Dynamic Modulus and Relaxation Branches

There are plenty of measurements of dynamic modulus of nearly monodisperse polymers starting with pioneering works of Onogi et al. (1970) and Vinogradov et al. (1972a). The more recent examples of the similar dependencies can be found in papers by Baumgaertel et al. (1990, 1992) for polybutadiene and for polystyrene and in paper by Pakula et al. (1996) for polyisoprene. 

To calculate the dynamic modulus, we turn to the expression for the stress tensor (6.46) and refer to the definition of equilibrium moments in Section 4.1.2, while memory functions are specified by their transforms as 

It means, according to the speculations in Chapter 3 that the environment of the chosen macromolecule is considered a viscoelastic medium, and, in addition, the internal resistance or the internal viscosity is taken into account. The latter was not considered in the previous section. 

We are calculating dynamic modulus and characteristic quantities for entangled systems, when the linear approximation of dynamic equation is used. 

To begin with, let us consider the simple case, when C can be neglected in comparison to ( B in equations (6.47), which can be done, if one considers low-frequency properties of the systems with long macromolecules - the strongly entangled systems. In this case, according to (4.32) and (4.33), we have 

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