In this work, we consider numerical methods for the Poisson–Nernst–Planck–Cahn–Hilliard (PNPCH) equations with steric interactions, which correspond to a non-constant mobility $H^{-1}$ gradient flow of a free-energy functional that consists of electrostatic free energies, steric interaction energies of short range, entropic contribution of ions, and concentration gradient energies. We propose a novel energy stable numerical scheme that respects mass conservation and positivity at the discrete level. Existence and uniqueness of the solution to the proposed nonlinear scheme are established by showing that the solution is a unique minimizer of a convex functional over a closed, convex domain. The positivity of numerical solutions is further theoretically justified by the singularity of the entropy terms, which prevents the minimizer from approaching zero concentrations. A further numerical analysis proves discrete free-energy dissipation. Extensive numerical tests are performed to validate that the numerical scheme is first-order accurate in time and second-order accurate in space, and is capable of preserving the desired properties, such as mass conservation, positivity, and free energy dissipation, at the discrete level. Moreover, the PNPCH equations and the proposed scheme are applied to study charge dynamics and self-assembled nanopatterns in highly concentrated electrolytes that are widely used in electrochemical energy devices. Numerical results demonstrate that the PNPCH equations and our numerical scheme are able to capture nanostructures, such as lamellar patterns and labyrinthine patterns in electric double layers and the bulk, and multiple time relaxation with multiple time scales. The multiple time relaxation dynamics with metastability take long time to reach an equilibrium, highlighting the need for robust, energy stable numerical schemes that allow large time stepping. In addition, we numerically characterize the interplay between cross steric interactions of short range and the concentration gradient regularization, and their impact on the development of nanostructures in the equilibrium state.

在这项工作中，我们考虑具有空间相互作用的泊松-能斯特-普朗克-卡恩-希拉德（PNPCH）方程的数值方法，这对应于非恒定迁移率 $H^{-\mathrm{1个}}$由静电自由能，短距离空间相互作用能，离子的熵贡献和浓度梯度能组成的自由能函数的梯度流。我们提出了一种新的能量稳定数值方案，该方案在离散水平上考虑了质量守恒和正性。通过显示该解决方案是在封闭凸域上的凸泛函的唯一极小值，来建立所提出的非线性方案的解的存在性和唯一性。数值解的正性在理论上进一步由熵项的奇异性证明，这可防止极小值接近零浓度。进一步的数值分析证明了离散的自由能耗散。进行了广泛的数值测试，以验证该数值方案在时间上是一阶精确的，在空间上是二阶的，并且能够在离散位置保留所需的特性，例如质量守恒，正性和自由能耗散。水平。此外，PNPCH方程和所提出的方案被用于研究电荷动力学和高浓度电解质中的自组装纳米图案，该电解质被广泛用于电化学能源设备中。数值结果表明，PNPCH方程和我们的数值方案能够捕获纳米结构，例如双电层和主体中的层状模式和迷宫式模式，以及多个时间尺度的多次时间弛豫。具有亚稳定性的多次时间弛豫动力学需要很长时间才能达到平衡，这突出表明了对鲁棒的，能量稳定的数值方案的需求，该方案允许较大的时间步长。此外，我们在数值上表征了短程交叉空间相互作用与浓度梯度规则化之间的相互作用，以及它们对处于平衡状态的纳米结构发展的影响。