Reptation in semidilute solutions

The final result was obtained from the fact that — t /96. Since 

The viscosity of a polymer melt is predicted to be proportional to molar mass for unentangled melts (the Rouse model) and proportional to the cube of molar mass for entangled melts (the reptation model). 

The deviations from the 3.4 power law at low molar masses (M MJ are because those chains are too short to be entangled (see Section 8.7.3). The deviations at very high molar mass are consistent with a crossover to pure reptation (see Section 9.4.5), 

The simple reptation model does not properly account for all the relaxation modes of a chain confined in a tube. This manifests itself in all measures of terminal dynamics, as the longest relaxation time, diffusion coefficient and viscosity all have stronger molar mass dependences than the reptation model predicts. Tn Sections 9.4.5 and 9.6.2, more accurate ana- 

Molar mass dependence of viscosity for polymer melts reduced by their critical molar mass. Open circles are polyisobutylene with M = MnnOgmor, from T. G. Fox 

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B. Tinland, G. Maret, and M. Rinaudo. Reptation in semidilute solutions of wormlike polymers. Macromolecules, 23 (1990), 596-602. 

D. W. Schaefer. Polymer reptation in semidilute solutions. /. Polymer Sci. Polymer 5ymp.,73(1985), 121-131. 

At the Rouse time of an entanglement strand tg, the chain in semidilute solution finds out that it is trapped in the confining tube. The stress relaxation modulus between and the reptation time r ep is almost con-... 

Hess W (1986) Self-diffusion and reptation in semidilute polymer solutions. Macromolecules 19(5) 1395-1404... 

Relaxation of Single and of Entangled Macromolecules. In the absence of hydrodynamic interactions (HI) the normal modes of a polymer are Rouse modes, which act as overdamped harmonic oscillators. With HI the Rouse modes are still nearly normal modes, but the relaxation spectrum is modified. The HI are screened in semidilute solutions. At higher concentrations and in bulk disentanglement by reptation and tube renewal dominate slow viscoelastic processes. 

In the subsequent 20 years (1960-80), the main principles of modern polymer physics were developed. These include the Edwards model of the polymer chain and its confining tube (Chapters 7 and 9), the modern view of semidilute solutions established by des Cloizeaux and de Gennes (Chapter 5), and the reptation theory of chain diffusion developed by de Gennes (Chapter 9) that led to the Doi-Edwards theory for the flow properties of polymer melts. 

The diffusion coefficient in semidilute polymer solutions is determined from the fact that the chain diffuses a distance of order of its own size in its reptation time ... 

Explain the length scales over which the reptation, Rouse, and Zimm models describe dynamics in semidilute entangled solutions of linear polymers. 

Polymer melts and semidilute and concentrated solutions of polymer are highly viscous. Even at a concentration of 1 wt %, solutions of polymer with a molecular weight greater than several million g/mol can flow only slowly. Their behaviors are even elastic like rubber at accessible time and frequency ranges. These exquisite properties had eluded researchers for decades until the tube model and the reptation theory elegantly solved the mystery. The tube model and the reptation theory were introduced by de Gennes." They were refined and applied to the viscoelasticity of semidilute solutions of polymers and polymer melts in the late 1970s by Doi and Edwards." Until then, there had been no molecular theory to explain these phenomena. We will leam the tube model and the reptation theory here. 

Instead of in concentrated solutions, the term cooperative diffusion coefficient is often used for long flexible chains because entangled chains should move together. This (or mutual) diffusion coefficient is an increasing function of c and becomes independent of A1 in the semidilute regime. As already noted, Dj exhibits opposite concentration dependence, and has strong Af-dependence in semidilute and concentrated solutions. The Af-dependence of this coefficient or the tracer diffusion coefficient, particularly in melt, was a hot issue in relation to experimental tests of the reptation theory " or the consistency with the 3.4 power law for melt viscosity in the 1980s (see Reference 5). 

The numerical values in Table 8-2 show that Ds for c above c o can be predicted accurately by eq 3.5 with the parameter values evaluated from dilute solution data. Given this fact, it is not unreasonable to suppose that there is no substantial difference between the modes of chain motion in the dilute and semi-dilute regimes. It is from this supposition that Phillies concluded that reptation does not occur in semi-dilute solutions. In fact, he ends up his paper by stating Since there is no major difference between diffusion in dilute and semidilute regimes, Ds measurements with c < c and c > c must be considered on an equal footing. ... 

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