d 5 structure of soft matter: small angle scatteringwpage.unina.it/lpaduano/phd lessons/neutron...

D 5 Structure of Soft Matter: Small AngleScattering

Henrich Frielinghaus

Julich Centre for Neutron Science

Forschungszentrum Julich GmbH

Contents

1 Introduction 2

2 Small Angle Scattering 22.1 Scattering Length Density . . . . . . . . . . . . . . . . . . . . . . . . .. . . 42.2 First Born Approximation . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 52.3 Independent Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 62.4 Spheres on a Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92.5 Short Summary of Part One . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11

3 Further Examples of Soft Matter Research 123.1 Polymer Blend Phase Diagrams . . . . . . . . . . . . . . . . . . . . . . .. . 123.2 Lattice Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 133.3 Scattering Functions of Polymer Blends . . . . . . . . . . . . . .. . . . . . . 143.4 Microemulsions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .173.5 Contrast Variation Experiments . . . . . . . . . . . . . . . . . . . .. . . . . . 173.6 Summary on Polymers and Microemulsions . . . . . . . . . . . . . .. . . . . 19

D5.2 Henrich Frielinghaus

1 Introduction

Soft Matter embraces many different substances which have to be described by manyfold con-cepts. These concepts are also taken from the neighboring material sciences. On the one handthe conventional solid state physics mainly focused on solid substances with many crystallineexamples. On the other hand the simple liquids describe the other border. In this sense softmatter describes materials between the crystalline and theliquid state. This can be understoodin terms of structure (long range order versus short range order), but also the dynamic responsesof soft matter include elastic and viscous properties. Thus, viscoelastic materials and complexfluids are typical soft matter materials. In this sense the materials aresoft. In the following, twoexamples should be discussed: Polymers are long chain molecules with interesting properties,and microemulsions are mixtures of oil and water mediated bya surfactant. Microscopicallystill domains of oil and water remain, but macroscopically these liquids appear homogenousand are thermodynamically stable.

The structural appearance of soft matter materials should be discussed on the basis of polymers.The smallest distance is represented by a chemical bond withthe typical length scale of 1.5A.The repeat unit of a polymer, the monomer, embraces already several atoms. The connectivityof a chain becomes visible on the length scale of several monomers, i.e. around 10A. The chain-like molecules usually form coils resembling a random walk.Thus the whole chain is visiblebetween approx. 30A and 500A. In polymeric blends consisting of several polymers demix-ing can occur. The typical domain size reaches from 300A to several 1000A. On even largerlength scales, superstructures can appear with regular or fractal properties. The driving forcefor domain and larger structure formation is a reduced miscibility of at least two components.Especially when the structure is regular one speaks of self assembly. Many samples appear ho-mogenous on much larger length scales (say at 1 to 10µm) and can be even transparent for light.This is the reason why scattering experiments are considered for the structure determination.

A typical small angle scattering experiment with neutrons or x-rays detects structural sizesof 10A to several 1000A. Especially ultra small angle methods extend this range toseveralµm. The small angle light scattering method begins in theµm-range due to the larger wavelength of the probe. Hierarchical structures often allow toseparate the length scales in distinctranges. So a single structural property becomes dominant for a certain lenght scale. For instancesmall angle scattering experiments do not detect the chemical structure of monomers anymore.Monomers appear as homogenous beads connected to a long chain molecule.

2 Small Angle Scattering

A typical small angle scattering experiment is schematically shown in Figure 1. The radiation ofthe source is prepared for the experiment by a monochromatorand a collimation line. Neutronsare generated by a reactor or a spallation source. The monochromator is a rotating cylinder withtilted lamellae, and selects neutrons of a certain wave length. The collimation consists of two

Structure of Soft Matter D5.3

Fig. 1: Scheme of a small angle scattering diffractometer. The neutrons pass fromthe left to the right. The incident beam is monochromated andcollimated beforeit hits the sample. Non-scattered neutrons are absorbed by the beam stop in thecenter of the detector. The scattered neutron intensity is detected as a function ofthe scattering angleθ.

apertures, which define the maximum divergence of the incident beam. Right after the secondaperture the sample is placed. Usually, many neutrons pass the sample unscattered. This highintensity has to be blocked by a beam stop close to the detector due to saturation problems ofthe detector. Only the scattered neutrons are detected by the detector. The scattered intensity isrecorded as a function of the scattering angleθ. The scattering vectorQ is related toθ by:

Q =4π

λsin

(

θ

2

)

≈2π

λθ (1)

If a (quasi) Bragg peak is detected at a scattering vectorQ the underlying length scaleof thestructure is given by:

` =2π

Q(2)

This concept is the reason why theQ-space is called reciprocal space. Large structures showscattering at smallQ, while small structures appear at large scattering vectors. Hierarchicalstructures appear with different scattering laws for differentQ-ranges. Often the scattering lawappears to be a power lawQ−α, indicating that the structure can be understood as a fractal.The transition points between the scattering laws indicatethe structural size (as given by eq. 2)between the two conceptually different structures.


2.1 Scattering Length Density

With discussing the different length scales of a polymer we already mentioned that the detailedmolecular structure of a monomer is not resolved by a typicalsmall angle scattering experi-ment. For neutrons the typical wavelength is 7A. Thus, a single bond between carbon atomscannot be resolved. The amplitude of the incident beam can beconsidered constant within 1Adistant points. The second argument is the consideredQ-range of a typical (neutron or x-ray)small angle scattering experiment. The maximal consideredscattering vector often is 0.3A−1,and according to eq. 2 this translates to length scales of at least 20A. So a simple saturatedcarbon chain (also called polyethylene) appears as a homogenous worm without distinctions ofhydrogen or carbon atoms.

When mixing two different substances, a scattering signal is only obtained for different scat-tering powers. The way of generating contrast for neutrons is the substitution of hydrogen bydeuterium. Even the otherwise identical protonated and deuterated polymer can be studied byneutron scattering. Undisturbed chain conformations [1] and chain dynamics [2] were studiedin this way. The scattering amplitude is proportional to thecontrast, which is given by thescattering length density difference. For neutrons one obtains:

ρA(neutrons) =1

VA

∑

i

bi (3)

The scattering length densityρA is the sum of all scattering lengthsbi of all atoms of onemonomer A divided by the monomer volumeVA. Examples for scattering lengths are given intable 1 and can be found on a web page [4].

For x-rays the electron density difference describes the contrast. The scattering length densityis calculated according to:

ρA(x-rays) =re

VA

∑

i

Zi (4)

The classical electron radius isre = e2/(4πε0mec2) = 2.82fm. The electron number of each

atomi is Zi. Equation 4 means that chemically different substances do have a contrast, but itcan be rather weak. Not only the sum of all electrons, but alsothe mass density finally playsa role for the x-ray contrast. The usually high intensity of the incident beam allows to collectscattered intensity even for low contrasts. The disadvantage of the high intensity is the potentialradiation damage.

For the present discussion of small angle scattering it doesnot play a role that the electronsoccupy a certain space around the nucleus, while for neutrons the scattering nucleus is a point-like particle. But when studying the crystalline structureof conventional salts for instance thisdifference has to be taken into account for the analysis of the scattering pattern and especiallythe intensities.


chemical scattering electron

element lengthbi [fm] numberZi

C 6.646 6

H -3.741 1

D 6.671 1

O 5.803 8

Table 1: Scattering lengths and number of electrons for some elements.

Another practical point is the transmission of the radiation. X-rays are rather strongly absorbedby usual samples. Thus, soft matter samples should have a thickness of 0.1mm. For neutronssample thicknesses of 1mm to 5mm or even more are possible. Ifmultiple scattering (morescattering events) plays a role the measured intensity is smeared out, and is usually hard to beinterpreted. The 1st Born Approximation is described in thefollowing section and assumessingle scattering processes only.

For light scattering experiments the polarizability playsan important role. Without going intodetails the final contrast is expressed by the refractive index incrementdn/dc:

ρA(light) =2πn

λ2·dn

dc(5)

The refractive index incrementdn/dc finally has to be determined separately when the absoluteintensity is of interest. The concentrationc is given in units volume per volume. The usedwavelength of light isλ.

2.2 First Born Approximation

The full presentation of the first Born approximation is quite lengthy. Thus, the essential pointsfor small angle scattering experiments are presented here only. For further reading reference [3]is recommended. The assumptions of the Born approximation are the following: A monochro-matic and non-divergent beam is assumed to irradiate (and pass) the sample. The sample poten-tial is a small perturbation for the incident beam, and thus the scattered outgoing waves carryjust a small fraction of the total radiation. Practically, as a rule of thumb, the total coherentlyscattered intensity should be less than 10% of the incoming intensity. The scattering occursonly once. That means that the outgoing waves arise from a single scattering process out ofthe incident wave. Multiple scattering would mean a redistribution of outgoing waves. Theoutgoing intensity is observed at large distances comparedto the sample diameter. The mayorresult of the first Born Approximation is that the amplitude of the outgoing waves is determinedby the Fourier transform of the scattering length density:


A(Q) =

∫

d3r ρ(r) · exp(iQr) (6)

The imaginary unit is calledi here. Please note that the argument carries the scalar product ofthe scattering vectorQ and the spatial vectorr. The scattering intensity is proportional to themacroscopic cross sectiondΣ/dΩ:

dΣ

dΩ(Q) =

1

Vtot|A|2(Q) (7)

One important property of this equation is the absolute square of the amplitude. This means thatthe observed intensity does not contain any information about the phases anymore. So everyscattering experiment interpretation has to be aware that the structure in real space cannot easilybe reconstructed. Nontheless most probable realizations can be calculated [5].

The macroscopic cross section is the instrument independent information about the sample,while a real measurement of the intensity carries also properties of the instrumental setup:

dΣ

dΩ(Q) =

∆I

∆Ω·

1

I0dSASTSε(8)

The intensity∆I is collected in a detector element covering the solid angle∆Ω. It has to benormalized to the incident intensityI0, the sample thicknessdS and sample areaAS, the sampletransmissionTS and the detector efficiencyε. By the use of a calibration standard with knownscattering power the geometrical and technical constants of the diffractometer machine can beremoved and the unique information about the sampledΣ/dΩ is obtained. This procedure isstandard for neutron scattering, but also possible for x-ray and light scattering.

2.3 Independent Spheres

As a first example a collection ofN independent homogenous spheres shall be considered. Arealistic sample could contain spherical colloids with an extremely narrow size distribution. Thescattering amplitude of a single sphere would be calculatedas the following:

A1(Q) =

∫

r<R

d3r ρ1 · exp(i ~Q~r) (9)

The coordinate system should be oriented in this way that theQ-vector points along the z-axis.The vector~r is represented in spherical coordinates. Then one finds:

A1(Q) = ρ1 ·

∫ 2π

0

dφ

∫ π

0

dθ

∫ R

0

dr exp(irQ cos θ)r2 sin θ (10)


Integration over the angles leads to:

A1(Q) = ρ1 · 2π · 2 ·

∫ R

0

drsin(Qr)

Qrr2 (11)

And finally one obtains:

A1(Q) = ρ14π

3R3 · 3

sin(QR) − QR cos(QR)

(QR)3(12)

The considered integral was carried out just for the colloidal particle. In reality it is embeddedin some environment with a scattering length densityρ2. The scattering amplitude would readthen:A1(Q) = (ρ1−ρ2)

4π3R3 ·3 sin(QR)−QR cos(QR)

(QR)3+ρ2δ(Q). Now the scattering length density

difference appears in the final expression as it is expected.Furthermore a Dirac delta func-tion δ describes the scattering from an infinitely large volume of the embedding environment.Mathematically, this term is correct, but physically it does not mean anything. Thisδ-functiondescribes some intensity at scattering vectorsQ = 0 where the transmitted, non-scattered beamis found anyways. So there is practically no distinction between the two intensities atQ = 0. Inthe following, thisδ-function is simply neglected. For the final macroscopic cross section, theamplitudes are squared and normalized to the total volume:

dΣ

dΩ(Q) =

N

VtotA2

1(Q) = φ1 · V1 · (∆ρ)2 ·

(

3sin(QR) − QR cos(QR)

(QR)3

)2

(13)

First of all,N scattered intensities of independent spheres were summed up independently. Thismeans that the superposition was taken out incoherently. Inreality many relative arrangementsare realized within the experimental exposure time, and so special arrangements are averagedout. Another argument aims at the coherence volume of the incident beam. The full volumerepresents many realizations of such sub-volumes.

The final result of eq. 13 shows some important properties: The first three factors contain theconcentration (volume fraction)φ1, the individual volumeV1 = 4π

3R3, and the contrast factor

(∆ρ)2. These factors describe the extrapolated intensity at small scattering angles (forward-scattering). Here, ‘extrapolated’ means a measurement close to the beam stop but neglectingthe non-scattered neutrons. The last factor is the formfactor of the particleF (Q), and is depictedin Figure 2. The limit at low scattering angles is unity. At larger anles many oscillations occurdue to the finite size of the particle. The first minimum occursatQmin = 4.493/R, which againshows the relation between real and reciprocal space: the larger the sphere the smallerQmin. Thegeneral structure of eq. 13 is kept for all dispersions and solutions with non-interacting particles:The forward-scattering tells about the concentration, theparticle volume and the contrast. Withknowing two of them, the third one is determined experimentally. The particle shape and sizeis determined by theQ-dependence of the intensity. Here, usually modelling helps to extractmany details of the experiment.


Fig. 2: The form factor of a solid sphere as a function of the dimensionless variableQR. The thin line with heavy oscillations represents the strict formfactor with asingle radiusR. The thick line with damped oscillations represents a formfactorwith a polydispersity ofR of 10%.

A second effective formfactor is plottet in Figure 2. Here a distribution of shperes with differentradii is assumed (polydispersity). The width of theR-distribution was assumed to be 10%. Theminima are not as sharp as for the monodisperse spheres. Especially the oscillations of higherorder are simply represented by a straight line. The straight line of this double logarithmicplot is connected with aQ−4 power law. It can be shown that all compact volumes have sucha scattering behaviour [6] (Porod scattering). Practically, all oscillations average out due tonon-negligible polydispersities, and one observes:

dΣ

dΩ(Q) = P · Q−4 (14)

The amplitude of the Porod scatteringP tells about the surface per volume. The general ap-pearance of the Porod constant isP = 2π(∆ρ)2Stot/Vtot. So, apart from the contrast, it mea-sures the total surfaceStot per total volumeVtot. For our shperes the Porod constant becomesP = 2π(∆ρ)24πR2/(4πR3/(3φ)) = 6πφ1(∆ρ)2/R. The surface to volume ratio is smallerthe larger the individual radiusR is. The remaining scaling with the concentrationφ1 and thecontrast(∆ρ)2 arises still from the prefactor which we discussed in context with eq. 13.


2.4 Spheres on a Lattice

Now the spheres shall be considered to have dependent positions. A typical soft matter ex-ample is a diblock copolymer. This is a linear polymer with two blocks consisting of differ-ent monomers. Once the two monomers become immiscible, a phase separation on macro-scopic scales cannot take place, but small micro-domains can form consisting of (almost) onemonomeric kind. This microphase separation can lead to spherical domains on a bcc (bodycentered cubic) lattice embedded in the other continuous domain.

If we calculate the amplitude now, the phases of all spheres have a relation. The integral overall spheres splits into a sum over all particles and an integral over the volume of one sphere:

A(Q) = (∆ρ)∑

h,k,l

exp(ia(

hkl

)

~Q)

∫

sphere

exp(iQr) (15)

The lengtha marks the basic distance of the lattice. The indicesh, k, l run over all positions,where a sphere is found. Now, the integral we have already solved (eq. 12), but the bcc latticestill needs to be Fourier transformed. From classical solidstate physics we know, that a bcc lat-tice transforms in a fcc lattice and vice versa. When calculating the macroscopic cross section,we basically have to calculate the quadrature. So:

dΣ

dΩ(Q) =

1

Vtot|A|2(Q) = (∆ρ)2 · S(Q) · F (Q) (16)

The form factor we already knowF (Q) = (3 sin(QR)−QR cos(QR)(QR)3

)2, while the structure factorstill is a sum over all points of the fcc lattice in reciprocalspaceS(Q) ∝ N

∑

h,k,l δ(Q0h −

Qx)δ(Q0k − Qy)δ(Q0l − Qz). The quadrature did not harm the peak shape, which appearsinfinitely sharp in this representation. In reality, the crystal has a finite size, and so the peakwidth is connected with its size. For polycrystals (powder samples) the average crystal domainsize determines the peak width. ThisS(Q) has the property that it is proportional to the numberof scatterers within a crystal domain, since all the waves ofone crystal domain superimposecoherently.

An example of such a soft matter system in shown in Figure 3. The real system [7] consistsof a polystyrene-polyisoprene block copolymer dissolved in squalane. The polystyrene blockis immiscible neither in polyisoprene nor in squalane, and forms the spherical domains, whilepolyisoprene and squalane mix well and form the embedding matrix. This scattering experi-ment is conducted with x-rays since the resolution is much better than with usual small angleneutron scattering diffractometers. Improvements for neutron small angle scattering are aboutto come [8]. The left part of Figure 3 shows the theoretical expectations ofS(Q), andF (Q)

independently, and the productdΣ/dΩ. One directly finds the 6th maximum suppressed dueto S(Q) and F (Q). This calculation is done for a powder sample with many orientations ofthe single crystal domains. Thus Debye-Scherrer rings are formed, the one-dimensional cut ofwhich we see here. The indices of the first seven peaks read (1,1,0), (2,0,0), (2,1,1), (2,2,0),


Fig. 3: On the left the theoretical structure and form factor (S(Q) andF (Q)), andthe product, the macroscopic cross sectiondΣ/dΩ are shown. The calculationsbase on a lattice constant of 380A and a spherical radius of 80A. On the right handside the x-ray experiment [7] shows peaks up to the 7th order.The real systemconsists of a polystyrene-polyisoprene diblock copolymerand squalane as a goodsolvent for the polyisoprene. Polystyrene spheres are formed in the matrix of theother two components.

(3,1,0), (2,2,2), and (3,2,1). The number of permutations and possible signs give rise to theintensity of a single peak. On the right hand side the experimental data are shown. The firstfour peaks are clearly indicated in a log-lin plot, while the5th and 7th order peak are indicatedin the inlay. The 5th order peak is really weak, but the 7th order peak is clearly visible.

To discuss the possible options of structure factors in softmatter science some examples areshown in Figure 4. The lowest one is the already discussed structure factor of a crystallinestructure. It shows clearly developed peaks indicating thepositions of the particles. The limit atsmall scattering angles is zero. This means the ideal crystal looks homogenous on large lengthscales, and no long range fluctuations can occur. This structure is formed due to the repulsiveinteraction of the particles (the polyisoprene corona looses entropy when two coronas intersect)and a particle conservation condition (no particles can leave the volume). A weaker case ofrepulsive interaction [9] is shown in the middle graph of Figure 4. Still some peaks occur dueto neigboured particles, but all correlations are lost in the largeQ-limit. On large lengh scales(low Q) the structure factor is suppressed (smaller 1), and the repulsive interaction becomesvisible. The interactions make the sample look more homogenous than a sample with randomlydistributed particles where the structure factorS(Q) = 1 is constant. The upper case of Fig-ure 4 shows the influence of attractive interaction (in combination with repulsive short rangeinteraction). Again some peaks are visible, indicating thecorrelations of neigboured particles,but now the attractive forces lead to an upturn of the structure factor at low scattering angles.The attraction can lead to cluster formation, and the largerentities appear less homogenous thanrandomly distributed spheres. Remember, the forward scattering can be interpreted by the prod-


Fig. 4: Structure factors of different soft matter examples. The lowest one is alreadydiscussed in Figure 3, and represents a bcc crystal structure. The middle caseappears due to hard sphere interactions of colloidal particles (two particles cannotintersect) [9]. The hard core radius was assumed to be 266A, and the concentrationwas 20%. If some attractive forces come into play, such as depletion forces, thestructure factor shows an upturn at low scattering angles. Depletion means, thatsmall and large particles like to separate due to the larger entropy of more freesmall particles. The weaker form of depletion considered here is an effectivelyattractive interaction of the large particles.

uct of (∆ρ)2φV ; while the first two factors remain constant, larger entities appear by a largervolumeV . The ultimate consequence of attraction are extremely large clusters [10], which canbe formed by colloidal filler particles in rubbers. These particles lead to the strengthening ofpure rubber. The formation of crystalline structures due toattractive interactions can be foundin colloidal dispersions [11]. While the extreme cases of interactions can lead to crystallinestructures, the weak cases of interaction lead to liquidlike structures. Now, we have seen struc-ture factors for the crystalline, the liquid-like and the gas-like (S(Q) = 1) states. Usually, phasetransitions occur between the different states.

2.5 Short Summary of Part One

Soft matter systems appear with many length scales, even hierarchical structures can form. Thetypical small angle scattering experiment smears the atomic structure out, and just an averagechemical structure is seen. The important variable of a small angle scattering experiment is thescattering vectorQ, the variable of the reciprocal space. Large structures appear at smallQ


and vice versa. The Fourier transformation of the scattering length density is connected withthe observed intensity. Due to the quardature the information about the phases is lost. Wehave developped the concept of structure and form factors with some essential examples. Theform factor can display the Porod law, which is typical for compact particles. Polydispersitysmears the oszillations out. The structure factor is dependent on the interactions, which can beattractive or repulsive. The strength of interactions opens a large variety between crystalline[12] and liquidlike structures [9].

3 Further Examples of Soft Matter Research

The concept of structure and form factor is important for soft matter science, but finally doesnot cover all possible examples. In the following some othersamples are shown which takeus away from this simple picture. However, these examples have important applications andappear in our daily life.

3.1 Polymer Blend Phase Diagrams

Polymer blends contain at least two different monomeric units, and are heated to moderatetemperatures where they are at least viscous. The simplest blend contains two homopolymers.Each of them contains just a single type of monomers. Just by mechanical mixing, homogenousblends may form. The mixing entropy is proportional to the reciprocal molar volumeV , andthus relatively small. The entropy supports mixing and has to overcome the enthalpy for ahomogenous blend. This is the explanation why many polymersdo not mix, but of course,positive examples can be found. A phase diagram of a polybutadiene(1,4)/polystyrene blendis shown in Figure 5. At high temperatures the two polymers mix, while at low temperaturesthey do not. The immiscible region is separated from the miscible region by the binodal. Thespinodal separates the metastable and the unstable region.In the metastable region the blendcan be supercooled. At the highest temperature where binodal and spinodal meet the criticalpoint is found.

While phase separation opens another interesting field of soft matter research [13], we focuson the one-phase region where the polymers do mix. As we have seen many times in the lastsection, the polymer blend looks homogenous by visual inspection but scattering experimentsreveal more information. Close to the critical point the blend ‘feels’ the vicinity to the phaseboundary. Energetically slightly demixed domains may occur due to fluctuations with the ther-mal energy. These thermal composition fluctuations are visible by scattering experiments. Thisscattering experiment is performed at several temperatures at the critical composition.

Before discussing further concepts, a similar phase diagram of a diblock copolymer is shownon the right side of Figure 5. This polymer is a linear chain with two blocks consisting of onemonomer species. The chemistry does not allow the polymer tobe phase separated on macro-


scopic length scales, but still microscopic domains of one species can be formed. The top lineobove the phase diagram shows possible structures: The bcc lattice of spheres we have alreadydiscussed. Important are also cylinders on a hexagonal lattice, the double diamond structure(Ia3d) and the lamellar structure. The double diamond structureis also called bicontinuousstructure since either domain forms a continous structure in space. If one domain is etchedaway chemically still a spongelike structure remains and keeps its shape. The temperature axis(ΓV ∝ T−1) is reverted. Otherwise the general structure is similar tothe phase diagram ofhomopolymers. At high temperatures the polymer is disordered, and at low temperatures thepolymer is micro-phase separated (ordered). The in the following discussed experiment alsochanges the temperature in the disordered region until micro-phase separation occurs.

3.2 Lattice Theories

While the form factor usually describes whole domains, we look on the structure of a polymerblend with higher accuracy. The extreme microscopic viewpoint would include all positions ofall atoms of the molecule. As we have discussed before, this viewpoint is too detailed in manycases. So we could embrace molecule parts as units, be it a CH2 unit or a whole monomer (saya styrene monomer for polystyrene). Now we assume that theseunits are space filling, and evenmore crudely they fit on a cubic lattice [15, 16]. This modelling oversimplifies the real system,but many interesting questions can be addressed in a straight way. At each point we can defineconcentration variableφ(r), which turns to 1 or 0 if a monomer A or B is found. If the polymerplacement is space filling (so we assume incompressibility), then we only need this variableφ(r). The scattering length density is directly connected to this variable:

ρA(r) = ρA · φ(r) (17)

The scattering amplitude will be calculated on the basis of the Fourier transformationAA(Q) =

ρA

∫

φ(r) exp(iQr), and the macroscopic cross section finally reads:

dΣ

dΩ(Q) = (∆ρ)2 · S(Q) (18)

The functionS(Q) is called scattering function from now on, and is related to the Fouriertransformed concentration variable according toS(Q) = 1

Vtot|φ(Q)|2.

These simple concepts allow to understand many properties of polymer blends, and are the basisfor understanding an even more complex behaviour if some of the restrictions above are relaxed.The compressibility can be introduced by allowing for lattice voids. This concept is connectedwith the term free volume. Of course one has to introduce two concentrations for each of thepolymer speciesφA andφB, sinceφA + φB < 1.

However, the Random Phase Approximation (RPA) [17] deals with the scattering of such poly-mer blends, and bases on the local concentrationφ of the polymer species. In a first step,


the scattering of a free polymer chain without interactionsis determined [1] assuming simpleproperties of chainlike molecules. Then interactions are introduced, mainly represented by theexchange free energyΓ (this energy is normalized to volume). The RPA describes then, how tocombine the scattering of a free chain and the interaction parameterΓ to obtain the scatteringof an interacting chain. Some predictions of this theory we are going to discuss now.

3.3 Scattering Functions of Polymer Blends

Here we are discussing general properties of scattering functions of the considered polymerblends. The fullQ-dependence gives us valuable details about the polymer kind. For the furtheranalysis, the scattering patterns are reduced to two or three parameters.

In Figure 6 the characteristic scattering patterns of homopolymer blends and diblock copoly-mers are shown. The homopolymer has a continuously decreasing intensity with increasingscattering vector. This dependence can be described with a simple Ornstein-Zernicke formulaaccording toS(Q) = S(0)/(1 + ξ2Q2). The forward scatteringS(0) describes the strengthof the thermal fluctuations. Large intensities mean strong fluctuations which are excited bythe thermal energy. This means that the free energy varies weakly as a function of the (local)composition. This relation is expressed by the Fluctuation-Dissipation theorem:

S−1(0) =1

kBT·d2(G/V )

dφ2(19)

The Gibb’s free energy appears here normalized to the VolumeV . The first derivative of theGibb’s free energy is connected with the osmotic pressure, while the second derivative is relatedto the osmotic compressibility. The inverse maximum intensity S−1(0) is also a susceptibility,describing the softness of the Gibb’s free energy with respect to external fields (while in theexperiment the thermal energy excites the fluctuations). The correlation lengthξ is inverselyproportional to the width of this curve. It describes over which distance the thermal fluctuationsare correlated, i.e. over which distance enrichments of onespecies remain of similar strength.With weak interactions between the two polymer species the correlation length is tightly con-nected with the polymer size, since this size is the only length scale of the system. With inter-actions and temperatures a little closer to the critical point the fluctuations and the correlationlength grow rapidly (as we will also see in Figure 7). The vicinity to the phase boundary inducesthis behaviour. As we will see below the fluctuations can growso strongly that the fluctuationsinfluence the Gibb’s free energy, and deviations from a simple model picture are visible.

The characteristic scattering pattern of a diblock copolymer shows a correlation peak at a finiteQ∗. At large length scales (smallQ) the chemical bond between the two blocks of the polymerinhibits fluctuations, and the ideal intensity grows likeS(Q) ∝ Q2. Polydispersity of theblock length ratio can lead to long range fluctuations, and finally the compressibility may allowfor density fluctuations on large length scales. Practically the polydispersity is much moreimportant. At small length scales far below the polymer sizethe chemical bond between the


blocks is not important anymore and only a mixture of chain parts is seen. ThusS(Q) ∝

Q−2 which reflects the fractal property of the chains, i.e. on several length scales the chainkeeps its chain properties. In thisQ-range homopolymers and diblock copolymers cannot bedistinguished. The maximum intensityS(Q∗) in between the two asymptotic scattering lawsreflects the strong fluctuations with the byQ∗ given periodicity. Again, this intensity is aninverse susceptibility. The periodicity is tightly connected with the polymer size [20]. Thepeak width∆Q ∝ ξ−1 is connected with a correlation length. Thisξ tells over which distancealternating enrichments can be expected. Again, with the vicinity of the phase boundary thefluctuations (and thus the intensity) and the correlation length grows. Strong fluctuations alsohave an influence on the Gibb’s free energy.

Now the temperature dependence of the susceptibilities shall be discussed (Figure 7). At hightemperatures the homopolymer blends show a linear temperature dependence which is moti-vated by the mean field theory:

1

2S−1

(

Q=0Q=Q∗

)

= (ΓS + Γσ) −ΓH

T(20)

There are two entropic terms. The first termΓS arises from the mixing of two polymers with agiven chain length; so this is an entropy of mixing of two particles with a given molecular size.The second termΓσ is a segmenal extropy of mixing, which considers the monomeric structureand the compressibility on the monomeric level. The enthalpic termΓh describes the enthalpy ofmixing on the monomeric level. While we developed this general scheme about scattering andsusceptibilities on the basis of the assumption of incompressability the generalization simplyconsiders pressure dependent segmental termsΓσ andΓh [18]. These effective interaction termscan be compared with more enhanced theories [19].

Focussing on the temperature behaviour of the susceptibility again, the mean field behaviour canbe extrapolated to a critical temperatureTMF , where the susceptibility is infinitely small. Thisidealized critical behaviour of the susceptibility can be described by a power law accordingto S−1(0) = CMF τ 1 where the reduced temperature is defined byτ = (T − TMF )/T . Themean field picture breaks down when the temperature is in the crossover region and deviationsbecome visible. Another asymptotic power law is observed close to the real critical temperatureaccording toS−1(0) = C+τ 1.24 [21]. The underlying theory is the 3-dimensional Ising theorydescribing two-component critical systems. The whole temperature range is described well bya crossover function [22, 23, 24]. So the two univerality classes, i.e. the mean field and the3d-Ising class, are connected by a single function.

The extrapolated mean field temperatureTMF indicates also the limit of the 3d-Ising region,which extends over the range betweenTMF andTC , the real critical temperature. This temper-ature difference is connected to an important parameter theGinzburg numberGi ∝ (TMF −

TC)/TMF . On the one hand this temperature difference is important for applications, becausethe real phase boundary has to be known well. On the other handthe Ginzburg number is impor-tant within the theoretical concept of homopolymer blends.The Ginzburg number is a measure


for the temperature range where thermal fluctuations influence the Gibb’s free energy and the3d-Ising behaviour becomes visible. Within classical concepts the Ginzburg numberGi ∝ N−1

was estimated to be rather small due to the dependence on the degree of polymerizationN .This term arises from the probability that two polymers meetin a given point (and interact).The probability to find one monomer of one species in a given point scales asN/R2

ee ∝ N−1/2.HereRee is the typical polymer size [1]. The product of two probabilities for two species scalesasN−1. Thus, the scaling of the Ginzburg number is similar to the entropy of mixingΓS. It wasan idea of Schwahn [22] that the segmental entropyΓσ has also an influence on the Ginzburgnumber. As a consequence the Ginzburg number (a) is much larger than previously estimated,(b) depends on the monomeric structure and is not universal with respect to the degree of poly-merisationN , and (c) depends on pressure [23].

For diblock copolymers one expects a similar temperature dependence at high temperaturesaccording to eq. 20 (forQ = Q∗). When regarding the measurements one observes a morelinear behaviour at high temperatures. At lower temperatures the behaviour is curved moreand more. The applied theory [25] describes the whole disordered phase region. Only theextrapolated mean field behaviour is found to be much different from the measured curves. Atlower temperaturesTODT the phase transition is indicated by a sudden increase of themeasuredintensity, while at even lower temperatures this intensitydoes not vary much anymore.

Again, the Ginzburg number can be related to the temperaturedifference of the mean field be-haviourTMF and the real phase transitionTODT . The Ginzburg number obviously is muchlarger than for homopolymer blends. The probability of interaction in a given point scales withN1/2 since now both blocks are connected. Therefore, the Ginzburg number should scale asGi ∝ N−1/2 according to this simple theory. The general trend is basically confirmed exper-imentally [26] but deviations are also visible. A theoretical concept for diblock copolymersincluding the segmental entropy is missing.

The whole understanding of phase boundaries and fluctuations is important for applications.Many daily life plastic products consist of polymer blends since the final product should havecombined properties of two different polymers. Therefore,polymer granulates are mixed athigh temperatures under shear in an extruder. The final shapeis given by a metal form wherethe polymer also cools down. This process involves a certaintemperature history which cov-ers the one-phase and two-phase regions. Thefore, the final product consists of many domainswith almost pure polymers. The domain size and shape are veryimportant for the final prod-uct. So the process has to be tailored in the right way to yieldthe specified domain structure.This tailoring is supported by a detailed knowledge of the phase boundaries and fluctuation be-haviour. Advanced polymer products also combine homopolymers and diblock copolymers foran even more precise and reproducable domain size/shape tailoring [27]. The diblock copoly-mer is mainly placed at the domain interfaces, and, therefore, influences the domain propertiesprecisely.


3.4 Microemulsions

Microemulsions are another important example of soft matter research [28]. They consist ofthree components: Water and oil are immiscible; but a surfactant mediates between oil andwater and macroscopically homogenous phases are formed. Ona microscopic scale oil andwater stay unmixed and form domains. The surfactant forms a film between oil and water andkeeps the system thermodynamically stable.

A typical phase diagram is shown in Figure 8 [29]. For this purpose equal volumes of oil andwater are mixed with a variable fraction of surfactantγ = msurf/(moil + mwater + msurf). Atlow surfactant concentration a three phase coexistence (3)is observed. Here a microemulsionphase coexists with excess oil and excess water. At high temperatures a two-phase coexistenceoccurs (2) with a microemulsion and an excess water phase. At low temperatures a two-phasecoexistence occurs (2) with a microemulsion and an excess oil phase. At higher sufactant con-centrations a one-phase microemulsion (1) is found. This microemulsion has a bicontinuousstructure. That means that the oil and water domains are continuous by themselves and havea sponge like structure which hosts the other domain. It is known that even higher surfactantconcentrations lead to ordered structures. Again, lamellar structures, cylinders on a hexagonallattice, and spheres on a cubic lattice appear [30]. Withoutthese liquid crystalline phases thephase diagram has a simple appearance of a fish. It is mainly found for non-ionic surfactants,for instance for CiEj surfactants.

When adding an amphiphilic polymer – a polymer which basically is a long surfactant molecule– new interesting effects are observed. Figure 8 also shows the one phase regions of microemul-sions with a certain polymer contentδ = mpol/(mpol + msurf). This contentδ is given with re-spect to the total amphiphile, while the total content is less then 0.5%. One directly observes thatmuch less surfactant is needed to form a single phase microemulsion. This dramatic increase ofthe efficiency is also called ‘polymer boosting effect’. In the following it is demonstrated that(a) the polymer decorates the surfactant membrane and (b) the polymer induces a larger rigidityof the membrane. This larger rigigity allows to form larger structures with a better surface tovolume ratio.

3.5 Contrast Variation Experiments

Up to now, we discussed the scattering of materials with two components only. For a fourcomponent system (oil, water, surfactant, and polymer) this simple concept does not hold any-more. As we discussed already, single components can be madevisible for neutron scatteringexperiments by the exchange of protons by deuterium (H-D-exchange). Thus, in principle everycontrast can be rendered. Figure 9 shows the principle of making single components and there-fore properties visible. The bulk contrast basically makesone domain visible against the otherdomain. Here, the film and the polymer play a minor role due to the much lower concentration.The film contrast makes the film visible against the two domains. Again, the polymer plays aminor role. The polymer contrast finally makes the polymer visible against all other compo-


nents. Due to the decreasing concentration in this series the intensity for the three contrasts isdecreasing by a factor of 100 with each step. This makes plausible why clean high intensityneutron scattering machines are desparately needed. In theory the macroscopic cross section ofa four component system reads:

dΣ

dΩ= (ρO − ρW )2SOO + (ρF − ρW )2SFF + (ρP − ρW )2SPP (21)

+ (ρO − ρW )(ρF − ρW )SOF + (ρF − ρW )(ρP − ρW )SFP

+ (ρO − ρW )(ρP − ρW )SOP

Again, incompressibility is assumed for this expression. The equation 21 is expressed in thisway that water is the matrix component; i.e. it is present at each position where no oil, nofilm and no polymer is found. The first three terms are already explained by the Figure 9. Thebulk contrast scattering is expressed bySOO(Q) = |φO(Q)|2, which is given by the Fouriertransformation of the local oil concentration. The film contrast scattering isSFF , and the poly-mer contrast scattering isSPP . The three latter terms hardly can be depicted in a simple way.Since the scattering intensity is a quadrature of the scattering amplitude all cross terms be-tween different components occur as well. In this manner theoil-film scattering is given bySOF (Q) = 2Re(φ†

O(Q)φF (Q)). The product of the two Fourier tranformations is a convolutionin real space, i.e.SOF (Q) = 2Re(

∫

φO(R − r)φF (R + r) exp(IQR)d3Rd3r). This means thatthe cross terms represent correlations between the components.

Practically, the six scattering functions are determined by measurements with many differentcontrasts. The three major contrasts (bulk, film and polymer) are selected plus a series of manycontrasts around the polymer contrast. In this way, usually15 to 20 scattering intensities aremeasured, which overdetermines the equation system and leads to a much higher accuracy forthe six scattering functions. The equation system of eq. 21 can be understood asd~Σ/dΩ =

∆ρ~S. The measured intensities are many more than the to be determined scattering functions,and so the matrix∆ρ is non-quadratic. When solving for the scattering functions this matrix∆ρ has to be inverted formally by the Singular Value Decomposition method and one obtains~S = ∆ρ−1d~Σ/dΩ.

An example for the polymer-film scattering function is shownin Figure 10. It describes thevaluable information about the relative position of the polymer with respect to the film. So thepure polymer and the pure film scattering do not provide this information. As indicated by thesolid line, this scattering can be described by a planar wallwhich is decorated by polymer in amushroom conformation.

The bulk contrast scattering also reveals valuable information about the domain structure. Anexample for the scattering is shown in Figure 11. The macroscopic cross section can be de-scribed by the Teubner-Strey theory which bases on a Ginzburg-Landau approach. This ap-proach considers long wavelength fluctuations of the oil andwater domains. The final formulareads:


dΣ

dΩ(Q) =

8πφW (1 − φW )/ξ

((2π/d)2 + ξ−2)2 − 2((2π/d)2 − ξ−2)Q2 + Q4(22)

The denominator contains the total fraction of waterφW . Much more interesting are the twostructural parametersξ andd which are obtained as fitting parameters from the measurement.The domain sized is to be understood as the average distance between two consecutive waterparts. The correlation lengthξ is a measure for the regularity of the alternating domains. Asindicated in Figure 11 the peak position is mainly given by the domain sized, and the peakwidth is mainly given by the correlation lengthξ. In comparison to diblock copolymers theforward scattering of microemulsions is clearly non-zero.Thermal fluctuations of oil and watercan take place easily on large length scales while for the diblock copolymer the bond betweenthe blocks makes sure that the polymer looks homogenous on large length scales.

The Gaussian random field theory allows for the calculation of the bending rigidityκ/(kBT ) =

0.85ξ/d from the structural parametersξ andd. This microscopic modulus depends linearlyon the polymer content according to advanced theories [31].The other important modulus ofthe microscopic theory by Helfrich [32] is the saddle splay modulusκ which has a very similardependence on the polymer content. While the bending rigidity is measured quite directly,the saddle splay modulus is measured indirectly. At the minimum surfactant concentrationΨ ≈ γmin (fish tail point) the saddle splay modulus is a small constant. Relative changes ofthe surfactant concentration are related to changes ofκ by the spatial renormalization (detailedinformation is discussed in reference [32]). The bottom line is that the two functionsln Ψ andκ vary in a similar manner as shown in Figure 12. Both variablesare plotted as a functionof the scaled polymer concentrationσR2

ee whereσ is the polymer grafting density andRee isthe typical polymer size. The minimum surfactant concentration Ψ decreases with the polymeramount which means that the amphiphile becomes more efficient. In a similar manner thebending rigidityκ increases. More rigid membranes allow to form larger structures with abetter surface to volume ratio. Both experimental magnitudes are connected on the level of theHelfrich model by Lipowsky’s theory about polymers. The speciality of Figure 12 is that thesevariables were measured for sticker polymers which consistof a short hydrophobic sticker (C10)and a long hydrophilic polymer chain (made of ethylene oxide). These stickers introduce someunsymmetry in comparison to diblock copolymers. On the one hand they change the preferredcurvature of the membrane; on the other hand they work as wellas efficiency boosters as diblockcopolymers. The major advantage of sticker polymers is thatthey are cheaper to produce andthey are well water soluble. Many applications need a quick solvation of powder components.

3.6 Summary on Polymers and Microemulsions

In summary we have learned about amphiphilic diblock copolymers that they enhance the effi-ciency of surfactants dramatically. The polymer decoratesthe membrane and makes the mem-brane more rigid. This rigidity allows to form larger structures with a better surface to volumeratio. The newly found sticker polymers have advantages in applications. The ongoing re-


search considers also a larger variety of polymer architectures which should allow for tailoredproperties of microemulsion.

In general we have learned about soft matter problems that they are close to applications. Themajor question is how to reach for tailor made properties. These questions can only be addressedby basic research knowledge. One important tool of basic research is the small angle scatteringmethod. While x-rays are available by small machines in the basement and with large intensi-ties by synchrotrons, neutrons are accessed through reactors and spallation sources. Reactorsprovide a continuously high intensity which is important for small angle neutron scatteringdiffractometers. Spallation sources work with extremly high intensity neutron pulses. Thesepulses can be used for SANS diffractometers in combination with a time of flight analysis. Themajor advantage of neutrons against x-rays is the free choice of cotrasts which is important formultiple component systems. Especially in the near future basic research will be expanded tomultiple component systems in order to mimick more complex functions of biological systemsfor instance. The common understanding of bio-physical principles is another major task forsoft matter research which is about to expand strongly.


Fig. 5: On the left a typical phase diagram of homopolymer blends is found. Usu-ally, the temperature is measured as a function of the composition. High temper-atures lead to homogenous mixing. The binodal separates themixing from themetastable region (upper curve), while the spinodal separates the metastable andthe unstable region. The point where binodal and spinodal meet is called criticalpoint. Close to the critical point thermal composition fluctuations are strong. Theblue arrow indicates a temperature series of the consideredexperiment. On theright a typical phase diagram of diblock copolymers is shown[14]. On top, thedifferent ordered structures are indicated. Below, the ordered structures are alsomarked in the phase diagram. The temperature axis (ΓV ∝ T−1) is reverted, whilethe x-axis shows the volume fraction of the polyisoprene block (chainlength ratio).The lower line is a theoretical spinodal line not supported by any phase transitions.


Fig. 6: The scattering functionS(Q) as a function of the scattering vector. On theleft the typical scattering of a homopolymer is shown. The sample is a polybuta-diene(1,4) / polystyrene blend at 104C and 500bar. The peak maximum intensityat Q = 0 is the inverse susceptibility. The peak width is inversely proportional tothe correlation lengthξ. On the right the typical scattering of a diblock polymeris shown which is a poly-ethylene-propylene–poly-dymethylsiloxane at 170C and1bar. The peak maximum is the inverse susceptibility. The peak position isQ∗ andthe peak width∆Q is connected with the reciprocal correlation lengthξ.


Fig. 7: The susceptibility as a function of the inverse temperature. On the leftthe typical temperature dependence of a homopolymer is shown. At high temper-atures a linear dependence is found which can be directly connected to mean fieldtheories; at medium temperatures there is a crossover to thelower temperatureswhere the 3-dimensional Ising behaviour is found. The 3d-Ising critical behaviouris curved, and the critical exponent is 1.24. On the right thecritical behaviour of adiblock copolymer is depicted. At higher temperatures the behaviour becomes quitelinear, but the extrapolated mean field behaviour is different as shown by the linesin the lower left corner. The whole temperature range is fitted by a function whichincludes fluctuations [25]. At lower temperatures the phasetransition is indicatedby a sudden jump of the susceptibility. For different pressures the phase transitionand the susceptibilities are different.


Fig. 8: The typical ‘fish’ phase diagram (red): temperature versus surfactant con-centrationγ. At low concentrationsγ there is a three phase coexistance (3) of twoexcess phases and a microemulsion. At high temperatures there is a two phase re-gion (2) of an excess water phase and a microemulsion. Al low temperatures (2)the excess phase is oil rich. At higher concentrationsγ there is a one phase mi-croemulsion which is considered in this paper. There are also ‘fish tails’ (blue andgreen) shown with some polymer additive [29]. The polymer amount is measuredrelatively to the total amphiphile. The decreased need of surfactant is a measurefor the dramatically increased efficiency (polymer boosting effect).


Fig. 9: The pricipal options of contrasts for a four component system, here water,oil, surfactant and polymer. In the bulk contrast the oil domains are mainly visibleagainst water. In the film contrast the surfactant is mainly highlighted against theoil and water domains. The polymer contrast focuses on the polymer alone.


Fig. 10: The polymer-film scattering function as a function of the scattering vector[29]. The fitted function describes a polymer in the presenceof a planar wall. Thepolymer shows a mushroom conformation.

Fig. 11: A typical scattering pattern of a bicontinuous microemulsion. The peakposition is connected to the domain spacingd. The peak width is inversely propor-tional to the correlation lengthξ. This kind of data is usually well described by theTeubner-Strey fitting function given in eq. 22.


Fig. 12: The minimum surfactant amountΨ as a function of the scaled poly-mer amountσR2

ee for microemulsions with increasing amount of a sticker polymerC12E90. The decrease shows that the microemulsion becomes more andmore effi-cient, i.e. the polymer boosting effect is successfully transferred to sticker polymers.On the right the bending rigidityκ is plotted as a function of the scaled polymeramount. The increased bending rigidity explains why largerdomain structures witha better surface to volume ratio can be formed. On a microscopic scale both dia-grams are related through the Helfrich [32] free energy. Thegeneral polymer effectwas described by Lipowsky [31].


References

[1] H. Frielinghaus in 35th Spring School 2004Physics meets Biology(ForschungszentrumJulich GmbH, Julich, 2004)

[2] M. Monkenbusch in 38th Spring School 2007Probing the Nanoworld(Forschungszen-trum Julich GmbH, Julich, 2007)

[3] Th. Bruckel, G. Heger, D. Richter, R. ZornNeutron Scattering – Lectures of the Labora-tory Course(Forschungszentrum Julich GmbH, Julich, 2001 and later)

[4] http://www.ncnr.nist.gov/resources/n-lengths/

[5] M.V. Petoukhov, D.I. SvergunBiophys. J.89, 1237 (1237) and http://www.embl-hamburg.de/ExternalInfo/Research/Sax

[6] R.J. RoeMethods of X-ray and Neutron Scattering in Polymer Science(Oxford UniversityPress, Oxford, 2000)

[7] Y. Liu, H. Nie, R. Bansil, M. Steinhart, J. Bang, T.P. LodgePhys. Rev. E73, 061803 (2006)and N. Sota, N. Sakamoto, K. Saijo, T. HashimotoMacromolecules36, 4534 (2003)

[8] Munich SANS

[9] X. Ye, T. Narayanan, P. Tong, J.S. Huang, M.Y. Lin, B.L. Carvalho, L.J. FettersPhys. Rev.E 54 6500 (1996)

[10] A. Botti, W. Pyckhout-Hintzen, D. Richter, V. Urban, E.Straube, J. Kohlbrecher,Polymer44, 7505 (2003)

[11] K.N. Pham, S.U. Egelhaaf, P.N. Pusey, W.C.K. PoonPhys. Rev. E69 011503 (2004)

[12] S. Forster, A. Timmann, M. Konrad, C. Schellbach, A. Meyer, S.S. Funari, P. Mulvaney,R. KnottJ. Phys. Chem. B109, 1347 (2005)

[13] D. Schwahn, S. Janßen, T. SpringerJ. Chem. Phys.97, 8775 (1992) and G. Muller, D.SchwahnPhys. Rev. E55, 7267 (1997)

[14] A.K. Khandpur, S. Forster, F.S. Bates, I.W. Hamley, A.J. Ryan, W. Bras, K. Almdal, K.MortensenMacromolecules28, 8796 (1995)

[15] J. Dudowicz, K.F. FreedMacromolecules24, 5076 (1991) and J. Dudowicz, K.F. FreedMacromolecules24, 5096 (1991) and J. Dudowicz, K.F. FreedMacromolecules24, 5112(1991)

[16] M.G. Bawendi, K.F. FreedJ. Chem. Phys.88, 2741 (1988)

[17] J.S. Higgins, H.C. BenoitPolymers and Neutron Scattering(Oxford University Press, NewYork, 1994)


[18] S. Janßen, D. Schwahn, K. Mortensen, T. SpringerMacromolecules26, 5587 (1993)

[19] J. Dudowicz, K.F. FreedMacromolecules28, 6625 (1995)

[20] L. Leibler Macromolecules13, 1602 (1980)

[21] D. Schwahn, K. Mortensen, H. Yee-MadeiraPhys. Rev. Lett.58, 1544 (1987)

[22] D. Schwahn, G. Meier, K. Mortensen, S. JanßenJ. Phys. II France4 837 (1994)

[23] H. Frielinghaus, D. Schwahn, L. WillnerMacromolecules34, 1751 (2001)

[24] M.Y. Belyakov, S.B. KieselevPhysica A190, 75 (1992) and M.A. Anisimov, S.B. Kiselev,J.V. Sengers, S. TangPhysica A188, 487 (1992)

[25] G.H. Fredrickson, E. HelfandJ. Phys. Chem.87, 697 (1987)

[26] D. Schwahn, H. Frielinghaus, K. Mortensen, K. AlmdalMacromolecules34, 1694 (2001)

[27] J.H. Lee, M.L. Ruegg, N.P.Balsara, Y. Zhu, S.P. Gido, R.Krishnamoorti, M.H. KimMacromolecules36, 6537 (2003) and B.J. Reynolds, M.L. Ruegg, N.P. Balsara, C.J.Radke, T.D. Shaffer, M.Y. Lin, K.R. Shull, D.J. LohseMacromolecules37, 7410 (2004)

[28] D. Byelov, H. Frielinghaus, O. Holderer, J. Allgaier, D. Richter Langmuir 20, 10433(2004)

[29] H. Endo, M. Mihailescu, M. Monkenbusch, J. Allgaier, G.Gompper, D. Richter, B.Jakobs, T. Sottmann, R. Strey, I. Grillo,J. Chem. Phys.115, 580 (2001)

[30] Ch. Frank, T. Sottmann, C. Stubenrauch, J. Allgaier, R.StreyLangmuir, 21, 9058 (2005)

[31] C. Hiergeist, R. LipowskyJ. Phys. II6, 1465 (1996)

[32] G. Gompper in 33th Spring School 2002Soft Matter(Forschungszentrum Julich GmbH,Julich, 2002) and W. HelfrichZ. Naturforsch., A: Phys. Sci.33, 305 (1978)

d 5 structure of soft matter: small angle scatteringwpage.unina.it/lpaduano/phd lessons/neutron...

Documents