water nucleation [m-2s-1] on a solid surface is given by,
where
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, Dc is the diameter of concave surface structure (e.g., pore, cavity, etc.), nw is the number of water molecules per unit volume at the liquid phase. For a hydrophilic surface with θw < 90°, p will be smaller than the bulk saturation vapor pressure (psat,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 pv = 100 kPa, supersaturation S = 1.5, water contact angle θw = 90°). (c) Water nucleation rate as a function of water contact angle α and structure parameter β* (pv = 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,
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 = Tsat) at t = t0, and further decreases to the subcooled condition, T < Tsat. Thus, Eq. 2.17 can be readily expressed as a function of the surface temperature T(t′),
where T(t0) = Tsat, T(t) = Ts, 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 (LN) can be expressed as,
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.
2.2.3 Spatial Control of Water Nucleation on Nanoengineered Surfaces
The ability to control nucleation is of great importance to many applications involving phase transitions, such as distillation, power generation, crystallization and anti-freezing. As indicated by the models of classical theory, the heterogeneous nucleation rate of single water embryo is dramatically affected by the intrinsic water contact angle and topography of the nucleation site (Eqs. 2.11 to 2.15). Considering the critical water nucleus radius (~10 nm) and nucleation density (~1010 m-2) for atmospheric condensation, the micro/nanoscale surface features can have an important influence on not only the single droplet nucleation process but also on the global condensation dynamics.
The recent development of functional surfaces and experimental techniques has led to exciting improvements in manipulating nucleation behaviors, and better understanding of the underlying physics. In the past decade, much attention has been paid
