Dilute Dispersions of Spherical Particles

Let us first consider a dilute system of rigid spherical particles dispersed at a volume fraction j m Newtonian liquid of shear viscosity rjo. According to Einstein,the Newtonian viscosity of the whole system, rj, is increased by the presence of the particles according to the equation 

The value of [t/] will tend to differ from 2.5 if any of the assumptions of Einstein s theory are violated. This may occur if particles are electrically charged, or if they are non-spherical or deformable, or if the particle surface is coated with a layer of molecules (solvent, surfactant or polymer) of thickness significant compared to the particle size. 

Particles dispersed in an aqueous medium invariably carry an electric charge. Thus they are surrounded by an electrical double-layer whose thickness k depends on the ionic strength of the solution. Flow causes a distortion of the local ionic atmosphere from spherical symmetry, but the Maxwell stress generated from the asymmetric electric field tends to restore the equilibrium symmetry of the double-layer. This leads to enhanced energy dissipation and hence an increased viscosity. This phenomenon was first described by Smoluchowski, and is now known as the primary electroviscous effect. For a dispersion of charged hard spheres of radius a at a concentration low enough for double-layers not to overlap (d 8a ic ), the intrinsic viscosity defined by eqn. (5.2) increases 

The presence of an adsorbed layer of polymer (or surfactant) of thickness d at the particle surface (or even a solvation layer of immobilized solvent molecules 

The effects of adsorbed layers or solvation layers become more important with decreasing size of the dispersed particles. 

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For dilute dispersions of spherical particles, the diffusion coefficient can be related to the hydrodynamic diameter of the particles by the Stokes-Einstein equation... 

Colloidal systems, because of their large number of dispersed particles, show non-Newtonian flow behavior. For a highly dilute dispersion of spherical particles, the following equation has been proposed by Einstein ... 

Diffusion coeffflcients can be readily measured by means of quasi-elastic or dynamic light scattering, also called photon correlation spectroscopy. For a description of the technique, we refer to sec. 1.7.8. In a dilute dispersion of spherical particles of bare radius a, the diffusion coefficient D can be directly related to d and a by the Stokes-Einsteln equation [1.6.3.321 ... 

Dynamic light-scattering, sometimes called quasi-elastic light scattering or photon correlation spectroscopy, can be used to measure the diffusion coefficients of polymer chains in solution and colloids, a kind of Doppler effect see Section 3.6.6. In a dilute dispersion of spherical particles, the diffusion coefficient D is related to the particle radius, a, through the Stokes-Einstein equation. 

In many colloid systems particles are covered with adsorbed layers (see Chapter 5). These too influence the viscosity since the effective radius, and hence the effective volume fraction, is greater than that of the core particles. In attempting to fit experimental data on dispersions of spherical particles to theoretical equations the effective volume fractions must be employed. If measurements are made on very dilute suspensions and at low shear rates, equation (8.7) (retaining only the first two terms) may be used to calculate the effective volume fraction and hence the particle size, and the thickness of the adsorbed layer if the size of the core particles is known. This is not, however, a very precise method and generally other methods of finding the adsorbed layer thickness are to be preferred. 

The viscosity of a fluid is increased by the presence of dispersed particles as they tend to disturb the flow patterns. For dilute suspensions of spherical particles in a Newtonian fluid, the viscosity is described by Einstein s relation. 

The relative viscosity of a dilute dispersion of rigid spherical particles is given by = 1 + ft0, where a is equal to [Tj], the limiting viscosity number (intrinsic viscosity) in terms of volume concentration, and ( ) is the volume fraction. Einstein has shown that, provided that the particle concentration is low enough and certain other conditions are met, [77] = 2.5, and the viscosity equation is then = 1 + 2.50. This expression is usually called the Einstein equation. 

Dilute Dispersions of Macromolecules in Nonelectrolytes The Stokes-Einstein equation has already been presented. It was noted that its validity was restricted to large solutes, such as spherical macromolecules and particles in a continuum solvent. The equation has also been found to predict accurately the diffusion coefficient of spherical latex particles and globular proteins. Corrections to Stokes-Einstein for molecules approximating spheroids is given by Tanford Physical Chemistry of Macromolecules, Wiley, New York, (1961). Since solute-solute interactions are ignored in this theory, it applies in the dilute range only. 

Effective Conductivity of Dilute Mixtures, The simplest, best defined case, is a cluster of spherical particles dispersed in a liquid and located in a... 

Dispersions. When particles are added to a liquid, the viscosity is increased. Near a particle the flow is disturbed, which causes the velocity gradient XF to be locally increased. Because the energy dissipation rate due to flow equals t] F2, more energy is dissipated, which becomes manifest as an increased macroscopic viscosity. The microscopic viscosity, as sensed by the particles, remains that of the solvent (pure liquid) rjs. For very dilute dispersions of solid spherical particles, Einstein derived... 

The dynamic electrophoretic mobility is a useful system characteristic. For instance, it provides a means for determining the isoelectric point from electroacoustic measurements. Other properties of colloidal dispersions, namely -potential and particle size, can also be obtained from measurements of dynamic electrophoretic mobility. For a dilute dispersion (less then a few weight percent) of spherical particles, the dynamic electrophoretic mobility is given by a Smoluchowski-type relationship [37,38] ... 

Effective Conductivity of Dilute Mixtures. The simplest best-defined case is a cluster of spherical particles dispersed in a liquid and located in a uniform electrical field. If the particles have the same conductivity as the liquid, the potential around the particles will not be distorted, and the mixture conductivity is equal to that of the liquid. If the particles have a lower conductivity, the streamlines will diverge away from the particles, and the mixture conductivity will be lower than that of the liquid. If the particles have a higher conductivity, the streamlines will converge into the particle, and the mixture conductivity will be higher than that of the liquid. 

The viscosity of a very dilute dispersion of rigid spherical particles in a Newtonian fluid is described by the Einstein equation (Equation 8.2) [26]. Where q is the viscosity of the dispersion, qi is the viscosity of the fluid alone, ( > is the volume fraction of particles and kg is the Einstein coefficient, which is 2.5 for spherical particles, k depends upon both particle shape and orientation. 

The particle-particle interactions lead to a dependence of the viscosity, q, of a colloidal dispersion on the particle volume fraction, (]). Einstein [892] showed that for a suspension of spherical particles in the dilute limit... 

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 relative viscosity of a dilute dispersion of rigid spherical particles is given by /j-ei = 1 + where a is equal to the limiting viscosity number (intrinsic viscosity) in terms of volume concentration, and (p is the volume... 

Then, the constitutive equation for a dilute dispersion of identical spherical particles is given by... 

In the case of dilute dispersions of rigid solid spherical particles, it can be demonstrated that the constitutive equation can be written as... 

The rheology of dispersions of liquid particles (emulsions) has also been largely investigated. If the droplets retain their spherical shape under shear, the continuous medium is Newtonian and the dispersion is diluted enough the emulsion behaves as Newtonian with viscosity given by Taylor equation [30]... 

Viscosity—Concentration Relationship for Dilute Dispersions. The viscosities of dilute dispersions have received considerable theoretical and experimental treatment, partly because of the similarity between polymer solutions and small particle dispersions at low concentration. Nondeformable spherical particles are usually assumed in the cases of molecules and particles. The key viscosity quantity for dispersions is the relative viscosity or viscosity ratio,... 

Stokes law is rigorously applicable only for the ideal situation in which uniform and perfectly spherical particles in a very dilute suspension settle without turbulence, interparticle collisions, and without che-mical/physical attraction or affinity for the dispersion medium [79]. Obviously, the equation does not apply precisely to common pharmaceutical suspensions in which the above-mentioned assumptions are most often not completely fulfilled. However, the basic concept of the equation does provide a valid indication of the many important factors controlling the rate of particle sedimentation and, therefore, a guideline for possible adjustments that can be made to a suspension formulation. 

This equation was derived for spherical particles and consists of a Taylor series in their volume fraction . This equation is valid for a mixture of different sized spheres at dilute concentrations. If a polymer is the disperse phase, it is convenient to convert the volume fraction to concentration, Cg, using the relationship where V2... 

Consider a dilute suspension of Np spherical soft particles moving with a velocity U exp(—/fflf) in a symmetrical electrolyte solution of viscosity r] and relative permittivity r in an applied oscillating pressure gradient field Vp exp(—imt) due to a sound wave propagating in the suspension, where m is the angular frequency 2n times frequency) and t is time. We treat the case in which m is low such that the dispersion of r can be neglected. We assume that the particle core of radius a is coated... 

When a dilute suspension of sufficiently large, mono-disperse, spherical particles is irradiated with white light, vivid colors appear at various angles to the incident beam. The angular positions of the spectra depend on m and D, hence they may be used to determine particle size in colloidal suspensions. 

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