Perturbation solutions for the nonlinear Poisson–Boltzmann equation with a high-order-accuracy Debye–Hückel approximation

• Published in: Zeitschrift für angewandte Mathematik und Physik Vol. 71; no. 4
• Main Authors: Zhao, Cunlu, Wang, Qiuwang, Zeng, Min
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.08.2020; Springer Nature B.V
• Subjects: Accuracy; Approximation; Boltzmann transport equation; Cartesian coordinates; Engineering; Exact solutions; Hyperbolic functions; Mathematical Methods in Physics; Nonlinearity; Order parameters; Perturbation methods; Robustness (mathematics); Spherical coordinates; Theoretical and Applied Mechanics; Trigonometric functions; Viscosity; Zeta potential; Electric double layer; Poisson–Boltzmann equation; Debye–Hückel approximation; Perturbation solution; 35Q60; 35Q92
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-020-01367-9

The Poisson–Boltzmann (P–B) equation is of fundamental importance in understanding solid–liquid electrolyte interfaces that are present in many fields. Due to the nonlinearity, it is usually challenging to find the explicit exact solutions of the P–B equation. The present work reports several perturbation solutions for the nonlinear P–B equation in the Cartesian and spherical coordinates. The new solutions contain a perturbation parameter from the high-order-accuracy approximation of the hyperbolic sine function and thus can apply to high zeta potential conditions. The comparison of the perturbation solutions with the traditional Debye–Hückel solutions and the full numerical solutions validates the robustness and accuracy of the perturbation solutions. The perturbation solutions are explicit and analytical and then can be used for a fast calculation of the EDL potential and interaction energy in versatile applications.