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In this paper, we study the contact angle hysteresis on a slowly moving rough boundary using a phase-field model with a relaxed boundary condition. In particular, we want to model the recent experi...

A dynamic wetting problem is studied for a moving thin fiber inserted in fluid and with a chemically inhomogeneous surface. A reduced model is derived for contact angle hysteresis by using the Onsager principle as an approximation tool. The model is simple and captures the essential dynamics of the contact angle. From this model, we derive an upper bound of the advancing contact angle and a lower bound of the receding angle, which are verified by numerical simulations. The results are consistent with the quasi-static results. The model can also be used to understand the asymmetric dependence of the advancing and receding contact angles on the fiber velocity, which was observed recently in the physical experiments reported in the work of Guan et al. [“Asymmetric and speed-dependent capillary force hysteresis and relaxation of a suddenly stopped moving contact line,” Phys. Rev. Lett. 116, 066102 (2016)].

We study the interface dynamics and contact angle hysteresis in a two dimensional, chemically patterned channel described by the Cahn-Hilliard equation with a relaxation boundary condition. A system for the dynamics of the contact angle and contact point is derived in the sharp interface limit. We then analyze the behavior of the solution using the phase plane analysis. We observe the stick-slip of the contact point and the contact angle hysteresis. As the size of the pattern decreases to zero, the stick-slip becomes weaker but the hysteresis becomes stronger in the sense that one observes either the advancing contact angle or the receding contact angle without any switching in between. Numerical examples are presented to verify our analysis.

We analyze the Cahn–Hilliard equation with a relaxation boundary condition modeling the evolution of an interface in contact with the solid boundary. An L ∞ estimate is established which enables us to prove the global existence of the solution. We also study the sharp interface limit of the system. The dynamic of the contact point and the contact angle are derived and the results are compared with the numerical simulations.

We provide rigorous justification for the classical Wenzel equation for the roughness enhanced effective contact angle. The minimization of the total surface energy is reformulated into a variational problem. As the size of the roughness becomes small, we show convergence of the minimizer. The limiting minimizer and effective contact angle are explicitly calculated to verify the Wenzel equation.

We study a stationary wetting problem on rough and inhomogeneous solid surfaces. We derive a new formula for the apparent contact angle by an asymptotic two-scale homogenization method. The formula reduces to a modified Wenzel equation for geometrically rough surfaces and a modified Cassie equation for chemically inhomogeneous surfaces. Unlike the classical Wenzel and Cassie equations, the modified equations correspond to local minimizers of the total interface energy in the solid-liquid-air system, so that they are consistent with experimental observations. The homogenization results are proved rigorously by a variational method.

In this paper, we derive a modified Cassie's equation for wetting on chemically patterned surfaces from a homogenization approach. The derivation re- veals that effective contact angle is a local average of the static contact angle along the contact line which de- scribes all possible equilibrium states including the local minimum of the free energy of the system. The usual Cassie's state which corresponds to the global minimum is only a special case. We then discuss the contact angle hysteresis on chemically patterned surfaces.

Wetting and contact angle hysteresis on chemically patterned surfaces in two dimensions are analyzed from a stationary phase-field model for immiscible two phase fluids. We first study the sharp-interface limit of the model by the method of matched asymptotic expansions. We then justify the results rigorously by the $\Gamma$-convergence theory for the related variational problem and study the properties of the limiting minimizers. The results also provide a clear geometric picture of the equilibrium configuration of the interface. This enables us to explicitly calculate the total surface energy for the two phase systems on chemically patterned surfaces with simple geometries, namely the two phase flow in a channel and the drop spreading. By considering the quasi-static motion of the interface described by the change of volume (or volume fraction), we can follow the change-of-energy landscape which also reveals the mechanism for the stick-slip motion of the interface and contact angle hysteresis on the che...

A new variational method is proposed to calculate the evolution of liquid film and liquid droplet moving on a solid substrate. A simple time evolution equation is obtained for the contact angle of a liquid film that starts to move on a horizontal substrate. The equation indicates the dynamical transition at the receding side and the ridge formation at the advancing side. The same method is applied for the evolution of a droplet that starts to move on an inclined solid surface, and again the characteristic shape change of the droplet is obtained by solving a simple ordinary differential system. We will show that this method has a potential application to a wide class of problems of droplets moving on a substrate.