Static Meniscus Research Articles

Mechanically forced meniscus water waves in a square container with functionalized wetting conditions

Capillary–gravity surface waves, called meniscus waves, are excited at the air–water interface near the lateral boundary of a mechanically forced square container. Experiments are conducted in two working containers: one is overall hydrophilic at four boundaries and the other is functionalized to have a hydrophobic boundary. The static meniscus structures and instantaneous wave topography are reconstructed at high resolution from the side-view and top-view high-speed camera recordings. Wave resonances arise from the interaction of four inward-traveling waves, with the frequency-response characteristics indicating the effect of wetting conditions on the selection of standing wave patterns. The most-resonant modes are identified as a superposition of dominant modes and higher-order components. An analytical model of the rest-state meniscus and resonant-state wave surface decomposes the wave responses into two groups of resonances along two transverse directions linked by a frequency-dependent parameter γ. Theoretical selection of the dominant wave components and the second-order harmonics is in close agreement with the spatiotemporal surface-fitting experimental results. Additionally, the wetting boundary conditions can be functionalized to excite specific standing wave patterns, which could provide a novel approach to precise pattern control in bioprinting technology for tissue engineering applications.

Physics of the Sub-Monolayer Lubricant in the Head-Disk Interface

This review presents a series of studies which have demonstrated that the diffusion characteristics of rarefied mobile lubricant films used in modern magnetic disks can be evaluated by a novel diffusion theory based on continuum mechanics, and that the meniscus force of the rarefied film is the major interaction force at the head-disk interface. The limitations of the conventional diffusion and disjoining pressure equations are first shown, and diffusion and disjoining pressure equations for rarefied liquid films are proposed, showing that the diffusion coefficient is in good agreement with the experiment. The experiment, in which glass spheres with radii of 1 and 2 mm collided with magnetic disks of different film thicknesses, showed that attraction similar to the pull-off forces of a static meniscus was measured only at the separation. Furthermore, mathematical analysis of the elastic meniscus contact between a sphere and a plane with a submonolayer liquid film showed that the maximum adhesion force is equal to the meniscus pull-off force and that the contact characteristics become similar to those of the JKR theory as the liquid film thickness decreases. A basic physical model of submonolayer liquid film is also proposed to justify the continuum mathematical equations.

Cavitation-induced microjets tuned by channels with alternating wettability patterns

A laser pulse focused near the closed end of a glass capillary partially filled with water creates a vapor bubble and an associated pressure wave. The pressure wave travels through the liquid toward the meniscus where it is reflected, creating a fast, focused microjet. In this study, we selectively coat the hydrophilic glass capillaries with hydrophobic strips along the capillary. The result after filling the capillary is a static meniscus which has a curvature markedly different than an unmodified capillary. This tilting asymmetry in the static meniscus alters the trajectory of the ensuing jets. The hydrophobic strips also influence the advancing contact line and receding contact line as the vapor bubble expands and collapses. We present thirteen different permutations of this system which includes three geometries and four coating schemes. The combination of geometry and coatings influences the jet breakup, the resulting drop size distribution, the trajectory of the jet tip, and the consistency of jet characteristics across trials. The inclusion of hydrophobic strips promotes jetting in line with the channel axis, with the most effective arrangement dependent on channel size.

Subharmonic parametric instability in nearly brimful circular cylinders: a weakly nonlinear analysis

In labscale Faraday experiments, meniscus waves respond harmonically to small-amplitude forcing without threshold, hence potentially cloaking the instability onset of parametric waves. Their suppression can be achieved by imposing a contact line pinned at the container brim with static contact angle $\theta _s=90^{\circ }$ (brimful condition). However, tunable meniscus waves are desired in some applications as those of liquid-based biosensors, where they can be controlled adjusting the shape of the static meniscus by slightly underfilling/overfilling the vessel ( $\theta _s\ne 90^{\circ }$ ) while keeping the contact line fixed at the brim. Here, we refer to this wetting condition as nearly brimful. Although classic inviscid theories based on Floquet analysis have been reformulated for the case of a pinned contact line (Kidambi, J. Fluid Mech., vol. 724, 2013, pp. 671–694), accounting for (i) viscous dissipation and (ii) static contact angle effects, including meniscus waves, makes such analyses practically intractable and a comprehensive theoretical framework is still lacking. Aiming at filling this gap, in this work we formalize a weakly nonlinear analysis via multiple time scale method capable of predicting the impact of (i) and (ii) on the instability onset of viscous subharmonic standing waves in both brimful and nearly brimful circular cylinders. Notwithstanding that the form of the resulting amplitude equation is in fact analogous to that obtained by symmetry arguments (Douady, J. Fluid Mech., vol. 221, 1990, pp. 383–409), the normal form coefficients are here computed numerically from first principles, thus allowing us to rationalize and systematically quantify the modifications on the Faraday tongues and on the associated bifurcation diagrams induced by the interaction of meniscus and subharmonic parametric waves.

Thin-Film Evaporation in a Mesh Screen Wick

Abstract Porous capillary wick structures are being employed in two-phase thermal management devices owing to their pumping capabilities and thermal performance enhancement during evaporation of the working fluid. Thin-film evaporation in a porous wick depends primarily on the shape of the liquid–vapor meniscus, especially near the wall. The primary objective of this paper is to study and investigate the thin-film evaporation of the liquid in a unit cell representation (UCR) of a single layer of a metallic wire mesh screen. The volume-of-fluid (VOF) method, which is an interface-capturing technique in multiphase flow modeling, is employed to obtain the steady-state meniscus shape under equilibrium conditions. This paper demonstrates the impact of the equilibrium contact angle (θ) and the initial meniscus height (H) on the steady-state interfacial pressure difference. It outlines a detailed process for estimating 3D interfacial surfaces, obtained from the VOF solution, to generate the final geometry for the thin-film evaporation analysis. A static meniscus heat-transfer model is subsequently solved using the commercial finite volume code, ansysfluent, to obtain the temperature and flow characteristics during evaporation. The relationship of parameters such as the average evaporation mass fluxes and heat transfer coefficients are estimated and presented in this paper. Finally, the relationship between the pressure drop across the liquid–vapor meniscus and the thin-film evaporation rate for screen mesh wicks is discussed.

Design of Under-Actuated Soft Adhesion Actuators for Climbing Robots.

Since climbing robots mainly rely on adhesion actuators to achieve adhesion, robust adhesion actuators have always been the challenge of climbing robot design. A novel under-actuated soft adhesion actuator (USAA) proposed in this paper for climbing robots can generate adhesion through robot’s load applied to the actuator. The actuator is composed of a soft film/substrate structure with an annular groove on the substrate and a cavity on the soft film. To fabricate the actuator, we first study the influence of the geometric parameters of the USAA on the maximum adhesion of the actuator by analysis and experiments, and then combine these parameters and the boundary conditions of the static meniscus in the mold to design the mold. Moreover, we fabricate a climbing robot equipped with USAAs and evaluate its performance on horizontal and inclined surfaces with a wide range of characteristics. The USAA can generate strong and controllable adhesion to various smooth and semi-smooth surfaces. Furthermore, the fabricated robot performs well on various surfaces under a certain load (at least 500 g) and speed (369 mm/min) through experiments. It’s adaptability to a variety of surfaces enables a wide range of applications and pushes the boundaries of soft adhesion actuators.

The effect of isolated ridges and grooves on static menisci in rectangular channels

We present theoretical and numerical results that demonstrate the sensitivity of the shape of a static meniscus in a rectangular channel to localised geometric perturbations in the form of narrow ridges and grooves imposed on the channel walls. The Young–Laplace equation is solved for a gas/liquid interface with fixed contact angle using computations, analytical arguments and semi-analytical solutions of a linearised model for small-amplitude perturbations. We find that the local deformation of the meniscus's contact line near a ridge or groove is strongly dependent on the shape of the perturbation. In particular, small-amplitude perturbations that change the channel volume induce a change in the pressure difference across the meniscus, resulting in long-range curvature of its contact line. We derive an explicit expression for this induced pressure difference directly in terms of the boundary data. We show how contact lines can be engineered to assume prescribed patterns using suitable combinations of ridges and grooves.

Space-time-resolved measurements of the effect of pinned contact line on the dispersion relation of water waves

Surface wave dynamics in small scale configurations can be substantially modified by edge constraints, in particular a pinned contact line, when the force it exerts is non-negligible with respect to gravity and surface tension. This work develops a hybrid approach, which combines a theoretical model with a complete space-time resolved measurement of the surface deformation and static menisci, allowing an accurate estimation of the contribution of a pinned contact line to shift the dispersion relation towards faster phase velocities.

Classification of static cylindrical menisci confined in a wedge

In this study, we analyze the shape of a puddle in a plane wedge with a translational symmetry. Under the assumption of equilibrium, where the viscous force is negligible, the shape of the puddle can be approximated as a curve represented as a particular case of Euler’s elastica. For specified geometrical parameter and wetting properties, the shape of the curve is solely determined by a single dimensionless parameter, i.e., either the shape factor or the dimensionless puddle height. The shape factor is a convex function of height. Our analysis shows that, the physically meaningful section of the curve can be classified into three types based on the wedge shape, contact angles, material properties, and the puddle size.

Dip coating of cylinders with Newtonian fluids

HypothesisThe Landau-Levich-Derjaguin (LLD) theory is widely applied to predict the film thickness in the dip-coating process. However, the theory was designed only for flat plates and thin fibers. Fifty years ago, White and Tallmadge attempted to generalize the LLD theory to thick rods using a numerical solution for a static meniscus and the LLD theory to forcedly match their numeric solution with the LLD asymptotics. The White-Talmadge solution has been criticized for not being rigorous yet widely used in engineering applications mostly owing to the lack of alternative solutions. A new set of experiments significantly expanding the range of White-Tallmadge conditions showed that their theory cannot explain the experimental results. We then hypothesized that the results of LLD theory can be improved by restoring the non-linear meniscus curvature in the equation. With this modification, the obtained equation should be able to describe static menisci on any cylindrical rods and the film profiles observed at non-zero rod velocity. ExperimentTo test the hypothesis, we distinguished capillary forces from viscous forces by running experiments with different rods and at different withdrawal velocities and video tracking the menisci profiles and measuring the weight of deposited films. The values of film thickness were then fitted with a mathematical model based on the modified LLD equation. We also fitted the meniscus profiles. FindingsThe results show that the derived equation allows one to reproduce the results of the LLD theory and go far beyond those to include rods of different radii. A new set of experimental data together with the White-Tallmadge experimental data are explained with the modified LLD theory. A set of simple formulas approximating numeric results have been derived. These formulas can be used in engineering applications for the prediction of the coating thickness.

Weierstrass’ variational theory for analysing meniscus stability in ribbon growth processes

We use the method of free energy minimization to analyse static meniscus shapes for crystal ribbon growth systems. To account for the possibility of multivalued curves as solutions to the minimization problem, we choose a parametric representation of the meniscus geometry. Using Weierstrass’ form of the Euler–Lagrange equation we derive analytical solutions that provide explicit knowledge on the behavior of the meniscus shapes. Young’s contact angle and Gibbs pinning conditions are analysed and shown to be a consequence of the energy minimization problem with variable end-points. For a given ribbon growth configuration, we find that there can exist multiple static menisci that satisfy the boundary conditions. The stability of these solutions is analysed using second order variations and are found to exhibit saddle node bifurcations. We show that the arc length is a natural representation of a meniscus geometry and provides the complete solution space, not accessible through the classical variational formulation. We provide a range of operating conditions for hydro-statically feasible menisci and illustrate the transition from a stable to spill-over configuration using a simple proof of concept experiment.

Calculation of Static Suspension Depth and Meniscus Shape of a Solid Spherical Inclusion at the Steel–Slag Interface

A method to calculate the meniscus shape around an inclusion on the steel–slag interface is presented. The meniscus shape is related to the properties of steel and slag and the size of the inclusion. The large-size inclusions have a large suspension depth and interaction range. At the small-size range of inclusions, the suspension depth increases slowly with the size, while the interaction range is sensitive to the size.

Contact angle hysteresis in a microchannel: Statics

We study contact angle hysteresis in a chemically heterogeneous microchannel by tracking static meniscus configurations in the microchannel upon varying the volume of liquid. We first construct a graphical force balance similar to a previous approach by Joanny and de Gennes for this system, though here with a straight contact line. It is shown that hysteresis is induced by wettability gradients above a finite threshold value. This is also visualized in a phase-plane plot enabling to easily predict stick-slip events of the contact line and the occurrence of hysteresis. Above the threshold and for nonoverlapping Gaussian defects, we find good agreement with the expressions by Joanny and de Gennes for the hysteresis amplitude induced by a dilute system of defects. In particular, the hysteresis amplitude is found to be proportional to the square of the defect force and to the defect concentration. For a model sinusoidal heterogeneity, decreasing the ratio between the heterogeneity wavelength and the microchannel gap size brings the system from a subthreshold regime, to a stick-slip dominated regime, and finally to a regime with a quasiconstant advancing and receding angle. In the latter case, the hysteresis amplitude is found to be proportional to the defect force. We also consider an unusual heterogeneity for which the gradients of increasing and decreasing wettability are different. In such a situation breaking the left/right symmetry, whether or not hysteresis is observed will depend on the side the liquid enters the microchannel.

Evolution of entrained water film thickness and dynamics of Marangoni flow in Marangoni drying.

As an ultra-clean wafer drying technique, Marangoni drying has been widely applied in the integrated circuits manufacturing process. When the wafer is vertically withdrawn from a deionization water bath, Marangoni stress along the meniscus, which is induced by the organic vapour, strips off the water film entrained on the wafer surface, and the wafer drying is thereby realized. In this work, a numerical model is presented that is comprised of the film, meniscus, and bulk regions for Marangoni drying. The model combines the transfer of organic vapour from air to water and the withdrawal of the wafer from the bath. The evolution of the entrained water film thickness, the tangential velocity, and the stress at the air–water interface are quantitatively investigated. The results reveal that the thickness of the entrained water film is reduced by more than one order of magnitude compared with the wafer withdrawn process without the Marangoni effect. In addition, owing to the receding of the contact line, it is found that the capillary pressure gradient dramatically increases, which contributes to the sudden increase in the tangential velocity in the dynamic meniscus. Moreover, the tangential velocity decreases in the static meniscus adjacent to the dynamic meniscus, which results from the redistribution of the interfacial concentration of the organic species driven by the Marangoni flow.

Discretization of three-dimensional free surface flows and moving boundary problems via elliptic grid methods based on variational principles

A new boundary-fitted technique to describe free surface and moving boundary problems is presented. We have extended the 2D elliptic grid generator developed by Dimakopoulos and Tsamopoulos (2003) [19] and further advanced by Chatzidai et al. (2009) [18] to 3D geometries. The set of equations arises from the fulfillment of the variational principles established by Brackbill and Saltzman (1982) [21], and refined by Christodoulou and Scriven (1992) [22]. These account for both smoothness and orthogonality of the grid lines of tessellated physical domains. The elliptic-grid equations are accompanied by new boundary constraints and conditions which are based either on the equidistribution of the nodes on boundary surfaces or on the existing 2D quasi-elliptic grid methodologies. The capabilities of the proposed algorithm are first demonstrated in tests with analytically described complex surfaces. The sequence in which these tests are presented is chosen to help the reader build up experience on the best choice of the elliptic grid parameters. Subsequently, the mesh equations are coupled with the Navier–Stokes equations, in order to reveal the full potential of the proposed methodology in free surface flows. More specifically, the problem of gas assisted injection in ducts of circular and square cross-sections is examined, where the fluid domain experiences extreme deformations. Finally, the flow-mesh solver is used to calculate the equilibrium shapes of static menisci in capillary tubes.

Wetting behaviors of the molten silicon on graphite surface

A theory which was proposed by Scheid et al. in 2010 (Scheid B, van Nierop E A, Stone H A 2010 Appl. Phys. Lett. 97 171906) suggests that very thin ribbons of molten material can be drawn out of a melt by adequately tuning the temperature gradient along the dynamic meniscus that connects the static meniscus at the melting bath to the region of the drawn flat film. Based on this theory, one-step manufacturing ultra-thin silicon wafer by pulling out from a molten silicon bath has attracted considerable attention in recent year due to its many attractive performances such as low cost, simple process, etc. By using this method, solar cell can have intensive applications due to its low cost and stable output efficiency. The results show that the thermal capillarity effect plays a great role in preparing the ultra-thin silicon. The thickness of the silicon wafer is sensitive to the capillary length and the strength of the surface tension variation as well. In order to reveal the mechanism for the effect of thermal capillary on the fabrication of ultra-thin silicon wafer, a thermal capillary finite element model is developed for the horizontal ribbon growth system to study the wetting behaviors of molten silicon on graphite. The mathematical model is established and simulated by using the commercial software; several parameters such as mass, viscous stress and capillary force are calculated. The wetting processes are tested by changing surface roughness (Ra=0.721 m and Ra=0.134 m), system temperatures (17371744 K), and durations (1030 s) at constant temperature on a high-temperature, high-vacuum contact angle measurement instrument. It is found that the wetting angle of silicon droplet on graphite decreases with surface roughness and temperature increasing; the wetting angle comes down with time going by (lasting 30 s) at constant temperature, which is consistent with the theoretical result of Wenzel. The influence of surface tension on wetting process is studied by analyzing the distributions of pressure and velocity field. It is shown that the differential pressure at the solid-liquid interfaces, induced by thermal capillary effect, decreases in the wetting process and reaches a balance which prevents the droplet from being wetted. At T=1700 K, the wetting angle and the shape of droplet change quickly within 0.4 ms and eventually become stable after 5 ms as shown in the simulation. The spreading length L and droplet height h at the steady-state are calculated with considering the influence of droplet radius on the wetting process. The results show that both L and h are directly related to the steady-state of wetting angle. The surface tension dominates the wetting process for droplet radius R0 5mm; while for R0 5 mm, the wetting process is dominated by gravity.

Multiscale level-set method for accurate modeling of immiscible two-phase flow with deposited thin films on solid surfaces

We developed a multiscale sharp-interface level-set method for immiscible two-phase flow with a pre-existing thin film on solid surfaces. The lubrication approximation theory is used to model the thin-film equation efficiently. The incompressible Navier–Stokes, level-set, and thin-film evolution equations are coupled sequentially to capture the dynamics occurring at multiple length scales. The Hamilton–Jacobi level-set reinitialization is employed to construct the signed-distance function, which takes into account the deposited thin-film on the solid surface. The proposed multiscale method is validated and shown to match the augmented Young–Laplace equation for a static meniscus in a capillary tube. Viscous bending of the advancing interface over the precursor film is captured by the proposed level-set method and agrees with the Cox–Voinov theory. The advancing bubble surrounded by a wetting film inside a capillary tube is considered, and the predicted film thickness compares well with both theory and experiments. We also demonstrate that the multiscale level-set approach can model immiscible two-phase flow with a capillary number as low as 10−6.

Dip-coating with prestructured substrates: transfer of simple liquids and Langmuir–Blodgett monolayers

When a plate is withdrawn from a liquid bath, either a static meniscus forms in the transition region between the bath and the substrate or a liquid film of finite thickness (a Landau–Levich film) is transferred onto the moving substrate. If the substrate is inhomogeneous, e.g. has a prestructure consisting of stripes of different wettabilities, the meniscus can be deformed or show a complex dynamic behavior. Here we study the free surface shape and dynamics of a dragged meniscus occurring for striped prestructures with two orientations, parallel and perpendicular to the transfer direction. A thin film model is employed that accounts for capillarity through a Laplace pressure and for the spatially varying wettability through a Derjaguin (or disjoining) pressure. Numerical continuation is used to obtain steady free surface profiles and corresponding bifurcation diagrams in the case of substrates with different homogeneous wettabilities. Direct numerical simulations are employed in the case of the various striped prestructures.The final part illustrates the importance of our findings for particular applications that involve complex liquids by modeling a Langmuir–Blodgett transfer experiment. There, one transfers a monolayer of an insoluble surfactant that covers the surface of the bath onto the moving substrate. The resulting pattern formation phenomena can be crucially influenced by the hydrodynamics of the liquid meniscus that itself depends on the prestructure on the substrate. In particular, we show how prestructure stripes parallel to the transfer direction lead to the formation of bent stripes in the surfactant coverage after transfer and present similar experimental results.

Creating controlled thickness gradients in polymer thin films via flowcoating.

Flowcoating is a popular technique for generating thin (5-200 nm), substrate-supported polymer films. In this process, a reservoir of coating fluid is held between the horizontal substrate and a nearly horizontal blade above the substrate; a film of fluid is drawn out of the reservoir by moving the substrate. Accelerating the substrate produces a film with a thickness gradient, particularly useful for high-throughput measurements where film thickness is an important parameter. The present work compares experimental film thickness profiles with a model based on a Landau-Levich treatment to identify the experimental parameters which govern film thickness. The key parameters are the capillary number and the radius of curvature of the reservoir's static meniscus, which is set by the blade angle, gap height, solution reservoir volume, and contact angles of the fluid with the blade and substrate. The results show excellent quantitative agreement with the first-principles model; the model thus provides a design approach which allows a user to produce polymer thin films of virtually any desired thickness profile.

The elastic Landau–Levich problem

AbstractWe study the classical Landau–Levich dip-coating problem in the case where the interface has significant elasticity. One aim of this work is to unravel the effect of surface-adsorbed hydrophobic particles on Landau–Levich flow. Motivated by recent findings (Vella, Aussillous & Mahadevan, Europhys. Lett., vol. 68, 2004, pp. 212–218) that a jammed monolayer of adsorbed particles on a fluid interface makes it respond akin to an elastic solid, we use the Helfrich elasticity model to study the effect of interfacial elasticity on Landau–Levich flow. We define an elasticity number, $\mathit{El}$, which represents the relative strength of viscous forces to elasticity. The main assumptions of the theory are that $\mathit{El}$ be small, and that surface tension effects are negligible. The shape of the free surface is formulated as a nonlinear boundary value problem: we develop the solution as an asymptotic expansion in the small parameter ${\mathit{El}}^{1/ 7} $ and use the method of matched asymptotic expansions to determine the film thickness as a function of $\mathit{El}$. The solution to the shape of the static meniscus is not as straightforward as in the classical Landau–Levich problem, as evaluation of higher-order effects is necessary in order to close the problem. A remarkable aspect of the problem is the occurrence of multiple solutions, and five of these are found numerically. In any event, the film thickness varies as ${\mathit{El}}^{4/ 7} $ in qualitative agreement with the experiments of Ouriemi & Homsy (Phys. Fluids, 2013, in press).