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.     

Modeling the Evolution and Rupture of Pendular Liquid Bridges in the Presence of Large Wetting Hysteresis

A model has been developed to predict the shape evolution, rupture distance and postrupture liquid distribution of a pendular liquid bridge between two unequally sized spherical particles in the presence of wetting hysteresis. Two different simplifications of the bridge geometry were considered: a toroidal and a parabolic approximation. The liquid bridge was assumed to rupture through its thinnest neck leaving liquid distributed on each sphere. Experimental measurements showed that the rupture distance was well predicted by both profile approximations by assuming that rupture occurred when the liquid-vapor interfacial area of the bridge and the postrupture droplets was equal. Both bridge profile approximations only correctly predicted the evolution of the apparent contact angle and the extent of postrupture liquid distribution when the solid-liquid interfacial area measured throughout the separation was included in the calculations. This is because during the pendular liquid bridge elongation, the three-phase contact line usually begins to slip on at least one of the spheres. The parabolic profile approximation was slightly more accurate than the toroidal one. The toroidal approximation is more difficult to use because one of the parameters passes through infinity as the bridge changes from convex to concave in shape. In some cases the toroidal approximation was also unable to generate a solution. Copyright 2000 Academic Press.     

A conductance study of reducing volume liquid bridges

This work demonstrates how electrical conductance measurements can be employed for the study of liquid bridge behavior when their volume varies with time while their separation distance remains constant. The liquid bridges are edge pinned between two vertical, identical rods (r-bridges) at varying separation distances. Liquid evaporation is used as a means of reducing the bridge volume in a continuous smooth fashion. A zero-order continuation sequence with respect to Bond number and liquid bridge volume is combined with the shooting method for the solution of the Young-Laplace equation to give the liquid bridge shape as a function of its instantaneous volume. A novel, very efficient computational scheme is developed based on singular perturbation expansion for the solution of the Laplace equation in the liquid bridge to compute its electrical conductance that proved faster by orders of magnitude compared to other alternative approaches. The potential for estimating the liquid bridge characteristics or the evaporation rate by matching the experimental and theoretical results is discussed extensively.     

Hydrodynamic interaction between spheres coated with deformable thin liquid films

In this article, we considered the hydrodynamic interaction between two unequal spheres coated with thin deformable liquids in the asymptotic lubrication regime. This problem is a prototype model for drop coalescence through the so-called "film drainage" mechanism, in which the hydrodynamic contribution comes dominantly from the lubrication region apart from the van der Waals interaction force. First, a general formulation was derived for two unequal coated spheres that experienced a head-to-head collision at a very close proximity. The resulting set of the evolution equations for the deforming film shapes and stress distributions was solved numerically. The film shapes and hydrodynamic interaction forces were determined as functions of the separation distance, film thickness, viscosity ratios, and capillary numbers. The results show that as the two spheres approach each other, the films begin to flatten and eventually to form negative curvature (or a broad dimple) at their forehead areas in which high lubrication pressure is formed. The dimple formation occurs earlier as the capillary number increases. For large capillary numbers, the film liquids are drained out from their forehead areas and the coated liquid films rupture before the two films "touch" each other. Meanwhile, for small capillary numbers, the gap liquid is drained out first and the two liquid films eventually coalesce.     

Different elution modes and field programming in gravitational field-flow fractionation: field programming using density and viscosity gradients

In previous papers, several approaches to programming of the resulting force field in GFFF were described and investigated. The experiments were dealing with flow-velocity and channel thickness, i.e. factors influencing hydrodynamic lift forces (HLF). The potential of density and viscosity of carrier liquid for field programming was predicted and demonstrated by preliminary experiments. This work is devoted to experimental verification of the influence of carrier liquid density and viscosity. Several carrier liquid density and simultaneously viscosity gradients using water-methanol mixtures are in this work implemented in the separation of a model silica mixture. Working with the water-methanol gradients, one is not able to separate the influence of density from the contribution of viscosity. However, we found experimental conditions to show the isolated effect of carrier liquid density (two water-methanol mixtures of equal viscosity differing in their densities). In order to demonstrate the isolated effect of viscosity, we implemented in this work a new system of (hydroxypropyl)methyl cellulose (HPMC) carrier liquids. Three different HPMC compositions enabled to vary the viscosity more than two times at almost constant density. With increasing carrier liquid viscosity, the focusing and elevating trend was clearly pronounced for 5 and 10 microm silica particles. By the isolated effect of increased viscosity, the centre of the 10 microm particle zone was elevated to the streamline at 16% of the channel height. These experiments have shown that the influence of carrier liquid viscosity on HLF should be taken into account even at higher levels above the channel bottom, i.e. beyond the near-wall region. Further, it is shown that higher value of carrier liquid viscosity improves the separation of the model mixture in terms of time and resolution.     

Viscoelastic Drag of Particles Settling in Wormlike Micellar Solutions of Varying Surfactant Concentration

Wormlike micellar octadecyl trimethyl ammonium chloride (OTAC) solution is a self-assembled fracturing fluid used to carry proppants into fractures in oil recovery. Slow settling velocity of proppant is desirably resulted from the viscoelastic drag with low viscosity of fracturing fluids for fracturing work. Steel spheres, as a substitute for proppants, fall into three semi-dilute OTAC solutions. The steady rheology demonstrates that OTAC solutions are divided into shear-thickening and shear-thinning regimes by the critical shear rate. The applied steel spheres always lie in the shear-thickening regime of the 2.8 wt% OTAC solution with aggregated micelles as their characteristic shear rates are less than the critical shear rate of the solution. Strong shear-thickening viscous drag results in lower settling velocity of steel spheres. Most of the applied steel spheres, on the other hand, lie in the shear-thinning regime of the 4 wt% OTAC solution with orientated micelles. Although the latter solution has small dissipation coefficient, high Weissenberg number, and consequently high elastic effect, the shear-thinning viscosity results in higher settling velocity of steel spheres.     

Atomic Force Microscopy Study of the Adhesion of Saccharomyces cerevisiae

The influence of the liquid properties and the operating variables on the intrinsic volumetric flow rate, q(0), of the droplets at the liquid-atmosphere interface in nonfoaming adsorptive bubble separation and that, W(f0), of the liquid in foam at the liquid-foam interface in foam separation was studied to estimate the enrichment ratio of surface-active substance in the two techniques. Each intrinsic flow rate was determined by the extrapolation method, which the authors proposed previously, and was mainly influenced by superficial gas velocity, liquid viscosity, and surface tension. Although the changes in the surface tension and the liquid viscosity were small, they greatly affected the intrinsic flow rates for both the droplets and the liquid in foam. The experimental results were applied to a dimensional analysis. Dimensionless volumetric flow rates, q(0)/Q and W(f0)/Q, were successfully correlated with a dimensionless number (Ohnesorge number, Z=&mgr;(L)/(rho(L)sigmad(b))(0.5)). q(0)/Q=6.66 Z(1.46) and W(f0)/Q=2.53Z(0.533) were obtained for the nonfoaming and the foaming regions within errors of 30 and 35%, respectively. The enrichment ratio of surface-active substance in the droplets and the liquid in foam to the bulk liquid was derived by using these correlations for the nonfoaming and the foaming regions, respectively. A good agreement of the experimental enrichment ratio and the calculated ones was obtained by using the correlations for nonfoaming and foaming. Copyright 2001 Academic Press.     

Viscosity of liquid mixtures: the Vesovic-Wakeham method for chain molecules

New expressions for the viscosity of liquid mixtures, consisting of chain-like molecules, are derived by means of Enskog-type analysis. The molecules of the fluid are modelled as chains of equally sized, tangentially joined, and rigid spheres. It is assumed that the collision dynamics in such a fluid can be approximated by instantaneous collisions. We determine the molecular size parameters from the viscosity of each pure species and show how the different effective parameters can be evaluated by extending the Vesovic-Wakeham (VW) method. We propose and implement a number of thermodynamically consistent mixing rules, taking advantage of SAFT-type analysis, in order to develop the VW method for chain molecules. The predictions of the VW-chain model have been compared in the first instance with experimental viscosity data for octane-dodecane and methane-decane mixtures, thus, illustrating that the resulting VW-chain model is capable of accurately representing the viscosity of real liquid mixtures.     

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.     

Hollow spheres of aromatic polyamide prepared by reaction‐induced phase separation

Hollow spheres of aromatic polyamide are obtained by the reaction‐induced phase separation during polymerization of 5‐hydroxyisophthalic acid and 1,4‐phenylene diamine in an aromatic solvent at a concentration of 1–2% at 320 °C without stirring. The hollow sphere has a dimple hole and the diameters of the hollow spheres are 3–4 μm. The droplets are initially generated via liquid–liquid phase separation and then rigid cross‐linked network structure formed the rigid skin layer on the surface of the droplets. The solidification of the droplets occurred owing to the further polymerization in them with maintaining the morphology to form the hollow spheres. The hollow spheres exhibit outstanding thermal stability. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2013     

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.     

Electrophoresis of two identical rigid spheres in a charged cylindrical pore

The electrophoresis of two identical spheres moving along the axis of a long cylindrical pore under the conditions of low surface potential and weak applied electric field is investigated. The geometry considered allows us to examine simultaneously the effects of boundary and the presence of a nearby entity on the behavior of a particle. The influences of the separation distance between two spheres, the thickness of a double layer, the ratio (radius of sphere/radius of pore), and the charged conditions on the surfaces of the spheres and the pore on the mobility of a particle are investigated. Several interesting results that are not reported in the literature are observed. For instance, although for the case of two positively charged spheres in an uncharged pore the qualitative behavior of a sphere depends largely on its size relative to that of a pore and the thickness of the double layer, this might not be the case when two uncharged spheres are in a positively charged pore. In addition, in the latter, the mobility of a sphere increases with the increases in the separation distance between two spheres, and this effect is pronounced when the ratio (radius of sphere/radius of pore) takes a medium value or the thickness of the double layer is either sufficiently thin or sufficiently thick.     

Cluster kinetics of pressure-induced glass formation

A prior correlation model for glass formation based on cluster-size distribution kinetics is here extended to account for pressure effects as well as temperature effects. The model describes how rapidly cooling or compressing a liquid or colloid leads to structural arrest and a consequent sharp rise in viscosity or dielectric relaxation time. In addition to activation energies, we include activation volumes in the rate coefficients for monomer-cluster addition and dissociation and cluster aggregation and breakage. The approach leads to scaled pressure correlations and plots for viscosity that reveal strong and fragile glass behavior, and agree with experimental data. A simple relationship among viscosity, attractive interparticle energy, and particle volume fraction displays how hard spheres with attractive forces can vitrify at small particle densities.     

Experiments on 2-D suspensions

The rheological properties (velocity profile, viscosity) and the statistical distibution of objects in quasi-two-dimensional suspension of macroscopic spheres submitted to a shear is reported. The statistical data are analyzed by analogy with percolation.     

Model for Plateau border drainage of power-law fluid with mobile interface and its application to foam drainage

A model for drainage of a power-law fluid through a Plateau border is proposed which accounts for the actual Plateau border geometry and interfacial mobility. The non-dimensionalized Navier-Stokes equations have been solved using finite element method to obtain the contours of velocity within the Plateau border cross section and average Plateau border velocity in terms of dimensionless inverse surface viscosity and power-law rheological parameters. The velocity coefficient, the correction for the average velocity through a Plateau border of actual geometry compared to that for a simplified circular geometry of the same area of cross section, was expressed as a function of dimensionless inverse surface viscosity and flow behavior index of the power-law fluid. The results of this improved model for Plateau border drainage were then incorporated in a previously developed foam drainage model [G. Narsimhan, J. Food Eng. 14 (1991) 139] to predict the evolution of liquid holdup profiles in a standing foam. Foam drainage was found to be slower for actual Plateau border cross section compared to circular geometry and faster for higher interfacial mobility and larger bubble size. Evolution of liquid holdup profiles in a standing foam formed by whipping and stabilized by 0.1% beta-lactoglobulin in the presence of xanthan gum when subjected to 16g and 45g centrifugal force fields was measured using magnetic resonance imaging for different xanthan gum concentrations. Drainage resulted in the formation of a separate liquid layer at the bottom at longer times. Measured bubble size, surface shear viscosity of beta-lactoglobulin solutions and literature values of power-law parameters of xanthan gum solution were employed in the current model to predict the evolution of liquid holdup profile which compared well with the experimental data. Newtonian model for foam drainage for zero shear viscosity underpredicted drainage rates and did not agree with the experimental data.     

Translational motion of a spherical particle near a planar liquid-fluid interface

The translational electrophoretic motion of a colloidal spherical particle parallel to a planar liquid-fluid interface is analyzed by using the reciprocal theorem developed by Yariv and Brenner [E. Yariv, H. Brenner, J. Fluid Mech. 484 (2003) 85]. Based on the thin electric double layers assumption, analytical solutions of the forces acting on the particle are obtained, and the influence of the liquid-fluid interface on the electrophoretic velocity of the particle is studied. It is found that the speed of the particle's electrokinetic motion will increase as the separation distance between the particle and the interface decreases. This enhancement of electrophoretic mobility becomes more significant when the viscosity of the fluid phase becomes larger.     

Liquid drainage in single plateau borders of foam

This paper reports on an investigation of the influence of the interfacial shear viscosity on the liquid drainage in single Plateau borders of foam. The simplified Navier-Stokes equation governing the liquid flow is solved for the liquid velocity by the numerical computational method. The numerical results show significant influence of the interfacial shear viscosity on the liquid velocity in the Plateau border. Comparison of the numerical results for the average velocity over the cross-section area of the Plateau border to the available analytical solution shows that the available analytical solution underestimates the average velocity. New, simple yet accurate correlations for the dependence of the average velocity on the radius of the cross section of the Plateau border, the pressure gradient, and the interfacial shear viscosity are obtained using the asymptotic analysis and the numerical data.     

The liquid flow force on a particle in the bubble-particle interaction in flotation

In this paper the problem of calculating the liquid flow force on a particle in interaction with an air bubble with a mobile surface in flotation as a function of the separation distance was solved. The force equation was obtained by first deriving the disturbed flow confined between the surfaces. The model for the force includes the separation distance between the bubble and the particle, the particle size, the bubble's Reynolds number, the bubble rise velocity, and the polar position of the particle on the bubble surface. The proposed equations provide an exact solution to the situation where the particle and the bubble are very close together. The attractive flow force and the surface forces are of similar orders of magnitude. Consequently, the models presented in this paper should provide a better estimate for calculating the forces on particles interacting with air bubbles in mineral flotation and other separation operations involving colloidal interactions.     

Sedimentation velocity and potential in a concentrated suspension of charged liquid drops

The sedimentation behavior of a concentrated suspension of charged liquid drops is analyzed theoretically at arbitrary surface potential and arbitrary double-layer thickness; that is, the effects of double-layer polarization and double-layer overlapping are taken into account. Kuwabara's unit cell model is employed to model the suspension system, and a pseudospectral method based on the Chebyshev polynomial is adopted to solve the governing electrokinetic equations numerically. Several interesting phenomena, which are of significant influence if the internal flow inside a liquid drop is taken into account, are observed. Key factors are examined such as the thickness of the electric double layer, the magnitude of the surface potential, the volume fraction of liquid drops, and the viscosity of the internal fluid. The results presented here add another dimension to the previous studies, which include concentrated suspensions of rigid particles and mercury drops under low zeta potential, with the consideration of the internal flow of liquid drops and double-layer polarization, characterized by its viscosity and the zeta potential respectively. It is found, among other things, that the smaller the viscosity of the internal fluid is, the higher the sedimentation velocity of liquid drops. The higher the zeta potential is, the larger the decrease in sedimentation velocity. In particular, the sedimentation velocity of an inviscid drop (gas bubble) is about three times higher than that of a rigid one. The decrease in sedimentation velocity resulting from the effect of double-layer polarization achieves about 50% if the zeta potential is sufficiently high.