excluded volume effects in the depletion attraction ......1.2 polymer solvents characteristics . . ....

Excluded Volume Effects in the

Depletion Attraction between

Nanoparticles

Dissertation

zur Erlangung des akademischen Grades

des Doktors der Naturwissenschaften

an der Universitat Konstanz

im Fachbereich Physik, Lehrstuhl Prof. Dr. Matthias Fuchs,

vorgelegt von

Maryam Naderian

Referenten: Prof. Dr. Matthias FuchProf. Dr. Rudolf Klein

Tag der mundlichen Prufung: 14. Dezember 2009

http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-123561

http://kops.ub.uni-konstanz.de/volltexte/2010/12356/

Contents

1 Introduction 1

1.1 The System: Mixtures of Polymers and Colloidal Particles . . . . . . . . . . . . . 41.1.1 Why Low Colloid Packing Fraction? . . . . . . . . . . . . . . . . . . . . . 5

1.2 Polymer Solvents Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Depletion Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Model 10

2.1 PRISM m-PY theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.2 PRISM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Dilute Colloid Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.1 Polymer-Colloid Radial Distribution Function . . . . . . . . . . . . . . . . 212.2.2 Colloid-Colloid Radial Distribution Function . . . . . . . . . . . . . . . . 242.2.3 Second Virial Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Thermodynamic consistency 26

3.1 Compressibility Route . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 Virial Route . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Numerical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 The Effective Interaction Length Scale . . . . . . . . . . . . . . . . . . . . . . . . 323.5 Excess Chemical Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4 Results 38

4.1 gcp(r), Polymer Segment-Colloid Pair Correlation Function . . . . . . . . . . . . 404.1.1 Density Profile of Polymers Close to an Isolated Colloid . . . . . . . . . . 404.1.2 Depletion Layer Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.1.3 Intermediate Separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.1.4 Long-Ranged Tail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 gcc(r), Colloid-Colloid Correlation Function . . . . . . . . . . . . . . . . . . . . . 464.2.1 Contact Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

ii

Contents

4.2.2 Induced Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Bc

2(r), Second Virial Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5 Experimental comparison 56

5.1 An introduction to small angle neutron scattering . . . . . . . . . . . . . . . . . 575.2 The experimental system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6 Summary and Outlook 62

7 Zusammenfassung und Ausblick 65

A Radial Fourier Transformation 68

B Thread-Like Chain 69

iii

List of Figures

1.1 Polymer-Colloid Mixture in 3D Space Based on Colloid Packing Fraction ϕc,Reduced Polymer Concentration ϕp/ϕ∗

p and Asymmetric Size Ratio q = Rg/R . . 21.2 Schematic Sketch to Illustrate Why long chains in Polymer-Colloid Mixture are

Interesting to Invertigate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Scattering Function for a Dilute Polymer Solution on Logarithmic Scales . . . . . 20

3.1 λ, Effective Interaction Length Scale . . . . . . . . . . . . . . . . . . . . . . . . . 343.2 Thermodynamic Consistency, the Real Polymer Excess Chemical Potential via

two Independent Thermodynamic Roots . . . . . . . . . . . . . . . . . . . . . . . 363.3 Excluded Volume Effects in the Excess Chemical Potential . . . . . . . . . . . . . 37

4.1 gcp(r), Polymer-Colloid Pair Correlation Function . . . . . . . . . . . . . . . . . . 414.2 Excluded Volume Effects in the Polymer-Colloid Pair Correlation Function . . . 424.3 Excluded Volume Effects in the Colloid Depletion Layer . . . . . . . . . . . . . . 444.4 Excluded Volume Effects in the Long-Ranged Tail . . . . . . . . . . . . . . . . . 454.5 gcc(r), Colloid Pair Correlation Function . . . . . . . . . . . . . . . . . . . . . . . 474.6 Excluded Volume Effects in the Colloid Pair Correlation Function . . . . . . . . 484.7 Potential of Mean Force between Particles . . . . . . . . . . . . . . . . . . . . . . 504.8 Excluded Valume Effects in the Induced Pair Potential between Particles . . . . 514.9 Second Virial Coefficient between Nanoparticles . . . . . . . . . . . . . . . . . . . 534.10 B, the Universal Combination Function . . . . . . . . . . . . . . . . . . . . . . . 544.11 Excluded Volume Effects in the Second Virial Coefficient between Nanoparticles 55

5.1 Second Virial Coefficient, Comparision of Numerical and Experimental Results . 61

iv

Acknowledgements

I would like to thank my supervisor, Prof. Matthias Fuchs, for giving me the opportunityto work on this interesting project and his permanent support during my PhD studies. I amparticularly thankful for his instructions and discussion, and furthermore for the opportunity toattend international meetings, which in overall made the time in his group very valuable.

I also would like to thank Dr. Michel Rawiso from the Institute Charles Sadron in Strasbourg,who helped me a lot to gain a deeper understanding about the experimental part of my project.Furthermore, I am grateful to Prof. Jorg Baschnagel, Strasbourg, He made this opportunityfor me to visit his group for four inspiring months and for help and support during my time inStrasbourg.

Special thanks go to all members of the Fuchs group at the University of Konstanz for thefriendly and scientifically stimulating atmosphere. I am particularly grateful to my colleaguesFabian Weysser and Christof Walz, as well as to Joe Brader, who helped me by correcting themanuscript and discussing several aspects of my work with me. I also would like to say thankyou to Martin Dauner for very nice friendship and for help with the correction of my thesis.

I acknowledge financial support from the International Research Training Group (IRTG) ’SoftCondensed Matter’ and from the Gleichstellungsrat of the University of Konstanz, who fundedthe last months of my doctoral project. Seminars, workshops and the exchange with laboratoriesin Strasbourg offered by the IRTG program contributed a lot to my personal and scientificdevelopment and therefore are gratefully acknowledged as well.

My sincerest thanks go to my parents, who, although being far away geographically, gave mea lot of support and advice throughout my stay abroad and always had an open ear for myproblems. I will also never forget all the gifts sent from home.

Finally, I would like to express my special gratitude to Bahram Kord for the many enjoyablemoments we had during the last years and his patience with me in times of trouble.

vi

1 Introduction

This thesis aims to shed light upon the equilibrium structural correlations and thermodynamicproperties of mixtures of nanoparticles with much larger non-adsorbing polymer coils. When thenon-adsorbing polymer is added to a suspension of colloids an effective ’depletion’ interaction isinduced between the colloids. In this thesis particular emphasis will be placed on the study ofsuch depletion interactions in the so-called ’protein limit’. An integral equation method calledPRISM (Protein Reference Site Interaction Model) based upon liquid state theory approacheswill be employed to treat a model system consisting of hard spheres (playing the role of nanopar-ticles) mixed with flexible coarse-grained thread polymers. Mixtures of colloidal particles andnon-adsorbing polymer coils are widely used in the chemical, pharmaceutical and food industriesas the macroscopic material properties can be tailored by tuning the microscopic interactions.There are three fundamental parameters:

The asymmetric size ratio q = Rg/R, the ratio of polymer radius of gyration to colloid radius,

where R2g =

1N

n∑k=1

(rk − rmean)2 and σc = 2R, the colloid diameter. ( rmean is the mean

position of the monomers.)

The reduced polymer concentration ϕp/ϕ∗p, the ratio of polymer concentration to chains over-

lap concentration. ϕ∗p is defined as the overlap polymer concentration, where the polymer

segments start to overlap.

The colloid packing fraction ϕc, the ratio of the total volume of the particles packed into aspace to the volume of that space.

Since we are interested in studying mixtures in the low colloidal packing fraction regime, ϕc 65%, two independent parameters remain. Based on these parameters the system is subdivided tofour sub-regions due to their extreme limits. These regions are schematically illustrated in Fig.1.1. The most important limits both historically and scientifically are i) The ’colloid limit’ Rg ≪R, sub-regions I & III, where the particles are much bigger than the polymer chains and has beenextensively studied for all ϕc and ϕp/ϕ∗

p parameters ranges. ii) The ’protein/nanoparticle limit’,sub-region II & IV, where the characteristic polymer size (radius of gyration Rg) is larger thanthe radius of the colloidal particles, R, leading to an asymmetry ratio q = Rg/R ≫ 1. The latter

1

1 Introduction

1

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Figure 1.1

is termed the ’protein limit’ because of its relevance to protein separation or crystallization andremains to be understood. It is of special interest because no satisfactory theoretical approachexists at finite density for ’real’ polymer statistics. To tackle this problem the subtle self-avoidingwalk statistics pioneered by Flory and de Gennes needs to be handled correctly in the presenceof colloidal particles.

The asymmetry size ratio, q, has a significant influence on the depletion attraction between twocolloids induced by the polymer fluid, leading to quantitative differences in the two limits. Theaddition of non-adsorbing polymer to a colloidal fluid causes depletion attractions that changeboth structure and phase behaviour. This was first investigated experimentally by Traube [1],but it took some decades before Asakura and Oosawa (AO) [2] and Vrij [3] could describe ittheoretically. The entropy-driven depletion attraction between two colloidal particles induced bythe polymer solution could be explained in terms of entropy. In the Asakura-Oosawa-Vrij modelthe following assumptions are made: i) R ≫ Rg. ii) Colloids behave as hard spheres dispersed inan ideal gas of polymer points (the internal conformational degrees of freedom were ignored). iii)The interactions in between colloids as well as polymers and colloids are repulsive. In conditionϕp < ϕ∗

p where polymer coils are dissolved in ’theta solvent’ and their second virial coefficientvanishes the second point is still acceptable. A large number of valuable studies based on thissimple model, tuning the particle pair potential [4], the asymmetry ratio [5] and polymer aswell as colloid concentrations [6], have been carried out. Additionally, much interest in the AO

2

model was generated by the interesting phase behaviour [7–12]. However, these approaches areapplicable only when the polymers are considered as spheres (without intramolecular structure)smaller than colloids, q ≪ 1. This model breaks down for large q values and the semidilutepolymer regime.

In the opposite limit to the AO model, there are some experimental [13–16] and computationalsimulation [17, 18] results investigating the mixture of long non-adsorbing real polymers andcolloids but there remains no satisfactory theoretical approach in the protein limit. Some re-cently performed theoretical investigations are scaling theory [19, 20], Density functional theory(DFT) [21–25] and field theory approach (FT) [26–28]. Scaling theory failed in protein limitdue to neglecting the polymer conformational degrees of freedom that lead to long-ranged tailcorrelations. In the based density functional theoretical approach the system properties arederived from grand canonical free energy of the system composed of polymer coils consistingof tangentially bonded segments and hard spheres. While this approach in dilute regime ne-glects the long-range intrachain correlations beyond the nearest neighbour that are expectedto be physically important, its results are only reliable in high monomer concentrations. Fieldtheory is based upon integrating out of polymer degrees of freedom in the presence of a fewsmall colloidal particles. The colloids play the role of perturbating points for the much biggerpolymer chains [29, 30]. This method leads to the exact asymptotic results for extreme limits inmixtures of non-adsorbing real polymers and nanoparticles in good solvent [26–28]. Its resultsare thus valid only in extreme limits. In addition to the mentioned disadvantages, both DFTand FT consider only one or two nanoparticles in polymer fluid and are not expendable to finitecolloidal packing fraction.

Recently, a specific integral equation method based on liquid state theory, the Polymer Refer-ence Interaction Site Model was suggested [31]. The appeal of PRISM is its ability to describethe structural correlations on all length scales [32–34]. It requires the form factors of both com-ponents, their interaction potentials and leads to the structural correlations of the mixture. It isapplicable for any kind of polymer with known intramolecular structure. This method employ-ing the PY closure also has been tested for good solvent [35]. However, in the protein limit thethermodynamic consistency which plays a very important role, was not achieved and predictionsfor the depletion layer were in considerable error. Later on, PRISM equations using a PY-likeclosure termed modified -PY suggested by Fuchs and coworkers have been employed to studymixtures of nonadsorbing ideal polymers with colloids [31, 36, 37] and highly accurate resultsfor the entire range of system parameters concerning densities and sizes and thermodynamicconsistency are obtained. In the m-PY closure a tunable length parameter to control the poly-mer segment-colloid direct interaction is introduced. This can also be expressed as penetrabilityof the polymer chain into the gap between two colloidal particles.

3

1 Introduction

Real polymers differ from the much simpler ’ideal’ model in their inability to intersect. Idealpolymers obey a simple Gaussian distribution of segments. Real ones, which in a good solventswell in order to prevent any self-crossing, require a more sophisticated distribution functionfor their mass density. It is characterized by the anomalous Flory exponent ν, which can onlybe computed through renormalization group techniques. Up to now, it has not been possibleto include real polymers in integral equation approaches to nanocolloid-polymer mixtures. Yet,especially as the colloid size is much smaller than the polymer one, the distinct polymer statisticsmatter and affect interactions, and therefore structural correlations.

Our goal is to describe the structure and scattering patterns of excluded volume interaction,nonadsorbing polymer-colloid mixtures in good solvent in the low colloidal particle volume frac-tion regime, which is theoretically poorly understood. The theoretical approach in this thesisis the only currently available theoretical method to handle at the molecular scale the struc-tural correlations of all species. We employ PRISM [32–34, 38–40] which is a macromolecularextension of the RISM theory of Chandler and Anderson [39]. Where the RISM theory is a gen-eralized Ornstein-Zernike equation for the mixture of polyatomic molecules [32–34]. This liquidstate approach (PRISM) is used to supplement the little existing information from experiment,theory, or computer simulation. In this way we also use the introduced closure in previous worksperformed by Fuchs and coworkers using PRISM for mixtures of ideal chains with hard spheresin theta solvent [31, 36, 37].

1.1 The System: Mixtures of Polymers and Colloidal Particles

The polymer colloid suspension under consideration is composed of three components: Impene-trable particles, flexible macromolecules, and solvent [8, 41, 42]. The size of the solvent particlesis much smaller than the smallest characteristic size of the system, theoretically it may there-fore be considered as a background continuum and its effect is incorporated into the effectiveinteractions between the other two components.

In our model system the rigid spheres (without any intramolecular structure) play the role ofthe colloidal particles and polymer chains are considered as non-adsorbing flexible polymers in’thread limit’. In the thread limit the hard core diameter of polymer segments is diminished toa point [43, 44]. The polymer solvent is a good solvent under athermal condition which will beexplained in Sec.1.2. We focus on real polymers where interactions between chain monomers canbe modelled as excluded volume. This causes a reduction in the conformational possibilities ofthe chain respect to the ideal chain, and leads to a self-avoiding random walk. In Self-avoidingrandom walks the chain never cross itself.

4

1.2 Polymer Solvents Characteristics

On the one hand, the steric interactions between colloid-polymer and polymer-polymer particlesare controlled by the polymer statistics. On the other hand, the colloidal particles can perturbthe polymer conformations and lead to a change in conformational entropy, in the case thatpolymers are sufficiently flexible. The strength of the perturbation included by the colloidsdepends on the asymmetric size ratio and polymer concentration.

We consider the simplest model: mixtures of hard spheres and flexible polymers in good sol-vent under athermal conditions. This is characterized by the hard-core repulsive interactionsbetween all species and is thus describable as a purely entropic statistical mechanics, ’packingproblem’. Although this model seems simple the study of its complex physical behaviour is notstraightforward.

1.1.1 Why Low Colloid Packing Fraction?

We employ a liquid state theory approach to investigate mixtures of non-adsorbing real polymerchains dissolved in good solvent with nanoparticles under athermal conditions, keeping a lowcolloidal packing fraction. The appeal of integral equation method is its applicability in allranges of the system parameters including asymmetric size ratios, colloidal packing fractionsand polymer concentrations. This enable us to investigate the system at low colloidal packingfraction limit and also:

• to study the polymer density profile near an isolated colloidal particle.

• to investigate the influence of excluded volume interactions between the chain monomerson the colloid-colloid interaction in a polymer fluid.

• to compare with existing theoretical results which are valid only at infinitely low colloiddensity.

• given a good agreement with results obtained via alternative approaches, to employ ourmethod to higher colloidal packing fractions which have not yet been investigated theoret-ically and which are not understood.

1.2 Polymer Solvents Characteristics

In this work we consider the ’excluded volume chain’, which is frequently referred to in theliterature as a real chain. We also often use the term the ’Gaussian chain’ which is also referredto as the ideal chain. An excluded volume chain dissolved in a solvent at a specific temperature

5

1 Introduction

called Θ-temperature can be approximated by the an ideal chain. At this temperature thesolvent is termed theta solvent [45]. In this section, it is clarified what one means by theseexpressions.

Consider ε(r), the energy cost for bringing two monomers with diameter σp from ∞ to withindistance r from each other in the presence of solvent. This energy contains of two parts, arepulsive hard-core and usually an attractive well. The second part depends on the difference ofthe interaction between monomer-monomer and monomer-solvent. Usually the monomers likeeach other more than they like the solvent; in this case the second part of ε(r) is an attractivepotential. In the case that the solvent and monomers have identical chemical structure theattraction disappears and the only remaining part for ε(r) is the hard-core repulsion. Thesystem is then called athermal. The last case is if monomers like each other less than they likesurrounding solvent (e.g. similarly charged monomers). On the other hand, in the low densitylimit of an imperfect gas system the interaction part of free energy density can be written as apower series called the virial expansion. Extending this idea to the polymer solution in infinitelylow polymer concentrations, the interaction part of the free energy density can be written as avirial series in powers of the monomer number density, ϱp, like:

βFint

V= B2ϱ

2p + B3ϱ

3p + ...

Where β = 1/kT and k is Boltzmann constant and T is the temperature. The coefficient of theϱ2

p term is called the second virial coefficient and is proportional to the excluded volume whenthe free energy is that of a single chain. The coefficient of ϱ3

p term is a three-body interactioncoefficient. The Second virial coefficient express the interaction energy of isolated pairs (in apolymer fluid monomer-monomer interaction energy) and is defined as:

B2 = −12

∫(e−βε(r) − 1)dr (1.1)

The integrand is called the Mayer f(r)-function. The excluded volume is defined as twice thesecond virial coefficient. According to treatment of excluded volume as a function of the typeand strength of ε(r) and temperature, the polymer solvents are classified as following:

• Athermal solvent. In the high temperature limit, the integrant and consequently the secondvirial coefficient become temperature independent and the excluded volume is maximal.

• Good solvents. In the athermal limit, the monomer-monomer attractions are slightlystronger than monomer-solvent ones. The net attraction cause a weak attractive wellε(r) < 0 which leads to the lower excluded volume than the athermal case, but still re-mains positive. The polymer chains dissolved into this solvent and the athermal one are

6

1.3 Depletion Layer

referred to as real or excluded volume interaction chain and their statistics described bythe self avoiding walk.

• Theta solvents. At some special temperature, called the Θ-temperature the attractivewell exactly cancels with the hard-core repulsive barrier and the net excluded volume iszero. In a theta solvent the conformation of the chains is nearly ideal and they are wellrepresented by ideal or Gaussian chains with statistics described by the random walk.

• Poor solvents. At the temperature below Θ-temperature where the attractive well domi-nates over the interaction potential, the excluded volume is negative and it is more likelyto find monomer close together. It looks that the chains shrink in such solvents.

• Non-solvents. The opposite point of athermal solvent is non-solvent which is the limitingcase of poor solvent. The attraction is negative and minimal and polymers nearly excludesall solvent from being within the coils.

When the interaction energy contains an attractive well and a repulsive barrier, the attractiondominates at low temperature and in higher temperature the repulsion dominates the interaction.In athermal solvent with no attractive well the excluded volume interaction does not dependon temperature. In the special case where the monomer-solvent attraction is stronger than themonomer-monomer one, a soft barrier occurs in addition to the hard-core one and the excludedvolume decreases to the athermal value at higher temperature.

1.3 Depletion Layer

When non-adsorbing polymers are added to stable colloidal suspensions the particles experiencean induced depletion attraction due to an unbalanced osmotic pressure arising from the exclusionof polymer molecules from the region between colloids. The strength of these attractions iscontrolled by the polymer concentration, Whereas the range of attractions depends on theasymmetric size ratio, q (the polymer to colloid size ratio).

We investigate the influence of the polymer to colloid size ratio and also excluded volume inter-action between a polymer segments on the induced depletion interactions and solution thermo-dynamics. Our studies involve the entire range of polymer to colloid size ratios, from the colloidlimit (Rg/R ≪ 1) to the protein/nanoparticle limit (Rg/R ≫ 1). For the colloid limit, wherethe colloidal particles are much bigger than the coils the induced attraction potential can beexplained via two physically intuitive arguments. First, the entropic point of view: the transla-tional entropy of polymer increases when particles cluster and their depletion layer surroundingthe excluded volume of particles overlap. Second, the osmotic pressure point of view: Pushing

7

1 Introduction

Figure 1.2

the colloidal particles together by the difference in osmotic pressure exerted by polymers on col-loids when the gap width between two particles approaches ∼ 2Rg and the polymer is squeezedout. Indeed, the question is what one means by ’depletion interaction’ if the polymer coils aremuch larger than the colloidal particle, Rg ≫ R, specially in the dilute limit where chains wraparound the particles (see Fig. 1.2)? Obviously, the first description via unbalanced osmoticpressure and squeezing polymer out is not very useful and the pair-decomposable effective po-tential loses meaning as many body effects become important. However, the fundamental ideathat the physical clustering of particles lead to the reduction in perturbation of polymer coilsby the hard spheres and increasing the polymer conformational entropy remains valid. In ourmethod the polymer coils internal conformational degrees of freedom is tackled directly. Ourresults describe how entropy loss due to excluded volume interaction with colloids depends onparticle-particle separation and how the single polymer site-site excluded volume interactioneffects the depletion attraction.

1.4 Thesis Structure

This thesis examines the effects of addition of nonadsorbing real polymers to colloidal suspen-sions. The structure of this thesis falls into five sections:

• The method employed and the considered model (Chapter 2).

8

1.4 Thesis Structure

• The thermodynamic consistency of the approach (Chapter 3).

• Comparing our numerical results with the existing theoretical ones in extreme limits (Chap-ter 4).

• Connection of our numerical works to experimental systems (Chapter 5).

• Summary and outlook (Chapter 6).

9

2 Model

2.1 PRISM m-PY theory

2.1.1 Introduction

Polymer Reference Interaction Site Model (PRISM) theory is a successful theory to describethe complex fluid structure. This model relates the intramolecular and intermolecular structurethrough a single integral equation. In 1914 Ornstein and Zernike (OZ) proposed to split the”influence” of one particle on the other particle in a simple liquid into two contributions, a directand an indirect part. The direct contribution is defined to be given by the direct correlationfunction, denoted c(r), and was first introduced in their investigation of density fluctuationsnear the critical point. The indirect part, g(r) = h(r) + 1, is due to the influence of the firstparticle on a third particle, which in turn affects the second one, directly and indirectly. Whereh(r) is the total correlation function and g(r) is the radial distribution function. Later on,(1970s) Chandler, Anderson and Co-workers extended the monoatomic OZ equation applicablein simple liquids to the polyatomic integral equation based on the Reference Interaction SiteModel (RISM) theory. In RISM theory each molecule is subdivided into bonded sphericallysymmetric interaction sites and intramolecular correlations play a role in the determinationof the intermolecular structure. In other words, the molecular shape and the rigidity, whichis described by intramolecular chemical bonding influence strongly the intermolecular packing.Schweitzer and coworkers in 1978 generalized this model to flexible and semiflexible polymersolutions, melts, alloys, self-assembling block copolymers and liquid-crystalline polymers. Thisapproach is referred to as polymer RISM or PRISM, theory. The first version of PRISM isbased on two simple ideas. First, the site-site direct correlation function does not depend on theposition of the monomers along a chain, that is, a ’preaveraging end effect’ is assumed. It shouldbe mentioned that, imposing this approximation some details of the results are eliminated butfor even very short linear polymer, N=3,4 the chain average correlation is studied accurately [34].The second assumption rests on the approximation that, the architecture of the macromoleculein presence of a solvent or other molecules does not change and a ’conformational preaverage’is considered. That means, the N body intramolecular structure is replaced by its ensemble

10


average pair correlation function.

In order to solve many kinds of integral equations including RISM and PRISM a fundamentalphysical approximation for the direct correlation function, that relates the bare intermolecu-lar potential, temperature, thermodynamic state and intra and intermolecular correlations isrequired. Such relations are called closure approximations. There are many different types ofclosures such as the mean-spherical approximation (MSA), rescaled MSA, hypernetted-chainapproximation (HNC) and Percus-Yevick (PY) approximation, but the fundamental question is,how a closure has to be chosen or constructed to yield reliable thermodynamic consistency andphysically meaningful structural properties. For the dense simple liquids with strong repulsionand weak attraction Percus-Yevick imposed an approximation that gives good agreement withcomputer simulation and experimental results for monoatomic liquids. Schweitzer and coworkeralso found that this approximation, the site-site PY closure, works well for athermal macro-molecular melts and mixtures in all molecular size. But for the binary system, mixtures ofmacromolecules and nanoparticles, classical PY closure does not lead to the correct gas liquidphase separation prediction [46]. As the accurate treatment of the ’cross correlation’ in suchmixtures is important to describe the packing and penetrability of polymer segments close tothe colloid, in 2000 Fuchs and coworkers [31] suggested a modified Percus-Yevick closure thatintroduces a length parameter to account for a non-local repulsive potential between segmentsand mesoscopic particles via thermodynamic consistency.

In this chapter we are not going to explain about general PRISM theory in technical detail.The goal is to present the PRISM theory and m-PY closure approach for this special problemand the numerical methods to determine radial distribution functions, second virial coefficient,and the enforcement of thermodynamic consistency. Since in this thesis we aim to investigatecolloid-polymer mixtures at the low colloidal volume fraction limit, in Sec. 2.1.2 m-PY PRISMintegral equations for ϕc → 0 are expanded and the m-PY employed closure is introduced. InSec. 2.2 the model used to investigate mixtures of nanoparticles and non-adsorbing polymers inlow density limits is clarified. The intramolecular form factors as the input functions in PRISMequations also described. In Sec. 2.2.1 the method of determination of the polymer segment-colloid correlation function, gcp(r), from m-PY PRISM equations at low colloidal density limitis explained. We explicate how one can obtain the colloid-colloid radial distribution function,gcc(r), and second virial coefficient, Bc

2(r), using the results of previous steps in Secs. 2.2.2 and2.2.3, respectively, and also the related equations are derived. It should be mentioned that inthis chapter the m-PY closure parameter λ still is kept arbitrary.

11

2 Model

2.1.2 PRISM

Employing a liquid state theory approach in a binary mixture requires that all the variables inthe equations appear as matrices. Consider Z particles composed of N segments in volume V,the local fluctuation density is:

ϱ(r) =Z∑

i=1

N∑α=1

δ(r − r(i)α (t)), (2.1)

with equilibrium average ⟨ϱ(r, t)⟩ = ϱ and its Fourier component ϱ(q) is (Carets denote theFourier-transformed quantities):

ϱ(q) =Z∑

i=1

N∑α=1

eiqr(i)α . (2.2)

The statistical average vanishes expect for zero wave vector, ⟨ϱ(q)⟩ = ϱδq,0. The density fluctu-ations are quantified by (matrix of partial) structure factors,

S(q) =1V

⟨ϱ(q)ϱ∗(q)⟩ − Z2N2

V⟨ϱ(q)⟩

=1V

Z∑i,j=1

N∑α,β=1

⟨eiq(r

(i)α −r

(j)β )

⟩− Z2N2

V⟨ϱ(q)⟩ .

(2.3)

The total structure factor, S(q) consists of two terms, (diagonal) intramolecular form factorωij(q) = ωjδij where i and j can be c and p respectively related to the contributions of colloids(radius R) and polymers (composed of N monomers or interaction sites). The intramolecularstructure factor illustrates the correlation of two different sites of the same molecule. The secondpart is the site-site intermolecular pair correlation or radial distribution function:

gij(q) =V

Z2N2

Z∑i,j=1,i =j

N∑α,β=1

⟨eiq(r

(i)α −r

(j)β )

⟩, (2.4)

which is trivially related to the total correlation function in real space, gij(r) = hij(r) + 1.

S(q) = ϱω(q) + ϱh(q)ϱ (2.5)

Weher ϱij(q) = ϱjδij is the number density of polymer segments (j = p) and colloids (j = c).While gij(r) describes the relative probability of finding site species i in distance r from site

12


species j located at the origin. The steric interaction or excluded volume to prevent the hardcores overlapping requires that

gij(r < (σi + σj)/2) = 0 (2.6)

where σp is the excluded volume diameter of a polymer segment and σc is the colloid hard corediameter σc = 2R. The generalized Ornstein-Zernike or Chandler-Anderson equation is givenby

S(q)−1 = ω(q)−1ϱ−1 + c(q). (2.7)

The intramolecular form factor, ω(q) is a known input in this approach and obviously dependson the statistics of the considered molecule, self avoiding walk, random walk, ring and etc.

ω(q) =1N

Z∑α,β=1

⟨eiq(rα−rβ)

⟩. (2.8)

The single polymer chain density fluctuation is assumed to be the same everywhere in thesolution, that is, a preaverage approximation neglecting the end effect is taken into account.The rigid colloid particles are described by one site molecule and act as point scatterers, soωc(q) = 1.

The effective interaction potential, cij(q), is the Fourier transform of the direct correlation func-tion which depends on the closure chosen based on the system under study. Liquid state theoryrests on the simple idea that even for long ranged fluctuation systems, cij(r) has a relativelysimple form and decays to zero beyond a few particle diameters. Taking into account the barepair interaction potentials, cij = −uij(r)/(kbT ), with Boltzmann constant kb and temperatureT, then Eqs. (2.5) and (2.7) would correspond to the random phase approximation(RPA). Thisis one of the simplest perturbation closures in liquid state theory for dense polymers and simplefluids which reduces to the mean spherical approximation (MSA) for pointlike particles. Since,in mixtures of nanoparticles and flexible chains, site-site excluded volume interaction in bothintramolecular and intermolecular form factors are considered, applying RPA results violate theoverlap constraints on g(r) (see Eq. (2.6)). For colloidal hard-spheres, colloid-colloid directcorrelation function is well-established by the Percus-Yevick approximation [47]

ccc(r > σc) = 0. (2.9)

13

2 Model

Implementation of this closure and site-site excluded volume condition, Eq. (2.6) in the gener-alized Ornstein Zernike equation, Eq. (2.7) leads to the coupled density fluctuations at differentwave vector, thus yielding a more complicated nonlinear integral equation than simple RPA.

The other component of the direct correlation function is cpp(r), which is proportional to pairwiseinteraction between two polymer segments given by a symmetric effective potential. Accordingto PRISM studies for polymer melts or pure homopolymer solutions this term vanishes beyondpolymer repeat unit size [33, 34], so one can enforce the corresponding simple closure,

cpp(r > σp) = 0 (2.10)

Here an ’athermal’ model is investigated, but a similar treatment attractive interactions is alsopossible [34, 48].

The colloid-polymer segment strict condition was introduced by Fuchs and coworker [31, 36]. Thefirst simple, but crude, guess for the colloid-polymer segment effective potential interaction couldbe the same as for the polymer-polymer segment or colloid-colloid one. It vanishes immediatelyin at a distance bigger than the particle size, (σc + σp)/2. Due to the penetrability of polymerchains and their conformation al changes near a colloidal particle, employing the simple PYclosure is not sufficient and leads to the overestimation of the structure of this binary mixturebecause of the overvaluation of depletion effects. The approximation consequently does not fulfilthermodynamic consistency [36, 48, 49]. Introducing a new length parameter is required, first, tocharacterize the interaction of two different kind of sites that extended beyond the overlap range,second, to be able to describe the change in polymer configuration close to colloid particles, andultimately, to lead to physically reliable results in the calculation. In this way, a parameter λ isintroduced in a modified Percus-Yevick (m-PY) relation, first pioneered by Fuchs [31].

ccp(q) =cscp(q)

1 + q2λ2, (2.11a)

cscp(r >

σc + σp

2) = 0. (2.11b)

From a physical point of view, since −kbTccp(r) expresses the segment-colloid effective potential,one expects cs

cp(r) to be negative to reflect the repulsive interaction and to show rapid variationon polymer segment range and slower behaviour connected with the colloidal size. Equivalentlyin real space:

ccp(r) =∫

14πλ

1|r − s|

e|r−s|/λcscp(s)ds. (2.12)

14


The segment-colloid interaction is blurred over the distance λ. That is, it does not turn offimmediately beyond the overlap range but decays to zero smoothly in the interval λ. In otherwords, cs

cp(r) can be interpreted as unconnected polymer segments and the (nonlocal) change inthe polymer conformations close to colloidal particle is captured by the spatial convolution.

In order to solve the three coupled PRISM integral equations, the length λ, which is an un-determined parameter in the m-PY closure, has to be specified. As it describes the polymerpenetrability near the nanoparticles and obviously influences the excluded volume interaction, itis expected that λ changes nontrivially with the physical parameters of the system like densitiesand asymmetry size ratio. Due to the same physical considerations, its magnitude should besmaller than or equal to the polymer correlation length or/and the colloid radius, e.g. whenthe polymer correlation length is comparative with Rg (radius of gyration) in the dilute limit orblob/mesh size for semidilute conditions. Furthermore the polymer configurational changes andtherefore λ depends on the space available between the colloids. The thermodynamic consis-tency should be enforced to yield an unique λ and to achieve a parameter-free priori description.Since the liquid state integral equation theories utilizing closures describe pair correlation func-tions approximately, deriving the unique answer for a thermodynamic quantity via differentapproaches is a requirement for physically reliable results which may be compared with experi-mental data. Implementation of this concept within a liquid state theory approach applying PYclosure is used for the first time in polymer melts [33] through different methods to calculatethe pressure in order to carry out the equation of state of the system. PRISM and PY (λ = 0)are used to describe the structure and thermodynamic properties of the mixture of colloids andpolymers in the colloid limit, where the colloid particles are much larger than the polymer coils.The free energy extracted from the compressibility theorem compares favourably with exact fieldtheoretic results [50]. Enforcing thermodynamic consistency via determination of unique λ waspioneered by Fuchs [31]. The excess chemical potential for inserting an ideal chain in a colloidfluid δµp, is calculated using two different methods, compressibility route and virial route. λ

has been chosen in such way that both approaches lead to the same result for δµp. Comparingwith field theoretic results less that 15 percentage error is observed [36]. In the present workthe excess chemical potential for inserting a single chain (ideal chain or real chain) into a hardsphere fluid, δµi, yields via the compressibility theorem,

∂δµi

∂ϱi= −kBTcij(0) and δµi =

∂Fex

∂ϱj,

where Fex is the excess free energy per unit volume. The limit of vanishing polymer concentrationshould be considered,

15

2 Model

Nβδµ(c)p |ϱp=0 = −N

∫ ϱc

0ccp(q = 0, ϱ′c)|ϱp=0dϱ′c, β ≡ 1/(kBT ) (2.13)

where this expression is for the chemical potential per molecule. The excess chemical potentialobtained via this route is not a sensitive function of λ as it highlights long wavelength corre-lations. Hence another independent and more local route to the insertion free energy, whichleads to strongly λ dependent results, is required to determine λ sensitively when equating bothexpressions. The approach of thermodynamic integration will be used to characterize δµp inde-pendently, as pioneered by Chandler for RISM approaches [39]. In this method the integration ofpair distribution functions, on a local distance corresponding to growing the particle from pointlike particle to the proposed one is related to the required thermodynamic property, the freeenergy. As a matter of fact the effect of turning on particle interaction on the variation of thefree energy is considered via the Mayer f function, f

(ζ)ij (r) = e−βuij(r) − 1 and ζ ∈ [0, 1], where

uij is the site-site potential, fij(r)(1) is the physical function of interest and fij(r)(0) belongs tosome known reference system. We are interested in the limit of vanishing polymer segment size,σp → 0. As the only interaction between particles is site-site excluded volume, the free energyof the mixture of colloid and polymer point like particles (ideal gas) is taken into account asF0 in a thermodynamic integration. Then, by growing the diameter of the colloid particle ζσc

from 0 (ζ = 0) to σc (ζ = 1), the true system, which is a mixture of hard spheres and athermalpolymers, will be achieved. From reference [39] one finds

Nβ(F − F0) = Nπϱcϱpσc

2∫ 10 dζ(σp + ζσc)2g

(ζ)cp (σp+ζσc

2 )

+2πNϱ2cσ

3c

∫ 1

0dζζ2g(ζ)

cc (ζσc)(2.14)

While the pair distribution function, g(ζ), depends on colloidal size, ζσc, via the colloidal volumepacking fraction, ϕc = (π/6)ϱcσ

3c , it needs to be λ are evaluated at the intermolecular distances

of closest approach. Only one extracts from the equation Eq. (2.14), that the growing col-loid particle has to push against the pressure imposed by the surrounding polymer and colloidparticles which in a virial theorem analogy is described by the probability of surface collision.

According to the thermodynamic relation, δµi =∂F ex

∂ϱi, one immediately derives from Eq. (2.14)

the second route to obtain a strongly λ dependent excess chemical potential.

16

2.2 Dilute Colloid Regime

Nβδµ(g)p |ϱp=0 = N

πϱcσc

2∫ 10 dζ(σp + ζσc)2g

(ζ)cp (σp+ζσc

2 )|ϱp=0

+2πϱcσ3c

∫ 1

0dζζ2 ∂g

(ζ)cc (ζσc)∂ϱp

|ϱp=0

(2.15)

The packing of polymer segments close to a colloid particle, gcp((ζσc + σp)/2), as mentionedalready, presents a sensitive λ dependence. On the one hand, a small amount of polymersdissolved in a colloidal fluid even in low colloidal packing fraction can strongly affect the colloidalsystem structure. On the other hand, according to scaling-law determination of λ as a functionof polymer parameters allow us to use the obtained unique λ in semidilute polymer regime aswill be shown in Sec.3.4. Ultimately, the two independent approaches, compressibility and virialtheorem, considered in case of vanishing polymer concentration have to be equal to yield thecorrect λ. In the next section our employed model and intramolecular form factors to describethe mixtures of non-adsorbing polymers and colloidal particles in low density limits are discussed.


The PRISM equation for the binary mixture of nonadsorbing polymers and nanoparticles indilute and semidilute conditions was already solved analytically [31, 36, 37]. The Gaussiandistribution for a single polymer chain form factor is considered, the results are compared withthe field theory approach [51–54] and good quantitative agreement is found. In later work,Fuchs suggests an ansatz for the colloid-polymer segment direct correlation function, ccp(q), andsolves the equations using Weiner-Hopf factorization analytically. Practically this analyticalmethod is not applicable when the polymer single molecular form factor, entered in PRISMequation as a priori, follows the self-avoiding walk statistics caused by intramolecular excludedvolume effects. While, our main purpose is to shed light on the determination of structuralcorrelations and interactions between the particles in a binary mixture for the all ranges ofasymmetry size ratios and concentrations the best way to deal with these complex equationsis a numerical method. Our computational approach is essentially applicable for any kind ofpolymer distribution function. This numerical method is split up into four main sequentialsteps and in each stage the results of the previous part is used. First we determine the colloid-polymer segment direct correlation function, ccp(r). Second, the colloid-polymer segment radialdistribution function gcp(r) is calculated. Third gcc(r), the colloid-colloid correlation functionand finally second virial coefficient are computed.

The particle dilute limit is important as the presence of a few particles can alter the structure

17

2 Model

of the mixture, since the single polymer form factor and correlation unaffected by the particles.Under this condition, definition of monomer density, ϱp, one can consider the system with avariety of polymer concentrations, from where polymer segments are well separated, the nameddilute limit, ϱp < ϱ∗p, to the concentration where polymer strands begin to overlap, ϱp = ϱ∗p, oreven interact strongly, called semidilute regime, ϱp > ϱ∗p, [20].

For the particle fluid the only thermodynamic parameter is the packing fraction, ϕc = (π/6)ϱcσ3c ,

where σc = 2R is the particle diameter. In this model the polymer chain fluid characterized bytwo physical parameters, the correlation or screening length ξ, and the reduced concentrationϕp. In the dilute polymer fluid the polymer correlation length reduces to ξ0 = Rg/

√2. The

reduced polymer concentration is defined as ϕp = 2π(ϱp/N)ξ30 , where N is the number of repeat

units per molecule. The reduced polymer concentration differs by a numerical factor from theoften used polymer packing fraction ηp = (4π/3)(ϱp/N)R3

g = ϱp/ϱ∗p ≈ ϕp/0.53 .

The treatment of the polymer chain correlation by changing the polymer concentration in thepolymer fluid is describable by the ’thread-like chain’ model [43, 44]. This model correspondsto the limit that all microscopic scales of the polymer solution approach zero but their ratiosremain finite. On one hand, statistical segment length lp/

√12, and segment diameter σp, are

taken to be negligibly small, σp ∝ lp → 0. On the other hand, in order to preserve the polymermolecule with finite radius of gyration, Rg, the degree of polymerization, N is increased beyondthe bounds, N → ∞, in such a way that ξ

1/ν0 = l

1/νp N = R

1/νg /2 remains fixed. Furthermore, it

is already investigated by the solution of PRISM integral equations for the the Gaussian chain,ν = 1/2, [55] that the intermolecular excluded volume stay active if the monomer density risesϱp → ∞ simultaneously, such that the number of polymer molecule per coils volume (∼ R3

g)remains finite: ϕp = 2π(ϱp/N)ξ3

0 stays fixed. Mathematically, the thread limit of PRISMequations describe the scaling law behaviour of polymer solutions and the dilute to semidilutecrossover, that is compared with field-theoretic scaling lows and results [35]. In both scalinglaw descriptions, all polymer fluid microscopic parameters like, lp, σp and reduces polymersegment density, ϱpσ

3, are dropped out and replaced by the polymer mesoscopic parameter,the chain size and the reduced polymer coil concentration. In addition, the polymer segment-segment direct correlation function is effectively regenerated by the Edward’s pseudopotentialmodel, density-dependent delta function, cpp(r) = cpp(0)δ(r), where the intermolecular excludedvolume parameter, cpp(0) follows selfconsistently the non-overlap condition, Eq. (2.6).

Two input single-polymer form factors are used. Gaussian (ideal) chains are described by randomwalk statistics [20] and excluded volume (real) chains by self-avoiding walk statistics [56].

ω(r) =1

4πΓ (1/ν)l1/νp

e−r/ξ0

rd−1/ν

{ν = 1/2 ideal chain

ν = 3/5 real chain(2.16)

18


where 1/ν classifies the polymer statistics, which will be used in PRISM equations and ξ is apolymer characteristic length, which strongly depends on the polymer concentration, ϕp. As amatter of fact, it reflects the correlation length between two monomers of a single chain in lowpolymer concentration , ϕp < ϕ∗

p and cross over from the radius of gyration of polymer, Rg, tothe density screening length or blob size in the semidilute regime, ϕp > ϕ∗

p, which describes thecorrelation of any pair monomers in polymer solution. ϕ∗

p is the polymer concentration, whenthe polymer coils begin to overlap.

ξ =ξ0

1 + 2ϕp. (2.17)

which is derived for the thread-like chain model using the Gaussian chain form factor (see Ap-pendix B). Note that the reduced polymer concentration is far smaller than the melt concentra-tion where the segments of different polymers start to pack densely and it is typically ≈ 30-40%of the melt density. In two extreme limits, very large distance (q → 0) and intramolecularregime (1 ≪ qRg ≪ qlp), two known and general scaling properties of the single polymer formfactor can also be derived from its Fourier transform, ω(q). It should be mentioned that the selfscattering term, qσp = O(1), presented in Eq. (2.8) is not accessible in the thread limit.

ω(q)

= N q → 0

∝ q−1/ν 1Rg

≪ q ≪ 1lp

In case ν = 1/2, the error of this structure factor is less than 15% for the entire region of q,while for ν = 3/5 this is valid only in the intramolecular (fractal) q regime presented in Fig. 2.1.

PRISM equations were already solved analytically for the mixtures of ideal chains and sphericalparticles, where there is no interaction between the polymer sites [31, 36, 37]. In order to showthe accuracy of our new approach the random walk statistics is inserted and the results arecompared with the analytical and field theoretical approaches [51–54]. In the next step theself avoiding walk chain is employed to illustrate the role of intramolecular excluded volumeinteraction in the structure and molecule-molecule interaction of the mixtures of non-adsorbingreal polymers and particles especially in protein limit. At the end and the outcomes are comparedwith the field theory results [26–28] in Chapter 4 and experimental data in Chapter 5. Throughout this work, the calculations, and the figure captions the dimensionless units are used withthe colloidal diameter as unit length, σc = 1. Hence, the only remaining physical parameters

19

2 Model

Figure 2.1

are, the asymmetry size ratio ξ in semidilute and ξ0 = Rg/√

2 in dilute polymer solution, thereduced polymer concentration ϕp, and the colloid packing fraction ϕc.

m-PY PRISM integral equations are nonlinear for the polymer and colloid structure even in thelimit of (semi) dilute polymer solutions. In order to gain to insight into the physics described bythese equations, the extreme limit of one component density should be considered. That is, inthe very dilute limit of the solvent, where the direct interaction between the solvent moleculesare at most pairwise, the equations in the correlation functions of the dilute species are linear.Thus the analysis of the integral equations will be simplified, as the correlation of the majority ofthe components are known. The colloid dilute limit and the related numerical approach will bestudied in the following step by step. Our numerical computation is divided into four sequentialmain stages.

• gcp(r), with an arbitrary λ. The colloid-polymer segment radial distribution function,gcp(r), gives the probability of funding a polymer in distance r from a colloid centred atthe origin.

• λ, to enforce the thermodynamic consistency. The excess free energy required for insertinga polymer chain in a colloid fluid via two different methods has to lead to the same result.This enables λ to be determined as a function of polymer size characteristic length ξ, and

20


colloid length σc.

• gcc(r), at the correct λ. In order to calculate the colloid pairwise interaction mediatedby the polymer solution, the colloid-colloid radial distribution function, gcc(r), calculatedwith the correct m-PY λ is needed. On the other hand, calculation of the colloid-colloidpair correlation function requires the colloid-polymer pair distribution function with thecorrect λ inserted into the equation as an input.

• Bc2. The colloid-colloid interaction as a function of polymer concentration ϕp, is accessible

through the determination of the second virial coefficient, Bc2 which is also experimentally

measurable.

2.2.1 Polymer-Colloid Radial Distribution Function

To achieve our goal, studying the polymer segment profile close to a single colloidal particleand the resulting force acting on two nanoparticles in the presence of long polymer strands in adilute colloid fluid, the first step is to determine the colloid-polymer direct correlation function,ccp(r). The function ccp(r) depends on a parameter λ which is a function of system lengthparameters, ξ and σc. A good guess is of order of these length parameters. In the limit of lowcolloid concentration, ϱc → 0, where small amounts of colloidal particles do not perturb thepolymer fluid structure, the generalized OZ equation implementation of m-PY closure leads tosimpler linear equations. The polymer colloid total distribution function in q space, hcp(q), isdescribed by the following set of linear equations

hcp(q) =cscp(q)

1 + q2λ2

1ϱp

Spp(q) (a)

hcp(r < (σc + σp)/2) = −1 (b)cscp(r > (σc + σp)/2) = 0 (c)

(2.18)

The proposed numerical method to solve PRISM integral equations is based on the one di-mensional Fourier back transformation of the PRISM equations. From the physical point ofview described by m-PY equations, these integrals split up into two parts: the inside distance,|r| < (σp +σc)/2, the distance between two species centres where the direct correlation functionplays a role and the total correlation function is constant. Outside distance, |r| > (σp + σc)/2,where the direct correlation function decays to zero, the penetrability of polymers is moreinteresting and all the structural properties and segment positions are described by the paircorrelation function. To follow this physical insight, it is assumed, that the direct correlation

21

2 Model

function is continuous when |r|⟨(σp + σc)/2 and becomes discontinuous at the contact pointsof polymer segments and colloids. It and decays to zero for distance larger than the contactvalue (not vanishing immediately). In addition, scaling all the length parameters by the colloiddiameter, σc, and considering the vanishing polymer repeat unit size, let us say σc = 1 andσp = 0. Therefore cs

cp(r) can be written as a sum of continuous and discontinuous parts:

cscp(r) = dcp(r) +

Cte(λ, ξ)2

(δ(r +

12) + δ(r − 1

2))

. (2.19)

Where Cte(λ, ξ) is an unknown parameter proportional to the slope of the radial distributionfunction at contact. It reflects the depletion layer around the colloidal particle. The depletionlayer is a function of both the polymer characteristic length ξ, and the penetrability of polymerchains near colloidal particles presented by λ. Both are scaled by diameter of the colloid.Obviously Cte(λ, ξ) should be determined from the PRISM m-PY equations evaluated at contactvalue, r ↘ 1

2 using the strict overlap condition, Eq. (2.18b).

We define the transform,

f(r) =∫ ∞

|r|sf(s)ds. (2.20)

This provides a way to simplify the three-dimensional Fourier transformation of f(r) only de-pending on radius r, to the one-dimensional ones [47],

f(q) = 2π

∫ ∞

−∞f(r)eiqrdr.

In our numerical computation the goal is to determine the intermolecular correlations includinghcp(r) and hcc(r), consequently the excess chemical potential and second virial coefficient. Thecorrelation functions can easily be derived from its tilde form functions f(r), presented in thischapter using Eq. (2.20).

The simplified three dimensional Fourier back transformation (see Appendix A) of Eq. (2.18a)together with Eq. (2.19) and considering the definition

1ϱp

Spp(q) = 2π

∫ ∞

−∞eiqrS(r)dr,

leads to the following equation,

(1 − λ2∂2r )hcp(r) = 2π

Cte

2[Spp(|r −

12|) + Spp(|r +

12|)] + 2π

∫ ∞

−∞dcp(s)Spp(|r − s|)ds. (2.21)

22


Regarding Eq. (2.18b) it is obvious that ∂rhcp(r) = r for |r| < 12 and considering that the

polymer segment-colloid direct correlation function cscp(r) is finite for overlaps in inside distance

and zero otherwise, Eq. (2.21) simplifies as following,

r = 2π∂r

[Cte2 (Spp(|r − 1

2 |) + Spp(|r + 12 |)) +

∫ 12

− 12

dcp(s)Spp(|r − s|)ds

]|r| < 1

2 . (2.22)

The partial structure factor, Spp(r), of a polymer fluid is required to determine the continuouspart of the polymer segment-colloid direct correlation function, dcp(r), Eq. (2.19). As argued,it is assumed that the presence of the small amount of colloid particles in polymer solution doesnot alter the single polymer form factor.

Spp(r) =1

4πΓ( 1ν )l1/ν

p

e−r/ξ

rd−1/ν=⇒ Spp(r) =

1

4πΓ( 1ν )l1/ν

p

∫ ∞

|r|s1+ 1

ν−de

−sξ ds (2.23)

Even replacing Eq. (2.23) in Eq. (2.22) is not yet enough to derive dcp(r) while Cte(λ, ξ) isan unknown parameter in Eq. (2.22). We thus have to employ a sophisticated method to solvethis equation, that is, as dcp(r) is considered a linear function of Cte(λ, ξ), Eq. (2.22) can bewritten,

d(r, c) = d0(r) + Cte(λ, ξ)d1(r). (2.24)

Using Eq. (2.19) we find

cscp(r) = d0(r) + Cte(λ, ξ)[d1(r) +

12(δ(r +

12) + δ(r − 1

2))]. (2.25)

In this case, Eq. (2.22) is a superposition of the inhomogeneous, imposing Cte(λ, ξ) = 0, andhomogeneous solution, where the left hand side of the Eq. (2.22) is considered to be zero (r = 0).Following this assumption hcp(r) is certainly a linear function of Cte(λ, ξ),

hcp(r) = h0(r) + C(ξ, λ)h1(r) (2.26)

23

2 Model

In the next step, in order to extract hcp(r), we need another independent equation valid for theentire range, −∞ < r < ∞. The simplified three dimensional Fourier back transformation ofEq. (2.18a) and a double convolution integral in right hand side of Eq. (2.18b) lead easily to:

hcp(r) =2π

λ

∫ 12

− 12

ccp(s)ds

∫ ∞

−∞e−

|r−s−t|λ Spp(t)dt (2.27)

The last step is the determination of Cte(λ, ξ). Implementing Eqs. (2.20) and (2.26) the constantCte(λ, ξ) is identified as

Cte(λ, ξ) =r − ∂rh0(r)

∂rh1(r)(2.28)

So C(λ, ξ) and finally the full expression for ccp(r) and gcp(r) = hcp(r) + 1 are obtained.

2.2.2 Colloid-Colloid Radial Distribution Function

In order to obtain the mean effective pair potential between two colloids in a polymer fluid,the colloid-colloid pair correlation function is required. gcc(r) describes the probability of twoisolated spheres in a polymer solution to be at the distance r is required. From PRISM equationsin low colloid dilute limit, ϕc → 0, it follows

hcc(q) = ccc(q) + ϱp

cscp(q)

1 + q2λ2hcp(q) (a)

hcc(r < σc) = −1 (b)cscc(r > σc) = 0 (c)

(2.29)

The Fourier transform of Eq. (2.24a) is

hcc(r) =ϕp

2λξ3−1/ν0 l

1/νp

∫ 1/2

−1/2cscp(s)ds

∫ ∞

−∞e

−|r−s−t|λ hcp(t)dt (2.30)

where the reduced polymer concentration is given by ϕp = 2π(ϱp/N)ξ30 and ν is the Flory expo-

nent. The numerical results for ccp(r) and gcp(r) from the previous stage and the strict overlapcondition Eq. (2.29b) are required to compute the colloid-colloid pair distribution function, gcc.Note that for semidilute polymer fluid, ϕ∗

p > ϕp, the polymer correlation length ξ0 ≈ Rg will be

replaced by the density screening length ξ =ξ0

1 + ϕp.

24


2.2.3 Second Virial Coefficient

The second virial coefficient Bc2, which describes the strength of the polymer-induced pair po-

tential and follows from the compressibility [57] is of interest, as it is measurable experimentally.

Bc2 =

−12

∫hcc(r)|ϱc=0dr = BHS

2 [1 − 3∫ ∞

1r2hcc(r)|ϱc=0dr] (2.31)

where BHS2 = 2π/3 is the second virial coefficient for the hard sphere fluid [47]. The sign of the

second virial coefficient is a signature of the kind of particle interactions. For a hard sphere fluidwhere ϕp = 0, its positive value exhibits a very strong repulsion, since the negative values of Bc

2

signals the presence of a net attractive effective interaction induced by the polymer (0 < ϕp).

25

3 Thermodynamic consistency

In investigating the structure of polymer-colloid mixtures, thermodynamic consistency playsan essential role. There are two independent ways to calculate the excess chemical potential,δµp, for inserting polymers in a hard sphere fluid. The compressibility root Eq. (2.13), whichemphasizes the long wavelength fluctuations and via local information captured by the paircorrelation function gcp(r) at contact, the so-called virial root Eq. (2.15). By tuning λ bothmethods can be made to yield the same result for δµ

(c)p . In other words, enforcing thermodynamic

consistency results in the determination of the effective interaction length scale λ.

In this section, first we introduce two roots of computation δµp then define the parameter λ

and at the end the method to be used to solve the equations is explained. To extract a uniqueparametric λ(ξ, σc) via equating the results of Eq. (2.13) δµ

(c)p and Eq. (2.15) δµ

(g)p a numerical

iteration method is required. This approach called Levenberg-Marquardt Method’ which itsalgorithm is available in ’Numerical recipes’ [58].

3.1 Compressibility Route

Via the compressibility theorem [47], Eq. (2.13),

Nβδµ(c)p |ϱp=0 = −N

∫ ϱc

0Ccp(q = 0, ϱ′c)|ϱp=0dϱ′c (3.1)

which at the colloid dilute limit, ϱc → 0, can be written as

Nβδµ(c)p |ϱp=0 = −NϱcCcp(q = 0, ϱc → 0)|ϱp=0 (3.2)

Applying the 3D Fourier transform of radial symmetric function,

ccp(q = 0) = 2π

∫ −∞

∞ccp(r)dr = 2π

∫ σc/2

−σc/2ccp(r)dr (3.3)

Using a well known scaling relation [20] between degrees of polymerization and polymer correla-

26

3.2 Virial Route

tion length, N = (ξ0

lp)

1ν , the definition ϕc = (π/6)ϱcσ

3c and using Eq. (3.3), Eq. (3.1) is written

as:

Nβδµ(c)p |ϱp=0 = −12Nπφc

∫ σc/2

−σc/2ccp(r)dr. (3.4)

3.2 Virial Route

The calculation of the excess chemical potential for insertion of a single polymer chain into acolloidal fluid requires a thermodynamic relation to extract the excess chemical potential fromthe excess free energy for a polyatomic fluid which is expanded by Chandler [39]. Consider uij(r),the potential energy associated with the pair interaction sites on different molecules when theyare separated by a distance r. Note that for a uniform system uij(r) and consequently, thepair distribution function, gij(r), only depend on |r|. Hence, for a uniform system for which allpoints in the volume V are identical,

V −1δF/δuij(r) = (1/2)NiϱiNjϱjgij(r)

= (1/2)ϱiϱj(hij(r) + 1),

(3.5)

where Ni is the number of sites i per molecule of type i and δF is the difference in Helmholtz freeenergy between the final and initial states of the system during a reversible process, where thetemperature, volume, and particle numbers, Ni(j) are fixed. The site-site interaction potentialcan be separated into reference and perturbation part. Let u

(0)ij (r) denote the unperturbed or

reference part of uij(r), and ∆uij(r) the perturbation:

uij(r) = u(0)ij (r) + ∆uij(r)

Using functional integration [47] in Eq. (3.5) we have

F/V = F (0)/V + ϱiϱjNiNj

∑i,j

∫ 1

0dζ

∫g(ζ)ij (r)∆uij(r)dr, (3.6)

where g(ζ)ij (r) = gij [r, u

(0)ij (r) + ζ∆uij(r)].

27


Another method to derive this results would be to differentiate the logarithm of the canonicalpartition function for a system with the pair potential u

(0)ij (r)+∆uij(r) with respect to ζ, which

leads to dF (ζ)/dζ where u(ζ)ij (r) = uij [r, u0

ij(r) + ζ∆uij(r)]. Integration of the result then givesEq. (3.6).

Considering the change by altering the Boltzmann factor instead of the potential directly leadsto the definition of the Mayer function or cluster function,

fij(r) = exp[−βuij(r)] + 1. (3.7)

Using the chain rule Eq. (3.5) yields

(δF/V )δfij(r)

= −(1/2)ϱiϱjNiNjYij(r), (3.8)

where

Yij(r) = eβuij(r)gij(r) (3.9)

is called cavity function or indirect correlation function. We now consider a scheme for ’growing’sites. Suppose

f(ζ)ij (r) = fij(r/ζσi) 0 ≤ ζ ≤ 1 (3.10)

with the growing parameter ζ and the site diameter σi(j). Using the functional integrationprocedures begun by parametrising length, Eq. (3.8) leads to

dβF (ζ)/V

dζ= −(1/2)NiϱiNjϱj

∑i,j

∫Y

(ζ)i,j dfij(r)/dζdr (3.11)

where Y(ζ)ij (r) and F (ζ)(r) refer to the function Yij(r) and the free energy F (r) evaluated for a

system with Mayer function f(ζ)ij (r). Integration of Eq. (3.11) over ζ yields

β(F − F0)/V = −(1/2)NiϱiNjϱj

∑i,j

∫ 1

0dζ

∫Y

(ζ)ij dfij(r)/dζdr. (3.12)

We employ this idea for the system under study, mixtures of nonadsorbing polymers and meso-scopic particles in a good solvent. As the limit of vanishing polymer segments size, σp → 0, is ofinterest and excluded volume is only interaction between the particles the considered referencesystem is the mixture of polymer sites and colloidal point particles. Since, only the site-site

28

3.2 Virial Route

interaction is considered and the interaction between sites is excluded volume, considering thatall the species in the system under study are identical leads to Ni = Nj = 1. On the onehand, increasing the size of species, here ζσc via ζ, causes a binary mixture with two differentpair interactions; the interaction between the growing size particles, colloid-colloid, and betweengrowing size and fixed size particles, polymer segment-colloid. On the other hand, as long asparticles grow very slowly, the species interaction are identical. So considering a system com-posed of a binary hard core particles, tunable core size, σi(j) ≡ ζσc, and fixed sites size, σi ≡ σp,for any pair sites (note that the polymer sites are ideal and there exist no interaction betweenthem) one can conclude that

fij(r) =

{−1 r < (σi + σj)/20 r > (σi + σj)/2

In this case the integration over r in Eq. (3.12) can be easily performed since the derivative ofa step function yields the delta function. Therefore, thermodynamic integration over growingparameter 0 < ζ < 1 via increasing the colloidal particles diameter, σ(ζ)

c → ζσc, to reach thetrue mixture leads to the following formula for excess free energy,

β(F − F0) =πϱpϱcσc

2

∫ 1

0(σp + ζσc)2g(ζ)

cp (σp + ζσc

2)dζ + 2πϱ2

cσ3c

∫ 1

0ζ2g(ζ)

cc (ζσc)dζ (3.13)

where the pair correlation functions g(ζ)(r), depends on the coupling parameter ζ via the volumefraction of the colloid particles and evaluated at the half diameter summation, the distances ofthe closest approach. The reference free energy, F0 is that of an ideal gas, noninteracting mixtureof point particles and point site polymers, at the same temperature, volume and density as thetrue system of interest; a mixture of nonadsorbing polymers and hard core colloidal particles.Equation (3.13) express the fact that the growing colloidal particles have to push against thepressure caused by the rest of the particles around, polymers and growing colloids. This pressureis described in an analogy to the virial theorem by the probability of contact at the surface.The differentiation of the free energy with respect to density of solute to obtain the chemicalpotential of a solute (polymer chain) at infinite dilution leads to


πϱcσc

2

∫ 1

0dζ(σp + ζσc)2g(ζ)

cp (σp + ζσc

2)|ϱp=0 + 2πϱ2

cσ3c

∫ 1

0dζζ2 ∂g

(ζ)cc (ζσc)∂ϱp

|ϱp=0

(3.14)

While the expression per molecule is of interest and all segments in a polymer are identical andindependent, both sides of Eq. (3.14) are multiplied by N , the monomers number per polymer

29


chain. The second term in right hand side of Eq. (3.13) comes into account in case of finitecolloid volume fraction. As the colloid dilute limit, ϱc → 0 is of interest the second on the termright hand side containing ϱ2

c is negligible.

βδµ(g)p |ϱp=0 =

πϱcσc

2

∫ 1

0dζ(σp + ζσc)2g(ζ)

cp (σp + ζσc

2)|ϱp=0 (3.15)

From Eqs. (2.20 - 2.22) it is concluded that gcp(r ↘ (σp+σc

2 )), the probability of finding onecolloid in a distance r from a polymer segment at contact point behaves like:

gcp(r) −→ A(ξ, λ(ξ

σc))(

r − 1/2ξ

)1/ν r ↘ 12. (3.16)

Where the length parameters are scaled by the colloid diameter σc = 1 and σp = 0. Theconstant A(λ, ξ) is a function of λ and ξ reflecting the slope of gcp(r) at contact point. Thisequation also expresses that the Gaussian chain, 1/ν = 2, penetrate easier than the excludedvolume interaction chain, 1/ν = 5/3, into a very small separation of two colloid particles, as itis predicted [26, 36]. In other words, the density profile of ideal chain in the gap between twocolloidal particles increases quadratically, which is faster than for a real chain.

On the other hand, imposing the condition q → ∞ in Eq. (2.20) evaluating at contact value(r ↘ (σp+σc

2 ) ≃ 12), the following equality between A(λ, ξ) and C(λ, ξ) is extracted:

C(λ, ξ) = −Γ(1ν

)1 − ν

ν2σcλ

2ξ−1ν l

1νp A(λ, ξ) (3.17)

replacing Eq. (3.16) and Eq. (3.17) in Eq. (3.15) leads to the following equation:


πϱcσc

2

∫ 1

0dζζ2A(ξ, λ(

ξ

ζσc))(

σp + ζσc − (σp + σc)2ξ

)1/ν . (3.18)

However, the solution of the full PRISM equations containing the microscopic length scales

is not yet achieved and for simplicity σp = (σp + ζσc − (σp + σc)

2ξ) ∼

√2lp has been chosen

[31, 36, 37]. By this choice the numerical results lead to λ. The effective length scale λ is adistance over which the polymer chain rearranges near to a colloid particle and for very smallpolymers becomes comparable to the polymer size, λ = ξ0 = Rg/

√2. Employing Eq. (3.18) in

Eq. (2.29) lead to the following formula for the polymer excess potential.


πϱcσ3c

2

∫ 1

0dζζ2A(ξ, λ(

ξ

ζσc))(

√2lpξ

)1/ν , (3.19)

30

3.3 Numerical Approach

using relation Eq. (3.17)

Nβδµ(g)p |ϱp=0 = −πNϱcσ

2c (√

2)1/ν

2Γ( 1ν )(1−ν

ν2 )

∫ 1

0dζζ2

C(ξ, λ( ξζσc

))

λ2l1νp , (3.20)

for the ideal chain, (ν = 1/2)

Nβδµ(g)p |ϱp=0 = −3Nφc

∫ 1

0dζζ2

C(ξ, λ( ξ0ζσc

))

λ2l2p (3.21)

and for the real chain, (ν = 3/5)

Nβδµ(g)p |ϱp=0 = −3(1.77638)Nφc

∫ 1

0dζζ2

C(ξ, λ( ξ0ζσc

))

λ2l53p . (3.22)

3.3 Numerical Approach

The exact calculation of the unknown λ considering all parameters via equating Eqs. (3.4)and (3.20) lead to a very complicated integral equation including thermodynamic integration,which is not tractable. In order to solve such an equation, a parameterized ansatz based oninsight into the physical role of the parameter λ is required to guess the initial value for theunknown parameters, a and b, (see Eq. (3.23)) and to achieve optimal results using the iterationmethod. For small polymers λ is proportional to the correlation length of the chain, ξ0 for thedilute polymer regime which crosses over to ξ for the semidilute polymer regime [31, 36]. Inthe opposite size ratio limit, for much bigger coils, as the effect of polymer size disappears, thevalue of λ depends only on the colloid size, σc = 2R. For much larger coils than particles thepenetrability no longer depends on the chain size or mesh size in semidilute polymer solutionand the colloid particles affect the polymer configuration entropy. Therefore, a simple, realisticansatz could take the form:

λ−1 =a

ξ+

b

σc, (3.23)

where for convenience dimensional units are restored. To compute a and b the ”Levenberg-Marquardt Method” [58] is employed and the numerical results for C(λ, ξ) and ccp(r) are used.χ2 is defined as the difference of δµp obtained via two identical methods, compressibility root,Eq. (3.4), and virial methodology theorem, Eq. (3.20):

31


χ2 = δµ(c)p − δµ

(g)p

= −12φcccp(q = 0, ϱc → 0)|ϱp=0

−

[−πϱcσ

2c (√

2)1/ν

2Γ( 1ν )(1−ν

ν2 )

∫ 1

0dζζ2

C(ξ, λ( ξζσc

))

λ2l1νp

]

or using the definition of ccp(r) employed in our numerical method, Eq. (2.25), χ2 can beexpressed as

χ2 = δµ(c)p − δµ

(g)p

= −12φc

[∫ σc/2

−σc/2d0(r)l

1νp dr + C(ξ, λ)l

1νp

∫ σc/2

−σc/2d1(r)dr + C(ξ, λ)l

1νp

]

−

[−3φc(

√2)1/ν

2Γ( 1ν )(1−ν

ν2 )

∫ 1

0dζζ2

C(ξ, λ( ξζσc

))

λ2l1νp

] (3.24)

While for Gaussian chains the analytical result for λ interpolate between known exact limits[31, 36], for real chains the following ansatz for λ could be used as a simple first attempt,

λ−1 =a

ξ+

b

σc. (3.25)

where the variables a and b are unknown parameters. Inserting Eq. (3.25) in Eq. (3.24), makinga good guess for initialization of a and b and minimizing χ2 respect to both, produce a set ofequations that using iteration to reach the limit of χ −→ 0 determine a and b with high accuracy.

The parameters a and b calculated via our numerical approach for ν = 1/2 are compared withthe analytically exact asymptotic results [31, 36]. The advantage of this method to determinea unique λ by forcing coincidence of δµp through both thermodynamical approaches will bepresented. Then by imposing Flory exponent ν = 3/5 the parameters a and b and consequentlyλ will be yielded for the mixtures of real polymer and nanoparticles. The effective interactionlength obtained for ideal chains and real chains will be compared.

3.4 The Effective Interaction Length Scale

As λ(ξ, σc) reflects the influence of strands near a colloid particle it is expected that it will dependon the colloid size σc (via parameter b) and the polymer characteristic length, ξ0 = Rg/

√2 in

dilute polymer solution which crosses over to ξ in the semidilute polymer fluid regime (viaparameter a). Minimizing Eq. (3.24) respect to a and b considering ν = 1/2 lead to λ−1 =1.05ξ

+3.19σc

. Furthermore, λ which reflects the penetrability of the polymer chain into the gap

32

3.4 The Effective Interaction Length Scale

space between the nanoparticles in first order is equal to ξ to first order. The relative error forparameter a is 5%. At the limit of small ξ0 when colloids are much bigger than polymers, forvanishing polymer concentration, ϕp → 0, the result ξ = ξ0 is valid.

As we approach the limit of ξ0 → ∞, the characteristic length of penetrability λ has to beindependent of polymer length for very large coils and will be dependent on the particle whichis represented by its coefficient b. Comparing numerical results for b with the exact asymptoticvalue λ1 =

√5 + 1 from [31, 36] the relative error is less than 1.5%. Calculation of the excess

chemical potential with the numerically computed λ shows the high accuracy of the thermo-dynamic consistency and the curves extracted via two different roots coincide. The maximumrelative error is less than 5%, since for the same graphs using the analytical results for λ therelative errors are about 15% [31, 36].

Recalculating the effective interaction length for the case of an excluded volume interactionchain in mixtures of polymers and colloids by inserting ν = 3/5 in Eq. (3.23), one expectsfor very small polymer the intramolecular structure will play no role and no difference betweenreal polymer or ideal polymer appears at ξ → 0. However, in protein limit as the penetrabilitydepends on the statistical properties, a significant difference in the value of parameter b isobserved, b = 4.85. So then the numerically obtained λ is:

λ−1 =1.05ξ

+4.85σc

. (3.26)

concerning a the coefficient of 1/ξ, as it is mentioned the relative error with the answer forthe point polymer is up to 5%. The parameter value b = 4.85 obtained for the real chain

computation is larger than b = 3.19 belong to the ideal chain calculation. Since limξ0→∞

λ =1b,

one can conclude that Gaussian chains penetrate easier than excluded volume interaction chainsdue to their noninteracting segments.

Although λ has been obtained for low colloid and polymer concentrations, we will use it in thesemidilute limit regime, ϕp > ϕ∗

p, but ϕc → 0 when ξ0 is replaced by the full density dependent

one, ξ =ξ0

1 + 2ϕp. This replacement means by increasing the polymer density, substitute the

single-polymer-molecule form factor, ω(r) by the collective one, Spp(r). It is a standard proce-dure from polymer scaling approaches suggested by de Gennes [20] to use the density screeninglength (blob size) in place of the chain size in semidilute regime. For Gaussian chains, also Fuchsand coworker [36] have shown that this extrapolation leads to the thermodynamic consistencyfor insertion free energy of a colloid particle into a polymer fluid which requires quite differentthermodynamic relations as the original ones used to obtain λ. The way of extrapolating λ tothe finite colloid or polymer concentration is also presented in their previous works [31] (theyhave analytically determined the λ values at the limits and match the suggested Pade-form to

33


2 4 6 8 10ξ

0.1

0.2

0.3λ(

ξ , σ

c)

Ideal chainReal chain

Figure 3.1: The effective interaction length scale λ is sketched for both flory exponents, ν = 1/2(the black carve) and ν = 3/5 (the red curve) as a function of polymer size ξ0.

λ) . This work was performed for more complicated case where ξ = ξ(ϕc, ϕp, ξ0) is a func-tion of all three fundamental parameters of the system ϕc, ϕp and ξ0. Here it is simplified toξ = ξ(ϕp, ξ0, ϕc = 0).

3.5 Excess Chemical Potential

The chemical potential gives a measure of tendency of polymer chains to dissolve in colloidalsphere fluids or energy which is required to solve the polymers. To determine the insertionenergy to add a polymer particle into a low density colloid fluid, δµp using the obtained λ,Eq. (3.26) is the last step to investigate the thermodynamical consistency of the system. Thesummarized procedure is as following

1. Getting the polymer segment colloid direct correlation function, ccp(r), via Eq. (2.22).

2. Using the numerical results of the first step in Eq. (2.27) to obtain the polymer segment-colloid radial distribution function, gcp(r).

3. Enforcing thermodynamic consistency by solving Eq. (3.24) and using the numericalresults of ccp(r) and gcp(r) gained from previous stages to obtain a unique λ, Eq. (3.26).

34


4. Employing the yielded λ in Eqs. (3.4) and (3.20) to derive the excess free energy forinserting a polymer in colloidal fluids via two roots, compressibility, δµ

(c)p and virial root,

δµ(g)p respectively.

Figure 3.2 shows double logarithmic plots of the numerically obtained excess chemical potentialvia two thermodynamical approaches using the unique λ, Eq. (3.26) as a function of polymercorrelation length for low colloidal packing fraction, ϕc = 5%. The inset shows that the numericalresults for λ satisfy the thermodynamic consistency condition from Eq. (3.4) given by the greenline and Eq. (3.20) plotted by the blue line up to relative errors of 15%.

The energy required to dissolve point polymer particles in hard sphere fluids in an ideal gasmodel follows is given by

Nβδµ(c)p = −ln(1 − ϕc) + O(ξ0). (3.27)

This relation connect the energy cost due to change in translational entropy by adding a pointparticle in a hard sphere solvent of packing fraction ϕc to the free volume fraction accessible tothe polymer. The free volume fraction concept used by Lekkerkerker et al. [10] for finite colloidpacking fraction but vanishing polymer concentration to describe the phase transition.

In the limit ξ0 → 0 our numerical results for δµp follows Eq. (3.27). The deviation observedfor ξ0 < 0.03 in δµ

(g)p (excess chemical potential gained via virial root ) is because of the

presence of the power= 1/3 in denominator of integrand, Eq. (2.27), due to the Flory exponentν = 3/5. Integration leads to the incomplete Gamma function. During multiple integration ina region between the limits of integrals a very large negative value of argument entering theincomplete Gamma function occurs which leads to a very large complex number which exceedscomputational limits (Note, the incomplete Gamma function diverges for very big argumentvalues). But for δµ

(c)p captured from long wavelength fluctuations, it is easy to approach ξ0 =

0.001 and the relative error is approximately 5%. The source of error is connected to thenumerical value of parameter a, which equals 1.05 and not 1 (the analytical result for a).

At the limit of much larger polymer, ξ0 → ∞, the power low treatment of the excess chemicalpotential is comparable with the prediction of the field theory results

NβF ex = AgϱpRd(Rg/R)1/ν . (3.28)

It expresses the excess free energy to insert one colloidal particle in the bulk solution. Ag is auniversal constant for a good solvent, ϱp is the monomer density and N is the density numberof segment per coil. In order to compare with our numerical results, first, the excess chemicalpotential using the well known thermodynamic relation δµi = (∂/∂ϱi)F ex must be calculatedand Eq. (3.28) has to be adapted for a system contain more than one colloidal particle with

35


0.1 1 10ξ

0

-0.1

0

0.1

0.1 1 10

Log10

(ξ0)

0.01

0.1

1

10

100L

og10

(Nβδ

µ p)Nβδµ

p

(g)

Nβδµp

(c)

FT

δµ/µ

Figure 3.2: The double logarithmic graphs of the real polymer excess chemical potentials in goodsolvent δµp|ϱp=0 versus size ratio ξ0 for colloidal packing fraction maximum ϕc = 0.05from two different thermodynamic roots is shown. The blue line is the result capturedfrom long wavelength, Eq. (3.4), the green line from the local packing, Eq. (3.20), andthe dashed line presents the field theoretical result for extremely long chains in goodsolvent. The inset highlights the relative error which is less than 15%.

al most ϕc = 0.05. While in this limit the colloidal particles are not yet correlated and can beconsidered independent, we multiply this equation with the proper prefactor.

The comparison m-PY PRISM results and those of field theory for δµp shows a very goodagreement for very large ξ and the relative error up to 5% is found. Replacing constants withtheir values in Eq. (3.28) leads to Nβδµp = 0.56ξ

1/ν0 which is sketched as a dashed line Fig. 3.2.

Since, the asymptotical fitted curve to our numerical results for ξ0 → ∞ using ’Least MeansSquares’ is Nβδµp = 0.61ξ

1/ν0 .

The numerical results for δµp for ideal chains (ν = 1/2) and real chains (ν = 3/5) using therelevant λ from thermodynamic consistency enforcing are plotted in Fig. 3.3. At the limit of verysmall polymer chains the excess chemical potentials obviously do not depend on the polymerchain intramolecular structure and curves coincide when ξ0 → 0. For much bigger polymeras it is also predicted by De Gennes and with agreement with the Field theoretic results δµp

follows δµ ∝ ξ1/ν0 , where 1/ν is equal to 2 for Gaussian chains and to 1.67 for excluded volume

interaction chains.

36


0.01 0.1 1 10

Log10

(ξ0)

0.01

0.1

1

10

100

Log

10(N

δµp)

δµp Real chain

δµp Ideal chain

Figure 3.3: For two polymer intramolecular structure factors, Gaussian chains, ν = 1/2 (thedashed line) and excused volume interaction chains, ν = 3/5 (the green line) the excesschemical potential as a function of polymer characteristic size, ξ0 = Rg/

√2 scaled by

the colloid size (σc). For too large polymer coil, δµp follows the De Gennes scaling law,F ex ∝ R

1/νg , the slopes show the relevant power.

37

4 Results

We have considered nanospheres embedded in a monodisperse solution of long, flexible, nonad-sorbing polymer chains and studied via an integral equation method called PRISM based onliquid state theory the structural correlation functions and related thermodynamic properties.The PRISM equations require the intramolecular correlation as an input and via the integralequation, yields the intermolecular correlations. It is thus important to know accurately theinput functions and how they change by varying the fundamental system parameters such as,polymer concentration, ϕp and colloid packing fraction, ϕc. It is assumed that a small numberof hard spheres dissolved in a polymer solution does not affect the form factor of the polymermolecules. The single polymer form factors, ω(r), or the single polymer density correlation func-tion of a polymer fluid which is well understood from field theoretic considerations [45, 56, 59],is characterized by the Gaussian chain (ν = 1/2) and the excluded volume chain (ν = 3/5).The intermolecular density-density correlation exhibits a soft repulsion between the polymermolecules which confirm the well known ’correlation hole’ arising for entropic reasons polymers,as they repel each other. The partial structure factor Spp(r) has nontrivial variation on thelength scale of the particle size (radius of gyration) and the mesh or blob scale characterized bythe density screening length, ξ.

As the polymer concentration crosses over from high dilution (ϕp ≪ ϕ∗p) to the semidilute regime

(ϕp ≫ ϕ∗p), the polymer correlation length, ξ0 = Rg/

√2 is replaced by the mesh or blob size

which is related to the radius of gyration and polymer concentration via ξ =ξ0

1 + 2ϕp. Since all

length parameters including ξ are scaled by the hard sphere diameter, σc = 2R, the polymerscreening lengths play the role of the asymmetric size ratio. The polymer concentration variesfrom dilute regime to semidilute regime, but the low concentration colloidal fluid, ϕc → 0, keptand ϕc = 5% is the maximum value which is used in computation.

Furthermore, this model is applicable to the whole range of asymmetric size ratios, concerningmuch smaller polymer chain Rg ≪ R which is named extreme colloid or AO extreme (ξ0 → 0)to much larger chains Rg ≫ R called nanoparticle extreme which in literature is also called theprotein limit (ξ0 → ∞).

As a main concern we investigate in a systematic and quantitative way how the excluded volume

38

interaction between the chain monomers affects on the depletion attraction between nanoparti-cles and the density profile of polymers near to an isolated particle. For this purpose, and tocheck the accuracy of our numerical methods based upon PRISM equations, we consider firstthe ideal chain, using ν = 1/2. The obtained results using our numerical method are comparedwell with the results from the analytical approach based on the same equations [31, 36, 37] andfrom the field theoretical approach. Although analysing the analytical results in extreme limitsis more tractable than the those obtained numerically, which include numerical error due todiscretization, it is not possible to handle the m-PY equations analytically considering the selfavoiding walk statistic for the polymer form factor captured by ν = 3/5. Besides the applicabil-ity of this integral equation method to get the full description for the structure and correlationsfunctions of the polymer-colloid mixtures over the entire polymer concentration range, the otheradvantage of this method is the ability to consider finite concentration of colloidal particles inthe mixture. Note that in the field theoretical approach for both ideal and real chains in verydilute colloid limit, just the presence of one or two hard sphere is taken into account. In ad-dition, solving the PRISM equations containing many body effect for the ideal chain leads usto highly accurate results [31, 36, 37]. This motivates us to test our numerical method on asystem containing the nonadsorbing flexible real chains in the presence of a few colloids to getthe physical insight to employ this method concerning many body effect when 0 ≪ ϕc < 0.53.

To go further, the correlation functions and related interesting thermodynamic properties ofsmall particles dissolved in a polymer solution are computed. In Sec. 4.1 the polymer segment-colloid radial distribution function which reflects the polymer density profile around a colloidfor various asymmetric size ratio is calculated. We discuss the colloid-colloid pair correlationfunction and polymer induced free energy of interaction or potential of mean force between twoparticles, U eff

cc , in Sec. 4.2. Section 4.3 presents results for the second virial coefficient Bcc2

of a dilute suspension of colloidal particles. The second virial coefficient, which is measurableexperimentally by small angle neutron or X-ray scattering, is of particular interest, while innanoparticle extreme its value appears to be correlated with the success on protein crystalliza-tion.

To make contact with the asymptotic exact results of Eisenriegler [26–28], based on a fieldtheoretical approach for dilute and semidilute solutions of nonadsorbing polymers, the amplitudeof collective polymer structure factor, Spp(r), and consequently all derived equations should bescaled properly. The difference in the amplitude of the polymer fluid structure factor reflectsthe different polymer density solution used in the system.

39

4 Results

4.1 gcp(r), Polymer Segment-Colloid Pair Correlation Function

The density profile of the polymer segment at a distance r from a colloid centred at origin isgiven by ϱpgcp(r). Representative results obtained by solving Eq. (2.27) numerically for theinteractive polymer segment-colloid radial distribution for different size ratio is given in Fig.4.1. The most important aspects of these results (for both Gaussian and Flory polymers) arelisted as follows. The qualitative behavior is independent of the value of ν (emphasizing thepolymer statistic) used in the mixture of polymers and hard spheres, but looking in more detailthe significant differences in polymer penetrability in void space, depletion layer and the mutualmean forces due to the single polymer site interaction are obvious. In other words, the excludedvolume interaction between a polymer monomers causes some variation in structural correlationsand consequently in the second virial coefficient and the excess chemical potential for insertinga colloid particle in a polymer solution respect to same functions for mixtures of noninteractingchain at the same conditions. Fig. 4.2 shows the polymer packing and layering at the distancer from the colloid center for the different type of polymer form factors and size ratios.

Using the validity of PRISM approach for the full parameter range, it is of interest to lookmore closely at the well studied colloid extreme in order to access more carefully our resultsand at the nanoparticle extreme which is not fully understand yet. At the nanoparticle extremeregime, Rg ≫ R, where the polymer-particle-density profile changes qualitatively, two lengthscales appear. A relative long length scale of order the polymer radius of gyration in dilutelimit , ξ0, and the blob size in semidilute limit, ξ. The other one is of order a local lengthscale, particle radius R, while particles grabbed by the polymer strands perturb the polymerconformational entropy.

4.1.1 Density Profile of Polymers Close to an Isolated Colloid

The depletion effect leads to a monomer density profile that decreases on approaching the surfaceof a particle with the distance from the surface to the power 1/ν,

gcp(r ↘ (σp + σc)/2) ∼ A(ξ, λ(ξ))(r − (σc + σp)/2

ξ0)1/ν . (4.1)

It expresses that close to a colloid particle the probability of finding a polymer at distance r

from a colloid particle centred at origin grows in a power-law regime, where A is an universalamplitude. The Flory exponent, 1/ν, in our mean-field like theory describes the Gaussian chain,ν = 1/2, and the site interaction chain, ν = 3/5, as required from the ’wall virial’ theorem[26–28, 60]. The monomer density near to the hard sphere is related to the force per area thatpolymer exerts on the particle. Equating this pressure to the chain osmotic pressure in the bulk

40


1 2 3 4 5r/ σ

c

0

0.5

1

gcp

(r)

0 1 2 3 4 5r/σ

c

-1

-0.5

0

r2h

cp

ξ0=.5

ξ0=1

ξ0=3

ξ0=5

ξ0=7

ξ0=10

ξ=.5

ξ=1

ξ=3

ξ=5 ξ=7ξ=10

Figure 4.1: Probability of finding a polymer segment at distance r (in units of 2Rc) from a colloidof radius Rc centred at the origin (gcp(r)) is sketched as a function of distance of amacromolecule from the colloid for various polymer correlation length ξ. The insetshows long-ranged tail of the depletion layer indicated by r2hcp(r) = r2(gcp(r) − 1) ofthe same data.

which via the ideal gas law is kbTNϱp, determines the power. This also suggests [19] that forvery large particles, the monomer density ϱpgcp(r) very close to a colloid must be independentof N, the unit number per coil.

Exponent ,

For real chains in good solvent under athermal condition, the segment density profilepower-law manner with the growth exponent ν = 3/5 near to a colloid particle for the bothcolloid and nanoparticle limits, are obtained. Considering both ideal and real polymers,the growth Flory exponent ν compares well with field theoretic calculations.

Amplitude ,

For polymer chains very small with respect to colloidal particles, ξ → 0, the PRISM resultsfor the amplitude A which is an universal coefficient and independent on the microscopic

41

4 Results

1 2 3 4 5r/σ

c

0

0.5

1

gcp

(r)

ξ=.5, 3, 10 Ideal ξ=.5 Real ξ=3 Real ξ=10 Real

ξ=.5 ξ=3

ξ=10

Figure 4.2: The polymer density profile for three different polymer length are presented. Thecolored lines are belong to the excluded volume chains and the dashed line chained tothe colorful lines are the same correlation length of noninteracting polymers.

details is given by A = (1/√

2)1/ν . This quantitatively agrees with the known field theoreticlimit [27] considering less than 5% relative error, which has its origin in the numericaldiscretization.

In the extreme nanoparticle limit, ξ → ∞, the amplitude A in Eq. (5.1) is proportional toξ1/ν0 , that is, the polymer density profile near to the particle surface surrounded by very

long polymers does not depend on the radius of gyration of the polymer. This also agreesqualitatively with the field theoretic result for a large polymer in region r − R ≪ R ≪ ξ,where R is radius of hard sphere and r is the distance from colloid centred at origin thedensity profile is described by

g(r) = −AgBg/Sd(d − 1/ν)(r − R

R)1/ν

where the unit sphere surface Sd = 4π in d = 3 spatial dimension and universal amplitudesAg ∼ 18.4 and Bg ∼ 0.99 are calculated in reference [27]. For very large polymer coils theincreasing the polymer size do not effect on the density profile of chains near to an isolatedparticle. This can also be concluded by extending the density profile of ideal chains to thereal ones near to a particle surface regarding that in numerical calculation for the effect

42


of the long Gaussian polymer chain is measurable for ξ > 10 while for the long excludedvolume chain the same computation should be calculated for at least ξ > 20.

Comparing the density profile of the real chain and the ideal chain in the protein limit neara colloidal particle, highlights that the ideal chain penetrates easier than the interaction chaininto the void space between two colloids. Put simply, the existence of the excluded volumeinteraction between real polymer chain segments prevents the accumulation of monomers whencompared with an ideal polymer with the same length, where there is no repulsion between themonomers, close to a particle surface. In the field theoretic approach for much larger polymer,considering the point particle as a polymer configuration perturbator, the real polymer chainin a good solvent is perturbed less than Gaussian chains. These results lead to a very niceexpression for the different excess chemical potential to immerse an ideal or real polymer chainin hard sphere fluid which is explained in Sec. 3.5

4.1.2 Depletion Layer Width

The difference in the polymer density close to a colloid from the one in bulk entails a depletionlayer with width altered by changing system parameters, asymmetric size ratio and speciesconcentrations. In the extreme colloid regime, where the colloid are much bigger than polymerchain, Rg ≪ R, the hard sphere appears as a flat wall to the polymers and the depletion layeris proportional to polymer correlation length, ξ. Only for dilute polymer solutions this widthis proportional to polymer size, ξ0 = Rg/

√2. In the semidilute polymer regime it is given by

the blob size or mesh width. In the opposite extreme limit, Rg ≫ R, the depletion layer widthcrosses over to the particle diameter. Fig. 4.3 presents the depletion layer induced by the realpolymer (the red line) and ideal polymer (black line) as a function of polymer correlation lengthin a double logarithmic plot. The depletion layer width, w is defined by the distance from thecolloidal particle surface where the cross correlation function becomes half, gcp(w + 1

2) = 12 .

In the case of very small polymer where the width is a linear function of ξ, ideal and real polymersin a good solvent must induce the same depletion width. The intersection point of two lines fittedto the log(w) in this sub-region placed in the limit of ξ → 0. In fact these are two asymptoticlines with different slopes which cross at ξ → 0. For both ideal and real chains, by increasingthe size ratio of polymer correlation length to the particle diameter the chains coil around theparticle, Fig. 4.1. The colloid diameter is the only parameter that plays the role of varying theinduced depletion and the depletion width will be independent of polymer characteristic size ξ.In this region the double logarithmic graph coincides asymptotically with horizontal line whichits y-intercept value are definitely different for the two different kind of chain form factors. Inthe field theory approach for real polymer there is no expression to describe the density profile

43

4 Results

-2 -1 0 1Log

10ξ

-1.5

-1

-0.5

0L

og10

w Real chainIdeal chain

Figure 4.3: The width of the depletion layer around a colloidal particle in a double logarithmicplot as a function of ξ for two polymer solvent qualities, good (red curve) and thetasolvent (black curve), are plotted. Thin dashed lines mark the asymptotic behaviorsof depletion layer for limits ξ → 0,∞.

of polymer in the sub-region R ≤ |r − R| ≪ Rg in three dimensions.

It is worth pointing out that our results are in line with the conjecture that the weaker depletionlayer and consequently thinner depletion layer effects arise from chains with excluded volumeinteraction than from ideal chains with the same radius of gyration.

4.1.3 Intermediate Separation

In the region σc = 1 ≪ r ≪ ξ the monomer density profile is independent of the polymercharacteristic length, ξ =

√2Rg and increases in a ∼ r1/ν−d power-law manner. This power-

law behavior reflects self similar chain connectivity correlations which is also given in the fieldtheoretic approach by the following equation which describes the asymptotic exact result in thementioned region for the extreme limit of very large polymer chain, ξ → ∞

g(r) = −AgCm,g(R/r)d−1/ν + 1.

44


0 1 2 3 4r/σ

c

-1

-0.5

0

r2h

cp

Real chainIdeal chain

ξ=0.5

ξ=1

ξ=3

ξ=5

ξ=7

Figure 4.4: r2hcp(r) = r2(gcp(r)− 1 which reveals the long-range tail aspect of polymer segment-colloid radial distribution function for real (purple curves) and ideal (orange curves)chains for different amounts of value of polymer correlation length scaled by the radiusof a colloid particle are presented. The longer range tail of the ideal chain respect withthe real one is observed.

Ag and Cm,g are constants which are given in Table(I) reference [53] and d is spacial dimension,d = 3. Comparing this with our numerical results gained by solving PRISM equations exceptsome percentage relative error (less than 5% relative error) due to the meaning of ξ → ∞ innumerical computation, show very good agreement.

4.1.4 Long-Ranged Tail

At the limit of ξ → ∞ the cross correlation function exhibits an additional power-law tail due tothe chain connectivity correlations. The Fig. 4.4 highlights a new longer-ranged tail aspect ofthe pair correlation function for two different kind of polymer solvency, good and theta solvent,which appears only in length scale ξ ≫ R.

As a matter of fact, the long polymer (for simplicity in the dilute case) cannot totally balance theperturbation due to a particle repulsion on length scales shorter than Rg, that is, the polymersegments rearrange by adding a small particle. The consequence of this effect which emerges as

45

4 Results

a narrow component of the depletion layer approximately of order radius of particle for Rg ≫ R

is discussed in several prior works [19, 61–67]. The footprint of this long-ranged tail also appearsin the second virial cross-coefficient given by the integral over the correlated part of segmentprofile, gcp(r) can approach unity only for r ∼ Rg,

Bcp2 = −1

2hcp(q = 0)

Concerning the two different polymer form factors, real and ideal chains, as are shown in Fig.4.4 the density profile of an excluded volume interaction chain approaches 1 faster than the non-interacting one. It reflects that the interaction between nanoparticles dissolved in real polymerfluids is shorter range than the same in Gaussian chain solvents.

Alternative long polymer based treatment have proved inconclusive and provided conflictingresults. Analysing de Gennes [19] earlier work indicates that in protein limit the depletion at-traction effect is not significant and neglected, so then all mixtures are miscible. This argument,is seemingly due to missing the long range aspect of cross correlation length. Tuinier [67] andOdijk [61–63] also conclude the same, that is, the range of the depletion attraction between twocolloidal particles only depend on the colloid size∼ 3R. They assume that the depletion layerof colloidal particles in suspension are uncorrelated and an uncontrolled ’superposition of oneparticle depletion layer approximation’ is used, which is not required in field theory or integraltheory approaches.

4.2 gcc(r), Colloid-Colloid Correlation Function

The colloid pair correlation function, gcc(r), describes the probability of two isolated spheres tobe at the a distance r. It can be considered as an example of the influence of the polymer onthe colloid packing. gcc(r), extracted from Eq. (2.29) using the result for the cross correlationfunction and the obtained value for λ is shown in Fig. 4.5. For a fixed polymer radius of gyrationRg/

√2 = ξ0 = 5 we plot the colloid correlation functions for increasing polymer concentration

at law colloid packing fraction ϕc = 5%.

Why is the low colloid packing fraction of interest? The pair distribution function depends onthe interaction force between the particles. In general, for a fairly dilute dispersion of particlesa good approximation is afforded by

gcc(r) = e−βUeffcc (r) (4.2)

So then, to investigate the polymer mediated effective interaction or induced pair potentialbetween two colloids, βU eff

cc (r), in either dilute or semidilute polymer fluid it is required to

46


1 2 3 4 5r/σ

c

1

1.1

gcc

(r)φ

p=2 Real

φp=1

φp=0.5

φp=0.1

ξ0=5

Figure 4.5: The colloid pair distribution functions for the real chains with fixed polymer correlationlength ξ0 = 5 and denoted polymer concentration are sketched.

determine the colloid pair distribution function at a very dilute colloid limit. The colloid secondvirial coefficient which is measurable at low density is a consequence of the integration overthe pair colloid correlation function. The influence of polymer solvency, good or theta solvent,on this effect is of most interest, that is, how the polymer intramolecular interaction effect ontwo immersed independent hard sphere interaction. In Fig. 4.6 the colloid pair correlationfunctions, gcc(r) for two isolated colloidal particles in two different kinds of polymer solution,real polymer and ideal polymer, for denoted polymer concentration are presented. In followingfor both polymer form factors which numerically means using ν = 1/2 for the Gaussian chainand ν = 3/5 for considering the site excluded volume interaction, some important characteristicaspects of the colloid radial distribution function are classified.

4.2.1 Contact Value

As can be seen in Fig. 4.5 the gcc(r) close to a isolated colloid particle is higher than one.The contact probability of two colloids is increased above the random value of unity then itdecreases monotonically to one after several colloid diameters and does not show any oscillationor layering features as can be observed for the finite colloid packing fraction fluid. The contactvalue, gcc(σc = 1), depends on the polymer concentration and bigger ϕp leads to the higher

47

4 Results

1 2 3 4 5r/σ

c

1

1.1

1.2

1.3

1.4

1.5

gcc

(r)

φp=2 Real

φp=1 Real

φp=0.5 Real

φp=0.1 Real

φp=2 Ideal

φp=1 Ideal

φp=0.5 Ideal

φp=0.1 Ideal

ξ0=5

Figure 4.6: The colloid pair distribution function for two polymer statistics, random walk (dashedlines) and self avoiding walk (solid lines) for various amount of ϕp = 0.1, 0.5, 1, 2 ispresented. The effect of excluded volume interaction on pair colloidal correlation isreducing the power and the amplitude of colloids-colloid correlations.

probability of colloid contact, which signals an effective attraction between the colloids, inducedby the polymer chains. This, which is a signature of the depletion effect and its dependency onpolymer solvent type in discussed below.

Comparing with the field theoretic approach [26] in case of Rg >> R the potential of mean forcebetween two colloids in contact which is given by the logarithm of colloid radial distributionfunction at contact, βU eff

cc = − ln gcc(1), (see Eq. (4.2)) follows,

(δF2

kBT

)r=2R

= −(2 − m)F1

kBT. (4.3)

Where, m is a constant approximately equals 1.5 and kB is Boltzmann constant and T istemperature. It connects the free energy of interaction, F2, to some coefficient of the excess freeenergy of insertion of one colloidal particle in a bulk real polymer solution, F1, at the contactpoint. The PRISM result are well compared with the asymptotically exact field theoretic oneand the relative error is less than 10%. As said previously, the field theoretic results are welldefined in the limit of ξ → ∞ where the colloids looks like a point particle beside of a polymermacromolecules.

48


Fig. 4.6 shows that the contact value reflecting colloid depletion layer exhibits a polymer solventdependence. In the presence of chains in good solvent gcc(1), the probability of finding a colloidin distance r = σc from the center of another colloidal particle, shows a sudden increase intheta temperature, when the monomer-monomer interaction balance with the monomer-solventinteraction. In other words, the polymer monomers excluded volume interaction induces lessdepletion effect than the ideal polymer with no monomer interaction. It agrees with the conclu-sions of the previous section from the width of the polymer segment-colloid correlation functionat half value, gcp(1

2 + ω) = 12 .

4.2.2 Induced Potential

The monotonically decreasing colloid pair distribution from the contact value (bigger than one)to one without any oscillation implies an attractive interaction between the colloidal particles anddoes not exhibit any repulsive potential barrier. The origin of this attractive induced potentialis the depletion effect between two colloidal particles in polymer solution. The potential of meanforce is related to the colloid-colloid radial distribution via the following relation

βU effcc (r) = − ln gcc(r) (4.4)

While, very low colloid packing fraction is considered, effective potential is only mediated bythe polymer solution either dilute or semidilute and the influence of another colloidal particlesare ignored. Within m-PY closure for direct correlation function Eq. (2.11) and using Eq.(2.29) the m-PY PRISM theory yields a result connects the total colloid correlation functionhcc(r) = gcc(r) − 1 to the polymer correlations via polymer-colloid direct correlation function:

hcc(r) = Θ(r − 2R)∫

e−iq.rϱpc2cp(q)Spp(q)

d3q(2π)3

(4.5)

Fig. 4.7 highlights the numerical results of m-PY PRISM approach for normalized potentialof mean force between two hard spheres, −kT log(gcc(r)), for two different types of polymersolvents at several polymer concentrations and low colloidal packing fraction. Although thedepletion potential increases by adding more polymer chains into both good and theta solventbut interestingly the spatial range of this potential in a non-monotonic function of polymerconcentration. This reflects the non-trivial dependence of the amplitude at contact and polymercorrelation length on ϕp. As it can be expected, the induced attraction approaches the maximum,-1 ( scaled by the contact value), in closest distance and vanishes when the particles are to farfrom each other. Interestingly, the mean force exerted on two colloidal particles by the polymerin theta solvent is stronger than the one exerted by polymers in good solvent under athermalconditions, that is the induced potential also depends on the polymer solvency characteristic.

49

4 Results

0 0.1 0.2 0.3 0.4 0.5(r-σ

c)/R

g

-1

-0.5

0

Ueff

cc/U

0φ

p=2 Real

φp=2 Ideal

φp=.5 Real

φp=.5 Ideal

φp=.1 Real

φp=.1 Ideal

ξ0=5

Real

Ideal

Figure 4.7: Normalized polymer induced potential of mean force between two colloidal particles,Ueff

cc (r)/U0 = − log(gcc(r))log(gcc(σc))

, versus reduced distance, (r − σc)/Rg for two differentpolymer solvents, good solvent (dashed lines) and theta solvent (solid lines). Thecurves are indicated in the limit of vanishing colloid concentration for the labelledpolymer concentrations and fixed polymer radius of gyration, Rg =

√2ξ0, which is

scaled by particle diameter,σc.

In the field theory approach [26], in the protein limit the potential of mean force between twoparticles immersed in a good polymer solvent is related to the square of the excess free energy forinsertion of a single particle into a bulk polymer fluid via the average polymer density correlationfunction which is termed here by Spp(r) in the solution without particles. The

βδF2 = −(NβF1

ϱp)2Spp(r).

where the single polymer excess free energy in given by

NβF1

ϱp= AgR

d−1/νR1/νg . (4.6)

The parameter Ag is introduced in Sec. 4.1. Replacing Spp from Eq. (2.23), andNβF1

ϱp, the

mean potential between two nanoparticles in a polymer fluid follows

50


2 4r/σ

c

-1

-0.5

0

Ueff

cc/U

0

φp=3 PRISM

φp=3 FT

φp=0.1 PRISM

φp=0.1 FT

ξ0=22

Figure 4.8: The normalized polymer induced potential of the mean force, Ueffcc (r)/U0 =

− log(gcc(r))log(gcc(σc))

for two parameter values ϕp = 0.1, 3 obtained via field theory approach(dashed cures) [26] and PRISM equations (solid lines) are compared.

βδF2 =(AgR

d−1/νR1/νg )2

4πΓ( 1ν )l1/ν

p

e−r/ξ

rd−1/ν(4.7)

It is concluded that for sub-region r ≪ ξ the potential of mean force is proportional to−(R/r)d−1/ν , which is independent of ξ in semidilute (ξ0 in dilute) polymer fluid and sug-gests that the range of the interaction between two point particles is of order of their radius R.In Fig. 4.8 Our numerical results are compared with the field theory one

On the other hand, in the limit of ultra-dilute polymer solution ϕp/ϕ∗p → 0, where hypernetted

chain (HNC) and Percus-Yevick closures in colloid-colloid direct correlations lead to the sameresults. The HNC based expression for the effective potential is given by U eff

cc = −hcc(r). Thiscan also be resulted by linearization (ln(1 + x) = x + O(x2) . . . for small x) of the potential ofmean force in case that the value of total correlation function, hcc ≪ 1.

Fig. 4.8 present the normalized potential of mean force determined via two different approaches,FT and m-PY PRISM for two parameter values ϕp = 0.1, 3 and polymer characteristic lengthξ0 = 22. the solid lines are belong to our numerical results from solving Eq. (2.29) and the dashedlines are the results of FT approach for the effective potential, Eq. (4.7). The qualification

51

4 Results

agreement between the FT results and m-PY PRISM ones at dilute polymer regime, ϕp = .1,and semidilute polymer regime, ϕp = 3, are observed. In case that the value gcc(r) is comparablewith 1 and consequently linearization, ln(1 + hcc(r)) ∼ hcc(r), is valid the curves coincide well.

4.3 Bc2(r), Second Virial Coefficient

To qualify the strength of the induced attraction potential induced by the polymer solutionthe colloid-colloid second virial coefficient Bc

2, which is measurable experimentally, is of mostinterest. Representative numerical results for the colloid-colloid second virial coefficient obtainedfrom solving Eq. (2.31) are shown in Fig. 4.7 for two different polymer to colloid size ratios,ξ0 = 10 (blue line) and ξ0 = 22 (green line), in a dilute suspension of colloidal particles asa function of polymer concentration for real polymers. As can be seen the effective colloidpair potential increase in negative half axis by adding polymer chains into the solution up toapproximately overlap concentration ∼ 0.53 which approaches to minimum and then in higherpolymer concentration, ϕp > ϕ∗

p the effective attraction decreases again. It is clear that thevalue of this function at zero polymer concentration must be identical(∼ BHS

2 = 2π/3σc) for allpolymer length scale.

In the field theoretic approach with self avoiding walk statistics [26] the universal function B

defined as

B =2bcc

2 Rdg

(NβF1/ϱp)2(4.8)

where bcc2 is related to the second virial coefficient Eq. (2.31) via

bcc2 = Bc

2 − BHS2 (4.9)

where NβF1 follows Eq. (4.6). Equation (4.8) expresses bcc2 in terms of the particle radius,

R = σc/2, the amplitude Ag and properties of polymer solution without particles. The comparedresults from field theoretic approach and m-PY PRISM ones are presented in Fig. 4.9, shiftingis due to this fact that the field theoretic asymptotic exact results are only valid at the limitof ξ0 → ∞ while our choice for ξ0 = 22 is not sufficiently big. The difference between theblue curve belong to ξ0 = 10 and the green one for ξ0 = 22 reflects this expression. Definitions = R3

gϱp/N to measure the overlap between chains, B approaches −s for small value of s andis proportional to −s/s1/(d−1/ν) = −s0.35 in the limit of s → ∞ and displays a minimum at(4/3π)s ≈ 1, where the chains begin to overlap. The value of the minimum of B is 0.049 [26]and implies the following relation to yield the minimum value of bcc

2 , Eq. (4.9), as a function ofsize ratio,

(bcc2 )min/BHS

2 = −0.5(Rg/R)2/ν−d (4.10)

52


0 0.5 1 1.5 2 2.5 3φ

p

-0.5

0

0.5

1

Bc 2/B

2HS

ξ0=10

ξ0=22

Figure 4.9: The reduced second virial coefficient, Bc2/BHS

2 , for two different polymer correlationlengths, ξ = 10, 22 which are scaled by the colloid diameter (σc = 1) as a function ofthe polymer concentration.

where d = 3. Our numerical results in case ν = 3/5 for bcc2 follows this relation up to 20%

relative error.

To investigate the effect of excluded volume interaction between the polymer segments on theinduced effective interaction between the pair particles, the second virial coefficient belong toreal and ideal polymers are sketched in Fig. 4.11. The colloid-colloid second virial coefficientwhen no polymer chain exists in the solvent obviously does not depend on polymer statistics.Also it is expected that for infinite polymer density keeping the ratio of polymer size to polymerdensity fixed the second virial coefficient should approaches to a constant. Considering theideal chain structure factor in m-PY PRISM equations in the limits ϕp → ∞, ξ0 → ∞ whereξ = constant the second virial coefficient behaves like Bc

2/BHS2 → −5 [36], which indicates a

finite effect of pairwise attraction independent of the size ratio. As a result of m-PY PRISMfor Gaussian chain statistics the minimum value of bcc

2 is given by −[3(2 + λ1)2/16λ21](ξ0/σc)

[36], where from the numerical computation, we obtained λ1 = 3.19 and the relative error inminimum value of second virial coefficient is less than 5%. The deeper value for the minimumof Bc

2 for Gaussian chains than the same for considered real chains is another significant effectof excluded volume effects in pair particles interactions and we find that the excluded volumeinteraction between the polymer segments reduces the amplitude of pair particle interaction of

53

4 Results

-1.5 -1 -0.5 0 0.5

Log10

(φp/φ

p

*)

-0.04

-0.02

0

ΒFT PRISM ξ

0=12

PRISM ξ0=22

Figure 4.10: Nonmonotonic overlap-dependence of the second virial coefficient bcc2 . The black

curve shows the field theoretical result for the universal combination B from Eq.(4.8) as a function of the logarithm of the reduced polymer concentration. The blueand green dots present the same function obtained by solving m-PY PRISM equationsusing self avoiding walk statistics for two polymer correlation lengths, ξ0 = 10 andξ0 = 22 respectively.

the order 10 approximately.

It should be mentioned that, in previous works performed by Tuinier et al [67] neglect of thelong-range correlation (see Sec. 4.1.4) makes the particle-particle second virial coefficient forideal coils positive under dilute polymer conditions ϕp < ϕ∗

p, in disagreement with the fieldtheoretic [26] and PRISM [36, 37, 65] predictions as well as experimental observations [68–70]for the dilute polymer regime.

In the next chapter we compare the obtained numerical results for the second virial coefficientwith the experimental results based on the mixture of star-polymers which play the role ofcolloidal particles and linear polymers dissolved good solvent in the extreme nanoparticle regime,where the star-polymers are much smaller than the linear one.

54


0 0.5 1 1.5 2 2.5φ

p

-10

-5

0

Bc 2/B

2HS

ξ0=10 Real chain

ξ0=22 Real chain

ξ0=22 Ideal chain

ξ0=10 Ideal chain

Figure 4.11: The scaled second virial coefficient, Bc2/BHS

2 , between two colloids for two differentpolymer solvents, theta and good. The dashed lines highlights the strength of inducedpair potential between two colloidal particles by the non interactive chains (thetasolvent), since the solid one displays the same function induced by the excludedvolume interaction chains (good solvent). For each solvent two polymer correlationlengths, ξ0 = 10, 22 (blue and green), are plotted.

55

5 Experimental comparison

I was in Strasbourg (Institute Charles Sadron) from November 2008 to February 2009, in orderto become familiar with the Small Angle Neutron Scattering (SANS) method and compare theresults of my theoretical approach with the experimental ones obtained from solutions of themodel soft nanoparticles based on fullerene (C60) molecules and long polystyrene chains usingsmall angle light and neutron scattering techniques.

Experimentally, building upon existing work performed in Strasbourg a system consisting ofnanoparticles of C60 molecules coated with low molecular weight PS chains (6 PS chains aregrafted on each C60 molecule) as model soft nanoparticles which are mixed with high molecularweight PS chains is considered. The neutron scattering (carried out on D22 at ILL Greno-ble) experiments are performed on the suspensions of these mixtures in the PS good solventtetrahydrofuren (THF). This experiment aims at determining the second virial coefficient ofnanoparticles using small angle neutron scattering in a contrast situation that allows to matchthe long PS chains. This function is accessible experimentally via Zimm extrapolation method[71] or Zimm plot and then will be analyzed and compared with the same one that is obtainedfrom numerical method in real space Sec. 4.3 in the same thermodynamical conditions and withthe same physical parameters. Obvious, only neutron scattering associated with the contrastvariation method allows studying the structure of such mixed systems.

Theoretically the mixture of nanoparticles and non-adsorbing linear polymers is a system con-taining macromolecules with a coarse grained complex form factor and hard spheres and theeffect of solvent is included in potential of interaction. Experimentally our system is describedby the mixture of polystyrene chains and star polymers in good solvent (THF). Plystyrenes playthe role of polymer chains and nanoparticles considered as fullerene C60 molecules coated withlow molecular weight polystyrene (PS) chain which is called star polymers. It should be notedthat the colloidal particles in experimental system are soft and the particle softness give riseto discrepancies when comparing the second virial coefficient of the nanoparticles (A2

c), fromtheory and experiment as a function of polymer concentration (cp) giving from the dilute regimeto the semidilute one.

Therefor, it is of our great interest to investigate the effect of softness of colloidal particles on

56

5.1 An introduction to small angle neutron scattering

their interaction mediated by polymer solution via comparing with the numerical results of m-PY PRISM equations. It has been noted that in theoretical hard spheres are used as a modelfor colloidal particles. Such a comparison will be carried out using the change in the secondvirial coefficient related to the nanoparticles Ac

2 as the polymer concentration is varied.

In this chapter first we give a short introduction about the SANS and then the experimentalmodel will be discussed. At the end, the experimental results and our numerical ones for thesecond virial coefficient will be compared.


The aim of scattering is determination of the average conformation and the organization ofparticles within size range from 10 to 1000 A dispersed in a homogeneous medium. The smallangel neutron scattering based upon nuclei interaction and concentration fluctuations.

The experiment consist of sending a well collimated beam of radiation of wavelength λ throughthe sample and of measuring the variation of the intensity as a function of the scattering angleθ. The physical parameter is the scattering vector q,

q =4π

λsinθ/2 (5.1)

The small angle scattering, small θ, leads to the measurement of small wave vector, 0 < q < q∗,where q∗ ≈ 0.6A−1 for liquids and homogeneous solids. Small q in Fourier space is correspondto the large r in real space. For q < q∗, far from critical point, the density fluctuations are veryweak and therefore negligible relative to the concentration fluctuations which appear when asolute is dispersed in a matrix.

Considering a two component system with the species, matrix (1), and particles (2) with thedensity distribution ϱi(r) at point r where i ∈ 1, 2, the partial stricture function, Sij(q), definedas the autocorrelation of Fourier component of the bulk density distribution,

Sij(q) =1v

< ϱi(q)ϱ∗j (q) > (5.2)

where < ... > indicates an averaging of the possible configurations and v is the volume of thesystem. The scattering intensity I(q) is connected to partial structure functions via scatteringlengths bi

57


I(q) =∑i,j

bibjSij(q)

= b21S11(q) + b2

2S22(q) + 2b1b2S12(q)(5.3)

The weight bi(cm) is given by the radiation to the ith elementary scatterer. Its value dependson the nature of atoms only and differs from one isotope to another and is sensitive to thestate of the nuclei spin. It is assumed that the total density distribution of system given byϱ(r) = ϱ1(r) + ϱ2(r) is constant, the total density fluctuation is negligible, δϱ(r) = 0, then thesystem is incompressible. This leads to δϱ1(r) = −δϱ2(r) and according to definition of partialstructure factor Eq. (5.2) we can write,

S11 = S22 = −S12 (5.4)

It is often convenient to use a contrast length defined as: ki = b2 − b1v1v2 , where vi is the volume

of the elementary scatterer i. Replacing k2 by k and S22(q) by S(q) and using the contrastlength definition in Eq. (5.4), for this system the intensity

I(q) = k2S(q), (5.5)

is reduced to the product of the square of the contrast length and scattering function S(q) of theparticles which only depends on the structure of scatterer particles. For a solution containing Zidentical (mono-dispersed) polymers made of N monomers, the structure factor is given by

S(q) =1V

Z∑i,j=1

N∑α,β=1

< e−iq.(rαi −rβ

j ) >

=1V

[ZN∑

α,β=1

< e−iq.(rα1 −rβ

1 ) > +Z(Z − 1)N∑

α,β=1

< e−iq.(rα1 −rβ

2 ) >](5.6)

as a function of position rαi of the αth scatterer of the ith polymer molecule. The first term reflects

the single molecule species correlation, named S′(q) and the second one describes the correlationof two different particle species, called S′′(q). Implementing the definition of concentrationc = ZN

V and the scaled functions:

P (q) =S′(q)S′(0)

, Q(q) =S′′(q)S′′(0)

in Eq. (5.5) lead to the following relation for scattering intensity

I(q) = k2c[NP (q) + cV Q(q)] (5.7)

58


where the factor 1/N2 renders S′(0) and S′′(0) equal to unity and approximation Z(Z−1) = Z2

are used. P (q) is referred to as the intramolecular or single chain structure factor and Q(q) isthe intermolecular correlation function. It is assumed that in dilute solution both correlationfunctions are not c dependent.

In case 0 < c ≪ 1, the Eq. (5.7) can be inverted to give the following equation [71]

1g(q, c)

=k2c

I(q, c)=

1NP (q)

+ 2cA2 + .... (5.8)

In dilute solution and for the neutral polymers, A2 is independent of c. Taking the low wavevector limit, q → 0, where q2R2

g < 1 (the Guinier range) P (q) leads to

P (q) = 1 +q2R2

g

3. (5.9)

Substituting for P (q) in Eq. (5.8) gives

g(q, c) = N(1 −q2R2

g

3) − 2cA2 + .... (5.10)

Using this relation between the intensity as a function of q and c in a Zimm plot, one can obtainthe second virial coefficient A2 and the particle formfactor P (q). Extrapolating to zero wavevector q = 0 in Eq. (5.10) leads to the second virial coefficient. In the limit of the infinite diluteparticle concentration, c → 0 in Eq. (5.7), the particle form factor, P (q), is determined.

In the experimental system, the mixture of linear polymers and star polymers, the star polymersplay the role of colloidal particles and are much smaller than the polymer chains. In order tocalculate the second virial coefficient between nanoparticles Eq. (5.10) is employed. As we areinterested in the limit of a very small q value (qRg < 1), and also Rg of coated polymers (arms)are much smaller than the same quantity of linear polymers, the second term in Eq. (5.10) couldbe also neglected.

As a matter of fact, each colloid particle consists of two parts, a) the core (C60) and b) the shell(6 coated polymer chains). The second virial coefficient between nanoparticles is consequenceof core-core, core-shell and shell-shell interactions. Considering Eq. (5.5),

I(q) = k2aaSaa(q) + 2k2

abSab(q) + k2bbSbb(q). (5.11)

In this equation, each term contains two parts, the intramolecular and the intermolecular struc-ture factors (see Eq. (5.7)) and from the intermolecular correlation function one determines the

59


second virial coefficient. Because i) experiments only probe for the Guinier range of q, where q iscomparable with the radius of gyration of the stars, and ii) the hydrodynamic radius of the coreis much smaller than Rg of the branches, cores can be considered like non-interacting point-likeparticles and the shell-shell interaction only plays a role in the second virial coefficient. Eq.(5.10) reflects this concept.

In the next section we will compare the obtained numerical result for the second virial coefficientdiscussed in Sec. 4.3 and the experimental one. A2 is connected to our notation of second virialcoefficient in theoretical part of thesis, Bc

2, via a constant. To avoid the complexity we normalizedboth results with their value at ϕp = 0.

5.2 The experimental system

The experimental system is composed of fullerene (C60) with the 6 coated polystyrene (PS)chains and much longer linear polystyrene chains in good solvent (THF). The arms connectedto C60 are short comparing with the polymer chain but they are long enough to yield a softcolloid particle and cancel the adsorbtion between the polymer chains (PS) and nanoparticles.In order to detect the scattered intensity only related to the stars, solutions of hydrogenatedstar-shape nano-particles and deuterated tetrahydrofuran (THFD) are considered.

5.3 Results

As it is already mentioned, the theoretical results are based on mixtures of hard sphere and longnon-adsorbing real polymers in a good solvent and of course does not depend on the chemicalstructure of particles. Hard spheres considered as a rigid bodies with constant value for formfactor in q space, while in experiment in order to cancel the attraction between the long linearchain and Fullerene core and getting non-adsorbing chains, the PS arms have to be enough long.On the other hand longness of these attached arms determine the softness of the particles, thatis, the longer arms entail the softer particle. Therefore make balance between these parameteris not an easy work and required to repeat the experiment with the different arms length. Onlyfor a fixed size ratio the results are presented in Fig. 5.1.

The Fig. 5.1 display the comparison between the experimental and numerical results for thesecond virial coefficient Bc

2 between two nano-particles versus various amount of the polymerconcentration. In the experimental system, the radius of gyration of the PS chains is 450 A andthe hydrodynamic radius of the star is equal to 20 A, that means, the asymmetric size ratio

60

5.3 Results

0 0.2 0.4 0.6 0.8 1 1.2 1.4φ

p

-0.5

0

0.5

1

B2c /B

2c(φ p=

0)

SANSNumeric

Figure 5.1: Second virial coefficient as a function of polymer concentration, ϕp the black line isdetermined from neutron scattering experiments performed on stars having 6 PS armsof molecular weight M=2500 g/mol dispersed in deuterated Tetrahydrofuran solutionsof deuterated PS chains of high molecular weight M=1,2106 g/mol. The red line isthe result of m-PY PRISM solution for mixture of hard sphere and non-adsorbingreal polymer in good solvent. Both measurement have been done for the size ratioapproximately q = 22 and they scaled by the Bc

2 at ϕp = 0.

q = Rg/Rh = 22. In the numerical calculation the same size ratio is used and both data arescaled by the second virial coefficient value at ϕp = 0

The second virial coefficient as the strength of the interaction between the particles, its sing isthe signal to distinguish the if it is the attraction or repulsion interaction. The negative value ofsecond virial coefficient shows the attraction induced by the solvent (here the polymer matrix)and the positive sign reflects the repulsion between the particles in present of polymer chain.The contact point which is the second virial between the particles without any polymers andboth curves are normalized by that Bc

2(ϕp = 0), equals one.

The experiment shows a positive second virial coefficient and its variation versus tuning polymerconcentration from 0 to 1 is not significant at the beginning the repulsion between the particlesreduce and this reduction continues even for the overlap concentration. Since the numericalresults are totally different and it presents an induced attraction by increasing polymer concen-tration and for ϕp > 0.53 the attraction interaction decrease. This difference in treatment fortwo systems with the same size ration can be interpreted as the effect of softness of nanoparticles,while the field theoretic results also predict the attraction between the nanoparticles induced bythe too large real polymers in good solvent [26, 27].

61

6 Summary and Outlook

In this thesis we calculated the correlation functions of mixtures of nonadsorbing real polymersand colloidal particles in both nanoparticle and colloid limits employing an extensive numericalcalculations within the frame work of the extended-PRISM approximation. Keeping the colloidalparticle volume fraction low and varying the polymer concentration from infinitely dilute tosemidilute polymer fluid. From the obtained radial distribution functions we investigated theeffect of excluded volume interactions between the polymer chains with respect to the case thatthe dissolved polymer chains are considered ideal (no monomer interactions). Furthermore theeffect of an asymmetric size ratio was studied and our results compared with exact asymptoticfield theoretic results in extreme limits.

The density profile of the polymer close to an isolated colloidal particle. It was determinedthat the cross correlation function, gcp(r), grows in a power-law regime Eq. (5.1) closeto a colloidal particles. Also we saw that the existence of the excluded volume interac-tion between real polymer chain segments prevents the accumulation of monomers whencompared with an ideal polymer with the same length close to a particle surface Fig. 4.2.

Long-tail range. In the protein limit, for distances far from the colloid centre the polymersegment-colloid pair distribution exhibits an additional power-law tail due to the chainconnectivity correlations. In the AO model for R 6 Rg this effect is ignored becausethe intramolecular degrees of freedom of the polymer are neglected [19]. It has beenhighlighted that the interaction between nanoparticles dissolved in real polymer fluids isof shorter range than the same in Gaussian chain solvents.

Depletion layer width. We showed the depletion layer is both weaker and thinner for chainswith excluded volume interaction than in the case of ideal chains with the same radius ofgyration, Rg.

Induced potential of mean force between two hard spheres. From Fig. 4.7, it is concludedthat the depletion potential induced by the polymers is a function of polymer concentrationand the type of polymer solvency. That is, a) the higher polymer concentration exertstronger force on two colloids b) with the same length, polymers in good solvent underathermal condition impose less pressure than the ones in theta solvent. It can be referred

62

from this fact that the accumulation of ideal polymers close to a hard sphere is more thanreal polymer near colloid surfaces.

Excess chemical potential for inserting a polymer chain in a hard sphere fluid. For much largerpolymer the excess chemical potential as a function of the polymer characteristic length,ξ shows a power-law behaviour with Flory exponent, 1/ν. It is determined that the inser-tion energy to add a real polymer into a low density colloid fluid is less than the same forinserting an ideal chain with the same length. That is, the real chains are perturbed lessthan the ideal one by the colloidal particles.

Second virial coefficient. As a signal of the strength of the polymer induced potential betweentwo colloidal particles the second virial coefficient is determined. It is important thatsolving PRISM equations predicts an attraction interaction due to adding real/ideal chainsinto colloidal suspension in agreement with the field theoretic predictions. In previousapproaches neglecting long-tail range in pair colloid distribution function the inducedpotential was repulsive. The minimum value of Bc

2(r) which appeared around the polymeroverlap concentration is approximately 10 times deeper than the minimum value of theself avoiding walk chain due to taking into account the random walk chain statistics.The second virial coefficient is of interest because it can be measured experimentally.The universal function giving the reduced second virial coefficient, B, Eq. (4.8) was alsodetermined and compared with the field theoretic results which are valid for extremelylarge polymer chains and qualitative agreement is observed (Fig. 4.10). In chapter 5our numerical results are compared with the experimental ones obtained for dissolvingstar polymers (6 PS chains are grafted on each C60 molecule) and PS polymers in goodsolvent. Seemingly because of softness of colloidal particles (star polymers) no quantitativeagreements was found Fig. 5.1.

All the above mentioned results compared well with the existing field theoretic results. Detailsof calculations and relative errors are explained in chapter 4. We also repeated the calculationusing our numerical method based upon m-PY PRISM for the ideal chain which has been alreadydone theoretically by Fuchs et al. [31, 36, 37]. Our result for the effective interaction lengthscale, λ, satisfies the thermodynamic consistency better than the theoretical ones obtained viainterpolation between two extreme limits by Fuchs (see Fig. 3.3). However in extreme limits,R ≫ Rg and R ≪ Rg a relative error in λ up to 5% comparing with the exact asymptotictheoretical ones is detected but the resulting functions computed using unique λ compared wellwith the field theoretical ones. It is worth pointing out that our numerical calculations inextremely small/big value of ξ (for ξ < 0.03 and ξ > 30) present numerical errors which have tobe improved by using better numerical programs.

63

6 Summary and Outlook

This system considered and the method developed in this thesis provide results in good agree-ment with the existing ones. The applicability of the PRISM approach to all ranges of densityand size motivates us to use it to treat more complicated mixtures, e.g. considering the samesystem at higher colloidal packing fraction or by replacing the hard sphere with the soft particlesand consequently altering the potential of interactions. In the next step it would be worth toinvestigate the mixtures of nonadsorbing real polymers and colloidal particles at higher colloidalpacking fraction which is both important fundamentally and has applications in industry. First,because λ which is obtained by enforcement of thermodynamic consistency can also be used athigher colloidal densities [36] and makes the description of phases and structures of this sys-tem easier, second at this limit because of the presence of many body interaction due to notinteracting colloidal particles it is not yet understood theoretically.

In industrial applications there exist a large variety of possible mixtures by choosing the poly-mers, colloidal particles, solvents and the different particle concentrations. In polymer science,by addition of solid particles to the polymer melt, gel and glass properties can be manipulated[72]. In colloid or nanoparticle science, by adding low concentration polymer chains into thecolloidal suspension, one can control the colloidal interactions and consequently manipulates thesystem properties [37, 42, 73]. The additive polymer chain influences the particle interactionand modifies the hard-core repulsive interaction to the attraction one. To investigate the com-positions of colloidal particles and polymer chains with comparable packing fractions is also oneof the most interesting topics which is not well understood yet.

64

7 Zusammenfassung und Ausblick

In dieser Arbeit wurden Korrelationsfunktionen von Mischungen nichtadsorbierender realer Poly-mere und kolloidaler Partikel sowohl, im Limes von Nanopartikeln als auch im kolloidalen Limesmit Hilfe Numerischer Methoden im Rahmen der extended-PRISM Naherung, berechnet. Dervon den Kolloiden eingenommene Volumenbruch wird dabei als klein angenommen und diePolymerkonzentration von infinitesimaler Verdunnung bis zur halb-verdunnten Polymerlosungvariiert. Mit der durch diese Methode erhaltenen radialen Verteilungsfunktion wurden dieAuswirkungen von ’exluded volume interaction’ zwischen den Polymerketten, unter der Vo-raussetzung, dass sich diese als ideal verhalten (keine Monomerinteraktion), untersucht. DesWeiteren wurden die Auswirkungen asymmetrischer Großenverteilungen untersucht, und dieResultate mit den exakten feldtheoretischen Ergebnissen im Limes extremer Verhaltnisse ver-glichen.

Das Dichteprofil eines Polymers, nahe einem isoliertem kolloidalen Partikel. Die Kreuzkorre-lationsfunktion gcp(r) verhalt sich in der Nahe eines Kolloids gemaß eines PotenzgesetzesEq. (5.1). Außerdem verhindert die ’excluded volume interaction’ zwischen den realenPolymerkettenabschnitten die Ansammlung von Monomeren im Vergleich zu einem ide-alen Polymer mit der selben Lange nahe der Oberflache eines Partikels.

Langreichweitiger Bereich. Im Limes von Proteinen, fur große Abstande vom Kolloid zeigt dasPolymersegmentkolloid ein weiteres Potenzgesetz welches seinen Ursprung in den ’chainconnectivity’ Korrelationen hat. Im AO-Modell fur R 6 Rg ist dieser Effekt nicht enthal-ten, da intramolekulare Freiheitsgrade der Polymere vernachlassigt werden [19]. Es wurdehervorgehoben, dass die Wechselwirkung zwischen den Nanopartikeln, welche in einemrealen Polymer gelost sind, kurzreichweitiger ist als in Losungen von Gaußschen Ketten.

Große der Verarmungszone. Es wurde gezeigt, dass die Verarmungszone fur Polymerketten mit’excluded volume interaction’ dunner und schwacher ist, als im Fall idealer Ketten mit demselben Gyrationsradius Rg

Induziertes Potenzial der mittleren Kraft zwischen zwei harten Kugeln. Aus Fig. 4.7 kannman schließen, dass das Verarmungspotential welches durch die Polymere verursacht wird,

65

7 Zusammenfassung und Ausblick

eine Funktion der Polymerkonzentration und der Loslichkeit des Polymers ist. Genauerwurde gezeigt, dass a) die hohere Polymerkonzentration eine großere Kraft auf die zweiKolloide ausubt und dass b) Polymere der selben Lange in guter Losung unter athermalenBedingungen weniger Druck ausuben als in einer Theta-Losung.

Chemisches Exzesspotenzial um eine Polymerkette in eine Hartkugelflussigkeit einzubringen.

Fur große Polymere verhalt sich das chemische Exzesspotential als Funktion der charak-teristischen Polymerlange ξ gemaß eines Potenzgesetzes mit dem Floryexponenten 1/ν. Eswird gezeigt, dass die Energie um ein reales Polymer in eine Kolloidlosung niedriger Dichteeinzubringen kleiner ist als die Energie, die fur das Einbringen einer idealen Polymerkettemit der selben Lange notwendig ist. Folglich werden die realen Polymere weniger von denKolloiden gestort als ideale Polymere.

Zweiter Virialkoeffizient. Der zweite Virialkoeffizient, welcher die Starke des gegenseitig in-duzierten Potentials zweier Polymere beinhaltet, wurde bestimmt. Wichtig ist hierbei,dass die Losungen der PRISM-Gleichungen bei Zugabe realer/idealer Polymerketten in dieKolloidlosung eine attraktive Wechselwirkung vorhersagt, was mit den feldtheoretischenVorhersagen ubereinstimmt. In fruheren Ansatzen, die ’long-tail’ Bereiche der Paarverteilungs-funktion der Kolloide vernachlassigen, ist das induzierte Potential abstoßend. Der mini-male Wert von Bc

2(r), der bei der Konzentration auftritt bei der die Polymere uberlappen,ist ungefahr 10 mal kleiner als der Minimalwert der ’self avoiding walk chain’. Da derzweite Virialkoeffizient auch experimentell bestimmt werden kann ist er von besonderemInteresse. Die universale Funktion, welche den reduzierten zweiten Virialkoeffizient B, Eq.(4.8) beschreibt, wurde berechnet und verglichen mit den feldtheoretischen Ergebnissen,die im Limes extrem großer Polymerketten gultig sind. Hierbei wurde eine qualitativeUbereinstimmung erreicht Fig. 4.10. Im Kapitel 5 werden die numerischen Ergebnisse mitexperimentellen Daten von gelosten Sternpolymeren (6 PS-Ketten befestigt auf einem C60-Molekul ) und PS Polymeren in guter Losung verglichen. Die zu weichen Kolloidpartikelscheinen die Ursache fur die nicht-quantitative Ubereinstimmung zu sein Fig. 5.1.

Alle oben genannten Resultate passen gut zu denen der feldtheoretischen Methoden. Die Einzel-heiten der Rechnungen und die relativen Fehler sind in Kapitel 4 erklart. Zusatzlich hierzuwurden die Rechnungen mit dem m-PY PRISM Modell fur ideale Polymerketten mit den selbennumerischen Methoden wiederholt. Dieses Modell wurde bereits von Fuchs et. al. [31, 36, 37]theoretisch gelost. Die Ergebnisse fur die effektive Wechselwirkungslange λ, zeigt eine besserethermodynamische Konsistenz als die Ergebnisse, die aus der Interpolation zweier extremer Lim-iten gewonnen wurden (s. Fuchs Fig. 3.3). In den extremen Grenzfallen R ≫ Rg und R ≪ Rg

jedoch ergibt sich ein relativer Fehler von 5% in λ verglichen mit theoretischen Vorhersagen ausder Asymptotik. Es sollte darauf hingewiesen werden, dass die numerischen Rechnungen fur ex-

66

trem kleine/große Werte von ξ (fur ξ < 0.03 und ξ > 30) numerische Fehler aufweisen, die durchbessere numerische Programme verkleinert werden mussen. Das betrachtete System zusammenmit der in dieser Arbeit entwickelten Methode liefern Ergebnisse die in guter Ubereinstimmungmit bestehenden Methoden sind. Die Anwendbarkeit der PRISM-Methode auf alle Bereiche derDichte und Großenverhaltnisse motiviert die Behandlung komplizierterer Mischungen, wie z.Bdas System bei hohen Kolloidvolumenbruchen, oder das Ersetzen der Harten Kugeln durch we-iche Partikel oder das verandern des Wechselwirkungspotentials. Im nachsten Schritt sollte eineMischung aus nicht-adsorbierenden realen Polymeren mit Kolloiden bei hohen Volumenbruchenuntersucht werden, da dies zum einen von grundlegender Wichtigkeit und zum anderen wichtigfur industrielle Anwendungen ist. Erstens kann λ, das durch Erzwingen der thermodynamischenKonsistenz gewonnen wird, bei hoheren Kolloiddichten benutzt werden [36] um die Phasen undStrukturen des Systems zu beschreiben. Fur industrielle Anwendungen gibt es eine große unter-schiedliche Zahl von Mischugen aus Polymeren, Kolloiden, Losungsmitteln und Partikelkonzen-trationen. In der Polymerforschung werden die Eigenschaften von Gelen und Glasern durchHinzufugen kleiner Konzentrationen von Polymerketten in die Kolloidlosung manipuliert [72].Bei Kolloiden oder Nanopartikeln lassen sich die Kolloidwechselwirkungen durch Hinzufugenkleiner Polymerkettenkonzentrationen in die Kolloidlosung kontrollieren, und folglich die Sys-temeigenschaften manipulieren [37, 42, 73]. Die zusatzlichen Polymerketten beeinflussen dieWechselwirkungen zwischen den Partikeln und verandert die Hard-core Abstoßung zu einer At-traktion. Die Untersuchung eines Systems mit Polymerketten und Kolloiden bei vergleichbarenVolumenbruchen ist ebenfalls ein sehr interessantes und unverstandenes Thema.

67

A Radial Fourier Transformation

The three dimentiobal Fourier transform, f(q), of any isotropic fucntion f(r), where r = |r| isdefined as:

f(q) =∫ ∞

−∞eiq.rf(r)dr =

4π

q

∫ ∞

0sin(qr)f(r)dr. (A.1)

Considering definition

f(r) =∫ ∞

|r|sf(s)ds, (A.2)

leads to the following relation for one dimentional Fourier transform of f(r):

f(q) = 4π

∫ ∞

0cos(qr)f(r)dr = 2π

∫ ∞

−∞eiqrf(r)dr (A.3)

and f(r) can be written as:

f(r) =1

4π2

∫ ∞

−∞e−iqrf(q)dq. (A.4)

The convolution of f and g is written f ∗ g and convolution of two functions f and g over isgiven by

f ∗ g =∫ ∞

−∞f(s)g(t − s)ds =

∫ ∞

−∞g(s)f(t − s)ds (A.5)

and its Fourier transformation follows:

F [f ∗ g] = F [f ]F [g]. (A.6)

68

B Thread-Like Chain

Considering the following relation for the single chain structure factor described by Gauaasianstatistics,

ω(q) =1

1/N + (qlp)2(B.1)

Solving PRISM equations for flexible polymer chain solution required to replace the site-sitedirect correlation fucntion between two monomres by a density-dependent delta-function. Usingabove simplification in equations lead to the following site-site radial distribution of the Yukawa,or screened Coulomb, form [34].

g(r) = 1 +2ϕp

ξ0

exp(−r/ξ) − exp(−r/ξ0)r

(B.2)

Two different screening lengths ξ0 and ξ are presented. The macromolecular length scale, ξ0,which is simply related to the radius of gyration, ξ0 = Rg/

√2. The other one, ξ, is short range

at high density and is determined from the core condition, Eq. (2.6) for two different polymersegments (i = j = p). In the thread limit the hard core monomers considered as points withvanished diameters and the core condition is replaced by gij(r = 0) = 0 which leads to

ξ =ξ0

1 + 2ϕp. (B.3)

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excluded volume effects in the depletion attraction ......1.2 polymer solvents characteristics . . ....

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