Stress relaxation comparison with experiments

The K-BKZ Theory Comparison with Experiment. The first data required to test the K-BKZ model is single-step stress relaxation data to determine the material parameters of interest. This is best seen from the following example for a simple shearing history. From equation 49, the shear stress for a simple shear deformation can be expressed as (see Ref 72)... 

The purpose of these comparisons is simply to point out how complete the parallel is between the Rouse molecular model and the mechanical models we discussed earlier. While the summations in the stress relaxation and creep expressions were included to give better agreement with experiment, the summations in the Rouse theory arise naturally from a consideration of different modes of vibration. It should be noted that all of these modes are overtones of the same fundamental and do not arise from considering different relaxation processes. As we have noted before, different types of encumbrance have different effects on the displacement of the molecules. The mechanical models correct for this in a way the simple Rouse model does not. Allowing for more than one value of f, along the lines of Example 3.7, is one of the ways the Rouse theory has been modified to generate two sets of Tp values. The results of this development are comparable to summing multiple effects in the mechanical models. In all cases the more elaborate expressions describe experimental results better. 

The method (27) can best be explained with reference to Figure 2. After stretching to 10, the force f is measured as a function of time. The strain is kept constant throughout the entire experiment. At a certain time, the sample is quenched to a temperature well below the glass-transition temperature, Tg, and cross-linked. Then the temperature is raised to the relaxation temperature, and the equilibrium force is determined. A direct comparison of the equilibrium force to the non-equilibrium stress-relaxation force can then be made. The experimental set-up is shown in Figure 4. 

Comparison of the dielectric and viscoelastic relaxation times, which, according to the above speculations, obey a simple relation rn = 3r, has attracted special attention of scholars (Watanabe et al. 1996 Ren et al. 2003). According to Watanabe et al. (1996), the ratio of the two longest relaxation times from alternative measurements is 2-3 for dilute solutions of polyisobu-tilene, while it is close to unity for undiluted (M 10Me) solutions. For undiluted polyisoprene and poly(d,/-lactic acid), it was found (Ren et al. 2003) that the relaxation time for the dielectric normal mode coincides approximately with the terminal viscoelastic relaxation time. This evidence is consistent with the above speculations and confirms that both dielectric and stress relaxation are closely related to motion of separate Kuhn s segments. However, there is a need in a more detailed theory experiment shows the existence of many relaxation times for both dielectric and viscoelastic relaxation, while the relaxation spectrum for the latter is much broader that for the former. 

The stress relaxation behavior of the Nafion system exhibits some unusual characteristics the relaxation master curves of the precursor as well as of Nafion in its acid and salt forms are very broad, and are characterized by a wide distribution of relaxation times. Figures 2, 3, and 4 show the individual stress relaxation curves and the master curves, with the reference temperatures indicated in the captions, for the precursor (42), Nafion acid and Nafion-K (31), respectively. Figure 3 also shows the master curves for styrene and two sytrene ionomers for the sake of comparison. Time-temperature superposition of stress relaxation data appears to be valid in the precursor and in the dry Nafion acid, at least over the time scale of the experiments. In the case of Nafion-K, time-temperature superposition is not valid, leading to a breakdown at low temperatures, but is reestablished at high temperatures (above 180°C). Similar behavior was also observed... 

The initial (maximum) film tension after the expansion in the film stress-relaxation experiments can also be used to determine the film elasticity (7). A plot of the initial film tension versus the logarithm of the relative film expansion is shown in Fig. 5. For comparison, the initial single interfacial tensions obtained in the experiments with the respective single oil/water interface are also plotted. The film elasticity obtained from the top of the curve is equal, within experimental error, to twice the interfacial elasticity of the single interface. 

FIGURE 4 Comparison of rheological model of Eqs. (47)-(49) with experiment for natural rubber, (a) Steady-state shear viscosity, (b) Transient shear viscosity at beginning of flow, (c) Stress, relaxation following now. 

The utility of the K-BKZ theory arises from several aspects of the model. First, it does capture many of the features, described below, of the behavior of polymeric melts and fluids subjected to large deformations or high shear rates. That is, it captures many of the nonlinear behaviors described above for steady flows as well as behaviors in transient conditions. In addition, imlike the more general multiple integral constitutive models (108,109), the experimental data required to determine the material properties are not overly burdensome. In fact, the information required is the single-step stress relaxation response in the mode of deformation of interest (72). If one is only interested in, eg, simple shear, then experiments need only be performed in simple shear and the exact form for U I, /2, ) need not be obtained. Furthermore, because the structure of the K-BKZ model is similar to that of finite elasticity theory, if a full three-dimensional characterization of the material is needed, some of the simplilying aspects of finite elasticity theories that have been developed over the years can be applied to the behavior of the viscoelastic fluid description provided by the K-BKZ model. One such example is the use of the VL form (98) of the strain energy function discussed above (110). The next section shows some comparisons of the material response predicted by the K-BKZ theory with actual experimental data. 

The majority of published work on extrusion behaviour deals with compounded stock. Those papers reporting work on raw rubbers have usually been on the use of capillary rheometers to determine extrusion properties at higher shear rates than are possible with Mooney viscometers. Capillary rheometers are, in principle, quite simple to use, and the application of electronic, minicomputer and laser technology has reduced the operation and data analysis to a routine task. There are no standard ASTM or other test procedures, but under a specific set of conditions, once a material is characterized, the data can be used as standard for comparison of all subsequent batches. It is readily possible to characterize a raw rubber by an extrusion experiment to determine the viscosity/shear rate curve, extrudate swell, and stress relaxation.Both Sezna and Karg have shown how the Monsanto Processability Tester (MPT), a modified, computerized extrusion rheometer, can be used in predicting mixing behaviour. The MPT (shown schematically in Fig. 7) is a most versatile instrument. It has a larger than conventional barrel for minimal pressure drop in the barrel, a pressure transducer at the entrance to the orifice, a microprocessor system, and a laser device for... 

IR Dichroism. Two types of IR-dichroism experiments were used in this study to follow segmental orientation. First, dynamic differential dichroism was used to follow chain orientation while the sample was elongated at a constant strain rate. This experiment was performed with different IR peaks which allowed a comparison of the molecular orientations for each blend constituent. Second, a cyclic experiment was used where the film was strained to a predetermined elongation, relaxed at the same strain rate until the stress was reduced to zero, and then elongated to a higher level of strain, and so forth. 

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