# A New Criterion for the Splashing of a Droplet on Dry Surface from High-Fidelity Simulations
###### Abstract  
In this study, a new criterion for the splashing of a droplet on a dry smooth surface is established from high-fidelity numerical simulations. The new criterion involves the Weber number, Reynolds number and contact angle. A new splashing mode, termed spreading splashing, is proposed, which predominates for contact angles below 120 degrees. For contact angles above 120 degrees, prompt splashing dominates. For contact angles above 90 degrees, there exists a critical Weber number of around 60, below which splashing does not occur. d pacs: 00.70.-a, 02.50.Jb, 02.50. The morphology of liquid droplets undergoes significant changes in splashing, often breaking into numerous secondary droplets that move far away from the point of impact. Hence, it is of significant importance to determine whether splashing occurs in engineering applications. For numerical simulations of the flow laden with spray droplets using the Euler-Lagrange method [7], the splashing criterion may be used to determine whether a droplet would deposit on the wall or splash into smaller droplets. Extensive studies on droplet splashing have been conducted. Over a century ago, [PERSON] [8; 9; 10] undertook pioneering studies on droplet splashing. He summarized and categorized a series of states that occur after droplets collide with a surface. He experimentally observed the splashing of large milk and mercury droplets impacting on a smooth glass surface, although he did not observe corona splashing, which is characterized by the formation of a crown-shaped lamella up in the air. The experimental results of [PERSON] et al. [11] and [PERSON] [12] showed that increasing surface roughness caused liquid droplets to be more prone to corona splashing, and splashing without a crown formation occurred only on extremely smooth surfaces. Based on their experiments, [PERSON] and [PERSON] [13] were the first to propose an empirical formula for the splashing threshold, expressed as \(K=We^{0.5}Re^{0.25}\). When \(K\) exceeds the critical value \(K_{c}\), splashing occurs. [PERSON] et al. [14] investigated the impact of small water droplets on relatively rough surfaces and obtained the critical value \(K_{c}\) of 57.7. [PERSON] and [PERSON] [15] observed the onset of splashing at a critical Weber number \(We_{c}=80\) for a number of different liquids impacting a polished gold surface, and correlated the splashing threshold with the critical Weber number. [PERSON] and [PERSON] [16] examined the effects of wall roughness on droplet impact at low Oh number, and correlated the critical Weber number for the onset of splashing with the following expression:  
\[We_{c}=a[ln(R_{a}^{*})]^{b}, \tag{3}\]  
where \(a\) and \(b\) are two parameters whose values depend on the droplet and solid surface properties, \(R_{a}^{*}\) is the non-dimensional average wall roughness defined as \(Ra/D_{0}\), and \(R_{a}\) is the dimensional average roughness. [PERSON] et al. [17] proposed a simple criterion, \(\sqrt{Ca}=We^{0.5}Re^{-0.5}=0.35\), to distinguish between the splashing and non-splashing modes. [PERSON] et al. [18] conducted a study on the splashing behavior of droplets impacting on various substrates with different wetting properties. They observed that the maximum dynamic advancing contact angle and the splashing ratio \(\beta\) effectively characterized the splashing behavior. [PERSON] et al. [19] conducted experiments to investigate the splashing behavior of liquid droplets impacting dry surfaces with varying roughness. The results indicated that the criterion \(K=We^{0.5}Re^{0.25}\) failed to accurately predict the splashing threshold for dry surfaces with different roughness levels. [PERSON] et al. [20] proposed a new splashing criterion \((WeRe)^{0.25}>25.15\) for a superhydrophobic surface with the contact angle \(CA=156^{\circ}\), based on their numerical simulation results. [PERSON] et al.