Spherical particles concentrically layered

When micelles are formed just above the cmc, they are spherical aggregates in which surfactant molecules are clustered, tails together, to form a spherical particle. At higher concentrations the amount of excess surfactant is such that the micelles acquire a rod shape or, eventually, even a layer structure. 

The mathematical model chosen for this analysis is that of a cylinder rotating about its axis (Fig. 2). Suitable end caps are assumed. The Hquid phase is introduced continuously at one end so that its angular velocity is identical everywhere with that of the cylinder. The dow is assumed to be uniform in the axial direction, forming a layer bound outwardly by the cylinder and inwardly by a free air—Hquid surface. Initially the continuous Hquid phase contains uniformly distributed spherical particles of a given size. The concentration of these particles is sufftcientiy low that thein interaction during sedimentation is neglected. 

Fig. 4. Onion-like graphitic particles formed by three concentric layers (C o, C240, Cs4o) polyhedral (marked P) and spherical (marked S) structures. For clarity, only a half pan of each shell is shown.

Equation (1) predicts that the rate of release can be constant only if the following parameters are constant (a) surface area, (b) diffusion coefficient, (c) diffusion layer thickness, and (d) concentration difference. These parameters, however, are not easily maintained constant, especially surface area. For spherical particles, the change in surface area can be related to the weight of the particle that is, under the assumption of sink conditions, Eq. (1) can be rewritten as the cube-root dissolution equation ... 

Carrique F, Arroyo FJ, Jimenez ML, Delgado Av. Influence of double-layer overlap on the electrophoretic mobility and DC conductivity of a concentrated suspension of spherical particles. J. Phys. Chem. B 2003 107 3199-3206. 

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 4-21 The concept of boundary layer and boundary layer thickness 5. (a) Compositional boundary layer surrounding a falling and dissolving spherical crystal. The arrow represents the direction of crystal motion. The shaded circle represents the spherical particle. The region between the solid circle and the dashed oval represents the boundary layer. For clarity, the thickness of the boundary layer is exaggerated, (b) Definition of boundary layer thickness 5. The compositional profile shown is "averaged" over all directions. From the average profile, the "effective" boundary layer thickness is obtained by drawing a tangent at x = 0 (r=a) to the concentration curve. The 5 is the distance between the interface (x = 0) and the point where the tangent line intercepts the bulk concentration.

Solution Equation (4.41) gives the Einstein relationship between [r/] and , the volume fraction occupied by the dispersed spheres. The volume fraction that should be used in this relationship is the value that describes the particles as they actually exist in the dispersion. In this case this includes the volume of the adsorbed layer. For spherical particles of radius R covered by a layer of thickness 8R, the total volume of the particles is (4/3) + 4ttR2 8R. Factoring out the volume of the dry particle gives Vdfy(1 + 38RJRS), which shows by the second term how the volume is increased above the core volume by the adsorbed layer. Since it is the dry volume fraction that is used to describe the concentration of the dispersion and hence to evaluate [77], the Einstein coefficient is increased above 2.5 by the factor (1 + 36/Vfts) by the adsorbed layer. The thickness of adsorbed layers can be extracted from experimental [77] values by this formula. ... 

We consider a spherical particle aggregate with radius r0 surrounded by a concentric boundary layer of thickness 8 (Fig. 19.16). Transport into the aggregate is described by the linear approximation of the radial diffusion model. Thus, the total flux from the particle to the fluid is given by Eq. 19-85 ... 

In the point orientation, concentric and radial alignments also have to be differentiated (Figure 2.27). The extreme case of concentric point orientation is found in the fullerene family. This orientation is also found in the spherical particles of carbon blacks, the diameter of which is from few tens to few hundreds of nanometers, minute hexagonal carbon layers being preferentially oriented along the surface (Figures 2.36a and 2.37a) [43], The concentric orientation of the carbon layers is more... 

In the basic model, put forward by Asakura and Oosawa (5), the hard spherical particles immersed in a solution of macromolecules are considered to be surrounded by depletion layers from which the polymer molecules are excluded. When two particles are far apart with no overlap of the depletion zones, the thermal force acting over the entire particle surface is uniform. However, when the particles come closer, such that their depletion zones begin to overlap, there is a region in which the polymer concentration is zero and the force exerted over the surfaces facing this region is smaller compared to that exerted over the rest of the surface. This gives rise to an attractive force between the two particles which is proportional to the osmotic pressure of the polymer solution. 

Quantitative measurements of electrokinetic phenomena permit the calculation of the zeta potential by use of the appropriate equations. However, in the deduction of the equations approximations are made this is because in the interfacial region physical properties such as concentration, viscosity, conductivity, and dielectric constant differ from their values in bulk solution, which is not taken into account. Corrections to compensate for these approximations have been introduced, as well as consideration of non-spherical particles and particles of dimensions comparable to the diffuse layer thickness. This should be consulted in the specialized literature. 

Boundary effects on the electrophoretic migration of a particle with ion cloud of arbitrary thickness were also investigated by Zydney [46] for the case of a spherical particle of radius a in a concentric spherical cavity of radius d. Based on Henry s [19] method, a semi-analytic solution has been developed for the particle mobility, which is valid for all double layer thicknesses and all particle/pore sizes. Two integrals in the mobility expression must be evaluated numerically to obtain the particle velocity except for the case of infinite Ka. The first-order correction to the electrophoretic mobility is 0(A3) for thin double layer, whereas it becomes 0(A) for thick double layer. Here the parameter A is the ratio of the particle-to-cavity radii. The boundary effect becomes more significant because the fluid velocity decays as r l when the double layer spans the entire cavity. The stronger A dependence of the first order correction for thick double layer than that obtained by Ennis and Andersion [45] results from the fact that the double layers overlap in... 

Electrokinetic equations describing the electrical conductivity of a suspension of colloidal particles are the same as those for the electrophoretic mobility of colloidal particles and thus conductivity measurements can provide us with essentially the same information as that from electrophoretic mobihty measurements. Several theoretical studies have been made on dilute suspensions of hard particles [1-3], mercury drops [4], and spherical polyelectrolytes (charged porous spheres) [5], and on concentrated suspensions of hard spherical particles [6] and mercury drops [7] on the basis of Kuwabara s cell model [8], which was originally applied to electrophoresis problem [9,10]. In this chapter, we develop a theory of conductivity of a concentrated suspension of soft particles [11]. The results cover those for the dilute case in the limit of very low particle volume fractions. We confine ourselves to the case where the overlapping of the electrical double layers of adjacent particles is negligible. 

For the next two types of theoretical models, the elementary reaction cell consists of a spherical particle of one reactant surrounded by a melt of the other reactant. In the first case, the product layers (C) grow on the surface of the more refractory particles (B) due to diffusion of atoms from the melt phase (A) through the product layer (see Fig. 20b). At a given temperature, the concentrations at the interphase boundaries are determined from the phase diagram of the system. Numerical calculations by Nekrasov et al. (1981, 1993) have shown satisfactory agreement with experimental results for a variety of systems. 

The catalyst plays a crucial role in the technology. A typical modern catalyst consists of 0.15-1.5 wt% Pd, 0.2-1.5 wt% Au, 4-10 wt% KOAc on silica spherical particles of 5 mm [8]. The very fast reaction takes place inside a thin layer (egg-shell catalyst). Preferred conditions are temperatures around 150 to 160 °C and pressures 8 to 10 bar. Hot spots above 200 °C lead to permanent catalyst deactivation. The excess of ethylene to acetic acid is 2 1 to 3 1. Because of explosion danger, the oxygen concentration in the reaction mixture should be kept below 8%. Small amount of water in the initial mixture are necessary for catalyst activation. The dilution of the reaction mixture with inert gas is necessary because of high exothermic effect. Accordingly, the reactor is designed at low values of the per-pass conversions, namely 15 - 35% for the acetic acid and 8-10% for ethylene. The above elements formulate hard constraints both for design and for plantwide control. 

SEM reveals the as synthesized ZGctab and ZGdtab to be composed of spherical particles with an average diameter between 2 and 10 micrometer (Fig.3a and b). Many spheres are twinned. Sometimes there is complete fusion of different spheres. Given the layered nature of Zeogrid (Fig.l), it is assumed that in these spheres, the layers are packed in a concentric manner. 

This results from the unfavourable mixing of the polymer chains, when these are in a good solvent condition this is shown schematically in Figure 8.2. Consider two spherical particles with the same radius and each containing an adsorbed hydrated layer with thickness 5. Before overlap, it is possible to define in each layer a chemical potential for the solvent, /t j , and a volume fraction for the chains in the layer, 4>2- In the overlap region (volume element dV), the chemical potential of the solvent is reduced to this results from the increase in A segment concentration in this overlap region. 

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