We present a phase-field model to study two-phase displacement with moving contact lines in a capillary tube. We construct a diffuse-interface formulation of solid-liquid surface energy by enforcing a consistent structure of the fluid-fluid interface between the bulk fluid and the solid surface. We first show, via simulation of equilibrium liquid slugs in a capillary tube, that this formulation allows prescribing arbitrary static contact angles and leads to the correct capillary pressure. We then propose a formulation to account for out-of-equilibrium dynamics near the contact line and demonstrate the ability of this generalized formulation to simulate spontaneous imbibition as well as viscously unstable, constant-rate displacements in a capillary tube. We show that our phase-field model captures the imbibition dynamics described by a theoretical model that combines classic Lucas-Washburn theory with Cox's law of dynamic contact angle. It also predicts wetting transition, thin-film formation, and interface pinch-off that quantitatively agree with recent experiments.

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• Contact line dynamics
• Interfacial flows
• Thin fluid films
• Wetting