Modelling polyelectrolyte solutions

Kalyuzhnyi, Yu.V., and Vlachy, V. Study of a model polyelectrolyte solution with directional attractive forces between the macroions. Journal of Chemical Physics, 1998,108, No. 18, p. 7870-7875. 

Harris, F. E. Rice, S. A. (1957). A model for ion binding and exchange in polyelectrolyte solutions and gels. Journal of Physical Chemistry, 58, 725-32. 

Hoagland, D.A., Arvanitidou E., and Welch C., Capillary Electrophoresis measurements of the free solution mobility for several model polyelectrolyte systems, Macromolecules, 32, 6180, 1999. 

A polyelectrolyte solution contains the salt of a polyion, a polymer comprised of repeating ionized units. In dilute solutions, a substantial fraction of sodium ions are bound to polyacrylate at concentrations where sodium acetate exhibits only dissoci-atedions. Thus counterion binding plays a central role in polyelectrolyte solutions [1], Close approach of counterions to polyions results in mutual perturbation of the hydration layers and the description of the electrical potential around polyions is different to both the Debye-Huckel treatment for soluble ions and the Gouy-Chapman model for a surface charge distribution, with Manning condensation of ions around the polyelectrolyte. 

We have now looked at two models for the second virial coefficient of uncharged colloidal solutes. In Section 3.5b we see that B depends on the magnitude of the particle charge for polyelectrolyte solutes. 

Strong evidence of ionic association was found by Stilbs and Lindman 69) in their PGSE study of aqueous polyelectrolyte solutions, polyacrylic acid and poly-methacrylic acid, neutralized by tetramethylammonium hydroxide, with or without sodium counterions. While polymer diffusion could not be detected since its T2 was too short, TMAOH and water diffusion was measured as function of degree of neutralization a, or Na+ content. A pronounced minimum of D(TMAOH) near a = 1 was interpreted in terms of a two-site model, leading to the determination that at a = 1, approximately half of the counterions are bound in both systems. Fourier transform techniques permitted the simultaneous measurement of diffusion of water and TMAOH. 

The cell model is a commonly used way of reducing the complicated many-body problem of a polyelectrolyte solution to an effective one-particle theory [24-30]. The idea depicted in Fig. 1 is to partition the solution into subvolumes, each containing only a single macroion together with its counterions. Since each sub-volume is electrically neutral, the electric field will on average vanish on the cell surface. By virtue of this construction different sub-volumes are electrostatically decoupled to a first approximation. Hence, the partition function is factorized and the problem is reduced to a singleparticle problem, namely the treatment of one sub-volume, called cell . Its shape should reflect the symmetry of the polyelectrolyte. Reviews of the basic concepts can be found in [24-26]. 

Abstract In this chapter we review recent advances which have been achieved in the theoretical description and understanding of polyelectrolyte solutions. We will discuss an improved density functional approach to go beyond mean-field theory for the cell model and an integral equation approach to describe stiff and flexible polyelectrolytes in good solvents and compare some of the results to computer simulations. Then we review some recent theoretical and numerical advances in the theory of poor solvent polyelectrolytes. At the end we show how to describe annealed polyelectrolytes in the bulk and discuss their adsorption properties. 

The PRISM (Polymer-Reference-Interaction-Site model) theory is an extension of the Ornstein-Zernike equation to molecular systems [20-22]. It connects the total correlation function h(r)=g(r) 1, where g(r) is the pair correlation function, with the direct correlation function c(r) and intramolecular correlation functions (co r)). For a primitive model of a polyelectrolyte solution with polymer chains and counterions only, there are three different relevant correlation functions the monomer-monomer, the counterion-counterion, and the monomer-counterion correlation function [23, 24]. Neglecting chain end effects and considering all monomers as equivalent, we obtain the following three PRISM equations for a homogeneous and isotropic system in Fourier space ... 

Our model of a polyelectrolyte solution consists of Np flexible bead-spring-chains which are located in a simulation box of length L with periodic boundary conditions. For each chain, a fraction / of the N monomers is monovalently charged (v=l), and fN oppositely charged monovalent counterions are added to obtain an electrically neutral system. In some cases Ns pairs of salt ions were added. The density is given in form of the charged... 

The theory of polyelectrolyte solutions was first developed for rigid macro-ions. The fundamental results obtained from this model are well discussed in the monograph published by Verwey and Overbeek in 1948 (29). Later, Levine, Booth, Kirkwood and a number of other investigators refined the fundamental work in various directions. ... 

In this section we introduce integral equation theories (IETs) and approximate closures applicable for various models of polyelectrolyte solutions. A theory for linear polyelectrolytes based on the polymer reference interaction site model has also been proposed [58, 59], but this approach will not be reviewed here. 

Before presenting numerical results, it is worth summarizing the main characteristics of the experimental results for the osmotic pressure of polyelectrolyte solutions [9, 17, 18, 57, 107], The measured osmotic coefficients most often exhibit strong negative deviations from ideality. The measured values are a) lower than it was predicted by the cylindrical cell model theory, b) rather (but not completely) insensitive to the nature of the counterions, and c) also insensitive to the polyelectrolyte concentration in a dilute regime and/or for... 

Das, T., Bratko, D., Bhuiyan, L.B., and Outhwaite, C.W. Polyelectrolyte solutions containing mixed valency ions in the cell model A simulation and modified Poisson-Boltzmann study. Journal of Chemical Physics, 1997,107, No. 21, p. 9197-9207. 

Vlachy, V., and Prausnitz, J.M. Donnan equilibrium - hypernetted-chain study of one-component and multicomponent models for aqueous polyelectrolyte solutions. Journal of Physical Chemistry, 1992, 96, No. 15, p. 6465-6469. 

Jiang, J.W., Blum, L., Bernard, O., and Prausnitz, J.M. Thermodynamic properties and phase equilibria of charged hard sphere chain model for polyelectrolyte solutions. Molecular Physics, 2001, 99, p. 1121-1128. 

Figure 8.5 shows that experimental data on unentangled polyelectrolyte solutions are described quite well by the Rouse model. Polyelectrolytes are charged polymers that have a wide range of concentrations where dynamics obey the Rouse model. 

A further application of time-resolved fluorescence measurements is in the study of conformational dynamics of polymer chains in solution. Fluorescence anisotropy measurements of macromolecules incorporating suitable fluorescent probes can give details of chain mobility and polymer conformation (2,14). A particular example studied in this laboratory is the conformational changes which occur in aqueous solutions of polyelectrolytes as the solution pH is varied (15,16). Poly(methacrylic acid) (PMA) is known to exist in a compact hypercoiled conformation at low pH but undergoes a transition to a more extended conformation at a degree of neutralization (a) of 0.2 to 0.3 (1 6). Similar conformational transitions are known to occur in biopolymer systems and consequently there is considerable interest in understanding the nature of the structures present in model synthetic polyelectrolyte solutions. 

---