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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,

(2.11)

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 θ_w 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 θ_w = (σ_s,w – σ_s,v)/σ_w,v is the wetting coefficient at the solid-water interface.

For a convex surface with curvature R_c (e.g., nanobumps shown in Figure 2.3b), we obtain

(2.13)

where x = R_c/r^* represents the size ratio of solid surface to the embryo, and g = (1 - 2mx + x²)^1/2.

For a concave surface with curvature R_c (e.g., nanopits shown in Figure 2.3c), we obtain

(2.14)

where x = R_c/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^*, the rate of heterogeneous water nucleation [m^-2s^-1] on a solid surface is given by,

(2.15)

where is the concentration of single molecules adsorbed on the surface, is the molecular mass of water, h_v is the latent heat of vaporization per molecule, υ is the vibration frequency of an adsorbed molecule normal to the surface (~10¹³ sec^-1). A correction factor, Zeldovich factor Z_s, denotes the probability that a nucleus of critical size will continue to grow instead of dissolving.

Computational results based on Eqs. 2.11 to 2.15 show that themicro and sub-microscale concave structures greatly accelerate the water nucleation, whereas the convex structures suppress the formation of water embryo. Xu et al. [59] quantitatively predicted the heterogeneous water nucleation rate J on different microstructures, including flat surface, conical apex and conical cavity, as shown in Figure 2.4. The computational results indicate that the nucleation energy barrier dramatically decreases when the surface morphologies transition from apexes to cavities. Although the kinetic pre-factor J₀ declines with ascending β of surface architecture, the heterogeneous water nucleation rate within the cavities is obviously higher than that on a planar surface and apexes. This means that during the condensation on a rough surface, water nucleates preferentially in the concave structures instead of the convex and flat areas.

It is worth to mention that the phenomenon of capillary condensation may further promote the water nucleation process in the nanoscale concave structures, as described by Kelvin equation,

(2.16)

where p is the equilibrium vapor pressure at the water embryo surface, D_c is the diameter of concave surface structure (e.g., pore, cavity, etc.), n_w is the number of water molecules per unit volume at the liquid phase. For a hydrophilic surface with θ_w < 90°, p will be smaller than the bulk saturation vapor pressure (p_sat,w), leading to water nucleation in cavity even in under-saturated conditions. A recent experimental study of capillary condensation revealed that at molecular scale, water with high negative (positive) curvature has surface tension higher (lower) than that of the bulk phase [96]. This implies that the nucleation of water molecules, or the critical water nucleus formation, is more prevalent than expected.

Figure 2.4 (a) Schematic showing an axisymmetric water nucleus on the microscale circular conical apex (β < 180°), flat (β = 180°), and microscale circular conical cavity (β > 180°). (b) ΔG*, J₀ and J as function of β (vapor pressure p_v = 100 kPa, supersaturation S = 1.5, water contact angle θ_w = 90°). (c) Water nucleation rate as a function of water contact angle α and structure parameter β* (p_v = 100 kPa, S = 1.5). Figure is reprinted with permission from [59].

The nucleation rate J also determines the number density of condensed droplets on a solid surface. If we assume that the nucleation events can be described in terms of a nonhomogeneous Poisson process [68, 85, 97], the probability that one water embryo nucleates at time t on a flat surface with homogeneous wettability is given by,

(2.17)

in which, A is the surface area of vapor-solid interface, the surface temperature variation T(t′) is regarded as a monotonically decreasing function of time i.e., T(t′) <0. The surface temperature first reaches the saturation point (T = T_sat) at t = t₀, and further decreases to the subcooled condition, T < T_sat. Thus, Eq. 2.17 can be readily expressed as a function of the surface temperature T(t′),

(2.18)

where T(t₀) = T_sat, T(t) = T_s, and C(T′) = –dT′ / dt′ represent the surface cooling rate as a function of surface temperature T. Rearranging the equation, we can obtain the critical surface area A* to form one water nucleus,

(2.19)

Because the spatial distribution of nucleation sites was in good agreement with the Poisson distribution [33], the mean nearest neighbor distance of water nuclei (L_N) can be expressed as,

(2.20)

where N = 1/A^* is the initial nucleation density of water embryos on a condensing surface. Eqs. 2.17 to 2.20 provide a fresh view on the heterogeneous nucleation process on surfaces, yet more experimental investigations are still required to provide direct evidence for validating the predictions of this models for the nucleation density on surface.