## 3 Results
However, near the supersaturation of S = 0.5, a gradual increase in temperature is observed. The gradual release of heat from the solution droplet implies that solute molecules were aggregating to form larger clusters. During this process, it was expected that the hydrogen bonding between water and urea molecules was broken while the bonding between urea molecules formed. The latter likely yielded the structural adjustment of lower energy states. The exothermic reaction became considerably more notable near the solubility limit (S = 1). Assuming that the heat release from the drying droplet was proportional to the number of urea molecules forming bonds, it was reasonable to expect that urea molecules were being added to an unidentified embryonic pre-nuclei, whose sizes were still small enough to be held soluble in the solution form. Approximately 100 s prior to the final crystallization, the temperature stagnated. This behavior indicates that the urea-urea bond-making process had ceased and the cluster size was no longer increasing. Soon after the urea solution reached a critical level of supersaturation (S = 2), an instantaneous release of heat occurred, followed by rapid crystallization. As depicted in Figure 2 (top), the initial crystallization site is indicated by the yellow arrow at \(\Delta t\) = 0.00 s, which coincides with the abrupt temperature rise shown in Figure 2 (bottom). This temperature increase (\(\Delta T\sim 25\) K) observed in the urea sample closely parallels the substantial temperature rise previously recorded in an experiment conducted on a high-temperature (\(\sim\)350 K) hydrophobic coated surface [33]. The transparent droplet turned into a visibly opaque white polycrystalline solid, as shown in Figure 2 (top). Afterward, a relatively slow radiative cooling of the solidified sample took place. Based on the evolution of temperature, the release of energy in the supersaturating urea solution occurred in two distinctive stages. Initially, the urea molecules existed in the solution form of mostly fully hydrated molecules. Commencing around S = 0.5, evidence emerges indicating the formation of microscopic pre-nucleation clusters, wherein approximately 10% of the entire heat energy was involved (assuming constant specific heat). Lacking any visible crystallization, the sample was still in a solution form. At S = 2, a very rapid solidification occurred, during which about 90% of the heat of crystallization was released. This was likely due to the large surface energy of the pre-nucleation clusters dispersion in the solution. Figure 3 presents the probability distribution of the crystallization events derived from 100 experimental trials. Notably, the frequency of the crystallization event increases with supersaturation because of the rapid increase in thermodynamic driving force (namely, the chemical potential difference between solution and crystal phases) until S = 2. A statistical methodology formulated by [PERSON] [34; 35], previously applied in the study of nucleation phenomena in metal alloys [36], is used to verify the experimental result. As nucleation is inherently stochastic, characterized by independent occurrences, the Poisson distribution is employed to describe the discrete probability of nucleation as a function of supersaturation. Assuming that a singular nucleation event is sufficient to trigger solidification, the probability of such an event transpiring within the sample volume \(V(t)\) between \(S\) and \(S+dS\) can be expressed as follows:  
\[P(1,S+dS)=\frac{J_{ss}V(t)}{dS/dt}\exp\left(-\int\limits_{S_{o}}^{S}\frac{J_{ss }V(t)}{dS/dt}dS\right). \tag{2}\]  
The steady-state nucleation rate \(J_{ss}\) per unit volume is given by \[J_{\rm ss}=\frac{k_{B}TN_{A}}{3a^{3}V_{m}\eta(C(t))}\exp\biggl{(}-\frac{\Delta G^{*} }{k_{B}T}\biggr{)}, \tag{3}\]  
where \(N_{A}\) and \(k_{B}\) represent [PERSON]'s number and [PERSON]'s constant, respectively. The concentration-dependent viscosity \(\eta(S)\) of the bulk urea solution is obtained by extrapolating experimental data [32; 37].