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.     

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.     

Probing the limits of the Derjaguin approximation with scanning force microscopy

We have measured the interaction force between a silicon nitride scanning force microscopy (SFM) probe and the basal plane of highly oriented pyrolitic graphite as a function of pH and ionic concentration in aqueous solutions. Forces in the range +/- 50 pN were reconstructed from measured signals using dynamical analysis of the cantilever. We modeled the force-separation data using a flat plate electric double-layer interaction and assumed the Derjaguin approximation to adapt the flat plate geometry for the SFM probe shape. Measured forces were well modeled by the theory at high ionic concentrations (10 and 100 mM), where Debye lengths were 3.0 and 0.96 nm, respectively. The theory failed to model forces at a lower ionic concentration (1 mM), where the Debye length was 9.6 nm. To investigate this, we calibrated the SFM probe geometry using blind reconstruction and obtained an apex radius of 7 nm. This value suggested that failure of the theory was due to an invalidation of the Derjaguin approximation at long Debye lengths, where the characteristic length scale for the interaction was larger than the size of the SFM probe. The errors were reduced by replacing the Derjaguin approximation with a surface element integration. The result experimentally demonstrates the limitations of the Derjaguin approximation for predicting interactions of nanoscale colloids.     

Molecular dynamics simulation of the forces between colloidal nanoparticles in n-decane solvent

Molecular dynamics is utilized to simulate solvation forces between two nanoparticles immersed in liquid n-decane. Three types of solvophilic nanoparticles are investigated with sizes in the 1-6 nm range: small and large amorphous spheres and crystalline cubes. We find that the solvation forces are negligible for the small spheres, which have diameters comparable to the end-to-end distance of all-trans decane, and we attribute this to the inability of the small spheres to induce decane ordering in the interparticle gap. The cubic nanoparticles (and to a lesser extent, the large spheres) are able to induce the formation of solidlike, n-decane layers in their gap for certain nanoparticle separations, and the transition between layered and disordered structures leads to solvation forces that oscillate between repulsion and attraction as the nanoparticle separation is varied. We find that the Derjaguin approximation [B. V. Derjaguin, Kolloid-Z. 69, 155 (1934)] is not effective at describing the dependence of the solvation forces on nanoparticle size and shape-contrasting results from a previous study involving these nanoparticles in Lennard-Jones solvent [Y. Qin and K. A. Fichthorn, J. Chem. Phys. 119, 9745 (2003)]. In particular, we find that for decane, the magnitude of the repulsive solvation forces is sensitive to nanoparticle size and shape, a phenomenon we attribute to the size and rigid-rod structure of n-decane, which makes its ordering in the interparticle gap sensitive to the size and the surface roughness of the nanoparticles.     

Surface integration approach: a new technique for evaluating geometry dependent forces between objects of various geometry and a plate

We present a new approach, which can be considered as a generalization of the Derjaguin approximation, that provides exact means to determine the force acting between a three-dimensional body of any shape and a half-space mutually interacting via pairwise potentials. Using it, in the cases of the Lennard-Jones, standard and the retarded (Casimir) van der Waals interactions we derive exact expressions for the forces between a half-space or a slab of finite thickness and an ellipsoid in a general orientation, which in the simplest case reduces to a sphere, a tilted fully elliptic torus, and a body obtained via rotation of a single loop generalized Cassini oval, a particular example of which mimics the shape of a red blood cell. The results are obtained for the case when the object is separated from the plane via a non-polar continuous medium that can be gas, liquid or vacuum. Specific examples of biological objects of various shapes interacting with a plate like substrates are also considered.     

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.     

Quasistatic detachment of a sphere from a liquid interface

In this paper the problem of removing a spherical particle initially attached to a liquid-gas interface is analytically treated. In particular, the Derjaguin equation for small radii is used to derive a closed-form approximate expression for the work of detachment of the sphere from the interface. Expressions corresponding to the prescribed displacement condition and the applied force condition, which seems to be the relevant condition for the flotation separation process, are presented. A special effort has been made to closely compare analytical results with data obtained through the exact numerical treatment of the detachment process. Results show that proposed expressions are sufficiently accurate to calculate the energy required to detach the sphere from the interface as soon as the sphere radius is small compared to the capillary length. Validity limits are specified.     

Capillary condensation as a morphological transition

The process of capillary condensation/evaporation in cylindrical pores is considered within the idea of symmetry breaking. Capillary condensation/evaporation is treated as a morphological transition between the wetting film configurations of different symmetry. We considered two models: (i) the classical Laplace theory of capillarity and (ii) the Derjaguin model which takes into account the surface forces expressed in terms of the disjoining pressure. Following the idea of Everett and Haynes, the problem of condensation/evaporation is considered as a transition from bumps/undulations to lenses. Using the method of phase portraits, we discuss the mathematical mechanisms of this transition hidden in the Laplace and Derjaguin equations. Analyzing the energetic barriers of the bump and lens formation, it is shown that the bump formation is a prerogative of capillary condensation: for the vapor-liquid transition in a pore, the bump plays the same role as the spherical nucleus in a bulk fluid. We show also that the Derjaguin model admits a variety of interfacial configurations responsible for film patterning at specific conditions.     

Exact analytical expressions for the potential of electrical double layer interactions for a sphere-plate system

Exact, closed-form analytical expressions are presented for evaluating the potential energy of electrical double layer (EDL) interactions between a sphere and an infinite flat plate for three different types of interactions: constant potential, constant charge, and an intermediate case as given by the linear superposition approximation (LSA). By taking advantage of the simpler sphere-plate geometry, simplifying assumptions used in the original Derjaguin approximation (DA) for sphere-sphere interaction are avoided, yielding expressions that are more accurate and applicable over the full range of κa. These analytical expressions are significant improvements over the existing equations in the literature that are valid only for large κa because the new equations facilitate the modeling of EDL interactions between nanoscale particles and surfaces over a wide range of ionic strength.     

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.     

Capillary forces: influence of roughness and heterogeneity

A formalism is described to calculate capillary forces between solid surfaces analytically. Assumptions are that the liquid menisci (1) have a much larger extension parallel to the gap than normal and (2) are formed by capillary condensation and are in equilibrium with the vapor. To calculate capillary forces, first the gap between the two surfaces is described by a height distribution function. Roughness is considered with an asperity distribution function. Both distributions can at least in principal be measured by light, electron, or atomic force microscopy or grazing incidence X-ray reflectivity. The total capillary force versus distance or vapor pressure is obtained by a convolution of both distributions and an integration. The formalism is applied to calculate the capillary force between rough spherical particles. In addition, a method to consider surface heterogeneity is suggested.     

球型胶粒间的相互作用能—Derjaguin法的改进

对求算球型胶体颗粒相互作用能的Ｄｅｒｊａｇｕｉｎ法进行了适当的修正，并由此得出简单的解析表达式，对低电位条件下等同和不等同球颗粒的计算结果表明，该法拓宽了Ｄｅｒｊａｇｕｉｎ公式的应用范围，提高了计算结果的精确度。     

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.     

Forces between a rigid probe particle and a liquid interface. III. Extraction of the planar half-space interaction energy E(D)

The deformation of a liquid drop (radius R0) under the probe particle (radius a) greatly complicates the interpretation by atomic force microscopy. For rigid interfaces, F(DeltaX) can be directly related to the interaction energy E(D) per unit area between planar half-spaces of probe material and drop material across a thickness D of the liquid medium by the Derjaguin approximation, [formula in text], where D(0) is the intersurface separation distance on the line of the centers of the bodies and DeltaX0 is a constant set by the somewhat arbitrary choice of origin for the separation distance DeltaX between the stage on which the drop rests and the lowest point on the probe particle. The problem of absolute intersurface separation distance is common to all surface force measurement techniques. For rigid interfaces, DeltaX0 may be established by bringing the surfaces into close (essentially hard) contact and making measurements in the constant compliance regime. For deformable interfaces, this is not possible and a general method of extracting the absolute separation distance has yet to be devised. In this paper we discuss a general algorithm for extracting E(D+DeltaX0) from F(DeltaX) data. We apply the method to constructed data to investigate the effect of data noise and to a set of real data for a sessile tetradecane droplet in water with an anionic surfactant and a bare silica probe.     

Capillary and van der Waals forces between uncharged colloidal particles linked by a liquid bridge

This work presents a theoretical study of the forces established between colloidal particles connected by means of a concave liquid bridge, where the solid particles are partially wetted by a certain amount of liquid also possessing a dry portion of their surfaces. In our analysis, we adopt a two-particle model assuming that the solids are spherical and with the same sizes and properties and that the liquid meniscus features an arc-of-circumference contour. The forces considered are the typical capillary ones, namely, wetting and Laplace forces, as well as the van der Waals force, assuming the particles uncharged. We analyze different parameters which govern the liquid bridge: interparticle separation, wetting angle, and liquid volume, which later determine the value of the forces. Due to the dual characteristic of the particles' surfaces, wet and dry, the forces are to be determined numerically in each case. The results indicate that the capillary forces are dominant in most of the situations meanwhile the van der Waals force is noticeable at very short distances between the particles.     

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.     

Interaction forces between adsorbed polymer layers

The interaction forces between adsorbed polymer layers were investigated. Two types of graft copolymers that were adsorbed on hydrophobic surfaces have been investigated: (i) a graft copolymer consisting of polymethylmethacrylate/polymethacrylic acid back bone (the B chain) on which several poly(ethylene oxide) chains are grafted (to be referred to as PMMA/PEO_n); and (ii) a graft copolymer consisting of inulin (linear polyfructose with degree of polymerization > 23) (the A chain) on which several C₁₂ chains are grafted (INUTEC SP1). In the first case adsorbed layers of the graft copolymer were obtained on mica sheets and the interaction forces were measured using the surface force apparatus. In the second case the interaction forces were measured using Atomic Force Microscopy (AFM). For this purpose a hydrophobically modified glass sphere was attached to the tip of the cantilever of the AFM and the glass plate was also made hydrophobic. Both the sphere and the glass plate contained an adsorbed layer of INUTEC SP1.In the surface forces apparatus one essentially measures the energy E(D)–distance D curves for the graft copolymer of PMMA/PEO_n between mica surfaces bearing the graft copolymer and this could be converted to interaction energy between flat surfaces. Using the de Gennes scaling theory, it is possible to calculate the interaction energy between the polymer layers. The same graft copolymer was used in latex dispersions and the high frequency modulus G′_∝ was measured as a function of the volume fraction ? of the dispersion. This high frequency modulus could be related to the potential of mean force. In this way one could compare the results obtained from rheology and those obtained from direct measurement of interaction forces.In the AFM method, the interaction forces are measured in the contact area between two surfaces, i.e. a spherical glass particle and a glass plate. Both glass spheres and plates were hydrophobized using dichlorodimethylsilane. Results were obtained for adsorbed layers of INUTEC SP1 in water and in the presence of various concentrations of Na₂SO₄ (0.3, 0.8, 1.0 and 1.5 mol dm^{− 3}). All results showed a rapid increase of force with a decrease of separation distance and the forces were still repulsive up to the highest Na₂SO₄ concentration. This explains the high stability of dispersions when using INUTEC SP1 as stabilizer.     

Controlling the formation of capillary bridges in binary liquid mixtures

We study the formation of capillary bridges between micrometer-sized glass spheres immersed in a binary liquid mixture using bright field and confocal microscopy. The bridges form upon heating due to the preferential wetting of the hydrophilic glass surface by the water-rich phase. If the system is cooled below the demixing temperature, the bridges disappear within a few seconds by intermolecular diffusion. Thus, this system offers the opportunity to switch the bridges on and off and to tune precisely the bridge volume by altering the temperature in a convenient range. We measure the bridge geometry as a function of the temperature from bright field images and calculate the cohesive force. We discuss the influence of the solvent composition on the bridge formation temperature, the strength of the capillary force, and the bridge volume growth rate. Furthermore, we find that the onset of bridge formation coincides with the water-lutidine bulk coexistence curve.     

Interfacial Forces in Dispersion Science and Technology

Interfacial forces determine many phenomena in dispersion science and technology. Eight types of interfacial forces are classified in this article. A general equation for all of them is derived here, with particular equation for each of them (being valid for simplified geometries, such as spheres, cylinders, etc.). As a new element, an interfacial anti-stretching force is introduced in this article, being equivalent to the definition of the interfacial energy in terms of tension as understood by Young. The differences and similarities between the interfacial gradient force and the interfacial spreading force (the Marangoni force) are shown. The well-known case of the liquid bridge induced interfacial force is supplemented by its less known version of a gaseous bridge induced interfacial force.     

Long-Range Electrostatic Interaction between a Charged Wall and Two Similarly Charged Colloidal Spheres at Low Surface Potentials

The long-range electrostatic interaction between a pair of similarly charged colloidal spheres and a charged planar wall at low surface potentials is theoretically investigated. The linear Poisson-Boltzmann equation (PBE) and the point charge approximation of the charged sphere are used. The electrical potential distribution in the electrolyte solution is found from the PBE at the constant surface potentials using the image charge method. The electrostatic forces acting on the spheres are then calculated. The results show that the repulsive interaction between a pair of similarly charged colloidal spheres clearly decreases when a charged wall appears nearby, but it is impossible for an attractive force to emerge at the scaled surface potentials less than 1. There is, however, an attractive force between the charged wall and the similarly charged colloidal spheres, when the surface potential zetap on the wall is sufficiently higher than the surface potential zetas on the spheres to make zetap > zetasexp(kappah) (h is the distance from the wall to the sphere center). In this case, there are negative surface charges on the spheres at positive surface potential zetas. It is these negative charges that produce the above attraction. Copyright 1999 Academic Press.