Non-dilute Suspensions Research Articles

Square root singularity in the viscosity of neutral colloidal suspensions at large frequencies

The asymptotic frequency $\omega$, dependence of the dynamic viscosity of neutral hard sphere colloidal suspensions is shown to be of the form $\eta_0 A(\phi) (\omega \tau_P)^{-1/2}$, where $A(\phi)$ has been determined as a function of the volume fraction $\phi$, for all concentrations in the fluid range, $\eta_0$ is the solvent viscosity and $\tau_P$ the P\'{e}clet time. For a soft potential it is shown that, to leading order steepness, the asymptotic behavior is the same as that for the hard sphere potential and a condition for the cross-over behavior to $1/\omega \tau_P$ is given. Our result for the hard sphere potential generalizes a result of Cichocki and Felderhof obtained at low concentrations and agrees well with the experiments of van der Werff et al, if the usual Stokes-Einstein diffusion coefficient $D_0$ in the Smoluchowski operator is consistently replaced by the short-time self diffusion coefficient $D_s(\phi)$ for non-dilute colloidal suspensions.

Protein−Macromolecule Interactions

Although the interactions between proteins and various types of macromolecules are of considerable technological interest,1 little theoretical work has been done that makes use of the fact that proteins are often comparatively small particles. Thus, our interest is in a regime opposite to that focused on in the usual depletion theories.2 Here, a protein will be viewed, perhaps naively, as a small hard sphere whose dielectric permittivity is negligible compared to that of water. Its interaction otherwise with some macromolecular segment will be inert. Some time ago, de Gennes already presented several, not so widely known, preliminary calculations of a spherical particle interacting with a semidilute solution of polymers.3 One supposition he made concerning the irrelevance of a certain scale, will be proved here. My aim is to present a scaling analysis of the interaction between a small sphere and a macromolecule, particularly in dilute solution. Though obviously of restricted validity, the expressions derived may prove helpful in qualitatively understanding phase separation phenomena occurring in nondilute suspensions. We first consider a protein sphere of radius a immersed in an aqueous solution containing a semidilute polymer which is well soluble. Its Kuhn length is AK, and the excluded volume between two segments is â ) AK. In a self-consistent field approximation, the polymer segment density ψ2(r b) at position r b is given by2

Numerical simulations of the effect of hydrodynamic interactions on diffusivities of integral membrane proteins

Proteins in a biological membrane can be idealized as disks suspended in a thin viscous sheet surrounded by a fluid of lower viscosity (Saffman 1976). To determine the effect of hydrodynamic interactions on protein diffusivities in non-dilute suspensions, we numerically solve the Stokes equations of motion for a system of disks in a bounded periodic two-dimensional fluid using a multipole expansion technique. We consider both free suspensions, in which all the proteins are mobile, and fixed beds, in which a fraction of the proteins are fixed. For free suspensions, we determine both translational and rotational short-time self-diffusivities and the gradient diffusivity as a function of the area fraction of the disks. The translational self- and gradient diffusivities computed in this way grow logarithmically with the number of disks owing to Stokes paradox; to obtain finite values, we renormalize our simulation results by treating long-range interactions in terms of a membrane with an enhanced viscosity in contact with a low-viscosity three-dimensional fluid. The diffusivities in fixed beds require no such adjustment because, at non-dilute area fractions of disks, the Brinkman screening of hydrodynamic interactions is more important that the viscous drag due to the surrounding three-dimensional fluid in limiting the range of hydrodynamic interactions. The diffusivities are determined as functions of the area fractions of both mobile and fixed proteins. We compare our results for diffusivities with experimental measurements of long-time protein self-diffusivity after adjusting our short-time diffusivities calculations in an approximate way to account for effects of hindered diffusion due to volume exclusion, and find very good agreement between the two.

Sound Wave Mobilities in a Nondilute Suspension of Spheres

The motion of particles due to the passage of a sound wave through a nondilute, monodisperse suspension of spheres is calculated correct to the first order in particle volume fraction using the macroscopic boundary integral technique (O'Brien, R. W., J. Fluid Mech. 91, 17 (1979)). This calculation uses functions describing the hydrodynamic interactions of pairs of spheres in a sound wave of arbitrary frequency. These functions are calculated using the multipole collocation technique (Weinbaum, S., Ganatos, P., and Yan, Z., Annu. Rev. Fluid Mech. 22, 275 (1990)). The results of the sound wave mobility calculation are compared with the high frequency limit which has previously been calculated by Sangani et al. (Sangani, A. S., Zhang, D. Z., and Prosperetti, A., Phys. Fluids A 3(12), 2955 (1991)). The agreement in the limit of infinitely high frequencies is found to be excellent for particle densities more than 0.52 times that of the suspending fluid. Discrepancies exist between the theories when the corrections to the high frequency limit given in Sangani et al. are used to calculate mobilities at finite frequencies. The results of this paper are designed to be the first step in developing a rigorous approach toward using ultrasonic techniques to determine the properties (size or density) of particles in a colloidal suspension.

Sedimentation equilibrium of suspensions of colloidal particles at finite concentrations

Equilibrium concentration profiles of non-dilute colloidal suspensions are calculated by means of the Carnahan-Starling expression for the osmotic compressibility of hard sphere liquids. The profiles depend on the average volume fraction of the suspension, 〈ϕ〉, and on the field interaction parameter, λ0 (reciprocal of the Peclet number at infinite dilution). Profiles are computed for values of 〈ϕ〉 and λ0 typical of those encountered in sedimentation field-flow fractionation experiments. It is found that, in most cases, the volume fraction at the depletion wall is negligibly small and that the volume fraction at the accumulation wall φ0, depends on the ratio 0/λ0 only. An inflexion point is found in the concentration profile if φ0 is larger than 0.13 whatever the value of λ0.

Application of the homogenization method to a suspension of fibres

The homogenization method is applied to the determination of a continuum model for an assembly of cylindrical parallel fibres suspended in the longitudinal flow of a viscous incompressible fluid. Whatever the solids concentration is, the homogenization gives the variations of the velocity and the pressure at the level of the microstructure and furnishes a rigorous deductive procedure for obtaining the governing equations of the homogenized bulk medium. The effective constitutive relation is that of a Newtonian incompressible anisotropic fluid. The viscosity is expressed by a second order tensor completely determined by the microstructure. Lower bounds of the viscosity coefficients are found when the microstructure exhibits symmetries. Numerical results are obtained for two cases of macroscopic isotropy. They emphasize the good approximation arising from the lower bounds and the importance of the microstructure geometry for nondilute suspensions. As a matter of fact the solids concentration is not able to describe correctly the influence of the microstructure on the bulk behaviour of concentrated suspensions.

The dynamic mobility of particles in a non-dilute suspension

When an alternating electric field is applied to a colloid the particles oscillate at a velocity proportional to the applied field strength. The complex constant of proportionality is termed the dynamic mobility (O'Brien 1988). Although this quantity can now be determined from electroacoustic measurements in suspensions of arbitrary concentration (O'Brien 1990), the theory for interpreting these measurements in terms of the size and charge of the particles is limited to dilute suspensions.In this paper we derive an expression for the O(ϕ) correction to the dynamic mobility in a random suspension of uniform spheres with volume fraction ϕ. It is assumed that the particle radius is much greater than the double layer thickness but much smaller than the sound wavelength. The mobility is calculated using O'Brien's 1979 macroscopic boundary integral technique. This method ensures a correct mathematical formulation of the problem, and yields an absolutely convergent expression for the average particle velocity. The evaluation of this expression to O(ϕ) involves the determination of the velocities of an isolated pair of particles at various separations and frequencies of oscillation. These velocities are computed using the collocation technique and the O(ϕ) correction to the dynamic mobility is then obtained by numerically integrating over all particle separations.

Fiber–fiber interactions in homogeneous flows of nondilute suspensions

A theoretical model for calculating the average interfiber spacing in nondilute short fiber suspensions is presented. The approach adopts the Doi–Edwards theory to model spacing between rigid rod‐like polymer molecules to determine the interfiber spacing. Modifications have been introduced to the Doi–Edwards theory in order to obtain accurate estimates of interfiber spacing at all orientation states and for fibers of finite diameter. A parametric study has been conducted to explore the influence of the relevant dimensionless parameters: the fiber volume fraction, the fiber aspect ratio, and the probability distribution function of the fiber orientation state on the average spacing between the fibers. An approach has been suggested to use this spacing to characterize fiber–fiber interactions in flowing suspensions. Numerical predictions of flow‐induced fiber orientation based on this approach are presented.

Velocity and concentration measurements of suspensions by nuclear magnetic resonance imaging

Nonuniform velocity and concentration distributions have been measured for suspensions of small, negatively buoyant particles flowing in horizontal tubes. All measurements were performed during statistically steady flow conditions with average fluid velocities up to 25 cm/s. The mean particle concentration φs ranged between 0.0≤φs≤0.39. Time‐averaged measurements were recorded noninvasively by using a three‐dimensional, flow‐compensating, nuclear magnetic resonance technique. This method spatially resolves both the fluid velocity and particle concentration distributions from a single imaging experiment and permits detailed analysis of the flow‐induced structure of nondilute suspensions.

Non-isothermal Extrusion of Composite Materials

Abstract Non-isothermal extrusion, of non-dilute fiber suspensions with solidification is analyzed by means of an integral constitutive equation for melt fiber suspension and a novel, streamlined inverse finite element method (IIFEM), to account for the solidification. Newton iteration is used to solve simultaneously the highly non-linear system of intergrodifferential equations for all the primary unknowns including the apriori unknown location of the solidification front and the free surface. The fiber orientation is computed using the orientation distribution function along selected streamlines of the melt suspension. It is shown that the final frozen-in orientation of the fibers in the resulting solid composite depends on the location of the solidification front. Thus by controlling the location of the solidification front by means of the flow, cooling rate and the capillary design may be possible to produce composites of several orientation patterns.

Constitutive equations for a nondilute dispersed fiber suspension

The description fiber suspension is usually applied to a two-phase medium one of whose phases is an incompressible fluid while the other consists of fibers suspended in the fluid [i, 2]. In the suspension at rest at concentrations c greater than the sedimentation concentration c s (for the fibers used in the pulp and paper industry the mass concentration c s is of the order of 0.5% [3]) the fibers interact to form a structure (so-called fiber network) which possesses the properties of a solid -elasticity and strength [1--3]o The fiber suspension as a whole is a saturated porous medium with an elastic skeleton. In this state it is said to be structured [2].

A similarity solution for steady-state crossflow microfiltration

A similarity solution is presented for the convective-diffusion equation governing the steady-state concentration—polarization boundary layer in crossflow microfiltration of the particles, under conditions where a thin stagnant layer of particles deposited on the microporous membrane surface provides the controlling resistance to filtration. The analysis employs concentration-dependent shear viscosities and shear-induced hydrodynamic diffusivities based on empirical correlations for suspensions of rigid spheres. The resulting permeate flux is v w( x) = τ w( a 4/3 x) 1 3 ν t- w/μ 0, where x is filter entrance, τ w is the wall shear stress exerted on the boundary layer by the tangential flow of bulk suspension through the filter channel, a is the particle radius, and μ 0 is the characteristic viscosity. The dimensionless permeate flux, ν − w (φ b), depends only on the particle volume fraction in the bulk suspension, φ b, and is given by ν − w = 0.0581φ b − 1 3 when the suspension is dilute (φ b < 0.10). The results for the permeate flux and for the concentration and velocity profiles show that the approximate solution of Davis and Leighton [ Chem. Engng Sci. 42, 275–281 (1987)] and Romero and Davis [ J. Membrane Sci. 39, 157–185 (1988)], which neglects axial convection in the differential particle mass balance but retains it when integrating across the entire boundary layer, is exact in the dilute limit and accurate to within a few percent for nondilute suspensions. The solution may easily be extended to other suspensions having different dependencies of viscosity and diffusivity on concentration.

The sedimentation of nondilute suspensions in inclined settlers

The base-state convective flows, which are set up when a nondilute sedimenting suspension is placed beneath an inclined wall, are analyzed theoretically using a two-fluid model. Their hydrodynamic stability and the corresponding spatial growth of small two-dimensional disturbances at the clear fluid–suspension interface are then determined over the entire range of the governing parameters through numerical solutions of the relevant Orr–Sommerfeld equations. Two mechanisms for the growth of instability waves at the interface are identified. The results demonstrate that the base-state flow becomes more unstable as inertial effects in the base state become more pronounced and thus, contrary to what has been suggested by earlier investigators, there is no restabilization as the base state approaches the inviscid limit. Increasing the concentration of the suspension is found to have a stabilizing effect on the two-phase interface, particularly when inertial effects dominate in the base state. Similarly, increasing the angle of inclination enhances the stability of the interface when viscous forces are dominant in the base flow.

The effective thermal conductivity and elongational viscosity of a nondilute suspension of aligned slender rods

The effective thermal conductivity and elongational viscosity of a nondilute suspension of infinitely conducting perfectly aligned slender rods of slenderness ratio ε≪1 are determined using an effective medium theory together with a self-consistent scheme. The expressions thereby obtained cover the range in particle concentration c from infinite dilution c/ε2≪1, to the semidilute regime, c/ε2≫1, but c≪1. In the latter case, the present results agree to leading order with a formula derived by Batchelor [J. Fluid Mech. 46, 813 (1971)] even though the analyses employ physical arguments that appear to be quite distinct.

Homogenization mechanics of a non-dilute suspension of magnetic particles

Using the technique of homogenization, the behavior of a non-dilute suspension of magnetic particles of arbitrary shape is obtained. It is shown that although the particles are suspended in a Newtonian fluid, its homogenized behavior is non-Newtonian and its viscosity moduli are depending on the discrete structure of the magnetic suspension. The results are given in a computational suitable form.

Isothermal extrusion of non-dilute fiber suspensions

The extrusion of a rod-like fiber suspension is a Newtonian solvent, as a first step to the fast and inexpensive production of composite materials, is investigated. The analysis is carried out by means of an integral constitutive equation for a non-dilute suspension, streamlined finite element for liquid with memory, and Newton iteration of nonlinear integro-differential equations. The predictions show substantial differences between dilute and nondilute fiber suspension regarding the processing conditions (pressure drop, velocity distribution, die-swell) and the resulting fiber orientation. Nondilute fiber suspensions exhibit substantial shear-thinning and negligible elasticity as evidenced by the small die-swell, and fiber concentration viscosity-thickening as evidenced by the large pressure drop. The fiber orientation is computed by solving the orientation distribution function along selected streamlines of the complex velocity field. It is shown that the fiber orientation far downstream can be made independent of the random fiber orientation at the inlet.

The equations of motion of a one-velocity, one-temperature bubbly fluid (revisited)

We present a continuum model for a non-dilute suspension of spheres with variable radius. To simplify the resulting equations and to stress the expansion dynamics only, we leave aside the problems of relative motion and temperature difference between the spheres and the surrounding liquid. It is shown that the state variable associated with the expansion degree of freedom is the scalar moment of momentum of the whole mixture, which is somewhat the analog of the usual angular momentum for the rotational degree of freedom. The equation of motion of this new variable leads to a non-dilute form of the Rayleigh-Plesset equation. Our simplified description of bubbly fluids involves six phenomenological functions of the bubble volume fraction. One of them is linked to the bubble resonant frequency and is attainable by sound velocity measurements. Among the five remaining functions are four viscosities. We detail their respective roles and correct a long-lived erroneous statement concerning the bulk or second viscosity of the bubbly fluid On presente un modele, style milieu continu, d'une suspension non diluee de spheres de rayon variable

On centrifugal separation of particles of two different sizes

We consider flow in a centrifugal force field of a non-dilute suspension with particles or droplets of two sizes. The volume fraction and the velocity fields are determined assuming small convection and shear terms. The resulting flow field is quite different from that in a gravitational settling of a similar mixture. In particular, the volume fraction is a function of time and radius in the sectors separated by kinematic shocks and the settling velocity is a non-monotonic function of the particle size.

On hindered settling of particles of different sizes

The flow of a non-dilute fluid suspension is considered in which the dispersed phase consists of particles or droplets of different sizes. A phenomenological two-phase flow theory is formulated for both continuous and discrete distributions of particle sizes and illustrated by considering the batch settling of such a mixture. The volume fractions and particle distribution functions are determined, as well as the composition of the sedimentary layer.

Rheological properties of nondilute suspensions of deformable particles

The creeping flow equations have been solved to account for effects of shear on drop shape when the influence of neighboring drops is considered. A cell model is used to account for finite concentrations of the dispersed phase. From these results a constitutive equation is derived and implications for several flows are discussed. The shape and orientation of drops are computed and compared with existing data. Finally, dynamic viscosity and rigidity are obtained for nondilute suspensions of viscoelastic particles in an oscillatory shear flow.