Concentrically Layered Spheres

For a concentrically layered sphere, the precedent relations simplify to those obtained in the framework of the Lorenz-Mie theory. In this specific case, we use the recurrence relation (2.107) with Qi given by (2.106). All matrices are diagonal and denoting by (T,.,Ar)n and (Ti,i+i. v) the elements of the matrix T we rewrite the recurrence relation as 

The computation of Lorenz-Mie coefEcients for concentrically layered spheres has been considered by Kerker [115], Toon and Ackerman [222], and Fuller [76], while recursive algorithms for multilayered spheres have been developed by Bhandari [13] and Mackowski et al. [154]. 

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This requirement implies that the same number of basis functions must be used to approximate the surface fields on each layer. For concentrically layered spheres, this requirement is not problematic because the basis functions are orthogonal on spherical surfaces. For nonspherical layered particles, we approximate the surface fields by a complete system of vector functions and it is natural to use fewer basis functions for smaller layer surfaces. However, for convergence tests it is simpler to consider a single trimcation index [181,248]. 

In Fig. 3.48, we show results for a concentrically layered sphere consisting of three layers of radii ksVi — 10, ksV2 = 7 and kgV = 4. The relative refractive indices with respect to the ambient medium are mri = 1.2 - - 0.2j, m,-2 = 1.5 + O.lj and mrs = 1.8 - - 0.3j. The scattering curves obtained with the TSPHERE and TLAY routines are close to each other. 

Figure 13-13. The glycogen molecule. A General structure. B Enlargement of structure at a branch point. The molecule is a sphere approximately 21 nm in diameter that can be visualized in electron micrographs. It has a molecular mass of 10 Da and consists of polysaccharide chains each containing about 13 glucose residues. The chains are either branched or unbranched and are arranged in 12 concentric layers (only four are shown in the figure). The branched chains (each has two branches) are found in the inner layers and the unbranched chains in the outer layer. (G, glycogenin, the primer molecule for glycogen synthesis.)...

Templates or SDAs formed by a single molecule or ion can be used in hydro-thermal methods to shape the final products. An assembly of molecules can also act as a template to direct the formation of new materials. The well-known silica MCM-41, with a unidimensional structure of hexagonal pores, is a good example. A surfectant (e.g., NR4 cations) is used as SDA, TEOS or a similar compound as a silicon source, and water as a solvent a catalyst (acid or basic) is also added. At concentrations above the critical micellar concentration, the surfactant molecules are ordered in micelles (layers, spheres, cylinders, etc.), where the molecules are weakly bonded by van der Waals or hydrogen bonds. Supramicel-lar interactions lead to a liquid crystal (LC) structure, on whose walls the inorganic species are formed, oligomerized, and finally polymerized [44]. 

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 ... 

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 ... 

Calculations of the capacitance of the mercury/aqueous electrolyte interface near the point of zero charge were performed103 with all hard-sphere diameters taken as 3 A. The results, for various electrolyte concentrations, agreed well with measured capacitances as shown in Table 3. They are a great improvement over what one gets104 when the metal is represented as ideal, i.e., a perfectly conducting hard wall. The temperature dependence of the compact-layer capacitance was also reproduced by these calculations. 

Encapsulation of small drugs in M S has also been demonstrated by a similar strategy [68]. In that work, the MS spheres were loaded with ibuprofen and then encapsulated within eight layers of PAH and PSS on the particle surface to cap the mesopore openings [68]. The encapsulated drug molecules were subsequently released from the MS particles under the influence of solution pH and salt concentration. 

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]. 

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... 

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