Particles isotropic

For a particle which is spherically isotropic (see Chapter 2), the three principal resistances to translation are all equal. It may then be shown (H3) that the net drag is — judJ regardless of orientation. Hence a spherically isotropic particle settling through a fluid in creeping flow falls vertically with its velocity independent of orientation. 

For a cube of side /, Eq. (4-50) gives the resistance as 12.70/, compared with experimental values of 12.58/ (P6), 12.63/ (H4), and 12.71/ (Cl). To the accuracy of the determinations, the resistance can be taken as 4ti / (DI). It is noteworthy that Eqs. (4-26) and (4-27) predict that a spherically isotropic cylinder with aspect ratio 0.812 should have a drag ratio of 1.050, while Eq. (4-50) gives Ag = 1.054. Agreement is so favorable that Eq. (4-50) may be useful for spherically isotropic particles other than the simple shapes for which it was developed. 

Setting the scattering angle 8 = 90° shows that scattering from an isotropic particle at this angle produces the same effect as an ideal polarizer. 

From FPE (4.27) it follows that in the absence of internal and external fields (U = 0) the parameter Tp exhaustively determines the response time of the system to a weak external excitation. For example, it is only xD that determines the dispersion of the magnetic susceptibility in a magnetically isotropic particle. 

Equations for macroscopic characteristic of magnetic particles can be obtained by averaging of the corresponding microscopic quantities with a distribution function satisfying Eq. (4.27). For example, for the dimensionless magnetization (e) of an assembly of magnetically isotropic particles, where... 

Compare this with Eq. (4.88). Coinciding in linear parts—a fact that has been mentioned in Ref. 14—the susceptibilities of a fluid and random solid assemblies differ in the cubic contributions unless one deals with magnetically isotropic particles for which Si = 0. This important fact has been overlooked in Refs. 64 and 65, where the authors have taken Eq. (4.89) as a starting point to study a solid system. Right from the comparison of the static formulas (4.88) and (4.89) for X 3, underestimation of the predicted values, when Eqs. (4.89) are used, becomes apparent. 

Note also that, when a contribution due to a uniaxial magnetic anisotropy is included in the particle energy [see Eq. (4.73)], the emerging parameter h, on changing, in a natural way guides the system through the variety of oscillation situations from the absence of any symmetric potential (an isotropic particle) to a twin pair of infinitely deep wells (a magnetically hard particle). In still other... 

In our model with the aid of parameter e we continuously pass from a zero bistable potential (magnetically isotropic particle) to a pair of symmetric wells of infinite depth (highly anisotropic particle). For the magnetic case, as for those of Refs. 21 and 22, a crucial circumstance enabling the harmonic suppression is that an antisymmetric contribution (bias) should be present in the potential. On the other hand, the presence of a symmetric contribution turns out to be an... 

Therefore, for the internal (Neel) relaxation the parameter, r m plays the same role as the fluid viscosity r in the mechanism of the external (Brownian) diffusion. Note that the density of the anisotropy energy K is not included in x. This means that xD can be considered as the internal relaxation time of the magnetic moment only for magnetically isotropic particles (where K = a = 0). The sum of the rotations—thus allowing for both the diffusion of the magnetic moment with respect to the particle and for the diffusion of the particle body relative to the liquid matrix—determines the angle ft of spontaneous rotation of the vector p at the time moment t ... 

This formula in an equivalent form (with Xg> instead of Xg because a magnetic moment diffusion inside an isotropic particle was considered) had been obtained in Section III.A.3 as Eq. (4.96). Besides that, similar formulas, with xg indeed, are well known in the theory of rotary Brownian diffusion in dipolar fluids [69]. 

Eq. (10.36), however, is correct only for optically isotropic particles, which are small compared to the wave-length. If the particle size exceeds 2/20 (as in polymer solutions), scattered light waves, coming from different parts of the same particle, will interfere with one another, which will cause a reduction of the intensity of the scattered light to a fraction P(6) given by... 

For a suspension of non-spherical, non-monosize non-adsorbing, isotropic particles, in the absence of multiple scattering ... 

Positions (coordinates) of atoms in the unit cell are the strongest contributors into the computed integrated intensities of Bragg reflections assuming that preferred orientation effects are weak. For this powder, preferred orientation was expected (and later found) to be minor due to small particle sizes and predominantly isotropic particle shapes. 

It is convenient to describe the free fall of nonspherical isotropic particles by using the sphericity parameter... 

To begin, focus on the MC estimation of the Helmholtz free energy A of a system of N classical particles. For simplicity of presentation we restrict ourselves for now to a one-component system of isotropic particles of mass m its configurations can be described by a set of coordinates = (qi, q2,. .., qN), say. Then we have, in the canonical ensemble,... 

In a certain sense these equations are the analogs of the fundamental single particle equations (38)-(39), at least in the isotropic case. [See also Eqs. (109)-(110) for the case where the particle neither translates nor rotates, bearing in mind that coy = x u, and that the shear-force and shear-torque triadics vanish for isotropic particles.] It is on the basis of this analogy that we have assumed the symmetry relation K = Kc". 

In the case of helicoidally isotropic particles it follows from Eqs. (73), (344), and (351) that... 

This should be compared with the comparable formula for the average resistance of a nonskew particle near the end of Section II,C, 1. For helicoidally isotropic particles the mean diffusivity is identical to D, given in Eq. (354a) (B25a). 

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