We construct a mean-field variational model to study how the dependence of dielectric coefficient (i.e., relative permittivity) on local ionic concentrations affects the electrostatic interaction in an ionic solution near a charged surface. The electrostatic free-energy functional of ionic concentrations, which is the key object in our model, consists mainly of the electrostatic potential energy and the ionic ideal-gas entropy. The electrostatic potential is determined by Poisson's equation in which the dielectric coefficient depends on the sum of concentrations of individual ionic species. This dependence is assumed to be qualitatively the same as that on the salt concentration for which experimental data are available and analytical forms can be obtained by the data fitting. We derive the first and second variations of the free-energy functional, obtain the generalized Boltzmann distributions, and show that the free-energy functional is in general nonconvex. To validate our mathematical analysis, we numerically minimize our electrostatic free-energy functional for a radially symmetric charged system. Our extensive computations reveal several features that are significantly different from a system modeled with a dielectric coefficient independent of ionic concentration. These include the non-monotonicity of ionic concentrations, the ionic depletion near a charged surface that has been previously predicted by a one-dimensional model, and the enhancement of such depletion due to the increase of surface charges or bulk ionic concentrations.

中文翻译：

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平均场理论和静电计算与离子浓度相关的电介质

我们构建了一个平均场变分模型来研究介电系数（即相对介电常数）对局部离子浓度的依赖性如何影响带电表面附近离子溶液中的静电相互作用。离子浓度的静电自由能泛函是我们模型中的关键对象，主要由静电势能和离子理想气体熵组成。静电势由泊松方程确定，其中介电系数取决于单个离子物质的浓度总和。假设这种依赖性与盐浓度的定性相同，实验数据可用，并且可以通过数据拟合获得分析形式。我们推导出自由能泛函的第一和第二变体，获得广义玻尔兹曼分布，并表明自由能泛函通常是非凸的。为了验证我们的数学分析，我们在数值上最小化径向对称带电系统的静电自由能函数。我们的大量计算揭示了几个与使用独立于离子浓度的介电系数建模的系统显着不同的特征。这些包括离子浓度的非单调性、先前由一维模型预测的带电表面附近的离子消耗，以及由于表面电荷或体离子浓度增加而导致的这种消耗的增强。并证明自由能泛函一般是非凸的。为了验证我们的数学分析，我们在数值上最小化径向对称带电系统的静电自由能函数。我们的大量计算揭示了几个与使用独立于离子浓度的介电系数建模的系统显着不同的特征。这些包括离子浓度的非单调性、先前由一维模型预测的带电表面附近的离子消耗，以及由于表面电荷或体离子浓度增加而导致的这种消耗的增强。并证明自由能泛函一般是非凸的。为了验证我们的数学分析，我们在数值上最小化径向对称带电系统的静电自由能函数。我们广泛的计算揭示了几个与使用独立于离子浓度的介电系数建模的系统显着不同的特征。这些包括离子浓度的非单调性、先前由一维模型预测的带电表面附近的离子消耗，以及由于表面电荷或体离子浓度增加而导致的这种消耗的增强。我们的大量计算揭示了几个与使用独立于离子浓度的介电系数建模的系统显着不同的特征。这些包括离子浓度的非单调性、先前由一维模型预测的带电表面附近的离子消耗，以及由于表面电荷或体离子浓度增加而导致的这种消耗的增强。我们的大量计算揭示了几个与使用独立于离子浓度的介电系数建模的系统显着不同的特征。这些包括离子浓度的非单调性、先前由一维模型预测的带电表面附近的离子消耗，以及由于表面电荷或体离子浓度增加而导致的这种消耗的增强。