Superposition of isotherms

The isothermal curves of mechanical properties in Chap. 3 are actually master curves constructed on the basis of the principles described here. Note that the manipulations are formally similar to the superpositioning of isotherms for crystallization in Fig. 4.8b, except that the objective here is to connect rather than superimpose the segments. Figure 4.17 shows a set of stress relaxation moduli measured on polystyrene of molecular weight 1.83 X 10 . These moduli were measured over a relatively narrow range of readily accessible times and over the range of temperatures shown in Fig. 4.17. We shall leave as an assignment the construction of a master curve from these data (Problem 10). 

Analysis of experimental data for some copolyesters (4,5,6) and syndiotactic poly(propylene)(6a) indicates a superposition of isotherms. However, in these examples the crystallization was conducted at relatively large undercoolings. Superposition is to be expected under these circumstances. 

Fig. 13.13 Examples of superposition of isotherms, (a) Data from Fig. 13.10, M = 3.0 X 10. (b) Data from Fig. 13.12, M = 3.1 x 10 . Solid curves derived Avrami equation with n = 4.

The exponent 1/n of isotherm (1) is linearly dependent on the temperature. At sufficiently high temperatures when, by (18), 1/n > 1, the theory gives simply a linear course of the isotherm at the beginning an isotherm of the form q = Cp1/", 1/n > 1, cannot be obtained by superposition of the Langmuir isotherm. Indeed, in this case a(b) still has the form of equation (9) with 1/n > 1, but we may no longer seek the asymptotic form of the beginning of the isotherm by formulas (11) and (12) since... 

The values of C and D are evaluated at the critical point and normal boiling point. U. is the vertical molecule-cation interaction energy and U isJthe corresponding molecule-anion term. U and w are calculated as the sums of all the appropriate dielectric and Lennard-Jones potentials. The actual calculation of an x/m isotherm is the superposition of several solution models. The principal one corresponds to the partial filling by molecules on the cation sites. The value of x/m is a constant times Xg, summed over all sites, where the constant is the molecular weight ratio. 

Equations 6.3-19 is the well-known isothermal Newtonian extrusion theory. Since it was obtained by the solution of a linear differential equation, it is composed of two independent terms, the first representing the contribution of drag flow Q,j, and the second, the pressure flow, Qp. The net flow rate is the linear superposition of the two. 

For an energetically heterogeneous surface, where centers of different adsorption intensity are scattered, a superposition of the Langmuir isotherm results in... 

Dielectric relaxation of complex materials over wide frequency and temperature ranges in general may be described in terms of several non-Debye relaxation processes. A quantitative analysis of the dielectric spectra begins with the construction of a fitting function in selected frequency and temperature intervals, which corresponds to the relaxation processes in the spectra. This fitting function is a linear superposition of the model functions (such as HN, Jonscher, dc-conductivity terms see Section II.B.l) that describes the frequency dependence of the isothermal data of the complex dielectric permittivity. The temperature behavior of the fitting parameters reflects the structural and dynamic properties of the material. 

Figures 1 and 2 show an increase in during cure brings about a decrease in 0 = T — in the isothermal regime. Assuming the Arrhenius dependence of the rate constants on T, one can get a good superposition of the kinetic cilrves in the region well above and a considerable retardation of the reaction gt reaction temperatures near or below (Fig. 2).

The maximum strain rate (e < Is1) for either extensional rheometer is often very slow compared with those of fabrication. Fortunately, time-temperature superposition approaches work well for SAN copolymers, and permit the elevation of the reduced strain rates kaj to those comparable to fabrication. Typical extensional rheology data for a SAN copolymer (h>an = 0.264, Mw = 7 kg/mol,Mw/Mn = 2.8) are illustrated in Figure 13.5 after time-temperature superposition to a reference temperature of 170°C [63]. The tensile stress growth coefficient rj (k, t) was measured at discrete times t during the startup of uniaxial extensional flow. Data points are marked with individual symbols (o) and terminate at the tensile break point at longest time t. Isothermal data points are connected by solid curves. Data were collected at selected k between 0.0167 and 0.0840 s-1 and at temperatures between 130 and 180 °C. Also illustrated in Figure 13.5 (dashed line) is a shear flow curve from a dynamic experiment displayed in a special format (3 versus or1) as suggested by Trouton [64]. The superposition of the low-strain rate data from two types (shear and extensional flow) of rheometers is an important validation of the reliability of both data sets. 

In Figure 3 we have plotted the results corresponding to waa=Wab 4RbT, A6a= -32kBT, and different values of Wbb- Isotherms and differential heats of adsorption can be understood by following the analysis of Figs.1-2, because the behavior observed in Fig. 3 is the superposition of those effects. 

The close superposition of C02/polymer isotherms at various tempo-atures, when plotted vs. activity, sugg ts that most of the apparent temperature dependence of the isotherms plotted vs. pressure is related to the activity change of CO2 with temperature. At constant activity, die actual mixing of CO2 with PMMA, PC, or PVBz appears to be nearly athermal i.e., the energy of interaction of CO2 with these polymers seems to be essentially that associated with the compression of the gas to its molar volume in the sorbed state. This aspect of polymer interactions with CO2 will also be consicfored further in forthcoming publications. 

Such comparisons typically yield varying results for different types of sorbents. This result is not unexpected in that observed behaviors for natural systems may in fact result from the superposition of different types of individual sorption phenomena and relationships. In such cases, simple limiting-condition isotherm models are rigorously applicable only on the local level. 

Abstract The present study demonstrates, by means of broadband dielectric measurements, that the primary a- and the secondary Johari-Goldstein (JG) /3-processes are strongly correlated, in contrast with the widespread opinion of statistical independence of these processes. This occurs for different glassforming systems, over a wide temperature and pressure range. In fact, we found that the ratio of the a- and P- relaxation times is invariant when calculated at different combinations of P and T that maintain either the primary or the JG relaxation times constant. The a-P interdependence is quantitatively confirmed by the clear dynamic scenario of two master curves (one for a-, one for P-relaxation) obtained when different isothermal and isobaric data are plotted together versus the reduced variable Tg(P)/T, where Tg is the glass transition temperature. Additionally, the a-P mutual dependence is confirmed by the overall superposition of spectra measured at different T-P combinations but with an invariant a-relaxation time. 

Equation 6.30 reproduces the familiar S-shaped crystallization isotherms that are often observed. The constant K contains a superposition of nucleation and growth parameters. It is related to the time for half conversion during crystallization (ti/2) by Equation 6.31, which can be obtained from Equation 6.30 by setting oc(ti/2)=0.5 and solving for K as a function of ti/2-... 

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