Storage modulus Strain

Both the storage modulus-strain amplitude measurements and the SEM observation indicated that compound 4 has somewhat poorer dispersion of fillers. However, there was no detectable difference in dispersion of compound 3 compared to the controls, compounds 1 and 2. 

As one example, in thin films of Na or K salts of PS-based ionomers cast from a nonpolar solvent, THF, shear deformation is only present when the ion content is near to or above the critical ion content of about 6 mol% and the TEM scan of Fig. 3, for a sample of 8.2 mol% demonstrates this but, for a THF-cast sample of a divalent Ca-salt of an SPS ionomer, having only an ion content of 4.1 mol%, both shear deformation zones and crazes are developed upon tensile straining in contrast to only crazing for the monovalent K-salt. This is evident from the TEM scans of Fig. 5. For the Ca-salt, one sees both an unfibrillated shear deformation zone, and, within this zone, a typical fibrillated craze. The Ca-salt also develops a much more extended rubbery plateau region than Na or K salts in storage modulus versus temperature curves and this is another indication that a stronger and more stable ionic network is present when divalent ions replace monovalent ones. Still another indication that the presence of divalent counterions can enhance mechanical properties comes from... 

A technique for performing dynamic mechanical measurements in which the sample is oscillated mechanically at a fixed frequency. Storage modulus and damping are calculated from the applied strain and the resultant stress and shift in phase angle. 

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. 

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... 

FIGURE 3.23 Plots of storage modulus against variable strain for the cross-linked mbber-silica hybrid nanocomposites (a) (ACM)-silica and (b) epoxidized natural mbber (ENR)-silica at different tetraethoxysilane (TEOS) concentrations. (Erom Bandyopadhyay, A., De Sarkar, M., and Bhowmick, A.K., J. Polym. Set, PartB Polym. Phys., 43, 2399, 2005. Courtesy of Wiley InterScience.)... 

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 deformation amplitude is varied [y (t) = yo sin (cot)] at a constant angular velocity. The resulting storage modulus (G ) is plotted versus the strain. 

The storage modulus (G ) was recorded at a frequency of IHz under 0.015 strain amplitude until stabilization of the protein network. In order to reduce stress in the sample, G recording started just before the gelation time which corresponds to the time at which G deviated from the baseline. Data were collected and rheological parameters were calculated using Carri-Med 50 software. For each system, the experiments were performed in triplicate. 

Figure 6. Dependence of storage modulus for PMMA and EPDM on temperature at a frequency of 1Hz and a strain of 0.1%.

Figure 2.11 Schematic of the storage modulus as a function of the frequency of the applied strain for a polymer melt. The plateau gives the network modulus, GN. The plateau extends down to lower frequencies as the molecular weight is increased because the relaxation time is proportional to M3... 

Now these expressions describe the frequency dependence of the stress with respect to the strain. It is normal to represent these as two moduli which determine the component of stress in phase with the applied strain (storage modulus) and the component out of phase by 90°. The functions have some identifying features. As the frequency increases, the loss modulus at first increases from zero to G/2 and then reduces to zero giving the bell-shaped curve in Figure 4.7. The maximum in the curve and crossover point between storage and loss moduli occurs at im. 

The term in phase with the rate of strain is related to the loss modulus and the term out of phase is related to the storage modulus ... 

Now the first term on the right is in phase with the applied strain, i.e. it has the form of a sine wave. This can be equated with the storage modulus. Conversely the phase difference between the second term on the right and the applied signal is the difference between sine and cosine waves which can be equated with the loss modulus ... 

Dynamic mechanical testers apply a small sinusoidal stress or strain to a small sample of the polymer to be examined and measure resonant frequency and damping versus temperature and forced frequency. Instrument software computes dynamic storage modulus (G ), dynamic loss modulus (G") and tan delta or damping factor. Measurements over a wide range of frequency and temperature provide a fingerprint of the polymer with sensitivity highly superior to DSC. 

Fig. 9. The effect of magnesium hydroxide filler type on the dynamic storage modulus G of polypropylene (PP) at 200 °C (strain amplitude 10%, filler level 60% by weight). Magnesium hydroxide fillers differed in origin particle size and treatment. Mean particle size (pm) type A ( ), 7.7 type B (+), 0.9 type C ( ), 4.0 type D ( ), 0.53 type E, stearate-coated version of type A, (X), 3.7 unfilled PP (O) [36]...

The dynamic response of polydimethylsiloxane (PDMS) reinforced with fused silica with and without surface treatment has been discussed in terms of interactions between the filler and polymer [54]. Since bound rubber measurements showed that PDMS chains were strongly attached to the silica surface, agglomeration due to direct contact between silica aggregates was considered an unlikely explanation for the marked increase in storage modulus seen with increasing filler content at low strains. Instead three types of flller-polymer-flller association were proposed which would cause agglomeration, as depicted in Fig. 15. 

Fig. 21 Plots of log dynamic storage modulus versus strain amplitude for ACM/silica hybrid nanocomposites at different silica concentrations at 50°C...

The interaction between two fillers particles can be investigated by measuring the Payne effect of a filled rubber compounds. In this measurement, dynamic properties are measured with strain sweep from a very small deformation to a high deformation. With the increased strain, the filler-filler network breaks and results in a lower storage modulus. This behavior is commonly known as the Payne effect... 

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