Dominant motion relaxation

We now discuss the translation of the MC time-step into physical time units. It is desirable to map the mobility of the lattice model (due to jumps of the effective monomers) onto the average jump rate of the torsional degrees of freedom, since these motions dominate the relaxation of the overall configuration of the chain. This means that we must allow for a temperature-depen-dent time unit tmc(T) which one attempted MCS per monomer corresponds to, via the formula ... 

Direct H—C dipole-dipole coupling dominating 13C relaxation (for protonated carbons) information about molecular shapes and motion are contained in the experimental spin-lattice relaxation times. 

In the theory of deuteron spin-lattice relaxation we apply a simple model to describe the relaxation of the magnetizations T and (A+E), for symmetry species of four coupled deuterons in CD4 free rotators. Expressions are derived for their direct relaxation rate via the intra and external quadrupole couplings. The jump motion between the equilibrium positions averages the relaxation rate within the same symmetry species. Spin conversion transitions couple the relaxation of T and (A+E). This mixing is included in the calculations by reapplying the simple model under somewhat different conditions. The results compare favorably with the experimental data for the zeolites HY, NaA and NaMordenite [6] and NaY presented here. Incoherent tunnelling is believed to dominate the relaxation process at lowest temperatures as soon as CD4 molecules become localized. 

In physisorbed systems, the electronic ground state of the adsorbate is only weakly perturbed upon adsorption. The physisorption potential is rather flat and shallow, i.e. the restoring force of the vertical motion of adatoms is weak, and thus, the corresponding adsorbate-substrate vibrations are low-frequency modes " Radiative phonon processes are expected to dominate the relaxation and coupling processes. 

Another disadvantage of the small, hydrophilic agents is that they tumble very rapidly in the extracellular fluid. In water at 25 °C, for example, Gd-DTPA has a rotational correlation time (rR) of 58 ps as determined by the fitting of NMRD data [3], and 105 ps by EPR simulation in the VO++-DTPA analog [4]. This very rapid motion dominates the relaxivity of the PCA in the frequency range of typical clinical interest (42-63 MHz). The reasons for this dominance of rR can be traced to the fact that small Gd3+ chelates like Gd-DTPA have a relaxivity at clinical frequencies that is determined predominantly by an inner sphere process for Gd-DTPA at 50 MHz, Chen et al. calculate that the relaxivity in water is 43 % inner sphere, 25 % second sphere, and 32 % outer sphere [4]. In turn, the inner sphere contribution to relaxivity is often modeled by the Solomon-Bloem-bergen equations [5,6]... 

This result suggests, if it is assumed that a C-H heteronuclear dipolar relaxation mechanism is operative, that methyl protons dominate the relaxation behavior of these carbons over much of the temperature range studied despite the 1/r dependence of the mechanism. The shorter T] for the CH as compared to the CH2 then arises from the shorter C-H distances. Apparently, the contributions to spectral density in the MHz region of the frequency spectrum due to backbone motions is minor relative to the sidegroup motion. The T p data for the CH and CH2 carbons also give an indication of methyl group rotational frequencies. 

The investigation of viscoelasticity of dilute blends confirms that the reptation dynamics does not determine correctly the terminal quantities characterising viscoelasticity of linear polymers. The reason for this, as has already been noted, that the reptation effect is an effect due to terms of order higher than the first in the equation of motion of the macromolecule, and it is actually the first-order terms that dominate the relaxation phenomena. Attempts to describe viscoelasticity without the leading linear terms lead to a distorted picture, so that one begins to understand the lack of success of the reptation model in the description of the viscoelasticity of polymers. Reptation is important and have to be included when one considers the non-linear effects in viscoelasticity. 

Which solvent motions represented in the bend force spectrum [see Equation (6)] are important in inducing the rate limiting transition from the OH stretch into the HOD bend For the same D2O model, Marti et al. (69) have found that the D20 librational spectrum is peaked at 400 cm 1 with a FWHM of 300 cm 1. Thus, the dominant motions in the bend force spectrum at 530 cm 1 responsible for the calculated relaxation are the solvent librations. 

For quadrupolar nuclei with dominant quadrupolar relaxation, the relaxation rates for longitudinal and transverse magnetization, Riq and R2q> are identical in the motional narrowing limit. This situation is met in many cases for Li, while for Li the relation R2>R will generally be found. For both nuclei, however, chemical exchange processes are quite common and the strongest contribution to Ri will then originate from this source. 

There is no straightforward and completely rigorous procedure for determining the relative combinations of the various relaxation mechanisms, except where one mechanism clearly dominates (e.g., if the maximum possible nuclear Overhauser effect (NOE) for a resonance is obtained, dipolar relaxation must dominate its relaxation or an increase in relaxation rate in proportion to the square of the applied field must be due to chemical shift anisotropy). Hence, the study of molecular motion in proteins from relaxation data is performed most readily on nuclei directly bonded to H, and so principally relaxed via dipole-dipole interactions (see Section 4(e)(iii)). 

The last term in Gilbert s equation above is the aligning term and consequently the term of interest in relaxation of the magnetization with respect to a magnetic field. For low damping (a< l), we see that precessional motion is the dominant motion and that the relaxation time is approximately given by... 

In the papers by Berk et al. (42) the EPR linewidths of triplet excitons in single-crystal pyrene at room temperature have been measured in experiments performed at 24 GHz. The data are fitted to a formula first presented by Reineker (43) in a theory based on the Haken-Strobl-Reineker model of exciton motion (the Haken-Strobl-Reineker model can be applied for triplet excitons because they have an exciton bandwidth small in comparison with the thermal energy ksT for more details see the review paper by Reineker (44)). This formula was rederived in the paper by Berk et al. (42) more directly from Blume s stochastic Liouville formalism (45). The agreement was excellent. This result again implied that the dominant spin-relaxation mechanism in pyrene, as in anthracene and presumably in similar molecular crystals, results from hopping between differently oriented molecules in the unit cell. 

The density and temperature dependence of some of the slow relaxations in PE were studied by Bharadwaj and Boyd (369) using NVT simulations with a relatively simple united atom model of the polsrmer. Runs of six nanoseconds for each of several conditions were performed, resulting in a comprehensive view of how relaxation processes depend on p and T. One insight provided by this work is that reorientation motion of the chains seems to require barrier crossings rather than free-volume adjustments. This result suggests that there is a need for analysis tools that are capable of cross-correlating different types of motions with structural or packing aspects of amorphous polymers so as to unravel the complex interplay of phenomena that determine the dominant motions in these systems. 

In any case, whatever the model, since tube-renewal is more important (compared to reptation) and accelerates the motion more efficiently for short chains than for long chains, it introduces additional molecular weight dependences of the dynamical quantities, and certainly contributes to the experimental deviation of the viscosity/molecular weight exponent from the reptation value 3. All treatments, including tube renewal, exhibit such deviations which vanish for asymptotically long chains. Detailed quantitative tests are, however, very difficult to perform when tube-renewal is taken into account, polydispersity becomes an essential parameter (the shortest mechanism, reptation and tube-renewal, dominates the relaxation process). No complete set of experiments, either for diffusion or for viscoelasticity, with constant polydispersity at all molecular weights, are presently available. 

The high degree of resolution in the CP/MAS NMR spectra also permits the analysis of the relaxation behaviour of individual carbon atoms. The T1 relaxation is seldom solely determined by a single motion of a single intermolecular interaction. Nevertheless, if the dominant source is known, T1 can provide useful information. As an example the relaxation behaviour of celluloseacetate in both acetone solution and in the solid state will be discussed. The experimental data in Table 2 give chemical shifts and relaxation T1 information for the above system 21 . 

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