Based on the phase-field theory, we present an improved lattice Boltzmann (LB) method for simulating droplet dynamics with soluble surfactants. This method takes advantage of three sets of particle distribution functions for solving the coupled system of two Cahn–Hilliard-like equations and incompressible Navier–Stokes equations. The phase-field model is formulated from the perspective of the Ginzburg–Landau free energy functional, where some modifications introduced circumvent unphysical behavior of the interfacial layer and improve the well-posedness of the model. We also give a comprehensive review on the existing surface tension force formulations and demonstrated that the popular potential form is artificial; instead, an alternative potential surface tension force is deduced. The equation of state accounting for the influence of the surfactant concentration on interfacial tension can be directly incorporated into the present approach, further improving the flexibility of the method. Besides, a linear equilibrium distribution function and a proper source term are introduced into the LB method for surfactants such that it can recover the correct physical formulations for a surfactant-laden multiphase system. An abundance of numerical experiments is carried out to validate the LB method, and the numerical performances of the tensor and potential surface tension forces are also evaluated. It is reported that the potential scheme achieves a better accuracy in solving interfacial dynamics at low surfactant concentrations and also is in favor of lower spurious velocities. In addition, the numerical predictions of surfactant-laden droplet dynamics show good agreement with the literature data.

中文翻译：

---

用格子 Boltzmann 方法模拟载有表面活性剂的液滴动力学

基于相场理论，我们提出了一种改进的格子玻尔兹曼 (LB) 方法，用于模拟具有可溶性表面活性剂的液滴动力学。该方法利用三组粒子分布函数来求解两个类 Cahn-Hilliard 方程和不可压缩 Navier-Stokes 方程的耦合系统。相场模型是从 Ginzburg-Landau 自由能泛函的角度制定的，其中引入了一些修改来规避界面层的非物理行为并提高模型的适定性。我们还对现有的表面张力公式进行了全面审查，并证明了流行的潜在形式是人为的；相反，可以推导出另一种潜在的表面张力。说明表面活性剂浓度对界面张力影响的状态方程可以直接纳入本方法，进一步提高方法的灵活性。此外，线性平衡分布函数和适当的源项被引入到表面活性剂的 LB 方法中，以便它可以恢复含有表面活性剂的多相体系的正确物理配方。进行了大量的数值实验来验证 LB 方法，并且还评估了张量和潜在表面张力的数值性能。据报道，该潜在方案在低表面活性剂浓度下求解界面动力学时获得了更好的准确性，并且有利于降低杂散速度。此外，