Spherical particles, suspension

According to the Einstein theory, the intrinsic viscosity of a spherical particle suspension is 2.5. However, for a colloidal suspension of nonspherical particles, [r ] > 2.5. Jeffery [112] obtained the viscosity of an ellipsoidal particle suspension under shear. Incorporating Jeffery s results of velocity fields around the particle, Simha [113] obtained expressions for two explicit limiting cases of ellipsoids. Kuhn and Kuhn [114] also obtained an expression for intrinsic viscosity for the full range of particle aspect ratio (p) by taking an approach similar to Simha s method. 

It is not surprising therefore that the optical properties of small metal particles have received a considerable interest worldwide. Their large range of applications goes from surface sensitive spectroscopic analysis to catalysis and even photonics with microwave polarizers [9-15]. These developments have sparked a renewed interest in the optical characterization of metallic particle suspensions, often routinely carried out by transmission electron microscopy (TEM) and UV-visible photo-absorption spectroscopy. The recent observation of large SP enhancements of the non linear optical response from these particles, initially for third order processes and more recently for second order processes has also initiated a particular attention for non linear optical phenomena [16-18]. Furthermore, the paradox that second order processes should vanish at first order for perfectly spherical particles whereas experimentally large intensities were collected for supposedly near-spherical particle suspensions had to be resolved. It is the purpose of tire present review to describe the current picture on the problem. 

Since DNA is a highly charged macromolecule surrounded by a layer of counter ions, it is more probable that the dielectric polarization of DNA arises from the polarization of the ion atmosphere. Various mechanisms of ionic polarization have been proposed. The theories of ionic polarization for a spherical particle suspension were reviewed and carefully discussed by Schwan (20, 21). Since DNA is a thin elongated molecule, those theories must be modified substantially. Various theories for ellipsoids are briefly reviewed here. 

Counter Ion Polarization. Schwan (20) attempted to explain the dielectric dispersion of a spherical particle suspension in terms of counter ion polarization, and Schwarz carried out the mathematical formulation (23), and found that the displacement of the counter ion in the double layer is equivalent to the existence of complex surface conductivity,... 

Particle Shape Effect. To this point, we have been dealing only with spherical particle suspensions. When the particles have irregular shapes, the rheological properties are expected to be very different from those of the spherical particle suspensions. Consider, for example, a simple system of cylindrical fibre suspensions. Because the particles are expected to align in the direction of the flow or shear, the viscosity needs to be treated as a second-order tensor, that is, the values of the viscosity under the same condition are different when different directions are referred. Only at the low (zero) shear limit may the particles be randomly distributed and have an isotropic rheological behavior. 

Equation 50 does not agree with the experimentally observed Quemada equation. However, by comparing equations 48 and 49 with 50, we may expect that the longitudinal viscosity of the cylindrical fiber suspension is smaller than the viscosity of spherical particle suspensions at a concentrated state, whereas the transverse viscosity is higher than the viscosity of spherical suspensions. 

Table 83 summarizes the equations derived hitherto for the specific viscosity of solutions and the names of the authors who devised them. It may be pointed out that only problems relating to spherical particle suspensions have been satisfactorily solved. All other expressions are valid only within certain limitations and cannot be formulated with accuracy. 

Kholodenko, A.L. and Douglas, J.F., Generalized Stokes-Einstein equation for spherical-particle suspensions, Phys. Rev. E 51,1081-1090 (1995). 

Altliough tire behaviour of colloidal suspensions does in general depend on temperature, a more important control parameter in practice tends to be tire particle concentration, often expressed as tire volume fraction ((). In fact, for hard- sphere suspensions tire phase behaviour is detennined by ( ) only. For spherical particles... 

In most colloidal suspensions tire particles have a tendency to sediment. At infinite dilution, spherical particles with a density difference Ap with tire solvent will move at tire Stokes velocity... 

Otlier possibilities for observing phase transitions are offered by suspensions of non-spherical particles. Such systems can display liquid crystalline phases, in addition to tire isotropic liquid and crystalline phases (see also section C2.2). First, we consider rod-like particles (see [114, 115], and references tlierein). As shown by Onsager [116, 117], sufficiently elongated particles will display a nematic phase, in which tire particles have a tendency to align parallel to... 

A somewhat similar problem arises in describing the viscosity of a suspension of spherical particles. This problem was analyzed by Einstein in 1906, with some corrections appearing in 1911. As we did with Stokes law, we shall only present qualitative arguments which give plausibility to the final form. The fact that it took Einstein 5 years to work out the bugs in this theory is an indication of the complexity of the formal analysis. Derivations of both the Stokes and Einstein equations which do not require vector calculus have been presented by Lauffer [Ref. 3]. The latter derivations are at about the same level of difficulty as most of the mathematics in this book. We shall only hint at the direction of Lauffer s derivation, however, since our interest in rigid spheres is marginal, at best. 

Figure 9.2 (a) Schematic representation of a unit cube containing a suspension of spherical particles at volume fraction [Pg.589]

Turbidity. Turbidity in water is removed by ozonation (0.5—2 ppm) through a combination of chemical oxidation and charge neutralization. GoUoidal particles that cause turbidity are maintained in suspension by negatively charged particles which are neutralized by ozone. Ozone further alters the surface properties of coUoidal materials by oxidizing the organic materials that occur on the surface of the coUoidal spherical particles. 

The electrokinetic effect is one of the few experimental methods for estimating double-layer potentials. If two electrodes are placed in a coUoidal suspension, and a voltage is impressed across them, the particles move toward the electrode of opposite charge. For nonconducting soHd spherical particles, the equation controlling this motion is presented below, where u = velocity of particles Tf = viscosity of medium V = applied field, F/cm ... 

Early suspension polymers, although less contaminated, were supplied as more or less spherical particles with a diameter in the range 50-100 p,m. Such materials had a much lower surface/volume ratio than the emulsion polymers and, being of low porosity, the materials were much slower in their gelation with plasticisers. The obvious requirement was to produce more porous particles and these became available about 1950 as easy-processing resins. 

The retentivity relative to solid particles (e.g., spherical particles of polystyrene of definite size) is found from experiments determining the amount of these particles in the suspension to be filtered before and after the filter media. The retentivity K is determined as follows where g, g" =amounts of solid particles in liquid sample before and after the medium, respectively. 

In 1906, Einstein worked out a theory of the viscosity of a liquid which contains, in suspension, spherical particles which are large compared with the size of molecules of the liquid. The predictions of the theory are found to be in good agreement with the measured values of the viscosity of liquids containing colloidal particles in suspension. The presence of these obstacles increases the apparent viscosity of the liquid, and Einstein found1 that the increment is proportional to the total volume v of the foreign particles in unit volume, that is to say, the sum of the volumes of the particles that are present in unit volume of the liquid thus,... 

How does yield stress depend on the size of particles We have mentioned above that increasing the specific surface, i.e. decreasing an average size of particles of one type, causes an increase in yield stress. This fact was observed in many works (for example [14-16]). Clear model experiments the purpose of which was to reveal the role of a particle s size were carried out in work [8], By an example of suspensions of spherical particles in polystyrene melt it was shown that yield stress of equiconcentrated dispersions may change by a hundred of times according to the diameter d of non-... 

This is obvious for the simplest case of nondeformable anisotropic particles. Even if such particles do not change the form, i.e. they are rigid, a new in principle effect in comparison to spherical particles, is their turn upon the flow of dispersion. For suspensions of anisodiametrical particles we can introduce a new characteristic time parameter Dr-1, equal to an inverse value of the coefficient of rotational diffusion and, correspondingly, a dimensionless parameter C = yDr 1. The value of Dr is expressed via the ratio of semiaxes of ellipsoid to the viscosity of a dispersion medium. 

Fig. 13. Pattern of variation of concentration dependence of suspension viscosity when the ratio between the length and diameter of aniso-diametricity of filler s particles increases. The arrow indicates the direction of growth of 1/d of filler s particles. The slop of the initial part of line A (for spherical particles) is 2.5...

In a study of the viscosity of a solution of suspension of spherical particles (colloids), suggested that the specific viscosity rjsp is related to a shape factor Ua-b in the following way ... 

Jeffrey [181] estimated the effective conductivity in a dilute suspension of spherical particles, and in terms of diffusion coefficient his solution was... 

Thus, the Stokes-Einstein equation is expected to be valid for colloidal particles and suspensions of large spherical particles. Experimental evidence supports these assumptions [101], and this equation has occasionally been used for much smaller species. 

The weight of the particles builds up with time and is proportional to 1/d. If we assume spherical particles, then we can convert the above curve to particle diameter from Stokes Law. Although we have added the pcurticle suspension to a "water cushion" as shown above, it might not seem that the settling of the particles would strictly adhere to Stokes Law, which assumes the terminal velocity to be constant. 

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. 

The second step in Ten Cate s two-step approach was to focus on crystal-crystal interaction by means of an explicit two-phase DNS of the turbulent suspension that completely resolves the translational and rotational motions and collisions of the spherical particles plus the turbulence of the liquid between the particles. The particle motions are driven by the turbulent flow and the particles affect the turbulent flow of the liquid in between. When particles approach one another down to a distance smaller than the grid spacing, lubrication theory is exploited to bridge the gap between them. 

A binary suspension consists of equal masses of spherical particles of the same shape and density whose free falling velocities in the liquid are 1 mm/s and 2 mm/s, respectively. The system is initially well mixed and the total volumetric concentration of solids is 0.2. As sedimentation proceeds, a sharp interface forms between the clear liquid and suspension consisting only of small particles, and a second interface separates the suspension of fines from the mixed suspension. Using a suitable model for the behaviour of the system, estimate the falling rates of the two interfaces. It may be assumed that the sedimentation velocity uc in a concentrated suspension of voidage e is related to the free falling velocity u0 of the particles by ... 

---