Analytical solution of Poisson?Boltzmann equation for interacting plates of arbitrary potentials and same sign

Abstract

a b s t r a c tEfficient 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.

Membrane Potential Generated by Ion Adsorption

“…However, the basically same idea—potential evaluation by employing Langmuir isotherm and Poisson-Boltzmann equation—has been often used in the research fields outside the membrane potential research field for quantitatively estimating the potential behavior in electrolytic solution around charged surface [17,18,19,20]. Moreover, the quantitative potential evaluation around the charged surface in aqueous solution primarily using the Poisson-Boltzmann equation has been very common research for more than several decades [17,21,22,23,24,25,26,27,28,29,30,31,32]. Hence, the foundation of the theory the authors propose here has been already widely accepted and well discussed outside the research field of membrane potential for decades.…”

Membrane potential generation without ion transport

“…20. K + is adsorbed to -SO 3 − of membrane of Selemion CMV at the interface between the KCl solution and membrane by obeying the Langmuir isotherm, and all the ions distribute according to the Boltzmann distribution [1,2,6,[9][10][11][12][13][14][18][19][20][21][22][23][24][25][26].…”

Determination of electrostatic potential distribution by atomic force microscopy (AFM) on model silica and alumina surfaces in aqueous electrolyte solutions

“…For highly-charged systems, usually numerical procedures are employed. In this work, a recent analytical solution of the Poisson-Boltzmann equation was employed to determine the electrostatic pressure force between two substrates [32]. The solution which allows analytical analysis of charge and potential regulated surfaces shows perfect agreement with numerical results for all levels of charging conditions under various charging regimes (constant potential and constant charge surfaces).…”