Reduced dynamic moduli

Fig. 5.2 Reduced dynamic moduli for undiluted narrow distribution polystyrene of low molecular weight. Data for a sample of Mw = 28900 were reduced to 160 °C, for which e= l.Ogm/ml and t)0 = 54500 poise (124). The solid lines were calculated from the Rouse...

FIG. 13.20 Double logarithmic plot of the reduced dynamic moduli of a Maxwell model vs. the angular frequency. 

FIG. 13.21 Double logarithmic plot of the reduced dynamic moduli of a Maxwell-Wiechert model with two relaxation times of 10,000 s and 1 s, vs. angular frequency. The corresponding spring constants are G and 100G, respectively. 

Reduced dynamic moduli for the ellipsoid models, eqs. 10.3.21 and 10.3.22 compai to experimental results on tobacco mosaic virus. From Nemoto et al. (1975). 

Figure 10 Reduced dynamic moduli for the 60/40 PS/PVME blend showing phase separation.

The reduced dynamic modulus of rigidity, obtained from the dispersion viscosity, is of the form (10) ... 

Rheological properties of filled polymers can be characterised by the same parameters as any fluid medium, including shear viscosity and its interdependence with applied shear stress and shear rate elongational viscosity under conditions of uniaxial extension and real and imaginary components of a complex dynamic modulus which depend on applied frequency [1]. The presence of fillers in viscoelastic polymers is generally considered to reduce melt elasticity and hence influence dependent phenomena such as die swell [2]. 

The above expressions confirm the known (Ferry 1980) method of reducing the dynamic modulus measured at different temperatures to an arbitrarily chosen standard temperature Tref, while offering a relatively insignificant improvement on the usual shift coefficient... 

Notwithstanding the simplifying assumptions in the dynamics of macromolecules, the sets of constitutive relations derived in Section 9.2.1 for polymer systems, are rather cumbersome. Now, it is expedient to employ additional assumptions to obtain reasonable approximations to many-mode constitutive relations. It can be seen that the constitutive equations are valid for the small mode numbers a, in fact, the first few modes determines main contribution to viscoelasticity. The very form of dependence of the dynamical modulus in Fig. 17 in Chapter 6 suggests to try to use the first modes to describe low-frequency viscoelastic behaviour. So, one can reduce the number of modes to minimum, while two cases have to be considered separately. 

Generation of Master Curves. Modulus and loss factor data were processed into a reduced frequency plot in the following manner modulus curves at different temperatures were shifted along the frequency axis until they partially overlapped to obtain a best fit minimizing the sum of the squares of a second order equation (in log modulus) between two sets of modulus data at different temperatures. This procedure was completely automated by a computer program. The modulus was chosen to be shifted rather than the loss factor because the modulus is measured more accurately and has less scatter than the loss factor. The final result is a constant temperature plot or master curve over a wider range of frequency than actually measured. Master curves showing the overlap of the shifted data points will not be presented here, but a typical one is found in another chapter of this book (Dlubac, J. J. et al., "Comparison of the Complex Dynamic Modulus as Measured by Three Apparatus"). 

Figure 13.13 Reduced storage modulus G and dynamic viscosity rj = G /w as functions of reduced frequency uto) for a cylinder-forming polystyrene-polybutadiene-polystyrene triblock copolymer with block molecular weights of 7000-43,000-7000. The curves are time-temperature-shifted to a reference temperature of 138°C the open symbols were obtained in the low-temperature ordered state the closed symbols were obtained in the high-temperature disordered state. (From Gouinlock and Porter 1977, reprinted with permission from the Society of Plastics Engineers.)...

Pellicle and tea-immersed pellicle were analyzed using nanoDMA (dynamic mechanical analysis) to see if the tannins had an effect on the viscoelasticity of the pellicle. NanoDMA is a technique used to study and characterize mechanical properties in viscoelastic materials. The method is an extension of nanoindentation testing [58, 59], An analysis of the nanoindentation load-depth curve gives the hardness (H) and reduced elastic modulus (E ), provided the area of contact, A, between the indenter tip and the sample is known [ 13]. By... 

FIGURE 5.11 Logarithmic comparison plot of the reduced dynamic loss modulus, G p (in dyne/cm = 0.1 Pa), against the logarithm of the reduced frequency, coaj-is ). The reduced reference temperatures give correspondence in the softening dispersion and match the positions of the loss tangent primary maxima. 

In attempting to predict the direction that future research in carbon black technology will follow, a review of the literature suggests that carbon black-elastomer interactions will provide the most potential to enhance compound performance. Le Bras demonstrated that carboxyl, phenolic, quinone, and other functional groups on the carbon black surface react with the polymer and provided evidence that chemical crosslinks exist between these materials in vul-canizates (LeBras and Papirer, 1979). Ayala et al. (1990, 1990) determined a rubber-filler interaction parameter directly from vulcanizatemeasurements. The authors identified the ratio a jn, where a = slope of the stress-strain curve that relates to the black-polymer interaction, and n = the ratio of dynamic modulus E at 1 and 25% strain amplitude and is a measure of filler-filler interaction. This interaction parameter emphasizes the contribution of carbon black-polymer interactions and reduces the influence of physical phenomena associated with networking. Use of this defined parameter enabled a number of conclusions to be made ... 

Fig. 2. Experimentally supa-posed results of (reduced) dynamic shear modulus obtained at various temperatures and frequencies...

Even a modest increase in strain amplitude can greatly reduce the dynamic modulus of a carbon-black-filled rubber [58,80,88-90]. Because the effect on the modulus of unfilled rubbers is very small by comparison, the effect has largely been attributed to the carbon black aggregate-aggregate network [58,90]. The difference between a modulus measured at low strain and that masured at very high strain (or a value extrapolated to infinite strain) has been used as a measure of carbon black network per se [58]. The loss of dynamic shear modulus of filled rubber which occurs with increases in strain amplitude is greater if the rubber is not vulcanized [80]. 

Cowman et al. [43] investigated the effect of temperature in the range of 25-65 °C on the dynamic rheological behaviour of salt-containing aqueous solution of hyaluronan, Hylan A (cross-linked HA) and a mixture of hylans (known under the trade name Synvisc ). The increase in temperature substantially reduces the modulus and complex viscosity for all three samples. 

The two components of the complex dynamic modulus (or of the complex compliance) are weighted quite differently with respect to the long-time and short-time contributions to H (or L). For example, as is evident from equations 26 and 27 of Chapter 3, J at a particular frequency o> is determined primarily by the spectral contributions for which o)t < 1, whereas J" is determined by those for which cot CSC 1. If for any reason ar is not the same for long retardation times and short retardation times, not only will the curves for J and J" fail to match in shape, but any attempt at a forced fit will provide one set of apparent ar values for J and a different set for J". Whenever this occurs, the method of reduced variables in its simple form as given above must be rejected no master curves can be drawn without subjecting the data to a more complicated analysis. 

Fig. 33. Effect of a tackifier on the dynamic modulus G of natural mbber as a function of reduced deformation frequency ojiit-. (Taken from Ref. 167, published by John Wiley Sons.)...

Once the values of T iock determined, one can calculate the linear viscoelastic properties of a block copolymer from the following expression for the reduced complex dynamic modulus G g(ty) ... 

Dynamic Modulus. The complex shear compliance is being obtained in a Periy-Fitzgerald transducer apparatus and preliminary results are illustrated here.f9) Figure 10 shows the real part of the compliance, reduced to 25 0 Figure 11, the imaginary part. 

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