In this paper, we consider numerical approximations for the two-component PFC models for binary colloidal crystals. In addition to the Cahn–Hilliard type two-component PFC model that is commonly used for considering mass conservation, we also derived a new Allen–Cahn type two-component PFC model by using the \(L^2\)-gradient flow and add two nonlocal Lagrange multipliers to the system to conserve the mass for each component. For these two types of two-component PFC models, the stabilized scalar auxiliary variable (SAV) approach is adopted to develop efficient, decoupled, second-order accurate, and linear numerical schemes, where a new SAV is introduced to reformulate the models, and two extra linear stabilization terms are added to improve the stability and keep the required accuracy thus allowing large time steps. These schemes are unconditionally energy stable, mass conservative and require solving only four linear equations with constant coefficients at each time step. Numerical examples are performed to demonstrate the accuracy and energy stability of the proposed schemes, and numerous 2D and 3D simulations are also presented to show a variety of complex binary ordered patterns of phase transformation.

在本文中，我们考虑二元胶体晶体的双组分 PFC 模型的数值近似。除了通常用于考虑质量守恒的 Cahn-Hilliard 型双组分 PFC 模型外，我们还通过使用\(L^2\)-梯度流并向系统添加两个非局部拉格朗日乘子以保存每个组件的质量。对于这两种双分量 PFC 模型，采用稳定标量辅助变量 (SAV) 方法来开发高效、解耦、二阶精确和线性数值方案，其中引入新的 SAV 来重新制定模型，并且添加了两个额外的线性稳定项以提高稳定性并保持所需的精度，从而允许大的时间步长。这些方案是无条件的能量稳定、质量保守的，并且在每个时间步只需要求解四个具有常数系数的线性方程。执行数值例子来证明所提出方案的准确性和能量稳定性，