Master curve representation

Fig. 7.17 Master curve representation using one, two and five Maxwell elements.

Fig. 4.3 Scaling representation of the spin-echo data at the first static structure factor peak Qmax- Different symbols correspond to different temperatures. Solid line is a KWW description (Eq. 4.8) of the master curve for 1,4-polybutadiene at Qmax=l-48 A L The scale r(T) is taken from a macroscopic viscosity measurement [130]. Inset Temperature dependence of the non-ergodicity parameter/(Q) near the lines through the points correspond to the MCT predictions (Eq. 4.37) (Reprinted with permission from [124]. Copyright 1988 The American Physical Society)...

Fig. 4.36 Scaling representation of NSE data (density correlation function) corresponding to PI at Q=1.92 A [second maximum of S(Q)]. Times have been divided by the KWW time Tpair to obtain a master curve. T=230 (cross), 240 (empty circle), 250 (plus), 264 (empty square), 280 (empty triangle), 320 K (empty diamond). The solid line indicates the fit with the KWW law for 250 K

Another often used representation of the viscoelastic flow behavior utilizes normal stress coefficients P/ = Ni/y. Figure 10 depicts flow curves of a family of PAA/water solutions differing in concentrations and therefore in their viscosities. Normalized by the zero-shear viscosity fiQ and by a constant shear rate /q shear stress value of to= 1 N/m they produce master curves for viscosity and the normal stress coefficient. The preparation... 

FIGURE 1.13. (a) k versus /q for PS films of various thicknesses and values of AT [32], (b) When plotted in a dimensionless representation, the data from (a) (plus additional data [36]) collapes to a single master curve described by Eq. (1.23). Adapted from [32] and [36]. 

In this representation all data collapses onto a single master curve. The 1 //q scaling of A., on one hand, and the master curve in Fig. 1.13b, on the other hand, are strong evidence for the model, which assumes the radiation pressure of propagating acoustic phonons as the main cause for the film instability. 

Fig. 8. Schematic representation of the construction of a master curve at T3. Data are slid horizontally either left or right until they overlap data at the temperature of interest. A master curve can be used to predict results at very long or short times, or equivalently, very short or long frequencies.

Viscoelastic data are commonly represented in the form of a master curve which allows the extrapolation of the data over broad temperature and frequency ranges. Master curves have, historically been presented as either storage modulus and loss modulus (or loss tangent) vs. reduced frequency. This representation requires a table of conversions to obtain meaningful frequency or temperature data. 

The effect of specific adsorption of anions (phosphate) on the electrokinetic behavior of alumina is shown in Figs. 4.15-4.18 (experimental data from Ref. [36]). All data points correspond to the same solid to liquid ratio. The electrokinetic curve obtained at initial phosphate concentration of 2 x 10" mol dm (Fig. 4.15) does not differ from the electrokinetic curve at pristine conditions (not shown). The presence of 10 mol dm phosphate induces a substantial shift in the lEP, and this shift is more pronounced at higher phosphate concentrations. This behavior is typical for specific adsorption of anions. The results from Fig. 4.15 and a few analogous sets of data points obtained at different initial phosphate concentrations (10" to 10 mol dm ) are re-plotted in Fig. 4.16 in the coordinates total phosphate concentration in solution - electrophoretic mobility. This representation gives a random cloud of points. Also the electrophoretic mobility plotted as the function of phosphate surface concentration (not shown) does not reveal any regularity. On the other hand the electrophoretic mobility plotted as the function of [HPOj ] (Fig. 4.17) or as the function of [PO ] (Fig. 4.18) produces one master curve containing all data points... 

As with the RT m, the parameter Tq can be used as a reference temperature to normalize the fracture toughness of RPV steels. Figures 10.5a, b and c (Sokolov, 1998) show the same fracture toughness data as in Fig. 10.2a, but demonstrate that, although these are valid plane strain Aic data from relatively large specimens, the Master Curve provides a good representation of the results when normalized to IT specimen size. 

Fracture toughness, data shown in Fig. 10.2, with (a) showing that the Master Curve provides a good representation of the data, while (b) shows the ASME K lower bound curve and (c) shows various tolerance bounds (Sokolov, 1998). 

There are cases for which the Master Curve does not provide a statistically valid representation of the experimental data and procedures have been developed to analyze the so-called inhomogeneous data. At this time, none of the procedures has been incorporated in the ASTM E1921 test standard, but they are utilized in some cases with example procedures described by Wallin et al. (2004), Scibetta (2012) and Choi et al. (2012). 

To illustrate the abihty of a generalized Maxwell Model (Prony Series) to fit long term data, consider the master curve data from Fig. 7.5 for polyisobutylene. A complete data set at 25°C was constructed as shown in Fig. 7.18. Thirty relaxation times evenly spaced in log time between 10 " and 10 were chosen and the sign control method used to calculate the Prony series representation seen in Fig. 7.19. The modulus E(t) calculated from... 

The master curve for a [90°]gs graphite/epoxy composite in uniaxial tension using TTSP is shown in Fig. 11.25. The following six-term Prony series representation of the data is also shown in Fig. 11.25 and as may be observed the agreement between the two is excellent. 

Fig. 11.26 Total stored and dissipated energy as calculated from Eqs. 11.58-11.60 normalized with respect to the initial total energy (t = 0 min) using the Prony series representation of the master curve in Fig. 11.25 Due to the log scale starting at t>0, the normalized total energy is slightly larger than one at the left end of the plot.

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