NATO ASI workshop on "Structure and Dynamics of Polymers and Colloidal Systems"
Les Houches, September 14-24, 1999
Lecture notes, draft copy
16/02/16 09:02
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
1
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 l B ; 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. l B = e² /  kBT = 7.12Å
in water at 20°C.

The Debye-Hückel screening length -1 is defined as ² = 4 l B 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.
2
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 excludedvolume 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
Nf  l B
R2
 k BT
2
R
Na
2
E F  k BT
(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 R02  Na 2 . The second term is the electrostatic energy due to
the Nfe charges. Minimisation with respect to R leads to

R  Nf 2 / 3 l B a 2

1/ 3
(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
3
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 D 2  g e a 2 . 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 L  N g e D . 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
 f 2lB 

D  a
 a 
1/ 3
 f 2lB 

g e  
 a 
2 / 3
(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.
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,
4
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 (La) and of linear charge density f  a A ,
where A is the distance between charges along the chain. The density of counterions at a
distance r from the chain (r<<L) is n(r )  n0 exp   (r) 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
r
r
0
0
m(r )   n(r ' )2r ' dr '   2 n0 r '(12lB / A) dr '
(3)
The behaviour of this integral depends very much on the value of the coupling parameter
u  l B A . 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 l B 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.
5
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 predicted 10.
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 liquidlike 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
c* 
N
N

 N 2
3
3
L
N g D
(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
6
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 c  c*  N / L3  B a  N 2 , then
3
  Lc * c 1 / 2  B ca 1 / 2  c 1 / 2 N 0
(5)
Keep in mind that  scales as c 1 / 2 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
R  a cB 
1/ 4
N 1/ 2
(6)
The concentration dependence ( c 1 / 4 ) is much stronger than for neutral polymers ( c 1 / 8 ).
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.
7
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 c s can be expressed in terms of
the same property without salt X 0 as X  X 0 1  2 Acs / c  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 distances 19.
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
S (q  0)  1 / f .

There is a broad peak in the salt-free structure factor at a finite wave vector
q*   1  c 1 / 2 . 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 q   1 .

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.
8

Some beautiful experiments on star polyelectrolytes 20 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.
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
27
or 
-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 1 29. There is a possibility that the
9
conflicting results are due to limitations of using Debye-Huckel screened interaction to
describe polyelectrolyte systems. Clearly experiments are necessary. Unfortunately direct
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
LP  c p  2c s 
1 3
, 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.
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.
10
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
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
11
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
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 beadcontrolled 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.
COMMENT
Recall 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 polymerpoor 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
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