# Long-Term Dynamics of Water Droplet Impact on Rotating Hydrophilic Disk
## 3 Results and Discussions
### Spreading Dynamics on Rotating Disks
#### 3.2.3 Maximum and Equilibrium Spreading Factor  
The maximum and equilibrium spreading diameters are important parameters related to the area of the ice region. It is thus necessary to summarize the dependence of the maximum and equilibrium spreading factors on the Weber number and bond number from the experimental results. Figure 10a shows the maximum spreading factor \(\beta_{\rm max}\) as a function of \(We\) for different bond numbers. It thus can be inferred that \(\beta_{\rm max}\) is primarily dependent on \(We\) while it is marginally dependent on \(Bo\). Moreover, the measured \(\beta_{\rm max}\) on the stationary surface is validated against published experimental data from [PERSON] et al. [26] and several theoretical models in the literature (see Table 2). The two experimental points of [PERSON] et al. [26] correspond to the cases of a water droplet impacting a stationary wall with \(Ra=0.4\,\mathrm{\SIUnitSymbolMicro m}\) at \(We=20\) and \(200\). The comparison, as shown in Figure 10a, demonstrates that the results of the present study are in good agreement with the [PERSON] model [48] and [PERSON] model [26]. In particular, the consistency between the present results and the [PERSON] model is better because the latter includes the influence of surface roughness. The surface roughness \(Ra=0.6\,\mathrm{\SIUnitSymbolMicro m}\), used in the present study, is slightly larger than that (\(0.4\,\mathrm{\SIUnitSymbolMicro m}\)) used in [PERSON] et al. [26], which is believed to be the reason for the slight discrepancy between our results and the [PERSON] model at high Weber numbers. The [PERSON] model predicts the experimental data well at low Weber numbers; however, the deviation becomes more pronounced at higher We numbers. Figure 10b shows \(\beta_{\rm max}\) as a function of \(Bo\) for different Weber numbers. It is clearly shown that, for a fixed Weber number, the influence of \(Bo\) is negligible except that a minor increase of \(\beta_{\rm max}\) is observed when \(Bo\) increases from \(0\) to \(0.3\). This is because the action of centrifugal force promotes slightly droplet spreading, but once the disk is set into rotation, the rotating speed does not change further \(\beta_{\rm max}\). Figure 9: Evolution of inner and exterior diameters of the ring-like droplets during the retraction phase. (**a**) \(Bo=0.3,1.2\) and \(2.6\) and the same \(We=166\); (**b**) \(We=133,166,200\) and the same \(Bo=1.2\). The ratio of the diameter at the beginning of the equilibrium regime over the initial diameter is defined as the equilibrium spreading factor \(\beta_{\text{eq}}\). The cases at \(We=20\) and \(We=33\) are dismissed since the droplets' oscillation at the equilibrium state makes it difficult to identify exactly the end of the retraction phase. For the stable and unstable cases without ultimate oscillation over the range \(We\in[67,200]\), the droplet spreading factor was measured every 1 ms, and the droplet spreading was considered to enter the equilibrium state when the change rate of \(\beta\) was less than 0.01 ms\({}^{-1}\). This value of spreading factor is considered as \(\beta_{\text{eq}}\) and the corresponding moment as the end of the retraction phase or the start of the equilibrium state. Figure 11 shows the relationship of \(\beta_{\text{eq}}\) with \(Bo\) and \(We\). As \(Bo\) increases, \(\beta_{\text{eq}}\) increases with respect to \(Bo\), as shown in Figure 11a. This is because the droplet keeps its equilibrium state under the balance of centrifugal force, adhesion force, and surface tension. Droplets at higher rotating speeds undergo higher centrifugal force, which needs to be balanced by higher surface tension and adhesion force, thus a larger spreading area. The experimental data at different Weber numbers are fitted to an empirical pow-law relation \(\beta_{\text{eq}}=aBo^{b}\) and the fitting curves for each \(We\) are also plotted in Figure 11a. The fitting parameters \((a,b)\) are listed in Table 3. Both \(a\) and \(b\) increase monotonically with \(We\).