HIGHLY CHARGED POLYELECTROLYTES Chain Conformation, Counterion Condensation and Solution Structure

NATO ASI workshop on "Structure and Dynamics of Polymers and Colloidal Systems" Les Houches, September 14-24, 1999 Lecture notes, draft copy tt/10/jj 13:25 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 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 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 ε is exactly compensated by the thermal energy kBT l B = e² / ε kBT = 7.12Å in water at 20°C • The Debye-Hückel screening length κ-1 is defined as κ² = 4π l B I where I is the total number of "free" charges in the solution Typically, κ-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 For strong polyelectrolytes, e.g with SO 3H units derived from a strong acid, all monomers are dissociated and the charges are said to be quenched 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 R Na 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 2 average end-to-end distance is R0 = Na The second term is the electrostatic energy due to the Nfe charges Minimisation with respect to R leads to R ≈ Nf 2/3 (l a ) 1/ B (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 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 θ-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; 2 if ge monomers are involved, then D = g e a 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/  f 2lB g e ≈   a    −2 / (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 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 (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