defined as the ratio of the actual surface area to the geometric surface area. As seen in Figure 1.7(b), Wenzel assumed a homogeneous state of wetting. Whereas the liquid completely fills the void space created by the roughness. Thus the liquid is in direct contact with, and adheres to, the solid surface while only being in contact with the surrounding air at the periphery [90]. The Wenzel contact angle, θw, is the modified Young’s contact angle, as calculated by Equation 1.33 [90]. It can be seen that roughness tends to increase a surface’s inherent wetting characteristics, as liquid is in contact with more area of the surface. Note that the equality fails to hold for large roughness factors.
Figure 1.7 (a) Water droplet on a theoretically perfectly flat, chemically homogeneous, rigid surface and the definition of Young’s contact angle, (b) Sessile water droplet on a rough surface in the homogeneous (Wenzel) wetting state completely penetrates the surface asperities, (c) The heterogeneous (Cassie-Baxter) wetting state traps air in the surface asperities leading to high contact angles and low solid-liquid contact areas, (d) The thermodynamic work of adhesion, Wa, for Young’s theoretical surface can be approximated by the surface tension of water, γwa, and the water-solid contact angle, θY. Adapted from [89]. (e) Ice frozen from a droplet in the Wenzel wetting state will have increased ice-solid contact area, and thus increased adhesion strength, (f) Ice frozen from a droplet in the Cassie-Baxter wetting state will have decreased ice-solid contact and thus decreased adhesion strength.
Cassie and Baxter (1944) and Cassie (1948) considered the wetting of chemically heterogeneous surfaces through an analysis of the energy expended in forming a unit geometrical area of interface between the solids and liquid. Cassie described a surface, made of n different materials, each with a respective material fraction, ƒi. Each of these materials have their own surface free energies/tensions, such that the weighted sum gives the free energy/tension of the entire system, as in Equation 1.34. These solid-gas and solid-liquid interfacial free energies for the system can be directly inserted into Young’s equation, yielding Equation 1.35 [91, 92].
Johnson and Dettre (1964) postulated that instead of impinging into the roughness, as in the Wenzel wetting state, liquid can rather sit on pockets of air. The actual area in contact with the solid surface is thus reduced, as shown in Figure 1.7(c). This wetting state yields much higher contact angles, regardless of intrinsic wettability of the solid material, and is thus a property of superhydrophobic materials. Applying Equation 1.35 to this two-component (that is, air and some solid) heterogeneous case, one notes that f1= fS and cosθY,1 = cosθY for the solid-liquid fraction. For the air-liquid fraction, f2= ƒA = 1 — fS and cosθY,2 = — 1 (contact angle of water and air is 180°). Incorporating this with the roughness factor, r, results in the Cassie-Baxter Equation, Equation 1.36 [93].
The three different theoretical wetting cases analogously have three different theoretical adhesion characteristics upon freezing. Consider the droplet placed on the theoretical surface which is chemically homogeneous, inert, perfectly flat, and completely rigid, as in Figure 1.7(a) which has frozen into solid ice. The thermodynamic work of adhesion, Wa, of this ice to the theoretical surface (on a per unit area basis) can be calculated as the work required to break the ice-solid (IS) bond and form two new interfaces (IA and SA).
Inserting γSA fromYoung’s relation, Equation 1.32, into Equation 1.37 yields:
Because the surface free energies of ice and water are approximately equivalent, and because we can assume that the water-solid interfacial free energy will also approximately be equivalent to the ice-solid interfacial free energy, the thermodynamic work of adhesion can be calculated by Equation 1.39 [89, 94].
Therefore, the thermodynamic work of ice adhesion can be approximated by the surface tension and intrinsic contact angle of water on the solid surface under study, as shown in Figure 1.7(d).
The Wenzel and Cassie-Baxter wetting states on rigid, chemically homogeneous, inert surfaces will theoretically manifest equivalent thermodynamic work of ice adhesion as Young’s surface (with the same intrinsic contact angle), on a per-area basis. Again, Wa can be approximated by the intrinsic contact angle of water on the surface. However, these three wetting states possess vastly different wetted areas. The Wenzel wetting state is associated with a wetted area greater than the geometric area of the surface, by a factor of r. The result being that the total thermodynamic work of adhesion of ice to a roughened surface where the water was in the Wenzel state upon freezing will be r times greater than that of a chemically equivalent, yet perfectly flat surface. The dislodgement of ice formed from a water droplet in the Wenzel wetting state is shown in Figure 1.7(e), with the ice-solid interface shown in red.
Contrarily, the Cassie-Baxter wetting state is associated high reductions in wetted surface areas compared to the geometric surface area, by a factor of fS. The total thermodynamic work of ice adhesion will
