Near a charged surface, counterions of different valences and sizes cluster; and their concentration profiles stratify. At a distance from such a surface larger than the Debye length, the electric field is screened by counterions. Recent studies by a variational mean-field approach that includes ionic size effects and by Monte Carlo simulations both suggest that the counterion stratification is determined by the ionic valence-to-volume ratios. Central in the mean-field approach is a free-energy functional of ionic concentrations in which the ionic size effects are included through the entropic effect of solvent molecules. The corresponding equilibrium conditions define the generalized Boltzmann distributions relating the ionic concentrations to the electrostatic potential. This paper presents a detailed analysis and numerical calculations of such a free-energy functional to understand the dependence of the ionic charge density on the electrostatic potential through the generalized Boltzmann distributions, the role of ionic valence-to-volume ratios in the counterion stratification, and the modification of Debye length due to the effect of ionic sizes.

中文翻译：

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离子尺寸效应：广义玻尔兹曼分布、反离子分层和修正德拜长度

在带电表面附近，不同价数和大小的反离子聚集在一起；并且它们的浓度分布是分层的。在距这种大于德拜长度的表面的距离处，电场被反离子屏蔽。最近通过包括离子尺寸效应的变分平均场方法和蒙特卡罗模拟的研究都表明，抗衡离子分层是由离子价与体积比决定的。平均场方法的核心是离子浓度的自由能泛函，其中通过溶剂分子的熵效应包括离子尺寸效应。相应的平衡条件定义了将离子浓度与静电势联系起来的广义玻尔兹曼分布。