Contour-length fluctuation

Fig. 3. 30 Apparent tube diameters from model fits with pure reptation (filled squares) and reptation and contour-length fluctuations (open squares) as a function of molecular weight. The dotted line is a guide for the eye (Reprinted with permission from [71]. Copyright 2002 The American Physical Society)...

It has been shown by Doi [11] that on the time scale of Tg contour length fluctuations may induce a rapid relaxation of chain ends, especially for moderately long chains. Indeed, wiggling motions involve forward and backward motions of the chain ends. Thus, chain length fluctuations in oriented materials lead to the creation of isotropic parts of tubes at each end. Their fractional length is roughly equal to 1.3(N e) > where N is the number of monomers per chain. An elaborate expression for the relaxation due to this mechanism, based on the Pearson-Helfand picture for star pol5uners [12], has been proposed by Viovy [13]. 

Furthermore, in Doi s formula (3-18) the numerical constant v accounting for the contribution of contour length fluctuations should vary as a function of N as... 

To sort out such a complicated dynamic situation, we first assume that the primitive chain is nailed down at some central point of the chain, i.e. the reptational motion is frozen only the contour length fluctuation is allowed. This is equivalent to setting rg —> oo while allowing the contour length fluctuation 5L(t) to occur with a finite characteristic relaxation time Tb- In this hypothetical situation, the portion of the tube that still possesses tube stress tt fa tb is reduced to a shorter length Lq, because of the fluctuation SL(t). Then, tt tube length that still possesses tube stress can be defined by... 

The effect of the primitive-chain contour length fluctuation was first considered by Doi. In an approach which was meant to be an approximation, Doi obtained the zero-shear viscosity and the steady-state compliance as... 

Taking the contour-length-fluctuation effect into account, Doi obtained another expression for the viscosity... 

Appendix 9.A — Contour Length Fluctuations of the Primitive Chain... 

While the [t, E) relaxation is going on, the relatively slow relaxation of (v v ) by the reptational motion also gets under way. At t when the primitive chain has recovered its equilibrium contour length, the effect of the contour length fluctuation on the terminal relaxation should basically be the same as that in the linear region. In other words, the relaxation of (v (t)vn(t)) should be described by Eqs. (9.11) and (9.12). Thus from Eq. (12.24), we write the stress relaxation after t as... 

Milner, S. T., and T. C. B. McLeish. 1998. Reptation and contour-length fluctuations in melts of linear polymers. Phys. Rev. Lett. 81 725-728. 

Theoretical treatment of the statistical properties of linear polymers (LPs), shown schematically in Figure 7.1a, have been in existence since the 1950s (Rouse, 1953), and have undergone continuous refinement. In particular, tube theories (viz. reptation) (de Gennes, 1971 Doi and Edwards, 1986), and subsequent refinements, like constraint release (Viovy et al., 1991) and contour length fluctuation (Frischknecht and Milner, 2000) are some of the greatest... 

Frischknecht, A. L., and Milner, S. T., 2000. Diffusion with contour length fluctuations in linear polymer melts. Macromolecules, 33 5273-5277. 

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