Entanglement distance

The different length scales involve different time scales with different types of motion. For short times corresponding to spatial distances shorter than the entanglement distance, we expect entropy-determined dynamics described by the so-called Rouse model [6, 35.]. As the spatial extent of motion increases and... 

This relation is the central result of the scaling model for the entanglement distance. 

In the packing model [50,62,68] the entanglement distance is interpreted by the gradual build-up of geometrical restrictions due to the existence of other chains in the environment or, more precisely, the entanglement distance is determined by a volume which must contain a defined number of different chains. This approach is based on the observation that, for many polymer chains, the product of the density of the chain sections between entanglements is... 

Me molecular mass of these entanglement strands) and the volume spanned by the entanglement distance is approximately a constant... 

Neglecting the influence of chain ends, Eq. (55) yields for the entanglement distance... 

Fig. 29. Double logarithmic representation of the entanglement distance in polyethylene at 509 K as a function of the polymer volume fraction >. (Reprinted with permission from [60]. Copyright 1993 American Chemical Society, Washington)...

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. 

Table 3.3 Fit results for the entanglement distance d in PE for various models. The reduced is also indicated...

Moreover, from Fig. 3.18 it is apparent that the model of des Cloizeaux also suffers from an incorrect Q-dependence of S(Q,f) in the plateau region, which is most apparent at the highest Q measured. It is important to note that the fits with the reptation model were done with only one free parameter, the entanglement distance d. The Rouse rate was determined earlier through NSE data taken for Kr. With this one free parameter, quantitative agreement over the whole range of Q and t using the reptation model with d=46.0 1.0 A was found. 

Richter D, Farago B, Fetters LJ, Fiuang JS, Ewen B, Lartique C (1990) Direct microscopic observation of the entanglement distance in a polymer melt. Phys Rev Lett 64 1389— 1392... 

The fluctuation forces also cause the curvilinear diffusion along the primitive path. However, because of the constraint effect of the tube, the positive and negative fluctuation forces perpendicular to the primitive path cancel each other out when averaged over a time period longer than required for a segment to travel over an entanglement distance, and thus make no net contribution to the curvihnear diffusion and the translational diffusion of the center of mass. Thus, in the study of the cmvilinear diffusion, Eq. (9.A.3 ) should be used, instead of Eq. (9.A.3). 

Number of Entanglement Strands per Cubed Entanglement Distance, rit... 

Corresponding to Eq. (13.4), the number of entanglement strands per cubed entanglement distance for the blend solution is now defined by... 

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