associated droplet base temperature for a substrate temperature Tsub equal to 267.75 K."/>
Figure 2.10 (a) Schematics of the thermal resistance networks of partial-Wenzel and hybrid-wetting droplets, indicating the resistances of the droplet curvature (Rc), water-vapor interface (Rv,w), and conduction resistances of droplet (Rd), hydrophobic coating (Rhc), liquid bridges (Rl), micropillars (Rm), nanopillars (Rn), and substrate (Rsub). The red dashed line represents the liquid-solid phase boundary of the supercooled droplet where the liquid water tends to freeze into ice. (b) Correlations of the freezing probability and droplet radius with the associated droplet base temperature for a substrate temperature Tsub = 267.75 K. Both reprinted with permission from [67].
where Tm is the melting temperature of ice (273.15K), Tb and A are the temperature and contact area of solid-water interface, respectively. Tb can be determined by the heat transfer model of a condensed droplet,
in which, Tsat – Tsub is the temperature difference between the saturated vapor and substrate (with a constant temperature), and Md, Mi, and Msub are the thermal insulances (reciprocal of heat transfer coefficient) associated with the droplet conduction, interface wetting morphology, and substrate.
For a condensed droplet, the conduction insulance Md = rθw/(4kw sin θw) depends on the droplet radius r, where kw is the thermal conductivity of water. Substituting the expression of Tb and Md into Eq. 2.22, the characteristic droplet freezing radius rf is given as,
where
2.3 Prospects
Despite significant improvements in experimental techniques over the past years, it remains challenging to characterize and control the nucleation dynamics of water and ice in an accurate manner. The recent improvements in simulation methods provide complementary approaches to study the water nucleation process in various idealized environments. It enriches our understanding of the kinetic evolution of nucleating embryos, as well as their volume and interfacial structures. However, the limited computational domain and timescale greatly restrict the application of simulations for real systems. We believe many opportunities exist for the further experimental exploration of nucleation at the very beginning stage by using higher resolution or three-dimensional viewing techniques (e.g., surface plasmons, confocal microscopy, etc.). New findings of nucleus properties and interfacial phenomena will lead to new approaches to control the nucleation process on solid surfaces, not only for water but also for other low surface tension liquids.
Figure 2.11 Critical droplet freezing radius as a function of the substrate temperature. The droplet freezing induced by frost propagation is excluded from the experimental statistics due to the different ice nucleation mechanism (Tsat = 298.05 ± 0.5 K). Figure is reprinted with permission from [67].
As mentioned above, the recent study of nanoengineered surfaces with hybrid wetting features suggests an increased interest in the spatial control of droplet condensation for anti-freezing. However, a long-term manipulation of droplet and ice nucleation has proven to be exceedingly difficult. Since the continuous phase change of water involves the nucleation at nanoscale and liquid transport at macroscale, new work in this area should emphasize attaining an optimized surface topography which can improve nucleation and water transport simultaneously. Although the current implementation of surface structures and chemistries encounters a number of degradation issues, fundamental studies of key interfacial phenomena should lead to a significant improvement in the condensation and anti-icing performance and might pave the way for long term durability.
2.4 Summary
In this chapter, the theoretical modeling framework of classical nucleation theory has been outlined. The fundamentals are applicable for predicting both water and ice nucleation processes due to the same controlling mechanism of embryo formation. In simple terms, the phase transition occurs when an embryo (cluster) of new phase forms with a critical size, which is related to a maximum energy of embryo formation as “nucleation barrier”. From classical theory, a temperature-dependent nucleation rate can be expressed:
The model demonstrates the relevant parameters of nucleation, including the nucleation barrier ΔG*, and the dynamical pre-factor J0 which depends on the diffusion coefficient for nucleation.
Classical nucleation theory provides a rational guideline to spatially control the heterogeneous nucleation of water and ice on solid surfaces. By altering the structural topography and intrinsic wettability on surfaces, the location and movement of nucleating water embryos can be manipulated. One example is enabling the condensate to form the suspended Cassie droplet during condensation. Such droplet wetting morphology can enhance the continuous condensation rate, while also suppressing ice nucleation in condensed water when the surface is in a supercooled condition. However, the understanding of interfacial properties for ice nuclei on solid surfaces remains inconclusive. Recent investigations on anti-freeze proteins and ionic surfaces indicate a strong influence of interface effects on the nucleation. These unresolved questions imply that more work needs to be conducted on the exact molecular mechanism underlying the nucleation processes. The answers will direct the future strategies for enhanced condensation and anti-icing applications.
Acknowledgement
