Containers spherical

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

Figure 20. (a) Volume containing spherical obstacles with volume fraction <(> = 0.15. (b) The... 

The macrotricyclic ligands (240) and (241) may be synthesized by multistep high-dilution procedures (Graf Lehn, 1975). They contain spherical cavities which are able to accommodate suitable guests whether they be cationic, neutral, or anionic. 

The microductile/compliant layer concept stems from the early work on composite models containing spherical particles and oriented fibers (Broutman and Agarwal, 1974) in that the stress around the inclusions are functions of the shear modulus and Poisson ratio of the interlayer. A photoelastic study (Marom and Arridge, 1976) has proven that the stress concentration in the radial and transverse directions when subjected to transverse loading was substantially reduced when there was a soft interlayer introduced at the fiber-matrix interface. The soft/ductile interlayer allowed the fiber to distribute the local stresses acting on the fibers more evenly, which, in turn, enhanced the energy absorption capability of the composite (Shelton and Marks, 1988). 

Figure 13. Critical stress (a) vs. Young s modulus of elastomers containing spherical inclusions. k= 0.74 minr1 T = 25°C.

According to the literature (26, 27), soot from diffusion flames contains spherical carbon particles from 0.01 to about 0.20 / m in diameter. The structure of such a flame shows a number of zones in the inner zones carbon is formed from gaseous fuel molecules the outer ones see the subsequent combustion of the particles. Where the latter process is incomplete, the flame produces soot. We will call such material combustion carbon to distinguish it from that derived by the pyrolysis of existing solid, involatile fragments, which we call char. A char consists of irregular particles of various sizes, some of which may... 

Electron microscopy provides direct evidence that the polymerized dispersions still contain spherical liposomes (Fig. 20). Laser light scattering measurements also demonstrate the presence of vesicles in the irradiated dispersions16). The size distribution of vesicles is not significantly altered by polymerization 13) (Fig. 21) as confirmed by gel filtration before and after irradiation of vesicles of (7). 

The easiest model to treat theoretically is the sphere, and many colloidal systems do, in fact, contain spherical or nearly spherical particles. Emulsions, latexes, liquid aerosols, etc., contain spherical particles. Certain protein molecules are approximately spherical. The crystallite particles in dispersions such as gold and silver iodide sols are sufficiently symmetrical to behave like spheres. 

Monodispersed sols containing spherical polymer particles (e.g. polystyrene latexes22"24, 135) can be prepared by emulsion polymerisation, and are particularly useful as model systems for studying various aspects of colloidal behaviour. The seed sol is prepared with the emulsifier concentration well above the critical micelle concentration then, with the emulsifier concentration below the critical micelle concentration, subsequent growth of the seed particles is achieved without the formation of further new particles. 

For a system containing spherical particles, D = RT/6rrr]aNA - i.e. D oc 1/m1, where m is the particle mass. For systems containing asymmetric particles, D is correspondingly smaller (see Table 2.3). Since D = k77/, the ratio D/D0 (where D is the experimental diffusion coefficient and D0 is the diffusion coefficient of a system containing the equivalent unsolvated spheres) is equal to the... 

Typically, micelles tend to be approximately spherical over a fairly wide range of concentration above the c.m.c., but often there are marked transitions to larger, non-spherical liquid-crystal structures at high concentrations. Systems containing spherical micelles tend to have low viscosities, whereas liquid-crystal phases tend to have high viscosities. The free energies of transition between micellar phases tend to be small and, consequently, the phase diagrams for these systems tend to be quite complicated and sensitive to additives. 

Figure 2.42 Viscosity increase as a function of volume fraction of filler for polystyrene and low density polyethylene containing spherical glass particles with diameters ranging between 36/.tm and 99.8 gm.

Aqueous suspension polymerization requires the usual additives, such as free radical initiators, colloidal dispersants (not always), and chain transfer agents to control molecular weight. After the process is completed, the suspension contains spherical particles approximately 100 pm in diameter. Suspension polymers are available as free-flowing powder or in pellet form for extrusion or injection molding.58... 

The following results refer to a bed 0.91 m deep containing spherical catalyst pellets of diameter 1.52 mm, with porosity 0.4 due to pores of diameter 75 A and tortuosity factor 3.5. 

Chondrite A stony meteorite that contains spherical bodies of pyroxene and olivine minerals... 

Suppose we have an electrolyte solution Initially containing spherical floes of radius aj, each composed of many elementary particles of radius a, so that fl/ o. Further, suppose that at t — 0 the electrolyte solution is suddenly diluted. The lowering of the ionic strength will increase tho double layer repulsion between particles in the floe and thereby tend to disrupt it. Of course, some time is required for Ions within the interior of the floe to diffuse out this time is on the order of I, where D, is... 

For the sake of simplicity, it will be assumed that the microemu15ion contains spherical globules of a single size. Their dispersion in the continuous phase is accompanied by an increase in the entropy of the system and the corresponding free energy change per unit volume of microemulsion is denoted by if. The Helmholtz free energy f per unit volume of microemulsion is written as the sum... 

Two other approaches have been taken to modelling the conductivity of composites, effective medium theories (Landauer, 1978) and computer simulation. In the effective medium approach the properties of the composite are determined by a combination of the properties of the two components. Treating a composite containing spherical inclusions as a series combination of slabs of the component materials leads to the Maxwell-Wagner relations, see Section 3.6.1. Treating the composite as a mixture of spherical particles with a broad size distribution in order to minimise voids leads to the equation ... 

A metallic crystal can be pictured as containing spherical atoms packed together and bonded to each other equally in all directions. We can model such a structure by packing uniform, hard spheres in a manner that most efficiently uses the available space. Such an arrangement is called closest packing (see Fig. 16.13). The spheres are packed in layers in which each sphere is surrounded by six others. In the second layer the spheres do not lie direotlv over those in the first layer. Instead, each one occupies an indentation (or dimple) formed by three spheres in the first layer. In the third layer the spheres can occupy the dimples of the second layer in two possible ways. They can occupy positions so that each sphere in the third layer lies directly over a sphere in the first layer (the aba arrangement), or they can occupy positions... 

CVP and ESA are techniques that appear to remain essentially valid for concentrated systems, with obvious advantages for applications. However, many systems are neither homodisperse nor do they contain spherical particles. Then, the computed -potentials are rather semlquantltatlve (but useful) electrokinetic characteristics. Loewenberg and O Brien have extended the theory to non-spherlcal particles ). 

We will limit our discussion here only to the films containing spherical particles of PB, as these are of all morphologies the ones most readily amenable to analysis. Materials with rod morphology or lamellar morphology exhibit qualitatively very similar behavior but quantitatively require more detailed analysis that is, however, unfortunately less exact. 

Forced Electric Dipole Transitions. In more recent work, Judd (15) has given further attention to the problem of intensities. According to this work, under certain symmetry restricted circumstances, the Hamiltonian for the interaction of a lanthanide ion with its neighbors can contain spherical harmonics with fc = 1 if the electrons of the rare-earth ion produce an electric field at the nucleus that exactly cancels that... 

P12-20b The catalytic hydrogenation of methyl linoleate to methyl oleate was carried out in a laboratory-scale slurry reactor in which hydrogen gas was bubbled up through the liquid containing spherical catalyst pellets. The pellet density is 2 g/cm. The following experiments were carried out at 25°C ... 

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