A FINITE DIFFERENCE SCHEME FOR SOLVING THE NONLINEAR POISSON-BOLTZMANN EQUATION MODELING CHARGED SPHERES

In this work, we propose an efficient numerical method for computing the electrostatic interaction between two like-charged spherical particles which is governed by the nonlinear Poisson-Boltzmann equation. The nonlinear problem is solved by a monotone iterative method which leads to a sequence of linearized equations. A modified central finite difference scheme is developed to solve the linearized equations on an exterior irregular domain using a uniform Cartesian grid. With uniform grids, the method is simple, and as a consequence, multigrid solvers can be employed to speed up the convergence. Numerical experiments on cases with two isolated spheres and two spheres confined in a charged cylindrical pore are carried out using the proposed method. Our numerical schemes are found efficient and the numerical results are found in good agreement with the previous published results. Mathematics Subject Classification: 65N06, 65B99.

[1]  Michael J. Holst,et al.  Adaptive multilevel finite element solution of the Poisson–Boltzmann equation I. Algorithms and examples , 2001 .

[2]  Michael J. Holst,et al.  The adaptive multilevel finite element solution of the Poisson-Boltzmann equation on massively parallel computers , 2001, IBM J. Res. Dev..

[3]  Bowen,et al.  Adaptive Finite-Element Solution of the Nonlinear Poisson-Boltzmann Equation: A Charged Spherical Particle at Various Distances from a Charged Cylindrical Pore in a Charged Planar Surface , 1997, Journal of colloid and interface science.

[4]  J. A. V. BUTLER,et al.  Theory of the Stability of Lyophobic Colloids , 1948, Nature.

[5]  Zhonghua Qiao,et al.  A Finite Difference Method and Analysis for 2D Nonlinear Poisson–Boltzmann Equations , 2007, J. Sci. Comput..

[6]  J. Neu.  Wall-Mediated Forces between Like-Charged Bodies in an Electrolyte , 1999 .

[7]  D. Chan,et al.  Long-Range Electrostatic Attractions between Identically Charged Particles in Confined Geometries: An Unresolved Problem. , 1999, Journal of colloid and interface science.

[8]  N. E. Hoskin.  The interaction of two identical spherical colloidal particles - The interaction of two identical spherical colloidal particles. I. Potential distribution , 1956, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[9]  Mikhail Shashkov,et al.  The Error-Minimization-Based Strategy for Moving Mesh Methods , 2006 .

[10]  Weizhang Huang.  Mathematical Principles of Anisotropic Mesh Adaptation , 2006 .

[11]  Xin Lu,et al.  Block Monotone Iterations for Numerical Solutions of Fourth-Order Nonlinear Elliptic Boundary Value Problems , 2003, SIAM J. Sci. Comput..

[12]  Chia-Ven Pao,et al.  Nonlinear parabolic and elliptic equations , 1993 .

[13]  Derek Y. C. Chan,et al.  Computation of forces between spherical colloidal particles : nonlinear Poisson-Boltzmann theory , 1994 .

[14]  W. Richard Bowen,et al.  Finite difference solution of the 2-dimensional Poisson–Boltzmann equation for spheres in confined geometries , 2002 .

[15]  C. V. Pao,et al.  Block monotone iterative methods for numerical solutions of nonlinear elliptic equations , 2022 .

[16]  Adel O. Sharif,et al.  Long-range electrostatic attraction between like-charge spheres in a charged pore , 1998, Nature.

[17]  Pavel Dyshlovenko,et al.  Adaptive mesh enrichment for the Poisson-Boltzmann equation , 2001 .

[18]  Michael J. Holst,et al.  Adaptive multilevel finite element solution of the Poisson-Boltzmann equation II. Refinement at solvent-accessible surfaces in biomolecular systems , 2000, J. Comput. Chem..

[19]  J. J. Gray,et al.  Colloidal particles: Origin of anomalous multibody interactions , 1999, Nature.

[20]  P. E. Dyshlovenko.  Adaptive numerical method for Poisson-Boltzmann equation and its application , 2002 .

[21]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.