Statistical Thermodynamics of Polyelectrolyte Solutions

Mandel, M. Statistical thermodynamics of polyelectrolyte solutions. In Selegny, E. (ed.) Polyelectrolytes, pp. 39-55. D. Reidel Publishing Company, Dordrecht (1974)... 

Considerable progress has been made in the solution theory of poly-electrolytes. However, for the condensed-phase analogs of polyelectrolytes, ionomers, this is not the case. Eisenberg (1) has put forth an initial theory of ionomer structure that contains conceptual formalisms of general use. His theory has been consulted extensively in the work reported here. Ponomarev and Ionova (2) have attempted to construct a sophisticated statistical mechanical model to describe the thermodynamics of ionomers. Recently, Gierke (3) has described a theory of ion transport in the Nafion ionomer based on a specific molecular organization. 

There are many well-established models for the Gibbs energy of nonelectrolyte solutions and also several methods to describe conventional polymer solutions. However, the state of the art for modeling thermodynamic properties of aqueous solutions of polyelectrolytes is far less elaborated. This is partly due to the particular features of such solutions but is also caused by insufficiencies in the knowledge of the parameters that characterize a polyelectrolyte, for example, the polydisper-sity and the different stmctures (primary, secondary etc.) of the polyelectrolytes. The development and testing of thermodynamic models has always been based on reliable experimental data for solutions for which all components are well characterized. Such characterization is particularly scarce for biopolymers and biopolyelectrolytes. Furthermore, such polymers are generally more complex than synthetic polymers. Therefore, the present contribution is restricted to a discussion of the thermodynamic properties of aqueous solutions of synthetic polyelectrolytes that consist of only two different repeating units that are statistically distributed. Furthermore, it is restricted to systems where sufficient information on the polyelectrolyte s polydispersity is available. 

An important factor that is not taken into account in the DLVO theory is adsorption, on the particle s surface, of long polymeric chains. The adsorption of a non-ionic polymer or a polyelectrolyte on the solid surface can cause, not only a modification of the zeta potential, but also a critical difference between the value of the zeta potential and the state of dispersion. Steric repulsion is associated with the obstmction effect of these polymers that are capable to form a sufficiently thick layer to prevent the particles from approaching one another in the distanee of influence of the Van der Waals attractive forces. Steric stabihzation will therefore depend on the adsorption of the polymeric dispersant and the thickness of the layer developed. Several interpretation models for stabilization by steric effect have been put forward. They rely either on a statistical approach, or on the thermodynamics of solutions. Steric stabilization is particularly useful in organic, fairly non-polar or non-polar environments, as in the case of tape casting (see section 5.4.3). 

It is the purpose of this paper to establish a statistical thermodynamical formalism for dilute polyelectrolyte solutions which may serve to discuss the assumptions which are usually introduced in the theoretical treatment of such systems. Use is made of a model which is kept as general as possible, and certainly is more realistic than a Kuhn-like chain, without being too complicated to be handled by ordinary statistical procedures. Canonical ensemble statistics are used, the external thermodynamic variables being the volume V, temperature T and composition of the system. Of course, for the problem thus outlined no exact solution of practical nature is presented which is in the present stage still beyond reach. It is hoped that such a formal treatment may help a better understanding of the problems underlying the theoretical approach to polyelectrolyte systems. It will also help to discuss the generality of theoretical description as presented by Marcus [7]. 

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