between the two positions is the molar Gibbs free energy for activating the diffusion of water molecules across the water-ice embryo interface. This energy barrier ∆F therefore leads to the expression for the diffusive flux,
(2.9)
where h is Planck constant and Nc is the number of water molecules in contact with a unit area of ice embryo surface. NA is the Avogadro constant. By replacing w↓ in Eq. 2.7 with
(2.10)
The homogeneous ice nucleation in supercooled water droplets is regarded as the controlling mechanism for the cirrus cloud formation in the atmosphere. In the past decades, the ice nucleus has been considered to form in the volume of supercooled droplets. However, recent studies reveal that the homogeneous ice nucleation initiating from the droplet surface should not be disregarded [9-11]. Tabazadeh et al. suggest that the total ice nucleation rate Jtotal in atmosphere should consist of both volume (Jv) and surface based rate (Js), Jtotal = JvV + JsS, where V and S are the volume and surface area of supercooled droplet in a unit volume of air, respectively [9]. It suggests that if the nucleation energy barrier at the droplet surface is significantly lower than that in the volume, the surface nucleation is favored over volume nucleation. Such competition between surface and volume nucleation also appears in scenarios of droplet icing on surfaces [31]. Jung et al. experimentally validated that the evaporative cooling of a supercooled droplet induced the homogeneous ice nucleation at the air-droplet interface rather than the expected heterogeneous nucleation at the solid surface [25] (see Figure 2.2). Similarly, Ehre et al. also reported that a positively charged pyroelectric surface initiated the heterogeneous ice nucleation at the solid-water interface of a deposited droplet, whereas a negatively charged surface can trigger the homogeneous ice nucleation at the air-droplet interface [72].
Figure 2.2 Snapshots (at 6ms interval) showing the ice nucleation of a 5µl droplet on a superhydrophobic surface in an unsaturated nitrogen flow field (N2 flow direction from left to right). The origin and direction of white arrows denote the instantaneous position and direction of motion of the freezing front, respectively. Figure is reprinted with permission from [25].
2.2.2 Heterogeneous Nucleation Rate
In daily life situations, phase transitions will follow a heterogeneous nucleation scheme. It can occur for any phase transition between two phases of vapor, liquid, or solid, such as condensation of vapor, bubble formation from liquid, solidification from liquid, etc. Heterogeneous nucleation originates from preferential sites of a third phase, for example, surfaces of a container or at impurities like dust. The interfacial energy of the new phase with these preferential sites is typically significantly lower than its surface energy relative to the original phase, thus diminishing the free energy barrier and facilitating nucleation.
2.2.2.1 Heterogeneous Water Nucleation on Solid Surfaces
Distinct from the homogeneous droplet formation that needs extremely high supersaturations (see Table 2.1), solid surfaces are capable of initiating water nucleation at a low supersaturation, which is normally below 10% or even below 1%. To predict the water nucleation rate on a water-insoluble solid surface, it is convenient to assume the water embryo nucleates on a surface with the shape of a spherical cap, as shown in Figure 2.3. By adopting the classical nucleation theory, the water nucleation barrier on a solid surface is expressed as,
f is the geometry factor which depends on the wetting feature (i.e., water contact angle) and curvature of solid surface roughness. For an ideal planar surface with intrinsic contact angle θw of the water droplet (see Figure 2.3a), we obtain
(2.12)
Figure 2.3 Schematic showing the volume and surface area of a spherical cap water nucleus on (a) a flat, partially wettable surface, (b) a convex, partially wettable surface, (c) a concave, partially wettable surface, and (d) a soft partially wettable surface.
where m = cos θw = (σs,w – σs,v)/σw,v is the wetting coefficient at the solid-water interface.
For a convex surface with curvature Rc (e.g., nanobumps shown in Figure 2.3b), we obtain
(2.13)
where x = Rc/r* represents the size ratio of solid surface to the embryo, and g = (1 - 2mx + x2)1/2.
For a concave surface with curvature Rc (e.g., nanopits shown in Figure 2.3c), we obtain
where x = Rc/r* and g = (1 + 2mx + x2)1/2.
When water vapor nucleates on solid elastic surfaces, however, the water nucleation barrier differs significantly from that of hard surfaces [93-95]. As shown in Figure 2.3d, for the droplet nucleating on a soft film, the surface tension of condensed water can pull the film surface upward along the periphery of the droplet. The Laplace pressure, meanwhile, compresses the soft film underneath the condensed droplet, forming a wetting ridge surrounding the droplet. Such deformation of soft surfaces apparently reduces the water nucleation barrier compared with the rigid surfaces. More investigations are still needed to quantitatively explore the effects of soft surface properties (e.g., viscosity) on the wetting feature of droplet embryos and associated water nucleation behavior.
Based on the prediction of nucleation barrier ∆G*,
