The surface of a liquid near a moving contact line is highly curved owing to diverging viscous forces. Thus, microscopic physics must be invoked at the contact line and matched to the hydrodynamic solution farther away. This matching has already been done for a variety of models, but always assuming the limit of vanishing speed. This excludes phenomena of the greatest current interest, in particular the stability of contact lines. Here we extend perturbation theory to arbitrary order and compute finite speed corrections to existing results. We also investigate the impact of the microscopic physics on the large-scale shape of the interface.
The moving contact-line problem is of both theoretical and practical interest. The dynamic contact angle changes with the capillary number defined by the contact-line speed, and the correlation also depends on the equilibrium contact angle measured at the static state. This correlation is usually required as an input to the traditional solvers based on the Navier-Stokes-like equations, but it is simulated as an output in the current study using the lattice Boltzmann method (LBM) in a displacement process of two-immiscible fluids. The macroscopic theory and the molecular dynamics (MD) simulatio…
The role of surface tension and wettability in the dynamics of air-liquid interfaces during immiscible fluid displacement flows in capillary tube driven by pressure has been investigated. The contact angle and capillary number drive the force wetting processes which is controlled by the balance between the capillary and the viscous lubrication forces. The dynamic wetting condition with the critical capillary number is studied analytically and validated experimentally, which demonstrates that the critical capillary number is associated with the contact angle, slip length and capillary radius.
Equilibrium and motion of a contact line are viewed as analogs of phase equilibrium and motion of an interphase boundary. This point of view makes evident the tendency to minimization of the length of the contact line at equilibrium. The concept of line tension is, however, of limited applicability, in view of a qualitatively different relaxation response of the contact line, compared to a two-dimensional curve. Both the analogy and qualitative distinction extend to a non-equilibrium situation arising due to coupling with reversible substrate modification. Under these conditions, the contact l…
Given the importance of dynamic contact angle, its numerous applications and the complexity and difficulty of use of available approaches, here we present a new simple semi-empirical models for estimation of dynamic contact angle's dependence on the wetting line velocity. These models should be applicable to any geometry, a very large range of capillary numbers and static contact angles and all solid-liquid-gas systems without requiring further experiments. Two simple equations are intuitively derived from the most promising theoretical dynamic contact angle models, the hydrodynamic and the mo…
We present a systematic study of capillary filling for a binary fluid by using a mesoscopic lattice Boltzmann model for immiscible fluids describing a diffusive interface moving at a given contact angle with respect to the walls. The phenomenological way to impose a given contact angle is analysed. Particular attention is given to the case of complete wetting, that is contact angle equal to zero. Numerical results yield quantitative agreement with the theoretical Washburn law, provided that the correct ratio of the dynamic viscosities between the two fluids is used. Finally, the presence of pr…
The surface free energy, or surface tension, of a liquid interface gives rise to a pressure jump when the interface is curved. Here we show that a similar capillary pressure arises at the interface of soft solids. We present experimental evidence that immersion of a thin elastomeric wire into a liquid induces a substantial elastic compression due to the solid capillary pressure at the bottom. We quantitatively determine the effective surface tension from the elastic displacement field, and find a value comparable to the liquid-vapor surface tension. Most importantly, these results also reveal …
The effect of confining walls on the fluctuation of a nanoscale thin film's free surface is studied using stochastic thin-film equations (STFEs). Two canonical boundary conditions are employed to reveal the influence of the confinement: (1) an imposed contact angle and (2) a pinned contact line. A linear stability analysis provides the wave eigenmodes, after which thermal-capillary-wave theory predicts the wave fluctuation amplitudes. Molecular dynamics (MD) simulations are performed to test the predictions, and a Langevin diffusion model is proposed to capture oscillations of the contact line…
We investigate the behaviour of accelerating contact lines in an unsteady quasi-capillary channel flow. The configuration consists of a liquid column that moves along a vertical 2D channel, open to the atmosphere and driven by a controlled pressure head. Both advancing and receding contact lines were analyzed to test the validity of classic models for dynamic wetting and to study the flow field near the interface. The operating conditions are characterized by a large acceleration, thus dominated by inertia. The shape of the moving meniscus was retrieved using Laser-Induced Fluorescence (LIF)-b…
The merging of immiscible drops differs significantly from the merging of miscible drops due to the formation of a liquid-liquid interface between drops. The immiscibility requires the formation of a four-phase contact point, where the drops, the gas and the substrate meet. We show that this point has its own unique dynamics, never studied beforehand. For very different scenarios, the propagation distance of this point follows scales with time like t^{0.5}. A model balancing the driving and dissipative forces agrees with our experiments.
We consider the phase equilibria of a fluid confined in a deep capillary groove of width $L$ with identical side walls and a bottom made of a different material. All walls are completely wet by the liquid. Using density functional theory and interfacial models, we show that the meniscus separating liquid and gas phases at two phase capillary-coexistence meets the bottom capped end of the groove at a capillary contact angle $θ^{\rm cap}(L)$ which depends on the difference between the Hamaker constants. If the bottom wall has a weaker wall-fluid attraction than the side walls, then $θ^{\rm cap}>…
We predict the volume of liquid recovered from different-shaped prismatic channels following gas displacement. This recovery factor depends strongly on the contact angle at which the gas-liquid interfaces meet the walls of the channel. We show that scaling the contact angle by a critical value, dependent on the channel shape, leads to a master curve for the recovery factor.} Our numerical simulations with the Surface Evolver are supported by theoretical calculations for the capillary entry pressure in a channel with a triangular or rectangular cross-section and for the interface shape in a "sc…
We propose a numerical approach to regularize the contact line singularity appearing in the computation of viscous capillary-gravity waves with moving contact line in cylindrical containers. The linearized Navier-Stokes equations are complemented by a macroscopic Navier-like slip condition on the container side wall, with a depth-varying slip-length bridging a free-slip condition at the contact line to a no-slip condition further away from the meniscus. In accordance with suggestions from the literature, this characteristic penetration depth is chosen as the typical Stokes layer thickness pert…
Capillary attraction between identical millimeter sized spheres floating at a liquid-air interface and the resulting aggregation is investigated at low Reynolds number. We show that the measured capillary forces between two spheres as a function of distance can be described by expressions obtained using the Nicolson approximation at low Bond numbers for far greater particle sizes than previously assumed. We find that viscous hydrodynamics interactions between the spheres needs to be included to describe the dynamics close to contact. We then consider the aggregates formed when a third sphere i…
Contact of a fluid with a solid or an elastic wall is investigated. The wall exerts molecular forces on the fluid which is locally strongly nonhomogeneous. The problem is approached with a fluid energy of the second gradient form and a wall surface energy depending on the value of the fluid density at the contact. >From the virtual work principle are obtained limit conditions taking into account the fluid density, its normal derivative to the wall and the curvature of the surface
The contact angle that a liquid drop makes on a soft substrate does not obey the classical Young's relation, since the solid is deformed elastically by the action of the capillary forces. The finite elasticity of the solid also renders the contact angles different from that predicted by Neumann's law, which applies when the drop is floating on another liquid. Here we derive an elasto-capillary model for contact angles on a soft solid, by coupling a mean-field model for the molecular interactions to elasticity. We demonstrate that the limit of vanishing elastic modulus yields Neumann's law or a…
We account for the presence of surface charges towards describing variations in the dynamic contact angle of an advancing liquid-gas meniscus. Starting from the thin-film based formalism, we present closed-form analytical expressions relating the dynamic contact-angle with the capillary number (essentially normalized contact-line speed) and other interfacial parameters. Specifically, our analysis presents, within the realm of hydrodynamic paradigm, a connection between the micro- and macro-scale physics at the vicinity of the contact-line region, taking the combined confluence of viscous and c…
The impact of solid and liquid objects (projectiles) onto liquids and soft solids (targets) generally results on the creation and expansion of an air cavity inside the impacted objects. The dynamics of cavity expansion and collapse depends on the projectile inertia as well as on the target properties. In this paper we study the impact of microfluidic jets generated by thermocavitation processes on a capillary bridge between two parallel planar walls. Different capillary bridge types were studied, Newtonian liquids, viscoelastic liquids and agarose gels. Thus, we compare the cavity formation an…
The dynamics of particles attached to an interface separating two immiscible fluids are encountered in a wide variety of applications. Here we present a combined asymptotic and numerical investigation of the fluid motion past spherical particles attached to a deformable interface undergoing uniform creeping flows in the limit of small Capillary number and small deviation of the contact angle from 90 degrees. Under the assumption of a constant three-phase contact angle, we calculate the interfacial deformation around an isolated particle and a particle pair. Applying the Lorentz reciprocal theo…
A sessile liquid drop can deform the substrate on which it rests if the solid is sufficiently "soft". In this paper we compute the detailed spatial structure of the capillary forces exerted by the drop on the solid substrate using a model based on Density Functional Theory. We show that, in addition to the normal forces, the drop exerts a previously unaccounted tangential force. The resultant effect on the solid is a pulling force near the contact line directed towards the interior of the drop, i.e. not along the interface. The resulting elastic deformations of the solid are worked out and ill…
A recent experiment showed that cylindrical segments of water filling a hydrophilic stripe on an otherwise hydrophobic surface display a capillary instability when their volume is increased beyond the critical volume at which their apparent contact angle on the surface reaches ninety degrees (Gau et al., Science, 283, 1999). Surprisingly, the fluid segments did not break up into droplets -- as would be expected for a classical Rayleigh-Plateau instability -- but instead displayed a long-wavelength instability where all excess fluid gathered in a single bulge along each stripe. We consider here…
We study the dynamics of the interface between two immiscible fluids in contact with a chemically homogeneous moving solid plate. We consider the generic case of two fluids with any viscosity ratio and of a plate moving in either directions (pulled or pushed in the bath). The problem is studied by a combination of two models, namely an extension to finite viscosity ratio of the lubrication theory and a Lattice Boltzmann method. Both methods allow to resolve, in different ways, the viscous singularity at the triple contact between the two fluids and the wall. We find a good agreement between th…
We show how the capillary filling of microchannels is affected by posts or ridges on the sides of the channels. Ridges perpendicular to the flow direction introduce contact line pinning which slows, or sometimes prevents, filling; whereas ridges parallel to the flow provide extra surface which may enhances filling. Patterning the microchannel surface with square posts has little effect on the ability of a channel to fill for equilibrium contact angle $θ_e \lesssim 30^{\mathrm{o}}$. For $θ_e \gtrsim 60^{\mathrm{o}}$, however, even a small number of posts can pin the advancing liquid front.
Local contact line pinning prevents droplets from rearranging to minimal global energy, and models for droplets without pinning cannot predict their shape. We show that experiments are much better described by a theory, developed herein, that does account for the constrained contact line motion, using as example droplets on tilted plates. We map out their shapes in suitable phase spaces. For 2D droplets, the critical point of maximum tilt depends on the hysteresis range and Bond number. In 3D, it also depends on the initial width, highlighting the importance of the deposition history.
We study the flow close to an advancing contact line in the limit of small capillary number. To take into account wetting effects, both long and short-ranged contributions to the disjoining pressure are taken into account. In front of the contact line, there is a microscopic film corresponding to a minimum of the interaction potential. We compute the parameters of the contact line solution relevant to the matching to a macroscopic problem, for example a spreading droplet. The result closely resembles previous results obtained with a slip model.
Wetting flows are controlled by the contact line motion. We derive an equation that describes the slow time evolution of the triple solid-liquid-fluid contact line for an arbitrary distribution of defects on a solid surface. The capillary rise along a partially wetted infinite vertical wall is considered. The contact line is assumed to be only slightly deformed by the defects. The derived equation is solved exactly for a simple example of a single defect. Introduction.
Many textbooks dealing with surface tension favor the thermodynamic approach (minimization of some thermodynamic potential such as free energy) over the mechanical approach (balance of forces) to describe capillary phenomena, stating that the latter is flawed and misleading. Yet, mechanical approach is more intuitive for students than free energy minimization, and does not require any knowledge of thermodynamics. In this paper we show that capillary phenomena can be unmistakably described using the mechanical approach, as long as the system on which the forces act is properly defined. After re…
Fast contact-line motion of a droplet spreading on a solid substrate under the electrowetting effect generates strong capillary waves on the droplet's surface. The capillary waves may be strong enough to induce ejection of a satellite droplet from the primary one. In this study, we show that the size of the satellite droplet and the ejection time are not only dependent on the contact-line velocity, which directly relates to the applied voltage enabling the electrowetting effect, but also affected by the ejection dynamics. We derive a theoretical model of the criteria for droplet ejection and e…
This paper presents a theoretical analysis of the liquid film dynamics during the oscillation of a meniscus between a liquid and its vapour in a cylindrical capillary. By using the theory of Taylor bubbles, the dynamic profile of the deposited liquid film is calculated within the lubrication approximation accounting for the finiteness of the film length, i.e. for the presence of the contact line. The latter is assumed to be pinned on a surface defect and thus immobile; the contact angle is allowed to vary. The fluid flow effect on the curvature in the central meniscus part is neglected. This c…
Capillary phenomena are involved in many industrial processes, especially those dealing with composite manufacturing. However, their modelling is still challenging. Therefore, a finite element setting is proposed to better investigate this complex issue. The variational formulation of a liquid-air Stokes system is established, while the solid substrate is described through boundary conditions. Expressing the weak form of Laplace's law over liquid-air, liquid-solid and air-solid interfaces, leads to a natural enforcement of the mechanical equilibrium over the wetting line, without imposing expl…
We investigate the transition to a Landau-Levich-Derjaguin film in forced dewetting using a quadtree adaptive solution to the Navier-Stokes equations with surface tension. We use a discretization of the capillary forces near the receding contact line that yields an equilibrium for a specified contact angle $θ_Δ$ called the numerical contact angle. Despite the well-known contact line singularity, dynamic simulations can proceed without any explicit additional numerical procedure. We investigate angles from $15^\circ$ to $110^\circ$ and capillary numbers from $0.00085$ to $0.2$ where the mesh si…
The breakup dynamics of a capillary bridge on a hydrophobic stripe between two hydrophilic stripes is studied experimentally and numerically using direct numerical simulations. The capillary bridge is formed from an evaporating water droplet wetting three neighboring stripes of a chemically patterned surface. By considering the breakup process in a phase space representation, the breakup dynamics can be evaluated without the uncertainty in determining the precise breakup time. The simulations are based on the Volume-of-Fluid (VOF) method implemented in Free Surface 3D (FS3D). In order to const…
We report experiments on the rapid contact line motion present in the early stages of capillary driven spreading of drops on dry solid substrates. The spreading data fails to follow a conventional viscous or inertial scaling. By integrating experiments and simulations, we quantify a contact line friction ($μ_f$), which is seen to limit the speed of the rapid dynamic wetting. A scaling based on this contact line friction is shown to yield a universal curve for the evolution of the contact line radius as a function of time, for a range of fluid viscosities, drop sizes and surface wettabilities.
We consider the effect of droplet geometry on the early-stages of coffee ring formation during the evaporation of a thin droplet with an arbitrary simple, smooth, pinned contact line. We perform a systematic matched asymptotic analysis of the small-capillary number, large-solutal Peclet number limit for two evaporative models: a kinetic model, in which the evaporative flux is constant across the droplet, and a diffusive model, in which the evaporative flux is singular at the contact line. For both evaporative models, solute is transported to the contact line by a capillary flow while, local to…
In this paper, we systematically investigate the wetting behavior of a liquid ring in a cylindrical capillary tube. We obtain analytical solutions of the axisymmetric Young-Laplace equation for arbitrary contact angles. We find that, for specific values of the contact angle and the volume of the liquid ring, two solutions of the Young-Laplace equation exist, but only the one with the lower value of the total interfacial energy corresponds to a stable configuration. The transition to an unstable configuration is characterized by specific critical parameters such as the liquid volume, throat dia…
We discuss the equilibrium condition for a liquid that partially wets a solid on the level of intermolecular forces. Using a mean field continuum description, we generalize the capillary pressure from variation of the free energy and show at what length scale the equilibrium contact angle is selected. After recovering Young's law for homogeneous substrates, it is shown how hysteresis of the contact angle can be incorporated in a self-consistent fashion. In all cases the liquid-vapor interface takes a nontrivial shape, which is compared to models using a disjoining pressure.
We report on an experimental investigation of the propagation of gravity-capillary waves in a narrow channel with a pinned contact line. By using Fourier Transform Profilometry (FTP) we measure the static curved meniscus as well as the surface perturbation. By varying the channel width, between 7 and 15 times the capillary length, we show how edge constraints modify the surface curvature and therefore the dispersion relation. From the space-time resolved field, we obtain a decomposition of the linear mode onto transverse modes satisfying the condition of pinned contact line. This approach, in …
Liquids in oil-bearing porous media assume complex shapes that depend on the reservoir characteristics and the wetting properties of the liquid. The wide variation in the geometry of rock formations makes it difficult to accurately predict the capillary pressure of small volumes of liquid and hence the likelihood of being able to move it. Here we consider the situation in which a small volume of liquid surrounds an upright cylinder on a flat substrate and predict the shape that the liquid takes and its capillary pressure. We validate our predictions by comparing with Surface Evolver simulation…
We study spreading of a droplet, with insoluble surfactant covering its capillary surface, on a textured substrate. In this process, the surfactant-dependent surface tension dominates the behaviors of the whole dynamics, particularly the moving contact lines. This allows us to derive the full dynamics of the droplets laid by the insoluble surfactant: (i) the moving contact lines, (ii) the evolution of the capillary surface, (iii) the surfactant dynamics on this moving surface with a boundary condition at the contact lines and (iv) the incompressible viscous fluids inside the droplet. Our deriv…
The effect of 'bridge splitting' is considered in the case of capillary adhesion: for a fixed total volume of liquid, does having more capillary bridges increase the total adhesion force? Previous studies have shown that the capillary-induced adhesion force between two planar surfaces is only substantially enhanced by bridge splitting in specific circumstances. Here this previous result is reconsidered, and it is shown that bridge splitting may significantly increase the adhesion forces when one of the surfaces is rough. The resistance to shear is also examined, and it is shown that bridge spl…
This article discusses the way the standard description of capillary filling dynamics has to be modified to account for liquid/solid slip in nanometric pores. It focuses in particular on the case of a large slip length compared to the pore size. It is shown that the liquid viscosity does not play a role, and that the flow is only controlled by the friction coefficient of the liquid at the wall. Moreover in the Washburn regime, the filling velocity does not depend on the tube radius. Finally, molecular dynamics simulations suggest that this standard description fails to describe the early stage…
When a very thin capillary is inserted into a liquid, the liquid is sucked into it: this imbibition process is controlled by a balance of capillary and drag forces, which are hard to quantify experimentally, in particularly considering flow on the nanoscale. By computer experiments using a generic coarse-grained model, it is shown that an analysis of imbibition forced by a controllable external pressure quantifies relevant physical parameter such as the Laplace pressure, Darcy's permeability, effective pore radius, effective viscosity, dynamic contact angle and slip length of the fluid flowing…
Wetting phenomena play a key role in flows through porous media. Relative permeability and capillary pressure-saturation functions show a high sensitivity to wettability, which has different definitions at the continuum- and pore-scale. At the continuum-scale, the state of wetting is defined as Amott-Harvey or USBM (United States Bureau of Mines) by capillary pressure drainage and imbibition cycles. At the pore-scale, the concept of contact angle is used, which until recently was not experimentally possible to determine within an opaque porous medium. Recent progress on measurements of pore-sc…
