Mellin Transform and Image Charge Method for Dielectric Sphere in an Electrolyte

We revisit the image charge method for the Green's function problem of the Poisson--Boltzmann equation for a dielectric sphere immersed in ionic solutions. Using finite Mellin transformation, we represent the reaction potential due to a source charge inside the sphere in terms of a one dimensional distribution of image charges. The image charges are generically composed of a point image at the Kelvin point and a line image extending from the Kelvin point to infinity with an oscillatory line charge strength. We further develop an efficient and accurate algorithm for discretization of the line image using Pade approximation and finite fraction expansion. Finally, we illustrate the power of our method by applying it in a multiscale reaction-field Monte Carlo simulation of monovalent electrolytes.

[1]  Ronald M. Levy,et al.  Dielectric and thermodynamic response of a generalized reaction field model for liquid state simulations , 1993 .

[2]  I. Lindell,et al.  Image solution for Poisson's equation in wedge geometry , 1995 .

[3]  William H. Press,et al.  Book-Review - Numerical Recipes in Pascal - the Art of Scientific Computing , 1989 .

[4]  Ari Sihvola,et al.  Electrostatic image theory for the layered dielectric sphere , 1991 .

[5]  G. Temple.  Static and Dynamic Electricity , 1940, Nature.

[6]  Arieh Warshel,et al.  A surface constrained all‐atom solvent model for effective simulations of polar solutions , 1989 .

[7]  Leon Poladian,et al.  GENERAL THEORY OF ELECTRICAL IMAGES IN SPHERE PAIRS , 1988 .

[8]  Zhenli Xu,et al.  Fast Analytical Methods for Macroscopic Electrostatic Models in Biomolecular Simulations , 2011, SIAM Rev..

[9]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[10]  Shaozhong Deng,et al.  Image charge approximations of reaction fields in solvents with arbitrary ionic strength , 2009, J. Comput. Phys..

[11]  W. Cai,et al.  Discrete Image Approximations of Ionic Solvent Induced Reaction Field to Charges , 2007 .

[12]  P. Linse.  Simulation of charged colloids in solution , 2005 .

[13]  Wei Cai,et al.  Extending the fast multipole method to charges inside or outside a dielectric sphere , 2007, J. Comput. Phys..

[14]  Guo-Wei Wei,et al.  Variational Multiscale Models for Charge Transport , 2012, SIAM Rev..

[15]  L. Greengard,et al.  A new version of the Fast Multipole Method for the Laplace equation in three dimensions , 1997, Acta Numerica.

[16]  R. Abagyan,et al.  Biased probability Monte Carlo conformational searches and electrostatic calculations for peptides and proteins. , 1994, Journal of molecular biology.

[17]  D. Jacobs,et al.  Ionic solvation studied by image-charge reaction field method. , 2011, The Journal of chemical physics.

[18]  Joseph Lipka,et al.  A Table of Integrals , 2010 .

[19]  A. Kornyshev,et al.  Double layer in ionic liquids: overscreening versus crowding. , 2010, Physical review letters.

[20]  Harold L. Friedman,et al.  Image approximation to the reaction field , 1975 .

[21]  I. Lindell.  Electrostatic image theory for the dielectric sphere , 1992 .

[22]  YunKyong Hyon,et al.  Energy variational analysis of ions in water and channels: Field theory for primitive models of complex ionic fluids. , 2010, The Journal of chemical physics.

[23]  Wai-Yim Ching,et al.  Long Range Interactions in Nanoscale Science. , 2010 .

[24]  Robert Krasny,et al.  An Ewald summation based multipole method , 2000 .

[25]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[26]  N. Metropolis,et al.  Equation of State Calculations by Fast Computing Machines , 1953, Resonance.

[27]  Haruki Nakamura,et al.  Non-Ewald methods: theory and applications to molecular systems , 2012, Biophysical Reviews.

[28]  Y. Levin,et al.  Effects of the dielectric discontinuity on the counterion distribution in a colloidal suspension. , 2011, The Journal of chemical physics.

[29]  Piet Hut,et al.  A hierarchical O(N log N) force-calculation algorithm , 1986, Nature.

[30]  J. Kirkwood,et al.  Theory of Solutions of Molecules Containing Widely Separated Charges with Special Application to Zwitterions , 1934 .

[31]  Berend Smit,et al.  Understanding molecular simulation: from algorithms to applications , 1996 .

[32]  Zhenli Xu,et al.  Effects of image charges, interfacial charge discreteness, and surface roughness on the zeta potential of spherical electric double layers. , 2012, The Journal of chemical physics.

[33]  Donald J. Jacobs,et al.  An image-based reaction field method for electrostatic interactions in molecular dynamics simulations of aqueous solutions. , 2009, The Journal of chemical physics.

[34]  Femke Olyslager,et al.  Closed form solutions of Maxwell s equations in the computer age , 2003 .

[35]  J. Ptaszycki,et al.  Extrait d'une lettre , 1880 .

[36]  F. A. Seiler,et al.  Numerical Recipes in C: The Art of Scientific Computing , 1989 .

[37]  R. Messina.  Electrostatics in soft matter , 2008, Journal of physics. Condensed matter : an Institute of Physics journal.

[38]  Bartosz A Grzybowski,et al.  Electrostatics at the nanoscale. , 2011, Nanoscale.

[39]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[40]  W. T. Norris.  Charge images in a dielectric sphere , 1995 .

[41]  I. M. Pyshik,et al.  Table of integrals, series, and products , 1965 .

[42]  D. Beglov,et al.  Finite representation of an infinite bulk system: Solvent boundary potential for computer simulations , 1994 .

[43]  D. Zorin,et al.  A kernel-independent adaptive fast multipole algorithm in two and three dimensions , 2004 .

[44]  P. P. Ewald.  Die Berechnung optischer und elektrostatischer Gitterpotentiale , 1921 .