Analytical solution of Poisson-Boltzmann equation for interacting plates of arbitrary potentials and same sign
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Efficient calculation of electrostatic interactions in colloidal systems is becoming more important with the advent of such probing techniques as atomic force microscopy. Such practice requires solving the nonlinear Poisson-Boltzmann equation (PBE). Unfortunately, explicit analytical solutions are available only for the weakly charged surfaces. Analysis of arbitrarily charged surfaces is possible only through cumbersome numerical computations. A compact analytical solution of the one-dimensional PBE is presented for two plates interacting in symmetrical electrolytes. The plates can have arbitrary surface potentials at infinite separation as long they have the same sign. Such a condition covers a majority of the colloidal systems encountered. The solution leads to a simple relationship which permits determination of surface potentials, surface charge densities, and electrostatic pressures as a function of plate separation H for different charging scenarios. An analytical expression is also presented for the potential profile between the plates for a given separation. Comparison of these potential profiles with those obtained by numerical analysis shows the validity of the proposed solution. © 2009 Elsevier Inc. All rights reserved.