#### Date of Award

August 2016

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dexuan Xie

#### Committee Members

Istvan Lauko, Hans Volkmer, Bruce Wade, Lei Wang

#### Abstract

Dielectric continuum models have been widely applied to the study of aqueous electrolytes since the early work done by Debye and Hückel in 1910s. Traditionally, they treat the water solvent as a simple dielectric medium with a permittivity constant without considering any correlation among water molecules. In the first part of this thesis, a nonlocal dielectric continuum model is proposed for predicting the electrostatics of electrolytes caused by any external charges. This model can be regarded as an extension of the traditional Debye Hückel equation. For this reason, it is called the nonlocal Debye-Hückel equation. As one important application, this dissertation considers the case of an ionic solution with fixed charges from the atoms of a biomolecule. To avoid the singularities caused by the fixed atomic charges in Dirac-delta distribution, a solution decomposition scheme is constructed such that the Debye-Huckel equation is split into two equations: one with the analytical solution and the other one becoming well defined without any singularity. Hence, the study of the Debye-Hückel equation is simplified remarkably. Furthermore, a linearized nonlocal Debye-Hückel equation is proposed and thoroughly studied. Its analytic solution is found in algebraic expressions.

In the second part of this dissertation, two linearized nonlocal Poisson-Boltzmann equations (PBE) are proposed by using new linearization schemes. The third part of this dissertation reports the finite element algorithms and software packages for solving both the nonlocal Debye-Hückel equation and the new linearized nonlocal PBE model. Numerical results validate the analytical solution of the nonlocal Debye-Hückel equation and the program packages, which are expected to be valuable in many electrolyte applications.

#### Recommended Citation

Jiang, Yi, "Nonlocal Debye-Hückel Equations and Nonlocal Linearized Poisson-boltzmann Equations for Electrostatics of Electrolytes" (2016). *Theses and Dissertations*. 1276.

https://dc.uwm.edu/etd/1276