Surface roughness effects on contact line motion with small capillary number

In this work, we investigate how surface roughness influences contact line dynamics by simulating forced wetting in a capillary tube. The tube wall is decorated with microgrooves and is intrinsically hydrophilic. A phase-field method is used to capture the fluid interface and the moving contact line. According to the numerical results, a criterion is proposed to judge whether the grooves are entirely wetted or not at vanishing capillary numbers. When the contact line moves over a train of grooves, the apparent contact angle exhibits a periodic nature, no matter whether the Cassie-Baxter or the Wenzel state is achieved. The oscillation amplitude of apparent contact angle is analyzed and found to be inversely proportional to the interface area. The contact line motion can be characterized as stick-jump-slip in the Cassie-Baxter state and stick-slip in the Wenzel state. By comparing to the contact line dynamics on smooth surfaces, equivalent microscopic contact angles and slip lengths are obtained. The equivalent slip length in the Cassie-Baxter state agrees well with the theoretical model in the literature. The equivalent contact angles are, however, much greater than the predictions of the Cassie-Baxter model and the Wenzel model for equilibrium stable states. Our results reveal that the pinning of the contact line at surface defects effectively enhances the hydrophobicity of rough surfaces, even when the surface material is intrinsically hydrophilic and the flow is under the Wenzel state.