The Jump Rotation Model

The assumption of free rotation about each C-C bond in an alkyl chain can give conformations of molecules that are precluded on grounds of excluded-volume effects. Following Tsutsumi, a jump model was employed [8.11] to describe trans-gauche isomerisms in the chain of liquid crystals by allowing jumps about one bond at any one time. To evaluate internal correlation functions gi t), not only the equilibrium probabilities of occupation given by Eq. (8.4) are needed, but also the conditional probability P(7, t 7o,0), where 7 and 70 denote one of the three equilibrium states (1, 2, 3) at times t and zero, respectively, 

It is noted that these conditional probabilities obey the relation 

By adopting the simplifying assumption of 6mm (i-e-i ignoring the off-diagonal elements of A, which is true in the free rotation limit), the following is found from Eq. (8.11)  

It is noted that in the limit cr 1, the above gi t) reduces to that given by the free rotation model [Eq. (8.13)]. In the fast motion limit and (3m,i = 0, it is found, for example, that 

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