Investigation of the Limits of the Linearized Poisson–Boltzmann Equation

Abstract

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–Hückel 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–Hückel equation is capable of representing the activity coefficients with a low divergence from the nonlinear Poisson–Boltzmann. The size-dissimilar full version of Debye–Hückel 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 (Θ) to gradually classify electrolytes in order of increasing influence of nonlinear ion–ion interactions, which differentiate the Debye–Hückel equations from the nonlinear Poisson–Boltzmann equation.

The Connection between the Debye and Güntelberg Charging Processes and the Importance of Relative Permittivity: The Ionic Cloud Charging Process

“…The Güntelberg charging process (GCP) is characterized by charging the central ion of interest in a solution where all other ions are already charged. This is done from a given considered electrostatic potential ψ i , which commonly is derived from the linearized Poisson–Boltzmann equation in the context of Debye–Hückel equations . It calculates the activity coefficient of an individual ion through italicRT nobreak0em0.25em⁡ ln nobreak0em.25em⁡ γ i normale normall = italicN normalA italicz i italice 0 ∫ 0 1 ψ i ( λ ) .25em normald λ …”

“…This is done from a given considered electrostatic potential ψ i , which commonly is derived from the linearized Poisson−Boltzmann equation in the context of Debye−Huckel equations. 36 It calculates the activity coefficient of an individual ion through…”

“…Other electrolyte theories which require numerical solutions, such as those developed for the Poisson–Boltzmann (PB) equation − and those using its several modifications − also require the charging processes to calculate thermodynamic properties.…”

Arbitrary-Shape Dielectric Particles Interacting in the Linearized Poisson–Boltzmann Framework: An Analytical Treatment

“…This, however, does not diminish the importance of studying the behavior of the LPBE also in the case of highly charged objects: their electrostatics may still be correctly described at sufficiently long distances (as compared to the Debye length) by the usual DH approximation provided that the sources of the electric field are properly renormalized. − (See also the recent ref for additional comments concerning the ranges of applicability of the DH theory.) This once again emphasizes the importance of a thorough study of the DH approximations, both theoretically and numerically, and justifies the constant stream of works related to the LPBE (see recent refs , , and − , and references therein).…”

Theory of charge asymmetric electrolytes. Onsager’s approach revisited

“…The description of thermodynamic properties of electrolytes has given rise to many developments since the establishment of limiting laws by Debye and Hückel (DH) [1][2][3]. This theory continues to inspire studies and extensions today [4][5][6][7][8]. Debye and Hückel have used the Poisson-Boltzmann (PB) equations as a starting point to describe the interactions between ions in the solution.…”