Interdisciplinary Applied Mathematics

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12.2.1 Modified Poisson-Boltzmann Equation

From the results presented above, we can conclude that the wall-ion, water-ion, and ion-ion interactions are important factors influencing the ion dis-

tribution in the channel and that the ion distribution can significantly influence the water concentration in the channel. The classical Poisson-Boltzmann equation considers these interactions only in a mean-field fashion    and    fails    to account    for    the    molecular    nature    of    the    ion,    water, and

the wall    (e.g.,    water    is    modeled    as a    continuum with    a    constant    permit

tivity) . In this section, we present a modified Poisson-Boltzmann equation that takes into account the wall-ion, water-ion, and ion-ion interactions in a lumped manner. The Poisson-Boltzmann equation can be modified to incorporate various effects that were neglected in the classical Poisson-Boltzmann equation, e.g., the finite ion size effects (Borukhov et al., 2000), nonelectrostatic interactions (Lue et al., 1999), dependence of the permittivity of the solution on the field strength (Woelki and Kohler, 2000), wall effects (Adamczyk and Warszynski, 1996) and discrete solvent effects (Burak and Andelman, 2001). Many of these modifications were based on statistical mechanics principles, and by incorporating all these effects into the Poisson-Boltzmann equation, it is possible to predict the ion distribution in the channel with good accuracy. However, due to the extremely complicated nature of the ion-wall, ion-water, and ion-ion interactions, a number of simplifications need to be made in the derivation of the modified Poisson-Boltzmann equation, and the calculation of the new terms can still be very difficult in many cases. Thus, though these existing modifications to the Poisson-Boltzmann equation can aid in the interpretation of various experimental observations, the development of a comprehensive, easy-to-calibrate, and accurate model is still an active area of research.

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