## 1 Introduction
### Transition for Secondary Atomization after Drop Impact on Liquid Films  
The two essential outcomes of drop impact on liquid films are the deposition of a liquid droplet in the film or the generation of secondary droplets through hydrodynamic mechanisms, such as a full or partial rebound of the initial droplet, or smaller droplets detaching from the bounding rim of an uprising crown associated with the splashing mechanism. However, the classification of these outcomes is not always unanimous. [PERSON] et al. [1] defined detailed terminology for the several morphological structures emerging from drop impact--prompt splash, crown splash, rebound, and partial rebound being the most relevant for this study of droplet deposition. Figure 1 presents illustrations of these phenomena organized by the required drop impact energy. Identifying the differences in the droplet hydrodynamic behavior depending on the liquid film thickness (\(h_{f}\)) was a concern. Namely, [PERSON] and [PERSON] [2] reported that the spreading of the droplet changed significantly due to the liquid film. Later, focusing their work on distinguishing the spreading behavior between the impact on dry and wetted surfaces, [PERSON] et al. [3] showed how maintaining all the impact parameters constant, except the surface conditions, resulted in entirely different drop impact morphological structures. Therefore, it is reasonable to include the effect of the liquid film characteristics in the empirical correlations (in a dimensionless form as \(\delta=h_{f}/D_{0}\), where \(D_{0}\) is the droplet diameter before impact) discerning the transition between deposition and the splash which generates secondary atomization. The empirical correlations that categorize the outcomes of impacts depend on dimensionless parameters, such as the Weber (\(We=\rho D_{0}U_{0}^{2}/\sigma\)) and Reynolds (\(Re=\rho U_{0}D_{0}/\mu\)) numbers, followed by the Ohnesorge (\(Oh=\sqrt{We}/Re\)) and Laplace (\(La=Oh^{-2}\)) numbers, dependent on the former, where \(\rho\)  
Figure 1: Drop impact mechanisms. \(\sigma\), and \(\mu\) are the density, surface tension, and dynamic viscosity of the liquid, respectively, and \(U_{0}\) is the droplet impact velocity. Most empirical correlations follow a relation between Weber and Reynolds numbers expressed as  
\[Re^{\alpha}We^{0.8}=K_{c} \tag{1}\]  
where \(K_{c}\) is the criterion between a droplet spreading and splashing. Table 1 summarizes the proposals for this criterion for wetted surfaces. There are several considerations about the origin of these criteria. The first relates to the criterion of [PERSON] and [PERSON] [4], which considers the impact of a droplet on a wetted surface equivalent to its impact on a very rough surface (\(r_{s}>12\)\(\upmu\)m), and the data can be traced back to the original contribution of [PERSON] and [PERSON] [8] with experiments within the range indicated in Table 1. The second consideration is the criterion of [PERSON] et al. [5] which considers the influence of the liquid film thickness in the impact outcome. To enlarge their experimental data, they used not only pure water but also mixtures of glycerol and water, thereby increasing the range of thermophysical properties of the fluids. Additionally, they performed experiments using different dimensionless film thicknesses (\(\delta=h_{f}/D_{0}\)). The validation range of this criterion can be consulted in Table 1. The third consideration refers to the correlation of [PERSON] and [PERSON] [6], which includes a random element in the criterion \(K_{c}\) through the mass ratio \(\eta\). This mass ratio can be above 1 in the sense that part of the liquid film may contribute to the mass of secondary droplets emerging from splash. In fact, [PERSON] and [PERSON] [4] consider a similar random element in the numerical model of spray impingement as \(\eta=0.2+0.9\)\(rnd(0,1)\). In Roisman and Tropea [9] a value for \(K_{c}=2800\) is suggested, which would correspond to \(\eta\approx 0.58\); however, no explanation for this value is provided.