## 3 Methodology  
The terms of surface energy (SE) and surface tension (ST) correspond physically. The SE is commonly used to describe energy for solid surfaces and ST for liquid surfaces. ST is defined as the work required to increase the surface area isothermally and reversibly by a unit amount, Embesajjad and Embesajjad [14]. However, the ST of solids is also incorrectly indicated. SE uses the unit of energy per area in mJ/m\({}^{2}\) (millijoule per square meter), with the equivalent unit mN/m (millinewton per meter) commonly used for ST, Embesajjad and Embesajjad [14]. The formula symbol is \(\sigma\) (less often \(\gamma\)), which, respectively, represents the tension at the solid/air, liquid/air, and solid/liquid interfaces. There are various ways to analyse surface energy: [PERSON]'s equation, [PERSON], [PERSON] equation, Lifshitz-van [PERSON] and Lewis acid-base, Girifalco [24]. As a first step, we use [PERSON]'s equation, which is the standard for simulation in COMSOL:  
\[\sigma_{s}=\sigma_{sl}+\sigma_{l}\cdot cos\theta \tag{1}\]  
Here, \(\sigma_{s}\) is the surface free energy of a solid, \(\sigma_{l}\) is the surface free energy of a liquid, \(\sigma_{sl}\) is the interfacial free energy between a solid and a liquid, \(\theta\) is the contact angle of a solid and a liquid. The following illustration (Figure 1) shows the droplet height \(h\), droplet diameter \(d\), and the contact (wetting) angle resulting from the equilibrium of forces during solid wetting according to [PERSON]'s Equation (1) in the stable phase:  
In a simple system such as that shown in Figure 1, the adhesion strength can be calculated through the work of adhesion \(W_{a}\), [PERSON] [26]:  
\[W_{a}=\sigma_{s}+\sigma_{l}-\sigma_{sl} \tag{2}\]  
Figure 1: A drop on a surface with an indication of the contact angle and surface tension for three media, respectively (according to [PERSON] [25]). The interaction of the droplet with the surface was simulated using the COMSOL Multiphysics software [27]. The droplet height and base were taken from the programming results, and the contact angle was obtained from [PERSON] et al. [28]:  
\[\theta=2\ tan^{-1}\bigg{(}\frac{2h}{d}\bigg{)} \tag{3}\]  
where \(h\) is the droplet height, \(d\) is the droplet diameter, and \(\theta\) is the contact (wetting) angle. Estimation of the surface energy of a solid by the contact angle ([PERSON] and [PERSON] [29])  
\[cos\theta=\frac{\gamma_{s}-\gamma_{sl}}{\gamma_{l}}. \tag{4}\]  
For displaying the interface and transferring ink in air on a smooth surface, we considered the Level Set method. Typically, inkjet ink is a Newtonian fluid that does not change viscosity when force is applied. The interface between the inkjet droplet and another substance is described by \(\phi\), where \(\phi\) = 1 represents ink and \(\phi\) = 0 is another substance. At the interface conversion level, \(\phi\) varies from 0 to 1. The conversion layer goes along with the jet droplet velocity u during interaction. The following COMSOL Equation (5) describes the convection of a reinitialized level set function, taking into account the interface between the two phases, ink and air:  
\[\frac{\partial\phi}{\partial t}+\textbf{u}\cdot\
abla\phi+\gamma\bigg{[} \bigg{(}\
abla\bigg{(}\phi(1-\phi)\frac{\
abla\phi}{|\
abla\phi|}\bigg{)} \bigg{)}-\varepsilon\
abla\cdot\
abla\phi\bigg{]}=0 \tag{5}\]  
where \(\varepsilon\) is the thickness of the transition layer, and it can be defined as \(\varepsilon\) = hc/2 (hc is the representative mesh size in the area passed by the falling droplet in the simulation, set to hc = 2.5 \(\mu\)m). \(\gamma\) is the amount of reinitialization or stabilization of the level set function and the current simulation is set to \(\gamma\) = 10 m/s. Density and dynamic viscosity depend on the interfacial coefficient \(\phi\):  
\[\rho=\rho_{1}+(\rho_{2}-\rho_{1})\phi \tag{6}\]  
where, \(\rho_{1}\), \(\rho_{2}\) denote air and ink density, respectively.