rel="nofollow" href="#ulink_04f0ff79-e675-5f23-abc1-73b0033aa8a4">(1.7)
Equations 1.5-1.7 can be expressed solely through θIW, Rs, and re by noting:
(1.8)
(1.9)
where
(1.10)
The equation for the critical ice nucleus radius,
Comparison of Equation 1.12, for the heterogeneous nucleation case, with Equation 1.3, for the homogeneous case, shows that the energetic difference between the two nucleation schemes lies in the geometric factor, f(θIW, Rs)- In the case of the foreign solid convex nucleating surface, which Fletcher solved in 1958, f(θ, Rs) is given by Equation 1.13.
where
Mahata built on the work of Fletcher by considering the case of a concave solid nucleating surface. A representative rendering of this heterogeneous nucleation case is shown in Figure 1.4(b). The surface areas and volume of the spherical cap of the growing embryo in this particular case were determined through an analogous geometric analysis. The geometry of the concave system is superimposed upon the cross section in Figure 1.4(b). Mahata determined the geometric factor for the concave heterogeneous nucleation case to be Equation 1.14 [76].
where
Further analysis of the geometric factors, fconvex and fconcave, can give insight into why surface roughness has been shown to both increase [79] and decrease [80] the rate of heterogeneous ice formation. Equations 1.13 and 1.14 have been plotted for a range of ice-water contact angles (i.e. temperatures) in Figure 1.5. Note that ƒ < 1 in all cases, and therefore
1 Have a minimized overall surface area, as ∆G* is a function of surface area.
2 Are exclusively decorated with nanoconvexities, as fconvex > fconcave.
3 Are decorated with nanoconvexities with minimized radii of curvature, as fconvex is maximized as Rs → 0.
Naturally, conclusions 1 and 2 are mutually exclusive. One cannot minimize surface area while also exclusively decorating a surface with nanoscale bumps. Thus, a compromise must be reached between decorating a surface with nanoconvex features and minimizing the overall surface area.
Taking conclusion 3 as the starting point for the thermodynamic surface design, one aims for a surface decorated with nanoscale spikes (that is, convex features with Rs → 0). Next, the distance between these nanospikes must be minimized while also minimizing the overall surface area. This can be realized by joining the nanospikes with spherical nanoconcavities. Figure 1.5(b) shows nanoconcavities with radii
Figure 1.5 Geometric factors, ƒ, for heterogeneous ice nucleation free energy barrier calculation on both (a) convex and (b) concave solid surfaces as a function of surface roughness radius of curvature, Rs. Adapted from [78].
1.2.3 Predicting Delays in Ice Nucleation
Often the experimental parameter of interest in anti-icing surface engineering is the rate at which critically-sized embryos of the new thermodynamic
