Isotropic rotational diffusion

Isotropic rotational diffusion with one internal degree of freedom... 

The nitroxide moiety of 16-SASL and 16-PC exhibits such a great deal of motion that the rotational correlation time can be calculated (Berliner 1978). The rotational correlation time (assuming isotropic rotational diffusion of the nitroxide fragment) can be calculated from the linear term of the line width parameter ... 

Note that in the entire discussion on spin labels, above, we have assumed isotropic rotational diffusion. In several real situations of biochemical relevance this is not a tenable assumption. Perhaps the most significant one occurs when the nitroxide... 

P(Qol, t) is the conditional probability of the orientation being at time t, provided it was Qq a t time zero. The symbol — F is the rotational diffusion operator. In the simplest possible case, F then takes the form of the Laplace operator, acting on the Euler angles ( ml) specifying the orientation of the molecule-fixed frame with respect to the laboratory frame, multiplied with a rotational diffusion coefficient. Dr. Equation (44) then becomes identical to the isotropic rotational diffusion equation. The rotational diffusion coefficient is simply related to the rotational correlation time introduced earlier, by tr = 1I6Dr. 

It can be shown4 for the simple case of isotropic rotational diffusion that the TCF is a single exponential function of time, decaying with a time constant, rc, the molecular correlation time ... 

It is instinctive to first consider isotropic rotational diffusion of an atom, which is a simple but somewhat accurate description of a molecular hquid like liquid methane just below rmeiting, a Situation wherein the centers of mass of the molecules remain stationary, but the molecules rotate freely about their centers of mass. If we use the polar angle — 9, (p) to define the orientation of the vector d, the probability distribution function, G (see equation 12), which... 

It has been recognized for a long time that the orientation dependence of a vector fixed to a polymer chain could not be represented by a simple isotropic rotational diffusion model. In such a model the orientation is assumed to follow a vector joining the center of a sphere to a point performing a random brownien diffusion on the surface of that sphere. According to this model which describes well the orientation of spherical objects or infinitely thin rigid rods, the OACF is an exponential function... 

With QENS, both the rotational and translational motions of CH4 can be observed. It was found that the rotational motion of CH4 in ZSM-5 can be described by an isotropic rotational diffusion model with a rotational diffusion time constant, Dr (72). The values of Dr for CH4 adsorbed at 250 K in ZSM-5 are of the order of 5 x 10 s . In MD simulations at 400 K, Dr was found to be of the order of 10 s (73). This difference is due to the fact that a radius of gyration of 0.15 nm was used in the computer fits of the QENS profiles. This radius is intermediate between a simple rotation model, with 0.11 nm for the distance between the protons and the center of mass of the methane molecule, and the radius of the channel in which the molecule performs oscillations. 

The Arrhenius plots of are given in Fig. 5. From the linear-relationship between In z and T-l(K-l), the activation energy of 1.4 kcal-mol-l was evaluated for the nearly isotropic rotational diffusion of NO2 in the X-type zeolite. 

For a molecule, macromolecule or supramolecular complex undergoing isotropic rotational diffusion, Eq. (7.18) reduces to [24]... 

Fig. 3.2.5 [Miill] Deuteron wideline spectra for different motional mechanisms and times tc-The angle between the axis of rotation and C—bond (principal axis Z of the quadfupole coupling tensor) is denoted by d. (a) Twofold jump with 9 — 60° (b) Twofold jump with 9 = 180° (flips of p phenylene). (c) Three-fold jump with 9 = 109° (rotation of a methyl group), (d) Rotational diffusion on a cone 9 = 109°. (e) Tetrahedral jump, (f) Isotropic rotational diffusion.

For many cases of molecular reorientation in liquids, the isotropic rotational diffusion model is applicable and the dipole-dipole interaction provides the dominant relaxation mechanism. Then, the spectral densities are given by [FARl]... 

The dependence of the cp line shape on the correlation time of isotropic rotational diffusion can be used to determine the rotational mobility of system components in... 

Fig. 9. Simulated MAS spectra under variation of the sample spinning rate ((jJj./2tt = 0-15 kHz). The calculations are based on given anisotropic chemical shielding data ((7ii = —15 kHz, (722 =—5 kHz, 0-33 = 20kHz) and isotropic rotational diffusion (r = 0.1 ms) (reproduced from Ref. [23] with permission).

Our data of T in the coil form in Figure 10 can nearly be put over the master curve for T of polystyrene in the organic solvents, as shown in Figure 11, with the vfflues of the scaling parameters p and q p = 1.305 and q = 0.045(K ), by which the data of T y P in the coil form (in Figure 10) can also be put over the master curve for T P of polystyrene in the organic solvents. Therefore, the molecular motion of (MA-St)p in the coil form may be described in terms of the conformationed jump model combined with isotropic rotational diffusion, when the ratio Xp/Xp seems to be nearly 0.07. 

It may be pointed out that Wallach s result in the limit of rapid internal rotation means that the quadrupole.interaction in a specific bond is effectively reduced by the factor (3 cos 0j - 1) while the correlation time remains unchanged i,e. in the absence of chemical exchange still equals Tr for isotropic rotational diffusion). Since there appears to be some uncertainty concerning this point it is reiterated here. 

Fig. 2. (a) Dependence of nitroxide spectra on the rotational correlation time, x, for isotropic rotation diffusion, (b) molecular axes system of a nitroxide radical. R is a hydrogen atom tor TEMPO or a functional group in other nitroxides. (c) Dependence of the extreme separation, 2A and the relative anisotropy, AArei, on the rotational correlation... 

Once is determined, the appropriate values of and may be set relative to it. For most nitroxide spectra, = 0.7 (rounded to the next highest odd integer) is sufficient, although this factor can be reduced somewhat at frequencies at or < 9 GHz, where the spectrum is dominated by the nearly axial " N hyper-fme anisotropy. The ratio between and depends on the rotational anisotropy parameter N. For an isotropic rotational diffusion tensor, it is generally sufficient to take ATnm=0.6 when > 10 (logjo one may generally use... 

For nitroxide spin labels, assuming isotropic rotational diffusion and near axial symmetry of the coupling tensor, i.e. A v A y, B and C are given by... 

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