Horng Tzyy-Leng
Department of Applied Mathematics, Feng Chia University, Taichung 40724, Taiwan.
Entropy (Basel). 2020 Jun 8;22(6):632. doi: 10.3390/e22060632.
The classical Poisson-Boltzmann model can only work when ion concentrations are very dilute, which often does not match the experimental conditions. Researchers have been working on the modification of the model to include the steric effect of ions, which is non-negligible when the ion concentrations are not dilute. Generally the steric effect was modeled to correct the Helmholtz free energy either through its internal energy or entropy, and an overview is given here. The Bikerman model, based on adding solvent entropy to the free energy through the concept of volume exclusion, is a rather popular steric-effect model nowadays. However, ion sizes are treated as identical in the Bikerman model, making an extension of the Bikerman model to include specific ion sizes desirable. Directly replacing the ions of non-specific size by specific ones in the model seems natural and has been accepted by many researchers in this field. However, this straightforward modification does not have a free energy formula to support it. Here modifications of the Bikerman model to include specific ion sizes have been developed iteratively, and such a model is achieved with a guarantee that: (1) it can approach Boltzmann distribution at diluteness; (2) it can reach saturation limit as the reciprocal of specific ion size under extreme electrostatic conditions; (3) its entropy can be derived by mean-field lattice gas model.
经典的泊松-玻尔兹曼模型仅在离子浓度非常稀时才有效，而这往往与实验条件不符。研究人员一直在致力于对该模型进行修正，以纳入离子的空间效应，当离子浓度不稀时，这种效应不可忽略。一般来说，空间效应是通过亥姆霍兹自由能的内能或熵来建模修正的，在此给出一个概述。基于通过体积排斥概念将溶剂熵添加到自由能中的比克曼模型，是目前相当流行的一种空间效应模型。然而，在比克曼模型中离子大小被视为相同，因此将比克曼模型扩展以纳入特定离子大小是很有必要的。在模型中直接用特定大小的离子取代非特定大小的离子似乎很自然，并且已被该领域的许多研究人员所接受。然而，这种直接的修正没有自由能公式来支持。在此，对比克曼模型进行了迭代开发以纳入特定离子大小，并且得到了这样一个模型，它保证：(1) 在稀溶液中能趋近玻尔兹曼分布；(2) 在极端静电条件下能达到作为特定离子大小倒数的饱和极限；(3) 其熵可以通过平均场晶格气体模型推导得出。
