Time-pressure superposition

The effect of pressure on linear viscoelastic properties can also be accounted for in terms of shift factors. One can define an isothermal time-shift factor flp(P) that accounts for the effect of pressure on the relaxation times at constant temperature, and it has been found that this factor follows the well-known Bams equation  

Ferry [l,p. 291] describes several equations that have been proposed to describe the combined effects of temperature and pressure like the WLF equation, these equations arise from assumptions regarding the dependence of free volume on pressure and temperature. The vertical shift factor b-j. can be easily generalized to account for the effect of pressure on density as shown by Eq. 4.78, but this effect is usually negligible. 

Tschoegl, N. W. Generating line spectra from experimental responses. Part IV Application to experimental data. RheoL Acta (1994) 33, pp. 60-70 

Brabec, C. J., Schausberger, A. An improved algorithm for calculating relaxation time spectra from material functions of polymers with monodisperse and bimodal molar mass distributions. Rheol. Acta (1995) 34, pp. 397- 05 

Brabec, C. J., Rogl, H., Schausberger, A. Investigation of relaxation properties of polymer melts by comparison of relaxation time spectra calculated with different algorithms. RheoL Acta (1997) 36, 

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The time-temperature and time-pressure superposition principles (TTSP and TPSP)... 

Since we are interested in this chapter in analyzing the T- and P-dependences of polymer viscoelasticity, our emphasis is on dielectric relaxation results. We focus on the means to extrapolate data measured at low strain rates and ambient pressures to higher rates and pressures. The usual practice is to invoke the time-temperature superposition principle with a similar approach for extrapolation to elevated pressures [22]. The limitations of conventional t-T superpositioning will be discussed. A newly developed thermodynamic scaling procedure, based on consideration of the intermolecular repulsive potential, is presented. Applications and limitations of this scaling procedure are described. 

The experimental ranges of strain rates (or strains) are summarized in Table 2 for the various types of experiments. Time-temperatiire superposition was successfully applied on the various steady shear flow and transient shear flow data. The shift factors were foimd to be exactly the same as those obtained for the dynamic data in the linear viscoelastic domain. Moreover, these were found to be also applicable in the case of entrance pressure losses leading to an implicit appUcation to elongational values. 

This method requires great precision in determining n , otherwise it can lead to contradictory results [5, 27]. In particular, the present authors have shown that it may be necessary to resort to time-temperature superposition to obtain variations in viscosity over a sufficiently wide range of shear rate [15]. It is also obvious that when pressures reach sufficiently high levels, the dependence of viscosity on pressure must also be taken into account [27]. 

In order to demonstrate convincingly that this is a general experimental fact of glass-formers, experimental data for many different materials and (for a particular material) experimental data for several dielectric relaxation times are presented herein. The glass-formers include both molecular liquids and amorphous polymers of diverse chemical structures. All show the property of temperature-pressure superpositioning of the dispersion of the structural a-relaxation at constant xa. 

Figure 41. The inset shows isothermal dielectric loss spectra of DOP at ambient pressure. The y-relaxation is the only resolved secondary relaxation. The main figure is obtained by time-temperature superposition.

The solution to equation (72) now depends upon the specific temperature (pressure) and structure dependence of the i,-. These dependencies are put into the KAHR model in a manner equivalent to the time-temperature superposition principles of viscoelasticity theory. Then KAHR assume that, by a change in temperature or S, each retardation time is shifted by the same amount and that... 

Experimental determination of Ay for a reaction requires the rate constant k to be determined at different pressures, k is obtained as a fit parameter by the reproduction of the experimental kinetic data with a suitable model. The data are the concentration of the reactants or of the products, or any other coordinate representing their concentration, as a function of time. The choice of a kinetic model for a solid-state chemical reaction is not trivial because many steps, having comparable rates, may be involved in making the kinetic law the superposition of the kinetics of all the different, and often unknown, processes. The evolution of the reaction should be analyzed considering all the fundamental aspects of condensed phase reactions and, in particular, beside the strictly chemical transformations, also the diffusion (transport of matter to and from the reaction center) and the nucleation processes. 

Fig. 3. Time distribution of pAu x rays measured with the 7 mm target at 16 hPa H2 pressure. For better clarity the fit is a superposition of only 6 time spectra with single up energies Ekin = 1/32, 1/8, 1/2, 2, 8, 32 eV. To obtain the final result we used a much finer grid of kinetic energies...

In turbulent flows, even when the mean flow is steady (see below), the flow variables, i.e-., velocity, pressure, and temperature, all fluctuate randomly with time due to the superposition of the turbulent eddies on the mean flow. For example, if the temperature at some point in the flow is measured by means of some device which has very good time response characteristics and the output of this device is displayed on a suitable instrument then a signal resembling that shown in Fig. 2.10 will be obtained in turbulent flow. 

We have presented experimental and theoretical results for vibrational relaxation of a solute, W(CO)6, in several different polyatomic supercritical solvents (ethane, carbon dioxide, and fluoroform), in argon, and in the collisionless gas phase. The gas phase dynamics reveal an intramolecular vibrational relaxation/redistribution lifetime of 1.28 0.1 ns, as well as the presence of faster (140 ps) and slower (>100 ns) components. The slower component is attributed to a heating-induced spectral shift of the CO stretch. The fast component results from the time evolution of the superposition state created by thermally populated low-frequency vibrational modes. The slow and fast components are strictly gas phase phenomena, and both disappear upon addition of sufficiently high pressures of argon. The vibrational... 

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