双电层内离子浓度分布：建模，分析与计算

Charged objects in aqueous solutions, such as biomolecules and soft matter, are surrounded by electric double layers (EDLs). Investigation of ion distributions in EDLs is fundamental to the understanding of the structure, function, and nature of complex charged systems. As a classical mean-field model, Poisson-Boltzmann (PB) theory is perhaps the most popular and efficient method for the calculation of ion distributions in EDLs. For highly charged systems, PB theory, however, is not able to predict ion distributions accurately. To remedy the defects, we propose a variational model by considering the ionic size effect and concentration dependent dielectrics within the PB framework. We include the size effect to the free-energy functional and employ the Bruggeman model to combine ionic sizes, concentration, and dielectrics inherently. We analyze the mathematical properties of the resulting variational model and illustrate the corresponding physical significance. We develop advanced numerical methods to solve the model efficiently. Applications to biomolecular systems of large scale demonstrate the effectiveness of the model and the efficiency of numerical methods. In addition, we develop Monte Carlo algorithms to simulate charged systems with concentration dependent dielectrics, and compare the results with that of the proposed variational model. As a problem-driven research, this project provides a powerful tool in the simulation of biomolecules and soft matter. It will be a valuable effort to the development of efficient, accurate theoretical models for complex charged systems of large scale.

双电层结构广泛存在于带电生物分子和软物质体系中。研究双电层内的离子浓度分布对正确认识复杂带电体系的结构，功能和性质具有重要的意义。泊松-玻尔兹曼（PB）理论是计算带电体系中离子浓度分布的经典平均场理论模型。当体系带电较强时，PB理论不能准确预测双电层内的离子分布。为了克服PB理论的不足，我们在PB理论框架下考虑离子的尺寸效应和离子浓度依赖的介电环境。我们从静电自由能出发，引入离子的尺寸效应，并采用Bruggeman模型将离子的尺寸、浓度与介电系数进行有机结合，导出变分模型。我们分析模型的数学性质，并解释其对应的物理意义。针对模型的计算困难，我们发展高效的数值计算方法，并把模型应用于大型生物分子和软物质体系。另外，我们发展蒙特卡洛模拟算法，并将模拟结果与修正的PB理论模型进行对比验证。该项目是科学问题驱动的应用数学研究，将为生物分子和软物质体系的模拟提供有力的计算工具，具有重要的实用价值。

双电层结构广泛存在于复杂带电体系。正确认识双电层结构对包括细胞膜离子通道、电化学能源器件等实际应用具有重要的意义，从而成为电化学、软物质科学和应用数学等领域共同关注的热门研究课题。基于平均场理论的泊松玻尔兹曼模型是描述双电层结构最为广泛应用的理论。由于忽略了离子的尺寸效应等因素，泊松玻尔兹曼理论不能很好地描述浓度较高的高价态离子溶液。在泊松玻尔兹曼理论框架下，项目发展了一类能刻画离子尺寸效应的变分模型。为了进一步能描述离子浓度依赖的介电环境，项目在变分模型中采用Bruggeman理论考虑离子局部浓度、尺寸大小、介电系数等因素。项目对模型的能量泛函以及模型解的数学性质作了系统的理论分析，并给出了相应的物理解释。针对模型的计算困难，项目发展了高效的数值迭代算法快速求解双电层内离子的稳态分布。另外，项目组将发展的模型与数值方法应用于研究离子尺寸效应对离子通道、电极板附近的双电层结构以及微分电容的影响。这一系列成功的应用进一步表明项目发展的模型与算法具有重要的实际应用价值。