In this work, we present a phase-field model that captures the evolution of ionic concentrations and phase fractions during solid-state metathesis (SSM) reactions where diffusion limits the rate of transformation. The evolution of the mole fraction of each ion is obtained via governing equations that describe the reduction of a free energy, which includes an energy landscape with local minima located at compositions corresponding to stable products. We utilized two Lagrange multipliers to impose constraints of electroneutrality as well as on the sum of mole fractions, which were then eliminated to derive set of two partial differential equations that describe the dynamics of the mole fraction evolution. From these governing equations, the expressions for effective mobilities for the cations and the anions were obtained. We first study the effect of mobilities of ions on the reaction kinetics, using a simple model considering the ions with an identical absolute value of charge numbers. The simulation results show that the overall characteristic mobility, defined as the sum of the two effective ionic mobilities, provides an excellent measure of the rate at which reaction progresses and that the ratio of the effective mobilities of the anions and the cations signifies the manner by which the reaction progresses. We then generalize the model to consider ions with different charge numbers and tuned the mobility of ions based on their diffusion coefficients reported in the literature and experimental data from a thin-film experiment for the synthesis of FeS₂ to demonstrate the capability of the model to predict the phase evolution during SSM reactions. In particular, the simulation predicts nonplanar phase evolution, which is recently observed in thin-film reactions for the synthesis of FeS₂ via transmission electron microscopy. The approach can serve as a basis for models for phase transformations in other multiphase ionic mixtures, such as in all-solid-state batteries and in ionic liquids.

在这项工作中，我们提出了一个相场模型，该模型捕获了扩散限制转化率的固态复分解 (SSM) 反应期间离子浓度和相分数的演变。每个离子的摩尔分数的演变是通过描述自由能减少的控制方程获得的，其中包括一个能量景观，其局部最小值位于与稳定产物相对应的成分处。我们利用两个拉格朗日乘数来施加电中性和摩尔分数之和的约束，然后将其消除以导出描述摩尔分数演变动力学的两个偏微分方程组。从这些控制方程中，获得了阳离子和阴离子的有效迁移率的表达式。我们首先研究离子迁移率对反应动力学的影响，使用一个简单的模型考虑具有相同电荷数绝对值的离子。模拟结果表明，总体特征迁移率（定义为两种有效离子迁移率的总和）提供了反应进行速率的极好度量，并且阴离子和阳离子的有效迁移率之比表示方式反应进行。然后，我们推广该模型以考虑具有不同电荷数的离子，并根据文献中报告的扩散系数和 FeS 合成薄膜实验的实验数据调整离子的迁移率 使用考虑具有相同电荷数绝对值的离子的简单模型。模拟结果表明，总体特征迁移率（定义为两种有效离子迁移率的总和）提供了反应进行速率的极好度量，并且阴离子和阳离子的有效迁移率之比表示方式反应进行。然后，我们推广该模型以考虑具有不同电荷数的离子，并根据文献中报告的扩散系数和 FeS 合成薄膜实验的实验数据调整离子的迁移率 使用考虑具有相同电荷数绝对值的离子的简单模型。模拟结果表明，总体特征迁移率（定义为两种有效离子迁移率的总和）提供了反应进行速率的极好度量，并且阴离子和阳离子的有效迁移率之比表示方式反应进行。然后，我们推广该模型以考虑具有不同电荷数的离子，并根据文献中报告的扩散系数和 FeS 合成薄膜实验的实验数据调整离子的迁移率 提供了反应进行速率的极好测量，并且阴离子和阳离子的有效迁移率之比表示反应进行的方式。然后，我们推广该模型以考虑具有不同电荷数的离子，并根据文献中报告的扩散系数和 FeS 合成薄膜实验的实验数据调整离子的迁移率 提供了反应进行速率的极好测量，并且阴离子和阳离子的有效迁移率之比表示反应进行的方式。然后，我们推广该模型以考虑具有不同电荷数的离子，并根据文献中报告的扩散系数和 FeS 合成薄膜实验的实验数据调整离子的迁移率_图2展示了模型预测 SSM 反应过程中相演化的能力。_{特别是，该模拟预测了最近通过透射电子显微镜在用于合成 FeS 2}的薄膜反应中观察到的非平面相演化。该方法可以作为其他多相离子混合物（例如全固态电池和离子液体）相变模型的基础。