The Modulus Curve

Dynamic mechanical experiments yield both the elastic modulus of the material and its mechanical damping, or energy dissipation, characteristics. These properties can be determined as a function of frequency (time) and temperature. Application of the time-temperature equivalence principle [1-3] yields master curves like those in Fig. 23.2. The five regions described in the curve are typical of polymer viscoelastic behavior. 

In the glassy region, the polymer is below its glass transition temperature, Tg, and typically has a modulus of 1010 dynes/cm2. The transition region includes the Tg, which is taken as the point of inflection of the modulus or the maximum in the damping curve. The modulus drops by a factor of 1000 in this region. The 

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Fig. 22 The strength versus the modulus curves of PpPTA fibres calculated with Eq. 58 for three different values of the critical shear strain...

Fig. 24 The strength versus the modulus curves for PBO fibres calculated for three different critical shear stress values and the observed strength of PBO (Zylon) given by the manufacturer... 

Figure 2.2 Illustrative plot of the Lennard-Jones-Devonshire interatomic potential showing the force and the modulus curve for the pair interaction. Positive values indicate repulsion and negative values indicate attraction...

The molecular weight between crosslinks (Me) was determined for each epoxy/amine ratio of the neat resin from the rubbery plateau region of the modulus curve following the Tg region. This can be seen in Figure 13 for several epoxy/amine ratios. The Me values were calculated from the following equation ... 

The value of the modulus and the shape of the modulus curve allow deductions concerning not only the state of aggregation but also the structure of polymers. Thus, by means of torsion-oscillation measurements, one can determine the proportions of amorphous and crystalline regions, crosslinking and chemical non-uniformity, and can distinguish random copolymers from block copolymers. This procedure is also very suitable for the investigation of plasticized or filled polymers, as well as for the characterization of mixtures of different polymers (polymer blends). 

The modulus curves of three blends of PEO E4000 (75, 50, and 25 wt. %) with PVN are shown in Figure 1 along with that for pure PVN. The 25% PEO blend... 

Figure 4 shows the modulus curves for two blends of PEO E4000 (75 and 50%) with poly-4-vinylbiphenyl and one blend of 50% PEO E4000 with polystyrene. None of these blends exhibited a well, and at high temperatures the samples became friable and broke. 

The rather unexpected properties described above seem to be peculiar to PVN, for none of the blends with polystyrene, poIy-4-vinylbiphenyl, and polyacenaphthylene contained significant amounts of amorphous PEO. The modulus curves for these systems are characteristic of blends of incompatible polymers. The photomicrograph in Figure 12 illustrates the different morphologies of PVB and PEO blends. The reason for the apparently different behavior for these polymers as compared with PVN is not yet understood. But there is strong evidence from dilute solution-studies that the conformational properties for these polymers differ markedly. 

With an increase in temperature from To to T the modulus curve will then be shifted along the log t axis over a distance ... 

Arrows indicate the rubber transition in the modulus curves... 

In Figure 6, the sharp cutting of the rubber transition in the modulus curves is accompanied by the shift in the tg8 or E peak to higher temperatures. Quite similar behavior was reported for block copolymers with up to 40% polybutadiene (22, 23). In Figure 7, the shift in the matrix E peak to... 

In both cases of copolymerization, there is a noticeable decrease in the slope of the modulus curve in the r on of the inflection point. This, in essence, means a decrease in the modulus in the rubbery region. This contrasts with the chemically cross-linked systems where the modulus in the rubbery region shows some increase with increasing temperatures. In the copolymer system, the molecules are inter-coimected by physical cross-hnks due to secondary forces. These cross-hnks can be disrupted reversibly by heating, and this forms the basis of the new class of copolymers referred to as thermoplastic elastomers. 

The rubbery state At approximately 30 K above the glass transition the modulus curve begins to flatten out into the plateau region C to D in the modulus interval 10 to lO Nm- and extends up to about 420 K. 

Figure 1.5 Change in modulus and mechanical damping in the region of the glass transition temperatures for (a) amorphous thermoplastic, (b) crosslinked thermoset, (c) a biend of two thermoplastic polymers. The 7g corresponds to the steepest slope in the modulus curves and (more approximately) to the peaks in the damping curves, assuming that the damping vibrations are of low frequency...

In the study of the effect of crystallinity on the 10 second tensile modulus, it was found that the modulus curve of the amorphous Nafion (prepared hy melting Nafion by heating and subsequently quenching it) was essentially parallel to that of the crystalline Nafion, but only moved along the temperatures ca 40°C lower than the latter (126,128). Thus, it was concluded that the small degree of crystallinity, ie, ca 7%, in perfluoroethylene ionomers shifted the modulus-temperature curve to ca 40° C higher. 

The forming window for a given polymer can be quantified by differential thermal mechanical analysis. Specifically, the temperature-dependent elastic modulus is key, as shown in Fig. 19, for typical thermoformable polymers (34). An adequate forming window is dictated if the modulus curve shows a flattening or... 

Three different resin-rubber blend systems were examined by frequency scan at room temperature. First, we examined blends of natural rubber with a compatible aliphatic oil. Figure 44 shows G vs. frequency at various loadings of oil to natural rubber. As we add oil, the modulus curve as a function of frequency is depressed. This is to be expected as the oil softens the composition which, in turn, causes the reduction of the modulus. In other words, oil does not change the tan 8 peak (7g) temperature of natural rubber it only reduces the room-temperature modulus value of natural rubber. Consequently, it does not function as an effective tackifying resin. 

Figure 5.11. Illustration of modulus versus time for a creep or stress relaxation experiment on a typical amorphous polymer (T = constant). Note the similarity to the plot of DMA storage modulus versus temperature. The reciprocal of creep compliance, l//(f), would be used to generate the modulus curve derived from the creep response.

Williams, Landel, and Ferry found that C and Ci were similar for many amorphous polymers with Q = 17.44 and C2 = 51.6 in the temperature range between Tg and Tg -i-100 °C. The equation is referred to as the universal WLF equation when Q and C2 assume these values. While this equation is not truly universal, it was developed from a large database for various polymers. When the equation is written in this form, it is clear that Tg serves as a corresponding state for viscoelastic behavior. A plot of log aj versus (T - T) for the data of Fig. 5.15 is shown in Fig. 5.17 here 7] is the inflection point in the modulus curve at Tg. Each DMA vendor has software available that automates the TTS operations of curve shifting, determining values of aj, and fitting the data to the WLF equation. 

For the glucomannan extracted from Amorphophallus Konjac, the modulus curves at 50°C at a load frequency of 1 Hz and a humidity rate of 0.1 %RH/min is compared to that of xylan in Figure 9. As in the xylan, a weak transition occurred at low relative humidity and a glass transition at 60 to 70 %RH. Also, for the glucomannan the transitions shifted to lower relative humidities with increasing temperature. 

It is important to note that, although the loss is a maximum at the point of inflection in the modulus curve, the tails extend considerably into both the glass and the rubber regions. It is this latter that gives rubbers their energy absorbing, vibration damping properties. 

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