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3.1 Ice Nucleation
ОглавлениеAs 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 shown, only at lower x values (<10), f(m,x) depends on x indicating nano-structuring can affect f(m,x) and as a result affect ice nucleation temperature and rate. Magnetic slippery surface (MAGSS) in which a selective ferrofluid is introduced on the surface to tune γSW − γSI, shows low value of m which results in the high value for f (m, x) [5].
Figure 3.3 The surface factor plotted versus different values of x and m for concave surfaces to show the effect of surface geometry and surface free energy on ice nucleation. As shown, only at lower x values (<1), f(m,x) depends on x indicating nano-structuring can affect f(m,x) and as a result affect ice nucleation temperature and rate. Magnetic slippery surface (MAGSS), in which a selective ferrofluid is introduced on the surface to tune (γSW –γSI), shows low value of m which results in the high value for f (m, x) [5].
On the effect of surface free energy on ice nucleation, according to Eq. (3.6), γIW only depends on temperature. Therefore, difference of solid-water interfacial free energy and solid-ice interfacial free energy, (γSW – γSI) is a determining factor in the value of m and as a result in ice nucleation phenomenon (see Eq. (3.2)). One of the widely used approaches in the literature is to reduce solid-air interfacial free energy, γsa. For example, adding functional groups which have high bond dissociation energies, e.g. -CF3 and -CHF2, to a surface leads to the reduction of surface free energy. The lowest possible γsa for a surface is achieved by a monolayer of -CF3 groups on a surface and γsa for such surface is in the range of 6-10 mJ/m2 [7, 8]. Generally, addition of the materials which contain C-F bonds to a surface reduces its surface free energy, γsa. For instance, grafting a surface with a monolayer of perfluorodecyltrichlorosilane (FTDS) reduced γsa and m value of -0.17 is achieved in this case [1]. As another example for implementing this approach, Irajizad and coworkers [2, 9, 10] used the concept of magnetic liquid surfaces and introduced magnetic slippery surfaces (MAGSS) in which a selective ferrofluid is introduced on the surface to tune (γSW – γSI). On such surface, a liquid-liquid interface is formed by a volumetric magnetic force. These MAGSS show low value of m, -0.95, which results in the value of 0.98 for f (m, x). This condition is pretty close to homogeneous ice nucleation limit, i.e. f (m, x) = 1 Thus, manipulating f (m, x) through the modification of surface structure and surface free energy results in an increase in energy barrier of ice nucleation and, as a result, reduction of ice nucleation temperature [5]. The role of f (m, x) in ice nucleation is illustrated in Figure 3.4. In the experiment shown in this figure, different surfaces are coated on a cold tube with a temperature of -30°C. The tube is exposed to water droplets and due to high value of ΔG*, i.e. high f (m, x), MAGSS showed lowest ice nucleation among other surfaces.
Figure 3.4 Different coatings are exposed to water droplets which shows the role of f (m, x) in ice nucleation. As it can be seen ice nucleation is suppressed for MAGSS coating. Slippery liquid infused porous surfaces (SLIPS) are icephobic surfaces which utilize the smooth nature of liquid surface to improve icephobicity [5].
Ice nucleation rate, J(T), which is reciprocal of ice nucleation delay time, τav, is another metric of ice nucleation. τav is defined as the average time required for a supercooled droplet, in equilibrium with its surrounding environment, to nucleate ice phase. In order to measure τav, icephobic coating is initially placed in a cold chamber. Chamber temperature is set to sub-zero temperature and after reaching equilibrium, a water droplet is placed on the coating. At a given temperature, the time required for ice nucleation to occur is recorded and the average time during a set of more than 10 experiments is reported as τav. Nucleation rate is defined as follows [3, 11, 12]:
where kB is Boltzmann constant and K is kinetic constant which is defined as [11]:
(3.12)
In which N denotes number of atomic nucleation sites per unit volume, β denotes the rate of addition of atoms or molecules to the critical nucleus and Z denotes Zeldovich non-equilibrium factor.
Ice nucleation delay time is an important metric in ice nucleation and depends on ΔG*, as is shown in Eq. (3.11), which can be tuned by modification of surface roughness and surface free energy. The other approach to increase τav is to increase hydrophobicity of surfaces. In fact, surfaces with higher water contact angle show higher ice nucleation delay times. Basically, by increasing hydrophobicity, m value is decreased [13]. Many studies conducted on hydrophobic and superhydrophobic surfaces indicate that τav value for such surfaces, especially for superhydrophobic surfaces, is high. For example, Tourkine et al. [14] grafted fluorinated thiols on a rough copper surface. By doing so, they made a superhydrophobic surface with increased ice nucleation delay time [14]. As another example, Alizadeh et al. [15] developed a superhydrophobic surface by grafting tridecafluoro-1,1,2,2-tetrahydrooctyl-trichlorosilane on a nanostructured silicone surface. By doing so, they boosted ice nucleation delay time [15].
We can rewrite J(T) in terms of chemical potential, µ.
(3.13)
Where ρ is the molar density of liquid and Δµ is the chemical potential difference between ice and liquid phase. Δµ depends on temperature and pressure of the system which can be defined as Eq. (3.14) in an isothermal condition:
in which vw is specific volume of liquid and vi is specific volume of ice. Patm is atmospheric pressure and PL is the liquid pressure which is obtained by Laplace equation (Eq. (3.15)). Note that this equation is valid down to few nm scale [16-18].
Where γLv denotes liquid-vapor surface tension and r is the average radius of curvature. In fact, the pressure can have either positive or negative effect on ice nucleation. If the (vw – vi) term, which is the slope of solid-liquid phase change line, is positive, pressure increases ice nucleation rate and if the slope is negative, e.g. for water, pressure reduces ice nucleation rate.
The second term on the R.H.S in Eq. (3.14) is negligible in micro-scale due to high radius of curvature, i.e. the differential between atmospheric and liquid pressure is close to zero (see Eq. (3.15)). In contrast, the second term is significant in nano-scale, due to the low radius of curvature. For example, the limit of ice nucleation of water in macro-scale, -38 to -40°C, shifted to lower temperatures at nano-scale. Thus, nano-confined geometry can suppress ice nucleation [19].
