High-frequency plateau modulus

Fig. 8.10 Comparison of the high-frequency plateau modulus of aqueous HEUR C16/35K solutions (o)...

A typical example of the master curve [2] is shown in Figure 9.3 for HEUR (polyethylene oxide) end-capped with -C16H33. The reference temperature is chosen at 5 C. From the horizontal shift factor, the activation energy is found to be 67kJmol . From the high-frequency plateau of the storage modulus, the number of elastically effective chains is found as a function of the polymer concentration, which was already studied in Section 8.2 (Figure 8.10). 

At low frequencies the loss modulus is linear in frequency and the storage modulus is quadratic for both models. As the frequency exceeds the reciprocal of the relaxation time ii the Rouse model approaches a square root dependence on frequency. The Zimm model varies as the 2/3rd power in frequency. At high frequencies there is some experimental evidence that suggests the storage modulus reaches a plateau value. The loss modulus has a linear dependence on frequency with a slope controlled by the solvent viscosity. Hearst and Tschoegl32 have both illustrated how a parameter h can be introduced into a bead spring... 

The contribution from the first term (reptation branch) has the same order of magnitude as the contribution from the second term at very high frequencies. However, one has to take into account that, due to distribution of relaxation times, the limit value of the first term is reached at higher frequencies than the limit value of the second term. At lower frequencies the plateau value of the dynamic modulus is determined by the second term and coincides with expression (6.52). 

Medalia [8,9] and Voet and Cook [10,11]. The effect is illustrated schematically in Fig. 1 for the compression mode. For a specific frequency and specific temperature the storage modulus decreases from a zero-amplitude plateau value, E0 (or G o in shearing) to a high-amplitude plateau value, E (or G j, with increasing amplitude, whereas the loss modulus shows a pronounced peak. 

Plateau storage modulus, G value at high frequencies. Pa... 

Emulsions with a high volume fraction of droplets (0 > 0.64) and foams show solidlike properties such as a yield stress and a low-frequency plateau value of G. The magnitudes of the yield stress and elastic modulus can be estimated using simple cellular foam models. These and related models show that at low shear rates where the shear stress is close to the yield value, the flow occurs by way of intermittent bubble-reorganization events. The dissipative processes that occur during foam and emulsion flows are still under active investigation. 

Another example of network formation is found in PEO (poly(ethylene oxide))-silica systems [58, 59]. At relatively small-particle concentrations, the elastic modulus increases at low frequencies, suggesting that stress relaxation of these hybrids is effectively arrested by the presence of silica nanoparticles. This is indicative of a transition from liquidlike to solidlike behavior. At high frequencies, the effect of particles is weak, indicating that the influence of particles on stress relaxation dynamics is much stronger than their influence on the plateau modulus. 

A discussion of this subject is presented below. The additional oMitribution GX is said to arise from permanently trapped (inter-chain) entanglements in the network. Linear polymers of high molecular wd t exhibit a storage modulus G (constant over a wide range of frequencies This plateau modulus... 

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