## 3 Results and Discussion
Figure 2: Temporal evolution of impacting water droplet on aluminum (**A**) and Teflon substrate (**B**) at a wide range of droplet temperature from \(20\ ^{\circ}\)C to \(-5.5\ ^{\circ}\)C. Droplet size and impact velocity are 2.6 mm and 1.6 m/s, respectively. \[\Delta G_{Het}(T,\theta_{wI})=\frac{T_{m}^{2}16\pi\
u_{wl}^{3}}{3H_{f}^{2}\Big{(}T _{m}-T_{Interfac}\Big{)}^{2}}f(\theta_{wI}) \tag{2}\]  
\[f(\theta_{wI})=\left[\frac{(2+cos\theta_{wI})(1-cos\theta_{wI})^{2}}{4}\right] \tag{3}\]  
Interface temperature of a liquid drop on a surface with various roughness is predicted by solving two transient conduction equations through the drop and coated (i.e., rough surface) domain. It is given by Equation (4). \[T_{w-interface}\left(h,t_{c}\right)=T_{drop}+[(T_{sub}-T_{d})]erfc\left(\frac{h _{pa}}{2\sqrt{\alpha_{pa}t_{c}}}\right) \tag{4}\]  
Furthermore, temperature distribution inside of spread water can be predicted from Equation (5) (see Supplementary Materials). \[T_{w}(x_{w},t_{c})=T_{drop}+[T_{sub}-T_{d})]\frac{erfc\left(\frac{h_{pa}}{2 \sqrt{\alpha_{pa}t_{c}}}\right)}{erfc\left(\frac{h_{pa}}{2\sqrt{\alpha_{w}t_{ c}}}\right)}.erfc\left(\frac{x_{w}>h}{2\sqrt{\alpha_{w}t_{c}}}\right) \tag{5}\]  
In equation above, \(T_{sub}\), is the temperature of semi post-air domain (i.e., coated surface) and \(T_{w}\) is water temperature, \(t_{c}\) is the total contact time of liquid drop on surface, and \(h_{pa}\) represents average or root mean square roughness of substrate. \(\alpha_{pa}\) and \(\alpha_{w}\) are thermal diffusivity of post-air domain and water respectively. Composite thermal diffusivity of air-pillar domain, \(\alpha_{pa}\), is found through equivalent thermal conductivity and specific heat of coated (i.e., semi air-pillar) domain [17]. At cooler conditions (\(T\leq-30\)\({}^{\circ}\)C) due to the presence of extreme cold interface temperature, which become close to substrate temperature, nucleation rate increases exponentially, resulting in instantaneous pinning when drop reaches its maximum spreading diameter on both aluminum and Teflon substrates. The change in the maximum spreading diameter is due to an increase of droplet internal bulk viscosity up to 8-fold [41; 42]. It was shown that an increase in viscous dissipation results in a decrease in the maximum spreading diameter, as can be inferred from droplet viscous dissipation function, \(\varphi\approx\mu\Big{(}\frac{U}{L}\Big{)}^{2}\)[43] where \(U\) and \(L\) are droplet impact velocity and near wall boundary layer thicknesses [44]. However, in order to show that reduction in droplet maximum spreading diameter is only related to viscosity variation and not by phase change and crystallization process, a classical energy-based model of maximum spreading diameter prediction derived by [PERSON] et al. [45] is presented by Equation (6). In this equation, the effect of phase change and solidification is also considered by assumption of all the kinetic energy stored in ice layer being lost. \[\zeta=\frac{D}{D_{\max}}=\sqrt{\frac{We+12}{\frac{3}{8}Wes^{*}+3(1-cos\theta _{a})+4\frac{We}{\sqrt{Re}}}} \tag{6}\]  
where \(s^{*}=\sqrt{2t^{*}\frac{Ste}{Pe}}\) is dimensionless growth rate of freezing front which follows from the Stefan problem soultion. It was showed that where \(\sqrt{\frac{Ste}{Pr}}<<1\) the effect of formed ice nuclei on the maximum spreading diameter is negligeble. The dimensionless Stefan number, \(\frac{C_{pJ}(T_{m}-T_{s})}{H_{J}}\), is increased by increasing the degree of sub cooling of surface (i.e., temperature difference between substrate and melting point of water, \((T_{m}-T_{s})\)). On the other hand, the ratio of droplet specific heat and latent heat of fusion is also increased when water droplet temperature is decreased. However, both the supercooled water droplet specific heat and latent heat of fusion are weak functions of temperature within the range of the present work (i.e., \(-10\)\({}^{\circ}\)C and \(-30\)\({}^{\circ}\)C).