Improving the staircase approximation for wettability implementation of phase-field model: Part 1 – Static contact angle

In this study, two schemes to treat fluid-solid interaction on complex boundaries are proposed and solved in the framework of lattice Boltzmann method. The schemes are applied on the phase-field-based wettability implementation methods for binary and ternary fluids to simulate static contact angle on a circular surface generated by the staircase approximation. For the binary system, surface-energy and geometric wetting conditions are adopted, while for the ternary system, surface-energy model with two discretization schemes resulting in explicit and implicit wetting conditions are developed and examined. It is proven that the implicit wetting condition approximating the value of order parameters on the boundary node, as the average of their value between the neighboring fluid node and solid node, enhances the accuracy of predicted contact angles. For modeling of contact angle on the circular surface and identifying the direction of normal vectors required for incorporating wetting condition, the gradient of fluid-solid field is calculated. Two methods namely, round-off and interpolation describing how the information of neighboring nodes given the direction of normal vector on the boundary nodes should be used, are discussed. It is illustrated that results of the interpolation method exhibit a good agreement with the analytical solution as opposed to the round-off when tested in the case of static contact angle on a circular surface for both binary and ternary systems. Finally, it is shown incorporating different wetting conditions with the appropriate way of handling normal vectors and in combination with the conservative phase-field model can model static contact angle at high-density ratios for both binary and ternary fluids with good accuracy.