NATO ASI workshop on "Structure and Dynamics of Polymers and Colloidal Systems"
Les Houches, September 14-24, 1999
Lecture notes, draft copy
25/09/00 14:09
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 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-1is defined as ² = 4I 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 copolymerof 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.
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 Ndoes 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]
1
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/geblobs 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.
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 Manning[7] and Oosawa[8], 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). Thetotal 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.
1
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 c1/2as 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 weights[14]. However the absolute values of c* are found to be higher than expected. This can be understood in terms of MC simulations which show 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.
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 Pfeuty[18] 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 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 .
- 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 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 Amis[21] 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 in[22] 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.