Investigation of the Limits of the Linearized Poisson-Boltzmann Equation

This work presents a comparison between a numerical solution of the Poisson-Boltzmann equation and the analytical solution of its linearized version through the Debye-Huckel equations considering both size-dissimilar and common ion diameters approaches. In order to verify the limits in which the linearized Poisson-Boltzmann equation is capable to satisfactorily reproduce the nonlinear version of Poisson-Boltzmann, we calculate mean ionic activity coefficients for different types of electrolytes as various temperatures. The divergence between the linearized and full Poisson-Boltzmann equations is higher for lower molalities, and both solutions tend to converge toward higher molalities. For electrolytes of lower valencies (1:1, 1:2, 2:1, and 1:3) and higher distances of closest approach, the full version of the Debye-Huckel equation is capable of representing the activity coefficients with a low divergence from the nonlinear Poisson-Boltzmann. The size-dissimilar full version of Debye-Huckel represents a clear improvement over the extended version that uses only common ion diameters when compared to the numerical solution of the Poisson-Boltzmann equation. We have derived a salt-specific index (Theta) to gradually classify electrolytes in order of increasing influence of nonlinear ion-ion interactions, which differentiate the Debye-Huckel equations from the nonlinear Poisson-Boltzmann equation.

Automated summary

This study compares the numerical solution of the Poisson-Boltzmann equation with the analytical solution of its linearized version using the Debye-Huckel equations. The activity coefficients of different types of electrolytes at different temperatures were calculated to verify the accuracy of the linearized equation. The results show that the linearized equation deviates more from the full Poisson-Boltzmann equation at lower molalities, and both solutions converge at higher molalities. For electrolytes with lower valencies and larger distances of closest approach, the full Debye-Huckel equation can accurately represent the activity coefficients with low divergence from the nonlinear equation. The size-dissimilar full Debye-Huckel equation is an improvement over the common ion diameters approach when compared to the numerical solution of the Poisson-Boltzmann equation.