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

  • Mohammad Azadi TabarEmail author
  • Farzad Barzegar
  • Mohammad Hossein Ghazanfari
  • Mohammad Mohammadi
Technical Paper


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.


Cassie–Baxter Wenzel Materials science Physical properties Contact angle Surface/length fraction Roughness factor 



Local length fraction


Local surface fraction


Global surface fraction


Contact angle


Roughness factor


Local roughness factor


Global roughness factor

List of symbols


Pillar width (µm)


Pillar edge-to-edge separation distance (µm)


Indicator function of drop base area (–)


Indicator function of drop apparent surface area (–)


Error function of area (–)


Error function of perimeter (–)


Local surface fraction (–)


Global surface fraction (–)


Length fraction (–)


Maximum cylindrical pillar diameter (µm)


Length of imaginary cell (µm)


Number of grids in x direction (–)


Square root of pillar number (–)


The pillar indicator function (–)


The number of faces of each block in the vicinity of the drop (–)


Drop radius (µm)


The ratio of actual surface area to apparent surface area (–)


The minimum drop radius after which |θsf − θlf| < 1 (µm)


Roughness factor (–)


Local roughness factor (–)


Global roughness factor (–)


The pillar indicator function inside the drop (–)

\(S_{C}^{\prime }\)

The indicator function of actual surface area (–)


Indicator function of pillar under drop perimeter (–)


The maximum number of faces in access to drop (–)

xc, yc

Drop center (µm)

x, y

Cartesian coordinates (µm)

(xp, yp)

The pillar center (µm)


Phase function of surface in Cassie–Baxter model (–)


Phase function of surface in Wenzel model (–)


Contact angle (°)


Contact angle of smooth surface in the Wenzel model (°)


Apparent contact angle (°)


Apparent contact angle of solid fraction of surface in Cassie–Baxter equation (°)


Apparent contact angle through involving local surface fraction in Cassie–Baxter equation (°)


Apparent contact angle through involving local length fraction in Cassie–Baxter equation (°)


Apparent contact angle through involving global surface fraction or global roughness factor in Cassie–Baxter or Wenzel equation, respectively (°)


Apparent contact angle through involving local roughness factor in Wenzel equation (°)



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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Chemical and Petroleum EngineeringSharif University of TechnologyTehranIran

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