Master curves, calculated

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.

Figure 9. Experimental and calculated master curves of the 25/75 PC-PST blend at 140°C...

Deduction of Shift Factors. Time-temperature shift factors for the blends were obtained by shifting the experimental relaxation isotherms to the calculated master curves (10). The temperature and time dependence of the shift factors of the 75/25 and 50/50 blends are represented in Figures 10a and 10b at t = 10 sec and t = 1000 sec for a reference temperature of 140°C. The empirically determined shift factors of the pure components are given in these figures by dotted lines their temperature dependence is of the WLF type. 

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. 

Master curves can also be constmcted for crystalline polymers, but the shift factor is usually not the same as the one calculated from the WLF equation. An additional vertical shift factor is usually required. This factor is a function of temperature, partly because the modulus changes as the degree of crystaHiuity changes with temperature. Because crystaHiuity is dependent on aging and thermal history, vertical factors and crystalline polymer master curves tend to have poor reproducibiUty. 

Though the functional form of the dynamic structure factor is more complicated than that for the self-correlation function, the data again collapse on a common master curve which is described very well by Eq. (28). Obviously, this structure factor originally calculated by de Gennes, describes the neutron data well (the only parameter fit is W/4 = 3kBT/2/C) [41, 44],... 

Master curves are important since they give directly the response to be expected at other times at that temperature. In addition, such curves are required to calculate the distribution of relaxation times as discussed earlier. Master curves can be made from stress relaxation data, dynamic mechanical data, or creep data (and, though less straightforwardly, from constant-strain-rate data and from dielectric response data). Figure 9 shows master curves for the compliance of poly(n. v-isoprene) of different molecular weights. The master curves were constructed from creep curves such as those shown in Figure 10 (32). The reference temperature 7, for the... 

Various factors govern autohesive tack, such as relaxation times (x) and monomer friction coefficient (Co) and have been estimated from the different crossover frequencies in the DMA frequency sweep master curves (as shown in Fig. 22a, b). The self-diffusion coefficient (D) of the samples has been calculated from the terminal relaxation time, xte, which is also called as the reptation time, xrep The D value has been calculated using the following equation ... 

In Fig. 16, the values of the exponent a are plotted against the barrier frequency this is clearly a kind of master curve that summarizes the essence of much of the work reported here. When the rate is calculated from Grote-Hynes theory, this curve depends only weakly on temperature. 

In the present case, all of our dynamic mechanical data could be reduced successfully into master curves using conventional shifting procedures. As an example, Figure 7 shows storage and loss-modulus master curves and demonstrates the good superposition obtained. In all cases, the shifting was not carried out empirically in order to obtain the best possible superposition instead the appropriate shift factors were calculated from the WLF equation (26) ... 

Calculation of Master Curves from Mechanical Models. The only way to obtain valid master curves for the thermorheologically complex systems (75/25 and 50/50 blends) is to calculate the moduli of the blends as a function of time, using an appropriate mechanical model. This method requires knowledge of the time and temperature dependence of the mechanical properties of the constituent phases. 

Fig. 14. Comparison of calculated [24] and measured [36, 37] master curves of dynamic properties of an epoxy resin...

The master curves and shift factors of transient and dynamic linear viscoelastic responses are calculated for linear, semi-crystalline, and cross-linked polymers. The transition from a WLF dependence to an Arrhenius temperature dependence of the shift factor in the vicinity of Tg is predicted and is related to the temperature dependence of physical aging rate. 

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... 

Figure 3. Master Curves of PMMA Multiplex DMA Data Represented on Reduced Frequency Nomograph. Master Curves Calculated Using "Universal" WLF Equation.

One method is simple visual inspection of the generated master curve. If the fit of the individual data sets is poor, subsequent extrapolation is open to significant error. Another method is to determine an activation energy plot (of the shift of T with frequency) for the data set and compare temperatures or frequencies calculated from the resulting Arrhenius equation to those read from the nomograph. 

The procedure by which the nomograph is generated is not limited to the WLF equation. Since it is based on the reduced variable concept, any superposition equation that results in the calculation of a temperature shift factor may be used to calculate the needed data to create the master curve and subsequent nomograph. The software can easily be modified to calculate and display a master curve on some other superposition equation. 

The following simple calculation illustrates the very significant temperature and time dependence of viscoelastic properties of polymers. It serves as a convenient, but less accurate, substitute for the accumulation of the large amount of data needed for generation of master curves. Suppose that a value is needed for the compliance (or modulus) of a plastic article for 10 years service at 25°C. What measurement time at 80°C will produce an equivalent figure We rely here on the use of a shift factor, aj, and Eq. (11 -39). Assume that the temperature dependence of the shift factor can be approximated by an Arrhenius expression of the form... 

We compare the orientational relaxation of central deuterated part of the PS HDH 188 copoljraer with that of the end part of the PS DH 184 copolymer. Both types of chains have almost the same molecular weight as well as deuterated blocks of comparable length. The master curves of the orientational relaxation average (calculated as the weight average of the measured orientation of each block) can be superimposed as shown in Figure 1. 

In this work we used polystyrene-based ionomers.-Since there is no crystallinity in this type of ionomer, only the effect of ionic interactions has been observed. Eisenberg et al. reported that for styrene-methacrylic acid ionomers, the position of the high inflection point in the stress relaxation master curve could be approximately predicted from the classical theory of rubber elasticity, assuming that each ion pah-acts as a crosslink up to ca. 6 mol %. Above 6 mol %, the deviation of data points from the calculated curve is very large. For sulfonated polystyrene ionomers, the inflection point in stress relaxation master curves and the rubbery plateau region in dynamic mechanical data seemed to follow the classical rubber theory at low ion content. Therefore, it is generally concluded that polystyrene-based ionomers with low ion content show a crosslinking effect due to multiplet formation. More... 

In calculating titration curves, separate equations for different regions of the curve ("before the equivalence point", "at the equivalence point", "after the equivalence point", etc.) are often employed. This section illustrates how to use a single "master" equation to calculate points on a titration curve. Instead of calculating pH as a function of the independent variable V, it is convenient to use pH as the independent variable and V as the dependent variable. The species distribution at a particular pH value is calculated from the [H ], and the volume of titrant required to produce that amount of each species is calculated. For example, in the titration of a weak monoprotic acid HA, we can calculate the concentration of A at a particular pH and then calculate the number of moles of base required to produce that amount of A . In general (J -j) moles of base per mole of acid are required to produce the species HjA from the original acid species HjA. 

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. 

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