Anisotropic Brownian rotations

Figure 2. Theoretical spectra of NO2 anisotropically rotating about its x axis (b), y axis (c), and z axis (d). The spectra were calculated for Brownian rotational diffusion model by using R// = 5.27 X 10 sec- Rj = 5.27 x 10 sec and T2 =3.0 G. For isotropic rotation (a), R// = Rj = 1.67 X 10 sec l and T2 l=3.0 G were used. The rotational diffusion values used through (a) to (d) correspond to a constant value of r =l x 10 sec v, 9 167 GHz...

In nitroxide free radicals characterized by a hyperfine tensor. A, and g-tensor, g, the interaction between the magnetic moment of an unpaired electron and the magnetic moment of nitrogen nucleus is highly anisotropic. The anisotropy determines the line shape of electron spin resonance (ESR) spectrum of completely immobilized nitrox-ides (Fig. 1), and can be completely averaged out for fast thermal Brownian rotational diffusion in low-viscous media. For the " N nucleus with 7=1 and no other interacting nuclei in the nitroxide, three equidistant lines of equal intensities and widths should be observed in ESR spectra in such media. In fact, even in media of... 

Molecular motions in low molecular weight molecules are rather complex, involving different types of motion such as rotational diffusion (isotropic or anisotropic torsional oscillations or reorientations), translational diffusion and random Brownian motion. The basic NMR theory concerning relaxation phenomena (spin-spin and spin-lattice relaxation times) and molecular dynamics, was derived assuming Brownian motion by Bloembergen, Purcell and Pound (BPP theory) 46). This theory was later modified by Solomon 46) and Kubo and Tomita48 an additional theory for spin-lattice relaxation times in the rotating frame was also developed 49>. 

Generation of the Brownian trajectories for rodlike molecules requires simulation of the anisotropic translational diffusion and rotational diffusion. The rotational and translational diffusion are coupled in this case, however, taking a sufficiently small time step enables the computation of the different components... 

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. 

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]). 

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. 

In real media, nitroxides change their orientations with respect to the external magnetic field due to the Brownian thermal rotational mobility usually characterized by the rotational correlation time, The anisotropic hyperfine interaction between the unpaired electron and nitrogen nucleus is modified by these changes with a frequency dependent on Xj. In this way, frequency-dependent perturbations are generated, which modify the energy levels and transition probabilities in the system. As a result, the line shape of ESR spectra of nitroxides (and of other free radicals in which anisotropic magnetic interactions occur) depends on the correlation time x. ... 

When the relaxation mechanism is the modulation of the magnetic g and A components due to the rotational diffusion of the paramagnetic group (mainly for the nitroxide radicals, and for 5 = y paramagnetic ions), the analysis of the spectra in the fast-slow motion regime provides the correlation time for the rotational motion. An increase in the correlation time corresponds to a decrease in mobility of the paramagnetic probe or label. The evaluation of the correlation time for the rotational motion was performed by simple methods or by computation of the spectra. Different diffusion models can be considered, such as Brownian or jump diffusion models, and the rotational mobility may be considered isotropic or anisotropic. In this latter case, for nitroxide radicals, the main information was obtained from the perpendicular component of the correlation time. Furthermore, a shift of the main rotational axis accounts for the compression of the labels due to other molecules approaching the label at the dendrimer surface. 

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. 

The dielectric spectroscopy of anisotropic fluids started in the 1970s by the extension of the Debye model from isotropic media (described in Appendix D) to uniaxial systems based on statistical mechanical Kubo formalism/ but no quantitative estimates about the critical frequencies or the susceptibilities were obtained. Quantitative estimates were given first on molecules with dipole moments along the long axis/ then for general dipole directions using the rotational Brownian picture in Maier-Saupe mean-field potential. This theory was subsequently refined in the 1990s.i ... 

Before discussing theoretical models for the rheology of fiber suspensions and its connection to fiber orientation, there are three topics that must be discussed Brownian motion, concentration regimes, and fiber flexibility. Brownian motion refers to the random movement of any sufficiently small particle as a result of the momentum transfer from suspending medium molecules. The relative effect that Brownian motion may have on orientation of anisotropic particles in a dynamic system can be estimated using the rotary Peclet number, Pe s y Dm, where y is the shear rate and Ao is the rotary diffusivity, which defines the ratio of the thermal energy in the system to the resistance to rotation. Doi and Edwards (1988) estimated the rotary diffusivity, Ao, to be... 

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