Equilibrium-Simulated Relaxation Modulus Curves

If there is no attractive interaction potential between two beads, the thermal fluctuations in an equilibrium state will eventually separate them far apart. Therefore, at equilibrium, the average distance (or the average distance over a long period of time) between the two beads of a dumbbell 

For the Fraenkel dumbbell, the Langevin equation in terms of the bond vector, b(t) = R2(i) - Ri(i), is given by 

Equation (17.3) can be transformed into the discrete form for simulation in the same way as described in Sec. 16.1. Defining 

As expected, the simulation results obtained for the Fraenkel dumbbell based on the combination of Eqs. (16.12) and (16.13) and on Eq. (17.7) are identical. During a simulation, the fluctuation in 5 t) as defined by Eq. (17.4) can be monitored separately allowing the time correlation function (J(O)J(t)) to be calculated. Physically, 5 t) approximately represents the deviation of b(t) from bp. Any small change in b(t) leads to a large 

In the short-time region where the process AS 0)AS t)) is dominant, the approximation as used in Eq. (17.8) is expected to be good. By contrast, over a long period of time, as the nonvanishing residual fluctuations in S t) are small and more comparable in (relative) magnitude to the slow fluctuations in bx t)by t), the approximation expressed as a product of two separate terms S and bx 0)by 0))(bx t)by t))) in Eq. (17.11) may not be well justified. Nevertheless, the approximate form as given by Eq. (17.8) helps us understand the coexistence of the fast and slow modes of motion as distinctly observed in the simulation results. To illustrate the results and at 

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