Master curve generation

Figure 14 Master curve generated from mean-square displacements at different temperatures, plotting them against the diffusion coefficient at that temperature times time. Shown are only the envelopes of this procedure for the monomer displacement in the bead-spring model and for the atom displacement in a binary Lennard-Jones mixture. Also indicated are the long-time Fickian diffusion limit, the Rouse-like subdiffusive regime for the bead-spring model ( ° 63), the MCT von Schweidler description of the plateau regime, and typical length scales R2 and R2e of the bead-spring model.

Figure 14J1 Two master curves generated from the type of data shown in Figure 14.19 for PBMA latex samples of = 7S000gmol (ujq>er curve) and = 500000 g mol (lowercurve)usingWLFparametersrqx>rtedbyFaTy [SI]. Note that the mean effective diffusion coefficients decrease with increased extent of mixing...

Fig. 12. Master curve generated by applying time-temperature shifting to the data given in Figure 11. (Note that k29s = l/ay.) After Tobolsky (25).

Figure 15.7 Dynamic mechanical analysis, (a) Frequency multiplexing results for an epoxy/fibreglass laminate (b) Master curve generated from fixed frequency multiplexing data for epoxy fibreglass laminate. Source Author s own files)...

Time-temperature superposition software generally has many user-friendly features such as toolbars, icons, and point-and-click mouse interactions. In addition, the automatic shifting capability enables even an inexperienced operator to rapidly generate master curves and evaluate the alternative equations for correlating the data. Plots of the shift factors calculated for the respective equations can be compared to the actual shift factors obtained from both the automatic or the manual curve-fitting routine in the software. Also, TTS can be coupled with the instrument control software to allow completely unattended experimental evaluation and master curve generation once a sample is loaded. 

The linear viscoelastic master curve of a material serves as an important fingerprint for its mechanical behavior and the fine features of these master curves correlate with the particular materials molecular details. For these reasons, master curves are widely generated in practice. Below, we illustrate an example [38] of master curve generation where the original data were taken under dynamic testing. Fig. 2 shows the data. Fig. 3 shows the master curve obtained by means of the shift factor calculated from the data in Fig. 2. Finally, Fig. 4 shows a plot of the shift factor that is seen to display WLF-type behavior. 

In order to exemplify the method of master curve generation, low-density polyethylene (LDFE) is diosen as a representative case and a step-by-step procedure is outlined, lowing four graphs before the final master curve is given. In the case of all other polymers, the master rheograms are directly presented. 

Allhougfi only three different types of polymers are presented, it can be seen oondusivdy fiiat the master curves generated hold go for virgin as well as r rocessed material. Thus, fire master curves r rted for virgin polymms in Figs. 4.11-4.43 could be used reliably as master rireograms of reprocessed materials. 

The dependence of x upon T raises questions about the exactitude of the time-temperature superposition approach, which is widely used to obtain viscoelastic parameters over a wide range of co. The value of v, at least in some systems, depends upon T, so time-temperature superposition faces the same difficulty that master curve generation faces for concentration reduction of the dynamic moduli the curves change their shapes and cannot be made to superpose exactly. There is... 

Time-temperature superposition (tTs) was carried out for these multi-temperature multi-frequency tests based on Williams-Landel-Ferry (WLF) relationship. It considers the equivalency of time and temperature in the context of free volume theory for an activated flow process in viscoelastic materials such as PET. It has been found the tTs holds for the whole temperature/frequenQr range. The master curve generated from tTs is shown in Figure 13 for the 10 wt. % bamboo-PET composite at the 25.0 C reference temperature. The time-temperature superposition shift factor follows Arrhenius temperature dqrendence according to the expression ... 

The term r Vf in Equation (3.71) can be interpreted as a reduced fiber-volume fraction. The word reduced is used because q 1. Moreover, it is apparent from Equation (3.72) that r is affected by the constituent material properties as well as by the reinforcement geometry factor To further assist in gaining appreciation of the Halpin-Tsai equations, the basic equation. Equation (3.71), is plotted in Figure 3-39 as a function of qV,. Curves with intermediate values of can be quickly generated. Note that all curves approach infinity as qVf approaches one. Obviously, practical values of qV, are less than about. 6, but most curves are shown in Figure 3-39 for values up to about. 9. Such master curves for various vaiues of can be used in design of composite materiais. 

Figure 14.10 Master curves of steady shear viscosity, r (at lower shear rates) and complex viscosity, r (at higher frequencies) for the first seven generations of PAMAM dendrimers at 40°C in the bulk state...

Some applications require the material to remain under constant stress for years, yet it is often not reasonable to conduct such extended time measurements. One approach which circumvents this employs time-temperature superposition. Measurements are obtained over a shorter time span at differing temperatures. A master curve of C as a function of a reduced time tl a where a is a shift factor, is generated, and this allows the results to be extended to longer times. The shift factor is obtained by employing the Williams, Landel, and Ferry (WLF) relationship... 

Figure 1.32 Plot of the shift factor as a function of temperature used to generate the master curve plotted in Fig. 1.30.

The master creep curve for the above data is generated by sliding the individual relaxation curves horizontally until they match with their neighbors, using a fixed scale for a hypothetical curve at 204°C. Since the curve does not exist for the desired temperature, we can interpolate between 208.6°C and 199.4°C. The resulting master curve is presented in Fig. 1.34. The amount each curve must be shifted from the master curve to its initial position is the shift factor, log (aT). The graph also shows the spring-dashpot models and the shift factor for a couple of temperatures. 

It was found that both normalizations yielded tear energy master curves over all the test temperatures investigated for all but the most highly crosslinked 828/DDS network. The fact that master curves can be generated over the entire range of test temperatures shows the important role that M,. plays in the rubbery fracture of these highly crosslinked epoxies. 

As before, once the form of the WLF equation has been determined a master curve is generated and plotted in the nomograph format. 

It should also be noted that while the nomograph program reported here uses the WLF equation to calculate the shift factor, the data reduction scheme is not limited to the WLF equation. That is, any curve fitting equation that results in the calculation of a temperature shift factor can easily be added to the program and used for the generation of the master curve in the nomograph... 

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