to ice bridging phenomenon which is a result of vapor source-sink behavior due to the vapor pressure gradient between a frozen droplet and adjacent liquid droplets.
Keywords: Ice nucleation, surface factor, Gibbs energy barrier, ice growth, surface free energy, ice bridging
3.1 Ice Nucleation
As a water droplet touches a subzero surface, it starts to freeze and adhere to the surface. Transformation of a water droplet to ice occurs through a two-step process: (1) Ice nucleation and (2) Ice growth.
Ice nucleation temperature, TN, is defined as the nucleation temperature of a sessile water droplet which is placed on a sub-zero surface where the total system of water droplet, surface and surrounding environment is cooled down in a quasi-equilibrium condition [1]. One could measure ice nucleation temperature, TN, through an isothermal chamber filled with inert gas, e.g. N2. The temperature of this chamber is set to 0°C and a surface is placed in the chamber. At this initial temperature, 30 µL of distilled water is placed on the surface. Temperature of the substrate is probed with a thermometer to assure isothermal condition. The chamber is cooled down at a rate of 1°C/min and ice nucleation of the droplet is monitored with a high-speed camera during the experiment. Ice nucleation temperature is obtained by recording the temperature at which sudden transparency change of the droplet occurs. TN is reported as the mean of nucleation temperatures measured during a set of more than 10 experiments [1, 2]. TN is a function of Gibbs energy barrier for heterogeneous ice nucleation which is defined as follows [3]:
In which γIw is interfacial tension of water-ice nucleolus, ΔGυ is the volumetric free energy of phase-change and surface factor, f (m, x), is the parameter that affects Gibbs energy barrier for heterogeneous ice nucleation, varies between 0 and 1, and its value is 1 for homogeneous nucleation. An ice nucleolus is a particle which acts as the nucleus for the formation of ice. The initial embryos of ice are formed from a supercooled mother phase, i.e. water droplet, that transform to ice nucleolus when reach to a critical size, rc. In this section the focus is mainly on f (m, x) which is governed by the interfacial free energy and geometry of the interfaces. In f (m, x), m is a function of interfacial free energies and is defined as:
Where γSW denotes the solid-water interfacial free energy, γSI denotes the solid-ice interfacial free energy and γIW denotes the ice-water interfacial free energy. These interfaces are illustrated in Figure 3.1.
Also, x which is a function of surface geometry is defined as follows:
(3.3)
Where R is radius of features at the surface and rc is the critical nucleolus radius. rc is defined in Eq. (3.4) and its typical value could vary from 1.53 to 4.47 nm for temperature range of -30 to -10°C [3, 4]:
(3.5)
As discussed, f (m, x) equal to 1 indicates homogeneous nucleation limit and f (m, x) equal to 0 indicates ice nucleation without sub-cooling. If m = 1 and x >1, f approaches zero in which case there is no sub-cooling. In order to achieve m = 1, γsw ≥ γsI + γwI should be satisfied. If m = 1 and x < 1, then 0 < f < 1. In this condition, suppression of ice nucleation which is a result
Figure 3.1 Ice nucleolus on a subzero substrate and the involved interfaces are shown. The value of m is equal to cos θ [5].
of nano-scale confinement occurs. f (m, x) is analytically derived for two types of surfaces. For convex surfaces, f (m, x) is defined as Eq. (3.7) and plotted in Figure 3.2.
(3.8)
Also, for concave surfaces f (m, x) is defined by Eq. (3.9) and plotted in Figure 3.3.
(3.10)
For x values larger than 10, f (m, x) becomes independent of x and only depends on m in contrast to x values less than 10, e.g. when R is of the order of rc, f (m, x) depends on x as well [5].
Ice nucleation on a surface depends on the roughness and structure of the surface, i.e. nano or micro surfaces. For example, for x < 10, ice nucleation on the surface depends on the roughness and structure of the surface, while for x > 10, surface structure has nothing to do with ice nucleation. In this case, ice nucleation only depends on m value, i.e., the interfacial free energies. As an example, nano-grooves on a surface can suppress ice nucleation [6]. Taking all the aforementioned arguments into account, it stands to reason that tuning surface free energy, m parameter, through different mechanisms is a way to increase ice nucleation energy barrier, especially where the geometry of surface does not affect ice nucleation energy barrier.
Figure 3.2 The surface factor plotted versus different values of x and m for convex surfaces to show the effect of surface geometry and surface free energy on ice nucleation. As
