Rotational Brownian motion

We discuss the rotational dynamics of water molecules in terms of the time correlation functions, Ciit) = (P [cos 0 (it)]) (/ = 1, 2), where Pi is the /th Legendre polynomial, cos 0 (it) = U (0) U (it), u [, Is a unit vector along the water dipole (HOH bisector), and U2 is a unit vector along an OH bond. Infrared spectroscopy probes Ci(it), and deuterium NMR probes According to the Debye model (Brownian rotational motion), both... 

It is assumed that the probe molecules undergo Brownian rotational motions with an angle-dependent ordering potential U(p)... 

Anisotropy can be measured when a fluorescent molecule is excited with polarized light. The ratio of emission intensity in each polarization plane, parallel and perpendicular relative to the excitation polarization plane, gives a measure of anisotropy, more commonly and incorrectly referred to in HTS as fluorescence polarization (FP). This anisotropy is proportional to the Brownian rotational motion of the fluorophore. 

Let us summarize the obtained results. From the expression for the viscosity of an infinite diluted suspension, it follows that the viscosity factor does not depend on the size distribution of particles. The physical explanation of this fact is that in an infinite diluted suspension W 1), particles are spaced far apart (in comparison with the particle size), and the mutual influence of particles may be ignored. Besides, under the condition a/h 1, we can neglect the interaction of particles with the walls. It is also possible to show that in an infinite diluted suspension containing spherical particles. Brownian motion of particles does not influence the viscosity of the suspension. However, if the shape of particles is not spherical, then Brownian motion can influence the viscosity of the suspension. It is explained by the primary orientation of non-spherical particles in the flow. For example, thin elongated cylinders in a shear flow have the preferential orientation parallel to the flow velocity, in spite of random fluctuations in their orientation caused by Brownian rotational motion. 

P NMR was used to study motion of tri-(2-ethylhexyl) phosphate, TOP, in polycarbonate at different temperatures and concentrations. Brownian rotational motion was observed in TOP but at two different time scales. If TOP was surrounded by other molecules of plasticizer it was capable of rotational diffusion with apparent activation energy of 56 kJ mol". Isolated TOP molecules (surrounded by polymer) showed a temperature dependent movement. These molecules do not diffuse below glass the transition tempera-... 

Recently, Coffey et al developed a new approach to the problem of Brownian rotational motions of a single-axis rotator in a uniaxial potential. Using very sophisticated mathematical procedures, they were able to obtain the exact analytic solutions for the retardation factors g and gjL in terms of a. The following formula for the parallel retardation factor renders a close approximation to the exact solution for all [Pg.163]

When the regular motion is simply uniform rotation of the absorption and emission dipoles with angular velocity to around the helix axis, one has p(t) - p(0) = cot. For the corresponding random motion, one might have m)2> = 2Dt, where D is the effective diffusion coefficient for Brownian rotation of the transition dipole around the helix axis. When these expressions are incorporated in Eqs. (4.31) and (4.24), the latter becomes a generalization of a relation recently derived using a more cumbersome approach/104-1... 

Abernathy and Sharp employed a similar idea, although in a more simplified form 130). They also worked in terms of a spin Hamiltonian varying with time in discrete steps and let the Hamiltonian contain the Zeeman and the ZFS interactions. They assumed, however, that the ZFS interaction was constant in the molecule-fixed (P) frame and that variation of the Hamiltonian originated only from fluctuation of the P frame with respect to the laboratory frame. These fluctuations were described in terms of Brownian reorientational motion, characterized by a time interval, x, (related to the rotational correlation time x ) and a Gaussian distribution of angular steps. 

When polymer melts, rubbers, or elastomers are cooled down below Tg, they may freeze to glasses (noncrystalline amorphous phases). The rotations motions of the chain segments (micro-Brownian motions) are almost stopped now, and the transparent materials become stiff and (in most cases) brittle. 

Not only do particles in Brownian agitation move rapidly about in the suspension medium, the magnitude of the movements being capable of exact calculation from the foregoing mathematical considerations, but they are likewise undergoing rotational motion due to an unequal distribution of molecular impacts upon the faces of the parts of a particle on each side of its axis of rotation. 

According to the theory developed by Smoluchowski and by Einstein, if a spherical particle of radius r rotates in a liquid of viscosity i), in a short time A/, by an angle Aa, then the mean value of angular rotation A is given by the Brownian equation for rotational motion ... 

In addition to translational Brownian motion, suspended molecules or particles undergo random rotational motion about their axes, so that, in the absence of aligning forces, they are in a state of random orientation. Rotary diffusion coefficients can be defined (ellipsoids of revolution have two such coefficients representing rotation about each principal axis) which depend on the size and shape of the molecules or particles in question28. 

Particles comprising an aerosol move randomly in brownian motion because of the gas molecules impacting on them. The random nature of the molecules striking the particles can also cause the particles to rotate, this brownian rotation being described by the equation (Fuchs, 1964)... 

Four different models for the molecular dynamics have been tested to simulate the experimental spectra. Brownian rotational diffusion and jump type diffusion [134, 135] have been used for this analysis, both in their pure forms and in two mixed models. Brownian rotational diffusion is characterized by the rotational diffusion constant D and jump type motion by a residence time t. The motions have been assumed to be isotropic. In the moderate jump model [135], both Brownian and jump type contributions to the motion are eou-pled via the condition Dx=. ... 

Above 100 K, motional effects on spectrum become pronounced with increasing temperature and, above 230 K, the spectra consist of essentially an isotropic and equally spaced hyperfine triplet, but with different relative intensities. The line shape simulations were carried out by adopting a Brownian rotational diffusion model in order to evaluate the associated (average) rotational correlation time, and its degree of anisotropy, = zpy, /... 

We illustrate this by referring to the motion of a fixed axis rotator, the normal Brownian rotation of which is described by the Langevin equation (see Ref. 8, Chapters 4 and 10) ... 

This technique relies on Brownian motion to cause very small particles suspended in a liquid to undergo the rapid incoherent rotational motion necessary to... 

The translational and rotational motion of a Brownian particle immersed in a fluid continuum is well described by the Stokes-Einstein and Debye equations, respectively. 

Classic Brownian motion has been widely applied in the past to the interpretation of experiments sensitive to rotational dynamics. ESR and NMR measurements of T and Tj for small paramagnetic probes have been interpreted on the basis of a simple Debye model, in which the rotating solute is considered a rigid Brownian rotator, sueh that the time scale of the rotational motion is much slower than that of the angular momentum relaxation and of any other degree of freedom in the liquid system. It is usually accepted that a fairly accurate description of the molecular dynamics is given by a Smoluchowski equation (or the equivalent Langevin equation), that can be solved analytically in the absence of external mean potentials. 

If the particles are not spherical, even in the very dilute limit where the translational Brownian motion would still be unimportant, rotational Brownian motion would come into play. This is a consequence of the fact that the rotational motion imparts to the particles a random orientation distribution, whereas in shear-dominated flows nonspherical particles tend toward preferred orientations. Since the excess energy dissipation by an individual anisotropic particle depends on its orientation with respect to the flow field, the suspension viscosity must be affected by the relative importance of rotational Brownian forces to viscous forces, although it should still vary linearly with particle volume fraction. 

This sort of thermal translation and rotational motion is what is responsible for the random, chaotic Brownian motion observable in microscopic particles. 

Rotational motion can be isotropic or anisotropic (e.g., when spin labels are attached to larger polymer molecules) and analysis of CW EPR spectra most often is quantified by spectral simulations assuming a rotational model of some sort (e.g., isotropic Brownian or uni-axial motion or more complicated models like microscopic order, macroscopic disorder, or MOMD see [19, 21]). 

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