Y.-H. Lin and A. K. Das Department of Applied Chemistry
National Chiao Tung University Hsinchu, Taiwan
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 deformation. While the fluctuationdissipation theorem is perfectly demonstrated in the Rouse-chain model, only a quasi version of the fluctuationdissipation theorem is observed in the Fraenkel-chain model. In both types of simulations on the Fraenkel-chain model, two distinctive modes of dynamics emerge in the relaxation modulus: a fast energetic interactions-derived mode and a slow entropy-derived mode, giving a GS(t) line shape typically observed experimentally. It has been shown through analysis that the fast mode arises from the segment-tension fluctuations or reflects the relaxation of the segment-tension arising
from segments being stretched by the applied step strain; and the slow mode arises from the fluctuating segmental-orientation anisotropy or represents the randomization of the segmental-orientation anisotropy induced by the step deformation.
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 dependence of the relaxation times. In other words, with 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 have been typically observed in the long-time region of the linear viscoelastic response of an entanglement-free polymer. 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 and the meaning of the Rouse segment as defined in the Rouse chain model. A comparison of the simulation result with experiment suggests that the Fraenkel-chain model, while being still relatively simple, has captured the basic element of the energetic interactions)the rigidity on the segment)in a polymer system.
Monte Carlo Simulations
of Stress Relaxations of Entanglement-Free Fraenkel Chains. 1:
Linear Polymer Viscoelasticity
Y.-H. Lin and A. K. Das Department of Applied Chemistry
National Chiao Tung University Hsinchu, Taiwan
1. Introduction
It has been extensively shown that the linear viscoelastic response of an entanglement-free polymer melt is well described by the Rouse theory.143,144,145,146,147
However, the agreement between theory and experiment is limited to the region below the modulus level corresponding to the molecular weight of a single Rouse segment that can be assigned to the polymer system)for instance, below ～
RT m
3.8×107143 Rouse, P. E. Jr. J. Chem. Phys. 1953, 21, 1271.
144 Bird, R. B.; Curtiss, C. F.; Armstrong, R. C.; Hassager, O. Dynamics of Polymeric Liquids, Vol. 2, Kinetic Theory, 2nded.; Wiley: New York, 1987.
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dynes/cm2corresponding to the Rouse-segmental molecular weight m850 in the case of polystyrene;148,149,150,151152,153,154,155,156,157,158
in other words, the agreement occurs only in the time or frequency region slower than the motion associated with a single Rouse segment or equivalently the relaxation rate of the highest Rouse mode.
Because of the entropic-force constant on the Rouse segment, this region may be referred to as the entropic region and the relaxation processes in it as entropy-derived dynamics. In the entropic region the entire viscoelastic response follows the same temperature dependence indicating that thermorheological simplicity is followed, as expected from the Rouse theory)the frictional factor associated with the Rouse segment carries the temperature dependence of the viscoelastic response. In the time or frequency region faster than the motion of a single Rouse segment, the modulus of
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the dynamic response is much higher)ranging from ～4×107 to ～1010dynes/cm2 for
polystyrene. The high modulus is due to the strong energetic interactions among segments, both intra-chain and inter-chain; the dynamics in this region may be properly referred to as energetic interactions-derived dynamics, which has also been referred to in the literature as the glassy relaxation or the structural relaxation or the relaxation. It has been widely observed that as the temperature approaches the glass transition temperature, Tg, from above, the energetic interactions-derived dynamics has a temperature dependence stronger than that of the entropy-derived.159 , 160 , 161 , 162 , 163 , 164 , 165 , 166
Thus, when the whole range of the viscoelastic response is included in the consideration, the thermorheological simplicity does not hold. Recently, the basic mechanism for the thermorheological complexity in polystyrene has been analyzed, showing that the effect as related to Tg
behaves in a universal way within the polystyrene system, entangled or not,15,167 and that the same basic mechanism is also responsible for the break-down of the
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StokesEinstein equation in relating the translational diffusion constant and viscosity or molecular rotational relaxation time in fragile glass-forming liquids,168 such as OTP169,170,171,172,173
and TNB.174,175 While it has been extensively demonstrated that the molecular theories: the Rouse theory15 for the entanglement-free region of molecular weight and the extended reptation theory (ERT)3,176,177,178
for the entangled region, describe the viscoelastic responses in the entropic region successfully in a quantitative way, the glassy-relaxation process can only be analyzed phenomenologically, often in terms of a stretched exponential form. In other words, we have quite limited understanding about the glassy-relaxation process at the molecular level. In this study, using the Monte Carlo simulation based on the Langevin equation,3,14 we compute the relaxation modulus curves of the model
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systems which contain elements of energetic interactions)the Fraenkel chains,179 shedding light on the coexistence of and the interrelation between the energetic interactions-derived and entropy-derived dynamic processes.
2. Monte Carlo Simulation Based on the Langevin Equation
In the Monte Carlo simulation, the continuous change in time, dt, in the Langevin equation is replaced by a small time-step, t. For a chain with the positions of the beads at time step i denoted by {Rn(i)}, the simulation form of the Langevin equation is expressed by
where Fn(i) is the force on the nth bead at the ith time step arising from the interaction potential; the random step vector dn(i) is characterized by the following first and second moments:
179 Fraenkel, G. K. J. Chem. Phys., 1952, 20, 642.