general, the classical nucleation theory is applicable for predicting the ice nucleation rate in a supercooled droplet [27]. However, an issue arises when seeking the value of intrinsic contact angle θi,s at ice nucleus and solid interface for calculating the ice nucleation barrier. Because of the difficulty in direct observation, θi,s has not been measurable in a macroscopic way up to now. In past few years, several experiments have indicated the presence of a thin, quasi-liquid layer on ice in contact with various surfaces [99-104]. By assuming such quasi-liquid layer existing between the ice nucleus and solid surface, Eberle et al. [47] extrapolated a modified Young’s equation to estimate the contact angle of ice embryo on quasi-liquid layer for a rough surface (see Figure 2.8),
where σi,ql, σs,ql, σs,i and σs,w denote the interfacial free energies between the ice and quasi-liquid layer, the surface and the quasi-liquid layer, the surface and ice, and the surface and liquid, respectively. d and ξ denote the thickness of quasi-liquid layer and a characteristic decay length of the interaction with bulk liquid [105].
Figure 2.8 Schematic showing the heterogeneous nucleation of an ice embryo with interfacial quasi-liquid layer in a nanoscale cavity. Figure is reprinted with permission from [47].
In principle, the effect of surface geometry on the nucleus formation (Eqs. 2.11 to 2.14) is applicable to the surface structures at all length-scales, yet particular phenomena arise at the molecular level. A recent Monte Carlo simulation showed that a convex surface can activate the surface nucleation only if the radius of curvature exceeds a minimum size [83]. Figure 2.9a shows the estimated nucleation barrier on spherical seeds with various radii. When the seed radius Rs ≤ 4σ (σ is the Lennard-Jones molecular diameter), the nuclei typically form in the bulk instead of on the seed surface. The surface nucleation starts with a spherical seed with radius Rs ≃5σ . As can be seen from Figure 2.9b, very small clusters grow on the seed surface, which tend to grow outward radially due to the large surface curvature. Before reaching the critical size, the clusters detach from the surface and move into the bulk vapor phase. In this case, the critical nucleus only forms in the bulk vapor phase, which is similar to the process of homogeneous nucleation. Although the detachment of nuclei clearly affects the nucleation process, it hardly affects the nucleation barrier (see Figure 2.9a). In comparison, larger seeds (Rs > 5σ) do lower the barrier and speed up the nucleation process.
Investigations of heterogeneous ice nucleation in nano-confinements help to broaden our horizon of nucleation behaviors [106–112]. Suzuki et al. experimentally reported that the ice nucleation mechanism could be precisely regulated by confinement within nanoporous alumina [106]. When supercooled water freezes inside a nanoporous aluminum oxide membrane with pore diameters ≤ 35 nm, the heterogeneous nucleation of hexagonal ice (Ih) is evidently suppressed. Instead, the homogeneous nucleation of cubic ice (Ic) dominates the water crystallization in the nanopores. Such transition of nucleation mechanism can be understood by comparing the critical ice nucleus radius r* with the pore diameter d. That is, only when r* < d is the associated crystalline phase stable within the nanoporous materials. Using numerical method, Koga et al. also demonstrated that water encapsulated in carbon nanotubes could form various new phases of ice which were not seen in the bulk configuration [110]. Using carbon nanotubes with diameters ranging from 1.1 ~ 1.4 nm and applied axial pressures of 50 ~ 500 MPa, confined liquid water can freeze to hexagonal and heptagonal ice nanotubes. The results suggest that the water structure modification imposed by the solid surface can play an important role in the heterogeneous ice nucleation mechanism.
Figure 2.9 (a) Free energy barriers for nucleation in a system of hard spheres with a smooth spherical seed. The seed has radii Rs = 5σ, 6σ and 7σ. The dashed curve represents the homogeneous nucleation barrier (Rs = 0σ). (b) Snapshots showing the nucleation process on spherical seeds. The seeds have radii Rs = 5σ (top) and 7σ (bottom). Parts (a) and (b) are reprinted with permission from [83].
To date, state of the art experimental techniques still do not have the spatial and temporal resolution to investigate the nanoscopic process of ice nucleation. However, the size-dependent droplet icing phenomenon reported by Hou et al. [67] offers another simple approach to analyze the ice nucleus formation on a solid surface. As expressed in Eq. 2.18, the heterogeneous ice nucleation probability is primarily a function of water temperature T(t). For a supercooled droplet condensed on a cold surface, the temperature at the droplet base (i.e., solid-water interface) is lower than at the droplet top due to the thermal resistance associated with droplet heat conduction [23, 24, 86]. Figure 2.10a shows the thermal resistance network of a partial-Wenzel droplet on a superhydrophobic surface and a hybrid-wetting droplet on a biphilic surface with hybrid wettability. If assuming the temperature of saturated vapor and rear-side of substrate is constant, the droplet base temperature (Tb) will keep on decreasing with the growing droplet size and increasing thermal conduction resistance. As a result, the ice nucleation rate and nucleation probability of supercooled droplet will increase with the growing droplet diameter, as shown in Figure 2.10b. This implies that during supercooled condensation on a flat surface, the ice embryo preferentially nucleates at the base of large droplets instead of smaller ones. Moreover, for the supercooled condensation on a liquid-repellent rough surface, partial-wetting droplet morphology is more likely to trigger the ice nucleation as compared to the suspended droplet morphology. The capillary liquid-bridges underneath PW droplet lead to not only a larger area of solid-water interface, but also a lower interface temperature due to contact with the bottom of surface structures.
Due to the exponential increase of J(T) at lower temperature, the ice nucleation rate can be estimated as J(T) = J(T0 + ∆T) = α exp(–λ∆T) in a narrow temperature interval ∆T = T–T0 around a reference temperature T0, where α = J (T0) and
