Equivalent sphere models

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),... 

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... 

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... 

For central forces, equivalent sphere models are often robust enough for assessing interparticle effects. Steric... 

Equivalent sphere model for a single aggregate [Source A.I. Medalia, /. Colloid Inter/ Sci, 32,115-131,1970]... 

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. ... 

Figure 4. Model of encounter of two polymer coils represented by equivalent spheres arrow indicates direction of diffusion of coil I relative to that of cod II after both cods have met.

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. 

The seas may also act as a receptor for depositing aerosol. Deposition velocities of particles to the sea are a function of particle size, density, and shape, as well as the state of the sea. Experimental determination of aerosol deposition velocities to the sea is almost impossible and has to rely upon data derived from wind tunnel studies and theoretical models. The results from two such models appear in Figure 4, in which particle size is expressed as aerodynamic diameter, or the diameter of an aero-dynamically equivalent sphere of unit specific gravity.If the airborne concentration in size fraction of diameter d is c then... 

IB. Hard Sphere Model. Here the molecule is assumed to be the equivalent of a billiard ball. That is, the molecule is presented as a rigid sphere of diameter or, mass m (the molecular weight), and the capability... 

An improvement of a classical repulsive expression (2) for one dimensional system of hard sphere by the very accurate presentation of the Liu s EoS (Figure 7) doesn t change a topologic picture of phase diagram in comparison with classical van der Waals expression for repulsive term. It seems that an improvement of repulsive term makes more plausible of isotherm behavior near second critical point. To analyze a qualitative behavior of thermod5mamic surface anomalies in whole via simpler model is preferable due to a topological equivalence of models under consideration. 

Other 3D Enclosures With Interior Solids. Warrington and Powe [278] showed that so far as the heat transfer is concerned, cubes and stubby cylinders behave similarly to equivalent spheres of the same volume. This appears to be the case for both the inner and outer body shape. So Eqs, 4.121,4.124, and 4.128 appear to be applicable to other inner and outer body shapes as well, it being understood that D0 = (6V0/7t)l/3 and D, = (6 Vz/Jt)1 3, where Va and U, are the inner and outer body volumes, respectively. Sparrow and Charmichi [258], using stubby cylinders for the inner and outer body shapes, confirmed the conduction layer model prediction that the heat transfer is independent of eccentricity E when Ra (based on inner cylinder diameter) is greater than about 1500. 

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.

We set the radius of the constituent particles equal to a = 0.1 pm, the same radius as was inferred in [66] from the arguments for cometary dust temperature and has long been used for modeling cometary dust [67]. We refer the reader to [68] for a discussion of the CP s size as well as for details of the computational techniques. The number N of the CPs is A = 64, 128, or 256 the larger numbers of N fall outside of the limitation of our computational resources for the selected refractive index, radius, and configuration of CPs. As a result, the aggregate with a = 0.1 pm has a radius of a volume-equivalent sphere Oy = 0.400, 0.504, or 0.635 pm. 

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). 

First, the shape and size distributions are determined by optical microscopy and image analysis. Simple rotatory-symmetric parametric shapes are identified and selected in Dauchet et al. (2015) and Charon et al. (2015), and the size distributions are modeled as log-normal ones for the radius of the volume-equivalent sphere. This is not a restriction analysis of more... 

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... 

Because it is likely that aggregates have significant internal flow through their structure, aggregate permeability must be considered. Fractal aggregates are expected to behave like objects that are smaller than equivalent spheres with reduced drag effects. Indeed, simulations of hydrodynamic friction using the Stokes model overestimate the friction of fractal objects. 

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