Asymptotic solutions are obtained to the Poisson–Boltzmann equation for large, highly charged spheres in an ionic solution. It is proved that as the size of the sphere is increased, keeping the surface charge density fixed, there is a critical value for the radius beyond which counterion condensation sets in. This critical radius is much larger than the Bjerrum length but small compared to the Debye length and depends on the ionic strength. An expression is derived for the effective charge. When the radius becomes much larger than the Debye length, it is shown that sufficiently many counterions condense in a shell of thickness small compared to the polyion radius to essentially neutralize the polyion charge.

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.

Starting from the BBGKY hierarchy and using the method described in paper I, the statistical mechanics of a polyion in a dilute ionic solution is studied. It is shown that if the length of the polyion is much larger than the distance of closest approach, ’’counterion condensation’’ occurs. If the length of the polyion is comparable to the Debye length or larger, then the critical charge density is shown to be independent of the ionic strength. If the length is much smaller than the Debye length, the critical value is shown to depend logarithmically on the ionic strength and the length of the polyion. Expressions are derived for the critical charge density and the distribution of the counterions around the polyion.

We apply the Mayer cluster expansion theory to the interaction of point ions with a polyion, modeling the polyion as a line charge of length 2L with suitable polyion-mobile ion distances of closest approach.By summing over all simple ring clusters we obtain limiting law expressions for contributions to the excess free energy and various colligative properties of the solution due to mobile ion-polyion interactions. These expressions include explicit end-effect corrections to the expressions derived by Manning for the case of an infinitely long polyion. We show that for a polyion of fixed dimensions, even though its length may be very large, cluster terms of higher order than the simple ring terms do not diverge in the limit of infinite dilution but instead disappear more rapidly than the ring terms. This indicates the absence of counterion condensation in this limit and validates the simple ring term expressions as limiting law expressions.

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