Particle rigid, spherical

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. 

The drag coefficient for rigid spherical particles is a function of particle Reynolds number, Re = d pii/ where [L = fluid viscosity, as shown in Fig. 6-57. At low Reynolds number, Stokes Law gives 24... 

Wall Effects When the diameter of a setthng particle is significant compared to the diameter of the container, the settling velocity is reduced. For rigid spherical particles settling with Re < 1, the correction given in Table 6-9 may be used. The factor k is multiplied by the settling velocity obtained from Stokes law to obtain the corrected set-... 

The methacrylic backbone structure makes the spherical Toyopearl particles rigid, which in turn allows linear pressure flow curves up to nearly 120 psi (<10 bar), as seen in Fig. 4.45. Toyopearl HW resins are highly resistant to chemical and microbial attack and are stable over a wide pH range (pH 2-12 for operation, and from pH 1 to 13 for routine cleaning and sanitization). Toyopearl HW resins are compatible with solvents such as methanol, ethanol, acetone, isopropanol, -propanol, and chloroform. Toyopearl HW media have been used with harsh denaturants such as guanidine chloride, sodium dodecyl sulfate, and urea with no loss of efficiency or resolution (40). Studies in which Toyopearl HW media were exposed to 50% trifluoroacetic acid at 40°C for 4 weeks revealed no change in the retention of various proteins. Similarly, the repeated exposure of Toyopearl HW-55S to 0.1 N NaOH did not change retention times or efficiencies for marker compounds (41). 

In Eq. (39), Cdo is a function of the particle Reynolds number, Rep — pedp V — Vp /p. For rigid spherical particles, the drag coefficient CD0 can be estimated by the following equations (Rowe and Henwood, 1961) ... 

Wall Correction Factor K for a Rigid Spherical Particle Moving on the Axis of a Cylindrical Tube in Creeping Flow... 

Fig. 10.15 Drag coefficient for rigid spherical particles in air as a function of Mach number with Reynolds number as parameter, for the case where the absolute temperatures of the particle and fluid are essentially the same.

The instantaneous drag on a rigid spherical particle moving with velocity Lp in a fluid whose instantaneous velocity in the vicinity of the particle is follows from an extension to Eq. (11-30) ... 

It is no wonder that the particles are spherical but crystalline, if one considers the formation mechanism. The rather smooth surface of the spherical magnetite may be due to the rapid contact recrystallization of the constituent primary particles (5), forming the rigid polycrystalline structure. Flowever, it must be noted that polycrystalline spheres are also prepared by normal deposition of monomeric solute, as shown in the formation of the uniform spherical polycrystalline particles of metal sulfides in Chapters 3.1-3.3. Thus, while we may be able to predict the final particle shape and structure from the formation mechanism, it is risky to conclude the formation mechanism only from characterization of the product. As a rule, scrupulous analyses are needed for concluding the growth mechanism in a particle system. 

The next topic is Einstein s theory of viscosity of dispersions of rigid, spherical particles. This theory is the starting point for most of the current approaches to flow properties of colloids and plays a practical, pedagogical, as well as historical role. 

The inner phase volume fraction determines many properties of an emulsion. One example is the viscosity r/em. For small volume fractions one can often regard the disperse phase as consisting of rigid, spherical particles instead of liquid, flexible drops. Then we can apply Einstein s3 equation [541], with rj being the viscosity of the pure dispersing agent ... 

In this section, collisions among rigid spherical particles are studied. Two simple cases, collinear and planar collisions, are described. For the general theory of stereomechanical impact of irregular-shaped rigid bodies in arbitrary motion, readers may refer to Goldsmith (1960). 

Verify that, for a collinear collision of two frictionless, nonspinning, rigid spherical particles, the total kinetic energy loss can be expressed by Eq. (2.6). 

For Stokes solution, it was necessary to assume a continuous, incompressible, viscous, and infinite medium with rigid particles and spherical particles. With these assumptions, Stokes found that the resisting force exerted by air on a moving particle, equivalent to the force exerted by moving air on a stationary particle, is... 

Base material provides mechanically stable rigid porous particles (mostly spherical) for reversed-phase HPLC adsorbents. Particle porosity on the mesoporous level (30 to 500-A diameter) is necessary to provide high specific surface area for the analyte retention. Surface of the base material should have specific chemical reactivity for further modification with selected ligands to form the reversed-phase bonded layer. Base material determines the mechanical and chemical stability—the most important parameters of future (modified) reversed-phase adsorbent. 

The value of the constants, a, b, c, etc., can either be determined experimentally or theoretically (107). For a colloidal dispersion of rigid spherical particles the value of a is 2.5. Therefore, Equation 13 equals the Einstein equation (Eq. 12) at low-volume fractions. A rigorous theoretical treatment of the interactions between pairs of droplets has established that b = 6.2 for rigid spherical particles. Experiments have shown that Equation 13 can be used up to particle concentrations of about 10% with a = 2.5 and b = 6.2 for colloidal dispersions in the absence of long-range colloidal interactions (107). It is difficult to theoretically determine the value of higher order terms in Equation 13 because of the mathematical complexities involved in describing interactions between three or more particles. 

The cross-linking process used for the poly(divinylbenzyl)styrene phase produces rigid, spherical particles with a well-controlled pore size. This makes them ideal for use in size exclusion chromatography and indeed they are amongst the most important materials in use for this separation technique. 

Einstein showed from hydrodynamic theory that for a dilute system of rigid, spherical particles... 

---