Modeling, Analysis And Computation Of The Ionic Correlation With Dielectric Interfaces

Understanding the properties of electrolytes is significantly important in chemical engineering,electro-physiology as well as theoretical statistical physics.The traditional mean field Poisson–Boltzmann theory ignores the correlation effects and becomes successful in many different areas.However,when there is a highly charged surface or dense electrolytes,the mean field theory fails to describe the system correctly and the correlation effects can not be neglected.We will investigate systematically the correlation effects of electrolytes,through building proper models to capture the main correlation,designing efficient and accurate numerical schemes to solve the equations and analyzing the property of the models.On one hand,the correlation effects come from the short ranged hard sphere interaction,which means two ions cannot get too close to each other.We present a detailed analysis on a variational mean-field approach that includes ionic size effects by taking into account the solvent entropic contribution.Through asymptotic analysis,the stratification phenomenon in near field is understood and a modified screen length in far field is proposed.On the other hand,the correlation effects come from the long range Coulomb interaction.One of the most important phenomenon is the counterintuitive like-charge attraction in electrolytes.We develop an efficient algorithm and perform the force calculation between two interfaces using a set of self-consistent equations which properly takes into account the electrostatic correlation and the dielectric-boundary effects.The like-charge attraction phenomenon is then investigated in detail,with different electrolytes property and material parameters.The correlation also affects the transport process of electrolytes.The self-energy is used to improve the traditional Poisson–Nernst–Planck model.Through the Debye charging process,we derived the system free energy and the self energy of an ion with finite radius.In order to solve the self energy equation,an analytical WKB approximation for point charges and the asymptotic expansions taking the ion radius as the small parameter are developed.Moreover,the variations of this energy functional give the self-energy-modified PNP equations which satisfy a proper energy law.Numerical results with a semi-implicit conservative numerical method is applied to investigate the effect of the Coulomb correlation.As an application in electrophysiology,we study the ion channel conformation dynamic problem.We generalize the Shockley–Ramo theorem into the electrolytes system,avoiding directly solve the inverse problem of the Poisson–Nernst–Planck type of equations.With the procedure of renormalizing of charge and dipole,a direct relationship between the induced currents and the macro charge velocity in electrolytes is obtained.With the help of computer science,analyzing and numerically computing for complex models instead of seeking for analytical solutions become powerful tools to study the electrolyte systems.This thesis mainly focus on the mathematical modeling,analysis and scientific computing we proposed for the ionic correlation with dielectric interfaces.