Wall energy relaxation in the Cahn–Hilliard model for moving contact lines

Abstract

The Cahn–Hilliard model uses diffusion between fluid components to regularize the stress singularity at a moving contact line. In addition, it represents the dynamics of the near-wall layer by the relaxation of a wall energy. The first part of the paper elucidates the role of the wall relaxation in a flowing system, with two main results. First, we show that wall energy relaxation produces a dynamic contact angle that deviates from the static one, and derive an analytical formula for the deviation. Second, we demonstrate that wall relaxation competes with Cahn–Hilliard diffusion in defining the apparent contact angle, the former tending to “rotate” the interface at the contact line while the latter to “bend” it in the bulk. Thus, varying the two in coordination may compensate each other to produce the same macroscopic solution that is insensitive to the microscopic dynamics of the contact line. The second part of the paper exploits this competition to develop a computational strategy for simulating realistic flows with microscopic slip length at a reduced cost. This consists in computing a moving contact line with a diffusion length larger than the real slip length, but using the wall relaxation to correct the solution to that corresponding to the small slip length. We derive an analytical criterion for the required amount of wall relaxation, and validate it by numerical results on dynamic wetting in capillary tubes and drop spreading.

Apparent and microscopic dynamic contact angles in confined flows

“…It should be noted that γ 2 γ 1 can be written as γ cos θ e with the fluid-fluid interfacial tension γ (=2 √ 2Kr/3) and the equilibrium contact angle θ e . In contrast to some of the formulations in the literature, 17,21,29,30,32 the present formulation recognizes the fluid slip velocity on the wall, which is necessary to predict the fluid flow in fine resolutions. In fact, it was shown to reproduce the results of molecular dynamics simulations 31 by adjusting the two parameters M and Γ, which are not identified just from the material properties, in some preliminary test cases.…”

“…The microscopic contact angle has been investigated theoretically 11 and by means of molecular dynamics (MD) simulation [12][13][14][15][16] and continuum mechanics; [16][17][18][19] however, the effect of the length scales involved in the contact angle measurement and the difference between the microscopic and apparent contact angles have not been explored. When it comes to the apparent contact angle, there have also been theoretical studies [20][21][22][23] and computational studies, 17,24,25 but there has been no attempt to investigate the universal behavior of the apparent contact angle claimed in the experimental studies.…”

“…[26][27][28] There have been several other approaches to give the wall boundary conditions for the flow with moving contact lines. Jacqmin 29 obtained the boundary condition for the Cahn-Hilliard equation by considering that the total fluid energy remains unchanged by the fluid advection, followed by Carlson et al 30 and Yue and Feng, 17 who additionally took into account the relaxation of the order parameter on the wall. Their approaches led to different set of wall boundary conditions including the no-slip boundary condition for the Navier-Stokes equation.…”

A multiscale diffuse-interface model for two-phase flow in porous media

“…The Navier-Stokes-Cahn-Hilliard (NSCH) system provides a complete model for binary-fluid flows and, in combination with appropriate wetting boundary conditions [18,19], gives a detailed description of flow and wetting phenomena. Numerical simulation of the NSCH equations represents many challenges, however.…”

“…Diffuseinterface (or phase-field ) models [10][11][12] in principle provide a cogent microscale modeling paradigm for two-phase flows in porous media. Diffuseinterface models have emerged over the past years as a class of comprehensive and versatile models for multi-component flows, enabling realistic descriptions of complicated physical phenomena such as evaporation and condensation [13,14], topological changes of fluid-fluid interfaces due to coalescence and fissuring [12,15,16], contact-line motion at fluid-solid interfaces [17][18][19], and elasto-capillary effects of complex fluids on elastic solid substrates [20]. Diffuse-interface models intrinsically bypass the contactline paradox of sharp-interface models [17,21,22], thereby enabling the description of preferential-wetting phenomena.…”