the icing delay time offered by a surface. Pruppacher and Klett (1980) extended the CNT to estimate the nucleation rate, J(T), via Equation 1.15 [81]
Figure 1.6 Rendering of the hypothetically best surface morphology to delay the nucleation of ice according to the heterogeneous classical nucleation theory. This surface is decorated with nano-scale concavities, with no nanoconvexities.
where K and kB denote the kinetic prefactor accounting for diffusive flux of water molecules across the ice embryo/water interface and the Boltzmann constant, respectively. The kinetic prefactor, K, can be related to the diffusion activation energy required for a water molecule to cross the water-ice interface, ∆Fdiff, as given by Equation 1.16 [81].
where h and n are the Planck constant and the number density of water molecules at the ice embryo/water interface (n ≈ 1019 m–2), respectively. The diffusion activation energy can be calculated using the empirical Vogel-Fulcher-Tammann diffusivity relationship to yield Equation 1.17 [82, 83].
where E and TR are empirical fitting parameters with values for liquid water of E = 892 K and Tr = 118 K in the temperature range of 150-273 K [83].
The stochastic process of water nucleating into ice follows the binomial distribution; that is, each ice embryo has the binary outcome of either reaching the critical radius to become stable, or not. Each growing ice embryo is one of n independent experiments within the binomial distribution. Further, each of these experiments has a low probability, p. Distributions such as these can be approximated well by a Poisson distribution. If the mean number of events that occur in one unit time, λ, is a constant then the phenomenon studied is deemed a homogeneous Poisson point process. The homogeneous Poisson distribution is given by Equation 1.18 [84].
Thus, the probability density function of the lifetime of a liquid droplet in contact with a surface can be found by setting x = 1 (i.e. ice embryos are not reaching the critical radius) and λ = J(T).
where the subscript c denotes that the process is occurring at constant temperature. The expected time delay in freezing water, τ, at a given constant temperature can thus be calculated using Equation 1.20.
1.2.4 Predicting Ice Nucleation Temperatures
In addition to determining the delay time for ice nucleation at a constant temperature, one can also determine experimentally and statistically the temperature at which the rate of critically-sized ice embryo formation becomes appreciable - the median nucleation temperature, TN.
Naturally, determination of a nucleation temperature involves decreasing the temperature of the studied system. If small changes in temperature, ∆T, are assumed around a reference temperature, T0, the free energy term in Equation 1.15 and the diffusion activation energy term in Equation 1.16 can be linearized according to Equations 1.21 and 1.22, respectively [78].
Equations 1.16, 1.21 and 1.22 can be inserted into Equation 1.15 to yield an approximation for the ice nucleation rate at the new temperature, given by Equation 1.23 [78].
where a and β are substrate-specific constants at temperature T0.
Experimentally, the nucleation temperature is the median temperature at which ice nucleates in a sessile water droplet placed on a studied surface when the entire liquid/surface/background gas system is cooled in a quasi-steady manner [75]. For ease of calculation, Equation 1.23 is best referenced to the experimental starting temperature, Ts, which is generally room temperature. For the change in temperature of ∆Ts around Ts, Equation 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
