Suspensions Of Monodisperse Spherical Particles Research Articles

Shear thickening and history-dependent rheology of monodisperse suspensions with finite inertia via an immersed boundary lattice Boltzmann method

Three-dimensional direct numerical simulations of dense suspensions of monodisperse spherical particles in simple shear flow have been performed at particle Reynolds numbers between 0.1 and 0.6. The particles translate and rotate under the influence of the applied shear. The lattice Boltzmann method was used to solve the flow of the interstitial Newtonian liquid, and an immersed boundary method was used to enforce the no-slip boundary condition at the surface of each particle. Short range spring forces were applied between colliding particles over sub-grid scale distances to prevent overlap. We computed the relative apparent viscosity for solids volume fractions up to 38% for several shear rates and particle concentrations and discuss the effects of these variables on particle rotation and cluster formations. The apparent viscosities increase with increasing particle Reynolds number (shear thickening) and solids fraction. As long as the particle Reynolds number is low (0.1), the computed viscosities are in good agreement with experimental measurements, as well as theoretical and empirical equations. For higher Reynolds numbers, we find much higher viscosities, which we relate to slower particle rotation and clustering. Simulations with a sudden change in shear rate also reveal a history (or hysteresis) effect due to the formation of clusters. We quantify the changes in particle rotation and clustering as a function of the Reynolds number and volume fraction.

Momentum balance and stresses in a suspension of spherical particles in a plane Couette flow

Non-Brownian suspension of monodisperse spherical particles, with volume fractions ranging between ϕ = 0.05 and 0.38 and particle Reynolds numbers ranging between Rep = 0.002 and 20, in plane Couette shear flows is investigated using three-dimensional particle-resolved numerical simulations. We examine the effects of volume fraction and particle Reynolds number on the macroscopic and microscopic stresses in the fluid phase. The effective viscosity of the suspension is in a good agreement with the previous empirical and experimental studies. At Rep = 20, however, the effective viscosity increases significantly compared to the lower particle Reynolds number simulations in the Stokes flow regime. Examining the stresses over the depth of the Couette gap reveals that this increase in wall shear stresses at high particle Reynolds numbers is mainly due to the significantly higher particle phase stress contributions. Next, we examine the momentum balance in the fluid and particle phase for different regimes to assess the significance of particle/particle interaction and fluid and particle inertia. At the highest particle Reynolds number and volume fraction, the particle inertia plays a dominant role in the momentum balance and the fluid inertia is non-negligible, while the short-lived contact forces are negligible compared to these effects. For all other regimes, the fluid inertia is negligible, but the particle inertia and contact forces are important in the momentum balance. Reynolds stresses originated from velocity fluctuations do not contribute significantly to the suspension stresses in any of the regimes we have studied, while the reduction in the shear-induced particle rotation can be a reason for higher wall shear stress at Rep = 20. Finally, we study the kinematics of particles, including their velocity fluctuations, rotation, and diffusion over the depth of the Couette gap. The particle diffusion coefficients in the cross flow direction exhibit an abrupt increase at Rep = 20.

On calculation of effective transport coefficients in monodisperse suspension of spherical particles

The analytical dependence of transport coefficients (heat conductivity, electric conductivity, and static magnetic and electric permeability) in monodisperse suspension of spherical particles on their concentration is obtained taking into account the distant interaction between the particles providing an arbitrary ratio between the transport coefficients of the disperse particles and the disperse liquid. Comparison with the experiments and the numerical simulations of other authors in the particular case in which electric conductivity of the disperse particles is far higher than the electric conductivity of the disperse liquid has shown that the formula fits the range of volume concentrations of the disperse particles from zero to 0.55.

The overall elastic dielectric properties of a suspension of spherical particles in rubber: An exact explicit solution in the small-deformation limit

A solution is constructed for the homogenization problem of the elastic dielectric response of rubber filled with a random isotropic distribution of polydisperse spherical particles in the classical limit of small deformations and moderate electric fields. In this limit, the overall elastic dielectric response is characterized by five (two elastic, one dielectric, and two electrostrictive) effective constants. Explicit formulas are derived for these constants directly in terms of the corresponding constants describing the elastic dielectric response of the underlying rubber and the filler particles, as well as the concentration of particles. By means of comparisons with finite-element simulations, these formulas are shown to also be applicable to isotropic suspensions of monodisperse spherical particles, provided that the particle concentration is sufficiently away from percolation. With the aim of gaining physical insight into the extreme enhancement in electrostrictive properties displayed by emerging dielectric elastomer composites, specific results are examined for the case of suspensions wherein the rubber is incompressible and the particles are mechanically rigid and of infinite permittivity.

Hydrodynamic instability of a suspension of spherical particles through a branching network of circular tubes

A numerical method is presented for computing the unsteady flow of a monodisperse suspension of spherical particles through a branching network of circular tubes. The particle motion and interparticle spacing in each tube are computed by integrating in time a one-dimensional convection equation using a finite-difference method. The particle fraction entering a descendent tube at a divergent bifurcation is related to the local and instantaneous flow rates through a partitioning law proposed by Klitzman and Johnson involv- ing a dimensionless exponent, q ≥ 1. When q = 1, the particle stream is divided in proportion to the flow rate; as q →∞ , the particles are channeled into the tube with the highest flow rate. The simulations reveal that when the network involves two or more generations, a supercritical Hopf bifurcation occurs at a critical value of q, yielding spontaneous, self-sustained oscillations in the segment flow rates, pressure drop across the network, and particle spacing in each tube. A phase diagram is presented to establish conditions for unsteady flow. As found recently for blood flow in a capillary network, oscillations can be induced for a given network tree order by decreasing the ratio of the tube diameter from one generation to the next or by decreasing the diameter of the terminal segments. The instability is more prominent for rigid than deformable particles, such as drops, bubbles, and cells, due to strong lubrication forces between the tightly fitting particles and tube walls. Variations in the local particle spacing, therefore, have a more significant effect on the effective viscosity of the suspension in each tube and pressure drop required to drive a specified flow rate.

Electric conductivity of a suspension of charged colloidal spheres with thin but polarized double layers

An analytical study of the effective electric conductivity of a homogeneous suspension of identical, charged, spherical particles in the solution of a symmetrically charged electrolyte is presented. The thickness of the electric double layers surrounding the particles is assumed to be small relative to the radius of each particle and to the gap width between two neighboring particles, but the effect of polarization of the mobile ions in the diffuse layer is taken into account. The effects of interactions among individual particles are taken into explicit account by employing a unit cell model that is known to provide good predictions for the sedimentation of monodisperse suspensions of spherical particles. The appropriate equations of conservation of electrochemical potential energies of ionic species are solved for each cell, in which a spherical particle is envisaged as surrounded by a concentric shell of the suspending fluid. Analytical expressions for the electric conductivity are obtained in closed form as functions of the volume fraction of the particles and other characteristics of the suspension. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made. The effects of particle interactions on the effective conductivity can be quite significant in general cases.

Concentration effects on photophoresis of aerosol spheres

The photophoretic motion of a homogeneous suspension of identical spherical particles of arbitrary thermal conductivity and surface properties exposed to a radiative flux is considered under conditions of small Knudsen, Reynolds, and Peclet numbers. The effects of interaction of the individual particles are taken into explicit account by employing a unit cell model which is known to provide good predictions for the sedimentation of monodisperse suspensions of spherical particles. The appropriate equations of conservation of energy and momentum are solved for each cell, in which a spherical particle is envisaged to be surrounded by a concentric shell of suspending fluid, and the photophoretic velocity of the particle is calculated for various cases. Analytical expressions of this mean particle velocity are obtained in closed form as monotonically decreasing functions of the non-vanishing volume fraction of the particles. The normalized photophoretic velocity increases with an increase in the relative conductivity of the particles for a given particle concentration.

On the solid-liquid transition of concentrated suspensions in transient shear flow

The transition from solid-like to liquid-like behavior of suspensions of monodisperse spherical particles in a Newtonian continuous phase with varying solid volume concentrations and particle sizes was investigated in rotational shear flows. It was found that the solid-liquid transition takes place in a relatively narrow shear stress range, where the corresponding shear strain γ and the shear rate \( \dot \gamma \) , respectively, increase over some orders of magnitudes. Below the transition region the material behaves like a nonlinear elastic solid with an initial linear range, i.e., \( \gamma = f\left( \sigma \right) \) . The transition depends on the type of the shear loading and on the solid volume concentration. At high shear rates a nonlinear viscous behavior (\( \dot \gamma = f\left( \sigma \right) \) ) with a terminal Newtonian range \( \dot \gamma = {\sigma \mathord{\left/ {\vphantom {\sigma \eta }} \right. \kern-\nulldelimiterspace} \eta } \) was observed, if the solid volume concentration is below a critical value. Above this value the material behaves like a moist bulk solid material.

Diffusiophoresis and Electrophoresis in Concentrated Suspensions of Charged Colloidal Spheres

The diffusiophoretic and electrophoretic motions of homogeneous suspensions of identical, charged, spherical particles in the solution of a symmetrically charged electrolyte are analyzed under conditions of small Reynolds numbers. The thickness of the electric double layers surrounding the particles is assumed to be small relative to the radius of each particle and to the gap width between two neighboring particles, but the effect of polarization of the mobile ions in the diffuse layer is taken into account. The effects of interaction among individual particles are taken into explicit account by employing a unit cell model which is known to provide good predictions for the sedimentation of monodisperse suspensions of spherical particles. The appropriate equations of conservation of electrochemical potential energies of ionic species and fluid momentum are solved for each cell, in which a spherical particle is envisaged to be surrounded by a concentric shell of suspending fluid, and the diffusiophoretic and electrophoretic velocities of the particle are calculated for various cases. Analytical expressions of these mean particle velocities are obtained in closed form as functions of the volume fraction of the particles. Comparisons between the approximate ensemble-averaged diffusiophoretic and electrophoretic velocities of a test particle in a dilute suspension and our cell-model results are made.

Some solutions of a cell model for a suspension of spherical vesicles in osmophoresis

The problem of the osmophoretic motion of a homogeneous suspension of identical spherical vesicles is analyzed in the limit of negligible Reynolds and Peclet numbers. The effects of interaction among individual vesicles are taken into explicit account by employing a unit cell model which is known to provide good predictions for the sedimentation of monodisperse suspensions of spherical particles. The appropriate equations of conservation of mass and momentum are solved for each cell, in which a spherical vesicle is envisaged to be surrounded by a concentric shell of suspending fluid, and the osmophoretic velocity of the vesicle is calculated for various cases. Analytical expressions of this mean vesicle velocity are obtained in closed form as functions of the volume fraction of the vesicles. Comparisons between the approximate ensemble-averaged osmophoretic velocity of a test vesicle in a dilute suspension and our cell-model results are made.

Diffusiophoresis in a concentrated suspension of colloidal spheres in nonelectrolyte gradients

The diffusiophoretic motion of a homogeneous suspension of identical spherical particles is considered under conditions of small Reynolds and Peclet numbers. The effects of interaction of the individual particles are taken into explicit account by employing a unit cell model which is known to provide good predictions for the sedimentation of monodisperse suspensions of spherical particles. The appropriate equations of conservation of mass and momentum are solved for each cell, in which a spherical particle is envisaged to be surrounded by a concentric shell of suspending fluid, and the diffusiophoretic velocity of the particle is calculated for various cases. Analytical expressions of this mean particle velocity are obtained in closed form as functions of the volume fraction of the particles. Comparisons between the ensemble-averaged diffusiophoretic velocity of a test particle in a dilute suspension and our cell-model results are made.

Concentration Effects on the Thermophoresis of Aerosol Spheres

The thermophoretic motion of a homogeneous suspension of identical spherical particles of arbitrary thermal conductivity and surface properties is considered under conditions of small Knudsen, Peclet, and Reynolds numbers. The effects of interaction of the individual particles are taken into explicit account by employing a unit cell model which is known to provide good predictions for the sedimentation of monodisperse suspensions of spherical particles. The appropriate equations of conservation of energy and momentum are solved for each cell, in which a spherical particle is envisaged to be surrounded by a concentric shell of suspending fluid, and the thermophoretic migration velocity of the particle is calculated for various cases. Analytical expressions of this mean particle velocity are obtained in closed form as functions of the volume fraction of the particles. Comparisons between the ensemble-averaged thermophoretic velocity of a test particle in a dilute suspension and our cell-model results are made. A parallel analysis for the sedimentation of aerosol spheres is also presented.

Extinction of Radiation in Sterically Interacting Systems of Monodisperse Spheres. Part 2: Experimental results

AbstractThe Bouguer‐Lambert‐Beer law (BLBL), commonly used to describe the extinction of radiation in disperse systems, is known to be valid in the limit of low concentrations only. In Part [1] it was shown that steric particle‐particle interactions lead to important effects in the range of high particle concentration and two versions of a more general extinction equation including steric interactions were derived for the regimes of fraunhofer diffraction and ray propagation in this part experimental evidence for the effects of steric interactions on the extinction of radiation in suspensions of monodisperse spherical particles is given. Various experiments on light extinction and on ultrasonic extinction are combined to cover a broad range of particle size parameters, mechanisms of particle‐wave interaction and receiver apertures.

Flow of suspensions with a highly disperse structure-forming fraction

A relation is obtained to describe the effect of the volume content of particles on the effective viscosity of a concentrated monodisperse suspension of spherical particles in a medium which can be regarded as a power-law fluid. It is shown that this effect becomes weaker as the nonlinearity of the medium increases. The analysis is performed on the basis of the proposition that the deciding role is played by the lubricant action of thin interlayers of the medium between the surfaces of the particles. These results are used to calculate the viscosity of a bidisperse suspension containing a highly dispersed colloidal fraction and a coarsely dispersed fraction (filler). The viscosity is calculated as a function of the ratio of the two fractions. The calculated results are compared with experimental data for model systems.

A two fluid model for sedimentation phenomena

Abstract The sedimentation of a homogeneous suspension of monodisperse spherical particles contained in an infinite vessel is described by means of a two fluid model. The motion of one of these fluids describes the global transport of matter (convention), and the motion of the second expresses the local averaged velocity of the particles in the local reference frame defined by the convective flow. By a gradient expansion, local equations coupling these two fields to the forces are derived. These equations are valid for arbitrary concentrations. In the case of a suspension contained in a finite vessel, wall effects are then phenomenologically taken into account by writing effective boundary conditions for the two velocity fields. The use of Beenakker and Mazur resummation techniques allows us to give results for the sedimentation velocity which agree well with experimental data up to the highest volume fractions investigated. Further developments are also examined.

CALIBRATION OF SPHERICAL PARTICLES BY LIGHT SCATTERING USING PHOTON CORRELATION SPECTROSCOPY

ABSTRACT Photon Correlation Spectroscopy, also known as Quasi-Elastic or Dynamic Light-Scattering, can be a convenient method of size measurement for suspensions of monodisperse spherical particles in the 20-500 nanometer (nm) size range [0.02 to 0.5 micrometer (μm)]. The results obtained vary widely, depending on the concentration and condition of the sample, as well as environmental factors. Techniques have been developed that improve the accuracy and precision of the measurements. These techniques include controlling the concentration, uniformity and dispersion of the sample. The improved method was verified by measuring particle-size standards.

Sedimentation and self-diffusion in suspensions of spherical particles

We study hindered settling in monodisperse suspensions of spherical particles in an incompressible viscous fluid. The sedimentation and self-diffusion coefficients are calculated in two-body approximation. We show that in a homogeneous and isotropic suspension in which the change of distribution function is not taken into account a transition to plug flow occurs at a volume fraction of about fifteen percent. At higher volume fractions the collective mobility coefficient vanishes. We discuss the relation to the experimental situation.

Effective conductivity of dispersions: The effect of resistance at the particle surfaces

The effective thermal or electrical conductivity of a suspension of monodisperse spherical particles embedded in a matrix of different conductivity is computed for the case when there exists a finite resistance at the particle-matrix interface. The calculation is carried out to second order in the volume fraction θ of the particle phase for a dispersion having the structure of equilibrium hard spheres. Both the first-order (Maxwell) and the second-order approximations are found to represent experimental data reasonably well for dispersions of low to moderate concentration. To first order in θ, the calculation of the conductivity of a dispersion with surface resistance is equivalent to the calculation of the conductivity of another dispersion with no surface resistance, but with a different particle conductivity. In the case of polydisperse systems, the effective conductivity is found to be sensitive to the particle size distribution.

Equilibrium properties of suspensions

This review gives a critical assessment of the various theoretical methods used to calculate, interpret and predict the equilibrium properties of stable particulate suspensions. The emphasis is on methods mainly used over the past decade to study osmotic pressure, static structure factor, radial distribution function, elastic constants and phase equilibria of suspensions of monodisperse spherical particles. The effects of polydispersity and non-sphericity of the particles are also briefly reviewed. Although most applications to data have used a pair-wise additive interparticle potential energy function of the DLVO type, a general discussion of the nature of this interparticle potential is given. An overview of the various statistical mechanical methods used is presented; this includes exact computer simulation methods and approximations such as perturbation theories an integral equation methods. Where possible direct comparisons are drawn between the results of accurate computer simulations and these approximate methods in order to indicate their relative merits, accuracy and range of validity. For situations where accurate experimental data exist, comparisons between theory and experimental results are made to assess the validity of models for the interparticle potentials.

Electroviscous effects in suspensions of monodisperse spherical particles

The electrophoretic mobility and viscosity of three monodisperse polystyrene latices were investigated in media of varying ionic strength. Electroviscous effects were marked for the smallest, 0.088 micron in diameter, which exhibited non-Newtonian flow at low ionic strengths. The viscosity-concentration relationships of all latices could be described by the empirical equation log ⁡ η γ = β α φ 1 − α φ . Both α and β were found to vary with the ionic strength of the suspension. The results for all three latices could be described by a single pair of straight lines by plotting α and β against the product of electrophoretic mobility and the Debye-Hückel reduced double layer thickness, on logarithmic coordinates.