Modulus shift factor

Fig. 9.10 (A) Frequency shift factors aj and (B) modulus shift factor bj as a function of temperature. Reprinted from [40],...

Fig. 25 Plot of activation energies calculated from storage modulus shift factors versus measured UPy content for linear random copolymers (plus signs) and for covalently cross-linked networks (circles). Reprinted with permission from [143], Copyright 2011 American Chemical Society...

Figure 6.28 Modulus shift factor as a function of extension ratio.

The temperature shift factors found in shifting the observed curves of G and G are given in Table 6.8. For the accuracy of superposition, the curves of G and G are shifted together. The time shift factor, aj and modulus shift factor, Pj are both based on 25 °C, Tq = 298 K. The WLF constants [6], Cj and C2 are also given in Table 6.8 ... 

Another importance of linear viscoelasticity is that it provides a reference for non-linear behaviour that is, the latter is expressed as a deviation from the former with use of appropriate parameters. First is the universal parameter, the elongation ratio, a, which reduces the time-scale of nonlinear behaviour to that of linear behaviour. Next is the modulus shift factor, T(a), which indicates the degree of strain-hardening or strainsoftening. Finally, comparison of linearised elongation data with that of shear data indicates, if they disagree, the presence of strain-induced crystallisation or strain-induced association. All these deviations from linearity are related to the structure of rubber. 

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. 

Polymers are a little more complicated. The drop in modulus (like the increase in creep rate) is caused by the increased ease with which molecules can slip past each other. In metals, which have a crystal structure, this reflects the increasing number of vacancies and the increased rate at which atoms jump into them. In polymers, which are amorphous, it reflects the increase in free volume which gives an increase in the rate of reptation. Then the shift factor is given, not by eqn. (23.11) but by... 

It was shown earlier that the variation of creep or relaxation moduli with time are as illustrated in Fig. 2.9. If we now introduce temperature as a variable then a series of such curves will be obtained as shown in Fig. 2.58. In general the relaxed and unrelaxed modulus terms are independent of temperature. The remainder of the moduli curves are essentially parallel and so this led to the thought that a shift factor, aj, could be applied to move from one curve to another. 

Thus all the different temperature related data in Fig. 2.58 could be shifted to a single master curve at the reference temperature (7 ). Alternatively if the properties are known at Tref then it is possible to determine the property at any desired temperature. It is important to note that the shift factor cannot be applied to a single value of modulus. This is because the shift factor is on the horizontal time-scale, not the vertical, modulus scale. If a single value of modulus 7, is known as well as the shift factor ar it is not possible to... 

Investigation of the linear viscoelastic properties of SDIBS with branch MWs exceeding the critical entanglement MW of PIB (about -7000 g/mol ) revealed that both the viscosity and the length of the entanglement plateau scaled with B rather than with the length of the branches, a distinctively different behavior than that of star-branched PIBs. However, the magnitude of the plateau modulus and the temperature dependence of the terminal zone shift factors were found to... 

FIGURE 34.3 Shift factor approach and fit to Equation 34.1 for typical field data (modulus). 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

Figure 5.65 Modulus-time master curve based on WLF-shift factors using data from Figure 5.63 with a reference temperature of 114°C. Reprinted, by permission, from F. Rodriguez, Principles of Polymer Systems, 2nd ed., p. 217. Copyright 1982 by Hemisphere Publishing Corporation.

In addition to knowing the temperature shift factors, it is also necessary to know the actual value of ( t ) at some temperature. Dielectric relaxation studies often have the advantage that a frequency of maximum loss can be determined for both the primary and secondary process at the same temperature because e" can be measured over at least 10 decades. For PEMA there is not enough dielectric relaxation strength associated with the a process and the fi process has a maximum too near in frequency to accurately resolve both processes. Only a very broad peak is observed near Tg. Studies of the frequency dependence of the shear modulus in the rubbery state could be carried out, but there... 

Fig. Z4 (a) Temperature ramp at a frequency a> = lOrads (strain amplitude A = 2%) for a nearly symmetric PEP-PEE diblock with Mn = 8.1 X 104gmol l, heating from the lamellar phase into the disordered phase. The order-disorder transition occurs at 291 1 °C, the grey band indicates the experimental uncertainty on the ODT (Rosedale and Bates 1990). (b) Dynamic elastic shear modulus as a function of reduced frequency (here aT is the time-temperature superposition shift factor) for a nearly symmetric PEP-PEE diblock with Mn = 5.0 X 1O g mol A Shift factors were determined by concurrently superimposing G and G"for w > and w > " respectively. The filled and open symbols correspond to the ordered and disordered states respectively. The temperature dependence of G (m < oi c) for 96 < T/°C 135 derives from the effects of composition fluctuations in the disordered state (Rosedale and Bates 1990). (c) G vs. G"for a PS-PI diblock with /PS = 0.83 (forming a BCC phase) (O) 110°C (A) 115°C ( ) 120°C (V) 125°C ( ) 130°C (A) 135°C ( ) 140°C ( ) 145°C. The ODT occurs at about 130°C (Han et at. 1995).

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) ... 

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