Concentric spheres model

The sizes were determined by small angle neutron scattering method (SANS), (Markovic et al., 1984). In order to interpret the SANS results, a concentric sphere model was used. A spherical homogeneous core particle of radius R and the total spherical units were taken as R2, giving the thickness of the adsorbed layer as (R2 - R,)... 

The model considered is of mass transfer for a liquid in motion between two concentric spheres of radii b and a, the latter being the radius of the bubble ... 

Figures 1 a and 1 b represent the two-phase and the three-phase models respectively in the representative volume element of the composite. In the modified model three concentric spheres were considered with each phase maintaining a constant volume 4). The novel element in this model is the introduction of the third intermediate hybrid phase, lying between the two principal phases.

Thus, in the three-layer model, with the intermediate layer having variable physical properties (and perhaps also chemical), subscripts f, i, m and c denote quantities corresponding to the filler, mesophase, matrix and composite respectively. It is easy to establish for the representative volume element (RVE) of a particulate composite, consisting of a cluster of three concentric spheres, that the following relations hold ... 

A better approach for the Rosen-Hashin models is to adopt models, whose representative volume element consists of three phases, which are either concentric spheres for the particulates, or co-axial cylinders for the fiber-composites, with each phase maintaining its constant volume fraction 4). 

Any fundamental study of the rheology of concentrated suspensions necessitates the use of simple systems of well-defined geometry and where the surface characteristics of the particles are well established. For that purpose well-characterized polymer particles of narrow size distribution are used in aqueous or non-aqueous systems. For interpretation of the rheological results, the inter-particle pair-potential must be well-defined and theories must be available for its calculation. The simplest system to consider is that where the pair potential may be represented by a hard sphere model. This, for example, is the case for polystyrene latex dispersions in organic solvents such as benzyl alcohol or cresol, whereby electrostatic interactions are well screened (1). Concentrated dispersions in non-polar media in which the particles are stabilized by a "built-in" stabilizer layer, may also be used, since the pair-potential can be represented by a hard-sphere interaction, where the hard sphere radius is given by the particles radius plus the adsorbed layer thickness. Systems of this type have been recently studied by Croucher and coworkers. (10,11) and Strivens (12). 

Solution-phase DPV of Au144-C6S dispersed in 10 mM [bis(triphenylpho-sphoranylidene)-ammoniumtetrakis-(pentafluorophenyl)-borate (BTPPATPFB)/ toluene] [acetonitrile] 2 1 revealed well-behaved, equally spaced and symmetric quantized double-layer charging peaks with AE - 0.270 0.010 V. Applying the classical concentric spheres capacitor model (8) reveals an individual cluster capacitance of 0.6 aF [334, 335]. 

Figure 2.3 Left, reduction models. In the shrinking core or contracting sphere model the rate of reduction is initially fast and decreases progressively due to diffusion limitations. The nucleation model applies when the initial reaction of the oxide with molecular hydrogen is difficult. Once metal nuclei are available for the dissociation of hydrogen, reduction proceeds at a higher rate until the system comes into the shrinking core regime. Right the reduction rate depends on the concentration of unreduced sample (1-a) as f(a) see Expressions (2-5) and (2-6).

In Chapter 3 (Section 3.5.2) the viscosity of a hard sphere model system was developed as a function of concentration. It was developed using an exact hydrodynamic solution developed by Einstein for the viscosity of dilute colloidal hard spheres dispersed in a solvent with a viscosity rj0. By using a mean field argument it is possible to show that the viscosity of a dispersion of hard spheres is given by... 

For the Coulomb attraction of reactants A and B (eA = — eB) present in equal concentrations n(t) and the black sphere model the kinetic equations read (d = 3)... 

Direct establishment of the asymptotic reaction law (2.1.78) requires performance of computer simulations up to certain reaction depths r, equation (5.1.60). In general, it depends on the initial concentrations of reactants. Since both computer simulations and real experiments are limited in time, it is important to clarify which values of the intermediate asymptotic exponents a(t), equation (4.1.68), could indeed be observed for, say, r 3. The relevant results for the black sphere model (3.2.16) obtained in [25, 26] are plotted in Figs 6.21 to 6.23. The illustrative results for the linear approximation are also presented there. 

As it is noted in Section 5.1, a distinctive feature of the linear approximation is the absence of back-coupling between the concentration n(t) and the correlation function Y (r, t) which is also independent of the initial reactant concentrations. Moreover, in the linear approximation the parameter k = D /D does not play any role at all. So, in the black-sphere model for the standard random distribution, Y(r > ro,t) = 1, one gets universal relations (4.1.69), (4.1.65) and (4.1.61) for d = 1,2 and 3 respectively. In contrast, in the superposition approximation the law K — K(t) loses its universality, since along with space dimension d it depends also on both the parameter k and the initial reactant concentrations. 

Trapping of particles A by B is described as earlier in terms of the black sphere model (3.2.16). A model of particle reproduction by division (8.2.6) along with a simplification of integral terms has also the following advantage. Creation of particles, as it was shown in Chapter 7 leads usually to the problem of the proper account of free volume available for particles A the superposition approximation is valid here only for small dimensionless particle concentrations. In our treatment of the reproduction this problem does not arise since prey animals A appear near other A s which are outside the... 

Fig. 39. Measurement of the apparent diffusion coefficient Dapp = 17q2 for two concentrations of a PVAc microgel in methanol 88 189. The full lines are theoretical curves for a soft sphere model with 7 branching shells93 ...

To our knowledge there have been no reported measurements of equilibrium defect concentrations in soft-sphere models. Similarly, relatively few measurements have been reported of defect free energies in models for real systems. Those that exist rely on integration methods to connect the defective solid to the perfect solid. In ab initio studies the computational cost of this procedure can be high, although results have recently started to appear, most notably for vacancies and interstitial defects in silicon. For a review see Ref. 109. 

This type of static quenching requires relatively high quencher concentrations and it follows the Perrin action sphere model [64]. According to this model, each emitter molecule is surrounded by an active volume (in the general case it needs not be a sphere), such that if there is one quencher molecule at least within this volume, then quenching takes place instantly but molecules which have no quencher within the active volume emit just like those in a sample devoid of quencher. The Perrin model leads to two observable results ... 

Single Sphere Model I The problem of diffusion of matter from a sphere initially at a uniform concentration when the surface concentration is maintained constant has been solved by Crank (7) and his equation (6.20) is (on substituting De for D) the same as the expression for the mass extracted as a function of time given by single sphere model I. 

It is interesting that Eq. (79) obtained in describing the dependence of viscosity of liquids whose molecules are modelled by solid spheres or their assemblies is very similar in its form to Eq. (69) and to equations describing the dependence of viscosity of disperse systems on concentration also modelled by assemblies of solid spheres. 

Therefore, the hard sphere model cannot be applied to a very concentrated dispersion without introducing a perturbation due to the van der Waals attraction. 

Monodiefperse Spheres The rheology of concentrated ceramic suspensions is very important for good mold filling. For concentrated suspensions that are colloidally stable (by steric means, giving a hard sphere model), there is a particle volume fraction (i.e., = 0.63 for... 

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