## 3 Results and Discussion
For the bulk solution, some initial small nuclei of crystals (with a partial but visible order of Na\({}^{+}\) and Cl\({}^{+}\) ions) started forming at 16.5 ns (Figure 4b). The nucleation of small visible crystals in PVDF started at 6-6.5 ns of simulation (not shown here), whilst 3.5-4 ns were necessary for PP systems (Figure 4d). Therefore, in the presence of the membrane surface, crystals formed in a shorter time than that observed for the bulk solution. Figure 3: Face-centered cubic lattice of NaCl. Na\({}^{+}\) ions in purple and Cl\({}^{-}\) ions in green. Similarly, the experimental data reported in Table 4 showed that the time for detecting and collecting crystals from the feed solution was lower for PP membranes with respect to PVDF membranes. These data are in accordance with the estimated Gibbs free energies' ratio (Equation (1) equal to 0.40 and 0.56 for PP and PVDF membranes, respectively. The lower the \(\Delta G_{heterogeneous}/\Delta G_{homogenous}\) ratio, the lower the energy barrier to overcome for forming a critical nucleus formation. For comparison, [PERSON] et al. [61] determined a nucleation time for NaCl crystals produced via the traditional evaporation method equal to 386 min. \begin{table}
\begin{tabular}{c c c} \hline  & **PVDF** & **PP** \\ \hline Minimum time for detecting and & 360 & from 165 to 283 1  \\ Crystals growth rate [mm/min] 2  \\ \hline \end{tabular}
\end{table} 
Table 4: Kinetic parameters of the NaCl crystallization tests. Figure 4: Snapshots showing the amorphous and crystal phases of Na+ and Cl- in supersaturated solutions at different times of simulations: (**a**,**b**) in NaCl bulk solution; (**c**,**d**) on PP surface. Blue circles indicate the region with higher density and partial order of NaCl ions. Na+ in purple, Cl- in green, water molecules in grey. Moreover, the experimental data reported in Table 4 prove that the crystal growth in the PVDF membrane is almost the same, just slightly higher than in PP membrane. The shape and dimension of crystals in both membrane systems are similar, as confirmed by the modelled crystals in Figure 2.  
After 100 ns of simulations, significant differences between the NaCl bulk solution and the system containing membrane surfaces were observed. PP and PVDF models produced multiple regularly structured face-centered cubic lattice NaCl crystals, suggesting a more controlled nucleation and higher growth of the crystals. These high numbers of crystal were of a smaller dimension compared to that obtained from the NaCl bulk solution (Figure 5a-c). This is in accordance with the fact that in the bulk solution, the nucleation is slower and the supersaturation in the solution decreases gradually because the nucleation of the formed crystals continues and different sizes can be obtained. Instead, with PVDF and PP surfaces, the nucleation is faster than in the bulk solution and many crystals form nearly simultaneously [12]. Moreover, the percentage of free ions after 100 ns of simulation in the bulk NaCl solution was about 14.3%, whereas in PVDF and PP, it was about 4.5% and 3.6%, respectively. The growth of crystals in PVDF and PP diminishes the solution of solute and leads to the termination of nucleation at the advanced stages of crystallization. Therefore, the majority of crystals grow to approximately identical sizes [12]. Figure 5: _Cont._  
In order to analyze the evolution of the NaCl crystals during simulation time, a quantitative overview is given in Figure 6, where we have plotted \(r^{2}h_{+-}(r)\) for times ranging from 5 to 100 ns. Here, \(h_{+-}(r)=g(r)^{-1}\), where g(r) is the radial distribution function and multiplied by \(r^{2}\) to magnify the structural details [31]. The function \(h_{+-}(r)\) is averaged over all ions in the system. However, ions in the growing crystal contributed to the long-range structure. At 5, 10, and up to 20 ns, \(r^{2}h_{+-}(r)\) is very short ranged and refers to the metastable solution with one discernable peak indicating few ordered ions.