Shape of menisci in terrestrial dewetted Bridgman growth

Dewetted Bridgman is a crystal growth technique in which the crystal is detached from the crucible wall by a liquid free surface at the level of the solid-liquid interface, called liquid meniscus, which creates a gap between the crystal and the ampoule. Dewetting phenomenon was first obtained spontaneously in spatial experiments during the Bridgman solidification, and opened the possibility of reproducing experiments on the earth--obtained by applying a gas pressure difference Delta P=P cold-P hot between the cold and the hot sides of the sample. In order to understand the process which leads to a crystal with a constant radius on the ground, analytical and numerical studies of axisymmetric meniscus shapes are made and the dependence of the meniscus shape on the pressure difference is established. For this aim, starting from the Young-Laplace equation of a capillary surface in equilibrium in the presence of gas pressure, a mathematical model able to describe the meniscus surface z=z(r) and the angle theta=theta(r) between the tangent to the meniscus and the horizontal axis is presented. On the basis of this model, inequalities of the pressure intervals for which dewetting is feasible are established. Numerical results are performed for InSb crystals.     

Transport mechanisms in capillary condensation of water at a single-asperity nanoscopic contact

Transport mechanisms involved in capillary condensation of water menisci in nanoscopic gaps between hydrophilic surfaces are investigated theoretically and experimentally by atomic force microscopy (AFM) measurements of capillary force. The measurements showed an instantaneous formation of a water meniscus by coalescence of the water layers adsorbed on the AFM tip and sample surfaces, followed by a time evolution of meniscus toward a stationary state corresponding to thermodynamic equilibrium. This dynamics of the water meniscus is indicated by time evolution of the meniscus force, which increases with the contact time toward its equilibrium value. Two water transport mechanisms competing in this meniscus dynamics are considered: (1) Knudsen diffusion and condensation of water molecules in the nanoscopic gap and (2) adsorption of water molecules on the surface region around the contact and flow of the surface water toward the meniscus. For the case of very hydrophilic surfaces, the dominant role of surface water transportation on the meniscus dynamics is supported by the results of the AFM measurements of capillary force of water menisci formed at sliding tip-sample contacts. These measurements revealed that fast movement of the contact impedes on the formation of menisci at thermodynamic equilibrium because the flow of the surface water is too slow to reach the moving meniscus.     

Electric field enhanced spreading of partially wetting thin liquid films

Equilibrium and dynamic electrowetting behavior of ultrathin liquid films of surfactant (SDS) laden water over silicon substrate (with native oxide) is investigated. A nonobtrusive optical method, namely, image analyzing interferometry, is used to measure the meniscus profile, adsorbed film thickness, and the curvature of the capillary meniscus. Significant advancement of the contact line of the liquid meniscus, as a result of the application of electric field, is observed even at relatively lower values of applied voltages. The results clearly demonstrate the balance of intermolecular and surface forces with additional contribution from Maxwell stress at the interline. The singular nature of Maxwell stress is exploited in this analysis to model the equilibrium meniscus profile using the augmented Young-Laplace equation, leading to the in situ evaluation of the dispersion constant. The electrowetting dynamics has been explored by measuring the velocity of the advancing interline. The interplay of different forces at the interface is modeled using a control volume approach, leading to an expression for the interline velocity. The model-predicted interline velocities are successfully compared with the experimentally measured velocities. Beyond a critical voltage, contact line instability resulting in emission of droplets from the curved meniscus has been observed.     

Effect of contact angle on capillary rise in annular meniscus

The shapes of axisymmetric annular menisci are calculated for a range of contact angles at the constraining cylindrical walls. From these shapes the capillary rise (or depression) Δh relative to an infinite external meniscus is obtained in terms of the radii of the inner and outer cylinders and physical properties of the liquids. This value of Δh may then be used to correct the measured rise in a concentric capillary tube relative to the annular meniscus. The capillary rise, Δh, has been experimentally measured for several liquids and their surface tensions directly calculated from these values. Excellent agreement with accepted values is obtained, even though the measured capillary rise is very small, thus confirming the accuracy of the theoretical calculations and experimental technique.     

Discontinuous liquid rise in capillaries with varying cross-sections

We consider theoretically liquid rise against gravity in capillaries with height-dependent cross-sections. For a conical capillary made from a hydrophobic surface and dipped in a liquid reservoir, the equilibrium liquid height depends on the cone-opening angle alpha, the Young-Dupré contact angle theta, the cone radius at the reservoir's level R(0), and the capillary length kappa(-)(1). As alpha is increased from zero, the meniscus' position changes continuously until, when alpha attains a critical value, the meniscus jumps to the bottom of the capillary. For hydrophilic surfaces the meniscus jumps to the top. The same liquid height discontinuity can be achieved with electrowetting with no mechanical motion. Essentially the same behavior is found for two tilted surfaces. We further consider capillaries with periodic radius modulations and find that there are few competing minima for the meniscus location. A transition from one to another can be performed by the use of electrowetting. Finite pressure difference between the two sides of the liquids can be incorporated as well, resulting in complicated phase-diagrams in the alpha-theta plane. The phenomenon discussed here may find uses in microfluidic applications requiring the transport small amounts of water "quanta" (volume < 1 nL) in a regular fashion.     

Effects of gas adsorption isotherm and liquid contact angle on capillary force for sphere-on-flat and cone-on-flat geometries

This paper explains the origin of the vapor pressure dependence of the asperity capillary force in vapor environments. A molecular adsorbate layer is readily formed on solid surface in ambient conditions unless the surface energy of the solid is low enough and unfavorable for vapor adsorption. Then, the capillary meniscus formed around the solid asperity contact should be in equilibrium with the adsorbate layer, not with the bare solid surface. A theoretical model incorporating the vapor adsorption isotherm into the solution of the Young-Laplace equation is developed. Two contact geometries--sphere-on-flat and cone-on-flat--are modeled. The calculation results show that the experimentally-observed strong vapor pressure dependence can be explained only when the adsorption isotherm of the vapor on the solid surface is taken into account. The large relative partial pressure dependence mainly comes from the change in the meniscus size due to the presence of the adsorbate layer.     

Finite-amplitude dynamics of coupled cylindrical menisci

The cylindrical meniscus is a liquid/gas interface of circular-cap cross-section constrained along its axis and bounded by end-planes. The inviscid motions of coupled cylindrical menisci are studied here. Motions result from the competition between inertia and surface tension forces. Restriction to shapes that are of circular-cap cross-section leads to an ordinary differential equation (ode) model, with the advantage that finite-amplitude stability can be examined. The second-order nonlinear ode model has a Hamiltonian structure, showing dynamical behavior like the Duffing-oscillator. The energy landscape has either a single- or double-welled potential depending on the extent of volume overfill. Total liquid volume is a bifurcation parameter, as in the corresponding problem for coupled spherical-cap droplets. Unlike the spherical-cap problem, however, axial disturbances can also destabilize, depending on overfill. For large volumes, previously known axial stability results are applied to find the limit at which axial symmetry is lost and comparison is made to the Plateau-Rayleigh limit.     

Simulations of novel nanostructures formed by capillary effects in lithography

High aspect ratio three-dimensional nanostructures are of tremendous interest to a wide range of fields such as photonics, plasmonics, fluid mechanics, and biology. Recent developments in capillary force lithography (CFL) have focused on taking advantage of the formation of menisci to enhance the functionality of small size-scale structures. In this study, simulations of the three-dimensional shapes of equilibrium menisci formed in capillaries with various cross-section geometries are studied. The capillary cross sections include regular polygons and equilateral star-shapes with sharp and rounded corners. The characteristic dimension of the physical lithography systems which are simulated is on the order of 100nm. At such size-scale, surface-tension-effects are predominant, and as a consequence, our simulations demonstrate that nanometer-sized structures with great application potentials can be fabricated. Specifically, this study demonstrates that surfaces with three-dimensional nanoscale structures can be fabricated from templates with micron or sub-micron features through the development of cusps in the corners of the polygonal capillaries. Quantitatively, the effects of contact angle, corner angle, meniscus confinement, and corner rounding radius are examined and scaling analyses are presented to describe the dependencies of the height variation across the meniscus on these parameters. These simulations serve as useful guides for extending the development and implementation of capillary force lithography.     

Menisci in a Diamond-Shaped Capillary

This paper presents a closed form analytical solution to the augmented Young-Laplace equation for the meniscus profile in a capillary formed between four equal-sized tangent cylinders centered on the vertices of a square. The solution is valid for a large class of disjoining pressure isotherms and contact angles.     

Phase transitions of capillary-held liquids in a slit-like pore

Dynamics of capillary held liquids plays important roles in a wide range of systems including adhesion, printing of paints and inks, the behavior of wet granular materials, and the mass transfer through porous media. Recent study suggested the presence of two distinct modes for the disappearance of capillary-held liquids in a slit-like pore of adjustable slit width that depended on the slit-opening rates. In contrast to the first mode that is well-documented in terms of the Young-Laplace equation, a novel and unexpected mode was observed when the liquid bridge was held in the vicinity of the thermodynamic phase boundary (equilibrium Kelvin length). Here we extended the study to three new compounds that have a wide range of vapor pressures. An evaporating liquid bridge developed large refractive index gradients that extended over a few micrometers from the edge of the meniscus once the slit width was increased beyond the equilibrium Kelvin length. This interfacial region with depleted refractive index retreated inward as the meniscus shrank with time, and the refractive index of the entire bridge subsequently fell from that of the liquid once the interfacial regions from the opposite sides of the shrinking bridge met at the center. The refractive index recovered to that of the liquid when the slit width was closed to below the Kelvin length and the vapor was allowed to recondense. The time scale of the evaporation and condensation depended on the size of the surface gap, and, when the surfaces were placed at a separation very close to the Kelvin length, it was possible to detect a stage in which the system was in an apparent kinetic equilibrium between two physical states--with and without the liquid connecting the two surfaces.     

Landau-Levich menisci

As shown by Landau, Levich and Derjaguin, a plate withdrawn out of a wetting bath at low capillary numbers deforms the very top of the liquid reservoir. At this place, a dynamic meniscus forms, whose shape and curvature select the thickness of the film entrained by the plate. In this paper, we measure accurately the thickness of the entrained film by reflectometry, and characterize the dynamic meniscus, which is found to decay exponentially towards the film. We show how this shape is modified when reversing the motion: as a plate penetrates the bath, the dynamic meniscus can "buckle" and present a stationary wavy profile, which we discuss.     

On the shape of a hydrostatic meniscus attached to a corrugated plate or wavy cylinder

The shape of a hydrostatic meniscus attached at a fixed contact angle to a vertical plate or circular cylinder with periodic corrugations is studied by analytical and numerical methods, and the effect of wall irregularities on the shape of the contact line and vertical component of the capillary force is discussed. An asymptotic analysis for a plate with small-amplitude sinusoidal corrugations is carried out to first order with respect to the corrugation amplitude, and a boundary-value problem is formulated and solved by a shooting method to determine the meniscus shape and elevation of the contact line. The meniscus attached to a corrugated plate with rounded corners produced by a Schwarz-Christoffel mapping function for a triangular wave is considered by numerical methods. The Laplace-Young equation determining the meniscus shape is solved in orthogonal curvilinear coordinates generated by conformal mapping using a finite-difference method. The numerical results are successfully compared with the predictions of the perturbation expansion for small amplitudes and discussed with reference to the rise of a meniscus inside a dihedral angle for large amplitudes. A companion asymptotic analysis is presented for a meniscus outside a vertical circular cylinder with small-amplitude sinusoidal corrugations. The analytical predictions are successfully compared with numerical solutions of the Laplace-Young equation for a meniscus outside an elliptical cylinder with aspect ratio near unity, regarded as a deformed circle.     

Parameters ruling capillary forces at the submillimetric scale

This paper reviews the way to compute capillary forces between two solids by numerically integrating the Laplace equation describing the shape of an axially symmetric meniscus at equilibrium. The numerical results of the proposed model have been experimentally validated with a test bed able to measure forces of about 1 mN with an accuracy of about 1 microN. Thanks to the simulation tool and the test bed, the influence of the following parameters has been studied: surface tension, solid geometry, volume of liquid, materials, separation distance between both solids, and surrounding environment. The way to compute the force from a given meniscus geometry has been clarified as far as the "Laplace" and "tension" contributions are concerned.     

Electrowetting-induced capillary flow in a parallel-plate channel

This paper investigated, theoretically and experimentally, the electrowetting-induced capillary rise in a parallel-plate channel. The measured equilibrium height of the meniscus was proportional to the square of the applied potential. A model, based on the kinetic equation of capillary flow with the consideration of an electrowetting dynamic contact angle, was established to simulate the capillary rise. The effects of the electrostatic charge and the contact-line friction were linearly added to describe the electrowetting dynamic contact angle. The model was found to be able to adequately describe the experimental data under different initial heights and applied voltages. The non-Poseuille flow effect had little influence in the meniscus rising phenomenon studied in this work.     

Shape analysis based critical Eotvos numbers for buoyancy induced partial detachment of oil drops from hydrophilic surfaces

Removal of oil drops from solid surfaces immersed in an aqueous medium is of interest in many applications. It has been shown that drop shape analysis can be used to predict conditions at which the stability limit of a lighter than water oil drop on a solid surface immersed in an aqueous bath is reached (Adv. Colloid Interface Sci. 98 (2002) 265). However the above analysis is restricted to cases where the contact angle made by the drop is below 90degrees and when the surface conditions result in a 'pinned' contact line. In this paper, it is shown that drop shape analysis can be used to predict the critical conditions at which drop stability limit is reached for drop contact angles of 90degrees and above, which is encountered with 'hydrophilic' surfaces. This critical condition can predict the occurrence of partial oil drop detachment, before complete removal due to 'roll-up', which occurs when the hydrophilic surface is adequately smooth which prevents 'pinning' of the contact line. The critical conditions at which partial drop detachment occurs can also be approximately predicted from simple force balances. It has been shown (Adv. Colloid Interface Sci. 98 (2002) 265) that for contact angles less than 90degrees, the critical limit based on shape analysis appears to resolve the differences that arise due to alternate expressions for capillary retention force. This paper shows that even for contact angles above 90degrees, the critical conditions predicted from the shape analysis resolves the differences in the predictions from the alternate force balances. Drop shape analysis used in this paper is based on the 'Arc-length' form of Young-Laplace or 'drop shape' equation, which is different from the 'Y vs X' form of the above equation that is used in Adv. Colloid Interface Sci. 98 (2002) 265. The above drop shape equation is solved by a fourth order Runge-Kutta technique and it is shown that for angles less than 90degrees, the two forms of the drop shape equation, predict almost identical values of the critical Eotvos number. This paper highlights the competing effects of interfacial tension lowering induced drop instability and 'roll-up', a term that is used to describe the retraction of the contact line of an oil drop on a surface, in being the primary c ause for drop detachment.     

Atomic-scale roughness effect on capillary force in atomic force microscopy

We study the capillary force in atomic force microscopy by using Monte Carlo simulations. Adopting a lattice gas model for water, we simulated water menisci that form between a rough silicon-nitride tip and a mica surface. Unlike its macroscopic counterpart, the water meniscus at the nanoscale gives rise to a capillary force that responds sensitively to the tip roughness. With only a slight change in tip shape, the pull-off force significantly changes its qualitative variation with humidity.     

On the thermodynamic theory of the strength of solids: 4. Capillary condensation and capillary evaporation in wedge-shaped crack

The processes of capillary condensation and capillary evaporation in a wedge-shaped crack are considered. Capillary evaporation is a comparatively new phenomenon that is opposite to capillary condensation and occurs upon the cleavage of a solid in a nonwetting liquid. For both cases, the positions of a meniscus inside a wedge-shaped crack have been calculated as functions of the meniscus curvature radius, liquid-contact angle, and crack-opening angle. The effect of temperature on the meniscus position has been analyzed; it has been established that the meniscus shifts from the gaseous toward the liquid phase as temperature rises. The regularities of meniscus displacements in the course of crack growth have been established: under the conformal mechanism of crack growth, the absolute position of the meniscus remains unchanged (i.e., the meniscus and the crack frontal line move at the same velocity), while, under the depth mechanism of growth, the relative position of the meniscus is retained.     

Wetting: Inverse Dynamic Problem and Equations for Microscopic Parameters

Movement of a liquid meniscus in a low-diameter capillary while it is being filled or emptied is considered. The liquid is nonvolatile. Assuming low Reynolds number and low capillary number, the liquid-gas interface shape is studied. Angles of inclination of this boundary to the solid near the contact line are small. Consideration is given to the inverse problem in wetting dynamics: to establish an analytic expression for the universal constant that influences the dynamics of a three-phase contact line. Inverse relations for microscopic parameters in terms of macroscopic measured values obtained in experiments with a meniscus moving through a capillary are derived. The inverse relations are substantiated independently. To do so, numerical experiments for a van der Waals liquid have been carried out, using the de Gennes model of partial wetting. General formulas for microparameters agree well with numerical experiments. The article provides the similarity criterion which influences the wetting in the case of a van der Waals liquid meniscus. The inverse dynamic problem for both an advancing and a receding meniscus is solved. A relation for the critical speed of meniscus recession is proposed. Two contact angles for a meniscus are discussed. Behavior of dynamic contact angles in the vicinity of the critical speed is studied. One of the angles is shown to vanish at less than the critical speed, and the other one, exactly at the critical speed. In the case of an advancing meniscus the equations for microparameters are valid for both partial and complete wetting. The proposed inverse expression for complete wetting allows determination of the maximum precursor film thickness and its dependence on the motion speed (also determination of the Hamaker constant in the case of a van der Waals liquid). Copyright 2000 Academic Press.     

Critical Eotvos numbers for buoyancy-induced oil drop detachment based on shape analysis

Critical values of the Eotvos number, which is half the Bond number, above which buoyancy induced drop detachment occurs, are estimated based on force balance equations available in the literature [Colloids Surf. A: Physicochem. Eng. Aspects 178 (2001) 249]. Since there are two significantly different expressions of the capillary retention force responsible for holding oil drops on a solid substrate in an aqueous phase, the critical dimensionless number is estimated with these two distinct equations. The differential equation defining the drop shape, with the constraints of the drop volume and the 'pinned' or 'receding' contact line, is numerically solved. The equilibrium drop shapes predicted are shown to match the experimentally observed variations in drop shape. From the numerical solution, it is observed that for interfacial tension (IFT) values lower than a certain limit for a given drop size, no numerically estimated drop shape can fulfil the drop volume constraint. Similarly, for the dimensionless number above a critical value, no shape can meet all the constraints. These critical Eotvos numbers are estimated, based on the above numerical approach, for initial contact angles measured in oil varying from 20 degrees to 90 degrees. It is found that the critical Eotvos numbers estimated from the numerical shape analysis are between the critical values estimated from the two force-balance equations. Near 90 degrees, the critical values estimated from the drop shape analysis matches the values from one of the force balance estimates, but merges with the critical values of the dimensionless number, estimated from the other force balance model near 10 degrees. From this analysis, it appears that a combination of the two equations for the capillary retention force is required, with one dominating when the contact angles are high, while the other applies for low values of the contact angle.