Ice Adhesion. Группа авторов
1.23 becomes Equation 1.24 [78].
where ∆T = Ts — T0 + ∆Ts. The value of as in Equation 1.24 is <<1. Further, because these experiments take place under quasi-steady cooling, ∆Ts can be described by a linear time relation, Equation 1.25
where α is the rate of temperature change per unit time.
As the temperature is continually decreased during experimentation aimed at the determination of the nucleation temperature, J(T) is no longer a constant. The rate of nucleation increases with decreasing system temperature. The Poisson point process in this scenario is nonhomogeneous as λ is a function of time. The nonhomogeneous Poisson distribution is given by Equation 1.26 [84].
where ζ and η are the bounds of the rate function λ(u). In the case of nucleation temperature determination, ζ = 0 and η = t, the length of time over which the change in nucleation rate takes place. Thus, the probability density function for the life-time of the liquid droplet under changing temperature conditions is given by Equation 1.27 [78, 85]
When integrated, Equation 1.27 yields the probability function of the frozen droplet, Equation 1.28.
Eberle et al. (2014) derive closed-form expressions for the nucleation temperature, TN, and the median freezing time, tmedian, by equating Equation 1.28 to 0.5 [78].
where T1 is the temperature at which the nucleation rate is equal to one embryo per second (i.e. J(T1) = 1 s–1). Note that the authors made use of the fact that (βα/as) >> 1 in deriving Equations 1.29 and 1.30. Equations 1.23 and 1.29 can be combined to determine the nucleation rate at the nucleation temperature, Equation 1.31.
1.3 The Adhesion of Ice to Surfaces
As was alluded to in Section 1.1.3, the transportation industry has spawned much of the rigorous research into surface icing. This is especially true of the aviation sector and the quest to understand the adhesion strength of ice on solid surfaces. In 1929 the Daniel Guggenheim Foundation commissioned Dr. William C. Geer to perform the first rigorous study of the problems encountered while flying in ice-forming conditions. He concluded his work by suggesting the idea that a device be built which can crack the ice formed on the leading edge of airplane wings - allowing the formed ice to be carried away by the air stream. This rubber device, known commercially as the “De-icer”, was developed by the engineers at BF Goodrich and quickly gained widespread adoption by aircraft manufacturers for use on lower speed (non-jet) aircraft [86].
Naturally BF Goodrich desired to improve their product, with the obvious goal of achieving a low value for the bond at the ice-rubber interface. This work was headed by Loughborough and Haas, leading to the first published study of the adhesion strength of ice in 1946. Loughborough and Haas comment on the effect of temperature, concluding that the shear strength of the bond between the ice and the De-icer increases linearly from 0 kPa at 0°C to 1.034 MPa at -25°C. Further, they implicitly discuss the consequences of ice formation onto a porous surface by noting that rubber which had been soaked in water presented ice adhesion shear strength 15% higher than dry rubber. The issue of surface roughness was also touched upon, with the authors suggesting that increased roughness affects ice adhesion strengths to the extent that rough substrates present greater overall surface areas. Finally, this article addressed the matter of surface free energy in determination of ice adhesion shear stress. The authors note that the addition of silicone to the rubber lowered the adhesion of ice to said surface by as much as 90 percent [87].
The seminal work of Loughborough and Haas introduced the main factors affecting ice adhesion, which are touched upon in every major subsequent publication in this area of research. These parameters include: temperature, surface roughness and porosity, and surface free energy. These factors affecting the adhesion of ice to solid surfaces are discussed in the following section.
1.3.1 Wetting and Icing of Ideal Surfaces
The formation of surface ice invariably involves the precursory step of liquid water coming into contact with the solid surface. Thus, it is important to understand the adhesion realities of the liquid water-solid surface-background gas system before one can begin to study the same system with solid ice.
Consider first a drop of water resting on a theoretical surface which is chemically homogeneous, perfectly flat, and completely rigid, as in Figure 1.7(a). The points along the rim of the water droplet, where the liquid, solid, and air meet, form the triple-phase contact line. Every point along this line is in a state of equilibrium under the balance of the solid-air, γSA solid-water, γSW, and water-air, γWA, surface free energies/tensions. The angle which is manifested between γSW and γWA is known as the water contact angle. Thomas Young resolved the force balance at the triple-phase contact line for this theoretical surface, some sufficient time after the water droplet has been placed and has reached equilibrium, in 1805. This force balance is presented