Effect of rotational Brownian motion

Quantitative information can be obtained only if the time-scale of rotational motions is of the order of the excited-state lifetime r. In fact, if the motions are slow with respect to r(r ro) or rapid (r 0), no information on motions can be obtained from emission anisotropy measurements because these motions occur out of the experimental time window. 

The emission anisotropy r0(A) at a wavelength of excitation X results from the addition of contributions from the JLa and b excited states with fractional contributions fa(X) and fh(X), respectively. According to the additivity law of emission anisotropies, ro(X) is given by 

The fractional contributions of the 1La and Lb excited states to the emission anisotropies are given by 

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Effect of rotational Brownian motion Box 5.2 Resolution of the absorption spectrum of indole ... 

However, for nonspherical particles, rotational Brownian motion effects already arise at 0(0). In the case of ellipsoidal particles, such calculations have a long history, dating back to early polymer-solution rheologists such as Simha and Kirkwood. Some of the history of early incorrect attempts to include such rotary Brownian effects is documented by Haber and Brenner (1984) in a paper addressed to calculating the 0(0) coefficient and normal stress coefficients for general triaxiai ellipsoidal particles in the case where the rotary Brownian motion is dominant over the shear (small rotary Peclet numbers)—a problem first resolved by Rallison (1978). 

Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker-Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

The first case, Eq. (8.2) corresponds to the magnetic moment being frozen or blocked as considered in [16]. Since M will maintain its direction relative to axes fixed in the particle for a long time compared with the Debye time Tp. The second case corresponds to the calculation in [17] where the effect of the rotational Brownian motion of the fluid on the magnetic susceptibility is ignored since the directional fluctuations of... 

Normal Brownian motion is a result of solvent molecules impacting on the solute particles, and these give both translational and rotational movement to the solute. An ion on its own will execute this Brownian motion. Since an ion has a charge which can interact with an external electric field, this interaction will perturb the translational Brownian motion, with a cation moving in the direction of the field while an anion will move in the opposite direction. The field will have a minor effect on the rotational Brownian motion, but this will not contribute to the translational mobility. 

Although the fluorophores are usually oriented randomly before the excitation (e.g., in solutions), the population of excited molecules with /lxa parallel with respect to excitation polarization dominates immediately after the short polarized excitation pulse. The anisotropic orientation of excited molecules starts to relax due to the rotational Brownian motion of fluorophores and the excitation energy migration among fluorophores. The rate of the latter process depends strongly on the distance between fluorophores, and an appropriate dilution suppresses its effect considerably. The relaxation can be monitored by measuring the time-resolved fluorescence anisotropy, which is deflned as r t) = [7n(/) - /x(0] / [7n(t)+2/L(t)], where 7n(t) is the paral-lely polarized and is the perpendicularly polarized fluorescence intensity with respect to the excitation pulse. 

A new approach of the bimolecular reaction theories is presented, which is based on the averaging of chemical anisotropy by translational and rotational Brownian motion of the particles.The effective steric factor change in reactions with only one anisotropic reagent was found. It is shown, that it can fall down to the values experimentally observed, only if the hopping mechanism of molecules approach and reorientation is realized. But if the motion is diffusive, then both particles should be chemically anisotropic to explain the experiment. 

Under the action of an applied acoustic field the suggestion was that there would be regions within the polymer where rotation (and vibration) of individual segments were able to take place freely, in phase with the rapid oscillatory movement of the solvent. This segmental movement (termed micro Brovmian motion) was in addition to the movement of the macromolecule as a whole (macro Brownian motion). However, in that segmental motion is a cooperative effect and depends upon the interaction... 

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. 

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