Entanglement distance tube diameter

In the temperature range between 400 and 550 K, NSE spectra on the same undiluted polyethylene melt were recorded. These data were analyzed with respect to the entanglement distance. The result for the temperature-dependent entanglement distance d(T) is shown in Fig. 30. An increase in the tube diameter from about 38 to 44 A with rising temperature is found. 

Since both the temperature dependence of the characteristic ratio and that of the density are known, the prediction of the scaling model for the temperature dependence of the tube diameter can be calculated using Eq. (53) the exponent a = 2.2 is known from the measurement of the -dependence. The solid line in Fig. 30 represents this prediction. The predicted temperature coefficient 0.67 + 0.1 x 10-3 K-1 differs from the measured value of 1.2 + 0.1 x 10-3 K-1. The discrepancy between the two values appears to be beyond the error bounds. Apparently, the scaling model, which covers only geometrical relations, is not in a position to simultaneously describe the dependences of the entanglement distance on the volume fraction or the flexibility. This may suggest that collective dynamic processes could also be responsible for the formation of the localization tube in addition to the purely geometric interactions. 

A preliminary estimate of which, according to (5.8), can be interpreted as the ratio of the square of the tube diameter (2 )2 to the mean square end-to-end distance (R2)o, shows that x AC 1 for strongly entangled systems. For large N, this enables us to replace summation by integration and, according to the rules of Appendix G, to obtain expressions for the characteristic quantities... 

This tube diameter can be interpreted as the end-to-end distance of an entanglement strand of A e monomers ... 

Linear polymers move a distance of order of their own size during their relaxation time, leading to a diffusion coefficient D R /r [Eq. (9.12)]. However, the diffusion of entangled stars is different because at the time scale of successful arm retraction, the branch point can only randomly hop between neighbouring entanglement cells by a distance of order one tube diameter a. For this reason, diffusion of an entangled star is much slower than diffusion of a linear polymer with the same number of monomers ... 

One of the examples of the scaling representation of macromolecules is the reptation model [48], according to which the tube diameter O, in which the macromolecule is confined (equal to the distance between entanglements nodes), can be estimated according to the relationship [49] ... 

With this picture, the diffusion coefficient of the branch point is approximately = x /(2 t ), where x is the hopping distance that the branch point moves every time the arm disentangles itself [50, 51]. An estimate of the hopping distance is the tube diameter a, since the tube diameter is the distance over which the branch point is localized by the entanglements with its neighbors. Hence, we can estimate that... 

This result, however, cannot be obtained directly by a scaling argument. In a semi-dilute 6 solution, there are two characteristic lengths the correlation length I/c (which represents the distance between three-body contacts and has been chosen here as the tube diameter) and the distance between entanglements 2 (distance between two body contacts) which is the mesh size of the transient network. These two lengths play a role in the viscoelastic properties of semi-dilute 6 solutions. Their relative importance is still a matter of controversy. 

The diameter, a, of the tube corresponds to the entanglement spacing, Mg. That is, a strand of polymer having molecular weight Mg spans a random walk end-to-end distance a (Fig. 3-24). Thus, = a M/Mg, and... 

A number of methods exist for the modelling of polymers at a molecular level. One of the most widely used methods is scaling presentation [54], according to which the diameter 0 of the tube in which the macromolecule is enclosed (it is equal to the distance between the entanglement points) can be estimated from the ratio [75] ... 

If we assume the existence of a tube, we can proceed as follows [4]. For very small t the beads of each chain can move only very short distances and do not feel effects from other chains. Hence, the motion of the chain is essentially the same as that of a Rouse chain in dilute solution. This situation is sustained until t increases to n at which g i) becomes comparable to dt, i.e., the beads begin to feel the effect of other entangling chains. Here, as before, dt denotes the diameter of the tube. Using the known expression for g t) of an isolated Rouse chain, we find... 

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