The Storage and Loss Moduli

As we move away from the terminal zone for a monodisperse polymer we come to a range of frequencies over which the storage modulus is essentially constant and the loss modulus has a marked maximum. This is the plateau or rubberyregion. As the frequency increases further, both moduli increase and move into a zone in which both are proportional to the square root 

Zeichner and coworkers [33,34] developed a measure of the breadth of the molecular weight distribution that is based on the curves of storage and loss moduli versus frequency. Based on data for a series of polypropylenes made by Ziegler-Natta catalysts and degraded by random chain scission, they found that the polydispersity index, i.e., the ratio Af /M , was related to the crossover modulus, G, as shown in Eq. 5.8 

The crossover modulus is the value of G (and G ) at the frequency where the two moduli are equal, i.e, where the curves of G co) and G (o) cross (see Fig. 5.5). An objective identification of the crossover point can be made by fitting the points on both curves in the vicinity of this point with cubic splines. Bafna [35] has pointed out that this empirical correlation is based on data for a single family of polypropylenes, i.e. a group of polymers having similar molecular weight distributions, and that it cannot be expected to be valid for other groups of polymers. 

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Plate-plate stress rheometer test. The resin is placed between the two steel plates of a stress-controlled rheometer, maintaining a gap larger than 0.5 cm. The upper plate oscillates at a given frequency whereas the lower plate is heated. The variation of the storage and loss moduli as a function of the temperature is monitored. Softening temperature can be estimated from the temperature at cross-over between the two moduli [26]. 

Example 2.17 Establish and plot the variation with frequency of the storage and loss moduli for materials which can have their viscoelastic behaviour described by the following models... 

The incorporation of reinforcing hllers into rubber results in most cases in an increase of the storage and loss moduli, G and G", and an increase in hysteresis, as quantihed by the loss angle 8, where tan 8 is C jG. When properly dispersed and coupled to the mbber matrix via a coupling agent, as represented by a low Payne effect, silica also shows less hysteretic loss at elevated temperatures. 

Figure 4.10 The storage and loss moduli for the viscoelastic solid in Figure 4.9...

The relaxation spectrum greatly influences the behaviour observed in experiments. As an example of this we can consider how the relaxation spectrum affects the storage and loss moduli. To evaluate this we need to change the kernel to that for a Maxwell model in oscillation and replace the experimental time by oscillation frequency ... 

Figure 4.12 The storage and loss moduli corresponding to the relaxation spectra in Figure 4.11...

These two experiments are fundamentally different in the nature of the applied deformation. In the case of the relaxation experiment a step strain is applied whereas the modulus is measured by an applied oscillating strain. Thus we are transforming between the time and frequency domains. In fact during the derivation of the storage and loss moduli these transforms have already been defined by Equation (4.53). In complex number form this becomes... 

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

The relaxation function has been calculated and is compared with experimental data in Figure 5.16. The agreement between the model and the data is reasonable. The storage and loss moduli for a polystyrene latex have also been measured and compared to the model for the relaxation spectra. The data was gathered for a dispersion in 10 2M sodium chloride at a volume fraction of 0.35 is shown in Figure 5.20. 

Figure 5.24 The storage and loss moduli for (a) a Zimm31 model and (b) a Rouse30 model...

Figure 5.32 Plot of the storage and loss moduli for a PVA gel (symbols). The data has been curve fitted using two log normal distributions (see Section 4.4.5)...

We can think of the strain as inducing melting. At the melting temperature TM we would expect the volume fraction of each phase to be equal to 0.5. We could argue that this happens where the storage and loss moduli are equal. Given these assumptions we can calculate the amount of solid and liquid-like material present as a function of strain. The apparent volume fraction of liquid is shown for a polyvinylidene fluoride latex in Figure 6.17. 

Note 4 The complex modulus is related to the storage and loss moduli through the relationships... 

Ferry and coworkers [118] extensively studied viscoelasticity of dilute solutions of stiff-chain polymers. Their results made clear that the stress or the storage and loss moduli for the solutions are sensitive to chain internal motions... 

The dynamic mechanical measurements were performed with a Rheometrics IV apparatus in a geometrical arrangement of parallel plates. The complex shear modulus G (= G + fG", where G and G", respectively, are the storage and loss moduli) at a constant frequency of 1 Hz was determined [30]. 

Fig. 22. Effect of the weight fraction of urea on the temperature dependence of the storage and loss moduli of poly(2-hydroxyethyl methacrylate) 1 = 0.00 2 = 0.021 3 = 0.039 4 = 0.0 5 = 0.115...

In all three equations Ex and E > are now the complex moduli the storage and loss moduli for the blend are obtained by direct substitution into these equations and separation of the real and imaginary parts to obtain separate mixture rules for each. Analytical expressions have been obtained for these, but they are lengthy and cumbersome. All the calculations described, therefore, were carried out by computer. The substitution of complex moduli into the solution of the equivalent purely elastic problem is justified by the correspondence principle of viscoelastic stress analysis (6). 

The behaviour and magnitude of the storage and loss moduli and yield stress as a function of applied stress or oscillatory frequency and concentration can be modelled mathematically and leads to conclusions about the structure of the material.3 For supramolecular gels, for example, their structure is not simple and may be described as cellular solids, fractal/colloidal systems or soft glassy materials. In order to be considered as gels (which are solid-like) the elastic modulus (O ) should be invariant with frequency up to a particular yield point, and should exceed G" by at least an order of magnitude (Figure 14.2). 

Figures 2.15 and 2.16 show the storage and loss moduli for poly(2-tert-butylcyclohexyl methacrylate) (P2tBCHM) and poly(4-tert butylcycloheyl methacrylate) P4tBCHM) [32],...

Soft viscoelastic solids and liquids of high viscosity, which cover a G region of 105-108 N /m2 in this region the storage and loss moduli have comparable values, so that tan 6 is neither very small nor very large test pieces are softer than the instruments they are confined in. 

This result for (193) is quite different from Fig. 13.56, where f(193 K) is approximately equal to 3 x 10 12 h. If we take for the WLF constants not the universal values of 17.44 and 51.6 but those from Table 13.10 for polyisobutylene (cf = 16.85 and cf = 102.4), we then calculate (193 K) = 3 x 10-10 h which is slightly better in agreement with Fig. 13.56. However, as said before, because of the flatness of the pseudo-rubber plateau the determination of log aT is rather inaccurate, so that the shift from temperatures below to above this plateau, thus from 80 to 25 °C is rather problematic. This means that the value of 3 x 10-12 h will also be wrong. The values of 16.85 and 102.4 have been determined with the aid of the storage and loss moduli, by which these problems were bypassed, so that the value of f(193 K) = 3 x 10-10 h will be more realistic than the other two values for (193 K). 

The quantities E and E , representing the storage and loss moduli, respectively, and tan 6, which equals E"/E , were obtained. 

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. 

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