Lattice evolution solution for the nonlinear Poisson–Boltzmann equation in confined domains

Abstract The lattice evolution method for solving the nonlinear Poisson–Boltzmann equation in confined domain is developed by introducing the second-order accurate Dirichlet and Neumann boundary implements, which are consistent with the non-slip model in lattice Boltzmann method for fluid flows. The lattice evolution method is validated by comparing with various analytical solutions and shows superior to the classical numerical solvers of the nonlinear Poisson equations with Neumann boundary conditions. The accuracy and stability of the method are discussed. This lattice evolution nonlinear Poisson–Boltzmann solver is suitable for efficient parallel computing, complex geometry conditions, and easy extension to three-dimensional cases.

[1]  D Frenkel,et al.  Lattice-Boltzmann method for the simulation of transport phenomena in charged colloids. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  J. A. McCammon,et al.  Solving the finite difference linearized Poisson‐Boltzmann equation: A comparison of relaxation and conjugate gradient methods , 1989 .

[3]  H. Zhou,et al.  Boundary element solution of macromolecular electrostatics: interaction energy between two proteins. , 1993, Biophysical journal.

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

[5]  O. Filippova,et al.  Grid Refinement for Lattice-BGK Models , 1998 .

[6]  R. Dror,et al.  Gaussian split Ewald: A fast Ewald mesh method for molecular simulation. , 2005, The Journal of chemical physics.

[7]  B. Honig,et al.  A rapid finite difference algorithm, utilizing successive over‐relaxation to solve the Poisson–Boltzmann equation , 1991 .

[8]  S. Kuyucak,et al.  Physics of Ion Channels , 2003, Journal of biological physics.

[9]  Dongqing Li,et al.  Electro-viscous effects on pressure-driven liquid flow in microchannels , 2001 .

[10]  Zhaoli Guo,et al.  A lattice Boltzmann algorithm for electro-osmotic flows in microfluidic devices. , 2005, The Journal of chemical physics.

[11]  Harold A. Scheraga,et al.  A fast adaptive multigrid boundary element method for macromolecular electrostatic computations in a solvent , 1997, J. Comput. Chem..

[12]  B. Chopard,et al.  Theory and applications of an alternative lattice Boltzmann grid refinement algorithm. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  A. Majumdar,et al.  Electrostatic control of ions and molecules in nanofluidic transistors. , 2005, Nano letters.

[14]  H. Chen,et al.  Theory of multicolor lattice gas: a cellular automaton Poisson solver , 1990 .

[15]  Richard A. Friesner,et al.  Numerical solution of the Poisson-Boltzmann equation using tetrahedral finite-element meshes , 1997, J. Comput. Chem..

[16]  Takaji Inamuro,et al.  A NON-SLIP BOUNDARY CONDITION FOR LATTICE BOLTZMANN SIMULATIONS , 1995, comp-gas/9508002.

[17]  Ernst,et al.  Simulation of diffusion in a two-dimensional lattice-gas cellular automaton: A test of mode-coupling theory. , 1989, Physical review letters.

[18]  Lin,et al.  Lattice boltzmann method on composite grids , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[19]  Benoît Roux,et al.  Ion conduction and selectivity in K(+) channels. , 2005, Annual review of biophysics and biomolecular structure.

[20]  Sauro Succi,et al.  Simulating two-dimensional thermal channel flows by means of a lattice Boltzmann method with new boundary conditions , 2004, Future Gener. Comput. Syst..

[21]  G. Hummer,et al.  Ion transport through membrane-spanning nanopores studied by molecular dynamics simulations and continuum electrostatics calculations. , 2005, Biophysical journal.

[22]  K. Sharp,et al.  Electrostatic interactions in macromolecules: theory and applications. , 1990, Annual review of biophysics and biophysical chemistry.

[23]  H. Scheraga,et al.  A fast adaptive multigrid boundary element method for macromolecular electrostatic computations in a solvent , 1997 .

[24]  Xiaoyi He,et al.  Lattice Boltzmann simulation of electrochemical systems , 2000 .

[25]  Jinchao Xu,et al.  Newton-Krylov-Multigrid Algorithms for Battery Simulation , 2002 .

[26]  Nathan A. Baker,et al.  Electrostatics of nanosystems: Application to microtubules and the ribosome , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[27]  B. Honig,et al.  Classical electrostatics in biology and chemistry. , 1995, Science.

[28]  P. Wong,et al.  Electrokinetics in micro devices for biotechnology applications , 2004, IEEE/ASME Transactions on Mechatronics.

[29]  Moran Wang,et al.  Lattice Poisson-Boltzmann simulations of electro-osmotic flows in microchannels. , 2006, Journal of colloid and interface science.

[30]  Q. Zou,et al.  On pressure and velocity boundary conditions for the lattice Boltzmann BGK model , 1995, comp-gas/9611001.

[31]  M. Stevens,et al.  Density Functional Theory of Ionic Screening: When Do Like Charges Attract? , 1990 .

[32]  N. Aluru,et al.  Ion concentrations and velocity profiles in nanochannel electroosmotic flows , 2003 .

[33]  Qisu Zou,et al.  N ov 1 99 6 On pressure and velocity flow boundary conditions and bounceback for the lattice Boltzmann BGK model , 2008 .

[34]  Miki Hirabayashi,et al.  The lattice BGK model for the Poisson equation , 2001 .

[35]  D. Draper,et al.  Ions and RNA folding. , 2005, Annual review of biophysics and biomolecular structure.

[36]  Chao-Yang Wang,et al.  Fundamental Models for Fuel Cell Engineering , 2004 .

[37]  Patrick B. Warren,et al.  Electroviscous Transport Problems via Lattice-Boltzmann , 1997 .

[38]  Howard A. Stone,et al.  ENGINEERING FLOWS IN SMALL DEVICES , 2004 .

[39]  SucciSauro,et al.  Simulating two-dimensional thermal channel flows by means of a lattice Boltzmann method with new boundary conditions , 2004 .

[40]  Michael J. Holst,et al.  Numerical solution of the nonlinear Poisson–Boltzmann equation: Developing more robust and efficient methods , 1995, J. Comput. Chem..

[41]  Richard A. Friesner,et al.  Numerical solution of the Poisson–Boltzmann equation using tetrahedral finite‐element meshes , 1997 .

[42]  Skordos,et al.  Initial and boundary conditions for the lattice Boltzmann method. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[43]  L. Luo,et al.  Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model , 1997 .

[44]  Sauro Succi,et al.  Lattice Boltzmann-Poisson method for electrorheological nanoflows in ion channels , 2005, Comput. Phys. Commun..

[45]  R. Flatt,et al.  Electrostatic repulsion between particles in cement suspensions: Domain of validity of linearized Poisson–Boltzmann equation for nonideal electrolytes , 2003 .

[46]  J. Freund.  Electro-osmosis in a nanometer-scale channel studied by atomistic simulation , 2002 .

[47]  D. Martínez,et al.  On boundary conditions in lattice Boltzmann methods , 1996 .

[48]  H. Ohashi,et al.  The Lattice BGK Solution of the QKPZ Equation : Universality and Scaling in Fluid Invasion of Porous Media , 2000 .

[49]  Arun Majumdar,et al.  Ion transport in nanofluidic channels , 2004 .

[50]  L. Luo,et al.  Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation , 1997 .

[51]  Shiyi Chen,et al.  LATTICE BOLTZMANN METHOD FOR FLUID FLOWS , 2001 .

[52]  Moran Wang,et al.  LATTICE BOLTZMANN SIMULATIONS OF MIXING ENHANCEMENT BY THE ELECTRO-OSMOTIC FLOW IN MICROCHANNELS , 2005 .

[53]  D. Sundholm,et al.  Universal method for computation of electrostatic potentials. , 2005, The Journal of chemical physics.

[54]  Shiyi Chen,et al.  A Novel Thermal Model for the Lattice Boltzmann Method in Incompressible Limit , 1998 .

[55]  C. Shu,et al.  Simplified thermal lattice Boltzmann model for incompressible thermal flows. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[56]  Robert S. Bernard,et al.  Boundary conditions for the lattice Boltzmann method , 1996 .