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Nonlinear Poisson‐Boltzmann equation in a model of a scanning tunneling microscope
Author(s) -
Chan KwongYu,
Henderson Douglas,
Stenger Frank
Publication year - 1994
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.1690100605
Subject(s) - poisson–boltzmann equation , superposition principle , mathematics , mathematical analysis , nonlinear system , plane (geometry) , boltzmann equation , scanning tunneling microscope , relaxation (psychology) , poisson's equation , boundary value problem , geometry , physics , ion , condensed matter physics , quantum mechanics , social psychology , psychology
The nonlinear Poisson‐Boltzmann equation is solved in the region between a sphere and a plane, which models the electrolyte solution interface between the tip and the substrate in a scanning tunneling microscope. A finite difference method is used with the domain transformed into bispherical coordinates. Picard iteration with relaxation is used to achieve convergence for this highly nonlinear problem. An adsorbed molecule on the substrate can also be modelled by a superposition of a perturbing potential in a small region of the plane. An approximate analytical solution using a superposition of individual solutions for plane, the adsorbed molecule, and the sphere is also attempted. Results for cases of different potential values on the boundary surfaces and different distances of the sphere from the plane are presented. The results of the numerical method, the approximate analytical method, as well as the previous solutions of the linearized equation are compared. © 1994 John Wiley & Sons, Inc.