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Conformational and phase behavior of comb copolymer brushesStepanyan, Roman

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Conformational andPhase Behavior of

Comb Copolymer Brushes

A Theoretical Study

R. Stepanyan

Conformational and Phase Behavior ofComb Copolymer BrushesR. StepanyanPh.D. thesisUniversity of GroningenThe NetherlandsSeptember 2003

ISBN: 90-367-1858-9MSC Ph.D.-thesis series 2003-09ISSN: 1570-1530

Rijksuniversiteit Groningen

Conformational andPhase Behavior of

Comb Copolymer Brushes

A Theoretical Study

Proefschrift

ter verkrijging van het doctoraat in deWiskunde en Natuurwetenschappenaan de Rijksuniversiteit Groningen

op gezag van deRector Magnificus, Dr. F. Zwartsin het openbaar te verdedigen op

maandag 29 september 2003om 14.15 uur

door

Roman Stepanyangeboren op 3 mei 1977

te Bakoe

Promotor:Prof. Dr. G. ten Brinke

Beoordelingscommissie:Prof. Dr. I. ErukhimovichProf. Dr. J. J. M. SlotProf. Dr. U. Steiner

ISBN: 90-367-1858-9

“Erst die Theorie entscheidet darüber,was beobachtet werden kann.” 1

A. Einstein.

1Only the theory decides on what can be observed

Contents

1 Introduction 11.1 Polymers and comb copolymers . . . . . . . . . . . . . . . . . . . 11.2 Comb copolymers: Experimental motivation . . . . . . . . . . . . . 3

1.2.1 Conformational behavior . . . . . . . . . . . . . . . . . . . 41.2.2 Phase behavior and microstructure formation . . . . . . . . 8

1.3 Conformational behavior: Theoretical results . . . . . . . . . . . . 131.3.1 Lyotropic behavior of the cylindrical polymer brushes . . . 131.3.2 Related problems . . . . . . . . . . . . . . . . . . . . . . 19

1.4 Phase behavior and microstructure formation: Theoretical results . 201.4.1 Comb copolymers with flexible backbone . . . . . . . . . . 211.4.2 Phase equilibria in hairy-rod system . . . . . . . . . . . . . 22

1.5 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Cylindrical Brushes of Comb CopolymerMolecules Containing Rigid Side Chains 252.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Straight cylindrical brush . . . . . . . . . . . . . . . . . . . . . . . 262.3 Bent cylindrical brush . . . . . . . . . . . . . . . . . . . . . . . . . 332.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Strongly Adsorbed Comb Copolymerswith Rigid Side Chains 393.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Straight comb copolymer molecule . . . . . . . . . . . . . . . . . . 40

i

ii CONTENTS

3.3 Bending elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 533.A Energy of attraction of two long rods . . . . . . . . . . . . . . . . . 55

4 Comb Copolymer Brush withChemically Different Side Chains 594.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Straight molecule with two types of side chains . . . . . . . . . . . 60

4.2.1 SCF approach . . . . . . . . . . . . . . . . . . . . . . . . . 624.2.2 Scaling approach – good solvent . . . . . . . . . . . . . . . 72

4.3 Bending effects – SCF approach. . . . . . . . . . . . . . . . . . . . 734.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 764.A Relation between v-parameters and Flory-Huggins χ-parameters. . . 77

5 Self-Organization of Hairy-Rod Polymers 795.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3 Phase equilibria between microphases and nematic . . . . . . . . . 82

5.3.1 Hexagonal H1 phase . . . . . . . . . . . . . . . . . . . . . 835.3.2 Hexagonal H2 phase . . . . . . . . . . . . . . . . . . . . . 845.3.3 Equilibrium between lamellar and hexagonal H2 phase . . . 855.3.4 Equilibrium between nematic and lamellar phases . . . . . . 88

5.4 Discussion and concluding remarks . . . . . . . . . . . . . . . . . 89

6 Phase Behavior and Structure Formationof Hairy-Rod Supramolecules 916.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.3 Nematic – isotropic liquid phase coexistence: Effect of association . 946.4 Equilibria between nematic, isotropic and microphases for κN > 1 . 100

6.4.1 Separation of the hexagonal phase H1 . . . . . . . . . . . . 1006.4.2 Separation of the hexagonal phase H2 . . . . . . . . . . . . 1056.4.3 Separation of the lamellar phase . . . . . . . . . . . . . . . 1086.4.4 Possible phase sequences . . . . . . . . . . . . . . . . . . . 114

6.5 Phase equilibria for κN < 1 . . . . . . . . . . . . . . . . . . . . . . 1166.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.A Expressions for the chemical potentials and partial pressures . . . . 120

CONTENTS iii

6.B Interaction between two cylindrical micelles . . . . . . . . . . . . 1246.C Free energy of the lattice formation . . . . . . . . . . . . . . . . . 125

Bibliography 135

Samenvatting 155

Dankwoord 159

CHAPTER 1

Introduction

1.1. Polymers and comb copolymers

Any polymer molecule is built up of a large number of repeating units. Each ofthese can have a very simple chemical structure, as a unit of polyethylene, or bemore complicated, as in the case of poly(p-phenylene-terephthalamide) also knownas ‘Kevlar’ or ‘Twaron’.

Polyethylene ‘Kevlar’

In both examples the macromolecule is linear implying that, if one abstracts from theconcrete chemical structure, it can be viewed as a linear thread, or chain, Figure 1.1a.One of the most important consequences of such an approach is that many resultsin polymer physics are independent of the chemical structure of the molecule butrely heavily on the fact that the degree of polymerization N, or, in other words, themolecule’s length, is a very large number, N 1, [1–5]. This is a very generalassumption, which covers a very diverse class of materials: from synthetic polymers,

1

2 Chapter 1

a b

c d

e

Figure 1.1: Examples of polymer architectures: (a) linear polymer, (b) star,(c) comb copolymer “bottlebrush”, (d) comb copolymer with rigid side

chains, (e) “hairy-rod”

containing typically from hundreds up to tens of thousand units, N ∼ 102 − 104, toDNA, where the number of links can reach a billion, N ∼ 109.

It is also possible to obtain architecturally more complex branched structurescombining linear chains together. The simplest example of such a molecule is astar-polymer, Figure 1.1b, built up from several chemically identical chains. In thisthesis we will be mainly interested in another type of branched polymers, so-calledcomb-shaped polymers: macromolecules, that contain relatively long side branchesspaced comparatively closely along the main chain (backbone). Experimentally it ispossible to construct such a molecule from chemically identical backbone and sidechain units. However, the presence of two types of units – those that form the mainchain and those that form the side chains – is a more common property of most

1.2. Comb copolymers: Experimental motivation 3

experimental comb copolymers studied.Several different types of comb-like molecules can be distinguished depending

on the stiffness of the main and side chains. Densely grafted comb copolymersconsisting of a flexible backbone with flexible side chains, Figure 1.1c, also knownas a polymeric “bottlebrushes”, are the best-studied systems. However, equally in-teresting representatives of this type of molecular architecture are side chain liquidcrystal polymers, i.e. flexible backbones with rigid side chains attached usually viaa spacer, Figure 1.1d, and hairy-rod polymers, i.e. a rigid backbone bearing flexibleside chains, Figure 1.1e.

Besides permanent covalent bonding of the side chains, physical, e.g. hydrogenor ionic, bonding can be used as well to construct comb-shaped complexes, alsocalled comb-like supramolecules. Noncovalent side chain attachment may play anon-trivial role, due to its dynamic nature – it can be broken at any moment and thenformed again.

1.2. Comb copolymers: Experimental motivation

Comb copolymers display many specific properties arising from their branched ar-chitecture [6–24]. First of all, the presence of the side chains strongly influencesthe conformational behavior of a single molecule. It offers a possibility to controlthe properties of dilute solutions, including lyotropic behavior in good solvent [13,14, 20] and adjustment of the molecular dimensions by varying the solvent quality[25, 26] or by quaternization [27]. The precise nature of the side chain attachment(permanent or reversible) is relatively unimportant in this case.

Another aspect worthy of attention is connected to the collective behavior ofcomb-shaped molecules and concerns the possibility of highly ordered microphaseseparated states in the melt. There is an opportunity to control the morphology,i.e. shape of the domains, in various ways through modification of the moleculararchitecture parameters. In contrast to the conformational properties, the melt phasebehavior is strongly influenced by the specific, covalent or non-covalent, type ofbonding between the main chain and the side chains.

In this section we shall present several examples of peculiar behavior of comb-like macromolecules both in dilute solution and in the melt. The experimentallyobserved phenomena suggest a variety of the possible applications and may serve asa motivation for our theoretical investigation.

4 Chapter 1

1.2.1. Conformational behavior

We focus first on the change in single molecule behavior induced by the side chains.As pointed out above, the precise nature of the attachment is not of crucial impor-tance in this case, as long as a densely grafted comb copolymer architecture can beachieved. On the other hand, the properties of the side chains, e.g. chemical struc-ture, dimensions, stiffness, mutual interaction, etc, are proven to be a determiningfactor for the conformational behavior of the complex. This opens an opportunity tocontrol the properties of the final materials by creating (synthesizing) comb-shapedmolecules with predefined side chains’ characteristics. Moreover, if it is possible toadjust these characteristics, e.g. interaction, by external stimuli, such as the solventquality, the pH or the ion concentration, then responsive materials, able to react tochanges in their environment, can be constructed.

Here we consider only a few of the possible experimental implementations andapplications of the concept presented above.

Lyotropic behavior

In a dilute solution the comb-like molecule adopts a conformation directly relatedto the solvent quality and side chain length. Under good solvent conditions theaffinity of the side chain units to the solvent molecules is higher than to the units ofits own kind [1, 2]. This leads to an effective repulsive interaction between the sidechains. As a result, if the side chains are long enough, the comb-shaped moleculesadopt the conformation of cylindrical brushes, forced by steric overcrowding of theside chains [9–18]. The effective “stiffness” of such a bottlebrush can be much higherthan of a bare backbone. In turn, the intermolecular repulsion of the “stiffened”molecules may induce a liquid crystalline ordering even in relatively dilute solutions.

Direct observation of lyotropic behavior of comb-like polymer solutions has beenreported by Wintermantel et al. [13]. The comb-shaped macromolecules were synthe-sized from so-called macromonomers, i.e. end-functionalized oligomers, by meansof radical homopolymerization reaction resulting in a main chain degree of polyme-rization up to 1000. From the X-ray patterns measured one can clearly observe apeak at the scattering vector values q ' 0.04 − 0.06Å, which cannot be explained bysingle-particle scattering but reflects the interparticle structure factor S (q) originatingfrom the intermolecular order of the molecules.

Besides the observation of liquid crystalline ordering, an indirect measurementof the molecule’s stiffness has been performed by a number of researchers [13, 14,

1.2. Comb copolymers: Experimental motivation 5

a b

Figure 1.2: Cylindrical core-shell brush molecule in (a) good for bothblocks and (b) good for PAA but poor for PS solvent.

20, 28]: fitting the measured radius of gyration versus molar mass curves by a worm-like chain model [29] one can extract the Kuhn segment length lK . Basically, twoimportant conclusions can be drawn from these measurements. First of all, as aresult of the side chains attachment the main chain indeed adopts an extremely stiffconformation, i.e. the Kuhn statistical segment length is of the order of the overallbackbone length. Secondly, the stiffness of the entire molecule strongly dependson the side chain length. Taking the above cited [14] polymacromonomers as acharacteristic example, one observes an increase in the effective Kuhn length fromlK = 890Å to lK = 2076Å if the side chains molar mass increases from 2780 g/molup to 4940 g/mol.

Responsive molecules

As explained above, the side chains repel each other under good solvent condi-tions. The “strength” of the repulsion is determined by the solvent quality. Thus,adjusting the latter, we can manipulate the properties of a single molecule.

The main chain can be considered as a statistical coil or, in other words, anentropic spring, which resists stretching. On the other hand, the repulsion between theside chains makes such stretching favorable. This ultimately leads to the length vari-ation of cylindrical brushes upon a change in the solvent quality for the side chains.Indeed, as has been reported by Fischer and Schmidt [25], variation in the solventquality leads to a change in the length of the rod-like molecule. Polymacromonomers(PS with a methacryloyl end group) were investigated by static and dynamic lightscattering in tetrahydrofuran and cyclohexane. The corresponding contour lengthsper monomer were extracted from the fitting procedure and showed an increase

6 Chapter 1

from 0.11 nm to 0.145 nm when going from cyclohexane to tetrahydrofuran. Note,however, that this is still far away from the all-trans 0.253 nm value.

Such polymers seem to be suitable as responsive materials for sensors or actuatorsif a directional orientation of the cylinders is achieved. This can certainly be done:the bottlebrushes are shown to form lyotropic phases (see the previous example).Fischer and Schmidt [25] also suggest an alternative way to control the length of thecombs – cylindrical polyelectrolyte brushes’ length can be controlled via the ionicstrength.

Another example of solvent-controlled molecular dimensions is presented byCheng et al. [26]. The polymer brushes synthesized have diblock side chains con-sisting of two chemically different blocks: polystyrene (PS) and poly(acrylic acid)(PAA). Upon the solvent quality change from good for both PS and PAA to goodfor the PAA (outer) and poor for the PS (inner) block, the PS core contracts drasti-cally, Figure 1.2, as follows from the 1H NMR spectrum. This allows to constructnanocylinders with well defined core-shell structure, the core (or shell if the diblocksare attached by the PAA end) size of which can be externally controlled.

Molecular nanowires

Due to the well-defined core-shell structure, comb copolymers with diblock sidechains can be utilized as nanocontainers or nanoreactors. The formation of nanowires[30] on the basis of a brush with polystyrene backbone plus poly(vinyl-2-pyridine)(PVP) side chains is one of the promising examples in this direction.

The PVP core of the cylindrical brush, Figure 1.3, was loaded with HAuCl4 intoluene or methylene chloride solution (analogous results were attained with cylin-ders loaded with CuCl2, H2PtCl6, and CoCl2). This process was followed by reduc-tion of the noble metal salt by an electron beam, by ultraviolet light, or by chemicalreducing agents. A continuous nanowire was formed under certain conditions mostprobably within the core of the cylindrical brushes. It should be noted that theobserved length of the wires was much higher (several micrometers!) than the lengthof an individual molecule – this effect resulted from a peculiar end-to-end aggregationof the brushes. The aggregates formed indeed resemble conventional wires verymuch: the conducting core is surrounded by a PS shell, which can serve as anelectrically insulating layer. Moreover, the properties of the conducting core canbe influenced, for instance, during the reduction process, leading to complexes withdifferent conductivities. However, the direct conductivity measurement still remainsa challenge.

1.2. Comb copolymers: Experimental motivation 7

Figure 1.3: Schematic illustration of a nanowire formation. Core–shellcylindrical brushes with a PVP core and PS shell are loaded with HAuCl4.Subsequent reduction yields a one-dimensional gold phase within the

macromolecular brush.

“Shaped” molecules: horseshoe- and meander-like structures

Not only the linear dimensions but also the shape of the comb-like polymer canbe controlled via adjustment of the side chain parameters, e.g. interaction. Peculiarhorseshoe- and meander-like conformations were observed by Stephan et al. [27]in the system consisting of two types of side chains, polyvinylpyridine (PVP) andpolymethylmethacrylate (PMMA), grafted to a common backbone. Depending on thesolvent, the copolymer brushes were shown to adopt different shapes – from worm-like to horseshoe- and meander-like structures – when spin-cast on mica, Figure 1.4.Due to the unfavorable interaction between the PVP and PMMA side chains (it wasadditionally boosted by quaternization) the side chains have been “phase separated”to different sides of the backbone. The inhomogeneous distribution of the differ-ent side chain types along the main chain and the presence of the selective solventmade the curved structures formation favorable. In principle, the curvature could bealso induced by different length, volume, and/or interaction of the PVP and PMMAchains.

The above example shows that the shape persistent synthetic macromoleculescan be chemically manipulated to change their conformations from worm-like to

8 Chapter 1

Figure 1.4: Sketch of the phase separated side chains in a horseshoe brush(left) and in a meandering brush (right). The selective solvent is good for

the thick and poor for the thin side chains.

regularly curved structures. Finally, it should be mentioned that peculiar spiral con-formations of pure PMMA brushes deposited from a dilute solution onto mica havebeen observed as well [31]. However, the adsorption process itself was assumed tobe responsible for the uneven side chain distribution making the “shaping” processmuch less controllable.

The number of the experimental results published (see, e.g. [15, 16, 18, 19, 23, 24,32–34]) goes far beyond and is far more diverse than the examples presented here.Two general conclusions about conformational properties of combs can be drawn:(i) Stiffness of the bottlebrushes is a result of the repulsive interaction between theside chains; the length of the side chains, their grafting density, and the quality of thesolvent play an important role.(ii) Dimensions and shape of the molecule are strongly influenced by changes in thestrength of the side chain repulsion, i.e. can be controlled by adjusting the solventquality, the ion concentration, using adsorption on surfaces, employing chemicallydifferent side chains, etc.

1.2.2. Phase behavior and microstructure formation

So far, we considered solutions of individual comb-like polymers emphasizing theside chain influence on the conformational behavior of the whole molecule. However,the melt state is equally interesting. As pointed out above, in many cases combcopolymers consist of a backbone and side chains, that are chemically different,generally disliking each other. This was not important under the conditions con-sidered above: the contact area between these two was negligible due to a good

1.2. Comb copolymers: Experimental motivation 9

Figure 1.5: Schematic representation of the three classical morphologies,from the left to the right: lamellar, cylindrical (hexagonal lattice), andspherical (body-centered cubic lattice). The diblock molecule consists of

a gray and a white blocks.

solvent. Apparently, this situations is reversed in the melt where, in principle, eachbackbone can be completely surrounded by its own and the neighbors’ side chains,giving rise to a highly unfavorable interaction. The strength of this interaction, or,in other words, the mixing free energy, is commonly described in terms of a Flory-Huggins interaction parameter [1], which describes the free energy cost per monomer(in kBT units) of contacts between two chemically different units. Correspondingly,the “penalty” for the entire molecule is proportional to χN, where N is the overalldegree of polymerization.

To explain the possible consequences of the repulsion, we will use a somewhatsimpler example of a copolymer system, namely a diblock copolymer , which con-sists of two types of homopolymers linked together. Driven by the mutual repulsionbetween chemically different monomers, but unable to macrophase separate due tothe covalent bond between blocks, they form mesoscopic (∼10nm) domains filledwith one or the other component, i.e. microphase separate [35–37]. Depending onthe domain shape one distinguishes different morphologies, e.g. Figure 1.5. Domainsthemselves are generally organized into a periodic structure, therefore the process isalso called self-organization.

The morphology of the microphase separated diblock melt is determined by thevalue of the χN parameter (note that χ is temperature dependent) and the moleculecomposition f and depends only weakly on the other parameters, like monomer sizeor shape, implying that their change influences the phase boundaries only a little. Onthe other hand, architectural changes can be much more effective at shifting phase

10 Chapter 1

boundaries [38] in the direction needed for a certain application without a changein the chemical composition. Apparently, this is the point where the comb-shapedpolymers become particularly interesting.

The precise nature of the comb-like molecules system is very important. In thecase of hairy-rods (see Figure 1.1e) [39–41] the stiff backbone, for instance, limitsthe set of accessible morphologies to hexagonal and lamellar. Furthermore, thechemical nature of the bond between side and main chains also strongly influencesthe phase behavior of the system. Here we will briefly discuss the peculiarities of the(micro)phase equilibria observed in the comb copolymer hairy-rod system, first withcovalently and then with reversibly bonded side chains.

Covalent hairy-rods

Self-organization in hairy-rod systems received a lot of attention, in particular, inthe context of electrical conductivity (see [42] and references therein), since there thebackbones consist of conjugated rigid polymers.

Unsubstituted rigid rod polymers have typically a high melting temperature, e.g.ca. 600C for the polyester poly(1,4-phenylene 2,5-di-n-alkoxyterephthalate) withside chain length n = 0. However, even a short alkyl tail of length 2 < n ≤ 6reduces substantially the melting temperature (to 250−300C) and a transition from acrystalline to a nematic phase is reported for the above mentioned polyester [39, 43].For longer side chains, the melting point is reduced further, for example for n =

12 it is 150C. However, as the nonpolar side chain becomes longer, i.e. n ≥ 8,the repulsion with the backbone becomes correspondingly larger. Therefore, oneenters into the regime of self-organization, where the alkyl tails and the backbonesmicrophase separate and a layered structure is formed [39, 41, 43–46].

A literature survey shows that in most cases the self-organization is in the formof lamellar structures. For example, the self-organized structures in bulk of alkylsubstituted polythiophenes have been systematically studied by Winokur et al. [47]and Chen and Ni [48], demonstrating the formation of self-organized lamellar phasesfor n ≥ 4. A further example concerns poly(p-phenylene) (PPP), which is an in-soluble and infusible conjugated polymer. Based on this polymer, Wegner et al.prepared several different types of hairy rods [49–51]. Poly(2,5-di-n-dodecyl-1,4-phenylene) contains two flexible alkyl tails at each aromatic ring [50]. In the solidstate a self-organized lamellar phase is obtained with a long period of ca. 35Å atroom temperature. Upon heating the long period slightly decreases until at ca. 190Can ODT to a disordered state occurs. A modification is reported where the alkyl

1.2. Comb copolymers: Experimental motivation 11

side chains have been replaced by alkoxy chains [51]. For octyloxy and dodecyloxyside chains (n = 8, 12), lamellar self-organized structures are obtained. However, forshorter butoxy or pentoxy tails (n = 4, 5) the distribution of side chains around thepolymer backbone was described as approximately cylindrical. These side chains areprobably too short to induce microphase separation.

Hairy-rod architectures involving polyaniline with covalently bonded side chainsis yet another example. The emeraldine base (EB) form of polyaniline consistsof alternating amine and imine repeat units. Side chains have been introduced bycovalent connection of short alkyl (methyl or ethyl) chains to the aromatic rings [52]or by a series of different long alkyl chains ranging from butyl to octadecyl to theaminic nitrogens [53]. Levon et al. [54] prepared N-alkylated polyaniline startingfrom leucoemeraldine, the completely reduced form of polyaniline. Products withoctyl or longer side chains showed strongly improved solubility in common organicsolvents such as chloroform, toluene, etc. In the bulk state self-organized layeredstructures are formed with a long period linearly increasing from ca. 20Å to 32Å asa function of the side chain length [54].

Many more examples [55] can be given, all demonstrating that bulk systemsconsisting of hairy-rod polymers self-organize due to microphase separation betweenthe backbone and the side chains if the side chains are long enough. In almost allcases a layered structure is found. It should, however, be realized that in all cases thelength of the side chains involved is relatively small.

Supramolecular hairy-rods

As discussed above, self-organization is an example how nanoscale structurescan be formed if different repulsive chemical groups are chemically connected tothe same molecules. By contrast, in supramolecular chemistry linking occurs viafunctional groups that are mutually connected by molecularly matching physicalinteractions, such as hydrogen bonding, π-stacking, charge transfer, steric match,interpenetrating ring-like structures etc [56, 57]. Using molecular recognition highlyspecific complexes can be build, which, in turn, are able to form a hierarchy ofstructures. Self-organization and supramolecular concepts can naturally be combinedto allow structuring [58–61].

The possibility of obtaining comb copolymer structures via the supramolecularroute, using physical matching interactions, such as ionic, coordination or hydrogenbonding has attracted a lot of attention lately. Most systems studied involve flexiblepolymers. However, since the synthesis is so simple, i.e. common precipitation in

12 Chapter 1

water for polyelectrolyte-surfactant complexes [59, 62, 63] or solvent casting froma common solvent in the case of hydrogen bonding [64], the natural question ariseswhether hairy-rod polymers can be prepared via a similar supramolecular route, i.e.can hairy-rod molecules be synthesized by simply connecting side chains by "rec-ognizing" driven physical bonds? In the case of rigid rod polymers, the drasticallyreduced solubility is a most complicating factor. Still a few examples, where thisconcept works, have been constructed recently.

Rod-like (conjugated) poly(2,5-pyridinediyl) (PPY), camphorsulfonic acid (CSA)and selected alkylphenolic type amphiphiles were combined to form supramoleculesthat form self-organized structures due to protonation, synergistic hydrogen bondsand polar nonpolar effects combined. PPY was first complexed with CSA to formPPY(CSA)x (x denotes number of CSA molecules per pyridinediyl). Amphiphilicmolecules, such as 5-pentyl-1,3-dihydroxybenzene (PRES), 4-hexyl-1,3-dihydroxy-benzene (HRES), octyl phenol (OP) or octyl gallate, i.e. 1-octyl-3,4,5-trihydroxyben-zoate (OG), were hydrogen bonded to PPY(CSA)x, resulting in hairy-rod complexesdenoted as PPY(CSA)x(amphiphile)y . The main observations were that for small xand y, e.g. x = 0.25, y = 0.25 or 0.5, between 25 − 200C a glassy birefringentmaterial was obtained. For increased values, x = y = 0.5, a glassy material wasobserved at 25C but at moderate temperatures, a birefringent fluid was formed. X-ray diffraction curves showed typically several very sharp reflections following thesequence q∗, 2q∗, 3q∗..., indicating a lamellar structure. Further increase of x andy resulted in a liquid crystalline state even at 25C and a transition to an isotropic(non-birefringent) disordered fluid at a higher temperature. If y/x > 2, a biphasicsystem was typically seen. Qualitatively the same behavior is observed for all theabove amphiphiles.

Slightly different system, consisting of rodlike PPY complexed with methane-sulfonic acid (MSA) and further with OG, was under the scope as well [65]. ThePPY(MSA)1.0(OG)y complexes were made in dilute solutions of formic acid fol-lowed by evaporation and drying in vacuum and then studied using small-angle X-rayscattering (SAXS). A strong tendency for self-organization has been observed and amicrophase separation occurred already when a little OG had been added. For smally the structure is relatively poor, but when y is increased the cylindrical cubic phaseis observed for y = 0.5 and y = 0.75. Increasing y further, at y = 1.0, an obliquelattice most likely comprising of elongated or elliptical self-organized domains isseen. A single cylindrical micelle contains now ca. 9 parallel polymer chains in thecross section. The lamellar phase appears for y > 1.5 (110C < T < 180C), whilst

1.3. Conformational behavior: Theoretical results 13

for y > 2.0 free OG starts to macrophase separate.These examples demonstrate that it is possible to form processable supramolecu-

lar hairy-rod polymer systems. Furthermore, the self-organization due to microphaseseparation between the backbone and the side chains (of moderate length) gives riseto a layered structure as in the case of covalent hairy-rod polymers. In addition,however, two other structures can be observed (cubic and oblique), and there is astrong tendency for macrophase separation.

1.3. Conformational behavior: Theoretical results

The conformational properties of the comb-like macromolecules have been studiedtheoretically by several authors [31, 66–87], using both analytical approaches [66–73,81, 82, 88] and computer simulations [31, 73–80, 83–87]. Apparently, the side chaininduced stiffness of the comb copolymers is one of the most studied problems [66–68, 73, 74, 76–80]. Recently, the peculiar conformations of comb-shaped polymermolecules received considerable attention as well [31, 81, 82, 84].

1.3.1. Lyotropic behavior of the cylindrical polymer brushes

The idea that the presence of branches should cause an increase in the excludedvolume effect and therefore the expansion of the branched polymer above that ofits linear counterpart, is quite old. Already in early sixties this conclusion was drawnby Kron and Ptitsyn [89] and Berry and Orofino [90]. They assumed the excludedvolume to be small and took it into account in the framework of a perturbation theory.Apparently, such an assumption fails in reality as shown, for instance, by recentexperiments [13, 14, 20, 28]. Accordingly, the deviations from ideal behavior, i.e.induced stiffness of the molecule and expansion of the side chains, are the mainconcerns of the more recent theories.

The stiffness of a macromolecule is traditionally described by its persistencelength λ 1 and the side chain attachment results in its increase. Another parameterof the bottlebrush, its diameter D, Figure 1.6, describes the dimensions of the sidechains. The ratio λ/D plays an important role in the polymer system: as discussedby Khokhlov and Semenov [91], lyotropic behavior of semiflexible chains becomes

1 The persistence length is a correlation length along the chain (the backbone in the case of combs),which is defined [2] as: 〈n(0)n(s)〉 = exp(−s/λ). Here n(s) is a tangent to the chain unit vector, s is acurvilinear coordinate along the chain [2].

14 Chapter 1

Figure 1.6: Model of a bottlebrush. The insets illustrate the representationof the backbone between two successive grafting points according to

Birshtein (a) and Fredrickson (b).

possible if this ratio is sufficiently large, say of the order of ten or more. Hence, theprediction of the λ/D parameter is one of theoretical challenges.

Scaling approach

One of the first predictions of the local conformational structure and dimensionsof long comb-like polymers in a good solvent was made by Birshtein et al. [66] onthe basis of scaling arguments. They considered a simple comb molecule consistingof side chains of degree of polymerization N grafted to a backbone, which has Nb

monomeric units. The grafting density is described by the number of main chainsegments m (see the inset (a) in Figure 1.6) between two successive side chains,Nb N m 1. First, a cylindrical brush with a fixed grafting distance b wasconsidered, yielding

D ∼ N3/4b−1/4 (1.1)

for the diameter of the molecule. Afterwards, the elasticity of the main chain betweentwo successive grafting point was taken into account and the distance b obtained inthe form

b ∼ m2/5 (N/m)3/25 . (1.2)

Combining (1.1) and (1.2) together one concludes that the side chains have an end toend distance D ∼ N18/25 with an exponent, which is only slightly smaller than that ofa 2D self-avoiding walk (0.72 vs 0.75). As the authors also argued, but not proved,that the increase in the main chain rigidity scales in the same way as the dimensions

1.3. Conformational behavior: Theoretical results 15

of the side chains, it followed that

λ

D∼ 1. (1.3)

This prediction implies that the side chain attachment has negligible influence on thelyotropicity – a result contradicting the more recent experimental data [13].

A somewhat different approach was employed by Fredrickson [67] who predictsD ∼ N3/4b−1/4 in the high grafting regime but considers the distance b to be constant(inset (b) in Figure 1.6). The persistence length λ is obtained as an elastic constantdescribing the free energy per unit length change ∆F upon bending of the molecule

∆FkBT

∼ λ

R2, (1.4)

where R is the radius of curvature. Fredrickson modified the Daoud–Cotton method[92] to analyze the toroidal geometry and, in particular, to take into account a slightredistribution of the side chains from the inner half to the outer half of the bent brush.The resulting prediction for the persistence length

λ ∼ N15/8b−17/8 (1.5)

leads to a positive exponent in the λ/D scaling law

λ

D∼ N9/8. (1.6)

Such an exponent value close to 1 means that the stiffening effect should be pro-nounced and the rigidity, required to enter the liquid crystalline regime, should beeasily achieved by making the side chains long enough.

Finally, a Birshtein-like approach was employed by Zhulina and Vilgis [69], butwith bending elasticity effects taken into account more accurately. The predicted ratioλ/D ∼ (N/m)9/10 is also in favor of lyotropicity in solution.

Self-consistent field approach

Recently Subbotin et al. [68] calculated the characteristics of a comb-like moleculein the framework of the analytical self-consistent field approach (SCF). The freeenergy of a side chain includes an entropic stretching term and the energy of ex-cluded volume interaction. These two effects are connected because the side chains

16 Chapter 1

surrounding the test chain are responsible for its stretching. The SCF approach takesthe stretching into account by introducing a field “induced” by the neighbors andcoupled to the mean monomer density (self-consistency condition). To obtain thepersistence length, first a straight cylindrical brush was considered and the bendingeffects are then addressed as a perturbation.

The starting point in the SCF theory is the side chain free energy written in theform

FkBT

=

∫dV

[− g(r) ln ZN(r | µ) − µ(r)c(r)

+ g(r) ln g(r) +v

2c2(r)

]. (1.7)

The side chain under consideration is supposed to be grafted at the origin r = 0and have its free end at point r with probability distribution g(r). The parameter vdescribes the excluded volume interaction between side chains, taken into account inthe second virial approximation (see the last term in (1.7), where c(r) is the monomerdensity). ZN(r | µ) is the partition function of a chain with the free end fixed at r inthe external (conjugated to the density) field µ(r). The elastic stretching is describedby the first two addenda in (1.7). The first of them is equal to the free energyof a chain stretched to the end-to-end distance |r| by the external field µ(r). Thesecond one subtracts the energy due to interaction with this field. Consequently, theircombination represents purely entropic stretching of the side chain. Finally, the thirdterm is responsible for the translational entropy of the free end.

The self-consistency condition δF/δc = 0 immediately yields: c = µ/v. Func-tions, such as g(r) and µ(r), have to be found from the extremum conditions for thefunctional (1.7), whereas the partition function ZN(r | µ) can be obtained from thewell-known relation [2]

∂Zm(r)∂m

=a2

6∆Zm(r) − µ(r)Zm(r) , (1.8)

where a is a statistical segment length. The authors employ the analogy between (1.8)and the Schrödinger equation to search its solution in the form of the semiclassicalapproximation [93]

Zm(r) ∼ exp (−mE − S (r)) . (1.9)

In fact, this means that the chain conformations are considered as trajectories ofclassical particles moving in an external potential µ. The constant E and function S (r)

1.3. Conformational behavior: Theoretical results 17

introduced in (1.9) are nothing but the energy of the particle and its action. In order tosimplify the calculation, one makes use of the Alexander–de Gennes approximation[94] g(r) ∼ δ(r − D/2), i.e. assumes all side chain ends to be located at the samedistance D/2 from the backbone. After an appropriate calculation the brush diameteris obtained as

D = 0.92a(v

a2b

)1/4N3/4 ∼ N3/4b−1/4 (1.10)

in complete agreement with the scaling prediction (1.1).Using the perturbation scheme with a small parameter D/R, the same approach

can be employed to analyze a bent brush with radius of curvature R. The main resultof such a calculation is the persistence length λ

λ = 0.047vN2

b2∼ N2b−2 . (1.11)

Note that this result is quite close to the Fredrickson’s prediction (1.5). Accordingly,

λ

D∼ N5/4 , (1.12)

which differs only slightly from the 9/8 exponent law (1.6). This relatively goodagreement can serve as an additional argument supporting both theories.

Computer simulation studies

Comb-like macromolecules have been extensively studied by means of computersimulations during the last decade. The role of the simulations is twofold. First ofall, they can be considered as a “clean” experiment in order to verify the proposedtheories since in many cases a real experiment would involve much more complicatedstructures than the analytically solvable models describe. Furthermore, computersimulations are very useful to predict the behavior of comb molecules taking intoaccount much more fine effects than is affordable in the theory (e.g. semi-flexibilityof side chains).

Due to relatively large structures to be studied, the researchers adopted mainlyMonte-Carlo techniques [95] to sample the phase space of the molecule. Still, simu-lation time remains one of the main computational problems restricting the systemsize. That is why long side chains were almost unaccessible in early simulation work[96, 97].

18 Chapter 1

In the mid-nineties Rouault and Borisov [73] performed a lattice MC simulationmainly to verify their own theory presented in the same paper. The largest systemstudied included a backbone of 800 monomers with 80 side chains of 35 monomerseach. Despite the quite impressive size of the system (in total 3600 monomers) it can-not be called densely grafted: a grafting distance of 10 units is not small if comparedto the Flory radius RF ∼ 353/5 ' 8.4 units. This explains the dependence D ∼ N3/5

obtained for the side chain size, which is specific for unperturbed three dimensionalSAW chains. The authors also pointed out that the traditional bond fluctuation model(BFM) led to extremely long relaxation times, around 1 CPU month. Due to thisreason a much more effective pivot [98] algorithm was employed in the subsequentpublication [74]. This allowed to increase the grafting density and D ∼ N 0.7 wasobtained, which is much closer to the predicted [66, 67] values. However, the elasticproperties were not considered in either of the two papers because of strong scatteringin the data needed for the persistence length calculation.

A quite thorough investigation of the influence of the side chain rigidity andthickness on the comb molecule elastic properties was presented by Saariaho andco-workers [78–80]. The induced rigidity was proven to be more pronounced in thecase of stiff side chains resulting in a λ ∼ N2 law. For rigid side chains the λ/D ratiothen obviously satisfies λ/D ∼ N in perfect agreement with a recent theory [70]. Itis important to note that the theoretically hardly accessible intermediate regime ofsemi-flexible side chains was also addressed [79]: the persistence length increased asa function of the side chains stiffness. Finally, the authors showed [80] that the ratioλ/D depends strongly on the side bead size, indicating the importance of the sidechain topology and the possibility to attain the liquid crystallinity regime by properside chain chemistry.

More simulation works are available [83, 99] generally employing a lattice BFMalgorithm to sample the configuration space. However, direct estimation of the per-sistence length remains outside their scope and they mainly focus on the cross sectiondiameter and radius of gyration of the comb-like molecule.

A few conclusions can be made on the basis of the simulation studies available.First of all, in the majority of papers the size of the system studied is quite far from theasymptotic regime described by theories. Thus, simulation of large enough structuresto make a direct comparison with the theory still remains a challenge. Anotherimportant point, which can be learned from the simulation data, is that the lyotropicitycan be induced not only by increasing the side chain length but also by changingintrinsic parameters of the side chain (flexibility, monomer size, etc) directly related

1.3. Conformational behavior: Theoretical results 19

to its chemical structure.

1.3.2. Related problems

Peculiarities of the bottlebrushes’ conformational behavior are not limited to elastic-ity and lyotropicity. Here we mention briefly some other issues concerning a singlemolecule conformation where the presence of side chains plays a determining role.

Generally, researches assumed that the comb molecule already reached the bot-tlebrush regime implying that the Flory radius strongly exceeds the grafting distance,RF = a2/5v1/5N3/5 b, where a is the statistical segment length and v is the excludedvolume parameter. However, the question about the crossover between star andcomb regimes is also interesting by itself. Very recently Denesyuk [100] presented arenormalized perturbation theory analysis supported by scaling arguments to identifytwo crossovers: between star and bottlebrush, and between bottlebrush and coil. Thefirst one appears when the backbone size becomes comparable to that of the sidechains. Beyond a certain point the swelling of side chains drastically slows down,and upon further growth of the backbone length the molecule starts to resemble a stiffcylinder, i.e., a bottlebrush. Further, when the backbone size exceeds the persistencelength λ, it starts to bend. This marks the second crossover, after which the brushadopts a coiled conformation when the length of the backbone is very large.

Another interesting aspect of the conformational behavior is that the cylindricallyshaped bottlebrush conformation is not always the most stable shape of the molecule.Experiments with strongly adsorbed molecular brushes by Möller, Sheiko and co-workers [15, 16, 18, 31–33] stimulated a series of theoretical works [81–83], aimingto explain the experimental results and, in particular, the peculiar curved conforma-tions observed [31]. As shown in [82], instability of the straight backbone can becaused solely by the entropic elasticity of the side chains. If they are allowed to flipover and change their position with respect to the backbone, a smaller extension canbe attained upon their uneven distribution. In turn, the induced asymmetry leads to aspontaneous curvature, as also explained in [81] and [83]. The instability is stronglyinfluenced by the length of the side chains N and the curved conformations shouldoccur for N > (λsλb/a2)1/3, where λs, λb, and a are the side chain and backbonepersistence lengths and statistical segment size respectively.

The effects of attraction between the side chains (e.g. hydrophobic side chainsand hydrophilic backbone in water, [101–104]) attracted an increasing interest aswell. Apparently, it results in the formation of much more compact structures thanbottlebrushes. One of the first computer simulation studies was presented by Rouault

20 Chapter 1

[74] and showed that the coil-globule collapse transition of the comb molecule resultsin a drastic decrease of its size in a narrow interval of attraction energies betweenmonomers. An analogous system was studied by Vasilevskaya et al. [84] who foundthat the resulting globule has a complex structure: in the case of a long backbone,the side chains form several spherical micelles while the main chain is wrapped onthe surface of these micelles and between them. Effects of the attraction in 2D havebeen studied by Flikkema and ten Brinke [86] for rigid side chains. A considerablecontraction of the comb was observed if the attraction energy per monomer wasincreased. If side chain flipping was allowed, the attraction between the side chainsled to aggregation of successive side chains at one side of the backbone resulting ina characteristic local spiraling of the backbone.

More examples of theoretical studies [71, 85, 87] can be given all supporting theidea that the comb-like copolymer molecule’s shape can be manipulated by architec-tural, chemical, or external factors allowing to attain the desired conformation.

1.4. Phase behavior and microstructure formation: Theo-retical results

Self-organization in block copolymer melts attracted enormous attention not onlybecause of the theoretically challenging problems but also due to its commercial andbiological importance. Despite the qualitatively rather simple physics behind thephenomenon (see section 1.2.2), an exact explanation of the observed effects requiresdeep insights and quite cumbersome calculations.

Diblock copolymers are at the moment the most thoroughly studied system: thetheoretical phase diagram calculated in the mean-field approximation reproduces theexperimentally observed one reasonably well. Coarse-grained mean-field theories,allowing to reduce the problem from a many-body to a one-body problem, proved tobe very productive in this area. They are generally categorized in one of three groups:weak segregation limit (WSL), strong segregation limit (SSL), and full self-consistentfield theories (SCFT). To some extent, the first two can be considered as analyticalapproximations to SCFT corresponding to weakly- and strongly-segregated melts.The WSL was introduced by Leibler [105] and is valid in the vicinity of the criticalpoint. Despite this serious limitation, WSL theory provides a nice explanation of thestructure of the mean-field phase diagram. The opposite SSL deals with a completelyseparated melt, where contacts between chemically different monomers take place

1.4. Phase behavior and microstructure formation: Theoretical results 21

only in a narrow region [106]. This corresponds to large values of χN, as commonlybelieved χN > 1000. The so-called full self-consistent field theory, introduced in theFourier-space formulation by Matsen and Schick [107], is more numerically drivenand, in principle, can achieve an arbitrary accuracy, thus, connecting weak and strongsegregation regions.

A number of valuable reviews [35–37, 108] are available discussing, in particular,advantages and disadvantages of the methods mentioned. Here we will focus on thecase of comb-like molecules.

1.4.1. Comb copolymers with flexible backbone

The phase equilibria in block copolymer melts is strongly influenced by the natureof the bond between chemically different blocks. In a weakly associating polymersystem macrophase separation prevails, whereas in the case of a covalent bondingthe system is restricted to a segregation on a microscopic level, leading to domainstructure formation. The intermediate regime, where homogeneous macrophases co-exist with microphase separated domains, is particularly interesting and correspondsto the phenomena observed in supramolecular systems. Here we first briefly discussthe covalently bonded comb-like polymers and then address the influence of thethermoreversible bond nature on the equilibrium behavior.

The mean-field WSL approach, similar to that developed by Leibler [105] for thediblock system, can be applied to graft copolymers as well [109–111]. In the case ofcomb copolymers this was done in particular by Dobrynin and Erukhimovich [110].In fact, the architecture mainly influences the calculation of the one-chain correlationfunctions, which, in turn, are used to compute the coefficients in the Landau freeenergy expansion (for details see [105, 111]). Generally, the correlation functions areobtained assuming Gaussian statistics of the side chains as well as of the backbonepieces between grafting points. This certainly imposes a certain limit on the graftingdensity and leaves out the bottlebrush-like regime, where the backbone is highlystretched.

In general, the phase diagrams obtained for (An-graft-Bm)k polymers are qual-itatively similar to those observed in the diblock AnBm case: classical lamellar,cylindrical, and spherical (see Figure 1.5) and complex gyroid structures are found tobe stable. However, the critical point shifts to values of the composition f = m/(m+n)smaller than 1/2, and, thus, in contrast to a diblock, the phase diagram is not mirror-symmetric anymore. Notably, this is a purely architectural effect, which is not presentin the analogous phase diagrams of the linear multiblock copolymer (AnBm)k.

22 Chapter 1

Supramolecular comb complexes, which appear in mixtures of flexible homopoly-mers and oligomers able of (hydrogen) bond formation with the monomer units ofthe polymer [112–115], lead to systems capable of both macro- and microphaseseparation. The temperature dependence of the hydrogen bond is an utmost importantfeature of the system. Apparently, a theoretical analysis should include the followingthree steps. First of all, the statistics of the bond formation should be addressedin order to answer the question about the possibility of comb-like supramoleculesformation. Secondly, the homogeneous state has to be analyzed revealing its stabilityagainst macro- and microseparation. Finally, in order to build a complete phasediagram, a direct comparison between ordered and homogeneous states has to beperformed to calculate their coexistence lines.

A partial analysis, including only the first two steps, was presented by Tanaka andIshida [116], who showed that microphase separation occurs at near-stoichiometricamounts of polymer and oligomer, whereas an excess of either component gives riseto macrophase separation. This agrees with a more complete picture developed in[117]. The degree of incompatibility of the components, the length of the oligomerchains, and the strength of the reversible bond are the main parameters controllingthe stability of the homogeneous state. Going further, Dormidontova and ten Brinke[117] constructed free energies and analyzed stabilities of the classical morphologies.They showed that microphase separation can take place if the tendency to segregate isnot very strong and the degree of association is large enough. The larger the incom-patibility of the components, the broader the regions of macrophase separation andthe smaller the probability to have stable ordered (lamellar or hexagonal) structuresin the whole volume of the system.

1.4.2. Phase equilibria in hairy-rod system

The coexistence of homogeneous and ordered phases is typical for the associatingpolymer system. So, it can be also expected in the supramolecular hairy-rod system,where one of the components (backbone polymer) is a stiff chain [115, 118]. How-ever, systems of this type are much less explored than their flexible counterparts.

Macrophase equilibria in the hairy-rods–solvent and hairy-rods–coils–solvent sys-tems have been studied by Ballauff [119, 120] in the framework of the Flory latticetheory. The primary goal of these works was to describe the nematic phase forma-tion in solution of hairy-rods as well as to address the problem of the compatibilitybetween hairy-rods and high molecular weight solvent (coils) of the same chemicalnature as the side chains. Hence, the question about microstructure formation was

1.5. Outline of this thesis 23

completely ignored until now.A hairy-rod system, in spite of being a close relative to comb copolymers with

a flexible backbone, has some distinctive features not encountered in other sorts ofcomb copolymers. The necessity to incorporate the possibility of liquid crystallineformation in the theoretical modeling can considerably complicate the latter. Thepresence of a stiff component makes orientational ordering effects an important in-gredient of the equilibrium behavior. This also means that application of the relativelysimple and well developed WSL or SSL techniques to the hairy-rod system is quitenontrivial.

1.5. Outline of this thesis

In this introductory chapter some relevant experimental results have been presentedin order to sketch the questions, potentially interesting for a theoretical consideration,as well as possible applications of the systems based on comb copolymers. We havealso outlined the necessary theoretical background for the theory presented furtheron.

The aim of this thesis is twofold. The first part of it, chapters 2–4, is devotedto single molecule behavior mainly in order to reveal the influence of the side chainproperties on the conformational behavior of the comb copolymer. Thus, the elastic-ity of a comb copolymer molecule with stiff side chains is addressed in chapter 2 andthe influence of the attraction between them studied in chapter 3. Next, our attentionis turned to a comb-like molecule with chemically different side chains in chapter 4,where the phase separation between chemically different side units is proven to leadto a peculiar bent conformation.

The second objective, pursued in the last two chapters, is related to the phasebehavior and self-organization in copolymer melts. In the work presented we developa theory describing phase equilibria and microstructure formation in a particular typeof comb copolymers, “hairy-rods”, characterized by a rigid backbone. In chapter 5we consider self-organization of covalently bonded hairy-rods, where side chainsare attached permanently. Finally, in chapter 6, the more complex problem aboutphase equilibria in associating rod–coil systems, able of supramolecular hairy-rodsformation, is addressed.

CHAPTER 2

Cylindrical Brushes of Comb CopolymerMolecules Containing Rigid Side Chains

An analysis of the cylindrical brush of an isolated comb copolymermolecule, consisting of a semiflexible backbone and rodlike side chains,is presented. Using a mean-field approach and a simplyfying assump-tion, which is tested by computer simulations, it is found that the per-sistence length of the brush, λ, scales as λ ∝ L2/ ln L for large valuesof the side chain length L. In the cylindrical brush regime the orderparameter of the rods is negative, implying that the rods orient normalto the cylindrical axis.

2.1. Introduction

In this chapter we consider theoretically the conformational behavior of cylindricalcomb copolymer brushes in dilute solution for the specific case of rigid rod sidechains. Cylindrical comb copolymer brushes are defined here as long chain moleculesconsisting of a flexible backbone densely grafted with relatively long side chains.

The primary objective of this investigation is to study the influence of the sidechain rigidity on the bottlebrush elastic properties. This question has been addressedearlier by means of computer simulations [78, 79] showing that the increase in the

25

26 Chapter 2

Figure 2.1: Model of comb copolymer molecule with rigid side chains.

persistence length λ is more pronounced in the case of semiflexible or rigid thancompletely flexible side chains (see also section 1.3.1).

In the framework of the mean-field approach we first analyze the properties ofa brush with straight backbone and calculate the angle distribution function of therigid side chains. Further, in the subsequent section, elastic properties are addressedand the persistence length is obtained as a function of the side chain length and thegrafting density.

Although the theoretical model considered here is slightly different from the caseinvestigated by recent simulations [78] (there is no excluded volume of the backbonein the theory) it is possible to compare the results in the regime of large side chainlength (or high grafting densities), where the properties of the backbone are relativelyunimportant.

2.2. Straight cylindrical brush

We consider a comb copolymer molecule having rodlike side chains of length L anddiameter d, L d, Figure 2.1, and assume that the backbone is a semiflexible chainof contour length Lc and persistence length λ0, containing N grafted rods with adistance b between two consecutive grafting points, satisfying d b < λ0, andLc = Nb. It is also assumed that the rod length L b.

Our considerations start with a straight cylindrical brush. First we calculate thefree energy per rod in this regime, and after that we will calculate the free energy dueto bending of the brush and thus obtain the persistence length.

The free energy of the rod consists of two parts, namely the orientational freeenergy and the steric free energy [2, 91, 121]. In order to find the steric free energywe use a mean-field approach. According to this approach the steric part of the freeenergy equals, Fster ' kBT ln(4π/Ω), where Ω is the average volume in orientation

2.2. Straight cylindrical brush 27

a b

Figure 2.2: Schematic illustration of the interaction between two rods for astraight brush.

space available for a test rod when the other rods are fixed in their average positions.Let us introduce a system of coordinates as illustrated in Figure 2.2, where the z-axis is directed along the axis of the cylinder and the (x,y)- plane corresponds to thecross-section. The corresponding spherical angles (θ, ϕ) are defined in the usual way,so that θ is the angle between the rod and the z-axis, and ϕ is the azimuth angle.If the test rod has polar angle θ1 and azimuth ϕ1, then it can interact with anotherrod having polar angle θ2 and azimuth ϕ2 only if ϕ1 ' ϕ2 (here we use the fact thatd/b 1 and assume that the angles θ1, θ2 > d/b) and if their distance is smaller thana critical value LB∗2(θ1, θ2) (Figure 2.2). Here B∗2(θ1, θ2) is a geometric factor, whichfor 0 < θ1 < π/2 is given by

B∗2(θ1, θ2) =

cos θ2 − sin θ2 cot θ1, if 0 < θ2 < θ1

cos θ1 − sin θ1 cot θ2, if θ1 < θ2 < π − θ1

sin θ2 cot θ1 − cos θ2, if π − θ1 < θ2 < π

(2.1)

and for π/2 < θ1 < π can be found by symmetry. The range of the interaction betweenthe rods in the brush is therefore of the order of L.

When the test rod, denoted by 1, and another rod, denoted by 2, are a distancez < LB∗2(θ1, θ2) apart, the excluded azimuth angle, ψz(θ1, θ2), for the test rod due to

28 Chapter 2

this second rod can be found using a simple geometric picture (Figure 2.2),

ψz(θ1, θ2) ' 2dz| cot θ1 − cot θ2 | . (2.2)

Let us introduce the distribution function of rod orientations, f (n), where n is theorientation vector of the rod. f (n) satisfies the normalization condition

∫dn f (n) = 1

and should be found after minimization of the free energy. The probability pz(θ1, ϕ1)that the test rod does not interact with the rod 2 is

pz(θ1, ϕ1) = 1 −∫

dn2 f (n2) δ(ϕ1 − ϕ2)ψz(θ1, θ2)

2π. (2.3)

Multiplying the probabilities pz(θ1, ϕ1) for different positions z of the given rods,which may interact with the test rod, we can find the probability P(θ1, ϕ1) that thetest rod does not interact with any rod,

P(θ1, ϕ1) =∏

z

(1 −

∫dn2 f (n2) δ(ϕ1 − ϕ2)

ψz(θ1, θ2)2π

). (2.4)

Averaging further the function P(θ1, ϕ1) with respect to the angles θ1, ϕ1 we find theaverage available free volume Ω for the test rod in orientation space from the formula

Ω ' 4π∫

dn1 f (n1)P(n1) . (2.5)

In the present formulation the problem is very complicted mathematically becausethe angular range for θ2 in the integral (2.3), (2.4) depends on the distance z and θ1.

In order to simplify the calculations we assume that θ2 = π/2, i.e. we estimate theexcluded azimuthal angle for the test rod by another rod by assuming the latter to beoriented perpendicular to the cylinder axis (computer simulation data supporting thisassumption are given further on). In this case the product in (2.4) should be takenover the positions z = kb, k = 1, .., n∗, of the different interacting rods, where n∗ =

(L/b) cos θ1 is the maximum number of rods interacting with the test rod. Thus, using(2.2)-(2.5) and taking into account that the distribution function does not depend onϕ due to symmetry, (2.5) can be written using the following approximation

2.2. Straight cylindrical brush 29

Ω

4π'

∫dn1 f (n1)

n∗∏

k=1

(1 − ∆ψk

2π

)' exp

−1

2π

∫dn1 f (n1)

n∗∑

k=1

∆ψk

, (2.6)

where

∆ψk ' 2dbk| cot θ1 | . (2.7)

Hence, the steric part of the free energy is

Fster ' kBT2π

∫dn1 f (n1)

n∗∑

k=1

∆ψk ' kBTπ

db

ln(n∗)∫

dn1 f (n1) | cot θ1 | . (2.8)

Because n∗ occurs as argument of the logarithmic function, we approximate n∗ asn∗ ' L/b. The total free energy of the rod includes also the orientational entropy andtherefore is given by

Frod

kBT=

∫f (n) ln f (n)dn +

1π

db

ln(L/b)∫

dn f (n) | cot θ | . (2.9)

Minimization of the free energy (2.9) using the normalization condition for thedistribution function f (n), which here due to symmetry does not depend on ϕ, givesrise to the following equation

ln( f (n)/Λ) = −ε | cot θ | , (2.10)

where

ε =dπb

ln(L/b) . (2.11)

Hence, the distribution function is given by

f (n) =Λ exp (−ε | cot θ |) , (2.12)

where Λ is the normalization constant. The free energy equation follows from (2.9)and (2.12)

Frod = kBT ln Λ . (2.13)

30 Chapter 2

Since Λ can not be found analytically in the general case, we will consider twolimiting cases. The first one corresponds to ε 1 (regime 1). In this case we use theperturbation theory and expand the distribution function in the series with respect tothe small parameter ε. In the first order of the perturbation scheme

f (n) ' Λ (1 − ε | cot θ |) (2.14)

and

Λ ' 14π

(1 + ε) . (2.15)

Note, the perturbation scheme does not work for the angles θ ≤ ε. Moreover, therods are repelled from this angle zone. The orientational order parameter can becalculated using the distribution function (2.14) excluding the smallest angles, and isslightly negative

η =12

⟨3 cos2 θ − 1

⟩' − ε

2' − 1

2πdb

ln(L/b) . (2.16)

The free energy per rod can be found from (2.11), (2.13), and (2.15) and equals

Frod ' kBTπ

db

ln(L/b) . (2.17)

For this mean-field picture to be correct, the fluctuations of the backbone orienta-tion, δθ, on the scales of the order of b and L should be smaller than ε. Generally thepersistence length is different on the scale ∼ b and ∼ L due to interactions between theside rods, therefore we should distinguish two cases. The persistence length on thescale ∼ b equals λ0, therefore the first inequality implies that δθ(b) ' √b/λ0 ε,or λ0 b(b/d)2 which we assume to be fulfilled. The second condition will beconsidered in the next section, after estimation of the corresponding persistencelength.

Next we proceed to the case ε 1 (regime 2). Here, the distribution function isgiven by

f (n) = Λ exp(−ε | π

2− θ |

)(2.18)

with the normalization constant Λ = ε/(4π). The orientational order parameter isnegative,

η =12

⟨3 cos2 θ − 1

⟩= −1

2+

3π2b2

d2 ln2(L/b). (2.19)

2.2. Straight cylindrical brush 31

Figure 2.3: Distribution function f (θ) for L = 200 and b = 2.

Hence, the rods have a tendency to orient perpendicular to the backbone. The freeenergy of the rod can again be estimated from (2.13)

F ' kBT ln [(d/b) ln(L/b)] . (2.20)

To support the approximation made in this section we will compare the theoreticalresult (2.12) to simulation results. Conformations of a straight brush were studiedby off-lattice Monte-Carlo simulations. The simulations algorithm is thoroughlydescribed in [76–78, 80]. The molecules consisted of a phantom straight backboneand rigid side chains modeled as a straight chain of hard spheres (beads). Thediameter d of the beads was taken as the unit of length. Side chain lengths of upto 300 beads were considered. To suppress end effects all parameters of interestwere computed by excluding 1/6 part of the backbone from each end (backbonelength was 300 for all simulation points except for L = 200 and 300 where thebackbone consisted of 600 and 900 beads). The initial conformation was formed asa 3D structure. The trial moves of the side chains consisted of choosing randomlynew orientations and was always accepted if the new conformation did not causean overlap between side chains. From the simulations the distribution function f (θ)

32 Chapter 2

0 100 200 300 400Length of side chains, L

−0.25

−0.2

−0.15

−0.1

−0.05

Ord

er p

aram

eter

theorysimulations

b=2

Figure 2.4: Order parameter as a function of the length of rods for b = 2.The solid line is the theoretical curve, and points represent the simulation

results.

and the order parameter η were obtained as a function of the distance b betweensuccessive grafting points and the length L of the side chain.

The distribution function for b = 2, L = 200 is shown in Figure 2.3. For thesevalues of grafting density and length of side chains the parameter ε ≈ 0.733, i.e. it isstill not in regime 2. Even though the simulation data show considerable scatter dueto the long side chain involved it is clear that the distribution function (2.12) and thesimulation results are already in rather good agreement. The rods are exluded fromthe angular ranges θ ≈ 0 and θ ≈ π due to their finite width. In the point θ = π/2 thetheoretical curve is not smooth as a result of the approximation.

Figure 2.4 presents the dependence of the order parameter η on the length of therods for b = 2. Even for the largest value of L, ε does not satisfy the strong inequalityε 1 (for L = 300, ε = 0.797) but extrapolation of the simulated data into the regionof larger values of L demonstrates a rather good agreement with the theoretical result.

2.3. Bent cylindrical brush 33

The results presented in Figures 2.3 and 2.4 show that the main assumption ofθ2 = π/2, which simplified the calculations considerably, does not cause too largedeviations. Also expression (2.19) implies that this assumption is valid for regime 2(η ≈ −1/2, i.e. all rods are oriented almost perpendicular to the backbone). Howeverit is not valid for the first regime where η ≈ 0 (see (2.16)). Therefore, from hereon in this chapter we will concentrate on the 2nd regime which corresponds to highgrafting densities or long side chains and regime 1 will be considered only brieflyusing scaling arguments.

2.3. Bent cylindrical brush

Now we proceed to the calculation of the persistence length λ of the cylindrical brush,which can be achieved following a standard procedure [67]. If the cylindrical brushis homogeneously bend with a radius of curvature R, the free energy change is relatedto R and the persistence length λ by:

∆F = kBTλLc

2R2. (2.21)

So, we will assume the brush to be bent with a radius of curvature R, and calculate∆F using the same methods as developed above. Let us consider two interacting rodsand introduce three coordinate systems, connected with these rods (Figure 2.5). Oneof the coordinate systems we denote as Z. The z-axis in this system is directed alongthe line connecting the grafting points of the rods, which we assume to be at a distancez = kb apart (actually this distance is kb(1 − (kb)2/(24R2)), however the correctionis numerically small and will be omitted). The x-axis is perpendicular to the z-axisand directed along the radius of curvature and the y-axis is perpendicular to the (xz)-plane. The other two coordinate systems are the local coordinate systems Z i, i = 1, 2defined in the following way. The origin of the local coordinate system coincideswith the grafting point of the rod under consideration. The zi−axis is directed alongthe tangential line to the cylinder axis, and the (xiyi)- plane is perpendicular to thisaxis and corresponds to the cross-section. The xi−axis is directed along the radiusof curvature. Knowing the transformations between the basis unit vectors of thecoordinate system Z, and the local coordinate systems Zi, i = 1, 2, we can express thespherical angles (θ1, ϕ1) and (θ2, ϕ2) of rod 1 and rod 2 in the coordinate system Zthrough their local spherical coordinates (θ1

1, ϕ11) and (θ2

2, ϕ22) in the coodinate system

34 Chapter 2

Figure 2.5: Interacting rods in a bent brush.

Z1 resp. Z2. This implies that the orientation vectors n1,n2 are given by

n1 = cos θ11e1

z + sin θ11 cos ϕ1

1e1x + sin θ1

1 sin ϕ11e1y

= cos θ1ez + sin θ1 cos ϕ1ex + sin θ1 sin ϕ1ey , (2.22)

n2 = cos θ22e2

z + sin θ22 cos ϕ2

2e2x + sin θ2

2 sin ϕ22e2y

= cos θ2ez + sin θ2 cos ϕ2ex + sin θ2 sin ϕ2ey . (2.23)

The transformations between the basis unit vectors (eiz, e

ix, e

iy) of the coordinate sys-

tem Zi, i = 1, 2, and the basis vectors (ez, ex, ey) of the coordinate system Z are thefollowing

e1z = cos(θ∗/2)ez + sin(θ∗/2)ex

e1x = − sin(θ∗/2)ez + cos(θ∗/2)ex

e1y = ey , (2.24)

e2z = cos(θ∗/2)ez − sin(θ∗/2)ex

e2x = sin(θ∗/2)ez + cos(θ∗/2)ex

e2y = ey , (2.25)

2.3. Bent cylindrical brush 35

whereθ∗(z) =

zR

(2.26)

is the angle between the axis z1 and z2. Using (2.22)-(2.26) with a small parameterθ∗(z) 1, the angles θ1, ϕ1, resp. θ2, ϕ2 can be expressed in terms of the anglesθ1

1, ϕ11, θ∗, resp. θ2

2, ϕ22, θ∗ in the following way

θ1 = θ11 +

θ∗

2cos ϕ1

1 +θ∗2

8cot θ1

1 sin2 ϕ11 ; ϕ1 = ϕ1

1 −θ∗

2cot θ1

1 sin ϕ11 ;

θ2 = θ22 −

θ∗

2cos ϕ2

2 +θ∗2

8cot θ2

2 sin2 ϕ22 ; ϕ2 = ϕ2

2 +θ∗

2cot θ2

2 sin ϕ22 . (2.27)

The interaction between the rods takes place when ϕ1 ' ϕ2.Now let us calculate the azimuth angle ∆ψ′k(θ1

1, ϕ11), which is excluded for the test

rod (1) due to the rod (2), when the last one is oriented perpendicular to the cylinderaxis (i.e. we assume that θ2

2 = π/2), and the number of rods, which interact with thetest rod is n∗′(θ1

1, ϕ11). Note that the excluded angles should be calculated in the local

coordinate system Z1 connected with the test rod. The functions ∆ψ′k and n∗′ can befound from geometrical arguments (Figure 2.5) and are given by

∆ψ′k =2dkb

(cos θ1 − sin θ1 cot θ2)

sin θ11

, (2.28)

n∗′ =Lb

(cos θ1 − sin θ1 cot θ2) . (2.29)

Here we assume that 0 ≤ θ11 ≤ π/2, the case π/2 ≤ θ1

1 ≤ π can be obtained bysymmetry. In (2.28), (2.29) we can eliminate the angles θ1, θ2 using equation (2.27).After that the same procedure as before is followed to calculate the free energy. Thecalculations show that in this case the free energy is given by

F′rod

kBT=

∫f (n) ln f (n)dn +

dπb

∫dn f (n) [ln(L/b) |cot θ|

−2LR

cos ϕ|cos θ| + L2

R2

(32

cos2 ϕ sin θ |cos θ| − 916

cos2 θ |cot θ|)], (2.30)

where the distribution function f (n) follows from minimization of the free energy.As it was mentioned before we consider region ε 1 (regime 2) where the approxi-mation is valid. Using this distribution function, we can estimate the order parameterη′ of the rods in the bent brush,

36 Chapter 2

η′ = η +9

4ε2 ln(L/b)

(LR

)2

. (2.31)

It is slightly increased compared the straight brush, η < η′ < 0, therefore rods becomemore disoriented.

After calculation of the integral in (2.30) in regime 2, we find that the correctionto the free energy of the rod due to the bending is given by

∆F′rod '3kBT

4 ln(L/b)

(LR

)2

. (2.32)

The persistence length equals

λ ' λ0 +3

2 ln(L/b)L2

b(2.33)

and scales as λ ∝ L2/ ln L for large L. This scaling of the persistence length as afunction of L is in good agreement with the recent computer simulations [78].

It is also possible to estimate the scaling behavior of the parameter λ/D: forε 1 D ∝ L and therefore

λ

D∝ L

ln L∝ L , (2.34)

which is in agreement with the computer simulations too [78].Our assumption that the rods orient normal to the backbone can not be used for

calculation of the bent brush free energy in regime 1 (ε 1), therefore let us usescaling arguments to consider this regime. Using (2.8) we obtain the potential energyof a rod in the straight brush, U(θ) ' ε |cot θ|, where θ is the angle between therod and the backbone. Upon bending this energy approximately equals U ′(θ, ϕ) 'ε | cot θ′(θ)|, where

θ′ ' θ +θ∗

2cos ϕ +

θ∗2

8cot θ sin2 ϕ

and θ∗ is the characteristic bending angle of the backbone on the distance of the orderof L, i.e. θ∗ ' L/R (see (2.26) and (2.27)). After expansion of the function U ′(θ, ϕ) upto terms of the order of 1/R2 and averaging using the equilibrium distribution function(2.12) we find for the increase of the steric part of the free energy ∆F ′rod ∼ kBT (L/R)2.

Therefore the persistence length scales as λ ∼ L2/b (more accurate considerationresults in an additional logarithmic factor as in (2.33)). Note, however, that this

2.4. Concluding remarks 37

scaling result is valid only when the fluctuations of the backbone orientation on thelengthscale of the order of L are small, i.e. δθ(L) ' √L/λ ε, or when L L∗ =

b(b/d)2 for b(b/d)2 λ0 b(b/d)4 and L L∗ = b√λ0/b for λ0 b(b/d)4 . The

fluctuations become very important when the rod length L < L∗. Thus, the mean fieldresult (2.33) is correct for side chain lengths L L∗.

2.4. Concluding remarks

In the present chapter we calculated the persistence length λ of a cylindrical brush of acomb copolymer molecule consisting of a semiflexible backbone having a persistencelength λ0 with rigid side chains of length L and diameter d. The linear grafting densityof the rods is 1/b so that d b L. Using a mean-field approach the free energyhas been calculated both for a straight and a bent brush showing that the persistencelength increases as a function of L and for large L scales as λ ∝ L2/ ln L. For shortrods satisfying L < L∗ = b(b/d)2, the fluctuations become important and the meanfield approach fails. Alternative approaches should be developed to calculate thepersistence length in this case.

For low grafting density, or equivalently for short side chains, L∗ L b exp(πb/d), in the straight brush regime the side chain rods are expelled from theangular range corresponding to parallel to the cylinder axis orientations and they arenearly isotropically distributed outside this range. With increasing grafting densityor rod length, the rods orient in the direction normal to the cylinder axis. However,on bending the rods start to disorient and penetrate to the range with strong stericinteraction and, therefore, the free energy increases. In this case the bending elasticityhas the same nature as the Frank elasticity in liquid crystals [122].

In the present model we use a semiflexible backbone with persistence length λ0.

However, in the computer simulations [78] a slightly different model (freely jointedhard sphere bead model) was used for the backbone. Following Birshtein et al. [66]we can approximate the free energy of this kind of brush by adding to the free energyof rods attached to a “cylinder” (see (2.20)), the free energy due to stretching ofthe backbone. A simple calculation shows that the spatial distance between twosuccessive grafting points is in a good approximation independent of the rod length,exactly as found in the computer simulations [78].

CHAPTER 3

Strongly Adsorbed Comb Copolymerswith Rigid Side Chains

We study the conformational behavior of a strongly adsorbed combcopolymer molecule, consisting of a semiflexible backbone and rigidside chains interacting via a van der Waals potential. Using a mean-field approach, two different regimes are distinguished depending on thestrength of the attraction between the side chains. In the weak attractionlimit the side chains are oriented preferably perpendicular to the back-bone. The persistence length λ of the comb copolymer molecule scalesas the second power of the side chain length L: λ ∝ L2. In the strongattraction limit all side chains become strongly tilted and the persistencelength scales as λ ∝ L4. The non-linear bending regime is also studiedand characterized by a change in structure and a decreasing moment ofbending force as a function of curvature, i.e. bending becomes easier.

3.1. Introduction

Due to recent development in SFM and AFM techniques an increased interest isattracted by strongly adsorbed high molecular weight complexes, including comb-like molecules [15, 16, 18, 20, 31–33]. Particularly interesting, the 2D confinement

39

40 Chapter 3

was shown to lead to unexpected spiral-like conformations (see section 1.2.1). Thisproblem was addressed theoretically [81, 82] and by computer simulations [83] andan asymmetric distribution of side chains was put forward as a possible explanation.Similar effects were examined a few years ago for linear chains, where, as an inter-mediate state in the coil-globule transition [2], toroidal structures can be formed in3D [123] or spirals if confined to a flat surface [124]. In this case attraction betweenthe monomeric units was held responsible.

The possibility that attraction between side chains might also lead to spiralingof comb copolymers has not been considered in detail yet. The present work is anattempt to study theoretically the influence of attraction between side chains on theconformational properties of comb copolymer molecules confined to a plane. Toobtain a tractable model we restrict ourselves to a semiflexible backbone denselygrafted with rigid side chains. The model should, however, equally well apply tothe case of semiflexible side chains with a length not significantly exceeding theirpersistence length.

In section 3.2 the effect of the strength of attraction between the side chains ontheir orientation will be discussed assuming a straight comb copolymer brush, i.e.a straight backbone. It is shown that a highly condensed state should appear forlarge attraction energies. In the subsequent section the flexibility of the backbone isintroduced and the stability of the straight conformation with respect to bending isexamined. Different regimes of persistence behavior are identified and the possibilityof globule-like (folded, spiraled, etc) conformations is discussed.

3.2. Straight comb copolymer molecule

We consider a comb copolymer molecule confined to a plane and model it as consist-ing of a semiflexible backbone with persistence length λ0 and rigid side chains. Theside chains are rigid rods of length L and width d, equidistantly grafted on both sidesof the main chain, alternately pointing “up” and “down” (Figure 3.1), with a distanceb between two consecutive rods at the same side of the backbone (d < b L).We do not allow flipping over from one side of the backbone to the other. Theeffect of flipping will be discussed briefly in the Concluding Remarks. Two typesof interaction between the rods are considered: steric repulsion and van der Waalsattraction. The attraction potential per unit length is modeled by the inverse sixth

3.2. Straight comb copolymer molecule 41

Figure 3.1: Schematic representation of a 2D comb copolymer molecule.

power law (in units of kBT )

u = −ε d4

r6. (3.1)

The energy parameter ε represents the energy of attraction between two small spheresof diameter d touching each other. If only steric interactions are important, thepresence of many rigid side chains leads to a stiff cylindrical comb copolymer brush,in particular in 2D [77–79]. Therefore, we will restrict the discussion first to a straightbackbone and devote the subsequent section to the bending elasticity of the molecule.

Generally, the free energy of this complex molecule can be written as a sum ofthree terms:

F = Fbb + Fsc + Fbb−sc . (3.2)

Here Fbb is the free energy of the backbone (i.e. of a semiflexible chain with persis-tence length λ0), Fsc refers to the side chains and includes both an entropy part and theinteraction between rods, and finally a cross term Fbb−sc representing the interactionbetween the backbone and the side chains. We will assume that Fbb−sc Fsc andhenceforth the third term in (3.2) will be neglected.

We assume each rod to have complete rotational freedom, apart from excludedvolume constraints, in the plane above or below the backbone. In practice, thiscan be realized by adding a spacer between the backbone and the mesogenic grouprepresenting the side chain [125, 126].

We start our theoretical consideration from the weak attraction limit where stericrepulsion plays the main role and the van der Waals attraction between the rods

42 Chapter 3

Figure 3.2: Test rod and its nearest neighbors for a straight comb copolymermolecule.

causes only small corrections to the system properties. Limits of applicability ofthis approximation will be discussed below.

Weak attraction limit. Let fi(θ), denote the orientation distribution function ofthe side chain. Here i = 1, 2 denotes the two sides of the backbone and θ is theorientation angle in the plane of the molecule (see Figure 3.1).

In the mean field approach the free energy per rod can be expressed as

F =12

∑

i

∫dθ fi(θ) ln[ fi(θ)] +

12

∑

i

∫dθ fi(θ)Ui(θ) , (3.3)

where Ui(θ) is the interaction energy. In the present model it comprises hard corerepulsion and van der Waals attraction between a test rod and its nearest neighborsfixed in their average positions ϕi (i = 1, 2) as depicted in Figure 3.2:

1kBT

Ui(θ) =

Uattr

i (θ) ϕ−i < θ < ϕ+i

∞ otherwise.(3.4)

From equations (3.3) and (3.4) we obtain the general expression for the freeenergy functional of a test rod in units of kBT

2F =

∫ ϕ+1

ϕ−1dθ f1(θ) ln f1(θ) +

∫ ϕ+2

ϕ−2dθ f2(θ) ln f2(θ)

+

∫ ϕ+1

ϕ−1dθ f1(θ)Uattr

1 (θ) +

∫ ϕ+2

ϕ−2dθ f2(θ)Uattr

2 (θ) . (3.5)

3.2. Straight comb copolymer molecule 43

The distribution functions fi(θ) are found by minimization, δF/δ fi(θ) = 0

fi(θ) =1Zi

exp[−Uattri (θ)] , (3.6)

where Zi is the normalization factor.The free energy per rod follows from (3.5) and (3.6) and for small ε (weak

attraction limit) can be written as:

F =12

(F1 + F2),

Fi = − ln(ϕ+i − ϕ−i ) +

1ϕ+

i − ϕ−i

∫ ϕ+i

ϕ−iUattr

i (θ) dθ . (3.7)

As mentioned above, the attractive part of the potential is modeled as the well-known van der Waals attraction. In the simplest case of two dielectric spheres itscales as the sixth power of the distance. Also the interaction energy of two parallelinfinitely long thin rods can be computed analytically. The intermediate case offinite non-parallel rods can be considered only asymptotically if their length stronglyexceeds the distance separating their central axes. This requires the strong inequalityb L, which is assumed to be satisfied in the present model.

Here we are interested in the attraction between two neighboring side chains,grafted on a straight backbone (Figure 3.2). The van der Waals attraction energy inthis particular case has the relatively simple form (see Appendix (3.43)):

Uattri (θ) = −3πε

8d4L

b5

1

sin5 ϕi

18xi

(1

(1 − xi)4− 1

(1 + xi)4

)(3.8)

with

xi =L sin(θ − ϕi)

b sin ϕi.

In (3.8) end effects are neglected and the potential therefore decreases as the fifthpower of the distance (see [127, 128]).

Next, the function Uattri (θ) will be expanded in a series around the point ϕi

Uattri (θ) = Uattr

i (ϕi) +12

∂2Uattri (ϕi)

∂θ2(θ − ϕi)

2 + ... , (3.9)

44 Chapter 3

where the first derivative term is absent because ∂U attri (ϕi)/∂θ = 0 by definition of ϕi.

Substituting (3.9) and (3.8) into the free energy (3.7) and taking into account that theangles ϕ+

i and ϕ−i for a straight backbone can be found from simple geometry (seeFigure 3.2)

ϕ+i − ϕi =

bL

(sin ϕi − d

b

)

ϕ−i − ϕi = − bL

(sin ϕi − d

b

), (3.10)

we obtain the following expression for the free energy in the weak attraction limit

Fi = − ln2bL− ln

(sin ϕi − d

b

)

− 3πε8

d4L

b5

1

sin5 ϕi

[1 +

5 (sin ϕi − d/b)2

sin2 ϕi+ . . .

]. (3.11)

At this point we should discuss the range of applicability of the expansion (3.11).In order that it can be truncated after the first term, sin ϕi − d/b should be small.This condition can be satisfied if we assume that d/b = 1 − δ2/2, where δ 1 isa dimensionless small parameter. In this case the angles ϕi should be close to π/2which implies that

sin ϕi = 1 − y2iδ2

2, (3.12)

where yi (−1 ≤ yi ≤ 1) is a new parameter. Now, the second term between squarebrackets in (3.11) is proportional to δ4 and can be omitted in a theory with accuracyup to δ2.

For the straight symmetric brush both sides of the backbone are equal implyingthat ϕ1 = ϕ2 (or y1 = y2). Therefore, combining this fact with (3.12), (3.11)and (3.44) one obtains the final expression for the free energy of the straight combcopolymer molecule

F0 = − lnb δ2

L− ln(1 − y2) − 3πε

8d4L

b5

[1 +

52y2δ2

]. (3.13)

The behavior of F0 can be studied by finding its minima. Depending on themagnitude of ε one or two minima are present. The minimum at y = 0 is important

3.2. Straight comb copolymer molecule 45

Figure 3.3: Behavior of tilting parameter y (dashed line) and tilting angle ϕ(solid line) as a function of the interaction strength ε.

in the region ε < ε∗ (weak attraction), where

ε∗ =8

15πb

L(1 − d/b). (3.14)

It corresponds to all rods oriented preferably perpendicular to the main chain.The existence of ε∗ clarifies the exact meaning of the weak and strong attraction

limits. For ε < ε∗ attraction can only shift slightly the quantitative characteristicsof the molecule whereas qualitatively (scaling laws, conformations, etc) they remainsimilar to the corresponding comb copolymer molecule with steric repulsion only.For ε > ε∗ (strong attraction limit) the picture changes qualitatively. Figure 3.3presents y found from minimization of (3.13) as a function of ε. The result is asecond-order phase transition at ε = ε∗ and a strong decrease in tilting angle (strongtilting towards backbone) for larger values of ε.

Since we are dealing with a 1-dimensional model with local interactions (weused the van der Waals interaction only between nearest neighbors) the mean fieldapproach does not work in the vicinity of the transition point and we will alwayshave y = 0 without phase transition at ε = ε∗. However, the way this is accomplishedin the strong interaction regime is by having alternating domains of oppositely tiltedside chains. Geometrical considerations show that the defect zone between thesedomains has to be large. Hence the extra free energy associated with a defect zone is

46 Chapter 3

Figure 3.4: Dense packing of rods in the straight molecule (the direction oftilting can also be opposite at opposite sides of the backbone).

large and consequently the domains with side chains strongly tilted in one directionwill be large. For a real comb copolymer of finite size this would almost certainlyimply a tilting of all side chains at one side to the same direction.

In order to obtain manageable analytical expressions (e.g. (3.13)), we had torestrict the final discussion to the very dense grafting limit, d/b ∼ 1. In practice, d/bwill be usually considerably smaller. For this case the above analysis remains valid,except that numerical factors will change. Here precise results can only be obtainedby numerical methods.

Strong attraction limit. In the strong attraction limit all conformational prop-erties of the comb copolymer molecule are dominated by the attraction part of thefree energy. It implies that the system will try to satisfy the condition of minimumattraction energy, which corresponds to the most densely packed state. All rods willlie down on the backbone as shown in Figure 3.4. The energy per rod can be estimatedfrom (3.8) as

E = −3πε8

Ld. (3.15)

However this result is correct only when the temperature T = 0.For T , 0 fluctuations will break the dense packing and conformations with some

free space between the rods will appear, so that the orientation angle will be slightlylarger than the minimal possible angle

ϕ = arcsindb

+ ϕ′ . (3.16)

3.3. Bending elasticity 47

Here ϕ′ is considered to be a small parameter which gives rise to a change in theattraction energy given by

E′ ' 15πεL8 d

√b2

d2− 1 ϕ′ . (3.17)

This energy should be of the order of thermal energy kBT (note that kBT ≡ 1 through-out this chater). This allows us to find the value of ϕ′ and associated with it thecharacteristic amplitude ψ of the fluctuations of the angle between two consecutivesegments of the backbone of length b

ψ =bL

√1 − d2

b2ϕ′ ' 8

15πd2

εL2. (3.18)

3.3. Bending elasticity

So far we limited ourselves to the consideration of a straight comb copolymer brush.The objective of the present section is to analyze the bending elasticity characteristicsof the molecule. This will be done by studying the behavior of the free energy as afunction of the curvature of the backbone. As before, we will start with the weakattraction limit.

Weak attraction limit. To examine theoretically the elasticity, we should gen-eralize the free energy (3.13) for the case of nonzero curvature. For our purposewe need an expansion of F as a function of 1/R up to the quadratic term only. Inother words, the limiting angles ϕ+

i , ϕ−i and the attraction energy Uattri have to be

recalculated for the case where the main chain is uniformly bent with a radius ofcurvature R (R L). The limiting angles can be found from simple geometricalarguments (see Figure 3.5)

ϕ+1 − ϕ−1 =

2bL

(sin ϕ1 − d

b+

LR

),

ϕ+2 − ϕ−2 =

2bL

(sin ϕ2 − d

b− L

R

). (3.19)

The attraction energy for the bent brush is given by (3.46) in the Appendix. Together(3.19) and (3.46) lead to the generalized expression for the free energy as a function

48 Chapter 3

Figure 3.5: The test rod and its nearest neighbors for a bent combcopolymer molecule.

of parameters y1 and y2

F = − lnbδ2

L− 1

2ln

[1 − y2

1 +2L

δ2R

]− 1

2ln

[1 − y2

2 −2L

δ2R

]

− 3πε8

d4L

b5

[1 + 5

L2

R2+

5δ2

4(y2

1 + y22)

]. (3.20)

The equilibrium values of the angles (i.e. yi) are found from minimization of the freeenergy.

First we focus on the solution for ε < ε∗ where the only stable value of the tiltangle corresponds to y1,2 = 0. In this limit the free energy has a very simple form

F = F0 +

(2

δ4− 15π ε L

8 b

)L2

R2, (3.21)

where F0 is the free energy of the straight brush. The main contribution is due to therepulsive part of the potential. The attraction contribution is negative and reduces thestiffness of the comb copolymer molecule. The persistence length of the moleculecan be calculated on the basis of the general relation

∆F =λb

2R2, (3.22)

where λ is the persistence length of the complex. Hence, it is given by

λ = λ0 +L2

b

(1

(1 − d/b)2− 15π εL

4 b

). (3.23)

3.3. Bending elasticity 49

Here λ0 is the persistence length of the bare backbone. The correction to λ0 scales asL2 and it is interesting to note that it decreases with increasing attraction strength.

Strong attraction limit. A qualitatively different behavior can be expected inthe region of large attraction parameter ε ε∗. As stated in the previous section,the attractive part of the potential dominates in this regime. For small bending thefluctuations are very important and this case can not be described by the mean fieldapproach. Therefore we use the scaling approach described in [2] and estimate thepersistence length for small ψ by

λ ' 2b

ψ2. (3.24)

Here ψ is given by (3.18). This leads to the following expression for the persistencelength

λ ' 225π2

32ε2bL4

d4. (3.25)

It strongly depends on the energy parameter ε. The scaling dependence on L alsodiffers from (3.23) and is much stronger. In the limit T → 0 (or ε → ∞) the moleculeis densely packed and the persistence length (3.25) becomes infinitely large.

Now we can estimate the characteristic radius of curvature separating the linearand non-linear bending regimes. The free energy per rod of the bent brush in thelinear regime is given by equations (3.22) and (3.25). Comparing this value to kBTgives

Rc ' 15πεbL2

8d2. (3.26)

This radius is very large for strong attraction and long side chains.To study the large bending regime (R < Rc), we start from the concave part of

the bent molecule. Since the fluctuations are not important in here we can safely putthe temperature T = 0. In this case the rods tend to be as close to each other aspossible and form the structure shown in Figure 3.6. The orientation angle ϕ2 for thisconformation is determined from geometry

ϕ2 = arcsin

(db

+LR

)− b

2R. (3.27)

Proceeding with the calculations one should take into account the approximationd/b = 1 − δ2/2, substitute (3.27) into the general expression (3.45) for the attraction

50 Chapter 3

Figure 3.6: Ordering of side chains in the concave part of the molecule inthe strong attraction limit.

energy and expand it into a series of the small parameter L/R. This leads to acorrection to the energy of the straight brush given by

∆Econcave =3πε8

Ld

[52

LR− 35

6L2

R2

]. (3.28)

Note that the linear term in the expansion is positive.For the convex part of the molecule the situation is quite different. Due to

bending, the available angle space increases. It makes the existence of a continuousstructure formed by rods impossible; inevitably some gaps should appear. The spacefilled by rods between two consecutive gaps will be called domain. Inside sucha domain rods form a densely packed system. The “first” rod in the domain (seeFigure 3.7a) will have the smallest angle allowed by steric repulsion

ϕ0 = arcsindb− b

2R(3.29)

and the positions of all other rods in the same domain can be calculated from the con-dition of touching. This leads to the following recursive relation for the orientationangle y(k) of the k-th rod

y(k + 1) = y(k) − aδ

[cos

(y(k)δ − b

2R

)−

(1 − δ

2

2

)]− b

Rδ. (3.30)

Knowing that the first rod is oriented according to (3.29)

y(1) = 1 +b

2Rδ

3.3. Bending elasticity 51

a b

c

Figure 3.7: Domain of n rods on the convex part of the molecule: (a) thegeneral case considered in (3.31); (b) the ordering corresponding to n = 1;

(c) the “complete” cluster (n = n∗).

and using the attraction energy between rods in the form (3.45) we obtain the changedue to bending of the energy per rod for a domain consisting of n rods

∆E(n) =3πε8

d4L

b5

[52

(1 + (n − 2)

bLδ

)LR− 5n

L2

R2

]. (3.31)

The domain size n can vary from 1 up to n∗ (the value of n∗ will be defined below).The equilibrium value of n can be found from minimization of (3.31). Two differentregimes are possible depending on the magnitude of curvature. For very large radiusof curvature R > R∗, where

R∗ =2L2

b δ, (3.32)

the energy (3.31) is a monotonously increasing function of n. This means that theminimal value of the energy will be attained for n = 1, a situation that is depicted in

52 Chapter 3

Figure 3.7b. In this case, the energy per rod is given by

∆E(1) =3πε8

d4L

b5

[52

(1 − b

Lδ

)LR− 5

L2

R2

]. (3.33)

For R < R∗, ∆E(n) decreases with increasing n. In this case the domain will grow untilthe maximal size n∗ allowed by geometry of the bent molecule (see Figure 3.7c). Thissize n∗ is to be found from (3.30) by integration

n∗ = 2∫ 0

y(1)

dyy(k + 1) − y(k)

. (3.34)

After direct calculation n∗ appears to be proportional to the logarithm of the radius ofcurvature R

n∗ ' 2Lb δ

ln2δ2R

L. (3.35)

Finally, the energy per rod in the domain shown in the Figure 3.7c can be found from(3.31) and (3.35)

∆E∗ =3πε8

Ld

[5

LR

ln2δ2R

L+ . . .

]. (3.36)

Now the first term in the expansion is proportional to ln L/R rather than L/R (3.33).The total energy of the brush is the sum of the concave (3.28) and the convex

part, where the convex part is given either by (3.33) or by (3.36) depending on thecurvature. For radius R > R∗ it reads

∆F =15πε

8Ld

[LR− 13

6L2

R2

]. (3.37)

As follows from the expansion for the energy of the comb copolymer molecule itsstiffness in the strong attraction limit has a non-persistent character. The presence ofa positive linear term implies that the brush will behave like a hard rod: a finite forceis needed to start bending. The moment of this force can be defined as the derivativeof the free energy with respect to the curvature M = ∂F/∂(1/R) and equals

M ' 15πε8

L2

d

(1 − 13

3LR

). (3.38)

Once the force applied to the straight molecule exceeds the critical value the cylin-drical brush will be “broken” and further bending will be much easier.

In the linear regime M ' λb/R increases with increasing curvature 1/R. Themoment of the force as a function of curvature 1/R is shown in Figure 3.8. It passesthrough a maximum value M ∼ εL2/d for R ∼ Rc.

3.4. Concluding remarks 53

Figure 3.8: Schematic representation of dependence of the bendingmoment on the curvature.

3.4. Concluding remarks

In the present chapter we described the conformational behavior of comb copolymermolecules with stiff side chains confined to a plane. A mean field approach was usedto examine the properties in different regimes. It was shown that attraction betweenside chains plays a crucial role and that depending on its relative strength differenttypes of behavior are possible.

In the weak attraction limit these comb copolymer molecules resemble persistentchains, although the corrections to the backbone’s persistence length λ0 are not neces-sarily small and scale as the second power of the molecular weight of the side chainsλ ' L2/b. In the preferable conformation all rods are uniformly distributed along thebackbone and stay in a perpendicular to the backbone position. This result coincideswith that predicted for comb copolymer molecules in three dimensions [70].

For relatively strong attraction the system switches to the tilting conformation andthe molecule is characterized by a different scaling law for the persistence length, firstλ ' ε2L4b/d4 and beyond a critical curvature Rc ' εL2b/d2, a non-linear bendingregime appears with a non-persistent mechanism of stiffness and decreasing bendingmoment of force as function of the curvature. In this regime the side rods arranged inthe convex part of the chain undergo the transition, when radius of curvature R = R∗ ∼2L2/b δ, from uniform ordering to non-uniform ordering with formation of domains

54 Chapter 3

consisting of n∗ ' 2L ln(2δ2R/L)/b δ rods. This transition connected with the factthat the minimum of the free energy attains for the domain structure rather than foruniform orientation which characterized by bigger free space. We also expect (if forsome values of parameters R∗ > Rc is satisfied) that the domain structure will be formbeyond the transition point.

In experiments the transition from the weak to the strong attraction limit, whichmay be induced by lowering the temperature, should show itself as an effectivestiffening of comb copolymer molecules strongly adsorbed on a surface. In practicethe transition from the second power law (3.23) to the fourth power law (3.25) maybe accompanied by an isotropic-nematic transition due to drastic stiffening of themolecule. This is of considerable interest as a possible way to adjust the molecularordering. For very strong attraction (or equivalently for low temperatures), wherenon-linear behavior is important, bending requires a critical value for the moment offorce (3.38) after which it becomes “softer”.

In the previous sections we were primarily interested in the dependence of theconformational characteristics of the molecule on the energy parameter ε and thelength of side chains L. We considered a completely symmetric and regular combstructure. In principle it is possible to imagine a system where flipping of side rodsover the backbone from one side to the other is possible. This can be realized, forinstance, by thermal fluctuations for a comb copolymer molecule confined to theinterface between two immiscible fluids. In this case the average value of b is a freeparameter and can be varied by flips of side chains. For the weak attraction limitthe free energy expression (3.13) shows that this will lead to an increase of the freeenergy. Thus even if flipping is possible, the rods will stay on different sides ofthe main chain to optimize the average distance between two neighbors. For strongattraction the state with the smallest value of b is preferable. This implies the possibleformation of domains of side chains all flipped to the same side of the backbonewith wall defects between two consecutive domains. The characteristic length ofsuch domains will be determined by interplay between energy and entropy of defects[129]. As a result inside one domain the molecule becomes asymmetric. Moleculeswith different grafting densities at both sides of the main chain where considered insome recent articles [31, 81, 83]. There the authors assumed a frozen asymmetry,whereas in our case it occurs spontaneously as a result of the attractive interaction.

Finally, note that the attraction can also result in spiraling of the comb copolymermolecule (as a part of coil-globule transition) if the contour length is large enough.

3.A. Energy of attraction of two long rods 55

a b

c d

Figure 3.9: Illustration for the calculation of the energy of attractionbetween two rods: (a) two rods of length L at a distance h from each other(h L); (b) the test rod between its nearest neighbors fixed in their averageposition on the straight backbone; (c) two arbitrary oriented neighboringrods on the straight brush; (d) two arbitrary oriented neighboring rods on

the bent brush.

Appendix

3.A. Energy of attraction of two long rods

Let us consider two rods of length L grafted as shown in Figure 3.9a on a distanceh L from each other, interacting with a van der Waals potential. The total energyof attraction can be obtained by an integration of (3.1) along both rods

E = −εd4∫ L

0

∫ L

0

ds1ds2[(h − s2 sin ∆ϕ)2 + (s1 − s2 cos ∆ϕ)2]3

. (3.39)

56 Chapter 3

Introducing new variables

y =s2

L

ω =s1 − s2 cos ∆ϕ

h − s2 sin ∆ϕ(3.40)

and performing the integration in (3.39) using the strong inequality L h, oneobtains for the energy

E = −3πε8

d4L

h5

14x

(1

(1 − x)4− 1

), (3.41)

where

x =L sin ∆ϕ

h. (3.42)

It is easy to apply expression (3.41) to the case depicted in Figure 3.9b in orderto get an expression for the energy of attraction between the test rod and its nearestneighbors

E = −3πε8

d4L

(b sin ϕ)5

18x

(1

(1 − x)4− 1

(1 + x)4

). (3.43)

Here h = b sin ϕ.In the limit d/b = 1 − δ2/2 with δ 1 considered in this chapter, it is easier

to expand the general expression (3.39) for the attraction energy into a series of thesmall parameter δ and then solve the integrals. If yθ and yϕ are orientation parametersassociated with the angles θ and ϕ (see Figure 3.9c) according to

y =1δ

(π

2− ϕ

)

the potential energy can be written in the form

U0(yϕ, yθ) = −3πε8

d4L

b5

[1 − 5

2Lδb

(yθ − yϕ)

+5L2δ2

b2(yθ − yϕ)2 +

5δ2

4(y2θ + y2

ϕ)

]. (3.44)

To generalize (3.44) for the case of the bent molecule one should take into accountthat the angles ϕ and θ in Figure 3.9c for the straight molecule correspond to the

3.A. Energy of attraction of two long rods 57

angles ϕ+γ/2 and θ−γ/2 in Figure 3.9d when the brush is bent (γ = b/R). Rewritingthese conditions in terms of yθ and yϕ we arrive at the expression

U(yϕ, yθ) = −3πε8

d4L

b5

[1 − 5

2

(yθ − yϕ +

γ

δ

) Lδb

+5(yθ − yϕ +

γ

δ

)2 L2δ2

b2+

54δ2

((yθ +

γ

2δ

)2+

(yϕ − γ

2δ

)2)]. (3.45)

Note that in (3.45) θ and ϕ are the angles between the rod and the tangent to thebackbone in the grafting point. Similar expression should be written for the concavepart.

Finally, the energy of attraction of the test rod oriented with an angle θ withrespect to its two neighbors (both are oriented with ϕ to the tangent) can be derivedfrom (3.44) and reads

Uneigh(yϕ, yθ) =12

[U0

(yϕ − γ

2δ, yθ +

γ

2δ

)+ U0

(yθ − γ

2δ, yϕ +

γ

2δ

)]. (3.46)

CHAPTER 4

Comb Copolymer Brush withChemically Different Side Chains

An investigation of side chain microphase separation within a singlecomb copolymer molecule containing chemically different A and B sidechains is carried out. Expressions for the transition point χ∗AB in a good(χ∗AB ∼ N−3/8), marginal (χ∗AB ∼ N−1/2), θ (χ∗AB ∼ N−2/3), and poor(χ∗AB ∼ N−1) solvent are derived both by a mean field calculation andby scaling arguments. Properties of the system below and above thetransition point are described. Some unusual “bow-like” conformationsare predicted for a single molecule in the microphase separated state ina good solvent.

4.1. Introduction

In the previous chapters we addressed the elastic properties of molecular bottle-brushes and, in particular, their stiffening, arising from the side chains attachment.The resulting conformations were always bottlebrush-like “straight” and generallycould be characterized by a persistence length λ. Now we turn our attention to thepossibility to attain non-straight stable conformations. The relevant experimentalresults were briefly discussed above, in the introductory section 1.2. As was observed,

59

60 Chapter 4

e.g., horseshoe and meander-like shapes of the comb complexes [27] can appear as aresult of the interaction between side chains. It would be also a challenge to preparea “switchable” system, where the transformation between the straight and curvedconformation could be controlled externally.

An example of such a system is considered theoretically in the present chapter.The curved conformations can be achieved by attaching chemically different (thus,disliking each other) side chains to a common backbone. In this case, unfavorableinteractions between the side chains lead to a micro-domain structure within a singlemolecule. Consequently, as a result of the appeared asymmetry in the side chaindistribution, curved conformations can be expected. Apparently, being able to controlthe interaction strength one can switch straight (homogeneous) to bent (domained)conformation and vice versa. It is the aim of this chapter to reveal the possibility ofsuch behavior and find the parameters controlling the process.

The chapter is organized as follows. The next section describes the self-consistentfield approach to a molecule with a straight backbone and chemically different sidechains. We show the possibility of side chain separation within the molecule anddiscuss the limits of the theory’s applicability. The subsequent section is devoted topossible unusual behavior of comb copolymer molecules with a flexible backbone andmicrophase separated side chains. Then all results are summarized and discussed inthe last section.

4.2. Straight molecule with two types of side chains

Chemically different polymeric chains, namely of the type “A” and “B”, are attachedto a main chain with grafting density 1/b. For simplicity, we choose the differentside chains alternatingly grafted to the backbone so that the linear density of A- (andB-) type chains is 1/2b (Figure 4.1). Furthermore, we assume that the side chainshave the same length NA = NB ≡ N and statistical segment length a. Although thecalculations can be done for arbitrary values of these quantities, we restrict ourselvesto this simpler case in order to reduce the number of free parameters in the model.

To attain the cylindrical brush regime the Flory radius of the side chain RF

should strongly exceed the distance b between two consecutive grafting points. Thiscondition can always be fulfilled by choosing long enough side chains so that acylindrical (i.e. straight) conformation will be realized. Therefore, we first considera molecule with a straight backbone. Possible deviations from this conformation willbe discussed further on.

4.2. Straight molecule with two types of side chains 61

Figure 4.1: Schematic representation of a comb copolymer molecule.Thick and thin lines represent two chemically different side chain.

To describe the interactions between the side chains and the solvent we need a setof three parameters vαβ (α, β = A, B), which are directly related to the experimentallymeasured Flory-Huggins χ-parameters (χAS , χBS , χAB; see Appendix). As far as weare interested in the properties of the cylindrical brush induced by the side chains weneglect any influence of the backbone.

Before presenting our calculations we discuss first what kind of effects mightbe expected. Let us start from a molecule in a dilute solution in a solvent that isequally good for both species with the additional condition that A chains do not feelthe presence of B chains in any other way than they feel the other A chains (repulsionbetween A and A, B and B, A and B is of the same strength). Therefore, they are bothhomogeneously distributed in the cross section of the brush molecule (Figure 4.2b;case RA ' RB will be considered throughout the text). With a gradual increase ofthe A-B (repulsive) interaction parameter, chains of different nature try to avoid eachother, but they are still mixed due to a certain entropic threshold. However, beyondsome value v∗AB one can expect a transition manifesting itself in the separation ofA chains from B chains within the comb copolymer molecule (Figure 4.2a). Thispoint v∗AB can be estimated by comparing the free energies of the mixed Fmix and theseparated Fsep states calculated on the basis of the side chain density distributionin the cross section of the molecule. Of course, this will not give us the exactbinodal point (in reality the transition will be smoother than assumed here) but rather

62 Chapter 4

2py

RA

RB

a b

Figure 4.2: Cross section of the molecule: (a) in the separated state; (b) inthe mixed state.

a good estimation knowing that this method gives an almost exact result in the caseof diblock copolymers [106, 130].

4.2.1. SCF approach

The free energy per length b in the straight brush regime (assuming a straight back-bone) can in the most general form be written as

F =12

(Fcon f ,A + Fcon f ,B

)+ Fint . (4.1)

The interaction part of the free energy Fint per chain in the second virial approxima-tion has the simple form

Fint =b2

∫ 2π

0dϕ

∫ ∞

0r dr

∑

α,β

vαβcα(r, ϕ)cβ(r, ϕ) . (4.2)

Here we use the polar coordinate system with the z-axis pointing along the straightbackbone and measure the free energy in the units of kBT (this makes all quantitieswith the dimension of energy dimensionless); cα(r, ϕ) are the concentration profiles(α, β = A, B). The conformational free energy of a type α chain Fcon f ,α corresponds

4.2. Straight molecule with two types of side chains 63

to the free energy of a stretched Gaussian chain [2, 68, 131] and can be written (seeSCF approach explanation in the introductory section 1.3.1) as

Fcon f ,α = b∫ 2π

0dϕ

∫ ∞

0r dr

[− gα(r, ϕ) ln Zα(r, ϕ; N | µα)

−µα(r, ϕ)cα(r, ϕ) + gα(r, ϕ) ln gα(r, ϕ)], (4.3)

where some new functions have been introduced. Zα is a partition function of a chainof length N with two fixed ends (one at the zero point and the other at (r, ϕ)) in theexternal field µα, so that − ln Zα is the corresponding free energy. To take into accountthe distribution of the side chain’s free end one should average the energy − ln Zα witha distribution function of the free end g(r, ϕ) and add the translational entropy termg ln g. Expression for the free energy (4.3) can be easily verified in the limiting caseof an ideal chain [2] when µ = 0 and − ln Zα = 3r2/(2Na2) so that minimization withrespect to g gives g = C exp

(− 3r2

2Na2

), where C is a normalization constant.

Following ref [68] we employ an analogy between a stretched Gaussian chainwhose partition function is to be found from a Schrödinger type equation [1, 2] (tenumerates monomeric units)

∂Z(r, ϕ; t | µ)∂t

=a2

6∆Z(r, ϕ; t | µ) − µ(r, ϕ)Z(r, ϕ; t | µ) (4.4)

and a quantum particle moving in the field µ. Using this analogy with the well knownsemi-classical WKB approximation one obtains the partition function in the form

Z = e−tE−S (r) , (4.5)

where S (r) '√

6a

r∫

0

dr1√µ(r1) − E and E is the analogy of the particle’s energy,

which will be implicitly calculated further on (for details see [68] and section 1.3.1).All the other functions in (4.2) and (4.3), i.e. cα, µα, gα, have to be determined

from the free energy (4.1) extremum. This requires some additional assumptionsabout the side chain alignment structure around the backbone.

4.2.1a. Separated stateSuppose the A-B “repulsion” parameter vAB is large enough and the system hasalready undergone side chain separation. This means that A-type side chains are

64 Chapter 4

concentrated inside some angle 2πψ (0 < ψ < 1) as depicted in Figure 4.2a and thenumber of B side chains in this region is exponentially small

cA(r, ϕ) =

cA(r) −πψ < ϕ < πψ0 otherwise.

(4.6)

The analogous fact is true for the B-type side chains in the other part of the anglespace πψ < ϕ < 2π − πψ.

Additionally we will use the Alexander-de Gennes approximation [1] for the freeend distribution functions implying that all A-chains’ free ends are located in somevery narrow region near RA (and RB for B-chains)

gα =1

4πψαbδ(r − Rα)

r. (4.7)

The numerical prefactor in (4.7) follows from the normalization condition. The

functions cA and cB have to be normalized as well: 4πψbRA∫

0

r drcA(r) = N and

4π(1 − ψ)bRB∫

0

r drcB(r) = N.

Equations. (4.6) and (4.7) together with the partition functions Zα taken in theform (4.5) significantly simplify the free energy for the separated state

Fsep =N2

(EA + EB) − 12

lnψ − 12

ln(1 − ψ)

+

√6

2a

∑

α=A,B

∫ Rα

0dr

√µα(r) − Eα

+2πb∑

α=A,B

ψα

∫ Rα

0r dr

(−µα(r)cα(r) +

12vααc2

α(r)

). (4.8)

Here ψA ≡ ψ and ψB ≡ 1 − ψ, and the energy of the A-B interface is neglected(consistency of this assumption will be verified further on). Together with (4.8), theself-consistency condition δFsep/δcα = 0 should be fulfilled. This gives the followingrelation between the concentration and the chemical potential

cα =µαvαα

. (4.9)

4.2. Straight molecule with two types of side chains 65

As the next step, functions µA and µB should be obtained from the extremumcondition of (4.8). Note that the minimum of the free energy as a function of con-centration implies a maximum as a function of the chemical potential as a conjugatedvariable [68]. Taking into account relation (4.9) we arrive at the equation for thechemical potentials √

64a

vαα√µα(r) − Eα

= 2πbrψαµα(r) , (4.10)

which together with the normalization condition for the concentration rewritten in theform

4πψαbNvαα

∫ Rα

0r dr µα(r) = 1 , (4.11)

gives the complete set of equations for Eα and µα(r).With reasonable accuracy, the solution of (4.10) can be approximated as [68]

µα(r) ' Eα +

√

6vαα8πψαabr

2/3

. (4.12)

Here EA and EB follow from the normalization (4.11).In this way the free energy (4.8) becomes a function of three parameters RA, RB

and ψ and can be easily minimized. It leads to the following expressions for the radiiof the comb copolymer cylindrical brush

RA ' 0.48

ψ1/40

(a2vAAN3

2b

)1/4

,

RB ' 0.48

(1 − ψ0)1/4

(a2vBBN3

2b

)1/4

, (4.13)

where 2πψ0 is the equilibrium value of the angle occupied by the A side chains inthe cross section of the cylindrical molecule, which should be determined from theequation √

1532π

( N

a2b

)1/2vAA

ψ3/20

− vBB

(1 − ψ0)3/2

=1

1 − ψ0− 1ψ0

. (4.14)

The free energy per side chain in this case reads

Fsep = 0.386( N

a2b

)1/2 [√vAA

ψ0+

√vBB

1 − ψ0

]− 1

2ln

[ψ0(1 − ψ0)

]. (4.15)

66 Chapter 4

Equation (4.14) can not be solved in the general case but obviously has a root ψ0 =

1/2 for vAA = vBB ≡ v. In this particular case the free energy (4.15) simplifies to

F(1/2)sep = 1.092

( Nv

a2b

)1/2

+ ln 2 . (4.16)

This result coincides with the one obtained in [68] and is very similar to [132] exceptfor the entropic ln 2 term, which is not present in the cited works due to the differencein the system studied.

Now let us estimate the thickness of the interface region. The typical blob size inthe interpenetration region should have energy of the order of 1(kBT ). If it consists ofg links, then its size may be estimated as a

√g and therefore A-B interaction energy

inside the blob readsgν

a3g3/2g χAB ∼ 1 . (4.17)

In order to use expression (4.16) for the free energy in the vicinity of the transitionpoint we have to make sure that g N. Together with (4.17) this gives a conditionfor the (4.16) applicability:

χAB a3

ν

1√N. (4.18)

4.2.1b. Mixed stateSo far we have calculated the free energy of the comb copolymer molecule in the“separated” state. The same method can be applied to the “mixed” state of the system,where A and B side chains are homogeneously mixed and therefore the concentrationprofiles have to be chosen in the form (α = A, B)

cα(r, ϕ) = cα(r), 0 < ϕ < 2π . (4.19)

Similar to the separated state (4.8), we can calculate the free energy Fmix in theframework of WKB and Alexander-de Gennes approximations

Fmix =N2

(EA + EB) +

√6

2a

∑

α=A,B

∫ Rα

0dr

√µα(r) − Eα

+2πb∑

α=A,B

∫ Rα

0r dr

(−µα(r)cα(r) +

12vααc2

α(r)

)

+2πb∫ min(RA;RB)

0r dr vAB cA(r)cB(r) . (4.20)

4.2. Straight molecule with two types of side chains 67

Note that the entropic logarithmic terms (see (4.8)) are absent and an integral respon-sible for the A-B interaction is added. Without loss of generality we further assumethat RA ≤ RB (but RA ' RB still holds).

The computations happen to be a bit more complicated now because of the non-trivial relation between concentrations and chemical potentials (compare with (4.9))

cA =µAvBB − µBvAB

vAAvBB − v2AB

, cB =µBvAA − µAvAB

vAAvBB − v2AB

. (4.21)

This leads to a set of two third order algebraic equations for the chemical potentialsinstead of (4.10), and the simple approximation (4.12) cannot be employed.

To simplify the equation we note that all the difficulties disappear for the case ofvAA = vBB ≡ v considered at the end of the previous subsection. It gives chemicalpotentials (now µA(r) = µB(r) ≡ µ(r)) in the form

µ(r) ' E +

√

6(v + vAB)8πabr

2/3

. (4.22)

where E ≡ EA = EB and the concentration profile is coupled to µ(r) as c(r) =

(v + vAB)−1µ(r). The constants EA,B are found from the normalization of c(r) andread

Eα =Nv

πbR2α

− 32

√

6(v + vAB)8πabRα

2/3

. (4.23)

Finally, substituting (4.22) and (4.23) into the free energy (4.20) and minimizing, oneobtains radii of the brush

RA,B ' 0.48

(a2(v + vAB)N3

2b

)1/4

(4.24)

and the free energy

F(1/2)mix = 0.772

(N(v + vAB)

a2b

)1/2

. (4.25)

As expected, the free energy increases with increasing strength of the A-B repul-sion. The interaction between A and B side chains also renormalizes the numericalprefactor in comparison with (4.16).

68 Chapter 4

4.2.1c. Transition from mixed to separated stateAt this point it is easy to estimate the value v∗AB of the A-B interaction parameterseparating mixed and segregated states. Comparing (4.16) to (4.25) one gets

v∗AB = v + 2.5

(a2bv

N

)1/2

+ 0.8a2bN

. (4.26)

Keeping in mind that v and vAB are effective parameters we can use (4.65) from theAppendix to express this point in terms of the χ-parameters (χAS = χBS ≡ χ)

χ∗ABN = 0.8ba2

ν+ 2.5

√Nba2

ν(1 − 2χ) . (4.27)

Here ν is the volume of one bead in the model. As follows from (4.27), only thesecond term is important well above the θ-point. The χ∗AB parameter appears to beproportional to N−1/2 as

χ∗AB ∼1√N. (4.28)

Comparing the result (4.27) with condition (4.18) one can see the interfacialterm is indeed important at the transition region. This implies that SCF calculationgenerally should be conducted incorporating compositional inhomogeneity effectsinto the free energy. Nevertheless the result (4.28) still holds giving correct scalingprediction for the transition point.

Now let us address the problem of the mean field approach applicability. Themethod itself implies that fluctuations are small. Thus we have to apply the result(4.28) rather to a marginal than to a “very good” solvent. It is also possible to estimatea range of parameters where all assumptions, made before, are fulfilled. First of all,note that the second virial approximation that we use is valid for small concentrationswhen additionally the second virial term is dominant over the third one, i.e. accordingto (4.64) for 1 − 2χ > νc or

1 − 2χ >(ν

ba2

)1/3N−1/3 . (4.29)

On the other hand, the mean field approach is correct if the correlation volumecontains many different chains; otherwise one should use the renormalization groupmethod [3, 133, 134] in order to describe the situation in a consistent way. Thecorrelation radius of the brush is of the order of [2]

ξ ∼ a√cν(1 − 2χ)

∼ a

(ba2

ν

)1/4N1/4

(1 − 2χ)1/4. (4.30)

4.2. Straight molecule with two types of side chains 69

Therefore the segments’ concentration in the blob of the size ξ is

c0 ∼ g(ξ)

ξ3∼ 1

a2ξ. (4.31)

This concentration should be less than the average concentration c inside the brush,c0 < c, which also implies that the correlation volume contains many thermal blobs.The last inequality yields

1 − 2χ <(ab

)4/3(ba2

ν

)N−1/3 . (4.32)

Range of the parameter values satisfying equations (4.29) and (4.32) corresponds tothe “marginal semidilute region” in the terminology of Schaefer et al. [135].

Thus using (4.29) and (4.32) we find that the mean field approach can be appliedif ν/a3 1, i.e. for chains with a large Kuhn segment.

Finally, note that the situation under θ-conditions can be considered using thescheme above starting from (4.1) with the interaction part including the third virialcoefficients

F(θ)int =

16w

3∑

i=0

ciAc3−i

B + vABcAcB .

The result for the transition point is

χ∗ABN ∼ N1/3 .

The exponent in this scaling law is less than that for the marginal solvent (4.27).

4.2.1d. Straight molecule in a poor solventHaving described the comb copolymer molecule in a good solvent, we now turn to thecase of a poor solvent. One can argue that even for high grafting densities the straightbrush regime may be inaccessible under poor solvent conditions. Nevertheless wewill restrict ourselves to the straight conformation of the molecule assuming thebackbone to be a quite stiff persistent chain.

Basically the same approach can be used except for the expression for the inter-action energy (4.2). Under poor solvent conditions solvent molecules practically donot penetrate inside the side chains corona around the backbone. This means that(4.2) should be modified accordingly

Fint =

∫ 2π

0

∫ ∞

0r dr νχABcA(r, ϕ) cB(r, ϕ) , (4.33)

70 Chapter 4

which resembles more the melt situation.

Separated state

Assuming as before that ψ-part of the angle space is occupied by A-chains andusing the normalization condition for cA(r, ϕ) given in the form

cA(r, ϕ) =

cA −πψ < ϕ < πψ, r0 < r < RA

0 otherwise,(4.34)

where cA = const and r0 is some cut-off parameter (corresponding to the radius ofthe backbone), one gets

cA =N

2πψbR2A

. (4.35)

The radius RA follows from the incompressibility condition cA = 1/ν as

RA =

(Nν

2πψb

)1/2

. (4.36)

The free energy as a functional of the chemical potentials and concentrations can bewritten in a form very similar to (4.8)

F′sep =N2

(EA + EB) − 12

lnψ − 12

ln(1 − ψ)

+∑

α=A,B

√

62a

∫ Rα

r0

dr√µα(r) − Eα − 2πψαb

∫ Rα

r0

r dr µα(r)cα

. (4.37)

Here we used again the WKB and Alexander-de Gennes approximations. Despitethe fact that the situation in a poor solvent is more similar to a melt than to the goodsolvent condition we can still employ the Alexander-de Gennes approximation (thehigh grafting density still ensures strong stretching of the side chains). In contrast to(4.8) the only free functions in (4.37) are µA and µB. They follow from δF′ /δµα = 0and have the form

µα(r) = Eα +38

R4α

a2N2r2. (4.38)

Substituting (4.35), (4.36) and (4.38) into the free energy (4.37), ψ = 1/2 is foundfrom the minimization. As a result the free energy reads

F′sep =3ν

8πa2bln

Nν

πbr2o

+ ln 2 . (4.39)

4.2. Straight molecule with two types of side chains 71

The parameter r0, which has been introduced to avoid singularities in the free energy,corresponds to the bare backbone radius and can be estimated as r0 ' b.

Mixed state

We now consider a homogeneous mixture of A and B chains inside the regionof radius RA around the backbone. Using the same arguments as before one obtainsconcentrations

cA = cB =N

2πbR2A

(4.40)

and radius of the brush

RA =

(Nνπb

)1/2

, (4.41)

which have to be plugged into the free energy

F′mix =N2

(EA + EB) + 2πbν∫ RA

r0

r dr χABcAcB

+∑

α=A,B

√

62a

∫ Rα

r0

dr√µα(r) − Eα − 2πb

∫ Rα

r0

r dr µα(r)cα

. (4.42)

Following the same procedure as before we end up with the free energy for the mixedstate

F′mix =3ν

8πa2bln

Nν

πbr20

+N4χAB (4.43)

Transition point

Now we are ready to compare free energies (4.39) and (4.43) to obtain the tran-sition point χ∗AB. In the case of a poor solvent χ∗AB is completely determined by theentropy threshold

Nχ∗AB = 4 ln 2 ' 2.8 . (4.44)

Of course the consideration above is valid in the framework of the assumption aboutapproximately circular shape of the molecule’s cross section, providing the smallestpossible area of the contact with the solvent. This obviously implies a very poorsolvent for both species.

The above result has a form which is typical for theories of microphase separationin melts [105, 106, 136], but the numerical value is somewhat smaller. It is quite easy

72 Chapter 4

to understand: the entropy loss due to the constraining motion of only one end ofthe side chain to get the segregated structure (the second one is fixed anyway in thegrafting point) is much less than the loss of the translational motion entropy, forinstance, of a diblock molecule restricted to be inside the domain (micelle, lamella,etc.).

4.2.2. Scaling approach – good solvent

At the end of this section we show how to estimate the critical χ∗AB-parameters usingsimple scaling arguments. The transition point from the mixed to the separated statecan be found from the condition that the energy of the A-B contacts per side chain∆E is of the order of kBT , ∆E ∼ 1(kBT ). This energy is given by

∆E ∼ N p(φ) χAB . (4.45)

Here p(φ) is the probability of the A-B contact, which depends on the average volumefraction φ of the monomers inside the brush. According to the mean field approachp(φ) ' φ where

φ ∼ νN/(bR2) (4.46)

and R is the radius of the brush. The last one is proportional to N3/4 for a marginalsolvent, (4.24), N2/3 for a θ−solvent (see [66], for instance) and N1/2 for a poorsolvent. Applying formula (4.45) for each of these three cases we obtain as scalinglaw for a marginal solvent

χ∗ (m)AB ∼ N−1/2 (4.47)

for a θ-solventχ∗ (θ)

AB ∼ N−2/3 (4.48)

and for a poor solventχ∗ (poor)AB ∼ N−1 . (4.49)

Clearly, all three results eqs (4.47), (4.48) and (4.49) are consistent with the mean-field calculations presented in the previous subsections.

However, the situation is different for a good solvent. The probability of contactin this case is smaller and is given by [1, 2]

p(φ) = φ5/4 ∼ N−5/8 . (4.50)

Note, here we used an equation for the radius of the brush in good solvent R ∼ N 3/4

(see [66]). Hence using (4.45) we find that the critical point for a good solvent is

χ∗ (good)AB ∼ N−3/8 . (4.51)

4.3. Bending effects – SCF approach. 73

4.3. Bending effects – SCF approach.

In this section we relax the straight backbone constraint and generalize the approachto calculate the free energy of a bent comb copolymer molecule. Our goal is to obtainthe free energy as a series in the small parameter 1/R, where R is the radius of thebackbone’s curvature. According to [121, 137], in the absence of the linear term in theexpansion, the persistence length of the molecule is determined by the second orderterm (the 0-th term corresponds to the straight comb). This is the normal situation fora symmetrical system where the persistent bending mechanism works.

However, for an asymmetric molecule (such as a microphase separated brush) thelinear term in the expansion is responsible for the presence of a spontaneous curvatureleading to the stability of bent conformations. Similar effects in two-dimensional con-formations of "simple" molecular bottle-brushes have attracted considerable attentionrecently [16, 31, 71, 81].

Let us consider therefore a bent comb molecule with completely separated sidechains (i.e. far above χ∗AB transition point). General ideas about how to constructthe free energy functional for this case can be found in [68]. We follow the sameapproach and introduce a new set of functions for the perturbed (i.e. bent) state of themolecule

c∗α(r, ϕ) = cα(r) + δcα(r, ϕ)

µ∗α(r, ϕ) = µα(r) + δµα(r, ϕ)

E∗α(ϕ) = Eα + δEα(ϕ)

g∗α(r, ϕ) =

θα(ϕ)

4πψ b rδ(r − Rα) −πψ < ϕ < πψ0 otherwise.

(4.52)

Here a system of coordinates has been introduced with z-axis along the backbone;(r, ϕ) are polar coordinates in the cross section; Eα, cα, and µα are given by theexpressions for the straight backbone.

Note that the free end distribution function is now written in the form of theperturbed Alexander-de Gennes approximation. This correction θα(ϕ) can be foundfrom the self-consistency equation

θα(ϕ) = Bαc∗α(Rα, ϕ) (4.53)

with constants Bα (α = A, B).

74 Chapter 4

Employing the WKB approximation one can write the free energy of the bentcylindrical brush molecule in the form F∗ = F∗A + F∗B

F∗A =1

4πψ

∫ πψ

−πψdϕ

(1 +

RA cos ϕR

)θA(ϕ)×

×NAE∗A(ϕ) +

√6

a

∫ RA

0dr

√µ∗(r, ϕ) − E∗(ϕ) + ln

θA(ϕ)ψ

+ 2πbψ∫ RA

0r dr

(1 +

r cos ϕR

) (−µ∗A(r, ϕ)c∗A(r, ϕ) +

vAA

2c∗2A (r, ϕ)

). (4.54)

and similar for F∗B.Using the self-consistency condition for the concentration c∗α in the form of (4.9),

the chemical potential has to be found from the equation√

64a

vAA√µ∗A(r, ϕ) − E∗A(ϕ)

=2πψθA(ϕ)

br1 + r cos ϕ/R

1 + RA cos ϕ/Rµ∗A(r, ϕ) , (4.55)

which is solved according to the perturbation scheme (4.52).Omitting the details of the calculation we give the result for the corrections to the

free energy due to the bending.

F∗ = F0 +F(I)

R+

F(II)

R2

F(I) ' − 0.004

(N5

a2b3v

)1/4

∆v

F(II) ' 0.1N2v

b. (4.56)

Here we give only the main corrections to the the straight brush free energy F0,assuming ∆v = vBB − vAA to be small. As one can see, the straight conformation isno longer favorable and a spontaneous curvature R0 = −2F(II)

/F(I) will occur (see

Figure 4.3). An estimation of R0 from (4.56) gives

R0 ' 50

(a2vN3

b

)1/4v

∆v. (4.57)

Finally, rewriting (4.57) in terms of χ-parameters results in

R0 ∼ (1 − 2χ)5/4

∆χN3/4 . (4.58)

4.3. Bending effects – SCF approach. 75

Figure 4.3: Schematic representation of a comb copolymer molecule in theseparated state.

Of course, (4.58) is valid well above the θ-point in the good solvent regime. Inprinciple the same derivation can be done under θ-conditions by introducing the thirdvirial coefficient.

Qualitatively (4.57) can be obtained from simple arguments too. Indeed uponbending the cylinder corresponding to the straight comb molecule gets distorted andassumes a toroid shape, Figure 4.3. This implies that the volume of the A chainsoccupying the outer part (we assume ψ ' 1/2) of the torus is (corresponding to a unitof the length b along the backbone)

Vout = b

πR2

A

2+

2R3A

3R

(4.59)

and similarly for the inner (occupied by B) part, but with the second term subtracted.Knowing that the free energy has to scale as

F ∼ vAAVout c2A + vBBVin c2

B (4.60)

with concentrations cA ∼ N/Vout and cB ∼ N/Vin, one finally obtains

F∗

F0∼ 1 +

vBB − vAA

vBB + vAA

RA

R+

R2A

R2. (4.61)

Now it is easy to see from the minimization of (4.61) that the spontaneous curva-ture scaling law is

R0 ∼ RAv

∆v, (4.62)

76 Chapter 4

which coincides with (4.57).In the closure of the present section we would like to emphasize that the spon-

taneous curvature (4.58) in the present model is the result of an interaction betweennearest neighbors (compare to [123] where toroidal shape of the conformation is theresult of an attraction between neighbors that are far apart, causing a coil-globuletransition). In some sense our situation resembles the one considered in [31] for 2D,where asymmetry of adsorbed comb copolymer conformations was claimed to becaused by stochastic processes during the adsorption. On the contrary, in our case thecurvature (4.58) is an intrinsic property of the molecule in the separated state, andthe only reason for it is the difference in the interaction parameters χAS and χBS oftwo chemically different species with the solvent.

4.4. Concluding remarks

In the present chapter we considered a comb copolymer molecule with two types ofside chains attached to a common backbone. We found that the spatial separation ofthe side chains within the molecule is possible if the A-B interaction parameter takesvalues larger than a predicted χ∗AB. This point has been calculated as a function of thesolvent quality resulting in the following scaling laws

χ∗ABN ∼ N5/8 good solvent,

χ∗ABN ∼ N1/2 marginal solvent,

χ∗ABN ∼ N1/3 θ-solvent,

χ∗ABN ∼ 1 poor solvent.

The exponent value gradually decreases from 5/8 to 0 when going from good topoor solvent conditions. The result for a poor solvent is typical for all microphaseseparation theories in melts [105, 106, 136]. It should be possible to observe thedescribed phenomena in an experiment choosing the appropriate length of the sidechains, because χ∗AB scales as N−3/8 under good solvent conditions, and hence, smallpositive values of χAB-parameter can already lead to the transition.

In the present work we considered a comb molecule with alternating chemicallydifferent side chains. In many cases the synthetic procedure will lead to chemicallydifferent side chains distributed statistically along the backbone. The appearanceof large groups of “like” chains is then possible, which may change the separationpicture drastically: a separation along the backbone may occur rather than in the cross

4.A. Relation between v-parameters and Flory-Huggins χ-parameters. 77

section. But, if the statistical distribution of the side chains is so that formation ofthe large groups (containing about RF/b side chains) is improbable, then the obtainedresults remain valid at least qualitatively.

It would be interesting to look at the system in a selective solvent (for instance,good for A and poor for B). In the "mixed" state, one can expect to find highlystretched A side chains and B chains collapsed onto the backbone. For large enoughχAB (corresponding to the "separated" state) one can expect the formation of evenmore peculiar shapes: for example, the B-chains can form a dense cylinder twinedby the backbone with A side chains pointing radially into the solvent. Of course,this system is out of the scope of this work in the sense that the methods and theapproximations used will not be valid in that region (it was assumed throughout thatthe radii of A and B brushes are close to each other).

In the previous section we described the comb copolymer molecule in the sepa-rated state. It was shown that a spontaneous curvature (see Figure 4.3)

R0 ∼ N3/4

∆χ

should occur as a result of the difference in the osmotic pressures in the spatiallyseparated A and B phase. In the our model the χ-parameters were responsible forthe bending, but in principle, the same effect can be achieved if other characteristics(molecular weight, grafting density) of the A and B side chains differ.

Appendix

4.A. Relation between v-parameters and Flory-Huggins χ-parameters.

Interaction parameters vαβ, (α, β = A, B) used in the current work have the meaningsimilar to the second virial coefficients in pair A-A, B-B and A-B interactions. Theycan be easily related to Flory-Huggins χ-parameters for polymer-solvent χAS , χBS

and polymer-polymer χAB interactions. In the framework of the Flory approach themixing energy should be written [1] in the form (kBT ≡ 1)

Emixing =

∫

V

dVν

[ΦS ln ΦS + χAS ΦAΦS + χBS ΦBΦS + χABΦAΦB

]. (4.63)

78 Chapter 4

Here Φα denotes the specific volume occupied by α-component (A-polymer, B-polymeror solvent) ν represents the volume of the monomeric unit (bead).

Substituting incompressibility condition

ΦS = 1 − ΦA − ΦB

and expanding (4.63) one gets (omitting constants and terms linear in ΦA and ΦB)

Emixing =

∫

V

dVν

[12

(1 − 2χAS )Φ2A +

12

(1 − 2χBS )Φ2B

+ (1 + χAB − χAS − χBS )ΦAΦB

]. (4.64)

Knowing that Φα = νcα one can easily obtain for vαβ

vAA = ν(1 − 2χAS )

vBB = ν(1 − 2χBS )

vAB = ν(1 + χAB − χAS − χBS ) . (4.65)

CHAPTER 5

Self-Organization of Hairy-Rod Polymers

Self-organized structure formation in the melt of hairy-rod polymersis analyzed theoretically. It is shown that the interplay between unfa-vorable repulsive rod–coil interactions and stretching of the side chainsis responsible for the appearance of three different microphases: onelamellar and two hexagonal. The first-order phase transitions betweenthese are considered in detail. If the side chains are long enough forthe elastic stretching free energy to dominate the repulsive interactions,hexagonally ordered domains of hairy-rod cylindrical brushes are formed.The lamellar phase is shown to be stable for shorter side chains andoccupies an important part of the phase diagram. In the intermediateside chain length regime another hexagonally ordered structure appears,characterized by cylindrical micelles with an elongated cross section,containing several hairy-rod polymers.

5.1. Introduction

A hairy-rod polymer [39–41] is one of the representatives of the comb copolymerfamily (see Figure 1.1e on page 2), consisting of a rigid backbone with a densesystem of side chains. As other block copolymers, they tend to self-organize and

79

80 Chapter 5

form nanoscale structures in bulk and in solutions.In particular in the context of electrical conductivity such architectures received

a lot of attention (see [42] and references therein), since there the backbones consistof conjugated rigid polymers. Besides block copolymeric self-organization, thereis an even more important reason why hairy-rods have attracted so much interest. Ingeneral rigid-rod-like polymers do not melt and dissolve only poorly, if at all, in com-mon solvents due to a strong aggregation tendency and a small gain of conformationalentropy upon dissolution or melting. By covalent connection of substituent groups(notably flexible alkyl chains) to the backbone, a system is achieved where the rigidpolymer can be regarded to dissolve in the background of the side chains due to the(infinite) attractive interaction between the solvent molecules and the backbone. Thiscauses melting point depression [138, 139] without loss of rigidity of the backbone[39]. In this way fusibility can be achieved and higher solubility in an organic solvent,to allow melt and solution processability. The nature and length of the covalentlybonded side chains have a large effect on the phase behavior. A wide variety ofdifferent modifications exists, depending on the selection of the backbone and sidechains (see the reviews by Ballauff [39] and Menzel [40]).

The microstructure formation itself has received considerable attention experi-mentally – a summary of the experimental results has been presented in section 1.2.2.Generally it is observed that the hairy-rod polymer systems tend to self-organizedue to microphase separation between the backbone and the side chains, if the sidechains are long enough. A layered structure is found in the majority of the cases,however, the length of the side chains involved was relatively small. Strange enough,almost no theoretical works are avalable to explain the observed effects relevant tothe microphase separation (see section 1.4.2 for a brief literature survey).

It is the objective of this chapter to address theoretically the microstructure for-mation driven by the repulsion between the stiff polymer backbone and the flexibleside chains of any length.

5.2. The model

We consider a melt of hairy-rod polymers with a backbone of length L and diameterd (L d) bearing M flexible side chains equidistantly grafted with a density 1/b(b = L/M) as depicted in Figure 5.1. Each side chain is a coil consisting of N beadsof volume ν and statistical segment length a, so that the unperturbed coil size isRc = a

√N (here we assume Rc L). This implies that the composition f , volume

5.2. The model 81

Figure 5.1: Model of the hairy-rod.

fraction of the backbone, is related to the side chain length by f = 1/(1 + κN), whereκ = ν/(πd2b/4) is the ratio between the volume of a side chain bead and the volumeof the backbone between two successive side chains.

In the limiting case of N = 0 (no side chains) the system obviously forms a ne-matic phase consisting of strongly oriented backbones, which have orientational freeenergy [140, 141] F Nem

or ' T ln(4π/Ω0), where Ω0 is the characteristic fluctuationangle.

In the presence of side chains each rod is surrounded by flexible coils. Theinteraction between rods and coils can be introduced in the following way. If bothwere not linked to each other, they would macrophase separate into almost purephases with a sharp interface in between, Figure 5.2, so that virtually no coils arepresent in the rod-rich phase and vice versa [142, 143].

The interface is characterized by interface tension γ. In order to define theinterfacial energy for one rod surrounded by flexible coils we use the followingapproximation

F (1)γ ' 2Ldγ. (5.1)

where the prefactor 2 reflects the fact that the rod’s contact area is twice as large asin the case of the flat interface (see Figure 5.2). The interfacial tension γ is relatedto the conventional Flory-Huggins χ-parameter and can be calculated analyticallyunder certain conditions [106, 144]. Here we treat γ as known and also assumelimT→∞ γ/T = Γ∞ > 0.

In general the free energy of the hairy-rod system, i.e. covalently bonded rodsand coils, can be written as a sum of three terms (per molecule)

F = Fγ + Fel + δFor. (5.2)

Here we take as a reference state the corresponding sum of the free energies of the

82 Chapter 5

Figure 5.2: Illustration of the interface free energy calculation.

pure nematic phase and the pure melt of flexible coils. Therefore, the last term canbe expressed as δFor ' T ln(Ω0/Ω). Since the new characteristic fluctuation angle Ω

is close to Ω0, δFor is small (in all phases we consider here backbones are aligned).In this way the free energy of the system consists of the rod-coil interfacial tensionFγ part and the elastic stretching energy of the side chains Fel:

F = Fγ + Fel. (5.3)

Due to unfavorable interaction between rods and coils the system tends to form rod-and coil-rich domains. Depending on the geometry of these domains also the energyof elastic stretching is changed. Thus the interplay between Fγ and Fel determineswhich geometry of the micelle (e.g. cylindrical or lamellar) is more favorable.

5.3. Phase equilibria between microphases and nematic

In this section we address the question about the different phases that can appear inthe melt. Let us start from a relatively small γ/T (or large N), such that the elasticstretching plays the dominant role. Then the cylindrical micelles will contain a singlehairy-rod molecule in its cross section. The cylinders should be closely packed andthus arranged in a hexagonal lattice, Figure 5.3a. In the opposite case of very highγ/T the elastic stretching is negligible. Therefore, to decrease the Fγ contribution,the system adopts a layered structure, Figure 5.3c, with each lamella being a doublelayer of rods. In the intermediate regime another hexagonal structure H2 with Q > 1molecules per micelle may be realized, Figure 5.3b.

In the following subsections we address the competition and coexistence betweenthese phases quantitatively.

5.3. Phase equilibria between microphases and nematic 83

a b c

Figure 5.3: Possible microstructures: (a) hexagonal H1; (b) hexagonal H2;(c) lamellar.

5.3.1. Hexagonal H1 phase

Here we start from the relatively small γ and large κN 1 case. Then the backboneoccupies only a small fraction of the total molecule’s volume, i.e. each hairy-rodforms a densely grafted cylindrical brush. To fill space, these brushes reside in ahexagonal lattice (Figure 5.3a). The free energy of the system equals FH1 = Fγ +

Fel where the first term is Fγ ' 2Ldγ due to the fact that each rod is completelysurrounded by coils.

The elastic stretching energy of the side chains can be calculated by the methodpresented in [68, 72] and reads

Fel = T3κd2

32a2M ln(κN) . (5.4)

In expression (5.3) we omitted the loss of translational energy and the interactionbetween the hairy-rod polymers because both are relatively small. The latter arisesfrom the free ends distribution, which leads to penetration of the side chain endsbelonging to one hairy-rod polymer into the coronas of its neighbors [106].

At this point it is useful to introduce a renormalized free energy F = F /(MT ).For the hexagonal phase consisting of separate hairy-rod cylindrical brushes, calledhexagonal H1, this free energy equals

FH1 = 2bdγT

+3d2

32a2κ ln(κN) . (5.5)

This expression allows us to identify the condition under which H1 will be found. It

84 Chapter 5

corresponds to the situation where the elastic tension of the side chains is larger thanthe “interfacial term” so that the rods are prevented from sticking together.

The lattice parameter ` can be estimated from the volume fraction condition f ∼d2/`2 and reads

` ∼ d/√

f . (5.6)

5.3.2. Hexagonal H2 phase

An increase of the γ/T parameter results in a tendency of the backbones to adopt apacking with a smaller total area of contact with side chains, Figure 5.3b. Assumingthat the coils can not penetrate into the core of a cylindrical H2 type micelle, thecore must have a double layer structure of closely packed backbones with surfaceQdL + 2dL (valid for Q ≥ 2), where the second addend is due to edges. For acore size smaller than the corona thickness, i.e. Q <

√N, the micelles still have

an approximately circular cross section so that in a good approximation (5.4) witha renormalized grafting density can be used for the elastic stretching. Therefore thefree energy per molecule in the H2 phase is given by

FH2 =bdγT

(1 +

2Q

)+

3d2

32a2κQ ln(κN) (5.7)

and the number of rods per unit cell Q follows from the minimization ∂FH2/∂Q = 0,i.e.

Q =

√2bdγ/T

3d2

32a2 κ ln(κN). (5.8)

This yields the following expression for the free energy of the hexagonal H2 phase,under the condition that the cylinders have a cross section that is still quite close tocircular

FH2 =bdγT

+ 2

√2bdγ

T3d2

32a2κ ln(κN) . (5.9)

At this point the first order H1-H2 transition can be identified and its temperatureobtained from the comparison of (5.5) and (5.9). It gives the following equation forthe H1-H2 coexistence line presented in Figure 5.4

bdγT

= (3 + 2√

2)3d2

32a2κ ln(κN) . (5.10)

5.3. Phase equilibria between microphases and nematic 85

Figure 5.4: Phase diagram of the melt of hairy-rod molecules. H1-H2coexistence line is given by (5.10); H2-L (solid part) corresponds to (5.19).

Using (5.8) we conclude that the average number of molecules per micelle oflength L is Q1 = 2 +

√2 ' 3.4 when H2 first appears. Hence, the cylindrical domain

has 3-4 backbones in its cross section. This small number is consistent with theapproximation used concerning the almost circular cross section. The period of thestructure is of the order of ` ∼ d

√Q/ f .

5.3.3. Equilibrium between lamellar and hexagonal H2 phase

Following the same reasoning as before one concludes that at high enough γ/T valuesa lamellar phase should appear. Apparently it is the one with minimal rod-coil contactarea. Its free energy is given by

FL =bdγT

+3d2

32a2π2κ2N , (5.11)

where the second term represents the elastic stretching of the side chains. It can beeasily obtained from the stretching free energy of a coil in a dense planar brush [1, 2]F1 = 3T H2/(2Na2). Here

H = πdκN/4 (5.12)

is the thickness of the side chain layer corresponding to one lamella.The lamellar-hexagonal H2 transition condition is obtained by equating the free

energies of the lamellar phase (5.11) and the hexagonal phase. However, (5.9) is

86 Chapter 5

Figure 5.5: Cross section of a hexagonal H2 phase cylindrical micelle inthe vicinity of the H2-Lam transition. Note the total cross section is still

close to circular.

not suitable for this purpose since it was written under the assumption that the H2micelle’s core has almost circular cross section. This is definitely not the case any-more in the vicinity of the L-H2 transition (see Figure 5.5). The elastic part of thefree energy can not be approximated by a (5.4) type expression but should rather becalculated starting from a planar sheet and introducing appropriate edge corrections.The essential element of our approach [144] is the assumption that in the frameworkof the Alexander–de Gennes approximation [94], all chains propagate outwards alongstraight lines. Near the edges of the micelle these lines are tilted with an angle α(x)(see Figure 5.5). The free end of the chain is located at the distance R(α) along thisline. The two functions are related through the incompressibility condition

dαdx

=2

R2(H − R sinα) . (5.13)

Here H is the end to end distance of the coil far from the edge, i.e. given by (5.12),and x is the distance from the edge of the core consisting of backbones.

From this the free energy of one stretched coil can be derived (see [144] fordetails)

F1 =3T2

ν

a2bd

ln(1 + R dα/dx

sinα

)

dα/dx(5.14)

5.3. Phase equilibria between microphases and nematic 87

and the total edge correction per one hairy-rod reads

Fedges =4MTQd

∫ Qd4

0(F1(x) − F1∞) dx , (5.15)

where F1∞ = limx→∞ F1 = 3νH/(2a2bd). Using equations (5.13)-(5.15) this correction

can be written in the form (c.f. equation (33) from [144])

Fedges =6MTQd

νH2

a2bd2

∫ π/2

0

ρ2 dα2(1 − ρ sinα)

ρ2 ln

(2

ρ sin α − 1)

2(1 − ρ sinα)− 1

(5.16)

with a new function ρ(α) = R(x(α))/H, which should be found from numerical min-imization. As it turns out, at the minimum the integral has a value of approximately1/2 and as a result

Fedges ' −T3d2

32a2

π3κ3N2 M2Q

. (5.17)

Therefore, the elastic free energy of the H2 phase near the H2 to lamellar phasetransition is approximately equal to

F′ (el)H2 =

3d2

32a2π2κ2N − 3d2

32a2

π3κ3N2

2Q(5.18)

and the total free energy now reads

F′H2 =bdγT

(1 +

2Q

)+ F′ (el)

H2 .

Using the equilibrium condition FL = F′H2 we arrive at the expression for the lamellar-hexagonal H2 coexistence line

bdγT

=3d2

32a2

π3κ3N2

4. (5.19)

Note that the number of rods per cylinder in the H2-phase dropped out from the result(5.19) without minimization. This fact is connected to the approximation (5.18),which in a sense can be viewed as an expansion in the small parameter 1/Q. So, toobtain the number of rods Q2 at the H2-Lam transition consistently one has to extendthe approximation (5.18). However Q2 can be also estimated from (5.18) when theedge correction becomes of the order of the main term

Q2 ∼ κN . (5.20)

88 Chapter 5

5.3.4. Equilibrium between nematic and lamellar phases

In the section 5.2 we started from the κN 1 case. The phase equilibrium linebetween the two hexagonal phases obtained, (5.10), can be extrapolated into the κN &1 region. This gives for the transition point in the T → ∞ limit (see Figure 5.4)

κN2 ' exp

[32ba2Γ∞

3(3 +√

2)κd

]. (5.21)

The H2-L coexistence line is prolonged into the region of κN & 1 as well, yielding inthe same limit

κN1 '√

128ba2Γ∞3π3κd

(5.22)

for Γ∞ > Γ∗∞ = 3π3κd/(128ba2 ). Unfortunately no simple extrapolation can be madein the Γ∞ < Γ∗∞ case.

Hence, only the case κN < 1 has not been considered yet. This regime ischaracterized by a small fraction of side chains implying that most probably a ne-maticly ordered structure should appear. We introduced the γ-part of the free energyassuming that a domain of backbones (cylindrical or lamellar) is immersed in aflexible surrounding (see (5.1) and similar). In the κN 1 case the situation isreversed: now the coils form the minority component, occupying the empty spaces inbetween rod-like backbones. Let us assume that one coil in the bulk of the nematicphase loses an energy

µc ' 2σNγ , (5.23)

where σ is a characteristic cross section area of a monomer, σ ∼ ν2/3. Then theenergy of the nematic phase can be written as

FN = 2σNγ

T, (5.24)

where we omitted the translational entropy term because of its smallness.Now we keep γ/T constant and gradually increase the length of the side chains,

so that κN approaches unity. Due to the increasing volume fraction of the coils in thesystem, at certain point around κN ∼ 1 a microscopically segregated state, notablylamellar, has to become favorable (see Figure 5.4).

It is hardly possible to make more a quantitative prediction within the frameworkof the simple model discussed. For this purpose expressions are needed for thenematic and the lamellar phase in the very vicinity of κN = 1. Therefore the lineseparating N and L regions in Figure 5.4 should be considered a qualitative estimationrather than a quantitative result.

5.4. Discussion and concluding remarks 89

5.4. Discussion and concluding remarks

In this chapter we have shown that a melt of hairy-rod polymers exhibits interestingphase behavior resulting from the balance between unfavorable rod-coil interactionand elastic stretching of the side chains. The main results are summarized in thephase diagram Figure 5.4.

We showed that for long enough (κN > 1) side chains the hairy-rod polymerstend to self-organize and form microstructures. Depending on the temperature, orequivalently on the strength of the rod-coil repulsion parameter, the system finds itselfin one of three possible microphases. At relatively high temperatures hexagonallyordered domains are formed (regions H1 and H2 in Figure 5.4) with one or severalmolecules per cylindrical micelle. This is explained to be a consequence of therelatively weak repulsion between rigid and flexible polymer, which cannot outweighthe elastic stretching part of the free energy. A prominent place in the phase diagrambelongs to the lamellar morphology. In the case of long side chains (κN 1) it isproven to be stable for low enough temperatures where the surface tension term startsto play the decisive role.

For somewhat shorter side chains the lamellar phase becomes the only possibleone, implying that no hexagonal structures can be found in this region. This servesas a possible explanation for the fact that the lamellar phase has been found exper-imentally in the majority of cases studied, where the side chains are relatively short[39, 41, 43, 46, 47, 50, 145, 146].

In the last part of the chapter we argued that a nematic phase should appear forκN < 1. In practice this might correspond to either a nematic or a crystalline phase.

Several experiments [50, 146] report order-disorder transitions. However, thistransition lies beyond our approach due to the assumption about absence of rotationaldisordering (see e.g. (5.2)) used in the discussion from the very beginning.

CHAPTER 6

Phase Behavior and Structure Formationof Hairy-Rod Supramolecules

Phase behavior and microstructure formation of rod and coil molecules,which can associate to form hairy-rod polymeric supramolecules, areaddressed theoretically. Association induces considerable compatibilityenhancement between the rod and coil molecules and various micro-scopically ordered structures can appear in the compatibility region. Theequilibria between microphase separated states, the coil-rich isotropicliquid and the rod-rich nematic are discussed in detail. In the regimewhere hairy-rod supramolecules with a high grafting density appearas a result of the association, three phase diagram types are possibledepending on the value of the association energy. In the low graftingdensity regime only the lamellar microstructure is proven to be stable.

6.1. Introduction

Supramolecular chemistry allows synthesis of highly specific complexes, includinghairy-rods, where chemically different groups are connected by means of a reversiblebond (see introductory section 1.2.2). In turn, the supramolecules, built in this way,are able to form a hierarchy of structures, resembling in this respect the considered

91

92 Chapter 6

above covalent hairy-rods. However, the supramolecular systems show much richerphase behavior involving competition between microstructures and homogeneousmacrosphases.

In the previous chapter we presented a theoretical analysis of structure formationin melts of hairy-rod polymers (see also [147]). We showed that three differentmicrophases are possible: one lamellar and two hexagonal. If the side chains are longenough for the elastic stretching free energy of the side chains to completely dominatethe repulsive interaction between the backbone and the side chains, hexagonallyordered domains of hairy-rod cylindrical brushes are formed. The lamellar stateis found to be stable for shorter side chains and occupies an important part of thephase diagram. In the intermediate side chain length regime a hexagonally orderedstructure appears characterized by cylindrical micelles with an elongated core crosssection and containing several hairy-rod polymers.

Supramolecular hairy-rods represent a more complicated system. Its phase be-havior is primarily determined by an interplay between association and chemicalincompatibility of the components. In a weakly associating system a macrophaseseparation should prevail over microstructure formation. In contrast, if the associ-ation is strong enough, the appearing supramolecules are expected to self-organizein microstructures, which can be described in terms similar to covalent hairy-rods.Thus, the first objective of this chapter is to address the complex equilibrium inthe associating rod – coil system and reveal the conditions, under which the self-organization becomes possible. Further, we aim to consider the competition betweenappearing micro- and macrophases in order to find their stability regions.

6.2. The model

We consider a melt consisting of rigid rods of length L and diameter d (L d)and flexible coils consisting of N beads of volume ν and statistical segment length a.The ideal coil size is Rc = a

√N, L Rc. We will assume that each rod contains

M associating groups (average distance between two successive groups is b = L/M)which can form bonds with the associating end of the coil (Figure 6.1a). It is assumedthat each coil has only one associating end. The energy of association between rodand coil equals −ε. The concentration of rods in the melt is c and their volumefraction is f = (π/4)Ld2c. The interaction between rods and coils can be introducedin the same way as in the previous chapter. It is well known that rods and polymercoils in the molten state are practically incompatible and separate into a nematic

6.2. The model 93

a b

Figure 6.1: (a) Model of the hairy-rod as a stiff backbone with reversiblyattached flexible side chains. (b) Flat interface between pure nematic and

isotropic phase.

phase consisting of rods and an isotropic phase consisting of the flexible polymer[142, 143]. The interface between the nematic and isotropic phase (Figure 6.1b),is assumed to be sharp and the interfacial tension γ corresponding to the planarorientation of rods at the interface is given by

γ = (w + sT )/d2 , (6.1)

where w is the energetic part of the surface energy and s > 0 is the entropic part(here T is temperature in energetic units). According to the definition (6.1), if a rodpenetrates into the polymer melt its free energy loss is approximately equal to (c.f.formula (5.1) from the previous chapter)

µr ' 2Ldγ . (6.2)

Therefore the free energy of the isotropic phase with a small amount of rigid rods isgiven by

F ∗I = TVc ln

(fe

)+ TV

1 − fNν

ln

(1 − f

e

)+ Vc µr . (6.3)

Here we omitted the interaction between the rods. V is the volume of the system. In(6.3) the first two terms represent the translational entropy of the rods and coils.

94 Chapter 6

The coils can also penetrate into the nematic phase. In order to write the freeenergy of the nematic phase with a small amount of coils we introduce the chemi-cal potential of the coil in the nematic phase µc which includes both energetic andentropic contributions. Further on we consider the limit

µc/T → ∞ (6.4)

for arbitrary T . This means that the coils practically do not penetrate into the nematicphase.

The free energy of the nematic phase contains also a term connected with orien-tational ordering of rods. It can be estimated [140, 141] as T ln(4π/Ω), where Ω isthe characteristic fluctuation angle, Ω ' 2π(d/L)2 . Thus the free energy is given by

F ∗N = TVc ln

(fe

)+ TV

1 − fNν

ln

(1 − f

e

)+ 2TVc ln

(Ld

)+ V

1 − fNν

µc. (6.5)

The equilibrium between the nematic and isotropic phase can be found in the usualway by equating the chemical potentials and osmotic pressures in both phases.

µ∗I = µ∗N ; µ∗I,N =1V

∂F ∗I,N∂c

P∗I = P∗N ; P∗I,N =1V

c∂F ∗I,N∂c

− F ∗I,N . (6.6)

The solution of these equations is given by

fN ' 1 − κN exp(−µc/T ) ' 1, fI '(L

d

)2

exp

(−2L

d

(w

T+ s

)) 1. (6.7)

This simple phenomenological model reflects the generic feature of the rod-coilmixtures to phase separate into almost pure components [120, 148]. The correspond-ing phase diagram is shown in Figure 6.2a.

6.3. Nematic – isotropic liquid phase coexistence: Effect ofassociation

In this section we study the influence of association between rods and coils on themacrophase separation described above. We start from the free energy of association

6.3. Nematic – isotropic liquid phase coexistence: Effect of association 95

a

b

Figure 6.2: Macrophase equilibria in the associating rod-coil system: (a) nocompatibility for small association energy; (b) enhanced compatibility in the

ε/w > 2b/d case.

96 Chapter 6

between rods and coils, Fbond, assuming that they are ideal (without excluded vol-ume). Let p denote the probability that an associating group of the rod has formeda bond with a flexible coil. The total number of bonds in the system is V Mcp andequals the number of associated coils. Therefore, the number of free coils in thesystem is (V/Nν)(1 − f − f κpN), where κ ≡ ν/(πbd2/4). The free energy of bondscan be expressed through the partition function Zbond as [117, 149–151]

Fbond = −T ln Zbond , (6.8)

where

Zbond = Pcomb

(vb

V

)VMcpexp

(ε V Mcp

T

)(6.9)

and Pcomb is the number of different ways to link rods and coils for a fixed probabilityp; vb is the bond volume. If we denote the number of rods in the system as Nr = Vc,and the number of coils as Nc = V(1 − f )/Nν, then the number of ways to chooseNr Mp coils for bond formation is the binomial coefficient

CNr MpNc

=Nc!

(Nr Mp)!(Nc − Nr Mp)!. (6.10)

Moreover, there are(Nr M)!

(Nr M(1 − p))!(6.11)

different ways to select Nr Mp bonds from Nr M associating groups. Therefore

Pcomb = CNr MpNc

(Nr M)!(Nr M(1 − p))!

(6.12)

and the free energy of bonds is given by

Fbond =V Mcp

[T ln

(Nνvb

)− ε

]+ TVcM

[p ln p + (1 − p) ln(1 − p)

]

+ TV(1 − f − f κN p)

Nνln

(1 − f − f κN p

e

)

− TV(1 − f )

Nνln

(1 − f

e

). (6.13)

The free energy of the isotropic phase can be written as

FI = F ∗I + Fbond + Fel , (6.14)

6.3. Nematic – isotropic liquid phase coexistence: Effect of association 97

where Fel is the elastic free energy of the side chains of the hairy-rod, which appearsas soon as the density of association is high enough. We approximate it by [68, 72]

Fel =

TVc 3κd2

32a2 Mp2 ln (κN p) , p > 1κN

0, otherwise.(6.15)

Hence, the final expression for the free energy of the isotropic phase is given by (pervolume of one rod (π/4)Ld2)

FI( f , p)T

= fµr

T+ M f p

[ln

(Nνvb

)− ε

T

]+ f M

[p ln p + (1 − p) ln(1 − p)

]

+ f ln

(fe

)+ M

(1 − f − f κN p)Nκ

ln

(1 − f − f κN p

e

)

+ f3κd2

32a2Mp2 ln (κN p) H

(p − 1

κN

), (6.16)

where

H(x) =

1, x ≥ 0

0, x < 0

is the Heavyside function. Similarly, the free energy of the nematic phase is

FN ( f , p)T

=2 f ln(L

d

)+ M

1 − fNκ

µc

T+ M f p

[ln

(Nνvb

)− ε

T

]

+ f M[p ln p + (1 − p) ln(1 − p)

]+ f ln

(fe

)

+ M(1 − f − f κN p)

Nκln

(1 − f − f κN p

e

). (6.17)

The probability of bonding in both phases can be found from the minimization of thecorresponding free energies

∂FI

∂p= 0;

∂FN

∂p= 0 (6.18)

and reads (N∗ ≡ Nν/vb)

p =1

2κN f

[1 − f + κN f − N∗ exp (−ε/T )

−√(

1 − f + κN f − N∗ exp (−ε/T ))2 − 4κN f (1 − f )

](6.19)

98 Chapter 6

for the nematic phase and for the isotropic phase when p < 1/κN . For p > 1/κN theprobability of bonding in the isotropic phase satisfies

ln

[pN∗e−ε/T

(1 − p) (1 − fI − fIκN p)

]+

3κd2 p

16a2ln (κN p e) = 0 . (6.20)

For a small volume fraction of rods, fI 1, it is approximately given by

p ' 1

1 + N∗e−ε∗/T, ε∗ = ε − 3κd2T

32a2

1

1 + N∗e−ε/Tln

( κN

1 + N∗e−ε/T

). (6.21)

Phase equilibrium between the isotropic and nematic phase can be obtained in astandard way from the equilibrium equations1

∂FI

∂ fI=

∂FN

∂ fN

fI∂FI

∂ fI− FI = fN

∂FN

∂ fN− FN . (6.22)

using (6.16) and (6.17) together with (6.19) and (6.21).At this point we have to consider two possible situations. The first one occurs

if κN > 1 so that both pI < 1/κN and pI > 1/κN are possible. The secondone corresponds to κN < 1 where the elastic tension of the associated coils is notimportant for any p.

Let us start from κN > 1. When the probability of bonding in the isotropic phasepI < 1/κN , or equivalently ε/T < ln((κN − 1)/N∗), expression (6.19) can be usedgiving the volume fraction of rods

fN ' 1,

fI '(L

d

)2

exp(−µr

T+

M

1 + N∗e−ε/T

(ε

T− ln N∗

)) 1 . (6.23)

However, for lower temperatures pI exceeds 1/κN implying that the rods are denselygrafted. Then the volume fraction of rods in the isotropic phase f I satisfies theequation

ln fI − MpI ln (1 − fI − fIκN pI ) ' 2 ln(L

d

)+

MNκ

− MpI ln N∗ − 3κd2

32a2Mp2

I ln(κN) − 2Lsd

+MT

(pIε − 2bw

d

), (6.24)

1Expressions for the chemical potentials µ = ∂F/∂ f and partial pressures P = f ∂F/∂ f − F for allthe considered phases can be found in the Appendix 6.A.

6.3. Nematic – isotropic liquid phase coexistence: Effect of association 99

where pI has to be determined from (6.20). Obviously, pI → 1 if T → 0 and thereforethe last term in (6.24) becomes dominant. Depending on its sign two characteristicasymptotics can be distinguished

fI → 0 if ε <2bw

d,

fI → 11 + Nκ

if ε >2bw

d. (6.25)

Thus for ε/w > 2b/d rods and coils become partially compatible. This fact has aclear physical meaning. A negative sign of −ε + (2bw/d) corresponds to a negative“total” energy (ε-part plus γ-part) due to the association of a coil to a rod, i.e. makingit favorable to keep all coils bonded (for T → 0, of course). In this case (6.24) givesthe fraction of rods in the coil-rich phase and the region of the phase diagram belowthe temperature

T ∗ ' ε − 2bwd

2bsd + ln N∗ + 3κd2

32a2 ln(κN)(6.26)

reveals compatibility enhancement with the asymptotic value of the rod fraction f I →1/(1 + κN) for T → 0. This situation is depicted in Figure 6.2b: a compatibilityregion occurs at low temperatures to the left of the stoichiometric point. For smallassociation energies ε/w < 2b/d the situation is qualitatively identical to the onewithout association (see Figure 6.2a).

Now we turn to the case κN < 1, when only low grafting densities are possible: p I

is always less than 1/κN . So the term responsible for the elastic stretching should beomitted according to approximation (6.16). This yields an equations for the isotropicphase density analogous to (6.24) but without the elastic 3κd2 ln(κN)/32a2 term. Inthe same way as above, the low temperature asymptotic fI → 0 appears for smallassociation energy values corresponding to Figure 6.2a. In the opposite case of thehigh energies ε > 2bw/d (6.24) shows the tendency of fI to increase approachingstoichiometric conditions 1/(1 + κN). However, it cannot be used in a quantitativemanner because it is valid only for an isotropic phase with small amount of immersedrods (note that for κN < 1 the value of 1/(1 + κN) is not small anymore).

100 Chapter 6

6.4. Equilibria between nematic, isotropic and microphasesfor κN > 1

As shown above, the association gives rise to a partial compatibility in the regionof volume fractions up to the stoichiometric point 1/(1 + κN). The existence of thisregion was proved above under the assumption that the phase is isotropic. However,due to attraction between hairy-rods immersed in a flexible chain melt, the formationof microdomained ordered structures may be expected.

In general, there are two mechanisms for attraction in hairy-rod systems. The firstone arises from the inhomogeneous distribution of the free polymer coils [130] whichcreates some additional loss of entropy of the coils compared to the homogeneousmelt [105] (see Appendix 6.B, where the interaction between two cylindrical micellesis analyzed). This mechanism ultimately results in the formation of microdomainlattice structures in the blend.

The second mechanism is connected to the incompatibility of rods and coils andis responsible for the selection between hexagonal and lamellar structures for certainvalues of the parameters. Furthermore, as in the case of the covalently bonded hairy-rods (see chapter 5 and [147]), we can distinguish two different hexagonal phases. Inthe first one, called H1, the ”cylinders” contain only one rod per unit cell (Q = 1),Figure 6.3a. In the second one, called H2, the surface term becomes more importantso that rods attract each other and the cylinders contain Q > 1 rods per unit cell,Figure 6.3b. On decreasing the temperature the cylinders transform to the lamellarphase.

Additionally we will prove that the tetragonal phase is unstable and is alwayssuppressed by the hexagonal one.

6.4.1. Separation of the hexagonal phase H1

The attraction energy of cylinders resulting in the lattice formation of the hexagonalphases (H1, H2) is mainly due to the inhomogeneous distribution of the free polymercoils. Using the Random Phase Approximation it can be expressed in terms of thefluctuations of the monomeric density in the polymer matrix as [105, 130]

∆FT

=ν

2N

∫ |∆φ f ree(k)|2g(

a2Nk2

6

) dk(2π)3

. (6.27)

6.4. Equilibria between nematic, isotropic and microphases for κN > 1 101

a b

Figure 6.3: Possible hexagonal phases: (a) hexagonal H1; (b) hexagonalH2.

Here ∆F is the free energy change relative to the homogeneous state and g(u) =

2(u − 1 + e−u) /u2 is the Debye scattering function. ∆φ f ree(k) denotes the Fourier

transform of the function φ f ree(r) − 1 ' −φassoc(r), where φ f ree and φassoc are thevolume fractions of the free and associated coils defined at the mesoscopic level.Assuming that all coils obey Gaussian statistics and adopting the superposition ap-proximation [130], where the overall density of the associated coils φassoc is a simplesum of corona’s densities of the individual cylinders fixed in the vertices of the lattice,we arrive at the interaction energy (per cylinder of unit length)

UH(Q)T

=Nν(Qp)2

2b2

2√3 `2

∑

br

h2( a2Nk2

6 )

g( a2Nk2

6 )− 1

4π2

∫dk

h2( a2Nk2

6 )

g( a2Nk2

6 )

, (6.28)

where ` is the period of the structure, br are the vectors of the reciprocal lattice, andh(u) =

(1 − e−u) /u.

102 Chapter 6

The first term in (6.28) arises directly from (6.27), whereas the second one is theenergy of the lattice with infinite period, which is used as a reference point. Theperiod of the structure can be easily related to the volume fraction of rods as

`2 =1

2√

3

πd2Qf

. (6.29)

After calculation of the sum and integral in (6.28), Appendix 6.C, we find theinteraction energy and renormalize it per volume (π/4)Ld2

UH(Q)T

= − 332κMQp2 f d2

a2

[3.457 + ln

(a2N f

Qd2

)]. (6.30)

Thus the free energy of the H1 phase (Q = 1) is given by

FH1

T=2 f Ld

γ

T− M f p

[ε

T− ln N∗

]+ f M

[p ln p + (1 − p) ln(1 − p)

]

+ 2 f ln(L

d

)+ M

(1 − f − f κN p)Nκ

ln

(1 − f − f κN p

e

)

+ f3κd2

32a2Mp2 ln (κN p) − 3

32κMp2 f d2

a2

[3.457 + ln

(a2N f

d2

)]. (6.31)

Here we approximated the loss of orientational energy of a rod by the term 2T f ln (L/d)and omitted the loss of its translational entropy because it is relatively small. Phaseequilibrium between the isotropic phase and the H1 phase can be found from theequilibrium equations

∂FI

∂ fI=∂FH1

∂ fH1,

∂FI

∂pI=∂FH1

∂pH1= 0

fI∂FI

∂ fI− FI = fH1

∂FH1

∂ fH1− FH1 . (6.32)

The probability of bonding and the binodal lines are

pI ' pH1 ' 1,

f (1)H1 '

316

d2

a2N,

fI '(L

d

)2

exp

(− 3

16d2κM

a2

)' 0 . (6.33)

6.4. Equilibria between nematic, isotropic and microphases for κN > 1 103

Similarly the phase equilibrium between the nematic and the H1 phase followsfrom equations

∂FN

∂ fN=∂FH1

∂ fH1,

∂FN

∂pN=∂FH1

∂pH1= 0,

fN∂FN

∂ fN− FN = fH1

∂FH1

∂ fH1− FH1 . (6.34)

Their solution is given by

pN '0, pH1 ' 1,

fN '1,

f (2)H1 '

11 + κN

[1 − exp

(− ε

T+ 2bd

γ

T+ ln N∗ +

3κd2

32a2ln (κN)

)]. (6.35)

The triple point T0, where isotropic, nematic and H1 phases coexist, can be obtainedfrom the intersection of the curves f (1)

H1(T ) and f (2)H1 (T ) and reads

ε

T0=

1

1 − 2bwεd

(2bsd

+ ln N∗ +3κd2

32a2ln (κN)

), (6.36)

where the probability of bonding p0 ' 1. Thus the hexagonal H1 phase is stable forf (1)H1 < f < f (2)

H1 below T0; for fI < f < f (1)H1 the system separates into the isotropic and

the H1 phase and for f (1)H1 < f < fN it separates into the H1 and the nematic phase

(see Figure 6.4).It is also interesting to note that at the triple point the period of the hexagonal

structure is ` =(8π/(3

√3)

)1/2a√

N ' 2a√

N so that the system has the structure ofalmost close-packed cylinders. As we have shown in Appendix 6.B, the two micellesinteraction energy reveals minimum at the distance r12 ' 3.3a

√N. Although it seems

natural to assume that the period ` of the appearing lattice is equal to r12, in realityit is much smaller ` ' 2a

√N . This becomes possible because repulsion between

closest neighbors is compensated by attraction between more distant micelles.Apparently, a question about the possible existence of a tetragonal (2D square

lattice) phase arises at this point. It would correspond to a lower density of rodsand might possibly appear at higher temperatures T > T0. In the framework of themethod employed, the only difference between the H1 and the tetragonal phase isthe interaction energy corresponding to the different lattices. The latter, equation

104 Chapter 6

Figure 6.4: Complete phase diagram in the case κN > 1

6.4. Equilibria between nematic, isotropic and microphases for κN > 1 105

(6.115), is calculated using the same method as explained above (6.27) in Appendix6.C. It can be shown that the critical point TT , at which the tetragonal structure wouldappear, lies below the point T0 for any values of the model’s parameters: TT < T0.This implies that the tetragonal phase is always suppressed by the hexagonal phase.

6.4.2. Separation of the hexagonal phase H2

Decreasing the temperature T , we effectively increase the repulsion between the rodsand coils due to the surface tension (see e.g. the first term in (6.31)). This resultsin the tendency of rods to adopt a packing with smaller total area of contact withcoils. It makes the H2 phase more favorable comparing to the H1, but at the sametime leads to an increase in the elastic energy of the side chains (see Figure 6.3). Thecompetition between these two factors results in the H1–H2 transition.

Let us follow along the binodal line f (1)H1 (T ) with the temperature going down

(Figure 6.4). At a certain moment phase H1 becomes unstable compared to separationinto the isotropic and the hexagonal H2 phase. The corresponding triple point can beobtained from the set of equations

∂FI

∂ fI=∂FH1

∂ fH1=∂FH2

∂ fH2,

∂FI

∂pI=∂FH1

∂pH1=∂FH2

∂pH2= 0 ,

fI∂FI

∂ fI− FI = fH1

∂FH1

∂ fH1− FH1 = fH2

∂FH2

∂ fH2− FH2 . (6.37)

In order to construct the free energy FH2 of the H2 phase one has to modify thesurface tension and the elastic energy terms in (6.31).

Assuming that coils cannot penetrate inside, the core of the cylindrical micelle thecore adopts a double layer structure of closely packed rods, as depicted in Figure 6.3b,with a surface per unit length along the cylinder Qd + 2d valid for Q > 2 (see also 5).For Q <

√N the cylinders still have an approximately circular cross section so that

106 Chapter 6

the approximation (6.15) with renormalized grafting density can be used. Thus

FH2

T= f Ld

γ

T

(1 +

2Q

)+ M f p

[ln N∗ − ε

T

]

+ f M[p ln p + (1 − p) ln(1 − p)

]+ 2 f ln

(Ld

)

+ M(1 − f − f κN p)

κNln

(1 − f − f κN p

e

)

+ f3d2κQ

32a2Mp2 ln (κN p)

− 332κMQp2 f d2

a2

[3.457 + ln

(a2N f

Qd2

)]. (6.38)

The number of rods per unit cell Q follows from the minimum condition ∂FH2/∂Q =

0,

Q '√

64ba2

3κp2d ln (κN)γ

T. (6.39)

In derivation of (6.39) we used the fact that Q is mainly determined by the interplaybetween the surface term and the elastic energy of the grafted coils, omitting therelatively small “lattice” term (the last one in (6.38)).

Hence, the solution of (6.37) is given by

pI 'pH1 ' pH2 ' 1,

Q1 '2 +√

2, fI ' 0,

f (t1)H1 '

316

d2

a2N, f (t1)

H2 '3

16Q1d2

a2N, (6.40)

and the critical temperature reads

w

T1' −s +

3κd3Q21

64ba2ln (κN) . (6.41)

Similarly the binodal line f (2)H2 (T ) finishes at the triple point, which can be found

from the system of equations

∂FN

∂ fN=∂FH1

∂ fH1=∂FH2

∂ fH2,

∂FN

∂pN=∂FH1

∂pH1=∂FH2

∂pH2= 0,

fN∂FN

∂ fN− FN = fH1

∂FH1

∂ fH1− FH1 = fH2

∂FH2

∂ fH2− FH2 (6.42)

6.4. Equilibria between nematic, isotropic and microphases for κN > 1 107

and is characterized by

pN ' 0, pH1 ' pH2 ' 1,

Q′1 ' Q1 ' 2 +

√2, fN ' 1,

f (t2)H1 ' 1

1 + κN

[1 − exp

(− ε

T1+

2bdγT1

+ ln N∗ +3d2κ

32a2ln (κN)

)],

f (t2)H2 ' 1

1 + κN

[1 − exp

(− ε

T1+

bdγT1

(1 +

2Q1

)

+ ln N∗ +3d2κQ1

32a2ln (κN)

)]. (6.43)

In a first approximation the corresponding critical temperature coincides withthe critical temperature (6.41). Note, the small difference between these criticaltemperatures, which we do not consider here, results in a small area of phase co-existence between H1 and H2 (see Figure 6.4). Also the triple points f (t2)

H1 and f (t2)H2

are exponentially close to each other. Using (6.40) and (6.43) we conclude that theaverage number of molecules per micelle of length L is Q1 = 2 +

√2 ' 3.4 when

H2 first appears. Hence, the cylindrical domain has 3-4 rods in its cross section.This small number is consistent with the approximation used concerning the almostcircular cross section.

The phase equilibrium between the isotropic and the hexagonal H2 phase isdetermined from the set of equations

∂FI

∂ fI=∂FH2

∂ fH2,

∂FI

∂pI=∂FH2

∂pH2= 0,

fI∂FI

∂ fI− FI = fH2

∂FH2

∂ fH2− FH2 , (6.44)

which has different asymptotic solutions near the triple point and far away from it. Asimple expansion in the vicinity of the f (t1)

H2 point gives the binodal line in the form

f (1)H2 (T ) ' 3

16d2

a2NQ(T ), (6.45)

whereas for 1 Q <√

N the solution of (6.44) reads

pI ' pH2 ' 1,

fI ' 0,

f (1)H2 '

11 + κN

[1 − exp

(− 3

16d2

a2NQ(T )

)]. (6.46)

108 Chapter 6

Here Q(T ) is given by (6.39) and is temperature dependent.Similar equilibrium conditions have to be applied to the nematic-H2 coexistence

yielding the result

pN ' 0, pH2 ' 1,

fN ' 1,

f (2)H2 ' 1

1 + κN

[1 − exp

(− ε

T+

bdγT

(1 +

2Q(T )

)

+ ln N∗ +3d2κQ(T )

32a2ln (κN)

)]. (6.47)

The results obtained for the binodal lines and the triple points are summarized inFigure 6.4.

With further decrease of the temperature the number of rods in the cross sectionQ increases going beyond the limit Q <

√N. The core of the cylinders containing

the rods becomes considerably elongated and the approximation (6.15) fails. It thiscase a planar rather than a circular cylindrical shape of the core should be takenas a reference state, reflecting the very high values of Q ∼ N in the vicinity of thehexagonal H2 to lamellar phase transition. We will address this in the next subsection.

6.4.3. Separation of the lamellar phase

Following the same procedure as before we proceed with the derivation of the freeenergy of the lamellar phase. Compared to the hexagonal phases, a number of termshas to be modified. Here we start from the interaction between lamellae.

Apparently, the total energy of the lamellae’s interaction is a sum of all the nearestneighbor interaction energies. We briefly outline the calculation scheme here (see[130, 152] for details). Let us consider first a single lamella immersed in a melt offlexible chains. The free energy per unit area of the lamella consists of the free energyof the attached ∆F(a) and free ∆F( f ) coils.

∆F = ∆F(a) + ∆F( f ) . (6.48)

The last one is given by

∆F( f ) =a2T24ν

∫(∇φ f )2

φ fdz,

6.4. Equilibria between nematic, isotropic and microphases for κN > 1 109

a

b

Figure 6.5: (a) Single lamella and its surrounding. (b) Two neighboringlamellae.

110 Chapter 6

where φ f (z) is the volume fraction of the free coils and the z axis points perpendicularto the lamella’s plane (see Figure 6.5a). The ∆F (a) term consists of two parts

∆F(a) =a2T24ν

∫(∇φa)2

φadz − T

ν

∫ ∞

0u(z)

[φa(z) − φ(box)

a

]dz , (6.49)

where the second term takes into account the additional energy of the attached coils’elastic elongation compared to its minimum value, corresponding to a box-like den-sity distribution φ(box)

a (z). Here

u(z) =3π2

8(z + H)2

N2a2(6.50)

is the parabolic molecular field responsible for the stretching of the attached coils[131, 152] and H is the width of the corona consisting of the attached coils. Thedeviation from the box-like distribution creates an interpenetration area ξ0 (see Fig-ure 6.5a), which in its turn is determined from the minimization of (6.48) with a trialfunction of the form

φa = 1 − φ f =12

(1 − tanh

zξ0

). (6.51)

After the minimization ξ0 is obtained

ξ0 =

(4

3π4

N2a4

H

)1/3

(6.52)

as well as the free energy of the lamella

∆F∞ = minξ0

∆F =πT8ν

(3πa2H

4N2

)1/3

. (6.53)

Using the same approach, the interaction between two lamellae can be consideredtoo. For this purpose we examine two planar micelles put on a distance D from eachother as depicted in Figure 6.5b. Their interaction energy (per unit area) reads

U(D) = ∆F(D) − 2F∞ , (6.54)

where similarly to the previously explained case

∆F(D) = minξ

[2∆F(a)(ξ) + ∆F( f )(ξ,D)

]. (6.55)

6.4. Equilibria between nematic, isotropic and microphases for κN > 1 111

Of course, ξ depends now on D; also an appropriate trial function has to be taken forthe minimization procedure

φ f (z) =12

[tanh

zξ

+ tanhD − zξ

]. (6.56)

Without going into details of the calculation we present the final result (see also[130, 152])

ξmin(D) ' ξ +49

D ,

ξ =

(2

3π4

N2a4

H

)1/3

(6.57)

and

∆F(D) ' a2T8νξ

+a2T D

18νξ2. (6.58)

The variables D and H can be easily related to the volume fraction of rods f andother characteristic quantities of the system as

D =πd2

f ∗ − f

f ∗2,

H =πd4κN p , (6.59)

where f ∗ = 1/(1 + κN p) is the volume fraction of rods if all coils in the system areattached.

Hence, using equations (6.54) and (6.57)–(6.59), and renormalizing the energyas per volume πd2L/4, the interaction energy is obtained

UL( f )T

= −0.227 f ∗M(

pa2

κ2d2N

)1/3

+ 1.312M

(p2d2

κa2N2

)1/3f ∗ − f

f ∗. (6.60)

The expression for the elastic free energy of the coils attached to a planar sur-face is trivial considering that the stretching free energy of one coil is [1, 2] F1 =

3T H2/(2Na2).

112 Chapter 6

So far the free energy of the lamellar phase is obtained in the form

FL

T= f Ld

γ

T+ M f p

[ln N∗ − ε

T

]+ f M

[p ln p + (1 − p) ln(1 − p)

]

+2 f ln(L

d

)+ M

(1 − f − f κN p)κN

ln

(Hξ

1 − f − f κN pe

)

+ f3π2d2κ2

32a2NMp3 +

UL( f )T

. (6.61)

where ξ according to (6.57) and (6.59) is given by

ξ =2aπ

(aN

3π2κpd

)1/3

. (6.62)

Here we have taken into account that any given free coil is confined in a space ofwidth 2ξ between a pair of two successive lamellae separated by a distance 2H.Therefore, the free coils additionally loose some translational entropy compared tothe hexagonal and isotropic phase.

The phase equilibrium between the isotropic and the lamellar phase can be de-scribed on the basis of the equations

∂FI

∂ fI=∂FL

∂ fL,

∂FI

∂pI=∂FL

∂pL= 0,

fI∂FI

∂ fI− FI = fL

∂FL

∂ fL− FL (6.63)

and the probability of bonding and the binodals are given by

pI ' pL ' 1,

fI ' 0,

f (1)L ' 1

1 + κN

1 −ξ

Hexp

−1.312

(κ2d2N

a2

)1/3 . (6.64)

Similarly the equilibrium between the nematic and the lamellar phase must fulfill theset of equations

∂FN

∂ fN=∂FL

∂ fL,

∂FN

∂pN=∂FL

∂pL= 0,

fN∂FN

∂ fN− FN = fL

∂FL

∂ fL− FL , (6.65)

6.4. Equilibria between nematic, isotropic and microphases for κN > 1 113

from which the corresponding probabilities and binodals are obtained

pN ' 0, pL ' 1,

fN ' 1,

f (2)L ' 1

1 + κN

[1 − exp

(−ε + bdγT

+ ln N∗ +3π2d2κ2N

32a2

)]. (6.66)

Now we would like to address the question about the hexagonal H2 to lamellarphase transition. As it was pointed out before, in the vicinity of the transition thecore of the H2-phase cylinders have a considerably non-circular shape (Figure 6.3b).Therefore, the calculation of the elastic energy and the interaction between this kindof cylinders becomes a nontrivial task. However, taking into account the highlyelongated shape of the cores of the H2-cylinders near the transition, the elastic freeenergy of the side chains can be approximated as that of the corresponding planarlamella plus some edge correction following the method developed in [144]. Herewe do not go into details of the calculation referring the reader to the previouschapter, where the following expression (see (5.17)) has been obtained for this edgecorrection

F ' −3 f TQ

νH2Mp2

a2bd2. (6.67)

Hence, the elastic free energy of the H2 phase near the H2 to lamellar phase transitionis approximated by

F′ (el)H2

T' 3π2d2κ

32a2f NMp3 − 3

(πκ

4

)3 d2N2Mp4

a2

fQ. (6.68)

In the very vicinity of the transition between H2 cylinders and lamellae the interactionenergy in the H2-phase can be approximated by the corresponding one for the lamella(6.60), yielding

F′H2

T= f Ld

γ

T

(1 +

2Q

)+ M f p

[ln N∗ − ε

T

]

+ f M[p ln p + (1 − p) ln(1 − p)

]+ 2 f ln

(Ld

)

+ M(1 − f − f κN p)

κNln

(1 − f − f κN p

e

)+

F′ (el)H2

T+

UL

T. (6.69)

114 Chapter 6

Comparing (6.69) to (6.61) one concludes that the H2–Lam transition is completelydetermined by the interplay between the elastic and the surface tension term. Thequantitative result follows from the equilibrium conditions

∂FL

∂ fL=∂F′H2

∂ fH2,

∂FL

∂pL=∂F′H2

∂pH2= 0,

fN∂FL

∂ fL− FL = fH2

∂F′H2

∂ fH2− F′H2 . (6.70)

The area of the lamellar-H2 phase coexistence is very narrow, like it was in the H1-H2 case, implying that the corresponding triple points (H2-Lam-Nem and H2-Lam-Iso; see Figure 6.4) are exponentially close to each other and characterized by thetemperature T2

w

T2' 3

2

(πκd4

)3 N2

ba2− s . (6.71)

Note that the number of rods per cylinder in the H2-phase dropped out from the result(6.71) without minimization. This fact is connected to the approximation (6.68),which in a sense can be viewed as an expansion in the small parameter 1/Q. So, toobtain the number of rods Q2 at the H2-Lam transition consistently one has to extendthe approximation (6.68). However, Q2 can be also estimated from (6.68) when theedge correction becomes of the order of the main term

Q2 ∼ κN . (6.72)

6.4.4. Possible phase sequences

In the previous subsections a complete phase diagram was described for the highlygrafted supramolecular hairy-rod system. However, an implicit assumption about thepresence of all the three microphases was used. Certainly, depending on the valuesof the model’s parameters (mainly the ratio ε/w) other sequences of microphases arepossible.

The prediction about the realization of a particular phase diagram can be madefrom a comparison of the temperatures where the microphases first appear. Forinstance, the simple hexagonal phase H1 is present if the temperature T0 given by(6.36) is higher than T1 (6.41), where a transition to H2 occurs. This leads to theconclusion that the phase diagram has the form shown in Figure 6.4 only if

εd2bw

> 1 +

3κd2

32a2 ln(κN) + ln N∗ + 2bsd

3κd2

32a2 (6 + 4√

2) ln(κN) − 2bsd

. (6.73)

6.4. Equilibria between nematic, isotropic and microphases for κN > 1 115

a

b

Figure 6.6: Two more possible phase diagrams in the κN > 1 case (see alsoFigure 6.4 and eqs (6.73), (6.74))

116 Chapter 6

In the absence of both hexagonal phases the lamellar phase would appear at the pointdetermined as the intersection of the curves f (1)

L (T ) and f (2)L (T )(equations (6.64) and

(6.66)). This point is lower than T2 (6.24) if the following condition is fulfilled

1 <εdbw

< 1 +4b

πdκN. (6.74)

Thus, only nematic, isotropic and lamellar phases coexist in this case and the corre-sponding diagram is shown in Figure 6.6b.

Finally, for values of ε/w lying in between that of the regimes (6.73) and (6.74)only H2 and lamellar structures are stable as depicted in Figure 6.6a.

6.5. Phase equilibria for κN < 1

In this section we consider the phase behavior of the same rod-coil system as before,but for small values of the parameter κN < 1. As explained above, this impliesa low grafting density of coils attached to a rod even for conversion p = 1. Adirect consequence in the framework of the presented model is the absence of theelastic stretching term in the free energies of all phases considered. In fact, the freeenergies of the isotropic liquid and both hexagonal phases can be borrowed from theprevious calculation, equations (6.16), (6.31) and (6.38), keeping in mind that theterm responsible for the stretching of coils has to be omitted everywhere.

However, the lamellar phase needs some additional attention. The expression forthe lattice free energy (6.60) was obtained by a method [130, 152] assuming highsurface density of the attached coils. Definitely, for κN < 1 this is not the caseanymore. Therefore, another method, based on (6.27), should be adopted for thispurpose. Using the same arguments as for the hexagonal phase (see Appendix 6.C)one arrives at the expression for the lattice energy of the lamellar structure:

U′L( f )

T' π2

128Mκp2d2

a2+

M f κp2d√

N8a

−1.728 +a f√

Nd

. (6.75)

6.5. Phase equilibria for κN < 1 117

Figure 6.7: Phase diagram of the system for κN < 1.

Therefore, the total free energy of the lamellar phase (per volume of a rod) reads

F′LT

= f Ldγ

T+ M f p

[ln N∗ − ε

T

]

+ f M[p ln p + (1 − p) ln(1 − p)

]+ 2 f ln

(Ld

)

+ M(1 − f − f κN p)

κNln

(Hξ

1 − f − f κN pe

)+

U′L( f )

T. (6.76)

Further analysis shows that, in contrast to the previously considered high graftingdensity situation, only the lamellar structure accompanied by nematic and isotropicliquid is present in the phase diagram Figure 6.7. It first appears at the temperature

1T2

=ln N∗ + bs/dε − bw/d

(6.77)

and takes a prominent place in the phase diagram totally suppressing both hexagonalphases.

This fact is also supported by our physical expectations. The actual selectionbetween H1, H2 and Lam phases in the case κN > 1 was performed by an interplay

118 Chapter 6

between the surface tension γ-term and the elastic stretching of the side chains. Atrelatively high temperatures the surface tension was small and therefore the systemadopted the hexagonal structure characterized by lower elastic energy. However, asthe temperature is decreased, the surface tension starts to play the dominant role andthe system transforms to the lamellar phase where the total contact area between rodsand coils is much smaller. In the current case of κN < 1 the elastic energy is notimportant at all. This implies the dominant position of the lamellar phase (comparedto hexagonal) for any value of the model parameters.

Proceeding with the binodal lines we write the standard equilibrium conditionsfor the isotropic–lamellar coexistence (6.63) yielding

fI '(L

d

)2

exp

(−Ldγ

T

),

f (1)L ' 1

4κd√

Na

. (6.78)

The same can be done for the equilibrium between the nematic and lamellar phases

fN ' 1 ,

f (2)L ' 1

1 + κN

[1 − exp

(−ε + bdγT

+ ln N∗)]. (6.79)

The resulting phase diagram is shown in Figure 6.7. A large part of it is occupied bythe lamellar microphase, which appears above the 1/T2 point.

6.6. Discussion

In this chapter we addressed the peculiarities of the phase equilibria and microstruc-ture formation of thermoreversible hairy-rod polymers. The main results are summa-rized in the phase diagrams Figure 6.4, 6.6 and 6.7.

We started from a blend of rods and coils, which are in general strongly incom-patible, Figure 6.2a. In has been shown that thermoreversible association induces asignificant compatibility enhancement: for the association energy beyond a certainvalue, ε > 2bw/d, a partial compatibility is achieved at temperatures below T ∗,(6.26), to the left from the stoichiometric point (see Figure 6.2b). The conversionparameter p is close to unity in the coil-rich phase and negligibly small in the rod-rich phase.

6.6. Discussion 119

Furthermore, the compatibility region was proven to be unstable against mi-crostructure formation. Due to mutual attraction between hairy-rod molecules in amelt of flexible coils, they tend to self-organize. Three types of microphases, namelyH1 and H2 hexagonal and lamellar, can appear and coexist with each other as wellas with the isotropic and nematic phase (see phase diagrams Figure 6.4, 6.6 and 6.7).The actual selection between the different microphases at any given temperature is aresult of the competition between the “surface tension” γ-part of the free energy andthe elastic stretching of the side chains: the first one is responsible for the tendencyof the rods to stick together, whereas the second one, if the hairy-rod is denselygrafted, prevents it. This competition leads to the hexagonal H1 phase being stableat relatively high temperatures. For somewhat lower temperatures it is followed byH2 and then by the lamellar microphases (Figure 6.4). Actually, depending on thevalue of the εd/bw parameter, equations (6.73) and (6.74), three different sequencesof microphases are possible. For quite small values, see (6.74), only a narrow strip ofthe lamellar microphase is present along with the nematic and isotropic phase, Fig-ure 6.6b. If εd/bw is large enough, (6.73), all three microstructures appear Figure 6.4.In the intermediate regime only two, H2 and lamellar, are possible. In many respectsthis situation resembles that of the covalently bonded system [147], where these threetypes of sequences were predicted as well.

A qualitatively different situation is observed in the κN < 1 case: the graftingdensity is always low and the elastic part of the free energy is negligibly small.This immediately results in the fact that only the lamellar structure can be foundFigure 6.7. Indeed, nothing prevents rods from the γ-driven tendency to keep thecontact area with coils as small as possible. The lamellar phase first appears atthe triple point T2, (6.77), and then occupies the lion share of the diagram (weremind that for κN < 1 the stoichiometric point 1/(1 + κN) > 1/2). Existence ofthe Figure 6.7 type phase diagram is in accordance with the few experimental dataavailable [65, 115, 153–156]. In these experimental systems, briefly discussed in theIntroduction, the κN parameter is definitely less than unity1. So, a lamellar structureis expected to be present along with the isotropic and nematic phase (in practice thelatter might correspond to either a nematic or a crystalline phase), which is indeedobserved experimentally [65, 154, 156].

1 Backbone “unit” includes (2.5-pyridinediyl)(MSA) plus OG’s head whereas a side chain consistsof a relatively short OG’s tail.

120 Chapter 6

Appendix

6.A. Expressions for the chemical potentials and partial pres-sures

We define chemical potential and pressure as

µ =∂F∂ f

,

P = f∂F∂ f− F .

Below we list the explicit expressions for the chemical potentials and partial pressuresof the isotropic liquid (I), nematic (N), hexagonal H1 and H2 and lamellar (L) phases.

Isotropic liquid phase

µI

T=

2Ld

(w

T+ s

)+ Mp

[ln N∗ − ε

T

]

+M[p ln p + (1 − p) ln(1 − p)

]+ ln f

−M (1 + κN p)Nκ

ln (1 − f − f κN p)

+3κd2

32a2Mp2 ln (κN p) H

(p − 1

κN

), (6.80)

PI

T= f − M

Nκln (1 − f − f κN p) +

MNκ

(1 − f − f κN p) . (6.81)

Nematic phase

µN

T= 2 ln

(Ld

)− M

Nκµc

T+ Mp

[ln N∗ − ε

T

]

+M[p ln p + (1 − p) ln(1 − p)

]+ ln f

−M (1 + κN p)Nκ

ln (1 − f − f κN p) , (6.82)

PN

T= f − M

Nκµc

T− M

Nκln (1 − f − f κN p) +

MNκ

(1 − f − f κN p) . (6.83)

6.A. Expressions for the chemical potentials and partial pressures 121

Hexagonal H1 phase

µH1

T=

2Ld

(wT

+ s)

+ Mp[ln N∗ − ε

T

]

+M[p ln p + (1 − p) ln(1 − p)

]+ 2 ln

(Ld

)

−M (1 + κN p)Nκ

ln (1 − f − f κN p)

+3κd2

32a2Mp2 ln (κN p)

− 332κMp2d2

a2

[3.457 + 1 + ln

(a2N f

d2

)], (6.84)

PH1

T= − M

Nκln (1 − f − f κN p) +

MNκ

(1 − f − f κN p) − 332

fκMp2d2

a2. (6.85)

Hexagonal H2 phase

In the case of the H2 phase first the minimization upon Q has to be carried out.

1f T

∂FH2

∂Q=

2L

Q2d

(w

T+ s

)+

3κd2

32a2Mp2 ln (κN p)

− 332κMp2d2

a2

[3.457 − 1 + ln

(a2N f

Qd2

)]= 0 . (6.86)

If we neglect the lattice formation energy, the result (6.39) can be obtained. Chemicalpotential of the H2 hexagonal phase can be obtained as

µH2 =

(∂F∂ f

)

Q+

(∂F∂Q

)

f

∂Q∂ f

=

(∂F∂ f

)

Q

122 Chapter 6

and reads

µH2

T=

Ld

(w

T+ s

) (1 +

2Q

)+ Mp

[ln N∗ − ε

T

]

+M[p ln p + (1 − p) ln(1 − p)

]+ 2 ln

(Ld

)

−M (1 + κN p)Nκ

ln (1 − f − f κN p)

+3κd2Q

32a2Mp2 ln (κN p)

− 332

QκMp2d2

a2

[3.457 + 1 + ln

(a2N f

Qd2

)], (6.87)

PH2

T= − M

Nκln (1 − f − f κN p)

+MNκ

(1 − f − f κN p) − 332

fκMp2d2

a2Q . (6.88)

Lamellar phase

µL

T=

Ld

(w

T+ s

)+ Mp

[ln N∗ − ε

T

]

+M[p ln p + (1 − p) ln(1 − p)

]+ 2 ln

(Ld

)

−M (1 + κN p)Nκ

ln

(Hξ

(1 − f − f κN p)

)

+3π2κ2d2

32a2NMp3 − 1.312M

f ∗

(p2d2

κa2N2

)1/3

, (6.89)

PL

T= − M

Nκln

(Hξ

(1 − f − f κN p)

)+

MNκ

(1 − f − f κN p)

+0.227 f ∗M(

pa2

κ2d2N

)1/3

− 1.312M

(p2d2

κa2N2

)1/3

. (6.90)

For H and ξ see (6.59) and (6.62).

6.A. Expressions for the chemical potentials and partial pressures 123

Hexagonal H2 phase in the vicinity of the H2-Lam transition

The free energy is given by the following expression

F′H2 = fLd

(w

T+ s

) (1 +

2Q

)+ M f p

[ln N∗ − ε

T

]

+ f M[p ln p + (1 − p) ln(1 − p)

]+ 2 f ln

(Ld

)

+M1 − f − f κN p

Nκln

(1 − f − f κN p

e

)+ F′ (el)

H2 + U′H2 . (6.91)

Here F′ (el)H2 is given by (6.22) and the energy of the lattice formation U ′H2 is approxi-

mated by the corresponding term for the lamellar phase.

µ′H2

T=

Ld

(wT

+ s) (

1 +2Q

)+ Mp

[ln N∗ − ε

T

]

+M[p ln p + (1 − p) ln(1 − p)

]+ 2 ln

(Ld

)

−M (1 + κN p)Nκ

ln (1 − f − f κN p)

+3π2κ2d2

32a2NMp3 − 3

(πκ

4

)3 d2N2 M

Qa2p4

−1.312Mf ∗

(p2d2

κa2N2

)1/3

, (6.92)

P′H2

T= − M

Nκln (1 − f − f κN p) +

MNκ

(1 − f − f κN p)

+0.227 f ∗M(

pa2

κ2d2N

)1/3

− 1.312M

(p2d2

κa2N2

)1/3

. (6.93)

As it has been pointed out before, the minimization upon Q cannot be performed.Indeed

∂F′H2

∂Q= −2 f L

d1

Q2+ 3

(πκ

4

)3 d2N2 M f

Q2a2p4

so ∂F′H2/∂Q = 0 does not have roots Q < ∞ in the approximation used. In order to

obtain it, a more precise approximation is needed for F ′ (el)H2 and U′H2.

124 Chapter 6

Figure 6.8: Two cylindrical micelles (a cross section)

6.B. Interaction between two cylindrical micelles

Let us consider the interaction between two cylindrical micelles in an otherwisehomogeneous system, Figure 6.8. We assume the distance between them to be largerthan the size of the corona, r12 & Rcorona. Therefore, the monomer density φassoc ofthe associated coils can be approximated as a sum of the densities φ0 of the isolatedmicelles

φassoc(r) = φ0(r) + φ0(r − r12) . (6.94)

Attached to a rod chains still approximately obey Gaussian statistics, so that a prob-ability to find the s-th monomer at the distance r from the 0-th one reads

p(r, s) =

(4πsa2

6

)−3/2

exp

[− 6r2

4sa2

].

Thus, the Fourier transform of φ0 has the form

φ0(k) = Ns1 − exp(−Nk2a2/6)

k2a2/6, (6.95)

where Ns is the total number of side chains per micelle. Note, the free energy ofthe micelles interaction arises from the inhomogeneous distribution φ f ree(r) = 1 −φassoc(r) of free coils in their surrounding. This energy can be calculated by the RPAmethod [105] and is given by the equation (6.27). Using (6.94) and (6.95) we obtain

∆φ(k) = φ0(k)(1 + e−ikr12

), (6.96)

6.C. Free energy of the lattice formation 125

x

F( )x

33 0.

Figure 6.9: The interaction free energy of two cylindrical micellesseparated by distance r12 = xa

√N

and therefore the interaction energy (per unit length along the “backbone” of themicelle)

∆Fint(r12)T

=3N2

s

a2Φ

(r12

a√

N

), (6.97)

where

Φ(x) =

∫ ∞

0

q dq

(2π)2

(1 − e−q2)2

q2 − 1 + e−q2J0(√

6xq) (6.98)

and J0(x) =∫ 2π

0 dϕ eix cos ϕ/(2π) is a Bessel function.The function Φ(x) is schematically depicted in Figure 6.9. The micelles at-

tract each other at large distances and the function reaches its minimum at x =

r12/(a√

N) ' 3.30. This attraction is responsible for the superlattice formation asshown in section 6.4.1.

6.C. Free energy of the lattice formation

Let us start from a general formula for the free energy of the whole lattice (see (6.27)and [105, 130])

Fint

T=

ν

2N

∫1

fD

(k2a2N

6

) |∆φ(k)|2 dk(2π)3

, (6.99)

126 Chapter 6

where fD(u) is the Debye scattering function

fD(u) =2

u2

(u − 1 + e−u) (6.100)

and ∆φ(k) is the Fourier transform of the function φ f ree(r)−1 = −φassoc(r), φ f ree andφassoc are monomeric densities of the free chains and chains attached to the rods.

Hexagonal lattice

We consider cylinders consisting of Q rods each and forming a hexagonal lattice(effectively 2D) with vertexes at the points rmn. Further, it is assumed [105, 130] thatthe density of the attached chains can be written as a sum of densities of virtuallyisolated cylinders, i.e. φassoc(r) =

∑m,n φ0(r − rmn), or, in other words,

∆φ(k) = −φ0(k)∑

m,n

e−ikrmn . (6.101)

This immediately gives

|∆φ(k)|2 = φ20(k)

∑

s,t

∑

m,n

e−ik(rmn+rst ) = φ20(k)Ncells

∑

m,n

e−ikrmn , (6.102)

where Ncells is the total number of cells (or, equally, cylinders) in the system.Function φ0 is calculated for Gaussian chains forming cylindrical brush and reads

φ0(k) = Ns1 − e−u

uN . (6.103)

Here Ns is the total number of the side chains forming a cylinder and u = Nk2a2/6.For the further derivation the identity

∑

m,n

e−ikrmn =∑

m,n

v′δ (k − bmn) (6.104)

is used [157]. Here bmn is a vector of a reciprocal lattice and v′ is volume of itselementary cell.

If one considers a sample in the form of a cube with side L, then the direct latticeelementary cell has dimensions ` × ` × L, where ` is the period of the lattice (Lwill drop out from all final expressions). The cross section of this cell is depicted inFigure 6.10.

6.C. Free energy of the lattice formation 127

Figure 6.10: An elementary cell of a hexagonal lattice.

The lattice basis vectors a1 and a2 have Cartesian coordinates a1 = (1; 0) anda2 = (1/2;

√3/2) and any vector of the lattice can be represented as their linear

combinationrmn = (ma1 + na2)`.

Volume of the direct lattice cell reads

v =

√3

2L`2 , (6.105)

which means that volume of the reciprocal cell is

v′ =(2π)3

v=

2(2π)3

√3

1

L`2. (6.106)

The reciprocal lattice basis vectors have to be built from the condition

a1b = 2πp1

a2b = 2πp2 (6.107)

for any vector b = p1b1 + p2b2 of the reciprocal lattice [157]. Here b1 and b2 are thebasis vectors of the reciprocal lattice and p1,2 are any integer numbers. From the setof equations (6.107) one obtains

b1 = 2π

(1;− 1√

3

)(6.108)

b2 = 2π

(0;

2√3

). (6.109)

128 Chapter 6

Now plugging expressions (6.100), (6.102) - (6.104), (6.106),and (6.108) into (6.99)we obtain the free energy of the system in the form

F (HEX)int

T=NcellsN2

s

2√

3

Nν

L`2

∑

m,n

(1 − e−umn

)2

umn − 1 + e−umn, (6.110)

where

umn =29

(2πN1/2a

`

)2

(m2 + n2 − mn) . (6.111)

Introducing x = (2/9)(2πN1/2a/`

)2one can rewrite (6.110)

F (HEX)int

T=

9NcellsN2s

16π2√

3

ν

a2L∑

m,n

x

(1 − e−x(m2+n2−mn)

)2

x(m2 + n2 − mn) − 1 + e−x(m2+n2−mn). (6.112)

The sum in (6.112) represents the gain in the free energy of the system due toinhomogeneity in the free coils distribution and corresponds to the first addend in(6.28). To obtain the interaction energy of the hexagonal lattice with a period `

one has to subtract from (6.112) the energy of the lattice with infinitely long period` → ∞, i.e. x = x∞ → 0.

Below we present the evaluation of the sum (6.112)

+∞∑

m=−∞

+∞∑

n=−∞

x(1 − e−x(m2+n2−mn))2

x(m2 + n2 − mn) − 1 + e−x(m2+n2−mn)=

2 + 4+∞∑

n=1

x(1 − e−xn2)2

xn2 − 1 + e−xn2+

2+∞∑

n=1

+∞∑

m=1

( x(1 − e−x(m2+n2−mn))2

x(m2 + n2 − mn) − 1 + e−x(m2+n2−mn)+

x(1 − e−x(m2+n2+mn))2

x(m2 + n2 + mn) − 1 + e−x(m2+n2+mn)

).

The last sum for x > 0 and n,m = 1, . . . ,∞ can be approximated as

+∞∑

n=1

+∞∑

m=1

(1

m2 + n2 − mn+

1

m2 + n2 + mn

)=

+∞∑

n=1

+∞∑

m=−∞

1

m2 + n2 − mn− 1

n2

.

6.C. Free energy of the lattice formation 129

The summation over m can be solved using the following formula [158] ( f (n) isan arbitrary function analytical everywhere in the complex plane , except a finitenumber of poles zi):

+∞∑

m=−∞f (m) = −π

∑

zi

res ( f (z) cot z) .

Therefore+∞∑

m=−∞

1

m2 + n2 − mn=

2π

n√

3

sinh(πn√

3)

cosh(πn√

3) − (−1)n.

Also considering that∑∞

n=1 1/n2 = π2/6 we obtain for the sum

2 +π2

3+

4π√3

∞∑

n=1

1n

sinh(πn√

3)

cosh(πn√

3) − (−1)n.

At this point one has to separate diverging part in the expression under the sum sign

2 +π2

3+

4π√3

∞∑

n=1

1n

sinh(πn

√3)

cosh(πn√

3) − (−1)n− 1

+4π√

3

∞∑

n=1

1n

= 5.23 +4π√

3

∞∑

n=1

1n.

The sum has to be computed for x → 0 as well and subtracted from the previousexpression. Taking into account that x∞ → 0, we can introduce new variables y1 =√

x∞m and y2 =√

x∞n and replace the summation by integral (c.f. the second addendin (6.28))

+∞∑

n,m=−∞

x∞(1 − e−x∞(m2+n2−mn))2

x∞(m2 + n2 − mn) − 1 + e−x∞(m2+n2−mn)=

2∫ +∞

0dy1

∫ +∞

−∞dy2

(1 − e−(y21+y2

2−y1y2))2

(y21 + y2

2 − y1y2) − 1 + e−(y21+y2

2−y1y2)=

2∫ 1

0dy1

∫ +∞

−∞dy2

(...)

+ 2∫ ∞

1dy1

∫ +∞

−∞dy2

(...).

The first integral can be solved numerically and is approximately equal to 5.05. After

130 Chapter 6

treating the last integral as

∫ ∞

1dy1

∫ +∞

−∞dy2

(...)

=

∫ ∞

1dy1

∫ +∞

−∞dy2

(1 − e−(y2

1+y22−y1y2))2

(y21 + y2

2 − y1y2) − 1 + e−(y21+y2

2−y1y2)− 1

y21 + y2

2 − y1y2

+

∫ ∞

1dy1

∫ +∞

−∞dy2

1

y21 + y2

2 − y1y2= 0.8074 +

∫ ∞

1dy1

2π√3y1

one can see that both the sum (appeared above) and the integral have the same typeof divergence. Subtracting one from the other we obtain

∞∑

1

1n−

∫ ∞

1

dyy

= limM∗→∞

M∗∑

1

1n−

∫ √xM∗

1

dyy

= γE − ln

√

23

2πN1/2a`

,

where γE ≈ 0.577 is Euler γ [158].To get rid of variables N and L we notice that Ns = QMpL/L and Ncells =

2V/(√

3L`2) (here V is the total volume of the sample).Finally we have to rewrite the total free energy of the lattice as energy per rod.

This can be done multiplying the free energy density by the rod’s volume. Also theperiod of the structure ` has to be expressed in terms of the volume fraction of rodsf as

`2 =1

2√

3

πd2Qf

.

All this together gives (6.30)

U(HEX)

T=

332

κQMp2d2

a2f

[−3.456 + ln

Qd2

f Na2

]. (6.113)

Tetragonal lattice

The interaction energy for a tetragonal lattice can be calculated using the sameapproach as described above. We just sketch all the necessary steps here.

A tetragonal 2D lattice is characterized by a set of vectors, Figure 6.11

a1 = (1; 0)

a2 = (0; 1)

6.C. Free energy of the lattice formation 131

Figure 6.11: An elementary cell of a tetragonal lattice.

so that an elementary cell’s volume reads v = L`2. A reciprocal lattice in this case isalso tetragonal with basis vectors

b1 = 2π(1; 0)

b2 = 2π(0; 1)

and volume v′ = (2π)3/(L`2). Using (6.99) one concludes that the free energy of thelattice has the form

F (T ET )int

T=

3NcellsN2s

2(2π)2

ν

La2

∑

m,n

x

(1 − e−x(m2+n2)

)2

x(m2 + n2) − 1 + e−x(m2+n2), (6.114)

where x =(2πN1/2a/`

)2/6.

The summation leads to

∑

m,n

x

(1 − e−x(m2+n2)

)2

x(m2 + n2) − 1 + e−x(m2+n2)'

2 + 2∞∑

n=1

1

n2+ 2

∞∑

n=1

∞∑

m=−∞

1

n2 + m2= 2 + 2

π2

6+ 2

∞∑

n=1

π

ncoth(πn) =

2 +π2

3+ 2π

∞∑

n=1

1n

[coth(πn) − 1] + 2π∞∑

n=1

1n

= 5.3134 + 2π∞∑

n=1

1n.

132 Chapter 6

The term for x→ 0 has to be subtracted from the expression above

+∞∑

n,m=−∞

x∞(1 − e−x∞(m2+n2))2

x∞(m2 + n2) − 1 + e−x∞(m2+n2)=

2∫ +∞

0dy1

∫ +∞

−∞dy2

(1 − e−(y21+y2

2))2

(y21 + y2

2) − 1 + e−(y21+y2

2

=

2∫ 1

0dy1

∫ +∞

−∞dy2

(...)

+ 2∫ ∞

1dy1

∫ +∞

−∞dy2

(...)

=

2 × 4.89 + 2

0.63 +

∫ ∞

1dy1

∫ +∞

−∞dy2

y21 + y2

2

= 11.05 + 2∫ ∞

1

π dyy

.

Finally, substituting Ncells,Ns and `2 = πd2/(4 f ) we get per volume of one rod

U(T ET )

T=

332

κMp2d2

a2f

[−2.798 + ln

d2

f Na2

]. (6.115)

As one can see from the comparison with (6.113), the lattice energy of the tetragonalstructure is always higher than for hexagonal H1 phase. This is the primary reasonwhy the tetragonal microphase is always suppressed by the hexagonal one.

Lamellar structure (κN < 1)

Let us start again from the expression (6.99). In the case of a lamellar structure wedeal with a 1D lattice. Following the outline of the previous section we first calculate|∆φ(k)|2, which has exactly the same form (6.101)-(6.103) as before. Cartesiancoordinates of the vectors of the direct a = (1; 0; 0) and reciprocal b = 2π(1; 0; 0)lattices can be easily found. Volume of the reciprocal cell reads v′ = (2π)3/(`L2).

Therefore, the free energy is obtained as

F (LAM)int

T=NcellsN2

s

4Nν

L2l

∑

n

(1 − e−xn2)2

xn2 − 1 + e−xn2=

NcellsN2s

4Nν

L2

√6

2πN1/2a

∑

n

√x

(1 − e−xn2)2

xn2 − 1 + e−xn2, (6.116)

6.C. Free energy of the lattice formation 133

where x =(2πN1/2a/`

)2/6.

Summation leads to

+∞∑

n=−∞

√x

(1 − e−xn2)2

xn2 − 1 + e−xn2= 2√

x + 2√

x+∞∑

n=1

(1 − e−xn2)2

xn2 − 1 + e−xn2'

2√

x + 2√

x+∞∑

n=1

1xn2

= 2√

x +π2

3√

x.

For x = x∞ → 0:

+∞∑

n=−∞

√x∞

(1 − e−x∞n2)2

x∞n2 − 1 + e−x∞n2=

∫ +∞

−∞dy

(1 − e−y2)2

y2 − 1 + e−y2' 5.644.

Rewriting (6.116) after summation one obtains for the free energy per rod (6.75)

U(LAM)

T=

π2

128Mκp2d2

a2+

M f κp2d√

N8a

−1.728 +a f√

Nd

. (6.117)

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[152] A. N. Semenov. “Theory of diblock-copolymer segregation to the interfaceand free surface of a homopolymer layer”. Macromolecules, 25:4967–4977,1992.

[153] M. Knaapila, O. Ikkala, M. Torkkeli, K. Jokela, R. Serimaa, I. P. Dolbnya,W. Bras, G. ten Brinke, L. E. Horsburgh, L.-O. Pålsson, and A. P.Monkman. “Polarized luminescence from self-assembled, aligned, andcleaved supramolecules of highly ordered rodlike polymers”. Appl. Phys. Lett.,81:1489–1491, 2002.

[154] M. Knaapila, J. Ruokolainen, M. Torkkeli, R. Serimaa, L. Horsburgh,A. P. Monkman, W. Bras, G. ten Brinke, and O. Ikkala. “Self-organizedsupramolecules of poly(2,5-pyridinediyl)”. Synthetic Metals, 121:1257–1258,2001.

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[155] H. Kosonen, J. Ruokolainen, M. Knaapila, M. Torkkeli, R. Serimaa,W. Bras, A. P. Monkman, G. ten Brinke, and O. Ikkala. “Self-organizedsupermolecules based on conducting polyaniline and hydrogen bondedamphiphiles”. Synthetic Metals, 121:1277–1278, 2001.

[156] M. Knaapila. “Review of some simple results of phase behaviour in associatedrod-like and flexible chains related to my work with PPY(MSA)1.0 complexedwith octyl gallate”. Private communication, 2002.

[157] P. M. Chaikin and T. C. Lubensky. Principles of condensed matter physics.Cambridge Univ. Press, Cambridge, 1995.

[158] G. A. Korn and T. M. Korn. Mathematical handbook for scientists andengineers: definitions, theorems, and formulas for reference and review.McGraw-Hill Book Co, New York, 1968.

List of Figures

1.1 Examples of polymer architectures: (a) linear polymer, (b) star, (c) combcopolymer “bottlebrush”, (d) comb copolymer with rigid side chains,(e) “hairy-rod” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Cylindrical core-shell brush molecule in (a) good for both blocks and(b) good for PAA but poor for PS solvent. . . . . . . . . . . . . . . 5

1.3 Schematic illustration of a nanowire formation. Core–shell cylindri-cal brushes with a PVP core and PS shell are loaded with HAuCl4.Subsequent reduction yields a one-dimensional gold phase within themacromolecular brush. . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Sketch of the phase separated side chains in a horseshoe brush (left)and in a meandering brush (right). The selective solvent is good forthe thick and poor for the thin side chains. . . . . . . . . . . . . . . 8

1.5 Schematic representation of the three classical morphologies, fromthe left to the right: lamellar, cylindrical (hexagonal lattice), andspherical (body-centered cubic lattice). The diblock molecule con-sists of a gray and a white blocks. . . . . . . . . . . . . . . . . . . 9

1.6 Model of a bottlebrush. The insets illustrate the representation ofthe backbone between two successive grafting points according toBirshtein (a) and Fredrickson (b). . . . . . . . . . . . . . . . . . . 14

2.1 Model of comb copolymer molecule with rigid side chains. . . . . 262.2 Schematic illustration of the interaction between two rods for a straight

brush. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3 Distribution function f (θ) for L = 200 and b = 2. . . . . . . . . . . 31

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152 LIST OF FIGURES

2.4 Order parameter as a function of the length of rods for b = 2. Thesolid line is the theoretical curve, and points represent the simulationresults. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Interacting rods in a bent brush. . . . . . . . . . . . . . . . . . . . 34

3.1 Schematic representation of a 2D comb copolymer molecule. . . . . 413.2 Test rod and its nearest neighbors for a straight comb copolymer

molecule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Behavior of tilting parameter y (dashed line) and tilting angle ϕ (solid

line) as a function of the interaction strength ε. . . . . . . . . . . . . 453.4 Dense packing of rods in the straight molecule (the direction of tilting

can also be opposite at opposite sides of the backbone). . . . . . . . 463.5 The test rod and its nearest neighbors for a bent comb copolymer

molecule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.6 Ordering of side chains in the concave part of the molecule in the

strong attraction limit. . . . . . . . . . . . . . . . . . . . . . . . . . 503.7 Domain of n rods on the convex part of the molecule: (a) the general

case considered in (3.31); (b) the ordering corresponding to n = 1;(c) the “complete” cluster (n = n∗). . . . . . . . . . . . . . . . . . . 51

3.8 Schematic representation of dependence of the bending moment onthe curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.9 Illustration for the calculation of the energy of attraction betweentwo rods: (a) two rods of length L at a distance h from each other(h L); (b) the test rod between its nearest neighbors fixed in theiraverage position on the straight backbone; (c) two arbitrary orientedneighboring rods on the straight brush; (d) two arbitrary orientedneighboring rods on the bent brush. . . . . . . . . . . . . . . . . . 55

4.1 Schematic representation of a comb copolymer molecule. Thick andthin lines represent two chemically different side chain. . . . . . . . 61

4.2 Cross section of the molecule: (a) in the separated state; (b) in themixed state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Schematic representation of a comb copolymer molecule in the sep-arated state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.1 Model of the hairy-rod. . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Illustration of the interface free energy calculation. . . . . . . . . . 82

LIST OF FIGURES 153

5.3 Possible microstructures: (a) hexagonal H1; (b) hexagonal H2; (c)lamellar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.4 Phase diagram of the melt of hairy-rod molecules. H1-H2 coexis-tence line is given by (5.10); H2-L (solid part) corresponds to (5.19). 85

5.5 Cross section of a hexagonal H2 phase cylindrical micelle in thevicinity of the H2-Lam transition. Note the total cross section is stillclose to circular. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.1 (a) Model of the hairy-rod as a stiff backbone with reversibly attachedflexible side chains. (b) Flat interface between pure nematic andisotropic phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2 Macrophase equilibria in the associating rod-coil system: (a) no com-patibility for small association energy; (b) enhanced compatibility inthe ε/w > 2b/d case. . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.3 Possible hexagonal phases: (a) hexagonal H1; (b) hexagonal H2. . . 1016.4 Complete phase diagram in the case κN > 1 . . . . . . . . . . . . . 1046.5 (a) Single lamella and its surrounding. (b) Two neighboring lamellae. 1096.6 Two more possible phase diagrams in the κN > 1 case (see also

Figure 6.4 and eqs (6.73), (6.74)) . . . . . . . . . . . . . . . . . . 1156.7 Phase diagram of the system for κN < 1. . . . . . . . . . . . . . . 1176.8 Two cylindrical micelles (a cross section) . . . . . . . . . . . . . . 1246.9 The interaction free energy of two cylindrical micelles separated by

distance r12 = xa√

N . . . . . . . . . . . . . . . . . . . . . . . . . 1256.10 An elementary cell of a hexagonal lattice. . . . . . . . . . . . . . . 1276.11 An elementary cell of a tetragonal lattice. . . . . . . . . . . . . . . 131

Samenvatting

De vraag, die we in dit proefschrift trachten te beantwoorden, is welke invloed dearchitectuur van een polymeermolecuul heeft op enerzijds de conformatie van hetmolecuul en anderzijds het fasegedrag vanzowel één enkel molecuul (intramolec-ulaire fasescheiding) als de polymeersmelt (microfasescheiding). Een polymeer-molecuul bestaat uit een lange keten opgebouwd uit kleine chemische eenheden –“monomeren”. Het eenvoudigste voorbeeld is een lineair homopolymeer dat uit éénsoort monomeren bestaat die in de vorm van een lineaire keten covalent gebondenzijn. Als twee of meer soorten homopolymeren chemisch aan elkaar vastzitten, danspreekt men over een blokcopolymeer. Soms vormen de gekoppelde homopolymeer-ketens, ook wel blokken genoemd, vertakte, e.g. kamvormige, structuren.

Dit proefschrift beperkt zich tot zulke kamvormige polymeren (kampolymeren)die bestaan uit een homopolymere backbone waaraan zijketens zijn vastgemaakt.Eerst beschouwen we de invloed van de zijketens op de conformatie, d.w.z. vorm, vanhet kampolymeer. Daarbij spelen de zijketeneigenschappen, zoals lengte, rigiditeit,enz., een belangrijke rol. Verder wordt er aandacht besteed aan het fasegedrag vanzogenaamde “hairy-rods” – copolymeren die uit een stijve backbone en flexibelezijketens bestaan. Als de zijketens met behulp van een chemische binding aan debackbone zijn vastgemaakt, dan spreken we van covalent hairy-rods. Anders, wan-neer de binding tussen de hoofd- en zijketens van fysische aard is (bijvoorbeeldvia waterstofbrugvorming), worden ze supramoleculaire hairy-rods genoemd. Indat laatste geval zijn de zijketens niet permanent gebonden, maar wordt de bindingdynamisch gebroken en gevormd.

In de introductie, hoofdstuk 1, wordt de experimentele en theoretische achter-grond besproken. Onder andere worden er voorbeelden gegeven van bijzondere con-formaties van kampolymeermoleculen en hoe die door de aanwezigheid van zijketens

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156 SAMENVATTING

gecontroleerd kunnen worden. Vervolgens presenteren we een beknopt overzicht vande relevante theorieën.

Tijdens de laatste decennia is er veel aandacht besteed aan de elastische eigen-schappen van kampolymeren in een verdunde oplossing. Wanneer de oplosmiddel-kwaliteit “goed” is met betrekking tot de zijketens, dan kan de interactie tussen dezijketenmonomeren met het “uitgesloten volume” model beschreven worden. Deaanwezigheid van lange zijketens, die elkaar afstoten, heeft vooral een sterke invloedop de molecuulconformatie. De effectieve “verstijving” van het molecuul wordtuitgedrukt met behulp van de persistentielengte λ: hoe stijver het molecuul hoe groterλ.

Het fasegedrag van een smelt van “hairy-rod” moleculen heeft veel minder aan-dacht gekregen. De drijvende kracht van de fasescheiding in dit systeem is de aan-wezigheid van twee verschillende soorten monomeren (de éne van de backbone ende andere van de zijketens) die elkaar afstoten. Omdat ze aan elkaar gebondenin hetzelfde molecuul zitten, is een volledige opsplitsing, “macroscheiding”, nietmogelijk. Echter, het aantal contacten tussen de chemisch verschillende monomerenkan worden verminderd door de scheiding op microscopische schaal. Dat leidt totde vorming van kleine gebiedjes die rijk zijn aan één van de monomeersoorten enwordt microfasescheiding genoemd. In het geval van hairy-rods nemen deze ge-biedjes meestal de vorm van laagjes aan die afwisselend rijk zijn aan backbones enzijketens. Ondanks de belangrijke toepassingsmogelijkheden van hairy-rods (bijv.op het gebied van geleidende polymeren), bestond er tot nu toe geen theorie over demicrofasescheiding van degelijke systemen. De ontwikkeling van deze theorie is éénvan de doelstellingen van dit proefschrift.

In hoofdstuk 2 wordt aandacht besteed aan de elastische eigenschappen van eenkamvormige polymeermolecuul met een semi-flexibele hoofdketen en stijve zijketens.De vraag over het verschil van invloed tussen stijve en flexibele zijketens wordt beant-woord. De interacties tussen de monomeren worden in de “mean-field” benaderingin rekening gebracht. De persistentielengte λ van het molecuul is afhankelijk van dezijketenlengte L en wordt groter voor langere zijketens: λ ∝ L2/ ln L. Dit bevestigtdat het verstijvingseffect voor stijve zijketens groter is dan voor flexibele zijketens.

In het daarop volgende hoofdstuk 3, wordt het effect van attractie tussen de zij-ketenmonomeren bestudeerd. Er wordt voornamelijk gekeken naar een sterk aan eenoppervlak of grenslaag geadsorbeerd kampolymeermolecuul. De beperking van hetsysteem tot twee dimensies maakt de invloed van stijve zijketens op de conformatienog sterker dan in 3d. Afhankelijk van de attractiesterkte ε kunnen twee verschillende

SAMENVATTING 157

regimes worden onderscheiden. Wanneer de attractie relatief zwak is, neemt depersistentielengte af naarmate ε groter wordt. In het regime van sterke attractie blijkthet dat de persistentielengte razendsnel stijgt als ε en L toenemen: λ ∝ ε 2L4.

Tot zover hebben we ons beperkt tot vertakte polymeren met slechts één soortzijketens. In het geval van twee chemisch verschillende A en B zijketens die aandezelfde backbone vastzitten, kan een ruimtelijke scheiding van zijketens binnen degrenzen van één molecuul worden verwacht. Het punt waarop zo’n fasescheidingplaatsvindt, wordt door de Flory-Huggins interactieparameter χAB bepaald. Omdathet molecuul in zo’n microfasegescheiden staat asymmetrisch is, kunnen bijzondereboogachtige conformaties stabiel worden. Door de temperatuurafhankelijkheid vanχAB kan de conformatie extern gecontroleerd worden: als de temperatuur wordtverlaagd schakelt het molecuul om naar de geboogde staat. Deze eigenschap kande basis voor potentiële toepassingen vormen, zoals nanoschakelaars.

Een belangrijke deel van dit proefschrift is gewijd aan “hairy-rod” systemen.Dergelijke architecturen kunnen zich in een ruime aandacht verheugen in verbandmet de verwerkbaarheid van stijve polymeren die zonder flexibele zijketens nauwe-lijks oplosbaar zijn. De aanwezigheid van flexibele zijketens maakt het molecuuloplosbaar in organische oplosmiddelen en is van groot industriëel belang aangeziende meerderheid van de geleidende polymeren stijf is. Omdat de hairy-rods uit tweemonomeersoorten zijn opgebouwd (backbone en zijketen), is microfasescheiding inde smelt mogelijk. In hoofdstuk 5 wordt het microfasegedrag van de covalentehairy-rod polymer smelt berekend. Afhankelijk van de temperatuur en de samen-stelling van het molecuul, kunnen cilindrische of gelaagde microstructuren ontstaan:de backbone-rijke gebiedjes kunnen de vorm van cilinders of laagjes aannemen dieruimtelijk in een periodieke structuur georganiseerd zijn.

In het laatste hoofdstuk 6 wordt opnieuw gekeken naar de hairy-rod polymeersmelt, echter nu wordt het supramoleculaire systeem onderzocht. De supramole-culaire hairy-rods ontstaan door eindstandige associatie tussen flexibele ketens enstaafvormige ketens (backbones), ketens die zonder associatie in essentie onmeng-baar zijn. Wanneer de associatie sterk genoeg is, kunnen de supramoleculen behan-deld worden op een vergelijkbare manier als de covalente hairy-rods. Een essentiëlecomplicatie is dat naast louter microfasescheiding ook macrofasescheiding mogelijkis waarbij een microfasegescheiden fase in evenwicht is met fasen die uit pure staafjesof flexibele ketens bestaan. Afhankelijk van de zijketenlengte en de sterkte van debinding zijn drie verschillende microfasen mogelijk: twee cilindrische waarbij decilinders resp. één dan wel meerdere backbones bevatten en een gelaagde. Wanneer

158 SAMENVATTING

de zijketens relatief kort zijn treedt alleen de gelaagde structuur op. Dat verklaarttevens het overheersende beeld van gelaagde structuren zoals dat uit de experimenteleliteratuur naar voren komt.

Dankwoord

Allereerst wil ik mijn promotor, professor Gerrit ten Brinke, bedanken voor zijnvoortdurende steun, inzicht en kundigheid. Zijn bijzondere persoonlijke eigenschap-pen hebben niet alleen voor een gerichte wetenschappelijke begeleiding gezorgdmaar ook voor de gezellige en vriendelijke atmosfeer in de groep. I also wish tothank Andrei Subbotin for all his ideas, patience, and energy during our four-yearcollaboration. Without him this thesis would have not seen the light of day in theform it is now.

The members of the reading committee, professor Igor Erukhimovich, professorHan Slot, and professor Ulli Steiner, are greatfully acknowledged for their carefulreading and useful comments.

A large part of this manuscript is based on the work done in collaboration withMatti Knaapila and professor Olli Ikkala. I really enjoyed our collaboration, espe-cially numerous email discussions with Matti.

Dan heb ik nog een flink aantal collega’s die ik wil bedanken voor hun directeen indirecte rol die ze hebben gespeeld in het afronden van dit proefschrift. Johande Jong: jouw kennis op gebied van Unix systemen is van onschatbare waardegeweest. Ook nog bedankt voor de leuke lange gesprekken tijdens onze gezamenlijketreinreizen naar Utrecht voor de RPK cursus in 2001 en voor het paranimfschap.Rikkert Nap wil ik bedanken voor het delen van zijn buitengewone historische kennisen voor de instemming om samen te promoveren. Mijn tweede paranimf ManuelReenders: voor de nuchtere kijk op de dingen en voor een goed voorbeeld vanhet consciëntieuse wetenschappelijke werk. Evgeny Polushkin: thanks for yourfriendship. Your support at diffucult moments was very important for me.

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160 DANKWOORD

Graag bedank ik mijn collega’s en kamergenoten Jelger Risselada, Joost de Witen Gerrit Gobius du Sart voor hun goede humeur en muziek welke voor een bijzon-dere gezelligheid op onze kamer zorgden. I also thank Edwin Flikkema, Karin deMoel, Christian Kok, Henk Angerman, Yulia Smirnova, and Sasa Bondzic for theirkind friendship.

Tenslotte wil ik nogmaals mijn paranimfen, Johan en Manuel, bedanken voor hunsteun tijdens mijn promotie.