Amorphous shift factor

All amorphous polymers show remarkably similar results when values of log ay are plotted versus T - Tq. Manipulation of these data shows that the empirically determined shift factors can be fitted by the expression... 

The shift factor required to superimpose a set of data for an amorphous polymer is described mathematically by the WLF equation (eq. 8) (24), which becomes... 

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... 

Master curves can often be made for crystalline as well as for amorphous polymers (33-38). The horizontal shift factor, however, will generally not correspond to a WLF shift factor. In addition, a vertical shift factor is generally required which, has a strong dependence on temperature (36-38). At least part of the vertical shift factor results from the change in... 

For transport in amorphous systems, the temperature dependence of a number of relaxation and transport processes in the vicinity of the glass transition temperature can be described by the Williams-Landel-Ferry (WLF) equation (Williams, Landel and Ferry, 1955). This relationship was originally derived by fitting observed data for a number of different liquid systems. It expresses a characteristic property, e.g. reciprocal dielectric relaxation time, magnetic resonance relaxation rate, in terms of shift factors, aj, which are the ratios of any mechanical relaxation process at temperature T, to its value at a reference temperature 7, and is defined by... 

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.

This transposition for amorphous polymers is accomplished using a shift factor ( x) calculated relative to a reference temperature T, which may be equal to Tg. The relationship of the shift factor to the reference temperature and some other temperature T, which is between Tg and T + 50 K, may be approximated by the Arrhenius-like Equation. 

For semi-crystalline polymers with melting points of more than 100 °C above the glass transition temperature and for amorphous polymers far above the glass transition temperature Tg (at around T = Tg + 190°C), the shift factors obtained from time-temperature superposition can be plotted in the form of an Arrhenius plot for thermally activated processes ... 

For amorphous polymers which melt above their glass transition temperature Tg, the WLF equation (according to Williams, Landel, Ferry, Eq. 3.15) with two material-specific parameters q and c2 gives a better description for the shift factors aT than the Arrhenius function according to Eq. 3.14. 

Figure 3.14 Left Shift factors aT from time-temperature superposition of two amorphous thermoplastics [91. Right Arrhenius plot a(T)=f(1/T). Lines WLF Eq. 3.15 with c pc = 4.2 and c2 pc=88 K for polycarbonate (PC) and with c,. PS=5.3 and c2 P = 113.4 K for polystyrene (PS)...

This bottom equation of Equations 13-98 is called the WLF equation, after Williams, Landel and Ferry, who found that for amorphous polymers the curve describing the temperature dependence of the the shift factor aT has the general form (Equation 13-99) ... 

It is common practice now to use the glass transition temperature measured by a very slow rate method as the reference temperature for master curve construction. Tlien the shift factor for most amorphous polymers is given fairly well by... 

A common practice is to reduce relaxation or creep data to the temperature Tg thus, the reference temperature is picked as the glass transition temperature measured by some slow technique such as dilatometry. The reason for choosing Tg as the reference temperature is founded on the idea that all amorphous polymers at their glass transition temperature will have similar viscoelastic behavior. This type of corresponding states principal is often expressed in terms of a hopefully universal mathematical relationship between the shift factor aT at a particular temperature and the difference between Tg and this temperature. Perhaps the most well known of these relationships is the WLF equation... 

As demonstrated before, the shifting involves three shift factors, one horizontal, usually expressed as aj, = b rip(T)/rip(Tp), where b = p T /pT is the hrst vertical shift factor that originates in the thermal expansion of the system (p is density). The subscript o indicates reference conditions, dehned by the selected reference temperature T, usually taken in the middle of the explored T-range. For homopolymer melts, as well as for amorphous resins, the two shift factors, aj, and b.j, are sufficient. However, for semi-crystalline polymers, a second vertical factor, v., has been found necessary — it accounts for variation of the crystallinity content during frequency scans at different temperatures [Ninomiya and Ferry, 1967 Dumoulin, 1988]. 

Quite independently, Williams, Landel and Ferry found an empirical equation, now called the WLF equation, which fits the dependence of the shift factor on temperature for a large number of amorphous polymers. The equation is usually written in the form C,(r- Tg)... 

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. 

If the values of shift factor logjo ut, obtained as above, are plotted against test temperature T, a smooth curve is obtained. Williams, Landel and Ferry (1955) showed that the same shift factor-temperature relationship was obtained from the experimental shifting of results from a large number of amorphous polymers. The empirical relationship thus obtained is known as the WLF equation, and can be used in one of the two forms ... 

The horizontal shift on a logarithmic time-scale is shown in Figure 6.14. Remarkably, Williams, Landel and Ferry [7] found an approximately identical shift factor-temperature relation for all amorphous polymers, which could be expressed as... 

Fig. 128. Temperature dependence of shift factor (a,) obtained from the time-temperature shift for an amorphous LassAhjNijo alloy.

To) (where is the empirical shift factor and is the reference temperature to which the data are shifted), shown in the inset, implies that the results are consistent with the Williams-Landel-Ferry (WLF) equation, known to describe viscoelastic behavior in many amorphous polymers (14) ... 

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