Reptation time

The tube is a construct which we might continue to sketch around an emerging chain as it diffuses out of the original sleeve. Instead, it is convenient to start with the tube initially in place and consider how long it takes for the molecule to escape. The initial entanglements which determine the contours of the tube comprise a set of constraints from which the molecule is relaxing, even if only to diffuse into another similar set. Accordingly, we identify this reptation time as a relaxation time r for the molecule. 

This simple argument can yield the expected molecular weight dependence of both the single chain diffusion constant (in three dimensions) D and the viscosity 77. For in one reptation time the chain has moved on average one chain end-to-end distance R (N/Nj a, so... 

The sum over weighted relaxation times is heavily dominated by the longest time (the reptation time) r gp=L /7T Dp. Because of this the frequency-dependent dissipative modulus, G"(cd) is expected to show a sharp maximum The higher modes do modify the prediction from that of a single-mode Maxwell model, but only to the extent of reducing the form of G"(a>) to the right of the maximum from ccr to In fact, experiments on monodisperse linear polymers... 

A naive application of dynamic dilution would introduce a dependence of the reptation time on the unrelaxed volume fraction. Since T gp L (aM/MJ this would imply the choice of in the case a=l for the dflution exponent. But... 

Fig. 12. The rheological functions G ((o) and G"(co) for an H-shaped PI of arm molecular weigh 20 kg mol and backbone 110 kg mol" [46]. The high-frequency arm-retraction modes can be seen as the shoulder from co 10 to co 10 together with a low-frequency peak due to the cross-bar dynamics at co 10. The smooth curves are the predictions of a model which takes Eq. (33) as the basis for the arm-retraction times and a Doi-Edwards reptation spectrum with fluctuations for the backbone. The reptation time is correctly predicted, as is the spectrum from the arm modes, though the low frequency form is more polydisperse than the simple theory predicts...

The simplest case of comb polymer is the H-shaped structure in which two side arms of equal length are grafted onto each end of a linear cross-bar [6]. In this case the backbones may reptate, but the reptation time is proportional to the square of Mj, rather than the cube, because the drag is dominated by the dumb-bell-like frictional branch points at the chain ends [45,46]. In this case the dependence on is not a signature of Rouse motion - the relaxation spectrum itself exhibits a characteristic reptation form. The dynamic structure factor would also point to entangled rather than free motion. 

Combining the predictions for the reptation time of the H cross-bar and the effective modulus when the arms are acting as solvent gives a prediction for the scaling of the viscosity of a melt of H-polymers on their structural parameters ... 

Unless the molecular weight of the star arm is rather low, this period of relaxation by fluctuation is generally interrupted by the reptation of the linear polymers. This happens when the centre of mass diffusion of the linear chains is able to renew the tube of all (central) portions of the chain not relaxed by the star-like modes. So the reptation time is given by... 

Tfr is transition zone relaxation time, rte is terminal relaxation time, is reptation time... 

Various factors govern autohesive tack, such as relaxation times (x) and monomer friction coefficient (Co) and have been estimated from the different crossover frequencies in the DMA frequency sweep master curves (as shown in Fig. 22a, b). The self-diffusion coefficient (D) of the samples has been calculated from the terminal relaxation time, xte, which is also called as the reptation time, xrep The D value has been calculated using the following equation ... 

The reptation time, Tr, or the time it takes for the chain to renew its configuration is related to Dt through a relation analogous to eq.3 DiTr = with Lt = d(N/Nd) the total length of the tube. Thus... 

This time is the longest relaxation time of a linear chain. We will refer to it as the "reptation time". 

Figure 7 Reptation the chain disengages fi-om its initial tube by back-and-forth motions the time necessary for a complete renewal of its initial configuration is the "reptation time" which is the longest relaxation time of the polymer.

Rouse time of an N-chain with an elementary time x directly connected to the reptation time Xc of each passing chain. 

The so-called tube renewal time Xf can be compared to the reptation time x if the prefactor is known in the above relation. Xr, /xc = (N/N ) / IStc for Klein... 

In a polydisperse sample, each N-chain is surrounded by chains of different lengths. Therefore, the constraint release time x. varies according to the reptation time of the passing chain. Some entanglements can be considered as permanent (P N), while others will disappear quickly (Ptime distribution will affect the tube renewal time and consequently the overall relaxation time of each chain. 

As the data of cOn, are straightforward and experimentally connected to the longest time by co = 0.7x [21], they will be used along with the reptation time in order to calculate the tube renewal time Xren of the diluted chains ... 

All the data confirm that the tube can be viewed as a Rouse chain (x e but with an elementary time which is not the reptation time of the passing... 

Figure 20 Experimental relation between the tube renewal time Xrg and the reptation time Xg of the matrix for different molecular weights of various polymers [ data from ref 19].

According to Doi-Edwards theory, the reptation of the long chains will occur in a tube whose diameter a veuies as Thus the number of monomers between entanglements will scale as < ) . Accordingly, the reptation time x (relation 3-14) should be proportional to (]) as a first approximation, the zero-shear viscosity tio and the steady-state compliance J should respectively scale as [Pg.133]

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