of heat transfer through convection mechanism. In other words, humidity affects convective heat transfer coefficient,
Figure 3.9 The plot of Eq. (3.40) which shows ice growth rate in an environment with external airflow for different environment temperatures and airflow speeds [5].
3.3 Ice Bridging Phenomenon
The theories discussed so far are for a single droplet without interference from other droplets. However, in reality, droplets can affect each other’s nucleation and cause ice bridging phenomenon. Ice bridging is a phenomenon in which suppression of vapor pressure due to freezing of droplets occurs. This creates a water-vapor gradient between ice and neighboring supercooled droplets. This water-vapor gradient leads to formation of an ice bridge between the frozen droplet and the neighboring supercooled droplet [26, 27]. In fact, the ice bridging between frozen droplet and liquid droplet depends on the length competition between liquid droplet diameter, D, and straight-line distance from liquid droplet to the frozen droplet, L, or ice-to-liquid droplet separation. Chen et al. [26] defined bridging parameter, S, as follows:
(3.41)
Smaller S leads to higher possibility of bridging. For example, Chen et al. [26] studied two droplets with the same diameter (D1 = D2 = 28 μm) placed in different distances from a frozen droplet (L1 = 9 μm, L2 =53 μm) as shown in Figure 3.10. They demonstrated that droplet (1) with smaller separation length, i.e. S1 < 1, was frozen after 12 s. In contrast, droplet (2) with larger separation length, i.e. S2 > 1, did not form an ice bridge and disappeared after 34 s [26].
As mentioned, ice bridge forms due to freezing of the vapor which is harvested from the liquid droplet. This vapor is transported due to vapor pressure gradient between ice droplet and neighboring supercooled liquid droplet, even if they have different radii. Thus, vapor pressure around the ice droplet should be lower than the vapor pressure around the liquid droplet to have source-sink behavior and form an ice bridge. In fact, the ice bridge forms when the mass of liquid droplet, ml, is at least equal to the mass of complete ice bridge, mbridge. Otherwise, the entire liquid droplet will evaporate and the formed ice bridge is not connected to the liquid droplet to freeze it. Thus, we will consider the case where ml ≥ mbridge. In this case, ice bridge length, L, reaches its maximum value, Lmax, which is the distance between the edge of the frozen droplet and center of liquid droplet. Here, we assume that the frozen and the liquid droplets are identical and the mass of liquid droplet and ice bridge scales as:
Figure 3.10 Ice bridging phenomenon for two similar liquid droplets with different ice-to-liquid droplet separation lengths [26].
Where d denotes initial diameter of liquid droplet, and ρl and ρi are densities of liquid and ice, respectively. From Eq. (3.42) and Eq. (3.43), one finds that in order for a complete ice bridge to form, Lmax < d should be satisfied, because ml should be higher than mbridge. Otherwise, ice bridge cannot connect to the liquid droplet and a complete ice bridge does not form. In Eqs. (3.42) and (3.43), it is assumed that the droplets are almost spherical. Thus, this scaling is good for hydrophobic and superhydrophobic surface on which contact angle is high and the droplet shape is close to a sphere.
Mass flow rate for the evaporation of a liquid droplet is defined by the following equation if it is assumed that droplet temperature is approximately equal to the substrate temperature:
(3.44)
Where Jl,e denotes mass flux of the vapor evaporating from liquid droplet, Ps,l denotes saturation pressure of the liquid droplet, Pn,i denotes vapor pressure near the surface of ice droplet and growing ice bridge, Tww is the substrate temperature, D denotes diffusivity of water vapor into the air,
(3.45)
Where vb is in-plane growth rate of the ice bridge and A|| is in-plane area of frozen droplet.
By mass conservation,
From this equation it is understood that ice bridge growth rate is independent of atmospheric pressure, P∞. However, high values of P∞ leads to the reduction in the evaporation rate of liquid droplet. Although the diameter of liquid droplet changes over time, the majority of vapor evaporated from the liquid droplet deposits somewhere close to the main base of ice bridge growth, due to the assumption Lmax < d. Thus, pressure gradient can be assumed to be constant during the ice bridge formation. Thus, from Eq. (3.46), vb can be assumed to be constant during ice bridge formation.
The length of ice bridge as a function of time can be obtained from
3.4 Summary
In this chapter, fundamentals of ice formation on an icephobic substrate are discussed. Ice formation is divided into two steps: ice nucleation and ice growth. As a droplet is placed on a sub-zero