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.     

A mathematical model for the interactions between non-identical rough spheres, liquid bridge and liquid vapor

Investigation of the interactions between the skeleton of an unsaturated particulate material and the contained liquid involves the general interaction model consisting of a liquid bridge in contact with two rigid smooth spherical particles of unequal size and dissimilar material, at a separation determined by their actual surface roughness, and surrounded by a gas with a vapor pressure at equilibrium with the liquid. The liquid retention and capillary force of the system are related to the capillary suction, liquid-solid contact angles, filling angles, roughness of the particles, and the ratio of particle radii in normalized terms by assuming a circular arc for the shape of the liquid profile. The normalized suction is also related to the corresponding relative humidity of the pore air. The calculated equilibrium relations are shown to possess non-uniqueness, which is interpreted in terms of mechanical properties of unsaturated particulate materials. The model is able to provide new insights into the behavior of an unsaturated particulate material.     

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.     

Intrusion pressure to initiate flow through pores between spheres

In this work, clusters of three or four spheres were used to examine intrusion pressure. Polytetrafluoroethylene (PTFE) or polyamide 66 (PA66) spheres were arranged horizontally to create a single pore. Liquid drops of water or ethylene glycol were gently introduced from above. If the spheres were too large, drops flowed through as soon as they were deposited. If the spheres were too small, liquid was suspended in the neck of the pore and could not pass through; drops became unstable and fell to one side. Alternatively, if spheres of a certain size were chosen, then capillary forces initially prevented drops of lesser stature from breaking though. However, as these drops grew taller, they eventually reached a height where the gravitational force exceeded the capillary force and the liquid flowed through the pore. A simple model for intrusion pressure was derived. Estimates from the model agreed well with experimentally measured values.     

Melt–vapor phase transition in the lead–selenium system at atmospheric and low pressure

The boiling temperature and the corresponding vapor phase composition in the existence domain of liquid solutions were calculated from the partial pressures of saturated vapor of the components and lead selenide over liquid melts in the lead–selenium system. The phase diagram was complemented with the liquid–vapor phase transition at atmospheric pressure and in vacuum of 100 Pa, which allowed us to judge the behavior of the components during the distillation separation.     

Vapor pressures and PVT properties of 1,1,1,3,3-pentafluoropropane (HFC-245fa)

The vapor pressures and PVT properties of superheated vapor and compressed liquid of 1,1,1,3,3-pentafluoropropane (HFC-245fa) were measured at wide range of temperature and pressure. The simple correlation for vapor pressures, compressibility factors of superheated vapor and specific volumes of liquid were developed on the basis of the present measurements. The critical pressure was calculated by extrapolating the developed vapor pressure equation to the critical temperature. Isothermal compressibility of liquid was calculated from the developed Tait equation. Specific volume data obtained show the good linearity in the Hudleston plots. Overall uncertainty in the vapor pressure, compressibility factor and specific volume measurements is estimated less than ±5 kPa, ±1.2% and ±0.09%, respectively.     

Capillary condensation of vapors in highly dispersed systems

Summary Exact calculations are made of the volumes of capillary condensate at various relative pressures in systems of two spheres of different radii in contact, and it is shown that a difference of 25% in the dimensions of the spheres in contact results in an appreciable increase in the volume of liquid, as compared with the case of spheres of equal radii.     

Fabrication of crystalline silicon spheres by selective laser heating in liquid medium

Micrometer and submicrometer crystalline silicon spheres were fabricated by selective laser heating of irregular silicon particles in liquid medium. TEM, SEM, XRD, and XPS characterized the structure and morphology of the prepared silicon spheres. The results suggested that they were spherical with a single crystalline structure. In this study, the formation mechanism of the spheres is analyzed, and the process parameters are optimized to obtain high-quality silicon spheres. A theoretical deduction regarding the relationship between critical laser energy density and particle size is also discussed, by which we can predict that larger spheres can be obtained at higher laser energy densities.     

Molar volume and adsorption isotherm dependence of capillary forces in nanoasperity contacts

The magnitude of the capillary force at any given temperature and adsorbate partial pressure depends primarily on four factors: the surface tension of the adsorbate, its liquid molar volume, its isothermal behavior, and the contact geometry. At large contacting radii, the adsorbate surface tension and the contact geometry are dominating. This is the case of surface force apparatus measurements and atomic force microscopy (AFM) experiments with micrometer-size spheres. However, as the size of contacting asperities decreases to the nanoscale as in AFM experiments with sharp tips, the molar volume and isotherm of the adsorbate become very important to capillary formation as well as capillary adhesion. This effect is experimentally and theoretically explored with simple alcohol molecules (ethanol, 1-butanol, and 1-pentanol) which have comparable surface tensions but differing liquid molar volumes. Adsorption isotherms for these alcohols on silicon oxide are also reported.     

Saturated vapor pressure in the thallium-cadmium system

Component vapor pressures in the thallium-cadmium system determined by the method of boiling points (isothermal variant) and the flow method were used to calculate the thermodynamic characteristics of the vapor and condensed phases in the region of the existence of liquid solutions. The temperature-concentration dependences of the thermodynamic values were determined. The phase diagram was augmented by liquid-vapor phase transitions at atmospheric pressure and in a vacuum (100 and 10 Pa) with the determination of the boundaries for the coexistence region of the liquid and vapor phases.     

Architecture of micron-scale hollow spheres composed of silica nanoparticles and approach to luminescent spheres

Micron-scale hollow spheres were successfully constructed with silica nanoparticles by templating of polymer spheres. Subsequently, the use of 3-aminopropyltriethoxysilane (APTES) introduces carbon and oxygen defects in the silica nanoparticles resulting from calcination of the aminopropyl group. In this approach, the template of micron-scale polymer spheres was prepared from dispersion polymerization. Subsequent St?ber process results in the formation of a silica layer attached to the polymer sphere surfaces. After calcination, the obtained micron-scale hollow silica spheres were then studied on the relationship between the particle diameter and the surface morphology. The luminescence of hollow spheres was prepared through using APTES in St?ber process, and which of related the appearance of luminescence to the APTES concentration and calcination temperature. The results of this study can provide useful information for the structure of micron-scale hollow spheres and their application to luminescent materials.     

Vitrification of polymer solutions as a function of solvent quality, analyzed via vapor pressures

Vapor pressures (headspace sampling in combination with gas chromatography) and glass transition temperatures [differential scanning calorimetry (DSC)] have been measured for solutions of polystyrene (PS) in either toluene (TL) (10-70 degrees C) or cyclohexane (CH) (32-60 degrees C) from moderately concentrated solutions up to the pure polymer. As long as the mixtures are liquid, the vapor pressure of TL (good solvent) is considerably lower than that of CH (theta solvent) under other identical conditions. These differences vanish upon the vitrification of the solutions. For TL the isothermal liquid-solid transition induced by an increase of polymer concentration takes place within a finite composition interval at constant vapor pressure; with CH this phenomenon is either absent or too insignificant to be detected. For PS solutions in TL the DSC traces look as usual, whereas these curves may become bimodal for solutions in CH. The implications of the vitrification of the polymer solutions for the determination of Flory-Huggins interaction parameters from vapor pressure data are discussed. A comparison of the results for TL/PS with recently published data on the same system demonstrates that the experimental method employed for the determination of vapor pressures plays an important role at high polymer concentrations and low temperatures.     

Structure and phase behavior of a confined nanodroplet composed of the flexible chain molecules

A polymer density functional theory has been employed for investigating the structure and phase behaviors of the chain polymer, which is modelled as the tangentially connected sphere chain with an attractive interaction, inside the nanosized pores. The excess free energy of the chain polymer has been approximated as the modified fundamental measure-theory for the hard spheres, the Wertheim's first-order perturbation for the chain connectivity, and the mean-field approximation for the van der Waals contribution. For the value of the chemical potential corresponding to a stable liquid phase in the bulk system and a metastable vapor phase, the flexible chain molecules undergo the liquid-vapor transition as the pore size is reduced; the vapor is the stable phase at small volume, whereas the liquid is the stable phase at large volume. The wide liquid-vapor coexistence curve, which explains the wide range of metastable liquid-vapor states, is observed at low temperature. The increase of temperature and decrease of pore size result in a narrowing of liquid-vapor coexistence curves. The increase of chain length leads to a shift of the liquid-vapor coexistence curve towards lower values of chemical potential. The coexistence curves for the confined phase diagram are contained within the corresponding bulk liquid-vapor coexistence curve. The equilibrium capillary phase transition occurs at a higher chemical potential than in the bulk phase.     

Particle-surface capillary forces with disjoining pressure

A new, atomic force microscopy (AFM) based experimental setup for the continuous acquisition of friction force data as a function of humidity has been developed. The current model of interactions between wet contacts under the influence of capillary effects, has been amended to include a vertical component due to the disjoining pressure and takes into account the influence of liquid films adsorbed on the surface. This is a 'switching' model, i.e. the contact between nanometer-sized sphere and a flat surface can exist in two distinct states due to capillary bridge formation/destruction as the humidity is varied. The model has been qualitatively verified on samples of differing wettability produced by UV-ozone treatment of polystyrene (PS). Results of AFM analysis of the friction vs. vapor pressure curves collected from the surface are presented. Correlation between important surface properties such as wettability, adsorption, and contact angle and friction force under varying humidity were found and discussed.     

Influence of surface topography on the interactions between nanostructured hydrophobic surfaces

Nanostructured particle coated surfaces, with hydrophobized particles arranged in close to hexagonal order and of specific diameters ranging from 30 nm up to 800 nm, were prepared by Langmuir-Blodgett deposition followed by silanization. These surfaces have been used to study interactions between hydrophobic surfaces and a hydrophobic probe using the AFM colloidal probe technique. The different particle coated surfaces exhibit similar water contact angles, independent of particle size, which facilitates studies of how the roughness length scale affects capillary forces (previously often referred to as "hydrophobic interactions") in aqueous solutions. For surfaces with smaller particles (diameter < 200 nm), an increase in roughness length scale is accompanied by a decrease in adhesion force and bubble rupture distance. It is suggested that this is caused by energy barriers that prevent the motion of the three-phase (vapor/liquid/solid) line over the surface features, which counteracts capillary growth. Some of the measured force curves display extremely long-range interaction behavior with rupture distances of several micrometers and capillary growth with an increase in volume during retraction. This is thought to be a consequence of nanobubbles resting on top of the surface features and an influx of air from the crevices between the particles on the surface.     

Packed uniform sphere model for solids: interstitial access opening sizes and pressure deficiencies for wetting liquids with comparison to reported experimental results

Mathematical relationships have been developed to describe the pressure deficiencies required for drainage and removal of wetting liquids through the access openings to the interstitial void space for a model comprised of uniform packed solid spheres. Access openings and associated pressure deficiencies are defined in terms of the packing and radius of the spheres, using a circular arc approximation for the liquid-vapor portion of the perimeter of the opening. This allows determination of equivalent particle radius rather than equivalent cylindrical pore radius within a porous solid sample by use of standard pressure, porosity and desorption data. For a known particle size and porosity, it allows comparison and prediction of drainage of wetting liquids and pressures required for removal of the liquid from compacted materials and collections of random packed spherical particles. Comparisons are made to experimental packing of spheres. Sorption isotherms for a volatile wetting liquid are presented, covering the access to the interstitial void space, the pendular liquid ring between adjacent touching spheres and the monolayer surface area. The larger size of the interstitial void space compared to the size of the access opening leads to lower imbibition pressures and hysteresis for both volatile and nonvolatile wetting liquids. The relationship to mercury porosimetry and the adjustment for contact angles other than 0 degrees and 180 degrees are discussed.     

Determination of the melting point of hard spheres from direct coexistence simulation methods

We consider the computation of the coexistence pressure of the liquid-solid transition of a system of hard spheres from direct simulation of the inhomogeneous system formed from liquid and solid phases separated by an interface. Monte Carlo simulations of the interfacial system are performed in three different ensembles. In a first approach, a series of simulations is carried out in the isothermal-isobaric ensemble, where the solid is allowed to relax to its equilibrium crystalline structure, thus avoiding the appearance of artificial stress in the system. Here, the total volume of the system fluctuates due to changes in the three dimensions of the simulation box. In a second approach, we consider simulations of the inhomogeneous system in an isothermal-isobaric ensemble where the normal pressure, as well as the area of the (planar) fluid-solid interface, are kept constant. Now, the total volume of the system fluctuates due to changes in the longitudinal dimension of the simulation box. In both approaches, the coexistence pressure is estimated by monitoring the evolution of the density along several simulations carried out at different pressures. Both routes are seen to provide consistent values of the fluid-solid coexistence pressure, p=11.54(4)k(B)T/sigma(3), which indicates that the error introduced by the use of the standard constant-pressure ensemble for this particular problem is small, provided the systems are sufficiently large. An additional simulation of the interfacial system is conducted in a canonical ensemble where the dimensions of the simulation box are allowed to change subject to the constraint that the total volume is kept fixed. In this approach, the coexistence pressure corresponds to the normal component of the pressure tensor, which can be computed as an appropriate ensemble average in a single simulation. This route yields a value of p=11.54(4)k(B)T/sigma(3). We conclude that the results obtained for the coexistence pressure from direct simulations of the liquid and solid phases in coexistence using different ensembles are mutually consistent and are in excellent agreement with the values obtained from free energy calculations.     

Creeping motion of an assemblage of composite spheres relative to a fluid

The sedimentation of a homogeneous distribution of spherical composite particles and the fluid flow through a bed of these particles are investigated theoretically. Each composite particle is composed of a spherical solid core and a surrounding porous shell. In the fluid-permeable porous shell, idealized hydrodynamic frictional segments are assumed to distribute uniformly. The effect of interactions among the particles is taken into explicit account by employing a fundamental cell-model representation which is known to provide good predictions for the motion of a swarm of nonporous spheres within a fluid. In the limit of a small Reynolds number, the Stokes and Brinkman equations are solved for the flow field in a unit cell, and the drag force exerted by the fluid on the particle is obtained in a closed form. For a distribution of composite spheres, the normalized mobility of the particles decreases or the particle interactions increase monotonically with a decrease in the permeability of their porous shells. The effect of particle interactions on the creeping motion of composite spheres relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solutions describing the drag force or mobility for a suspension of composite spheres reduce to those for suspensions of solid spheres and of porous spheres. The hydrodynamic behavior for composite spheres may be approximated by that for permeable spheres when the porous layer is sufficiently thick, depending on the permeability.     

Nanospheres in a nematic liquid crystal solvent: the influence of particle size

We use Monte Carlo simulations to investigate a simple lattice model for nematic liquid crystals containing nanospheres. The influence of particle size on the phase behaviour is studied using two different sized particles. The phase diagram is found to be topologically equivalent for both particle sizes, with a large biphasic region corresponding to coexistence between a rod-rich nematic and a rod-poor isotropic phase. For small spheres, the rod-rich nematic phase is stable for relatively large volume fractions of spheres (up to a maximum of about 16%). In contrast, the nematic phase for the system with larger spheres is constrained to a much narrower region of the phase diagram.     

Sedimentation of spheres at small Reynolds number

The effect of fluid inertia on the settling of spheres in a viscous incompressible fluid is studied in the limit of small Reynolds number. The kinetic energy of flow depends on the positions of the spheres, and gives rise to forces on the spheres. In the dilute limit it suffices to study the corresponding pair interaction. The interaction is calculated from the Stokes flow for two spheres settling between plane walls in the point particle limit. The dissipative interaction between a pair of spheres is calculated from the Proudman-Pearson [I. Proudman and J. R. A. Pearson, J. Fluid Mech. 2, 237 (1957)] solution of the Navier-Stokes equations for flow about a sphere in unbounded geometry. The combination of kinetic and dissipative interaction gives rise to a repulsive force of range of the order of the sphere diameter divided by the Reynolds number.