Correlated internal bond rotations

The second term in the above expression represents a cross-term between the two types of motion, but is zero except when rriL = 0. Unless it is necessary to calculate Jo (a ), or the spin-spin relaxation time, the overall correlation functions will be approximated by linear combinations of the products of the correlation functions for each motion [i.e., retain only the first term in Eq. (8.10)]. To discuss the superimposed rotations model, it is assumed that internal rotations about different C-C bonds are independent and use additional coordinate frames to carry out successive transformations from the local a frame to the molecule-fixed frame. Free rotational diffusion will be used to describe each bond rotation in the following section. 

Here a simple extension of the master equation method developed for macromolecules in solution [8.4, 8.22] is used to model correlated internal motions in liquid crystals. By explicitly generating all of the possible conformations in a mesogen and weighing these conformers according to their equilibrium probabilities imposed by the nematic mean field [8.12, 8.14], those improbable conformations that were obtained based on the assumption of independent rotations about different C-C bonds may be effectively eliminated. Thus, internal rotations about different axes are considered to be highly correlated. A similar approach has been used to model correlated internal motions in lamellar mesophases of lyotropic liquid crystals [8.20]. All of the studies still retain the simplifying assumption of decoupling internal rotations from the reorientation of the whole molecule. First, the decoupled model of correlated internal motions is considered. 

Chachaty and co-workers [8.20, 8.37, 8.38] were first to describe correlated internal motions in alkyl chains of surfactant molecules that form lyotropic liquid crystals. The last section described an extension of the master equation method of Wittebort and Szabo [8.4] to treat spin relaxation of deuterons on a chain undergoing trans-gauche jump rotations in liquid crystals. This method was also followed by Chachaty et al. to deal with spin relaxation of nuclei in surfactants. However, they assumed that the conformational changes occur by trans-gauche isomerization about one bond at a time. In their spectral density calculations (see Section 8.3.1), they used a transition rate matrix that was constructed from the jump rate Wi, W2, and Ws about each bond. Since W3 is much smaller than Wi and W2, the time scale of internal motions was practically governed by Wi and W2 of each C-C bond. Since... 

In the previous discussion, the electron-nucleus spin system was assumed to be rigidly held within a molecule isotropically rotating in solution. If the molecule cannot be treated as a rigid sphere, its motion is in general anisotropic, and three or five different reorientational correlation times have to be considered 79). Furthermore, it was calculated that free rotation of water protons about the metal ion-oxygen bond decreases the proton relaxation time in aqua ions of about 20% 79). A general treatment for considering the presence of internal motions faster than the reorientational correlation time of the whole molecule is the Lipari Szabo model free treatment 80). Relaxation is calculated as the sum of two terms 8J), of the type... 

The parameter t is given in Eq. A-9 in the Appendix, as a function of the correlation time, t associated with internal motion. One of the input parameters is the angle j3, formed between the relaxation vector (C—H bond) and the internal axis of rotation (or jump axis), namely the C-5—C-6 bond. The others are correlation times t0 and r, of the HWH model, obtained from the fit of the data for the backbone carbons. The fitting parameters for the two-state jump model are lifetimes ta and tb, and for the restricted-diffusion model, the correlation time t- for internal rotation. The allowed range of motion (or the jump range) is defined by 2x for both models (Eqs. A-4 and A-9). 

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