Relaxation correlation times

The movements capable of relaxing the nuclear spin that are of interest here are related to the presence of unpaired electrons, as has been discussed in Section 3.1. They are electron spin relaxation, molecular rotation, and chemical exchange. These correlation times are indicated as rs (electronic relaxation correlation time), xr (rotational correlation time), and xm (exchange correlation time). All of them can modulate the dipolar coupling energy and therefore can cause nuclear relaxation. Each of them contributes to the decay of the correlation function. If these movements are independent of one another, then the correlation function decays according to the product... 

Fig. 30. The local segmental relaxation correlation time Tc of high molecular weight PS as a function of temperature obtained by two-dimensional exchange nmr (154,155) up to long times exceeding 100 s, and compared with the shift factor ot,s from time-temperature superposition of recoverable creep compliance Jr( ) curves of another high molecular weight...

Figure 9.3 Mip-mediated surface water displacement from polystyrene (PS). PS beads were labeled with tethered nitroxide spin labels. Upon excitation during electron spin resonance, polarization was transferred to the surrounding water molecules within 15 A. Polarized water molecules present relaxation correlation times (t in Table inset) that reflect the local environment. Longer times denote more confinement and lower diffusivily. Only one adhesive protein, mfp-3s, is capable of increasing the surface water relaxation time, presumably by adsorbing directly onto the PS surface. Mfp-3s is notable for having a high hydrophobicity (inset table), as indicated by the negative hydropathy value, which denotes a normalized per average free energy of transfer from water to the non-polar solvent Data from ref. 22.

For spin-spin (scalar) coupling between spins I and S, where I = j and S >, the interaction Hamiltonian involves the scalar coupling tensor J. The scalar relaxation of the nucleus I can arise from either a time-dependent 5 or a time-dependent J ( first kind ). If the relaxation time (T (S)) of the nucleus S is short compared with 1//, where J is the scalar coupling constant, the nucleus I sees the average of the spin-spin interaction and does not show the expected multiplet, but a single line. The scalar relaxation correlation time tg is then equal to T (S), and the scalar spin-spin contributions to the longitudinal and transverse relaxation times of the I nucleus are given by... 

Figure Bl.13.2. Spin-lattice and spin-spm relaxation rates (R and/ 2> respectively) for a carbon-13 spin directly bonded to a proton as a fiinction of correlation time at the magnetic fields of 7 and 14 T.

Small molecules in low viscosity solutions have, typically, rotational correlation times of a few tens of picoseconds, which means that the extreme narrowing conditions usually prevail. As a consequence, the interpretation of certain relaxation parameters, such as carbon-13 and NOE for proton-bearing carbons, is very simple. Basically, tlie DCC for a directly bonded CH pair can be assumed to be known and the experiments yield a value of the correlation time, t. One interesting application of the measurement of is to follow its variation with the site in the molecule (motional anisotropy), with temperature (the correlation... 

N-protonation the absolute magnitude of the Ad values is larger than for Af-methylation <770MR(9)53>. Nuclear relaxation rates of and have been measured as a function of temperature for neat liquid pyridazine, and nuclear Overhauser enhancement has been used to separate the dipolar and spin rotational contributions to relaxation. Dipolar relaxation rates have been combined with quadrupole relaxation rates to determine rotational correlation times for motion about each principal molecular axis (78MI21200). NMR analysis has been used to determine the structure of phenyllithium-pyridazine adducts and of the corresponding dihydropyridazines obtained by hydrolysis of the adducts <78RTC116>. 

Figure 8 Effects of spin diffusion. The NOE between two protons (indicated by the solid line) may be altered by the presence of alternative pathways for the magnetization (dashed lines). The size of the NOE can be calculated for a structure from the experimental mixing time, and the complete relaxation matrix, (Ry), which is a function of all mterproton distances d j and functions describing the motion of the protons, y is the gyromagnetic ratio of the proton, ti is the Planck constant, t is the rotational correlation time, and O) is the Larmor frequency of the proton m the magnetic field. The expression for (Rjj) is an approximation assuming an internally rigid molecule.

In spin relaxation theory (see, e.g., Zweers and Brom[1977]) this quantity is equal to the correlation time of two-level Zeeman system (r,). The states A and E have total spins of protons f and 2, respectively. The diagram of Zeeman splitting of the lowest tunneling AE octet n = 0 is shown in fig. 51. Since the spin wavefunction belongs to the same symmetry group as that of the hindered rotation, the spin and rotational states are fully correlated, and the transitions observed in the NMR spectra Am = + 1 and Am = 2 include, aside from the Zeeman frequencies, sidebands shifted by A. The special technique of dipole-dipole driven low-field NMR in the time and frequency domain [Weitenkamp et al. 1983 Clough et al. 1985] has allowed one to detect these sidebands directly. 

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. 

Usually, nuclear relaxation data for the study of reorientational motions of molecules and molecular segments are obtained for non-viscous liquids in the extreme narrowing region where the product of the resonance frequency and the reorientational correlation time is much less than unity [1, 3, 5]. The dipolar spin-lattice relaxation rate of nucleus i is then directly proportional to the reorientational correlation time p... 

Cole and Davidson s continuous distribution of correlation times [9] has found broad application in the interpretation of relaxation data of viscous liquids and glassy solids. The corresponding spectral density is ... 

The measurement of correlation times in molten salts and ionic liquids has recently been reviewed [11] (for more recent references refer to Carper et al. [12]). We have measured the spin-lattice relaxation rates l/Tj and nuclear Overhauser factors p in temperature ranges in and outside the extreme narrowing region for the neat ionic liquid [BMIM][PFg], in order to observe the temperature dependence of the spectral density. Subsequently, the models for the description of the reorientation-al dynamics introduced in the theoretical section (Section 4.5.3) were fitted to the experimental relaxation data. The nuclei of the aliphatic chains can be assumed to relax only through the dipolar mechanism. This is in contrast to the aromatic nuclei, which can also relax to some extent through the chemical-shift anisotropy mechanism. The latter mechanism has to be taken into account to fit the models to the experimental relaxation data (cf [1] or [3] for more details). Preliminary results are shown in Figures 4.5-1 and 4.5-2, together with the curves for the fitted functions. 

Table 4.5-1 gives values for the fit parameters and the reorientational correlation times calculated from the dipolar relaxation rates. 

Fig. 7. Theoretical line shapes resulting from an interchange between two NMR frequencies

Up to now it has been tacitly assumed that each molecular motion can be described by a single correlation time. On the other hand, it is well-known, e.g., from dielectric and mechanical relaxation studies as well as from photon correlation spectroscopy and NMR relaxation times that in polymers one often deals with a distribution of correlation times60 65), in particular in glassy systems. Although the phenomenon as such is well established, little is known about the nature of this distribution. In particular, most techniques employed in this area do not allow a distinction of a heterogeneous distribution, where spatially separed groups move with different time constants and a homogeneous distribution, where each monomer unit shows essentially the same non-exponential relaxation. Even worse, relaxation... 

Fig. 13. Calculated 2H solid echo spectra for log-Gaussian distributions of correlation times of different widths. Note the differences of the line shapes for fully relaxed and partially relaxed spectra. The centre of the distribution of correlation times is given as a normalized exchange rate a0 = 1/3tc. For deuterons in aliphatic C—H bonds the conversion factor is approximately 4.10s sec-1...

Polycarbonate (PC) serves as a convenient example for both, the direct determination of the distribution of correlation times and the close connection of localized motions and mechanical properties. This material shows a pronounced P-relaxation in the glassy state, but the nature of the corresponding motional mechanism was not clear 76 80> before the advent of advanced NMR techniques. Meanwhile it has been shown both from 2H NMR 17) and later from 13C NMRSI) that only the phenyl groups exhibit major mobility, consisting in 180° flips augmented by substantial small angle fluctuations about the same axis, reaching an rms amplitude of 35° at 380 K, for details see Ref. 17). 

Fig. 24. Fully and partially relaxed methyl-deuteron spectra of polycarbonate, proving the heterogeneous nature of the distribution of correlation times...

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