## 3 Results and Discussion  
Wetting dynamics of an impacting supercooled water droplet on a cold substrate combined with cold stagnation flow is evaluated in the current study. In order to evaluate the net effect of airflow on an impacting supercooled water drop, the maximum air velocity was chosen based on the conception of no droplet deformation of a free falling drop exposed to gas flow. The dimensionless number describing this phenomenon is the relative Weber number, \(\frac{\rho_{air}\left(U_{g}-U_{d}\right)^{2}D_{d}}{\sigma_{lv}}\), which is defined based on density of the gas, which is air in this experiment (\(\rho_{air}\)), relative velocity of impinging water drop and air jet velocity, (\(U_{g}-U_{d}\)), water droplet size \(D_{d}\), and surface tension of water drop (\(\sigma_{lv}\)). For air velocity up to 10 m/s and with constant droplet diameter up to 2.6 mm the evaporation effect on droplet size during free falling (at height of 0.15 m in the current study) is negligible (1% of initial droplet size) [36]. Calculation of relative Weber number shows that it becomes close to the suggested value (i.e., 3) for having a non-deformed drop [37]. Deformation of the water drop is almost negligible and the droplet remains spherical. Dynamics of a water drop in supercooling conditions fundamentally changes that of room temperature. When drop impact condition is no longer isothermal, phase change might happen based on the difference between droplet and target surface temperature [38]. In order to demonstrate the effect of droplet degree of super cooling conditions on wetting dynamics behavior, temporal spreading factor versus dimensionless time is illustrated in Figure 2 for both aluminum and Teflon substrates. Increasing relaxing phase time rapidly increases rate of heat transfer within the drop through the substrate. Large interfacial area results in significant decrease in receding velocity compared to the room temperature one due to the increase in heat transfer cooling rate. As depicted in Figure 2, it is clear that at the maximum spreading diameter, instantaneous freezing (i.e., no retraction, pinning) was observed for coldest droplet and substrate temperature condition (i.e., \(-5.5\ ^{\circ}\)C and \(-30\ ^{\circ}\)C, respectively). However, slight retraction was observed at the initial stage of the retraction phase for the remaining scenarios. In fact, due to both the liberation of latent heat of fusion and molecular diffusivity [15] there is a sufficient freezing delay time when droplet and substrate temperature are not sufficiently cold, resulting in retraction initiations. However, after 25 ms, due to the large heat transfer cooling rate (i.e., large contact area which is almost not varied by time), the retraction force becomes almost zero, signaling the presence of heterogeneous ice nucleations between the impacting water drop and underneath the substrate. It becomes strong enough so that the droplet is pinned on the surface. This phenomenon can be further elaborated through classical ice nucleation theory combined with thermal transport analysis. As shown in Equation (1), nucleation rate [39; 40] is related to the multiplication of kinetic rate (i.e., activation energy of diffusion) to the exponential function of Gibbs free surface energy and interface temperature of an impacting water drop on substrates along with solid-liquid interfacial area and Boltzmann constant (i.e., \(k_{B}\)). \[J(T)=K(T)A_{s-l}\exp\left(\frac{-\Delta G_{Het}(T,\theta_{wI})}{k_{B}T_{ interface}}\right) \tag{1}\]  
where \(A_{s-l}\) is the solid-liquid surface area. Additionally Gibbs free surface energy is defined based on the melting temperature of water, \(T_{m}\), water-ice surface tension, \(\gamma_{wI}\), latent heat of fusion, \(H_{f}\), solid-liquid or gas-liquid interface temperature, \(T_{Interface}\), and geometrical factor, \(f_{(\theta_{wI})}\), as illustrated in Equations (2) and (3) [17; 40].