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Computer simulations and scaling concepts inpolymer physics

Dr. Ralf EveraersMax Planck-Institut fur Polymerforschung, Mainz

Habilitationsschrift

September 5, 2002

Contents

I Summary of scientific work 1

1 Introduction 3

2 Statistical Physics of Charged Polymers 9

2.1 The electrostatic persistence length of intrinsically flexible poly-mers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Polyampholytes . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 From single chains . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 . . . to solutions of finite concentration . . . . . . . . . . 17

3 Dynamic Fluctuations of Semi-flexible Filaments 19

4 Statistical physics of polymer networks 25

4.1 Entanglement effects in rubber-elastic systems . . . . . . . . . 26

4.1.1 Summary of results from my PhD thesis . . . . . . . . 28

4.1.2 The constrained mode model (CMM) . . . . . . . . . . 29

4.1.3 Quantitative tests in simulations of defect-free modelpolymer networks . . . . . . . . . . . . . . . . . . . . . 33

4.1.4 Tube Models for Rubber-Elastic Systems . . . . . . . . 33

4.2 Chain conformations in rubber at equilibrium swelling . . . . . 34

5 Acknowledgements 39

II Publications 49

A The electrostatic persistence length of polymers beyond theOSF limit 51

i

ii CONTENTS

B Conformations of Random Polyampholytes 65

C Complexation and precipitation in polyampholyte solutions 71

D Polyampholytes: From single chains to solutions 79

E Dynamic Fluctuations of Semiflexible Filaments 101

F Constrained Fluctuation Theories of Rubber Elasticity: Gen-eral Results and an Exactly Solvable Model 107

G Tube Models for Rubber-Elastic Systems 119

H Self-Similar Chain Conformations in Polymer Gels 133

List of publications

1. “The Electrostatic Persistence Length of Polymers beyond the OSFLimit”, Ralf Everaers, Andrey Milchev and Vesselin Yamakov, Eur.Phys. J. E 8, 3 (2002).

2. “Tube Models for Rubber-Elastic Systems”, Boris Mergell and RalfEveraers, Macromolecules 16, 5675 (2001)

3. “Conformations of Random Polyampholytes”, Vesselin Yamakov, An-drey Milchev, Hans Jorg Limbach, Burkhard Dunweg and Ralf Ever-aers, Phys. Rev. Lett. 85, 4305 (2000).

4. “Self-similar chain conformations in polymer gels”, Mathias Putz, KurtKremer and Ralf Everaers, Phys. Rev. Lett. 84, 298 (2000).

5. “Dynamic Fluctuations of Semiflexible Filaments”, R. Everaers, F.Julicher, A. Ajdari, and A. C. Maggs, Phys. Rev. Lett. 82, 3717(1999).

6. “Constrained Fluctuation Theories of Rubber Elasticity: General Re-sults and an Exactly Solvable Model”, R. Everaers, Eur. Phys. J. B4, 341-350 (1998).

7. “Polyampholytes: From Single Molecules to Solutions”, R. Everaers,A. Johner, and J.-F. Joanny, Macromolecules 30, 8478 (1997).

8. “Complexation and Precipitation in Polyampholyte Solutions”, R. Ev-eraers, A. Johner, and J.-F. Joanny, Europhys. Lett. 37, 275 (1997).

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Part I

Summary of scientific work

1

Chapter 1

Introduction

Polymers are chain molecules consisting of a large number of covalently linkedatomic groups. Macromolecules of this type are of great importance not onlyfor our everyday life but also for life itself. From a technological point ofview, natural and synthetic polymeric materials offer a wide assortment ofuseful properties (mechanical, thermal, electric, optical, etc.) which explainstheir ubiquity as fibers, films, processing aids, rheology modifiers, and asstructural or packaging materials with applications ranging from plastic bagsto Formula 1 chassis and tires. From a biological point of view, the eminenceof polymers is evident from the fact that proteins (poly amino acids) andDNA (poly nucleic acids) also belong to this category.

The physical properties of polymers can be grouped into two classes:electronic and conformational. The former are determined by the state ofthe electron shells. Examples are electric conductivity and optical properties.This thesis is concerned with the latter type of properties which are relatedto, i.e. the spatial arrangement and motion of atoms and atomic groupsin a macromolecule which determine the basic behavior of both biologicaland synthetic polymers. The conformational phenomena characteristic ofpolymers occur on length and times scales substantially larger than atomicscales and are related to their (compared to ordinary molecules) enormouslength. Polymeric materials can be distinguished according to

• Properties of the constituting monomers (the presence of one or severaltypes of monomers, monomer interactions with the solvent and eachother, the presence of ionizable groups, the local chain flexibility)

• Connectivity (linear, ring, and branched chains, networks)

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4 CHAPTER 1. INTRODUCTION

• Concentration regimes

– Dilute solutions (dominated by single chain properties)

– Semi-dilute solutions (characterized by a strong overlap betweencoils at very low monomer concentrations, a state which only existsin polymeric systems). Depending on chain stiffness and concen-tration semi-dilute solutions can be isotropic or nematic.

– Concentrated polymers (with low or vanishing solvent content)exist in a semi-crystalline, glassy or melt state.

Within each of these classes typically polymeric properties exhibit a certainuniversality (i.e. molecular details are hidden behind simple concepts suchas virial coefficients, persistence lengths, ..) 1 and are most appropriatelystudied theoretically within the framework of classical statistical physics. Ingeneral [GK94], polymeric systems are characterized by

1. long-range correlations that manifest themselves in strong fluctuationeffects and long relaxation times, anomalously small enthalpies, andaccordingly, anomalously high susceptibilities to external forces;

2. a long-term memory for the preparation conditions where the system’sconnectivity (and, in the case of systems containing closed loops, alsothe topology) become quenched.

Unfortunately, the dominance of entropic effects is not only responsible foruniversal behavior but also for the sensitivity of soft matter to impuritiesand artifacts introduced during the preparation.

Experimental methods used to study polymers include light, x-ray, andneutron scattering, nuclear magnetic resonance, local force techniques such assurface-force measurements or atomic force microscopy, video and magneticresonance imaging and rheology. Theoretically, polymers are investigated us-ing self-consistent mean field theories [Edw65, Hel75, Lei80, Sch98], integral

1A partial exception are biopolymers such as DNA or proteins. They differ from syn-thetic heteropolymers by their precisely determined primary structure or sequence, theirability to locally form helices or fold into specific motives (secondary structure) and, inthe case of proteins, adopt a specific overall or tertiary structure. This particular shape isresponsible for their specific (e.g. enzymatic) properties and largely determined by theirprimary structure. However, on a higher level, fiber forming proteins such as actin ortubulin exhibit again polymeric behaviour.

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equations from liquid state theory [SC94], as well as renormalization grouptechniques. As often in physics, progress is mainly due to the discovery ofanalogies to other topics and the simultaneous application of complementarymethods to a given problem. In soft matter physics, the classical examplesare the description of polymers using path integrals [Edw65, Wie86, Vil96] isde Gennes’ discovery [dG72] of the relation between critical phenomena andself-avoding polymers and the subsequent introduction of critical exponents,scaling functions and renormalization group techniques into the field. Com-bined with (neutron) scattering these methods have lead to a quantitativeunderstanding of neutral polymer solutions [dCJ87].

For various reasons the application of these methods to the topics treatedin the present collection has met with limited success. Often theories aremathematically extremely complex, even though they are based on relativelysimple and intuitive ideas and the ubiquitous assumption of Gaussian chainstatistics. Even then the prediction of experimental observables may requirethe use of uncontrolled approximations as well as the introduction of ad-justable parameters which are difficult to measure independently. In caseswhere deviations between theory and experiments appear, it is often difficultto decide if they are due to problems with the preparation of the experimen-tal system, weaknesses of the employed model or errors in the theoreticalanalysis of the model.

Over the last thirty years, two new methods have revolutionized soft mat-ter physics, which approach these problems from very different perspectives.Scaling theories [dG79] sketch the essence of an existing (or yet to be de-veloped) theory by concentrating on the identification and interdependenceof relevant time and length scales. Stripping away mathematical complica-tions often leads to an intuitive qualitative understanding of complex sys-tems [dG79]. On the other hand, computer simulations [Bin95] are on theway to become a third, independent branch of research complementary toanalytical theories and experiments. Bearing in mind the limitations in theaccessible time and length scales, they offer a couple of advantages: Greaterfreedom in and control over the preparation of the systems and the micro-scopic interactions, direct access to the microscopic structure and dynamics,and (within the statistical error) exact results for well-defined model systems.Moreover simulations sometimes allow the realization of “Gedankenexperi-ments” such as the comparison of otherwise identical systems where polymerchains are either allowed or forbidden to freely intersect in order to quan-tify the importance of entanglements. In particular, computer simulations


can provide an important bridging function between theory and experiment:On the one hand, the comparison to expriments allows a verification of theemployed models and the identification of the relevant details. On the otherhand, simulations can assist the systematic development of theories describ-ing models.

In spite of the mentioned advantages, one has to remain aware of the limi-tations of the numerical approach. From a practical point of view, there is thedesire to study ever larger systems for longer times. Even with the presentlyavailable computer power there may be severe problems in the equilibrationor the measurement of non self-averaging quantities. This holds in particularfor disordered systems. From a fundamental point of view, simulations canonly be used to refute theories and to gain insight into the mechanism con-trolling a system. The “solution” of a problem requires a theory. However,in the absence of an exact solution, a “statistically” exact solution generatedon a computer may be preferable to theories making uncontrolled physicaland/or mathematical approximations.

Over the last years I have tried to contribute to the understanding of anumber of fundamental problems in polymer physics using a combination of(scaling) theories and computer simulations. In my experience, the combinedapproach leads to important synergetic effects: Theoretical insight helps todesign simulations and to rationalize the abundantly available microscopic in-formation. On the other hand, simulations provide the necessary independentevidence for the justification of phenomenological models or the relevance ofcertain length or time scales.2 They can supply benchmark results for the

2Scaling theories are most successful for problems which can be discussed in terms of asingle length and/or time scale. Typical examples are the polymer radius of gyration or thecorrelation length in semi-dilute solutions and the corresponding Zimm or Rouse relaxationtimes. In some cases, proper theories can do little more than provide analytical expressionsfor crossover functions. To some extent the situation resembles classical hydrodynamics,where Reynolds, Mach, Froude, etc. numbers are not only used to classify problems butto quantitatively relate experimental model data to real-world prototype data.

However, in hydrodynamics the underlying Navier-Stokes equation is known, so that therelevant dimensionless parameters can be identified by dimensional analysis. In polymerphysics, the relevant length scales may not be apprarent from the bare equations describ-ing the systems. Rather they appear in the course of a suitable renormalization process,which in most cases is difficult to implement analytically. Thus the mere identificationof the length scales governing the behavior of a systems is often already a major success.This holds in particular for solutions of charged polymers which are discussed in section 2.Scaling arguments provide the most economic tool for this purpose, but require confirma-tion by independent experimental, theoretical or numerical methods. For example, field

7

systematic development of theories, in particular via independent checks ofthe various approximations necessary for the calculation of experimentallyaccessible observables. Moreover, numerical results for experimentally inac-cessible limiting cases (such as defect-free network architectures or the singlechain limit of infinite dilution) may help to provide a sound basis for theoriesdescribing experimentally relevant situations.

With respect to scaling one may, of course, wonder, if there are anyproblems left for “normal” polymer systems, which were not already treated20 years ago in de Gennes’ book [dG79]. The articles in the present collectionexplore theoretical concepts such as

1. an electrostatic persistence length for intrinsically flexible polymers,

2. a Rayleigh instability for polyampholytes chains carrying a net chargeand,

3. complexation and precipitation in polyampholyte solutions of finiteconcentration,

4. a dynamic length scale characterizing tension propagation in semi-flexible polymers,

5. the local chain structure in polymer networks at equilibrium swellingin good solvent, and

6. a deformation dependent tube diameter describing constrained fluctu-ations in polymer networks.

In my opinion, both the swelling exponent ν = 7/10 in gels (section 4.2,Ref. H) and the exponent 1/8 which characterizes the tension propagation insemi-flexible polymers (section 3, Ref. E) are excellent examples for qualita-tively new features in classical problems of polymer physics, whose discoveryis due to the combination of scaling arguments with computer simulations.This combination is even more promising for studies of polyelectrolytes wheresimple scaling fails due to the presence of a large number of (potentially) rel-evant length scales. The persistence length problem (section 2.1, Ref. A)

theoretic calculations reveal how the deceptive success of the classical Flory argument forthe gyration radius of self-avoiding walks arises from the fortuitous cancellation of un-controlled errors. Nevertheless scaling arguments provide extremely useful guidelines forexperimental, numerical or theoretical investigations.


is a nice example for a problem which had previously resisted a number oftheoretical, experimental and numerical solution attempts. In the case ofpolyampholytes (section 2.2) it is hard to see how the single chain problemcould have been settled without our (Ref. B) and, in particular, the earlysimulations by Kantor and Kardar. In turn, our theoretical considerationsfor the many-chain systems (Refs. C and D) are based on these single chainresults. The emerging, rather complex picture provides a framework for theinterpretation of experiments, which would have been very difficult to extractfrom a purely numerical approach. Finally, our recent results for polymernetworks (section 4.1) illustrate the “learning process” where the scrupulousanalysis of the behavior of a model system (Ref. [Eve99]) leads to simplifica-tions and crucial modifications of a theory which quantitatively addresses theexperimentally relevant situation of randomly cross-linked networks (Ref. G).

Chapter 2

Statistical Physics of ChargedPolymers

In polar solvents such as water polymers with ionizable side-groups can dis-sociate into charged macroions and small counterions. Macromolecules ofthis type are commonly referred to as polyelectrolytes [BJ96], a class whichcomprises proteins and nucleic acids as well as synthetic polymers such assulfonated polystyrene and polyacrylic acid. Most polyelectrolytes are water-soluble due to the gain of translational entropy of the dissociated counteri-ons, an effect which probably contributes significantly to their prominencein biological systems. At the same time, the behaviour of polyelectrolytes issignificantly less well understood than that of neutral polymers.

Barrat and Joanny [BJ96] argue that the difficulties in understandingpolyelectrolyte solutions can be traced back (a) to “the difficulty in applyingrenormalization group theories and scaling ideas to systems with long-range(Coulombic) forces” and (b) to the ensuing necessity to introduce such a largenumber of adjustable parameters in the interpretation of experiments as torender them inconclusive. Here I wish to demonstrate that the combinationof computer simulations and scaling arguments can help to find a way out ofthis impasse.

It is instructive to follow the arguments of Barrat and Joanny [BJ96] inmore detail. Our good understanding of solutions of neutral polymers is dueto the fact that the range of interactions between monomers is much smallerthan the scale determining the physical properties of the solution, i.e. thesize of the polymer coil or the correlation length. As a consequence, molecu-lar details only affect adjustable prefactors in theories predicting the depen-

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10 CHAPTER 2. STATISTICAL PHYSICS OF CHARGED POLYMERS

dence of solution properties on molecular weight. In contrast, polyelectrolytesolutions are controlled by a intricate interplay of short- and long-range in-teractions. The screening of the electrostatic interactions (i.e. the tendencyof oppositely charged objects to spatially arrange in such a way as to renderthe effective interactions between any two charges short-ranged) introducesyet another intermediate length scale in the problem, which may be compa-rable to the chain size or to the correlation length. However, the distributionof counterions around macroions limits not only the range of their effectiveinteractions but can also renormalize the macroion charges as a result of“counterion condensation”. Details of this process are controlled by the localchain architecture, which therefore indirectly influences the long-range partof the interactions. Last but not least, the long-range interactions have anontrivial influence on the local structure (stiffness) of the polymer chains.

The most general case of a polyelectrolyte solution is therefore of intim-idating complexity. As two examples in this thesis show, even single chainproblems are so subtle that they are the subject of considerable controversiesin the current literature.

2.1 The electrostatic persistence length of in-

trinsically flexible polymers

The first example concerns polyelectrolytes with Debye-Huckel interactions.Compared to the complex situation outlined above, we address the well-defined limiting case of a dilute solution of weakly charged polyelectrolytesin the presence of mono-valent salt. In this case, no counter ion (Manning)condensation occurs and the salt degrees of freedom can be integrated outusing the Debye-Huckel theory of simple electrolytes. As a result, chargedgroups on the polymer chain interact via a Yukawa-like, screened Coulombinteraction

UDH/kBT = (q2lB/r) exp(−κr) (2.1)

where the screening length κ−1 ∼ (lBc)−1/2 is related to the salt concentra-tion. The Bjerrum length lB is defined as the distance between two unitcharges where their Coulomb energy is equal to kBT and provides a con-venient measure of the strength of the bare electrostatic interactions. Thepolymers are characterized by their persistence length l0 and the line charge

THE ELECTROSTATIC PERSISTENCE LENGTH 11

density fq/A where q is the valence of the ionizable groups, A the contourlength distance between them, and f the fraction of ionized groups. We ne-glect all non-electrostatic interactions along the chain, i.e. assume a so-calledΘ-solvent.

On first sight, the problem seems trivial. We have a single type of inter-action which, due to screening, is effectively short ranged. One is thereforeinclined to treat the situation in strict analogy to the classical excluded vol-ume problem for polymers in a good solvent. The result is a characteristicswelling of the polymers relative to the neutral case with a gyration radiuswhich scales as Rg ∼ N ν with ν ≈ 3/5. However, in contrast to the classicalexcluded volume problem, the range of the interactions can be considerablelarger than the monomer size. In a 1mM salt solution κ−1 ≈ 10nm and forpure water (the small ion concentration can only be reduced to the dissocia-tion equilibrium of water) the screening length can grow to 1 µm. Thus thereare situations where polymers of considerable length experience no differencebetween the long range Coulomb interaction and the nominally short rangeDebye-Huckel interaction. The absence of screening leads to qualitativelydifferent behaviour: the effect of unscreened Coulomb interaction is akin toa stretching force acting on the chain ends, leading to a linear scaling of thegryration radius with chain length: Rg ∼ N .

Nevertheless, both effects are essentially treated correctly in the origi-nal Katchalsky theory [KKK48] from 1948 and more recent elaborations byMuthukumar [Mut87, Mut96, GCM01]. However, electrostatic interactionscan have another effect which was first discussed by Odijk [Odi77] and bySkolnick and Fixman [SF77] (OSF) for semi-flexible polymers in the limitwhere the screening length is smaller than the intrinsic persistence length. Inthis case, it is appropriate to consider small bending fluctuations and to cal-culate the increase of the electrostatic energy relative to the straight groundstate to lowest order in the local curvature. OSF found that the Debye–Huckel interaction makes an additive contribution to the bending rigidityand introduced the notion of an “electrostatic persistence length”:

lOSF =q2f 2lB4A2κ2

(2.2)

In the OSF limit κ−1 � l0 the intrinsic and the electrostatic persistencelength are simply additive, but ever since the publication of their resultsthere has been a lively debate on how to extend the theory to the case of


intrinsically flexible polyelectrolytes. Barrat and Joanny [BJ93] have shownthat the original OSF derivation breaks down, if the chains start to bendsignificantly on length scales comparable to the screening length. Insteadtheir variational theory suggested a qualitatively different dependence le ∼κ−1 of the electrostatic persistence length on the screening length, a resultwhich is (i) compatible with the Katchalsky/Muthukumar theory and seemed(ii) to be in better agreement with the evidence from all available experimentsand computer simulations.

The alternative view by Khokhlov and Khachaturian [KK82] is basedon renormalization group ideas. They argue that the OSF theory can beapplied to a “stretched chain of polyelectrolyte blobs”, a concept introducedby de Gennes et al. [dGPVB76] to describe the behavior of weakly chargedflexible polyelectrolytes in the absence of screening. The persistence length ofthe blob chain is then calculated from Eq. (2.2) using suitably renormalizedparameters.

The purpose of Ref. A was to shed some new light on this problem bycombining a scaling analysis with large scale Monte Carlo simulations. Inthe scaling analysis we consequently adopt the renormalization group viewpoint of Khokhlov andKhachaturian, which allows us to hide the differencebetween strongly and weakly charged polyelectrolytes behind a suitably de-fined system of “blob” units. As a result, observables such as the chainextension can be expressed as a function of only two independent variables:The effective chain length X and the effective screening length Y . In par-ticular, it is possible to express and compare the predictions of the two setsof theories in this way (see Fig. 2.1). It turns out that the differences aresurprisingly small for the parameter ranges studied in previous numerical orexperimental investigations. This is due to the effects of electrostatically ex-cluded volume. Accounting for these effects is crucial, since local stiffeningreduces the number of non-local contacts responsible for the excluded volumeswelling.

From the scaling analysis we extraced the necessary guidance for thedata analysis and the choice of simulation parameters. Data were generatedwith a substantial effort (a generous grant of 1.5 × 105 hours of CPU timeallowed us to simulate chains which were about ten times longer than thoseanalyzed in previous simulations). In reduced units, the measured intrachaindistances cover ten orders of magnitude and allow us to rule out the theoriesof Katchalsky [KKK48], Muthukumar [Mut87, Mut96, GCM01] and Barratand Joanny [BJ93]. In contrast, the estimates provided by the scaling theory

THE ELECTROSTATIC PERSISTENCE LENGTH 13

Figure 2.1: Theoretical predictions for the spatial extension of intrinsicallyflexible polyelectrolytes as a function of reduced chain length and reducedscreening length. Electrostatic interactions are relevant in the upper rightpart of the map which is limited by solid black lines. The dashed and dottedlines correspond to crossover lines in the Odijk-Skolnick-Fixman-Khokhlov-Khachaturian and the Katchalsky-Muthukumar-Barrat-Joanny scaling the-ories respectively. For details see Ref. A

of Khokhlov and Khachaturian [KK82] which neglects all prefactors agreewith the simulation results within a factor of three over the entire parameterrange. The largest deviations occur around the anticipated crossover lineswhere knowledge of the proper crossover functions is particularly important.

Future theoretical work has to concentrate on a quantitative descriptionof the multiple crossovers; our data provide benchmark results to assess thequality of proposed approximation schemes. The extent to which experimentsat finite concentration can be described using single chain theories with effec-tive Debye-Huckel intrachain interactions is quite another matter. Stevensand Kremer [SK96] have shown that the Debye-Huckel approximation workswell for dilute solutions of model polyelectrolytes with weak electrostatic in-


teractions. In a number of cases, the Lund group [JPS95, UJP+97] foundexcellent agreement between experiments and the results of the single-chainsimulations (this statement does not contradict our opinion that the interpre-tation of the numerical and experimental results in terms of various apparentelectrostatic persistence lengths creates more confusion than insight; thispoint is discussed at length in Ref. A). In general, additional effects such assolvent quality or counter-ion condensation need to be taken into account aswell.

2.2 Polyampholytes

Polyelectrolytes which carry charges of both signs are commonly referred toas polyampholytes [CJ96, E.K99]. Depending on the method of synthesis, thecharge sequence can either be alternating or random. Furthermore, one oftendistinguishes between quenched and annealed polyampholytes depending onwhether the chains comprise strong or weak acids and bases. In the first(quenched) case the charge distribution along the chain is solely determinedby the chemistry. In the second (annealed) case the degree of dissociationcan be controlled via the pH, leading to a change of the sign of the chain netcharge at the so-called iso-electric point. For sufficiently large net charges,polyampholytes show the typical behavior of polyelectrolytes: they are water-soluble and adopt highly expanded conformations. The situation changesdrastically in the vicinity of the iso-electric point. The chains adopt compactconformations and often precipitate from solution. This effect is particularlystriking for crosslinked polyampholyte gels and has applications in sewagetreatment where chains which precipitate with pollutants can be recycledsubsequently by a simple change of pH.

2.2.1 From single chains . . .

Apart from one early paper by Edwards, King and Pincus [EKP80] which de-scribes the isoelectric state of polyampholytes as a microelectrolyte of Debye-Huckel type, the statistical physics of polyampholyte has only come into fo-cus in the 1990s. While the behaviour of alternating polyampholytes wasrationalized relatively quickly in analogy to a theta collapse due to effec-tively short–ranged dipole-dipole interactions [VI93, WJJ93], the statisticalmechanics of quenched random polyampholytes turned out to be quite sub-

2.2. POLYAMPHOLYTES 15

tle. As in the case of randomly cross-lined polymer networks (section 4) orspin-glasses [MPV87] one has to deal with quenched disorder: the behavioris dominated by the frustrated attempts of the charges to find an optimalarrangement. The ”quenching” prevents charge rearrangements along thechain which ultimately would produce alternating polyampholytes. Conse-quently, random polyampholytes show a much stronger tendency to fold backonto themselves in the three-dimensional embedding space as the only pos-sibility to bring chain sections with opposite net charges into closer contact.While the search for an optimal charge arrangement is frustrated by thequenched sequences, it is not clear a priori to which extent this frustrationaffects ensemble averages over the disorder.

The first two investigations of the single chain properties by Higgs andJoanny [HJ91] and by Kantor and Kardar [KK91] came to drastically differ-ent conclusions with repect to the scaling of the mean-square radius of gyra-tion with chain length: 〈R2

g〉 ∼ N2/3 (globules) versus 〈R2g〉 ∼ N2 (stretched

chains). The paradox was resolved by Kantor and Kardar [KLK92] whoused computer simulations to demonstrate the sensitivity of random polyam-pholytes to small disparities in the number of positively and negativelycharged monomers per chain. In an ensemble of statistically neutral PAs oflength N , the typical net charge is |Q| ∝ N1/2. Chains with a net charge upto this value behave as globally neutral (Q ≡ 0) polyampholytes and form di-lute globules of spherical shape as predicted by Higgs and Joanny [HJ91]. Incontrast, the more strongly charged members of the ensemble adopt stronglyelongated conformations leading to a situation where ensemble averages forquantities such as the gyration radius for statistically neutral random PAs aredominated by the untypical, extended chains in the wings of the net chargedistribution [KLK92].

Most importantly, Kantor and Kardar [KK94, KK95a] proposed a “pearl-necklace” model based on the analogy to the Rayleigh instability of chargedliquid droplets, which desintegrate beyond a critical net charge. In thepolyampholyte case, the chain connectivity prevents complete fission leadingto a structure of pearls (dilute globules behaving according to the Higgs-Joanny theory for polyampholytes with vanishing net charge) connected bystrings (Fig. 2.2.1). The size of the pearls is given by the Rayleigh crite-rion (a balance between surface and electrostatic energy); the length of thestrings follows similarly from the balance of a line tension and the electro-static pearl-pearl repulsion.

The pearl-necklace model was successfully adapted to the qualitatively


Figure 2.2: Snapshot from a Monte Carlo simulation of long polyampholytechains in the necklace regime (from Ref. B)

similar problem of polyelectrolytes in poor solvent [Kho80, DRO96, LDBD99].One aspect which became clear through the availability of simulation datais the dynamic character of the pearl formation [LHK02]. The free energydifferences between states with similar numbers of pearls are quite small sothat the chains oscillate between these states under the influence of thermalfluctuations.

However, in the case of polyampholytes large question marks remained,since the scaling theory completely neglects the quenched disorder in thecharge sequence. Kantor and Kardar have argued that in the PA case thecharge inhomogeneities should drastically modify the necklace picture. In-deed, computer simulations of PAs reveal a rich variety of conformations [LO98]and it is unclear if the disorder is relevant for ensemble averages of quanti-ties as the gyration radius 〈Rg〉. Ignoring all details of the charge sequenceexcept the net charge, the necklace model predicts 〈Rg〉 ∝ N1/2, while theevidence from Monte Carlo simulations [KK95a] (〈Rg〉 ∝ N0.6) and exactenumerations [KK95b] (〈Rg〉 ∝ N2/3) rather suggests a faster growth.

Ref. B settles this controversy on the shape of isolated random PAs ingeneral and the effect of the quenched disorder in the charge sequence inparticular. For this purpose we first work out a complete scaling theory forthe Kantor–Kardar necklace model. The result provides a framework for theanalysis of simulation data for various ensembles of quenched random PAs:Fixed (zero or nonzero) net charge, and randomly charged chains with a

2.2. POLYAMPHOLYTES 17

typical net charge of order N1/2. The long length of our chains (N ≤ 4096)and the large number of independent charge sequences (between 512 and1024) turned out to be crucial, since our results suggest that deviations fromthe predictions of the necklace model for ensemble averages are merely finitesize effects.

2.2.2 . . . to solutions of finite concentration

During my post-doc time at the Institut Charles Sadron in Strasbourg Iinvestigated polyampholyte solutions of finite concentration. Our analysisof the phase equilibrium between a homogeneous dense phase and a dilutesupernatant is based on the description of polyampholytes in solution aselongated globules or necklaces [CJ96, GS94, DR95]. Previously the solubil-ity of polyampholytes was estimated from analogies to polymers in a poorsolvent and to polyelectrolytes (i.e. polymers that carry charges of only onesign), implying (i) that solutions precipitate under conditions where individ-ual chains collapse and (ii) that charged chains are better soluble than neutralchains. These arguments are, however, questionable as the water-solubilityof polyelectrolytes is mostly due to the gain in translational entropy of thecounter-ions in the water phase [Kho80]. The polymers are dissolved in spiteof their high electrostatic self energies, which they minimize by adoptingstretched conformations. In the following, I disuss some of the simple exam-ples from Refs. C and D in order to outline the behaviour of polyampholytesolutions.

Precipitation. In the case of random polyampholytes with vanishing netcharges on each polymer, the phase equilibrium between a dilute supernatantof spherical globules and a dense precipitate can be understood along the linesof the poor solvent analogue: The local chain structure in the dense phaseand in the interior is identical; the excess free energy of chains in solutionis proportional to their surface area. As a consequence, the ratio of themonomer concentrations in the two phases obeys log(cdil/cdense) ∼ N−2/3.

Complexation. In bimodal solutions of random polyampholytes with op-posite net charges of equal magnitude, dissolved unimers are in chemicalequilibrium with neutral dimers. The situation therefore resembles mixturesof oppositely charged polyelectrolytes [BE90]. Translational entropy favorsunimers while dimerization lowers the electrostatic self energy. Quite inter-estingly, the criterion for the dominance of either species is the same as forthe formation of elongated pearl-necklace conformations. Unimers dominate


for net charges below the Rayleigh instability, while chains with higher netcharge prefer the formation of charge neutral dimers. As a consequence andin marked contrast to the situation at infinite dilution, there are practicallyno elongated chains in a solutions of finite concentration of an ”ideal” sampleof polyampholytes with Gaussian net charge distribution.

Donnan equilibrium. For non-neutral samples, on the other hand, the freecounter-ions accumulate in the supernatant together with the most stronglyoppositely charged unimers, if the system is concentrated beyond the onset ofphase separation. The dilute phase consists of elongated globules and, as forordinary polyelectrolytes, has a concentration proportional to the total poly-mer concentration. This is due to (i) the formation of a Donnan equilibriumwith an electrostatic potential difference between the phases which ensurestheir charge neutrality and (ii) sublinear dependence of the excess free energyon the chain net charge in the necklace or elongated globule regime.

In general, non-self-neutralizing samples behave like mixtures of a sol-uble and a non-soluble component. Concentrating the sample leads to aseparation of an almost self-neutralizing polyampholyte precipitate and apolyelectrolyte-like supernatant consisting of the most strongly charged chainsand counter-ions. These results are in good agreement with experimentalobservations [CSC93, SMC+94, OCMC96], but sheds some doubts on theiroriginal interpretation within a single chain picture.

Chapter 3

Dynamic Fluctuations ofSemi-flexible Filaments

Among the classical models of polymer physics are the Kratky-Porod worm-like chain (WLC)[KP49] for the conformations of semi-flexible filaments andthe Rouse model [Rou53] for the dynamics of flexible polymers. Ref. E dis-cusses some subtle aspects of the dynamics of semi-flexible filaments, whichis shown to be dominated by two independent dynamic length scales for theequilibration of thermal fluctuations and the propagation of tension. The ideato the existence of these length scales was conceived on a scaling level. Thesimulations established and illustrated the phenomenon and subsequent ana-lytical efforts [LM02] have lead to a quantitative understaning. The followingdiscussion is more detailed than those in the preceding sections. Hopefullythis will facilitate the reading of Ref. E which, due to the letter format, iswritten in a fairly dense way.

Consider an elastic rod of diameter a of a material with Young’s modulusY . Its elastic properties can be characterized by a compression modulusE ∝ a2Y and a bending modulus κ ∝ a4Y . As a consequence of the strongerdependence of the bending modulus on the rod diameter, it is much easier tobend a thin filament than to stretch it. Semiflexible polymers such as DNAor actin are effectively incompressible and their conformational statistics canbe derived by considering a line with a bending Hamiltonian

H =1

2κ∫

ds

(d2~r

ds2

)2

(3.1)

19

20 DYNAMIC FLUCTUATIONS

=1

2κ∫

ds

(d~t

ds

)2

(3.2)

where ~t = d~rds

denotes the tangent vector. The incompressibility constraint∣∣∣~t∣∣∣ ≡ 1 is very difficult to implement exactly in analytical theories. In terms ofthe tangent vector, the static problem corresponds to the diffusion problemon the surface of a sphere. However, the Langevin equation for the timeevolution of the filament conformation in real space is non-linear due to thenecessity to locally fulfill the incompressibility constraint at all times.

In spite of the difficulties in the analytical solution of this model it ispossible to deduce the main aspects of the filament behavior from simplescaling arguments.

The energy necessary to bend a filament of length L into a half-cirle isgiven by 1

2κL(π/L)2 and becomes comparable to the thermal energy kBT

for chains with a contour length of the order of the persistence length lp =κ/kBT . Filaments with a contour length L smaller than lp behave to a firstapproximation like stiff rods, i.e. their root-mean-square end-to-end distanceis identical to their contour length:

⟨~r2⟩

= L2 L � lp (3.3)

On large length scales, WLCs form random coils:

⟨~r2⟩

= 2Llp L � lp (3.4)

In the second limit, the filament dynamics is described by the Rouse model,which makes two assumptions. The polymers are treated as (Gaussian) flex-ible random coils and their monomers are supposed to move against a localStokes friction in a medium with viscosity η. Quite curiously this single chainmodel is appropriate for polymers in a dense (unentangled) melt of otherpolymers, while dilute solutions are better described by the Zimm modelwhich includes hydrodynamic interactions [DE86]. However, for the rod-likemolecules we are eventually interested in, the differences between the Rouseand the Zimm model reduce to logarithmic corrections which are henceforthneglected.

The Rouse model can be solved by dynamic scaling. For large times, thechains move by simple diffusion. Within the local friction approximation thechain friction coefficient is given by ηL, so that

21

〈(xCM(t)− xCM(0))2〉 ∼ kBT

ηLt t � τRouse (3.5)

The crossover time below which one has to distinguish between monomer andcenter-of-mass motion is given by the time necessary for a chain to diffuseover the distance of its own diameter:

τRouse ∼ηlpL

2

kBT(3.6)

This is also the time scale which characterizes the internal relaxation of thechain conformations.

For the following considerations it is useful to transform this relation intoa time-dependent (contour) length over which long chains are equilibrated

l1(t) ∼

√√√√kBT

ηlpt (3.7)

and to express the motion of individual monomers as

〈(x(t)− x(0))2〉 ∼ lp l1(t) (3.8)

Within a linear response picture, this translates into a displacement δx(t) ofa monomer due to a force f acting on it for t > 0:

〈δx(t)〉 = lp l1(t)f

kBT(3.9)

The part l2(t) of the chain which follows the motion of the pulled monomercan be estimated by equating the pulling force and the total friction forceacting on a chain of length l2(t) moving at a velocity 〈δx(t)〉/t:

f = ηl2(t)lp l1(t)

t

f

kBT↔ l2(t) =

kBTt

ηlpl1(t)= l1(t) (3.10)

Thus, for flexible polymers the tension propagation length l2(t) is equal tothe equilibration length l1(t). There is a single dynamic length scale as thereis also only a single static length scale, the gyration radius.

Ref. E analyses the analogous problem for semi-flexible polymers in thelimit L � lp where L and lp are independent static length scales. Thediscussion simplifies considerably, if one considers the discretized version of


Eq. (3.2) in two dimensions. In this case, the Hamiltonian can be written asa quadratic potential for bond angles:

Hb =kBT

2

L/b−1∑i=1

lpb

Θ2i (3.11)

so that equilibrated chain conformations can be generated by sampling bondangles from appropriate Gaussian distributions with 〈Θ2

i 〉 = b/lp. Unfortu-nately, the transformation to angular space does not help to understand thedynamics. Formulating a set of independent Langevin equations for the bondangles would produce some kind of pivot dynamics where, to use a Germanproverb, the tail wags the dog. Instead one has to assume that in the dis-cretized model the viscous drag is concentrated in ”beads” at the segmentjunctions. The corresponding friction coefficient is of the order of ζ = ηb.While the implementation of the contour length constraint |~ri − ~ri−1| ≡ b inthe simulation leads to non-trivial complications, it is nevertheless possible todeduce most of the scaling relation which characterize semi-flexible polymersby considering a trimer with b = L/2 and 〈Θ2〉 ∼ L/lp.

A characteristic difference between semi-flexible and flexible polymersis the anisotropy of the chain conformations and fluctuations. To a firstapproximation, bending fluctuations only lead to transverse displacementsrelative to the chain axis. They are of the order of 〈δr2

⊥〉 ∼ L2〈Θ2〉 = L3/lp.Due to the contour length constraint, bending fluctuations also induce muchsmaller, longitudinal fluctuations of the order of 〈δr2

||〉 ∼ L2(〈Θ4〉 − 〈Θ2〉2) =

L2(L/lp)2.

In order to estimate the equilibration time for the trimer, we assume thatthe first segment is fixed in space, while the the second endpoint moves ina quadratic potential with ”spring constant” k⊥ = kBT/〈δr2

⊥〉. The char-acteristic time scale of this motion is of the order τ⊥(L) = ζ(L)/k⊥(L) ∼ηL4/(kBT lp), a result which matches Eq. 3.6 for filaments with crossoverlength L = lp. A given time t is thus sufficient to equilibrate filament (sec-tions) of length

l1(t) = (kBT lp

ηt)1/4 (3.12)

with time-dependent transverse fluctuations of the order of

〈δr2⊥(t)〉 ∼ l1(t)

3/lp ∼ t3/4. (3.13)

23

The equilibration and the transverse fluctuations are independent for neigh-boring sections of length l1 of long filaments with L � l1. However, theequilibration of each segment also causes a longitudinal displacement of theorder of l1(t)

4/l2p. In the absence of longitudinal friction, these longitudinaldisplacements would be simply additive due to the connectivity of neigh-boring sections: 〈δr2

||(t)〉 ∼ (L/l1(t))(l1(t)4/l2p). In reality, the exchange of

contour length is limited to sections within the tension propagation lengthl2(t)

f = ηl2(t)(l2(t)/l1(t))(l1(t)

4/l2p)

t

f

kBT↔ l2(t) =

(kBT

ηl5p t

)1/8

, (3.14)

restricting the longitudinal fluctuations to

〈δr2||(t)〉 ∼ (l2(t)/l1(t))(l1(t)

4/l2p) ∼ t7/8. (3.15)

While these considerations are fairly convincing on a scaling level, their cor-rectness nevertheless needs to be established by independent means. In Ref. Ewe have used Brownian Dynamics simulations of linear and crosslinked fila-ments for this purpose. I would like to close by pointing to two interestingaspects of the simulation work: (i) the generation of data which cover a dy-namic range of ten orders of magnitude by varying the spatial discretizationof the simulated WLC and (ii) the “cloud” analysis of the dynamic fluctua-tions.

Chapter 4

Statistical physics of polymernetworks

Polymer networks [Tre75] are created by randomly cross-linking solutions ormelts of linear polymers. They are the basic structural element of systemsas different as tire rubber and gels and have a wide range of technical andbiological applications. While they have been a subject of statistical me-chanics for more than sixty years, their rigorous treatment still presents achallenge. The main difficulty is the quenched disorder introduced by chem-ically cross-linking linear precursor chains (vulcanization). Thermodynamicvariables need to be averaged over the conserved random connectivity andtopology of the resulting networks [DE76], introducing similar complicationsas in the theory of spin glasses [MPV87].

Section 4.1 describes my contributions to the systematic development ofa tube model for the localization of polymer chains due to crosslinking andentanglements. The present collection contains two theoretical papers. To-gether with the detailed verification of this phenomenological ansatz with thehelp of computer simulations in Ref. [Eve99], they continue the main subjectof my PhD thesis. In order to facilitate the comparison and to emphasizethe methodological difference, section 4.1.1 presents a short summary of ourprevious results. Section 4.2 discusses the swelling process of a piece of rub-ber in good solvent and presents an additional example for the combinationof computer simulations and scaling ideas.

25

26 CHAPTER 4. STATISTICAL PHYSICS OF POLYMER NETWORKS

4.1 Entanglement effects in rubber-elastic sys-

tems

A proper statistical theory of polymer networks has to account for the con-servation of both connectivity and topology after the preparation of the net-works.

The first point was essentially solved in the 1930s and 1940s after thediscovery of the nature of polymeric molecules and their high degree of con-formational flexibility. In a melt of identical chains polymers adopt randomcoil conformations [Flo49] with mean-square end-to-end distances propor-tional to their length, 〈~r2〉 ∼ N . A simple statistical mechanical argumentthen suggests that flexible polymers react to forces on their ends as linear,entropic springs. The spring constant, k = 3kBT

〈~r2〉 , is proportional to the tem-

perature. The phantom model [Jam47, JG47, Flo76] treats a piece of rubberas a random network of non-interacting entropic springs and qualitativelyexplains the observed behavior, including — to a first approximation — theshape of the measured stress-strain curves. The Hamiltonian of the phantommodel has a simple quadratic form

Hph =k

2

∑〈i,j〉

r2ij , (4.1)

where 〈i, j〉 denotes a pair of nodes which are connected by a polymer chainand ~rij(t) = ~ri(t) − ~rj(t) the distance between them. In spite of the simpleform of Eq. (4.1), the phantom model is far from trivial, since one has tofind ways to represent the complicated connectivity of a randomly cross- orendlinked melt of linear precursor chains. In statistical mechanical termsone has to deal with a system containing quenched disorder [DE76]. Thiscomplication does not seem to be critical for highly crosslinked networks (i.e.with many crosslinks per precurser chain) [Flo76, HB88]. In contrast, thevulcanization transition from liquid to solid behaviour bears strong analogiesto spin glasses [GCZ96].

While for a given connectivity the phantom model Hamiltonian for non-interacting chains formally takes a simple quadratic form, it seems almostimpossible to fully characterize the topological constraints or entanglements,which are due to the mutual impenetrability of the polymer backbones. Inprinciple, this requires an infinite set of topological invariants from mathe-matical knot theory [Edw67b, Edw68, Rol96]. Moreover, all but the most

ENTANGLEMENT EFFECTS 27

primitive invariants are algebraic so that their statistics cannot be calculatedanalytically for entangled random walks.

So far no rigorous solution of the statistical mechanics of entangled poly-mer networks exists. Topological theories of rubber elasticity represent themost fundamental approach, but encounter serious mathematical difficultiesalready on the level of pair-wise entanglement between meshes. Most theo-ries therefore omit such a detailed description in favor of a mean-field ansatzwhere the different parts of the network are thought to move in a deformation-dependent elastic matrix which exerts restoring forces towards some rest po-sitions. The classical theories [Tre75, RA75, Flo77, EF78, FE82, K81] assumethat such forces only act on the cross-links or junction points. There are, how-ever, good reasons to suspect that this approach overlooks important aspectsof the physics of an entangled network. The evidence comes from the studyof non-cross-linked polymer melts, which show extremely slow relaxation forchains whose lengths exceeds a phenomenological “entanglement length”,Ne. A simple and very successful explanation of these effects is provided bythe tube model of Edwards [Edw67a] and the reptation theory [dG71] of deGennes. The idea is that the presence of the other polymers restricts a chainto fairly small fluctuations inside a tube-like region with a cross-section ofthe order of R2

g(Ne) along its coarse-grained contour. A polymer can loosethe memory of its initial conformation only by a one-dimensional, curvilineardiffusion along and finally out of its original tube (“reptation”). In a networkthe chemical cross-links suppress this relexation mechanism and isolate theconfinement aspect of the problem.

When these theories were worked out in the 1970s, the only way to as-sess their success was a comparison to experimentally measured stress-straincurves. Unfortunately, this does not even allow to discriminate betweenthe classical and the tube models, since formally identical stress-strain rela-tions can be derived from the two approaches [VE93]. More interesting isa comparison of the absolute values of measured and predicted moduli or,less specifically, the extrapolation of the measured moduli to the limit ofvanishing cross-link density where the classical contribution to the modulusvanishes. Such investigations have indicated from early on that the classicaltheories underestimate the modulus [MW56, L.M59]. However, it is quitedifficult to prepare model networks with a well defined density of elasticallyactive chains. This holds in particular in the limit N � Ne.

A detailed test of the theoretical models requires access to microscopic in-formation not available in rheological experiments. Using small-angle neutron-


scattering [RFF+90, RWZ+93, SUPH+95] it is, at least indirectly, possibleto detect and quantify the tube-like confinement of the chain motion in poly-mer melts and networks and to investigate the effect of shear deformationson the tube. An alternative are large scale computer simulations of suitablycoarse-grained polymer models [KG95]. My PhD thesis documents some ef-forts along these lines. However, in order to benefit as much as possible fromthe abundantly available information, I had to reformulate and extend theconstrained fluctuation theories of rubber elasticity.

4.1.1 Summary of results from my PhD thesis

The main part of my PhD work with Prof. Kremer were Molecular Dynam-ics simulations of defect-free model polymer networks with diamond latticeconnectivity. The idea behind the investigation of these model networks wasthe isolation of the effects of a quenched random topology from those ofa quenched random connectivity. After developing an efficient simulationprogram [EK94] and preparing the idealized network structures [EK96a],an important algorithmic progress consistet in the measurement of normalstresses for networks in strongly deformed simulation boxes. As a result, wewere able to obtain the first reliable numerical estimates for the elastic prop-erties of a polymer network. The simultaneous access to this ”macroscopic”observable and all microscopic details allows a parameter free evaluation ofall statistical mechanical theories of rubber elasticity which are based on awell-defined microscopic picture. During my PhD work, I carried out twosuch tests. The first [EK95] was devoted to a quantitative test of the ideaunderlying the classical theories of rubber elasticity, which neglect all en-tanglement effects on the chain conformations between chemical crosslinks.By calculating stresses (respectively deformation dependent free energies) forensembles of entropic springs with extensions corresponding to those of thenetwork strands in the original simulation, we were able to show that thisansatz cannot account for the observed stresses. The second test [EK96b]was devoted to the ideas underlying topological theories of rubber elastic-ity. By evaluating the topological degree of linking for all mesh pairs, wewere able to show that the entanglement contributions to the shear moduluspredicted by the Graessley and Pearson theory are in good agreement withthose observed in our simulations.

Taken together, these results lend strong credibility to the basic ideas de-veloped by Edwards and his co-workers. Unfortunately, there is no systematic


extension of the topological ansatz to a quantitative theory describing mi-croscopic deformations and macroscopic restoring forces. Another idea goingback to Edwards’, the tube model [Edw67a], is more promising in this re-spect. However, the variants of the tube model available at the time werenot directly applicable to artificial network connectivities. The followingsections document an attempt to reformulate the tube model in a languagewhich simplifies the calculations and allows the application of the idea toarbitrary network connectivities.

4.1.2 The constrained mode model (CMM)

In introducing the tube model, Edwards’ idea was to consider individualchains in regular arrays of obstacles representing the other chains. Their maineffect is to block transverse fluctuations of a given polymer around its coarsegrained contour. In other words, topological constraints divide the configu-ration space of individual chains into momentarily accessible and inaccessibleregions. The important point to note is that obstacles which represent tem-porary topological constraints are not part of the Hamiltonian controlling thestatistical weight of particular polymer conformations. Thus as long as thesystem is ergodic, they do not affect ensemble averages. Crosslinking breaksthis ergodicity and permanently confines the chains to tube-like regions inspace.

A closer look at the tube model therefore has to address details such as:

1. the form and deformation dependence of the localizing potential,

2. suitable variables for formulating the theory, and

3. the calculation of disorder averages.

In the original problem as well as in the obstacle model the nature of theconstraints is topological. Chain conformations are accessible or inaccessibledepending on whether or not they can be reached by continuous deformationsof a primitive path without crossing of other chains or obstacles. In constrast,tube models introduce geometrical constraints by localizing monomers inspace. While it is unclear to which extent this approximation is justified,it does have the advantage to considerably simplify calculations. This holdsin particular, if the constraints are represented as harmonic springs attachingmonomers to particular points in space:


Hconstr =∑

i

1

2

(~ri − ~ξi(λ)

)tl(λ)

(~ri − ~ξi(λ)

). (4.2)

The variable λ parametrizes the deformation of the sample. Since rubber-like materials are effectively incompressible with a Poisson ratio close to 1/2,one typically considers volume conserving, uni-directional elongations withλ⊥ = λ

−1/2|| . ”Affine deformations” on a microscopic scale imply that micro-

scopic distances or positions are rescaled by the same factor as the macro-scopic dimesions of the sample. In the present case, it is generally assumed

that (ξx, ξy, ξz)(λ) = (λξx, ξy/√

(λ), ξz/√

(λ)). The tensorial form allows fordifferent, strain dependent spring constants in different spatial directions. Aanisotropic localizing potential represents non-spherical “cavities”.

In the phantom model without constraints the problem separates in Carte-sian co-ordinates x, y, z. Furthermore, a conformation of a network of har-monic springs can be analyzed in terms of either the bead positions ~ri(t) or

the deviations ~ui(t) of the nodes from their equilibrium positions ~Ri. The

latter are characterized by a force equilibrium∑

j~Rij ≡ 0, where j indexes all

nodes which are connected with node i. In this representation, the Hamil-tonian separates into two independent contributions from the equilibriumextensions of the springs and the fluctuations, which can be written as a sumover independent normal modes or phonons ~up [DS65, Eic72]:

Hph =k

2

∑〈i,j〉

~R2ij +

k

2

∑〈i,j〉

~u2ij (4.3)

=k

2

∑〈i,j〉

~R2ij +

kp

2

∑p

~u2p (4.4)

If a sample is deformed, the equilibrium positions of the junction pointschange affinely. The fluctuations, on the other hand, depend only on theconnectivity but not on size and shape of the network. The shear modulus ofthe phantom model can therefore be calculated without having to integrateout the dynamic eigenmodes of the network.

When constraining potentials of the type Eq. (4.2) are introduced into thephantom model, it is useful to measure all positions relative to the phantommodel equilibrium position ~R in the absence of constraints. Defining ~vi ≡~ξi − ~Ri, Eq. (4.2) contains terms of the form ~ui − ~vi instead of ~ri − ~ξi. In


Ref. G B. Mergell and I have shown thatHconstr is diagonal in the generalizedRouse modes up of the phantom network without constraints:

Hconstr =∑p

l(λ)

2(up − vp)

2 . (4.5)

Thus, the confinement of polymers in random walk like tubes can be discussedin terms of constrained modes. In fact, this result provides a direct linkbetween two diverging developments in the theory of polymer networks: theideas of Edwards, Flory and others on the suppression of fluctuations due toentanglements and the considerations of Eichinger [Eic72], Graessley [Gra75],Mark [KME90], and others on the dynamics of (micro) phantom networks. Inthe following I will illustrate two advantages of the formulation in mode space:Firstly, the calculation of disorder averages can be performed without havingrecourse to uncontrolled approximations such as the replika trick. Secondly,the mode analysis provides a convenient framework for a quantitative testof the ideas underlying the constrained junction and tube models of rubberelasticity.

The formal aspects of the calculation of disorder averages were also firstdiscussed by Edwards. He pointed out that averages over annealed andquenched variables have to be taken in different ways. In the present case,the quenched variables characterize the shape of the random walk-like tubewhile the annealed variables describe the polymer conformations. Statisticalaverages for observables such as the deformation dependent free energy F (λ)of the confined chains are calculated in two steps. In the first step, thermalfluctuations (i.e. the annealed variables) are integrated out for a given tube.In the second step, the logarithm of the partition function is averaged overdifferent tube realizations. Ref. F illustrates that terms such as ”quencheddisorder”, ”non-Gibbsian statistical mechanics” or ”strain dependent local-ization” have a particularly simple interpretation for the constrained modemodel. For constrained harmonic oscillators the deformation dependent freeenergy is given by:

F (λ, vp) = −kBT log(∫

dup e− kp

2kBTu2

p−l(λ)

2kBT(up−λvp)2

)(4.6)

〈F (λ)〉 =∫

dvpF (λ, vp) p(vp) (4.7)

In general, problems of this type are extremely difficult to treat. However,


the present case can be solved exactly by noting that the oscillators performreduced thermal fluctuations

〈δu2p〉 ≡ 〈(up − Up)

2〉 =kBT

kp + l. (4.8)

around non-vanishing equilibrium excitations

Up =l

kp + lvp (4.9)

The calculation of the free energy is straightforward for a given disorderrealization vp and the disorder average becomes trivial if vp is sampled froma Gaussian distribution.

A subtle point is the choice of the width of this distribution. As pointedout above, we are trying to emulate constraints which do not affect ensem-ble averages in the state of preparation. In the present case, the choice〈v2

p〉 = (kp+l)kBTkpl

ensures that static ensemble averages over all oscillators

remain unaffected by the confinement, since in this case 〈u2p〉 = 〈U2

p 〉 +

〈δu2p〉 = kBT

kp. However, dynamic expectation values such as the autocor-

relation function 〈up(t)up(0)〉 reveal the presence of the constraints, sincelimt→∞〈up(t)up(0)〉 = 〈U2

p 〉 6= 0.

Static ensemble averages are affected by confinement effects under strain.In particular, they contain information about the strain dependence of theconfining potentials. The analogy to the obstacle model suggests an affinedeformation of a tube which represents topological constraints. This choicecorresponds to vp(λ) = λvp and l(λ) = l/λ2. Ref. F discusses the subtle sideeffect that in this case the constraints do not contribute to the stresses eventhough they store elastic energy. In particular, we make the point that thefamiliar Doi-Edwards expression for the stress tensor only holds for affinelydeforming confining potentials. In Ref. G we refer to this case as ”modelB”, while we denote the ansatz of a deformation independent strength of theconfining potential as ”model A”. Model A predicts deformation independentfluctuations 〈δu2

p〉 and affine deformations of the mean extensions Up, i.e.the behavior expected in phantom networks without topological constraints.Furthermore, Ref. G considers the simultaneous presence of both types ofconfinements (”model C”).


4.1.3 Quantitative tests in simulations of defect-freemodel polymer networks

The constrained mode model can be used to cast the ideas of Flory [Flo77]and Edwards [Edw67a] into a form which is particularly suited for the anal-ysis of simulation data. Treating the motion of cross-links and networksstrands as independent Einstein and Rouse modes respectively, one can di-rectly distinguish the two types of entanglement effects discussed in the con-strained junction and the tube model.

In Ref. [Eve99] I have applied this analysis to the simulation data frommy PhD thesis. In agreement with the tube model and in contradiction tothe assumptions of the classical theories of rubber elasticity, it turns outthat both Einstein and Rouse modes are affected by the entanglements. Thequantitative success of the CMM in predicting the chain conformations understrain from an analysis of the confined fluctuations in the unstrained networksstrongly supports the idea underlying all constrained fluctuation theories ofrubber elasticity. Furthermore, the evaluation of the Doi-Edwards expressionfor microscopic conformations under strain yields reasonable normal stresses.In particular, both results support the modelization of topological constraintsusing affinely deforming potentials (model B).

In summary, (i) the constrained mode model, when applied to the gen-eralized Rouse modes of the corresponding phantom network without entan-glements, provides a formally exact solution of the constrained fluctuationmodel for rubber elastic systems; (ii) there is strong support for this ansatzfrom simulations of model networks.

4.1.4 Tube Models for Rubber-Elastic Systems

As point out above, small angle neutron scattering (SANS) experiments areparticularly interesting, since they allow to measure single chain propertiesby deuterating part of the polymers. If such a system is first crosslinkedinto a network and subsequently subjected to a macroscopic strain, one canobtain information on the microscopic deformations of labeled random pathsthrough the network. In Ref. G we make the point that the experimen-tally observed localization is due to two, qualitatively different and to a firstapproximation independent effects: the incorporation of the chains into anetwork and the presence of topological constraints (entanglements).

In order to interpret SANS data, they need to be compared to the pre-


dictions of theories of rubber elasticity. However, even for the fundamentalcase of a randomly crosslinked phantom network, the relevant structure fac-tors cannot be calculated exactly [dC94, Ull79, HB88]. It is therefore notsurprising that similar difficulties prevent the direct application of the con-strained fluctuation models to experimental systems. In essence, the problemis related to the unknown eigenvalue spectrum of the Kirchhoff matrix forrandomly crosslinked networks and the subsequent evaluation of ensembleaverages for random paths through these networks.

To overcome this problem, Warner and Edwards [WE78] proposed toreplace the ensemble average over labeled precursor chains which are inte-grated into the network at uncorrelated random cross-link positions by anaverage over chains where all monomers experience an equally strong local-izing potential. Formally, the “smearing out” of the crosslinking effect alongthe chain leads to Eq. (4.2), i.e. a tube model for a linear chain with aknown, integrable Rouse eigenmode spectrum. However, in order to modelconfinement due to crosslinking they used (in our notation) Model A, sincethis ansatz reproduces the essential properties of phantom models (affinedeformation of equilibrium positions and deformation independence of fluc-tuations). In contrast, Heinrich and Straube and Rubinstein and Panyukovtreated confinement of linear chains due to entanglements using Model B.Obviously, both effects are present simultaneously in polymer networks. InRef. G we develop the idea that in order to preserve the qualitatively dif-ferent deformation dependence of the two types of confinement, they shouldbe treated in a “double tube” model based on our Model C. From the solu-tion of the generalized tube model we obtain expressions for the microscopicdeformations and macroscopic elastic properties which can be compared toexperiments and simulations. First preliminary simulation results (Fig. 4.1)strongly support the double tube model.

4.2 Chain conformations in rubber at equi-

librium swelling

As a last point, I present some results for a related, but formally even morecomplicated problem [PR96]: the equilibrium swelling of a piece of rubberin good solvent. This is another classical problem of polymer physics which,quite surprisingly, is not even fully understood on a scaling level. In gen-

EQUILIBRIUM SWELLING 35

100

101

102

n

100

101

<R

2 (n;λ

α)>/<

R2 (n

;λα=

1)>

100

101

102

n10

010

110

2

n

A B C

Figure 4.1: Length scale dependend deformation of random paths throughend-linked polymer networks parallel and perpendicular to the elongation.Simulation results (symbols) for uniaxial elongation with 0.8 ≤ λ ≤ 4.0in comparison to best fits based on the Warner and Edwards theory(A), Heinrich-Straube/Rubinstein-Panyukov theory (B), and the Mergell-Everaers double tube model (C). The fits have one (A,B) respectively two(C) tube diameters as adjustable parameters. (Svaneborg, Grest, Kremerand Everaers, unpublished). Note that theories which are based on the de-scription of polymers as Gaussian chains can only be expected to describethe large length scale behavior.


eral, the present system is a simple example for a polymer gel [Flo53, dG79,DKOY91, Dus93, Add96]. Gels are soft solids governed by a complex in-terplay of the elasticity of the polymer network and the polymer/solventinteraction. They can be quite sensitive to the preparation conditions andcan undergo large volume changes in response to small variations of a controlparameter such as temperature, solvent composition, pH or salt concentra-tion.

The two basic theories addressing the swelling of a piece of rubber, theclassical Flory-Rehner theory [Flo53] and de Gennes’ c∗–theorem [dG79],have opposite views on the nature of the swelling process, local deformationsand the dependence of the macroscopic degree of swelling on the networkstructure. Both theories characterize the network structure very crudely viaa mean strand length Ns.

The classical Flory-Rehner theory [Flo53] writes the gel free energy F asa sum of two independent terms: a free energy of mixing with the solvent (fa-voring swelling and estimated from the Flory-Huggins theory of semi-dilutesolutions of linear polymers) and an elastic free energy (due to the affinestretching of the network strands which are treated as Gaussian, concentration-

independent, linear entropic springs). Minimizing F yields Qeq ∝ Ns3/5

forthe equilibrium degree of swelling which is defined as the ratio of the sam-ple volumes in the swollen and in the dry state. The Flory-Rehner theoryimplies that the structure factor of long paths through the network is of theform S(q) ∼ q−2 both locally, where the chains are unperturbed, and onlarge scales, where they deform affinely (〈~r2〉eq ∝ 〈~r2〉dry Q2/3

eq ) with the outerdimensions of the sample.

More recent treatments are based on the scaling theory of semi-dilutesolutions of linear polymers [dG79] and the idea that locally, inside of socalled “blobs”, the chains behave as isolated, self-avoiding walks with ν ≈3/5. On larger scales, the solution behaves as a dense melt of Gaussian blobchains. In the case of swollen networks, a controversy exists whether thesize of the network strands is determined by the global connectivity or bythe local swelling. Quite interestingly, the first view [PR96, Pan90, ORC94]

leads again to the Flory-Rehner result Qeq ∝ Ns3/5

. In contrast, de Gennes’c∗–theorem [dG79] asserts that the macroscopic swelling is only limited bythe local connectivity, which begins to be felt at the overlap concentrationc∗ ∝ Ns/(bNs

ν)3 of a semi-dilute solution of linear polymers of average length

Ns, corresponding to Qeq ∝ Ns4/5

. The c∗–theorem predicts S(q) ∼ q−5/3 for

EQUILIBRIUM SWELLING 37

q > 2π/(bNs3/5

) as well as unusual elastic properties due to the non-linearelasticity of the network strands [HTG97, Dao77].

Since both pictures are supported by part of the experimental evidencefrom combinations of thermodynamic and rheological investigations withneutron or light scattering [THB73, CBD82, PMCG92, Hil97, BC96], therewas some hope that computer simulations with their better access to andcontrol over microscopic details of the network structure might help to clearthe issue. However, instead of simply confirming one of the established theo-ries our simulation results suggested (i) a stronger swelling of network strandsthan predicted by the c∗–theorem (ν ≈ 0.72 compared to ν ≈ 3/5 respectivelyν = 1/2 in the Flory-Rehner picture), but (ii) relatively small macroscopicdegrees of swelling in qualitative agreement with the Flory-Rehner theory.

Quite interestingly, the scaling argument we developed a posteriori torationalize the simulation results turned out to be a rederviation of the Flory-Rehner theory based on a Flory argument. The c∗–theorem is based onthe assumptions that unconnected chains which share the same volume inthe dense state can freely desinterpenetrate. In this case, their size can beestimated by the standard Flory argument which considers the equilibriumbetween an elastic stretching energy and a repulsive energy due to binarycontacts between monomers for a single chain. In contrast, we argue that apolymer network is a solid. As a consequence, it is impossible to substantiallymodify spatial neighbor relations between different chains. We thereforepropose to consider a Flory argument for a group of N1/2 interpenetratingchains which share the same volume in the crosslinked melt. The result is aswelling exponent of ν = 7/10 in three dimensions on length scales up to the(effective) strand length. On larger scales the deformations become affine,leading to a macroscopic degree of swelling proportional to N3/5 in agreementwith the Flory-Rehner theory. Both results are in excellent agreement withthe main findings from our simulations.

Chapter 5

Acknowledgements

This is probably the perfect place to express my gratitude to two scientificmentors whose influence can be felt in most of my scientific work. I amtruely grateful to Prof. Jean-Francois Joanny and to Prof. Kurt Kremerfor their continuous encouragement and support, for discussions and ideasand for having achieved the almost impossible: being there when they wereneeded and, equally important, not being there when they were not needed.Similarly, I am indebited to Prof. J. Prost, Prof. K. Binder and Prof.E. Sackmann for their support during all these years. I have benefittedenormously from interactions with collaborators, colleagues, postdocs andstudents and I am grateful to all of them. In particular, I would like tothank Albert Johner (Strasbourg), Armand Ajdari, Frank Julicher, and TonyMaggs (Paris), Burkhard Dunweg, Doris Kirsch, Boris Mergell, and HelmutSchiessel (Mainz), and Fred MacKintosh, David Morse, and Tom McLeish(Santa Barbara).

I would also like to thank the Institut Charles Sadron, the Institut Curie,the Max-Planck-Insitut fur Polymerforschung and the ITP in Santa Barbarafor their hospitality and financial support. I am particularly grateful to theDFG for an Emmy-Noether grant which allowed me to establish a “Nach-wuchsgruppe.”

Last but not least, I would like to thank my family, friends, and, inparticular, Claire Loison for love & life in real space & time.

39

40 CHAPTER 5. ACKNOWLEDGEMENTS

Bibliography

[Add96] J.P. Cohen Addad, editor. Physical Properties of Polymeric Gels.Wiley, Chichester, 1996.

[BC96] J. Bastide and S. J. Candau. Structure of gels as investigatedby means of static scattering techniques. In J.P. Cohen Addad,editor, Physical Properties of Polymeric Gels. Wiley, Chichester,1996.

[BE90] V. Yu. Borue and I. Ya. Erukhimovich. Macromolecules, 23:3625,1990.

[Bin95] 1995.

[BJ93] J.-L. Barrat and J.-F. Joanny. Persistence length of polyelec-trolyte chains. Europhysics Letters, 24:333–338, 1993.

[BJ96] J.-L. Barrat and J.-F. Joanny. Theory of polyelectrolyte solu-tions. Advances in Chemical Physics, 94:1–65, 1996.

[CBD82] S. Candeau, J. Bastide, and M. Delsanti. Adv. Polym. Sci.,44:27, 1982.

[CJ96] F. Candau and J.-F. Joanny. Synthetic polyampholytes in aque-ous solutions. In Joseph C. Salamone, editor, Polymeric Mate-rials Encyclopedia. CRC Press, Boca Raton, 1996.

[CSC93] Jean-Marc Corpart, Joseph Selb, and F. Candau. Polymer,34:3873, 1993.

[Dao77] S. Daoudi. J. Phys. (France), 38:1301, 1977.

[dC94] J. des Cloizeaux. J. Phys. (France), 4:539, 1994.

41

42 BIBLIOGRAPHY

[dCJ87] J. des Cloizeaux and G. Jannink, editors. Les Polymeres enSolution. Les Editions de Physique, Les Ulis, 1987.

[DE76] R. T. Deam and S. F. Edwards. Phil. Trans. R. Soc. A, 280:317,1976.

[DE86] M. Doi and S. F. Edwards. The Theory of Polymer Dynamics.Claredon Press, Oxford, 1986.

[dG71] P. G. de Gennes. Reptation of a polymer chain in presence offixed obstacles. J. Chem. Phys., 55:572, 1971.

[dG72] P. G. de Gennes. Exponents for excluded volume problem asderived by wilson method. Phys. Lett. A, 38:339, 1972.

[dG79] P. G. de Gennes. Scaling Concepts in Polymer Physics. CornellUniversity Press, Ithaca und London, 1979.

[dGPVB76] P. G. de Gennes, P. Pincus, R. M. Velasco, and F. Brochard.Remarks on polyelectrolyte conformation. Journal de Physique(France), 37:1461–1473, 1976.

[DKOY91] D. Derosi, K. Kajiwara, Y. Osaka, and A. Yamauchi. PolymerGels. Plenum Press, New York, 1991.

[DR95] Andrey V. Dobrynin and Michael Rubinstein. J. de Physique II,5:677, 1995.

[DRO96] Andrey V. Dobrynin, Michael Rubinstein, and S. P. Obukhov.Macromolecules, 29:2974, 1996.

[DS65] J. A. Duiser and A. J. Stavermann. In J.A. Prins, editor, Physicsof non-crystalline solids, page 376, Amsterdam, 1965. North-Holland.

[Dus93] K. Dusek, editor. Responsive Gels: Volume Transistions I&II,volume 109 and 110 of Adv. Polym. Sci. Springer, Berlin, 1993.

[Edw65] S. F. Edwards. Statistical mechanics of polymers with excludedvolume. Proc. Phys. Soc., 85:613, 1965.

BIBLIOGRAPHY 43

[Edw67a] S. F. Edwards. Statistical mechanics of polymerized material.Proc. Phys. Soc., 92:9, 1967.

[Edw67b] S. F. Edwards. Statistical mechanics with topological constraints.i. Proc. Phys. Soc., 91:513, 1967.

[Edw68] S. F. Edwards. Statistical mechanics with topological constraints.ii. J.Phys. A, 1:15, 1968.

[EF78] B. Erman and P. J. Flory. J. Chem. Phys., 68:5363, 1978.

[Eic72] B. E. Eichinger. Macromolecules, 5:496, 1972.

[EK94] R. Everaers and K. Kremer. Comp. Phys. Comm., 81:19, 1994.

[EK95] R. Everaers and K. Kremer. Test of the foundations of classicalrubber elasticity. Macromolecules, 28:7291, 1995.

[EK96a] R. Everaers and K. Kremer. Elastic properties of polymer net-works. J. Mol. Mod., 2:293, 1996.

[EK96b] R. Everaers and K. Kremer. Topological interactions in modelpolymer networks. Phys. Rev. E, 53:R37, 1996.

[E.K99] S. E.Kudaibergenov. Recent advances in the study of syntheticpolyampholytes in solutions. Adv. Pol. Sci., 144:115–197, 1999.

[EKP80] S. F. Edwards, P. R. King, and P. Pincus. Ferroelectrics, 30:3,1980.

[Eve99] R. Everaers. Entanglement effects in defect-free model polymernetworks. New J. Phys., 1:12.1–12.54, 1999.

[FE82] P. J. Flory and B. Erman. Macromolecules, 15:800, 1982.

[Flo49] P. J. Flory. J. Chem. Phys., 17:303, 1949.

[Flo53] P. J. Flory. Principles of Polymer Chemistry. Cornell UniversityPress, Ithaca N.Y, 1953.

[Flo76] P. J. Flory. Proc. Royal Soc. London Ser. A., 351:351, 1976.

[Flo77] P. J. Flory. J. Chem. Phys., 66:5720, 1977.

44 BIBLIOGRAPHY

[GCM01] K. Ghosh, G. A. Carri, and M. Muthukumar. Configurationalproperties of a single semiflexible polyelectrolyte. Journal ofChemical Physics, 115:4367–4375, 2001.

[GCZ96] P. Goldbart, H. E. Castello, and A. Zippelius. Adv. Mod. Phys.,45:393, 1996.

[GK94] Alexander Yu. Grosberg and Alexei R. Khokhlov. SatisticalPhysics of Macromolecules. American Institute of Physics, NewYork, 1994.

[Gra75] W. W. Graessley. Macromolecules, 8:186 and 865, 1975.

[GS94] A. Gutin and E. Shakhnovich. Phys. Rev. E, 50:R3322, 1994.

[HB88] P. G. Higgs and R. C. Ball. J. Phys. (France), 49:1785, 1988.

[Hel75] E. Helfand. Theory of inhomogeneous polymers - fundamentalsof gaussian random-walk model. J. Chem. Phys., 62:999, 1975.

[Hil97] G. Hild. Interpretation of equilibrium swelling data on modelnetworks using affine and ‘phantom’ network models. Polymer,38:3279, 1997.

[HJ91] P. Higgs and J.-F. Joanny. J. Chem. Phys., 94:1543, 1991.

[HTG97] Th. Holzl, H. L. Trautenberg, and D Goritz. Phys. Rev. Let.,79:2293, 1997.

[Jam47] H. James. J. Chem. Phys., 15:651, 1947.

[JG47] H. James and E. Guth. J. Chem. Phys., 15:669, 1947.

[JPS95] B. Jonsson, C. Peterson, and B. Soderberg. Variational approachto the structure and thermodynamics of linear polyelectrolyteswith coulomb and screened coulomb interactions. Journal ofPhysical Chemistry, 99:1251–1266, 1995.

[K81] S. Kastner. Colloid Polym. Sci., 259:499 and 508, 1981.

[KG95] K. Kremer and G. S. Grest. In K. Binder, editor, Monte Carloand Molecular Dynamics Simulations in Polymer Science. Ox-ford University Press, New York and Oxford, 1995.

BIBLIOGRAPHY 45

[Kho80] A. Khokhlov. J. Phys. A, 13:979, 1980.

[KK82] A. R. Khokhlov and K. A. Khachaturian. On the theory ofweakly charged poly-electrolytes. Polymer, 23:1742–1750, 1982.

[KK91] Yacov Kantor and Mehran Kardar. Europhys. Lett., 14:421,1991.

[KK94] Yacov Kantor and Mehran Kardar. Europhys. Lett., 27:643,1994.

[KK95a] Yacov Kantor and Mehran Kardar. Phys. Rev. E, 51:1299, 1995.

[KK95b] Yacov Kantor and Mehran Kardar. Phys. Rev. E, 52:835, 1995.

[KKK48] W. Kuhn, O. Kunzle, and A. Katchalsky. A. Helv. Chim. Acta,31:1994, 1948.

[KLK92] Yacov Kantor, Hao Li, and Mehran Kardar. Phys. Rev. Lett.,69:61, 1992.

[KME90] A. Kloczkowski, J. Mark, and B. Erman. Macromolecules,23:3481, 1990.

[KP49] O. Kratky and G. Porod. Rec. Trav. Chim., 1106:2211, 1949.

[LDBD99] A. Lyulin, B. Dunweg, O. Borisov, and A. A. Darinskii. Com-puter simulation studies of a single polyelectrolyte chain in poorsolvent. Macromolecules, 32:3264–3278, 1999.

[Lei80] L. Leibler. Theory of microphase separation in block co-polymers. Macromolecules, 13:1602, 1980.

[LHK02] H.J. Limbach, C. Holm, and K. Kremer. Structure of polyelec-trolytes in poor solvent. cond-mat, page 0206274, 2002.

[L.M59] L.Mullins. J. appl. Polymer Sci., 2:1, 1959.

[LM02] T. B. Liverpool and A. C. Maggs. Dynamic scattering fromsemiflexible polymers. Macromolecules, 34:6064–6073, 2002.

[LO98] N. Lee and S. Obukhov. Multiple conformations in polyam-pholytes. Europ. Phys. J. B, 1:371–376, 1998.

46 BIBLIOGRAPHY

[MPV87] M. Mezard, G. Parisi, and M. V. Virasoro. Spinglas Theory andBeyond. World Scientific, Singapore, 1987.

[Mut87] M. Muthukumar. Adsorption of a polyelectrolyte chain to acharged surface. Journal of Chemical Physics, 86:7230–7235,1987.

[Mut96] M. Muthukumar. Double screening in polyelectrolyte solutions:Limiting laws and crossover formulas. Journal of ChemicalPhysics, 105:5183–5199, 1996.

[MW56] C. G. Moore and W. F. Watson. J. Polymer Sci., 19:237, 1956.

[OCMC96] A. Ohlemacher, F. Candau, J. P. Munch, and S. J. Candau. J.Polym. Sci. Phys. Ed., 34:2747, 1996.

[Odi77] T. Odijk. Polyelectrolytes near the rod limit. Journal of PolymerScience: Polymer Physics Edition, 15:477–483, 1977.

[ORC94] S. P. Obukhov, M. Rubinstein, and R. H. Colby. Network mod-ulus and superelasticity. Macromolecules, 27:3191, 1994.

[Pan90] S. Panyukov. Scaling theory of high elasticity. Sov. Phys. JETP,71:372, 1990.

[PMCG92] S. K. Patel, S. Malone, C. Cohen, and J. R. Gillmor. Macro-molecules, 25:5241, 1992.

[PR96] S. Panyukov and Y. Rabin. Phys. Rep., 269:1, 1996.

[RA75] G. Ronca and G. Allegra. J. Chem. Phys., 63:4990, 1975.

[RFF+90] D. Richter, B. Farago, L. J. Fetters, J. S. Huang, B. Ewen, andC. Lartigue. Phys. Rev. Let., 64:1389, 1990.

[Rol96] D. Rolfsen. Knots and Links. Publish or Perish Inc., Berkley,1996.

[Rou53] P. E. Rouse. J. Chem. Phys., 21:1273, 1953.

[RWZ+93] D. Richter, L. Willner, A. Zirkel, B. Farago, L. J. Fetters, andJ. S. Huang. Phys. Rev. Let., 71:4158, 1993.

BIBLIOGRAPHY 47

[SC94] K. S. Schweizer and J. G. Curro. Adv. Polym. Sci., 116:319,1994.

[Sch98] F. Schmid. Self-consistent-field theories for complex fluids. J.Phys. - Cond. Mat., 10:8105, 1998.

[SF77] J. Skolnick and M. Fixman. Electrostatic persistence length ofa wornlike polyelectrolyte. Macromolecules, 10:944–948, 1977.

[SK96] M. J. Stevens and K. Kremer. Structure of salt-free linearpolyelectrolytes in the debye-huckel approximation. Journal dePhysique II, 6:1607, 1996.

[SMC+94] M. Skouri, J. P. Munch, S. J. Candau, S. Neyret, and F. Candau.Macromolecules, 27:69, 1994.

[SUPH+95] E. Straube, V. Urban, W. Pyckhout-Hintzen, D. Richter, andC. J. Glinka. Small-angle neutron-scattering investigation oftopological constraints an tube deformation in networks. Phys.Rev. Let., 74:4464, 1995.

[THB73] T. Tanaka, L.O. Hocker, and G.B. Benedek. J. Chem. Phys.,59:5151, 1973.

[Tre75] L. R. G. Treloar. The Physics of Rubber Elasticity. ClarendonPress, Oxford, 1975.

[UJP+97] M. Ullner, B. Jonsson, C. Peterson, O. Sommelius, andB. Soderberg. The electrostatic persistence length calculatedfrom monte carlo, variational and perturbation methods. Jour-nal of Chemical Physics, 107:1279–1287, 1997.

[Ull79] R. Ullman. J. Chem. Phys., 71:436, 1979.

[VE93] T. A. Vilgis and B. Erman. Macromolecules, 26:6657, 1993.

[VI93] J. M. Victor and J. B. Imbert. Europhys. Lett., 24:189–195, 1993.

[Vil96] T. A. Vilgis. Polymer theory: Path integrals and scaling. Macro-molecules, 29:6165–6174, 1996.

48 BIBLIOGRAPHY

[WE78] M. Warner and S. F. Edwards. Neutron-scattering from strainedpolymer networks. J.Phys. A, 11:1649, 1978.

[Wie86] F. W. Wiegel. World Scientific, Philadelphia, 1986.

[WJJ93] J. Wittmer, A. Johner, and J.-F. Joanny. Europhys. Lett.,24:263, 1993.

Part II

Publications

49

Appendix A

The electrostatic persistencelength of polymers beyond theOSF limit

51

DOI 10.1140/epje/i2002-10007-3

Eur. Phys. J. E 8, 3–14 (2002) THE EUROPEANPHYSICAL JOURNAL Ec©

EDP SciencesSocieta Italiana di FisicaSpringer-Verlag 2002

The electrostatic persistence length of polymers beyond theOSF limit

R. Everaers1, A. Milchev1,2, and V. Yamakov1,2,a

1 Max-Planck-Institut fur Polymerforschung, Postfach 3148, D-55021 Mainz, Germany2 Institute for Physical Chemistry, Bulgarian Academy of Sciences, G. Bonchev Street, Block 11, 1113 Sofia, Bulgaria

Received 12 February 2002

Abstract. We use large-scale Monte Carlo simulations to test scaling theories for the electrostatic persis-tence length le of isolated, uniformly charged polymers with Debye-Huckel intrachain interactions in thelimit where the screening length κ−1 exceeds the intrinsic persistence length of the chains. Our simulationscover a significantly larger part of the parameter space than previous studies. We observe no significantdeviations from the prediction le ∝ κ−2 by Khokhlov and Khachaturian which is based on applying theOdijk-Skolnick-Fixman theories of electrostatic bending rigidity and electrostatically excluded volume tothe stretched de Gennes-Pincus-Velasco-Brochard polyelectrolyte blob chain. A linear or sublinear depen-dence of the persistence length on the screening length can be ruled out. We show that previous resultspointing into this direction are due to a combination of excluded-volume and finite chain length effects.The paper emphasizes the role of scaling arguments in the development of useful representations for ex-perimental and simulation data.

PACS. 64.60.-i General studies of phase transitions – 36.20.-r Macromolecules and polymer molecules –87.15.By Structure and bonding

1 Introduction

The theoretical understanding of macromolecules carry-ing ionizable groups is far from complete [1,2]. In spiteof the long range of the interactions, the systems areoften discussed using analogies to neutral polymers. Aprominent example is the concept of an electrostaticpersistence length, which was introduced more than 20years ago by Odijk [3] and by Skolnick and Fixman [4](OSF). They considered a semi-flexible polymer or worm-like chain (WLC) with intrinsic persistence length l0and Debye-Huckel (DH) screened electrostatic interactionsUDH/kBT = (q2lB/r) exp(−κr) between charges q/e. TheBjerrum length lB characterizes the strength of the elec-trostatic interactions and is defined as the distance wherethe Coulomb energy of two unit charges e is equal to kBT .Ionizable groups are assumed to be spaced at regular in-tervals A along the chain. For experimental systems, Ais controlled by the chemistry, but the fraction f ≤ 1of ionized charged groups needs to be determined inde-pendently. Setting f = A/lB is a rough way of account-ing for Manning condensation [1] of counter-ions in caseswhere lB > A. Due to the presence of mobile ions the bareCoulomb interaction is cut off beyond the screening length

a Present address: Argonne National Laboratory, MaterialsScience Division, Build. 212, 9700 S. Cass Avenue, Argonne,IL-60439, USA. e-mail: [email protected]

κ−1. OSF were interested in bending fluctuations and con-sidered the resulting increase of the electrostatic energyrelative to the straight ground state. In the (“OSF”) limitκ−1 � l0, where the screening length is smaller than theintrinsic persistence length of the chain and to lowest or-der in the local curvature, the Debye-Huckel interactionmakes an additive contribution to the bending rigidity. Asa consequence, a WLC with DH interactions (DHWLC)behaves in this limit on large length scales like an ordinaryWLC with renormalized persistence length:

lp = l0 + lOSF, (1)

lOSF =q2f2lB4A2κ2

. (2)

Ever since, there has been a lively debate on how to ex-tend the theory to parameter ranges beyond the OSF limitκ−1 � l0. Barrat and Joanny (BJ) [5] have shown that theoriginal OSF derivation breaks down, if the chains startto bend significantly on length scales comparable to thescreening length. As a consequence, equation (2) cannotsimply remain valid beyond the OSF limit as was some-times speculated [6]. Two main scenarios, which we denoteby “OSFKK” and “KMBJ” after the initials of the mainauthors, have been discussed in the literature:

– OSFKK. According to Khokhlov and Khachaturian(KK) [7] the OSF theory can be applied to a “stretched

4 The European Physical Journal E

chain of polyelectrolyte blobs”, a concept introducedby de Gennes et al. [8] to describe the behavior ofweakly charged flexible polyelectrolytes in the absenceof screening. The persistence length of the blob chain isthen calculated from equation (2) using suitably renor-malized parameters.

– KMBJ. Refinements by Muthukumar [9–11] of the orig-inal theory of Katchalsky [12] treat electrostatic in-teractions in strict analogy to short-range excluded-volume interactions. Quite interestingly, the resultsare consistent with the scaling picture of de Genneset al. [8] in the two limits of strong and vanishingscreening. Moreover, they are supported by recent cal-culations by BJ and others [5,13] who determined thepersistence length of the blob chain in a variationalprocedure and found le ∼ κ−1.

In addition, there is a number of recent theories whichfall into neither of the two classes outlined above [14–19].While there is a growing consensus among theoreticiansthat the OSFKK result is asymptotically correct [20–22],experiments [23–27] as well as computer simulations [28–34] have consistently provided evidence for a compara-tively weak κ-dependence of the electrostatic persistencelength. However, it should be noted that researchers havesuspected early on that these results might be related tothe employed definitions of the persistence length and thedifficulties in accounting for excluded-volume interactionsin the data analysis [22–24,28,33,34].

The purpose of the present paper is to shed some newlight on this problem by combining a scaling analysis withlarge-scale Monte Carlo simulations. We re-examine theKMBJ and the OSFKK theory in order to extract guid-ance for the data analysis and the choice of simulationparameters. In particular, we take into account a secondset of results [6,35], also due to OSF, which treat the ef-fects of electrostatically excluded volume. As a result, weare able i) to rule out the KMBJ theory, ii) to providebenchmark results for analytical solutions of the DHWLCmodel as well as iii) for a comparison to experiments inorder to clarify if the systems under consideration are ac-tually described by the DHWLC model or if additionaleffects such as solvent quality or counter-ion condensationneed to be taken into account as well.

The paper is organized as follows: in Section 2 we re-view the predictions of the KMBJ and OSFKK theories,followed by a discussion in Section 3 of how experimentsand simulations should be set up and analyzed in order todiscriminate between the two scaling pictures. Details ofour Monte Carlo simulations can be found in Section 4.We present our results in Section 5, compare to previousexperimental and numerical results in Section 6 and closewith a brief discussion.

2 Scaling theories of intrinsically flexibleDebye-Huckel chains

The DHWLC is characterized by the following set of pa-rameters: q, lB, κ, A, f , l0, and Ltot, where Ltot denotes

the total chain length. Correlation functions can be cal-culated for chain segments with arbitrary contour lengthL < Ltot. We focus on the non-OSF limit lB, A, l0 �κ−1 � √〈r2〉 (Ltot), where the screening length is largerthan all microscopic length scales of the polymer modelbut smaller than the size of the entire chain.

For an understanding of the physics, some of theselength scales and parameters are less relevant than others.For example, the actual distribution of the charges onthe chain should be unimportant as long as A � κ−1.In our simulations we use discrete charges spaced by adistance equal to the intrinsic persistence length, whilethe scaling arguments assume a continuous charge distri-bution. Similarly, in the non-OSF limit with l0 � κ−1

the physics should not depend on the details of the WLCcrossover from rigid rod to random coil behavior at L ≈ l0.In our simulations we therefore use freely jointed chains(FJC) whose (Kuhn) bond length b corresponds (up toa henceforth neglected factor of two) to the persistencelength of a WLC. Finally, it is convenient to consider thelimit of infinite total chain length in order to eliminate theLtot-dependence. Again the practical limitation to Ntot =Ltot/b = 4096 segments should be unimportant, since ourchains always fulfill the condition κ−1 �√〈r2〉 (Ltot).

The remaining independent parameters (the linecharge density fq/A, lB, κ, l0 and L) can be reduced fur-ther using the notion of a “polyelectrolyte blob” which wasintroduced by de Gennes et al. [8] to describe the crossoverfrom locally unchanged chain statistics to stretching onlong length scales.

Consider first weakly charged flexible polyelectrolytes,where the electrostatic interactions are irrelevant onthe length scales comparable to the intrinsic persis-tence length l0. On larger length scales an undis-turbed WLC with a contour length L has a spatial ex-tension

⟨r2⟩

= 2Ll0. Neglecting prefactors, the electro-static energy of such a chain is given by Ue/kBT �q2(fL/A)2lB/

√〈r2〉. Electrostatic interactions becomerelevant for Ue/kBT ≥ 1 or chain lengths L exceeding

lg = l1/30

(A2

f2q2lB

)2/3

(3)

and whose spatial extension is given by

ξ = l2/30

(A2

f2q2lB

)1/3

. (4)

However, this derivation breaks down for strongly inter-acting systems where the contour length per blob becomessmaller than the intrinsic persistence length. In this case,a similar argument can be made for a WLC with L < l0and

⟨r2⟩

= L2 yielding

lg = ξ =A2

q2f2lB. (5)

Both definitions match for lg = l0, hiding a subtlecrossover [21] behind a crudely renormalized system of

R. Everaers et al.: The electrostatic persistence length of polymers beyond the OSF limit 5

100

102

104GPVB BJ

F

10−2

100

102

10410−2

100

102KK1

KK2

KK3

F

KMBJ

OSFKK

<r2>/ξ2 = X

<r2>/ξ2 = X

X2

X2

Y2 X

Y4/5 X6/5

Y

X=N/g

Y=(

κξ)-1

Y

Fig. 1. Spatial extension of DHWLC beyond the OSF limitas a function of reduced chain length and reduced screeninglength. Electrostatic interactions are relevant in the upper rightpart of the map which is limited by solid black lines. Thedashed and dotted lines correspond to crossover lines in theOSFKK and KMBJ scaling theories, respectively. We have alsoincluded the scaling predictions equations (8) to (13) for thechain radii. For details see the main text.

units. Throughout the paper all quantities will be ex-pressed using these natural units of contour length andspatial distance. On a scaling level, they become a func-tion of only two dimensionless parameters: the reducedchain or segment length

X = L/lg, (6)

and the reduced screening length

Y = (κξ)−1 . (7)

In the model under consideration the electrostatic in-teractions are purely repulsive. Therefore, the chains arealways extended relative to the neutral case. For κlB � 1the details of the process are quite involved and dif-ficult to treat from first principles. The approximationschemes used are often based on mechanical analogies suchas stretching, bending and swelling due to short-rangeexcluded-volume interactions and have been reviewed inreferences [1,2]. We have summarized the various predic-tions in terms of a schematic map of the XY parameterspace (Fig. 1). Before we discuss the controversial parts,we first present those aspects which seem well understood:

– The Debye-Huckel interaction is irrelevant inside theelectrostatic blob, i.e. for weakly charged chains

⟨r2⟩/ξ2 � X for X < 1. (8)

For strongly charged chains, this regime does not exist.

– In the absence of screening, when the monomers inter-act via an infinite-range Coulomb potential, the chainsare stretched into a “blob pole”:⟨r2⟩/ξ2 � X2 for 1 < X < ∞ and Y →∞. (9)

In Figure 1 we have marked the line dividing these tworegimes as “GPVB” after the initials of the authors ofreference [8] where the notion of the electrostatic blobwas introduced.

– For sufficiently long chains, the Debye-Huckel interac-tion becomes effectively short-ranged, leading to self-avoiding walk (SAW) behavior:⟨

r2⟩/ξ2 ∼ X2ν for 0 < Y < ∞ and X →∞, (10)

where ν ≈ 3/5 is the usual Flory exponent.– For strong screening with q2lB < κ−1 < A the Debye-

Huckel interaction reduces to an ordinary excluded-volume potential with a second virial coefficientv � q2lBκ−2 between charges. Using a conventionalFlory argument to balance the two-body repulsionv(fL/A)2/R3

F with the entropic elasticity of a Gaus-sian chain R2

F/(Ll0), one obtains

⟨r2⟩/ξ2 �

{X for X < Y −4,Y 4/5X6/5 for X > Y −4, for Y � 1.

(11)In Figure 1 the corresponding line, beyond which theshort-range excluded-volume interaction becomes rel-evant, is marked as “F”.

The controversial parts of the phase diagram con-cern the crossover from the blob pole to the self-avoidingwalk regime. The problem is often treated in analogyto a simple WLC. With an Onsager virial coefficientv � l2pd between rigid segments of length lp and diam-eter d, excluded-volume effects become relevant beyonda “Flory length” lF = l3p/d2 leading to a Flory radius of

R2F � d2/5l

8/5p (L/lp)6/5. In the case of the blob chain, the

diameter of the electrostatically excluded volume is givenby d � κ−1 [6,35]. However, there is disagreement with re-spect to the κ-dependence of the electrostatic persistencelength le.

– Variational approaches such as the theory of Barratand Joanny (BJ) often predict le � d � κ−1. Asa consequence, lF = lp = κ−1, so that there is adirect crossover from the stiff blob chain to a SAWregime when the contour length ξ N

g of the blob chainreaches the screening length κ−1. Using dimensionlessunits this corresponds to X = N/g = lF/ξ = lp/ξ =1/(κξ) = Y (the dotted line in Fig. 1 marked “BJ”).The result⟨

r2⟩/ξ2 � Y 4/5X6/5 for X > Y and Y � 1 (12)

is identical to equation (11). On a scaling level, thepredictions of the BJ theory coincide with those ofthe excluded-volume theories of Katchalsky [12] andMuthukumar [9–11].


– Most theories favour the relation le � ξ−1κ−2 firstobtained by Khokhlov and Khachaturian (KK). KKargued that the OSF result equation (2) should alsoapply to a stretched chain of blobs with line chargedensity fqlg/(Aξ) so that le/ξ = 1/(κξ)2 = Y 2. Sincele � d, the resulting phase diagram is considerablymore complicated. The blob chain starts to bend forreduced segment lengths X exceeding lp/ξ = le/ξ =Y 2 (the “KK1” line in Fig. 1), while excluded-volumeeffects become relevant beyond lF/ξ = 1/(κξ)4 = Y 4

(“KK2”):

⟨r2⟩/ξ2 �

{Y 2X for Y 2 < X < Y 4,Y 6/5X6/5 for X > Y 4,

for Y �1.

(13)Finally, and in contrast to the KMBJ theory, theOSFKK approach implies another crossover (“KK3”)within the SAW regime at Y = 1 from equation (13)to equation (11).

3 Implications for data production andanalysis

In general, scaling theories make two kinds of predictions:i) about the existence of characteristic length scales orcrossover lines in conformation space and ii) about theasymptotic behavior of observables in the areas betweenthese crossover lines. In principle, attempts at refutationcan aim at either type of prediction. However, in thepresent case the identification of asymptotic exponentsturns out to be particularly difficult. Apart from numer-ical prefactors and logarithmic corrections 1, one is facedwith four problems:

– Although the various regimes predicted by theKMBJ and the OSFKK theory are characterized bydifferent combinations of powers of X and Y , the expo-nents are often similar and the absolute differences be-tween the predicted chain extensions relatively small.

– At least on a scaling level all crossover lines meet atX = Y = 1 for chain and screening length of the order

1 For example, in the absence of screening the long-rangeCoulomb interactions along the blob pole create a tensionwhich grows logarithmically with the chain length N . As aconsequence, the blob diameter is reduced and the chains growwith Rg ∼ N log1/3(N). In particular, an N monomer segmentof a longer chain will always be more extended than an Nmonomer chain whereby the deformation is strongest for seg-ments located near the center of the longer polymer [8,36]. Inthe presence of screening, this effect leads, on the one hand,to an increase of the contour length of the blob chain. Onthe other hand, the correspondingly reduced line charge den-sity results in a reduction of the OSFKK persistence length.Similarly, but neglecting the stretching, there are logarithmiccorrections to the electrostatically excluded volume around theblob chain [35]. Moreover, although fairly robust, the Flory ar-gument used to estimate the excluded-volume effects is far fromexact. In addition to the aforementioned crossovers, a completetheory will have to account for all of these effects.

of the diameter of the polyelectrolyte blob. In the caseof the OSFKK theory some of the predicted regimesare extremely narrow in the sense that chain lengthsof X = 104 blobs are required for a width of one orderof magnitude in Y -direction. This validity range wouldseem to be the absolute minimum for identifying powerlaw behavior.

– The crossover lines are neither parallel to each othernor to the “natural” X and Y directions of variationsof chain and screening length, respectively.

– Results will be influenced by the finite total lengthLtot of polyelectrolyte chains studied in experimentsor simulations. The importance of these effects varieswith the ratio of the screening length Y and the con-tour length X of the blob chain. As a consequence,they risk to mask the asymptotic Y -dependence of ob-servables such as the electrostatic persistence length,if they are evaluated for chains with fixed Ltot.

Quite obviously, the discrimination between the two scal-ing pictures requires the investigation of chains whose ef-fective length X is as large as possible. In addition, oneshould rely on those observables and data representationswhich are most sensitive to the differences between thetheories and least sensitive to the omitted constants, cor-rections and crossovers. In the following we discuss theanalysis of data for internal distances and for the tangentcorrelation function. In particular, we will argue that it isrelatively easy to discard the KMBJ picture using simplescaling plots, while the verification of some of the predic-tions by the OSFKK theory requires astronomical chainlengths.

Consider first the scaling predictions equations (8) to(13) for the mean-square internal distances at reducedscreening lengths Y > 1. Describing the GPVB crossoverto the blob pole within the chain-under-tension model [8,29], equations (8) and (9) can be combined as

⟨r2⟩/ξ2 =

X + X2. Taking this into account, the KMBJ theory sug-gests that all data points should collapse when plotted inthe following manner as a function of the KMBJ persis-tence length:

〈r2(Z = X/Y )〉/ξ2 −X

X2�{

1 for Z < 1,Z−4/5 for Z > 1. (14)

In contrast, the OSFKK theory predicts data collapse, ifthe segment lengths are rescaled with the OSFKK persis-tence length Y 2, and a breakdown of scaling for segmentlengths approaching the Flory length Y 4:

〈r2(Z =X/Y 2)〉/ξ2−X

X2�

1 for Z < 1,Z−1 for 1<Z <Y 2,Y −2/5Z−4/5 for Y 2 < Z.

(15)The predictions of the two scaling theories differ moststrongly for chain radii along the Y = X1/2 KK1 line forsegment lengths equal to the OSFKK persistence length.In the KMBJ theory, this line is already deep in the SAWregime. Equations (12) and (13) imply

〈r2(X,Y = X1/2)〉Xξ2

�{

X3/5 (KMBJ),X (OSFKK). (16)


Thus the ratio⟨r2⟩OSFKK

/⟨r2⟩KMBJ

= X2/5 is fairlysmall and X = 105/2 blobs are required for this ratio tobecome of order 10. For comparison, both theories predictfull extension along the KMBJ line:

〈r2(X,Y = X)〉Xξ2

�{

X (KMBJ),X (OSFKK), (17)

and differ only by a factor of⟨r2⟩OSFKK

/⟨r2⟩KMBJ

=X1/10 along the KK2 line:

〈r2(X,Y = X1/4)〉Xξ2

�{

X2/5 (KMBJ),X1/2 (OSFKK).

(18)

While segment lengths of the order of X = 103

blobs are thus sufficient to discriminate between theKMBJ and the OSFKK proposals for the electrostaticpersistence length, the requirements for resolving the ad-ditional crossovers predicted by the OSFKK theory aremuch higher. Consider again the KK2 line where excluded-volume effects are expected to become relevant for the un-dulating blob chain. Equation (13) can be rewritten in theform

〈r2(Z = Y −4/5X1/5)〉Y 2Xξ2

�{

1 for Z < 1,Z for Z ≥ 1, (19)

where Z is again a suitable variable measuring the effec-tive distance from the crossover line at Z = 1. In order toidentify the asymptotic behavior, one needs at least datacovering the interval Z ∈ [0.1, 10]. Since the validity rangeof equation (19) is limited by the KK1 and KK3 lines (sothat 1 < Y < X1/2 or X1/5 > Z > X−1/5), this impliesa minimum segment length of X = 105 blobs for estab-lishing the KK2 crossover. Similarly, the KK3 crossoverbetween equations (11) and (13) at Y = 1 becomes rele-vant for chains of X = 1010 blobs!

The mean-square internal distances and the tangentcorrelation function (TCF) obey a Green-Kubo–like rela-tion:

〈bN · b0〉 =12

d2

dN2〈r2(N)〉 =

12

ξ2

g2

d2

dX2

〈r2(X)〉ξ2

. (20)

For a WLC the TCF is simply given by

〈b(s) · b(0)〉 = b2 exp(−s/lp), (21)

so that the persistence length can be read off directlyfrom a semi-logarithmic plot. Numerical studies of poly-electrolytes [30,33,34] have therefore often focused on thisquantity in spite of two intrinsic problems: i) the TCF isconsiderably more difficult to measure with the same rel-ative precision than internal distances and ii) the TCFis particularly sensitive to finite chain length effects (acharacteristic sign is the faster than exponential decay ofthe TCF on length scale approaching the chain length).In contrast to the case of ordinary SAWs [37], nothing isknown about the functional form of the corrections. In thefollowing discussion we will focus on a third aspect: the

sensitivity of the TCF to the neighborhood of crossoverlines.

On a scaling level, the behavior of the TCF can beobtained by applying equation (20) to equations (8) to(13). For Y > 1 the KMBJ theory predicts

g2

ξ2〈b(Z = X/Y ) · b(0)〉 �

{1 for Z < 1,Z−4/5 for Z > 1. (22)

Within the OSFKK theory, simple predictions can onlybe made for segment lengths below the persistence lengthand beyond the Flory length:

g2

ξ2〈b(Z = X/Y 2) · b(0)〉 �

{1 for Z < 1,Y −2/5Z−4/5 for Y 2 < Z.

(23)However, since both theories are based on the analogy to amechanical WLC, they are often associated with the muchmore detailed prediction

g2

ξ2〈b(X) · b(0)〉 � (24){

exp(−X/Y ) for X < Y (KMBJ),exp(−X/Y 2) for X < Y 2 (OSFKK),

for the functional form of the decay of the tangent corre-lations. Measuring this quantity for DHWLC, therefore,seems to be the most direct way of justifying or refutingthis analogy and its exploitation. In particular, numer-ical work [30,33,34] has concentrated on i) establishingthe existence of a single exponential decay of the TCFover a certain range of length scales and ii) extractingthe κ-dependence of the measured decay length. In thefollowing we will re-examine this approach by taking acloser look at equations (22) and (23), since they containadditional crossovers neglected in equation (24).

The situation should be uncritical for the GPVBcrossover where the chain-under-tension models [8,29]suggests that equation (24) remains valid for X < 1. Incontrast, nothing is known in detail about the way theTCF crosses over to the slow power law decay character-istic for the SAW behavior on large length scales. How-ever, matching equation (24) (which only accounts for thelocal bending rigidity) with the asymptotic behavior inequations (22) and (23) shows that the tangent correla-tion function is much more sensitive to excluded-volumeeffects than the chain radii. This is most obvious for theOSFKK theory where the two limits match close to theOSFKK persistence length X = Y 2 instead of the Florylength X = Y 4. While the scaling of the TCF with theOSFKK persistence length should start to break downaround X/Y 2 ≈ 1, one can nevertheless expect equa-tion (24) to hold up to this point. In the case of theKMBJ theory the situation is quite different, since thepersistence length and the Flory length coincide. As a con-sequence, equation (24) effectively breaks down as soon asthe tangent correlation function starts to deviate from one.On the other hand, in the absence of other relevant lengthscales the TCF should scale with the KMBJ persistencelength for arbitrary segment length!


In our opinion, these arguments shed some doubts onattempts to identify the electrostatic persistence lengthwhich are based too closely on equation (24). Scaling plotstesting equations (22) and (23) may offer a simpler andsafer alternative.

4 Simulation model, method and parameters

As already mentioned in Section 2, we model the poly-mers as freely jointed chains (FJC) with unit chargesq = 1 at each joint. Lengths were measured in units ofthe bond length b. We varied the the Bjerrum lengthlB = 42, 1, 1/42, 1/162, 1/1002b and the screening lengthκ−1 = 1, 2, 4, 8, 16, 32, 64, 128 b. As in our previous studyon polyampholytes [38], the chains have a length of up toN = 4096 monomers.

– Since we study the conformations of isolated chains, weemploy the efficient technique of pivot rotations due toSokal et al. [39,32]. We use two types of pivot moves:Either we rotate the part between the free end of thechain and a randomly selected monomer around anaxis, defined by the bond between this monomer andits nearest neighbor; or we rotate a segment betweentwo randomly selected monomers around an axis join-ing them. The latter provides better efficiency in thecase of a stretched chain with large excess charge. OneMC step consists of N attempted rotations at ran-dom positions along the chain. Chain conformationsare stored at intervals of 8–32 MC steps. For each pa-rameter set we simulate 8 independent Markov chainsin parallel. We typically store 8 × 60 conformationsrepresenting a total of 1.5 × 107 attempted rotationsfor our longest chains.

– Instead of the slower procedure of Stellman andGans [40] which corrects the accumulating numericerror in off-lattice implementations of the pivot algo-rithm, we regularly reconstruct the chains with thecorrect bond length.

– For calculating the long-range electrostatic interac-tions we use a direct summation whereby the energyof the system is obtained by direct counting of all thepair energies of the beads. This method is still efficientfor macromolecules of up to few thousand monomers.The DH potential is tabulated in two arrays for shortand long distances, respectively.

– For better efficiency starting configurations of thechains are created by means of the configurationalbiased [41] MC method although one should keep inmind that due to the long-range interactions the firstpart of the newly grown chain does not experience thecumulative field of the rest of the chain and a numberof rotational moves are still needed before the chainis well equilibrated. Measurements are performed andconformations stored only after the chain end-to-enddistances are well equilibrated.

– Since statistics is gathered both with respect to chainconformations as well as to different Bjerrum length

10−2

100

102

104

X=N/g

10−2

100

102

KK1

KK2

KK3

F

Y=(

κξ)-1

100

102

104 GPVB BJ

KK1

KK2

KK3

F

(a)

(b)

Fig. 2. Areas of the conformation diagram Figure 1 for whichthere are experimental and simulation data available. In (a) thegreen and blue lines indicate the parameter ranges investigatedin previous numerical studies: Reed and Reed [28] (dashed,blue), Barrat and Boyer [29] (solid, blue), Seidel [31] (solid,green), Jonsson, Ullner et al. [32–34] (dotted, green), Mickaand Kremer [30] (dashed, green). Experiments (shown in red)have access to longer chains, but the reduced screening lengthsY are typically smaller than ten: Reed et al. [24] (dashed, red),Beer et al. [27] (solid, red). The colored grids in (b) denote theparameter ranges covered by different sets of our MC simu-lations. Note that the predictions of the KMBJ and the OS-FKK theory differ most strongly along the KK1 line and areidentical outside of the the gray shaded area.

lB and screening length κ−1, we use a simple paral-lelization where different processors of a CrayT3E su-percomputer perform independent simulation of singlechains. The total CPU time used for this project is ofthe order of 1.5× 105 single processor hours.

The simulation parameters translate into our blob unitsas

g =(

b

lB

)2/3

(g > 1) (25)

ξ = b

(b

lB

)1/3

(26)

and

g =b

lB(g ≤ 1) (27)

ξ = bb

lB, (28)

where we now use the number g of monomers per blobinstead of the corresponding contour length lg = bg.


<10−1

5−1

2−1

1

2

5

>10

10−2

1 102

10410

−2

1

102

104 GPVB

X=N/g

10−2

1 102

104

GPVBBJ

KK1

KK2

KK3

F

X=N/g

BJ

1

102

104

X=N/g X=N/g

KK1

KK2

KK3

FF

Y=1/κξY=1/κξ

GPVB GPVB

ξ2 X

<r2(X,Y)>__________

κξ1__Y=

(a) (b)

(c) (d)

ξ2 X

<r2(X,Y)>__________

Fig. 3. Comparison of measured internal distances 〈r2(X, Y )〉 to the predictions of the KMBJ ((a) and (c)) and OSFKK ((b)and (d)) scaling theories. In the top row we show log-log-log representations of 〈r2(X, Y )〉/(ξ2X), where all distances arenormalized to the undisturbed random walk. The colored areas were generated by interpolation between the results of allsimulations for a given coupling strength. The supporting grids and the crossover lines show the two sets of scaling predictionsequations (8) to (13) as an extension of Figure 1 to three dimensions. In the plots of the bottom row the colors indicate theratios 〈r2(X, Y )〉/

⟨r2⟩OSFKK

and 〈r2(X, Y )〉/⟨r2⟩KMBJ

, respectively.

Figure 2 shows where our own data are locatedwithin the XY -conformation space. The effective chainand screening lengths studied cover a range of seven andfive orders of magnitude, respectively. The reduced mean-square internal distances vary over ten orders of magni-tude. Along the KK1 line our data extend on a logarith-mic scale about a factor of two further into the asymptoticregime than previous studies. While this allows us to dis-criminate between the KMBJ and the OSFKK predictionsfor the electrostatic persistence length, our chains are stilltoo short to resolve the different RW and SAW regimespredicted by the OSFKK theory.

Note that only by studying strongly stretched chainswe are able to push the effective chain length X close to105 and that our unified description of strongly and weaklycharged flexible polyelectrolytes needs to be confirmed bythe data analysis. To facilitate the comparison, all figuresmake use of the same color code to indicate data obtainedfor a particular coupling strength lB/b ranging from bluefor g = 100002/3 ≈ 460 over different shades of violet forg = 2562/3 ≈ 40 and g = 162/3 ≈ 6.4 to red for g = 1 andorange for g = 1/16. The first three systems can safely beregarded as Gaussian chains, while the last two are at andbeyond the crossover to the strong-stretching regime.

5 Results

In the data analysis we mainly concentrate on identify-ing the scaling behavior: i) Do different data sets overlapwhen rescaled according to our extension equation (5) ofthe definition of the electrostatic blob? ii) How do the re-sults of our simulations compare to the predictions of theKMBJ and OSFKK scaling theories? In terms of observ-ables we start by presenting data for internal distancesaveraged along our chains of total length N = 4096. Inthe second part, we discuss results for the tangent corre-lation function. While the TCF was also averaged alongthe chain, we only take into account distances up to halfthe chain length in order to reduce finite chain length ef-fects [37]. Except for the most weakly charged system,the chain extensions are much larger than the screeninglength, so that we do not expect finite chain length ef-fects to be very important. At the end, we briefly presentresults for shorter chain lengths.

Figures 3(a) and (b) are three-dimensional log-log-logplots giving an overview of all data. We show reducedinternal distances 〈r2(X,Y )〉/((N/g)ξ2) normalized to thesize of the undisturbed random walk as a function of thereduced chain and screening lengths equations (6) and (7),respectively. Results for different coupling constants are


10− 2

1 102

104

1

102

104

106

X

X

X

X3/5

X1/2

X2/5

BJ: Y=X

KK1: Y=X1/2

KK2: Y=X1/4

<r2(X,Y)> _________ξ2 N/g

Fig. 4. Extension of chain segments along the BJ, KK1, andKK2 crossover lines in comparison to the predictions of theOSFKK (solid line) and the KMBJ (dashed line) scaling the-ories (see Eqs. (17, 16), and (18); note that we have not usedadditional prefactors for this comparison). Different data setsare shifted by factors of

√1000.

combined into colored surfaces, while the supporting gridsshow the two sets of scaling predictions equations (8) to(13) as an extension of Figure 1 to three dimensions.

The complementary Figures 3(c) and (d) showthe ratios 〈r2(X,Y )〉/〈r2(X,Y )〉KMBJ and 〈r2(X,Y )〉/〈r2(X,Y )〉OSFKK of the interpolated simulation results tothe scaling predictions in a color coding where green, redand blue indicate agreement, under- and over-estimationby a factor of three or more, respectively. The advantageof this representation is the localization of deviations inour schematic map of the parameter space.

Qualitatively, the interpretation of Figure 3 seemsclear. There are neither indications for a failure of theblob scaling nor for significant deviations from the pre-dictions of the OSFKK theory. (We emphasize again thatwe have neglected all numerical prefactors and that equa-tions (8) to (13) treat crossovers in the crudest manner.)In particular, there is no evidence that the chains start tobend on length scales comparable to the screening lengthas predicted by KMBJ. In the relevant part of confor-mation space, the KMBJ theory systematically underes-timates the observed chain extensions.

Nevertheless, Figure 3 could be misleading, since therejection of the KMBJ theory is mainly based on datafalling into the strong-stretching regime, while the theoryis meant to apply to weakly stretched Gaussian chains.Thus so far our conclusions rest on the assumption thatthe extension equation (5) of the blob scaling to stronglycharged chains can be used to extrapolate the behavior ofweakly charged systems to segment lengths inaccessible bysimulation. How well this assumption is fulfilled is hard tojudge from Figure 3. Definite conclusions require a moredetailed analysis.

Figure 4 presents chain radii measured along the BJ,KK1 and KK2 crossover lines. The first point to note isthat in all three cases we observe almost perfect scalingof data obtained for different coupling constants. Equa-

10−410−2 100 102 10410−2

10−1

100

101

102

103

104

10−2 100 102 10410−2

10−1

100

101

102

103

104

10−410−2 100 102 104

−2 100 102 10410

<r2

(X,Y

)>

____

____

ξ2 X

2

<r 2(X

,Y)>

- N b

2 ______________

ξ2 X

2

<r2

(X,Y

)>

____

____

ξ2 X

2

<r 2(X

,Y)>

- N b

2 ______________

ξ2 X

2

X/Y2 X/Y2

X/Y X/Y

(a) (b)

(c) (d)

Fig. 5. Crossover scaling for internal distances versus segmentlength. In the top (bottom) row segment lengths X are nor-malized to the KMBJ (OSFKK) persistence length Y (Y 2),respectively. Figures (a) and (c) on the left-hand side showinternal distances 〈r2(X, Y )〉 normalized to the mean-squareextension ξ2X2 of the blob pole. The grid is the same as inFigure 2(b) and shows (const−X) and (const− Y ) lines. Fig-ures (b) and (d) on the right-hand side are inspired by thechain-under-tension model for Gaussian chains equations (14)and (15). Only (const−X) lines are shown. Data points fallinginto the range 1 < Y < X < Y 2 are marked using the colorcode indicating the coupling strength. Results for different cou-pling constants are shifted by factors of ten.

tions (16) to (18) predict simple crossovers at the blob sizearound X = 1. In agreement with both scaling pictures,we observe stretched blob chains along the BJ line. Themost important set of data are the radii measured alongthe KK1 line for chains with a contour length X = Y 2

equal to the OSFKK persistence length. In agreement withthe OSFKK theory, we find a simple crossover aroundX = 1 to stretched blob chains. Contrary to the predic-tions of the KMBJ theory, the radii are essentially identi-


103

102

101

100

10-1

10-2

10-2 100 102 10-4 10-2 100 102 0 1 2

(a) (b) (c)

X/Y2X/Y X/Y2

g2 / ξ2

< b

(X)

. b(0

) >

→→

Fig. 6. Scaled tangent correlation functions. Results for differ-ent coupling constants are shifted by factors of ten. The coloredlines mark the results of our fits to a simple exponential decayin the range Y < X < Y 2. (a) log-log plot using KMBJ scaling,(b) log-log plot using OSFKK scaling, (c) semi-log plot usingOSFKK scaling.

cal to those observed along the BJ line and do not showSAW behavior. In particular, the asymptotic slope pre-dicted by the OSFKK theory is already observable forweakly charged chains to which the KMBJ theory can beapplied directly. The last set of data is taken along theKK2 line which marks the onset of excluded-volume ef-fects in the OSFKK theory. Here our results are consistentwith the predictions of both theories. This observation isin agreement with the estimate of a minimum segmentlength of X = 1010 blobs for the difference to becomerelevant (see Eq. (18)).

The screening length dependence of the effective bend-ing rigidity of the blob chain can also be determined fromcrossover scaling of internal distances normalized to thesize of the stretched blob pole (Fig. 5). In contrast to Fig-ures 3 and 4, this representation is independent of theprefactors neglected in deriving the chain extensions andthe location of the crossover lines. It therefore providesan efficient way of eliminating the possibility of an elec-trostatic persistence length scaling like κ−1 but with anunusually large prefactor.

The disadvantage of plotting 〈r2(X,Y )〉/(ξ2X2) di-rectly (Figs. 5 (a) and (c)) is the occurence of a 1/X di-vergence of results for segment lengths smaller than theblob size. Correcting for this in the manner suggested bythe chain-under-tension model (Eqs. (14) and (15)) as inFigures 5 (b) and (d) largely eliminates effects due to theGPVB crossover, but introduces some artifacts for smallN , where 〈r2(N = 1)〉 − b2 ≡ 0 for a FJC. In agreementwith our previous results and independently of the cou-pling constant, we observe extremely poor scaling when

the data are plotted as a function of the ratio X/Y of chainlength over KMBJ persistence length. In constrast, thedata superimpose considerably better, if the OSFKK scal-ing is used as in the corresponding Figures 5(c) and (d).

Similar conclusions can be drawn from an analysis ofthe tangent correlation function (Fig. 6). In fact, our dis-cussion in Section 3 shows that Figures 5 (b), (d) and thelog-log plots in Figures 6 (a), (b) are directly comparable.Figure 6 (c) shows the same data in the semi-logarithmicrepresentation commonly used to identify a simple ex-ponential decay of the correlation function. Clearly, theOSFKK scaling does not work perfectly up to the OS-FKK persistence length, but, at least qualitatively, weobserve the expected slow-down of the decay of the corre-lations.

6 Comparison to previous experimental andnumerical results

The discrepancy between our conclusion, le ∼ κ−y withy = 2, and the results of previous numerical and experi-mental investigations, y � 2, can be traced back to thedefinition of the electrostatic persistence length, le. Weprefer an indirect approach, where simulation results forobservables such as intrachain distances or TCFs are com-pared to the predictions of the OSFKK and KMBJ scal-ing theories over as large a parameter range as possible(Fig. 3). More direct methods proceed by i) defining anobservable apparent electrostatic persistence length le,app,ii) calculating le,app for numerical or experimental data,iii) plotting le,app as a function of κ−1, and (tentatively)iv) extracting effective values for y.

According to the theories sketched in Section 2 and inagreement with our simulation results, intrinsically flex-bile polyelectrolytes show a rich behavior due to an intri-cate interplay of effective stretching, bending and swellinginteractions. Attempts to measure le aim at isolating thebending contribution. Unfortunately, this is not straight-forward. Although the excluded-volume interactions canbe estimated for a given bending stiffness, it is difficultto invert this process. In fact, the definition of a properfunctional le(Y ) = F [

⟨r2⟩(X,Y )] may require as much

theoretical understanding as the solution of the originalproblem. Current definitions of le,app (for a recent reviewsee Ullner and Woodward [34]) are invariably inspired bythe analogy to a simple WLC in the absence of excluded-volume interactions. As a consequence, they can only beexpected to agree with our approach in cases where the(effective) Hamiltonian describing the system contains noother interactions. Unfortunately, this situation is ratheran exception than the rule. According to OSFKK the-ory, the bending energy only dominates on length scalesbetween the persistence and the Flory length. WithinKMBJ theory the corresponding regime does not evenexist (see Fig. 1). The following examples illustrate ourimpression that focusing on le,app may sometimes createmore confusion than insight.

Doubts about the general validity of the OSF re-sult were originally raised in experimental papers [24,27],


100 102 104 106

100

102

KK1

KK2

KK3

100

102

GPVB BJKK1

KK2

KK3

X=N/g

Y=(

κξ)-1

(a)

(b)

Fig. 7. Location of apparent electrostatic persistence lengthsle,app in our schematic map of the XY parameter space. Weshow results for the reduced crossover distance Xcd = lcd/ξ (a)and the reduced orientational correlation length Xoc = locξ (b)for chains of length Ntot = 4096 (solid lines) and Ntot = 256(dashed lines). In the first case, we compare

⟨r2⟩

(X, Y ) tothe size of the stretched blob pole equation (9) and defineXcd = lcd/ξ implicitly via

⟨r2⟩

(Xcd, Y ) ≡ ξ2X2cd/3. The re-

sults for Xoc presented in (b) are decay lengths extracted fromfits of TCFs to simple exponentials. For the fits we used datafrom the segment length interval X < Y, 1

2Ntot/g. However,

for Y < 10 the decay of the TCFs ceases to be well de-scribed by a simple exponential (see also Fig. 6). Being stronglydependent on the data range selected for the fits (data notshown), the values presented in (b) for Y < 10 thus have to betaken with a grain of salt. We note that the results for datasets with different coupling constant scale quite well and thatthere is good qualitative agreement between the two meth-ods. The results nicely follow the OSFKK prediction for re-duced screening lengths Y > 10, but are strongly influencedby excluded-volume effects for smaller values of Y . In particu-lar, the extracted persistence lengths systematically exceed theOSFKK estimate but scale roughly linearly in Y . With respectto finite chain length effects, the first method turns out to bemore robust than the second.

where light scattering was used to determine the influ-ence of the concentration of mono-valent salt on the gyra-tion radius of long polyelectrolytes in the coil regime. Theresults were interpreted in terms of an apparent persis-tence length derived from the relation 〈R2

g〉 ∝ le,appL fora WLC in the absence of excluded-volume interactions.Since the experimental data are actually located in a re-gion which is dominated by excluded-volume effects (seeFig. 2) with 〈R2

g〉 ∝ κ−2/5l2/5e L6/5, this definition leads

to the somewhat unfortunate relation le,app ∝ κ−2/5l2/5e

for chains of constant length. Even though the observeddependences 〈R2

g〉 ∝ κ−y with y ∈ [0.9, 1.1] are exactly inthe expected range, the inappropriate comparison created

a puzzle, which both groups resolved correctly by compar-ing their data to theories which do account for excluded-volume effects. In fact, Schmidt et al. [27] even observeda breakdown of their KMBJ theory for samples with thelowest salt concentration which turn out to be locatedclose to the KK2 crossover line. However, it would prob-ably be difficult to discriminate between the OSFKK andthe KMBJ theories on the basis of the available experi-mental data, since the differences between the two theo-ries for the parameters in question are fairly minor (seethe discussion in Sect. 3 on chain length requirements).

The comparison of subsequent numerical results withexperimental data has provided important evidence forthe validity the Debye-Huckel approximation [28,33].However, apart from the agreement with experimental re-sults, the evaluation of more sophisticated definitions ofle,app has provided only limited insight [28–34]. Figure 7illustrates the problems by locating our results for the“orientational correlation length” loc and the “crossoverdistance” lcd in our map of the parameter space. The firstlength is defined as the decay length of a simple exponen-tial fitted to the TCF, while the second tries to identifythe crossover from the blob pole equation (9) to an un-dulating blob chain equations (12) or (13). Clearly, theextracted length scales can only be identified with the elec-trostatic persistence length le as long as le is well separatedfrom other relevant length scales, i.e. for sufficiently largechain and screening lengths. Figure 7 shows that our re-sults closely follow the KK1 line for N = 4096 and Y > 10.However, the violation of either of the two conditions leadsto deviations. Relative to the OSFKK prediction, the ex-tracted persistence lengths– increase for small reduced screening lengths due to

excluded-volume effects and– decrease in the opposite limit due to finite chain length

effects which are particularly strong for the TCF andquantities related to it.

Depending on the definition of le,app and the range of chainand screening lengths studied, the combination of thesetwo effects can lead to the observation of effective expo-nents le,app ∼ κ−y which are much smaller than y = 2.However, since the weak κ-dependence of le,app is an arti-fact of the definition of the quantity, there seems to be nocontradition to the OSFKK theory.

7 Discussion

In this paper we have combined a scaling analysis of theconformational properties of intrinsically flexible polyelec-trolytes with Debye-Huckel interactions with extensiveMonte Carlo simulations of isolated chains. Our studywas focused on the controversial case of polyelectrolytesbeyond the OSF limit, i.e. on the case where the electro-static screening length κ−1 exceeds the bare persistencelength of the polymers in the absence of electrostatic in-teractions.

Our main result is the refutation of theories which ei-ther predict [5,13] or implicitly assume [9–12] an electro-static persistence length scaling as κ−1. In contrast, we


have observed no significant deviations from the scenarioproposed by Khokhlov and Khachaturian [7] who com-bined the idea by de Gennes et al. [8] of a stretched chainof polyelectrolyte blobs with the Odijk-Skolnick-Fixmantheories of the electrostatic persistence length [3,4] andthe electrostatically excluded volume [6,35] between chainsegments. Our results suggest that it is indeed possible tounderstand intrinsically flexible polyelectrolytes by con-sidering a hierarchy of effects due to interactions betweendifferent classes of monomer pairs:

– Stretching due to the (effectively unscreened) Coulombrepulsion between neighboring monomers into a chainof blobs which has a finite

– Bending rigidity due to the screening of interactions be-tween monomers with a distance larger than g/(κξ)along the chain. As a consequence, the blob chainremains straight up to the electrostatic persistencelength le = κ−2/ξ. Beyond le the chain behaves likea random walk, before

– Swelling due to the electrostatically excluded vol-ume between chain segments with a distance largerthan le becomes relevant beyond the Flory lengthlF = κ−4/ξ3.

It may be worthwhile to compare this scenario to the sit-uation originally considered by OSF, i.e. a WLC whoseintrinsic persistence length l0 exceeds the screening lengthκ−1. In the OSF limit the expansion of the electrostaticenergy around a straight ground state is justified a pri-ori, since the chain is known to be essentially straighton length scales where the interaction is relevant. In con-trast, the expansion can only be justified a posteriori forstrongly charged, intrinsically flexible polyelectrolytes bythe following two observations: i) in the absence of screen-ing, the chain would be straight on all length scales, sincelg < l0 (see Eq. (5) ); ii) the predicted persistence lengthlOSF = 1/(ξκ2) = Y 2ξ is larger than the screening length,so that the chain is indeed found to be straight on thoselength scales where the interaction is relevant. This pointseems to be implicit in a remark made by Odijk at theend of reference [6] and is confirmed by our simulation re-sults for systems with lB/b = 42 and lB/b = 1. However,in the case of weakly charged, intrinsically flexible poly-electrolytes with lg < l0 (see Eqs. (3) and (4) ) the firstcondition no longer holds and the expansion of the elec-trostatic energy breaks down for the original chain withits bare parameters [5]. KK argued that the previous casecould be recovered by renormalizing to the stretched blobchain. While there are theoretical justifications for the un-derlying assumption of the irrelevance of longitudinal andtransverse fluctuations within the blob chain [20], we feelthat we have presented the strongest evidence so far forthe legitimacy of the renormalization scheme by show-ing that the difference between strongly (lB/b = 42, 1)and weakly (lB/b = 1/42, 1/162, 1/1002) charged polyelec-trolytes can indeed be hidden behind a suitably chosensystem of units (see Eqs. (3) to (5) ). While the applica-bility of the OSFKK theory follows from the success of thismapping, we emphasize that we have also presented direct

evidence for a κ−2 scaling of the electrostatic persistencelength for weakly charged chains.

Clearly, scaling arguments cannot do justice to the fullcomplexity of the problem. Omitting all numerical prefac-tors, the ubiquituous logarithmic corrections, finite chainlength effects and, in our opinion most importantly, a re-fined description of the crossovers between narrow neigh-boring regimes, they cannot hope (and should not be ex-pected) to describe numerical or experimental data in de-tail. Quite obviously, these features call for a quantita-tive explanation. While our numerical results can serveas benchmarks for the development of theories, it is asobering thought that the simplest model of a single, iso-lated polyelectrolyte chain is still unsolved. Compared tothe much better understood neutral polymers, the OS-FKK theory represents the equivalent of the standardFlory argument for the excluded-volume effect. Neverthe-less, we believe to have shown that the OSFKK theoryprovides the indispensable “big picture” needed for thedesign and analysis of experiments and computer simula-tions.

We gratefully acknowledge helpful discussions with B. Dunweg,J.-F. Joanny, K. Kremer and H. Schiessel and are particularlyindebted to the latter and to M. Ullner for critical readings ofthe manuscript. RE is supported by an Emmy-Noether fellow-ship of the DFG.

Additional remark

After this work was finished, we learned of a comparable(N = 4096 and lB/b = 1) simulation study by Nguyenand Shklovskii [42]. The data analysis is complementaryto the one presented here and led the authors to similarconclusions.

References

1. J.-L. Barrat, J.-F. Joanny, Adv. Chem. Phys. 94, 1 (1996).2. M. Ullner, Polyelectrolyte models in theory and simulation,

in S. Tripathy, J. Kumar, H.S. Nalwa (Editors), Handbookof Polyelectrolytes and Their Applications (American Sci-entific Publishers, Los Angeles, 2002).

3. T. Odijk, J. Polym. Sci., Polym. Phys. Ed. 15, 477 (1977).4. J. Skolnick, M. Fixman, Macromolecules 10, 948 (1977).5. J.-L. Barrat, J.-F. Joanny, Europhys. Lett. 24, 333 (1993).6. T. Odijk, A.C. Houwaart, 1-1 electrolyte solution. J.

Polym. Sci., Polym. Phys. Ed. 16, 627 (1978).7. A.R. Khokhlov, K.A. Khachaturian, Polymer 23, 1742

(1982).8. P.G. de Gennes, P. Pincus, R.M. Velasco, F. Brochard, J.

Phys. (Paris) 37, 1461 (1976).9. M. Muthukumar, J. Chem. Phys. 86, 7230 (1987).

10. M. Muthukumar, J. Chem. Phys. 105, 5183 (1996).11. K. Ghosh, G.A. Carri, M. Muthukumar, J. Chem. Phys.

115, 4367 (2001).12. W. Kuhn, O. Kunzle, A. Katchalsky, Helv. Chim. Acta

31, 1994 (1948).


13. B.-Y. Ha, D. Thirumalai, Macromolecules, 28, 577 (1995).14. D. Bratko, K.A. Dawson, J. Chem. Phys. 99, 5352 (1993).15. T.A. Vilgis, J. Wilder, Comput. Theor. Polym. Sci. 8, 61

(1998).16. J. Wilder, T.A. Vilgis, Phys. Rev. E 57, 6865 (1998).17. T.B. Liverpool, M. Stapper, Europhys. Lett. 40, 485

(1997).18. T.B. Liverpool, M. Stapper, Eur. Phys. J. E 5, 359 (2001).19. P.L. Hansen, R. Podgornik, J. Chem. Phys. 114, 8637

(2001).20. H. Li, T.A. Witten, Macromolecules 28, 5921 (1995).21. B.-Y. Ha, D. Thirumalai, J. Chem. Phys. 110, 7533 (1999).22. R.R. Netz, H. Orland, Eur. Phys. J. B 8, 81 (1999).23. M. Tricot, Macromolecules 17, 1698 (1984).24. W.F. Reed, S. Ghosh, G. Medjahdi, J. Francois, Macro-

molecules 24, 6189 (1991).25. S. Forster, M. Schmidt, M. Antonietti, J. Phys. Chem.,

96, 4008 (1992).26. S. Forster, M. Schmidt, Adv. Polym. Sci. 120, 50 (1995).27. M. Beer, M. Schmidt, M. Muthukumar, Macromolecules

30, 8375 (1997).28. C.E. Reed, W.F. Reed, J. Chem. Phys. 94, 8479 (1991).29. J.-L. Barrat, D. Boyer, J. Phys. II 3, 343 (1993).

30. U. Micka, K. Kremer, Phys. Rev. E 54, 2653 (1996).31. C. Seidel, Ber. Bunsenges. Phys. Chem. 100, 757 (1996).32. B. Jonsson, C. Peterson, B. Soderberg, J. Phys. Chem. 99,

1251 (1995).33. M. Ullner, B. Jonsson, C. Peterson, O. Sommelius, B.

Soderberg, J. Chem. Phys. 107, 1279 (1997).34. M. Ullner, C.E. Woodward, Macromolecules 35, 1424

(2002).35. M. Fixman, J. Skolnick, Macromolecules 11, 863 (1978).36. M. Castelnovo, P. Sens, J.-F. Joanny, Eur. Phys. J. E 1,

115 (2000).37. L. Schafer, A. Ostendorf, J. Hager, J. Phys. (Paris) 32,

7875 (1999).38. V. Yamakov, A. Milchev, H.-J. Limbach, B. Dunweg, R.

Everaers, Phys. Rev. Lett. 85, 4305 (2000).39. N. Madras, A.D. Sokal, J. Stat. Phys. 50, 109 (1988).40. S.D. Stellman, P.J. Gans, Macromolecules 5, 516 (1972).41. D. Frenkel, B. Smit, Understanding Molecular Simulation:

From Algorithms to Application (Academic Press, NewYork, 1996).

42. T.T. Nguyen, B.I. Shklovskii, Persistence length of a poly-electrolyte in salty water: a Monte Carlo study, submittedto Phys. Rev. E (2002) cond-mat/0202168.

Appendix B

Conformations of RandomPolyampholytes

65

VOLUME 85, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 13 NOVEMBER 2000

Conformations of Random Polyampholytes

Vesselin Yamakov,1,2,* Andrey Milchev,1,2 Hans Jörg Limbach,1 Burkhard Dünweg,1 and Ralf Everaers1

1Max-Planck-Institut für Polymerforschung, Postfach 3148, D-55021 Mainz, Germany2Institute for Physical Chemistry, Bulgarian Academy of Sciences, G. Bonchev Street, Block 11, 1113 Sofia, Bulgaria

(Received 3 March 2000)

We study the size Rg of random polyampholytes (i.e., polymers with randomly charged monomers)as a function of their length N . All results of our extensive Monte Carlo simulations can be rationalizedin terms of the scaling theory we develop for the Kantor-Kardar necklace model, although this theoryneglects the quenched disorder in the charge sequence along the chain. We find �Rg� ~ N1�2. Theelongated globule model, the initial predictions of both Higgs and Joanny (~ N1�3) and Kantor andKardar (~ N), and previous numerical estimates are ruled out.

PACS numbers: 64.60.– i, 36.20.–r, 87.15.By

Polyampholytes (PAs) are heteropolymers comprisingneutral, positively, and negatively charged monomers.Such molecules are often water soluble, offer numerousapplications [1], and can be regarded as simple modelsystems for electrostatic interactions in proteins and otherbiopolymers. Depending on the method of synthesis, thecharge sequence can be either alternating or random. Thefirst case is well understood in terms of a theta collapsedue to effectively short-ranged interactions [2,3]. Incontrast, the statistical mechanics of random PAs [4–14]has turned out to be surprisingly complex. The purpose ofthis Letter is to settle a long-standing controversy on theshape of isolated random PAs in general and the effect ofthe quenched disorder in the charge sequence in particular.

The interest in this question was triggered by the dis-covery of Kantor, Li, and Kardar [7] that random PAs aresensitive to small disparities in the number of positivelyand negatively charged monomers per chain. In an en-semble of statistically neutral PAs of length N , the typicalnet charge is jQj ~ N1�2. Chains with a net charge upto this value behave as globally neutral (Q � 0) PAsand form dilute globules of spherical shape as predictedby Higgs and Joanny [5]. In contrast, the more stronglycharged members of the ensemble adopt strongly elon-gated conformations leading to a situation where en-semble averages for quantities such as the gyration radiusfor statistically neutral random PAs are dominated by theuntypical, extended chains in the wings of the net chargedistribution [7].

To explain this behavior Kantor and Kardar have pro-posed a model where, as a function of their net charge andin analogy to the Rayleigh instability of charged droplets,the PA globules split into a pearl-necklace–like sequenceof smaller globules connected by thin strings [8,9]. Whilepolyelectrolytes (PEs) in poor solvent [15] are welldescribed by the necklace concept [16,17], Kantor andKardar have argued that in the PA case the charge in-homogeneities should drastically modify the necklacepicture. Indeed, computer simulations of PAs reveal arich variety of conformations [18], and it is unclear if thedisorder is relevant for ensemble averages of quantities

as the gyration radius �Rg�. Ignoring all details of thecharge sequence except the net charge, the necklace modelpredicts �Rg� ~ N1�2, while the evidence from MonteCarlo simulations [9] (�Rg� ~ N0.6) and exact enumera-tions [10] (�Rg� ~ N2�3) rather suggests a faster growth.Nevertheless, the effect seems to be weaker than predicted[6] by Kantor’s and Kardar’s original renormalizationgroup argument (�Rg� ~ N).

In the following we present a complete scaling theory forthe Kantor-Kardar necklace model as well as large scalecomputer simulations of various ensembles of quenchedrandom PAs: Fixed (zero or nonzero) net charge, and ran-domly charged chains with a typical net charge of orderN1�2. With respect to the length of our chains N # 4096 aswell as the number of independent charge sequences (be-tween 512 and 1024) we by far exceed previous simulationstudies [9,18,19]. The good statistics for large chains turnsout to be crucial, since our results suggest that deviationsfrom the predictions of the necklace model for ensembleaverages are merely finite size effects.

We consider isolated, flexible chains of N monomers ofdiameter b in a good solvent with no added salt. Thiscorresponds to the limit of infinite dilution, where thechains do not form complexes [13] and where counter-ions, which may be necessary to balance the net chargeof the considered PA ensemble, can be considered as in-finitely far away. For a particular chain, a fraction f �f1 1 f2 of the monomers at quenched random positionscarries charges 6e, resulting in a net charge per monomerof edf � e� f1 2 f2�. The strength of the unscreenedelectrostatic interactions is characterized by the Bjerrumlength, lB � e2�ekBT .

Globally neutral chains (df � 0) assume a globularconformation if they are sufficiently long [5], while forshorter chains a smooth crossover to self-avoiding walks(SAWs) occurs. Within the framework of mean fieldtheory, the attraction energy is estimated via the Debye-Hückel polarization energy density [4] fDH ~ k3kBT ,where the inverse squared screening length k2 � lBfc isproportional to the monomer concentration c. Thus theattraction will be important on length scales larger than

0031-9007�00�85(20)�4305(4)$15.00 © 2000 The American Physical Society 4305


the so-called blob diameter ja � k21. On length scalesbelow ja the conformation is described as an unperturbedSAW, so that ja � bgn

a , where ga is the number ofmonomers in the blob, and n � 0.59. Since c � ga�j3

a,the blob size is given as ga � �b��lBf�1��12n�. The PAchain is then envisioned as a spherical droplet of blobs;this minimizes the surface energy which is estimated as�N�ga�2�3kBT (each surface blob contributes kBT ). Thegyration radius hence scales as

R2g�df � 0, f, lB�b, N�

R2SAW

~

(1 N�ga ø 1 ,� N

ga�2�1�32n� N�ga ¿ 1 .

(1)

In analogy to Khokhlov’s description of PEs in poorsolvent [15], a first understanding of the effect of a nonzeronet charge density df can be gained from an elongatedglobule model [1,11,12]. A globule becomes extended assoon as the Coulomb energy kBT �dfN�2lB��N�ga�1�3jaexceeds the surface energy kBT �N�ga�2�3; i.e., for N .

gR � f�df2, the number of monomers in a “Rayleighblob,” whose size is given by jR � ja�gR�ga�1�3. Theglobule is stable for ga , N , gR , while for N . gR

the elongated globule model predicts an object of diameterjR and length �N�gR�jR , whose relative extension is thusgiven by

R2g�df, f, lB�b, N�

R2g�df � 0, f, lB�b, N�

~

(1 N�gR ø 1 ,� N

gR�4�3 N�gR ¿ 1 . (2)

However, Kantor and Kardar [8,9] argued that the elec-trostatic repulsion should rather result in necklace-likeconformations, where spherical regions with net chargesbelow the instability threshold alternate with thin strings.Based on this concept, Dobrynin et al. [16] developed ascaling theory for PEs in poor solvent, which has beenrecently shown in computer simulations to describe thedata much better than the earlier Khokhlov picture [17].Applied to PAs, one expects the pearl and string diame-ters to be given by jR and ja, respectively. The lengthl � jR�jR�ja�1�2 ¿ jR of the strings is then again de-termined by the equilibrium between the additional surfaceenergy of the strings, kBTl�ja, and the electrostatic repul-sion between the pearls, kBT �dfgR�2lB�l. Note that eventhough the strings make up for most of the length of thenecklace, they contain only a negligible fraction of the PAvolume, with Rg�df, N� � �N�gR�l. Hence the necklacemodel predicts a different scaling for the chain dimensions,

R2g�df, f, lB�b, N�

R2g�df � 0, f, lB�b, N�

~jR

ja

µNgR

∂4�3

, N�gR ¿ 1 ,

(3)

than the elongated globule model, Eq. (2). It should benoted that in the necklace case no universal scaling func-tion of just one scaling argument N�gR occurs; i.e., acomplete data collapse is possible only for either the globu-lar regime or the necklace regime, the reason being that

the regimes are separated by a first-order phase transition[16,17]. Conversely, the elongated globule model predictsjust a smooth crossover, such that only a single scalingfunction occurs.

Finally, we consider the ensemble treated by Kantor andKardar, randomly charged PAs with a Gaussian net chargedistribution of zero mean and width �df2� � f�N . Thecalculation of averages such as �Rg� is somewhat subtle.For N ø ga one will, of course, observe SAW behavior,while for N ¿ ga the ensemble comprises contributionsfrom both the globular and the necklace phases. Indeed,each charge realization implies a certain value of gR �f�df2, the typical value being gR � N . Hence therewill always be a finite (N-independent) fraction of chainswhose gR is small enough that they are in the extendednecklace phase. This fraction will asymptotically domi-nate the average value of Rg. Thus the average stretchingrelative to the globule is found by just using Eq. (3), wheregR is replaced by N . Since then jR�ja � �gR�ga�1�3 ��N�ga�1�3, we find

R2g��df� � 0, f, lB�b, N�

R2g�df � 0, f, lB�b, N�

~

(1 N�ga ø 1 ,� N

ga�1�3 N�ga ¿ 1 . (4)

One thus finds �Rg� ~ N1�2 for the random PA necklace[8,9], which is formally a random walk (RW) exponent,while the underlying structure is completely different.Note that within the elongated globule model (i.e., dis-regarding the possibility of a Rayleigh instability) thenet charge fluctuations are predicted to be irrelevant asoriginally assumed by Higgs and Joanny [5].

In our Monte Carlo simulations we studied a bead-springmodel with short-ranged potentials to model connectivityand excluded volume. All monomers are charged � f � 1�and interact via an unscreened Coulomb potential. Thisyields the largest amount of charge fluctuations with thesmallest number of monomers, while we are not inter-ested in details of the chain structure below the distancebetween neighboring charges. We varied the blob size ga

by studying different values of lB�b � 1�64, . . . , 4. Notethat in order to reach the O �103� blobs necessary for theformation of well-defined globules and necklaces, we hadto push the strength of the electrostatic interaction to oreven slightly beyond the validity limit [lB�b � O �1�] ofthe blob picture.

Further factors which facilitated the feasibility of theinvestigation were the use of a large parallel computer,exploiting the inherent parallelism resulting from the dis-order realizations, plus the application of a very efficienthybrid algorithm which combines local moves with thepivot technique [20], while starting off from a configu-ration that was generated via the enhanced configurationalbiased Monte Carlo method [21] with already equilibratedbond lengths. At each state point we studied 512 or 1024different realizations of the disorder, each of which wasobserved for a fixed run time. For the shorter chains andsmaller charges, this run time was long enough to yield

4306


101

102

103

N10

110

210

3

N

10−1

100

<Rg

2 >/N

l2

SAW

GLO

BULE

SAW

RW

(a) (b)

FIG. 1. Radius of gyration as a function of chain length N for(a) globally (full symbols) and (b) statistically (open symbols)neutral random PAs with lB�b � 1�64 �±�, 1�16 ��, 1�4 ��,1�2 ��, 1 ��, 2 ��. The data are normalized to the sizeof random walks, while the straight lines indicate the slopesexpected for self-avoiding walk (Rg ~ Nn , n � 0.588), randomwalk (Rg ~ N1�2), and globular (Rg ~ N1�3) conformations.

a few hundred statistically independent configurations perrealization (as estimated via the autocorrelation function ofthe end-to-end vector). On the other hand, the long glob-ally neutral chains at strong charging were very difficult toequilibrate in their dense globular state. Reasonable sta-tistics (with at least a few ten independent configurations)

10−2

100

102

104

N/ga

10−1

100

101

2(1/3−ν)

2(1/2−1/3)

2(0.6−1/3)Rg

2(⟨δf⟩=0)

_________

Rg

2(δf=0)

Rg

2(δf=0)_________

Rg

2(SAW)

FIG. 2. Scaling plot of the data presented in Fig. 1. The fullsymbols show the shrinking of globally neutral PAs relative touncharged SAWs, while the open symbols represent the swellingof statistically neutral relative to globally neutral PAs. Thedata are plotted as a function of the reduced chain length N�gaand support Eq. (1) and, in particular, the prediction Eq. (4) ofthe necklace model. The dashed line corresponds to the earliernumerical estimate Rg ~ N0.6 [9], which is clearly not supportedby the data.

is available up to lB�b � 1, while for lB�b � 2 only thedata up to N � 512 are reliable. For lB�b � 4 the globu-lar state was practically inaccessible, and only necklacescould be studied. For further details we refer the reader toRef. [22], where the analogous model was simulated withthe same methods to study PE adsorption. All in all, weneeded roughly 5 3 104 hours single-processor CPU timefor the calculation.

Figure 1 shows the chain length dependence of the gyra-tion radii of globally (full symbols) and statistically (opensymbols) neutral random PAs. The data show unequivo-cally that sufficiently long random PAs with a globalneutrality constraint adopt globular conformations (Rg ~

N1�3), while unconstrained random PAs are on the averagesignificantly more extended, with Rg ~ N1�2. Clearly, agrowth of Rg with N which is even faster than that of theSAW, as was suggested by Refs. [9,10], can be ruled out.

The corresponding scaling plot (Fig. 2) supports theHiggs and Joanny [5] picture of the behavior of globallyneutral chains as well as our formulation of the necklacemodel for PAs carrying a net charge. For the SAW data wetook those with the weakest charge lB�b � 1�64, which isvery close to the true SAW behavior for our chain lengths.One also sees that the crossover from the SAW into theglobule is subject to considerable corrections to scaling,which are probably mainly due to the rather small ga val-ues of our simulation.

10-2

10-1

100

101

102

N/gR

10-2

10-1

100

101

102

Φ ×

Rg

2 (δf≠

0)/R

g

2 (δf=

0)

100

101

102

103

anecklace model: Φ=ξ /ξR

elongated globule model: Φ=1

FIG. 3. Swelling of random PAs due to a nonzero net chargedensity df . 0. We show three data sets lB�b � 1 ��, 2 ��,4 �� for systems with fixed asymmetry df � 1�32, and varyingchain lengths 32 # N # 4096 and one data set with lB�b �1 ��, fixed chain length N � 1024 and charge asymmetries1�128 # df # 1�2. The shaded areas indicate the parameterregions where the scaling forms Eqs. (2) and (3) break down,corresponding to the elongated globule and the necklace model,respectively. The dashed line represents the function �N�gR�4�3

predicted by both models for N�gR ¿ 1. The insets showtypical conformations for chains with N � 1024 and lB�b � 4with df � 0 and N�gR � 4, respectively.

4307


1

10

100

1000

1 10 100 1000

<R

(|i-j

|)2 >

|i-j|

0.5

1

2

1 5 10 50 100

(k*R

g)2 *

S(k

) / N

k*Rg

FIG. 4. Ensemble averages for the internal structure of statis-tically neutral random PAs with N � 2048 and lB�b � 1. The1 symbols in the inset show the mean square distance �r2

ij� oftwo monomers as a function of their distance ji 2 jj along thechain in comparison to a ji 2 jj231�2 power law. The figureshows the structure factor S�k� (3) in the Kratky representa-tion as a function of kRg in comparison to the Debye function(solid line) for random-walk– like fractal objects and the resultone obtains in the Gaussian approximation from the data in theinset (1).

In order to better characterize the Rayleigh instabilitywe also investigated ensembles of random PAs with a fixednonzero net charge. Figure 3 demonstrates that the elon-gated globule model describes the onset of the deformationup to elongations by about a factor of 2. The Rayleigh in-stability occurs around N�gR � 2, while for N�gR . 3the data are in excellent agreement with the predictionEq. (3) of the necklace model.

Quite interestingly, the exponent 1�2 even seems tocharacterize the ensemble averages for the mean squareinternal distances �r2

ij� � ��ri 2 �rj�2� in statistically neu-tral PAs (see the inset of Fig. 4). That random PAs are,however, far from being random-walk– like fractal objectsis demonstrated by the structure factor S�k� (3 in Fig. 4)which clearly deviates from the Debye function. For com-parison, we have also calculated S�k� in the Gaussianapproximation �exp�i �k ? �rij� � exp�2k2�r2

ij��2 showingthat the distribution function p�rij� cannot be specified byits second moment alone.

In summary, our results have demonstrated a remarkablesuccess of the simple necklace model for random polyam-pholytes. In particular, the scaling of the average extensionof the chains is not affected by the quenched disorder ofthe charge positions along the chains. Nevertheless, a moredetailed description of the relation between the charge se-

quence on individual chains and their typical conforma-tions remains a challenge.

This work was supported by collaboration GrantNo. I�72 164 from the Volkswagen foundation. We thankthe Rechenzentrum Garching for the generous allocationof Cray T3E CPU time.

*Present address: Argonne National Laboratory, MaterialsScience Division, Building 212, 9700 S. Cass Avenue, Ar-gonne, IL 60439.

[1] F. Candau and J.-F. Joanny, in Polymeric Materials En-cyclopedia, edited by J. C. Salamone (CRC Press, BocaRaton, 1996).

[2] J. M. Victor and J. B. Imbert, Europhys. Lett. 24, 189(1993).

[3] J. Wittmer, A. Johner, and J.-F. Joanny, Europhys. Lett. 24,263 (1993).

[4] S. F. Edwards, P. R. King, and P. Pincus, Ferroelectrics 30,3 (1980).

[5] P. Higgs and J.-F. Joanny, J. Chem. Phys. 94, 1543 (1991).[6] Y. Kantor and M. Kardar, Europhys. Lett. 14, 421 (1991).[7] Y. Kantor, H. Li, and M. Kardar, Phys. Rev. Lett. 69, 61

(1992); Y. Kantor, M. Kardar, and H. Li, Phys. Rev. E 49,1383 (1994).

[8] Y. Kantor and M. Kardar, Europhys. Lett. 27, 643 (1994).[9] Y. Kantor and M. Kardar, Phys. Rev. E 51, 1299 (1995).

[10] Y. Kantor and M. Kardar, Phys. Rev. E 52, 835 (1995).[11] A. Gutin and E. Shakhnovich, Phys. Rev. E 50, R3322

(1994).[12] A. V. Dobrynin and M. Rubinstein, J. Phys. II (France) 5,

677 (1995).[13] R. Everaers, A. Johner, and J.-F. Joanny, Europhys. Lett.

37, 275 (1997); Macromolecules 30, 8478 (1997).[14] D. Ertas and Y. Kantor, Phys. Rev. E 53, 846 (1996); 55,

261 (1997); S. Wolfling and Y. Kantor, Phys. Rev. E 57,5719 (1998).

[15] A. Khokhlov, J. Phys. A 13, 979 (1980).[16] A. V. Dobrynin, M. Rubinstein, and S. P. Obukhov, Macro-

molecules 29, 2974 (1996).[17] A. Lyulin, B. Dünweg, O. Borisov, and A. A. Darinskii,

Macromolecules 32, 3264 (1999).[18] N. Lee and S. Obukhov, Eur. Phys. J. B 1, 371 (1998).[19] M. Tanaka et al., Phys. Rev. E 56, 5798 (1997); J. Chem.

Phys. 110, 8176 (1999); Langmuir 15, 4052 (1999).[20] N. Madras and A. D. Sokal, J. Stat. Phys. 50, 109 (1988).[21] D. Frenkel and B. Smit, Understanding Molecular Simu-

lation: From Algorithms to Application (Academic Press,New York, 1996).

[22] V. Yamakov, A. Milchev, O. Borisov, and B. Dünweg,J. Phys. Condens. Matter 11, 9907 (1999).

4308

Appendix C

Complexation and precipitationin polyampholyte solutions

71

EUROPHYSICS LETTERS 1 February 1997

Europhys. Lett., 37 (4), pp. 275-280 (1997)

Complexation and precipitation in polyampholyte solutions

R. Everaers, A. Johner and J.-F. Joanny

Institut Charles Sadron (CNRS UPR022)6 rue Boussingault, F-67083 Strasbourg Cedex, France

(received 9 October 1996; accepted in final form 23 December 1996)

PACS. 61.25Hq – Macromolecular and polymer solutions; polymer melts; swelling.PACS. 64.75+g – Solubility, segregation, and mixing; phase separation.PACS. 87.15Da – Physical chemistry of solutions of biomolecules; condensed states.

Abstract. – We discuss multichain effects in polyampholyte solutions of finite concentrationand find that the existing single-chain theories are limited to exponentially small concentra-tions, if the sample contains chains with net charges of both signs. We show that chargedpolyampholytes have a strong tendency to form neutral complexes and to precipitate. Stretchedchains almost play no role in the ideal case of a neutral sample with nearly symmetric chargedistribution. However, for a non-neutral sample, the free counterions accumulate in the super-natant together with charged, elongated chains and the behavior of the solution is dominatedby “impurities.”

Polyampholytes are polymers that carry positively as well as negatively charged groups.Being often water-soluble these molecules offer numerous applications besides providing simplemodel systems for electrostatic interactions in proteins and other biopolymers [1]. A specialclass are quenched polyampholytes, where the charges are predetermined by the chemistry andindependent of the pH. Over the last few years much progress has been made in synthesizingsuch polymers [1] and in understanding their conformations at infinite dilution [2]-[9]. Theapplication of these single-chain theories to experiments on random polyampholytes has,however, led to apparently contradictory results [10], [11].

In the present letter we discuss the solubility of polyampholytes and the composition ofsolutions at finite concentration. For ordinary polyelectrolytes, which carry charges of only onesign, the water-solubility is mostly due to the gain in translational entropy of the counter-ionsin the water phase. The polymers are dissolved in spite of their high electrostatic self-energies,which they minimize by adopting stretched conformations. In contrast, polyampholyte samplescan be self-neutralizing, thus resembling mixtures of oppositely charged polyelectrolytes [12].One can, therefore, expect the formation and precipitation of neutral complexes at finiteconcentrations. For not perfectly neutral samples the free counter-ions have a tendency tostay in the dilute phase, even though the majority of the polyampholytes precipitates. Whatis not clear a priori is the nature of the chains which accumulate in the supernatant to ensurecharge neutrality.

c© Les Editions de Physique

276 EUROPHYSICS LETTERS

Our analysis of the phase equilibrium between a homogeneous bulk and a dilute supernatantis based on the description of polyampholytes in solution as elongated globules [1], [7], [8]. Herewe mostly consider polyampholyte samples with uni- and bimodal net charge distributions, forwhich the main effects can be worked out by simple arguments. We also present some resultsfor randomly co-polymerized samples. A detailed treatment of this more realistic case will begiven elsewhere [13].

Model polyampholyte samples. – For simplicity we only consider net charge polydispersityand treat the chain length N and the fraction f of charged monomers as constants. Samplesare then characterized by a normalized distribution p(δf) for the excess charge per monomerδf . For non-neutral samples with δf0 ≡

∫δf p(δf)dδf 6= 0 the net charge is balanced by a

counter-ion density ci,tot = δf0ctot. Concentrations cδf refer to monomer concentrations inthe dilute phase. If there is phase separation, we use capital letters Cδf to indicate bulkconcentrations. Chain concentrations are given by cδf/N and Cδf/N , respectively.

We consider three different model distributions: (I) Random co-polymerization: p(δf) =√(N/2πf) exp

[−((δf − δf0)2N)/2f

]. Note that the unbiased case with δf0 = 0 is an ex-

ception rather than the rule and that depletion during synthesis leads to broader chargedistributions [14]. (II) Unimodal samples where all the chains have the same net chargedensity δf0. This is a reasonable approximation for (I), if δf0 is larger than the width

√f/N

of the distribution. (III) Bimodal samples which are a mixture of two types of polyampholytes(indicated by subscripts ±) with opposite net charges ±δf of equal magnitude and totalconcentrations ctot

+ = ( 12 + ε)ctot and ctot

− = ( 12 − ε)ctot.

Bulk properties. – We concentrate on the generic case of a neutral ensemble of polyam-pholytes in a Θ-solvent with no added salt. The bulk can be viewed as a dense liquidof blobs of ga = b2/l2Bf

2 monomers [2]. Inside each blob the Debye-Huckel polarizationenergy FDH = (−κ3/12π)kBTV is of order −kBT , where κ2 = 4πlBfCbulk, lB = e2/εkBTthe Bjerrum length, and b the monomer size. We usually measure the chain length inunits of these polyampholyte blobs: N = N/ga À 1. The bulk density is of the orderof Cbulk ∼ ga/ξ

3a ≡ ga/b

3g3/2a = lBf/b

4. The bulk chain chemical potential is given byµbulk = Fbulk + kBT log

(Cδfb

3/N)

with a bulk free energy per chain Fbulk of −kBT per blobindependent of the chain net charge [15].

Single-chain properties. – The essential features of the conformations of polyampholytesin solution seem to be captured in the elongated-globule model [7], [8], [1]. Here we follow thenotation in ref. [1]. In a globular state the internal monomer concentration is still approxi-mately given by Cbulk , so that three-body repulsion and polarization contribute Fbulk to thefree energy. The competition between surface tension, Fsurf ∼ N2/3kBT , and electrostatic re-pulsion between the excess charges, Fcoul ∼ (δfN)2(lB/ξaN1/3)kBT , determines the size of theelectrostatic blob, g′e = f/δf2. The molecules resemble a sequence of N/g′e electrostatic blobs,each consisting of g′e/ga polyampholyte blobs. Up to logarithmic corrections the total surface

and Coulomb energies of an elongated globule can be written as Fsurf + Fcoul ∼ N δf2/3kBT

with δf = δfb/lBf3/2. Note that for an ensemble of random polyampholytes N = f/〈δf2〉, so

that a finite fraction of the chains forms elongated globules with a size proportional to N [4], [5].For high net charges, when g′e ≤ ga or δf ≥ 1, the polarization energy can be neglected andthe chains behave as polyelectrolytes. They form a linear sequence of N/ge electrostatic blobs

with ge =(b/lBδf

2)2/3

and Fcoul = (N/ge)kBT = N δf4/3kBT . Summarizing these arguments,

the excess free energy of a single chain in solution relative to the bulk may be approximated

r. everaers et al.: complexation and precipitation in polyampholyte etc. 277

as

Fex(δf , N)kBT

=

N2/3

(1 + δf

2N), δf

2< 1/N (spherical globules),

2δf2/3N , 1/N < δf

2< 1 (elongated globules),

2δf4/3N , δf

2> 1 (polyelectrolytes).

(1)

The chain chemical potential is the sum of its free energy and the ideal gas term for thetranslational entropy µuni(δf , N , cδf ) = Fbulk(N)+Fex(δf , N)+kBT log(cδfb

3/N )− δfN Ψ . Forasymmetric samples, an electrostatic potential difference Ψ = (b/lbf1/2)Ψ develops betweenthe bulk and the dilute phase. In general, Ψ has to be determined from the condition ofcharge neutrality of both phases. Only in simple cases can this tedious procedure be avoidedby considering solubility products.

Dimerization and precipitation for a sample with symmetric charge distribution. – Westart by considering a bimodal sample (III) with ε = 0. In dilute solution the unimers are inchemical equilibrium with neutral dimers and at high enough ctot the sample phase separates.

For a globular state the dimer free energy is equal to that of a neutral unimer of twicethe original length. The chemical potential of the dimers is given by µdim(±δf , N , cdim) =Fbulk(2N) + Fex(0, 2N) + kBT log(cdimb

3/2N). The condition µdim ≡ µ+ + µ− of chemicalequilibrium leads to the law of mass action:

cdim / c+c− = (2b3/N) exp[2Fex(δf , N)− Fex(0, 2N)

]≡ KA . (2)

Precipitation sets in, when the concentration is so high that the bulk chemical potential isreached, µdim = µ+ +µ− = 2µbulk. As a consequence, the unimer concentrations in the dilutephase are limited by a solubility product

c+c− ≤ (1/4)C2bulk exp

[−2Fex(δf , N)

]≡ LP (3)

and the dimer concentration by

cdim ≤ (b3/2N)C2bulk exp

[−Fex(0, 2N)

]= KALP . (4)

Note that eqs. (2) to (4) are not restricted to ε = 0.For weakly charged chains the formation of dimers for ctot > 1/KA is preceded by precip-

itation at ctotphase sep = KALP + 2

√LP. The composition of the supernatant is independent of

ctot: cco-ex± =

√LP and cco-ex

dim = KALP. Dimers dominate due to the reduced unimer solubility

for√LP > 1/KA, or, up to logarithms, δf

2> 1/N . Since the criterion is the same as for the

crossover between the spherical and elongated globule regimes, the supernatant contains onlyspherical globules—either of neutral or pairs of charged polyampholytes. To observe elongatedglobules one has to reduce the polymer concentration to ctot < 1/KA ∼ exp[−4δf

2/3N ].

Precipitation sets in for exp[−(2N)2/3] < cco-exdil /Cbulk < exp[−N2/3]. Thus, a symmetric

ensemble of polyampholytes with high chain length N is practically insoluble. Experimentally,such systems always consist of a supernatant in co-existence with a bulk.

Non-neutral samples. – Although these considerations have the benefit of simplicity, theyare not in agreement with most experimental data [10], [11]: i) many random polyampholytesamples are water-soluble at finite concentrations, 2) the supernatant is not always dominatedby neutral chains, and 3) its concentration increases with the total concentration.


In the following we argue, that these effects are due to the presence of free counter-ions. Theextreme case is a sample where all polymers have a net charge of equal sign, e.g. the unimodaldistribution (II). In such a case, a polyampholyte molecule can only precipitate together with allits counter-ions. The respective concentrations in the dilute phase are limited by a solubilityproduct, c+(δf0c+)Nδf ≤ Cbulk(δf0Cbulk)Nδf exp[∆Fbulk − Fex(δf , N)], so that

c+ ≤ L1/(Nδf+1)I with LI ≡ CNδf+1

bulk exp[∆Fbulk − Fex] . (5)

The effect is quite dramatic. For δf0N = 1, i.e. a single counter-ion per chain, ∆Fbulk

can be neglected [15]. By effectively reducing Fex by a factor of two the solubility of thesample corresponds to neutral chains which are three times shorter. The counter-ions, on theother hand, experience an electrostatic potential barrier Ψ corresponding to one half of theglobule surface energy. For larger δf0, L1/(Nδf+1)

I becomes independent of the chain length andreaches a minimum of order unity for chains at the crossover between the elongated globuleand the polyelectrolyte regimes. Without discussing the corresponding semi-dilute solutionsany further one may, for all practical purposes, regard such samples as water-soluble.

Impurities and fractionation. – We now turn to the situation where a sample containsa small proportion of free counter-ions. As an example we consider again a bimodal sample(III), but now with a slightly higher proportion of positively charged polyampholytes (ε > 0).

Up to the onset of precipitation the extra chains and counter-ions do not qualitatively changethe behavior of the system. If unimers dominate, there is a slight excess c+ − c− = 2εctot ofpositively charged chains, in the opposite case the concentrations approach cdim = (1−2ε)ctot,c+ = 2εctot and c− = (1/2ε − 1)/KA. However, the composition and concentration of thesupernatant are no longer fixed by establishing a phase equilibrium. At co-existence, theunimer concentrations, not being limited individually but by a solubility product, becomedependent on the total concentration: c+ → 2εctot, while c− = LP/2εctot → 0. Since thedimer concentration is still given by eq. (4) and a constant, the dilute phase may, dependingon ctot, either be dominated by charged or neutral globules (fig. 1).

The increase of the unimer concentration in the supernatant is limited by the solubilityproduct eq. (5) for the excess chains and the counter-ions. Asymptotically, c+ − c− ≈ c+,C+ = (1/2 + ε)Cbulk, and C− = (1/2− ε)Cbulk , so that

c+ ≤ 2εL1/(Nδf+1)I , (6)

where ∆Fbulk = 0 for small ε.

Discussion. – Our results suggest that care has to be taken in the interpretation ofexperiments on samples which contain polyampholyte chains with net charges of both signs:At finite concentrations it is not possible to identify the composition of the dilute phase withthe composition of the sample.

It is worthwhile to illustrate the consequences using the example of randomly chargedchains treated by Kantor and Kardar [4] (case (I) with δf0 = 0). At infinite dilution theaverages for quantities such as the hydrodynamic radius are dominated by the extended chainsin the wings of the sample charge distribution. However, already for total concentrationsas low as Cbulk exp[−N2/3] most of the material is precipitated, so that bulk and samplecomposition coincide. By equating the chemical potentials in the two phases one obtains forthe concentrations in the dilute phase

c(δf) ∼ exp[−δf2N(1 + 2N2/3)/2] . (7)

r. everaers et al.: complexation and precipitation in polyampholyte etc. 279

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1

1.2

1.4

δf N1/2~ ~

spherical globules

elongatedglobules

p(δf

N1/

2 )~

~

elongatedglobules

-17.5 -15 -12.5 -10 -7.5 -5 -2.5 0-20

-17.5

-15

-12.5

-10

-7.5

-5

-2.5

0

log10 (ctot / cbulk)

log 1

0 (c

dil /

cbu

lk)

cdim

c+

c-

Fig. 1. Fig. 2.

Fig. 1. – Composition of the dilute phase vs. total concentration for asymmetric (ε = 0.01) bimodal

distributions of chains with N = 20, ga = 10 and net charge δfN1/2 = 3. The gray shaded areasindicate concentrations for which the dilute phase is dominated by elongated globules.

Fig. 2. – Probability distribution for the excess charge on the unimers: in the supernatant when mostof the chains are precipitated (——); at infinite dilution (- - - - - -).

Compared to the bulk, the distribution of net charges on the unimers in the supernatant isN1/3 times narrower, i.e. there are practically no elongated globules dissolved (fig. 2). Forthe Gedankenexperiment to work, where random polyampholytes with an overall neutralityconstraint are cut into two pieces, the concentrations have to be even lower: the least charged,elongated halves start to form dimers for ctot > Cbulk exp[−2N2/3]. Thus while Kantor andKardar concluded that statistical fluctuations in the net charge density play a dominant role,we believe that for experiments the opposite view is equally relevant: pairs of oppositelycharged chains have a strong tendency to form neutral complexes and to precipitate.

Nevertheless, we find that the extended states play an important role in experiments.The reason is i) the sublinear dependence of the excess free energy on the chain net chargein the elongated globule regime (eq. (1)) and ii) the fact that samples are never perfectlyself-neutralizing and therefore contain free counter-ions. At co-existence, an electrostaticpotential difference Ψ establishes between the phases which determines the composition ofthe supernatant.

For biased random co-polymerization (case (I) with 0 < δf0 ¿ N−1/2) we find two limitingcases [13]. If the bias is exponentially small or if ctot is close to the onset of phase separation,Ψ is so small that the composition of the supernatant is similar to the situation depicted infig 2: the Gaussian peak is shifted, so that spherical, but now on the average slightly chargedunimers dominate. In most cases, however, the potential difference exceeds the critical valueof Ψ = 2 with the consequence that i) the solubility of elongated globules with δf > ( 3

4 Ψ)−3

increases with their net charge, ii) those with the highest charge have the strongest tendencyto accumulate in the supernatant where iii) their concentrations may exceed those in the bulk.To balance the charges on the counter-ions with the limited number of highly charged globules,Ψ adapts its value, so that all chains from the tail of the sample net charge density distributionwith δf > δf

∗are dissolved, where δf

∗is determined by the condition δf0 =

∫∞δf∗ dδf δf p(δf).

In general, non–self-neutralizing samples behave like mixtures of a soluble and a non-solublecomponent. Concentrating the sample leads to a separation of an almost self-neutralizingbulk and a supernatant containing the counter-ions and the most strongly charged chains.The fractionation process can be repeated by redissolving the bulk until the dilute phase isdominated by weakly charged spherical globules.


Clearly, our understanding of the effects discussed in this article can only be as good asour understanding of the properties of polyampholytes in general. While the Debye-Huckelapproximation for the description of dense states and the elongated globule model for chainsin dilute solution capture the essential physics, they can certainly be refined [12], [6], [5]. Forexample, future work on the necklace model [5] in combination with arguments along the linesof those presented here may allow predictions on whether or not particular charge sequenceslead to more soluble molecules than others.

***

The authors thank F. Candau and J. Selb for helpful discussions. RE gratefully acknowl-edges financial support by the French Ministry of Foreign Affairs.

REFERENCES

[1] Candau F. and Joanny J.-F., in Polymeric Materials Encyclopedia, edited by J. C. Salamone

(CRC Press, Boca Raton) 1996.

[2] Higgs P. and Joanny J.-F., J. Chem. Phys., 94 (1991) 1543.

[3] Kantor Y. and Kardar M., Europhys. Lett., 14 (1991) 421.

[4] Kantor Y. And Kardar M., Phys. Rev. Lett., 69 (1992) 61; Kantor Y., Kardar M. and Li

H., Phys. Rev. E, 49 (1994) 1383.

[5] Kantor Y. and Kardar M., Europhys. Lett., 27 (1994) 643; Phys. Rev. E, 51 (1995) 1299.

[6] Wittmer J., Johner A. and Joanny J.-F., Europhys. Lett., 24 (1993) 263.

[7] Gutin A. and Shakhnovich E., Phys. Rev. E, 50 (1994) R3322.

[8] Dobrynin A. V. and Rubinstein M., J. Phys. II, 5 (1995) 677.

[9] Levin Y. and Barbosa M. C., Europhys. Lett., 31 (1995) 513.

[10] Skouri M., Munch J., Candau S., Neyret S. and Candau F., Macromolecules, 27 (1994)69.

[11] Ohlemacher A., Candau F., Munch J. and Candau S., J. Polym. Sci. Phys. Ed., 34 (1996)2747.

[12] Borue V. Y. and Erukhimovich I. Y., Macromolecules, 23 (1990) 3625.

[13] Everaers R., Johner A. and Joanny J.-F., submitted to Macromolecules.

[14] Corpart J.-M., Selb J. and Candau F., Polymers, 34 (1993) 3873.

[15] The bulk properties may change due to the presence of counter-ions. It is important to note thatthe gain in polarization energy of a counter-ion in the bulk is of the order Fbulk/fN and negligiblecompared to the polymer bulk and excess energies. As a consequence, counter-ions accumulate inthe bulk only for want of easily dissolvable polyampholytes with opposite charge. A homogeneousbulk is inflated due to their osmotic pressure with corrections ∆Fbulk/Fbulk ∼ δf0ga which may beneglected as long as the number of counter-ions per blob is small. A higher counter-ion contentmight, as in the case of polyelectrolytes in a poor solvent [16], [17], lead to the formation ofmesophases. In any case we expect |∆Fbulk|/Fbulk < 1.

[16] Borue V. Y. and Erukhimovich I. Y., Macromolecules, 21 (1988) 3240.

[17] Joanny J.-F. and Leibler L., J. Phys. (Paris), 51 (1990) 545.

Appendix D

Polyampholytes: From singlechains to solutions

79

Polyampholytes: From Single Chains to Solutions

R. Everaers,* A. Johner, and J.-F. Joanny

Institut Charles Sadron (CNRS UPR022), 6 rue Boussingault,F-67083 Strasbourg Cedex, France

Received June 27, 1997; Revised Manuscript Received October 9, 1997X

ABSTRACT: We discuss multichain effects in salt-free polyampholyte solutions of finite concentrationand find that the existing single-chain theories are limited to exponentially small concentrations, if thesample contains chains with net charges of both signs. We show that charged polyampholytes have astrong tendency to form neutral complexes and to precipitate. Neutral samples, where the average netcharge of the chains is exactly zero, phase separate at exponentially small concentrations. The supernatantis dominated by neutral unimers or clusters that form spherical globules and its polymer concentrationcdil ∼ exp(-N2/3) is independent of the total polymer concentration. For non-neutral samples, on theother hand, the free counterions accumulate in the supernatant together with the most strongly oppositelycharged unimers, if the system is concentrated beyond the onset of phase separation. The dilute phasethen consists of elongated globules and has a concentration proportional to the total polymer concentration.

1. Introduction

Polyampholytes are polymers comprising neutral,positively charged, and negatively charged monomers.Such molecules are often water-soluble and offer nu-merous applications.1 In addition, they can be regardedas simple model systems for electrostatic interactionsin proteins and other biopolymers. Over the last yearsmuch theoretical effort has been devoted to understand-ing the conformations of intrinsically flexible, isolated,quenched random polyampholytes in solution.2-11 Forthis special class of polyampholytes the charges arerandomly distributed along the chain and insensitiveto changes of the pH: they are predetermined by thechemistry.The conformations of dissolved polyampholyte chains

result from a competition between attraction due tofluctuations in the density of oppositely charged mono-mers and repulsion between excess charges. Higgs andJoanny2 showed that neutral random polyampholytescollapse into spherical, dilute globules, while Kantor andKardar demonstrated that already the typical netcharge of order xN for a random copolymer of length Nis sufficient to stretch the chains.4,5 They suggestedthat the averages for quantities such as the hydrody-namic radius can be dominated by the extended chainsin the wings of the charge distribution and thereforegrow like N as opposed to N1/3 for the majority ofcollapsed chains. Although there are still open ques-tions concerning the behavior of polyampholyte chainsin solution (see, e.g., the necklace model of Kantor andKardar6,7), the essential features seem to be capturedin the elongated globule model introduced by Gutin andShakhnovich.1,9,10

Even though the single chain picture is valid only atinfinite dilution, the results have been tentativelyapplied to experiments on quenched random polyam-pholyte solutions.1 Candau et al.12 use copolymerizationin a microemulsion consisting of varying compositionsof neutral and charged monomers and non-polymeriz-able inorganic counterions to synthesize quenchedrandom polyampholytes with controlled charge and net

charge densities. Typically, already the inclusion of afew percent of charged monomers is sufficient to inducephase separation and precipitation in dilute polymersolutions, even if the quality of the solvent is good forthe neutral backbone of the polymer.13,14 As for poly-electrolytes, the solubility increases with the averagenet charge of a sample. But mixtures of oppositelycharged chains precipitate, even if the average netcharge is much higher than for a perfectly solublesample with a narrower net charge distribution.14Moreover, Candau et al. reported an increase of theconcentration of the supernatant with the total polyam-pholyte concentration13 and apparently conflicting evi-dence for the predominance of either spherical14 orelongated chains13 in the dilute phase.The purpose of the present paper is the investigation

of polyampholyte solutions of finite concentration. Ouranalysis of the phase equilibrium between a homoge-neous dense phase and a dilute supernatant is basedon the description of polyampholytes in solution aselongated globules.1,9,10 Up to now the solubility ofpolyampholytes has been estimated from analogies topolymers in a poor solvent and to polyelectrolytes (i.e.,polymers that carry charges of only one sign), implying(i) that solutions precipitate under conditions whereindividual chains collapse and (ii) that charged chainsare more soluble than neutral chains. These argumentsare, however, questionable, as the water solubility ofpolyelectrolytes is mostly due to the gain in transla-tional entropy of the counterions in the water phase.15The polymers are dissolved in spite of their highelectrostatic self-energies, which they minimize byadopting stretched conformations. In contrast, a sampleof polyampholyte chains can be globally neutral, evenif individual molecules carry net charges. The situationtherefore resembles mixtures of oppositely chargedpolyelectrolytes.16 Indeed, we find17 that multichaineffects such as complexation and selective precipitationplay an important role and that the solubility ofpolyampholytes is mostly due to the presence of the freecounterions. Neutral samples, where the net chargeson the chains cancel each other exactly, phase separateat exponentially small concentrations. The supernatantis dominated by neutral unimers or clusters which formspherical globules, and its polymer concentration cdil ∼exp(-N2/3) is independent of the total polymer concen-tration. For non-neutral samples, on the other hand,

* To whom correspondence should be addressed. New address:Institut Curie, Section de recherche, 11 rue Pierre et Marie Curie,75231 Paris Cedex 05, France.

X Abstract published in Advance ACS Abstracts, December 15,1997.

8478 Macromolecules 1997, 30, 8478-8498

S0024-9297(97)00947-9 CCC: $14.00 © 1997 American Chemical Society

the free counterions accumulate in the supernatanttogether with the most strongly oppositely chargedunimers, if the system is concentrated beyond the onsetof phase separation. The dilute phase then consists ofelongated globules and has a concentration proportionalto the total polymer concentration.Some of our results have already appeared as a short

note.17 Here we provide more details, in particular forthe case of randomly copolymerized samples. The paperis organized as follows: In the second section weintroduce the notations, review the elongated globulemodel, and outline the treatment of association andphase equilibria. In the third section we considerpolyampholyte samples with uni- and bimodal netcharge distributions, for which many of the effects canbe worked out by simple arguments. Sections 4 and 5deal with the more realistic case of randomly copoly-merized samples. We conclude by a discussion of theinterpretation of experiments on polyampholyte samplescontaining chains with net charges of both signs. TheAppendix discusses the validity of the main assumptionwe made in order to calculate the composition of thephases.

2. Theoretical Background

In the following we review the scaling picture of theconformation of a random polyampholyte with a certainnet charge density in solution. We introduce the modelpolyampholyte samples with idealized net charge dis-tributions that we study in this paper and formallydiscuss association and phase equilibria in polyam-pholyte solutions of finite concentration.2.1. Properties of the Dense Phase. Our model

polyampholytes are flexible chains of N monomers ofdiameter b. A fraction f ) f+ + f- of the monomers atquenched random positions is charged. The net chargedensity of a particular chain is δf ) f+ - f-. Thestrength of the electrostatic interactions is characterizedby the Bjerrum length, lB ) e2/εkBT. We do not treatthe case of ionomers15 where oppositely charged groupson the chains are tightly bound to each other. Byassuming that b J lB, we ensure that the attractionbetween any two charges is smaller than kBT. Concen-trations are always given as monomer concentrations.We use small (capital) letters to indicate concentrationsin the dilute (dense) phase. The respective chainconcentrations of molecules with net charge density δfare then cδf/N (Cδf/N).We focus on the generic case of a neutral ensemble of

polyampholytes in a Θ-solvent with no added salt. Fora homogeneous dense solution of concentration Cdense,the contribution of the electrostatic interactions to thefree energy density is given by the Debye-Huckelexpression fDH/kBT ) -(1/12π)κ3, where κ2 ) 4πlBfCdenseis the inverse squared screening length.8 Higgs andJoanny2 have argued that the dense phase can beviewed as a liquid of blobs of ga ) b2/lB2f2 monomers.18We usually measure the chain length in units of thesepolyampholyte blobs: N ≡ N/ga . 1. The monomerdensity is of the order of Cdense ∼ ga/êa3 ) ga/b3ga3/2 )lBf/b4 with a free energy Fdense ≈ -NkBT per chainindependent of the chain net charge.In the Appendix we discuss the inflation of the dense

phase due to the osmotic pressure of free counterionsand the formation of mesophases. Throughout thepaper we assume that the counterion concentrations areso small that these effects can be safely neglected.

2.2. Single-Chain Properties. To a first approxi-mation the conformation of a polyampholyte moleculein solution depends on its length N and its excess chargedensity δf. For N . 1, three characteristic regimes canbe distinguished for the conformations of isolated mol-ecules in solution (Figure 1):1,10 (i) chains which areoverall neutral collapse into spherical globules (polyam-pholyte effect), while (ii) highly charged chains arestretched (polyelectrolyte effect). The crossover betweenthe two extremes was investigated by Kantor andKardar.3,4,5,6,7 Using RG arguments and Monte Carlosimulations, they showed that (iii) the typical net chargeof order xfN for a random copolymer of length N issufficient to elongate the globule.The essential features of these extended states seem

to be captured in the elongated globule model. It wasintroduced by Gutin and Shakhnovich9 and worked outas a Flory theory by Dobrynin and Rubinstein.10 Herewe summarize the formulation as a scaling theory byCandau and Joanny.1 In the globular states the inter-nal monomer concentration is still approximately givenby Cdense so that three-body repulsion and polarizationcontribute Fdense to the free energy of the globule. Theshape is determined from a competition between surfacetension and electrostatic repulsion between the excesscharges. The surface energy of a spherical globule is ofthe order of kBT times the number of blobs on thesurface: Fsurf ∼ N2/3kBT. In addition the globule con-tains δf N excess charges at a typical distance of theorder of the globule radius êaN1/3, which repel eachother. The electrostatic energy is thus of the order Fcoul∼ (δf N)2(lB/êaN1/3)kBT. It exceeds the surface energyfor N > g′e ) f/δf2. In this case the molecule can beviewed as a linear sequence of N/g′e electrostatic blobs,each consisting of g′e/ga densely packed polyampholyteblobs. Note that for an ensemble of random polyam-pholytes f/⟨δf2⟩ ) N. As a consequence, for all chainlengths a finite fraction of the chains form elongatedglobules with a size proportional to N.Up to logarithmic corrections19 the total surface and

Coulomb energy of an elongated globule is N/g′e timesthat of an electrostatic blob and can be written as Fsurf+ Fcoul ∼ Nδf2/3kBT with

The crossover to the polyelectrolyte regime occurs atg′e ≡ ga/δf2 ) ga or δf ) 1. For chains with δf > 1, thepolarization energy can be neglected and the chainsform a linear sequence of N/ge electrostatic blobs eachcontaining ge ) ga/δf4/3 monomers. The free energy isFcoul ) (N/ge)kBT ) N δf4/3kBT. In summary the excess

Figure 1. Diagram of states for polyampholyte chains.

δf ) δfxgaf

(2.1)

Macromolecules, Vol. 30, No. 26, 1997 Polyampholytes: From Single Chains to Solutions 8479

free energy of a single chains in solution relative to thedense phase may be approximated as

The Debye-Huckel approximation for the descriptionof dense states and the elongated globule model forchains in dilute solution capture the essential physicsand therefore provide a sensible base for the followingdiscussion. Nevertheless, the resulting expressions forthe free energies of polyampholytes in the dense and inthe dilute phase can be refined (see the Appendix andrefs 6, 8, and 16). For example, future work on thenecklace model6 may allow us to go beyond the rathercrude characterization of a polyampholyte chain onlyby its net charge.2.3. Model Polyampholyte Samples. For simplic-

ity we only consider net charge polydispersity and treatthe chain length N and the fraction f of chargedmonomers as constants. Polyampholyte samples arethen characterized by a normalized distribution p(δf)for the excess charge per monomer δf. For non-neutralsamples with a non-vanishing average net chargedensity δf0 ≡ ∫ δf p(δf) dδf * 0, the net charge on thechains is balanced by a counterion density ci,tot ) δf0ctot.We consider three different model distributions.(I) Biased Random Copolymerization. Chemi-

cally, it is very difficult to find a pair of monomers withopposite residual charges and equal reactivities in thepolymerization process. For this reason, there often isa surplus of, e.g., positive monomers in polyampholytemolecules formed in the early stages of the reaction. Aslong as the depletion of monomers of this type in thereaction bath is not felt, the reaction can still bedescribed as a random process, however, with a biastoward the formation of non-neutral chains:

The Gaussian distribution eq 2.3 is a good approxima-tion for the binomial distribution for (δf - δf0)2N <xNf. Note that the unbiased case with δf0 ) 0 is anexception rather than the rule and that monomerdepletion during the synthesis leads to broader netcharge distributions.12

(II) Unimodal Samples. We briefly discuss the casewhen all chains have the same net charge density δf0.This is a reasonable approximation for eq 2.3, if δf0 islarger than the width 1/xN of the distribution.(III) Bimodal Samples. Most finite concentration

effects can be worked out qualitatively by consideringa mixture of two types of polyampholytes with differentnet charges and otherwise identical characteristics. Wediscuss in detail the case of opposite net charges (δf ofequal magnitude and of a mixture of neutral andcharged chains. In the former case, the total concentra-tions of the species in the sample are given by c+

tot and

(1/2 + ε)ctot and c-tot ) (1/2 - ε)ctot. If ε * 0, the sample

also contains counterions to ensure charge neutrality.2.4. Outline of the Problem: Association and

Phase Equilibria. At infinite dilution, all moleculesare dissolved as unimers. In a dilute phase of finiteconcentration, we have to consider an association equi-librium between unimers, dimers, and higher orderaggregates. Formally, this corresponds to a chemicalequilibrium. In addition, solutions of polyampholytesestablish over a wide range of the total concentrationctot a phase equilibrium between a dense phase ofprecipitated molecules (or precipitate) and a dilutesupernatant.Neglecting the interaction between the dissolved

molecules and clusters (see the Appendix) the freeenergy density of the dilute phase can be obtained byminimizing the free energy

under the constraint of mass conservation:

Here, cδfdil indicates the total monomer concentration of

molecules with net charge density δf in the dilute phase,cδf is the concentration of such molecules dissolved asunimers, cδfδf′ is the concentration of dimers composedof molecules with net charge densities δf and δf ′, andci is the counterion concentration. The dots stand forhigher order clusters that usually do not play animportant role. Note that for a globular state the dimerfree energy equals the free energy of a neutral unimerof twice the unimer length.The free energy density of a phase-separated system

depends on the volume fraction Φ of the dense phaseand the compositions {Cδf} of the precipitate and{cδf

dil} and the supernatant. Since we are dealing withmulticomponent systems, the composition of the phasesis not fixed by establishing a phase equilibrium. It oftenturns out to be useful to use

instead of the total concentration ctot ) ΦCdense+ (1-Φ)cdil as an auxiliary variable in the treatment of phase-separated systems. If ctot is not too close to the onset ofphase separation and cdil is negligible compared to Cdense,Φ ) ctot/Cdense.Quite similarly to the association equilibrium (which

needs to be solved as a subproblem), the phase equilib-

Fex(δf,N)kBT

)

{N2/3 (1 + δf 2N) δf 2 < 1/N (spherical globules)

2δf 2/3N 1/N < δf 2 < 1 (elongated globules)

2δf 4/3N δf 2 > 1 (polyelectrolytes)

(2.2)

p(δf) )x N2π

e-[(δf-δf0)2N/2] (2.3)

fdil({cδfdil},ci)

kBT) ci(log(cib

3) - 1) +

∑δf

cδf

N (log cδf b3

N- 1 + Fdense(N) + Fex(δf,N)) +

∑δf,δf ′

cδfδf ′

N (log cδfδf ′ b3

2N- 1 + Fdense(2N) +

Fex(12(δf++δf-),2N)) + ... (2.4)

cδf +1

2∑δf ′

cδfδf ′ + ... ) cδfdil (2.5)

f ) Φ fdense({Cδf},Ci) + (1 - Φ) fdil({cδfdil},ci) (2.6)

Φ )ctot - cdilCdense - cdil

(2.7)

8480 Everaers et al. Macromolecules, Vol. 30, No. 26, 1997

rium is given by the minimization of the free energy fsubject to the constraints of mass conservation andcharge neutrality:

The constraints are conveniently included in theminimization procedure as Lagrange multipliers relatedto the osmotic pressure and the chain and counterion(electro-) chemical potentials. The calculation of themonomer density in the precipitate in section 2.1implicitly made the assumption that the osmotic pres-sure of the dilute phase and the ideal gas contributionof the chains in the dense phase can be neglected. Thisreduces the osmotic pressure equilibrium to the condi-tion

where Cdense ) lBf/b4 is a constant that is independentof the sample composition. This point is further dis-cussed in Appendix A.1.The chemical potentials are given by

Equating the chemical potentials leads to relations like

In the general case, the chemical potentials have to bechosen in such a way that the mass balances arefulfilled. In addition, for asymmetric net charge distri-butions, an electrostatic potential difference20 Ψ )Ψ/xfga develops between the dense and the dilutephase, which has to be determined from the conditionof charge neutrality of both phases. In section 3 weconsider simple cases where these tedious procedurescan be avoided by considering solubility products andlaws of mass action for the association.A limiting case, which greatly simplifies the calcula-

tions, is a supernatant in equilibrium with an infinitedense phase: the composition of the dense phase is

identical with the sample composition and the chemicalpotentials are known from eq 2.13. For a given system,the assumption of an infinite dense phase is justified,if there is no depletion, i.e., if for all δf the number ofmolecules in the supernatant is negligible compared tothe dense phase. If the supernatant is dominated byunimers, this amounts to the condition (1 - Φ)cδf ) (1- Φ)CδfXδf , ΦCδf for any δf. In this case, we can alsowrite (1 - Φ)cdil + ΦCdense ≈ ΦCdense ) ctot, so that thetotal concentration has to exceed

Equation 2.16 is straightforward to generalize for morecomplicated compositions of the supernatant.

3. Simple Model DistributionsIn general, the calculation of the chemical and phase

equilibria introduced in the preceding section is adifficult task. To gain some insight, we investigate inthis section simple model distributions. We start byconsidering the bimodal sample (III). In solution, theunimers are in chemical equilibrium with neutraldimers and, already for very low total concentrationsctot, the sample phase separates. We show that in thesymmetric case the supernatant is dominated by spheri-cal globules, formed either from neutral or from pairsof charged polyampholytes. In section 3.2 we discussthe formation of higher order clusters; they becomeimportant for overall neutral, but asymmetric bimodalnet charge distributions, where the positive and nega-tive chains have net charges of different magnitudes.We then turn to non-neutral ensembles, where the

charges on the polymers do not exactly cancel each otherbut are neutralized by free counterions. We discuss twocases: the bimodal sample (III) with a small asymmetryin the fraction of chains with positive and negative netcharge and a mixture of neutral chains with a smallproportion of charged chains. In both cases the coun-terions and excess chains accumulate in the superna-tant, if the sample is concentrated beyond the onset ofphase separation.3.1. Dimerization and Precipitation. We start by

discussing dimerization and precipitation for the bimo-dal sample (III) with net charges of equal magnitudebut opposite signs on the polyampholyte chains. Wewrite the equations for the general case, where we allowfor a small asymmetry ε in the fraction of chains withpositive and negative net charge. The discussion of theresults for ε * 0 is, however, postponed to section 3.3.Within the dilute phase there is an association or

chemical equilibrium between unimers and dimers.Using eqs 2.14 and 2.15, the condition µdim) µ+ + µ-leads to the law of mass action:

where c( ) c(,tot - cdim/2. The dimer concentration isgiven by

ΦCδf + (1 - Φ)cδfdil ) p(δf)ctot (2.8)

ΦCi + (1 - Φ)ci ) δf0ctot (2.9)

∫ δf Cδf dδf ) Ci (2.10)

∫ δf cδfdil dδf ) ci (2.11)

∫ Cδf dδf ) Cdense (2.12)

µdense ) Fdense + kBT log(Cδf b3

N ) (2.13)

µuni(δf,N,cδf) ) Fdense(N) + Fex(δf,N) +

kBT logcδf b

3

N- δf NΨ (2.14)

µdim(δf,δf ′,N,cδfδf ′) ) Fdense(2N) + Fex(12(δf +

δf ′),2N) + kBT logcδfδf ′b

3

2N- (δf + δf ′)NΨ (2.15)

cδf

Cδf≡ Xδf ) exp(-Fex(δf,N) + δf NΨ)

cδfδf ′

cδfcδf ′) 2b3

Nexp(-Fex(δf,N) - Fex(δf ′,N) +

Fex(12(δf + δf ′),2N))

ctot,minCdense

) 1(max(Xδf))

-1 + 1)

{max(Xδf) if max(Xδf) , 11 - (max(Xδf))

-1 if max(Xδf) . 1 (2.16)

cdimc+c-

) 2b3

Ne2Fex(δf,N)-Fex(0,2N) ≡ KA (3.1)

cdim ) ctot + 2KA

(1 - xε2KA2ctot

2 + (1 + KA ctot)) (3.2)


Thus, for ctotKA , 1 the degree of association is verysmall: cdim/ctot∼ KActot2 f 0. On the other hand, dimersdominate if ctot > 1/KA before the onset of precipitation.In equilibrium with a dense phase, the chemical poten-tials have to fulfill µdim ) µ+ + µ- ) 2µdense. As aconsequence, the unimer concentrations in the dilutephase are related to each other by a solubility product21

the dimer concentration is given directly by

Note, that cdimcoex is independent of ε and δf.

At coexistence, the solubility product eq 3.3 has to besolved for c( ) c(,tot - cdim/2 - ΦCdense/2, where Φ isthe volume fraction of the precipitate. The result is

Phase separation sets in at a total concentration of

At this point we only consider the case ε) 0. If unimersdominate the dilute phase at the onset of phase separa-tion cdim ) 1/4KAcphase sep

tot 2 ) KALP or cphase septot ) 2xLP. In

the opposite limit where dimers dominate cdim )cphase septot ) KALP. In the general case, eq 3.6 reduces tocphase septot ) KALP + 2xLP. For symmetry reasons thecomposition of the dilute phase remains unchangedbeyond the onset of phase separation: c(

coex ) xLP. Asa consequence, dimers dominate in the supernatant forxLP > 1/KA, or, up to logarithms,

The result is quite suggestive. As soon as the netcharge forces unimers to elongate they start to formdimers. Note that the domination of dimers overcharged unimers is a consequence of the reduced unimersolubility. Figure 2 illustrates the change in thecomposition of the dilute phase for symmetric bimodaldistributions of polyampholytes with different netcharges. The different curves end on a line whichindicates the total concentration at coexistence. Forneutral chains cphase sep

tot is of order exp(-N2/3)Cdense anddecreases for highly charged chains where the dilutephase consists predominately of neutral dimers toctotcoex ∼ exp(-(2N)2/3)Cdense. Thus, a symmetric en-semble of polyampholytes with high reduced chainlength N is practically insoluble. Experimentally, sucha system always consists of a supernatant in coexistencewith a precipitate. The dilute phase contains onlyspherical globules, either formed from neutral or pairsof charged polyampholytes.3.2. Formation of Higher Order Clusters. In

general, the tendency toward association increases withconcentration. In view of the low solubility of neutralsamples, we only consider the case of a phase equilib-rium where the vast majority of the polyampholytes is

precipitated into a dense phase of known compositionp(δf). The chemical potentials of the polyampholytesin the dense phase are given by eq 2.13.A polyampholyte cluster in solution can be character-

ized by a set of numbers {νδf}, which indicate thenumber of polyampholyte molecules with net charge δfthat it contains. The cluster size, the net chargedensity, and the chemical potential are ν ) ∑νδf, ∆f )(1/ν)∑δf νδf, and µ({νδf}) ) Fdense(νN) + Fex(∆f,νN) + kBTlog(c({νδf})b3/νN) + νN∆fΨ. In general, the electrostaticpotential difference Ψ does not vanish for samples withan asymmetric net charge density distribution. Atcoexistence µ({νδf}) ) ∑νδfµdense(δf), so that the concen-tration in the dilute phase is given by

For symmetric net charge distributions Ψ ) 0 and fordimers eq 3.8 reduces to eq 3.4. Since the excessinternal free energy of a cluster relative to the densephase is at least equal to its surface energy, andtherefore of order (νN)2/3, there is no association beyondthe formation of neutral dimers. This may, however,change for strongly asymmetric net charge distributions.Asymmetric net charge distributions occur when

equal amounts of positive and negative monomers arecopolymerized, but the reaction constants for the twotypes differ. Polyampholytes produced in the earlystages of the reaction contain a surplus of, e.g., posi-tively charged groups with the consequence that poly-mers formed toward the end of the reaction predomi-nantly consist of the remaining negatively chargedmonomers. Here we consider again globally neutralbimodal distributions with -δf- g δf+ g 0. Chargeneutrality determines the concentration ratio of the twotypes of molecules in the dense phase:

c+coex c-

coex ) 14Cdense

2 e-2Fex(δf,N) ) LP (3.3)

cdimcoex ) b3

2NCdense

2 e-Fex(0,2N) ) KALP (3.4)

Φ )ctot - KALP - 2 xε2ctot2 + LP

Cdense(3.5)

cphase septot )

KALP + 2xLP + ε2(KA

2LP2 - 4LP)

(1 - 4ε2)(3.6)

δf 2 > 1N

(3.7)

Figure 2. Fraction of unimers in the dilute phase versus totalconcentration for symmetric bimodal distributions of chainswith N ) 20, ga ) 10 and different net charges. For polyam-pholytes in the polyelectrolyte regime (δf g 1) the solutioncontains practically no unimers. The dashed line indicates theonset of phase separation (eq 3.6). For ctot > cphase sep

tot the ratiocuni/(cuni + cdim) remains constant.

ccoex({νδf}) ) νNb3

(Cdenseb3

N )ν

×

(∏p(δf)νδf) e-Fex(∆f,νN)-νN∆fΨ (3.8)

C+

Cdense) 11 - δf+/δf-

(3.9)

C-

Cdense) 11 - δf-/δf+

(3.10)


In fact, the same ratios also hold in the dilute phasefor c(

dil/cdil. However, the actual polymer concentra-tion, cdil, and the nature of the formed clusters dependon the net charge densities of the polyampholyte mol-ecules. To determine them, one has to calculate theelectrostatic potential difference Ψ between the denseand the dilute phase, which is implicitly determined bythe condition

of charge neutrality. Equation 3.11 is difficult toevaluate in general, since it extends over all types ofclusters.We solved eqs 3.8 and 3.11 numerically for molecules

with N ) 100 and ga ) 20. Figures 3 and 4 show thevariation of the average degree of association ⟨ν⟩ andthe total concentration cdil of the dilute phase fordifferent combinations of 0 < -δf- < 1 and 0 < δf+/δf-< 1. The plateaus in Figure 3 at values of ⟨ν⟩ ) 1, 2, 3,

... correspond to unimer, dimer, trimer, etc. dominance.The clusters form at charge ratios δf+/|δf-| ≈ 1 fordimers, δf+/|δf-| ≈ 1/2 for trimers, etc.; i.e., chargecommensurability plays an important role. In general,the solubility of the sample decreases for higher chargeson the unimers. Note, however, that the solubility ofneutral clusters is independent of the charge densitiesof their constituents. At intermediate charge ratios theconcentration cdil of the dilute phase drops by severalorders of magnitude. In general, larger clusters aredisfavored due to their surface energy and the reducedgain in translational entropy in the dilute phase. Forthis reason, they dominate the supernatant only, if theelectrostatic energy of any two fragments is too high.The arguments are the same as in the preceding section,where we found that unimers only form dimers if δf2 >1/N.Unimers dominate not only for weakly charged

polyampholytes with δf 2 < 1/N but also for very asym-metric samples. Whenever the solution is dominatedby only two components, one can evaluate a solubilityproduct and avoid the calculation of the potentialdifference Ψ. For unimers with different charge densi-ties eq 3.3 takes the form

and with eqs 3.9 one obtains for the concentration ofthe supernatant

For weakly charged chains in the spherical globuleregime the solubility corresponds to that of a sym-metrical sample with δf2 ) δf+|δf-|.The case of an almost symmetrical sample where the

unimers like to form dimers (δf(2 > 1/N) can be treatedlikewise. The slight negative excess charge of dimersin solution is compensated for by a correspondingamount of positively charged unimers. The solubilityproduct and the polymer concentration in the superna-tant are now given by

Equations 3.13 and 3.15 faithfully reproduce the nu-merical results in the regions of unimer and dimerdominance, respectively. A similar analysis can beperformed for the plateaus corresponding to higherorder clusters. However, since the values for cdil easilybecome astronomically low, there seems little point inelaborating on these effects. The main conclusion isthat higher order complexes may play a role for asym-

Figure 3. Average degree of association ⟨v⟩ in the dilute phaseat coexistence for an asymmetric bimodal distribution with N) 100, ga ) 20, and different combinations of 0 < -δf- < 1and 0 < δf+/|δf-| < 1. Integral values correspond to thepredominance of unimers, dimers, trimers, etc.

Figure 4. Total concentration of the dilute phase at coexist-ence for an asymmetric bimodal distribution with N ) 100, ga) 20, and different combinations of 0 < -δf- < 1 and 0 < δf+/|δf-| < 1.

∑{νdft}

∆f ccoex({νδf}) ) 0 (3.11)

c+|δf-| c-

δf+

C+|δf-| C-

δf+) exp(-|δf-|Fex(δf+) - δf+Fex(δf-)) (3.12)

cdil ) Cdense exp(-|δf-|Fex(δf+) + δf+Fex(δf-)

|δf-| + δf+) (3.13)

c+|δf++δf-| c+,-

δf+

C+|δf-| C-

δf+ ( N2b3)δf+

) exp(-|δf+ + δf-|Fex(δf+) -

δf+Fex(12(δf++δf-),2N)) (3.14)

cdil ) Cdense ( Nb3Cdense

)δf+/δf- |δf-|(δf+ - δf-)δf+/δf-

|δf+ + δf-|1+δf+/δf-×

(-δf+ + δf-

δf-

Fex(δf+) +δf+

δf-

Fex(12(δf+ + δf-),2N))(3.15)


metric net charge distributions without, however, en-hancing the solubility of a globally neutral polyam-pholyte sample.3.3. Non-neutral Charge Distributions: Sym-

metric Charges. We now discuss non-neutral samplescontaining free counterions that cannot be removed bydialysis. We assume that there are so few counterionsthat they modify the properties neither of the densephase nor of the dissolved globules. These points arefurther addressed in the Appendix. In this section wereturn to the example of section 3.1, a bimodal samplewith net charges of equal magnitude on all chains, buta slightly higher proportion of positively charged polyam-pholytes (ε > 0). In the following section, we considera mixture of neutral and charged chains.Qualitatively, the association equilibrium in the dilute

phase expressed via the law of mass action (eq 3.1) isnot affected by the presence of the counterions. Thebehavior of a phase-separated sample is, however,fundamentally different. In a three- or more-componentsystem the composition of the phases is not fixed by aphase rule at phase equilibrium. In the present case,for example, the unimer concentrations are not limitedindividually but are limited by a solubility product 22

(eq 3.3) and become dependent on the total concentra-tion.We first discuss the limit xLP < 1/KA, where dimers

may be neglected. Before the onset of phase separationthere are no significant changes compared to the ε ) 0case (Figure 5): c+ - c- ) 2εctot and the changes in cdimand cphase sep

tot are of order ε2. At coexistence, however,the unimer concentrations are limited by the solubilityproduct eq 3.3. For large ctot the volume fraction of theprecipitate is Φ ) (1 - 2ε)ctot/Cdense, so that the unimerconcentration in the supernatant grows like c+ ) c+,tot- CdenseΦ/2 ) 2εctot, while c- ) LP/c+ f 0.In the opposite limit, where the unimers are elongated

globules, the solution is dominated by dimers for KActot> 1 (Figure 6). For KActot > 1/4ε2 one finds cdim ) (1 -2ε)ctot and c+ ) c+,tot - cdim/2 ) 2εctot. The latter resultsalso hold at coexistence, but now the dimers start toprecipitate and their concentration in the supernatantis independent of ctot. The concentration of the minorityunimers can be calculated from the law of mass action,c- ) cdim/c+KA. Before phase separation, c- ) (1/2ε -1)/KA, i.e., independent of ctot, and at coexistence, c- )LP/2εctot f 0.In both cases the composition and concentration of

the supernatant is no longer fixed with the onset of

phase separation. Rather, the “impurities”, i.e., theadditional polyampholyte molecules with their counter-ions, accumulate in the dilute phase and dominate itsproperties for high enough total concentrations. For netcharges of δfN1/2 . 1 on the polyampholyte chains thereare three different regimes: (i) for ctot < 1/KA the dilutephase predominantly consists of unimers of both chargeswhich form elongated globules, (ii) around the onset ofphase separation for 1/KA< ctot < KALP/2ε one findsneutral dimers which form spherical globules, and (iii)finally, at even higher concentrations the supernatantagain contains mostly elongated globules, but this timeonly the majority unimers.Of course, the concentration of the supernatant can-

not increase indefinitely as the extra chains can pre-cipitate together with their counterions. The respectiveconcentrations are related by a solubility product

where we have used the fact that ci ) δf(c+ - c-), Ci )δf(C+ - C-), and the properties of the precipitate remainunaffected for small enough ε. Asymptotically, c+ - c-≈ c+, C+ ) (1/2 + ε)Cdense, and C- ) (1/2 - ε)Cdense sothat

Only for ctot > Cdense exp(-Fex/(Nδf + 1)) does thecomposition of the supernatant become independent ofthe total concentration. The effect is quite dramatic.For δfN ) 1, i.e., a single counterion per chain, Fex iseffectively reduced by a factor of 2 and c+

max corre-sponds to the c+

coex of neutral chains which are 3 timesshorter. For larger δf0, c+

max becomes independent ofchain length and reaches a maximum of order Cdense forchains at the crossover between the elongated globuleand the polyelectrolyte regimes. Qualitatively, onewould thus expect that for a sample that consists ofpolyampholyte chains with different charge densitiesthose with the highest charge accumulate in the super-natant.3.4. Non-neutral Charge Distributions: Impuri-

ties and Fractionation. In this section we considera model sample consisting of neutral polyampholytesplus a small fraction ε of charged chains together with

Figure 5. Composition of the dilute phase versus totalconcentration for asymmetric (ε ) 0.01) bimodal distributionsof chains with N ) 20, ga ) 10, and net charge δfN1/2 ) 1.

Figure 6. Composition of the dilute phase versus totalconcentration for asymmetric (ε ) 0.01) bimodal distributionsof chains with N ) 20, ga ) 10, and net charge δfN1/2 ) 3. Thegray shaded areas indicate concentrations for which the dilutephase is dominated by elongated globules.

c+(c+ - c-)Nδf

C+(C+ - C-)Nδf

e-Fex(δf,N) (3.16)

c+max ) 2εCdense e

-Fex/(Nδf+1) (3.17)


their counterions. We write the respective concentra-tions as (1-ε)ctot for the neutral chains, εctot for thecharged chains, and δfεctot for the counterions. Atinfinite dilution the solution is dominated by the neutralchains. With increasing concentration, a point is reachedwhere the neutral chains start to precipitate and thesample phase separates. We are interested in thecomposition of the supernatant, i.e., in the precipitationof the charged chains with the counterions into a largedense phase of neutral chains.Since we have only two charged components, the

counterion concentration is fixed by the neutralityconstraint: ci ) δfcδf and Ci ) δfCδf. From the solubilityproduct of charged chains and counterions ci

δfN cδf )Ci

δfNCδf exp(-Fex(δf,N)) one obtains the ratio cδf/Cδf )Xδf ) exp(-Fex(δf)/(Nδf + 1)) of the concentrations ofthe charged chains in the two phases. For the neutralchains, the usual relation c0/C0 ) X0 ) exp(-Fex(0))holds. In a phase separated system, the mass balancesand the condition for phase equilibrium take the forms

The concentrations c0 and cδf in the dilute phase arewritten in terms of ε, ctot, and Φ:

and the condition eq 3.20 for phase equilibrium can beused to eliminate Φ:

with Φ ) 0 at Cdense/ctot ) (1 - ε)X0-1 + εXδf

-1.While Figure 7 shows a plot of the concentrations in

the dilute phase obtained from these equations, themain features can be deduced from very simple argu-ments: At the onset of precipitation Φ ) 0 and ctot ≈X0Cdense, while in the opposite limit, when the systemis almost dense, Φ ) 1 and ctot ≈ Cdense. Inserting thesetwo limits into eqs 3.21 and 3.22 yields c0 ) (1 - ε)X0Cdense independent of the total concentration, whilecδf can be seen to increase from εX0Cdense to εXδfCdense.As a consequence, the composition cδf/c0 of the super-natant varies between the sample average ε/1 - ε and(ε/1 - ε)XδfX0

-1, while its concentration cdil ) cδf + c0

increases from cdil ≈ X0Cdense to cdil/Cdense ) (1 - ε)X0 +εXδf. Since we showed in the preceding section that, dueto the counterion translational entropy, Xδf can be ordersof magnitude larger than X0, the effect is quite drastic.In order to estimate the composition of the dilute

phase between these two limits, we replace the exactsolution eq 3.23 for Φ by the expression Φ0 ) (ctot/Cdense- X0)/(1 - X0)≈ ctot/Cdense obtained for ε) 0. This yields

In agreement with Figure 7 this ansatz predicts thatthe concentration of the neutral chains remains constantafter phase separation sets in, while the concentrationof charged chains cδf ) εctot keeps growing until ctot ≈XδfCdense, i.e., until they reach their solubility productwith the counterions. At higher total concentration, cδfsaturates at a value cδf ) εXδfCdense. Thus, not only isthe composition of the supernatant different from thecomposition of the sample, but it also changes signifi-cantly with the total concentration. In the present case,the supernatant is dominated by neutral chains onlyup to a total concentration of X0/ε. Beyond that, mostof the dissolved chains are charged.

4. Randomly Copolymerized Polyampholytes

A simple model for a continuous net charge densitydistribution are randomly copolymerized polyampholytes(I). We first discuss the symmetric case where chargesof both types are included with equal probability so thatthe sample is globally neutral. The case of biasedcopolymerization, where the polyampholytes have anaverage net charge, is treated along similar lines insection 5.We begin with the simplest, and as it turns out, also

the most important concentration regime: a superna-tant in equilibrium with a precipitate that is largeenough to have the same composition as the sample. Insection 4.1 we show that this regime is reached for totalconcentrations exceeding Cdense exp(-N2/3) and that thesupernatant contains practically no unimers outside thespherical globule regime. Sequential dimerization inextremely dilute solutions and the onset of precipitationare discussed in sections 4.2 and 4.3, respectively.4.1. Composition of a Supernatant in Equilib-

rium with a Large Precipitate. In this section, we

(1 - ε) ctot ) Φc0 X0-1 + (1 - Φ)c0 (3.18)

εctot ) Φcδf Xδf-1 + (1 - Φ)cδf (3.19)

Cdense ) c0 X0-1 + cδf Xδf

-1 (3.20)

c0 )(1 - ε)ctot

1 + Φ(X0-1 - 1)

(3.21)

cδf )εctot

1 + Φ(Xδf-1 - 1)

(3.22)

Φ ) 12 (ε ctot

Cdense- Xδf

1 - Xδf+(1 - ε)

ctotCdense

- X0

1 - X0) +

12[(ε ctot

Cdense- Xδf

1 - Xδf+(1 - ε)

ctotCdense

- X0

1 - X0)2 -

4X0Xδf -

ctotCdense

((1 - ε)Xδf + ε X0)

(1 - Xδf)(1 - X0)]1/2 (3.23)

Figure 7. Concentration of charged and neutral chains in thesupernatant versus total concentration.

c0 ≈ (1 - ε)X0Cdense cδf ≈εctot

1 +ctot

CdenseXδf

(3.24)


assume that for a sample of randomly copolymerizedpolyampholytes the vast majority of the chains hasprecipitated into a dense phase. The composition of thisdense phase is therefore C(δf) ) Cdensep(δf) with thesample composition given by eq 2.3 for δf0 ) 0. Insections 4.1.1 and 4.1.2 we discuss the unimer anddimer content of the supernatant. We conclude thissection by discussing the limit of validity of the assump-tion that the dense phase can be considered as infinite.4.1.1. Unimers. Equating the chemical potentials

of the polyampholytes in the dense phase (eq 2.13) tothe unimer chemical potentials in the supernatant (eq2.14) one obtains

The net charge density distribution for the dissolvedunimers is therefore

Compared to the dense phase (or to the case of infinitedilution), the net charge density distribution is N1/3

times narrower (Figure 8); i.e., its width is N-5/6

compared to N-1/2. For large N there are practically nodissolved unimers outside the spherical globule regime.4.1.2. Dimers. Besides the unimers, the superna-

tant also contains polyampholyte clusters. For example,the concentration of a particular type of dimers is givenby

The net charge distribution for the dimers can beobtained by fixing δf ′ ) 2δfdim - δf and integrating outδf in the previous expression

and has the same form as for the unimers. The actualconcentrations are, however, much smaller due to theprefactor exp(-(2N)2/3). The total concentration ofpolyampholytes with net charge δf dissolved in thedilute phase as part of a dimer is thus given by

Comparing this to the unimer concentration

we recover the result from section 3.1: for large N,polyampholyte molecules with a net charge density δflarger than the critical value N-1/2 are preferentiallydissolved as part of neutral dimers. However, theseaggregates play no role for the properties of thesupernatant: according to eq 4.2 there are more dimersformed by pairs of neutral chains than by pairs ofoppositely charged chains, and even the former are bya factor of the order of exp(-N2/3) less frequent thanneutral unimers. Thus, the supernatant is dominatedby unimers in the spherical globule regime. The deter-mination of the total concentration of charged chainsin the supernatant may nevertheless be of interest forexperiments, where the supernatant is first separatedfrom the precipitate and afterward diluted so stronglythat the dimers split into (elongated) unimers.4.1.3. Condition for a Quasi-Infinite Precipitate.

Our assumption that we know the composition of theprecipitate requires that for any charge δf, the amountof dissolved polymer in the dilute phase is much smallerthan the amount in the dense phase. In the presentcase, the supernatant is dominated by unimers and theneutral chains have the highest solubility: max(Xδf) )X0 , 1. Depletion effects due to dimer formation canbe neglected, since, according to eq 4.4, it is again theneutral chains that have the highest solubility as dimersbut with cdim(0,*) ∼ X0

2. For strictly symmetric randomcopolymerization the dense phase can therefore beregarded as infinite as soon as the total concentrationexceeds ctot,min ) Cdense exp(-N2/3) (see section 2.4).4.2. Sequential Dimerization. After having dis-

cussed the limit of total concentrations exceeding ctot,min) Cdense exp(-N2/3) we now turn to even more dilutesystems, where none or only a small portion of thematerial has precipitated. To calculate the compositionof a solution of polyampholytes of different excess

Figure 8. Probability distribution for the excess charge onthe unimers: in the supernatant when most of the chains areprecipitated (s); at infinite dilution (- - -).

c(δf) )x N2πCdense ×

exp(- δf 2N2

(1 + 2N 2/3) - N 2/3 if δf 2 < 1/N

- δf 2N2

- 2δf 2/3N if δf 2 < 1

- δf 2N2

- 2δf 4/3N if δf 2 > 1) (4.1)

puni(δf) ∼ (- δf 2N2

(1 + 2N 2/3)) δf 2 < 1/N

c(δf,δf ′) ) Cdense2p(δf) p(δf ′)

Nga3/2

×

exp(-(δf + δf ′2 )2 (2N)5/3 - (2N)2/3 if (δf + δf ′

2 )2 < 1/2N

-2(δf + δf ′2 )2/3(2N) if (δf + δf ′

2 )2 < 1 )(4.2)

cdim(δfdim) ) Cdensex 2πNga

3×

exp(-δfdim

2(2N)2

(1 + 2(2N)2/3) - (2N)2/3 if δfdim2 < 1/2N

-δfdim

2(2N)2

- 2δfdim2/3(2N) if δfdim

2 < 1 )(4.3)

cdim(δf,*) ) 12 ∫ dδf ′ c(δf,δf ′)

≈ Cdense21/6xπN11/6ga

3/2p(δf) p(-δf) e-(2N)2/3 (4.4)

cdim(δf,*)

c(δf)) 21/6xπN11/6ga

3/2p(-δf) e-(2N)2/3 + Fex(δf,N) (4.5)


charge, one has to consider the formation of all types ofdimers and higher order clusters. In the precedingsection this task was relatively simple due to ourassumption that a large dense phase of known composi-tion acted as a reservoir for the chains and thereby fixedtheir chemical potentials. In contrast, in a very dilutesystem, one has to include explicitly the conservationof the total concentrations of the various species. Wefirst consider the case of a homogeneous solution (priorto phase separation) where molecules can either bedissolved as unimers or dimers.23 In principle, all theconcentrations are coupled in the mass balances:

Due to the chemical equilibrium between dimers andunimers their concentrations are related by the law ofmass action:

Inserting eq 4.7 into eq 4.6 and evaluating the integralby the method of steepest decent show that neutraldimers are predominantly formed:

Thus, using this approximation, the concentrationsof unimers with excess charge (δf and of all dimerscontaining at least one of the chains are related by alaw of mass action with association constant

The mass balances eq 4.6 now couple only the concen-trations of molecules with opposite net charge density(δf and reduce to the same quadratic equations as inthe case of bimodal distributions, i.e.

The interpretation of these equations remains un-changed. At infinite dilution all molecules are dissolvedas unimers. At finite concentrations, however, those forwhich Kδf p(δf)ctot . 1 form dimers.4.3. Sequential Precipitation. The composition of

the dilute phase changes between the onset of phaseseparation and the limit where most of the polymer isprecipitated. After phase separation has occurred, thefollowing relations hold between the volume fraction Φof the precipitate, the total monomer concentration ctot,the polyampholyte concentrations in the precipitateC(δf), the unimer and dimer concentrations in thesupernatant c(δf) and Kδf c(δf)2, and the ratio Xδf )e-Fex(δf) of the unimer concentrations in the dilute andthe dense phase:

The unimer concentration c(δf) can be written in termsof p(δf), ctot, and Φ:

The relation between the parameter Φ and the totalconcentration ctot follows from the constraint that theindividual concentrations in the dense phase have toadd up to Cdense. It can either be expressed directly interms of the concentrations in the dense phase orindirectly via eq 4.11 in terms of the concentrations inthe supernatant:

For 1 . Φ . X0 ) exp(-N2/3) we recover the case treatedin section 4.1: eq 4.12 simplifies to C(δf) ) p(δf)ctot/Φ,since KδfXδf

2/Φ2 ∼ exp(-(2N)2/3)/Φ2 , 1 and the condi-tion for phase equilibrium, eq 4.13, yields Φ ) ctot/Cdense.The total concentration where phase separation sets

in can be obtained by setting Φ ) 0 in eq 4.12. At thispoint, the unimer concentrations in the dilute phase are

Weakly charged polyampholytes are predominantlydissolved as unimers and, accordingly, have a concen-tration of

Highly charged polyampholytes are predominantly dis-solved as dimers, and the unimer concentration arestrongly reduced:

The total concentration, ctotphase sep, at which phase sepa-

ration sets in, is determined by the condition eq 4.13for the polymer density in the dense phase. We calcu-late the individual polyampholyte concentrations in thedense phase from the unimer concentrations in thedilute phase, C(δf) ) c(δf)Xδf

-1, and approximate eq 4.15by the two limiting expressions; i.e., we assume thatmolecules with a lower (higher) net charge density thana certain δf* are exclusively dissolved as unimers(dimers) and calculate the two contributions to thedensity of the precipitate separately. Rather thancharacterizing the onset of phase separation by the totalconcentration ctot ) (p(δf*)Kδf*)-1 we use the parameterδf*. The contribution of precipitated unimers to thedense phase density, C(δf) ) p(δf)ctotXδf

-1, is monotoni-cally increasing for δf < 1, while the contribution of

precipitated dimers, C(δf) ) xp(δf)ctotKδf-1Xδf

-1 ∼xp(δf), is largest at δf ) 0. The polyampholyte con-centration in the dense phase has a sharp maximum atδf* and the two approximations match at this point. Asa consequence, the precipitate is dominated by chains

cδf + 12 ∫ dδf ′ cδfδf ′ ) p(δf)ctot (4.6)

cδfδf ′

cδfcδf ′) 2CdenseNga

3/2eFex(δf,N)+Fex(δf ′,N)-Fex((δf+δf ′)/2,2N)

(4.7)

12 ∫ dδf ′ cδfδf ′ )

x21/6πN11/6ga

3/2

cδf c-δf

Cdensee2Fex(δf,N)-(2N)

2/3(4.8)

Kδf ≡12∫dδf ′ cδfδf ′

cδfc-δf(4.9)

cδf )p(δf))p(-δf) -1 + x1 + 4Kδf p(δf) ctot

2Kδf(4.10)

ctot p(δf) ) ΦC(δf) + (1 - Φ)(c(δf) + Kδf c(δf)2) (4.11)

c(δf) ) 12Kδf ( Φ

Φ(1 - Φ)Xδf+ 1)(-1 +

x1 + 4p(δf) ctot Kδf (1 - Φ)

(ΦXδf-1 + (1 - Φ))2 ) (4.12)

∫ dδf C(δf) ) Cdense (4.13)

ctotΦ

- 1 - ΦΦ ∫ dδf c(δf) + Kδf c(δf)

2 ) Cdense (4.14)

c(δf) ) 12Kδf

(-1 + x1 + 4p(δf) ctotphase sepKδf) (4.15)

c(δf) ) p(δf) ctotphase sep p(δf) ctot

phase sepKδf , 1

c(δf) ) xp(δf) ctotphase sep Kδf-1 p(δf) ctot

phase sepKδf . 1


with a net charge density around δf* contributing, withlogarithmic accuracy, C(δf*) ) Kδf

-1Xδf-1 to the polymer

concentration. Up to prefactors, eq 4.13 therefore readsKδf

-1Xδf-1 ) Cdense or Fex(δf*) ≈ (2N)2/3. The result δf*

∼ N-1/2 implies that phase separation occurs forctotphase sep ∼ exp(-2N2/3) with the precipitate mainlyconsisting of chains with net charges around the cross-over to the elongated globule regime. Note that neglect-ing dimer formation leads to a qualitatively differentresult. Phase separation is then predicted to start withchains in the polyelectrolyte regime with a net chargeof δf ) (8/3)3/2 and to occur at much lower concentrationsof the order of exp(-N)Cdense.

5. Biased Random Copolymerization

We now discuss to the most general case of a polyam-pholyte sample with a continuous net charge distribu-tion with non-vanishing mean. As a concrete example,we consider samples synthesized by biased randomcopolymerization ((I) with 0 < δf0, i.e., a bias towardthe inclusion of positively charged monomers into thechains with the consequence that the sample alsocontains negatively charged counterions), but the resultsshould not sensitively depend on this choice. After abrief outline of the problem, we show in section 5.1 thatonly for exponentially small δf0 is the asymptoticcomposition of the phases reached for finite volumefractions 1 - Φ of the supernatant. As a consequence,most experimental systems should be in a crossoverregime between the onset of precipitation and theasymptotic regime corresponding to Φ ) 1, which isinvestigated in section 5.2. Before we discuss theconsequences for the solubility of a sample of randomlycopolymerized polyampholytes in section 5.4, we brieflyconsider complexation before the onset of precipitationin section 5.3.We have already shown in section 3 that counterions

strongly influence the composition of the phases. Quali-tatively, this is due to the fact that their behavior islargely governed by translational entropy, since theirenergy gain in precipitating into the dense phase isminute (of the order Fdense/fN) and negligible comparedto the polymer free energies. The counterions havetherefore the tendency to remain in the dilute phase.In contrast to the simple cases discussed before, thesystem now contains more than one type of chargedchains. The main question to be addressed is whichchains precipitate and which accumulate together withthe counterions in the supernatant to ensure electro-neutrality.Formally, the electroneutrality constraint leads to an

electrostatic potential difference Ψ ) Ψ/xfga (eq 2.14),which modifies the concentration ratios between thephases:

In addition, the concentrations are coupled by massbalances eqs 2.8 and 2.9. Dimers and higher order

aggregates can be neglected, because (i) beyond theonset of phase separation practically all unimers in thesupernatant have net charges of equal sign and (ii) forctot . X0Cdense the dissolution of predominately neutralaggregates (see section 5.3) cannot significantly reducethe concentrations in the dense phase. In this case

Usually one can neglect the small difference betweenthe ion concentrations in the precipitate and in thesupernatant: Fdense

ion ) -(3/211/3)(1/fga) , 1 and Ψ <1/fga , 1, since Ψ is often of order one or smaller. Infact, Φ has to be larger than Xi before the ion concentra-tion in the supernatant is appreciably reduced due toion accumulation in the dense phase. We have checkedthat in the numerical examples in section 5.2 deviationsoccur only in the physically less interesting case of analmost dense system with Φ of order 1. Furthermore,these effects do not qualitatively change the behavior,but considerably complicate the discussion of an alreadyquite intricate system. For the sake of simplicity wetherefore assume in the following that ci ) δf0ctot.For the following discussion it is useful to introduce

the polyampholyte monomer and charge densities in thesupernatant:

For a given volume fraction Φ of the dense phase thepotential difference Ψ can be calculated from theelectroneutrality condition (eq 2.11), since both cpc andci are proportional to ctot:

In a second step, the volume fraction Φ of the densephase can be related to the total concentration ctot usingthe condition that the concentrations in the dense phasehave to add up to Cdense (eqs 4.13 or 4.14).5.1. Composition of a Supernatant in Equilib-

rium with an Infinite Dense Phase. As in section 4we start with the case of a dense phase composition thatis identical with the sample composition eq 2.3. Thisreduces the problem to finding the electrostatic potentialdifference Ψ, for which the polyampholyte chargedensity in the supernatant neutralizes the counteriondensity. Figure 9 illustrates the influence of Ψ on the

Xδf ≡c(δf)

C(δf)) e-Fex(δf)+δfNΨ (5.1)

Xi ≡ciCi

) eFdenseion -Ψ ≈ 1 (5.2)

C(δf) )p(δf) ctot

Φ + (1 - Φ)Xδf(5.3)

c(δf) )p(δf) ctot

ΦXδf-1 + (1 - Φ)

(5.4)

Ci )δf0 ctot

Φ + (1 - Φ)Xi≈ δf0ctot (5.5)

ci )δf0 ctot

ΦXi-1 + (1 - Φ)

≈ δf0ctot (5.6)

cdil ) ∫ dδf c(δf) (5.7)

cpc ) ∫ dδf δf c(δf) (5.8)

cpc ) δf0 ctot (5.9)


unimer concentrations in the supernatant. In the plotwe used a symmetric Gaussian net charge distributionwith δf0 ) 0. The deviations for small 0 < δf0 ,1/xN are negligible. For all values of Ψ, where thesupernatant is dominated by chains from a well-definedpeak, one can approximate the polyampholyte (charge)density by the method of steepest decent. Up toprefactors, cpc ) δfpeakcdil and cdil ) c(δfpeak). From theneutrality condition eq 5.9 one obtains the simple result

Compared to the precipitate, the polyampholyte con-centration in the supernatant is smaller by a factor ofδf0/δfpeak. At the same time, the net charge density onthe dissolved chains is higher by a factor of δfpeak/δf0.Both phases have therefore the comparable polyam-pholyte charge densities, which are required to neutral-ize the nearly homogeneous counterion charge density.In general, an electrostatic potential difference Ψ >

0 leads to an increase of the concentrations of positivelycharged polyampholytes. The details of this process are,however, quite intricate, since the concentrations de-pend on both the gain in electrostatic potential energyδfΨN and the free energy penalty Fex(δf,N) for dissolvedchains relative to the dense phase. Most importantly,the interplay between these two terms is different inthe spherical and the elongated globule regimes.The concentrations in the supernatant show a peak

in the spherical polyampholyte regime, which dominates

for values of Ψ up to order 1 (see Figure 9). The heightand the position of this peak increase slowly with thepotential Ψ. However, the net charges on sphericalglobules can only neutralize an exponentially smallcounterion content. Using Ψmax ) 5/2 (see below) onecan estimate that δf0 has to be smaller than exp(-N2/3

+ 25/16N1/3). As a consequence, elongated globules playan important role in samples containing counterions.For values of Ψ larger than 7/3 a second maximum in

the unimer concentrations cδf appears independent ofN at the crossover between the elongated polyampholyteand the polyelectrolyte regimes at δfpeak )1. This is dueto the sublinear dependence of the excess free energyon the chain net charge density in the elongated globuleregime (eq 2.2). The maximum therefore exists quitegenerally, although its particular shape and position arean artifact of our free energy function. Note that thesolubility of elongated globules increases with their netcharge for 1 > δf > (3/4Ψ)-3 and that for values of thepotential beyond Ψ ) 2 the concentrations in thesupernatant of chains with net charge density 1 > δf >(1/2Ψ)-3 exceed those in the precipitate. In this range,small changes of Ψ have an enormous effect on theconcentrations in the dilute phase. While Ψ ) 1corresponds to an exponentially dilute supernatantdominated by spherical globules, the other extreme isreached for Ψmax ) 5/2, where c(δf ) 1) ≈ Cdense.According to eq 5.10, the density of the supernatant forδfpeak ) 1 is cdil ) δf0Cdense and therefore independentof N. Since cdil ∼ c(δfpeak) ) Cdense p(δfpeak) exp(-Fex(δfpeak,N) + δfpeakNΨ) ) Cdense exp((Ψ - 5/2)N) one findsfor the electrostatic potential difference Ψ ) 5/2 + (1/N)log(δf0). Finally, we compare the polyampholyte chargedensity in the supernatant due to the dissolution ofchains from the two peaks. The crossover to thedominance of strongly elongated chains occurs for Ψ )5/2 + 25/16N-2/3 - N-1/3, corresponding to the exponen-tially small bias of δf0 ∼ exp(-N2/3 + N1/3) , 1/xN wehad already estimated in the preceding paragraph.With respect to the applicability of the results to

experiments, one has to realize that in a realistic samplethe total concentration of chains with δf ) 1 is of orderctot exp(-N/2). Consequently, the above considerationscan only hold if these chains accumulate in an expo-nentially small volume fraction of the system. To bemore specific, we have to evaluate the criterion eq 2.16.The assumption of an infinite dense phase is justified,if there is no depletion, i.e., if the number of moleculeswith the highest solubility in the supernatant is negli-gible compared to the dense phase. In the present case(max(Xδf)) ) X1 ) exp((Ψ - 2)N). Since Ψ > 7/3 > 2,one needs 1 > ctot/Cdense > 1 - exp((2 - Ψ)N), corre-sponding to the expected exponentially small volumefractions of the supernatant. Nevertheless, the consid-erations illustrate the fact that the dissolution of highlycharged globules offers the best compromise betweenthe gain in counterion translational entropy and theincrease in the polyampholyte free energy. In thefollowing section we discuss the balance between thistendency to dissolve highly charged globules and therestrictions on their total number.5.2. Selective Precipitation. We now consider

precipitation for a sample with Nδf02 , 1 where thedense phase may be treated as infinite only for expo-nentially small volume fractions of the supernatant. Wemostly restrict ourselves to the experimentally relevantconcentration range X0Cdense , ctot , Cdense where Φ ≈ctot/Cdense. In this limit eqs 5.3 and 5.4 for the polyam-

Figure 9. Concentration c(δf) of polyampholyte unimers inthe supernatant (eq 5.1) imposing different electrostaticpotentials Ψ and a composition of the dense phase correspond-ing to a randomly copolymerized sample (eq 2.3 with δf0 ) 0).Shifting the composition to a nonzero 0 < δf0 , 1/xN hasnegligible effects on the drawn curves. Note the different scaleson the y-axis for chain lengths of N ) 20 and N ) 100,respectively. The gray shaded area indicates the elongatedpolyampholyte regime.

cdil )δf0

δfpeakCdense (5.10)


pholyte concentrations can be simplified to

showing that chains start to precipitate as soon as ctot≈ XδfCdense. Using this argument, one has, of course,to keep in mind that Ψ and therefore the Xδf depend onthe total concentration ctot.In the following we study the phase equilibrium

numerically for several cases. We also employ a simpleapproximation where chains with a particular netcharge density δf are assumed to be either completelyprecipitated or dissolved depending on the above crite-

rion. This ansatz has the advantage that the monomerconcentration cdil and the charge concentration cpc canbe expressed in terms of simple functions and allowsus to rationalize the numerical results.To illustrate the general behavior, we have solved eq

5.9 numerically for N ) 100, δf0 ) 0.01, and totalconcentrations ctot that vary over 10 orders of magni-tude. Figures 10-12 show how the composition of thephases, the electrostatic potential difference betweenthem, and the concentration of the dilute phase dependon the total concentration. We find the following:(i) At low concentrations with precipitate volume

fractions around Φ ≈ X0 the situation is very similar tothe unbiased case with δf0 ) 0 treated in section 4: thesupernatant is composed of spherical globules, whilemost of the chains with higher net charges are precipi-tated. The difference is that now on average thespherical globules in the supernatant are charged.Close to the onset of phase separation the compositionof the phases can be approximated by saying that allchains with net charge density δf < 0 and δf > δf* areprecipitated, while the dilute phase consists of thosechains with 0 < δf < δf*, i.e. c(δf) ) ctotp(δf) Θ(δf)

Figure 10. Composition of the phases for a sample with N ) 100, δf0 ) 0.01, and volume fractions of the dense phase e-N2/3 ≈2-32 e Φ ≈ ctot/Cdense e 2-4 from a numerical solution of eqs 5.1-5.9 setting ci ) δf0ctot. We show the sample composition p(δf) (- - -,eq 2.3), the composition of the dense phase C(δf)/Cdense (s, eq 5.3), and the polyampholyte concentrations c(δf)/ctot in the supernatant(- - -, 5.4).

C(δf) )X0,Φ,1 p(δf)

Cdense-1 + Xδfctot

-1 )

{p(δf) Xδf-1 ctot if ctot , Xδf Cdense

p(δf) Cdense if ctot . Xδf Cdense(5.11)

c(δf) )X0,Φ,1 p(δf)

Cdense-1 Xδf

-1 + ctot-1 )

{p(δf) ctot if ctot , Xδf Cdense

p(δf) Xδf Cdense if ctot . Xδf Cdense(5.12)


Θ(δf* - δf). To be specific, we have chosen Φ ) X0 ,1. (For a neutral sample with δf ) 0 both phasescontain at this point roughly equal amounts of polyam-pholyte.) For δf0,δf* , 1/xN the condition of chargeneutrality eq 5.9 requires δf* ) xδf0(8π/N)1/4 and

The potential difference Ψ follows from the conditionXδf* ) Φ/(1 - Φ) that both phases contain equalamounts of the chains with δf*: Ψ ) δf*N2/3.(ii) With increasing concentration and in spite of a

growing potential difference between the phases, ctotexceeds XδfCdense for the spherical globules, i.e. thesechains precipitate and their concentration in the super-natant becomes independent of the total concentration.As a consequence, their contribution to the polyam-pholyte charge density in the supernatant can no longerbalance the counterion charge density δf0ctot. (Other-wise, one would reach the asymptotic compositionsdiscussed in the previous section.)(iii) For Ψ of order 1 the most strongly charged chains

redissolve as Xδf grows faster than ctot and finallyexceeds ctot/Cdensesinitially for chains at the crossoverbetween the elongated polyampholyte and the polyelec-trolyte regimes, for larger Ψ also for chains withprogressively smaller δf. The potential difference Ψ

adapts itself to such a value that the charge density inthe tail of the distribution cancels the counterioncharges. To approximate this behavior, we assume thatall chains with a net charge density larger than acertain δf* are in the dilute phase, while all other chainsare precipitated. The polyampholyte concentrations inthe supernatant are then given by c(δf) ) ctot/(1 - Φ)p(δf) Θ(δf - δf*). For small δf0 eq 5.9 yields

xN/2(δf* ) δf0) ) -xlog(x2πN(1 - Φ)δf0). As long asΦ , 1 the supernatant has a concentration

proportional to δf0 and the total concentration ctot. Thecomposition of the supernatant is, however, only weaklydependent on ctot. The potential difference Ψ followsfrom the condition Xδf* ) Φ/(1 - Φ) that both phasescontain equal amounts of the chains with δf*:

(iv) For total concentrations of order of the concentra-tion of the dense phase and net charges where (1 - ctot/Cdense)Xδf . 1, eq 5.4 for the concentrations in thesupernatant reduces to c(δf) ) p(δf)ctot/(1 - ctot/Cdense).The relative increase occurs because chains with a totalconcentration p(δf)ctot become confined to the smallvolume (1 - ctot/Cdense) of the supernatant. Due to this“leverage”, the potential difference can actually decreasetoward the value calculated in the preceding section assmaller and smaller parts of the tail are sufficient tobalance the counterion charges. These effects can alsobe understood in the framework of our simple ap-proximation. For Φ f 1, δf* starts to grow. Due to thehigher average charge on the dissolved chains, cdildecreases. In eq 5.15 ctot≈ Cdense and cdil∼ δf0/δf*Cdense.The aymptotic case of a quasi-infinite dense phase isreached for δf* ) 1. Inserting this into eqs 5.15 and5.16 yields cdil ) δf0Cdense and Ψ ) 5/2 + (1/N)log(x2πNδf0) in agreement with what we found in thepreceding section.5.3. Complexation before the Onset of Precipi-

tation. In very dilute solutions with ctot , exp(-N2/3)charged polyampholyte molecules have a tendency toform neutral complexes. In section 3.1 we consideredbimodal distributions of polyampholytes with oppositenet charges (δf of equal magnitude. We found thatchains with charge densities δf > N-1/2 have a tendencyto form dimers at concentrations ctot ∼ Kδf

-1 much lowerthan the onset of phase separation, where Kδf is theassociation constant in the law of mass action eq 3.1.Beyond Kδf

-1 the dimer and unimer concentrations werefound to converge to cdim ) 2p(-δf)ctot, cδf ) ∆pδfctot )(p(δf) - p(-δf))ctot, and c-δf ) 2p(-δf)/Kδf∆pδf, where wehave assumed an excess of positively charged chains.In section 4.2 we showed that for a slowly varyingdistribution of polyampholyte net charges the dimer andunimer concentration were still related to each otherby a law of mass action with an effective associationconstant eq 4.9. This result, and the predominance ofneutral dimers, is not affected by a small asymmetry

Figure 11. Electrostatic potential difference between thephases for the same system as in Figure 10. The small dots inthe left half of the figure correspond to the plots in Figure 10,while the large dot and the solid line indicate our approxima-tions for Φ ) e-N2/3 and for large Φ.

Figure 12. Ratio of the supernatant and the total concentra-tion for the same system as in Figure 11. The small dotscorrespond to the plots in Figure 10, while the large dot andthe solid line indicate our approximations for Φ ) e-N2/3 andfor large Φ.

cdil/ctot ) ( N2πδf0

2)1/4 (5.13)

cdil )ctot

1 - Φ12Erfc(xN/2(δf* - δf0)) (5.14)

≈ ctotxN/2δf0

x-log(x2πN(1 - Φ)δf0)(5.15)

Ψ )log( Φ

1 - Φ) + Fex(δf*)

Nδf*(5.16)


in eq 2.3. Since for a symmetric ensemble it is sufficientto consider dimers (section 4.2, one might hope that asmall asymmetry δf0 does not fundamentally changethis result.Unfortunately, the relative excess of positively charged

chains ∆pδf/pδf approaches unity for δf > δf0-1 and islargest for highly charged chains which, in spite of theirhigh excess free energy, are predicted to be dissolvedas unimers. This suggests that the chains mightinstead prefer to form trimers by associating with twochains of half the opposite charge.24 The law of massaction for these trimers25 has the form

and the trimer formation competes with the formationof dimers from chains with charge δf/2 as well as δf/2.If we imagine concentrating the sample from infinitedilute, we can calculate threshold concentrations for theformation of these different complexes:

Comparison shows that the dimerization occurs firstand that the formation of trimers of the type {δf, -δf/2,-δf/2} competes with that of dimers composed of chainswith (δf/2. The problem becomes solvable if ∆pδf , p(-δf/2), i.e., if the trimers do not deplete the -δf/2 unimers.At the onset of phase separation, the dilute phase wouldthen, as in the symmetric case, consist of unimers withcharges between (N-1/2 with a concentration of cδf )p(δf)ctot, those with a negative charge between -N-1/2

and -1/2N-1/2 slightly depleted due to the formation oftrimers. All polyampholytes with higher charges weredissolved in complexes with cδf,-δf ) 2p(-δf)ctot andcδf,-δf/2,-δf/2 ) ∆pδfctot. Unfortunately, this condition isnot so obviously fulfilled, which will probably lead tothe formation of higher order clusters, although it seemslikely that dimers and trimers should dominate.5.4. Solubility of Polyampholyte Samples. In the

experiments of Candau et al.13,14 solutions appearedperfectly transparent and homogeneous even thoughphase separation had, in fact, occurred. Only if thesolutions were subjected to several hours of centrifuga-tion, the precipitatesrepresenting roughly half of thetotal polymer contentsbecame visible. Thus, for practi-cal purposes, exponentially small volume fractions Φ ofthe dense phase are of little consequence for the ques-tion of the solubility of a sample. In the following, wedefine solubility as the total polymer concentration ctot,where a precipitate becomes observable. As a threshold,we use quite arbitrarily a volume fraction Φ ) 10-3.As an example, we consider samples synthesized by

biased random copolymerization, but now with a biasxNδf0 of order 1 or larger. Note that this case is anidealization where the width of the net charge densitydistribution can be controlled by varying the chainlength. In experiments other factors certainly play arole as well.For the example in Figures 13 and 14 we numerically

solved eqs 5.1-5.9 including the differences in thecounterion concentrations in the two phases for fga )

10. The results are in very good agreement with whatwe obtain from the approximation introduced in section5.2, indicating that the relevant mechanism is theprecipitation of polyampholyte chains with mutuallycanceling net charge. We therefore discuss this aspectfirst, before considering the precipitation of polyam-pholyte chains together with their counterions.We assume that all chains with a net charge density

larger than a certain δf* are dissolved, while all otherchains are precipitated. The polyampholyte concentra-tions in the supernatant are then given by c(δf) ) ctot/(1 - Φ)p(δf) Θ(δf - δf*), so that

and, using ctot ) ΦCdense + (1 - Φ)cdil,

cδf,-δf/2,-δf/2

cδfcδf/22

) Lδf ∼ eFex(δf)+2Fex(δf/2)-(3N)2/3

(5.17)

1 )cδf,-δf

cδf≈ Kδf p(-δf) ctot (5.18)

1 )cδf,-δf/2,-δf/2

cδf≈ Lδf p(-δf/2)2 ctot

2 (5.19)

Figure 13. Total polymer concentration ctot of a solution witha volume fraction Φ ) 10-3 of the precipitate versus the scaledbias δf0xN for randomly copolymerized polyampholytes. Thesolid line represents the numerical solution of the approximatescheme, while the points were generated by solving numeri-cally eqs 5.1-5.9 including the differences in the counterionconcentrations in the two phases for fga ) 10.

Figure 14. Ratio of the polyampholyte concentration in thedilute phase cdil and the total polymer concentration for thesame systems as in Figure 13.

cdil )ctot

1 - Φ12Erfc(xN/2(δf* - δf0)) (5.20)

cpc )ctot

1 - Φ(δf02Erfc(xN/2(δf* - δf0)) +

1x2πN

exp(-N/2(δf* - δf0)2)) (5.21)

ctot )ΦCdense

1 - 1/2Erfc(xN/2(δf* - δf0))(5.22)


Assuming a homogeneous counterion distributionthroughout the system, δf* is determined from thecondition ci ) δf0ctot ) cpc ) ∫δf*

∞ dδf δf p(δf)ctot. Herewe discuss three points:

δf0 ) 0: For comparison we repeat the results for theunbiased case (see section 4). Such a sample is practi-cally insoluble as the concentration of the dilute phaseis limited to cdil ) exp(-N2/3)Cdense ≈ 0. For finitevolume fractions Φ of the dense phase, practically allmolecules are precipitated: ctot ) ΦCdense and cdil/ctot )0.

δf* ) δf0: The net charge density distribution is cutat its peak for xNδf0 ) x2/π(1 - Φ)/(1 - 2Φ) ≈x2/π. In this case, already half of the polymers aredissolved cdil/ctot ) 1/2, but the solubility is still extremelysmall ctot ) 2ΦCdense.xNδf0 > 1: There is a dramatic increase in solubil-

ity, if the bias is larger than the width of the net chargedensity distribution. For Nδf02 > -2 log(2xπΦ) pre-cipitation becomes observable only for solutions with atotal polymer concentration of the order of ctot ) cdil ∼Cdense. Thus, if we compare systems at the onset ofphase separation with Φ ) 10-3, then by changing thebias δf0 from zero to 3 times the standard deviation 1/xN, the concentration of the dilute phase increases byN2/3 orders of magnitude! In the same way, the solubil-ity of a sample with fixed non-zero average net chargedensity δf0 increaseswith chain length as a consequenceof the reduced width of p(δf) (Figure 15).It is instructive to compare this scenario to a uniform

sample, where all chains have the same net chargedensity δf0. In this case, one can obtain the concentra-tion ratios in the two coexisting phases from a solubilityproduct for the chains and their counterions (see section3.4):

For δf0xN < 1 the solubility increases much fasterthan for a randomly copolymerized sample. Not sur-prisingly, the uniform distribution is only a reasonablemodel for samples where the mean of the net chargedensity distribution p(δf) is much larger than the width,i.e., in the present case for δf0xN . 1. In this limit,one obtains an upper bound for the solubility due to theprecipitation of polyampholyte chains with their coun-terions:

where we have taken δf0 < 1 to be in the elongatedpolyampholyte regime. Note that around δf0 ) 3N-1/2

the exponential takes the form exp(-2(3N)1/6/xfga),becoming much smaller than 1 in the limit of large N.This sheds some doubt on our earlier remark thatrandomly copolymerized samples should become solubleas soon as their mean is larger than a few standarddeviations. However, as the number of charges perpolyampholyte blob, fga, is typically of order 10, the

chains need to have a length exceeding N ) 103 blobsfor the exponential factor to become relevant. Note thatthis implies a chain length of N ) (fga)4/f and a bias assmall as δf0 ) f/(fga)2. For all practical purposes, theprecipitation of polyampholyte chains together withtheir counterions requires concentrations of the dilutephase of the order of Cdense. In this sense, a sample canbe said to be soluble, if the width of the net chargedensity distribution p(δf) is much smaller than the meanand vice versa.

6. Discussion and Conclusion

To summarize, our results suggest that care has tobe taken in the interpretation of experiments on sampleswhich contain polyampholyte chains with net chargesof both signs. Pairs of oppositely charged chains havea strong tendency to form neutral complexes and toprecipitate. At finite concentrations it is therefore notpossible to identify the composition of the dilute phasewith the composition of the sample. In addition we finda striking contrast in the behavior of neutral and non-neutral samples. In the first case, samples are almostinsoluble with a dilute phase consisting of sphericalglobules. In the second case, the systems behave like amixture of a non-soluble, neutral part with a solublepart consisting of the most strongly charged chains andthe counterions. If the system is concentrated beyondthe onset of phase separation, the soluble part ac-cumulates in the dilute phase where the dissolvedpolyampholyte chains have the form of elongated glob-ules.It is worthwhile to illustrate the consequences using

the example of randomly charged chains (case I withδf0 ) 0). Kantor and Kardar4,5 showed that at infinitedilution the averages for quantities such as the hydro-dynamic radius are dominated by the extended chainsin the wings of the sample charge distribution. Toillustrate the effect, they presented a set of simulationsof randomly charged, but overall neutral chains, whichwere subsequently cut in the middle: The neutralchains collapsed into spherical globules, while thehalves, when separated from each other, stretched onaverage due to their non-vanishing net charge. How-ever, our analysis in section 4 shows that the leastcharged, elongated halves start to form dimers for ctot> cdense exp(-2N2/3). Neutral samples of polyampholytesare for large N practically insoluble. For total concen-trations as low as Cdense exp(-N2/3) most of the materialis precipitated, so that the compositions of the precipi-tate and the sample coincide. The supernatant has aconcentration cdil ) Cdense exp(-N2/3) independent of thetotal concentration and contains, except for highlyasymmetric samples, practically no elongated globules.Nevertheless, we find that elongated globules play an

important role in experiments. The reason is (i) thesublinear dependence of the excess free energy on thechain net charge in the elongated globule regime (eq2.2) and (ii) the fact that samples are never perfectlyself-neutralizing and therefore contain free counterions.The behavior of these counterions is dominated bytranslational entropy, and the condition of electroneu-trality (via an electro-static potential difference betweenthe phases) leads to the dissolution of charged globules.At coexistence, the composition of the supernatant isdetermined by the interplay between the the gain inelectrostatic potential energy and the free energy pen-alty for dissolved chains.

Xδf0) exp(-Fex(δf,N)/(δfNxfga + 1)) (5.23)

cdil ) CdenseXδf0(5.24)

ctot ) Cdense (Φ + (1 - Φ)Xδf0) (5.25)

ctot ) Cdense exp(-2δf0

-1/3

xfga ) (5.26)


We have shown (section 5), that in the case of randomcopolymerization an exponentially small bias leads tothe accumulation of the most strongly charged chainsin the supernatant, while the counterion concentrationremains almost uniform throughout the system. Tobalance these charges, the potential difference adaptsits value so that all chains from the tail of the samplenet charge density distribution with δf > δf* aredissolved, where δf* is determined by the condition δf0) ∫δf*

∞ dδf δf p(δf). The arguments leading to this resultare fairly general, and we therefore expect any non-neutral sample to behave like a mixture of a soluble anda non-soluble component (Figure 15). Solutions of finiteconcentration separate into an almost self-neutralizingdense phase (containing all chains with net charges ofthe same sign as the counterions plus a correspondingamount of weakly charged chains with net charges ofthe opposite sign) and a supernatant consisting ofcharged, elongated globules and counterions with aconcentration cdil ∼ ctot.Clearly, our understanding of the effects discussed in

this article can only be as good as our understanding ofthe properties of single chains in solution and of thedense phase. For example, we show in the Appendixthat for a sample with on the average less than onecounterion per polyampholyte blob, precipitated coun-terions do not change the properties of the dense phase.Higher values may, however, lead to an inflation of thedense phase and eventually to the formation of meso-phases.Our treatment of the dilute phase, on the other hand,

is based on the elongated globule model. This descrip-tion is rather crude as it characterizes a polyampholyteonly by its net charge (nevertheless, the sublinearregime in the excess free energy seems, at least on ascaling level, rather robust). In addition, long-rangeelectrostatic interactions between dissolved globules ofa solution become relevant at fairly low concentrations.Our crude arguments in the Appendix show that theresulting corrections to the chemical potential of dis-solved polyampholyte chains do not change the composi-tion of the phases. However, as a consequence of thestrong repulsion between the macroions the polyam-pholytes may arrange in structures similar to colloidalcrystals. At the same time, the globules have due totheir size relatively small surface potentials, so that theinteraction with the counterions remains weak.From an experimental point of view, it would be very

interesting to extend the calculations to solutions

containing salt, even though our considerations showthat this is not necessary in order to prepare polyam-pholyte solutions of finite concentration. Including salt,chain stiffness, or solvent effects requires work on thesingle chain level before solutions can be discussed alongthe lines of the present paper. A theory for solutions ofrodlike polyampholytes could, for example, be based onthe results by Barrat and Joanny26 for the pair interac-tion; salt and solvent effects have been discussed in theframework of the elongated globule model.1,10 For thesystems treated here, future work on the necklacemodel6,7 will hopefully lead to a more detailed pictureand, in combination with arguments along the lines ofthose given here, allow predictions on whether or notparticular charge sequences lead to more soluble mol-ecules than others.In conclusion, we believe that the surprising proper-

ties of single polyampholyte chains in solution, whichhave been, at least partially, understood over the lastyears, give rise to equally interesting multichain effectsin solutions of finite concentration. In this paper, wehave mainly addressed the composition of dilute solu-tions. We have shown that pairs of oppositely chargedchains have a strong tendency to form neutral com-plexes and to precipitate and that the solubility ofpolyampholytes is due to the translational entropy ofcounterions. The preferential dissolution of the moststrongly charged chains in a sample turns out to be adirect consequence of the way single, dissolved polyam-pholyte chains deform into elongated globules in orderto minimize the electrostatic repulsion between theirexcess charges and to preserve the energy gain fromfluctuations in the charge density.

Acknowledgment. The authors thank F. Candau,I. Erukhimovich, A. Grosberg, J. M. Mendez Alcaraz,and J. Selb for helpful discussions. R.E. gratefullyacknowledges financial support by the French Ministryof Foreign Affairs.

Appendix

A. Dense Polyampholyte Phase ContainingHigher Concentrations of Free Counterions.Throughout the paper we have assumed that theconcentrations of free counterions are too small to affectthe properties of the dense or the dilute phase. Weessentially treated the counterions as non-interactingparticles whose distribution was coupled to the polyam-pholyte concentrations through the electroneutralitycondition for the phases and otherwise dictated bytranslational entropy. In this section we discuss thepossibilities of an inflation of the dense phase due tothe counterion osmotic pressure and the formation ofmesophases.A.1. Inflation of the Dense Phase Due to the

Osmotic Pressure of Free Counterions. Considera dense phase of polyampholytes and counterions withconcentrations Cδf and Ci. Charge neutrality requiresCi ) ∫δfCδf dδf, and we assume that the counterionconcentration is much smaller than the polymer con-centration Cdense ) ∫Cδf dδf. The free energy densityconsists of a Debye-Huckel term for the polarizationenergy, which is proportional to the total charge density(fCdense + Ci)3/2, a term for the three-body repulsion, and

Figure 15. Net charge density distribution for two randomlycopolymerized polyampholyte samples with δf0 ) x2/π100and chain lengths N ) 100 and N ) 1000, respectively. Theshaded areas indicate precipitated chains.


the translational entropies of the different species:

The chemical potential for chains with net charge δf isgiven by µδf ) N(∂f/∂Cδf):

while the counterion chemical potential is calculatedfrom µion ) ∂f/∂Cion:

Finally, we obtain the osmotic pressure of the densephase Π ) ∑δfCδfµδf + Ciµi - fdense:

Throughout the paper we neglected the O(1/N) chainand the counterion osmotic pressure contributions to eqA.4 as well as the osmotic pressure of the dilute phase.Under these assumptions eq A.4 becomes independentof the composition of the dense phase. The osmoticpressure equilibrium can then be solved independentof the phase equilibrium, yielding, up to prefactors, theresult derived from scaling arguments in section 2.1:Cdense,0 ) 2-4/3b-3ga-1/2 and Fdense,0 ) -(3/28/3)NkBT.

We now include the counterion contribution to theosmotic pressure. Treating the dilute phase as an idealgas and writing the density of the precipitate as Cdense) γCdense,0 give the pressure equilibrium in the form:

Thus, the relevant measure for the counterion osmoticpressure is the number πi of excess counterions in thedense phase per blob. For πi , 1, γ tends to 1 (thebranch of solutions converging to zero is unphysical).Since the osmotic pressure of the counterions tends toswell the dense phase, γ is in general smaller than 1.

The difference of the counterion densities in the twophases for a given volume fraction Φ can be calculated

from the chemical equilibrium and the mass balance forthe counterions.

The counterion distribution in the system becomes non-uniform for Xi < 1. Since the gain in polarization energyof the counterions in the dense phase is negligible (eqA.7), the ratio ci/Ci is controlled by the electrostaticpotential barrier, Ψ, between the phases. The resultingexcess osmotic pressure in the dense phase is propor-tional to the total counterion concentration, δf0ctot.Since Ψ is a function of the sample composition and

concentration, eq A.8 is difficult to discuss in detail. Ingeneral, Ψ is large, if the counterions are forced to stayin the dense phase for want of easily dissolvablepolyampholytes with opposite charge. In that case itis justified to neglect the osmotic pressure of the dilutephase and

In many cases, we expect polyampholyte solutions tobe in the opposite limit, where the counterion concen-trations in precipitate and supernatant are of the sameorder. In section 5 we saw that the scaled potential Ψ) xfgaΨ is usually of order 1. On the other hand, fga,the number of charges per polyampholyte blob, isusually of order 10, so that Xi ≈ (1 - Ψ) can beexpanded. In this case

Independent of Ψ and Φ, the number πi of excesscounterions in the dense phase per blob is smaller thanthe ensemble average δf0ga. Thus, for a sample withon the average less than one counterion per polyam-pholyte blob, precipitated counterions do not change theproperties of the dense phase. Note that this criterionis not related to the crossover between spherical andelongated globules.In the most general case, however, the electrostatic

potential difference Ψ between the phases has to bedetermined from the condition of electroneutrality of thetotal system. Unfortunately, the polyampholyte con-centration ratios between the phases also depend on thedegree of inflation of the dense phase, since the freeenergies entering the chemical potential are changed:

Thus, the pressure and chemical equilibria cannot besolved independently.A.2. Formation of Mesophases. It has been ar-

gued27,28 that polyelectrolytes in a poor solvent may formmesophases, and this raises the question whether the

fdense

kBT) -((fCdense + Ci)lB)

3/2 + b6Cdense3 +

∑δf

Cδf

N (log(Cδfb3

N ) - 1) + Ci(log(Cib3) - 1) (A.1)

µδf

kBT) - 3

2((fCdense + Ci)lB)

1/2lBfN + 3b6Cdense2N +

log(Cδfb3

N ) (A.2)

µionkBT

) - 32((fCdense + Ci)lB)

1/2lB + log(Cib3) (A.3)

Πdense

kBT) - 1

2((fCdense + Ci)lB)

3/2 + 2b6Cdense3 +

Cdense

N+ Ci (A.4)

γ3/2 - γ3 ) πi ) 25/3gaCi - ciCdense,0

(A.5)

ci ) CiXi ≡ CieFdenseion -Ψ (A.6)

Fdenseion ∼ - 1

fgaγ1/2 , 1 (A.7)

Ci - ci )δf0ctot

Φ + (1 - Φ)Xi(1 - Xi) (A.8)

πi ∼Xi,1

δf0gaγ1

1 + XiΦ-1 < δf0ga (A.9)

πi ∼Ψ<1δf0gaγΦΨ < δf0ga (A.10)

Fdense(γ) ) Fdense,0 + ∆Fdense

∆Fdense ) - 325/3

(γ1/2 - 1)N + 328/3

(γ2 - 1)N (A.11)


effect should also occur in a dense phase of chargedpolyampholytes.In the case of polyelectrolytes (with charge density

fPE per monomer) mesophases form as a result of thecompeting effects of counterion entropy and monomer-monomer attraction. The preferred lengthscale êæ fordensity fluctuations δæ can be estimated by equatingthe elastic energy b2(δæ)2êæ

3/æêæ2 to the electrostatic

energy lB(δæ)2fPE2êæ6/êæ (the counterion distribution is

assumed to be undisturbed): êæ-4 ∼ lBb-2fPE2æ. While

êæ does not depend on the quality of the solvent for theuncharged backbone of the chains, the instability (cor-responding to a divergence of the structure factor at q) 2π/êæ) occurs only in poor enough solvents. It isimportant to note that the tendency to formmesophasesis diminished in the presence of salt, i.e., screening ofthe electrostatic interactions. The effect vanishes if theDebye-length κ-1 becomes shorter than êæ.For the purpose of screening at a given density, there

is no difference between free salt ions and the boundcharges on random polyampholytes.8 A dense phase ofcharged polyampholytes can therefore be mapped on apolyelectrolyte solution with fPE ) δf, κ2 ) lB fæ, and æ) γCdense. Mesophases can form only if δf2 > γ, i.e., fora non-inflated dense phase only for chains that are onaverage in the polyelectrolyte regime.B. More Concentrated Dilute Polyampholyte

Phases Containing Free Counterions. So far wetreated the dilute phase as an ideal gas composed ofunimers, dimers, etc. and free counterions. We nowdiscuss the leading corrections in more concentrateddilute phases as they can occur for samples containingfree counterions. In particular, we address two ques-tions (counterion condensation into dissolved globulesand corrections to the ideal gas chemical potential dueto electrostatic interactions) and find that these pro-cesses can be neglected for the purposes of the presentpaper.B.1. Counterion Condensation into Dissolved

Globules. One of the main conclusions of the paper isthat even in dilute solutions oppositely charged polyam-pholytes have a strong tendency to associate. In prin-ciple, the electrostatic energy of charged globules canalso be lowered by counterion condensation.Formally, the following treatment is analogous to the

dimer formation discussed in section 3.1; i.e., we con-sider a chemical equilibrium between dissolved globules,free counterions, and globules with a bound counterion.We need to determine the change in the globule freeenergy ∆Fex(δhf,N) to be able to write down a law of massaction. In general, this quantity is not easy to deter-mine. Here we assume that the main effect is thereduction of the net charge density of the globule fromδf to δf - 1/N or, in our scaled variables, from δf to δf- 1/xfga/N. This implies29

When the polyelectrolyte regime is disregared, ∆Fex(δf,N) has a maximum of order -N1/6/xfga for globules

at the crossover between the spherical and the elongatedpolyampholyte regimes. Following the arguments insection 3.1 and assuming a uniform sample with ci )δf0cδf mean that the globule concentrations would haveto be of the order of cion cond ) Cdense(xf/δf0) exp(-N1/6/xfga) for counterion condensation to occur. This canbe compared to the concentration where the samplephase separates (see section 5.4):

The exponent has, except for the surface energy whichdominates Fex in the spherical polyampholyte regime,the same scaling form as eq B.1, but a comparison ofthe prefactors suggests that phase separation occurs atlower concentrations than the condensation of counte-rions onto the globules. Although in a multicomponentsystem there is no simple upper limit for the concentra-tions in the dilute phase, our arguments in section 5.4show that the argument of the exponential is usuallysmaller than 1. Furthermore, we neglected the inflationeffects discussed in the preceding section as well aspossible changes in the surface energy. Both would tendto destabilize the complex so that our estimate shouldbe regarded as a lower bound on the concentration ofthe dilute phase where counterion condensation be-comes relevant.B.2. Electrostatic Interactions. The systems in-

vestigated in the paper were assumed to be salt-free,and as a consequence the deviations from the ideal gasbehavior due to long-range electrostatic interactionsbetween charged constituents become relevant at fairlylow concentrations. Since the paper mostly discussesphase equilibria, we concentrate on corrections to thechemical potential of dissolved polyampholyte chains,which might influence the composition of the phases.At sufficiently low concentrations of the dilute phase,

one can again invoke the Debye-Huckel theory for pointcharges. At higher concentrations, the strong repulsionbetween the macro-ions leads to an ordered structuresimilar to charge-stabilized colloids, which we describeusing a simple cell model. In both cases we find thatdissolved charged globules are stabilized relative to theideal gas. Within the Debye-Huckel theory, this is dueto a gain in polarization energy, at higher concentrationsdue to the reduction of the electro-static self-energy ofthe chain as the central charge is neutralized bycounterions over the distance of the cell radius.In section 5 we have argued that for non-neutral

samples the electrostatic potential difference betweenthe precipitate and the supernatant is usually highenough to lead to the dissolution of the most stronglycharged chains in spite of their high excess free energies.Since the corrections discussed in this Appendix reducethese excess free energies, our simple approximation forthe composition of the phases remains valid: all chainsthat have a net charge density beyond a certain valueδf* determined by charge neutrality are dissolved; therest precipitate. In particular, the proportionalitybetween cdil, the concentration of the dilute phase, andctot, the total polymer concentration, remains unaffected.Only the estimate for the necessary potential barrierΨ is lowered by the additional stabilization of thecharged chains in solution.B.2.1. Debye-Hu1ckel Theory. Within Debye-

Huckel theory the square of the inverse screening length

∆Fex(δf,N)kBT

)Fex(δf - 1/xfga/N,N) - Fex(δf,N)

kBT)

1

xfga{-2δfN 2/3 δf 2 < 1/N spherical globules-4/3δf

-1/3 1/N < δf 2 < 1 elongated globules-8/3δf

1/3 δf 2 > 1 polyelectrolytes(B.1)

Cdense exp(-Fex(δf,N)/(δfNxfga))


κ for a solution containing charged polyampholytechains and counterions is given by

where the dots represent dimers and higher orderclusters. Taking into account the polarization energyyields correction terms to the free energy density fdil,DH) -κdil3, the chain chemical potential µδf,DH )-3/2κlBfgaδf2N2, the ion chemical potential µi,DH )-3/2κlB, and the osmotic pressure of the dilute phaseΠdil,DH ) -1/2κdil3.It is instructive to calculate κ in the three character-

istic concentration regimes for randomly copolymerizedsamples treated in the paper.(1) At Infinite Dilution or for δf0 xN . 1.

Note, that one would expect the same result for thecorresponding simple electrolyte solutions, where all thepolymers are cut into monomers.(2) At Coexistence for a Neutral Sample.

For volume fractions Φ ≈ exp(-N2/3) this result alsoholds approximately for a sample with 0 < δf0 < 1/N1/2

and can then be written in the form κdil2 ) lBfctot/N.

(3) For ctot . exp(-N2/3) and for δf0xN , 1 wherethe most strongly charged chains of a non-neutralsample are dissolved together with the counterions:

In all cases, κdil2 is proportional lB fctot. In the singlechain regime at extremely low concentrations the pre-factor is of order 1. If the sample contains polyam-pholytes with net charges of both signs, this value isstrongly reduced due to the dimerization and precipita-tion of charged globules at small, but finite, concentra-tions and of order 1/N around the onset of phaseseparation. For higher total concentrations, when themost strongly charged chains are redissolved, the pre-factor increases again to xNδf0.The results have to be compared to the validity range

of the Debye-Huckel theory. For interaction energiesbetween globules with a net charge density δf 2 ∼ 1/Nto be smaller than kBT, the concentration of the dilutephase has to be smaller than Cdense/N2.31 At this point,the corrections to the chain chemical potential ∆µδf ∼-kBT are still negligible.Although the Debye-Huckel theory is restricted to

fairly low concentrations (the overlap concentration forthe corresponding neutral polymer solution is Cdense/xN), this is sufficient for neutral samples. The con-centrations of the dilute phase of the order of exp(-N2/3)Cdense are so low that the polarization effects can beneglected altogether. However, for non-neutral samples

the concentration of the dilute phase becomes propor-tional to the total polymer concentration and can exceedthe validity range of the Debye-Huckel theory.B.2.2. Cell Model. At concentrations F, where the

interaction between the macroions at their mean dis-tance d ) F-1/3 is much larger than kBT, the probabilityto find two macroions at distances smaller than d isstrongly reduced. This situation is typical for charged-stabilized colloids, and we adopt a simple cell model30,32to calculate the mean-field free energy for a system offixed globules and mobile counterions.The spherical globules with diameter σ and a net

charge Z homogeneously distributed over its volume areassumed to be located in the center of a spherical cellwith diameter d together with Z mobile counterions toensure charge neutrality. Instead of solving the non-linear Poisson-Boltzmann equation, we use a simplevariational ansatz along the lines of ref 32. We assumethat Z - Z* counterions are inside the globule with thevariational parameter Z* being the renormalized centralcharge. Furthermore, we assume that the counteriondistribution inside and outside the globule is uniformand that the globule radius remains unchanged.The variational free energy per cell is then given as

the sum of the counterion translational entropies in thetwo regions and the electrostatic field energy:

where Φ ) σ3/d3 is the globule volume fraction. Mini-mizing with respect to Z* yields

Since Λ ∼ ZlB/σ is the surface potential and for polyam-pholytes Φ ) cdil/Cdense, we actually recover the resultsfrom Appendix B.1 for the onset of counterion condensa-tion at small Φ. There we argued that the surfacepotential is often small and in this case Z* ) Z(1 - Φ);i.e., the counterions remain uniformly distributedthroughout the entire cell.The chemical potential of the polyampholyte globules

is nevertheless reduced relative to the case of infinitedilution or Φ ) 0, since the electric field outside theglobule decays to zero at the cell boundary (Gauss’ law).In section 2.2, we argued that the excess free energy ofa dissolved globule relative to the dense phase is thesum of the globule’s surface and electrostatic self-energies. In the elongated globule regime, both termsare of the same order and we can write:

Equation B.8 implies that the globules become morespherical with increasing concentration and fully so atΦ ) (1/2(1 - Nδf 2)-1)). However, the above consider-ations certainly break down when the long axis of theglobules is of the order of the cell diameter d, i.e., atthe much smaller overlap concentration Φ* )c*dil/Cdense ) (Nδf 2)-2. Addressing this point thereforerequires a more careful approach. The cell model alsobreaks down if with increasing density the interactionsbetween neighboring macro-ions become smaller than

κdil2 ) lBci + lB∫dδf (δfN)2

cδf

N+ ... (B.2)

κdil2 ) lBδf0ctot + lBfctotx N

2π∫dδf δf 2Ne-(δf-δf0)2N/2 )

(1 + δf 2N +|δf0|xfga)lBfctot (B.3)

κdil2 ) lBfCdensee

-N2/3x N2π∫dδf δf 2Ne-δf2N5/3

)

lBfCdense

Ne-N2/3

(B.4)

κdil2 ) x2πNδf0(1 - Φ) lBfctot (B.5)

Fvar

kBT) 310

Z*2lBσ

1 - 3/2Φ1/3 + 1/2Φ

(1 - Φ)2+ Z log(Z - Z*

Φ ) +

Z* log( Z*Z - Z*

Φ1 - Φ) - Z log(d3b3) (B.6)

Z - Z*Z

) {ΦeΛ Φ , e-Λ

Φ Φ f 1(B.7)

Fex(Φ,δf,N) ) Fex(Φ ) 0,δf,N)(1 - 2Φ1/3 + Φ) (B.8)


kBT. There are two possible mechanisms for this tohappen: (i) the counterion screening length can reducefaster than the typical macroion distance and (ii) theinteractions can be weakened by the renormalizationof the central charges. Both limits can be expressed interms of the surface potential Λ ) ZlB/σ and the centralcharge Z: the screening becomes efficient for Φ > Λ-3

and charge renormalization for Φ > 1 - 1/xZΛ. Forpolyampholytes Λ < 1 < ZΛ, so that from this point ofview the cell model remains valid for practically allconcentrations of the dilute phase.In summary, the present spherical cell model can be

trusted upto the overlap concentration of the elongatedglobules Φ ) xc*dil/Cdense ) (Nδf 2)-2, i.e., for concentra-tions of the dilute phase that are a finite fraction of theconcentration of the dense phase.

References and Notes

(1) Candau, F.; Joanny, J.-F. Synthetic polyampholytes in aque-ous solutions. In Polymeric Materials Encyclopedia; Salam-one, J. C., Ed.; CRC Press: Boca Raton, FL 1996.

(2) Higgs, P.; Joanny, J.-F. J. Chem. Phys. 1991, 94, 1543.(3) Kantor, Y.; Kardar, M. Europhys. Lett. 1991, 14, 421.(4) Kantor, Y.; Li, H.; Kardar, M. Phys. Rev. Lett. 1992, 69, 61.(5) Kantor, Y.; Kardar, M.; Li, H. Phys. Rev. E 1994, 49, 1383.(6) Kantor, Y.; Kardar, M. Europhys. Lett. 1994, 27, 643.(7) Kantor, Y.; Kardar, M. Phys. Rev. E 1995, 51, 1299.(8) Wittmer, J.; Johner, A.; Joanny, J. Europhys. Lett. 1993, 24,

263.(9) Gutin, A.; Shakhnovich, E. Phys. Rev. E 1994, 50, R3322.(10) Dobrynin, A. V.; Rubinstein, M. J. Phys. II 1995, 5, 677.(11) Levin, Y.; Barbosa, M. C. Europhys. Lett. 1995, 31, 513.(12) Corpart, J.-M.; Selb, J.; Candau, F. Polymer 1993, 34, 3873.(13) Skouri, M.; Munch, J.; Candau, S.; Neyret, S.; Candau, F.

Macromolecules 1994, 27, 69.(14) Ohlemacher, A.; Candau, F.; Munch, J. P.; Candau, S. J. J.

Polym. Sci., Phys. Ed. 1996, 34, 2747.(15) Khokhlov, A. J. Phys. A 1980, 13, 979.(16) Borue, V. Y.; Erukhimovich, I. Y. Macromolecules 1990, 23,

3625.(17) Everaers, R.; Johner, A.; Joanny, J.-F. Europhys. Lett. 1997,

37, 275.(18) For example, one recovers consistent estimates for the

screening length κ2 ) 4πlBcionz2 for a solution of ions withvalence z by considering (i) the interaction between individualcharges inside a blob (cion ) fCdense and |z| ) 1) and (ii) theinteraction between blobs (cion ) cdense/ga) with a typical netcharge |z| ) xfga.

(19) These correction terms become important, when one considersthe formation of dimers from unimers with net charges ofthe same sign. In the framework of the formalism developedbelow (see, e.g., eq 3.1), the difference of the unimer anddimer free energies 2Fex(δf,N) - Fex(δf,2N) would seem tovanish in the elongated globule regime. However, taking intoaccount the log(N) corrections, the difference is of the orderof -Fex(δf,N), i.e., dimer formation is strongly prohibited.

(20) The product fga is the number of charges per blob and, ingeneral, much larger than 1.

(21) Here we assume that the precipitate contains only a negli-gible fraction of counterions and therefore equal amounts ofchains with positive and negative net charge. This is reason-able until the concentrations in the supernatant exceed thelimit eq 3.17.

(22) Equation 3.3 is correct only as long as the counterions andthe excess chains do not start to precipitate into the densephase. In this section we estimate the validity range fromthe asymptotic case where the total concentration is so highthat precipitate and sample composition become identical (seeeq 3.17). The crossover will be treated in more detail for theexample in the following section.

(23) In light of our results for bimodal distributions, it seemsreasonable to neglect higher order clusters for (almost)symmetric charge distributions.

(24) It is not obvious that the negatively charged chains shouldhave equal charges δf/2, but other combinations only makethe problem more complicated without changing the basicphysics.

(25) As in the case of dimers the trimer free energy dominatesand leads to the formation of predominantly neutral com-plexes.

(26) Barrat, J.-L.; Joanny, J.-F. Adv. Chem. Phys. 1996, 94, 1.(27) Borue, V. Y.; Erukhimovich, I. Y. Macromolecules 1988, 21,

3240.(28) Joanny, J.-F.; Leibler, L. J. Phys. (Fr.) 1990, 51, 545.(29) On a scaling level, one obtains identical results by considering

a model where the counterions can condense into the globuleswith a homogeneously distributed net charge density (seeAppendix B.2.2) or (following Alexander et al.30) by estimatingthe counterion condensation onto the globules by equatingthe electrostatic potential Λ at the surface to the counteriontranslational entropy at large distances.

(30) Alexander, S.; et al. J. Chem. Phys. 1984, 80, 5776.(31) Charges can be treated as point-like, if their diameter is much

smaller than κ-1. This is the case as long as cdil , Cdense/N2/3;i.e., within the validity range of the Debye-Huckel theory,the globules can indeed be regarded as point-like.

(32) Safran, S.; et al. J. Phys. (Fr.) 1990, 51, 503.

MA970947U


Appendix E

Dynamic Fluctuations ofSemiflexible Filaments

101

VOLUME 82, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 3 MAY 1999

Dynamic Fluctuations of Semiflexible Filaments

R. Everaers,1,2 F. Jülicher,1 A. Ajdari,3 and A. C. Maggs31Institut Curie, Physico-Chimie Curie, UMR CNRS/IC 168, 26 rue d’Ulm, 75248 Paris Cedex 05, France

2Max-Planck-Institut für Polymerforschung, Postfach 3148, D-55021 Mainz, Germany3Physico-Chimie Théorique, Esa CNRS 7083, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France

(Received 4 August 1998)

We study both the longitudinal and transverse fluctuations of a semiflexible filament. Using scalingarguments and numerical simulations, we find several regimes for the longitudinal fluctuations which,for short times, scale askdr2

k l , t7y8 and are correlated over a lengthl2 , t1y8. Our results arepertinent to experiments on cross-linked filament systems and motor-filament assays. The techniqueswe develop for the analysis of dynamic correlations should have wide applications in the study ofpolymer systems. [S0031-9007(99)09052-3]

PACS numbers: 83.80.Lz, 83.10.Nn, 83.20.Jp

Advances in manipulation techniques of biopolymersallow the experimentalist to study and visualize the mo-tion of semiflexible filaments such as DNA, actin and mi-crotubules under the influence of thermal noise, solventflows, and forces generated by motor proteins [1,2]. Afirst step towards the understanding of mechanical proper-ties of elastic filaments are linear response functions whichare related to dynamic fluctuations, via the fluctuation-dissipation theorem. For flexible polymers, the dynam-ics is well described by the Rouse and Zimm models andgoverned by the evolution of a single intrinsic length scale[3]. For semiflexible polymers a complete theory of theanisotropic motion is still lacking.

In this Letter, we present a full characterization of thedynamic fluctuations of semiflexible filaments. We showthat the dynamics do not scale in the conventional man-ner and that two competing length scales govern the trans-verse and longitudinal fluctuations of stiff filaments. Thissurprising prediction is clearly demonstrated by numericaldata which we have generated over 12 decades in time us-ing a new simulation scheme and data-analysis techniques.Finally, we discuss situations where the newly predictedregimes for longitudinal motion are dominant.

A standard model of semiflexible polymers is thewormlike chain (WLC),H ky2

RL0 dsf≠2 $rssdy≠s2g2,

of an incompressible elastic rod with bending modulusk and arclengths. For many biopolymers the contourlengthsL of interest are of the order of the persistencelengthL . k where thermal fluctuations begin to induceappreciable deviations from a straight rod. (Here andin the following, we use units for whichkBT 1, aswell as h 1, whereh is the friction coefficient of thefilament per unit length. In these units time is a volume.For actin,k . 20 mm and1 s ø 1 mm3.) WLCs behavequalitatively different from ordinary flexible polymerswith L ¿ k, which are controlled by entropic tensions.In particular, the conformations and fluctuations of WLCsare highly anisotropic.

The static anisotropy in the WLC model is character-ized by the exponent for the growth of transverse fluc-

tuations, kdr2'l , L3yk, as a function of the filament

length L , k [4]. Because of the conservation of fila-ment length the transverse fluctuations result in lon-gitudinal fluctuationskdr2

k l , L4yk2. The fluctuationdissipation theorem relates the anisotropic fluctuations toa tensorial response function with eigenvalueslk kdr2

k landl' kdr2

'l, characterizing the effective bending andcompressional elasticities, respectively. We shall gener-alize this statement to the dynamic response.

At linear order, the dynamics of a filament is commonlydescribed by a Langevin equation for the transversefluctuations alone:

≠r'

≠t 2k

≠4r'

≠s4 1 f'ss, td . (1)

Here, f' denotes a transverse stochastic force and hy-drodynamic interactions are neglected on the ground thatthey induce only logarithmic corrections. After timet,the filament is equilibrated over a lengthl1std , sktd1y4.This results in a scaling law for the transverse motion of amonomer [5] easily observed in dynamic light scattering[6] or microrheology [7]:

kdr'std2l , l1std3yk , t3y4yk1y4. (2)

The longitudinal fluctuations of a filament are moresubtle. Two different approaches exist in the litera-ture: Approximating the incompressibility via a globalLagrange multiplier for the total length of the filament[8,9] leads to the prediction (2) for both the longi-tudinal and transverse motions, while taking the localconstraint into account leads again to scaling int3y4,however, with an amplitude smaller by a factorLyk thanin Eq. (2) [10,11]. The latter result can be found with asimple scaling argument: Each section of lengthl1std ofthe filament is independent and contributes the static valuekdr2

k lsl1d , l41yk2 to the mean square longitudinal fluc-

tuations of the end of a filament:

kdrkstd2l , sLyl1d sl41yk2d Lt3y4yk5y4. (3)

0031-9007y99y82(18)y3717(4)$15.00 © 1999 The American Physical Society 3717


The derivation of Eq. (3) neglects longitudinal friction.As demonstrated in [11], this is appropriate in a shearedsample where the longitudinal friction drops out ofthe dynamic equations leading to a scaling int3y4 inthe high frequency shear modulus. For a filament ina quiescent solvent, however, longitudinal friction andlocal incompressibility cannot be neglected [12,13]. Theshortening (or extension) of a filament section of lengthl1std also requires the longitudinal motion of its neighbors.As a consequence, longitudinal friction limits the numberof sections which can contribute within a finite time:Consider the response of a filament to a weak constantlongitudinal forcefk applied to the end. Equation (3)together with the fluctuation dissipation theorem predictsthat the end drifts asdrkstd , fkLt3y4yk5y4. However,not the whole filament is set into motion at once:The velocity scales asyk , drkyt , fkLyk5y4t1y4, butclearly the total dragLyk cannot be larger than the appliedforce. For a long filament we can resolve this paradox byassuming that the tension propagates a distancel2std ,t1y8k5y8 , L. Only a lengthl2 is set into motion witha velocity of the order ofyk , fkyl2 and the enddrifts a distancedrkstd , fkt7y8yk5y8. The fluctuation-dissipation theorem relates this response to the amplitudeof longitudinal fluctuations:

kdr2k stdl , t7y8yk5y8. (4)

Our surprising conclusion is that longitudinal and trans-verse dynamics are governed by two different dynamiclength scales [14]. This suggests that the fluctuations ofthe filament ends obey a pair of scaling relations with dif-ferent scaling arguments:

kdrkstd2l t7y8

k5y8 Q

√t1y8k5y8

L

!, (5)

kdr'std2l t3y4

k1y4 W

√t1y4k1y4

L

!. (6)

The scaling functionsQ and W are constant for smallarguments. For large argumentsx, Qsxd , x, W sxd ,x if the filament can diffuse freely.

In order to test these predictions, we have performedsimulations of a semiflexible polymer in two dimensionsimposing a constraint on the contour length using thetechnique described in [15]. The polymer is discretizedas a sequence of beads with positions$ri, i [ 0, . . . , n,with fixed distanceb j$ri 2 $ri21j and normalized bondvectors$di s$ri 2 $ri21dyb. The anglesui characterizingthe bond directions are coupled by simple angular springs:E

12

Pn21i1

k

b sui 2 ui11d2. The beads move against anisotropic friction 2b≠$riy≠t under the influence of theforces due to the angular springs,2≠Ey≠$ri and stochasticforces $Fra

i :

b≠$ri

≠t 2

≠E≠$ri

1 $Frai 1 Ti11

$di11 2 Ti$di . (7)

The tensionsTi play the role of Lagrange multiplierswhose values are calculated at each time step from thecondition that the bond lengths are equal tob.

The shortest characteristic time of the model is approxi-matelytsbd b4yk, while the relaxation time of a chainof n Lyb segments varies astn , n4tsbd. The totalsimulation time needed to equilibrate a chain is propor-tional to n5. The equilibration of long chains becomesquickly impossible.

We get around the problem of generating independentconfigurations by simulating long chains for a timefarshorter than the equilibration time but then performing en-semble averages over many short runs (see Fig. 1). Thesesimulations are useful because we can easily preparefullyequilibratedinitial conformations by drawing bond anglesdu ui 2 ui21 randomly from a Gaussian distributionPsdud , exp

°2

k

2b du2¢. The choice of the segmentation

then determines a window of accessible time scales. Theelementary time step of the integrator is1022tsbd; timesshorter thantsbd are affected by discretization errors. Wegenerated data for times up to103tsbd with a computa-tional effort proportional to onlyn1. We chose a sequenceof segmentationsbj 22jk and studied chains of lengthky8, ky4, ky2, k with the number of segments varyingbetweenn 8 and n 512. The overlap between ad-jacent time windows (tn 24tny2) provides a convenientcheck on the coarse graining procedure and the scaling ofthe parameters.

Imposing a constraint on the bond lengths providesaccess to longer times than simulations done with stifflongitudinal springs without changing the results beyondthe relaxation time of the high frequency Rouse modes(see Fig. 2) [16]. Indeed, we avoid simulating these fastbut uninteresting modes which limit the integration stepin any explicit integration scheme. There are neverthelesssevere complications [15]; we must add a pseudopotential[17], 2

12 logsDd to E and include the corresponding forces

in (7): D is the determinant of the Jacobian describing thetransformation from Cartesian to bond angle coordinates.A proper calculation of these forces is essential to ensure

FIG. 1. Cloud of end points generated by simulatingN 100realizations of the dynamics starting from the illustrated initialcondition. The moments of this cloud are used to distinguishthe longitudinal and transverse dynamics. The transversefluctuations are larger in amplitude than the longitudinalfluctuations.

3718


10-10

10-8

10-6

10-4

10-2

1

10-12 10-8 10-4 10-8 10-4 1

< δ

r2 >

/ κ2

t/κ3

t1/2 t1/2

t3/4 t7/8 t7/8

(a) (b)

FIG. 2. (a) Amplitude of a filament end fluctuations as afunction of time for an incompressible filament (black symbols,L k) determined from the momentsl1std, l2std of the 2Dclouds. The transverse fluctuations scale askdr2

'l l1std ,t3y4, while the longitudinal modes obey initiallykdr2

k l l2std , t7y8. The scaling of the crossover to free diffusionat L l2std is treated in Fig. 3. For compressible filaments(grey symbols,L ky4, modulus K 107k21) a Rouse-like scaling l2std kdr2

k l , t1y2 is found for short times.(b) Isotropic fluctuations of a cross-linked filament pair scale askdr2

3l , t7y8. Different symbols correspond to different levelsof coarse graining.

that we start our short runs from initial conformationswhich are properly equilibrated.

The objective of our simulations is the characterizationof the transverse and longitudinal motion and the fulltensorial response of the chain ends. For this purpose weperformN simulations (typicallyN 1000) starting froman identical preequilibrated conformation; each simulationuses an independent series of random forces. We thenrecord theN coordinates of one end of the chain as afunction of time which form an evolving two-dimensionalcloud in thesx, yd plane. The moments,l1 . l2, and axisof inertia of the point cloud characterize the amplitudeand direction of transverse and longitudinal movement,respectively, for the given initial conformation (see Fig. 1).

We prepare a total ofM random realizations (typicallyM 100) of the initial chain over which we can calcu-late average properties of the cloud performing a total ofMN 105 simulations [see Fig. 2(a)]. For short times,the evolution of the cloud is very anisotropic, and thetransverse dynamics corresponds to the larger momentl1std kdr2

'l which scales according to Eq. (2). Thesmaller momentl2std kdr2

k l characterizes the longitu-dinal motions of the filament and for short times varies inagreement with our prediction [Eq. (4)]. For long timeswith l2 . L, a length-dependent crossover to free dif-fusion of the whole filament occurs (see Fig. 3 for thecrossover scaling). In addition, the analysis shows thatthe direction of the longitudinal motion is initially parallelto the local tangent and relaxes only with time towards

0.01

0.1

1

0.1 1

Q{3

,4}

l2(t)/L

1

-1

Λ3

Λ4

FIG. 3. Scaling behavior of the two smallest momentsL3,4 ofthe 4D clouds. The scaled amplitudeQh3,4j Lh3,4jk

5y8t27y8

is shown as a function of the tension propagation lengthl2std t1y8k5y8 for L k (e), ky2 (1), ky4 (h), ky8 (3).The plot confirms the scaling form of Eq. (5).

the average orientation of the whole filament. This tiltangle between the initial tangent and the average ori-entation of the chain is of the order ofkc2l , Lyk.Relaxation to the average orientation occurs only whenl1std , L. Fluctuations projected onto the average ori-entation of the whole filament,kdr2

k l 1 kc2l kdr2'l, are

dominated by the contribution oftransversemotion forshort times. They are an incorrect measure of the under-lying longitudinal dynamics. However, fluctuations pro-jected on the initial local tangent vector are dominated bythe longitudinal dynamics and scale ast7y8.

To better understand the tension propagation, we exam-ined the joint motion of the two end points of a chain. Ina series ofM simulations, one generates a distribution ofpoints in four dimensions (4D):hfxs0d, xsLd, ys0d, ysLdgj.We are interested in the evolution of the four momentshL1 . L2 . L3 . L4j of the 4D cloud which character-ize the dynamics of the whole chain.

Our picture of the propagation of tension fluctuationsas introduced above suggests the following scenario:As long as l2 , L, the movement of the ends areuncorrelated. The 4D distribution factorizes and reducesto a product of the 2D case discussed above: Thetwo smaller momentsL3 . L4 . kdr2

k l and scale ast7y8yk5y8; the two larger momentsL1 . L2 . kdr2

'lscale ast3y4yk1y4. For longer timesl2std . L the twoends see each other as tension propagates along thefilament. This lifts the degeneracy betweenL3 and L4,with L4 now characterizing the end to end fluctuationsso that according to (3)L4 , Lt3y4yk5y4, and L3 ,tyL characterizing the longitudinal free diffusion of thechain. Figure 3 shows the momentsL3 and L4 plottednormalized byt7y8yk5y8 as functions oft1y8k5y8yL, sothat they clearly follow the scaling form Eq. (5) [18].

How can one observe the motion corresponding toL3,4

which scales ast7y8? Naively, one might expect it to

3719


be subdominant compared toL1,2 in most experimen-tal situations. However, a case where it dominates is apair of cross-linked filaments confined to a quasi-two-dimensional region (a standard experimental setup forfluorescence microscope studies of actin filaments). Themean-square displacements of the cross-link are deter-mined by the sum of theinverseresponse functions of thetwo chains (the sum of the effective elastic moduli) and isthus dominated by the stiff longitudinal response. Cross-link motion should therefore scale askdr2

3l , t7y8. Wehave checked this argument by simulating filament pairsof length L in two dimensions cross-linked perpendicu-larly at their midpoints [see Fig. 2(b)]. The motion of thecross-link is isotropic, and its amplitudekdr2

3l is indeednumerically almost identical tokdr2

k l for a single filament.This example of the cross-link demonstrates that the fullresponse function is needed to estimate effective elastici-ties in cross-linked geometries, the knowledge of a singlecomponent of the tensor being insufficient.

Anisotropic response functions are also expected to beimportant for understanding filament pulling experimentsusing micromanipulation techniques or for studies of theaction of myosin molecular motors. Myosin moleculescan generate forces and displacements parallel to thelocal tangent of actin filaments. The resulting filamentmotion is then determined by the longitudinal responsefunction discussed above. In some micromanipulationexperiments on myosin function, actin filaments are usedas force transducers [2]: The motion of one of the endsis an indicator for forces induced somewhere along thefilament. The first response should be due to tensionpropagation and our prediction forl2std should thuscharacterize the transmission of information in a dynamicfilament. Indeed tension propagation leads to an effectivelow-frequency filter since filament motion observed at adistanceL is characterized by the longitudinal responsefunction which is zero for times shorter thanL8yk5.

The anisotropic scaling of the dynamics predicted herecould also be observable in incoherent neutron scatteringexperiments on stiff synthetic polymers. Recent workhas shown that effects of semiflexibility are observablein these systems [9]. We expect that for long times theincoherent structure factor shows a behavior characterizedby longitudinal modes.

Finally, we have presented a novel scheme of simu-lation and data analysis. Rather than following a singletrajectory in configuration space for a very long time, weperform short, independent runs from equilibrated initialconfigurations and gain access to a wide range of timescales by varying the discretization of the model. Toproperly account for the anisotropy, we obtain the dy-namic correlations by averaging over independent runsfrom identical initial configuration,beforecalculating en-

semble averages. These techniques have the advantage ofdirectly separating the nature of the different relaxationprocesses in our simulations and will also be useful forother machine studies of polymer dynamics.

We would like to thank Paul Chaikin, Fred Gittes,Fred Mackintosh, David Morse, and Jacques Prost fordiscussions on this work.

[1] F. Gittes, B. Mickey, J. Nettleton, and J. Howard, J. CellBiol. 120, 923 (1993); T. T. Perkins, D. E. Smith, andS. Chu, Science264, 819 (1994); J. Kaset al., Biophys. J.70, 609 (1996).

[2] J. T. Finer, R. M. Simmons, and J. A. Spudich, Nature(London)368, 113 (1994).

[3] M. Doi and S. F. Edwards,The Theory of PolymerDynamics(Clarendon Press, Oxford, 1986).

[4] T. Odjik, Macromolecules16, 1340 (1983).[5] E. Farge and A. C. Maggs, Macromolecules26, 5041

(1993).[6] C. F. Schmidt, M. Bärmann, G. Isenberg, and E. Sack-

mann, Macromolecules22, 3638 (1989).[7] F. Amblard, A. C. Maggs , B. Yurke, A. Pargellis, and

S. Leibler, Phys. Rev. Lett.77, 4470 (1996).[8] J. Harris and J. Hearst, Chem. Phys.44, 2595 (1966).[9] L. Harnau, R. G. Winkler, and P. Reineker, J. Chem. Phys.

106, 2469 (1997).[10] F. Gittes and F. Mackintosh, Phys. Rev. E58, R1241

(1998); R. Granek, J. Phys. II (France)7, 1761 (1997).[11] D. Morse, Phys. Rev. E58, R1237 (1998); Macro-

molecules 31, 7030 (1998); Macromolecules31, 7044(1998).

[12] U. Seifert, W. Wintz, and P. Nelson, Phys. Rev. Lett.77,5389 (1996).

[13] A. Ajdari, F. Jülicher, and A. C. Maggs, J. Phys. I (France)47, 1823 (1997).

[14] The scaling with two length scales is related to theexpansion about an ordered (straight) state in a systemwith no real long range order. An analogous situationexists in the dynamics of the one- and two-dimensionalHeisenberg model [see G. Reiter, Phys. Rev. B21, 5356(1980)].

[15] E. J. Hinch, J. Fluid Mech.271, 219 (1994); P. S. Grassia,E. J. Hinch, and L. C. Nitsche, J. Fluid Mech.282, 373(1995); P. S. Grassia and E. J. Hinch, J. Fluid Mech.308,255 (1996).

[16] In three dimensions there is also a torsional mode whichproduces even richer behavior and will be treated in afuture publication.

[17] M. Fixman, J. Chem. Phys.69, 1527 (1978).[18] The short time splitting betweenL3 and L4 in Fig. 3 is

partly due to fluctuations of order of1yN1y2 and partlyintrinsic due to the nonharmonicity of the correspondingelasticity.

3720

Appendix F

Constrained FluctuationTheories of Rubber Elasticity:General Results and an ExactlySolvable Model

107

Eur. Phys. J. B 4, 341–350 (1998) THE EUROPEANPHYSICAL JOURNAL Bc©

EDP SciencesSpringer-Verlag 1998

Constrained fluctuation theories of rubber elasticity:General results and an exactly solvable model

R. Everaersa

Institut Charles Sadron, 6 rue Boussingault, 67083 Strasbourg Cedex, FranceandMax-Planck-Institut fur Polymerforschung, Postfach 3148, 55021 Mainz, Germany

Received: 28 November 1997 / Received in final form and accepted: 31 March 1998

Abstract. We present a new model of rubber elasticity where linear forces act to constrain the fluctuationsof the eigenmodes of the phantom model. The model allows us to treat the constrained junction andthe tube model within the same, transparent formalism, does not require any further approximations,and is particularly suited for the analysis of simulation data for (strained) model polymer networks. As aninteresting side result we show that in order for the model to be consistent, the constraints (but not the meanpolymer conformations!) have to deform affinely, a severe restriction that might also apply to other models.Complementary, we prove in analogy to the derivation of the virial theorem that introducing constraintsinto the phantom network Hamiltonian leads to extra terms in addition to the usual Doi-Edwards formulasfor the polymer contribution to the stress tensor which vanish only for affinely deforming constraints.

PACS. 05.40.+j Fluctuation phenomena, random processes, and Brownian motion – 62.20.Dc Elasticity,elastic constants – 61.41.+e Polymers, elastomers, and plastics

1 Introduction

Most current models of rubber elasticity are based on thephantom model [1–3] combined with the idea that entan-glements between the polymer chains reduce the fluctu-ations [4–21]. Being deformation dependent, the effectiveconstraints then contribute to the elastic properties of thenetwork. What divides the community, however, is thequestion which fluctuations are affected by the mutual im-penetrability of the network strands.

The classical theories [1–9] date back more than halfa century and can qualitatively explain many aspects ofthe physics of rubber elasticity. They are based on theobservation that a flexible, randomly coiled polymer in amelt with a mean-square end-to-end distance

⟨r2⟩

can beviewed as a linear entropic spring with spring constant

k =3kBT

〈r2〉and assume that the elastic response of rub-

ber has its sole origin in the elongation of the networkstrands. In the two original classical models, the phan-tom model and the junction affine model, the junctionpoints are considered to be free to fluctuate or fixed inspace respectively. Later refinements considered partiallyrestricted fluctuations.

Edwards [10] argued in 1967 that the average effect ofthe complicated topological constraints on the conforma-

a Present address: Institut Curie, Section de recherche, 11rue Pierre et Marie Curie, 75231 Paris Cedex 05, France.e-mail: [email protected]

tion of a network strand is to confine the strand to theneighborhood of a mean conformation. The tube modelprovides on a mean-field level a unified view on networksand entangled polymer melts [11–14]. There is convincingevidence from simulations [22–25] as well as from neu-tron scattering experiments [26–30] for the existence ofthe tube, but it still remains to be shown that it is possi-ble to calculate the elastic response from this ansatz.

That the importance of entanglements is still some-times disputed is due to the inconclusiveness of rheologi-cal experiments [31–37]. It is very complicated to preparewell-characterized model systems and to relate experimen-tal data to structural properties or to analytical theory ina unique way. Due to the lack of microscopic informationtheories can only be tested indirectly by comparing mea-sured and predicted stress-strain curves or by extrapolat-ing the moduli to the limit of vanishing cross-link density.This has lead to ambiguous and contradictory results. Al-though SANS investigations of deformed networks havepartially closed this gap [29,30], we have argued [38], thatdue to the direct and simultaneous accessibility of micro-scopic and macroscopic information, computer simulations(for a recent review see Ref. [39]) are in a much better po-sition to decide these issues. In the ideal case, it shouldbe possible to test statistical mechanical models of rubberelasticity without using any adjustable parameters. Twoquestions are of principal interest:

1. How do the microscopic chain conformations changein response to a macroscopic deformation?

342 The European Physical Journal B

2. What is the relation between the (change of the) mi-croscopic conformations and macroscopically observedstresses and deformation free energies?

Theories worked out for a comparison to experimentaldata are not necessarily well-suited for a quantitative testof the underlying ideas. For example, many theories de-scribe limiting cases such as networks with strand lengthsmuch smaller or larger than the melt entanglement length,as it is the case for the constrained junction and tube mod-els respectively. Simulations and experiments, on the otherhand, often fall into the crossover region between these twoextremes. The purpose of the present paper is the intro-duction of a conceptually transparent framework for theanalysis of constrained fluctuations and their contributionto the shear elastic properties of a polymer network and toillustrate some subtleties concerning the second questionrelated to the presence of constraints.

The paper is organized as follows. First we review thephantom model which treats the network strands as idealentropic springs and completely ignores entanglements. Inthe phantom model, the eigenmodes only depend on theconnectivity and not on the size and shape of the net-work and therefore do not contribute to the shear modu-lus. In Section 3 we introduce the constrained mode model(CMM) where linear forces act to constrain the fluctua-tions of the eigenmodes of the phantom model. The modelis exactly solvable and makes no assumptions on the lengthof the network strands. In the discussion (for details seethe appendix) in Section 4 we point out how the con-strained junction and the tube model can be obtainedas simple limiting cases of the CMM. We close by somegeneral considerations concerning the deformation depen-dence of the constraints. Quite surprisingly, one is withinthe CMM almost restricted to the simplest possible choice,affinely deforming constraints. In particular, most of thefunctional forms which were used in other constrained flu-cutation models of rubber elasticity in order to explainnon-affine chain deformations and to improve the agree-ment with measured stress-strain curves lead to incon-sistencies in the present case. While we have no generalarguments against the use of non-affinely deforming con-straints, we show that they generate extra terms in thestress tensor in addition to the usual Doi-Edwards ex-pression. A quantitative test of the CMM and of the Doi-Edwards formula in computer simulations of model poly-mer networks will be presented in a future publication [40].

2 The phantom model

The Hamiltonian of the phantom model [1–3] is given by

Hph =k

2

∑〈i,j<i〉

r2ij , (1)

where 〈i, j < i〉 denotes a pair of nodes which are con-nected by a polymer chain and rij(t) = ri(t) − rj(t) thedistance between them. Due to the linearity of the springsthe problem separates in Cartesian co-ordinates x, y, z.

Furthermore, a conformation of a network of harmonicsprings can be analyzed in terms of either the bead posi-tions ri(t) or the deviations ui(t) of the nodes from theirequilibrium positions1 Ri. The latter are characterized bya force equilibrium

∑j Rij ≡ 0, where j indexes all nodes

which are connected with node i. In this representation,the Hamiltonian separates into two independent contribu-tions from the equilibrium extensions of the springs andthe fluctuations, where the latter can be written as a sumover independent normal modes or phonons up [41,42]:

Hph =k

2

∑〈i,j<i〉

Rij2 +

k

2

∑〈i,j<i〉

uij2 (2)

=k

2

∑〈i,j<i〉

Rij2 +

kp

2

∑p

up2. (3)

In the second form, the calculation of the shear modulusof the phantom model from the deformation dependenceof the free energy is formally straightforward:

Fph(λ) =k

2

∑〈i,j<i〉

Rij2(λ)

− kBT∑p

log

(∫d3upe

−1

kBTkp2 up

2). (4)

Elastic properties

In this paper we always consider uni-axial elongations

←→λ =

λ

1/√λ

1/√λ

. (5)

This volume-conserving deformation (det(←→λ ) = 1) is the

standard choice [4] for rubber-like materials, since theycan be considered to be incompressible with a Poissonratio of 1

2 . The corresponding Lagrangian strain tensor is

←→ε =1

2

(←→λt←→λ − 1

), (6)

where the superscript t indicates matrix transpose. Thedeformation dependent free energy and the shear modulus

1 Formally, a phantom network collapses to the size of onenetwork strand when the Hamiltonian (1) is used with openboundary conditions. In reality, this collapse is prohibited byvolume interactions between the chains. Within the phantomnetwork model one can avoid the collapse by either fixing somejunction points on the sample surface [1,2], or, more conve-niently, by spanning the network over a fixed volume with thehelp of periodic boundary conditions. See R.T. Deam, S.F. Ed-wards, Phil. Trans. R. Soc. A 280, 317 (1976).

R. Everaers: Constrained fluctuations in rubber elasticity theory 343

of the phantom model are then given by

Fph(λ) = E0(λ) +3kBT

2

∑p

log

(kp

kBT

)(7)

E0(λ) =k

2

∑〈i,j<i〉

(←→λ Rij

)2

(8)

Gph =1

3

1

V

d2Fph

dλ2

∣∣∣∣λ=1

=1

3

1

V

d2E0

dλ2

∣∣∣∣λ=1

=〈R2

strand〉

〈r2〉ρstrand kBT (9)

where⟨r2⟩

the mean square end-to-end distance of the

un-cross-linked strands in a melt, and 〈R2strand〉 = 〈R2

ij〉the expectation value of the square of the mean extensionof the network strands. The interpretation of equations(7, 9) is simple. If a sample is deformed, the equilibriumpositions of the junction points change affinely (Eq. (8)).The fluctuations, on the other hand, depend only on theconnectivity but not on size and shape of the network.The shear modulus of the phantom model can thereforebe calculated without having to integrate out the dynamiceigenmodes of the network.

It is easy to see that dividing the network strandsinto Gaussian sub-strands (i.e. formally introducing addi-tional, two-functional junction points along the strands)changes nothing for a phantom network. The spring con-stant is doubled for a strand of half the original length,i.e. if an entropic spring of spring constant k is replacedby a linear sequence of N springs the latter have a springconstant of Nk. Furthermore, the equilibrium positions ofthe new cross-links are along the line connecting the equi-librium positions of the original endpoints. Since 〈R2

N 〉 =1N2 〈R2〉, one finds kN

∑Ni=1〈R

2N 〉 = k〈R2〉 and the pre-

dicted modulus remains unchanged.

Finite deformations change the free energy density ofthe phantom model by

1

V∆Fph =

1

2

(λ2 +

2

λ− 3)Gph, (10)

and give rise to a normal tension 〈σT 〉 =⟨σxx −

12 (σyy + σzz)

⟩, where the σαα are the di-

agonal elements of the microscopic stress tensor

〈←→σ 〉 = 1V

←→λ ∂F

∂←→ε

←→λt. This can be shown in general

in analogy to the virial theorem (see Sect. 4). For thespecial case of classical rubber elasticity, equation (10),one finds

〈σT 〉 (t→∞) =(λ2 −

1

λ

)G (11)

by writing the change in the free energy density for a de-formation of a sample of size L3

0 in terms of normal forcesFαα acting on the side walls of the sample and expressing

the forces via pressures:

L30df = GL3

0(λ− λ−2)dλ = Fxxdx+ Fyydy + Fzzdz

=(Fxx − (Fyy + Fzz)/(2λ

3/2))L0dλ

=

(σxxL

20/λ−

1

2(σyy + σzz)λ

1/2L20/λ

3/2

)L0dλ

= σTL30λ−1dλ.

Fluctuations

Although the eigenmodes do not contribute to the shearmodulus of the phantom model, they have neverthelessreceived considerable attention [3,41–43]. This was due tothe problem of how to calculate 〈Rij

2〉 for a randomlycross-linked network. Here we re-derive the result fromsome simple and general considerations.

Consider a randomly cross-linked melt and a melt ofthe unconnected strands. For instantaneous cross-linkingthe statistics of the strand conformations and, in particu-lar, 〈r2

ij〉 is the same in both cases [4]. The precise valuefollows from applying the equipartition theorem to themelt of unconnected strands: k

2 〈r2ij〉 = 3

2kBT . The inter-nal “energy” of the network, on the other hand, is given bythe energy of the equilibrium conformation E0 = k

2 〈R2ij〉

plus 12kBT per fluctuating mode. The number of modes

equals three times the number of junction points and lat-ter is for an f -functional network given by 2

f times the

number of strands. This corresponds to a thermal energyof 3 × 2

f ×12kBT per strand, i.e. k

2 〈u2ij〉 = 3

f kBT and

k2 〈R

2ij〉 =

(32 −

3f

)kBT :

〈u2ij〉 =

2

f

⟨r2⟩

(12)

〈R2ij〉 =

(1−

2

f

)⟨r2⟩· (13)

As a result, the shear modulus of the phantom model canbe written in a form, where it only depends on the stranddensity and the functionality of the network [3]:

Gph =

(1−

2

f

)ρstrand kBT. (14)

In principle, one can calculate the exact eigenmodes orgeneralized Rouse modes from the knowledge of the con-nectivity matrix for a particular network [42,44]. In theappendix we propose an intuitive approximation, wherethe movements of the junction points and of the strandsbetween them are considered to be independent.

3 The constrained mode model

The phantom model completely neglects “entanglements”between network strands due to their mutual impene-trability. The classical view of the problem, associated


with the name of Flory, is to assume that their main ef-fect is a partial suppression of the junction fluctuations[5–9]. The non-classical theories of rubber elasticity dis-cuss constraints like tubes [11–14] or slip-links [15,21,45]that in addition also restrict the fluctuations of the strandsbetween the junction points.

Often these constraints are introduced as additional,single-node terms into the phantom model Hamiltonian,which constrain the movement of the monomers and junc-tion points. The standard choice are anisotropic, harmonicsprings of strength

←→l (λ) between the nodes and points

ξi(λ) which are fixed in space:

Hconstr =∑i

1

2(ri − ξi(λ))

t ←→l (λ) (ri − ξi(λ)) . (15)

Equivalently one can restrict the nodes to box-like regionsof width δ(λ) = (δx(λ), δy(λ), δz(λ)):

exp (−Hconstr/kBT ) =∏i

Θ(ri − (ξi(λ)

−1

2δ(λ))

)Θ

((ξi(λ) +

1

2δ(λ)) − ri

), (16)

where Θ(r) ≡ Θ(x)Θ(y)Θ(z) is the product of the stepfunctions of the components of the vector.

Here we study a slightly different constraint model,where instead of restricting the motion of individualnodes, deformation dependent generalized forces coupleto the eigenmodes up of the phantom network:

Hconstr=∑p

1

2(up−vp(λ))t

←→lp (λ) (up−vp(λ)). (17)

The idea to base the analysis on (single chain) eigenmodesis not new, but has apparently not been fully exploited.Edwards and Vilgis [13] proposed a tube model where theprimitive path is given by constrained long-wavelengtheigen (Rouse) mode of the chain and the fluctuations aredue to unimpeded short-wavelength Rouse modes. In theappendix we derive analoguous results in the framework ofour “constrained mode model” (CMM). In a similar spiritDuering, Kremer, and Grest [25,46] tried to determine theshear elastic properties of rubber-like materials from thepartial relaxation of (approximate) eigenmodes in the un-deformed state. Here we provide a more formal base forthis ansatz and as a result propose a modified formula forthe shear relaxation modulus.

As for the comparison of the confinement generatedby the two types of constraining Hamiltonians, equations(15, 17), we note that both can be used to model straighttubes with unimpeded fluctuations of the chain parallelto the tube axes. And while both models allow one to ar-bitrarily displace the average positions of the monomersfrom their equilibrium positions in the phantom model,both also loose the anisotropic character of the monomerdiffusion parallel and perpendicular to a twisted, ran-dom walk-like tube. In what they differ is that equa-tion (15) neglects correlations between the restoring forces

acting on different monomers (and therefore suppressesalso short-wavelength fluctuations), while equation (17)neglects that correlations between different Rouse modesallow to displace the chain along the tube axes withoutgenerating restoring forces. However, while neither modelis exact, the CMM equation (17) has the big advantagethat it can be solved without further approximations.

As the sum of the phantom and constrained modemodel Hamiltonians equations (1, 17) is diagonal andquadratic in the modes, the model is straightforward tosolve. Consider one Cartesian component α of a particu-lar mode up. Under the influence of the constraining force,the mode has a non-zero mean excitation

Upα(λ) =vpα(λ)

kp/lpαα(λ) + 1(18)

and the fluctuations around this mean value are reduced

〈δu2pα〉 ≡ 〈(upα − Upα(λ))2〉 =

kBT

kp + lpαα(λ)· (19)

In the most general case the deformation dependent freeenergy has the form

Fpα(λ) =kp

2Upα(λ)2 +

lpαα(λ)

2(vpα(λ) − Upα(λ))

2

+kBT

2log

(kp + lpαα(λ)

kBT

), (20)

while the contribution of the pth mode to the shear mod-

ulus is given by1

3

d2

dλ2(Fpx + Fpy + Fpz). The first two

terms in equation (20) represent elastic energies storedin the mean excitations of the mode and the constrain-ing spring respectively. In contrast to the phantom model,equation (7), the log-term, representing the fluctuations,is deformation dependent.

The parameters of any constrained fluctuation modelhave to be chosen in such a way that the average strandconformations in the unstrained state remain unchangedcompared to the phantom model. This condition is veryeasy to express for the constrained mode model. Whilein the absence of constraints a mode undergoes thermal

fluctuations of width 〈u2pα〉 =

kBT

kp, the excitations in

the constrained mode model are the sum of the non-vanishing mean excitations equation (18) and the (re-duced) thermal excitations equation (19). By demandingthat 〈u2

pα〉 = 〈U2pα〉+ 〈δu2

pα〉 one obtains

〈v2pα〉 =

1

γp〈u2pα〉

〈U2pα〉 = γp〈u

2pα〉

〈δu2pα〉 = (1− γp)〈u

2pα〉

γp =lp

kp + lp·

The parameter γp measures how strongly the fluctuationsof the pth mode are confined. The extreme cases are γp = 0


and γp = 1 corresponding to completely free and com-pletely frozen fluctuations respectively. Note, that aver-aging over the quenched disorder (free energies etc. arecalculated with fixed vectors vp(λ) assigned to each modeand averaged afterwards over the (Gaussian) distributionof constraints characterized by 〈v2

pα〉) does not require anyfurther approximations.

The limiting cases impose certain restrictions on the

deformation dependence of the parameters←→lp (λ) and

vp(λ). The following arguments apply, strictly speaking,only to the constrained mode model equation (17), sinceprevious constrained junction and tube models introducedconstraints on the motion of individual monomers, equa-tions (15, 16). However as we will show that relationswhich have been used in order to improve the agreementwith experimentally measured stress-strain curves lead toinconsistencies in the present case, one might wonder, ifthese problems are more general and not just an artifactof the slightly different manner the constraints are intro-duced in equation (17).

As already noted by Flory, the centers of the con-straints have to move affinely with the sample

vp(λ) =←→λ vp. (21)

This is, for example, necessary in order to ensure thatcompletely frozen (i.e., in general, long-wavelength)modes with γp = 1 deform affinely with the sample. Inthe limit of γp = 0 equation (20) can be rewritten as

Fpα(λ, γp = 0) =KBT

2

lpαα(λ)

lpαα

(vpα(λ)

vpα

)2

. Since uncon-

strained fluctuations must not contribute to the shearmodulus, the deformation dependence of vpα(λ) andlpα(λ) may not be chosen independently. The simplestchoice consistent with this requirement is:

←→lp (λ) =

(←→λ)−2←→

lp , (22)

which can be visualized as cavities that move and de-form affinely with the sample. In particular, it is notpossible to let the strength of the constraints vary witha different power as in some tube models [14]. Floryand Erman proposed more complicated relations of thetype lp/lp(λ) = λ2 (1 + γζ(λ− 1)) in our notation. Byvarying the parameter ζ they were able to improve theagreement of their constrained junction model with mea-sured stress-strain curves. However, at least in the presentcase, this ansatz leads to inconsistencies. Demanding thatlimγ→1G = 1, limits one’s freedom in choosing to ζ = 0or ζ = 1, and in the remaining non-trivial case one findsd

dγG

∣∣∣∣γ=0

< 0 or negative contributions of weakly con-

strained modes to the shear modulus. Nevertheless, thereseem to be other possibilities. At least, we have not foundany theoretical arguments that would exclude, for ex-ample, lp/lp(λ) = λ2

(1 + γ2(λ− 1)

). As a consequence,

equation (22) involves an assumption and we will latercome back to the point how it may be tested in computersimulations.

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

γ = 50/51= 20/21

= 10/11

= 5/6

= 3/4

____

__ /

(λ-

λ-2)

d F p

d λ

1/λ

Fig. 1. Contribution of a partially constrained mode to therestoring forces of a deformed network in Mooney-Rivlin rep-

resentation. The values of γ =κ

κ+ 1were chosen to corre-

spond to Figure 2 in Flory’s first paper on constrained junctionmodels [6].

Some general conclusions can be drawn with respectto the microscopic deformations in the constrained modemodel with equations (21, 22). The mean-excitations,

Upα(λ)

Upα(1)=

λαα

(1− γ)λ2αα + γ

, (23)

of partially frozen modes deform sub-affinely. Only in thelimit γ → 1 one finds Up(λ) = λUp(1). A similar resultholds for the width of the fluctuations:

〈δu2pα(λ)〉

〈δu2pα(1)〉

=λ2αα

(1− γ)λ2αα + γ

· (24)

While the fluctuations are deformation independent forunconstrained modes, their width increases sub-affinelyfor 0 < γ < 1. For completely frozen modes with γ ≡1, 〈δu2

pα(λ)〉 ≡ 0 independent of λ. Finally, one obtainsfor the deformation dependent free energy and the shearmodulus of the constrained mode model:

Fpα(λ) =KBT

2

1

1 + γp(λ−2αα − 1

)+kBT

2log

(1 +

γp

1− γpλ−2αα

)(25)

Gcm = Gph +kBT

V

∑p

γ2p . (26)

Inspection of equation (25) shows that one recovers theideal stress-strain behavior equation (11) only in the twolimiting cases of γ = 0 and γ = 1. In Figure 1 we have plot-ted some results for intermediate values in the Mooney-Rivlin form often used to represent experimental data.Qualitatively, the curves are very similar to those obtainedfrom Flory’s constrained junction model [6] and we cantherefore expect the same, good agreement with exper-imental data. However, as we have made no particularassumptions about which modes are constrained by the


entanglements, it is clear that nothing is to be learnedfrom the shape of the stress-strain curves about the valid-ity of the classical models of rubber elasticity.

4 Discussion

While the constrained mode model as presented above isformally very simple, an obvious difficulty is hidden in theemployment of the eigenmodes of the phantom model. Inprinciple, it is possible to calculate these so-called general-ized Rouse modes from the knowledge of the connectivitymatrix for a particular network [42,44]. In the appendixwe work out an intuitive approximation which in a simi-lar form is employed by all constrained fluctuation modelsof rubber elasticity, where the movements of the individ-ual junction points and network strands are consideredto be independent. For the fluctuations of the junctionpoints around their mean positions we follow Flory anduse an Einstein-model. For the network strands we followthe community insofar as we employ single chain Rousemodes, but insist on the apparently uncommon choice ofusing boundary conditions which correspond to fixed asopposed to free chain ends. Such Rouse modes can be usedto describe the fluctuations of monomers around their re-spective equilibrium positions. Together the Einstein andthe Rouse modes form a complete and orthogonal basisset. In particular, they are like the true eigenmodes inde-pendent of size and shape of the network. In contrast to theRouse modes for long paths through the network, whichwere used by Duering et al. [25,46] to analyze their sim-ulation data, the autocorrelation functions of the presentmodes decay to zero for non-entangled systems. In fact,the two types of modes allow a much clearer distinctionbetween classical and non-classical entanglement effects.In the appendix we show how (i) by constraining the Ein-stein modes one obtains the crossover from the phantomto the affine model described by the constrained junctionmodels of classical rubber elasticity [5–9] and how (ii) byadditionally constraining the Rouse modes one recoversthe tube model [11–14] with a strand length independentshear modulus. In the general case, the CMM allows oneto study the whole crossover from networks with strandlenghts N � Ne, which are well-described by classicalrubber elasticity, to entanglement dominated systems withN � Ne.

Even with the simplified mode spectrum, the CMM isprobably still too detailed (i.e., has too many free param-eters γp) to be tested quantitatively in experiments. Aswe have already pointed out, the worst approach is prob-ably to rely on a fit of rheological data to the stress-straincurves that follow from equation (25) regarding γ as a pa-rameter characterizing the average confinement of what-ever modes. Judging from the inconclusiveness of previ-ous investigations along these lines [31–37], it is clear thatmore microscopic input is needed. But even neutron scat-tering experiments [29,30,47] are limited to comparing thedata to structure factors calculated from models. Clearly,more information on, for example, the wave-length depen-dence of the parameters γp of the constrained mode model

is needed, before a fit could be attempted. We note thatrecent data are in very good agreement with a variant ofthe tube model which tries to account for non-affine de-formations of both, the tube axes and diameter [29,30].As we have pointed out, such effects appear rather natu-rally in the framework of the constrained mode model asa consequence of partially frozen modes with 0 < γp < 1.

While the great number of parameters γp of the con-strained mode model is disadvantageous for a compari-son with experiments, the opposite is true for the inter-pretation of simulation data. Current molecular dynamicssimulations of model polymer networks cover the relevanttime and length scales and allow to directly determine therestoring forces in strained systems [38,39,48–50]. At thesame time, and in contrast to experiments, one has ac-cess to the actual chain conformations. This allows one tomeasure the parameters γp of the CMM in a simulation ofan undeformed network. In subsequent runs of the samesystem under strain one can directly test the predictionsof the CMM for the microscopic deformations, equations(23, 24), and the elastic properties, i.e., for the deforma-tion dependence of the network free energy equation (25),the corresponding restoring forces, the small strain shearmodulus equation (26), and the modified mode analysisexpression (B.1) for the shear relaxation modulus G(t)[25,46].

As a last point we address a general aspect of all con-strained fluctuation models of rubber elasticity, namelythe relation between the polymer contribution to thestress tensor and the deformation dependence of the con-straints. Within the CMM, equation (25), the normal ten-

sion σT =λ

V

dF

dλinduced by a deformation equation (5)

can be written in the form

〈σT 〉 =1

V

∑〈i,j<i〉

k

(X2ij(λ) −

1

2

(Y 2ij(λ) + Z2

ij(λ)))

+1

V

∑p

kp

⟨u2px(λ)−

1

2

(u2py(λ) + u2

pz(λ))⟩

.

(27)

One the one hand, one might well have expected thisresult. Equation (27) is the equivalent of the Doi andEdwards [11] formula for the contribution of free poly-mers to the stress tensor and has the same form as thevirial theorem (see below) for harmonic springs. On theother hand, it is quite surprising that the CMM shouldyield the same (measurable!) expression as the phantommodel without constraints, i.e. that the constraints in theCMM do not contribute directly to the restoring forces,even though equation (20) shows that they store elasticenergy.

To better understand this apparent contradiction were-derive in the following the virial theorem for deforma-tion dependent interactions. It turns out that the valid-ity of the Doi-Edwards stress formulas is bound to non-trivial conditions on the deformation dependence of theconstraints. In fact, one recovers equation (27) for anytheoretical model that augments the phantom model


Hamiltonian by terms which become deformation invari-

ant under affine transformations r→←→λ r.

Consider a general Hamiltonian H =∑i<j U(|rij |)

(e.g. the phantom model Hamiltonian Eq. (1)) without

constraints. Using scaled co-ordinates ri =←→λ si, dis-

tances are calculated by contraction with the metric tensor←→Λ =

←→λ

t←→λ . By this change of variables the deforma-

tion dependence of the partition function is transferredfrom the integration bounds to the Hamiltonian:

Z = det(←→λ )

N

×

∫ds3N exp

− 1

kBT

∑i<j

U

(√s tij←→Λ sij

) .(28)

Differentiation of the free energy F = −kBT log(Z) withrespect to the deformation parameter λ shows that sys-tems responds to a deformation equation (5) by develop-ing a normal tension:

1

V

d F

d λ=

1

V

⟨d H

d λ

⟩=

1

λ

⟨σxx −

1

2(σyy + σzz)

⟩=σT

λ(29)

where

σαβ =2

V

←→λ

d H

d Λαβ

←→λt

=1

V

∑i<j

U ′(rij)

rij(rij)α (rij)β (30)

is the instantaneous stress tensor. For harmonic interac-tions, as in the phantom and Rouse model Hamiltoniansfor polymer melts and networks, equation (30) reduces tothe Doi and Edwards [11] formula for the polymer con-tribution to the stress tensor. Differentiating twice andtaking the limit λ → 1 yields an expression for the smallstrain elastic modulus:

3G = 〈Tr←→σ 〉 −V

kBT(⟨σ2T

⟩− 〈σT 〉

2)

+1

V

⟨∑i<j

(U ′′ −

U ′

rij

)(x2ij −

12 (y2

ij + z2ij))

2

r2ij

⟩.

(31)

Similar expression for other types of deformation havebeen discussed in the literature and are widely used insimulations [49,51,52]. For the phantom and Rouse modelHamiltonians the last, so-called hyper-virial term van-ishes. In these cases equation (31) describes the polymercontribution to the shear modulus in analogy to the Doiand Edwards [11] formula for the stress tensor.

Now consider the constraint-Hamiltonian equa-tion (17). If one uses the scaled variables, it becomes in-dependent of the deformation provided the constraints de-form affinely as in equations (21, 22). The same holds for

the constraint-Hamiltonians equations (15, 16). In such acase, differentiating the free energy with respect to λ yieldsthe same expressions as for the phantom model withoutconstraints. The expectation values do, of course, changeas the averages are taken over a different ensemble. Fortheories which employ non-affinely deforming constraintsadditional terms appear, for example, in the stress-tensorequation (30). This general result is interesting, since ina computer simulation equations (30, 31) can be evalu-ated for the true microscopic as well as for the effectiveentropic interactions2. A comparison of equations (26, 31)for the shear modulus, on the other hand, provides a directtest for the importance of correlations between differentmodes.

To summarize, with the constrained mode model(CMM) equation (17) we have introduced an exactly solv-able constrained fluctuation model of rubber elasticitywhich is particularly suited for the analysis of simulationdata. It differs from most other models in that deforma-tion dependent linear forces couple to the eigenmodes ofthe phantom network instead of restricting the motion ofindividual monomers or junction points. Nevertheless, thetube model as well as the constrained junction modelsof classical rubber elasticity can be recovered as limitingcases in an approximation where the true eigenmodes arereplaced by a combination of Einstein and Rouse modesfor the movements of the junction points and networkstrands respectively. In the general case, the CMM al-lows one to study the whole crossover from networks withstrand lengths N � Ne which are well-described by clas-sical rubber elasticity to entanglement dominated systemswith N � Ne. In the CMM the confinement of the mo-tion of each mode is characterized by a single parame-ter γ and sub-affine microscopic deformations, equations(23, 24), and deviations from the ideal stress-strain behav-ior (Fig. 1) are predicted as a result of partial confinementof modes with 0 < γ < 1. The small strain shear modulusof the CMM, equation (26), has a particularly simple formand suggests a modification of the mode analysis expres-sion equation (B.1) for the shear relaxation modulus G(t)proposed by Duering et al. [25,46]. Quite surprisingly, wefind that the CMM can only be formulated consistently foraffinely deforming constraints, equations (21, 22). Whilethis is, of course, no formal proof that the same restric-tions also apply to other constrained fluctuation models,we have shown in analogy to the derivation of the virialtheorem that for a system with constraints the usual Doi-Edwards formulas for the polymer contribution to thestress tensor also require affinely deforming constraints.In this case, one is in the interesting situation that theconstraints only contribute explicitely to the deformationfree energy but not to the tensions in a strained system.

2 Such a comparison has already been carried out for an en-tangled melt, but not for a cross-linked melt with permanenttopological constraints. Gao and Weiner [53] showed that theDoi-Edwards formula does work in the reptation regime, i.e.on the time-scales where there is no fundamental differencebetween a melt and a network.


I gratefully acknowledge intense discussions with K. Kremerand G.S. Grest on how we should analyze our simulationdata. Without their input, their criticism and their patiencethis work would not have been possible. I have also benefit-ted from discussions with J. Baschnagel, B. Dunweg, M. Got-tlieb, J.-F. Joanny, A. Johner, E. Straube, W. Wintz, and H.-P.Wittmann.

Note added

After this work was finished Rubinstein and Panyukov [54]published a paper where they develop a complementaryview of the relation between non-affine chain deformationsand corrections to the neo-hookian elasticity. These resultsare obtained using a constraint-Hamiltonian of the type ofequation (15) for virtual chains coupled to the backboneof long network strands and effectively confining them totube-like regions. Note that as in the CMM the strengthof the confining potential is assumed to vary affinely withthe deformation.

Appendix A: Classical and tube models aslimiting cases of the constrained mode model

In principle, one can calculate the exact eigenmodes orgeneralized Rouse modes from the knowledge of the con-nectivity matrix for a particular network [42,44]. Here wepropose an approximation, where the movements of thejunction points and of the strands between them are con-sidered to be independent. The ansatz has the advantagethat it provides an intuitive distinction between the clas-sical and tube contributions to the shear modulus.

A.1. Constraining the motion of the junction points

Classical rubber elasticity only considers the extension ofthe network strands as a whole and can therefore be for-mulated in terms of the motion of the junction points.The following approximation corresponds to the Einstein-model in solid-state physics and was, for example, alsoused by Flory to estimate the contribution of the defor-mation dependent fluctuations to the shear modulus of hisconstrained junction model.

Consider a particular junction point i of an f -functional network. If one assumes that its topologicalneighbors are fixed at their equilibrium positions Rj , a dis-placement uiα of node i in one spatial direction requires anenergy (fk/2)u2

iα. Treating these displacements as inde-pendent eigenmodes (“Einstein modes”) with spring con-stant kp = fk, one can use the equipartition theorem to

obtain 〈u2iα〉 =

kBT

fk. As a consistency check, we note

that this result implies that the extension of a particu-lar network strand should undergo thermal fluctuations

of a width 〈|uij |2〉 = 2〈|ui|2〉 = 2 × 3kBT

fk= 2

f

⟨r2⟩

in

agreement with equation (12).To demonstrate that constraining these modes has the

same effects as predicted by the conventional constrained

junction models [5–9] we show that the shear moduluscalculated from the constrained mode model interpolatesbetween the phantom and junction affine network lim-its. That partially constrained modes lead to the typicalMooney-Rivlin deviations from the ideal stress-strain re-lation equation (11) was already discussed in Section 3.

While it is obvious from equation (26) that Gcm re-duces to Gph for γp ≡ 0, one needs to know the actualnumber of modes to calculate Gcm in the opposite limit ofγp ≡ 1. Using our result from Section 2 that there are 2/f(three component) modes per network strand and equa-tion (14) for the shear modulus of the phantom model onefinds

Gcm(γ ≡ 1) = Gph +2

fρstrandkBT

= ρstrandkBT

= Gaff .

The junction affine model [4] is, in fact, the oldest modelof rubber elasticity. The assumption, that the surroundingmolecules suppress the movements of the junction pointsso strongly that the latters’ instantaneous positions (andnot only their mean positions as in the case of the phantommodel) change affinely with the shape of the sample, leadsimmediately to the above results for Gaff .

In reference [48] we quantitatively tested the classicaltheories in computer simulations of model polymer net-works. We determined the true shear moduli by measur-ing the restoring forces in deformed networks and calcu-lated parameter-free predictions from the phantom andthe junction affine model. By showing that the measuredshear moduli exceeded Gaff , we provided the first quanti-tative proof that the classical theories overlook importantcontributions to the entropy change in a deformed net-work.

However, in the light of the present results our analy-sis of the constrained junction models appears incorrect.In reference [48] we calculated the classical modulus fromthe observed deformation dependence of single strand en-tropies. In the context of the constrained mode model thiscorresponds to analyzing quantities of the type

E(λ) =kp

2

⟨rij

2(λ)⟩

=kp

2

(⟨Rij

2(λ)⟩

+ 2⟨U2pα(λ)

⟩+ 2

⟨δu2pα(λ)

⟩)which are identical to the proper free energies equa-tion (20) only in the limits γ = 0 and γ = 1. Whileevaluating the free energy of the constrained mode modelis quite cumbersome (the constraining springs are not di-rectly observable!), we show in Section 4 that one can el-egantly circumvent this problem by comparing measuredand calculated normal tensions.

A.2. Constraining the strand fluctuations betweenthe junction points

The non-classical theories of rubber elasticity discuss con-straints like tubes, slip-links [15,45] or the presence of filler


material, that restrict the fluctuations of the strands be-tween the junction points. Treating the strands as inde-pendent, these fluctuations are most naturally analyzedin terms of single chain Rouse-modes [11]. Consider a lin-ear Gaussian chain of Nstrand + 1 beads with a Hamil-

tonian H =kNstrand

2

∑Nstrandi=1 (ri(t)− ri−1(t))

2and the

spring constants k Nstrand chosen as discussed in Sec-tion 2. For the present purposes, it turns out to be conve-nient to regard the end points as fixed at r0(t) ≡ R0

and rNstrand(t) ≡ RNstrand , instead of the usual openboundary conditions. One can then write the the devi-ations ui(t) = ri(t) − Ri from the equilibrium positionsRi = R0 + i/Nstrand (RNstrand −R0) in terms of sin-Rouse-modes:

up(t) =1

Nstrand + 1

Nstrand∑i=0

ui(t) sin

(pπi

Nstrand

)(A.1)

H =k

2(R0 −RNstrand)2 +

∑p

kp

2up

2 (A.2)

kp =2π2k

Nstrandp2. (A.3)

As the Einstein modes describe the fluctuations of thejunction points, the Rouse modes equation (A.1) describethe fluctuations of the monomers around their respec-tive equilibrium positions. Together the Einstein and theRouse modes form a complete and orthogonal basis set. Inparticular, they are like the true eigenmodes independentof size and shape of the network and allow an intuitive dis-tinction between classical and non-classical contributionsto the shear modulus.

The correspondence with the tube model is mostconveniently discussed in the limit of long strands ofNstrand → ∞ Gaussian units, where the classical contri-bution to the shear modulus becomes negligible: Gclass ≤

Gaff =ρ

NstrandkBT → 0 (ρ here denotes the number

density of the Gaussian units). Consider, for example, theansatz

γp =

1 for p <

Nstrand

Ne

0 for p >Nstrand

Ne

(A.4)

which freezes all modes with a wavelength largerthan a certain “entanglement length” Ne. The mean-square distance between the equilibrium positionsof neighboring Gaussian units, 〈(Ui −Ui+1)2〉 =

8∑p γp

kBT

kpsin2

(pπ

Nstrand

), as calculated from the

first Nstrand/Ne frozen modes is of the order of〈r2(Ne)〉/N2

e . The corresponding tube axes has a length

ofNstrand

Ne

√〈r2(Ne)〉 and deforms affinely with the sam-

ple. The tube width, on the other hand, is given by thefluctuations of a monomer around its equilibrium position,

〈δu2i 〉 = 2

∑p(1− γp)

kBT

kp, and of the order of R2

g(Ne) in-

dependently of the deformation. Finally, the shear modu-lus is obtained by multiplying the number of frozen modesper chain, Nstrand/Ne, with the chain density ρ/Nstrand:

Gtube =Nstrand

Ne

ρ

NstrandkBT =

ρ

NekBT. (A.5)

As already discussed in Section 3, partially frozen modeswith 0 < γp < 1 lead to a weaker than affine deforma-tions of both, the tube axes and the tube diameter, and tothe characteristic Mooney-Rivlin corrections to the idealstress-strain curves.

Appendix B: A mode analysis expressionfor the shear relaxation modulus

In this section we comment on the mode analysis as it wasused by Duering et al. [25,46] to estimate the shear relax-ation function of model polymer networks in computersimulations of undeformed samples. A heuristic general-ization of the shear modulus equation (26) of the con-strained mode model for finite times is

Gcm(t) = Gph +kBT

V

∑p

(〈up(t) · up(0)〉

〈up2〉

)2

, (B.1)

where the sum over the modes includes both, the Einsteinand the Rouse modes (A.1) for the fluctuations of thejunctions points and the monomers around their respec-tive equilibrium positions. Equation (B.1) has the correctlimiting behavior for large times and reduces to the Rouse-model result [11]

GRouse(t) = Gph +kBT

V

∑p

e−2t/τp (B.2)

for γ = 0. The factor of two in the exponential betweenthe auto-correlation function of a mode 〈up(t) · up(0)〉 =⟨up

2⟩

exp(−t/τp) and its contribution to the shear relax-ation modulus lead Duering et al. [25,46] to the slightlydifferent ansatz

G(t) =kBT

V

∑p

〈Xp(2t) ·Xp(0)〉⟨Xp

2⟩ t→∞

−→kBT

V

∑p

γp,

(B.3)

where the Xp are now the usual free chain Rouse modes

Xp(t) =1

Npath

N∑i=1

ri(t) cos

(pπ(i− 1)

Npath − 1

)−

1

2Npath

[r1(t) + (−1)prNpath(t)

](B.4)

used for the analysis of the conformations of long pathsthrough the network. Equation (B.3) does not agree withequation (26), but reduces by construction to the Rouseresult equation (B.2).


Why do we now propose to use the network strandRouse modes, up, equation (A.1), instead of the free chainRouse modes, Xp, equation (B.4)? First of all, we notethat even for a phantom network the auto-correlationfunctions of the free chain Rouse modes, 〈Xp(t) ·Xp(0)〉,do not decay to zero. This is simply due to the fact thateach network strand has in contrast to a free chain a non-vanishing mean extension. As a consequence, the signatureof permanent entanglements in polymer networks is muchless clear, when the conformations are analyzed in termsof the free chain Rouse modes, than when they are ana-lyzed in terms of the network strand Rouse modes. Therethe incomplete decay of a mode auto-correlation func-tion 〈up(t) · up(0)〉 directly signals the existence of effectswhich are ignored by the classical theories of rubber elas-ticity. Also, since the free chain Rouse modes when appliedto a phantom network become deformation dependent,they are only of limited use for the analysis of the con-formations of strained networks. While these are practicalreasons for the use of the network strand Rouse modes,equation (A.1), the difference between equation (B.1) andequation (B.3) suggests that the question of which type ofmodes is used is not just a matter of convenience. Ac-cording to the results we obtained for the constrainedmode model there is a subtle difference in how the equilib-rium extensions of the network strands contribute to theshear modulus. The part which is due to the cross-linkingand treated by the phantom model contributes linearly(Eq. (14)), while the part that is due to the constraining

of fluctuations, γ =〈U2pα〉〈u2pα〉

, contributes only quadratically

(Eq. (26)). Using the free chain Rouse modes this differ-ence is lost and the modulus over-estimated. In order tomake the distinction and to use equation (26), it is nec-essary to employ modes which as the true eigenmodes ofthe phantom model depend only on the connectivity ofthe network.

References

1. H. James, J. Chem. Phys. 15, 651 (1947).2. H. James, E. Guth, J. Chem. Phys. 15, 669 (1947).3. P.J. Flory, Proc. Royal Soc. Lond. A. 351, 351 (1976).4. L.R.G. Treloar, The Physics of Rubber Elasticity (Claren-

don Press, Oxford, 1975).5. G. Ronca, G. Allegra, J. Chem. Phys. 63, 4990 (1975).6. P.J. Flory, J. Chem. Phys. 66, 5720 (1976).7. B. Erman, P.J. Flory, J. Chem. Phys. 68, 5363 (1978).8. P.J. Flory, B. Erman, Macromol. 15, 800 (1982).9. S. Kastner, Colloid Polym. Sci. 259, 499 and 508 (1981).

10. S.F. Edwards, Proc. Phys. Soc. 92, 9 (1967).11. M. Doi, S.F. Edwards, The Theory of Polymer Dynamics

(Claredon Press, Oxford, 1986).12. W.W. Graessley, Adv. Pol. Sci. 47, 67 (1982).13. S.F. Edwards, T.A. Vilgis, Rep. Progr. Phys. 51, 243

(1988).

14. G. Heinrich, E. Straube, G. Helmis, Adv. Pol. Sci. 85, 34(1988).

15. R.C. Ball, M. Doi, S.F. Edwards, M. Warner, Polymer 22,1010 (1981).

16. G. Marrucci, Macromol. 14, 434 (1981).17. R.J. Gaylord, J. Poly. Bull. 8, 325 (1982).18. S.F. Edwards, T.A. Vilgis, Polymer 27, 483 (1986).19. D. Adolf, Macromol. 21, 228 (1988).20. M. Kosc, Colloid Polym. Sci. 266, 105 (1988).21. P.G. Higgs, R.C. Ball, Europhys. Lett. 8, 357 (1989).22. K. Kremer, G.S. Grest, J. Chem. Phys. 92, 5057 (1990).23. W. Paul, K. Binder, D.W. Heermann, K. Kremer, J. Phys.

II France 1, 37 (1990).24. W. Paul, K. Binder, D.W. Heermann, K. Kremer, J. Chem.

Phys. 95, 7726 (1991).25. E.R. Duering, K. Kremer, G.S. Grest, J. Chem. Phys. 101,

8169 (1994).26. D. Richter et al., Phys. Rev. Lett. 64, 1389 (1990).27. D. Richter et al., Macromol. 25, 6156 (1992).28. D. Richter et al., Phys. Rev. Lett. 71, 4158 (1993).29. E. Straube, V. Urban, W. Pyckhout-Hintzen, D. Richter,

Macromol. 27, 7681 (1994).30. E. Straube et al., Phys. Rev. Lett. 74, 4464 (1995).31. M. Gottlieb et al., Macromol. 14, 1039 (1981).32. M. Gottlieb, R.J. Gaylord, Polymer 24, 1644 (1983).33. M. Gottlieb, R.J. Gaylord, Macromol. 17, 2024 (1984).34. M. Gottlieb, R.J. Gaylord, Macromol. 20, 130 (1987).35. B. Erman, P.J. Flory, Macromol. 15, 806 (1982).36. S.K. Patel, S. Malone, C. Cohen, J.R. Gillmor, Macromol.

25, 5241 (1992).37. M.A. Sharaf, J.E. Mark, Polymer 35, 740 (1994).38. R. Everaers, K. Kremer, G.S. Grest, Macromol. Symposia

93, 53 (1995).39. K. Kremer, G.S. Grest, in Monte-Carlo and Molecular

Dynamics Simulations in Polymer Science, edited by K.Binder (Oxford University Press, New York and Oxford,1995).

40. R. Everaers, K. Kremer, in preparation; G.S. Grest, R.Everaers, K. Kremer, in preparation.

41. J.A. Duiser, A.J. Stavermann, in Physics of non-crystalline solids, edited by J. Prins (North-Holland, Am-sterdam, 1965), p. 376.

42. B.E. Eichinger, Macromol. 5, 496 (1972).43. W.W. Graessley, Macromol. 8, 186 and 865 (1975).44. J.U. Sommer, M. Schulz, H.L. Trautenberg, J. Chem.

Phys. 98, 7515 (1993).45. W.W. Graessley, D.S. Pearson, J. Chem. Phys. 66, 3363

(1977).46. E.R. Duering, K. Kremer, G.S. Grest, Phys. Rev. Lett. 67,

3531 (1991).47. T.A. Vilgis, F. Boue, Polymer 27, 1154 (1986).48. R. Everaers, K. Kremer, Macromol. 28, 7291 (1995).49. S.J. Barsky, M. Plischke, Phys. Rev. E 53, 871 (1996).50. S.J. Barsky, M. Plischke, Z. Zhou, B. Joos, Phys. Rev. E

54, 5370 (1996).51. J.R. Ray, Comp. Phys. Rep. 8, 109 (1988).52. Z. Zhou, B. Joos, Phys. Rev. B 54, 3841 (1996).53. J. Gao, J.H. Weiner, J. Chem. Phys. 103, 1614 (1995).54. M. Rubinstein, S. Panyukov, Macromol. 30, 8036 (1997).

Appendix G

Tube Models forRubber-Elastic Systems

119

Tube Models for Rubber-Elastic Systems

Boris Mergell and Ralf Everaers*

Max-Planck-Institut fur Polymerforschung, Postfach 3148, D-55021 Mainz, Germany

Received December 29, 2000; Revised Manuscript Received April 6, 2001

ABSTRACT: In the first part of the paper, we show that the constraining potentials introduced to mimicentanglement effects in Edwards’ tube model and Flory’s constrained junction model are diagonal in thegeneralized Rouse modes of the corresponding phantom network. As a consequence, both models canformally be solved exactly for arbitrary connectivity using the recently introduced constrained mode model.In the second part, we solve a double tube model for the confinement of long paths in polymer networkswhich is partially due to cross-linking and partially due to entanglements. Our model describes a nontrivialcrossover between the Warner-Edwards and the Heinrich-Straube tube models. We present results forthe macroscopic elastic properties as well as for the microscopic deformations including structure factors.

I. Introduction

Polymer networks1 are the basic structural elementof systems as different as tire rubber and gels and havea wide range of technical and biological applications.From a macroscopic point of view, rubberlike materialshave very distinct visco- and thermoelastic properties.1,2

They reversibly sustain elongations of up to 1000% withsmall strain elastic moduli which are 4 or 5 orders ofmagnitude smaller than those for other solids. Maybeeven more unusual are the thermoelastic propertiesdiscovered by Gough and Joule in the 19th century:when heated, a piece of rubber under a constant loadcontracts, and conversely, heat is released duringstretching. This implies that the stress induced by adeformation is mostly due to a decrease in entropy. Themicroscopic, statistical mechanical origin of this entropychange remained obscure until the discovery of poly-meric molecules and their high degree of conformationalflexibility in the 1930s. In a melt of identical chains,polymers adopt random coil conformations3 with mean-square end-to-end distances proportional to their length,⟨rb2⟩ ∼ N. A simple statistical mechanical argument,which only takes the connectivity of the chains intoaccount, then suggests that flexible polymers react toforces on their ends as linear, entropic springs. Thespring constant, k ) (3kBT)/⟨rb2⟩, is proportional to thetemperature. Treating a piece of rubber as a randomnetwork of noninteracting entropic springs (the phan-tom model4-6) qualitatively explains the observed be-havior, includingsto a first approximationsthe shapeof the measured stress-strain curves.

Despite more than 60 years of growing qualitativeunderstanding, a rigorous statistical mechanical treat-ment of polymer networks remains a challenge to thepresent day. Similar to spin glasses,7 the main difficultyis the presence of quenched disorder over which ther-modynamic variables need to be averaged. In the caseof polymer networks,8-10 the vulcanization process leadsto a simultaneous quench of two different kinds ofdisorder: (i) a random connectivity due to the introduc-tion of chemical cross-links and (ii) a random topologydue to the formation of closed loops and the mutualimpenetrability of the polymer backbones. Since forinstantaneous cross-linking monomer-monomer con-tacts and entanglements become quenched with aprobability proportional to their occurrence in the melt,ensemble averages of static expectation values for the

chain structure etc. are not affected by the vulcanizationas long as the system remains in its state of preparation.

For a given connectivity the phantom model Hamil-tonian for noninteracting polymer chains formally takesa simple quadratic form,4-6 so that one can at leastformulate theories which take the random connectivityof the networks fully into account.11-13 The situation isless clear for entanglements or topological constraints,since they do not enter the Hamiltonian as such butdivide phase space into accessible and inaccessibleregions. In simple cases, entanglements can be charac-terized by topological invariants from mathematicalknot theory.8,9 However, attempts to formulate topologi-cal theories of rubber elasticity (for references see ref14) encounter serious difficulties. Most theories there-fore omit such a detailed description in favor of a mean-field ansatz where the different parts of the networkare thought to move in a deformation-dependent elasticmatrix which exerts restoring forces toward some restpositions. These restoring forces may be due to chemicalcross-links which localize random paths through thenetwork in space15 or to entanglements. The classicaltheories of rubber elasticity1,16-20 assume that entangle-ments act only on the cross-links or junction points,while the tube models2,21-26 stress the importance of thetopological constraints acting along the contour ofstrands exceeding a minimum “entanglement length”,Ne. Originally devised for polymer networks, the tubeconcept is particularly successful in explaining theextremely long relaxation times in non-cross-linkedpolymer melts as the result of a one-dimensional,curvilinear diffusion called reptation27 of linear chainsof length N . Ne within and finally out of their originaltubes. Over the past decade, computer simulations14,28-30

and experiments31-33 have finally also collected mount-ing evidence for the importance and correctness of thetube concept in the description of polymer networks.

More than 30 years after its introduction and despiteits intuitivity and its success in providing a unified viewon entangled polymer networks and melts,2,23-26 thereexists to date no complete solution of the Edwards tubemodel for polymer networks. Some of the open problemsare apparent from a recent controversy on the inter-pretation of SANS data.32-37 Such data constitute animportant experimental test of the tube concept, sincethey contain information on the degree and deformationdependence of the confinement of the microscopic chainmotion and therefore allow for a more detailed test of

5675Macromolecules 2001, 34, 5675-5686

10.1021/ma002228c CCC: $20.00 © 2001 American Chemical SocietyPublished on Web 06/28/2001

theories of rubber elasticity than rheological data.38-40

On the theoretical side, the original approach ofWarner and Edwards15 used mathematically ratherinvolved replica methods26 to describe the localizationof a long polymer chain in space due to cross-linking.The replica method allows for a very elegant, self-consistent introduction of constraining potentials, whichconfine individual polymer strands to random-walk liketubular regions in space while ensemble averages overall polymers remain identical to those of unconstrainedchains. Later Heinrich and Straube25,32 recalculatedthese results for a solely entangled system where theyargued that there are qualitative differences betweenconfinement due to entanglements and confinement dueto cross-linking. In particular, they argued that thestrength of the confining potential should vary affinelywith the macroscopic strain, resulting in fluctuationsperpendicular to the tube axis which vary only like thesquare root of the macroscopic strain.

Replica calculations provide limited insight into physi-cal mechanism and make approximations which aredifficult to control.12 It is therefore interesting to notethat Flory was able to solve the, in many respectssimilar, constrained-junction model17 without usingsuch methods. Recent refinements of the constrained-junction model such as the constrained-chain model41

and the diffused-constrained model42 have more or lessconverged to the (Heinrich and Straube) tube model,even though the term is not mentioned explicitly.Another variant of this model was recently solved byRubinstein and Panyukov.43 In particular, the authorsillustrated how nontrivial, subaffine deformations of thepolymer strands result from an affinely deformingconfining potential.

While tube models are usually formulated and dis-cussed in real space, two other recent papers havepointed independently to considerable simplifications ofthe calculations in mode space. Read and McLeish35

were able to rederive the Warner-Edwards result in aparticularly simple and transparent manner by showingthat a harmonic tube potential is diagonal in the Rousemodes of a linear chain. Complementary, one of thepresent authors introduced a general constrained modemodel (CMM),44 where confinement is modeled bydeformation dependent linear forces coupled to (ap-proximate) eigenmodes of the phantom network insteadof a tube-like potential in real space. This model caneasily be solved exactly and is particularly suited forthe analysis of simulation data, where its parameters,the degrees of confinement for all considered modes, aredirectly measurable. Simulations of defect-free modelpolymer networks under strain analyzed in the frame-work of the CMM14 provide evidence that it is indeedpossible to predict macroscopic restoring forces andmicroscopic deformations from constrained fluctuationtheories. In particular, the results support the choiceof Flory,17 Heinrich and Straube,25 and Rubinstein andPanyukov43 for the deformation dependence of theconfining potential. Despite this success, the CMM inits original form suffers from two important deficits: (i)due to the multitude of independent parameters it iscompletely useless for a comparison to experiment, and(ii) apart from recovering the tube model on a scalinglevel, ref 44 remained fairly vague on the exact relationbetween the approximations made by the Edwards tubemodel and the CMM, respectively.

In the present paper, we show that the two modelsare, in fact, equivalent. The proof, presented in sectionIIB is a generalization of the result by Read andMcLeish to arbitrary connectivity. It provides the linkbetween the considerations of Eichinger,11 Graessley,45

Mark,46 and others on the dynamics of (micro) phantomnetworks and the ideas of Edwards and Flory on thesuppression of fluctuations due to entanglements. As aconsequence, the CMM can be used to formally solvethe Edwards tube model exactly, while in turn theindependent parameters of the CMM are obtained as afunction of a single parameter: the strength of the tubepotential. Quite interestingly, it turns out that theentanglement contribution to the shear modulus de-pends on the connectivity of the network. To explore theconsequences, we discuss in the second part the intro-duction of entanglement effects into the Warner-Edwards model, which represents the network as anensemble of independent long paths comprising manystrands. Besides recovering some results by Rubinsteinand Panyukov for entanglement dominated systems, wealso calculate the single chain structure factor for thiscontroversial case.32-37 Finally we propose a “doubletube” model to describe systems where the confinementof the fluctuations due to cross-links and due to en-tanglements is of similar importance and where botheffects are treated within the same formalism.

II. Constrained Fluctuations in Networks ofArbitrary Connectivity

A. The Phantom Model. The Hamiltonian of thephantom model4-6 is given by Hph ) k/2∑⟨i,j⟩

M rij2, where

⟨i, j⟩ denotes a pair of nodes i, j ∈1, ... , M which areconnected by a polymer chain acting as an entropicspring of strength k ) (3kBT)/⟨rb2⟩, and rbij(t) ) rbi(t) -rbj(t) the distance between them. To simplify the notation,we always assume that all elementary springs have thesame strength k. The problem is most convenientlystudied using periodic boundary conditions, which spanthe network over a fixed volume10 and define theequilibrium position RBi ) (Xi, Yi, Zi). A conformation ofa network of harmonic springs can be analyzed in termsof either the bead positions rbi(t) or the deviations ubi(t)of the nodes from their equilibrium positions RBi. In thisrepresentation, the Hamiltonian separates into twoindependent contributions from the equilibrium exten-sions of the springs and the fluctuations. For thefollowing considerations it is useful to write fluctuationsas a quadratic form.11 Finally, we note that the problemseparates in Cartesian coordinates R ) x, y, z due tothe linearity of the springs. In the following we simplifythe notation by writing the equations only for onespatial dimension:

Here u denotes a M-dimensional vector with (u)i ≡ (ubi)x.K is the connectivity or Kirchhoff matrix whose diagonalelements (K)ii ) fik are given by the node’s functionality(e.g., a node which is part of a linear chain is connectedto its two neighbors, so that fi ) 2 in contrast to a four-functional cross-link with fi ) 4). The off-diagonalelements of the Kirchhoff matrix are given by (K)ij ) -k, if nodes i and j are connected and by (K)ij ) 0otherwise. Furthermore, we have assumed that allnetwork strands have the same length.

Hph )k

2∑⟨i, j⟩

XijR2 +

1

2ut K u. (1)

5676 Mergell and Everaers Macromolecules, Vol. 34, No. 16, 2001

The fluctuations can be written as a sum overindependent modes ep which are the eigenvectors of theKirchhoff matrix: Kep ) kpep where the ep can bechosen to be orthonormal ep‚ep′ ) δpp′. The transforma-tion to the eigenvector representation u ) S u and backto the node representation u ) S-1u is mediated by amatrix S whose column vectors correspond to the ep.By construction, S is orthogonal with St ) S-1. Fur-thermore, the Kirchhoff matrix is diagonal in theeigenvector representation (K)pp ) (S-1 K S)pp ) kp. TheHamiltonian then reduces to

Since the connectivity is the result of a randomprocess, it is difficult to discuss the properties of theKirchhoff matrix and the eigenmode spectrum in gen-eral.11,45 The following simple argument44 ignores thesedifficulties. The idea is to relate the mean squareequilibrium distances ⟨Xij

2⟩ to the thermal fluctuationsof the phantom network.

Consider the network strands before and after theformation of the network by end-linking. In the meltstate, the typical mean square extension ⟨rb2⟩ is entirelydue to thermal fluctuations, while ⟨rb⟩ ) 0. In the cross-linked state, the strands show reduced thermal fluctua-tions ⟨ubij

2⟩ around quenched, nonvanishing mean exten-sions ⟨RBij

2⟩. However, the ensemble average of the totalextension ⟨RBij

2⟩ + ⟨ubij2⟩is not affected by the end-linking

procedure. The fluctuation contribution ⟨ubij2⟩ depends

on the connectivity of the network and can be estimatedusing the equipartition theorem. The total thermalenergy in the fluctuations, Ufluc, is given by (3/2)kBTtimes the number of modes and therefore Ufluc )(3/2)kBTNnodes ) (2/f)(3/2)kBTNstrands, where Nnodes andNstrands are the number of junction points and networkstrands, which are related by Nstrands ) (f/2)Nnodes in anf-functional network. Equating the thermal energy permode to (k/2)⟨ubij

2⟩, one obtains6,44,45

Using these results, one can finally estimate theelastic properties of randomly cross- or end-linkedphantom networks. Since the fluctuations are indepen-dent of size and shape of the network, they do notcontribute to the elastic response. The equilibriumpositions of the junction points, on the other hand,change affinely in the macroscopic strain. The elasticfree energy density due to a volume-conserving, uniaxialelongation with λ| ) λ⊥

-1/2 ) λ is simply given by

where Fstrand is the number density of elastically activestrands. For incompressible materials such as rubber,the shear modulus is given by 1/3 of the second derivativeof the corresponding free energy density with respectto the strain parameter λ. In response to a finite strain,

the system develops a normal tension σT:

Experimentally observed stress-strain curves showdeviations from eq 8. Usually the results are normalizedto the classical prediction and plotted vs the inversestrain 1/λ, since they often follow the semiempiricalMooney-Rivlin form

B. The Constraint Hamiltonian. Most theoriesintroduce the entanglement effects as additional, single-node terms into the phantom model Hamiltonian, whichconstrain the movement of the monomers and junctionpoints. The standard choice are anisotropic, harmonicsprings of strength l6(λ)between the nodes and pointsêBi(λ) which are fixed in space:

While all models assume that the tube position changesaffinely with the macroscopic deformation

there are two different choices for the deformationdependence of the confining potential:

Since this choice of Hconstr leaves the different spatialdimensions uncoupled, we consider the problem againin one dimension and express Hconstr in the eigenvectorrepresentation of the Kirchhoff matrix of the uncon-strained network. Using vb (λ) ) êBx(λ) - XB (λ) ) λ6vb (λ )1) one obtains

Hph ) k2 ∑

⟨i, j⟩Xij

2 + ∑p

kp

2up

2 (2)

⟨ubij2⟩ ) 2

f⟨rb2⟩ (3)

⟨RBij2⟩ ) (1 - 2

f )⟨rb2⟩ (4)

∆Fph(λ) ) (λ2 + 2λ

- 3) ⟨Rstrand2⟩

⟨rb2⟩Fstrand

) (λ2 + 2λ

- 3)(1 - 2f )Fstrand (5)

Gph ) 1V

13

d2∆Fph(λ)

dλ2 |λ)1

(6)

) (1 - 2f ) FstrandkBT (7)

σT ) (λ2 - 1λ)Gph (8)

σT

λ2 - 1λ

≈ 2C1 +2C2

λ(9)

Hconstr ) ∑i

1

2(rbi - êBi(λ))t l6(λ) (rbi - êBi(λ)) (10)

êBi(λ) ) λ6êBi(λ ) 1) (11)

l6(λ) ) l6(λ ) 1) (12)

l6(λ) ) λ6-2 l6(λ ) 1) (13)

Hconstr )l(λ)2

(u - v)t(u - v)

)l(λ)2

(S-1u - S-1v)t(S-1u - S-1v)

)l(λ)2

(u - v)tSS-1(u - v) (14)

)l(λ)2

(u - v)t(u - v)

) ∑p

l(λ)

2(up - vp)

2

Macromolecules, Vol. 34, No. 16, 2001 Tube Models for Rubber-Elastic Systems 5677

Thus, the introduction of the single node springs doesnot change the eigenvectors of the original Kirchhoffmatrix. The derivation of eq 14, which is the Hamilto-nian of the Constrained Mode Model (CMM),44 is acentral result of this work. It provides the link betweenthe considerations of Eichinger,11 Graessley,45 Mark,46

and others on the dynamics of (micro) phantom net-works and the ideas of Edwards and Flory on thesuppression of fluctuations due to entanglements.

C. Solution and Disorder Averages: The Con-strained Mode Model (CMM). Since the total Hamil-tonian of the CMM

is diagonal and quadratic in the modes, both the exactsolution of the model for given vbp and the subsequentcalculation of averages over the quenched Gaussiandisorder in the vbp are extremely simple.44 In the follow-ing we summarize the results and give general expres-sions for quantities of physical interest such as shearmoduli, stress-strain relations, and microscopic defor-mations.

Consider an arbitrary mode ubp of the polymer net-work. Under the influence of the constraining potential,each Cartesian component R will fluctuate around anonvanishing mean excitation UBp with

Using the notation δuBp ≡ ubp - UBp, the Hamiltonian forthis mode reads

Expectation values are calculated by averaging overboth the thermal and the static fluctuations, which aredue to the quenched topological disorder (in order tosimplify the notation, we use l ≡ l(λ ) 1), ⟨vpR

2⟩ ≡ vpR2(λ

) 1)⟩ etc.)

Both distributions are Gaussian and their widths

follow from the Hamiltonian and the condition that therandom introduction of topological constraints on thedynamics does not affect static expectation values in thestate of preparation. In particular

Eq 21 relates the strength l of the confining potentialto the width of P(vpR). The result, ⟨vpR

2⟩ ) (1/γp)(kBT/kp), ⟨UpR

2⟩ ) γp(kBT/kp), ⟨δupR2⟩ ) (1 - γp)(kBT/kp) can

be expressed conveniently using a parameter

which measures the degree of confinement of the modes.As a result, one obtains for the mean square staticexcitations

Quantities of physical interest are typically sums overthe eigenmodes of the Kirchhoff matrix. For example,the tube diameter is defined as the average width ofthe thermal fluctuations of the nodes:

In particular

More generally, distances between any two monomersrnmR ) rnR - rmR in real space are given by

For the discussion of the elastic properties of thedifferent tube models it turns out to be useful to definethe sum

Using eq 27, the confinement contribution to the normaltension2,44 and the shear modulus can be written as

D. Model A: Deformation independent strengthof the Confining Potential. To completely define themodel, one needs to specify the deformation dependenceof the confining potential. One plausible choice is

i.e., a confining potential whose strength is strainindependent. The following discussion will make clearthat this choice leads to a situation which mathemati-cally resembles the phantom model without constraints.

Using eq 30 the thermal fluctuations (and thereforealso the tube diameter eq 25) are deformation indepen-dent and remain isotropic in strained systems. Themean excitations, on the other hand, vary affinely withthe macroscopic strain. This leads to the followingrelation for the deformation dependence of the total

0 e γp≡ lkp + l

e 1 (22)

⟨UpR2(λ)⟩ ) λR

2( lR(λ)

kp + lR(λ))2kp + ll

kBTkp

(23)

dTR2(λ) )

1

M∑

p

⟨δupR2(λ)⟩ (24)

dTR2 )

kBT

Ml∑

p

γp (25)

⟨rnmR2(λ)⟩ ) ∑

p

⟨upR2(λ)⟩Sp,nm

2 + λR2 RnmR

2 (26)

g(λ) )kBT

V

1

1 - λR2∑

p (⟨upR2⟩(λ)

⟨upR2⟩

- 1) (27)

σT(λ) )1

V∑

p

kp (⟨up|2(λ)⟩ - ⟨up⊥

2(λ)⟩)

) (λ2 - 1)g(λ) + (1 - λ-1)g(λ-1/2)(28)

Gconstr ) g(1) (29)

l6A(λ) ) l6A(λ ) 1) (30)

H ) Hph + Hconstr (15)

UpR(λ) )lR(λ)

kp + lR(λ)vpR(λ) (16)

HpR[vpR] )kp

2UpR

2(λ) +lR(λ)

2(UpR(λ) - vpR(λ))2 +

lR(λ) + kp

2δupR

2 (17)

⟨Ap(λ)⟩ ) ∫ dvp ∫ dδup Ap[vp,δup] P(vpR)P(δupR) (18)

⟨δupR2(λ)⟩ )

kBT

kp + lR(λ)(19)

⟨vpR2⟩ )

kp + lkpl

kBT (20)

⟨upR2⟩ ) ⟨δupR

2⟩ + ⟨UpR2⟩ ≡ kBT

kp(21)


excitation of the modes:

Using eqs 27-29, one obtains via

a classical stress-strain relation:

E. Model B: Affine Deformation of the ConfiningPotential. The ansatz

goes back to Ronca and Allegra16 and was used by Flory,by Heinrich and Straube,25 and by Rubinstein andPanyukov.43 It corresponds to affinely deforming cavitiesand leads to a more complex behavior including correc-tions to the classically predicted stress-strain behavior.

Using eq 35, the mean excitations of partially frozenmodes as well as the thermal fluctuations, becomedeformation dependent. The total excitation of a modeis given by

Only in the limit of completely frozen modes, γp f 1,does one find affine deformations with upR(λ) ) λRupR(λ ) 1).

Concerning the elastic properties, eq 27 takes theform

while the shear modulus can be written as

Note the different functional form of eqs 34 and 38.Since 0 e γ e 1, the contribution of confined modes tothe elastic response is stronger in model A than in modelB. Furthermore, within model B, the interplays betweenthe network connectivity (represented by the eigenmodespectrum {kp} of the Kirchhoff matrix) and the confiningpotential l are different for the shear modulus eq 38 andthe tube diameter eq 25.

F. Model C: Simultaneous Presence of BothTypes of Confinement. Finally, we can discuss asituation where confinement effects of type A and B arepresent simultaneously. Coupling each node to two extrasprings lA(λ) ) lA and lBR(λ) ) lBR/λR

2 leads to thefollowing Hamiltonian in the eigenmode representation:

Model A and model B are recovered by setting lA andlB, respectively, equal to zero. Furthermore, we assume,that both types of confinement can be activated anddeactivated independently. This requires

In the presence of both types of confinement, the meanexcitation of the modes is given by

while the thermal fluctuations are reduced to

Finally the condition that the simultaneous presenceof both constraints does not affect ensemble averagesin the state of preparation requires

From eqs 40-44, one can calculate the deformationdependent total excitation of the modes:

so that

In the present case, the shear modulus can be writtenas

Note that the shear modulus is not simply the sum of

Hp )kp

2up

2 +lA

2(up - vAp(λ))2 +

lB(λ)2

(up - vBp(λ))2

(39)

⟨vAp2⟩ )

lA + kp

lAkp(40)

⟨vBp2⟩ )

lB + kp

lBkp(41)

⟨Up2⟩(λ) )

lA2⟨vAp

2⟩(λ) + lB2(λ)⟨vBp

2⟩(λ)

(lA + lB(λ) + kp)2

+

2lAlB(λ)⟨vApvBp⟩(λ)

(lA + lB(λ) + kp)2

(42)

⟨δup2⟩(λ) )

kBT

lA + lB(λ) + kp(43)

⟨vApvBp⟩ )kBTkp

(44)

⟨up2⟩(λ)

⟨up2⟩

) 1 + (λ2 - 1)( lA

kp + lA +lB

λ2

+

lB

λ2(lA +lB

λ2)(kp + lA +

lB

λ2)2)(45)

gC(λ) )kBT

V∑

p ( lA

kp + lA +lB

λ2

+

lB

λ2(lA +lB

λ2)(kp + lA +

lB

λ2)2) (46)

GC )kBT

V∑

p

γAp(1 - γBp)

(1 - γApγBp)+

γBp(1 - γAp)(γBp(1 - γAp) + γAp(1 - γBp))

(1 - γApγBp)2

(47)

⟨upR2⟩(λ)

⟨upR2⟩

) 1 + (λR2 - 1)

lA

kp + lA(31)

gA(λ) )kBT

V∑

p( lA

kp + lA) (32)

σT(λ) ) (λ2 - λ-1)GA (33)

GA )kBT

V∑

p

γp (34)

l6B(λ) ) λ6-2 l6B(λ ) 1) (35)

⟨upR2⟩(λ)

⟨upR2⟩

) 1 + (λR2 - 1)( lB(λ)

kp + lB(λ))2

(36)

gB(λ) )kBT

V∑

p( lB(λ)

kp + lB(λ))2

(37)

GB )kBTV ∑

pγp

2 (38)


the contributions from the A and B confinements. Whileeqs 34 and 38 are reproduced in the limits γBp ) 0 andγAp ) 0, respectively, eq 47 reflects the fact that a modecan never contribute more than kBT to the shearmodulus. Thus, for γBp ) 1 (respectively γAp ) 1) thepth mode contributes this maximum amount indepen-dent of the value of γAp (respectively γBp).

An important point, which holds for all three models,is that it is not possible to estimate the confinementcontribution to the shear modulus from the knowledgeof the absolute strength lA, lB of the confining potentialsalone. Required is rather the knowledge of the relativestrengths γAp,γBp which in turn are functions of thenetwork connectivity.

G. Discussion. It is not a priori clear, whetherentanglement effects are more appropriately describedby model A or model B. While model A has the benefitof simplicity, Ronca and Allegra proposed model B,16

because it leads (on length scales beyond the tubediameter) to the conservation of intermolecular contactsunder strain. Similar conclusions were drawn by Hei-nrich and Straube25 and Rubinstein and Panyukov.43

In the end, this problem will have to be resolved by aderivation of the tube model from more fundamentaltopological considerations. For the time being, an em-pirical approach seems to be the safest option. Fortu-nately, the evidence provided by experiments36 and bysimulations14 points into the same direction.

Since details of the interpretation of the relevantexperiments are still controversial (see section III.D.3),we concentrate on simulation results where the straindependence of approximate eigenmodes of the phantommodel was measured directly.14 Figure 1 shows acomparison of data obtained for defect-free model poly-mer networks to the predictions eq 31 of model A andeq 36 of model B. The result is unanimous. We thereforebelieve eq 35 and model B to be the appropriate choicefor modeling confinement due to entanglements. Theshear modulus of an entangled network should thus begiven by44

where in contrast to ref 44 the various γp are no longerfree parameters but depend through eq 22 on a singleparameter: the strength l of the confining potential,which is assumed to be homogeneous for all monomers.

The difficulty of this formal solution of the generalizedconstrained fluctuation model for polymer networks ishidden in the use of the generalized Rouse modes of thephantom model, which are difficult to obtain for realisticconnectivities.46,47 A useful ansatz for end-linked net-works is a separation into independent Flory-Einsteinrespectively Rouse modes for the cross-links and net-work strands.14,44 In fact, the simulation results pre-sented in Figure 1 are based on such a decomposition.

For randomly cross-linked networks with a typicallyexponential strand length polydispersity, the separationinto Flory-Einstein and single-chain Rouse modesceases to be useful. In this case, we can think of tworadically different strategies.

• To keep the network connectivity in the analysis.For example, there is no principle reason why themethods presented by Sommer et al.47 and Everaers14

could not be combined, to investigate the strain depen-dence of constrained generalized Rouse modes in com-puter simulations. Note, however, that this completelydestroys the self-averaging properties of the approxima-tion used in ref 14. Analytic progress in the evaluationof, for example, eq 38 for the entanglement contributionto the shear modulus requires information on thestatistical properties of the eigenvalue spectra of net-works generated by random cross-linking. To our knowl-edge, the only available results were obtained numeri-cally by Shy and Eichinger.48 Note that model C isirrelevant, if one is able to carry out calculations withthe proper network eigenmodes.

• To average out the connectivity effects in tubemodels for polymer networks.15 In the second part ofthe paper, we will consider linear chains under theinfluence of two types of confinement: network con-nectivity and entanglements.

III. Tube Models

In SANS experiments of dense polymer melts, it ispossible to measure single chain properties by deuter-ating part of the polymers.49 If such a system is firstcross-linked into a network and subsequently subjectedto a macroscopic strain, one can obtain information onthe microscopic deformations of labeled random pathsthrough the network.49 To interpret the results, theyneed to be compared to the predictions of theories ofrubber elasticity. Unfortunately, for randomly cross-linked networks it is quite difficult to calculate therelevant structure factors even in the simplest cas-es.12,50,51 Because the cross-link positions on differentprecursor chains should be uncorrelated, Warner andEdwards15 had the idea to consider a tube model, wherethe cross-linking effect is “smeared out” along the chain.To model confinement due to cross-linking, they used(in our notation) model A, since this ansatz reproducesthe essential properties of phantom models (affinedeformation of equilibrium positions and deformationindependence of fluctuations). In contrast, Heinrich andStraube25 and Rubinstein and Panyukov43 treated con-finement due to entanglements using model B. Obvi-ously, both effects are present simultaneously in poly-mer networks. In the following, we will develop the ideathat in order to preserve the qualitatively differentdeformation dependence of the two types of confinement,they should be treated in a “double tube” model basedon our model C.

Before entering into a detailed discussion, we wouldlike to point out a possible source of confusion related

Figure 1. Excitation of constrained modes parallel andperpendicular to the elongation at λ ) 1.5 as a function of themode degree of confinement 0 e γ e 1. The dashed (dotted)lines show the predictions eq 31 of model A and eq 36 of modelB respectively for generalized Rouse modes of a phantomnetwork with identical connectivity. The symbols represent theresult of computer simulations of defect-free model polymernetworks.14 The investigated modes are single-chain Rousemodes for network strands of length N ≈ 1.25Ne.

G ) Gph +kBT

V∑

p

γp2 (48)


to the ambiguous use of the term “tube” in the literature(including the present paper). A real tube is a hollow,cylindrical object, suggesting that in the present contextthe term should be reserved for the confining potentialdescribed by quantities such as êBi,vbp, l. It is in this sensethat we speak of an “affinely deforming tube”. However,a harmonic confining tube potential is a theoreticalconstruction which is difficult to visualize. For example,in the continuum chain limit used below, the forcesexerted “per monomer” become infinitely small corre-sponding to êBi f ∞, l f 0. On the other hand, the termtube is often associated with the tube “contents”, i.e.,the superposition of the accessible polymer configura-tions characterized via a locally smooth tube axis (theequilibrium positions UBp) and a tube diameter dT(defined via the fluctuations δBup). This second definitionrefers to measurable quantities.49 Which kind of tubewe are referring to, will hopefully always be clear fromthe context and the mathematical definition of theobjects under discussion.

In the case of linear polymers, the phantom modelreduces to the Rouse model with vanishing equilibriumpositions RBi ≡ 0. As a consequence, there are no straineffects other than those caused by the confinement ofthermal fluctuations. In particular, the “intrinsic” phan-tom modulus vanishes (see eq 5). Since the networksare modeled as superpositions of independent linearpaths, we have to introduce confinement of type A inorder to recover the phantom network shear modulusGph in the absence of entanglements.

In the Rouse model, the Kirchhoff matrix takes thesimple tridiagonal form

and, depending on the boundary conditions, is diago-nalized by transforming to sin or cos modes using thetransformation matrix

The eigenvalues of the diagonalized Kirchhoff matrix(K)pp ) (S-1 K S)pp ) kp are given by

If we consider a path with given radius of gyration Rg2,

the basic spring constant is given by k ) (N kBT/2Rg2).

In the continuous chain limit (N f ∞), sums overeigenmodes can be approximated by integrals. Forexample, one obtains from eq 25 an expression for thetube diameter

which could be further simplified, since in this limit thesprings representing a chain segment between twonodes are much stronger than the springs realizing thetube, i.e., k . l.

For normally distributed internal distances rbxx′ be-tween points x ) n/N, x′ ) m/N on the chain contourthe structure factor is given by

In the present case, eq 26 reduces to

In the undeformed state

so that the structure factor is given by the Debyefunction:

A. The Warner-Edwards Model. Warner andEdwards15 used the replica method to calculate theconformational statistics of long paths through ran-domly cross-linked phantom networks. The basic ideawas to represent the localization of the paths in spacedue to their integration into a network by a coarse-grained tube-like potential. Recently, it was shown byRead and McLeish34,35 that the same result could beobtained along the lines of the following, much simplercalculation, where we evaluate model A for linearpolymers.

Evaluation of the integrals in eqs 25 and 26 yieldsfor the deformation independent tube diameter and theinternal distances

We note that the latter equation can be rewritten in theform

with a universal scaling function fA(y) which does notdepend explicitly on the deformation. Equation 59measures the degree of affineness of deformations ondifferent length scales. Locally, i.e., for distances insidethe tube with Rg

2| x - x′| , dT2 corresponding to y , 1,

the polymer remains undeformed. Thus, limyf0 f(y) )0. Deformations become affine for Rg

2| x - x′| . dT2 and

y . 1, where f(y) tends to one.

K ) k(-2 1 0 ... 01 -2 1 0 ...

···0 ... 1 -2

) (49)

S ) (S)jp ) 1xN

exp(iπ jpN) (50)

kp ) 4k sin2(pπ2N) (51)

dTR2 ) 1

N ∫0

Ndp 1

kp + l

)kBT

xl(4k + l)≈ kBT

2xlk(52)

S(qb, λ) ) ∫0

1dx ∫0

1dx′ ×

exp (-1

2∑

R ) 1

3

qR2⟨rxx′R

2(λ)⟩) (53)

⟨rxx′R2(λ)⟩ ) 1

N ∫-∞

∞dp ⟨upR

2(λ)⟩|eiπpx - eiπpx′|2 (54)

⟨rxx′R2(λ ) 1)⟩ ) 2Rg

2| x - x′| (55)

S(qb, λ ) 1) ) 2Nq4Rg

4(exp(-q2Rg

2) - 1 + q2Rg2) (56)

dTAR2 )

kBT

2 xklA

(57)

⟨rxx′R2(λ)⟩

2Rg2

) λR2|x - x′| +

(1 - λR2)

dTAR2

Rg2

(1 - e-(Rg2|x-x′|)/(dTAR

2)) (58)

⟨rxx′R2(λ)⟩ - ⟨rxx′R

2(1)⟩

(λR2 - 1)⟨rxx′R

2(1)⟩) fA(Rg

2|x - x′|dTA

2 )fA(y) ) 1 + e-y - 1

y(59)


Furthermore, one obtains for the shear modulus andthe stress-strain relation

so that the Mooney-Rivlin parameters are simply givenby

B. The Heinrich-Straube/Rubinstein-Panyuk-ov Model. Heinrich and Straube25 and Rubinstein andPanyukov43 have carried out analogous considerationsfor model B, i.e. an affinely deforming tube. The relationbetween the strength of the springs lB and the tubediameter in the unstrained state is identical to theprevious case. However, the tube diameter now becomesdeformation dependent:

Thus, the typical width of the fluctuations changes onlywith the square root of the width of the confiningpotential. Using equations eqs 54 and 36, one obtainsfor the mean square internal distances:

Again, we can rewrite this result in terms of a universalscaling function for the degree of affineness of thepolymer deformation:

Equation 67 shows that Straube’s conjecture31-33 fA(y)) fB(y) is incorrect. However, the two functions arequalitatively very similar.

For the shear modulus and the stress-strain relation,we find

in agreement with Rubinstein and Panyukov.43 Toaccount for the network contribution to the shearmodulus, these authors add the phantom networkresults to eqs 69 and 70. This leads to the followingrelations for the Mooney-Rivlin parameters:43

Note that eq 70 holds only for λ ≈ 1. For largecompression or extension the approximation k . l (λ)breaks down and one regains the result of Heinrich andStraube:25

C. The “Double Tube” Model. In the following, wediscuss a combination of two different constraints, onerepresenting the network (model A) and thereforedeformation independent and the other representing theentanglements (model B). Thus, we use model C tocombine the Warner-Edwards model with the Hein-rich-Straube/Rubinstein-Panyukov model.

Evaluating eq 25 one obtains for the tube diameter

The deformation dependent internal distances are givenby

In this case, it is not possible to rewrite the result interms of a universal scaling function, because therelative importance of the two types of confinement isdeformation dependent. Introducing Φ(λ) ) dTCR

4(λ)/dTBR

4(λ), eq 75 can be rewritten as

For the elastic properties of the double tube model wefind

gA(λ) )Fb2xklA

6) GA (60)

GA ) 14

Fb2kBT

dTA2

(61)

σT(λ) ) (λ2 - 1λ)GA (62)

2C1 ) GA (63)

2C2 ) 0 (64)

dTBR2(λ) ) λR

dTB2

3(65)

⟨rxx′R2(λ)⟩

2Rg2

) λR2|x - x′| +

12

(λR2 - 1)|x - x′|e-(Rg

2|x-x′|)/(dTBR2(λ)) -

32

(λR2 - 1)

dTBR2(λ)

Rg2

(1 -e-(Rg2|x-x′|)/(dTBR

2(λ))) (66)


2(1)⟩


2(1)⟩) fB(Rg

2|x - x′|dTBR

2(λ) )fB(y) ) 1 + 1

2e-y + 3

2e-y - 1

y(67)

gB(λ) ) 18

Fb2kBT

λdTB2

)GB

λ(68)

GB ) 18

Fb2kBT

dTB2

(69)

σT(λ) ) (xλ - 1xλ

+ λ - 1λ)GB (70)

2C1 ) Gph + 12

GB (71)

2C2 ) 12

GB (72)

σT(λ) ) (λ - 1xλ)GB (λ , 1, λ . 1) (73)

1dTCR

4(λ)) 1

dTAR4

+ 1dTBR

4(λ)(74)

⟨rxx′R2(λ)⟩

2Rg2

) λR2|x - x′| +

12

(λR2 - 1)|x - x′|dTCR

4(λ)

dTBR4(λ)

e-(Rg2|x-x′|)/(dTCR

2(λ)) -

32

(λR2 - 1)

dTCR2(λ)

Rg2

(1 - e-(Rg2|x-x′|)/(dTCR

2(λ))) ×

dTCR4(λ)(dTAR

4 + 23

dTBR4(λ))

dTBR4(λ)dTAR

4(75)


2(1)⟩


2(1)⟩ (Rg2|x - x′|

dTCR2(λ)

, Φ(λ)) )

fA(y) + Φ(λ)(fB(y) - fA(y)) (76)


Again, eq 78 only holds for moderate strains. Shearmodulus and the Mooney-Rivlin parameters are givenby

D. Comparison of the Different Tube Models. Inthe following, we compare the predictions of the differ-ent models for the microscopic deformations and themacroscopic elastic properties from two different pointsof view.

1. As a function of the network connectivity, i.e., theratio of the average strand length Nc between cross-linksto the melt entanglement length Ne. For this purpose,we identify GA with the shear modulus of the corre-sponding phantom network Gph:

where we use f ) 4 for our plots. Similarly, we choosefor GB a value of the order of the melt plateau modulusGe:

2. Assuming that the system is characterized by acertain tube diameter dTC or shear modulus GC, wediscuss its response to a deformation as a function ofthe relative importance 0 e Φ e 1 of the cross-link andthe entanglement contribution to the confinement

where Φ is of the order (1 + (Ne/Nc)2)-1.1. Elastic Properties. Figure 2 shows the shear

modulus dependence on the ratio of the network strandlength Nc to the melt entanglement length Ne. As

expected GC crosses over from Gph for short strands toGe in the limit of infinite strand length. For comparisonwe have also included the prediction of Rubinstein andPanyukov, Gph + Ge. The shear moduli predicted by ouransatz are always smaller than this sum. In particular,we find G ) Gph for Nc , Ne. The physical reason isthat in a highly cross-linked network the typical fluc-tuations are much smaller than the melt tube diameter.As a consequence, the network does not feel the ad-ditional confinement and the entanglements do notcontribute to the elastic response. Figure 3 showsanalogous results for the Mooney-Rivlin parameters C1and C2 again in comparison to the predictions ofRubinstein and Panyukov. Note that C2 is not predictedto be strand length independent.

Figure 4 shows the reduced force in the Mooney-Rivlin representation for different entanglement con-tributions Φ to the confinement. For moderate elonga-tions up to λ ≈ 2 the curves are well represented by theMooney-Rivlin form. For a given shear modulus, C1 andC2 are a function of the entanglement contribution Φ

Figure 2. Langley plot of the shear modulus. The solid linecorresponds to the “double tube” model, the dotted line to theHeinrich-Straube/Rubinstein-Panyukov model and the dashedline to the phantom model. Ne represents the entanglementlength and Nc the cross-link length.

Figure 3. Plot of the parameters 2C1 and 2C2 of the Mooney-Rivlin equation f(λ-1) ) 2C1 + 2C2λ-1 for the Rubinstein-Panyukov model (dotted) and the “double tube” model (solid).

Figure 4. Mooney-Rivlin representation of the reduced forcefor different values of Φ (from top to bottom: the Phantommodel (dashed line, Φ ) 0), the “double tube” model (solid lines,Φ ) 1/3, 1/2, 3/4) and the Heinrich-Straube/Rubinstein-Panyukov model (dotted line, Φ ) 1)).

gC(λ) )2gB

2(λ) + gA2

x4gB2(λ) + gA

2(77)

σT(λ) ) (λ2 - 1)gC(λ) + (1 - λ-1)gC(λ-1/2) (78)

GC )2GB

2 + GA2

x4GB2 + GA

2(79)

2C1 )GA

4 + 6GB2GA

2 + 4GB4

(4GB2 + GA

2)3/2(80)

2C2 )4GB

4

(4GB2 + GA

2)3/2(81)

14

Fb2kBT

dTA2

) GA ) Gph ) (1 - 2f )FkBT

Nc(82)

dTA2 ) f - 2

4fb2Nc (83)

18

Fb2kBT

dTB2

) GB ) Ge ) 34

Fb2kBTNe

(84)

dTB2 ) 1

6b2Ne (85)

Φ )dTC

4

dTB4

(86)

1 - Φ )dTC

4

dTA4

(87)


to the confinement:

2. The Tube Diameter. Since eq 47 can be writtenin the form

a plot of dTC-2 vs Ne/Nc looks very similar to Figure 2.

The deformation dependence of the tube diameter(Figure 5) takes the form:

In the parallel direction, the entanglement contributionto the confinement vanishes for large λ so that limλf∞dTC|(λ) ) dTA|. On the other hand, the entanglementsbecome relatively stronger in the perpendicular direc-tion with limλf∞ dTC⊥(λ) ) dTB⊥(λ).

3. Microscopic Deformations and StructureFunctions. Figure 6 compares the universal scalingfunctions of the Warner-Edwards and Heinrich-Straube/Rubinstein-Panyukov model defined by eqs 59and 67.

More important for the actual microscopic deforma-tions than the difference between these two functionsis the fact, that the distances are scaled with thedeformation dependent tube diameter. As a conse-quence, deformations parallel to the elongation aresmaller in model B than in model A, while the situationis reversed in the perpendicular direction. In the general

case (eq 76 of model C), the results are further compli-cated by the deformation dependent mixing of the twoconfinement effects. Nevertheless, eqs 59, 67, and 76should be useful for the analysis of simulation datawhere real space distances are directly accessible.

Experimentally, the microscopic deformations canonly be measured via small-angle neutron scatter-ing.31,32 Unfortunately, there seems to be no way tocondense the structure functions eq 53 which resultfrom eqs 58, 66, and 75 for different strains into a singlemaster plot. Figures 7 and 8 show a comparison forthree characteristic values of λ. Qualitatively, theresults for the three models are quite similar. Inparticular, they do not predict Lozenge-like patterns forthe two-dimensional structure functions as they were

Figure 5. Tube diameter dTC(Φ,λ) ) ((1 - Φ) + Φ/λ2)-1/4 inparallel (upper curves) and perpendicular stretching directionfor different elongation ratios λ whereas Φ can be expressedby the entanglement length Ne and the cross-link length Ncby Φ ) dTC

4/dTB4 ) 1/(1 + (Ne/Nc)2) using dTB

2/dTA2 ) Ne/Nc.

The dashed curve corresponds to the Warner-Edwards model,i.e., dTC(Φ ) 0,λ), the dotted curve corresponds to the Hein-rich-Straube/Rubinstein-Panyukov model, i.e., dTC(Φ ) 1,λ),and the solid line represents the “double tube” model with Φ) 3/4.

Figure 6. Comparison of the universal scaling functions ofeqs 59 and 67 for the Warner-Edwards model (dashed) andthe Heinrich-Straube/Rubinstein-Panyukov model (dotted)with y ) (Rg

2|x - x′|)/dTA/B2.

Figure 7. Kratky plots of the different structure factors inparallel and perpendicular stretching direction with Rg/dT )6: Warner-Edwards model (dashed line), Heinrich-Straube/Rubinstein-Panyukov model (dotted line), and “double tube”model with Φ ) 3/4 (solid line). The upper curves correspondto the perpendicular stretching direction.

2C1 ) GC(1 - Φ2

4 - 2Φ) (88)

2C2 ) GCΦ2

4 - 2Φ(89)

GC ) 18

Fb2kBT

dTC2

(2 - Φ) (90)

dTCR4

dTCR4(λ)

) (1 - Φ) + ΦλR

2(91)

limλf∞

dTC|(λ)dTC|

) (1 - Φ)-1/4 (92)

limλf∞

dTC⊥(λ)dTC⊥

) (Φλ)-1/4 (93)


observed by Straube et al.32 In particular, we agree withRead and McLeish34 that the interpretation of Straubeet al.31,32,36 is based on an ad hoc approximation in thecalculation of structure functions from model B. Inprinciple, their alternative idea, to investigated theinfluence of dangling ends on the structure-functionwithin models A and B,34 can be easily extended tomodel C. Judging from the small differences betweenthe models (Figures 7 and 8) and the results in ref 34,this would probably allow one to obtain an excellent fitof the data and to correctly account for the deformationdependence of the tube.36 However, since the lozengepatterns were also observed in triblock systems whereonly the central part of the chains was labeled,33

dangling ends seem to be too simple an explanation. Atpresent it is therefore unclear, if the lozenge patternsare a generic effect or if they are due to other artifactssuch as chain scission.36,37 Simulations14,28-30 might helpto clarify this point.

IV. ConclusionIn this paper, we have presented theoretical consid-

erations related to the entanglement problem in rub-ber-elastic polymer networks. More specifically, wehave dealt with constrained fluctuation models ingeneral and tube models in particular. The basic ideagoes back to Edwards,21 who argued that on a mean-field level different parts of the network behave, as ifthey were embedded in a deformation-dependent elasticmatrix which exerts restoring forces toward some restpositions. In the first part of our paper, we were able toshow that the generalized Rouse modes of the corre-sponding phantom network without entanglement re-main eigenmodes in the presence of the elastic matrix.In fact, the derivation of eq 14, which is the Hamiltonianof the exactly solvable constrained mode model (CMM),44

provides a direct link between two diverging develop-ments in the theory of polymer networks: the ideas ofEdwards, Flory, and others on the suppression offluctuations due to entanglements and the consider-ations of Eichinger,11 Graessley,45 Mark,46 and otherson the dynamics of (micro) phantom networks. Analmost trivial conclusion from our theory is the observa-tion, that it is not possible to estimate the entanglementeffects from the knowledge of the absolute strength ofthe confining potentials alone. Required is rather theknowledge of the relative strength which in turn is afunction of the network connectivity in eq 38.

Unfortunately, it is difficult to exploit our formallyexact solution of the constrained fluctuation model forarbitrary connectivity, since it requires the eigenvaluespectrum of the Kirchhoff matrix for randomly cross-linked networks. In the second part of the paper we havetherefore reexamined the idea of Heinrich and Straube25

to introduce entanglement effects into the Warner-Edwards model15 for linear, random paths through apolymer network, whose localization in space is modeledby a harmonic tube-like potential. In agreement withHeinrich and Straube,25 and with Rubinstein andPanyukov43 we have argued that in contrast to confine-ment due to cross-linking, confinement due to entangle-ments is deformation dependent. Our treatment of thetube model differs from previous attempts in that weexplicitly consider the simultaneous presence of twodifferent confining potentials. The effects are shown tobe nonadditive. From the solution of the generalizedtube model we have obtained expressions for the mi-croscopic deformations and macroscopic elastic proper-ties which can be compared to experiments and simu-lations.

While we believe to have made some progress, we donot claim to have solved the entanglement problemitself. For example, it remains to be shown how thegeometrical tube constraint arises as a consequence ofthe topological constraints on the polymer conforma-tions. However, even on the level of the tube model, weare guilty of (at least) two possibly important omis-sions: (i) we have neglected fluctuations in the localstrength of the confining potential, and (ii) we havesuppressed the anisotropic character of the chain motionparallel and perpendicular to the tube. In the absenceof more elaborate theories, computer simulations alongthe lines of refs 14, 28, 29, 30, and 47 may present thebest approach to a quantification of the importance ofthese effects.

Acknowledgment. The authors wish to thank K.Kremer, M. Putz, and T. A. Vilgis for helpful discus-

Figure 8. Contour plots of the different structure factors withRg/dT ) 6: Warner-Edwards model (dashed), Heinrich-Straube/Rubinstein-Panyukov model (dotted line), “doubletube” model with Φ ) 3/4 (solid line). The upper curvescorrespond to the perpendicular stretching direction.


sions. We are particularly grateful to E. Straube forrepeated critical readings of our manuscript and forpointing out similarities between our considerations andthose by Read and McLeish.

References and Notes

(1) Treloar, L. R. G. The Physics of Rubber Elasticity; ClarendonPress: Oxford, England, 1975.

(2) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics;Clarendon Press: Oxford, England, 1986.

(3) Flory, P. J. J. Chem. Phys. 1949, 17, 303.(4) James, H. J. Chem. Phys. 1947, 15, 651.(5) James, H.; Guth, E. J. Chem. Phys. 1947, 15, 669.(6) Flory, P. J. Proc. Royal Soc. London Ser. A. 1976, 351, 351.(7) Mezard, M.; Parisi, G.; Virasoro, M. V. Spinglas Theory and

Beyond; World Scientific: Singapore, 1987.(8) Edwards, S. F. Proc. Phys. Soc. 1967, 91, 513.(9) Edwards, S. F. J. Phys. A 1968, 1, 15.

(10) Deam, R. T.; Edwards, S. F. Philos. Trans. R. Soc. A 1976,280, 317.

(11) Eichinger, B. E. Macromolecules 1972, 5, 496.(12) Higgs, P. G.; Ball, R. C. J. Phys. (Fr.) 1988, 49, 1785.(13) Zippelius, A.; Goldbart, P.; Goldenfeld, N. Europhys. Lett.

1993, 23, 451.(14) Everaers, R. New J. Phys. 1999, 1, 12.1-12.54.(15) Warner, M.; Edwards, S. F. J. Phys. A 1978, 11, 1649.(16) Ronca, G.; Allegra, G. J. Chem. Phys. 1975, 63, 4990.(17) Flory, P. J. J. Chem. Phys. 1977, 66, 5720.(18) Erman, B.; Flory, P. J. J. Chem. Phys. 1978, 68, 5363.(19) Flory, P. J.; Erman, B. Macromolecules 1982, 15, 800.(20) Kastner, S. Colloid Polym. Sci. 1981, 259, 499 and 508.(21) Edwards, S. F. Proc. Phys. Soc. 1967, 92, 9.(22) Marrucci, G. Macromolecules 1981, 14, 434.(23) Graessley, W. W. Adv. Polym. Sci. 1982, 47, 67.(24) Gaylord, R. J. J. Polym. Bull. 1982, 8, 325.(25) Heinrich, G.; Straube, E.; Helmis, G. Adv. Pol. Sci. 1988, 85,

34.

(26) Edwards, S. F.; Vilgis, T. A. Rep. Prog. Phys. 1988, 51, 243.(27) de Gennes, P. G. J. Chem. Phys. 1971, 55, 572.(28) Duering, E. R.; Kremer, K.; Grest, G. S. Phys. Rev. Lett. 1991,

67, 3531.(29) Duering, E. R.; Kremer, K.; Grest, G. S. J. Chem. Phys. 1994,

101, 8169.(30) Everaers, R.; Kremer, K. Macromolecules 1995, 28, 7291.(31) Straube, E.; Urban, V.; Pyckhout-Hintzen, W.; Richter, D.

Macromolecules 1994, 27, 7681.(32) Straube, E.; Urban, V.; Pyckhout-Hintzen, W.; Richter, D.;

Glinka, C. J. Phys. Rev. Lett. 1995, 74, 4464.(33) Westermann, S.; Urban, V.; Pyckhout-Hintzen, W.; Richter,

D.; Straube, E. Macromolecules 1996, 29, 6165-6174.(34) Read, D. J.; McLeish, T. C. B. Phys. Rev. Lett. 1997, 79, 87.(35) Read, D. J.; McLeish, T. C. B. Macromolecules 1997, 30, 6376.(36) Westermann, S.; Urban, V.; Pyckhout-Hintzen, W.; Richter,

D.; Straube, E. Phys. Rev. Lett. 1998, 80, 5449.(37) Read, D. J.; McLeish, T. C. B. Phys. Rev. Lett. 1998, 80, 5450.(38) Gottlieb, M.; Gaylord, R. J. Polymer 1983, 24, 1644.(39) Gottlieb, M.; Gaylord, R. J. Macromolecules 1984, 17, 2024.(40) Gottlieb, M.; Gaylord, R. J. Macromolecules 1987, 20, 130.(41) Erman, B.; Monnerie, L. Macromolecules 1989, 22, 3342.(42) Kloczkowski, A.; Mark, J.; Erman, B. Macromolecules 1995,

28, 5089.(43) Rubinstein, M.; Panyukov, S. Macromolecules 1997, 30, 8036.(44) Everaers, R. Eur. J. Phys. B 1998, 4, 341.(45) Graessley, W. W. Macromolecules 1975, 8, 186 and 865.(46) Kloczkowski, A.; Mark, J.; Erman, B. Macromolecules 1992,

23, 3481.(47) Sommer, J. U.; Schulz, M.; Trautenberg, H. L. J. Chem. Phys.

1993, 98, 7515.(48) Shy, L. Y.; Eichinger, B. E. J. Chem. Phys. 1989, 90, 5179.(49) Higgins, J. S.; Benoit, H. C. Polymers and Neutron Scattering;

Clarendon Press: Oxford, England, 1997.(50) des Cloizeaux, J. J. Phys. (Fr.) 1994, 4, 539.(51) Ullman, R. J. Chem. Phys. 1979, 71, 436.

MA002228C


Appendix H

Self-Similar ChainConformations in Polymer Gels

133

VOLUME 84, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R S 10 JANUARY 2000

Self-Similar Chain Conformations in Polymer Gels

Mathias Pütz,* Kurt Kremer, and Ralf EveraersMax-Planck-Institut für Polymerforschung, Postfach 3148, D-55021 Mainz, Germany

(Received 10 June 1999)

We use molecular dynamics simulations to study the swelling of randomly end-cross-linked polymernetworks in good solvent conditions. We find that the equilibrium degree of swelling saturates at Qeq �N3�5

e for mean strand lengths Ns exceeding the melt entanglement length Ne. The internal structureof the network strands in the swollen state is characterized by a new exponent n � 0.72 6 0.02. Ourfindings can be rationalized by a Flory argument for a self-similar structure of mutually interpenetratingnetwork strands, agree partially with the classical Flory-Rehner theory, and are in contradiction to deGennes’ c�-theorem.

PACS numbers: 61.41.+e, 64.75.+g, 82.70.Gg

Polymer gels [1–5] are soft solids governed by a com-plex interplay of the elasticity of the polymer networkand the polymer/solvent interaction. They are sensitiveto the preparation conditions and can undergo large vol-ume changes in response to small variations of a controlparameter such as temperature, solvent composition, pH,or salt concentration. In this Letter we reexamine a clas-sical but still controversial problem of polymer physics[1,2], the equilibrium swelling of a piece of rubber in goodsolvent.

We discuss below the two basic theories addressing thissituation, the classical Flory-Rehner theory [1] and deGennes’ c�-theorem [2]. Both are supported by part ofthe experimental evidence gathered by combining thermo-dynamic and rheological investigations with neutron orlight scattering [6–10]. Here we use computer simula-tions [11–15], since they offer some advantages in theaccess to and the control over microscopic details of thenetwork structure. We concentrate on the role of entan-glements in limiting the swelling process of defect-freemodel networks and, in particular, the structure of the net-work strands in the swollen gel [16]. Questions relatingto the structural heterogeneity on larger length scales andthe butterfly effect [10,17] will be addressed in a futurepublication.

For networks prepared by cross-linking a dry (i.e., sol-vent-free) melt of linear chains, the strands have Gaussianstatistics, i.e., the mean-square end-to-end distance is re-lated to the average length, Ns, by �r2�dry ~ b2N2n

s , wheren � 1�2 and b is the monomer radius. The same relationalso holds for all internal distances, leading to the charac-teristic structure factor S�q� � q21�n for the scattering atwave vector q from a fractal object.

The classical Flory-Rehner theory [1] writes the gelfree energy F as a sum of two independent terms: a freeenergy of mixing with the solvent (favoring swelling andestimated from the Flory-Huggins theory of semidilute so-lutions of linear polymers) and an elastic free energy (dueto the affine stretching of the network strands which aretreated as Gaussian, concentration-independent, linear en-tropic springs). Minimizing F yields Qeq ~ N3�5

s for the

equilibrium degree of swelling. The Flory-Rehner theoryimplies that the structure factor of long paths through thenetwork is of the form S�q� � q22 both locally, where thechains are unperturbed, and on large scales, where theydeform affinely (�r2�eq ~ �r2�dryQ2�3

eq ) with the outer di-mensions of the sample. The stretching should be visi-ble in the crossover region around q � 2p��bN1�2

s � withS�q� � q21.

More recent treatments are based on the scaling theoryof semidilute solutions of linear polymers [2] and the ideathat locally, inside of so-called “blobs,” the chains behaveas isolated, self-avoiding walks with n � 3�5. On largerscales, the solution behaves as a dense melt of Gaussianblob chains. In the case of swollen networks, a controversyexists whether the size of the network strands is determinedby the global connectivity or by the local swelling. Quiteinterestingly, the first view [17–19] leads again to theFlory-Rehner result Qeq ~ N3�5

s . In contrast, de Gennes’c�-theorem [2] asserts that the macroscopic swelling islimited only by the local connectivity, which begins tobe felt at the overlap concentration c� ~ Ns��bNn

s �3 of asemidilute solution of linear polymers of average lengthNs, corresponding to Qeq ~ N

4�5s . The c�–theorem pre-

dicts S�q� � q25�3 for q . 2p��bN3�5s � as well as un-

usual elastic properties due to the nonlinear elasticity ofthe network strands [14,20].

As in earlier investigations of polymer melts andnetworks [21–25], we used a coarse-grained polymermodel where beads interacting via a truncated, purelyrepulsive Lennard-Jones (LJ) potential are connected byanharmonic springs. With e, s, and t as the LJ units ofenergy, length, and time, the equations of motion wereintegrated by a velocity-Verlet algorithm with a weak localcoupling to a heat bath at kBT � 1e. The potentials wereparametrized in such a way that chains were effectivelyuncrossable, i.e., the network topology was conservedfor all times. In our studies we did not simulate thesolvent explicitly, but rather used vacuum which can beconsidered as a perfect solvent for our purely repulsive(athermal) network chains. The relevant length and timescales for chains in a melt are the average bond length,

298 0031-9007�00�84(2)�298(4)$15.00 © 2000 The American Physical Society


p�l2 � � 0.965�5�s, the mean-square end-to-end distance

�r2� �N�dry � 1.74�2�l2N [21], the melt entangle-ment length, Ne � 33�2� monomers, and the Rousetime tRouse�N� � 1.35tN2 [26]. In dilute solutions,single chains adopt self-avoiding conformations with�r2� �N� � 1.8l2N3�5.

Using this model, it is possible to study differentnetwork structures including randomly cross-linked,randomly end-cross-linked [22,23], and end-linked melts[24], as well as networks with the regular connectivity ofa crystal lattice [25]. Here we investigate end-cross-linkedmodel networks created from an equilibrated monodis-perse melt with M precursor chains of length N at ameltlike density rdry � 0.85s23 by connecting the endmonomers of the chains to a randomly chosen adjacentmonomer of a different chain. This method yieldsdefect-free trifunctional systems with an exponentialdistribution of strand lengths Ns with an average ofNs � N�3. The Gaussian statistics of the strands remainsunperturbed after cross-linking [22,27]. The systems stud-ied range from M�N � 3200�25 (i.e., the average strandsize Ns � 8.3) up to M�N � 500�700 (Ns � 233),some systems being as large as MN � 5 3 105. Allsimulations used periodic boundary conditions in a cubicbox and were performed at constant volume. Startingfrom Vdry � MN�rdry , the size of the simulation box wasincreased in small steps alternating with equilibration pe-riods of at least five entanglement times tR�Ne� � 1400t.The isotropic pressure P was obtained from the micro-scopic virial tensor and the condition Peq 0 was used todefine equilibrium swelling with Qeq � Veq�Vdry . Testswith a part of the networks using open boundaries did notshow any significant changes of the results.

We investigated the equilibrium swelling of our modelnetworks as a function of the average strand length Ns.Figure 1 shows Q21

eq N3�5e as a function of the average

strand length �Ne�Ns�23�5. In agreement with experimen-tal results for highly cross-linked networks [8,9], our re-sults for short strands are compatible with the Flory-Rehner[1] prediction Qeq ~ N3�5

s . They do, however, not allowfor an independent determination of the exponent. Incontradiction to the original theory, we observe a satu-ration of the equilibrium swelling degree for large Ns

[8]. The crossover occurs for Ns � Ne. The extrapolatedmaximal degree of equilibrium swelling Qmax�Ns ! `� �6.8�3� is close to the swelling degree of an ideal Flory-gelwith average strand length Ne : 1.15N3�5

e � 9.5, wherethe prefactor is empirically obtained from the slope of thestraight line in Fig. 1. In contrast, the corresponding esti-mate based on the c�-theorem, Qeq � b3�s3N4�5

e � 36,is clearly too high (b � 1.3s is the stastical segmentlength in good solution). Our interpretation is that to a firstapproximation entanglements act as chemical cross-linksin limiting the swelling of polymer networks. The situa-tion is analogous to an “olympic gel” [2] of topologically

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2 2.5 3 3.5

Ne3/

5 Qeq

-1

(Ne/Ns)3/5

FIG. 1. Strand length Ns dependence of the equilibrium degreeof swelling Qeq. The straight line going through zero representsFlory’s prediction Qeq ~ N3�5

s . The melt entanglement lengthNe was used to normalize the axis in order to show that devi-ations from Flory’s theory occur around Ns � Ne and that theasymptotic value Qeq�Ns ! `� is of the order of N3�5

e .

linked ring polymers. In contrast to solutions of linearpolymers, systems containing trapped entanglementscannot be arbitrarily diluted.

The chain conformations at equilibrium swelling arebest characterized by their structure factor S�q�. Figure 2shows S�q� of the precursor chains within the network forour most weakly cross-linked N � 700 sample. We havechosen the Kratky representation [q2S�q� vs q] to showthe deviation from the Gaussian case [S�q� ~ q22] moreclearly. The observed power law form S�q� ~ q21�n ischaracteristic of fractals and common in polymeric sys-tems. However, the observed exponent n � 0.72�2� is un-expected. Furthermore, the fractal structure is observedfor a q range of 2s &

2p

q & 15.5s � bN0.72e , suggesting

that the mean extension of the effective strands of lengthNe is the only relevant length scale in the problem. For

1

10

0.1 1

q2 Ssc

(q)

q [σ-1]

FIG. 2. Single chain structure function in networks at equi-librium swelling in Kratky representation. The straight linecorresponds to a power law q221�n (n � 0.72). The figurecontains scattering data for the precursor chains (≤) of lengthN � 700 (Ns � 233) and for network strands of lengths Ns �10 (�), Ns � 40 (�), Ns � 70 (Ø), and Ns � 100 (�) withina different network with precursor chains of length N � 100(Ns � 33).

299


smaller q, we see the onset of the expected scattering ofa Gaussian chain consisting of randomly oriented parts oflength bN0.72

e . Our precursor chains (even N � 700) aretoo short to see it clearly developed.

Since the scattering from the precursor chains couldbe affected by polydispersity effects, we have investigatedthe conformations of the network strands as a function oftheir contour length Ns. For high q, all structure functionsfall on top of each other and show the same fractal struc-ture with S�q� � q21�0.72�2� (Fig. 2). The complementaryFig. 3 shows a log-log plot of the mean-square strand ex-tension �r2�eq�Ns� versus their length. In agreement withthe results for the structure functions, we find a power law�r2�eq ~ b2N230.72

s for strands which are shorter than theeffective strand length Ne and therefore subaffine defor-mations. Long strands, on the other hand, deform affinelywith �r2�eq � �r2�dryQ

2�3eq .

Clearly, the results of our simulations do not agree withthe predictions of any of the theories presented in the in-troduction. While the neglect of entanglements seems tobe fairly simple to repair by treating them as effectivecross-links (with Ne supplanting the average strand lengthNs [19]), the fractal structure of the strands and the ex-ponent n � 0.72�2� come as a surprise. In the follow-ing, we discuss a possible explanation for the strongerswelling of network strands (n � 0.72) than of singlechains (n � 3�5) under good solvent conditions.

We begin by recalling Flory’s argument [1] for the typi-cal size RF ~ bNn of a single polymer chain oflength N and statistical segment size b in a goodsolvent. The equilibrium between an elastic energy~ R2

F��b2N� of a Gaussian chain stretched to RF anda repulsive energy ~ bdRd

F�N�RdF�2 due to binary

contacts between monomers in d dimensions leads ton � 3��d 1 2�.

FIG. 3. Log-log plot of the mean-square end-to-end distance�r2�eq of the individual network strands within a single network(Ns � 33) at equilibrium swelling Q � 5.8 versus strand lengthNs. The straight line corresponds to a power law �r2�eq � N2n

swith n � 0.72. The data are normalized to an affine deformation�r2�eq � Q2�3

eq �r2�dry .

The simplest models for swollen networks have theregular connectivity of a crystal lattice. In agreement withthe c�-theorem, they adopt equilibrium conformations withstrand extensions of the order of RF [28]. However, thesesystems are hardly good models for the swelling process ofnetworks prepared in the dry state, since the hypotheticalinitial state at melt density has an unphysical local struc-ture with average strand extensions RFQ21�3 ~ bN1�3

s asin dense globules. In contrast, if the corresponding semidi-lute solution is compressed, the chains shrink only weaklyfrom RF to the Gaussian coil radius R ~ bN1�2

s . Instead,they become highly interpenetrating with nF ~ N1�2

s (theFlory number) of them sharing a volume of R3. More-over, at least the simplest model for highly cross-linkednetworks prepared in the dry state, nF � N1�2

s mutuallyinterpenetrating regular networks with strand extensionsof the order of R [25], cannot possibly comply with thec�-theorem, if one disregards macroscopic chain separa-tion: Either the strands extend to RF , leading to internalconcentrations of c�N1�2

s , or the systems swell to c�, inwhich case the strands are stretched to RFN1�6

s . The sameconclusions should hold for any network without too manydefects, where the global connectivity forces neighboringchains to share the same volume independent of the degreeof swelling.

We now consider a Flory argument for a group ofchains which can swell but not desinterpenetrate, i.e.,nF � Ndn21 chains of length N which span a volume Rd

FR .The equilibrium between the elastic energy ~ n2

FR��b2N�and the repulsive energy ~ bdRd

FR�nFN�RdFR�2 leads to

n �4 1 d4 1 2d

. (1)

Quite interestingly, this local argument reproduces inthree dimensions with Qeq � Nd��d12� � N3�5 andRFR � Q1�dbN1�2 � N7�10 the results of the classicalFlory-Rehner theory of gels. However, in analogy tothe Flory argument for single chains, Eq. (1) shouldalso apply to subchains of length G with 1 ø G , Nwhich share their volume with a correspondingly smallernumber of other subchains. In particular, the local degreeof swelling, G1�5, should be subaffine and the exponentn � 7�10 should characterize the entire local chainstructure up to the length scale of the effective strandlength, Ne. This is in excellent agreement with the mainfindings from our simulations (see Figs. 2 and 3).

Before we conclude, some additional remarks are in or-der: (i) For swelling in a theta solvent, the analogous scal-ing argument yields Q � N3�8 in agreement with previoustheories and experiments [8,9,19] and predicts local chainstructures characterized by n � 5�8. (ii) Equation (1) canalso be derived along the lines of [17–19] from an equi-librium between the elastic energy of blob chains and theosmotic pressure of a semidilute polymer solutions. Notethat the appropriate blob size is a function of the size G ofthe subchains under consideration and that isolated chain

300


behavior is expected only below the original correlationlength jprep for systems prepared by cross-linking semidi-lute solutions. (iii) Sommer, Vilgis, and Heinrich [29] haveargued that the effective inner fractal dimension di of apolymer network is larger than di � 1 for linear chains,leading to stronger swelling with n � �di 1 2��d 1 2�.While the correction goes into the right direction, it is dif-ficult to explain a strand length independent effective in-ner fractal dimension of di � 1.5 as an effect of the localconnectivity. (iv) However, such effects may well be im-portant in systems with a sufficient number of defects suchas dangling ends or clusters. If the global connectivity isweak, the chains may locally desinterpenetrate, leading toa behavior which agrees much better with the c�-theorem[7,12].

In summary, we have used large scale computer simu-lations and scaling arguments to investigate the equilib-rium swelling of defect-free model polymer networksprepared at melt density. We find that the chain structureon short scales is independent of the network connectivityand characterized by an exponent n � 7�10, while themacroscopic degree of swelling is controlled throughan (entanglement limited) effective strand length. Thepredicted chain structure should be directly observable inneutron scattering experiments.

We acknowledge the support of the Höchstleis-tungsrechenzentrum Jülich and the Rechenzentrum of theMPG in München and thank G. S. Grest for discussionsand a careful reading of the manuscript.

Note added.—After this work was finished, we learnedof unpublished theoretical work by Rabin [30] andby Erman et al. [31] predicting n � 7�10 in swollennetworks.

*Present address: Sandia National Laboratories, Albu-querque, NM 87185-1349.

[1] P. J. Flory, Principles of Polymer Chemistry (Cornell Uni-versity Press, Ithaca, 1953).

[2] P. G. de Gennes, Scaling Concepts in Polymer Physics(Cornell University Press, Ithaca, 1979).

[3] D. Derosi, K. Kajiwara, Y. Osaka, and A. Yamauchi, Poly-mer Gels (Plenum Press, New York, 1991).

[4] Responsive Gels: Volume Transitions I&II, Advances inPolymer Science Vols. 109 and 110, edited by K. Dusek(Springer-Verlag, Berlin, 1993).

[5] Physical Properties of Polymeric Gels, edited by J. C. Ad-dad (Wiley, Chichester, 1996).

[6] T. Tanaka, L. Hocker, and G. Benedek, J. Chem. Phys. 59,5151 (1973).

[7] S. Candau, J. Bastide, and M. Delsanti, Adv. Polym. Sci.44, 27 (1982).

[8] S. K. Patel, S. Malone, C. Cohen, and J. R. Gillmor, Macro-molecules 25, 5241 (1992).

[9] G. Hild, Polymer 38, 3279 (1997); Prog. Polym. Sci. 23,1019 (1998).

[10] J. Bastide and S. J. Candau, in Ref. [5].[11] K. Kremer and G. S. Grest, in Monte Carlo and Molecu-

lar Dynamics Simulations in Polymer Science, edited byK. Binder (Oxford University Press, New York, 1995).

[12] J. U. Sommer, Macromol. Symp. 81, 139 (1994); H. L.Trautenberg, J. U. Sommer, and D. Göritz, J. Chem. Soc.Faraday Trans. 91, 2649 (1995).

[13] R. Everaers and K. Kremer, J. Mol. Model. 2, 293 (1996).[14] T. Hölzl, H. L. Trautenberg, and D. Göritz, Phys. Rev. Lett.

79, 2293 (1997); R. Everaers and K. Kremer, Phys. Rev.Lett. 82, 1341 (1999).

[15] F. Escobedo and J. de Pablo, J. Chem. Phys. 104, 4788(1996); J. Chem. Phys. 106, 793 (1997); Mol. Phys. 90, 437(1997); J. Chem. Phys. 110, 1290 (1999). N. R. Kenkare,C. K. Hall, and S. A. Khan, J. Chem. Phys. 110, 7556(1999).

[16] To our knowledge, all previous SANS studies have con-centrated on the low q Guinier regime (i.e., the radiusof gyration of the strands) and no particular attention hasbeen paid to the local chain structure. See, for example,M. Beltzung, J. Herz, and C. Picot, Macromolecules 16,580 (1983); N. S. Davidson and R. W. Richards, Macro-molecules 19, 2576 (1986).

[17] S. Panyukov and Y. Rabin, Phys. Rep. 269, 1 (1996).[18] S. Panyukov, Sov. Phys. JETP 71, 372 (1990).[19] S. P. Obukhov, M. Rubinstein, and R. H. Colby, Macro-

molecules 27, 3191 (1994).[20] S. Daoudi, J. Phys. (Paris) 38, 1301 (1977); R. Everaers, J.

Phys. (Paris) II 5, 1491 (1995); R. Everaers, I. S. Graham,M. J. Zuckermann, and E. Sackmann, J. Chem. Phys. 104,3774 (1996).

[21] K. Kremer and G. S. Grest, J. Chem. Phys. 92, 5057 (1990).[22] G. S. Grest and K. Kremer, Macromolecules 23, 4994

(1990).[23] E. R. Duering, K. Kremer, and G. S. Grest, Phys. Rev. Lett.

67, 3531 (1991).[24] E. R. Duering, K. Kremer, and G. S. Grest, J. Chem. Phys.

101, 8169 (1994).[25] R. Everaers and K. Kremer, Macromolecules 28, 7291

(1995); Phys. Rev. E 53, R37 (1996); R. Everaers, NewJ. Phys. 1, 12.1–12.54 (1999).

[26] M. Pütz, K. Kremer, and G. S. Grest (to be published).[27] M. Pütz, and K. Kremer (to be published).[28] H. L. Trautenberg, Ph.D. thesis, University of Regensburg,

Germany, 1997.[29] J. U. Sommer, T. Vilgis, and G. Heinrich, J. Chem. Phys.

100, 9181 (1994).[30] Y. Rabin (unpublished).[31] B. Erman (unpublished).

301

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