For solutions containing rod‐shaped polyelectrolytes and their counterions only, the Poisson‐Boltzmann equation can be solved explicity without the usual assumption of the Deybe‐Hückel theory. At charge densities corresponding to half‐neutralized polyacrylic acid a large proportion of monovalent counterions is located close to the polyion even in solutions as dilute as 1/800 base molar.

In any theory of the titratioii behavior of polyelectrolytes, a knowledge of the counterion distribution fuiiction is required for the calculation of the electrical free energy of ionization. The problem has been examined by Hermans and Overbeeli' for flexible polyelectrolyte molecules at low extension of the polymer coil. Kunzle suggested a trc~;itnient2 which can he applied over a wider range of molecular configurations.In both of these treatnieiits the procedure of the Debye-Huckel theor) is followed in that the ratio qb/kT (where t is the electronic charge, + the electrostatic potential, k Boltzmann's constant, and T the temperature) is assumed to be small compared with unity. This assumption is hardly applicable to highly charged polyions, and it will be shown that for a special case an explicit solution ran be obtained without the Debye-Hurk~I approximation. Physical iModelThe present contribution deals with the couuterion distribution of partially neutralked polymeric acids in solutions containing no electrolytes except for the polyanions and their monovalent counterions. The polyanions are represented by thin rods which are so long that end effects can be neglected, and the charges are considered to be spread uniformly over their surfaces. The rod is the skis of a cylindrical region containing all the counterions belonging to it. Neighboring chains are considered to be essentially parallel, and the time-average charge distribution is assumed to have cylindrical symmetry. General Expression for Counterion DistributionWe shall use the following notation: a = radius of the rod representing the polyanion b = distance between ionizable groups on the polyanion r = distance from the axis of the polyanion * H. Morawetz, Doctoral Dissertation, Polytechnic Institute of Brooklyo, 1950. t Present address: Dow Chemical Co., Midland, Michigan. 543