Wenzel s equation

Wentworth process, for ethanol separation from water, 8 834-835 Wenzel s equation, 22 111-112 Wenzel contact angle, 22 111 Werner nomenclature scheme, 17 391-392 Werner-Pfleiderer (WP) Compounder, 22 44 Werner-type inclusion compounds, 14 171-172... 

Wenzel s relation has been confirmed in terms of the first two laws of thermodynamics. Huh and Mason, in 1976, used a perturbation method for solving the Young-Laplace equation while applying Wenzel s equation to the surface texture. Their results can be reduced to Wenzel s equation for random roughness of small amplitude. They assume that hysteresis was caused by nonisotropic equilibrium positions of the three-phase contact line, and its movement was predicted to occur in jumps. On the other hand, in 1966, Timmons and Zisman attributed hysteresis to microporosity of solids, because they found that hysteresis was dependent on the size of the liquid molecules or associated cluster of molecules (like water behaves as an associated cluster of six molecules). 

From these assumptions and the condition that dF = 0 for a minimum, Wenzel s equation,... 

Approximation 1 is not true, but it does not lead to a serious error when the roughness is small compared with the drop size. Condition 2 is not true except when the differential of A corresponds to exactly one period of the surface. Condition 3 demands that the surface be noncomposite. Condition 4 is true only in the absence of a gravitational field. (This is an indication, but not proof that Wenzel s equation may not be valid in a gravitational field.)... 

Equation of Cassie and Baxter. Cassie and Baxter [2] extended Wenzel s treatment to composite surfaces. The same assumptions necessary for the derivation of Wenzel s equation are necessary, except that Equation 66 is replaced by 70, and Equation 68 is replaced by 71. These two changes are direct consequences of the difference between the geometry of composite and noncomposite surfaces. 

Maximum and minimum possible angles Most probable contact angles, 0 Lower. Calculated from Wenzel s equation Upper. Calculated from Cassie and Baxter s equation Possible curve of receding angles Possible curve of advancing angles... 

As has been pointed out several times, the surface of paper is far from being a smooth ideal surface. This has great consequences on the wetting behaviour that is observed. For random rough surfaces, the influence of surface roughness can be described by using Wenzel s equation ... 

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... 

Grzelak EM, Errington JR (2010) Nanoscale limit to the applicability of Wenzel s equation. Langmuir 26(16) 13297-13304... 

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... 

Wenzel s equation applies to what is called homogeneous wetting, and it can only be applied to homogeneous, rough surfaces. Surfaces having heterogeneous character can be modeled with the Cassie-Baxter model [43] ... 

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]. 

This equation is the Wenzel s equation with a roughness parameter... 

Figure 10.1 Schematic of a drop partially wetting (a] a smooth, defect-free solid (Young s equation], (b] a homogenous, rough surface (Wenzel s equation], and (c] a heterogeneous, rough surface (the Cassie-Baxter equation],...

Thus, employing Wenzel s equation, the Young equation can be revised using the experimental contact angle. In this case, the roughness factor must be known. Calculation of this factor from the topography of the surface is possible but rather controversial. 

Two equations due respectively to Wenzel and to Cassie and Baxter 0 are frequently quoted to determine the contact angle on a rough and on a heterogenous surface. Both equations can be derived thermodynamically (e.g. reference 10) by considering average values of the surface tensions. Wenzel s equation relates the observed contact angle, 0, measured in relation to the mean surface, to the roughness ratio r of the surface (true area divided by projected area) ... 

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