Storage viscoelastic functions

An extensive study by Koppelmann (1958) of viscoelastic functions through one of the secondary mechanisms in Poly(methyl methacrylate) is shown in Fig. 13.15. Dynamic storage moduli and tan <5 s near 25 °C are plotted vs. angular frequency. It shows first, that a secondary mechanism is present, secondly that E and G are not completely parallel, because the Poisson constant is not a real constant, but also dependent on frequency (the numbers in between both moduli are the actual, calculated Poisson constants) and thirdly that tan <5E is practically equal to tan <5G. 

The appropriate viscoelastic functions are the dynamic rheological properties (storage modulus G and the loss modulus G", and the dynamic viscosities f and T ") extrapolated to infinite dilution and are called the intrinsic dynamic rheological properties ... 

Because the relaxation spectra are similar for transient and dynamic relaxation viscoelastic functions, H t) can also be obtained from the storage relaxation modulus. The plot of the kernel of the integral of Eq. (9.8), x /(l + (o x ), versus logcax is a sigmoidal curve that intercepts the ordinate axis at 0.5 and reaches the value of 1 in the limit cox oo (see Fig. 9.5). The kernel can be approximated by the step function... 

It has been remarked that time (frequency) - temperature reduced data on carbon black filled rubbers exhibit increased scatter compared to similar data on unfilled polymers. Payne (102) ascribes this to the effects of secondary aggregation. Possibly related to this are the recent observations of Adicoff and Lepie (174) who show that the WLF shift factors of filled rubbers giving the best fit are slightly different for the storage and loss moduli and that they are dependent on strain. Use of different shift factors for the various viscoelastic functions is not justified theoretically and choice of a single, mean ar-funetion is preferred as an approximation. The result, of course, is increased scatter of the experimental points of the master curve. This effect is small for carbon black... 

Thus, once the four parameters of Eq 7.42 are known, the relaxation spectrum, and then any linear viscoelastic function can be calculated. For example, the experimental data of the dynamic storage and loss shear moduli, respectively G and G , or the linear viscoelastic stress growth function in shear or uniaxial elongation can be computed from the dependencies [Utracki and Schlund, 1987] ... 

The dynamic tests at small amplitude in parallel plates or cone-and-plate geometry are simple and reproducible. From the experimental values of storage and loss shear moduli, G and G", respectively, first the yield stress ought to be extracted and then the characteristic four material parameters in Eq. (2.13), rjo, r, mi, and m2, might be calculated. Next, knowing these parameters one may calculate the Gross frequency relaxation spectrum (see Eqs. (2.31) and (2.32)) and then other linear viscoelastic functions. 

A fundamental quantity relating the basic viscoelastic functions (i.e., storage, loss modulus and compliance, shear viscosity) is the monomeric friction coefficient, which is a measure of the frictional resistence per monomer unit encountered by a moving chain segment. This co-... 

The appropriate viscoelastic functions are ordinarily the complex modulus or the complex viscosity, and the corresponding quantities extrapolated to infinite dilution are the intrinsic storage and loss moduli... 

A qualitative discussion could be pursued in terms of any of the viscoelastic functions surveyed in Chapter 2. It is convenient for the moment to choose the components of the dynamic modulus, G and G". For a periodic strain, the energy storage per cycle depends on G, and is contributed by the polymeric solute molecules alone the energy dissipation depends on G", and has contributions from both solute and solvent. The relative contributions of the solute to G and G" depend on the extent to which the Brownian motions are correlated with the external forces. Force in phase with displacement corresponds to energy storage, but force in phase with velocity corresponds to energy dissipation. 

Rheometric Scientific markets several devices designed for characterizing viscoelastic fluids. These instmments measure the response of a Hquid to sinusoidal oscillatory motion to determine dynamic viscosity as well as storage and loss moduH. The Rheometric Scientific line includes a fluids spectrometer (RFS-II), a dynamic spectrometer (RDS-7700 series II), and a mechanical spectrometer (RMS-800). The fluids spectrometer is designed for fairly low viscosity materials. The dynamic spectrometer can be used to test soHds, melts, and Hquids at frequencies from 10 to 500 rad/s and as a function of strain ampHtude and temperature. It is a stripped down version of the extremely versatile mechanical spectrometer, which is both a dynamic viscometer and a dynamic mechanical testing device. The RMS-800 can carry out measurements under rotational shear, oscillatory shear, torsional motion, and tension compression, as well as normal stress measurements. Step strain, creep, and creep recovery modes are also available. It is used on a wide range of materials, including adhesives, pastes, mbber, and plastics. 

For a viscoelastic liquid (7(0) = 0. These expressions transform the stress relaxation function to the storage and loss moduli. Being Fourier trans-... 

Fig. 6 Variation of viscoelastic properties as a function of strain amplitude of uncrosslinked and dynamically vulcanized blends at 180°C (a) storage modulus, (b) loss modulus. CD2 TPV prepared by preblending, PD2 TPV prepared by phase mixing, SD2 TPV prepared by split addition...

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