Particle spherical

Let us now turn to spherical particles and let us ask for their reaction to the external magnetic or electric multipole potential 

Considering first the magnetic modes s = 1, it is convenient to shift the curl operator appearing in the vector potential dA r) to the orbitals 

By dissecting the orbitals i , /c into a radial function times a spherical harmonic according to Eq. (7.35) we obtain 

Substituting Eq. (7.63) into J(r, t) according to Eq. (7.55), we may again sum over the orientation of orbitals i , /c independent of the radial behavior and independent of energy. Just as in the case of the scalar orthogonality relation (7.39), (7.40), we may prove the vector orthogonality relation 

The induced current density is parallel to the external vector potential 5A r), whereas no oscillations of the charge density arise. 

It is interesting to note that formula (64) is applicable to two cylindrically symmetric particles, such as between two nanotubes [43, 61], if fhe applied field is the only source of radiation. If fhis is fhe case, only the principal axis (diagonal elements) of the polarizability tensors contribute, corresponding to the component aligned in the same direction as the laser polarization. 

According to a simple theory due to Einstein (1906) for spherical particles at very low concentrations, the relative viscosity, T)/r]s, where r]s is the viscosity of the pure liquid medium, is related to the volume fraction, p, of particles by the simple equation 

This may also be expressed in terms of the relative viscosity increment, r)i (the term specific viscosity is now discouraged by 

This term is also used for rjfp, where p is the mass fraction of particles. 

Positions are shown at successive time intervals (a) when the 

Increase in concentration, or of interparticle forces, will lead to larger aggregates which will enhance the viscosity even further. Systems under these conditions are usually non-Newtonian since these effects are sensitive to shear rate. Increase in the shear rate 

An interesting feature of the null-field method is that all matrix equations become considerably simpler and reduce to the corresponding equations of the Lorenz-Mie theory when the particle is spherically. For a spherical particle of radius R, the orthogonality relations of the vector spherical harmonics show that the QP matrices are diagonal 

The transition matrix of a spherical particle is diagonal with entries 

Equations (2.46) and (2.47) relating the transition matrix to the size parameter and relative refractive index are identical to the expressions of the Lorenz-Mie coefficients given by Bohren and Huffman [17]. 

In addition to familiar phenomena such as these, new phenomena arise because of the synergism between the properties of the polymer and those of the particle filler. One such synergistic effect is enhanced shear thinning, which occurs because the shear rate experienced by the polymer confined between two particles can be much larger than the overall imposed shear rate (Khan and Prud homme 1987). Another general observation is that the filled melt is often effectively less elastic than the polymer alone, evidently because the filler enhances the viscosity more than it does the first normal stress difference Ni (Han 1981 Han et al. 1981). Thus, Fig, 6-38 shows that at fixed shear stress the first normal stress difference Nj for polypropylene decreases upon addition of CaC03 particles.  

In a filled melt at low particle loadings (0 0.05), even particles that interact strongly 

There are a large number of methods for producing metallic colloids in solution.  

The selection of recipes that produce particle populations with narrow size and shape distributions is critical for developing labels with uniform brightness and eolor. 

Fabrication methods that utilize a small amount of smaller colloid as a seed for subsequent particle growth typically produce narrow size distributions of particles  

As an example, fabrication procedures for the production of homogeneous populations of gold and silver spheres in the 10 - 100 nm size range are listed in Table 1. The final particle size can be adjusted by changing the initial concentration of the 5 nm seed gold. 

Fabrication of 80 nm diameter gold colloid Fabrication of 60 nm silver colloid 

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A. L. Loeb, J. Th. G. Overbeek, and P. H. Wiersema, The Electrical Double Layer Around a Spherical Particle, MIT Press, Cambridge, MA, 1961. 

The basic phenomenon involved is that particles of ore are carried upward and held in the froth by virtue of their being attached to an air bubble, as illustrated in the inset to Fig. XIII-4. Consider, for example, the gravity-free situation indicated in Fig. XIII-5 for the case of a spherical particle. The particle may be entirely in phase A or entirely in phase B. Alternatively, it may be located in the interface, in which case both 7sa nnd 7sb contribute to the total surface free energy of the system. Also, however, some liquid-liquid interface has been eliminated. It may be shown (see Problem XIII-12) that if there is a finite contact angle, 0sab> the stable position of the particle is at the interface, as shown in Fig. XIII-5Z>. Actual measured detachment forces are in the range of 5 to 20 dyn [60]. 

Show that the stable position of a spherical particle is, indeed, that shown in Fig. Xni-5b if 0SAB is nonzero. Optional by what percent of its radius should the particle extend into phase A if 7ab cos 9 sab is -34 erg/cm and 7ab is 40 ergs/cm. ... 

It has been shown that spherical particles with a distribution of sizes produce diffraction patterns that are indistingiushable from those produced by triaxial ellipsoids. It is therefore possible to assume a shape and detemiine a size distribution, or to assume a size distribution and detemiine a shape, but not both simultaneously. 

Figure B3.3.10. Contour plots of the free energy landscape associated with crystal niicleation for spherical particles with short-range attractions. The axes represent the number of atoms identifiable as belonging to a high-density cluster, and as being in a crystalline environment, respectively, (a) State point significantly below the metastable critical temperature. The niicleation pathway involves simple growth of a crystalline nucleus, (b) State point at the metastable critical temperature. The niicleation pathway is significantly curved, and the initial nucleus is liqiiidlike rather than crystalline. Thanks are due to D Frenkel and P R ten Wolde for this figure. For fiirther details see [189].

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

Altliough tire majority of studies on model colloids involve (quasi-) spherical particles, tliere is a growing interest in the properties of non-spherical colloids. These tend to be eitlier rod-like or plate-like. 

Altliough tire behaviour of colloidal suspensions does in general depend on temperature, a more important control parameter in practice tends to be tire particle concentration, often expressed as tire volume fraction ((). In fact, for hard- sphere suspensions tire phase behaviour is detennined by ( ) only. For spherical particles... 

In most colloidal suspensions tire particles have a tendency to sediment. At infinite dilution, spherical particles with a density difference Ap with tire solvent will move at tire Stokes velocity... 

In electrophoresis, the motion of charged colloidal particles under the influence of an electric field is studied. For spherical particles, we can write... 

Otlier possibilities for observing phase transitions are offered by suspensions of non-spherical particles. Such systems can display liquid crystalline phases, in addition to tire isotropic liquid and crystalline phases (see also section C2.2). First, we consider rod-like particles (see [114, 115], and references tlierein). As shown by Onsager [116, 117], sufficiently elongated particles will display a nematic phase, in which tire particles have a tendency to align parallel to... 

Hamaker H C 1937 London-van der Waals attraction between spherical particles Physica 4 1058-72... 

Molecular graphics representation ofihe paths generated by 32 hard spherical particles in the solid (left) and ht) phase. (Reproduced from Alder B J and T E Wainwright 1959. Studies in Molecular Dynamics. I. Method. Journal of Chemical Physics. 31. 459-466.)... 

Stiffness analysis of polymer composites filled with spherical particles... 

In Figure 5.24 the predicted direct stress distributions for a glass-filled epoxy resin under unconstrained conditions for both pha.ses are shown. The material parameters used in this calculation are elasticity modulus and Poisson s ratio of (3.01 GPa, 0.35) for the epoxy matrix and (76.0 GPa, 0.21) for glass spheres, respectively. According to this result the position of maximum stress concentration is almost directly above the pole of the spherical particle. Therefore for a... 

Fig. 1.10 An aggregate of spherical particles, having a very open structure.

It will be convenient to deal first with the distribution aspect of the problem. One of the clearest ways in which to represent the distribution of sizes is by means of a histogram. Suppose that the diameters of SOO small spherical particles, forming a random sample of a powder, have been measured and that they range from 2-7 to 5-3 pm. Let the range be divided into thirteen class intervals 2-7 to 2-9 pm, 2-9 to 3-1 pm, etc., and the number of particles within each class noted (Table 1.5). A histogram may then be drawn in which the number of particles with diameters within any given range is plotted as if they all had the diameter of the middle of the... 

An exactly similar expression is obtained for spherical particles, where L and A now refer to the diameters of particles rather than their edge length. 

Fig. 3.15 (a) A pore in the form of an interstice between close-packed and equal-sized spherical particles. The adsorbed him which precedes capillary condensation is indicated, (b) Adsorption isotherm (idealized). 

In Unger and Fischer s study of the effect of mercury intrusion on structure, three samples of porous silica were specially prepared from spherical particles 100-200 pm in diameter so as to provide a wide range of porosity (Table 3.16). The initial pore volume n (EtOH) was determined by ethanol titration (see next paragraph). The pore volume u (Hg, i) obtained from the first penetration of mercury agreed moderately well with u fEtOH),... 

For spherical particles of radius R moving through a medium of viscosity 17, Stokes showed that the friction factor is given by... 

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. 

Figure 9.1 Distortion of flow streamlines around a spherical particle of radius R. The relative velocity in the plane containing the center of the sphere equals v, as r ...

A somewhat similar problem arises in describing the viscosity of a suspension of spherical particles. This problem was analyzed by Einstein in 1906, with some corrections appearing in 1911. As we did with Stokes law, we shall only present qualitative arguments which give plausibility to the final form. The fact that it took Einstein 5 years to work out the bugs in this theory is an indication of the complexity of the formal analysis. Derivations of both the Stokes and Einstein equations which do not require vector calculus have been presented by Lauffer [Ref. 3]. The latter derivations are at about the same level of difficulty as most of the mathematics in this book. We shall only hint at the direction of Lauffer s derivation, however, since our interest in rigid spheres is marginal, at best. 

Figure 9.2 (a) Schematic representation of a unit cube containing a suspension of spherical particles at volume fraction [Pg.589]

In the last section we noted that Simha and others have derived theoretical expressions for q pl(p for rigid ellipsoids of revolution. Solving the equation of motion for this case is even more involved than for spherical particles, so we simply present the final result. Several comments are necessary to appreciate these results ... 

Since f is a measurable quantity for, say, a protein, and since the latter can be considered to fail into category (3) in general, the friction factor provides some information regarding the eilipticity and/or solvation of the molecule. In the following discussion we attach the subscript 0 to both the friction factor and the associated radius of a nonsolvated spherical particle and use f and R without subscripts to signify these quantities in the general case. Because of Stokes law, we write... 

All that can be concluded from the data given in the preceding example is that the particle is not an unsolvated sphere. However, when an appropriate display of contours is examined for f/fo (e.g.. Ref. 2), the latter is found to be consistent with an unsolvated particle of axial ratio about 4 1 or with a spherical particle hydrated to the extent of about 0.48 g water (g polymer). Of course, there are a number of combinations of these variables which are also possible, and some additional experimental data—such as the intrinsic viscosity—are needed to select that combination which is consistent with all experimental observations. 

The particle can be assumed to be spherical, in which case M/N can be replaced by (4/3)ttR P2, and f by 671770R- In this case the radius can be evaluated from the sedimentation coefficient s = 2R (p2 - p)/9t7o. Then, working in reverse, we can evaluate M and f from R. These quantities are called, respectively, the mass, friction factor, and radius of an equivalent sphere, a hypothetical spherical particle which settles at the same rate as the actual molecule. 

Figure 9.15 Schematic illustration of size exclusion in a cylindrical pore (a) for spherical particles of radius R and (b) for a flexible chain, showing allowed (solid) and forbidden (broken) conformations of polymer.

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