The Modified Cassie Equation

For the above system, we assume the liquid-vapor interface is given by 

Here /c is a constant, e is the period of the pattern along y-axis. The equation (4.13) represents an interface which differs from the plane x kz only near the bottom surface z = 0. 

---

Suppose that the contact line is almost straight as shown in Figure 4.6. From the modified Cassie equation, it is easy to compute the maximum and minimum effective contact angles for the two configurations. For the left one, the minimum effective contact angle, which is also the receding contact angle, is... 

The two configurations have very different wetting properties, although their area ratios are the same. This is consistent, in principle, with some existing analytical and experimental results [18,19,31]. This example shows clearly how the modified Cassie equation can explain the contact angle hysteresis phenomenon, while the classical Cassies equation cannot do this. 

There are some other very interesting unsolved problems. Notice that we only derived the modified Cassie equation for chemically patterned surfaces. We expect that there is a similar relation for chemically homogeneous rough surfaces. Rigorous verification of the modified Cassie equation is also important. One needs to consider the convergence of local minimizers, which is a difficult mathematical problem and requires further study. 

The modified Cassie-Baxter equation was successfully used to interpret some of the contact angle data reported for heterogeneous surfaces. 

In the present work, we aimed to examine how the structure of a nanosphere array affected the contact angle between a liquid droplet and fluorinated surfaces. Four types of silica nanospheres with various diameters were stacked to form different organized arrays using a self-assembly technique. To clarify the superhydrophobicity, we proposed a modified parameter into the conventional Cassie-Baxter equation. This mathematical model presented in this study can probably shed some light on how the variation of particle size would induce the superhydrophobicity of nanosphere surfaces. 

Equation (15) gives an expression for the apparent contact angles of drops on surfaces with cavities. It is a modified form of the Cassie-Baxter formula. 

Here, r = 470 nm and R = 500 nm. We thus obtain the value of /2 = 0.8013. The 0 value, for the flat silica surface without micro-pores, after modification, is experimentally about 114°. So the CA for our modified micro-porous sample can be estimated to be about 152.5° by Cassie s equation, equation (2), which is in good agreement with our result. 

For the second case, Cassie and Baxter modified Wenzel s equation by introducing the fractions f and /2, where f ccxresponds to the area in contact with the liquid divided by the projected area, and, to the area in contact with the air trapped... 

At first the BET equation was derived from the kinetic considerations analogous to those proposed by Langmuir while deriving the monomolecular adsorption isotherm. First, statistical thermodynamic derivation was carried out by Cassie [123]. Lately, a slightly modified derivation has been proposed by HiU [124-126], Fowler and Guggenheim [127]. 

---