Clear signatures of reptation in entangled linear polymers

P. Schleger, G. Ehlers, B. Farago (ILL), D. Richter, A. Wischnewski (IFF, FZ Jülich).

The dynamics of a dense polymeric system is dominated by entanglement effects which give rise to a temporary network of topological constraints severely restricting the motional degrees of freedom of each chain. The reptation hypothesis of de Gennes is the most successful phenomenological approach to date for describing the dynamics of long polymer chains, but the model must compete with a variety of counter ideas, since a strict experimental test on a molecular level is still missing, and existing data is inconclusive.

The concept of reptation is particularly appealing in its simplicity : The motion of an individual polymer is postulated to be restricted by the surrounding chain conformations within a tube defined by the overall chain contour (Fig. 1). During the lifetime of this tube, any lateral diffusion of the chain is thus quenched. Initially, the chain relaxes freely until the mean-squared displacement becomes comparable to the tube diameter d. Once the topological constraints are reached, the excitations may relax only along the tube profile (local reptation). The time scale for a collective longitudinal creep along the tube (reptation) is much larger than the transverse relaxation (the reptation time is cubic in the chain length). The result is a plateau in the decay of the single-chain correlation function in time, due to the confinement of the chain within its tube (see Fig. 2). The appearence of a plateau is a classic signature of the topological constraints hindering the relaxation in visco-elastic fluids (independent of the reptation hypothesis).

The physical quantity which is sensitive to these density fluctuations is the single-chain dynamic structure factor S(Q,t), which has been calculated for a variety of models, including the reptation model of de Gennes. S(Q,t) can be measured using a dilute mixture of fully protonated chains within a background of deuterated, but otherwise identical, chains. One is thus sensitive to the single-chain correlations as a result of the scattering length contrast between the protonated (i.e. labeled) polymer and the deuterated background.

However, until now, no measurement of S(Q,t) has been able to separate the reptation model from other phenomenological entanglement theories, such as the rubber-like model of des Cloizeaux, where the polymer is assumed to be spatially fixed at transient (but long-lived) cross-links, and undergoing Rouse-relaxation inbetween. Also under consideration is the model of Ronca, which is a generalised Rouse model where the influence of the neighbouring chains is approximated via a memory function of an elastic medium. Finally, a recent model of Chatterjee and Loring has appeared, where the entanglements are treated as fluctuating obstacles hindering individual chain relaxation. All these models are more or less consistent with the existing neutron spin-echo data, despite representing different basic relaxation mechanisms.

The same polyethylene (PEB-2) sample of molecular weight M_w = 36 k as used in previous neutron spin-echo experiments was remeasured at T = 509 K on IN15. The experimental results and fits using the reptation model as well as the models of des Cloizeaux and Ronca are shown in Figs. 2 and 3. It is apparent that these data clearly favour reptation: The models of Ronca and des Cloizeaux produce a plateau which is too flat, whereas, the model of Chatterjee and Loring [7] relaxes too quickly (not shown).

Figure 1: Illustration of the reptation concept. The red polymer chain is confined through the topology of its environment, restricting its motion within a tube. Only when the chain has reptated out of the confinement have the density correlations decayed to zero. The large separation between the timescales for transverse and longitudinal fluctuations give rise to a well pronounced plateau in the dynamic structure factor.

Figure 2: Plot of S(Q,t) vs t at Q = 0.050 Å-1, and 0.077 Å-1, with a comparison between the predictions of reptation (dark blue lines), local reptation (pale blue lines), the model of des Cloizeaux (green lines), and the Ronca model (red lines). The vertical line and arrow indicate the upper Fourier-time limit of previous experiments.

It is important to note that the fits with the reptation model were done with only one free parameter, the entanglement distance d. The remaining parameters are fixed from other measurement techniques. With this one free parameter, we find quantitative agreement over the whole range of Q and t using the reptation model with d = 46.0 ± 0.1 Å. The tube diameter deduced here is essentially unchanged from the previous neutron spin-echo data for t < 20 nsec (d = 43.5 ± 0.7 Å), where it was already found to agree with the tube diameter deduced from the plateau modulus in rheological measurements (d = 42 Å).

One might ask if these data are actually sensitive to the reptational diffusion via a creeping motion along the tube, or if only local reptation is important on this time-scale. Local reptation effectively corresponds to the behaviour of a chain confined to a tube in the limit of infinite chain length, and indeed a difference between global and local reptation on the time scales accessible on IN15 only becomes apparent at sufficiently small Q (Fig. 2). More measurements are needed to verify if this small extra decay is due to a collective reptational diffusion. Incidently, preliminary results on the mass-dependence show that the reptation model holds for chains with 24k < M_w < 190k, but that for shorter chains, the Q dependence of the plateau is significantly modified. This indicates that one really must be in the highly entangled limit for the reptation model in its present form to hold.

Figure 3: Semi-log plot of S(Q,t) vs t for various Q. The red lines are the fit of the reptation model. The pale blue lines are a fit using the model of des Cloizeaux.

In summary, the quantitative experimental technique of neutron spin-echo spectroscopy has seen an unambiguous signature of reptation in a flexible linear polymer. The data now covers a substantial region of the time-domain where the reptation concept is in principle applicable. Compared with other phenomenological entanglement models, reptation is now the only approach with provides a consistent description of all the neutron spin-echo data. This is compelling evidence that on the time scales measured here, reptation is indeed the principle relaxation mechanism in entangled linear-chain systems. This result will hopefully encourage the development of a microscopic theory, perhaps based on a mode-coupling approach, from which reptation emerges as a characteristic feature.