Pressure-dependent shift factor

The inclusion of values in Table 1 l-III derived from dynamic bulk viscoelastic measurements implies the concept that the relaxation times describing time-de-pendent volume changes also depend on the fractional free volume—consistent with the picture of the glass transition outlined in Section C. In fact, the measurements of dynamic storage and loss bulk compliance of poly(vinyl acetate) shown in Fig. 2-9 are reduced from data at different temperatures and pressures using shift factors calculated from free volume parameters obtained from shear measurements, so it may be concluded that the local molecular motions needed to accomplish volume collapse depend on the magnitude of the free volume in the same manner as the motions which accomplish shear displacements. Moreover, it was pointed out in connection with Fig. 11 -7 that the isothermal contraction following a quench to a temperature near or below Tg has a temperature dependence which can be described by reduced variables with shift factors ay identical with those for shear viscoelastic behavior. These features will be discussed more fully in Chapter 18. 

The relaxation spectrum can also be affected by changes in pressure, p, especially if these pressure changes are large— that is, hundreds of atmospheres (Ferry 1980 Tanner 1985). As with temperature, a simple shifting procedure can be used to account for pressure effects. Thus the shift factor aj p is both temperature-and pressure-dependent. 

While an increase in temperature speeds up the viscoelastic response, an increase in pressure slows it down. In the so-called piezorheologically simple systems, all the response times have the same dependence on pressure, and the generalized shift factor is expressed by the Fillers-Moonan-Tschoegl equation (17)... 

Two differences are noticeable between the profiles at 0 and 1.2 GPa. First, at higher pressure all the peaks shift to higher Q, indicating a reduction in lattice constant. Furthermore, the relative intensities of (111), (220), and (311) reflections exhibit significant pressure dependence. This can be understood in terms of the unusual structure factor (described below) and a decrease in a with fixed Cgo diameter. A similar change in relative intensity occurs at low temperature and P = 0 GPa due to thermal contraction (8). Scans at intermediate torque values indicate that both of these changes arc monotonic with increasing pressure and that the P-induced shift is reversible upon release of the torque. 

A master curve can be constructed as indicated in Figure 22.8, where the zero-shear-rate viscosity t]q has to be evaluated for each one of the indicated viscosity curves. Both, the effect of temperature and pressure on the viscosity versus shear rate curve can be addressed by considering a shift factor that may be related, for instance, to the free volume of the system by means of the Williams, Landel, and Ferry (WLF) equation [9, 15, 23, 24]. With the aid of this shift factor, the new viscosity curve can be constructed from known viscosity values and the reference curve at the prescribed values of temperature and a pressure. The use of shift factors to take into account the temperature dependence on the viscosity curve was also used by Shenoy et al. [19-21] in their methodology for producing viscosity curves from MFI measurements. 

Interrante and Bundy (206) have reported the pressure dependence of conductivity of a single crystal of K2Pt(CN)4Bro.3(H20) r. Up to 25 kbar the conductivity increases by a factor of four. This trend is consistent with the recently reported shift of the plasma edge with pressure vide infra. In the pressure range of 25 to 1(X) kbar the conductivity drops. The absolute value of the conductivity at 25 kbar is suspect as the conductivity reported for the sample at ambient pressure is quite low. 

The frequency shift factor, ar, has been related to the free-volume fraction,/ [Ferry, 1980]. There is a direct correlation between/and the Simha-Somcynsky (S-S) hole fraction, h [Utracki and Simha, 2001b]. Under ambient pressure, h depends on the reduced temperature [Utracki and Simha, 2001a] ... 

In addition to the free volume [36,37] and coupling [43] models, the Gibbs-Adams-DiMarzo [39-42], (GAD), entropy model and the Tool-Narayanaswamy-Moynihan [44—47], (TNM), model are used to analyze the history and time-dependent phenomena displayed by glassy supercooled liquids. Havlicek, Ilavsky, and Hrouz have successfully applied the GAD model to fit the concentration dependence of the viscoelastic response of amorphous polymers and the normal depression of Tg by dilution [100]. They have also used the model to describe the compositional variation of the viscoelastic shift factors and Tg of random Copolymers [101]. With Vojta they have calculated the model molecular parameters for 15 different polymers [102]. They furthermore fitted the effect of pressure on kinetic processes with this thermodynamic model [103]. Scherer has also applied the GAD model to the kinetics of structural relaxation of glasses [104], The GAD model is based on the decrease of the crHiformational entropy of polymeric chains with a decrease in temperature. How or why it applies to nonpolymeric systems remains a question. 

The shift factor ap can be used to combine time-dependent or frequency-dependent data measured at different pressures, exactly as ap is used for different temperatures in Section A above, and with a shift factor ar,p data at different temperatures and pressures can be combined. It is necessary to take into account the pressure dependence of the limiting values of the specific viscoelastic function at high and low frequencies, of course, in an analogous manner to the use of a temperature-dependent Jg and the factor Tp/Topo in equations 19 and 20. The pressure dependence of dynamic shear measurements has been analyzed in this way by Zosel and Tokiura. A very comprehensive study of stress relaxation in simple elongation, with the results converted to the shear relaxation modulus, of several polymers was made by Fillers and Tschoegl. An example of measurements on Hypalon 40 (a chlorosulfonated polyethylene lightly filled with 4% carbon black) at pressures from 1 to 4600 bars and a constant temperature of 25°C... 

Although the effect of combined temperature and pressure changes on the shift factor ot,p is complicated if the nonlinear pressure dependence of/ is taken into account, as seen in equation 57, a simple formulation is obtained in the linear range where 8/ is constant the difference between and fi in equation 49 is given by... 

At this juncture, it is very important to emphasize that Eq. (13.20), or any modified version of Eq. (13.20), is not appropriate for calculating the shift factor for shear-rate dependent viscosities of polymer/diluent mixtures in the capillary section where the deformation of fluid is significant (i.e., 0) and information on pressure p in... 

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. 

The relative intensities of the T-shaped and linear He ICl features in the LIE and action spectra have a strong dependence on the downstream distance where the spectra are recorded, and thus on local temperature [39, 67], as is evident in Eig. 8. A shift by a factor of 20 in intensity from the T-shaped to the linear He I Cl feature was observed as the downstream distance was varied when using a He backing pressure of 155 psi. While precise Eranck-Condon... 

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