The Wenzels and Cassies Equations

The roughness-enhanced wetting is described by two famous equations that illustrate the relation between the apparent contact angle 6 and the 

We now compute the total interface energy in a bounded domain (e/j(y e, z / s),l) x (0, e) x (-M, M). Suppose the liquid-vapor interface 

In [27], we proved that if Gy is a constant, when the scale e becomes smaller and smaller, the global minimizers of the energy converge to the function 

This equation is the Wenzel s equation with a roughness parameter 

In addition, our method can also be generalized to the case when Gy varies on the surface. In this case, the apparent contact angle satisfies 

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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). 

Nosonovsky M (2007) On the range of applicability of the Wenzel and Cassie equations. Langmuir 23 9919-9920... 

From the thermodynamic point of view, wetting of a rough solid surface is governed by Young, Wenzel, and Cassie equations that relate the contact angle between liquid and solid to interface free energies and to surface roughness. 

The Young, Wenzel, and Cassie equations wete deduced on the basis of simple semi-empirical considerations. Later investigations put these equations into a thermodynamic framework, however, they also showed that thete is no one single... 

E. Bormashenko, Young, Boruvka-Neumann, Wenzel and Cassie-Baxter equations as the transversality conditions for the variational problem of wetting. Colloids Surfaces 345,163-165, (2009). 

The effect of roughness on the wettability of an idealized sinusoidal surface has been studied with a digital computer. The equations of Wenzel and of Cassie and Baxter are discussed in relation to the model. The heights of the energy barriers between metastable states of a drop are seen to be of utmost importance in determining the magnitude of contact angle hysteresis. 

Note that is the change in energy corresponding to the transition from the Cassie-Baxter state to the Wenzel state. Since all air in both the Cassie-Baxter and Wenzel states is at the ambient conditions, there is no contribution to from the pressure terms in equation (14). The only contribution comes from the surface energy change. 

To probe to the superhydrophobic behavior, Wenzel [18] and Cassie [19] models have placed emphasis on the geometrical structure of solid surfaces which acts as a crucial factor in determining the hydrophobicity. First, a rough material has a higher surface area than a smooth one, which increases its hydrophobicity (Wenzel model) second, air may be trapped in the rough surface, which also enhances the hydrophobicity because the drop is then partially sitting on air (Cassie model) [20, 21]. Typically, the Wenzel equation is a simple model to characterize the influ-... 

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... 

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... 

In the simplest case, the wettability of a solid surface is commonly evaluated by the contact angle given by Young s equation. Then Wenzel proposed a model describing the contact angle on a rough surface, and Cassie proposed an equation for the surface composed of solid and air. The concepts of these equations are outlined in fig. 7.18 [106]. 

Wetting on rough or chemically patterned surfaces has also been studied for a long time. Two well-known equations, the Wenzel s equation [11] and the Cassie s equation [12] are introduced to characterize the macroscopic apparent contact angle in terms of surface roughness and inhomogeneity (see Equations (4.6) and (4.7) in Section 4.3). There have been many works on the derivation and validity of these two equations [10,13-17], despite some controversies on their correctness [18-22]. In particular, they cannot describe the contact angle hysteresis and cannot explain some of the experimental observations [18,23]. Various modifications or variations of these two equations are proposed and studied [10,20,24-26]. 

The Young equation cannot be used directly to explain the effect of surface roughness on the wettability of a material because it is valid only for ideal smooth solid surfaces. There are two wetting models that are proposed when a water droplet sits on rough surfaces, these are the Wenzel model and the Cassie-Baxter model. 

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