**4. Three-phase contact line enlargement**

After the rupture of the liquid film, the liquid phase begins to retreat from the solid surface due to an uneven distribution of the liquid-gas interfacial tension. Generally, the movement of the three-phase contact line is involved in both the wetting and dewetting mechanisms. The contact line movement is driven by fluid dynamics and molecular interactions of the contacting phases. Surface tension and inertial and viscous forces influence the expansion of the TPC line. The resultant of the force action influences the curvature of the liquid-gas interface and therefore affects the shape of the attached bubble and contact angle. The dynamic process of either wetting or dewetting can be described by the velocity of the contact line movement *UTPC* which is defined as the time derivative of three-phase contact line radius *rTPC*:

$$U\_{\rm TPC} = \frac{dr\_{\rm TPC}}{dt} \tag{2}$$

If we focus on the wetting process (drop spreading on the surface of the solid particle), several theoretical models have been developed to describe the TPC line expansion, relating the velocity dependence on dynamic contact angle to measurable properties such as surface and interfacial tension, liquid viscosity, and static contact angle. The two main approaches prevail during the modeling of TPC line expansion: the hydrodynamic models are rather macroscopic and disregard the role of the solid surface, whereas the molecular-kinetic models involve also the properties of solid surface.

Cox [27] established the basics of the hydrodynamic model for the wetting mechanism in pure liquids. It suggests that the process is dominated by the fluid viscous dissipation. Thus, the bulk viscous friction is the main resistance force for the TPC line contact motion [28]. The model solves the equations governing the fluid dissipation, the continuity, and the Navier-Stokes equations and relates the expansion velocity *UTPC* to the dynamic contact angle *θ*. The main disadvantage of this model is the inadequate description of the fluid motion very near to the contact line. The molecular-kinetic model eliminates the viscous dissipation but includes the solid surface characteristics. The theory is based on a statistical treatment of the transport mechanism of molecules and ions in pure liquids [29]. This model assumes the energy dissipation to occur only at the moving contact line, where adsorption and desorption processes occur. This idea is commonly applied to dynamic wetting. The movement of the three-phase contact line is ruled by the statistic kinetics of molecular events arising at the adsorption sites of the solid surface [26]. The dependence of expansion velocity on the dynamic contact angle is due to the disturbance of adsorption equilibrium, which is driven by the changes in the local interface tensions [30]. A combined molecularhydrodynamic approach is currently recommended [28, 31]. The dewetting hydrodynamics is used to describe the effect of fluid flow on the interface deformation far from the three-phase contact line. Molecular kinetics is then used for the description of dewetting process close to the contact line. A comprehensive review of wetting and spreading problematics is reported by Bonn [32], Blake [33], or De Coninck [34].
