Cylindrically symmetric particles

The simplest form for the chiral interaction between two cylindrically symmetric particles is [72]... 

The polarizability of cylindrically symmetric particles (rod-shaped or ellipsoidal particles) can be characterized by isotropic (a) and anisotropic (P) parts of the polarizability tensors ... 

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. 

Quite probably the answer to the second question will look not too much different from the expressions for the models that have been thoroughly analyzed here, but the establishment of this result may turn out to be tedious. We have seen in Section II how to handle rigid nonpolarizable particles of arbitrary symmetry using the formalism of Hoye and Stell. The addition of fluctuating polarizability has been considered by those authors only for molecules of cylindrical symmetry, but its extension to molecules of arbitrary symmetry is unlikely to raise fundamental problems. On the other hand, particles lacking cylindrical symmetry even in the nonpolarizable case are substantially more awkward to deal with than cylindrically symmetric particles. In treating the constant-polarizability case, Wertheim excludes all permanent multipoles beyond the dipole clearly the quadrupole at least must also be included to provide a realistic model for many real fluids of interest. 

The first attempt to develop a theory for non-rigid mesogens was made by Marcelja who extended the Maier-Saupe theory for nematics composed of cylindrically symmetric particles to include molecular flexibility. The advent of studies of the variation of the orientational order... 

The molecules in a nematic liquid crystal tend to be parallel to a unique direction known as the director which is identical with the optic axis of the phase. The molecular orientation fluctuates with respect to the director and the extent of these fluctuations is reflected by the orientational order parameters they are defined in Section 6. As we shall see the order parameters for cylindrically symmetric particles are chosen to be unity in a perfectly ordered phase or crystal while for the disordered phase or liquid the order parameters vanish. In a nematic phase the order parameters are intermediate between these two extremes. The magnitude of the orientational fluctuations are controlled by the energy of the molecule as it changes its orientation with respect to the director. For a cylindrically symmetric molecule this energy is determined entirely by the angle between the director and the molecular symmetry axis. 

To locate the nematic-isotropic transition it is necessary to determine the temperature dependence of the free energy but this is not possible without making further approximations concerning the temperature variation of the segmental interaction parameters X. and This variation with temperature results primarily through their dependence on the orientational order of the system. A rigorous derivation of this dependence is extremely difficult and so we adopt a semi-intuitive approach. As we have seen, the orientational order in a mesophase is characterized by an infinite set of order parameters but the most important of these are the second-rank order parameters, at least close to the nematic-isotropic transition. Indeed for cylindrically symmetric particles both theory and experiment agree that the potential of mean torque is proportional to When the mesophase... 

The deformation can be very complicated to describe in a single-particle framework, but a good understanding of the basic behavior can be obtained with an overall parameterization of the shape of the whole nucleus in terms of quadmpole distortions with cylindrical symmetries. If we start from a (solid) spherical nucleus, then there are two cylindrically symmetric quadmpole deformations to consider. The deformations are indicated schematically in Figure 6.10 and give the nuclei ellipsoidal shapes (an ellipsoid is a three-dimensional object formed by the rotation of an ellipse around one of its two major axes). The prolate deformation in which one axis is longer relative to the other two produces a shape that is similar to that of a U.S. football but more rounded on the ends. The oblate shape with one axis shorter than the other two becomes a pancake shape in the limit of very large deformations. 

Consider a cylindrical soft particle, that is, an infinitely long cylindrical hard particle of core radius a covered with an ion-penetrable layer of polyelectrolytes of thickness d in a symmetrical electrolyte solution of valence z and bulk concentration (number density) n. The polymer-coated particle has thus an inner radius a and an outer radius b = a + d. The origin of the cylindrical coordinate system (r, z, cp) is held fixed on the cylinder axis. We consider the case where dissociated groups of valence Z are distributed with a uniform density N in the polyelectrolyte layer so that the density of the fixed charges in the surface layer is given by pgx = ZeN. We assume that the potential i/ (r) satisfies the following cylindrical Poisson-Boltz-mann equations ... 

Because the lattice is two-dimensional, all translations commute with reflection in the plane of the lattice, so any electron (or vibrational) eigenstate can be chosen to be either even or odd under this reflection. For this reason, the single-particle electron states are rigorously separated into two classes, called a and 7t. The even a states are derived from carbon s and p, py orbitals (that is, their hybridized sp orbitals), while the odd Jt states are derived from carbon p orbitals. These latter are cylindrically symmetric in the x—y plane, lie near the Fermi level (half-filled) and are the electrically active states of interest in low energy experimental probing of graphene. 

A magnetic field was utilized as a driving force for split-flow thin (SPLITT) fractitMiation by Fuh and co-workers. This technique allowed the complete separation of paramagnetic icm-labeled particles from non-labeled particles. Zborowski et al. constmcted a cylindrically symmetric SPLITT fractionation channel that used a quadmpole magnetic field. In SPLITT fractionation, the separation axis is the thin dimension of the channel. 

For the ease of discussion, we restrict ourselves from now on to the interaction of molecules with a permanent electric dipole moment with electric fields, but the same arguments and principles hold of course for the interaction of particles with a magnetic dipole moment with magnetic fields. In a quadrupole or hexapole focuser, the magnitude of the electric field is zero on the symmetry axis, and this axis is normally made to coincide with the molecular beam axis. Close to this axis, the electric field strength is — to a good approximation — cylindrically symmetric, and it increases with... 

A particle-optical approach can also be used for the treatment of focusing of atoms by laser light (McClelland and Scheinfein 1991). In this approach the atoms are treated as classical particles that move in the potential field of a laser beam. This method was originally developed for charged particle optics, for calculation of trajectories in a cylindrically symmetric potential field. The equation of motion can be derived from the Lagrangian L = Mv /2 — U p, z), where U p, z) is the potential energy eqn (6.1) and 2 is the axis of symmetry. In cylindrical coordinates, the radial equation of motion is... 

In 1965 Lindhard demonstrated that a charged particle moving closely parallel to an atomic row in a crystal experiences a continuum ("string ) potential made up from he atomic potentials in the row. The two dimensional potential, Vr(p) Is cylindrically symmetric and is function only of p, the distance from the row. For any atomic screened Coulomb potential... 

The porous nature of the catalyst particle gives rise to the possible development of significant gradients of both concentration and temperature across the particle, because of the resistance to diffusion of material and heat transfer, respectively. The situation is illustrated schematically in Figure 8.9 for a spherical or cylindrical (viewed end-on) particle of radius R. The gradients on the left represent those of cA, say, for A(g) + product(s), and those on the right are for temperature T the gradients in each case, however, are symmetrical with respect to the centerline axis of the particle. 

In addition, assumptions for the reaction probability of the hard spheres—which strictly speaking cannot react—are introduced. That is, the reaction probability is not calculated from the actual potentials or dynamics of the collisions but simply postulated based on physical intuition. Note that the assumption of a spherically symmetric (hard-sphere) interaction potential implies that the reaction probability P cannot depend on (j> (see Fig. 4.1.1), since there will be a cylindrical symmetry around the direction of the relative velocity. In addition, the assumption of structureless particles implies that the quantum numbers that specify the internal excitation cannot be defined within the present model. 

Let us consider a shallow fluidized bed combustor with multiple coal feeders which are used to reduce the lateral concentration gradient of coal (11). For simplicity, let us assume that the bed can be divided into N similar cylinders of radius R, each with a single feed point in the center. The assumption allows us to use the symmetrical properties of a cylindrical coordinate system and thus greatly reduce the difficulty of computation. The model proposed is based on the two phase theory of fluidization. Both diffusion and reaction resistances in combustion are considered, and the particle size distribution of coal is taken into account also. The assumptions of the model are (a) The bed consists of two phases, namely, the bubble and emulsion phases. The voidage of emulsion phase remains constant and is equal to that at incipient fluidization, and the flow of gas through the bed in excess of minimum fluidization passes through the bed in the form of bubbles (12). (b) The emulsion phase is well mixed in the axial... 

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