Mode-localized motion

Turning from chemical exchange to nuclear relaxation time measurements, the field of NMR offers many good examples of chemical information from T, measurements. Recall from Fig. 4-7 that Ti is reciprocally related to Tc, the correlation time, for high-frequency relaxation modes. For small- to medium-size molecules in the liquid phase, T, lies to the left side of the minimum in Fig. 4-7. A larger value of T, is, therefore, associated with a smaller Tc, hence, with a more rapid rate of molecular motion. It is possible to measure Ti for individual carbon atoms in a molecule, and such results provide detailed information on the local motion of atoms or groups of atoms. Levy and Nelson " have reviewed these observations. A few examples are shown here. T, values (in seconds) are noted for individual carbon atoms. 

Despite the difficulty cited, the study of the vibrational spectrum of a liquid is useful to the extent that it is possible to separate intramolecular and inter-molecular modes of motion. It is now well established that the presence of disorder in a system can lead to localization of vibrational modes 28-34>, and that this localization is more pronounced the higher the vibrational frequency. It is also well established that there are low frequency coherent (phonon-like) excitations in a disordered material 35,36) These excitations are, however, heavily damped by virtue of the structural irregularities and the coupling between single molecule diffusive motion and collective motion of groups of atoms. 

There were different generalisations of the reptation-tube model, aimed to soften the borders of the tube and to take into account the underlying stochastic dynamics. It seems that the correct expansion of the Doi-Edwards model, including the underlying stochastic motion and specific movement of the chain along its contour - the reptation mobility as a particular mode of motion, is presented by equations (3.37), (3.39) and (3.41). In any case, the introduction of local anisotropy of mobility of a particle of chain, as described by these equations, allows one to get the same effects on the relaxation times and mobility of macromolecule, which are determined by the Doi-Edwards model. 

To construct the model, it has been assumed that the amorphous polymer regions posses an approximately paracrystalline order with chain bundles locally parallel. A penetrant molecule may diffuse through the matrix of the polymer by two modes of motion, Fig. 5-2. 

Fig. 9 Eigenvalues of the energy-difference Hessian computed at the Franck-Condon point of benzene in the 28-dimensional space orthogonal to the pseudo-branching plane. The labels refer to the most similar normal modes of So benzene (Wilson s convention). The dominant local motions are indicated in boxes (reprinted with permission from [31])...

Secondary relaxations occur in the sub-glassy region and it is thought that it originates from localized motions of some parts of the glass structure. The y5-relaxation mode itself gets decoupled from the modes responsible for or-relaxation which is associated with the glass transition. y9-relaxations are often associated with local ionic motions, particularly in... 

One of the principal advantages of CPMAS experiments is that resolution in the solid state allows individual-carbon relaxation experiments to be performed. If a sufficient number of unique resonances exist, the results can be interpreted in terms of rigid-body and local motions (e.g., methyl rotation, segmental modes in polymers, etc.) (1,2). This presents a distinct advantage over the more common proton relaxation measurements, in which efficient spin diffusion usually results in averaging of relaxation behavior over the ensemble of protons to yield a single relaxation time for all protons. This makes interpretation of the data in terms of unique motions difficult. 

Photophysical and photochemical processes in polymer solids are extremely important in that they relate directly to the functions of photoresists and other molecular functional devices. These processes are influenced significantly by the molecular structure of the polymer matrix and its motion. As already discussed in Section 2.1.3, the reactivity of functional groups in polymer solids changes markedly at the glass transition temperature (Tg) of the matrix. Their reactivity is also affected by the / transition temperature, Tp, which corresponds to the relaxation of local motion modes of the main chain and by Ty, the temperature corresponding to the onset of side chain rotation. These transition temperatures can be detected also by other experimental techniques, such as dynamic viscoelasticity measurements, dielectric dispersion, and NMR spectroscopy. The values obtained depend on the frequency of the measurement. Since photochemical and photophysical parameters are measures of the motion of a polymer chain, they provide means to estimate experimentally the values of Tp and Tr. In homogeneous solids, reactions are related to the free volume distribution. This important theoretical parameter can be discussed on the basis of photophysical processes. 

Figure 1 The normal modes of motion for the three stretch modes of DCCH. (Adapted from T. A. Holme and R. D. Levine, Chem. Phys. Lett. 150 393 (1988).) The displacement of the atoms in each mode is shown by an arrow. Note that while all atoms contribute to all modes, the respective contributions do vary and the v, mode is almost a localized CH stretch. For recent studies of the overtone spectroscopy of HCCH and its isotopomers see J. Lievin, M. Abbouti Temsamani, P. Gaspard, and M. Herman, Chem. Phys. Lett. 190 419 (1995) M. J. Bramley, S. Carter, N. C. Handy, and 1. M. Mills, J. Mol. Spectrosc. 160 181 (1993) B. C. Smith and J. S. Winn, J. Chem. Phys. 89 4638 (1988) K. Yamanouchi, N. Ikeda, S. Tuschiya, D. M. Jonas, J. K. Lundberg, G. W. Abramson, and R. W. Field, J. Chem. Phys. 95 6330 (1991).)...

For an individual system D depends on the parameters in the Hamiltonian. One expects D to increase with increasing coupling strength between the molecular modes, all else being the same. The relevant rate near the transition is a microscopic frequency of local motion, o>, that depends on the local density of states. Thus in this regime,... 

When classical trajectories are calculated for the H2O model with two O—H stretches discussed in section 4.3.2 (p. 76), local-mode type motion is found for which energy is trapped in individual O—H bonds (Lawton and Child, 1979, 1981). The trajectories are quasiperiodic and application of the EBK semiclassical quantization condition [Eq. (2.72)] results in pairs of local-mode states in which there are n quanta in one bond and m in the other or vice versa. The pair of local-mode states (n,m) and (m,n) have symmetry-related trajectories which have the same energy. The local-mode trajectory for the (5,0) state is depicted in Figure 4.6d. 

---