Subject Particle radius

In this paper it is shown that the rate of deposition of Brownian particles on the collector can be calculated by solving the convective diffusion equation subject to a virtual first order chemical reaction as a boundary condition at the surface. The boundary condition concentrates the surface-particle interaction forces. When the interaction potential between the particle and the collector experiences a sufficiently high maximum (see f ig. 2) the apparent rate constant of the boundary condition has the Arrhenius form. Equations for the apparent activation energy and the apparent frequency factor are established for this case as functions of Hamaker s constant, dielectric constant, ionic strength, surface potentials and particle radius. The rate... 

One of the most attractive features of colloidal semiconductor systems is the ability to control the mean particle size and size distribution by judicious choice of experimental conditions (such as reactant concentration, mixing regimen, reaction temperature, type of stabilizer, solvent composition, pH) during particle synthesis. Over the last decade and a half, innovative chemical [69], colloid chemical [69-72] and electrochemical [73-75] methods have been developed for the preparation of relatively monodispersed ultrasmall semiconductor particles. Such particles (typically <10 nm across [50, 59, 60]) are found to exhibit quantum effects when the particle radius becomes smaller than the Bohr radius of the first exciton state. Under this condition, the wave functions associated with photogenerated charge carriers within the particle (vide infra) are subject to extreme... 

Thus, it may be seen that, by reducing the particle radius, it is possible to obtain systems where transit from the particle interior to the surface occurs more rapidly than recombination, implying that quantum efficiencies for photoredox reaction of near unity are feasible. However, the achieving of such high quantum efficiencies depends very much upon the rapid removal of one or both types of charge carrier upon their arrival at the semiconductor surface, underlining the importance of the interfacial charge-transfer kinetics. This is the subject of the next section. 

In what follows, the equation of diffusion derived in Chapter 2 is generalized to take into account the effect of flow. For point particles (dp = 0), rates of convective diffusion can often be predicted from theory or from experiment with aqueous solutions because the Schmidt numbers are of the same order of magnitude. There is an extensive literature on this subject to which the reader is directed. For particle diffusion, there is a difference from the usual theory of convective diffusion because of the special boundary condition The concentration vanishes at a distance of one particle radius from the surface. This has a very large effect on particle deposition rates and causes considerable difficulty in the mathematical theory. As discussed in this chapter, the theory can be simplified by incorporating the particle radius in the diffusion boundary condition. 

Some metals are soluble as atomic species in molten silicates, the most quantitative studies having been made with Ca0-Si02-Al203(37, 26, 27 mole per cent respectively). The results at 1800 K gave solubilities of 0.055, 0.16, 0.001 and 0.101 for the pure metals Cu, Ag, Au and Pb. When these metal solubilities were compared for metal alloys which produced 1 mm Hg pressure of each of these elements at this temperature, it was found drat the solubility decreases as the atomic radius increases, i.e. when die difference in vapour pressure of die pure metals is removed by alloy formation. If the solution was subjected to a temperature cycle of about 20 K around the control temperamre, the copper solution precipitated copper particles which grew with time. Thus the liquid metal drops, once precipitated, remained stable thereafter. 

According to the classical Stokes law, a spherical particle, i, of radius rif moving with velocity V through a static fluid of viscosity t] is subjected to a force f as shown by Eqs (7.11) or (7.12) ... 

Closely related to the precise measurement of detonation velocity is the subject of detonation front curvature. Front curvature of Tetryl was examined at the Naval Ordnance Laboratory (NOL) (Refs 31 39). For point-initiated charges of 1.51g/cc, it was found that the radius of curvature of the detonation front increases with charge length in the manner expected for spherical expansion of the front. The radius of curvature is also a function of the chemical nature of the expl, its particle size and its packing density... 

The mechanism of particle filtration by screen filters has been the subject of many studies because it is relatively easily described mathematically Bungay has published a review of this work [49], Ferry [50] was the first to model membrane retention by a screen filter in his model pores were assumed to be equal circular capillaries with a large radius, r, compared to the solvent molecule radius. Therefore, the total area of the pore is available for transport of solvent. A solute molecule whose radius, a, is an appreciable fraction of the pore radius cannot approach nearer than one molecular radius of the pore overall. The model is illustrated in Figure 2.32. 

Now consider a case of collision of a fixed single sphere with a cloud of particles as shown in Fig. 5.8(b). The sphere of radius r is subjected to a shear flow of a particle cloud with a velocity gradient of dUp/dy. The particle number density is denoted as n, and the mass of a particle is m. Select the velocity Up as zero in the center plane of the sphere. The relative velocity is estimated by... 

Lanthanides doped into nanocrystalline semiconductors have been the subject of numerous investigations in the past decades. If the size of a semiconductor particle is smaller than the Bohr radius of the excitons, the so-called quantum confinement occurs. As a result, the band gap of the semiconductor increases and discrete energy levels occur at the edges of the valence and conduction bands (Bol et al., 2002 Bras, 1986). These quantum size effects have stimulated extensive interest in both basic and applied research. 

The shape of the metal specimen considered is obviously related to the kind of system to be modelled. For SERS and the others SE phenomena, the presence of curved surfaces, with a curvature radius on the nanometric scale, is fundamental for the enhancement. Thus, spheres, ellipsoids, ensembles of spheres, spheres close to planar metal surfaces and planar metal surfaces with random roughness have been considered. We refer to the review by Metiu [57], which describes most of these analytically solvable models. More recently, the modelling of the electric field acting on the point molecule has moved to more realistic shapes (including fractal metal specimen) [59] which require numerical methods to be tackled. The aim of these approaches is usually to calculate the total electric field around the metal particle, and the molecule does not even appear explicitly in the calculations. Interested readers are referred to some recent reviews on the subject [60] (see also Chapters 2 and 5 of ref. [56]). 

The most widely used unsteady state method for determining diffusivities in porous solids involves measuring the rate of adsorption or desorption when the sample is subjected to a well defined change in the concentration or pressure of sorbate. The experimental methods differ mainly in the choice of the initial and boundary conditions and the means by which progress towards the new position of equilibrium is followed. The diffusivities are found by matching the experimental transient sorption curve to the solution of Fick s second law. Detailed presentations of the relevant formulae may be found in the literature [1, 2, 12, 15-17]. For spherical particles of radius R, for example, the fractional uptake after a pressure step obeys the relation... 

Landfester et al. [ 143] studied the miniemulsion polymerization of styrene using hexadecane as the costabilizer. When styrene miniemulsions were subjected to varying sonication times (see Table 5), very similar trends are seen as for the MMA miniemulsions. The particle size and the polydispersity of miniemulsion droplets rapidly polymerized after sonication either do not depend on the amount of the costabihzer, or are very weak functions of the amount of costabilizer (see Table 6). It was found that doubhng the amount of costabilizer does not decrease the radius nor have any effect on the polydis-... 

Near rich limits of hydrocarbon flames, soot is sometimes produced in the flame. The carbonaceous particles—or any other solid particles— easily can be the most powerful radiators of energy from the flame. The function k(t) is difficult to compute for soot radiation for use in equation (21) because it depends on the histories of number densities and of size distributions of the particles produced for example, an approximate formula for Ip for spherical particles of radius with number density surface emissivity 6, and surface temperature is Ip = Tl nrle ns) [50]. These parameters depend on the chemical kinetics of soot production—a complicated subject. Currently it is uncertain whether any of the tabulated flammability limits are due mainly to radiant loss (since convective and diffusive phenomena will be seen below to represent more attractive alternatives), but if any of them are, then the rich limits of sooting hydrocarbon flames almost certainly can be attributed to radiant loss from soot. 

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