Apparent and microscopic dynamic contact angles in confined flows

Abstract

An abundance of empirical correlations between a dynamic contact angle and a capillary number representing a translational velocity of a contact line have been provided for the last decades. The experimentally obtained dynamic contact angles are inevitably apparent contact angles but often undistinguished from microscopic contact angles formed right on the wall. As Bonn et al. [“Wetting and spreading,” Rev. Mod. Phys. 81, 739–805 (2009)] pointed out, however, most of the experimental studies simply report values of angles recorded at some length scale which is quantitatively unknown. It is therefore hard to evaluate or judge the physical validity and the generality of the empirical correlations. The present study is an attempt to clear this clutter regarding the dynamic contact angle by measuring both the apparent and the microscopic dynamic contact angles from the identical data sets in a well-controlled manner, by means of numerical simulation. The numerical method was constructed so that it reproduced the fine details of the flow with a moving contact line predicted by molecular dynamics simulations [T. Qian, X. Wang, and P. Sheng, “Molecular hydrodynamics of the moving contact line in two-phase immiscible flows,” Commun. Comput. Phys. 1, 1–52 (2006)]. We show that the microscopic contact angle as a function of the capillary number has the same form as Blake’s molecular-kinetic model [T. Blake and J. Haynes, “Kinetics of liquid/liquid displacement,” J. Colloid Interface Sci. 30, 421–423 (1969)], regardless of the way the flow is driven, the channel width, the mechanical properties of the receding fluid, and the value of the equilibrium contact angle under the conditions where the Reynolds and capillary numbers are small. We have also found that the apparent contact angle obtained by the arc-fitting of the interface behaves surprisingly universally as claimed in experimental studies in the literature [e.g., X. Li et al., “An experimental study on dynamic pore wettability,” Chem. Eng. Sci. 104, 988–997 (2013)], although the angle deviates significantly from the microscopic contact angle. It leads to a practically important point that it suffices to measure arc-fitted contact angles to make formulae to predict flow rates in capillary tubes.