Lattice-Boltzmann simulations of the dynamics of liquid barrels

We study the relaxation towards equilibrium of a liquid barrel—a partially wetting droplet in a wedge geometry—using a diffuse-interface approach. We formulate a hydrodynamic model of the motion of the barrel in the framework of the Navier-Stokes and Cahn-Hilliard equations of motion. We present a lattice-Boltzmann method to integrate the diffuse-interface equations, where we introduce an algorithm to model the dynamic wetting of the liquid on smooth solid boundaries. We present simulation results of the over-damped dynamics of the liquid barrel. We find that the relaxation of the droplets is driven by capillary forces and damped by friction forces. We show that the friction is determined by the contribution of the bulk flow, the corner flow near the contact lines and the motion of the contact lines by comparing simulation results for the relaxation time of the barrel. Our results are in broad agreement with previous analytical predictions based on a sharp interface model

Theoretical and Computational Modelling of Wetting Phenomena in Smooth Geometries

Capillarity and wetting are the study of the interfaces that separate immiscible fluids and their interaction with solid surfaces. The interest in understanding capillary and wetting phenomena in complex geometries has grown in recent years. This is partly motivated by applications, such as the micro-fabrication of surfaces that achieve a controlled wettability, but also because of the fundamental role that the geometry of a solid surface can play in the statics and dynamics of liquids that come into contact with it. In this work, the statics and dynamics of liquids in contact with smooth, but non-planar geometries are studied. The approach is theoretical, and include mathematical modelling and numerical simulations using a new lattice-Boltzmann simulation method. The latter can account for solid boundaries of arbitrary geometry and a variety of boundary conditions relevant to experimental situations. The focus is directed to two model systems. First, an analysis on the statics and dynamics of a droplet inside wedge is performed, this is accomplished by proposing the shape of the droplet, a new shape that will be referred in this document as a “liquid barrel”. Using this assumption, the static position and shape of the droplet in response to an external body force is predicted. Then, the analysis is extended to include to dynamical situations in the absence of external forces, in which the translational motion of the liquid barrel towards equilibrium it is described. The proposed analytical model was validated by comparison with full 3D lattice-Boltzmann simulations and with recent experimental results. The applicability of these ideas is materialised with the purpose of achieving energy-invariant manipulation of a liquid barrel in a reconfigurable wedge. As a second model system, the evaporation of a sessile droplet in contact with a wavy solid surface was studied. Due to the non-planar solid topography, the droplet position in equilibrium is restricted to a discrete set of positions. It is shown that when the amplitude of the surface is sufficiently high, the droplet can suddenly readjust its shape and location to a new equilibrium configuration. These events occur in a time-scale much shorter than the evaporation time-scale, a “snap”. With numerical simulations and theoretical analysis, the study reveals the causes for the snap transitions, which lie in shape bifurcations of the droplet shapes, The analysis and results are compared against recent experiments of droplets evaporating on smooth sinusoidal surfaces. With the advent of low-friction surfaces, in which static friction is practically absent, the mobility of droplets is close to ideal, and with this, predicting and controlling them in static cases becomes a challenge. The analysis and results presented in this work can be used for manipulating the position and defining the shape of droplets via the geometry of their confinements

Slippery when wet: mobility regimes of confined drops in electrowetting

The motion of confined droplets in immiscible liquid-liquid systems strongly depends on the intrinsic relative wettability of the liquids on the confining solid material and on the typical speed, which can induce the formation of a lubricating layer of the continuous phase. In electrowetting, which routinely makes use of aqueous drops in ambient non-polar fluids that wet the wall material, electric stresses enter the force balance in addition to capillary and viscous forces and confinement effects. Here, we study the mobility of droplets upon electrowetting actuation in a wedge-shaped channel, and the subsequent relaxation when the electrowetting actuation is removed. We find that the droplets display two different mobility regimes: a fast regime, corresponding to gliding on a thin film of the ambient fluid, and a slow regime, where the film is replaced by direct contact between the droplet and the channel walls. Using a combination of experiments and numerical simulations, we show that the cross-over between these regimes arises from the interplay between the small-scale dynamics of the thin film of ambient fluid and the large-scale motion of the droplet. Our results shed light on the complex dynamics of droplets in non-uniform channels driven by electric actuation, and can thus help the rational design of devices based on electrowetting-driven droplet transport

Statics and dynamics of liquid barrels in wedge geometries

We present a theoretical study of the statics and dynamics of a partially wetting liquid droplet, of equilibrium contact angle , confined in a solid wedge geometry of opening angle . We focus on a mostly non-wetting regime, given by the condition , where the droplet forms a liquid barrel – a closed shape of positive mean curvature. Using a quasi-equilibrium assumption for the shape of the liquid–gas interface, we compute the changes to the surface energy and pressure distribution of the liquid upon a translation along the symmetry plane of the wedge. Our model is in good agreement with numerical calculations of the surface energy minimisation of static droplets deformed by gravity. Beyond the statics, we put forward a Lagrangian description of the droplet dynamics. We focus on the overdamped limit, where the driving capillary force is balanced by the frictional forces arising from the bulk hydrodynamics, the corner flow near the contact lines and the contact-line friction. Our results provide a theoretical framework to describe the motion of partially wetting liquids in confinement, and can be used to gain further understanding on the relative importance of dissipative processes that span from microscopic to macroscopic length scales

Energy invariance in capillary systems

We demonstrate the continuous translational invariance of the energy of a capillary surface in contact with reconfigurable solid boundaries. We present a theoretical approach to find the energy-invariant equilibria of spherical capillary surfaces in contact with solid boundaries of arbitrary shape and examine the implications of dynamic frictional forces upon of a reconfiguration of the boundaries. Experimentally, we realise our ideas by manipulating the position of a droplet in a wedge geometry using lubricant-impregnated solid surfaces, which eliminate the contact-angle hysteresis and provide a test bed for quantifying dissipative losses out of equilibrium. Our experiments show that dissipative energy losses for an otherwise energy-invariant reconfiguration are relatively small, provided that the actuation timescale is longer than the typical relaxation timescale of the capillary surface. We discuss the wider applicability of our ideas as a pathway for liquid manipulation at no potential energy cost in low-pinning, low-friction situations

The long cross-over dynamics of capillary imbibition

Spontaneous capillary imbibition is a classical problem in interfacial fluid dynamics with a broad range of applications, from microfluidics to agriculture. Here we study the duration of the cross-over between an initial linear growth of the imbibition front to the diffusive-like growth limit of Washburn's law. We show that local-resistance sources, such as the inertial resistance and the friction caused by the advancing meniscus, always limit the motion of an imbibing front. Both effects give rise to a cross-over of the growth exponent between the linear and the diffusive-like regimes. We show how this cross-over is much longer than previously thought - even longer than the time it takes the liquid to fill the porous medium. Such slowly slowing-down dynamics is likely to cause similar long cross-over phenomena in processes governed by wetting

Lattice-Boltzmann Simulations of Electrowetting Phenomena

When a voltage difference is applied between a conducting liquid and a conducting (solid) electrode, the liquid is observed to spread on the solid. This phenomenon, generally referred to as electrowetting, underpins a number of interfacial phenomena of interest in applications that range from droplet microfluidics to optics. Here, we present a lattice-Boltzmann method that can simulate the coupled hydrodynamics and electrostatics equations of motion of a two-phase fluid as a means to model electrowetting phenomena. Our method has the advantage of modelling the electrostatic fields within the lattice-Boltzmann algorithm itself, eliminating the need for a hybrid method. We validate our method by reproducing the static equilibrium configuration of a droplet subject to an applied voltage and show that the apparent contact angle of the drop depends on the voltage following the Young-Lippmann equation up to contact angles of ≈ 50°. At higher voltages, we observe a saturation of the contact angle caused by the competition between electric and capillary stresses, similar to previous experimental observations. We also study the stability of a dielectric film trapped between a conducting fluid and a solid electrode and find a good agreement with analytical predictions based on lubrication theory. Finally, we investigate the film dynamics at long times and report observations of film breakup and entrapment similar to previously reported experimental results