Macroscopic Theory for Capillary-Pressure Hysteresis

Abstract

In this article, we present a theory of macroscopic contact angle hysteresis by considering the minimization of the Helmholtz free energy of a solid-liquid-gas system over a convex set, subject to a constant volume constraint. The liquid and solid surfaces in contact are assumed to adhere weakly to each other, causing the interfacial energy to be set-valued. A simple calculus of variations argument for the minimization of the Helmholtz energy leads to the Young-Laplace equation for the drop surface in contact with the gas and a variational inequality that yields contact angle hysteresis for advancing/receding flow. We also show that the Young-Laplace equation with a Dirichlet boundary condition together with the variational inequality yields a basic hysteresis operator that describes the relationship between capillary pressure and volume. We validate the theory using results from the experiment for a sessile macroscopic drop. Although the capillary effect is a complex phenomenon even for a droplet as various points along the contact line might be pinned, the capillary pressure and volume of the drop are scalar variables that encapsulate the global quasistatic energy information for the entire droplet. Studying the capillary pressure versus volume relationship greatly simplifies the understanding and modeling of the phenomenon just as scalar magnetic hysteresis graphs greatly aided the modeling of devices with magnetic materials.

On a two-point boundary value problem for the 2-D Navier-Stokes equations arising from capillary effect

“…We assume that γ SL take the values in the set [γ SLmin , γ SLmax ]. Justification for this assumption is provided in detail in [4,9].…”

“…Each cycle of wetting and drying involves a motion of the contact line and subsequent change in the curvature of the capillary interface. These two phenomena correspond to two different mechanisms for energy converted to heat -(i) hysteresis that has its origins in the adhesion between liquid and solid and is quasi-static or rate-independent [4]; (ii) viscous friction that opposes the motion of neighboring fluid molecules. In earlier work [4], we studied the quasi-static hysteresis phenomenon due to capillary effect and showed that the energy required to overcome adhesion while completing a cycle is simply obtained from the area of the graph of capillary pressure versus volume of the liquid.…”

“…These two phenomena correspond to two different mechanisms for energy converted to heat -(i) hysteresis that has its origins in the adhesion between liquid and solid and is quasi-static or rate-independent [4]; (ii) viscous friction that opposes the motion of neighboring fluid molecules. In earlier work [4], we studied the quasi-static hysteresis phenomenon due to capillary effect and showed that the energy required to overcome adhesion while completing a cycle is simply obtained from the area of the graph of capillary pressure versus volume of the liquid. In this article, we consider a special case of the more general theory in [4] that is amenable to computation of the energy lost due to viscosity when the capillary surface is deformed without motion of the solid-liquid contact line.…”

“…In earlier work [4], we studied the quasi-static hysteresis phenomenon due to capillary effect and showed that the energy required to overcome adhesion while completing a cycle is simply obtained from the area of the graph of capillary pressure versus volume of the liquid. In this article, we consider a special case of the more general theory in [4] that is amenable to computation of the energy lost due to viscosity when the capillary surface is deformed without motion of the solid-liquid contact line.…”

Realization of arbitrary hysteresis by a low-dimensional gradient flow

“…Various models of complex hysteresis relationships have been proposed and used by physicists and engineers. A few most prominent models include the Preisach model of magnetic materials, the Prandtl-Ishlinskii model of plasticity and friction and the Ising spin-interaction model of phase transitions in statistical physics (that have been also adapted to model sorption hysteresis [21][22][23][24]; damage accumulation [25]; constitutive laws of "smart" materials; hysteresis in economics [26][27][28], social dynamics and population biology [29][30][31][32]; see [33,34] for further examples). Mathematical tools that have been developed in the area of modeling hysteresis phenomena and systems with hysteretic components are diverse and include the method of hysteresis operators [35], differential inclusions [36], variational inequalities [37], variational approach based on Γ-convergence [38] and switched systems [39].…”