Spin Relaxation by Correlated Internal Motions

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. 

Evaluation of this expression requires knowing the conditional probability Piio[ LM t QLM (0))0] that, at time t, the molecule has configuration i and orientation Q lm and when t = 0, the molecule has configuration I and orientation lm 0)- Under the decoupling assumption, the conditional probability can be expressed as the product of configuration and orientation 

it is necessary to find the configuration conditional probability p(i,t /,0) = Piioit) in Eq. (8.30) in order to obtain the internal correlation functions. This is achieved by following the master equation method of Wittebort and Szabo [8.4]. Conformational transitions between N distinct configurations occur via one-bond, two-bond, or three-bond motion [8.24] in the chain. These bond motions are characterized by phenomenological rate constants and fcs, respectively. In general, there is more than 

Rij must also satisfy the detailed-balance principle, 

Because of this requirement and Vij = rji Rij = Peq i)rij, Table III of Ref. [8.23] provides an example of how to construct an R matrix for 5CB using a diamond lattice for the chain. The master equation can be solved [8.4] as an eigenvalue problem to give the conditional probability 

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