On the applicability range of Cassie–Baxter and Wenzel equation: a numerical study

Abstract

In this study, the range of applicability for Cassie–Baxter and Wenzel equations for estimating apparent contact angle on rough surfaces is numerically discussed. To do this, circular drops with different sizes are simulated on rough surfaces with a square pillar pattern and randomly distributed cylindrical pillar. With the aid of numerical method, the local surface fraction, local length fraction and local roughness factor for drops with different sizes on the surface are computed. Then, the global surface fraction and global roughness factor have been compared with the local surface fraction and local roughness factor, respectively. Local surface and local length fractions, as well as local roughness factor, behave oscillatory. It has been found that when drop radius is nearly 3 times greater than summation of pillar width and pillar edge-to-edge separation distance, the contact angles calculated through involving local and global surface fractions in Cassie–Baxter equation for the case of square pillar pattern, provided differences lower than ± 1° which is in agreement with the results of Marmur and Bittoun (Langmuir 25(3):1277–1281, 2009). The similar differences limit for contact angles calculated from local and global roughness factors are observed for square pillar pattern at Wenzel state when drop radius is nearly two times greater than the summation of pillar width, pillar edge-to-edge separation distance, and pillar height. Also, when drop size is almost 1000 times greater than the summation of pillar width and pillar edge-to-edge separation distance, the contact angle calculated through involving local surface and length fractions in Cassie–Baxter equation are different by ± 1°. Results of this work determine the minimum drop radius after which, different forms of Cassie–Baxter, as well as Wenzel equations, tend to the answer of their general equations.