The minimum condition of Planck's diffusion potential and Goldman's theory

Abstract

According to Nernst, the differential equation of the diffusion potential is dependent on the concentrations of the ions as well as on their gradient within the diffusion layer itself. Regarding that a potential depends only on the boundary values and must be independent of the gradient, the general solution of this differential equation yields no potential function. In contrast with the general solution, Planck's singular solution of Nernst's equation yields a potential representation. While the general solution can be applied to ions of arbitrary electrochemical valency as well as to arbitrary boundary conditions, Planck's solution must be restricted to ions of equal valency in isotonic solutions.

The difficulty of obtaining this potential representation with the help of the classic method of calculus of variations results from the fact that Nernst's arrangement is homogeneous of degree one in the concentration gradients. Thus, Hamilton–Jabobi's fundamental method of the elimination of the slope function in order to obtain a potential representation yields only an identity. Therefore special methods must be considered. Based on Caratheodory's theorem of arbitrary slope functions it could be shown that isotonicity is the necessary condition in order to obtain a potential representation of a diffusion potential. Finally the influence of the osmotic flux on this potential will be discussed with the help of the Onsager relation.

Though Planck's theory dealing with the classic exception to the rule presents a singular solution only, this case is realized in biological membranes as shown by Goldman. In this outstanding case the dissipation of energy assumes an extremum value with respect to time and position, and the condition of the constant field is true.