The Equivalent Sphere Model

The Equivalent Sphere Model.—The free-draining approximation leads to the prediction that the frictional coefficient /o should increase with the first power of the molecular weight and that the intrinsic viscosity should increase with M raised to a power a little greater than unity. Experiments show that both quantities increase with a power of M less by about 0.4 to 0.5 than predicted. Closer inspection of the problem casts serious doubt on the free-draining approximation. Even with a frictional coefficient f less by a factor of ten than the value to be expected for a spherical bead comparable in size to a chain unit, the motion of the solvent must be markedly altered deep within a random-coiled molecule of high molecular weight. 

Solvent toward the center of an actual polymer molecule will acquire a velocity more nearly that of the molecule than the velocity which 

This remarkably simple treatment suffers one serious deficiency the value of remains quantitatively undefined. More or less intuitively it has been suggested by various investigators that should increase as the root-mean-square end-to-end distance /for a linear chain, or, more generally, as the root-mean-square distance /s2 q beads from the center of any polymer molecule, linear or branched. Accepting this postulate unquestioningly, we should then have/o proportional to and [rj] proportional to These conclusions happen to 

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In the equivalent sphere model, the volume = 4pii 3/3. For spheres, for random coils, Rg = M1/2 for rods, Rg M (Ross-Murphy, 1994). Specifically from the Zimm plot (Cowie, 1991),... 

At the 0-temperature, the excluded volume effect is avoided (a = 1.0, see Sect. 1.3.). Making some solvent to rotate with the molecule leads to the equivalent sphere model. In this case one assumes that a certain inner solid sphere of solvent is... 

The most frequently calculated property is the mean square unperturbed end-to-end distance, (r )o. Other properties susceptible to rapid computation include the average of the end-to-end vector, (r)o, and the mean square unperturbed radius of gyration, 5 )0. The viscosity of a dilute solution in a solvent can be estimated from 5 )0 or (r )o via the equivalent sphere model for hydrodynamic properties. Several higher even moments, (r )o and (s )o, p = 2,3,..., provide information about the shape (width, skewness) of the distribution functions for and When combined with information about the electronic charge distribution within individual rigid units, the RIS model can calculate the mean square dipole moment, (m )o-Also accessible are optical properties that depend on the anisotropy of the... 

The parameters R and Rj in equation (5.32) are the radii of the equivalent hard spheres representing biopolymers i and y, respectively (where i = j for interactions between the same macroions). The equivalent hard sphere corresponds to the space occupied in the aqueous medium by a single biopolymer molecule (or particle) which is completely inaccessible to other biopolymers. In practice, the hard sphere model is a highly satisfactory description for many globular proteins. 

Stokes s law and the equations developed from it apply to spherical particles only, but the dispersed units in systems of actual interest often fail to meet this shape requirement. Equation (12) is sometimes used in these cases anyway. The lack of compliance of the system to the model is acknowledged by labeling the mass, calculated by Equation (12), as the mass of an equivalent sphere. As the name implies, this is a fictitious particle with the same density as the unsolvated particle that settles with the same velocity as the experimental system. If the actual settling particle is an unsolvated polyhedron, the equivalent sphere may be a fairly good model for it, and the mass of the equivalent sphere may be a reasonable approximation to the actual mass of the particle. The approximation clearly becomes poorer if the particle is asymmetrical, solvated, or both. Characterization of dispersed particles by their mass as equivalent spheres at least has the advantage of requiring only one experimental observation, the sedimentation rate, of the system. We see in sections below that the equivalent sphere calculations still play a useful role, even in systems for which supplementary diffusion studies have also been conducted. 

In this example, about one-third of the solids are coarse enough to have settled out in 5 minutes. The remaining particles have velocities of 6.7 10 4 m s-1 (= 20 10 2 m/300 s) or less, corresponding to equivalent spheres with radii of 12 /xm or less. Incidentally, the equivalent sphere is a poor model for these clay particles, as can be seen from the electron micrographs in Figure 1.12b. ... 

We have discussed the relative sizes of P, O, and T sites (Section 3.3), but these are based on the hard sphere model. For the bcc structure all sites for atoms are equivalent. We can use any atom for the origin of the cube or the center of the cube. We have used the 3 2PTOT notation for the bcc structure because it describes the relative spacings of all sites. Figure 3.8 is the model for a ccp structure, with all P, O, and T sites shown by different balls for each type of layer. We can start a cube with any type of ball. The model is bcc if all balls are the same. 

Low Temperature and Higher Water Content. A few neutron scattering spectra are shown in figure 4. Experimental (absolute) intensities are represented by open circles, and concern the sample A3 at 20°C. Four deuteration rates (0 %, 40 %, 50 % and 80 % in w/w) of the decane component have been used, which give rise to rather different spectra, both in level of intensity and in shape. The first maximum is due to interparticle effects which can be roughly taken into account by the hard equivalent sphere model (7,8). Here the scattering curves become practically independent of the interparticle interactions for q>qm -0.03 Two theoretical results are shown while... 

Fig. 12. Values of the reduced diffusion coefficient for the soft-sphere model as a function of the reduced volume from molecular dynamics simulations circles, from Cape and Woodcock squares, from Hiwatari et al. triangles, from Ross and Schofield. The scale at the right shows the equivalent diffusivities for argon-like soft-spheres.

Note that the opposite choices are usually made in oceanographic research, during analysis of oceanic albedo. In this case, the backscattered photons have significant effects therefore, a description of the phytoplankton heterogeneity is required. In order to numerically solve Maxwell s equations for the heterogeneous particles, such models usually simplify the description of the shapes by means of the equivalent sphere approximation (see Bernard et al., 2009 for an example of core-shell model). 

As previously mentioned, nonspherical cells have been modeled as spheres with equivalent radius and effective complex index of refraction. This approximation can be justified by the fact that they are typically well mixed and randomly oriented in the PBRs. Then, the equivalent radius r q can be approximated such that either the volume or the surface area of the equivalent sphere is identical to that of the actual cell. The radius of the volume-equivalent sphere can be expressed as... 

In his theory of polymer dilute solutions, Flory (1945, 1949c, 1953) (Orofino and Flory, 1957 Flory and Fisk, 1966) (see also Morawetz, 1965 Tanford, 1961 Tsvetkov el al., 1964 Bresler and Yerusalimski, 1965 Yamakawa, 1971 Casassa, 1976, 1977 Rafikov et al., 1978) modelled a macromolecule with a cloud of disconnected segments distributed about the centre of mass by the Gaussian law (Equation 104). The equivalent sphere is subdivided into elementary spherictd layers for which the Gibbs potential (Equation 31) Af7,n.j is realized. 

A schematic diagram showing the details of the proposed model for the equivalent sphere representing the microbial floe is presented in Figure 6.34. A number of simplifying assumptions are implicit in the lengthy manipulation of the model equations. 

The cylinder/cylinder geometry (for nozzles in cylindrical vessels) is much more difficult to analyze than the nozzle sphere, which can be treated as an axisymmetric structure. To obtain a suitable stress concentration factor for a nozzle in a cylindrical vessel, an approximate axisymmetric model is sometimes used. A popular approximation used is where the equivalent sphere has twice the diameter of the shell. The general trend of the experimental results is shown in Figure 7.8, which is for a flush nozzle in a cylindrical shell. 

Some of the defects of the hard-sphere model for the reaction cross section are apparent. It assumes that all the line-of-centers kinetic energy can be used for overcoming the threshold. It does not account for the dependence of Q( , /, j) on the internal states of the reactants. It does not consider any effect due to molecular structure or to the details of the collision process. Improvement requires either direct experimental measurement of g( , Uj) or equivalently, a calculation which takes into account the interaction potential between colliding molecules in specific internal states. 

We saw following (5.60) that it was the repulsion between unlike molecules in the model that was responsible for the form of (5.60), so it is that which is now also responsible for the form of (5.65). In the two-component (primitive) version of the penetrable-sphere model, which we treat in 5.S-5.7, we see the equivalent of (5.65) again in (5.133), where it has a similar physical origin. The negative mass m in that case is (in dimensionless form) -l/2(s+2), with s the dimensionality of the potential see (5.134) below. 

At the present time, the most successful correlations of dense-fluid transport properties are based upon consideration of the hard-sphere model. One reason for this is that, as discussed in Chapter 5, it is possible to calculate values from theory for this model at densities from the dilute-gas state up to solidification. Second, this is physically a reasonably realistic molecular model because the van der Waals model, which has been successfully applied to equilibrium properties of dense fluids, becomes equivalent to the hard-sphere model for transport properties. 

As is seen from Table 8, there is no correlation between Porod radii and the cross-Unk density. Over most of the composition range studied, the equivalent sphere radii derived from the Hosemaim analysis were significantly larger than the Porod radii. For high PDMS contents, the average domain sizes derived from Hosemann analysis are substantially lower than the Porod radii. The discrepancy between the two methods was ascribed to the invaUd-ity of the models for certain composition ranges. 

Equation 5.32 is quite different from the equation published by Ouyang for at least two reasons first we have used the Medalia s aggregate equivalent sphere diameter in exploiting the two spheres model of Gent et al., second misprints may be suspected in Ouyang s publications because several of his equations are suffering from unit inconsistencies. 

Tn general, the. solvent-accessible surface (SAS) represents a specific class of surfaces, including the Connolly surface. Specifically, the SAS stands for a quite discrete model of a surface, which is based on the work of Lee and Richards [182. They were interested in the interactions between protein and solvent molecules that determine the hydrophobicity and the folding of the proteins. In order to obtain the surface of the molecule, which the solvent can access, a probe sphere rolls over the van der Waals surface (equivalent to the Connolly surface). The trace of the center of the probe sphere determines the solvent-accessible surjace, often called the accessible swface or the Lee and Richards surface (Figure 2-120). Simultaneously, the trajectory generated between the probe and the van der Waals surface is defined as the molecular or Connolly surface. 

For a liver alcohol dehydrogenase (LADH) model an NS2O coordination sphere is required. The chelating aldehydes are ideal for the formation of this donor set when combined with bis(pentafluoro-thiophenolato)zinc. Structural data on the complexes with one equivalent of 6-methylpyridine-2-carbaldehyde, 6-methoxypyridine-2-carbaldehyde, 2-(dimethylamino)benzal-dehyde) demonstrate that the coordination sphere for LADH has been reproduced to a close approximation and the corresponding alcohol complexes have also been characterized.354 Other thiophenols have been used to form such complexes but have not been structurally characterized.304... 

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