Ice Adhesion. Группа авторов
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 The dependence of the Gibbs free energy change ΔG on the nucleus radius r. The blue curve has a maximum free energy ΔG* at a critical nucleus radius r*, which defines the critical nucleation barrier.
2.2.1 Homogeneous Nucleation Rate
The classical nucleation theory provides a rate equation for the emerging embryos of a new phase. The standard form of homogeneous nucleation rate J can be written as,
where J0 is a pre-exponential factor, which depends on the rate at which molecules strike a unit area of the embryo surface. For homogeneous nucleation of water, including condensation and icing, J0 can be expressed as,
in which, csat,2 is the concentration of water molecules at saturation with respect to the new phase, w↓ is the flux of water molecules to the nucleus surface.
Despite its simplicity, the classical nucleation theory provides a remarkably good prediction for most materials in a certain range of temperatures. Though the theoretical temperature dependence of J is usually higher than that found in experiments, various correction functions have been developed to modify the classical model and bring it into better agreement with the experimental observations [89-91].
2.2.1.1 Homogeneous Nucleation of Water Droplets and Ice from Vapor
Homogeneous nucleation of water droplets in supersaturated vapor may be the most fundamental example. The pre-exponential factor in the rate equation for homogeneous nucleation of water was developed by Becker and Doring in 1935 [4] as
using csat,w = psat,w/kBT,
v As indicated in Eqs. 2.3 to 2.6, J is very sensitive to the supersaturation since the exponential term varies with S-2. This result is clearly revealed by the numerical evaluation of homogeneous nucleation rate J for droplets in water vapor (see Rows 1 and 2 in Table 2.1 [7]). For the nucleation of water vapor at –12°C, J (water) increases 5 orders of magnitude as Sv,w increases from 5 to 6. The ascending nucleation rate indicates a critical supersaturation Sv,w(cr) at which the droplets suddenly emerge from the water vapor. From an experimental point of view, Sv,w(cr) are normally defined to correspond to J = 1 cm-3 sec-1 in most instances. Considering that the required critical supersaturation is extremely high, the homogeneous nucleation of water droplets can only be realized under laboratory conditions and does not occur in the atmosphere.
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