(557c) Using Fundamental Gibbsian Thermodynamics to Revisit the Cassie–Baxter and Wenzel Equations

For over 80 years, the Cassie–Baxter and Wenzel equations have been used to predict the contact angles of liquid drops on heterogeneous and rough surfaces, respectively. Over a decade ago, however, a debate emerged on their applicability and limitations. In the Cassie–Baxter equation, area fractions of each solid constituent in the surface are used, and in the Wenzel equation, an area roughness is used. This suggests that the whole solid–liquid interfacial area influences the contact angle. Is this true? Herein, we introduce Gibbsian composite-system thermodynamics as a tool to rigorously determine the equilibrium conditions (thermal, mechanical, and chemical) of a liquid drop in its vapor on a solid surface that is either heterogeneous or rough. Through these derivations, we demonstrate that the properties at the three-phase (solid–liquid–vapor) contact line dictate equilibrium contact angle instead of the two-phase (solid–liquid) contact area. That is, a line fraction needs to be used in the Cassie–Baxter equation, and a line roughness needs to be used in the Wenzel equation. We derive a new form of the Wenzel equation and introduce a dimensionless number that compares the amount of liquid penetrating the rough features to the size of the drop. These insights into the thermodynamic equilibrium contact angles of drops in the Cassie–Baxter (Langmuir 2018 34 12191–12198) and Wenzel states (Langmuir 2020 36 435–446) may help guide future studies, both experimental and theoretical, to investigate the magnitude of pinning forces that are known to dictate nonequilibrium advancing and receding contact angles. This research was supported by funding from the Natural Sciences and Engineering Research Council of Canada, Alberta Innovates and Alberta Advanced Education, and the University of Alberta. JAWE holds a Canada Research Chair in Thermodynamics.