A domain decomposition method for the directional contact angle hysteresis interval on doubly periodic rough surfaces

RESUMO

We study wetting on a doubly periodic rough surface in three dimensions. Although the liquid-vapor interface meets the solid at the local Young's angle, microscale roughness can cause the macroscopic apparent angle to differ substantially from this value. We formulate the directional contact angle hysteresis (CAH) interval in terms of apparent angles associated with pinned microscopic configurations. To approximate its receding and advancing endpoints, we evolve capillary mean curvature flow (CMCF) toward extremal stationary states. Computing these states is difficult because the pinning that produces hysteresis is generated at the scale of the roughness, whereas the apparent angle is only meaningful at the macroscopic scale, so a single uniform grid must resolve both. We therefore introduce a two-scale alternating (TSA) method based on a Schwarz decomposition: a Merriman-Bence-Osher (MBO) diffusion-generated scheme resolves the contact-line near region, while a linearized minimal-surface problem updates the far region. For an idealized reference iteration, we prove decay of an approximate interfacial energy. Numerical experiments on a representative doubly periodic surface show a strongly anisotropic CAH interval whose width varies by more than a factor of three with contact-line orientation and changes sharply near the diagonal directions. Because a stationary droplet must meet the solid at an apparent angle inside this interval, the computed anisotropy constrains which macroscopic wetted regions the surface can support; it is consistent with a square-like stationary droplet whose sides align with the diagonal directions.