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PB-steric equations: A general model of Poisson-Boltzmann equations

Jhih-Hong Lyu, Tai-Chia Lin

Abstract

When ions are crowded, the effect of steric repulsion between ions (which can produce oscillations in charge density profiles) becomes significant and the conventional Poisson-Boltzmann (PB) equation should be modified. Several modified PB equations were developed but the associated total ionic charge density has no oscillation. This motivates us to derive a general model of PB equations called the PB-steric equations with a parameter $Λ$, which not only include the conventional and modified PB equations but also have oscillatory total ionic charge density under different assumptions of steric effects and chemical potentials. As $Λ=0$, the PB-steric equation becomes the conventional PB equation, but as $Λ>0$, the concentrations of ions and solvent molecules are determined by the Lambert type functions. To approach the modified PB equations, we study the asymptotic limit of PB-steric equations with the Robin boundary condition as $Λ$ goes to infinity. Our theoretical results show that the PB-steric equations (for $0\leqΛ\leq\infty$) may include the conventional and modified PB equations. On the other hand, we use the PB-steric equations to find oscillatory total ionic charge density which cannot be obtained in the conventional and modified PB equations.

PB-steric equations: A general model of Poisson-Boltzmann equations

Abstract

When ions are crowded, the effect of steric repulsion between ions (which can produce oscillations in charge density profiles) becomes significant and the conventional Poisson-Boltzmann (PB) equation should be modified. Several modified PB equations were developed but the associated total ionic charge density has no oscillation. This motivates us to derive a general model of PB equations called the PB-steric equations with a parameter , which not only include the conventional and modified PB equations but also have oscillatory total ionic charge density under different assumptions of steric effects and chemical potentials. As , the PB-steric equation becomes the conventional PB equation, but as , the concentrations of ions and solvent molecules are determined by the Lambert type functions. To approach the modified PB equations, we study the asymptotic limit of PB-steric equations with the Robin boundary condition as goes to infinity. Our theoretical results show that the PB-steric equations (for ) may include the conventional and modified PB equations. On the other hand, we use the PB-steric equations to find oscillatory total ionic charge density which cannot be obtained in the conventional and modified PB equations.
Paper Structure (14 sections, 12 theorems, 72 equations, 1 figure)

This paper contains 14 sections, 12 theorems, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Analysis of nonlinear terms under (A1)--(A2) and (A3)--(A4)
  3. Analysis of ciΛ and fΛ under (A1) and (A2)
  4. Analysis of c* and f* under (A1) and (A2)
  5. Analysis of ciΛ and fΛ under (A3) and (A4)
  6. Analysis of ci* and f* under (A3) and (A4)
  7. Proof of Theorems \ref{['thm:1.1']} and \ref{['thm:1.5']}
  8. Uniform boundedness of and
  9. Convergence of phi-Lambda under (A1) and (A2)
  10. Proof of Theorem \ref{['thm:1.5']}
  11. Numerical methods
  12. Conclusions
  13. Derivation of (1.10) and (1.11)
  14. An example of oscillatory f-lambda for Remark 3

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb R^d$ be a bounded smooth domain, $\varepsilon\in\mathcal{C}^\infty(\overline\Omega)$ be a positive function, $\rho_0\in\mathcal{C}^\infty(\overline\Omega)$, $\phi_{bd}\in\mathcal{C}^2(\partial\Omega)$, $z_0=0$, and $z_iz_j<0$ for some $i,j\in\{1,\dots,N\}$. Assume that (A1) where $\phi^*$ is the solution of with the Robin boundary condition eq:1.15.

Figures (1)

  • Figure 1: The profiles of $f_\Lambda(\phi)$ and $f_\Lambda(\phi_\Lambda(x))$ under assumptions (A1)$"'$ and (A2). In (a), curves 1--4 are profiles of function $f_\Lambda=\sum_{i=1}^3z_ic_{i,\Lambda}$ with $\Lambda=0.5$, $1$, $2$, $4$. In (b), curves 1--4 are the profiles of function $f_\Lambda\circ\phi_\Lambda$ with $\Lambda=0.5$, $1$, $2$, $4$.

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 2.1
  • proof
  • Claim 2.2
  • ...and 25 more