Statistical mechanics of electrolytes and polyelectrolytes. III. The cylindrical Poisson–Boltzmann equation

Abstract

The Poisson–Boltzmann equation for the potential due to an infinitely long, cylindrical polyion in a dilute ionic solution is studied and certain properties of its solution are proved. It is shown that when the charge density on the polyion exceeds a critical value, the surrounding ionic solution separates into two essentially independent phases. The inner phase which is close to the polyion consists predominantly of counterions. An explicit solution to the distribution function in this phase is obtained to the leading order and its validity is proved. This function depends only on the polyion charge density, and not on the concentration of the ionic solution. The distribution function in the outer phase depends upon the ionic strength but, to the leading order, not on the polyion charge density as long as it exceeds the critical value. It is shown that as the charge density on the polyion is increased, a proportionate number of counterions appears in the inner phase so as to neutralize the effect of this increase on the outer phase. It is proved that the potential, to its leading order, in the outer phase for any polyion charge density higher than the critical value, is the same as that due a polyion carrying a charge density equal to the critical value. The nonlinear dependence of the surface value of the potential on the polyion charge density is proved.