The cell model for polyelectrolytes with added salt

Using a comparison theorem from the theory of elliptic equations, we obtain rigorous bounds for the exact solution to the Poisson–Boltzmann equation for the cylindrical cell model, and error bounds for approximations to the exact solution. We present a method for obtaining perturbative solutions, formally to any order, from the exact solution to the salt-free case, and we obtain an explicit solution for the potential correct to the first order. We derive formulas for thermodynamic quantities and compare them with experimental values. We find that the predicted counterion activity coefficients agree well with experimental results; however, agreement of the predicted coion activity coefficient with experiment is only qualitative. We rigorously show that as the concentration of excess salt tends to zero, and as the polyion’s radius becomes very small in comparison to the cell’s radius, the coion activity coefficient given by the Poisson–Boltzmann equation approaches 16/13 for polyion charge densities above a critical value; for subcritical charge densities the limiting value depends on the charge density and lies between 1 and 16/13.