Length fluctuation

In order to convert this fluctuation into an equivalent length fluctuation, we have to fix the working point of the interferometer. The simplest choice is to work at the point where dPout/d5L is maximum. At that point Pout = Pin/ and... 

The fluctuation of the photon number is interpreted as a length fluctuation asL, and... 

A second appealing feature of tube model theories is that they provide a natural hierarchy of effects which one can incorporate or ignore at will in a calculation, depending on the accuracy desired. We will see how, in the case of linear polymers, bare reptation in a fixed tube provides a first-order calculation more accurate levels of the theory may incorporate the co-operative effects of constraint release and further refinements such as path-length fluctuation via the Rouse modes of the chains. 

Fig.3. G (co) and G"(co) for monodisperse linear polymers of PI, PB and PS. The curves have been shifted so that the plateau moduli and terminal times coincide. The dashed line indicates the Doi-Edwards prediction for G"((o) in the absence of path-length fluctuations...

Fig. 10. Data for G (co) on three monodisperse linear polystyrenes from [15] with values of M/Mg of 22, 57 and 191. The theoretical curves account for path length fluctuations calculated as for star polymers [41] choosing values for Gq and Tg consistent with published data...

The mathematical treatment that arises from the dynamic dilution hypothesis is remarkably simple - and very effective in the cases of star polymers and of path length fluctuation contributions to constraint release in Hnear polymers. The physics is equally appealing all relaxed segments on a timescale rare treated in just the same way they do not contribute to the entanglement network as far as the unrelaxed material is concerned. If the volume fraction of unrelaxed chain material is 0, then on this timescale the entanglement molecular weight is renormalised to Mg/0 or, equivalently, the tube diameter to However, such a... 

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)...

Egami (2001) has developed pair-density-function analysis of pulsed neutron-diffraction data to determine the structure in a time interval that is short relative to a bond-length fluctuation. This method provides a direct observation of the pattern of bond-length fluctuations. Several other measurements have been shown to provide indirect signatures of the presence of bond-length fluctuations (Goodenough and Zhou, 2001). For example,... 

The phonon contribution to the thermal conductivity is strongly suppressed by bond-length fluctuations. 

Figure 61 shows the ZFC and FC M(T) curves for the FMM phase II, which show a first-order transition at 7c where the bond-length fluctuations become frozen out to restore the phonon contribution to the thermal conductivity, fig. 62. In the I + II two-phase samples t = 0.973 and t = 0.974, phase II is suppressed and phase fluctuations inhibit phonon formation below Tco or 7c. Tokura et al. (1996) applied hydrostatic pressure to the FM phase II of (Ndo.i25Smo.875)o.5Sro.5Mn03 having / /c they reported the appearance of a CE AFI second phase having a volume fraction that increased with pressure below Too = 7n- Since pressure... 

Schieber JD, Neergaard J, Gupta S (2003) A full-chain temporary network model with sliplinks, chain-length fluctuations, chain connectivity and chain stretching. J Rheol 47(l) 213-233... 

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]. 

The influence of various relaxation processes (retraction alone, reptation alone, chain-length fluctuation alone) on the relative relaxation of an end-block of size equal to that of PS DH 184 is compared to experimental data in Figure 5. It can be seen that none of these processes alone can accomit for the observed behavior. 

These results, as compared to a previous study [29], illustrate a better agreement between theory and experiments. This can be attributed to a better approximation of the mode distribution involved in the chain fluctuation process and to which the end of the chains are quite sensitive. These data also prove that under our experimental conditions, the relaxation of the chain is dominated by the chain retraction and chain-length fluctuation process. The... 

The spectroscopic data have been compared with the theoretical predictions of the Doi- Edwards model. In the time scale of our experiments, a quantitative agreement between experiment and theory is obtained if chain length fluctuations, retraction and reptation are taken into account. In the case of star polymers, the large scale fluctuation mechanism as proposed by Pearson and Helfand associated with the retraction process is accounting for... 

The refinement introduced by Doi [9] who considers tube length fluctuations is more relevant to experimental scaling laws as far as the viscosity/molecular weight dependence is concerned. Following that concept, Gc(t) can be cast into the integral form ... 

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... 

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