NATO ASI workshop on "Structure and Dynamics of Polymers and Colloidal Systems" Les Houches, September 14-24, 1999 Lecture notes, draft copy

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HIGHLY CHARGED POLYELECTROLYTES :

Chain Conformation, Counterion Condensation and Solution Structure

Claudine E. Williams

Physique des Fluides Organisés (CNRS URA 792)Collège de France

Paris, France

1. Introduction

Polyelectrolytes are polymer chains containing a variable amount (usually large) of ionisable

monomers. Once dissolved in a polar solvent such as water, the ions pairs dissociate. The

electrostatic charges of one sign are localised on the chain whereas the large number of

oppositely charged counterions are scattered in the solution. Polyelectrolytes are everywhere

around us and in us. Most biopolymers, including DNA and proteins, are polyelectrolytes and

many water soluble polymers of industrial interest are charged. Thus phenomena specific to

polyelectrolytes have strong implications in molecular and cell biology as well as technology.

Despite more than 50 years of continuing interest, the unique properties of charged polymers

are still poorly understood, in contrast to their neutral counterparts. The complexity stems

primarily from the simultaneous presence of long range electrostatic interactions and short

range excluded volume interactions and to the crucial role of the counterions. The fifties have

been a golden era when most of the physical and chemical properties of the single chain have

been understood (the contribution of the school of Katchalsky is rather outstanding in that

context). A second leap forward came with the isotropic model for semi-dilute solutions of de

Gennes and collaborators. During the last decade many new theoretical approaches, both

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analytical and computational, have appeared and a large amount of experimental information

has been collected which have led to a deeper understanding of these complex systems.

In this lecture I will try to give you a flavour of some of the interesting questions

raised, limiting myself to the static properties of linear flexible and highly charged synthetic

polyelectrolytes and selecting topics on which I have been personally active. A word of

caution is in order: because of the short time available, the tutorial will be a bit sketchy. My

only hope is to get you interested enough to search the reviews and detailed articles listed in

the references.

2. Some characteristic lengths and definitions

Most of the flexible polyelectrolytes have a vinylic backbone. The monomer size is about

2.5Å.

The solvent is characterised by the Bjerrum length ; it is the distance over which the

electrostatic energy between two elementary charges e in a solvent of dielectric

permittivity is exactly compensated by the thermal energy kBT. = e² / kBT = 7.12Å

in water at 20°C.

The Debye-Hückel screening length -1 is defined as ² = 4 I where I is the total number

of "free" charges in the solution. Typically, -1 is of the order of 100Å for a 10-3 M solution.

Polyelectrolytes are said to be weakly charged when a small fraction of the monomers are

charged; Coulomb interactions interplay with usual Van der Waals interaction. They are

highly charged when a large fraction of monomers are charged; in this case Coulomb

interactions dominate.

The latter definitions should not be confused with the notion of weak and strong

polyelectrolytes. In the weak case, the charged monomer units are derived from a weak

acid, e.g., monomers with COOH groups. In solution, not all groups are dissociated and

the degree of dissociation depends on the pH of the solution; each chain can be viewed as

a random copolymer of monomers with COO- and COOH groups which fluctuate; the

charges are said to be annealed 1. For strong polyelectrolytes, e.g. with SO3H units derived

from a strong acid, all monomers are dissociated and the charges are said to be quenched.

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3. Two important properties of the single chain

3.1. At infinite dilution, a polyelectrolyte chain is highly extended

It was recognised very early on that polyelectrolyte chains are very large objects. The large

increase in reduced viscosity as concentration decreases, was interpreted as evidence of chain

stretching for highly dilute salt-free solutions and in the 50's highly charged polyelectrolytes

were commonly pictured as rigid rods. Chain stretching is indeed very dramatic: for instance,

a chain of neutralised polyacrylic acid of degree of polymerisation 1000 has a radius of about

200Å in its coiled state (uncharged at low pH) but reaches almost 2000Å when fully charged

(fully stretched = 2500Å)

The effect of the repulsive interaction between like charges on the chain conformation

can be understood by a Flory-type calculation, due to Kuhn, Künzle and Katchalsky in 1948

(before Flory published his own calculation for neutral chains with short-range excluded-

volume interactions !)2. It relies on simplifying assumptions but gives a simple physical

picture. Let us consider a chain with N monomers and assume that a fraction f of those are

ionisable. Thus, in solution, the chain contains fN charged monomers and (1-f)N neutral

monomers, all randomly distributed. In a mean-field approach, the Flory-type energy for a

chain of size R is

(1)

The first term is the elastic energy where we assume that the chain has a gaussian

configuration when all electrostatic interactions are switched off, i.e. the mean squared

average end-to-end distance is . The second term is the electrostatic energy due to

the Nfe charges. Minimisation with respect to R leads to

(2)

It is important to stress that the linear dependency of R with N does not imply that the chain is

fully extended ; it may retain some local molecular flexibility and still R would scale as N.

The flexibility is clearly seen in Monte-Carlo simulations.3

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COMMENTS

In (1), the electrostatic term should contain a numerical factor which depends on the distribution of charges in the volume of the chain

Taking a more realistic rod like shape would only introduce a logarithmic term in (2) Counterions are not taken into account

A « blob » picture, as introduced by de Gennes et al. 4, is useful to get a better image of

the chain conformation. It also allows us to introduce some basic concepts of the statistical

physics of polymers.5 We assume here that the chain is weakly charged and that the backbone

(chain without charges) is in a -solvent. We now look at the spatial monomer-monomer

correlations  and find that there is one important length which we call D, the electrostatic blob

size. On length scales smaller than D, the electrostatic interaction is only a weak perturbation,

the chain statistics are determined by the solvent quality and thus remain gaussian in our case;

if ge monomers are involved, then . On length scales larger than D, the electrostatic

repulsion between blobs dominates and the chain has the conformation of a rod of N/ge blobs

of size D. The total length is . The size of the electrostatic blob and the number

of monomers involved depend on the linear charge density of the chain but not on its size.

Indeed, using the fact that on a length scale D the electrostatic interaction is of the same order

as the polymer fluctuations and that the subchain has a gaussian configuration, one finds that

(3)

COMMENT

The same reasoning can be applied for a chain in good solvent. The case of a bad solvent is more subtle and a globule/solvent surface tension contribution has to be included in the energy; this will be briefly treated in the last section.

3.2. The effective charge of highly charged polyelectrolytes is renormalized by counterion

condensation.

When we looked at the chain conformation, we implicitly assumed that the entropy of mixing

was driving the counterions to distribute uniformly in the solution. However when the chain is

highly charged, the electrostatic interactions attract the counterions to the oppositely charged

polymer chain. The potential close to the chain can be so high that for some counterions the

entropy of mixing is dominated by the electrostatic interaction and they remain bound to the

chain, so reducing the effective charge of the chain compared to the nominal (or chemical)

charge. This phenomenon is known as counterion condensation.

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The distribution of charges around a single infinitely long rod has been first calculated

using Poisson-Bolzmann theory.6 In an alternate approach, due to Manning7 and Oosawa8,

which we will develop here, the counterions are assumed to be divided in two species, free in

the solution or condensed in a sheath around the chain. There is chemical equilibrium between

the two species.

Imagine the chain as a rod of length L (La) and of linear charge density ,

where A is the distance between charges along the chain. The density of counterions at a

distance r from the chain (r<<L) is where (r)is the electrostatic

potential calculated by Gauss theorem as (r) = 2lB/A ln(r). The total number of counterions

m(r) per unit length of rod within a cylinder r becomes

(3)

The behaviour of this integral depends very much on the value of the coupling parameter

. When u is small, i.e. when f is small, the integral is dominated by its upper bound

and the total number of counterions decreases as r decreases. On the other hand, when u is

larger than 1 (highly charged chain), the integral diverges at its lower bound. There is a

condensation of counterions until the value of u reaches 1, at which point the average distance

between charges is equal to the Bjerrum length.

What does this imply in practical terms? For a vinylic polymer in water at room

temperature, a is 2.5Å and is 7.12Å; the onset of counterion condensation corresponds to a

critical f equal to 0.35, i.e. when about every third monomer is charged. Let us imagine that

chemistry allows us to gradually introduce more and more ionizable monomers in a neutral

polymer. The effective (or net) charge fraction of the chain will increase as the chemical (or

bare) charge fraction up to 0.35, then it will remain constant as more and more counterions

are condensed. Thus in this regime the condensed counterions partially neutralise the bare-rod

charge density uniformly to a net charge density. The results of various techniques

(osmometry, electrophoresis, conductivity...) which are sensitive to the number of free

counterions in the solution have validated, at least qualitatively, the existence of counterion

condensation. They are discussed in details by Manning.9 In this lecture I will focus on recent

osmotic pressure measurements to determine the amount of osmotically active counterions in

various conditions.

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COMMENTS

Manning-Oosawa theory is strictly valid for an infinite, uniformly charged rod at zero concentration. Various attempts to take into account the effect of finite chains at nonzero concentration have predicted deviations to MO theory.

The coupling between CC and chain conformation (flexible chain) has also been considered theoretically and in Monte Carlo simulations. A collapse of the stretched chain is even predicted10.

The condensed counterions are confined to a sheath around the chain but they still retain some mobility along the chain. Ion pairing takes place in region where the dielectric constant is too low for the charges to be dissociated (ionomer effect). These points are discussed extensively and very clearly in Oosawa's book.

4. Semi-dilute solutions of flexible, hydrophilic polyelectrolytes

Most realistic experiments with polyelectrolytes are done at concentrations where the chains

are interacting and the single chain behaviour is no longer relevant. In the fifties, when

polyelectrolytes were the focus of intensive studies, it was firmly believed that the molecules

were retaining their rod-like conformation and as concentration increases, they would form a

lattice of rods.11 No evidence for such a structure was found except for a single broad peak in

scattering experiments whose position varies as c1/2 as expected for a 2d (short range) order of

rods. A breakthrough came in the seventies when de Gennes applied to polyelectrolytes the

techniques of statistical physics that had been successful for neutral polymers and introduced

the idea that a semi-dilute solution of polyelectrolytes remains isotropic at any concentration,

the chains forming an entangled network. In what follows, I will describe the main results of

the scaling theory in its simplest form. More details about the theory of polyelectrolytes can

be found in the review article of Barrat and Joanny.12

4.1. The overlap concentration c*

In dilute solution, the chain are elongated (size L) and their average separation is d; as c

increases one reaches a situation where d is still larger than L but becomes smaller than -1,

the Debye screening length. A peak appears in the scattering profile, characteristic of a liquid-

like order; its position q* is of the order of 2/d (c/N) 1/3 , as observed experimentally.13

c* can be defined as the concentration when d = L , then

(4)

Boris has verified this scaling law by compiling various literature results on poly(styrene

sulphonate) of different molecular weights14. However the absolute values of c* are found to

be higher than expected. This can be understood in terms of MC simulations which show

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clearly that well before c* the chains begin to be less extended, shifting c* away as it is

approached. The experimental determination of c* can be a tricky. For shorter chains, it is the

concentration where the q* versus c exponent changes from 1/3 to 1/2 (semidilute behaviour,

see next paragraph); for longer chains it is slightly arbitrarily set as the concentration when

the solution viscosity is twice the solvent viscosity (see 15 )

4.2. Isotropic model for semidilute solutions

We give here the static scaling picture of de Gennes et al., as revised by Rubinstein et al.16, 17

Consider an ensemble of chains as defined in section 3.1. We will introduce a parameter

B=Na/L which depends on solvent quality (i.e. good or theta solvent). Once again we will

focus on salt-free solutions and look at the monomer-monomer correlations. The model

postulates that the chains overlap and form a transient network above c*. There is an

important correlation length such that, for distances r<, the electrostatic forces are

dominant and the section of the chain has the same extended configuration as in dilute

solution; for distances r>, both electrostatic and excluded volume interactions are screened

and the chain follows random walk statistics since both electrostatic and excluded volume

interactions are screened. If we assume that depends on c as a power law, that it should be

independent of N and that = L at , then

(5)

Keep in mind that scales as and is thus proportional to -1 the Debye screening length.

Each correlation volume (or blob) 3b contains g monomers and one chain amongst others is a

random walk of N/g monomers and has a size

(6)

The concentration dependence ( ) is much stronger than for neutral polymers ( ).

COMMENT

We have assumed here that there is only one characteristic length in the problem, i.e. that the electrostatic persistence length of an intrinsically flexible polyelectrolyte is proportional to the Debye screening length. This is still a disputed fact. See section 4.3 and, for instance, ref. 11.

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Adding salt screens the electrostatic interactions. Eventually the polyelectrolyte reverts

to neutral chain behaviour when the salt screening length is smaller than the mesh size. For

the sake of brevity, I will not discuss this here but refer the reader to reference 15, the paper of

Dobrynin et al. (The original paper of Pfeuty18 may also be consulted.) Briefly, any property X

of a polyelectrolyte solution with added (monovalent) salt can be expressed in terms of the

same property without salt as with the appropriate scaling exponent .

Experimentally, the static monomer-monomer correlations are best measured in

Fourier space, i.e. using the techniques of small-angle scattering, principally neutron (SANS)

and x-ray (SAXS) scattering but also static light scattering for probing very large distances19.

There is a large body of experimental data on various polymers by a large number of groups

and also Monte-Carlo simulations that show qualitative agreement with the predictions of the

isotropic model for semi-dilute solutions. In the lecture, I will show and discuss some of what

I believe are the most spectacular experimental results. However I will not detail them here

but refer the interested reader to a few publications listed at the end of the Reference list.

Evidently the choice of these is highly personal.

First let us summarise the predictions.

In the absence of salt, the structure factor at small wave vectors, which is related to the

osmotic compressibility and is dominated by the small ions, is very small and given by

.

There is a broad peak in the salt-free structure factor at a finite wave vector .

At this value, it can be shown that S(q) is larger than S(0) and therefore the profile is an

increasing function of q at .

At large q-values, corresponding to distances smaller than the mesh size , the chain has a

rod-like behaviour and S(q) should decrease as 1/q.

At high salt concentrations such that the counterions are localised in a sheath of size -1<,

the solution behaves similarly to a neutral solution, the peak disappears and S(q) decays

monotonically from S(q=0)=2cs / f²c.

COMMENTS

The peak in the structure factor is not due to some order in the solution but it is related to the very small value of S(q=0) due to the constraint of electro-neutrality.

Some beautiful experiments on star polyelectrolytes20 illustrate the difference between a peak related to 3-d order of the dense star cores and the polyelectrolyte peak when the star arms overlap and form a semi-dilute background.

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4.3. Puzzles and riddles

Although the isotropic model for semi-dilute solutions has been found adequate to describe

the physical properties of a large number of flexible polyelectrolytes, many unsolved

problems still exist and they are often at the heart of heated controversies.

Let us mention first the presence of "large aggregates" in salt free solutions, evidenced by

an upturn in the scattering profiles at very small angles whose origin is still unknown. A slow

mode in dynamic light scattering is likely to be related to the same phenomenon.

Experimentally, the phenomena are difficult to catch and although many articles are published

every year on the subject, the physical origin of the attractive forces that would produce the

aggregates is still difficult to ascertain. Recent experiments by Brett and Amis21 using coupled

SANS and SLS give the most quantitative static picture of the aggregates to date. A sensible

description of the experimental scene is available in22 and enlightened comments can be found

in 23.

There have been many theoretical approaches predicting attractive forces between like

charged chains but these are concerned with ideal chains and limited to very diluted systems.

Indeed a quantitative treatment of the interactions in these highly charged systems, taking into

account the counterions explicitely requires to use simplified models. These models give

directions for further research but they are still far from real systems. For this particular

problem the gap between theory and experiment is still wide open.

What is the effect of electrostatic interactions on the rigidity of intrinsically flexible

polyelectrolytes? This is another simple question with no simple answer either theoretical or

experimental. Odijk24 25 and, independently, Skolnick and Fixman26 were the first to tackle the

theory of the electrostatic persistence length. They used perturbative methods, starting from a

rod-like chain. They predict that the total persistence length LT is the sum of a bare persistence

length L0 (chain without charges) and of an electrostatic one LOSF that scales as the square of

the Debye screening length has been rather well verified for stiff PE such as DNA. This

finding was in direct opposition to scaling theories which assumed that the chain is flexible at

all scales and predict that Lel scales linearly with -1. Further theoretical predictions, using

various methods fall randomly into two categories, according to the -1 27or -2 28 dependence

of Lel on the screening length. MC simulations do not give a clear answer either and a recent

investigation indicate an exponent smaller than 129. There is a possibility that the conflicting

results are due to limitations of using Debye-Huckel screened interaction to describe

polyelectrolyte systems. Clearly experiments are necessary. Unfortunately direct

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measurement of a persistence length at finite concentrations are difficult to perform. A recent

SANS determination of the electrostatic rigidity of a fully charged PSS, using sophisticated

contrast variation techniques showed that, for the conditions of the experiment,30

, indicating that all ions contribute to the chain stiffness, including the

condensed ones, and showing still another scaling law!

It is fair to mention that there are some indefectible tenants of an ordered model for

flexible polyelectrolytes, in analogy to the situation of charged rods and globules.

1References? Raphael E., Joanny J.F. Eur. Phys J. B 1990 13 6232 Kuhn W., Kunzle O., Katchalsky A. Helv. Chem. Acta 1948 31 1994 3 Stevens M.J.and Kremer K. J. Chem. Phys. 1995 103 16694 de Gennes P.G., Pincus P., Velasco R.M., Brochard F. J. Phys. (Paris) 1976 37, 14615 de Gennes P.G. Scaling Concepts in Polymer Physics, Cornell Univ. Press, Ithaca, NY 19796 Fuoss R.M., Katchalsky A., Lifson S. Proc. Natl. Acad. Sci. USA 1951 37 5797 Manning G.S. J. Chem. Phys. 1969 51, 924 and 9348 Oosawa F. Polyelectrolytes 1971 M. Dekker, New York9 Manning G.S., Ber. Bunsenges. Phys. Chem. 1966 100, 909-92210 Brilliantov N.V., Kuznetsov D.V., Klein R., Phys. Rev. Lett. 1998 81, 1433-143611 Katchalsky A., Alexandrowicz Z., Kedem O. in Chemical Physics of Ionic Solutions, Wiley, New York 196612 Barrat J.L. and Joanny J.F. Adv. Chem. Phys. 1996 100, 909-922 13 Kaji K., Urakawa H., Kanaya T. and Kitamaru R. J. Phys. France 1988 49, 993-100014 Boris D.C., PhD thesis, Univ. of Rochester, New York 199615 Boris D.C., Colby R.H. Macromolecules 1998 31 5746-575516 Dobrynin A., Colby R. H., Rubinstein M. Macromolecules 1995 28 185917 Rubinstein M, Colby R. H., Dobrynin A. Phys. Rev. Lett. 1994 73 277618 Pfeuty P.J. J. Phys. France Coll C2 1978 39, 14919 Higgins J.S., Benoit H.C. Polymers and Neutron Scattering, Oxford University Press 1996 20 Heinrich M. PhD thesis, Strasbourg 1998; Heinrich M., Rawiso M. to be published21 Brett D.E., Amis E. J. Macromolecules 1998 31, 7378-738422 Sedlak M. Macromolecules 1993 26, 1158-116223 Schmitz K.S. 1994 in Macro-ion Characterization, Chapter 1, ACS Symposium Series24 Odijk T. J. Polym. Sci. Polym. Phys. Ed. 1977 15, 477-48325 Odijk T. , Houwaart A.C. J. Polym. Sci. Polym. Phys. Ed. 1978 16, 627-63926 Skolnick J. , Fixman M. Macromolecules 1977 10, 944-94827 Barrat J.L., Joanny J.F. Europhys. Lett. 1993 24, 333; Ha B.Y., Thirumalai D. Macromolecules 1995 28, 577;

Witten T., Pincus P. Europhys. Lett. 1993 3, 31528 Witten T., Li H. Macromolecules 1995 28, 5921 29 Micka U., Kremer K. J. Phys. Condens. Matter (UK) 1995 8, 9463

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These are but a few of the complex problems that anyone working on polyelectrolytes is

bound to encounter. It is these questions that make the field challenging and exciting.

5. Highly charged hydrophobic polyelectrolytes

The description of the statistical physics of polyelectrolyte chains that we have outlined in the

preceding section assumes implicitly that the chain backbone is in a -solvent and that its

properties are entirely determined by the electrostatic interactions (strong coupling limit).

Numerous experiments have validated this assumption, using very often fully charged

poly(styrene sulphonate), PSS, as a model compound. In all these studies, the fact that the

uncharged polymer is highly hydrophobic seemed irrelevant in agreement with the theoretical

predictions. The question we might ask at this point is the following: what happens if the

hydrophobic character of the chain is increased by introducing some styrene monomers in

the chain, still keeping the charge fraction in the strong coupling regime? According to what

we have learned earlier, the effective charge of the chain should be renormalised by Manning

condensation and the structure of the solution should remain unchanged since the electrostatic

interactions are assumed to be dominant31. As you can guess this is not the case and

hydrophobic effects have drastic effects on the chain properties. Note that in this case the

experiments have been leading theory.

The tutorial discussion will be based on an experimental investigation using small

angle X-ray and neutron scattering, osmometry and fluorescence emission spectroscopy32, 33, 34. The polyelectrolyte considered is a random copolymer of various compositions of styrene,

a hydrophobic monomer, and styrene sulphonate, an ionizable water soluble monomer. The

charge fraction was varied between 0.3 (limit of solubility in water) and 1 (fully charged) i.e.

in a charge range where the average linear spacing between charges b is smaller than the

Bjerrum length. The experiments have shown that:

the expected renormalised effective charge is found only at f=1 and is continuously

reduced as f decreases; at f=0.38 the reduction is about 75% of Manning's value as

measured at f=1 (as shown by osmometry and confirmed by SAXS and SANS). This

30 Spiteri M.N., Boué F., Cotton J.P., Lapp A. Phys. Rev. Lett. 31 Essafi W., Lafuma F.and Williams C.E., Eur. Phys J. B 1999 9 26132 Essafi W., Lafuma F.and Williams C.E., J. Phys II France 1995 5 126933 Essafi W. PhD thesis, Paris VI 199634 Spiteri M.N., PhD thesis, Paris XI 1997

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indicates that CC is not the only process involved and that some charges are trapped in the

solution where they are no longer osmotically active.

there are some diffuse hydrophobic regions in the solution (as shown by fluorescence

emission of a pyrene probe) a fact which invalidates the current description of the chain

imbedded in as a continuous medium of dielectric constant

the chain conformation gets more and more compact as f decreases. Around f=0.3, SAXS

data can be interpreted as due to interacting weakly charged hard spheres, the effective

charge being obtained by osmometry. This is also confirmed by SANS data (using

contrast variation techniques) which show that the single chain form factor in semidilute

solution is that of a wormlike chain when fully charged whereas it evolves towards a more

spherical dense object as f decreases.

addition of a good solvent for the backbone to the aqueous solution reverts to "normal"

behaviour

all properties are continuous as f is varied and there is no evidence for any sharp

transition.

The following qualitative picture has emerged from these studies. Close to longer

sequences of hydrophobic monomers, the local is much smaller than the solvent value and

the neighbouring ionic monomers are not dissociated and remain as ion pairs (recall that PSS

is an ionomer at low charge contents) . This has two consequences: a proportion of the

counterions are bound on site and are not osmotically active, reducing even more the

counterion contribution to the osmotic pressure; dipolar attractive forces produce aggregation

in the diffuse hydrophobic regions, stabilising the effect.

At the same time and not quite independently, theorists36 have revived the problem of

a single (weakly charged) chain in a poor solvent where the overall shape is determined by the

balance of the electrostatic repulsion and the surface tension. It was first suggested by

Khokhlov35 that the chain would minimise its energy if it took the shape of an elongated

cylinder. However Dobrynin and Rubinstein36 have shown that a more favourable shape

would be that of a necklace with compact beads joined by narrow strings. This configuration

results from a Rayleigh charge instability, similar to the breaking up of a charged drop into

smaller ones. Changing the charge on the chain results in a cascade of abrupt transitions

between necklaces with different numbers of beads. Above c*, there are two important

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regimes37 38, labelled string controlled and bead controlled, depending on whether the

concentration is sufficient for the inter-bead repulsion to become important. It is predicted

that scales as c-1/2 in the first regime but as c-1/3 in the second regime, a departure to the

seemingly universal law for the polyelectrolyte peak! This finding is certainly comforting the

experimentalists who were trying desperately to scale their data with an exponent 0.5... There

is clearly qualitative agreement between the pearl necklace model of Dobrynin and

Rubinstein. Although the theory cannot handle the complexities of real systems it provides

trends and effects to look for. For instance recently the string-controlled and the bead-

controlled regimes have been clearly identified in a model system of a polyelectrolyte in

various non aqueous solvents of varying dielectric constant and quality 39. Globules have also

been seen in various simulations, the most extensive being the molecular dynamics

simulations of Micka et al at high concentrations and using discrete ions40.

COMMENTRecall that we are dealing here with a highly charged system. The situation is different with solutions of weakly charged polyelectrolytes in a poor solvent which form mesophases with alternating polymer-rich and polymer-poor regions. In these regions, the condition of electroneutrality is violated locally at a cost in electrostatic energy but the gain in translational entropy of the counterions is sufficient to stabilise the structure.41 42 43

35 Khokhlov A.R. J. Phys. A 1980 13 97936 Dobrynin A.V., Rubinstein M., Obukhov S.P. Macromolecules 1996 29 2974 37 Dobrynin A.V., Rubinstein M. Macromolecules 1999 32 915-92238 Chatellier X. PhD thesis, Strasbourg 199839 Waigh T.A., Galin J.C., Ober R., Williams C.E. 199940 Micka U., Holm C., Kremer K., Langmuir 1999 15 4033-404441 Borue V., Erukhimovich I. Macromolecules 1988 21 324042 Joanny J.F., Leibler L. J. Phys II (France) 1990 51 54543 Moussaid A., Schosseler F., Munch J.P., Candau J.S. J. Phys II (France) 1993 3 573

Some general references pertaining to the static conformation of hydrophilic, flexible polyelectrolytes

- Förster S., Schmidt M. Adv. Polym. Sci. 1995120, 51

- Jannink G. Makromol. Chem., Makromol. Symp. 1986 1, 67

- Schmitz K.S. Macroions in Solution and Colloidal Suspensions, VCH, New York, 1993

- Proceedings of meetings on Polyelectrolytes such as ref. 23 are regularly published

- The first critical tests of the isotropic model for semi-dilute polyelectrolyte solutions, using polystyrene

sulfonate as a model polymer can be found in the following publications: Nierlich M. et al, J. Phys.

(France)1979 40, 701; Williams C.E. et al, J. Polym. Sci. Polym. Lett. 1979 17 379; Nierlich, Boué F.,

Lapp A., Oberthür R., Colloid Polym. Sci. 1985 263 955

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