On the modeling of hydrophobic contact angles on rough surfaces

The apparent contact angle of a drop on a rough surface is often modeled using either Wenzel's or Cassie's formulas. Previous experiments are not conclusive regarding which formula to use and when. This information is critical in designing a superhydrophobic substrate for applications in microscale devices. A drop on a rough substrate can occupy multiple equilibrium states. These equilibrium states denote respective local minima in energy. The particular shape that a drop attains depends on how the drop is formed. We propose a design procedure to develop a rough superhydrophobic substrate that accounts for the multiple equilibrium drop shapes. The theory is expected to work well to maximize the advancing contact angle of a drop. It is noted in the end that appropriate models for the receding contact angles on rough substrate must be investigated further before appropriate design procedures, which will maximize the receding contact angle or minimize hysteresis (i.e., minimize the difference between the advancing and receding contact angles), are developed. We discuss a model for the receding contact angle, based on the limited data in the literature.