Assumptions of the Poisson-Boltzmann Equation

The simplicity of the Poisson-Boltzmann approach to electrolyte solutions belies several approximations used in the derivation of Eqs. [3] and [4]. The first of two parts of the derivation begins with Maxwell s equation for the electric displacement D(r)  

The second part of the derivation requires that a particular ion (i.e., a single fixed or mobile charge) be chosen and its position fixed. The distribution of ions of type i at position r with respect to the chosen ion is 

Debye and Hiickel were concerned with treating a system of hard-sphere cations and anions of identical size and opposite charge in an isotropic environment without explicit boundaries. By linearizing the charge density with 

Having outlined the derivation of the Poisson-Boltzmann equation, we now turn to a discussion of the main assumptions used, under which circumstances they become invalid, and how these problems might be remedied. A comprehensive treatment of the PB equation in which most of the approximations are addressed in a detailed manner was given by Bell and Levine. In their work they derived a modified PB equation which provided corrections for most of the deficiencies in the original theory. Their results have been further extended by Outhwaite and co-workers in an attempt to place the PB equation on par with other more elaborate theories. We discuss here only the generalities of the basic DH assumptions for specific details the reader is referred to the Bell-Levine paper ° and others cited below. 

Garrett and Poladian demonstrate the uniqueness of the nonlinear PB equation for the case of a constant dielectric coefficient. The extension for a variable (and positive) dielectric coefficient is readily shown by a suitable modification of Green s theorem based on Eq. [3]. However, care must be taken in the numerical solution of some modified PB theories as nonuniqueness has been observed.  

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