Fraenkel chain

In this chapter, the simulation scheme will be introduced and illustrated by applying it to the Rouse chain. The computer simulation results of the Rouse-chain model can be directly compared with the analytical solutions to demonstrate the validity of the simulations. This will serve to validate simulations by the same scheme on systems of which analytical solutions are not available. In Chapters 17 and 18, simulations on entanglement-free Fraenkel chains " are studied giving new understandings at the molecular level of polymer viscoelasticity. 

We can picture an A -bead Fraenkel chain as the chain shown in Fig. 3.2 just as we have pictured an A -bead Rouse chain in Chapters 3 and 7. Here, the only difference is that the spring connecting two beads, instead of being entropic (Eq. (3.30)), is characterized by the Fraenkel potential ... 

Monte Carlo Simulations of Stress Relaxation of Fraenkel Chains... 

The above analysis can be extended to Fraenkel chains longer than a dumbbell. As detailed in Appendix 17.B, by comparing the simulation results obtained with and without the contribution from couplings between different segments, the above conclusion has been shown to be applicable to a multiple-segment Fraenkel chain as well. The basic natme of the fast mode and that of the slow mode are further confirmed by the studies of the step strain-simulated Gs t) curves as detailed below. 

The step strain-simulated Gs t) curves obtained at A = 0.2 and 0.5 are compared with equilibrium-simulated Gs t) ones in Fig. 17.5 for the two-, five-, ten- and twenty-bead Fraenkel chains. There are clear differences between the equilibrium-simulated curves and the step strain-simulated at A = 0.2 in the cases of two- and five-bead chains, indicating that the fluctuation-dissipation theorem is not fulfilled totally as in the Rouse-chain case. This may be due to A = 0.2 not being in the linear region yet as some small differences can be observed between the Gs t) curves at A = 0.2 and 0.5. In fact, the numerically calculated Gs(0) values as a function of the strain A... 

Fig. 17.5 Comparison of the step strain-simuiated Gs t) curves for the two-bead, five-bead, ten-bead and twenty-bead Fraenkel chains at A = 0.2 (a) and 0.5 (o) with the equilibrium-simulated curves ( ). To avoid overlapping between different sets of curves, the results of = 5, 10 and 20 have been multiplied by 10, 10 and 10 , respectively.

Note For the mean square end-to-end vector of a Fraenkel chain, which is a static property, there is no correlation between different segments, just as in the case of the freely jointed chain (Chapter 1). 

Consider a volume Vq containing n Fraenkel chains, each with N beads. Right after the application of a step shear deformation E to such a system in equilibrium, the shear stress, —[Pg.388]

The second normal-stress difference as a function of time obtained from the simulations on the five-bead Fraenkel chain is nonzero as shown in Fig. 18.2. By averaging over all orientations, the initial value of the second normal-stress difference calculated from... 

Fig. 18.7 Comparison of the time dependences of —Sxy t, )(o) 4 x (bx t)by(t)) (solid line) and 4 x ux t)uy t)) (dashed line) obtained from simulations on the five-bead Fraenkel chain at A = 0.5, 1, 2 and 4. To avoid overlapping between shown curves, the results at different A values have been shifted along the vertical axis by the indicated factors.

As shown in Fig. 18.10, the Fraenkel-chain Gs(t,X) curves at different A values can be superposed on one another over the entropic region closely by allowing a vertical shift. Thus from these simulation results, one can... 

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