Bidisperse Suspensions Research Articles

Osmotic depletion, non-additivity and phase separation

A brief discussion is given of the subtle interplay between non-additivity of hard sphere diameters and the osmotic depletion effect, and their influence on phase separation in bidisperse colloidal suspensions.

Shear-induced radial segregation in bidisperse suspensions

In this paper, we discuss experimental evidence for radial particle segregation in a parallel-plate geometry. The motion of coloured tracer particles of a size different from the bulk suspension is followed as a function of time. The tracer particles are seen to experience a constant drift velocity independent of their radial position. This is in addition to the random-walk motion arising from their interactions with other particles in the suspension. These observations are found to be consistent with the shear-induced migration model of Leighton & Acrivos (1987 a,b) as well as the tracer diffusivity measurements of Phan & Leighton (1996). In the experiments of Abbott et al. (1991), it was observed that larger particles migrated radially outward to regions of lower shear stress in a wide-gap Couette device. In our experiments large tracers were also observed to migrate radially outward. In this case, however, the radial migration resulted in migration to regions of higher shear stress, contrary to expectations. This apparent discrepancy is explained in terms of a model that incorporates both the stress-induced migration of earlier studies and a curvature-induced migration flux (which in turn is shear-induced) as an opposing effect.

Classification of concentrated suspensions using inclined settlers

Sedimentation vessels with parallel inclined walls are able to clarify or classify particle suspensions more rapidly than conventional settlers due to their increased surface area and decreased settling distances. The use of continuous inclined settlers to classify or separate particles by size and/or density from concentrated suspensions is considered. Mass balances are applied to each particle species in order to predict its concentration in the settler overflow (fine fraction) and settler underflow (coarse fraction), given the settler geometry, operating conditions, fluid and particle properties, and the particle c concentration and size distribution in the feed. A hindered settling function with no adjustable parameters is employed for bidisperse and polydisperse suspensions. Unlike many processes, it is shown that the efficiency of the classification process increases with increasing particle concentration in the feed, as measured by an increasing fraction of smaller or slower-settling particles partitioning to the fine fraction rather than contaminating the coarse fraction. This is a direct result of hindered settling, which reduces the settling rate and increases the relative concentration of slower-settling particles in the upper portion of the vessel. Experimental data are presented for bidisperse suspensions of glass and acrylic beads and of coal particles and yeast cells, and for polydisperse suspensions of polystyrene latex beads. The results are in good agreement with the model predictions.

Dynamic viscous resuspension of bidisperse suspensions—I. Effective diffusivity

In this paper, we describe an experimental investigation into the dynamic viscous resuspension of bidisperse suspensions in an annular parallel plate device. We use the model of Chapman & Leighton to determine the effective diffusivity of the particulate phase (51.0 and 119.0 μm glass spheres) in the plane of shear as a function of overall concentration (0.3<Φ<0.5) as well as the relative size fraction of the large particles ( 0.0<Φ L (Φ L +Φ S )<1.0 ). The effective diffusivity of a mixed suspension was found to be somewhat less than the volume average of the diffusivities of monodisperse large and small sphere suspensions.

On the modeling and investigation of polydispersed rotating suspensions

The flow field associated with the centrifugal separation of non-colloidal polydispersed (in particular, bidispersed) suspensions is considered. The “mixture” model framework is developed, and some indicative solutions are obtained and discussed. It is shown that the treatment of rotating polydispersions, in particular for non-small values of particle Taylor number, encounters idiosyncratic physical and mathematical complications. Therefore, the state of knowledge in the centrifugal case lags much behind the gravity analog.

A study of the sedimentation of noncolloidal bidisperse, concentrated suspensions by an acoustic technique

This paper uses an acoustic technique to determine the concentration profile developing during the sedimentation of noncolloidal bidisperse suspensions of glass beads in a Newtonian fluid. Various bead diameter ratios have been used and a wide range of relative concentrations is covered. From the shock front velocities and the concentrations in different zones, the sedimentation velocities of small and large particles in a homogeneous suspension of respective concentrations cs0 and cl0 have been determined. The semidilute regime (c0=ss0+cl0&amp;lt;20%) has many similarities with the dilute regime, where large particles provide the dominant hydrodynamic hindrance to settling. In the concentrated regime (c0≳35%), a mutual hindrance leads to a velocity reduction of large particles and to an enhancement of small ones, as compared to a monodisperse suspension. The data clearly demonstrate that size segregation in the concentrated regime disappears at a critical concentration, which for the size ratio 1.68 is equal to c0=45%.

The Huggins Coefficient for the Square-Well Colloidal Fluid

The Huggins coefficient, which characterizes the exact, dilute limiting viscosity of colloidal suspensions with square-well interactions, is calculated. An overview of the method and numerical results are presented for both monodisperse and bidisperse suspensions of spherical colloids. The results demonstrate that attractions increase the viscosity much more than repulsions of similar magnitude. Further, the viscosity of polydisperse suspensions of these particles is very sensitive to the smaller size fraction. These calculations complement other theoretical work on the equilibrium thermodynamic properties and colloidal diffusion to provide a more complete reference state for analyzing the role of colloidal potentials on suspension properties

Bimodal model of concentrated suspension viscosity for distributed particle sizes

A polydisperse suspension with a particle size distribution from submicrometer to hundreds of micrometers is treated as bimodal, wherein it is made up of a colloidal fine fraction and a noncolloidal coarse fraction. According to the bimodal model, the fine fraction imparts to the suspension its non-Newtonian characteristics and behaves independently of the coarse fraction, whereas the coarse particles only raise the apparent viscosity through hydrodynamic interactions. It is shown that when the particle microstructure is random, the contribution to the viscosity of the coarse fraction, which is generally polymodal, is characterized by lubrication concepts with the maximum packing fraction φm as the scaling parameter. It is found that φm is equal to the dry random packing fraction divided by a filler dilatancy factor of 1.19. The bimodal model and lubrication concepts have also been successfully applied to bidisperse suspensions with a very large particle size ratio. An inverse procedure is described that makes use of the results from viscosity and dry random packing measurements to divide a continuous size distribution into a colloidal and a noncolloidal fraction.

Shear-Induced Size Segregation Phenomena in Bidisperse Suspensions

The behavior of tracer particles in a concentrated suspension of non-colloidal spheres sheared in a parallel plate viscometer at low Reynolds number was investigated. The tracer particles were shown to migrate radially, with the tracer particles larger than those making up the suspension migrating radially outward. The migration velocity of the large tracer particles was measured, and found to be consistent with a shear-induced migration scaling (Leighton and Acrivos, 1987b), however the direction of the migration was the reverse of that suggested by the experiments of Abbot, et al. (1991) in a wide gap Couette device. The particle migration and segregation described here are interpreted in terms of the curvature of the flow geometry.

Thermocapillary migration of a bidisperse suspension of bubbles

We consider the thermocapillary motion of a well-mixed suspension of non-conducting spherical bubbles of negligible viscosity in a viscous conducting liquid under conditions of vanishingly small Reynolds and Marangoni numbers. Recently, Acrivos, Jeffrey &amp; Saville (1990) showed that when all the bubbles are of identical size, the ensemble-averaged migration velocity $\bar{U}_{1}$ of a test bubble of radius a1 within the suspension equals $U^{(0)}_1[1-\frac{3}{2}c_1+O(c^2_1)]$, where c1 is the volume fraction of the bubbles and U1(0) is the thermocapillary velocity of single bubble given by Young, Goldstein &amp; Block (1959). Here we extend this result to a bi-disperse suspension containing bubbles of radii a1 and a2 ≡ λa1 in which case $\bar{U}_1 = U^{(0)}_1[1-\frac{3}{2}c_1 - S(\lambda)c_2+\ldots]$, where c1 and c2 are the corresponding volume fractions of the two sets of bubbles. Values for S(Λ) are presented for some typical size ratios Λ, and asymptotic expressions for S(Λ) are derived for Λ → 0 and for Λ → ∞.

Sedimentation of solid particles in oil-in-water emulsions

This study reports experimental results on the sedimentation of spherical glass beads in oil-in-water emulsions. The oil droplets had a volume/surface mean diameter of about 10μm. Three different sizes of glass beads were used, namely, 29, 57 and 157 μm. The solids volume fraction, based on the total volume, was varied up to 0.4; and the oil volume concentration, solids free basis, was varied up to 40%. The fingering phenomenon was observed for the case of the small solids of 29 μm. As a result, both the settling velocity of the solids and the creaming velocity of the oil droplets were significantly enhanced. For the case of the intermediate solids (57 μm), no fingering was observed when the solids settled in a 20% oil emulsion. The settling velocity of the solids can be predicted by assuming the emulsion to form a continuous phase. However, fingering was observed when the solids settled in a more concentrated (40%) emulsion. For the case of the large solids (157 μm) no structure of any kind was observed. The solids settling velocity can be predicted by assuming the emulsions to form a continuous phase. For the low oil concentration (20%) emulsion system, the conditions for the onset of internal structures (fingering) for the different solids were consistent with previous experimental observations for the case of bidisperse solids suspensions.

Effective viscosity theory of a random concentrated suspension of polydisperse hard spheres

Abstract We consider the calculation of the effective viscosity of high-concentrated suspensions with a random distribution of large rigid spheres of different radii in an incompressible liquid. The analysis is based on the study of statistics of tetrahedra that vertices are in the centers of the spheres and distribution of gaps between particles. The dependence of the effective viscosity of bidisperse suspensions on the mixture composition and ratio of sphere radii is considered as a particular case.

Colloidal suspensions: an example of inhomogeneous, complex liquids

A brief review is presented of some recent statistical mechanics work and numerical simulations of concentrated colloidal suspensions. The topics under consideration include the phase diagram of sterically stabilized, bidisperse suspensions, sedimentation equilibrium of interacting colloids, and an 'ab initio' description of charge-stabilized suspensions.

Velocity distribution functions for a bidisperse, sedimenting particle-gas suspension

The velocity distribution function of a dilute bidisperse particle-gas suspension depends on the relative magnitudes of the viscous relaxation time, τ v, and the time between successive collisions, τ v. The The distribution functions in the two asymptotic limits, τ c ⪡ τ v and τ v ⪡ τ c, which were analysed previously are qualitatively very different. In the former limit, the leading-order distributions are Gaussian distributions about the mean velocity of the suspension, whereas in the latter case the distributions for the two species are singular at their respective terminal velocities. Here, we calculate the properties of the suspension for intermediate values of τ v/τ c by approximating the distribution function as a composite Gaussian distribution. This distribution reduces to a Gaussian distribution in the limit τ v ⪡ τ c, in agreement with previous asymptotic analysis. In the intermediate regime, however, the composite Gaussian has a non-zero skewness, which is a salient feature of the distribution in the limit τ v ⪡ τ c. We have also performed numerical calculations using the direct-simulation Monte Carlo method. The approximate values for the moments of the velocity distribution obtained using the composite Gaussian compare well with the full numerical solutions for all values of τ v/τ c.

The viscosity of semidilute, bidisperse suspensions of hard spheres

The viscosity of semidilute bimodal suspensions of sterically stabilized silica particles was measured as a function of the total volume fraction and the size ratio of the dispersed particles, and of the composition of the mixtures. The unimodal components of which the bimodal suspensions were composed showed hard sphere behavior. In the mixtures, a minimum in the viscosity was observed at small size ratios (al/as<2) when plotted against the composition. For larger size ratios a maximum was found. The origin of the viscosity maximum is uncertain but might be explained by depletion forces.

Density profiles of concentrated colloidal suspensions in sedimentation equilibrium

Sedimentation equilibria in concentrated colloidal suspensions are studied within the framework of density functional theory for inhomogeneous fluids. The density profile in the gravitational field is calculated exactly for the one-dimensional model of hard rods. The predictions of the local density approximation agree very well with coarse-grained density profiles derived from extensive Monte Carlo simulations of three-dimensional systems of hard spheres and charge-stabilized colloidal particles. The simulations show that the shape of the density profiles is very sensitive to the nature of the interactions between colloidal particles, and allow a direct test of a simple inversion procedure, which extracts the osmotic equation of state of colloidal suspensions from a measurement of their density profile. The method is sufficiently sensitive to allow the crystal-fluid tie line to be located at high colloid concentrations. The local density approximation is generalized to calculate density profiles of bidisperse hard sphere suspensions; the competition between gravity and entropy leads to nonmonotonic density profiles which are sensitive to the mass ratio, for a given size ratio. The density functional approach and Monte Carlo simulations are finally extended to study the effect of interparticle attractions on the density profiles, which exhibit a region of rapid variation corresponding to the ‘‘liquid-gas’’ interface for sufficiently strong attractive interactions.

The effect of hydrodynamic interactions on the average properties of a bidisperse suspension of high Reynolds number, low Weber number bubbles

The hydrodynamic interaction between a pair of nondeformable bubbles (low Weber number limit) in potential flow (high Reynolds number limit) was analyzed. The velocity potential was determined using twin spherical expansions, and the equations of motion were calculated by enforcing the zero net force condition on the surface of the bubbles. The acceleration due to the interaction is expressed in a perturbation series in the parameter (ai/R), where ai is the radius of bubble i, R is the distance between the bubbles, and the leading-order acceleration was found to decrease as (ai/R)4. The effect of potential flow interactions on the trajectory of a pair of bubbles of different sizes (size ratio greater than 1.07 at a Reynolds number of 200) rising due to gravity was studied. A salient feature of the trajectories is that the surfaces of the bubble do not come into contact during the interaction, except when the smaller bubble radius is less than 0.233 times the larger bubble radius when the Reynolds number based on the larger bubble is 200. In the latter case, however, the Reynolds number based on the radius of the smaller bubble is not large enough to justify the potential flow approximation. For interactions where collisions do not occur, the mean-square fluctuating velocity in a uniform suspension and the hydrodynamic diffusivities in a nonuniform suspension were calculated by performing an ensemble average over pair interactions. The pair averaging procedure is valid for dilute suspensions (V≪18/Re, where V is the volume fraction of the bubbles and Re is the Reynolds number based on the bubble radius and its terminal velocity).

Dynamic modeling of the sedimentation of concentrated bidispersed suspensions

A model based on the principle of continuity and the assumption that sedimentation results in a stratification of particles was proposed to describe the sedimentation of concentrated bidispersed suspensions. Using the slip velocity correlation of Fessas and Weiland (1984) the predicted movement of the settling interfaces was in good agreement with experiments. The model also predicted increases in concentrations of the monodispersed suspensions formed from the settling of suspensions consisting of heavy and buoyant particles when sedimentation was enhanced by the convective fingering effect.

Structure factors of charged bidispersed colloidal suspensions

In the present study, the effects of charge bidispersity on static structure factors of dilute charged aqueous colloidal suspensions, for compositions covering the full molar fraction range, are investigated using static light-scattering experiments and integral equation theory. The suspensions under study are made from different charged polystyrene particles with an average diameter of 100 nm. The experimentally-observed static structure factors are then compared with theoretical results obtained from the hypernetted chain approximation. The values of both charges are determined by fitting the scattering data of the two pure components and are kept constant for the mixtures. In order to fully describe the experimental data, the effects of the intrinsic polydispersity have to be taken into account. It is found that the height of the main peak of the measured static structure factor, SM(q), changes monotonically as a function of the composition.

Random packings of spheres and fluidity limits of monodisperse and bidisperse suspensions.

The transition from a fluid to solid state is investigated by studying the viscosity of concentrated noncolloidal monodisperse and bidisperse suspensions of hard spheres. The notion of a range of fluidity limits, analogous to glass transition densities, is introduced. Lower bounds to this range are determined and compared with dry random close packings of the spheres. The ratio of the random-close-packing fraction to the lower bound fluidity limit of the bidisperse suspensions is found to be the same as for monodisperse spheres, a constant of 1.19, suggesting a random structure at this limit.