Master curves, calculated experimental data

Figure 3. Comparison of experimental data of network D 0.4 and calculated master curves, using any of the curves of Fig. 2 and Eqs. 5 and 6 with appropriate adjustable constant and integration limits.

Another method for checking the consistency of the data and subsequent superposition is to plot a fixed frequency graph (corresponding to one or more frequency that is represented in the experimental data set) generated from the calculated master curve and compare it with the actual experimental data. Figures 4 and 5 show E and E" (respectively) vs. temperature for a fixed frequency of 0.1 Hz. The solid line represents the actual experimental data determined by DMA at 0.1 Hz. From inspection one can see there is excellent agreement between the calculated data and the experimentally measured data. 

Fokkink et al. [35] argue that for divalent metal cations the proton stoichiometry coefficient as a function of (PZC-pH) should give one bell shaped master curve for all oxides and all metal cations, with a maximum r value slightly below 2 for PZC-pH = 0 and r = 1 at PZC-pH = 4. They support their calculations by experimental data from six sources covering seven adsorbents and six metal cations. 

Fig. 21 Steady state incoherent intermediate scattering functions d> (r) as functions of accumulated strain yt for various shear rates y the data were obtained in a col loidal hard sphere dispersion at packing fraction

Assuming that the WLF equation does indeed describe the time-temperature shifts, the complete viscoelastic response of any polymer under any experimental conditions may be obtained from knowledge of any two of the following three functions the master curve at any temperature, the modulus-temperature curve at any time, and the shift factors relative to some reference temperature. For example, suppose we are given the constants Cj, and C2 for a polymer whose master curve is known. (The values given for C, and C2 are those that result from fitting equation (4-6) to the aT vs. T data.5) For simplicity, we can assume that the master curve is at the same reference temperature as that in the WLF equation, perhaps Tg. Suppose it is desired to calculate the 10-second modulus-versus-temperature curve for this polymer. 

Partial master curves of 10 g.dL"l solutions of a,o)-alkaline earth dicarboxylato PBD in xylene at 297 K are reported in Figure 10, and result from a good frequency-temperature superposition of the experimental data.l7 Only the G" master curve of the solution of Be-based HTP is ill-defined due to the poor accuracy in the determination of the very small values of G". The shift factors support an apparent Arrhenius-type of dependence (Figure 11), from which the activation energy of the observed secondary ionic relaxation process was calculated and found to decrease as the radius of the alkaline earth cations increases (Figure 12). One also observes that the relaxation spectrum calculated by the first order approximation of Ninomiya and Ferry S is displaced along the time scale in relation with the cation size (Figure 13). The dynamic behavior of the 10 g.dL solution is obviously... 

The scaling exponent A is found in the 0.65-0.75 range, independently of the experimental conditions, the copolymers composition and the nature of the midblock. Modulus-frequency master curves have been built up by using appropriate reaction time dependent renormalisation factors for the individual frequency and modulus data. The scaling of these factors with reaction time has allowed to calculate the static scaling exponents for the increase observed in both modulus and viscosity. 

The renormalized horizontal and vertical factors used to build up the master curve follows the longest relaxation time, Xz, and the steady-state creep compliance Je, respectively (d, 19, 4S). Therefore, the scaling relationships expressed by eqs.4 and 5 can be used to calculate the static exponents s and t. Figure 10 shows the log-log plot of ah and ay versus log e, in case of sample G1 (Table III), for which t=1.95 and s=0.69. The values of s, t and A are listed in Table III not for all the samples, but only when enough experimental data were available in the sol-gel transition so as to construct accurately the master curves. 

Figure 6. Master loss curve for a 4.2% lr/p rJ solution of anthrone in o-terphenyl. Dashed line indicates the experimental data, continuous line is that calculated for 8 = 0.55.

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. 

Figure 5.59. Isochronal (fixed frequency) data for a frequency of 0.1 Hz generated from the master curve for PMMA of Fig. 5.58 the soUd fine is from experimental data, and the crosses (x) are calculated from the master curve (from Weissman and Chartoff, 1990, reprinted with permission from ACS Symposium Series 424, 111. Copyright 1990 by the American Chemical Society).

The scaled probability densities are calculated using the data presented in Fig. 18 and plotted against t in Fig. 19. The scaled probability densities at all t fell nicely onto single master curves both for H and K, demonstrating that the dynamical scaling law holds for interfacial curvatures within experimental accuracy. The results shown in Fig. 19 stand out by the fact that they demonstrate that not only the global structure but also the local shape of the spinodal interface evolves with the dynamical self-similarity. In addition, the probability densities experimentally obtained in the present study are expected to be universal in a variety of condensed matter systems. 

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