Electrostatic Interactions In Charged Systems:Modeling,Analysis,and Computation

Electrostatic interactions play a key role in many complex charged systems. Accurate and efficient modeling and computations of such interactions have been challenging due to the inhomogeneity, complicated geometry, multiple scales, and the nature of the many-body interaction of an underlying charged system. The Poisson-Boltzmann (PB) theory is one of the most successful theories on studying the electrostatic interactions in complex charged systems. It has been applied to many areas, such as biomolecular systems, colloidal science, and electrochemistry. Despite its success in many applications, the classical PB theory is known to fail in capturing well the ionic size effects and ion-ion correlation. The first part of this thesis focuses on the improvement and modification of PB theory to incorporate ionic size effects within a mean-field theoretical framework. The modified PB model is applied to study the distribution of ionic concentration near a charged surface. In the following part, Monte Carlo simulations are also reported. Such simulations confirm that the modified PB model is capable of capturing the ionie size effects accurately and efficiently, and that the conclusions made by the modified PB model are true. Finally, the effect of the electrostatics on the motion of the dielectric interface of a biomolecule solvation system is concerned.In Chapter2, the classical PB is modified to consider the ionic size effects of an ionic solution. It begins with a variational formulation of the continuum electrostatics of the ionic solution. The free energy functional of the system is minimized under some constraints that describe the mass conservation of the ions and charge neutrality of the whole system. A Lagrange multiplier method is developed to solve this constrained optimization problem. To incorporate size effects of the ions and solvent molecules, the entropy part of the free energy is modified based on the lattice model of ideal gas. A nonlinear transcendental sys-tem for ionic concentrations is obtained by minimizing the modified free energy functional respect to ionic concentrations. It turns out the nonlinear system is not solvable when the ionic sizes are different; therefore, the Boltzmann-like dis- tributions are no longer available in this case. An augmented Lagrangian method is proposed to solve the constrained optimization problem for non-uniform ionic sizes. Numerical results illustrate that the modified model and the correspond-ing numerical optimization methods are able to compute the ionic concentrations and electric potential accurately and efficiently. Also, the modified model is ap-plied to study the effect of the charge density of the surface and ionic sizes on the ionic concentrations near the charged surface. Obvious stratifications of the counterions are observed in the vicinity of the highly charged surface. The ionic valence-to-volume ratio is found to be a key parameter to determine the structure of the stratifications. This is a joint work with Professor Zhongming Wang and Professor Bo Li.To further confirm the accuracy of the modified model and the conclusions made in Chapter2, Monte Carlo simulations are reported in Chapter3to in-vestigate the ionic distribution of the same system. This electrolyte system is studied by an unrestrictive primitive model, in which the ions are treated as hard spheres and the solvent is modeled through its dielectric permittivity. The canonical ensemble Monte Carlo simulations are conducted with Metropolis cri-terion. The whole Monte Carlo simulation consists of three stages:acceleration, equilibration, and statistics. Extensive simulations indicate that the ionic sizes play an important role in determining the structure of ionic distribution. For a weakly charged surface, the counterions with larger valences adsorb closer onto the charged surface. For a highly charged surface, the results of Monte Carlo simulations and that of modified PB theory both show counterions stratify near the surface. Moreover, the structure of the stratifications are closely related to values of the valence-to-volume ratio, rather than the valences of counterions alone. To be more specific, the counterions with largest value of the valence-to-volume ratio distribute closest to the charged surface; then, the counterions with second largest value of the valence-to-volume ratio come to the next layer, and so on. The result of the adsorption of counterions onto the charged surface is the process of the competition between electric energy and entropy. The values of the valence-to-volume ratio exactly reflect this competition. Qualitative agreements between the results of Monte Carlo simulation and that of the modified Poisson- Boltzmann theory confirm our conclusion that the ionic valence-to-volume ratio is a key parameter to determine the structure of the stratifications. This is a joint work with Jiayi Wen, Professor Zhenli Xu, and Professor Bo Li.Chapter4is devoted to the study of the motion of a dielectric interface us-ing a variational solvation model. In cylindrical coordinates, such motion of the dielectric interface is governed by the steepest descent of a free energy functional that consists of surface area energy and electrostatic energy. The effective inter-face force is defined and an explicit formula of such force is obtained. Represented by a function graph, the interface solves a time-dependent nonlinear system. It can be found that the interface force always pointing from the solvent region to solute region. In the case that the interior of the interface is of a lower dielectrics, the interface is initially flattened by the surface force, and then is driven inward fast by both the surface and the electrostatic forces. When the dielectric co-efficient of interior region is greater than that of outer region, the competition between geometrical and electrostatic contributions leads to the existence of a cylindrical steady-state solution of the nonlinear system. Linear stability anal-ysis shows that the steady-state solution is only stable for a perturbation with a wavenumber larger than a critical threshold. Efficient and accurate numerical schemes are developed for the nonlinear system. Numerical results confirm our analysis on the role of each component of the effective interface force. This is a joint work with Professor Li-Tien Cheng, Professor Bo Li, and Michael White.