Cassie's Law Reformulated: Composite Surfaces from Superspreading to Superhydrophobic.

In 1948, Cassie provided an equation describing the wetting of a smooth, heterogeneous surface. He proposed that the cosine of the contact angle, θ c , for a droplet on a composite surface could be predicted from a weighted average using the fractional surface areas, f i , of the cosines of contact angles of a droplet on the individual component surfaces, i.e., cos θ c = f 1 cos θ 1 + f 2 cos θ 2 . This was a generalization of an earlier equation for porous materials, which has recently proven fundamental to underpinning the theoretical description of wetting of superhydrophobic and superoleophobic surfaces. However, there has been little attention paid to what happens when a liquid exhibits complete wetting on one of the surface components. Here, we show that Cassie's equation can be reformulated using spreading coefficients. This reformulated equation is capable of describing composite surfaces where the individual surface components have negative (droplet state/partial wetting) or positive (film-forming/complete wetting) spreading coefficients. The original Cassie equation is then a special case when the combination of interfacial tensions results in a droplet state on the composite surface for which a contact angle can be defined. In the case of a composite surface created from a partial wetting (droplet state) surface and a complete wetting (film-forming) surface, there is a threshold surface area fraction at which a liquid on the composite surface transitions from a droplet to a film state. The applicability of this equation is demonstrated from literature data including data on mixed self-assembled monolayers on copper, silver, and gold surfaces that was regarded as definitive in establishing the validity of the Cassie equation. Finally, we discuss the implications of these ideas for super-liquid repellent surfaces.