Particle rotation

Case 2. The particles rotate in small packets ( coherently or in phase ). Obviously, the first-order rate law no longer holds. In chapter B2.1 we shall see that this simple consideration has found a deeper meaning in some of the most recent kinetic investigations [21]. 

Figure 9.10 Schematic relationship between the radius Rq of an unsolvated sphere and the effective radius R of a solvated sphere or of a spherical volume excluded by an ellipsoidal particle rotating through all directions.

Press, W., 1981, Single-Particle Rotations in Molecular Crystals. Springer Tracts in Modem Physics, Vol. 92 (Springer, Berlin). Punnkinen, M., 1980, Phys. Rev. B 21, 54. 

As an illustration, consider again the case of the earth, rotating with constant angular velocity, Fig. 2.3b, and suppose that the particle p on its surface remains at rest in the system of coordinates moving together with the earth. Since in this case the particle rotates with Earth in the horizontal plane perpendicular to the z-axis, we can use Equation (2.41), (two-dimensional case), and this gives... 

Peng, X., Tomita, Y., and Tashiro, H., Effect of Particle-Particle Collision and Particle Rotation upon Floating Mechanism of Coarse Particles in Horizontal Pneumatic Pipe, JSME Inti. J., Series B, 37(3) 485-490 (1994)... 

The large ribosomal subunit at 2.4 A resolution. (A) The particle rotated with respect to the crown view so that its active site cleft can be seen. (B) The crown view. (C) The back view of the particle, i.e., the crown view rotated 180° about its vertical axis. Reprinted with permission from Ban et al., Science 289, 905 (2000). Copyright 2000 American Association for the Advancement of Science. 

Rotational diffusion of particles occurs in polymer much slowly than in liquids. Therefore, the observed difference in liquid (k ) and solid polymer (ks) rate constants can be explained by the different rates of reactant orientation in the liquid and polymer. The EPR spectra were obtained for the stable nitroxyl radical (2,2,6,6-tetramethyl-4-benzoyloxypiperidine-l-oxyl). The molecular mobility was calculated from the shape of the EPR spectrum of this radical [14,15], These values were used for the estimation of the orientation rate of reactants in the liquid and polymer cage. The frequency of orientation of the reactant pairs was calculated as vor = Pvrot> where P is the steric factor of the reaction, and vIol is the frequency of particle rotation to the angle equal to 4tt. The results of this comparison are given in Table 19.2. 

Figure 3.10 The dilation of the flow field around a spherical particle. The shear field has a vorticity equal to y/2 and the particle rotates with this constant angular velocity...

For particles of arbitrary shape held in a flow, Eqs. (6-34) and (6-37) should be used for Re > 1000. For particles in free fall the only data available (P2) show that the transfer is little affected by particle rotation with rotational velocities less than 50% of the particle velocity. The correlation for fixed particles was adequate provided that the equivalent diameter was used in place of L. For particles of arbitrary shape falling in the Newton s law regime, Eq. (6-35) should be used with replacing L and Sho taken as 2. 

It is convenient to distinguish between particle or fluid rotation about axes normal and parallel to the direction of relative motion. These two types of motion may be termed respectively top spin and screw motion (Til). Top spin is of more general importance since this corresponds to particle rotation caused by fluid shear or by collision with rigid surfaces. Workers concerned with suspension rheology and allied topics have concentrated on motion at low Re, while very high Reynolds numbers have concerned aerodynamicists. The gap between these two ranges is wide and uncharted, and we make no attempt to close it here. 

Much of the work on particle rotation at low Rqq follows from the early work of Jeffery (J2) who considered a rigid, neutrally buoyant spheroid subject to the uniform shear field defined by Eq. (10-30). Jeffery showed that the particle center moves with the velocity which the continuous fluid would have at that point in the absence of the particle, while the axis of the spheroid undergoes rotation in one of a family of periodic orbits with angular velocities... 

For a sphere where a = b, the particle rotates with an angular velocity of G/2 and a period of rotation of 4ti /G. 

Effects such as lift due to particle rotation or fluid velocity gradients can readily be included in Eq. (11-70) if appropriate. The resulting equation of motion is... 

In suspensions of particles with an aspect ratio (length to diameter) greater than 1, particle rotation during flow results in a large effective hydrodynamic volume, and Kh > 2.5 (see Figure 4.7). At particle volume fractions above about 5-10%, interaction between particles during flow causes the viscosity relationship to deviate from the Einstein equation. In such instances, the reduced viscosity is better described by the following relationship ... 

The relative orientation of the two particles is shown in Fig. 13.11 note that the pairwise correspondence between the matrix (13.12) for the particle in position (a) and the matrix (13.13) for the same particle rotated into position (b) is specific to a given direction of observation es. If the direction of observation changes, then so, in general, does the orientation of the rotated particle so the matrix (13.13) does not apply to a single rotated particle in a fixed orientation but rather to a set of identical particles in different orientations, one for every direction of observation. 

These values were used for the estimation of the orientation rate of reactants in the liquid and polymer cage. The frequency of orientation of the reactant pairs was calculated as vor = Pvrot, where P is the steric factor of the reaction, and vIol is the frequency of particle rotation to the angle equal to 4tt. The results of this comparison are given in Table 19.2. 

This zone may be created in an air vortex for example in a cylindrical chamber with inlet and outlet placed on the opposite sides of the chamber axis (Fig. Id). The fed particles enter with the air stream through whirl blades, which create a screw-type flow. The blade profiles should be designed such that the radial component of the air velocity everywhere vanishes. Particles rotate while drifting radially towards the chamber walls with size-dependent velocities. As they simultaneously proceed along the chamber axis they reach the wall in different locations. Larger particles arrive... 

Saltation of solids occurs in the turbulent boundary layer where the wall effects on the particle motion must be accounted for. Such effects include the lift due to the imposed mean shear (Saffman lift, see 3.2.3) and particle rotation (Magnus effect, see 3.2.4), as well as an increase in drag force (Faxen effect). In pneumatic conveying, the motion of a particle in the boundary layer is primarily affected by the shear-induced lift. In addition, the added mass effect and Basset force can be neglected for most cases where the particle... 

Applications of optical methods to study dilute colloidal dispersions subject to flow were pioneered by Mason and coworkers. These authors used simple turbidity measurements to follow the orientation dynamics of ellipsoidal particles during transient shear flow experiments [175,176], In addition, the superposition of shear and electric fields were studied. The goal of this work was to verify the predictions of theories predicting the orientation distributions of prolate and oblate particles, such as that discussed in section 7.2.I.2. This simple technique clearly demonstrated the phenomena of particle rotations within Jeffery orbits, as well as the effects of Brownian motion and particle size distributions. The method employed a parallel plate flow cell with the light sent down the velocity gradient axis. 

Features of PEPT of particular benefit to engineering studies include the fact that the actual particles of interest may be used as tracers, rather than dissimilar materials of unknown behaviour, and that y-rays are sufficiently penetrating that location is unimpaired by the presence of metal walls, for example. In recent developments, the minimum size of particles which can be tracked has been reduced to approximately 60 pm. It is now possible to track multiple particles, to determine particle rotation and to track motion within real industrial equipment by use of a mobile modular positron camera. These developments are described later. 

A specific feature of the operator (Je +J ) is that when it acts at an arbitrary function of the product (e n), the result is zero. Therefore, in the absence of an external field, when U = —KVm(e nf, one finds from (4.310) that e> = 1, that is, the particle rotates in unison with the surrounding liquid. In this case neither the torque acting on the yolk (—JeU) nor the one exerted on the eggshell (—J U) equals zero. However, their sum vanishes identically. This is quite natural since the equilibrium of a system as a whole cannot be disturbed by the internal forces (or torques). 

Brownian motion effects are weak, then we may transit from the statistical description to the dynamical one. In such a situation the particle rotation is determined simply by the balance of viscous and field-induced torques and thus is governed by the equation... 

Press W (1981) In Single particle rotations in molecular crystals. Springer, Berlin Heidelberg New York... 

For thin disks, the rise and decay curves of optical retardation d(t) are related to the particle rotational motion by (3)... 

Although the electron spin is often referred to as an intrinsic angular momentum, it should be emphasized that spin has no classical analog. In particular, one should not imagine a spinning electron as a particle rotating about an inner axis (e.g., like the earth does). Spin has to be regarded as a pure... 

Magnetic particles in a solution undergo two types of relaxation Brownian relaxation, in which the entire particle rotates, and Neel relaxation, in which the moment rotates while the particle remains still. The Brownian relaxation time xB is... 

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