Relaxation of the continuum approximation in the theory of electrolytes. I. Formal results

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Abstract
In this paper, the role of the continuum approximation in the theory of electrolytes is re−examined, and a possible generalization of this approximation is considered. For several reasons (cited in the manuscript), the model of Debye and Hückel is used as the starting point in our analysis. We proceed by postulating a functional form for the dependence of the permittivity ε (r) on the distance from the central ion. When this expression for the permittivity is used within the context of Poisson−Boltzmann theory, there results a nonlinear differential equation whose analytic properties are investigated in detail; in particular, it is proved that solutions of this equation exist and are unique. By construction of the Green’s function, we obtain the associated nonlinear integral equation and solutions to this equation for a choice of parameters corresponding to an electrolyte system considered previously by Guggenheim. Our main conclusions follow from a comparison of our results with those obtained previously using the continuum approximation. We find that the relaxation of this approximation leads to a significant enhancement of the potential felt by a counterion in the immediate neighborhood of the central ion, with an attendent accelerated damping of the potential as one moves away from the central ion. A second, rather unexpected result which emerges from our study is that the computed potential is surprisingly insensitive to the explicit down−range behavior of the function postulated to describe the change in permittivity as a function of distance. This paper concludes with some remarks on future problems to be studied. In the following paper detailed ion−distribution profiles are reported, and the question of the internal consistency of the augmented Poisson−Boltzmann equation, introduced in this paper, is examined.