Concentric spherical shell particles

A very similar approach used to study the properties of concentric spherical shell particles with size distributions is discusses in Sect. 4.3.2 below. 

Fig. 13.8 Sketches of distributions of crazes among concentric-spherical-shell particles resulting in plastic strain (a) at initiation of crazing and (b) in well-developed craze flow (after Piorkowska et al. (1990) courtesy of the American Chemical Society (ACS)).

The filled polymer is considered as a collection of repesentative volume elements (RVE) of many spherical or cylindrical composites of various sizes. Each of these contains a filler particle and two concentric spherical shells, a thin one corresponding to the mesophase, and another thicker, representing the matrix respectively. The volume fraction of the filler in each composite is the same, as the total volume fraction of the filler in the filled polymer. 

Theoretically, SPR absorption can be estimated by solving Maxwell s equations. Gustav Mie rationalized this for spherical particles in 1908. Nowadays these equations can be solved to predict the corresponding SPR bands for spheres, concentric spherical shells, spheroids and infinite cylinders, and an approximation is required for other geometries. The routine measurement of the SPR absorption of most reported processes of synthesis of Au NPs is, indeed, one of the key points for the characterization of new nanomaterials [183]. 

For most particulate composites the mismatch between the particles and the matrix is more important than the anisotropy of either component (though alumina/aluminium titanate composites provide a notable exception and are described below). The main features of the stresses can therefore be understood in terms of a simple elastic model assuming thermoelastic isotropy and consisting of a spherical particle in a concentric spherical shell of matrix with dimensions chosen to give the appropriate volume fractions. The particles are predicted to be under a uniform hydrostatic stress, ap after cooling. If the particles have a larger thermal expansion coefficient than the matrix, this stress is tensile, and vice versa. For small particle volume fractions the stress... 

The model was developed to show that if the proper boundary conditions are used,one should not expect at low Reynolds numbers that the Nusselt and Sherwood numbers approach the limiting value of two, which is valid for a sphere in an infinite static medium. Since the particles are members of an assemblage, they assume in their model that there is a concentric spherical shell of radius... 

Consider a concentrated suspension of charged spherical soft particles moving with a velocity 17 in a liquid containing a general electrolyte in an applied electric field E. We assume that the particle core of radius a is coated with an ion-penetrable layer of polyelectrolytes with a thickness d. The polyelectrolyte-coated particle has thus an inner radius a and an outer radius b = a + d. We employ a cell model [4] in which each particle is surrounded by a concentric spherical shell of an electrolyte solution, having an outer radius c such that the particle/cell volume ratio in the unit cell is equal to the particle volume fraction 4> throughout the entire dispersion (Fig. 22.1), namely. 

Effect of Particle Size in Concentric Spherical Shell Materials. . 335... 

Fig. 19a-d. Morphologies of four compliant particles in PS matrix a) KRO-1 resin particle b) HIPS particle c) concentric spherical shell (CSS) particle d) pools of low molecular weight polybutadiene as pure rubber particles (from Ref. [34] courtesy of Pergamon Press)... 

Fig. 13.3 Stress-strain curves of homo-PS and three toughened blends containing particles having KRO-1 diblock morphology, HIPS particles, and particles with concentric-spherical-shell (CSS) morphology having the highest elastic compliance (from Argon et al. (1987) courtesy of Pergamon Press).

The tensile toughness Wp exhibited by homo-PS and the three PS blends containing particles of KRO-1 resin, HIPS, and concentric spherical shells (CSSs) is presented in Fig. 13.5. It shows the very substantial improvement in toughness achievable by lowering the eraze-flow stress <7oo by incorporation of compliant particles, but demonstrates also that the peak toughness achievable eventually plateaus when the craze-flow stress becomes too low as in the case of blends with the CSS particles. 

Intense in situ electron irradiation of particles of carbon soot for tenths of minutes in the electron microscope produces spherical particles consisting of onion-like concentric spherical shells of graphene-like layers [208]. The microscope allows the concentric shell structure to be imaged during its formation. 

Mampel extended the treatment to include due allowance for three-dimensional growth of product into the particles by considering the latter to consist of a series of concentric thin spherical shells. The fractional reaction within each such shell was calculated and the total reaction found by integration to include all such shells. This analysis, which includes the effects of overlap, ingestion and also particle size of the reactant, is not amenable to general solution, but the following special cases are of interest. 

These spherical nano-particles about 55 nm in diameter have a fluorescent material of ruthenium pyridine inside, and the shell of silicon dioxide, as shown in Fig. 36. The excitation wavelength of the ruthenium pyridine is 480 nm and the emission wavelength is 592 nm [81]. In order to get a clear image of nano-particles, the mass concentration of the fluorescent particles should be limited to a very low level. 

The first component h2/Z) is the period of time required to traverse a distance b in any direction, whereas the second term/ (alb) strongly depends on the dimension-ahty. Adam and Delbrtlck define appropriate boundary conditions and equations describing the concentration of molecules in the diffusion space in terms of space coordinates and time. They treated four cases (1) onedimensional diffusion in the linear interval a < jc < h (2) two-dimensional diffusion on the circular ring a < r < b (3) three-dimensional diffusion in a spherical shell a < r < b, and (4) combined three-dimensional and surface diffusion. They provide a useful account of how reduced dimensionahty of diffusion can (a) lower the time required for a metabolite or particle originating at point P to reach point Q, and (b) improve the likelihood for capture (or catch) of regulatory molecules by other molecules localized in the immediate vicinity of some target point Q. 

Equqtion (51.15) provides the extinction cross section for spherical particles in a dielectric medium. When the particles are coated by a surface layer, the optical properties of both the core and shell materials must be considered. The extinction cross section of a concentric core-shell sphere is given by [144],... 

Braithewaite and Madsen and Wadsworth used the simplified continuity equation for copper sulfide ores using finite difference approximations. The spherical ore matrix was divideid into j concentric shells of thickness Ar, with 1 corresponding to the center and = n to the outer edge of the spherical matrix particle. The term (R may be evaluated using the equation... 

---