process, taking almost 100 years. Based on Gibbs’ thermodynamic works from the end of 19th century, Volmer and Weber formulated the first kinetic model of nucleation for the vapor-to-liquid phase transition in 1926 [1, 2], which was further developed by Farkas [3], Becker [4], and Zeldovich [5]. Later, Turnbull and Fisher extended this theory to the crystal nucleation from liquid by redefining the kinetic model in the liquid phase [6]. These works constitute the “classical nucleation theory”, which provides a ready explanation for various nucleation processes [7-12]. It was originally developed for the ideal case of homogeneous nucleation, which involves the two phases of one material only. In practical situations, however, one has typically to deal with heterogeneous nucleation, where the presence of a third phase of another material (e.g., aerosol particles and solid surfaces) acts as nucleation site. This opens the way to control the nucleation in phase transitions by designing corresponding surfaces of the third phase. Recently, water condensation [13-24] and icing [25-32] involving heterogeneous nucleation on functional surfaces, has attracted much attention due to its significance in energy and environmental applications. To improve the condensation and anti-icing performance, durable super-hydrophobic [33-45] and super-icephobic surfaces [46-56] are urgently required in multiple industries, such as power generation, water harvesting, aviation, etc. Guided by the classical nucleation theory, considerable progress has been achieved in manipulating heterogeneous nucleation via surface topography [57-61] and chemistries [62-71]. Studies of interfacial effects on water molecules, e.g., surface charge [72-77] and recrystallization inhibitors [78-81], also enrich the understanding of water properties and nucleus formation at interfaces. To date experimental exploration of nucleation embryos remains inadequate, although considerable research effort using numerical investigations has been made [82-87].
In this chapter, classical nucleation theory is introduced and applied to derive nucleation rates for the two cases of homogeneous and heterogeneous nucleations. The results are used for analyzing the initial steps of both water condensation and ice crystallization. On that basis, we subsequently discuss the recent progress in the field of controlling heterogeneous nucleation on solid surfaces for achieving enhanced condensation or ice inhibition.
2.2 Classical Nucleation Theory
Up to now, classical nucleation theory (CNT) is still the theory most widely used to quantitatively describe the kinetics of nucleation. This theory was established based on two major assumptions: (1) the nuclei (i.e., embryos) can be regarded as spherical clusters, which possess the macroscopic densities and surface tensions, and (2) their distribution follows a Boltzmann statistics.
In water vapor there are always clusters of a few molecules which exist due to random agglomeration of molecules. These agglomerates constantly form and decay. When the system is in a metastable, supersaturated state, such clusters should, in principle, be energetically favored because the chemical potential of a molecule within the cluster is lower than in the vapor phase. However, cluster formation also leads to the existence of a liquid-vapor interface, which inevitably leads to an energy penalty – the interfacial energy – for forming this interface. Therefore, whether such a cluster can develop into a nucleation embryo depends on the relative magnitudes of these two energy contributions. The change of Gibbs free energy ∆G for forming a cluster containing n molecules can be expressed as,
(2.1)
in which, Δµ is the change in chemical potential for a molecule on moving from the vapor phase (denoted by subscript 1) to the liquid phase (denoted by subscript 2). A and σ1,2 denote the area of interface and the surface tension between liquid and vapor phases, respectively. σ1,2 A is the energy required to create the interface between condensed and vapor phases.
For simplicity, it is more convenient to express the properties of a cluster in terms of its radius, r. Assuming that the cluster is spherical due to surface tension, its volume is
where
where kB is Boltzmann constant, S1,2 is supersaturation ratio between original phase and new phase, p1 is the actual pressure of original phase, and p2 is the saturation pressure of new phase at temperature T.
The total free energy change ∆G(r) is a balance between by two ‘competitive’ factors, the volume free energy and the interfacial energy due to the formation of new phase, as demonstrated in Eq. 2.2. If the system temperature is below the saturation point T < Ts, i.e., the original phase is supersaturated, the chemical potential difference ∆μ < 0, while the interfacial free energy is always positive. As the volume of cluster scales with r3 and the interfacial area scales with r2, the interfacial free energy dominates ∆G(r) in the limit of small clusters. For a sufficiently large cluster, however, the volumetric term is dominant, leading to a negative ∆G(r) < 0. Therefore, the maximum energy penalty as a function of cluster radius can be readily obtained by solving d∆G(r)/dr = 0, giving the critical radius (r*) of a cluster, which on further growth leads to decrease in ∆G(r) and will then lead to nucleation:
(2.4)
And the critical nucleation barrier ∆G* is given by,
(2.5)
The nucleation process shown in Figure 2.1 can be understood in two thermodynamic regimes: (1) the free molecules in the supersaturated original phase form small clusters. As the clusters grow, ∆G(r) increases (being dominated by the rapid increase in surface energy), implying that the cluster growth or the continuous nucleation in this regime is not thermodynamically favorable, i.e., most of the molecules return back into the original phase. This is why homogeneous nucleation needs high supersaturation or supercooling, making, e.g., possible the existence of supercooled water at 150 K [88]. (2) Once the clusters reach the critical nucleus size of r* and pass the barrier of ∆G(r*), further growth of the nucleus will lead to decrease in ∆G(r). Thus, the further nucleus growth will become thermodynamically favored and will eventually lead to new phase formation in bulk.
Figure 2.1
