The Storage and Loss Compliances

C) entangled, polydisperse with M. The monodisperse,entangled sample (B) shows a plateau, but this is smeared out for C, due to the presence of small molecules. Jg is the glassy compliance, which only comes into play at very high frequencies. 

10 (logarithmic scales).Time-temperature superposition was achieved by using separate shift factors for high and low frequency ranges. The dashed line is A while the data points have had the viscous contribution subtracted. The features apparent in the adjusted loss compliance data have the same significance as those in the /.(f) data shown in Fig. 10. From Plazek [31], 

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These expressions show that, for example, any value of the storage modulus depends upon the values of both the storage and loss compliance. This is true for these dynamic properties, only the ratios interrelate simply ... 

Figure 5.29 The storage and loss compliance for a Mooney39 model...

Note 4 The eomplex complianee is related to the storage and loss compliances through the relationships... 

Other types of linear viscoelastic experiments may be used. Dynamic shear compliance measurements provide the storage and loss compliances J (co) and J"(co). An equation analogous to Eq.(3.12) is available for determining the initial modulus from J"(co) ... 

Develop equations relating the storage and loss compliances to the instantaneous and transient compliances analogous to Eqs. 6.53 and 6.54. 

Oscillatory shear can also be performed by varying the stress sinusoidally and measuring the resulting strain as a function of time. The results can then be interpreted in terms of the real and imaginary components of the complex compliance, ] =J -i J". These components, the storage and loss compliances, are simply related to the storage and loss moduli as shown below. 

The storage and loss compliances are closely related to the retardation spectrum function I(T)as follows ... 

The storage and loss compliances can also be expressed in terms of a discrete spectrum of retardation times ... 

In the frequency domain, the storage and loss compliance are given as... 

Figure 3.16a shows the storage and loss components of the compliance of crystalline polytetrafluoroethylene at 22.6°C. While not identical to the theoretical curve based on a single Voigt element, the general features are readily recognizable. Note that the range of frequencies over which the feature in Fig. 3.16a develops is much narrower than suggested by the scale in Fig. 3.13. This is because the sample under investigation is crystalline. For amorphous polymers, the observed loss peaks are actually broader than predicted by a...

Dynamic storage and loss compliance function for the cubic array are presented in Eqs. (T 11) and (T 12), respectively. 

The linear viscoelastic properties of all samples were characterized by dynamic shear measurements in the parallel-plate geometry. Experimental details have been previously published [9]. Using time-temperature equivalence, master curves for the storage and loss moduli were obtained. Fig. 1 shows the master curves at 140°C for the relaxation spectra and Table 3 gives the values of zero-shear viscosities, steady-state compliances and weight-average relaxation times at the same temperature. 

Although creep-compliance (Kawabata, 1977 Dahme, 1985) and stress-relaxation techniques (Comby et al., 1986) have been used to study the viscoelestic properties of pectin solutions and gels, the most common technique is small-deformation dynamic measurement, in which the sample is subjected to a low-amplitude, sinusoidal shear deformation. The resultant stress response may be resolved into an in-phase and 90° out-of-phase components the ratio of these stress components to applied strain gives the storage and loss moduli (G and G"), which can be related by the following expression ... 

FIGURE 24.22. Graphical representation of the storage and loss moduli and E" as components of a vector Ecomplex plane. d is the absolute value of the dynamic modulus. The corresponding compliances are shown in the right hand part of the figure. 

The storage and loss shear moduli, G and G", vs. oscillation frequency ta, and the creep compliance J vs. time t, measured at each concentration and tanperature, were temperature shifted with respect to frequency or time. These temperature master curves at each concentration were then shifted to overlap one another along the frequency or time axis. The dynamic shear moduli master curves as a function of reduced frequency (oa ac are shown in fig. 4.4, and the shear creep compliance master curves as a function of reduced time tlajUc are shown in fig. 4.5. Master curves... 

As an example of bulk viscoelastic behavior, data for a poly(vinyl acetate) of moderately high molecular weight are shown in Fig. 2-9. Measurements by McKinney and Belcher of the storage and loss bulk compliance B and B" at various temperatures and pressures are plotted after reduction to a reference temperature and pressure of 50°C and 1 atm respectively (see Chapter 11). The complex bulk compliance is formally analogous to the complex shear compliance, but the two functions present several marked contrasts. 

Bueche applied the bead-spring normal coordinate treatment to a cross-linked network to calculate the retardation spectrum and the creep, storage, and loss compliances, L, J t), J, and J", rather than H, (7(f), G, and C".The results..jvere equivalent to equations 34 to 37 except for numerical factors close to unity. A similar calculation was made by Nakada, and by Ham, who considered different models of lattice connectivity. W... 

FIG. 18-12. Dynamic bulk storage and loss compliances of poly(vinyl acetate) reduced to I atm and 1000 Hz by equation 6 and plotted against reduced temperature defined by equation 12. Bq and correspond to B, and Bg in the text. (McKinney and Belcher. )... 

In Chapter 4 it was explained that the linear elastic behavior of molten polymers has a strong and detailed dependency on molecular structure. In this chapter, we will review what is known about how molecular structure affects linear viscoelastic properties such as the zero-shear viscosity, the steady-state compliance, and the storage and loss moduli. For linear polymers, linear properties are a rich source of information about molecular structure, rivaling more elaborate techniques such as GPC and NMR. Experiments in the linear regime can also provide information about long-chain branching but are insufficient by themselves and must be supplemented by nonlinear properties, particularly those describing the response to an extensional flow. The experimental techniques and material functions of nonlinear viscoelasticity are described in Chapter 10. 

Figure 5.15 Storage and loss compliances versus reduced frequency for the polyfvinyl acetate) of Rgs.

Figure 7.17 Storage and loss compliance shifted for the UHMWPE, at 37 C. From [12].

The corresponding relationships are noted for the storage and loss moduli, zero-shear viscosity, steady-state recoverable compliance, and average relaxation times (cf. eqns [32]-[35]) ... 

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