The Storage Modulus

At long times for the viscoelastic liquids on the left of the figure, G (co) approaches 0 with decreasing frequency, just as G t) does with increasing / macro-scopically, this means that the phase angle between stress and strain approaches 90° as the stored energy per cycle of deformation becomes negligible compared with that dissipated as heat. However, the shape of the curve is somewhat different and G ( /t) G(t) at all times. At the end of the terminal zone at the left, G becomes proportional to co instead of exponentially dependent on t this relation is 

Storage modulus plotted against frequency, with logarithmic scales, for the eight systems identified as in Fig. 2-1. 

At intermediate times, the behavior is very similar to what has already been described for G(t), except that G (l/t) always exceeds G(t) to some extent. On a molecular basis, the magnitude of G depends on what contour rearrangements can take place within the period of the oscillatory deformation. 

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The storage modulus rather than compliance is plotted. This is a trivial difference, but note that the modulus is measured at a single frequency, 1 Hz. 

Fig. 18. Resolution of the complex modulus G into two vectors, G the storage modulus, and G the loss modulus the phase angle is 5.

Figure 17 Variation of the storage modulus (E ) and loss tangent (tan5) at 3 Hz, for two PDEB specimens freshly quenched, and O aged for 14 months.

In conclusion, the different thermal histories imposed to PTEB have a minor effect on the /3 and y relaxations, while the a. transition is greatly dependent on the annealing of the samples, being considerably more intense and narrower for the specimen freshly quenched from the melt, which exhibits only a liquid crystalline order. The increase of the storage modulus produced by the aging process confirms the dynamic mechanical results obtained for PDEB [24], a polyester of the same series, as well as the micro-hardness increase [22] (a direct consequence of the modulus rise) with the aging time. 

Experimentally DMTA is carried out on a small specimen of polymer held in a temperature-controlled chamber. The specimen is subjected to a sinusoidal mechanical loading (stress), which induces a corresponding extension (strain) in the material. The technique of DMTA essentially uses these measurements to evaluate a property known as the complex dynamic modulus, , which is resolved into two component parts, the storage modulus, E and the loss modulus, E . Mathematically these moduli are out of phase by an angle 5, the ratio of these moduli being defined as tan 5, Le. 

The value of this latter parameter is proportional to the energy dissipated as heat per cycle, and is known as the loss modulus. The former quantity, Gj, is proportional to the recoverable energy, and is called the storage modulus. The two are combined to form the complex modulus, G related by the equation... 

Consider a deformation consisting of repeated sinusoidal oscillations of shear strain. The relation between stress and strain is an ellipse, provided that the strain amplitude is small, and the slope of the line joining points where tangents to the ellipse are vertical represents an effective elastic modulus, termed the storage modulus /r. The area of the ellipse represents energy dissipated in unit volume per cycle of deformation, expressed by the equation... 

Dynamic mechanical properties of the nanocomposites are shown in Figure 4.6. There is 10% improvement of the storage modulus at 20°C by incorporating only 4 wt% of the nanombe. 

Researchers [37] also compared the storage modulus of a 40 phr carbon black-filled compound and a 10 phr SWNT-NR nanocomposite. The different properties between carbon black- and SWNTs-filled NR nanocomposites can be explained in terms of two different filler morphology, particularly surface area, aspect ratio, and stmcture. It can be observed from Figure 28.22 that... 

Dynamic measurements of the uncured compounds were also performed with the aid of the RPA 2000 at 100°C and a frequency of 0.5 Hz. The Payne effect was measured as the storage modulus G at a low strain of 0.56%. 

The exceptionally strong influence of calcium-ions on pectin solutions especially made with HM citrus pectins can be shown by a frequency sweep. The addition of calcium leads to an increase of the complex viscosity. Additionally we can observe a stable trapping of air bubbles in the solution. This effect can not be caused by the increase of viscosity. The frequency sweeps of the solutions give the answer. The storage modulus curves show the significant increase of the elastic shares caused by the addition of calcium-ions. 

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