Equivalent spheres

The particle can be assumed to be spherical, in which case M/N can be replaced by (4/3)ttR P2, and f by 671770R- In this case the radius can be evaluated from the sedimentation coefficient s = 2R (p2 - p)/9t7o. Then, working in reverse, we can evaluate M and f from R. These quantities are called, respectively, the mass, friction factor, and radius of an equivalent sphere, a hypothetical spherical particle which settles at the same rate as the actual molecule. 

Because of the diversity of filler particle shapes, it is difficult to clearly express particle size values in terms of a particle dimension such as length or diameter. Therefore, the particle size of fillers is usually expressed as a theoretical dimension, the equivalent spherical diameter (esd), ie, the diameter of a sphere having the same volume as the particle. An estimate of regularity may be made by comparing the surface area of the equivalent sphere to the actual measured surface area of the particle. The greater the deviation, the more irregular the particle. 

The external surface area of the filler can be estimated from a psd by summing the area of all of the equivalent spheres. This method does not take into account the morphology of the surface. It usually yields low results which provide Htde information on the actual area of the filler that induences physical and chemical processes in compounded systems. In practice, surface area is usually determined (5) from the measured quantity of nitrogen gas that adsorbs in a monolayer at the particle surface according to the BET theory. From this monolayer capacity value the specific surface area can be determined (6), which is an area per unit mass, usually expressed in m /g. 

Beeause of the uniqueness and simplieity of the sphere, the eharaeteristies of non-spherieal partieles (whieh most real ones are) are often related baek in some way to the size of an equivalent sphere whieh has some shared ehar-aeteristie, sueh as the same volume or surfaee area. 

Thus the Stokes diameter of any partiele is that of an equivalent sphere having same terminal settling veloeity and is a useful additional partiele eharaeteristie for partieulate systems involving fluid motion. 

NPei and NRtt are based on the equivalent sphere diameters and on the nominal velocities ug and which in turn are based on the holdup of gas and liquid. The Schmidt number is included in the correlation partly because the range of variables covers part of the laminar-flow region (NRei < 1) and the transition region (1 < NRtl < 100) where molecular diffusion may contribute to axial mixing, and partly because the kinematic viscosity (changes of which were found to have no effect on axial mixing) is thereby eliminated from the correlation. 

P 0 is the apparent limiting anisotropy and P is the anisotropy (17). The slope is read as the straight line portion of the curves in Figures la-f and applied in equation 10 to obtain the equivalent-sphere molar volume. The difference between the extrapolated intercept of the linear portion of the line on they ordinate and the extrapolated intercept of the curved line is attributed to the internal rotation of the fluorophores in the molecule (5). 

To = temperature of the solvent at which tan A goes through a maximum. These values are presented in Table II. VSE (the Stokes-Eeinstein volume) is calculated for a spherical molecule if the molecule is aspherical this calculation (VSE) is called Vapparent The Vapparent can be smaller or larger than the Stokes-Einstein volume and varies from the equivalent sphere volume obtained by solution of equations 3,4 and 5. 

By relating the endpoint of crushed DBF absorption to the void space within and between equivalent spheres of aggregates, and assuming the spheres to be packed at random, Wang et al. obtained the following equation for the effective volume fraction of carbon black ... 

According to Eq. (18), the high polymer molecule should exhibit the frictional coefficient of an equivalent sphere (compare Eq. 15) having a radius proportional to the root-mean-square end-to-end distance (r ) (or to (s ) / ). Similarly, according to Eq. (23) its contribution to the viscosity should be that of an equivalent sphere (compare Eq. 16) having a volume proportional to (r ) / In analogy with Eq. (17 ), we might write... 

Another method of describing particle size is in terms of equivalent diameter or the equivalent sphere dpe, which is the diameter of a sphere possessing the same ratio of surface to volume as the actual particle. Thus, from the equation, Vp/Sp = dp/6/ the equivalent diameter (dp e) is... 

The size of a spherical particle is readily expressed in terms of its diameter. With asymmetrical particles, an equivalent spherical diameter is used to relate the size of the particle to the diameter of a perfect sphere having the same surface area (surface diameter, ds), the same volume (volume diameter, dv), or the same observed area in its most stable plane (projected diameter, dp) [46], The size may also be expressed using the Stokes diameter, dst, which describes an equivalent sphere undergoing sedimentation at the same rate as the sample particle. Obviously, the type of diameter reflects the method and equipment employed in determining the particle size. Since any collection of particles is usually polydisperse (as opposed to a monodisperse sample in which particles are fairly uniform in size), it is necessary to know not only the mean size of the particles, but also the particle size distribution. 

To determine the settling characteristics of a sediment, you drop a sample of the material into a column of water. You measure the time it takes for the solids to fall a distance of 2 ft and find that it ranges from 1 to 20 s. If the solid SG = 2.5, what is the range of particle sizes in the sediment, in terms of the diameters of equivalent spheres ... 

The simplest shape of a particle is the sphere in that, because of its symmetry, any question of orientation does not have to be considered, since the particle looks exactly the same from whatever direction it is viewed and behaves in the same manner in a fluid, irrespective of its orientation. No other particle has this characteristic. Frequently, the size of a particle of irregular shape is defined in terms of the size of an equivalent sphere although the particle is represented by a sphere of different size according to the property selected. Some of the important sizes of equivalent spheres are ... 

The next step consists of the determination of the size of the macromolecules in space. Two equivalent sphere radii can be measured directly by means of static and dynamic LS. Another one can be determined from a combination of the molar mass and the second virial coefficient A2. Similarly, an equivalent sphere radius is obtained from a combination of the molar mass with the intrinsic viscosity. This is outlined in the following sections. 

Fig. 8. Representation of the interaction functions O and R in terms of equivalent sphere radii and Rj respectively. Both interaction functions depend on the segment density but small solvent molecules can easier penetrate into a coil (left) than two of such coils penetrate into each other (right)...

The results of this consideration may be summarized as follows. The study of global properties of macromolecules in dilute solutions by means of static and dynamic LS and by viscometry allows the determination of the molar mass and four differently defined equivalent sphere radii, R, and (see... 

Table 2). All the radii have a certain molar mass dependence. The magnitudes of these radii, however, can deviate strongly from each other. These differences result from the fact that they are physically differently defined. The radius of gyration, R, is solely geometrically defined the thermodynamically equivalent sphere radius, R-p is defined by the domains of interaction between two macromolecules, or in other words, on the excluded volume. The two hydrodynamic radii R and R result from the interaction of the macromolecule with the solvent (where the latter differs from R by the fact that in viscometry the particle is exposed to a shear gradient field).

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