Mode Analysis and Generalized Rouse Model

For Gaussian chains the spatial structure of the eigenmodes is given by the Rouse form 

How can one hope to extract the contributions of the different normal modes from the relaxation behavior of the dynamic structure factor The capability of neutron scattering to directly observe molecular motions on their natural time and length scale enables the determination of the mode contributions to the relaxation of S(Q, t). Different relaxation modes influence the scattering function in different Q-ranges. Since the dynamic structure factor is not simply broken down into a sum or product of more contributions, the Q-dependence is not easy to represent. In order to make the effects more transparent, we consider the maximum possible contribution of a given mode p to the relaxation of the dynamic structure factor. This maximum contribution is reached when the correlator xp(t)xp(0) in Eq. (32) has fallen to zero. For simplicity, we retain all the other relaxation modes xs(t)xs(0) = 1 for s p. 

In order to be able to evaluate data with a reasonable number of parameters, the mode analysis assumes, as a first approximation, that the exponential correlation of the correlations [Eq. (18)] is maintained, and only the relaxation rates 1/tp are allowed to depend on a general form on the mode index 

Some years ago, on the basis of the excluded-volume interaction of chains, Hess [49] presented a generalized Rouse model in order to treat consistently the dynamics of entangled polymeric liquids. The theory treats a generalized Langevin equation where the entanglement friction function appears as a kernel 

Equations (35) and (36) define the entanglement friction function in the generalized Rouse equation (34) which now can be solved by Fourier transformation, yielding the frequency-dependent correlators xp(co)xp-(co ) . In order to calculate the dynamic structure factor following Eq. (32), the time-dependent correlators xp(t)xp(0) are needed. 

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