Comparison of the Poisson-Boltzmann Model and the Donnan Equilibrium of a Polyelectrolyte in Salt Solution.

A sequence of charged parallel flat plates immersed in a salt solution is used to model the swelling behavior of a polyelectrolyte. The solution of the nonlinear Poisson-Boltzmann equation gives rise to an analytical expression of the plate distance as a function of swelling pressure (p), plate charge density (varsigma), and salt concentration (n). The exact treatment of the system also yields the connection between polyelectrolyte concentration nP, p, varsigma, and n. A formula for the electric potential difference between polyelectrolyte and salt solutions is derived. On the other hand, the relation between nP, p, and n is known from the Donnan equilibrium. It is shown that the Donnan potential, UD(nP, n), and swelling pressure, pD(nP, n), are always larger than the equivalent quantities in the Poisson-Boltzmann theory (UPB(varsigma, nP, n) and pPB(varsigma, nP, n), respectively). The transition from the Poisson-Boltzmann theory to the Donnan model is achieved by the limiting process varsigma --> 0 which reveals the intrinsic linkage between the two theories. pD(nP, n) = limvarsigma-->0 pPB(varsigma, nP, n) and UD(nP, n) = limvarsigma-->0 UPB(varsigma, nP, n). Copyright 1999 Academic Press.