Distance between entanglements

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... 

The reason why simple scaling considerations do not lead to a unique result for the exponent a is due to the fact that the entanglement problem as a geometrical phenomenon contains two independent lengths the lateral distance between the chains s = (L/V) 1/2 and the step width of the random walk... 

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. 

Fig. 1.2 Richness of dynamic modulus in a bulk polymer and its molecular origin. The associated length scales vary from the typical bond length ( A) at low temperatures to interchain distances ( 10 A) around the glass transition. In the plateau regime of the modulus typical scales involve distances between entanglements of the order of 50-100 A. In the flow regime the relevant length scale is determined by the proper chain dimensions...

Continuity In the limit of vanishing distance between two density matrices, the difference between their entanglement should tend to zero. 

The De Gennes method can be applied directly to Bueche s model without the defect diffusion formalism. Consider a chain with n main chain atoms, mean square end-to-end distance , and a large number of entanglement obstacles along its contour. For simplicity, suppose the number of main chain atoms between successive obstacles and the scalar distance between obstacles are... 

A rep < 1, Des < 1, the nucleation dynamics is stochastic in nature as a critical fluctuation in one, or more, order parameters is required for the development of a nucleus. For DeYep > 1, Des < 1 the chains become more uniformly oriented in the flow direction but the conformation remains unaffected. Hence a thermally activated fluctuation in the conformation can be sufficient for the development of a nucleus. For a number of polymers, for example PET and PEEK, the Kuhn length is larger than the distance between two entanglements. For this class of polymers, the nucleation dynamics is very similar to the phase transition observed in liquid crystalline polymers under quiescent [8], and flow conditions [21]. In fast flows, Derep > 1, Des > 1, A > A (T), one reaches the condition where the chains are fully oriented and the chain conformation becomes similar to that of the crystalline state. Critical fluctuations in the orientation and conformation of the chain are therefore no longer needed, as these requirements are fulfilled, in a more deterministic manner, by the applied flow field. Hence, an increase of the parameters Deiep, Des and A results into a shift of the nucleation dynamics from a stochastic to a more deterministic process, resulting into an increase of the nucleation rate. 

The E-modulus appeared to be inversely proportional to the distance between crosslinks, so, approximately, proportional to the number of cross-links. If no permanent cross-links are present, i.e. in the unvulcanized condition, then E is governed by the number of physical entanglements. Light vulcanization does not contribute significantly to total number of cross-links therefore, in lightly crosslinked vulcanizates, such as technical rubbers, the E modulus is hardly dependent on the degree of vulcanization (Qu. 5.4 en 5.5). 

The important features of filaments in absence of entanglement of their branches are the total length of the filament, the length of the main hypha, the number of extremities or tips, the number of branching points and the distance between two branching points and between branching points and tips. 

Unfortunately, no study has yet been published on the carbon nanotube network 3D structure, probably because the thickness of the reconstructed volume is small compared with the expected distance between two successive entanglements. 

Both the questions of the transition from Rouse to reptation dynamics and of what fixes the average distance between entanglements in polymer liquids has been the subject of a number of recent theoretical and experimental investigations. 

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