Reptation Subject

Both the questions of the transition from Rouse to reptation dynamics and of what fixes the average distance between entanglements in polymer liquids has been the subject of a number of recent theoretical and experimental investigations. 

Because of the interaction of the two complicated and not well-understood fields, turbulent flow and non-Newtonian fluids, understanding of DR mechanism(s) is still quite limited. Cates and coworkers (for example, Refs. " ) and a number of other investigators have done theoretical studies of the dynamics of self-assemblies of worm-like micelles. Because these so-called living polymers are subject to reversible scission and recombination, their relaxation behavior differs from reptating polymer chains. An additional form of stress relaxation is provided by continuous breaking and repair of the micellar chains. Thus, stress relaxation in micellar networks occurs through a combination of reptation and breaking. For rapid scission kinetics, linear viscoelastic (Maxwell) behavior is predicted and is observed for some surfactant systems at low frequencies. In many cationic surfactant systems, however, the observed behavior in Cole-Cole plots does not fit the Maxwell model. 

At present, no reported data on ring self-diffusion in polymer concentrates are available other than those of Mills et al. and no theory of this subject exists other than Klein s. Thus we see a virgin field of research open before us. What seems most needed is experimental data for self-diffusion in the melt and concentrated solutions of rings. Diffusion of linear chains in ring chain matrices should also be instructive, as pointed out by Mills et al. The reptation idea now dominating the study of polymer self-diffusion will face crucial tests when accurate and systematic diffusion data on these systems become available. 

The subject of this paper is limited to the dynamics of a single entangled polymer chain, just as in the original de Gennes paper on reptation. In a melt or concentrated solution, the dynamics of any chain would be affected by the motion of surrounding polymers (by constraint release), and this effect has to be self-consistently taken into account. In order to do that, one has to start from a reliable model of the single-chain dynamics, such the reptation model, Doi s fluctuation theory,or the repton modeldescribed in the present paper. 

The dynamics of the polymer chain is naturally slowed down by entanglements, in comparison with the results for dilute and semidilute conditions (Table 7.1). There is a vast literature on the subject of viscoelasticity of polymer melts and how close the experimental results are with the predictions of the reptation model (Doi and Edwards 1986). Since conditions such as polymer melts are not of common place in the experiments exploring polymer translocation, we do not dwell on this subject here. 

It should be pointed out that Eq. (4.124) was derived without invoking fluctuations of contour length (i.e., without considering the Rouse motion in a reptating chain). The main idea behind the derivation of Eq. (4.124) is that since experimental data for r)Q is usually obtained from shear flow measurements, stress effects must be included in the reptation model that is, when a polymer is subjected to shear flow, a relaxation of polymer chains to reptate around the entangled junctions must be taken into consideration, in addition to the reptation of the overall center-of-mass motion. 

The models based on double reptation that do not take other relaxation mechanisms into account have the decided advantage of being readily subject to mathematical manipulation. 

General Regimes of Response. The nonlinear viscoelastic response of polymers, of course, follows some of the same classifications as does the linear response. Hence, the behavior above the glass temperature and into the terminal zone is fluid behavior, and often follows time-temperature superposition. The phenomenology of polymer melts and solutions is commonly described by constitutive laws that relate the stress and strain histories to each other (59,69). A brief description of the K-BKZ model (70-72) is provided as it seems to capture most of the behaviors of polymer melts and solutions subjected to large deformations or high deformation rates. At the same time the nonlinear form of the reptation... 

The reptation model outlined here has been studied in great detail by Doi and Edwards, who considered in particular the influence of the tube diameter (or the so-called entanglement mass) and the viscoelastic behavior of polymer melts this is the subject of Chapter 8. 

Two different scenarios for chain-end dynamics have been suggested. Doi [142] introduced the so-called contour length fluctuation (CLF) as a modification of the tube/reptation model. Due to the stochastic nature of chain modes in the tube, the chain ends are fluctuating back and forth a length proportional to the square root of the chain length and, hence, are subject to tube constraints to a much lower degree than the central part of the chain. The chain-end blocks are therefore expected to have a molecular mass obeying... 

---