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The original Classical Nucleation Theory developed in the 1920’s and 1930’s considered only cases where a new thermodynamic phase was formed within a homogeneous solution. It is known, even intuitively, that nucleation is more likely to occur at a heterogeneous solid-liquid interface than within the homogeneous solution. Thus, it is imperative that the free energy barrier, ∆G*, for the heterogeneous case be determined if one wishes to thermodynamically design anti-icing surfaces. Fletcher extended the Classical Nucleation Theory to the heterogeneous case in 1958 by noting that a foreign solid surface introduces a low-energy solid-solid interface between the forming ice embryo and foreign surface. His work considered the case where an embryo of radius re is forming on a convex nucleating particle of radius Rs within a larger (liquid) parent phase, as in Figure 1.4(a) [75]. The free energy of formation of such an embryo is given by Equation 1.4.

(1.4)

where VI is the volume of the spherical cap of ice embryo. SAIW and SAIS are the areas of the interfaces between the ice embryo and the water, and the ice embryo and the solid nucleator, respectively. γIS and γSW are the interfacial energies between the ice and solid nucleator, and solid nucleator and water, respectively.

The geometry of a spherical cap of ice growing on a convex nucleating surface is shown superimposed upon the cross section of such a system in Figure 1.4(a). The interface between the ice embryo and the solid nucleator is known to possess a disordered, quasi-liquid layer of water molecules [77]. The spherical cap ice embryo, the quasi-liquid layer, and the liquid parent phase meet at a triple phase contact line (point A in Figure 1.4(a)), forming an ice-water contact angle, θ . As the thickness of the quasi-liquid layer is a function of temperature,IWso too is θ[77, 78].

IW The surface areas and volume of the spherical cap in Equation 1.4 can be expressed as functions of θIW, Rs, and re through a geometric analysis. Let d be the distance from the centre of the spherical nucleator, Cs, to the centre of the spherical ice embryo cap, Ce. Drawing a radial line, Rs, from the centre of the nucleator to point A (the triple phase contact line) yields an angle ϕ between Rs and d which allows for the calculation of the ice-solid interfacial area, SAIS.

Figure 1.4 Heterogeneous nucleation of an embryo growing on a foreign solid surface which is (a) 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.

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].