Electrical force between two permeable planar charged surfaces in an electrolyte solution

Numerical calculations of the pressure and electrical contribution to the Helmholtz interaction free energy for two uniformly charged sheets interacting across 1: 1 electrolyte solution are presented. The model differs from that considered by most other workers in that the electrolyte is allowed to penetrate the charged surfaces. Further, in addition to the cases of constant charge and constant potential, the case where the surfaces contain reversibly ionised groups is considered. In order that the results may have some relevance to cell-membrane interactions, all the numerical calculations are done using parameters which represent approximately the conditions in physiological saline solution. The results for constant potential are identical to those obtained for impenetrable interacting charged sheets. The results for constant charge differ in an essential way. Specifically, the pressure in the limit of zero separation tends to a constant value whereas for impermeable surfaces it is, for small separations, inversely proportional to the square of the separation and thus diverges as the separation approaches zero. Consequently, the difference between constant potential and constant charge for this model is not as marked as differences obtained by other workers who studied impermeable surfaces. As one might anticipate, the results for reversibly ionised surfaces lie between the two extremes of constant charge and constant potential. Present calculations differ from those of Gingell in that the complete Poisson-Boltzmann equation is solved numerically.