Identical Spherical Particles Research Articles

Improvement on the Derjaguin's method for the interaction of spherical particles

A correction to the classical Derjaguin's method has been given for the electrical double layer interaction of two spherical particles. Simple analytic expressions are given. Results obtained respectively for identical and dissimilar spherical particles show that the expressions open up the usable area of familiar Derjaguin's formulas.

Interphase interaction in fine suspension flow

A self-consistent model of moderately concentrated fine suspensions of identical spherical particles is developed. It provides for a completely closed fluid dynamic scheme of suspensions neglecting random particle fluctuations. Above all, it brings forward a set of conservation equations governing the mean flow of both phases of a suspension in the continuum approximation. The model allows also to obtain all constitutive rheological relations (“equations of state”) which determine terms of those equations as functions of unknown variables and physical parameters. The rheological relations are found in an explicit form for effective stresses and different constituents of the interphase interaction force in weakly unsteady flow of a moderately concentrated suspension. The stresses are shown to take shape as a result of a certain relaxation process. The force constituents are of the same origin and have basically the same meaning as those for a single particle in an unbounded fluid. In particular, the contents of the paper bring to an end a recent scholastic and rather fruitless debate as to how the drag and buoyancy constituents have to be singled out from the total force acting on a particle of a uniform fluidized bed at steady conditions. It is demonstrated in an unequivocal way that it is the density of the fluidized bed as a whole that must be used while expressing the buoyancy force, but not that of the fluidizing fluid alone.

Small-angle neutron-scattering investigation of long-range correlations in silica aerogels: Simulations and experiments.

Numerical simulations of gel formation by off-lattice diffusion-limited cluster-cluster aggregation (DLCA) of identical spherical particles in a cubic box are performed. Both the correlation function g(r) and the small-angle scattering function S(q), which is related to the Fourier transform of g(r)-1, are calculated for the resulting gel structure which is made of connected fractal clusters. In addition to the short-range features already described in a previous paper, it is found that g(r) goes through a minimum before tending to unity for large r values. As a consequence the scattering function S(q) exhibits a maximum at small q values. After multiplying S(q) by the form factor P(q), the intensity curve I(q) is calculated and its behavior is found to be in good agreement with small-angle neutron-scattering experiments on silica aerogels of various densities. In contrast with previous approaches, the low-q maximum in the I(q) curve is here directly quantitatively accounted for, without introducing any phenomenological ad hoc term to describe the intercluster correlations.

Fluid dynamics of coarse dispersions

Dispersions of identical spherical particles in a fluid are considered with allowance for random fluctuations that both, particles and the fluid, are involved in. The particles are assumed to be sufficiently large in the sense that the interparticle exchange by momentum and energy of the particle fluctuations is carried out mainly through direct collisions, and the role of interphase interaction in this process is negligible. Then the particles may be regarded as statistically independent, and the fluctuation energy is almost uniformly distributed over their translational degrees of freedom. Equations of mass and momentum conservation for flow of the dispersed phase as well as an equation of fluctuation energy transfer are stated similarly to those obtained by the classical method by Enskog in the kinetic theory of gases. The equations involved additional terms due to the interphase interaction and the energy dissipation by collisions. These equations, together with those of mass and momentum conservation of the continuous phase, constitute a complete set of equations which govern the macroscopic flow of disperse mixtures. A closure problem for this set is discussed in detail. It is reduced, in fact, to the determination of a single quantity characterizing the particle fluctuation intensity in a macroscopically homogeneous state of a disperse mixture.

Small-angle neutron-scattering investigation of short-range correlations in fractal aerogels: Simulations and experiments.

The center-to-center interparticle-distance distribution function f(r) and its Fourier transform, the scattering function S(q), are computed for simulated fractal aggregates made of identical spherical particles using several three-dimensional off-lattice cluster-cluster algorithms. As expected f(r) exhibits a \ensuremath{\delta} peak at the particle diameter followed by a discontinuity at twice this distance as a consequence of the nonoverlapping character of the spherical particles. As a result the curve ${\mathrm{log}}_{10}$S(q) versus ${\mathrm{log}}_{10}$q goes through a broad minimum followed by damped oscillations at large-q values. These simulations are compared with experimental small-angle neutron-scattering results on colloidal silica aerogels. The experimental scattering function S(q) is derived from the ratio between the scattered intensity I(q) and the form factor P(q) determined from measurements on the diluted colloidal solution. The agreement between simulations and experiments is qualitatively good and the influence of aerogel density is well accounted for. The departure of the experimental curves from the theoretical S(q) curve, which is stronger for larger particle sizes, is attributed to short-wavelength corrections to the simple scattering theory.

Evaluating Porosity Parameters for Porosity–Property Relations

Porosity parameters are considered from the standpoint of a map of the volume fraction porosity (P) and the ratio of normalized radii of the pore (rp) and the solid particles, webs, or ligaments (rs) defining the pores. For pore structures derived from uniform stacking of identical spherical particles or bubbles, this “map” turns out to be a single S‐shaped curve showing (1) very significant constraints on porosity variations achievable, and (2) rp/rs contains no information not already given by P, but (3) P or rp/rs do not fully characterize the porosity. Of other interrelated porosity factors, i.e, open vs closed or inter‐ vs intragranular nature, coordination number of the pores or particles, the shape (e.g., curvature) of the pores, and the minimal solid area (i.e., minimum web cross sections or the bond area between particles) representing some porosity characteristics independent of P or rp/rs, it is concluded that the minitics solid area is the most appropriate one. For more complex ideal and real pore structures, (1) the above map starts from the ideal P vs rp/rs curve at the two extremes of P (0 and 1), but becomes a moderate band in‐between, with the ideal P vs rp/rs curve at, or near, the mean of this band, and (2) a second parameter is still needed, and the minimum solid area is still the most appropriate one.

Interaction Free Energy between Identical Spherical Colloidal Particles: The Linearized Poisson-Boltzmann Theory

The linearized Poisson-Boltzmann theory is used to calculate the electrical double-layer interaction free energy between identical spherical colloidal particles. Results are given for interaction under conditions of constant surface potential, constant surface charge, and for the case in which charge regulation due to the dissociation of surface groups may be modeled by a linear relationship between the surface charge and the surface potential. Accurate results are obtained using a two-center expansion for the solution of the linearized Poisson-Boltzmann equation and numerical implementations of the algorithm are given for a full range of particles sizes, κ a, and particle separations, κ h.

Quasistatic deformation of a granular medium

Static deformations of simple shear of a material composed of identical spherical particles are considered on the basis of maximal configurational entropy of the system macrostates. It is established theoretically and experimentally that the same principle holds also in slow shear flow of a material characterized by a universal value of the particle concentration which is independent of the initial concentration.

Breakup of Latex Doublets by Impaction

The breakup by impaction of a simple agglomerate consisting of two identical spherical particles has been studied. The agglomerates were doublets produced by nebulizing suspensions of latex particles. The doublet concentrations before and after impaction were determined from the particle size spectra obtained with an optical particle counter. Preliminary experiments showed that the latex doublets did not break up in the acceleration nozzle of the impactor. The doublet fraction (DF) is defined as the doublet/singlet concentration ratio after impaction to that before impaction. The DF of 2.99-μm Dow latex measured at √Stk = 1.62 decreased by about a factor of 2 as the relative humidity (RH) was varied from 8% to 85%. The remaining experiments were made at low RH with seven different surfactant-free suspensions of polystyrene latex stabilized by the addition of charged surface functional groups. All of the data, for particle sizes ranging from 1.62 to 4.07 μm and impact velocities from 10 to 80 m/s, could be fitted by the expression, DF = 18.2 v−0.58 D−1.60, where the impact velocity v is in m/s and the particle diameter D is in μm. The van der Waals adhesion energy is calculated to be less than 1/3000 of the kinetic energy needed to break up half of the doublets. The repulsive electrostatic energy of the surface functional groups is estimated to be about 1.25 times larger than the van der Waals adhesion energy. It is concluded that the doublets are probably bound by bridges of residue left after droplet evaporation.

Equilibrium properties of ferrocolloids

We consider a thermodynamic model of a stable macroscopically homogeneous ferrocolloid containing identical spherical particles with permanent moments suspended in a non-dissociated liquid on the basis of the hard sphere perturbation theory. Magnetostatic properties and the phase diagram of the ferrocolloid are shown to agree very well with experimental evidence. Dipole interparticle interactions result in an effective attraction between the particles which increase, as the strength of an externally applied magnetic field grows, favours the phase separation and cause a reduction in the coefficient of mutual Brownian diffusion of the particles.

Capillary interaction of spherical particles adsorbed on the surface of an oil/water droplet stabilized by the particles. Part II

We develop a conical-cell model for determining the capillary interaction between identical spherical particles forming a monolayer adsorbed at variable packing on the surface of an oil-in-water (Pickering) emulsion droplet. Each adsorbed particle is assigned a cone-shaped section of the droplet volume. The vertex of the cone is located at the center of the droplet and the axis of the cone passes through the center of the adsorbed particle. A typical adsorbed particle is surrounded by an oil/water interfacial shell having circular symmetry about the axis of the cone. The shape of the oil/ water interface is obtained by solving the Young—Laplace equation. It is required that the volume of the dispersed phase in the droplet, which is contained in the cone, does not change on adsorption of the monolayer of particles. The interfacial energy assigned to a single adsorbed particle and its surrounding oil/water interface situated within its cone is determined. The capillary interaction is obtained by subtracting the corresponding interfacial energy when capillary interaction between the adsorbed particles is ignored. One method of obtaining interfacial energy without capillary interaction between the particles is based on the model of Menon, Nagarajan and Wasan, described in part I (S. Levine and B.D. Bowen, Colloids Surfaces, 59 (1991) 377). With this choice of interfacial energy without particle interactions, the capillary interaction is small and attractive. The leading term is identical with that obtained in Part I, by developing further the model of Menon, Nagarajan and Wasan. This term is proportional to the fourth power of the particle radius and diminishes as the inverse square of the separation between the particles. The use of the conical-cell model yields an additional term which is expressed in terms of the difference between the so-called effective (macroscopic) interfacial tension, due to the layer of adsorbed particles, and the conventional (microscopic) tension of the oil/water interface without particles. Although our result is consistent with the order of magnitude of the capillary interaction found with an earlier cylindrical-cell model, which was intended to apply to a very large droplet, the latter gave an incorrect capillary repulsion between the adsorbed particles.

An investigation of the structure of colloidal aerogels

Silica aerogels were obtained by destabilization of colloidal silica solutions. Samples of different densities, obtained from identical elementary spherical particles of known distribution, were prepared. Transmission electron microscopy images of the aerogels show the very tenuous structure of an aggregate of spheres, whose average diameter was measured. This structure has been analyzed by small-angle neutron and X-ray scatterings. The power-law dependence of the scattered intensity, a characteristic of fractal structures, is observed for the lightest sample. A model of the density-density correlation function gives a satisfactory account of the experimental result. The different materials have the same fractal dimension, and are thus mutually self-similar. The particle sizes measured in both real and reciprocal space are in good agreement.

Capillary interaction of spherical particles adsorbed on the surface of an oil/water droplet stabilized by the particles. Part I

The capillary interaction energy between identical spherical particles adsorbed as a monolayer on the surface of an oil/water emulsion droplet stabilized by the particles is determined. Use is made of the theoretical work by Menon, Nagarajan and Wasan on the free energy change due to the adsorption of the particles on the droplet from the continuous phase. It is assumed that the oil/water interfaces joining neighboring adsorbed particles are portions of spheres with a common radius and a common center. The capillary interaction between the particles is indirectly taken into account through the change in radius of the droplet on the adsorption of the monolayer of particles. The capillary interaction is obtained as the difference between two large energies of the order of 105kT per particle which are almost equal. Hence that part of the adsorption energy attributed to a particle which excludes its interactions with neighboring adsorbed particles must be clearly identified. The capillary interaction in attractive and is comparable with a multiple of kT, diminishing as the inverse square of the separation between the particles and proportional to the fourth power of the particle radius.

Numerical exploration of planetary arc dynamics

Ground-based observations have revealed in 1984 and 1985 the existence of interrupted ring-like structures, or “arcs” around Neptune. Recent observations by the Voyager 2 spacecraft in August 1989 have confirmed the presence of stable, dense arcs, embedded in a faint continuos ring. These structures would be destroyed in a few years if they followed unperturbed Keplerian motions. We study here the stabilizing effect of corotation and Lindblad resonances on planetary arcs, along the lines of previous analytical work by Lissauer ( Nature (London) 318, 1985, 544–545) and Goldreich, Tremaine, and Borderies ( Astron. J. 92, 1986, 490–494). We first describe analytically the responce of a test particle to the combined effects of corotation and Lindblad resonances, caused by Neptunian satellites. The evolution of the particle is shown to be described by two coupled dynamical systems, the coupling depending on collective effects in the arc. This yields a formula for the energy provided by the Lindblad resonance, which is actually used to confine the arc material around the corotation resonance radius. We show in particular that the gradient of the torque density across the arc must be negative for the latter to be stable. This confirms in a general framework the constraints, previously given by Lin, Papaloizou, and Ruden ( Mon. Not. R. Astron. Soc. 227, 1987, 75–95), on the relative positions of the corotation and the Lindblad resonances for the arc to be stable. We test our results with a direct numerical simulation, which takes into account inelastic collisions between identical spherical particles. Two configurations are studied: (1) an arc at a L 4 Lagrange point of a satellite, further perturbed by an isolated Lindblad resonance with a second satellite, and (2) an arc at an isolated corotation resonance with a single satellite on an eccentric orbit. Our main results are: (a) the corotation points alone are unstable against dissipative collisions, (b) a stable arc must be submitted to a negative gradient of torque density, and (c) such a stable arc reaches a limit cycle where the energy provided by the resonant satellite is balanced by the energy dissipated by collisions.

Hysteresis phenomena in ensemble of dipole-interacting particles without intrinsic anisotropy

It is shown that hysteresis can be observed in an assembly of chaotic distributed particles without intrinsic magnetic anisotropy due to interparticle magnetostatic interactions. To take this interaction into account, pair approximation and computer simulation of the remagnetization process are used. The investigation is restricted to the case of identical spherical particles with constant magnetic moment. Consideration is given to the case of zero temperature and large dissipation, which results in the quasistatic remagnetization process, so that the moment of the particle is always aligned in the field direction. >

Effective Diameter of Agglomerates for Radiative Extinction and Scattering

This study examines the extinction and scattering characteristics of randomly oriented agglomerates consisting of closely-packed small identical spherical particles such as soot in flames. An equivalent sphere is introduced as one that exhibits similar scattering or extinction cross-section as the agglomerate and has the same refractive index as the primary particles. A simple analytical reasoning is presented that establishes the ratio of the diameter of this sphere to the diameter of the primary particles to be proportional to the cube root of the number of particles in the agglomerate. Simple expressions for the proportionality constants for various morphologies are developed. Previous studies in literature have used extensive numerical computations to empirically correlate the above proportionality without considering the proportionality constants. Scattering patterns for various morphologies are also predicted which have significance for optical diagnostic techniques.

Peristaltic transport of a particle-fluid suspension.

Peristaltic pumping by a sinusoidal traveling wave in the walls of a two-dimensional channel filled with a viscous incompressible fluid in which are distributed identical rigid spherical particles, is investigated theoretically. A perturbation solution is obtained which satisfies the momentum equations for the case in which amplitude ratio (wave amplitude/channel half width) is small. The results show that the fluid phase mean axial velocity decreases with increase in the particle concentration. The phenomenon of reflux (the mean flow reversal) is discussed. A reversal of velocity in the neighborhood of the centerline occurs when the pressure gradient is greater than that of the critical reflux condition. It is found that the critical reflux pressure is lower for the particle-fluid suspension than for the particle-free fluid. It is further observed that the mean flow reversal is strongly dependent on the particle concentration and the presence of particles in the fluid favors the reversal flow. A motivation of the present analysis has been the hope that such a theory of two-phase flow process is very useful in understanding the role of peristaltic muscular contraction in transporting bio-fluid behaving like a particle-fluid mixture. Also the theory is important to the engineering applications of pumping solid-fluid mixtures by peristalsis.

Stabilization of emulsions by fine particles II. capillary and van der Waals forces between particles

The effective surface tension (free energy per unit area) of a planar oil/water interface completely covered with a closely-packed monolayer of identical spherical particles is determined as a function of contact angle. The surface tension diminishes as contact angle increases from 0° to 90° or decreases from 180° to 90°. Preliminary experimental results show this general trend. A theoretical study is made of the capillary forces acting between the interfacial spherical particles which stabilize oil/water emulsion droplets. Applying a two-dimensional cell model, the oil/water meniscus immediately surrounding a given particle of a closely-packed monolayer structure is assumed to have circular symmetry. Changes in interfacial areas between the oil, water and solid occur when an isolated surface particle becomes a member of the monolayer structure. For the simplified model used here the accompanying energy change due to interfacial tension yields a repulsion between the surface particles for all contact angles. By applying the well-known Derjaguin method of determining the interaction of particles at close separation, the van der Waals attraction between adjacent spheres in the monolayer is calculated as a function of the contact angle. The magnitudes of the capillary and van der Waals energy per particle are smaller by three or four orders of magnitude than the depth of the energy well in which an isolated solid sphere is trapped at the oil/water interface.

Light scattering by fractal aggregates : a numerical investigation

A numerical method is described which is able to calculate the scattering of polarized light by a large assembly of identical spherical particles, of complex refractive index, such as a colloid or aerosol aggregate. The method takes care of multiple scattering inside the aggregate, except that the criterium for the Rayleigh approximation must be satisfied for the individual particles. It is applied to computer-generated aggregates containing up to 512 particles with various fractal dimensions. The scattered intensities for various polarization directions are studied as a function of the scattering angle and size of the aggregate Calcul par methode numerique de la diffusion de la lumiere polarisee par une collection de particules spheriques identiques, d'indice de refraction complexe, tels que des agregats d'aerosols ou de colloides

Wave Propagation Through Fluid Saturated Porous Rocks

A sedimentary rock is modeled by a random packing of identical spherical particles. The connected pore space is filled with an inviscid, compressible fluid. A low-frequency expansion technique is used to calculate the effective wave speeds explicitly in terms of the microstructural properties of the rock considered. The effect of both the pore fluid and the initial confining pressure to which the rock is subjected can be included in the calculations.