Diffusional correlation time

The diffusional correlation time to depends on the size of both the metal and the ligand-containing moieties, according to their diffusion coefficients, Dm and Di, and on the minimal distance that can be achieved between the ligand and the metal ion, called distance of closest approach, d [8,9]. 

Thus, the line shape determining R2 correlates with the rotational diffusional correlation time rc. 

For a globular protein of approximately spherical shape, the isotropic tumbling rate can be characterized by the rotational diffusional correlation time, tc, as described above. Assuming that the protein fits in a sphere of radius r, then the viscosity (rf) and temperature (T) of the sample determines rc. 

Measurements of Tj have been made (269,270) to probe the structure of the second coordination sphere around Cr(acac)3. Acetone, chloroform, and methylene chloride were chosen as second sphere ligands. The shortening of their Tj values is found to be independent of solute concentration, and the value measured is determined by the diffusional correlation time of the solute molecule. No detectable second coordination sphere therefore exists in these solvents. However, methanol forms a discrete second sphere, and above a certain concentration the measured Ty values vary linearly with solute concentration. The observed Tj values are compatible with a model having a coordination number of 8, a Cr-CH3 separation of 700 pm, and an equilibrium constant for displacing solvent (CHCI3) of ca. 10. This equilibrium constant value is consistent with values previously... 

When both dipolar and scalar interactions are present, p, the nuclear-electron coupling parameter, has to be calculated as a function of Because p depends also on the spectral density near the electron Larmor frequency, p can be plotted as a function of frequency for various values of In practice, it proves to be convenient to present p as a function of (cDsT,) where t,, the diffusional correlation time, is given by—... 

Inequality (6.67) is the softest criterion of perturbation theory. Its physical meaning is straightforward the reorientation angle (2.30) should be small. Otherwise, a complete circle may be accomplished during the correlation time of angular momentum and the rotation may be considered to be quasi-free. Diffusional theory should not be extended to this situation. When it was nevertheless done [268], the results turned out to be qualitatively incorrect orientational relaxation time 19,2 remained finite for xj —> 00. In reality t0j2 tends to infinity in this limit [27, 269]. 

In the diffusion-controlled regime different equations should be derived, taking into account that the interaction eneigy is now modulated by fluctuations in r between d and infinity. In this case the kind of integration to be performed depends on the model assumed for the diffusional behavior of the system. According to one of the most commonly used models for diffusion [11,12], the following equations have been derived when to is the dominant correlation time ... 

The mean diffusional structure (structure D) is that obtained with an exposure time of the order of the orientational correlation time (or of the translational counterpart) of the molecule. This structure has also been explored by computer simulation experiments,but the agreement between the results and experiment is not always satisfactory. 

Here, tr and tj are the correlation times for the diffusional rotation and the isotropic random motion, respectively. 0r is the angle between the C—H internuclear vector and the z axis. 

Here, tr, tj and ti are the correlation times for the diffusional rotation, libration and isotropic motion, respectively. and are halves of the vertical angles of the cones associated with the corresponding motions. The librational motion is defined here as time-dependent reorientation in which the zi axis randomly changes in direction within the larger cone with the Z2 axis as shown in Fig. 3.6. 

One characteristic of in situ NMR experiments is that there is typically a wide range of correlation times characterizing molecular motion. Some species will be essentially immobile as a result of strong chemisorption to the catalyst surface or physical entrapment, as in the case of a coke molecule. Other species may reside exclusively in the gas phase or else be partitioned into adsorbed and gas phase populations in slow exchange on the NMR time scale owing to diffusional constraints. Figure 7 shows an example. At high temperatures, methanol and dimethyl ether are partitioned between the gas phase and adsorbed phase on zeolite HZSM-5 [31 ]. For many adsorbates on zeolites, especially at reaction... 

Finally it is interesting to point out the good agreement between correlation times 2 and 3 in Figure 3. Correlation time 3 has been computed from the diffusional contribution to the proton spin-lattice relaxation time measured for the CD OH - X OH system, after the proton exchange contribution has been removed, whereas correlation time 2 has been obtained, in a straight forward manner, for the CH OD-X-OD system. 

In polymers, due to the constraint resulting from the connectivity of the chain, the local motions are usually too complicated to be described by a single isotropic correlation time x, as discussed in chapter 4. Indeed, fluorescence anisotropy decay experiments, which directly yield the orientation autocorrelation function, have shown that the experimental data obtained on anthracene-labelled polybutadiene and polyisoprene in solution or in the melt cannot be represented by simple motional models. To account for the connectivity of the polymer backbone, specific autocorrelation functions, based on models in which conformational changes propagate along the chain according to a damped diffusional process, have been derived for local chain... 

If the mode of the time-fluctuation of the C-H vector under consideration is designated, the spectral densities appearing in eqns (33) and (34) can be formulated. For example, if the C-H vector r undergoes random diffusional rotation while the internuclear length is held constant, i.e. if the vector r undergoes rotational diffusional motions around the x, y, and z-axes with a common rotational diffusion constant R, all of the correlation functions of the orientation functions of r can be described by a correlation time Tc as... 

These relations are together generally referred to as single correlation time theory and used to correlate the relaxation phenomena for monomeric substances in solution to their molecular motion. Nevertheless, in the case of macromolecules, the C-H vectors in the molecular structure are not thought to undergo such isotropic spherical diffusional rotation. In fact, the relaxation phenomena of macromolecules seldom follow these relations and particular modes of motion must be assumed for the internuclear vectors considering the detailed molecular structure. 

As mentioned above, the relaxation phenomena of macromolecules seldom follow the single correlation time theory dictated by eqn (36). In such cases, a wide distribution is usually introduced in the correlation time. However, as discussed elsewhere, the distribution of correlation time not only fails to explain the temperature dependencies of Ti, T2 and the NOE of the non-crystalline components observed by scalar decoupled NMR on linear polyesters and polyethylene, but also overlooks the intrinsic motion of long-chain molecules. On the contrary, the 3r theory dictated by eqn (41) was found to be very effective to describe such temperature dependencies of the relaxation parameters. Irrespectively of whether the motional mode assumed in the 3t model for the C-H vector is really true, the concept that the C-H vector in macromolecules involves plural independent diffusional motions with discretely different correlation times is very useful to explain the magnetic relaxation phenomena of macromolecules, as will be shown later. 

EXAMPLE 18.6 Autocorrelation times. The correlation time m/ is the time required for the velocity autocorrelation function to reach 1/e of its initial value. The correlation time is short when the mass is small or when the friction coefficient is large. The correlation time for the diffusional motion of a small protein of mass m = 10,000 g moP is in the picosecond time range (see Equation (18.56)) ... 

It is customary to describe the reorlentational kinetics by a time reflecting the rate at which the CF decays to its long time limit. In the case of a purely diffusional motion the best choice is the relaxation time T which appears in the onent of the equation 19 and 21 for the translational and the rotational CF s. For CF s which are not exponential we define a correlation time X as the area under the normalized CF ... 

Intcrmolecular dipole-dipole relaxation depends on the correlation time for translational motion rather than rotational motion. Intermolecular dipole-dipole interactions arise from the fluctuations which are caused by the random translational motions of neighboring nuclei. The equations describing the relaxation processes are similar to those used to describe the intramolecular motions, except is replaced by t, the translation correlation time. The correlation times are expressed in terms of diffusional coefficients (D), and t, the rotational correlation time and the translational correlation time for Brownian motion, are given by the Debye-Stokes-Einstein theory ... 

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

Confined flows typically exhibit laminar-flow regimes, i.e. rely on a diffusion mixing mechanism, and consequently are only slowly mixed when the diffusion distance is set too large. For this reason, in view of the potential of microfabrication, many authors pointed to the enhancement of mass transfer that can be achieved on further decreasing the diffusional length scales. By simple correlations based on Fick s law, it is evident that short liquid mixing times in the order of milliseconds should result on decreasing the diffusion distance to a few micrometers. 

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