Ice Adhesion. Группа авторов

ice. Next, we show the extension of the homogeneous case to the more pertinent heterogeneous nucleation scenario of ice forming on a solid surface. In doing so, a method of rational thermodynamic design for anti-icing surfaces is presented.

      1.2.1 Homogeneous Classical Nucleation Theory

The Classical Nucleation Theory models a growing ice embryo of clustered H₂0 molecules as a sphere of radius re, as shown in Figure 1.3(a). The change in free energy of a liquid water system due to spherical ice embryo growth is a result of the competing energy effects of: (i) creating a new lower energy (solid) phase, and (ii) creating a new higher energy (solid-liquid) interface. The balance of these energy terms is given in Equation 1.1.

(1.1)

      where ∆GIW(T) and γIW(T) denote the free energy difference between ice and water (per unit volume) and the interfacial energy between ice and water (per unit area), respectively. The first term in Equation 1.1 quantifies the decrease in free energy of the volume occupied by the ice embryo when the phase changes from liquid to solid, whereas the second term quantifies the increased contribution of the newly formed interface. At low embryo radii, the interfacial energy term dominates. Conversely, at higher ice embryo radii, the volume energy term dominates. The balance between interfacial and volume energy change during ice embryo growth is shown schematically in Figure 1.3(b).

Figure 1.3(b) shows how the balance between interfacial and volume energy change yields a local maximum in Gibbs free energy, ∆G*. Therefore, it is energetically favourable for ice embryos with radii less than some critical radius, , to decay back into the liquid phase, while those with radii greater than are energetically favoured to continue on to the growth phase. The critical ice embryo radius can thus be calculated from Equation 1.2.

      Figure 1.3 (a) The Classical Nucleation Theory models a growing embryo of clustered molecules for the new thermodynamic phase as a sphere of radius re. (b) Schematic representation of the free energy associated with the growth of the embryo due to the interfacial energy and volume energy. ∆G^* and are the free energy barrier for continued nucleus growth and critical nucleus radius, respectively.

(1.2)

      Determination of a critical ice embryo radius for a given temperature therefore necessitates collecting the pertinent thermodynamic data. The free energy difference between ice and water can be calculated using the Gibbs-Helmholtz equation (i.e. ∆GIW = ∆HIW(Tfusion – T)/Tfusion) [72]. Whereas the latent heat of fusion, ∆HIW and the ice-water interfacial energy can be estimated using the empirical relations of Angell et al. (1982), and Hacker and Dorsch (1951), respectively [73, 74]. If, for example, a desired nucleation temperature of -25°C is set, the critical ice embryo radius is calculated to be ≈ 1.7 nm (with ∆HIW = 278 MJ/m³ and γIW = 2.17 mJ/m²).

      The free energy barrier for the homogeneous formation of a stable ice embryo can be found by combining Equations 1.2 and 1.1.

(1.3)

1.2.2 Heterogeneous Classical Nucleation Theory

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