Ice Adhesion. Группа авторов

convex, and (b) concave in shape. The geometry of heterogeneous nucleation of the spherical cap is shown superimposed for both cases. Adapted from [75, 76].

(1.5)

      Similarly, drawing a radial line, re, from the centre of the ice embryo to point A yields an angle ψ between re and an extended line d which allows for the calculation of the ice-water (parent phase) interfacial area, SAIW.

      (1.6)

Angles ϕ and ψ can also be used to calculate the volume of the ice embryo spherical cap, as in Equation 1.7.

(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)

(1.11)

The equation for the critical ice nucleus radius, for the heterogeneous case is equivalent to that of the homogeneous case, Equation 1.2. This is necessarily true because the embryo surface in its entirety must be in equilibrium with the (water) parent phase [75]. The free energy barrier for the heterogeneous formation of a stable ice embryo can thus be found by combining Equation 1.2 with Equations 1.4-1.11.

(1.12)

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.

(1.13)

      where and m = cos(θIW). Note that as x approaches 0, fconvex approaches unity and Equation 1.12 agrees exactly with that for the homogeneous nucleation case.

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].

(1.14)

      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 agreeing with the intuitive assessment that ice nucleates more readily on a foreign solid surface. Equations 1.12-1.14 show that the free energy for the heterogeneous formation of a stable ice embryo is maximized for surfaces which:

      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 should be avoided in order to maximize fconcave (and thus ∆G*). One should strive to reach this limit of for the nanoconcavities of the engineered surface in order to maximize the occurrence of the desired nanospikes. In the case of the 1.5 nm critical nuclei radius calculated for -25°C supercooling, a roughness radius of curvature for the nanoconcavities of 15 nm is desired. Such a hypothetical surface for the suppression of ice nucleation is shown in Figure 1.6.