Ice Adhesion. Группа авторов
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].
3.2 Ice Growth
Although ice nucleation is governed by the thermodynamics of ice-water-surface system, further ice growth is controlled by heat transfer. As ice nucleation occurs, release of freezing enthalpy leads to the temperature increase at water-ice interface. In fact, heat transfer at water-ice interface controls ice growth rate. Here, in order to obtain a theory of ice growth, it is assumed that water-ice interface is flat and the curvature at this interface is ignored. Considering this assumption, Gibbs-Thomson undercooling effect becomes negligible. Gibbs-Thomson undercooling effect is the effect of ice-water interface curvature on the temperature of freezing front. This effect causes the temperature of freezing front to be different from equilibrium melting temperature [20, 21]. We consider that the temperature of freezing front stays at equilibrium temperature of Tf. In case one considers the undercooling effect (ΔT), the equilibrium temperature at water-ice interface should be replaced by Tf – ΔT. Also, airflow around the droplet is a parameter that should be taken into account. To consider airflow effect, two extreme cases of ice growth are defined. The first one is a droplet in an environment without airflow and the second one is a droplet in an environment with airflow surrounding the droplet.
3.2.1 Scenario I: Droplet in an Environment without Airflow
In this scenario, tip singularity formation is a common phenomenon that occurs during ice growth. In this phenomenon, when a droplet is placed on a cold plate, it freezes and turns to an ice drop with a pointy tip (Figure 3.5). Tip singularity formation is mainly due to the water expansion after freezing and is governed by the quasi-steady heat transfer at the later stages of ice formation. Marín et al. [21] stated that the freezing front is convex at earlier stages of ice growth and at the final stages it becomes concave. They also reported that the freezing front is almost perpendicular to the ice-air interface, i.e. γ = ϕ + θ ≈ 90° (see Figure 3.5), due to the fact that latent heat cannot transfer across the solid-air interface due to low thermal conductivity of air. The shape of solid-liquid front is obtained through the assumption of constant front temperature at the equilibrium melting temperature, Tf , i.e. neglecting Gibbs-Thomson effect.
Figure 3.5 Geometry of droplet at later stages of ice formation and after complete ice formation. θ denotes the angle between liquid-air interface and horizontal, ϕ denotes the angle between freezing front and horizontal and α denotes the final tip angle [21].
For obtaining geometric theory for tip formation, the first step is to write mass conservation with respect to z, as temporal dynamics is not significant.
Where Vl and Vs denote liquid and solid volumes, respectively, v is density ratio, and z is height of trijunction (Figure 3.5).
The liquid at the top of freezing front is divided into two parts. The upper part is like a spherical cap with angle of θ and the lower part has a volume of Vd. Thus,
(3.17)
(3.18)
Considering the geometries of upper and lower parts of the liquid, one finds,
(3.19)
Based on (Eqs. 3.16–3.20) and the fact that
Eq. (3.21) can give volumes of liquid part before and after freezing. If this equation is multiplied by r3, the left side gives the volume of liquid. Also, the right side gives the volume of this liquid when it is frozen where expansion factor is considered. According to Eq. (3.21), we have α = π–2θ regardless of v value and as mentioned before γ ≈ 90°. Therefore, from Eq. (3.21), a constant value of α = 131° for the tip angle is obtained which is in a great harmony with the experimental results [21].
Now, we determine the growth rate of ice in scenario one (Figure 3.6). In this case, as the thermal conductivity of air is low, convective heat transfer is low and the generated enthalpy of phase-change is transferred through the ice by thermalconduction mechanism and subsequently, through the substrate,
Where l denotes temporal height and r denotes the radius of freezingfront. The heat transfer away from the interface to the substrate,