The Poisson–Boltzmann equation and its application to polyelectrolytes

TLDR

The Poisson–Boltzmann equation’s validity is reexamined using functional expansion and mean spherical approximation, particularly for strong Coulomb fields from polyelectrolytes acting on salt solutions. A two‑phase condensation model approximates the PB equation and maintains fair accuracy even at high salt concentrations where traditional limiting laws break down. Although high local ion densities can cause substantial PB errors, a small potential adjustment compensates, as shown for DNA where the approximation remains adequate up to about 0.1 M salt but fails beyond that.

Abstract

The validity of the Poisson–Boltzmann (PB) equation is reconsidered on the basis of functional expansion techniques supplemented by the mean spherical approximation. In the application of greatest interest a strong Coulomb potential originating in an external source, such as a polyelectrolyte molecule, acts on a salt solution of small mobile ions. Where the local charge density of mobile ions is high, substantial errors may occur in the PB approximation that relates charge density to mean potential. However, the solution to the PB equation is nevertheless a good approximation in the indicated application because a quite small percentage change in the electrostatic potential can compensate large errors in the Boltzmann distribution. An application to DNA illustrates this compensation, and also its impending failure at bulk salt concentrations in excess of 0.1M. A two phase (or condensation) model is derived as an approximation to the PB equation and retains fair accuracy even at substantial salt concentrations, where the limiting laws lose theoretical validity.