Superpositioned storage moduli

Figure 8 - Superpositioned storage moduli for pure 97% 1,2-PBD, pure PIP, and a mixture containing 25% by volume of the PIP. Different reference temperatures (+5°C and -50°C for the pure 1,2-PBD and PIP respectively and -20°C for the blend) were employed to allow depiction of the data on an expanded scale. The transition temperatures of all blends were, of course, intermediate to the pure component T s (-63°C and 0°C for the PIP and 1,2-PBD respectively).

Figure 11. Superpositioned storage moduli for two blends of PBD that had 97% 1,2- units with 50% PIP and 25% PIP. The reference temperature is -20 C in both cases, with the shift factors given in Figure 7.

Fig. 5.8. Storage modulus vs frequency for narrow distribution polystyrene melts, reduced to 160° C by temperature-frequency superposition. Molecular weight range from Mw = 8900 (L9) to Mw= 581000 (L18) (124). [Reproduced from Macromolecules 3, 111 (1970).]...

Fig. 21. Storage modulus (G ) for PCL based silicate nanocomposites. Silicate loadings are indicated by percentual values in the figure. Master-curves were obtained by application of time-temperature superposition and shifted to T0=55 °C. From Ref. [54].

Figure 13.14 Storage modulus G as a function of reduced frequency for a symmetric PEP-PEE sample at temperatures above Toot = 96°C (open symbols), and below it (filled symbols), time-temperature shifted to obtain superposition atcu > o)c with a reference temperature of 35 °C. (Reprinted with permission from Rosedale and Bates, Macromolecules 23 2329. Copyright 1990, American Chemical Society.)...

The Boltzmann superposition principle can be used to show that the storage modulus is related to the sine transform of G t),... 

Use the Boltzmann superposition integral to derive the storage modulus of a viscoelastic liquid as a sine transform of the stress relaxation modulus G(t) [Eq. (7.149) with 6 eq = 0)]. Also derive the loss modulus as a cosine transform of G(t) [Eq. (7.150) with Ggq = 0] for a viscoelastic liquid. 

While this paper reports only preliminary findings. It does Illustrate the usefulness of photocalorimetry to define optimum cure conditions for UV curable adhesives. In addition, once the mechanical spectrum of fully cured adhesive has been mapped, mechanical spectroscopy can be used to monitor cure efficiency. In this paper we have not explored the molecular weight Implications of Incomplete polymerization. Preliminary evaluation of loss and storage modulus data would suggest that time-temperature superposition may be necessary to evaluate molecular welght/degree of cure relationships and terminal, plateau, and transition zones (4). 

Fig. 6.12 Illustration of (a) the storage modulus, the loss modulus and the loss factor as a function of frequency across the glass transition temperature of amorphous polymers (b) the loss factor as a function of temperature according to the time-temperature superposition principle. Below the a peak for glass transition, there are secondary relaxation peaks...

The investigation of the Han plots, which is the log-log plot of storage modulus versus loss modulus, is another effective method to determine the onset of phase separation. This method is more sensitive to concentration fluctuations than data obtained from time-temperature superposition. The Han plot of homogeneous phases shows two main features temperature independence and terminal slope of two (Han et al. 1990, 1995). Deviations from these two criteria were reported only for Han plots above the LCST and below the UCST (Kim et al. 1998 Sharma and Clarke 2004). Therefore, it has been suggested to use this method to infer the phase-separation (binodal) temperature rheologicaUy. 

Yu et al. (2011) studied rheology and phase separation of polymer blends with weak dynamic asymmetry ((poly(Me methacrylate)/poly(styrene-co-maleic anhydride)). They showed that the failure of methods, such as the time-temperature superposition principle in isothermal experiments or the deviation of the storage modulus from the apparent extrapolation of modulus in the miscible regime in non-isothermal tests, to predict the binodal temperature is not always applicable in systems with weak dynamic asymmetry. Therefore, they proposed a rheological model, which is an integration of the double reptation model and the selfconcentration model to describe the linear viscoelasticity of miscible blends. Then, the deviatirMi of experimental data from the model predictions for miscible... 

The composite natm e of polyurethane elastomers strongly affects their linear viscoelastic properties. It is known that for most polymers, linear viscoelastic moduli (storage modulus, E u,T), and loss modulus, E" u,T)) are characterized by the so-called time-temperature superposition (TTS) (see, e.g. Ferry [74]). Such behavior can be understood if one assumes that E (and E") is always a function of the product ut T), where t(T) is effective relaxation time. 

Storage modulus versus frequency for narrow distri-totion polystyrene melts of increasing molecular weight, reduced to I60°C by temperature-frequency superposition. Molecular weight ranges from - 8900 (L9) to s 581,000 (L18). From Onogi et al. 1970). 

The time-temperature superposition principle has been applied to the loss and storage moduli. For the homogeneous blend (one phase at temperature equal to 115 C), the superposition method works very well. Typical low frequency behaviours of G and G are shown by the lines in Figure 10. For temperatures close to (125, 135 and 140 C), a shoulder develops in the low frequency region for the storage modulus and becomes more important as the temperature is closer to T. This behaviour is similar to that observed by Bates et al. [19] for block copolymers near in the homogeneous region (disordered zone). In fact, these temperatures are well below as determined... 

For temperature above i.e. 150 and 160 C, that is in the two-phase region, the low-frequency storage modulus presents a plateau and the superposition no longer holds. In fact, the results obtained are typical of those of an immiscible blend, similar to the results presented above for... 

Figure 46. Storage modulus, C/, of gluten - water mixtures in a time sweep test 10% strain, I Hz frequency). The master curve has been obtained by applying the time-temperature superposition principle. Temperatures considered 40 ( ), 50 ( ), 60 (0), 70 (O), 80 (A), and 90 ( ) °C (modified from [263])...

FIGURE 14.1 Time-temperature superposition of storage modulus. 

Master curves were generated from the dynamic mechanical data by employing only horizontal shifts to the tan 5 data. After these shifts were obtained, vertical shift factors were then employed to the storage modulus, G , data. The best superposition was obtained in the glassy... 

In the present case, all of our dynamic mechanical data could be reduced successfully into master curves using conventional shifting procedures. As an example, Figure 7 shows storage and loss-modulus master curves and demonstrates the good superposition obtained. In all cases, the shifting was not carried out empirically in order to obtain the best possible superposition instead the appropriate shift factors were calculated from the WLF equation (26) ... 

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