Dilute Dispersions of Spheres

Here r x is the viscosity of the disperse phase. It is interesting to look at various limiting cases  

As the concentration is increased above q 0.01, hydrodynamic interactions between particles become important. In a flowing suspension, particles move at the velocity of the streamline corresponding to the particle centre. Hence particles will come close to particles on nearby streamlines and the disturbance of the fluid around one particle interacts with that around passing particles. The details of the interactions were analysed by Batchelor17 and we may write the viscosity in shear flow as 

There are three important points about Equation (3.47). Firstly the viscosity is the low shear limiting value, rj(0), indicating that we may expect some thinning as the deformation rate is increased. The reason is that a uniform distribution was used (ensured by significant Brownian motion, i.e. Pe 1) and this microstructure will change at high rates of deformation. Secondly there is a difference between the result for shear and that for extension. Thirdly the equation is only accurate up to cp 0.1 as terms of order / 3 become increasingly important. If we write the equation in the form often used for polymer solutions we have for Equation (3.47 a)  

However, we should seek a more reliable solution that will describe the full range of volume fractions at which flow can occur and give some guidance as to the shear thinning behaviour. 

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Because wet foams contain approximately spherical bubbles, their viscosities can be estimated by the same means that are used to predict emulsion viscosities (14). In this case, the foam viscosity is described in terms of the viscosity of the continuous phase ( 0) and the amount of dispersed gas through an empirical extension of Einstein s equation for a dilute dispersion of spheres ... 

We consider the limit of a dilute dispersion of spheres so that a. In addition, we consider the high charge density limit, X a. Using the variational ansatz for the charge density, the free energy per charge, F/Z (see Chapter 5) is written in these limits as... 

Figure 4.1.8. Hypothetical microstmctures that may be described by the effective medium model (a) Continuous matrix of phase 1 containing a dilute dispersion of spheres of phase 2. (f>) A grain boundary shell of phase 1 surrounding a spherical grain of phase 2.

Batchelor GK. Sedimentation in a dilute dispersion of spheres. J Fluid Mech 52(2) 245-268, 1972. 

Batchelor, Sedimentation in a Dilute Dispersion of Spheres, J. Fluid Mech., 52, 245, 1972. 

For dilute dispersions of hard spheres, Einstein s viscosity equation predicts... 

In 1906 Albert Einstein (Nobel Prize, 1921) published his first derivation of an expression for the viscosity of a dilute dispersion of solid spheres. The initial theory contained errors that were corrected in a subsequent paper that appeared in 1911. It would be no mistake to infer from the historical existence of this error that the theory is complex. Therefore we restrict our discussion to an abbreviated description of the assumptions of the theory and some highlights of the derivation. Before examining the Einstein theory, let us qualitatively consider what effect the presence of dispersed particles is expected to have on the viscosity of a fluid. 

These latices consist of submicroscopic, water-swollen, hydrophilic polymer spheres colloidally suspended in the continuous xylene phase. A typical electron micrograph of a diluted dispersion of a sodium poly (p-vinylbenzene sulfonate) latex which had been treated to remove water is shown in Figure 3. The inverse... 

In Section 5.1.2 the effect of solute molecules and particles on viscosity is briefly discussed. It follows that the intrinsic viscosity [t/] is a measure of the extent to which a certain solute can increase viscosity. (Remember that t] equals specific viscosity — 1) divided by concentration for infinitesimally small concentration.) According to the Einstein equation (5.6) the specific viscosity of a dispersion of spheres is 2.5q>, where

volume fraction. This means that [t/] = 2.5

0, where c is concentration in units of mass per unit volume. For a very dilute polymer solution the effective volume fraction can be given as the number of molecules per unit volume N times (4/3)jir, where ty, is the hydrodynamic radius see Eq. (6.5). Furthermore, N = c- M/Nav- For the amylose mentioned in the question just discussed, rh x 25 nm and M = 106 Da. It follows that [//] would equal... 

In this section, the kinetic aspects of aggregation will especially be discussed. Most of the theory derived is valid for the ideal case of a dilute dispersion of monodisperse hard spheres. Most food dispersions do not comply with these restrictions. Where possible, the effects of deviations from the ideal case will at least be mentioned. Some consequences of aggregation are also discussed. 

Fig. 9. Normalized UV-visible spectra of dilute dispersions of 640 nm latex spheres coated with five monolayers of Au Si02 nanoparticles. The thickness of the corresponding silica shells is indicated. The trends are consistent with the predictions of Eq. (15) but quantitative agreement is not possible due to the higher volume fraction of the shells in experiments...

Modification of the surfaces of coUoidal silica spheres with silane coupling agents enables transfer of the particles to nonpolar solvents. With 3-methacryloxypropyltri-methoxysilane bonded to the surface, the particles have been transferred from water to the polymerizable monomer, methyl acrylate. Electrostatic repulsion due to a low level of residual charge on the particle surfaces cause the dilute dispersions of particles to form a non-close packed colloidal crystalline array (CCA). Polymerization of the methyl acrylate with 200 nm diameter silica spheres in a CC fixes the positions of the spheres in a plastic film by the reactions shown in Figure 11.14. The difriaction... 

Figure 4. Temperature dependence of the PNIPAM colloid diameter and turbidity. The diameter was determined using a commercial quasielastic light scattering apparatus (Malvern Zetasizer 4). The turbidity was measured for a disordered dilute dispersion of these PNIPAM colloids by measuring light transmission through a 1.0 cm pathlength quartz cell with a UV-visible-near IR spectrophotometer. Solids content of the sample in the turbidity experiment was 0.071%, which corresponds to a particle concentration of 2.49 x 10 spheres/cc. Also shown is the temperature dependence of the turbidity of this random colloidal dispersion. The light scattering increases as the particle becomes more compact due to its increased refractive index mismatch from the aqueous medium (76) (Adapted from ref 16).

In assuming a dilute dispersion of uniform, rigid, noninteracting spheres, Einstein (69,70) derived an equation expressing the increase in viscosity of the dispersion. 

Explanation for the existence of order in these dilute dispersions of isotropic spheres requires the existence of forces whose range is long compared to those of chemical valence bonds or van der Waals forces. The polymer spheres possess bo md sulfate radicals on their surfaces, which can dissociate even in media of moderate polarity. The resulting coulombic forces, even when partially shielded by an atmosphere of free coimterions, possess the required long range. This range is drastically shortened, however, when neutral electrolyte is added, thereby producing an order-disorder transition. 

Consider a dilute dispersion of uniform spherical polymer particles as shown in Fig. 19. These spheres experience Brownian motion and therefore diffuse in all directions, causing collisions between the particles. If an adhesion bond forms between the surface molecules, then a collision has a chance of creating a doublet, that is. two particles adhering together at the single molecular bond which forms at the point of contact. If the adhesive bond is weaker than kT, then thermal collisions can break this bond in a period of time. The spheres will then separate and move apart. Thus there is a dynamic equilibrium between joining and separation, giving a certain number of doublets in the suspension at equilibrium. 

The experimental setup is shown schematically in Fig. 23(a). Uniformly sized microspheres with diametey ther 10, 25 or 96 pm, were dispersed in kerosene-based ferrofluid " and confined between two glass plates. The spacing between the plates were several times the diameter of the spheres. Two pairs of Helmholtz coils were used to produce two sinusoidal fields, H sinui t and H sin(w t+i/2) in the plane with amplitude H. The sample cell"(20x20 mm ) contained a very dilute dispersion of polystyrene spheres. This produces only a few pairs of spheres which were far apart and thus not interacting. The frequencies of the various modes were low (< Hz) and could easily be measured manually using a stop-watch. 

We present an improved model for the flocculation of a dispersion of hard spheres in the presence of non-adsorbing polymer. The pair potential is derived from a recent theory for interacting polymer near a flat surface, and is a function of the depletion thickness. This thickness is of the order of the radius of gyration in dilute polymer solutions but decreases when the coils in solution begin to overlap. Flocculation occurs when the osmotic attraction energy, which is a consequence of the depletion, outweighs the loss in configurational entropy of the dispersed particles. Our analysis differs from that of De Hek and Vrij with respect to the dependence of the depletion thickness on the polymer concentration (i.e., we do not consider the polymer coils to be hard spheres) and to the stability criterion used (binodal, not spinodal phase separation conditions). 

Vincent et al.(3) used a simplified configurational entropy term Ass = -k ln(4>f /4>.). For a dilute dispersion, the In 4>d term is probably correct, but for the floe phase, with of the order of 0.5, a term In 4>f certainly can overestimate the entropy in the floe, because hard spheres with finite volume have at high concentration much less translational freedom than (volumeless) point... 

In Chapter 3 (Section 3.5.2) the viscosity of a hard sphere model system was developed as a function of concentration. It was developed using an exact hydrodynamic solution developed by Einstein for the viscosity of dilute colloidal hard spheres dispersed in a solvent with a viscosity rj0. By using a mean field argument it is possible to show that the viscosity of a dispersion of hard spheres is given by... 

Following Einstein s work (Einstein, 1906, 1911) (Equation 2.24) on dilute rigid sphere dispersions, models for estimating viscosity of concentrated non-food dispersions of solids are based on volume fraction (p) of the suspended granules and the relative viscosity of the dispersion, % = (vIVs), where t] is the viscosity of the dispersion rjs is the viscosity of the continuous phase (Jinescu, 1974 Metzner, 1985). 

To parameterize the new quantities occurring in these equations a few semi-empirical relations from the literature were adopted. The asymptotic value of bubble induced turbulent kinetic energy, fesia, is estimated based on the work of [3]. By use of the so-called cell model assumed valid for dilute dispersions, an average relation for the pseudo-turbulent stresses around a group of spheres in potential flow has been formulated. Prom this relation an expression for the turbulent normal stresses determining the asymptotic value for bubble Induced turbulent energy was derived ... 

A model for the bulk effective resistivity of a dilute suspension (disperse phase) of noninteracting conducting spheres (not necessarily mono-dispersed) of material resistivity 9id and void fraction ad suspended in a continuous medium of material resistivity 9ic was derived by Maxwell (1954). His result is... 

Drop Deformability When a neutrally buoyant, initially spherical droplet is suspended in another liquid and subjected to shear or extensional stress, it deforms and then breaks up into smaller droplets. Taylor [1932,1934] extended the work of Einstein [1906, 1911] on dilute suspension of solid spheres in a Newtonian liquid to dispersion of single Newtonian liquid droplet in another Newtonian liquid, subjected to a well-defined deformational field. Taylor noted that at low deformation rates in both uniform shear and planar hyperbolic fields, the sphere deforms into a spheroid (Figure 7.9). 

It is frequently desirable to be able to describe and/or predict dispersion viscosity in terms of the viscosity of the continuous phase (i/q) and the amount of dispersed material. A very large number of equations have been advanced for estimating emulsion, foam, suspension or aerosol viscosities. Most ofthese are empirical extensions of Einsteins equation for a dilute suspension of non-interacting spheres ... 

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