Theory of capillary-induced interactions beyond the superposition approximation

Within a general theoretical framework, we study the effective, deformation-induced interaction between two colloidal particles trapped at a fluid interface in the regime of small deformations. In many studies, this interaction has been computed with the ansatz that the actual interface configuration for the pair is given by the linear superposition of the interface deformations around the single particles. Here, we assess the validity of this approach and compute the leading term of the effective interaction for a large interparticle separation beyond this so-called superposition approximation. As an application, we consider the experimentally relevant case of interface deformations owing to the electrostatic field emanating from charged colloidal particles. In mechanical isolation, i.e., if the net force acting on the total system consisting of the particles plus the interface vanishes, the superposition approximation is actually invalid. The effective capillary interaction is governed by contributions beyond this approximation and turns out to be attractive. For sufficiently small surface charges on the colloids, such that linearization is strictly valid, and at asymptotically large separations, the effective interaction does not overcome the direct electrostatic repulsion between the colloidal particles.     

Capillary forces between spherical particles floating at a liquid-liquid interface

We study the capillary forces acting on sub-millimeter particles (0.02-0.6 mm) trapped at a liquid-liquid interface due to gravity-induced interface deformations. An analytical procedure is developed to solve the linearized capillary (Young-Laplace) equation and calculate the forces for an arbitrary number of particles, allowing also for a background curvature of the interface. The full solution is expressed in a series of Bessel functions with coefficients determined by the contact angle at the particle surface. For sub-millimeter spherical particles, it is shown that the forces calculated using the lowest order term of the full solution (linear superposition approximation; LSA) are accurate to within a few percents. Consequently the many particle capillary force is simply the sum of the isolated pair interactions. To test these theoretical results, we use video microscopy to follow the motion of individual particles and pairs of interacting particles at a liquid-liquid interface with a slight macroscopic background curvature. Particle velocities are determined by the balance of capillary forces and viscous drag. The measured velocities (and thus the capillary forces) are well described by the LSA solution with a single fitting parameter.     

The role of capillary and surface forces in the crossover behavior of solid nanoparticles at liquid interfaces

We investigate the interaction between a nanoparticle and an oil-water interface with particular emphasis on the particle crossing through the interface. The formation of a three-phase contact line is investigated in two cases, namely in the presence and in the absence of surface forces. We carefully examine the interplay between capillary and surface forces in such systems. Two instabilities of the interface (snap-in/snap-out) as the particle is moved through the interface are identified and quantitatively described. While the snap-in instability was observed in some AFM studies, the precise interface position and configuration relative to the particle at the instability depends on the nature of the surface forces present in the system. After the snap-in, the particle is adsorbed and must overcome an energy barrier due to the interface deformation in order to cross-over to the other liquid. We make quantitative predictions on the interface configuration at the instabilities and the free energy barrier height. The roles of particle size and different interaction parameters characterizing the system in determining the magnitude of the energy barrier for crossing and in the formation of a three-phase contact line are discussed. Ultimately, this study will enable us to make quantitative predictions on capillary effects in nanoparticle-microemulsions mixtures and other colloidal systems. For particles in the micrometer range and larger the capillary forces dominate over the surface forces and dictate how the snap-in occurs. However, the situation becomes different for particle sizes smaller than about 100 nm. The presence of surface forces modifies the interface configuration and the free energy jump at the snap-in instability.     

Electrodipping force acting on solid particles at a fluid interface

We report experimental results which show that the interfacial deformation around glass particles (radius, 200-300 microm) at an oil-water (or air-water) interface is dominated by an electric force, rather than by gravity. It turns out that this force, called for brevity "electrodipping," is independent of the electrolyte concentration in the water phase. The force is greater for oil-water than for air-water interfaces. Under our experimental conditions, it is due to charges at the particle-oil (instead of particle-water) boundary. The derived theoretical expressions, and the experiment, indicate that this electric force pushes the particles into water. To compute exactly the electric stresses, we solved numerically the electrostatic boundary problem, which reduces to a set of differential equations. Convenient analytical expressions are also derived. Both the experimental and the calculated meniscus profile, which are in excellent agreement, exhibit a logarithmic dependence at long distances. This gives rise to a long-range electric-field-induced capillary attraction between the particles, detected by other authors. Deviation from the logarithmic dependence is observed at short distances from the particle surface due to the electric pressure difference across the meniscus. The latter effect gives rise to an additional short-range contribution to the capillary interaction between two floating particles. The above conclusions are valid for either planar or spherical fluid interfaces, including emulsion drops. The electrodipping force, and the related long-range capillary attraction, can engender two-dimensional aggregation and self-assembly of colloidal particles. These effects could have implications for colloid science and the development of new materials.     

Measurement and modeling on hydrodynamic forces and deformation of an air bubble approaching a solid sphere in liquids

The interaction between bubbles and solid surfaces is central to a broad range of industrial and biological processes. Various experimental techniques have been developed to measure the interactions of bubbles approaching solids in a liquid. A main challenge is to accurately and reliably control the relative motion over a wide range of hydrodynamic conditions and at the same time to determine the interaction forces, bubble–solid separation and bubble deformation. Existing experimental methods are able to focus only on one of the aspects of this problem, mostly for bubbles and particles with characteristic dimensions either below 100 μm or above 1 cm. As a result, either the interfacial deformations are measured directly with the forces being inferred from a model, or the forces are measured directly with the deformations to be deduced from the theory. The recently developed integrated thin film drainage apparatus (ITFDA) filled the gap of intermediate bubble/particle size ranges that are commonly encountered in mineral and oil recovery applications. Equipped with side-view digital cameras along with a bimorph cantilever as force sensor and speaker diaphragm as the driver for bubble to approach a solid sphere, the ITFDA has the capacity to measure simultaneously and independently the forces and interfacial deformations as a bubble approaches a solid sphere in a liquid. Coupled with the thin liquid film drainage modeling, the ITFDA measurement allows the critical role of surface tension, fluid viscosity and bubble approach speed in determining bubble deformation (profile) and hydrodynamic forces to be elucidated. Here we compare the available methods of studying bubble–solid interactions and demonstrate unique features and advantages of the ITFDA for measuring both forces and bubble deformations in systems of Reynolds numbers as high as 10. The consistency and accuracy of such measurement are tested against the well established Stokes–Reynolds–Young–Laplace model. The potential to use the design principles of the ITFDA for fundamental and developmental research is demonstrated.     

Capillary attraction of colloidal particles at an aqueous interface

We study the capillary forces arising from charged colloidal particles trapped at an oil-water interface. Since it is quadratic in the electric field, the electric stress acting on the interface cannot be written as the superposition of one-particle terms. Indeed, we find that the interfacial pressure is dominated by two-particle terms, which induce capillary forces involving one, two, three, or four particles. The dominant interaction is attractive and varies with the inverse cube of the particle distance.     

Protrusion and deformation of an elastic particle at an advancing front interface

The response of an elastic particle to an extensional field close to an advancing liquid–gas interface in a capillary was examined both from an experimental and theoretical viewpoint. Experimental evidence is given on both the protrusion and deformation of an elastic particle at an advancing front interface. To interpret our experimental results, we followed and extended Hoffman's approach to the case of deformable particles. A hybrid approach was followed where the height of the bump was calculated theoretically as an equilibrium between capillary and drag forces for a given deformation of the particle, whereas the latter was supplied by an independent experimental measurement. For an elastic particle, the height of the bump goes through a maximum and further on decay almost linearly when plotted against the dimensionless parameter introduced by Hoffman. The difference between the theoretical prediction and the experimental data is discussed in relation to the asymmetry of the bump profiles observed in the experiment that could be responsible for a larger contribution of capillary forces from those calculated. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 41: 1362–1374, 2003     

London potential-energy surfaces for symmetrical alkali trimers

Semiempirical valence bond calculations have been performed on the symmetrical alkali trimers in order to obtain their potential-energy surface, binding energy, and equilibrium geometry. For these calculations original and generalized Rosen–Morse potentials for the ground state and two similar types of new potentials for the triplet state of alkali dimers have been used. At small internuclear distances the potential surfaces show a shallow well which extends into the entrance and exit valleys without an energy barrier. The trimers are found to be stable both in linear and bent configurations, but the most stable configuration in each case is linear and symmetric. The force constants corresponding to stretching and bending deformations of the trimers have been calculated for this configuration. It has been observed that unlike the stretching, the bending deformation does not sensitively affect the energy variation of the ground state of the trimers.     

Prediction and measurement of the interparticle depletion interaction next to a flat wall

A theoretical and experimental study was performed to investigate the depletion interaction between two colloidal particles next to a solid wall in a solution of nonadsorbing macromolecules. By calculating the change in free volume available to the macromolecules upon approach of the two particles, a relatively simple expression was developed for the interparticle depletion attraction in hard sphere systems as a function of the particle-particle and particle-plate spacing. Perhaps the most useful result obtained from this analysis was that the wall has no effect whenever the ratio of the particle radius to the macromolecule radius is greater than four. (In charged systems, this ratio would apply to the effective particle and macromolecule sizes.) A series of experiments was then performed in which the hydrodynamic force balance (HFB) apparatus was used to measure the shear force needed to separate a colloidal doublet consisting of two particles trapped in a secondary energy well formed by a repulsive electrostatic force and an attractive depletion force. The macromolecules used here were small, nanometer-sized spheres of either silica or polystyrene. Agreement between the measured separation forces and those predicted using the force balance model of J. Y. Walz and A. Sharma (J. Colloid Interface Sci. 168, 485 (1994)) was within a factor of 1.3 using no adjustable parameters and accounting for polydispersity and uncertainty in the macromolecule size. It is shown that this remaining discrepancy could be caused by the Brownian (stochastic) nature of the doublet breakup process.     

A single particle model to simulate the dynamics of entangled polymer melts

We present a computer simulation model of polymer melts representing each chain as one single particle. Besides the position coordinate of each particle, we introduce a parameter n(ij) for each pair of particles i and j within a specified distance from each other. These numbers, called entanglement numbers, describe the deviation of the system of ignored coordinates from its equilibrium state for the given configuration of the centers of mass of the polymers. The deviations of the entanglement numbers from their equilibrium values give rise to transient forces, which, together with the conservative forces derived from the potential of mean force, govern the displacements of the particles. We have applied our model to a melt of C(800)H(1602) chains at 450 K and have found good agreement with experiments and more detailed simulations. Properties addressed in this paper are radial distribution functions, dynamic structure factors, and linear as well as nonlinear rheological properties.     

Self-assembled shells composed of colloidal particles: fabrication and characterization

We construct shells with tunable morphology and mechanical response with colloidal particles that self-assemble at the interface of emulsion droplets. Particles self-assemble to minimize the total interfacial energy, spontaneously forming a particle layer that encapsulates the droplets. We stabilize these layers to form solid shells at the droplet interface by aggregating the particles, connecting the particles with adsorbed polymer, or fusing the particles. These techniques reproducibly yield shells with controllable properties such as elastic moduli and breaking forces. To enable diffusive exchange through the particle shells, we transfer them into solvents that are miscible with the encapsulant. We characterize the mechanical properties of the shells by measuring the response to deformation by calibrated microcantilevers.     

Colloidal rings in a liquid mixture

We investigate the self-assembly of colloidal particles on microscopic decane droplets in water and show that, by use of paramagnetic colloids, it is possible to assemble ringlike structures that can be controlled with a magnetic field. Moreover, the use of paramagnetic colloids allows us to determine the attractive forces between the colloids located at the three-phase contact line between decane, water, and air. The attractive force is in the femtonewton range and is attributed to capillary interactions due to interface deformations. When the liquid emulsion dries on a glass slide, we observe solid deposits in the form of microscopic rings of varying diameters.     

Quantitative measurement of friction between single microspheres by friction force microscopy

The sliding friction between single silica microspheres was examined by applying friction force microscopy to probe the interaction between spherical silica particles glued to a tipless atomic force microscopy (AFM) cantilever and another particle glued to a glass slide. A three-dimensional model handling the complex contact geometry between spherical particles was established to compute friction and normal forces at the sliding interface from measured deflections of the AFM cantilever. Results obtained at different loads show a linear relationship between friction and normal force, with a friction coefficient of 0.4 between silica spheres. Friction in this system occurs at multi-asperity contacts. The results show that the macroscopic friction law of Amontons can be used to model the friction behavior of micrometer-sized granular matter. For plasma-treated silica particles, increased friction as well as wear could be observed during sliding.     

Nonperiodic stochastic boundary conditions for molecular dynamics simulations of materials embedded into a continuum mechanics domain

A scheme is described for performing molecular dynamics simulations on polymers under nonperiodic, stochastic boundary conditions. It has been designed to allow later the embedding of a particle domain treated by molecular dynamics into a continuum environment treated by finite elements. It combines, in the boundary region, harmonically restrained particles to confine the system with dissipative particle dynamics to dissipate energy and to thermostat the simulation. The equilibrium position of the tethered particles, the so-called anchor points, are well suited for transmitting deformations, forces and force derivatives between the particle and continuum domains. In the present work the particle scheme is tested by comparing results for coarse-grained polystyrene melts under nonperiodic and regular periodic boundary conditions. Excellent agreement is found for thermodynamic, structural, and dynamic properties.     

Capillary force required to detach micron-sized particles from solid surfaces—Validation with bubbles circulating in water and 2 μm-diameter latex spheres

The adhesion forces holding micron-sized particles to solid surfaces can be studied through the detachment forces developed by the transit of an air–liquid interface in a capillary. Two key variables affect the direction and magnitude of the capillary detachment force: (i) the thickness of the liquid film between the bubble and the capillary walls, and (ii) the effective angle of the triple phase contact between the particles and the interface. Variations in film thickness were calculated using a two-phase flow model. Film thickness was used to determine the time-variation of the capillary force during transit of the bubble. The curve for particle detachment was predicted from the calculated force. This curve proved to be non-linear and gave in situ information on the effective contact angle developing at the particle–bubble interface during detachment. This approach allowed an accurate determination of the detachment force. This theoretical approach was validated using latex particles 2 μm in diameter.     

Theoretical consideration on preparing silver particle films by adsorbing nanoparticles from bulk colloids to an air-water interface

In our previous paper, a method for preparing enormous surface-enhanced Raman scattering (SERS) active substrates through the aggregation of silver particles trapped at an air-water interface was reported. Here, further efforts were devoted to investigate the origin of assembling silver particle films by adsorbing nanoparticles from bulk colloids to the air-water interface. It was revealed that it is thermodynamically favorable for a colloidal particle in bulk colloids to adsorb to the air-water interface; however, a finite sorption barrier between it and the nearby particles usually restrains the adsorption process. When an electrolyte such as KCl, which is commonly used as an activating agent for additional SERS enhancement, was added into silver colloids, it largely reduced the sorption barrier. Thus, silver nanoparticles can break through the sorption barrier, pop up, and be trapped at the air-water interface. The trapped silver particles are more inclined to aggregate at the interface than those in bulk colloids due to the increase of van der Waals forces and the reduction of electrostatic forces. The morphology of the as-prepared silver particle films was characterized by scanning electron microscope, and their SERS activity was tested using NaSCN as a probe molecule. The surface enhancement of the silver particle films is about 1-2 orders of magnitude higher compared with that of silver colloids, because most of the silver particles in the films are in the aggregation form that provides enormous SERS enhancement. Furthermore, the stability of such type of films is much better that of colloid solutions.     

One-dimensional trapped atoms: critical coupling parameter and critical number of particles

In this paper we consider the case of a Bose gas in low dimension in order to illustrate the applicability of a method that allows us to construct analytical relations, valid for a broad range of coupling parameters, for a function which asymptotic expansions are known. The method is well suitable to investigate the problem of stability of a collection of Bose particles trapped in one- dimensional configuration for the case where the scattering length presents a negative value. The eigenvalues for this interacting quantum one-dimensional many particle system become negative when the interactions overcome the trapping energy and, in this case, the system becomes unstable. Here we calculate the critical coupling parameter and apply for the case of Lithium atoms obtaining the critical number of particles for the limit of stability.     

Modeling adhesion forces between deformable bodies by FEM and Hamaker summation

A new numerical approach is presented for predicting adhesion forces of particles at flat and rough surfaces. The new hybrid method uses the finite element method (FEM) for the determination of elastic and plastic particle deformation combined with numerical Hamaker summation. In the numerical approach, the influence of the plastic deformation can be fully included. We show how the adhesion force depends on the contact geometry and the material properties. For easy comparison with other models, the force-displacement behavior of the systems is presented. The numerical approach is supported by atomic force microscopy (AFM) measurements. The experimentally observed adhesion force hysteresis is described very well by the new approach. Although calculations in this article are focused on spherical particles, our approach can be extended to particles of arbitrary shapes.