# Generalized Wenzel and Cassie-Baxter equations

A researcher from Bangladesh asked me about my “2007 work with a generalized formulation of the Cassie-Baxter and Wenzel equations.”

The Wenzel equation was introduced in 1936, and it deals with wetting of a rough surface by a water droplet placed on the surface. According to the Wenzel equation, the contact angle (CA) on a rough surface is given by cosX=RfcosY, where Y is the CA with a smooth surface and Rf is the “roughness factor” of the surface.

The Cassie-Baxter (or just Cassie) equation was introduced in 1944, and it deals with a heterogeneous surface consisting of two fractions, with the fractional areas f1 and f2 (so that f1+f2=1). The Cassie equation says that cosX=f1cosY+f2cosZ, where Y and Z are contact angles with the two fractions.

There are several ways of how these equations can be generalized. My attempt is only one of many. In my comment
M. Nosonovsky, 2007, “On the Range of Applicability of the Wenzel and Cassie Equations,” Langmuir 2007, 23, 9919-9920 I suggested the situation when the Wenzel roughness factor can change with the coordinate, Rf(x, y). In the paper
M. Nosonovsky & B. Bhushan, 2008, “Patterned Nonadhesive Surfaces: Superhydrophobicity and Wetting Regime Transitions,” Langmuir, 24, 1525-1533, I further extended this idea for the cases of f1(x, y) and f2(x, y).

The rigorous introduction of the concept requires the definition of three different length scales, since the averaged roughness and heterogeneity parameters, Rf(x, y), f1(x, y), and f2(x, y) should be defined at the length scale larger than the actual roughness details but smaller than the macroscale water droplet. .Wetting involves interactions at different scale levels: macroscale (water droplet size), microscale (surface texture size), and nanoscale (molecular size).

The discussion in my comment is also related to the range of applicability of the Wenzel and Cassie-Baxter equations and to whether wetting is a 1D or 2D phenomenon. This is what I wrote:

However, it is still not clear under what circumstances these equations are valid. This question is related to the old discussion of whether the wetting of superhydrophobic surfaces is a 1D process, which is determined by thermodynamic equations for the free surface energies, or a 2D process, which is more adequately described by the kinetics of the triple line (the solid-liquid-vapor contact line) and surface tension. A large body of evidence has been presented for support of the validity of the Wenzel and Cassie equations based upon the energy approach as well as evidence that support the line tension approach. Gao and McCarthy showed experimentally that the contact angle of a droplet is defined by the triple line and does not depend upon the roughness under the bulk of the droplet. A similar result for chemically heterogeneous surfaces was obtained by Extrand…

Forces in physics are defined as derivatives of the energy by corresponding generalized coordinates, so the question of whether the forces or the energy governs a physical phenomenon is similar to the chicken and egg problem. To be more precise, there is a similarity with the classical Zeno’s paradox of the arrow (whether the moving material point is a 1D or 0D object).

UPDATE. The statement about “the classical Zeno’s paradox” requires a clarification. Today we all know that velocity is a vector. Vector is a 3D object (or 1D in the case of the 1D motion along a path). However, until Newton and Leibnitz, the nature of the velocity was not well understood because the concept of the derivative was unknown. This is the reason for Zeno’s paradox: you cannot explain the motion, if you are thinking about it in terms of the position points (which are 0D objects) rather than in terms of the velocity. In a somewhat similar manner, you can think in terms of the energy or in terms of the forces (which are derivatives of the potential energy). When you deal with the surface tension forces, you only consider the 1D contact line, however when you are thinking in terms of the surface energy, you deal with the 2D area. Surface tension force is measured in N/m (i.e., force per unit length), whereas the surface energy is in J/m2 (i.e., energy per unit area). These two quantities are very similar (although not always identical). Please see also the discussion at “Is surface tension a real force? A new geometric interpretation of surface tension for the superhydrophobicity”