Concentric spheres

Here are the results using the Merz-Kollman-Singh scheme, which fits th electrostatic potential to points selected on a set of concentric spheres around eaci atom ... 

At the same time it will be useful to look at the way in which this energy is distributed in the space around an ion in a vacuum. It follows from (4) that, if we imagine a concentric sphere of any radius R, larger than a, drawn round the ion, and if we integrate from R to infinity, we find that the amount of energy associated with the field outside this sphere is equal to... 

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). 

Kerker, M. and Blatehford, C.G. (1982) Elastic scattering, absorption, and surface-enhanced Raman scattering by concentric spheres comprised of a metaUic and a dielectric region. Physical Remeu> B — Condensed Matter, 26, 4052—4063. 

Once the atomic contributions FA are determined, the corresponding integrals IA are subsequently computed on grids that consist of points on concentric spheres around each atom. Switching to polar coordinates r, 0, and < ) the radial and angular integrations are separated according to... 

In order to obtain the velocity v, we consider two concentric spheres, the first representing the drop surface with radius r, and the second one having a Active radius r2 as the liquid is not compressible, we can choose r2 (although altering) in such a way that the volume difference AV of the spheres is constant, i.e. AV = V2 - Vi = jn(r - rj), so that for a very slight difference r2 - r, = x we find... 

The magnification of the image (M) obtained for the idealised case where the tip of the specimen and the screen consist of two concentric spheres of separation a is given by... 

Equation (20) indicates that the total heat flow (Q) is constant over each concentric sphere within the insulating layer. Application of Eq. (20) to the problem described in Figure 5 gives Boundary conditions ... 

Liposomes are artificial structures composed of phospholipid bilayers exhibiting amphiphilic properties (Chapter 22). In complex liposome morphologies, concentric spheres or sheets of lipid bilayers are usually separated by aqueous regions that are sequestered or compartmentalized... 

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]. 

Shell Extranuclear electrons are arranged in orbits at various distances from the nucleus in a series of concentric spheres called shells. In order of increasing distance from the nucleus, the shells are designated the K, L, M, N, O, P, and Q shells the number of electrons that each shell can contain is limited. 

A scaled-up version of this central template-concentric sphere surface assembly approach has been demonstrated for the growth of multi-layer core-shell nano- and microparticles, based upon the repeated layer-by-layer deposition of linear polymers and silica nanoparticles onto a colloidal particle template (Figure 6.8) [60]. In this case, the regioselective chemistry occurs via electrostatic interactions, as opposed to the covalent bond formation of most of the examples in this chapter. The central colloidal seed particle dictates the final particle... 

Fig. 24. The heat source function for concentric spheres with an outer diameter of 10.08 laa. The core, consisting of coal, has a diameter of 9.24 pm, and the outer layer is water. The incident wavelength is 1,450 nm. (From Sitarski, 1987.)...

The outer and core radii were determined from optical resonance measurements using Mie theory solutions (Aden and Kerker, 1951 Bohren and Huffman, 1983) for concentric spheres to interpret the resonance spectra. Figure 36 presents some of the data of Ray et al. for a pure component glycerol droplet and for a coated droplet having an initial coating thickness given by yo = 0-321. Here y is a reduced thickness defined by y = (a - aj/a. 

Another procedure for determining s would consist in spreading the total volume Vl of the ligand around the cation of radius n, and calculating the thickness s of the layer between two concentric spheres whose difference in volume is equal to the volume Vl of the ligand. The value of s is then given by equation (VIII). 

A melt inclusion and its olivine host are concentric spheres of radius R- and / 2. Consider H2O in the melt inclusion and its re-equilibration with ambient melt outside olivine. Ignore anisotropy of olivine. If / i=0.1mm, / 2 = 1 mm, H2O diffusivity in olivine is D = 10 " m /s, K=0.000, find the reequilibration timescale. 

Figure 5 shows small multilamellar vesicles under the electron microscope, visible as concentric spheres. Note the fascinating texture of the spheres - the actually fluid bilayer appears structured, frozen in time at the moment of preparation. 

Aden, A. L., and M. Kerker, 1951. Scattering of electromagnetic waves from two concentric spheres,/. Appl. Phys., 22, 1242-1246. 

The surfaces being considered are not planar, and therefore instead of Helmholtz-Perrin parallel-plate condensers, one has concentric-sphere capacitors Gouy-Chapman regions show radial instead of planar symmetry. All such points complicate the mathematics, but lead to few new truths. Hence, such details will be ignored in this very simple account of the dominating role of double layers in colloid chemistry. 

Exercise. A particle obeys the ordinary diffusion equation in the space between two concentric spheres. Find the splitting probability and the conditional mean first-passage times. 

The droplet and flame surfaces are represented by concentric spheres. 

Liposomes are artificial structures composed of phospholipid bilayers exhibiting amphiphilic properties (chapter 12). In complex liposome morphologies, concentric spheres or sheets of lipid bilayers are usually separated by aqueous regions that are sequestered or compartmentalized from the surrounding solution. The phospholipid constituents of liposomes consist of hydrophobic lipid tails connected to a head constructed of various glycerylphosphate derivatives. The hydrophobic interaction between the fatty acid tails is the primary driving force for creating liposomal bilayers in aqueous solutions. 

Olaofe and Levine (4) investigated the scattering for concentric spheres and spherically symmetric inhomogeneous spheres with Cauchy and parabolic distributions of the radial profile of refractive index. Comparisons were with the volume-weighted refractive index... 

They found, that for a concentric sphere with a variable coating, the scattering was not sensitive to the form of the variation in the shell whenever there was a constant amount of refractive material (m = 1.0738). For inhomogeneous spheres with either a Cauchy or parabolic distribution, the forward scattering... 

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