Time dependent shear modulus

Using the fluctuation-dissipation theorem [361, which relates microscopic fluctuations at equilibrium to macroscopic behaviour in the limit of linear responses, the time-dependent shear modulus can be evaluated [371 ... 

Figure 3.8 Time-dependent shear modulus [as G(t)/Gol versus time (as t/r) (a) linear coordinates and (b) log-log coordinates.

Time dependent shear modulus in the linear response regime denoted as g (f) of the quiescent system generalized one if including dependence on shear rate... 

Maxwell was the first to describe the viscoelastic response at the fluid-to-glass transition phenomenologically. He introduced a time-dependent shear modulus describing the response of a viscoelastic fluid to a time-dependent shear... 

Predictions of the (idealized) MCT equations for the potential part of the equilibrium, time-dependent shear modulus ( ) of hard spheres for various packing fractions 0 are shown in Fig. 3 and calculated from the limit of (lid) for vanishing shear rate ... 

According to the viscoelastic theory of Doi and Edwards (24), the time dependent shear modulus, G(t), is related to the plateau modulus, G, as... 

In correspondence to this, we write for the time dependent shear modulus... 

Fig. 6.6. Time dependent shear modulus of PVC. Master curve set up for Tg = 65 °C as the reference temperature. The dashed line indicates the slope predicted by the Rouse-model. Data from Eisele [66]...

The neutron scattering experiments indicate that the Rouse-model remains valid for chain sequences below a critical length. This suggests we represent the time dependent shear modulus of an entangled polymer melt as being composed of three parts... 

Doi and Edwards analysed the described disentangling process of the primitive chain in more detail. As in the case of the Rouse-motion, the dynamics of the disentangling process can also be represented as a superposition of independent modes. Again, only one time constant, the disentangling time Td, is included, and it sets the time scale for the complete process. In the Doi-Edwards treatment, ra is identified with the longest relaxation time. Calculations result in an expression for the time dependent shear modulus in the terminal flow region. It has the form... 

The result is a power law that is characteristic for the relaxation of Rouse chains and it may be compared with experiments on poljmier melts. Indeed, one finds good agreement for several systems. The center of the glass-rubber transition, as observed in stress relaxation experiments, is often well-described by Elq. (8.81). Figure 8.6 depicts, as an example, the time-dependent shear modulus of poly(vinylchloride) (PVC), presented as a master-curve referred to the glass transition temperature (Tg = 65 °C). The slope of the log log plot corresponds to an exponent —1/2. 

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