## 3 Results and Discussions
### Empirical Formula of \(\beta_{\text{max}}\)  
The \(\beta_{\text{max}}\) of droplets impingement is one of the most important parameters of droplet impact dynamics, which directly represents the spreading ability of droplets under different working conditions. Studying \(\beta_{\text{max}}\) has many applications in inkjet printing, spray cooling, and other fields. The maximum spreading factor \(\beta_{\text{max}}\) is closely related to _We_ and _Re_ numbers, as demonstrated by previous studies. [PERSON] et al. [37] proposed a model to describe the variation of \(\beta_{\text{max}}\) with _We_ and _Re_ based on energy conservation of kinetic energy, surface energy, and viscous dissipation, considering droplets at maximum spreading diameter as cylindrical geometry. Compared with experimental data, the model showed a mean error of 8.70%. [PERSON] and [PERSON] [50] proposed a semi-empirical model  
Figure 8: Variation of “finger” numbers with impact velocity.  
considering the flow within the impacting droplet and showed the largest mean error of 14.64%. [PERSON] and [PERSON] (1995) [51] proposed an empirical formula based on regression analysis where the mean error is 14.33%. [PERSON] et al. [14] found that existing models did not perform well for low-viscous liquids and modified the [PERSON] and [PERSON] model by changing its coefficients, which showed better agreement with different liquids.  
Figure 9 shows the experimental data of \(\beta_{\text{max}}\) falling from different heights on the surface with \(Ra=0.8\) um and the comparison with the data predicted by the different models mentioned above. [PERSON]'s model shows the smallest error (6.81%) for low-viscous R1336 mzz(Z) impingement. However, to improve the predictive capability further, a new empirical model based on [PERSON]'s regression analysis has been proposed in this study, as expressed in equation (6), adopting the method of [PERSON] et al. Table 2 summaries the predictive performance of different models. Our new empirical model has the smallest mean error (5.53%) for the prediction of \(\beta_{\text{max}}\) in R1336 mzz(Z) impingement. To further examine the sensitivity of the new model on the surface roughness, Figure 10 plots the experimental data of \(\beta_{\text{max}}\) and the predictive curve by the new model based on regression analysis under four roughness surfaces. In general, the predicting curve by this model is able to represent \(\beta_{\text{max}}\) variation in practice reasonably.  
\[\beta_{\text{max}}=0.923\Big{(}\frac{We}{Oh}\Big{)}^{0.143} \tag{7}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Expression** & **Mean-Error (\%)** \\ \hline [PERSON] et al. [37] & \(\beta_{\text{max}}=\sqrt{\frac{W_{G}+12}{3(1-\cos\theta)+4\big{(}\frac{W_{G}} {\text{V}Ke}\big{)}}}\) & 8.70 \\ [PERSON] and [PERSON] [50] & \(\beta_{\text{max}}=\big{(}0.87 Re^{\frac{1}{5}}{-}0.4 Re^{\frac{2}{5}}We^{- \frac{1}{2}}\big{)}\) & 14.64 \\ [PERSON] and [PERSON] [51] & \(\beta_{\text{max}}\)\(=0.61\big{(}\frac{W_{G}}{Oh}\big{)}^{0.166}\) & 14.33 \\ [PERSON] et al. [14] & \(\beta_{\text{max}}\)\(=1.27\big{(}\frac{W_{G}}{Oh}\big{)}^{0.122}\) & 6.81 \\ New empirical model & \(\beta_{\text{max}}\)\(=0.923\big{(}\frac{W_{G}}{Oh}\big{)}^{0.143}\) & 5.53 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of models and their predictive mean errors.  
Figure 9: Comparison of experimental data with the predictive data by previous models.