Equation Cassie

An interesting application of the Cassie equation (X-27) arises in the case of a screen or woven material where now /2 is the fraction of open area, ys2V is zero and ys2L is simply YLV- The relationship then becomes... 

Why do you think the Cassie equation Eq. X-27 might work better than Eq. X-28 for predicting the contact angle as a function of surface polarity ... 

Fig. 9.10 Comparison of the formation and wetting behavior of the aliphatic HUT/DDT (a,c,e) and the aromatic HMB/MMB (b,d,f) mixed monolayer system on Au(lll). (a,b) Composition of the solution and surface composition of the resulting SAM. (c,d) Plot of the cos 6>= (7sv-7si)/7iv) of the advancing (and additionally in d) receding) water contact angle as a function of the surface OH concentration. The straight line represents the Cassie equation [104], in c) the grey line is calculated after the equation from Israelachvili [105] describing the contact angle on heterogeneous surfaces. (e,f)...

For a chemically heterogeneous surface the same averaging approach directly yields the Cassie equation ... 

In this equation, 9c is the MSAPCA, x is the area fraction of the solid surface that is characterized by a given chemistry i, and 9yi is the YCA for each chemistry i, had it covered the whole surface. This form generalizes the original Cassie equation that was based only on two chemistries. The summation can be done in a form of an integral when the chemical heterogeneity is continuously varying over the surface. As demonstrated by numerical simulations, the Cassie equation also becomes a good approximation when the drop... 

Contact angles for water on partially methylated quartz plates were found to be in close agreement with the Cassie equation by R. Crawford, L, K. Koopel, and J. Ralston Colloids and Surfaces, in press (1987)... 

Similarly, CA measurements were used (in conjunction with the Cassie equation) [136] to estimate the corresponding surface coverages. For conversions of < 50%, FTIR and CA data are in quantitative agreement [ 128]. 

Fig. 11 Arrhenius plots for a NHS - CIO SAM and b PNHSMA. The solid lines correspond to linear least squares fits of the data insets linearized kinetics for different temperatures for SAM and PNHSMA evaluated based on CA measurements linearization according to pseudo first order kinetics of the NHS ester surface coverage data calculated from the corresponding CA data using the Cassie equation). (Adapted with permission from [128], copyright (2003), American Chemical Society)...

According to Wenzel s equation, obviously, high roughness can enhance the hydrophilicity of a native hydrophilic surface. In our study, such porous film is much rougher than relatively flat surface obtained without using template, so this model can explain why our as-synthesized films show superhydrophilicity with a CA about 5°. After modification with a lower surface free energy material, the air can be trapped in such ordered pore array, and hence a composite surface composed of air and pore array is formed. The measurements show that the surface is superhydrophobic with water CA of 154°. In this case, the Cassie equation is applicable. Based on a simplified model, as schematically illustrated in Fig. 6, the relationship... 

The contact angle of a liquid drop on a composite surface consisting of material A and material B can be described by the Cassie equation [13,14] ... 

By contrast, real surfaces can be rough and/or heterogeneous in composition [24]. In the following we will briefly discuss as an illustrative example how the wettability and the contact angles measured will depend on the surface composition. For instance, the surface can be composed by domains of different composition. The effect of a patchy structure on the Young equation was described by Cassie, who proposed the following equation (Cassie equation) for a two component, heterogeneous surface [32] ... 

Keywords Wenzels equation, Cassies equation, contact angle hysteresis,... 

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. 

X. Xu and X. P. Wang, Derivation of the Wenzel and Cassie equations from a phase field model for two phase flow on rough surface. SIAM J. Appl. Math., 70,2929-2941, (2010). 

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