Thermodynamically consistent phase field model for liquid-gas phase transition with soluble surfactant

Despite the enormous potential in facilitating natural development and migration of interfaces during multiphase simulation, the phase-field method remains restricted to low-density ratios, owing to inherent thermodynamic inconsistency, especially for multiphase flow systems with surfactants. The present paper first constructs a liquid-vapor phase transition phase-field model with soluble surfactants using the second law of thermodynamics as the original model. Then, a simplified liquid-vapor phase transition model with soluble surfactants that satisfies thermodynamic consistency is proposed to simulate pool boiling at higher-density ratio. A novel numerical algorithm for the simplified model that satisfies semi-discrete thermodynamic consistency is also developed. Compared with the original model, the thermodynamically consistent characteristics of the simplified numerical model proposed in this paper can significantly reduce the spurious velocity on the interface of a static droplet and thus enable the numerical model to simulate liquid-vapor transition at higher liquid/vapor density ratios. Vapor-liquid coexistence, Laplace's law, and multiple bubble coalescence are used to validate the accuracies and effectiveness of the mathematical model and numerical algorithm. The liquid/vapor density ratio can reach 6776:1 under saturation temperature 0.3T_c (T_c is the critical temperature). The approach is then used to model pool boiling at a low saturation temperature (0.5T_c) with and without soluble surfactants, significantly lower than reported in comparable literature. The results demonstrate that surfactants significantly influence the dynamics of bubbles, and a critical concentration can be identified. In addition, soluble surfactants can also suppress coalescence between adjacent bubbles and prevent the formation of larger bubbles during pool boiling.