Estimations of Entanglement Spacings

If G were absolutely independent of frequency in the plateau zone—corresponding to a horizontal segment in the curves of Fig. 13-2—its value here, which may be called G%, the pseudoequilibrium modulus of the entanglement network, could be related by analogy with equation 33 of Chapter 10 to the density of entanglement network strands Vg  

In Chapter 3, it was pointed out that the contribution of a certain class of retardation mechanisms to the equilibrium compliance can be obtained by integrating over the appropriate portion of the retardation spectrum if a maximum appears there which can be separated from neighboring contributions the integration is performed in accordance with equation 33 of Chapter 3, and it is illustrated in Fig. 3-7. Alternatively, an integration can be performed over the corresponding maximum in J by equation 53 of Chapter 3 with finite limits, as illustrated in Fig. 3-8. 

The same procedure can be used for an uncross-linked polymer with sufficiently high molecular weight so that the maximum in L or J associated with the entanglement network can be reasonably separated from the contributions at longer times or lower frequencies. The appropriate equations are 

the terms Jg and 1 /ojtjo in equation 53 of Chapter 3 have been omitted because they are negligible. The limits a and b are chosen to encompass the maximum. A similar integration can be performed over the appropriate maximum in G (which can be seen in curves III and IV of Fig. 2-4, also Fig.. 13-3 and is related to the maximum in H at long times seen in Fig. 13-5), by the following equation cf. equation 55 of Chapter 3 and associated discussion)  

There is convergence at the lower limit but the upper must be adjusted to terminate before the transition zone is entered. Corresponding to equation 3, it is also possible 

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