)

Gσ (83)

where Gi(

φi)

/G is a normalized modulus, for which no explicit form wasgiven, of component i. Equations 81, 82 and 83 can then be substituted intoEq. 56 and a similar procedure as outlined above results in a slightly modifiedform of the shear dependent diffusion equation. When then considering theconsequences of their model, they did not impose the condition that Tr σ = 0,and also assumed the existence of an isotropic stress. This latter was derivedon the basis of rubber elasticity theory rather than viscoelasticity and henceits applicability to polymer blends is questionable.

Criado-Sancho et al. [38] (CSJCV) extended the DO and CM formulism,but also assumed that Tr σ = 0, arguing that such an assumption is valid forCouette type flows, if not cone and plate type flows. In addition they includedan isotropic stress as a direct contribution to the free energy, and hence thechemical potential, in an analogous way to the theory of Wolf [23] discussedearlier, but within the framework of EIT. Their excess free energy took theform,

Es = Ψ1γ2 = (xAVA + xBVB)J0

eσ2xy (84)

which is similar to Eq. 11, but does not include the non-linear shear effects.The double reptation mixing rule, rather than the more phenomenologicalmixing rule of Eq. 17, was used to determine the shear stress Eq. 67 and the

156 N. Clarke

steady state compliance,

J0e =

φ2AGAτ2

A + 8φA(1 – φA)(GAGB)1/2[

τAτB/(

τA + τB)]2 + φ2

BGBτ2B

φ2AGAτA + 4φA(1 – φA)(GAGB)1/2

[

τAτB/(

τA + τB)]

+ φ2BGBτB

(85)

This latter expression follows from Eq. 14 relating the first normal stress andthe shear stress. Unlike SBJ, directional dependence of the shear-inducedspinodal shifts was incorporated by invoking the consequences of Eq. 69; i.e.,in the flow-gradient direction the constraint of a constant shear stress wasimposed, whereas in the vorticity direction the constraint of constant shearrate was imposed. This leads to expressions for ∆χc(qy) and ∆χc(qz) that areeven more complex than those from the CM model, and so will not be re-produced here. However, a comparison of the predictions of the two differentapproaches was made as can be seen in Fig. 4. In the flow gradient directionCSJCV predict a more complex range of shear-induced stability changes thanthat predicted by the CM model.

In reference [52] the important role that the prefactor α plays in deter-mining the apparent phase behaviour was neglected, which results in aninconsistency in the theory. From Eqs. 78 and 79, it is clear that whether ∆χcis negative or positive depends also on the sign of α. For blends in whichNeA = NeB, α is positive for NA > NB, and negative otherwise. This ensuressymmetry; in other words the effects of shear are not dependent on the wayin which the components are labelled. However, for NeA = NeB, the situationis modified; α is positive if,

NA

NeA>

NB

NeB(86)

It is useful to rewrite this condition in terms of the component relaxationtimes and plateau moduli. If the component relaxation times are given by thetube model, then,

τ ′ =N3

BNeA

N3ANeB

(87)

and the plateau modulus per monomer volume is given by,

Gi

kBT≈ b2

i

a2i

=1

Nei(88)

where b is the statistical segment length and a is the tube diameter, and wehave used the relation a2 ≈ Neb2. In other words, the energy arising fromshear is kBT per entanglement segment, and the energy per monomer vol-ume is kBT divided by the number of statistical segments per entanglementsegment. An important point to recognize is that the use of Eq. 88 ensuresconsistency between the magnitudes of the Flory- Huggins and the shearstress contributions to the thermodynamic driving force for the dynamics of


Fig. 4 Borders of instability in the flow-gradient direction for G′ = 1, according to themodel of Clarke and McLeish (dotted line), and Criado-Sancho et al. (solid line). The CMmodel predicts shear-induced suppression of fluctuations, i.e., ∆χc > 0, to the left of thedotted line and shear-induced enhancement to the right, i.e., ∆χc < 0. The CSJVC modelpredicts shear-induced suppression of fluctuations to the left of the left-most solid curveand to the right of the right-most solid curve and shear-induced enhancement betweenthe two curves. Reproduced with permission from reference [38]

concentration fluctuations. Hence, the sign of α is determined by,

α < 0 if τ ′ > 1/G′4; α > 0 if τ ′ < 1/G′4 (89)

Combining equations the relevant results enables predictions of the condi-tions for which ∆χc(qy) < 0 and ∆χc(qy) > 0 as a function of τ ′, concentrationand G′. Figure 4 (dotted line) and Fig. 5 illustrate the behaviour in the flow-gradient direction for G′ = 1 and G′ = 2, respectively. In each case there areregions in which shear-induced mixing, ∆χc(qy) > 0, can occur and regionsin which shear-induced de-mixing, ∆χc(qy) > 0, can occur. Although the be-haviour is slightly less varied in the vorticity direction, there are again regionsin which shear-induced mixing can occur and regions in which shear-inducedde-mixing, can occur. In particular we noted that in order to observe shear-induced de-mixing, a necessary (but not sufficient) condition is that, if thecomponents are labelled such that G′ is greater than one, we must have τ ′less than one. Of course, it is not simply the sign of ∆χc which determineswhether concentration fluctuations will grow or decay, it is also its magni-tude with respect to the quiescent driving force, which is determined by thequantity, χc – χ.

158 N. Clarke

Fig. 5 Regions of shear-induced enhancement of concentration fluctuations, ∆χc < 0, forG′ = 2, according to the CM model. Reproduced with permission from reference [60]

3.2.5Non-linear Flow

Since for most entangled polymeric systems, the relaxation times are suchthat the condition γ τ > 1 is easily attained experimentally, it is natural to askwhat will happen at higher shear rates, for which the Maxwell constitutiveequation no longer applies. The only extension to date that includes such ef-fects within the dynamic framework was introduced by CM and consideredin detail by Clarke [60]. The theory reviewed in the previous section predictsthat it is only the magnitude of the shift which changes with a change in shearrate; the qualitative effect remains the same, either shear-induced mixing orshear-induced de-mixing, depending on the particular set of values of τ ′, G′and φA. It is important to note that although the theory has been developedassuming small concentration fluctuations, it is not restricted to low shearrates only.


Doi and Edwards derived a constitutive equation for polymer melts, by ap-plying the tube model for the regime of strong flow. They assumed that afterdeformation a polymer recovers its equilibrium length quickly by retractingwithin its tube; whereas the stress associated with the orientation of the tubewith the flow remains after this initial retraction, and is relaxed more slowlyby reptation. The stress relaxation function of Eq. 4, is the same as before; itrepresents the time dependent correlation function of the end to end vectorof a reptating polymer chain. In other words, it is also possible to apply themixing rule of des Cloiseaux to the strong flow regime, from which,

σ totxy = φ2

AGAσAAxy + 2φAφB

√

GAGBσABxy + φ2

BGBσBBxy (90)

and,

Ntot1 = φ2

AGANAA1 + 2φAφB

√

GAGBNAB1 + φ2

BGBNBB1 (91)

In order to determine the three dimensionless shear stresses and dimension-less first normal stress differences, which are functions of the shear rate andrelaxation times, that contribute to Eqs. 90 and 91 we utilised a model due toLarson [63], who showed that a differential approximation to Eq. 5 is given by,

τ

[∂

∂tσ + 2D : σσ – ∇vT ·σ – σ ·∇v

]

+ σ = δ (92)

At low shear rates; i.e., γ τ < 1, the second term on the left hand side of Eq. 92can be neglected, and the expression becomes an exact differential version ofthe upper convected Maxwell model. In steady state shear flow Eq. 92 giveseach component first normal stress as,

Nii1 =

2γ τσ iixy

1 + 2γ τσ iixy/3

(93)

and the individual shear stresses as the solution to the cubic equation,

49

(γ τ ii)2(

σ iixy

)3+

23γ τ ii

(

σ iixy

)2+(

1 +23γ τ ii

)

σ iixy = γ τ ii (94)

where τAA ≡ τA, τBB ≡ τB and τAB ≡ 2τAτB/(

τA + τB)

. The resultant shearstresses and first normal stress can then be substituted into Eq. 62 to give ef-fective diffusion coefficients in the qy and qz directions. In the qy direction thequantity ∂σxy/∂γ is also required; this is easily found from Eq. 94.

A significant problem with the Doi-Edwards formulation (and Larson’s ap-proximation) is that a shear stress which decreases with shear rate, abovesome critical value of γ τ , is predicted. This is physically unrealistic, and is be-lieved to be due to the model assuming that in strong flow the tube segmentsbecome too highly oriented in the direction of flow. Models which attempt toovercome this problem have only very recently been proposed [64, 65]. Conse-quently, it is necessary to restrict modelling in the qy direction to shear rates

160 N. Clarke

for which shear thinning does not occur; i.e.,

∂σxy(max[τA, τB])

∂γ> 0 (95)

The model also predicts a first normal stress which is constant in the limit ofvery strong flows. Again this represents too strong a degree of shear thinning.However, the model does allow us to examine, at least qualitatively, the effectsof shear flow, in the shear thinning regime, on the behaviour of concentrationfluctuations. The effect in the qy direction for G′ = 1 was illustrated in Fig. 2of reference [52], whilst in reference [60] we presented results for the exper-imentally more accessible qz direction. In the limit of very strong shear, forwhich the first normal stress difference becomes a constant, dependent onlyon the plateau modulus,

limγ→∞ Nii

1 = 3Gii (96)

Fig. 6 The magnitude of ∆χc as a function of shear rate for G′ = 2 and φA = 0.5,for various values of τ ′: (– · – · –) τ ′ = 1.0, (— — —) τ ′ = 0.5, (– – –) τ ′ = 0.1,(· · · · ··) τ ′ = 0.05, and (—) τ ′ = 0.01. As can be seen for τ ′ = 1.0, ∆χc is always positive;for τ ′ = 0.5 and τ ′ = 0.1, ∆χc changes from being negative at low shear rates to positive athigher shear rates; and for τ ′ = 0.05, and τ ′ = 0.01, ∆χc changes from positive at low shearrates to negative at higher shear rates. Reproduced from permission from reference [60]


where GAA ≡ GA, GBB ≡ GB and GAB ≡ √GAGB, the shift in the spinodal is

simply given by,

∆χc(qz) =2αGA

kBT

(

φA + φB√

G′) (

1 –√

G′)

(97)

As a consequence, whether shear-induced mixing or de-mixing occurs in thestrong shear limit depends only on G′ and α. For G′ = 1, ∆χc(qz), which is al-ways positive at low shear rates, becomes zero in the strong shear limit. ForG′ = 1, the result of Eq. 97 can be combined with the predictions of the previ-ous section. A range of complex shear dependent behaviours were predicted.

In Fig. 6, we illustrate the magnitude of the shear-induced shift as a func-tion of shear rate for G′ = 2 and φA = 0.5. Various values of τ ′ are chosen toillustrate the range of possible behaviours, including shear-induced mixing atlow shear rates but shear-induced de-mixing at higher shear rates, and visaversa. In each case it can be seen that the magnitude increases dramaticallyabove γ τA ≈ 0.1, showing that a consideration of non-linear flow effects isimportant.

3.2.6Model Systems

It is desirable to find a model system which overcomes the problem of someof the assumptions made in the theory. Probably the most serious problemarises from the assumption that the monomeric friction coefficients are equalfor both components and that they do not vary as a function of composition.The most appealing candidates for which this assumption can be expectedto be reasonable are blends of chemically identical polymers, in which onecomponent has been deuterated. As shown by Wiltzius et al. [66] for blendsof polybutadiene (PB) and deuterated PB, and Beaucage et al. [67] for blendsof polydimethylsiloxane (PDMS) and deuterated PDMS, such polymers stillpossess an interaction parameter and can undergo phase separation. The ad-vantage of both these candidates is that they possesses a phase separationtemperature above their respective glass transition temperatures, for highenough molecular weights. Both Clarke [60] and more recently CSJCV haveapplied their respective theories to such model blends.

First, we note that in order to predict a phase diagram as a functionof shear rate, we must account for the variation with temperature of themolecular dynamics. It is well established that the terminal relaxation timechanges rapidly with temperature, due to changes in the monomeric frictioncoefficient, and a suitable description of the behaviour is provided by thephenomenological WLF formula [68],

ζ0(T) = ζ0(T0)aT (98)

162 N. Clarke

where aT is the shift factor,

log aT =– c0

1(T – T0)

c02 + T – T0

(99)

T0 is some reference temperature and c01 and c0

2 are empirical constants. Asnoted previously, the terminal relaxation time is related to the Doi-Edwardsrelaxation time of a polymer in a fixed matrix; hence [21],

τi =5ζ0b2N3

i

8kBTπ2Ne(100)

and τ ′ is given by Eq. 87. The quiescent spinodal curve is determined byEq. 27, and for a PB/dPB blend the relation between χ and temperature wasfound to be [66],

χ ≈ 0.198

T(101)

The statistical segment length of PB is [69] b = 6.9 angstroms; the degree ofpolymerisation of an entanglement segment is [68] Ne = 35; the monomericfriction coefficient at a reference temperature, T0 = 298 K, is [68] ζ0 = 1.607×10–10 dynes-sec/m; the WLF coefficients, at the same reference temperatureare [68] c0

1 = 3.64 and c02 = 186.5 K.

The resultant phase diagrams for a particular molecular weight combina-tion are shown in Fig. 7 for the qz direction. The most striking observationwas that the phase boundary in the qz direction is much more sensitive toshear than that in the qy direction. This is because the variation of the shearrate which occurs in the shear gradient direction reduces, to a certain degree,the gradient in the stress.

The effect of there being a maximum in the shear stress as γ increases, wasshown in reference [60], where at higher shear rates, and lower temperatures(where the monomeric friction coefficient is greatest), the effective spinodalcurve diverged away from the quiescent spinodal. From Eq. 71, it is clear thatsuch a maximum would lead to a singularity in the effective diffusion coef-ficient. Since a maximum in the shear stress is not in line with experimentalobservations [62], we should also be aware that the prediction of the spinodaldiverging from the quiescent curve is an artefact of the constitutive equationused to model the behaviour of polymers in strong flows.

From Fig. 7, we see that it is possible to produce “islands” of immiscibil-ity, and even for the region of immiscibility to disappear completely (withinthe illustrated shear range) at intermediate shear rates. Since G′ = 1, we cansee from Eq. 97 that in the limit of strong flow, the model predicts that the ef-fect of shear will vanish. This occurs because in this limit there are no stressgradients; the stress is independent of the degree of polymerisation, and isdependent only on the plateau modulus.


Fig. 7 Phase diagram for a polybutadiene/deuterated polybutadiene blend with NPB =3500 and NdPB = 3200, showing the regions in which concentration fluctuations will growin the vorticity direction as a function of absolute shear rate. The solid line is the quies-cent spinodal curve, and the unlabelled curves correspond to (- - -) γ = 0.5 s–1, (– – –)γ = 10 s–1, and (– · – · –) γ = 100 s–1. Reproduced from permission from reference [60]

In reference [38] it was assumed that the relaxation time was independentof temperature over a range of 100 C which is unrealistic. A second differ-ence between the predictions is that Clarke allowed non-linear rheological be-haviour, which becomes especially important as the temperature is decreaseddue to a corresponding increase in the relaxation time, whereas CSJCV onlyconsidered weakly non-linear rheological behaviour. Nonetheless the predic-tions of reference [38] are shown in Fig. 8 for a blend of PDMS/dPDMS inwhich the degree of polymerisation of each component is 964 and 957, respec-tively. Significantly the predictions are compared with those due to the CMmodel, and it is found that whereas the latter predicts shear-induced mixingin the vorticity direction, the CSJCV model leads to shear-induced de-mixing.Such differences highlight a potential method for experimentally determin-ing the most appropriate model. However, the predictions according to the

164 N. Clarke

Fig. 8 Phase diagram for a polydimethylsiloxane (PDMS) and deuterated PDMS blend,with NPDMS = 964 and NdPDMS = 957, showing the regions in which concentration fluc-tuations will grow in the vorticity direction as a function of absolute shear rate. Thedotted curve corresponds to the quiescent spinodal. The continuous curves are calcu-lated according to the CSJCV model, and the dashed curves according to the CM model.Curves 1 and 5: γ = 10 s–1, curves 2, 3 and 4: γ = 5 s–1. Curve 3 was calculated byCSJVC using the CM model with a ratio of entanglement degree of polymerisation ofNePDMS/NedPDMS = 0.9, whereas for curves 4 and 5, NePDMS/NedPDMS = 0.5 was used. Re-produced with permission from reference [38]

CM model for this blend should be treated with some caution. From Eqs. 87and 88, it is clear that one of the important ratios is that of the entanglementdegrees of polymerisation. For isotopic blends this ratio is most probably∼ 1, since the ratio is the inverse of the ratio of the plateau modulae, whichshould be identical for each component. In such a case if both componentsalso have very similar degrees of polymerisation, the CM model predicts thatshear will have no effect on stability. In order to overcome this problem intheir comparison between the models, CSJVC calculated shifts according tothe CM model with the ratio of the entanglement degrees of polymerisation= 1. This illustrates another difference between the models. The CM modelrelies on rheological asymmetry between components, whereas the CSJCVmodel predicts shear-induced phenomena even without such asymmetry.

3.2.7Temperature Effects

In the above model calculations it was assumed that the various dimen-sionless rheological parameters are equal at all temperatures. Clarke andMcLeish [70] extended their model to include different temperature depen-dencies of the rheological parameters. Although many of the characteristics


of the blend vary with temperature, it was proposed that the most significantfactor is the temperature dependence of the monomeric friction coefficient,ζ i

0. In pure melts, ζ0 changes rapidly with temperature, and a suitable descrip-tion of the behaviour is given by Eqs. 98 and 99.

The situation in blends of different polymers is much less well understood,despite the attention that has recently been devoted to the subject [71–76].It is generally observed that blends in the miscible state are thermorheo-logically complex; i.e., it is not possible to determine a single set of WLFparameters that will lead to time-temperature superposition at all frequen-cies. Colby [71] suggested that a reasonable description of the monomericfriction coefficient for each component in the blend can be obtained by usingc0

1 and c02 as measured for the pure components, and Tg as measured for the

blend. Consequently, the local dynamics of the two components are coupledthrough the use of a single Tg. However, each component friction factorhas its own distinct temperature dependence, which is in agreement withobservations [71, 77, 78]. More sophisticated theoretical schemes have beensuggested [79, 80], which propose that local concentration fluctuations resultin a range of local dynamic environments. Such an idea is successful in de-scribing the broad range of glass transition temperatures that are observed inpolymer blends. However, these models remain phenomenological, in that itis still assumed that each component retains its own WLF parameters uponblending [80]. We used the original idea proposed by Colby since it capturesthe underlying physical behaviour in a manner suitable for quantitative an-alysis.

The important consequence of applying Eq. 99, using pure componentvalues for c0

1 and c02, and the blend Tg, is that the ratio of the monomeric fric-

tion coefficients is temperature dependent. This in turn leads to temperaturedependence of both α (see Eq. 61), and τ ′. If we assume the statistical seg-ment lengths of each component to be equal, then the response of the blendto shear flow is characterised by three independent parameters for each com-ponent, the degree of polymerisation of an entanglement segment, Nie, thedegree of polymerisation of a chain, Ni, and the monomeric friction coeffi-cient, ζ i

0. Equivalently, we may use Ni, Nie, and the terminal relaxation timeτi, fixed at some arbitrary temperature, as our independent parameters. Thetemperature dependence of τi is then governed by the WLF equation, with theappropriate parameters.

We illustrated the possible consequences of such behaviour for a modelsystem, based on the SCPE/EVA blend studied in reference [81], with NSCPE =3300 and NEVA = 1960. This blend phase separates upon heating; that is itexhibits a lower critical solution temperature. We estimated the following re-lationship between χ and temperature, χ = 0.011 – (3.8/T), using experimen-tally determined spinodal temperatures for two different 50/50 SCPE/EVAblends [4]. The quiescent spinodal curve is found from the Flory-Huggins freeenergy. We also included a concentration dependence of the glass transition

166 N. Clarke

temperature, by use of the Fox-Flory relation [68],

1Tg

=φA

TgA+

φB

TgB(102)

where Tg is the glass transition temperature of the blend, and Tgi is the glasstransition temperature of pure component i.

The outcome of substituting this temperature dependence into the effect-ive diffusion coefficient in the vorticity direction given by equation is illus-trated in Fig. 9, with the relevant model parameters given in reference [70].For the stated WLF parameters τ ′ increases with temperature. For each curve,γ is constant for all temperatures and compositions, and the relaxation timesof each component are fixed (arbitrarily) at T = 355 K. Since the magnitudeof the shear dependent shift is proportional to (γ τA)2, and since τA decreasesrapidly with temperature the effect of shear is less noticeable at higher tem-peratures. This is illustrated by the smaller shifts in the phase boundarywhich occur at higher temperatures. The results clearly illustrate that fora small range of shear rates, it is possible to induce miscibility gaps, if the rela-

Fig. 9 The effect of shear on first-order fluctuations, for a blend of SCPE/EVA. The threeunlabelled curves are γ = 4.75 s–1 (for the inner most miscibility gap and the shifted spin-odal closest to the γ = 0 s–1 curve), γ = 5 s–1 and γ = 5.25 s–1. The behaviour for shearrates of γ τA > 0.75 s–1 was not calculated since at such reduced shear rates the weaklyNewtonian theory we have used is not applicable. Reproduced with permission fromreference [70]


tive relaxation time, τ ′, and the relative plateau modulus, G′, satisfy certaincriteria. As discussed earlier, a necessary, but not sufficient, condition beingthat if G′ > 1, then τ ′ < 1. A significant difference between our results andthe experimental observations is the temperature range over which closedloop miscibility gaps occur. Experimentally [81], the range is only 10 de-grees below the quiescent spinodal, our model suggests a typical range of20–30 degrees. This difference may be due to a lack of accurate rheologicaldata.

Whilst we accounted for a concentration dependence of the glass transitiontemperature, we did not include the additional contributions to the gradi-ent of the stress which arise from gradients in the monomeric friction factor.Once we have determined the temperature dependence of the relaxation timesusing the appropriate blend Tg, we assume the relaxation times to be fixedand not a function of concentration. Hence, our theory is strictly only ap-plicable to blends in which each component has a similar glass transitiontemperature. Despite attempts to understand and explain the consequencesthat such behaviour will have on blend rheology, a clear picture has yet toemerge.

Another possible factor that we neglected is the effect of shear flow onthe glass transition temperature. Since the phase boundaries are well abovethe glass transition temperature, we may expect that this does not play tooimportant a role.

Our results indicated that closed loop miscibility gaps are probably an ex-ception, requiring a delicate balance of the governing parameters. In mostblends, shear will simply shift the spinodal. The delicacy of the criteria neces-sary to explain closed loop miscibility gaps may go some way to explainingwhy only a few experimental systems have been found to display them. Inrelaxing the constraint of the ratio of the relaxation time being temperatureindependent, the theory represents an important step in developing a phys-ical picture that accounts for the important underlying mechanisms drivingthe response of polymer blends to shear flow.

3.3Scattering Under Shear

We have seen that scattering techniques are useful for probing shifts in phaseboundaries; however they can provide more detailed information about or-ganisation particularly as a phase boundary is approached. As already noteda phase separating blend often scatters light very strongly and becomes opti-cally opaque. Even near a phase boundary the fluctuations, which provide theinitial structures for growth once inside the two phase region, are very large.The theory presented in the previous section can be used to predict scatter-ing patterns that are anisotropic, and thus measuring such patterns can testthe theory further. The starting point is identical to that presented in the pre-

168 N. Clarke

vious section: the equation of motion for the concentration fluctuations thatincludes convection and coupling to the stress, and a dynamic equation forthe stress. Since we are now interested in scattering patterns we will no longerconfine attention to the qy = 0 and qz = 0 directions.

As an illustration, we note that Eq. 38, can be solved through the methodof the integrating factor,

S(q, t) = exp

⎧

⎨

⎩

t∫

0

D(q, t′)dt′⎫

⎬

⎭(103)

×⎡

⎣S0(q) +

t∫

0

C(q, t′′)exp

–∫

D(q, t′)dt′

dt′′⎤

⎦

where the structure factor at time t = 0 is defined as,

S –10 (q) = 2

(

χS – χ + κq20

)

(104)

Clearly numerical techniques are necessary to calculate the structure factorfrom Eq. 103, which is why to date such predictions (and indeed measure-ments) have been mostly confined to polymer solutions [49, 82]. An exceptionis due to Lai and Fuller [43], who by neglecting the thermal noise contri-bution were able to derive an analytical expression for the time dependenceof the structure factor. As shown in Fig. 10, they showed that in the flow –

Fig. 10 Time resolved structure factor in the flow – flow gradient plane following theimposition of shear calculated according to the model of Lai and Fuller. tRτrepresentsa scaled time


flow gradient plane the contours of the structure factor are ellipsoidal andoriented at an angle of 135 to the direction of flow immediately followingthe imposition of shear. The orientation angle then rotates with time to-wards the direction perpendicular to the flow. This was qualitatively in-linewith what they observed experimentally, suggesting that a model which onlyincludes the convective effects of shear is able to capture some importantphysical processes. However, as they noted, the mixture that they studied,polystyrene/polybutadiene/dioctylphthalate probably had a negligible vis-coelastic nature.

The model of CM has recently been adapted by Miroshnychenko andClarke [83] to predict enhanced scattering from entangled polymer solutionssubjected to oscillatory flow, and a reasonable agreement between predictionsand experiments [84] was found. This highlights the importance of scatter-ing techniques in this field – they permit a quantitative comparison betweentheory and experiment.

3.4Numerical Modelling

A natural progression from the linear stability analysis and the calculationof scattering curves is to use numerical techniques to solve the complete setof equations of motion including that for the evolution of the stress – anadvantage of numerical modeling is that the adiabatic approximation is nolonger necessary. Although a large number of simulations of binary mixturesunder flow have been carried out (see for example references [85–96]), whichhave simultaneously solved diffusion and flow equations, very few have in-corporated the effect that stress has on the diffusive flux, as discussed above.Hence the simulations are only able to predict the effects that shear has onmorphology development, and the consequences of morphology on rheolog-ical response. Since it is clear from experiments that the coupling betweenstress and diffusion is vital to understand the behaviour of high molecularweight polymer blends, such works seem likely to have limited applicabilityto the viscous mixtures of interest in this review, and the results are restrictedto systems where viscous effects are minimal or there is negligible viscositydifference between the components.

To date, only a limited number of studies incorporating the coupling ofEq. 59 have been carried out for either polymer solution or blends. Onukiet al. [97] used a time-dependent Ginzburg-Landau model to describe the twofluid dynamics of polymer and solvent, which is similar in physical contentto Eq. 59. Stress evolution was determined using a form of Eq. 92 withoutthe non-linear term (i.e., they used the upper convected Maxwell model).They found that above the coexistence curve; i.e., within the one phase re-gion, a dynamical steady state was reached, with enhanced fluctuations. Atrelatively large shear elongated polymer-rich regions formed a transient net-

170 N. Clarke

work supporting most of the stress. An interesting, and experimentally veri-fiable, feature of their results was that because such a network is continuouslydeformed in shear flow, the shear stress and the normal stress differenceexhibited large fluctuations. Such predictions are only possible using numer-ical techniques; the linear stability analysis assumes that the magnitude offluctuations of shear and normal stresses are given once the concentrationfluctuations are known. Okuzono [98] also numerically studied the dynam-ics of polymer solutions in two dimensions in the one-phase region underthe influence of shear flow. Viscoelastic effects were incorporated some-what differently into the model by introducing particles which had memoriesof their own past history. They found that shear-induced phase separationcan change qualitative features of rheological response, which is unsurpris-ing. Jupp et al. [99] have investigated the model introduced by Clarke andMcLeish, in particular using the proposed relation, Eq. 66, between compo-sition and stress. It is somewhat surprising given the extent to which theirmodelling relies on the theory of references [52] and [60], that they refer-ence the former only in passing and the latter not at all. As well as modellingstructural evolution and finding similar results to Onuki as regards the be-haviour of the stresses, they numerically investigated the conditions for aninstability. Since this can be done exactly using the analytical approach dis-cussed in previous sections, this seems a somewhat cumbersome approach.A disadvantage of numerical techniques is that whilst they are often ableto qualitatively reproduce observations, quantitative comparison is difficultsince it is difficult to determine whether morphological differences are dueto simply choosing the wrong parameters or the model being insufficientlyphysically realistic.

4Conclusions

The effect of shear on entangled polymer blends is both qualitatively andquantitatively different from that on polymer solutions. Two different ap-proaches to understanding the origins of such behaviour have emerged, onebased purely on thermodynamic considerations, the other on dynamic con-siderations. That there is common ground between the models is indicatedby the fact that within the framework of either, the controlling factors are therelaxation times and the plateau modulae. Importantly these are parametersthat can be determined by independent rheological experiments. Recently thetwo approaches have begun to converge, as highlighted by the work of CSJCV.Although the various models have been able to qualitatively reproduce thevaried experimental observations, much work is still necessary before a clearpicture of the most suitable theoretical approach emerges. From this view-


point, the most promising experimental observations are the shift in stabilityconditions and scattering patterns as instabilities are approached. Blendswith components that have the most closely matched molecular parameters(e.g., the friction coefficient) and glass transition temperatures are those thatare most likely to provide a strong test of the theoretical ideas.

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Author Index Volumes 101–183

Author Index Volumes 1–100 see Volume 100

de, Abajo, J. and de la Campa, J. G.: Processable Aromatic Polyimides. Vol. 140, pp. 23–60.Abe, A., Furuya, H., Zhou, Z., Hiejima, T. and Kobayashi, Y.: Stepwise Phase Transitions

of Chain Molecules: Crystallization/Melting via a Nematic Liquid-Crystalline Phase.Vol. 181, pp. 121–152.

Abetz, V. see Förster, S.: Vol. 166, pp. 173–210.Adolf, D. B. see Ediger, M. D.: Vol. 116, pp. 73–110.Aharoni, S. M. and Edwards, S. F.: Rigid Polymer Networks. Vol. 118, pp. 1–231.Albertsson, A.-C. and Varma, I. K.: Aliphatic Polyesters: Synthesis, Properties and Applica-

tions. Vol. 157, pp. 99–138.Albertsson, A.-C. see Edlund, U.: Vol. 157, pp. 53–98.Albertsson, A.-C. see Söderqvist Lindblad, M.: Vol. 157, pp. 139–161.Albertsson, A.-C. see Stridsberg, K. M.: Vol. 157, pp. 27–51.Albertsson, A.-C. see Al-Malaika, S.: Vol. 169, pp. 177–199.Al-Malaika, S.: Perspectives in Stabilisation of Polyolefins. Vol. 169, pp. 121–150.Améduri, B., Boutevin, B. and Gramain, P.: Synthesis of Block Copolymers by Radical

Polymerization and Telomerization. Vol. 127, pp. 87–142.Améduri, B. and Boutevin, B.: Synthesis and Properties of Fluorinated Telechelic Monodis-

persed Compounds. Vol. 102, pp. 133–170.Amselem, S. see Domb, A. J.: Vol. 107, pp. 93–142.Anantawaraskul, S., Soares, J. B. P. and Wood-Adams, P. M.: Fractionation of Semicrystalline

Polymers by Crystallization Analysis Fractionation and Temperature Rising ElutionFractionation. Vol. 182, pp. 1–54.

Andrady, A. L.: Wavelenght Sensitivity in Polymer Photodegradation. Vol. 128, pp. 47–94.Andreis, M. and Koenig, J. L.: Application of Nitrogen–15 NMR to Polymers. Vol. 124,

pp. 191–238.Angiolini, L. see Carlini, C.: Vol. 123, pp. 127–214.Anjum, N. see Gupta, B.: Vol. 162, pp. 37–63.Anseth, K. S., Newman, S. M. and Bowman, C. N.: Polymeric Dental Composites: Properties

and Reaction Behavior of Multimethacrylate Dental Restorations. Vol. 122, pp. 177–218.Antonietti, M. see Cölfen, H.: Vol. 150, pp. 67–187.Aoki, H. see Ito, S.: Vol. 182, pp. 243–281.Armitage, B. A. see O’Brien, D. F.: Vol. 126, pp. 53–58.Arndt, M. see Kaminski, W.: Vol. 127, pp. 143–187.Arnold Jr., F. E. and Arnold, F. E.: Rigid-Rod Polymers and Molecular Composites. Vol. 117,

pp. 257–296.Arora, M. see Kumar, M. N. V. R.: Vol. 160, pp. 45–118.Arshady, R.: Polymer Synthesis via Activated Esters: A New Dimension of Creativity in

Macromolecular Chemistry. Vol. 111, pp. 1–42.

176 Author Index Volumes 101–183

Auer, S. and Frenkel, D.: Numerical Simulation of Crystal Nucleation in Colloids. Vol. 173,pp. 149–208.

Auriemma, F., De Rosa, C. and Corradini, P.: Solid Mesophases in Semicrystalline Polymers:Structural Analysis by Diffraction Techniques. Vol. 181, pp. 1–74.

Bahar, I., Erman, B. and Monnerie, L.: Effect of Molecular Structure on Local Chain Dynam-ics: Analytical Approaches and Computational Methods. Vol. 116, pp. 145–206.

Ballauff, M. see Dingenouts, N.: Vol. 144, pp. 1–48.Ballauff, M. see Holm, C.: Vol. 166, pp. 1–27.Ballauff, M. see Rühe, J.: Vol. 165, pp. 79–150.Baltá-Calleja, F. J., González Arche, A., Ezquerra, T. A., Santa Cruz, C., Batallón, F., Frick,

B. and López Cabarcos, E.: Structure and Properties of Ferroelectric Copolymers ofPoly(vinylidene) Fluoride. Vol. 108, pp. 1–48.

Baltussen, J. J. M. see Northolt, M. G.: Vol. 178, (in press).Barnes, M. D. see Otaigbe, J. U.: Vol. 154, pp. 1–86.Barshtein, G. R. and Sabsai, O. Y.: Compositions with Mineralorganic Fillers. Vol. 101,

pp. 1–28.Barton, J. see Hunkeler, D.: Vol. 112, pp. 115–134.Baschnagel, J., Binder, K., Doruker, P., Gusev, A. A., Hahn, O., Kremer, K., Mattice, W. L.,

Müller-Plathe, F., Murat, M., Paul, W., Santos, S., Sutter, U. W. and Tries, V.: Bridging theGap Between Atomistic and Coarse-Grained Models of Polymers: Status and Perspec-tives. Vol. 152, pp. 41–156.

Bassett, D. C.: On the Role of the Hexagonal Phase in the Crystallization of Polyethylene.Vol. 180, pp. 1–16.

Batallán, F. see Baltá-Calleja, F. J.: Vol. 108, pp. 1–48.Batog, A. E., Pet’ko, I. P. and Penczek, P.: Aliphatic-Cycloaliphatic Epoxy Compounds and

Polymers. Vol. 144, pp. 49–114.Baughman, T. W. and Wagener, K. B.: Recent Advances in ADMET Polymerization. Vol. 176,

pp. 1–42.Becker, O. and Simon, G. P.: Epoxy Layered Silicate Nanocomposites. Vol. 179, pp. 29–82.Bell, C. L. and Peppas, N. A.: Biomedical Membranes from Hydrogels and Interpolymer

Complexes. Vol. 122, pp. 125–176.Bellon-Maurel, A. see Calmon-Decriaud, A.: Vol. 135, pp. 207–226.Bennett, D. E. see O’Brien, D. F.: Vol. 126, pp. 53–84.Berry, G. C.: Static and Dynamic Light Scattering on Moderately Concentraded Solutions:

Isotropic Solutions of Flexible and Rodlike Chains and Nematic Solutions of RodlikeChains. Vol. 114, pp. 233–290.

Bershtein, V. A. and Ryzhov, V. A.: Far Infrared Spectroscopy of Polymers. Vol. 114,pp. 43–122.

Bhargava, R., Wang, S.-Q. and Koenig, J. L: FTIR Microspectroscopy of Polymeric Systems.Vol. 163, pp. 137–191.

Biesalski, M. see Rühe, J.: Vol. 165, pp. 79–150.Bigg, D. M.: Thermal Conductivity of Heterophase Polymer Compositions. Vol. 119, pp. 1–30.Binder, K.: Phase Transitions in Polymer Blends and Block Copolymer Melts: Some Recent

Developments. Vol. 112, pp. 115–134.Binder, K.: Phase Transitions of Polymer Blends and Block Copolymer Melts in Thin Films.

Vol. 138, pp. 1–90.Binder, K. see Baschnagel, J.: Vol. 152, pp. 41–156.Binder, K., Müller, M., Virnau, P. and González MacDowell, L.: Polymer+Solvent Systems:

Phase Diagrams, Interface Free Energies, and Nucleation. Vol. 173, pp. 1–104.

Author Index Volumes 101–183 177

Bird, R. B. see Curtiss, C. F.: Vol. 125, pp. 1–102.Biswas, M. and Mukherjee, A.: Synthesis and Evaluation of Metal-Containing Polymers.

Vol. 115, pp. 89–124.Biswas, M. and Sinha Ray, S.: Recent Progress in Synthesis and Evaluation of Polymer-

Montmorillonite Nanocomposites. Vol. 155, pp. 167–221.Blankenburg, L. see Klemm, E.: Vol. 177, pp. 53–90.Blumen, A. see Gurtovenko, A. A.: Vol. 182, pp. 131–242.Bogdal, D., Penczek, P., Pielichowski, J. and Prociak, A.: Microwave Assisted Synthesis,

Crosslinking, and Processing of Polymeric Materials. Vol. 163, pp. 193–263.Bohrisch, J., Eisenbach, C. D., Jaeger, W., Mori, H., Müller, A. H. E., Rehahn, M., Schaller, C.,

Traser, S. and Wittmeyer, P.: New Polyelectrolyte Architectures. Vol. 165, pp. 1–41.Bolze, J. see Dingenouts, N.: Vol. 144, pp. 1–48.Bosshard, C.: see Gubler, U.: Vol. 158, pp. 123–190.Boutevin, B. and Robin, J. J.: Synthesis and Properties of Fluorinated Diols. Vol. 102,

pp. 105–132.Boutevin, B. see Améduri, B.: Vol. 102, pp. 133–170.Boutevin, B. see Améduri, B.: Vol. 127, pp. 87–142.Boutevin, B. see Guida-Pietrasanta, F.: Vol. 179, pp. 1–27.Bowman, C. N. see Anseth, K. S.: Vol. 122, pp. 177–218.Boyd, R. H.: Prediction of Polymer Crystal Structures and Properties. Vol. 116, pp. 1–26.Bracco, S. see Sozzani, P.: Vol. 181, pp. 153–177.Briber, R. M. see Hedrick, J. L.: Vol. 141, pp. 1–44.Bronnikov, S. V., Vettegren, V. I. and Frenkel, S. Y.: Kinetics of Deformation and Relaxation

in Highly Oriented Polymers. Vol. 125, pp. 103–146.Brown, H. R. see Creton, C.: Vol. 156, pp. 53–135.Bruza, K. J. see Kirchhoff, R. A.: Vol. 117, pp. 1–66.Buchmeiser, M. R.: Regioselective Polymerization of 1-Alkynes and Stereoselective Cy-

clopolymerization of a, w-Heptadiynes. Vol. 176, pp. 89–119.Budkowski, A.: Interfacial Phenomena in Thin Polymer Films: Phase Coexistence and Seg-

regation. Vol. 148, pp. 1–112.Bunz, U. H. F.: Synthesis and Structure of PAEs. Vol. 177, pp. 1–52.Burban, J. H. see Cussler, E. L.: Vol. 110, pp. 67–80.Burchard, W.: Solution Properties of Branched Macromolecules. Vol. 143, pp. 113–194.Butté, A. see Schork, F. J.: Vol. 175, pp. 129–255.

Calmon-Decriaud, A., Bellon-Maurel, V., Silvestre, F.: Standard Methods for Testing theAerobic Biodegradation of Polymeric Materials. Vol. 135, pp. 207–226.

Cameron, N. R. and Sherrington, D. C.: High Internal Phase Emulsions (HIPEs)-Structure,Properties and Use in Polymer Preparation. Vol. 126, pp. 163–214.

de la Campa, J. G. see de Abajo, J.: Vol. 140, pp. 23–60.Candau, F. see Hunkeler, D.: Vol. 112, pp. 115–134.Canelas, D. A. and DeSimone, J. M.: Polymerizations in Liquid and Supercritical Carbon

Dioxide. Vol. 133, pp. 103–140.Canva, M. and Stegeman, G. I.: Quadratic Parametric Interactions in Organic Waveguides.

Vol. 158, pp. 87–121.Capek, I.: Kinetics of the Free-Radical Emulsion Polymerization of Vinyl Chloride. Vol. 120,

pp. 135–206.Capek, I.: Radical Polymerization of Polyoxyethylene Macromonomers in Disperse Systems.

Vol. 145, pp. 1–56.


Capek, I. and Chern, C.-S.: Radical Polymerization in Direct Mini-Emulsion Systems.Vol. 155, pp. 101–166.

Cappella, B. see Munz, M.: Vol. 164, pp. 87–210.Carlesso, G. see Prokop, A.: Vol. 160, pp. 119–174.Carlini, C. and Angiolini, L.: Polymers as Free Radical Photoinitiators. Vol. 123, pp. 127–214.Carter, K. R. see Hedrick, J. L.: Vol. 141, pp. 1–44.Casas-Vazquez, J. see Jou, D.: Vol. 120, pp. 207–266.Chandrasekhar, V.: Polymer Solid Electrolytes: Synthesis and Structure. Vol. 135,

pp. 139–206.Chang, J. Y. see Han, M. J.: Vol. 153, pp. 1–36.Chang, T.: Recent Advances in Liquid Chromatography Analysis of Synthetic Polymers.

Vol. 163, pp. 1–60.Charleux, B. and Faust, R.: Synthesis of Branched Polymers by Cationic Polymerization.

Vol. 142, pp. 1–70.Chen, P. see Jaffe, M.: Vol. 117, pp. 297–328.Chern, C.-S. see Capek, I.: Vol. 155, pp. 101–166.Chevolot, Y. see Mathieu, H. J.: Vol. 162, pp. 1–35.Choe, E.-W. see Jaffe, M.: Vol. 117, pp. 297–328.Chow, P. Y. and Gan, L. M.: Microemulsion Polymerizations and Reactions. Vol. 175,

pp. 257–298.Chow, T. S.: Glassy State Relaxation and Deformation in Polymers. Vol. 103, pp. 149–190.Chujo, Y. see Uemura, T.: Vol. 167, pp. 81–106.Chung, S.-J. see Lin, T.-C.: Vol. 161, pp. 157–193.Chung, T.-S. see Jaffe, M.: Vol. 117, pp. 297–328.Clarke, N.: Effect of Shear Flow on Polymer Blends. Vol. 183, pp. 127–173.Cölfen, H. and Antonietti, M.: Field-Flow Fractionation Techniques for Polymer and Colloid

Analysis. Vol. 150, pp. 67–187.Colmenero, J. see Richter, D.: Vol. 174, (in press).Comanita, B. see Roovers, J.: Vol. 142, pp. 179–228.Comotti, A. see Sozzani, P.: Vol. 181, pp. 153–177.Connell, J. W. see Hergenrother, P. M.: Vol. 117, pp. 67–110.Corradini, P. see Auriemma, F.: Vol. 181, pp. 1–74.Creton, C., Kramer, E. J., Brown, H. R. and Hui, C.-Y.: Adhesion and Fracture of Interfaces

Between Immiscible Polymers: From the Molecular to the Continuum Scale. Vol. 156,pp. 53–135.

Criado-Sancho, M. see Jou, D.: Vol. 120, pp. 207–266.Curro, J. G. see Schweizer, K. S.: Vol. 116, pp. 319–378.Curtiss, C. F. and Bird, R. B.: Statistical Mechanics of Transport Phenomena: Polymeric

Liquid Mixtures. Vol. 125, pp. 1–102.Cussler, E. L., Wang, K. L. and Burban, J. H.: Hydrogels as Separation Agents. Vol. 110,

pp. 67–80.

Dalton, L.: Nonlinear Optical Polymeric Materials: From Chromophore Design to Commer-cial Applications. Vol. 158, pp. 1–86.

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Gubler, U. and Bosshard, C.: Molecular Design for Third-Order Nonlinear Optics. Vol. 158,pp. 123–190.

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Ise, N. see Matsuoka, H.: Vol. 114, pp. 187–232.Ishikawa, T.: Advances in Inorganic Fibers. Vol. 178, (in press).Ito, H.: Chemical Amplification Resists for Microlithography. Vol. 172, pp. 37–245.Ito, K. and Kawaguchi, S.: Poly(macronomers), Homo- and Copolymerization. Vol. 142,

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Enhancement by Hydrogen Bonding. Vol. 146, pp. 121–194.Jin, J. see Shim, H.-K.: Vol. 158, pp. 191–241.Jinnai, H., Nishikawa, Y., Ikehara, T. and Nishi, T.: Emerging Technologies for the 3D Analysis

of Polymer Structures. Vol. 170, pp. 115–167.Jo, W. H. and Yang, J. S.: Molecular Simulation Approaches for Multiphase Polymer Systems.

Vol. 156, pp. 1–52.Joanny, J.-F. see Holm, C.: Vol. 166, pp. 67–111.Joanny, J.-F. see Thünemann, A. F.: Vol. 166, pp. 113–171.Johannsmann, D. see Rühe, J.: Vol. 165, pp. 79–150.Johansson, M. see Hult, A.: Vol. 143, pp. 1–34.Joos-Müller, B. see Funke, W.: Vol. 136, pp. 137–232.Jou, D., Casas-Vazquez, J. and Criado-Sancho, M.: Thermodynamics of Polymer Solutions

under Flow: Phase Separation and Polymer Degradation. Vol. 120, pp. 207–266.

Kaetsu, I.: Radiation Synthesis of Polymeric Materials for Biomedical and BiochemicalApplications. Vol. 105, pp. 81–98.

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Effects of Thermodynamics and Rheology. Vol. 106, pp. 31–86.Kanaya, T. and Kaji, K.: Dynamcis in the Glassy State and Near the Glass Transition of

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Kaneko, M. see Ramaraj, R.: Vol. 123, pp. 215–242.Kang, E. T., Neoh, K. G. and Tan, K. L.: X-Ray Photoelectron Spectroscopic Studies of Elec-

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Vol. 161, pp. 87–156.Kirchhoff, R. A. and Bruza, K. J.: Polymers from Benzocyclobutenes. Vol. 117, pp. 1–66.Kishore, K. and Ganesh, K.: Polymers Containing Disulfide, Tetrasulfide, Diselenide and

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Köhler, W. and Schäfer, R.: Polymer Analysis by Thermal-Diffusion Forced Rayleigh Scat-tering. Vol. 151, pp. 1–59.

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Vol. 168, pp. 247–290.Kuchanov, S. I.: Modern Aspects of Quantitative Theory of Free-Radical Copolymerization.

Vol. 103, pp. 1–102.Kuchanov, S. I.: Principles of Quantitive Description of Chemical Structure of Synthetic

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Subject Index

n-Alkyl methacrylate 115

Alloys 64

Basic lattice cluster theory (BLCT) 96, 122Bending energies 89Berthelot combing rule 107Binodal 19BLCT (basic lattice cluster theory) 96, 122– free energy expression 97Blend miscibility,

structural asymmetries 78Brasovskii critical behavior 46

Cahn-Hillard theory 140, 150Chain stiffness, SLCT 88Clark and McLeish 158Clausius-Clapeyron equation 35Coherent scattering length 13Combining rules 68Concentration fluctuations,

shear-induced 157Copolymer, statistical 19, 64, 95, 96Correlation length 18, 39– susceptibility 39Couette type flow 155Critical amplitudes/exponents 21– of susceptibility 54Critical point 19Cross section, macroscopic 11Crossover 1– function 21CSJCV 155, 161

Debye form factor 12Diblock copolymer 1, 3Doi-Edwards relaxation time 152, 159Doi-Onuki model 147Double critical point 48, 49

DPB 111

Energy dissipation function,minimization 147

Entanglements 134Entropic contributions 96, 101, 121Entropy of mixing,

non-combinatorial 29Ethylene/norbornene, BLCT 101EVA-SCPE 131

Field crossover 1Finger tensor 135Fisher exponent η 45“Flory” lattice 11Flory-Huggins (FH) theory/parameter

1, 15, 64, 65, 95, 135Fluorescence, PS* 129Friction 147

Gaussian approximation 21Gibbs condition of stability 18Gibbs free energy of mixing 16Gibbs-DiMarzio theory 123Ginzburg criterion 1, 23Ginzburg number 23, 81Ginzburg-Landau Hamiltonian 21,49Guggenheim surface fraction 77

Helmholtz free energy of mixing 97“Hidden” variables 44Homopolymer blend-diblock copolymer

structure factor 50Horst and Wolf model 137HPB 111

Ideal mixture 11Ising critical behavior 1, 81Isotropic Lifshitz critical behavior 47

198 Subject Index

Lattice ccordination number 97Lattice cluster theory (LCT) 28, 64, 65, 120Lattice cluster theory, polyolefins 69LCST 79, 111, 121, 128– polyolefin blends, SLCT 86Lifshitz line/critical behavior, isotropic 47

Maxwell model 152Mean-field theory 81Miscibility, monomer molecular

structure 63Monomer structural asymmetry 64Monte Carlo simulation, LCT 73

Neutron contrast factor/variation 9, 12, 13Neutron guides 9Noise, thermal 140Norbornene/ethylene, BLCT 101

Order-disorder transition 115Ornstein-Zernicke approximation 18Ortho-dichloro-benzene 39

Partial structure factor 12PB/PS 42, 47, 52PBA-SCPE 131PDMS 161, 164– /PEMS 37PE/PEP 46PEE/PDMS 52PEP 75, 86, 91– /PDMS 37PIB-DhhPP 86PMMA 119, 132Polarizability model 29Poly ε-caprolactone 133Poly(isobutylene) 86Polybutabiene 161– unsaturated,

unsaturated/perdeuterated 111– saturated (sPB) 105Polyethylene, solution-chlorinated 131Polymer alloys 64Polymer blends 2Polymeric microemulsions 48Polyolefins 68– binary blends, weakly inreacting 95Polystyrene-acrylonitrile 132PP 86, 91PP-hhPP 93

PP-PEP 91, 123PPMS/dPS 34PRISM theory 69PS* fluorescence 129PS-b-PnAMA 123PS-PMMA 119PS-PVME 85, 129

Random copolymer FH theory 99Random phase approxiamtion

(RPA) 16, 143Rayleigh’s variational principle 147Renormalized 3D-Ising behavior 42Reptation 151Rouse model 134RPA 16

SALS 131SAN/PMMA 132Scattering, shear 167SCPE/EVA 165Shear, first-order fluctuations 166– phase separation, partially miscible

blends 130Shear flow 127Shear thinning 160SLCT (simplified lattice clluster theory)

74, 120Small angle neutron scattering (SANS)

8, 76, 78, 98, 106Solvent, nonselective 38Spinodal 19Stability, Gibbs condition 18Stability analysis 150Stiffness/steric interactions 103Stress relaxation 146Stress tensor 134Structure factor 11, 38Structure-property theory, polyolefins 65Styrene 117Surface fraction, Guggenheim 77Susceptibility 50– critical exponents 54Swapping, alternate methyl side groups

91

TEM 47Temperature, molecular dynamics 161Temperature/pressure external fields 3Thermal composition fluctuations 2

Subject Index 199

Thermal noise 140Tube model 134, 154

UCST 79, 111, 128– polyolefin blends, SLCT 91United atom group model 70United residue 123

Van der Waals interaction 30, 31, 66Viscosity 133

Zimm approximation 16

phase behaviour of polymer blends

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