Monte Carlo simulations of stress relaxation of entanglement-free Fraenkel chains. I. Linear polymer viscoelasticity

Abstract

Shear stress relaxation modulus GS(t) curves of entanglement-free Fraenkel chains have been calculated using Monte Carlo simulations based on the Langevin equation, carrying out both in the equilibrium state and following the application of a step shear deformation. While the fluctuation-dissipation theorem is perfectly demonstrated in the Rouse-chain model, a quasiversion of the fluctuation-dissipation theorem is observed in the Fraenkel-chain model. In both types of simulations on the Fraenkel-chain model, two distinct modes of dynamics emerge in GS(t), giving a line shape similar to that typically observed experimentally. Analyses show that the fast mode arises from the segment-tension fluctuations or reflects the relaxation of the segment tension created by segments being stretched by the applied step strain-an energetic-interactions-driven process-while the slow mode arises from the fluctuations in segmental orientation or represents the randomization of the segmental-orientation anisotropy induced by the step deformation-an entropy-driven process. Furthermore, it is demonstrated that the slow mode is well described by the Rouse theory in all aspects: the magnitude of modulus, the line shape of the relaxation curve, and the number-of-beads (N) dependence of the relaxation times. In other words, one Fraenkel segment substituting for one Rouse segment, it has been shown that the entropic-force constant on each segment is not a required element to give rise to the Rouse modes of motion, which describe the relaxation modulus of an entanglement-free polymer over the long-time region very well. This conclusion provides an explanation resolving a long-standing fundamental paradox in the success of Rouse-segment-based molecular theories for polymer viscoelasticity-namely, the paradox between the Rouse segment size being of the same order of magnitude as that of the Kuhn segment (each Fraenkel segment with a large force constant HF can be regarded as basically equivalent to a Kuhn segment) and the meaning of the Rouse segment as defined in the Rouse-chain model. The general agreement observed in the comparison of the simulation and experimental results indicates that the Fraenkel-chain model, while being still relatively simple, has captured the key element in energetic interactions--the rigidity on the segment--in a polymer system.