Modeling Pendular Liquid Bridges with a Reducing Solid-Liquid Interface

Liquid bridges formed between particles of dissimilar surface properties are important in many processes involving the handling of powders in mixtures. For instance, growth kinetic models for wet granulation frequently incorporate the evolution and resistance to breakage of individual liquid bridges between particles in a statistical form. These models generally propose a confusing definition of liquid-to-solid contact angles. Taken as a single thermodynamic value, they typically neglect the influence of wetting hysteresis on the liquid bridge. In this paper, a simple model based on the interfacial energies is proposed for the evolution of liquid bridges when one solid-liquid interface reduces. This receding process is well described by a balance between the adhesion energy of the bridge liquid on the particle surface and the capillary energy stored by the liquid free surface. The extent of solid-liquid interfacial area reduction can hence be predicted from the initial liquid bridge configuration. The liquid bridge shape is approximated by a parabolic curve, which is validated from the good agreement between measured and calculated contact angles or liquid-vapor interfacial area. Copyright 2000 Academic Press.     

Capillary forces between two spheres with a fixed volume liquid bridge: theory and experiment

Capillary forces are commonly encountered in nature because of the spontaneous condensation of liquid from surrounding vapor, leading to the formation of a liquid bridge. In most cases, the advent of capillary forces by condensation leads to undesirable events such as an increase in the strength of granules, which leads to flow problems and/or caking of powder samples. The prediction and control of the magnitude of capillary forces is necessary for eliminating or minimizing these undesirable events. The capillary force as a function of the separation distance, for a liquid bridge with a fixed volume in a sphere/plate geometry, was calculated using different expressions reported previously. These relationships were developed earlier, either on the basis of the total energy of two solid surfaces interacting through the liquid and the ambient vapor or by direct calculation of the force as a result of the differential gas pressure across the liquid bridge. It is shown that the results obtained using these methodologies (total energy or differential pressure) agree, confirming that a total-energy-based approach is applicable, despite the thermodynamic nonequilibrium conditions of a fixed volume bridge rupture process. On the basis of the formulas for the capillary force between a sphere and a plane surface, equations for the calculation of the capillary force between two spheres are derived in this study. Experimental measurements using an atomic force microscope (AFM) validate the formulas developed. The most common approach for transforming interaction force or energy from that of sphere/plate geometry to that of sphere/sphere geometry is the Derjaguin approximation. However, a comparison of the theoretical formulas derived in this study for the interaction of two spheres with those for sphere/plate geometry shows that the Derjaguin approximation is only valid at zero separation distance. This study attempts to explain the inapplicability of the Derjaguin approximation at larger separation distances. In particular, the area of a liquid bridge changes with the separation distance, H, and thereby does not permit the application of the "integral method," as used in the Derjaguin approximation.     

Electrical conductance study of theta-liquid bridges

This work investigates the behavior of small liquid bridges that are formed between two horizontal supporting surfaces, aligned at the vertical direction. The contact lines of the liquid bridges are not edge-pinned but free to move across the supporting surfaces with the contact angle as a parameter (theta-bridges). An a.c. electrical conductance technique coupled with high resolution optical images is used to characterize the geometrical details of constant volume liquid bridges when their length is increased gradually until rupture. A mathematical framework is developed for the identification of the geometrical characteristics of theta-liquid bridges explicitly from conductance data. Theoretical predictions show good agreement with measurements for most of the bridge lengths (separation distance between supports) except close to the rupture point where the bridge is highly stretched. It is further shown that for short and moderate separation distances the present model can be used with confidence to determine the bridge volume and neck radius from the electrical signal.     

Pore-scale modelling and tomographic visualisation of drying in granular media

Spatio-temporal evolution of liquid phase clusters during drying of a granular medium (realised by random packing of cylindrical particles) has been investigated at the length-scale of individual pores. X-ray microtomography has been used to explicitly resolve the three-dimensional spatial distribution of the solid, liquid, and gas phases within the wet particle assemblies. The propagation of liquid menisci through the granular medium during drying was dynamically followed. The effect of contact angle on the degree of dispersion of the drying front has been studied by observing drying in a layer of untreated (hydrophilic) and silanised particles; the drying front was found to be sharper in the case of the silanised (less hydrophilic) particles. This observation was confirmed by direct numerical simulations of drying in a digitally encoded porous medium identical in structure to the experimental one. The simulations also revealed that the average gas-liquid interfacial area in a given porous microstructure strongly depends on the contact angle.     

The Dynamics of Marangoni-Driven Local Film Drainage between Two Drops

A study of Marangoni-driven local continuous film drainage between two drops induced by an initially nonuniform interfacial distribution of insoluble surfactant is reported. Using the lubrication approximation, a coupled system of fourth-order nonlinear partial differential equations was derived to describe the spatio-temporal evolution of the continuous film thickness and surfactant interfacial concentration. Numerical solutions of these governing equations were obtained using the Numerical Method of Lines with appropriate initial and boundary conditions. A full parametric study was undertaken to explore the effect of the viscosity ratio, background surfactant concentration, the surface Péclet number, and van der Waals interaction forces on the dynamics of the draining film for the case where surfactant is present in trace amounts. Marangoni stresses were found to cause large deformations in the liquid film: Thickening of the film at the surfactant leading edge was accompanied by rapid and severe thinning far upstream. Under certain conditions, this severe thinning leads directly to film rupture due to the influence of van der Waals forces. Time scales for rupture, promoted by Marangoni-driven local film drainage were compared with those associated with the dimpling effect, which accompanies the approach of two drops, and implications of the results of this study on drop coalescence are discussed. Copyright 2001 Academic Press.     

Determination of surface tension and contact angle from the shapes of axisymmetric fluid interfaces without use of apex coordinates

Drop shape techniques, such as axisymmetric drop shape analysis, are widely used to measure surface properties, as they are accurate and reliable. Nevertheless, they are not applicable in experimental studies dealing with fluid configurations that do not present an apex. A new methodology is presented for measuring interfacial properties of liquids, such as surface tension and contact angles, by analyzing the shape of an axisymmetric liquid-fluid interface without use of apex coordinates. The theoretical shape of the interface is generated numerically as a function of surface tension and some geometrical parameters at the starting point of the interface, e.g., contact angle and radius of the interface. Then, the numerical shape is fitted to the experimental profile, taking the interfacial properties as adjustable parameters. The best fit identifies the true values of surface tension and contact angle. Comparison between the experimental and the theoretical profiles is performed using the theoretical image fitting analysis (TIFA) strategy. The new method, TIFA-axisymmetric interfaces (TIFA-AI), is applicable to any axisymmetric experimental configuration (with or without apex). The versatility and accuracy of TIFA-AI is shown by considering various configurations: liquid bridges, sessile and pendant drops, and liquid lenses.     

Validity of the "sharp-kink approximation" for water and other fluids

The contact angle of a liquid droplet on a solid surface is a direct measure of fundamental atomic-scale forces acting between liquid molecules and the solid surface. In this work, the validity is assessed of a simple equation, which approximately relates the contact angle of a liquid on a surface to its density, its surface tension, and the effective molecule-surface potential. This equation is derived in the sharp-kink approximation, where the density profile of the liquid is assumed to drop precipitously within one molecular diameter of the substrate. It is found that this equation satisfactorily reproduces the temperature-dependence of the contact angle for helium on alkali metal surfaces. The equation also seems be applicable to liquids such as water on solid surfaces such as gold and graphite, on the basis of a comparison of predicted and measured contact angles near room-temperature. Nevertheless, we conclude that, to fully test the equation's applicability to fluids such as water, it remains necessary to measure the contact angle's temperature-dependence. We hypothesize that the effects of electrostatic forces can increase with temperature, potentially driving the wetting temperature much higher and closer to the critical point, or lower, closer to room temperature, than predicted using current theories.     

Rupture of draining foam films due to random pressure fluctuations

A generalized formalism for the rupture of a draining foam film due to imposed random pressure fluctuations, modeled as a Gaussian white noise, is presented in which the flow inside the film is decomposed into a flow due to film drainage and a flow due to imposed perturbation. The evolution of the amplitude of perturbation is described by a stochastic differential equation. The rupture time distribution is calculated from the sample paths of perturbation amplitude as the time for this amplitude to equal one-half the film thickness and is calculated for different amplitudes of imposed perturbations, film thicknesses, electrostatic interactions, viscosities, and interfacial mobilities. The probability of film rupture is high for thicker films, especially at smaller times, as a result of faster growth of perturbations in a thick film due to a smaller disjoining pressure gradient. Larger viscosity, larger surface viscosity, higher Marangoni number, and smaller imposed pressure fluctuation result in slower growth of perturbation of a draining film, thus leading to larger rupture time. It is shown that a composite rupture time distribution combining short time simulation results with equilibrium distribution is a good approximation.     

Rupture work of pendular bridges

Capillary bridging can generate substantial forces between solid surfaces. Impacted technologies and sciences include micro- and nanomachining, disk drive interfaces, scanning probe microscopy, biology, and granular mechanics. Existing calculations of the rupture work of capillary bridges do not consider the thermodynamics relating to the evaporation that can occur in the case of volatile liquids. Here, we show that the occurrence of evaporation decreases the rupture work by a factor of about 2. The decrease arises from heat taken from the surroundings that is converted into work. The treatment is based on a thermodynamic control-volume analysis of the pendular bridge geometry. We extend the mathematical formulation of Orr et al., solving the meniscus problem exactly for non-wetting surfaces. The extension provides analytical results for conditions at the rupture point and at a possible inflection point and for the rupture work. A simple equation (eq 32) is shown to fit the rupture work for the two cases over a meniscus curvature range of 3 orders of magnitude. Coefficients for the equation are given in tabular form for different contact angle pairs.     

Solution of the mixed Dirichlet–Neumann problem for molecular orientation in liquid crystals

The orientation of the nematic director field under the action of an external time‐dependent field is theoretically investigated as a mixed Dirichlet–Neumann boundary‐value problem. This mathematical problem represents the situation in which a nematic liquid crystal sample is limited by two inhomogeneous flat surfaces, separated by a distance d, on which the anchoring is weak. By considering the one‐constant approximation and a parabolic approximation for the surface energy, the initial conditions and boundary‐value problem for the profile of the tilt angle can be analytically solved even in the case in which the surfaces are not identical, which represents the more general situation. The results are valid for small deviations from the homeotropic orientation and for θ?Θ?1, where θ is the actual tilt angle and Θ characterizes the easy direction imposed by the surface, and can be relevant to investigation of the molecular orientation in a nematic cell submitted to a small external voltage.     

Prediction of solid-fluid interfacial tension and contact angle

Two simple equations have been developed using the lattice theory and the regular solution assumption to predict the solid-vapor and solid-liquid interfacial tension. The required parameters are the liquid critical temperature and volume, the solid melting temperature and the molar volume of liquid and solid compounds. To confirm the models, the predicted solid-fluid interfacial tension values have been used to predict the contact angle of the liquid drop on the solid surface applying Young's equation. Agreement of the predicted contact angle with the experimental data reveals the reliability of the developed models.     

Nonlinear instabilities and pathways of rupture in thin liquid bilayers

A long-wave nonlinear analysis of dewetting of thin (<100 nm) liquid bilayers on solid substrates is presented. The short and the long time dynamics, interfacial morphologies, and the pathways of rupture and dewetting are studied to assess the roles of interfacial energies, film thicknesses, and viscosities. The twin interfaces (liquid-liquid and liquid-air) of bilayers under the influence of attractive van der Waals forces show a variety of dewetting pathways which, depending on the interfacial energies and film thicknesses, initially start with one of the two basic modes of instability--in-phase bending and out-of-phase squeezing. These short time modes of evolution and the extent of relative deformations at the interfaces are predicted from the linear stability analysis and verified by the nonlinear simulations. Simulations also show that in the later nonlinear regime, the intermolecular and viscous forces can profoundly modify the initial mode of instability and its growth rate leading to different pathways of dewetting and late stage morphologies. The complex late time patterns such as embedded droplets, inversion of top and bottom phases, and encapsulation of one fluid into the other are also engendered by tuning the intermolecular forces.     

Rupture kinetics of liquid bridges during a pulling process: a kinetic density functional theory study

Capillary bridge is a common phenomenon in nature and can significantly contribute to the adhesion of biological and artificial micro- and nanoscale objects. Especially, it plays a crucial role in the operation of atomic force microscopy (AFM) and influences in the measured force. In the present work, we study the rupture kinetics and transition pathways of liquid bridges connecting an AFM tip and a flat substrate during a process of pulling the tip off. Depending on thermodynamic conditions and the tip velocity, two regimes corresponding to different transition pathways are identified. In the single-bridge regime, the initial equilibrium bridge persists as a single one during the pulling process until the liquid bridge breaks. While, in the multibridge regime the stretched liquid bridge transforms into an intermediate state with a collection of slender liquid bridges, which then break gradually during the pulling process. Moreover, the critical rupture distance at which the bridges break changes with the tip velocity and thermodynamic conditions, and its maximum value occurs near the boundary between the single-bridge regime and the multibridge regime, where the longest range capillary force is produced. In this work, the effects of tip velocity, tip size, tip-fluid interaction, and humidity on rupture kinetics and transition pathways are also systematically studied.     

Influence of the Type of Contact between Particles Joined by a Liquid Bridge on the Capillary Cohesive Forces

The influence of the particle dimensions and type of interparticle contact on the magnitude of the capillary forces between the powder particles is studied on the basis of a model describing a capillary interaction of two particles joined by a liquid bridge. Various contact types were implemented using combinations of different particle shapes: spherical, conical, or plane. The meniscus of the bridge is described using a circular approximation; experimental results confirm that its use is justified. A method is developed for calculating the capillary forces and the amount of the liquid in the bridge with allowance for various parameters of the powder. The calculated results show that the dimensions of the particles and the type of their contact significantly affect the magnitude of the capillary forces.     

Dynamics of partially obstructed breakup of bubbles in microfluidic Y‐junctions

For revealing the dynamics of partially obstructed breakup of bubbles in microfluidic Y‐junctions, the combination of dimensionless power‐law and geometric model was applied to study the effects of capillary number, bubble length, and channel angle on the bubble rupture process. In the squeezing process, the gas‐liquid interface curve follows the parabolic model. For the evolution of the bubble neck during breakup, the increase of the bubble length, the channel angle, and the capillary number leads to the decrease of the focus distance α. The chord m increases with the increase of the capillary number and the decrease of the bubble length, and it would reach the maximum value when the channel angle is 90°. In the fast pinch‐off stage during bubble breakup, the bubble's neck curve no longer conforms to the parabolic model so the focus and chord no longer exist. For the evolution of the bubble head during breakup, the value of γ approaches 1 with the increase of the capillary number and the bubble length, and with the close of the channel angle to 90°. It is found that the quadrilateral model can be applied for the partially obstructed rupture of bubbles in the symmetrical microfluidic Y‐junction.     

Phase behavior of sucrose monoalkanoate in a water-long-chain alcohol system

Phase diagram of a water/sucrose monododecanoate (SE)/hexanol system was determined at 30°C. Aqueous micellar, reverse micellar, normal hexagonal liquid crystalline, and lamellar liquid crystalline phases appear in the phase diagram. The change in interlayer spacing and interfacial section area of surfactant in the liquid crystalline phases was investigated by small-angle x-ray scattering. Upon addition of water, the section area and the radius of cylindrical aggregates are almost constant in a hexagonal liquid crystal, whereas the distance between each cylinder is separated on the water-SE axis. The interlayer spacing slightly decreases or is almost unchanged on the surfactant-hexanol axis, because alcohol molecules penetrate into the palisade of bilayers. Although the average section area decreases with increasing alcohol content, each section area of SE and alcohol molecules are kept constant. Since the interfacial section area of alcohol is less than the section area of hydrocarbon chain, the phase transition from lamellar liquid crystal to reverse micelle occurs in an alcohol-rich region.     

Deblurred observation of the molecular structure of an oil-water interface

We simulated the interface between liquid water and a stationary phase of tethered n-C18 alkyl chains at a thermodynamic state of low pressure and water vapor-liquid coexistence. The interfacial water (oxygen atom) density profile so obtained is compared with a precisely defined proximal density of water molecules (oxygen atoms) conditional on the alkyl chain configurations. Though the conventional interfacial density profile takes a traditional monotonic form, the proximal radial distribution of oxygen atoms around a specific methyl (methylene) group closely resembles that for a solitary methane solute in liquid water. Moreover, this proximal radial distribution function is sufficient to accurately reconstruct the water oxygen density profile of the oil-water interface. These observations provide an alternative interpretation to collective drying or vaporization interpretations of commonly observed oil-water interfacial profiles for which water penetration into the interfacial region plays a role.     

Equilibrium morphologies and effective spring constants of capillary bridges

We theoretically study the behavior of a liquid bridge formed between a pair of rigid and parallel plates. The plates are smooth, they may either be homogeneous or decorated by circular patches of more hydrophilic domains, and they are generally not identical. We calculate the mechanical equilibrium distance of the liquid bridge as a function of liquid volume, contact angle, and radius of the chemical domain. We show that a liquid bridge can be an equilibrium configuration as long as the sum of the contact angles at the two walls is larger than 180°. When comparisons are possible, our results agree well with recent analytical and molecular dynamics simulation results. We also derive the effective spring constant of the liquid bridge as it is perturbed from its equilibrium distance. The spring constant diverges when the sum of the contact angles is 180° and is finite otherwise. The value of the spring constant decreases with increasing contact angle and volume, and the rate at which it decreases depends strongly on the properties of the two plates.     

Liquid Bridge between Two Moving Spheres: An Experimental Study of Viscosity Effects

The effects of viscosity on the mechanical response of a liquid bridge are investigated in the case of small amounts of liquid axially strained between two moving spheres. An experimental setup allows the measurement of capillary and viscous forces exerted on the spheres as a function of the spheres separation distance and the spheres velocity. The experimental results are found to be accurately described over a large range in spheres velocity and liquid viscosity by a simple closed-form expression. In addition, the bridge rupture distance is found to increase like the square root of the separation velocity. Copyright 2000 Academic Press.     

Nodoids and toroids: comparison of two geometries for the meniscus profile of a wetting liquid between two touching isolated spheres and extensions to the model of a collection of packed spheres

In the mid 1960s the present authors published two papers dealing with penetration of nonwetting liquids such as mercury into the interstitial void spaces using the model of uniform packed spheres. A circular arc was used to approximate the liquid-vapor interface in both papers. However, our circular arc-toroid values for the pressure-volume relationship in the pendular ring which exists between two touching spheres was criticized. The authors concluded that our approximation led to unacceptably large differences compared to the values calculated from the exact nodoid shape. This incorrect conclusion was never rebutted and has, in fact, been misinterpreted by subsequent workers to include values calculated for the shape of the access opening and the associated pressure for penetration into the void space of a collection of spheres. This leaves a cloud of uncertainty, not only over our original work on nonwetting fluids, but on the application of our procedures to the field of wetting fluids. The contrast in the geometrical shapes of the toroid and nodoid is depicted and the pressure values are compared at equal volumes. In contrast to the claim of excessive error, we show the toroid geometry, in conjunction with a pressure-volume work derivation, to have a maximum error of 0.06% as compared to the nodoid at all liquid-solid contact angles. The toroid also has the advantage of using a readily derived work versus surface free energy balance rather than requiring the use of incomplete elliptic integrals to evaluate the nodoid. Attempts to use radii of curvature to evaluate the toroid shape are shown to give extremely poor approximations of the exact values for the pressure. Values reported for access to the interior void space of a collection of spheres still need adjustment for the effect of contact angles between 0 degrees and 180 degrees for characterizing assemblies of real solids by computing "equivalent spherical" particle size from porosity and mercury penetration data. However, there is no anticipation that use of the circular arc will introduce large errors in the results. This gives confidence to us and others working with wetting media to test the potential applicability of the packed sphere model to various diverse fields.