Time-temperature equivalence

3 TIME-TEMPERATURE EQUIVALENCE 16.3.1 Time- Temperature Equi valence 

In fact, if you compare the results obtained at a constant temperature, where (o is varied, to those obtained by oscillating the sample 

FIGURE 13-80 Schematic representation of the dependence of G G , and tan 5 with temperature measured at constant frequency. 

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The preceding example of superpositioning is an illustration of the principle of time-temperature equivalency. We referred to this in the last chapter in connection with the mechanical behavior of polymer samples and shall take up the... 

The kinetic nature of the glass transition should be clear from the last chapter, where we first identified this transition by a change in the mechanical properties of a sample in very rapid deformations. In that chapter we concluded that molecular motion could simply not keep up with these high-frequency deformations. The complementarity between time and temperature enters the picture in this way. At lower temperatures the motion of molecules becomes more sluggish and equivalent effects on mechanical properties are produced by cooling as by frequency variations. We shall return to an examination of this time-temperature equivalency in Sec. 4.10. First, however, it will be profitable to consider the possibility of a thermodynamic description of the transition which occurs at Tg. 

We shall presently examine the physical significance of the shift factors, since they quantitatively embody the time-temperature equivalence principle. For the present, however, we shall regard these as purely empirical parameters. The following Ust enumerates some pertinent properties of a ... 

A well-known example of this time-temperature equivalence is the steady-state creep of a crystalline metal or ceramic, where it follows immediately from the kinetics of thermal activation (Chapter 6). At a constant stress o the creep rate varies with temperature as... 

Fig. 23.5. Schematic of the time-temperature equivalence for the modulus. Every point on the curve for temperature T, lies at the same distance, log (07), to the left of that for temperature Tq.

Dynamic mechanical experiments yield both the elastic modulus of the material and its mechanical damping, or energy dissipation, characteristics. These properties can be determined as a function of frequency (time) and temperature. Application of the time-temperature equivalence principle [1-3] yields master curves like those in Fig. 23.2. The five regions described in the curve are typical of polymer viscoelastic behavior. 

Dynamic measurements can be made using either free or forced vibrations, and at resonance or outside of resonance conditions. Since the material depends on time and temperature, characterization over wide frequency ranges may be simplified by applying the WLF (118) equation for time-temperature equivalence. This relationship is generally useful in many propellant studies when used with caution under conditions which have established validity. 

Time-Temperature Equivalence (Steady-Stale Phenomena)... 

It is well established that between Tg and about Tg + 50 K, the relaxation kinetics obeys the WLF law (Williams et al., 1955). If Pr is a property depending on the macromolecular mobility (relaxation modulus, complex modulus, viscosity, diffusion rate, etc.), the time-temperature equivalence principle may be formulated as... 

It was initially stated that Cf are Cf were universal constants (Cf 17 Cf 50 K), but Cf can vary between 2 and 50 and Cf between 14 and 250 K (Mark, 1996). Epoxy values have been found in the low part of these intervals Cf 10, Cf 40 15 K (Gerard et al., 1991), whereas unsaturated polyester values can be relatively high Cf/Cf = 15-55 = 73-267 K (Shibayama and Suzuki, 1965). There is, to our knowledge, no synthetic study on the ideality and crosslinking effects on Cfand Cf. The time-temperature equivalence principles will be examined in detail in Chapter 11, which is devoted to elasticity and viscoelasticity. 

As discussed in Chapter 10, network polymers - as linear polymers - obey the time-temperature equivalence principle in the domain where they are stable, both chemically (no postcure, no thermal degradation), and physically (no orientation relaxation, water desorption, physical aging, etc.). 

Existence of a time-temperature equivalence that takes a different mathematical form in the glassy state (Arrhenius) and in the glass transition and rubbery regions (WLF). 

From a time-temperature equivalence principle (see below), any material history may be represented by an isothermal equivalent, corresponding to a point U (tu, Tu) in the (t, T) graph. If the point U is below the TSC curve, the material will not undergo failure in the particular conditions. In contrast, if the point U is above the TSC curve, the material will undergo failure because its index % will change sign. 

After what has been said about the T-t equivalence, it is not surprising that the time dependency of E resembles the T-de pendency, which we have considered in detail before. In (log t) we see, indeed, the same phases and transitions as in E(T) (Figure 6.17). It should be remarked that this time-temperature equivalence only holds for amorphous polymers or for the amorphous part in semi-crystalline polymers. 

Semi-crystalline polymers, such as PE an PP, are tough at temperatures above Tg, though for PP (Tg -15 °C) the critical temperature limit is about room temperature here also the time-temperature equivalence plays a role. Below Tg, semi-crystalline polymers have a low impact strength (unless secondary transitions occur). 

The time-temperature equivalence principle can also be applied to other viscoelastic functions in a similar way. Again, this leads to shift factors that are identical with those obtained from stress relaxation ... 

When the rate of elongation is increased, the tensile strength and the modulus also increase the elongation to break generally decreases (except in rubbers). Normally an increase of the speed of testing is similar to a decrease of the temperature of testing. To lightly cross-linked rubbers even the time-temperature equivalence principle can be applied. The rate dependence will not surprise in view of the viscoelastic nature and the influence of the Poisson ratio on the ultimate properties. 

Lightly cross-linked elastomers follow a simple pattern of ultimate behaviour. Smith (1958) has shown that the ultimate properties of this class of polymers follow a time-temperature equivalence principle just as the viscoelastic response to small non-destructive stresses does. 

Crystallisation accompanying stretching invalidates the simple time-temperature equivalence principle. 

If stress relaxation curves are obtained at a number of different temperatures, it is found that these curves can be superimposed by horizontal shifts to produce what is called a master curve .42 This concept of time-temperature equivalence is very important to understanding and predicting polymer behavior. As an example, a polymer at very low... 

In this section we are going to examine such viscoelastic properties in some detail and we will start by examining in turn three important mechanical methods of measurement creep, stress relaxation, and dynamic mechanical analysis. This will lead us to interesting things like time-temperature equivalence and a discussion of the molecular basis of what we have referred to as relaxation behavior. 

Note that Equation 8 or 9 represents an equivalence between frequency and temperature, which can be expressed as a time-temperature equivalence. The Arrhenius equation is found to be most applicable at lower temperatures. At higher temperatures, a better representation of the equivalence between frequency and temperature is given by the WLF (Williams-Landel-Ferry) equation, which can be written as... 

The major features of linear viscoelastic behavior that will be reviewed here are the superposition principle and time-temperature equivalence. Where they are valid, both make it possible to calculate the mechanical response of a material under a wide range of conditions from a limited store of experimental information. 

The linear viscoelastic properties of all samples were characterized by dynamic shear measurements in the parallel-plate geometry. Experimental details have been previously published [9]. Using time-temperature equivalence, master curves for the storage and loss moduli were obtained. Fig. 1 shows the master curves at 140°C for the relaxation spectra and Table 3 gives the values of zero-shear viscosities, steady-state compliances and weight-average relaxation times at the same temperature. 

The use of this time-temperature equivalence allows one to obtain "master curves" at a reference temperature, which enlarges considerably the experimental window. For glass-forming materials such as polystyrene, polymethylmetacrylate, polycarbonate, polymerists describe the shift factor aj in terms of the WLF equation ... 

The time-temperature equivalence implies that the viscosity (or relaxation times) of polymers may be written as the product of two functions ... 

The above phenomenological description of the viscoelastic behaviour of polymer melts and concentrated solutions leads to the following important conclusions if one focuses on the behavioiu- in the terminal region of relaxation, what is usually done for temperature (time-temperature equivalence) may also be done for the concentration effects and the effects of chain length one may define a "time-chain length equivalence" and "time-concentration equivalence"[4]. For monodisperse species, the various shifts along the vertical (modulus) axis and horizontal (time or frequency axis) are contained in two reducing parameters the... 

The temperature dependence of modulus data can be quite accurately described using an Arrhenius equation with the concept of time-temperature equivalence. An equivalent time, t may thus be deduced for a process at an absolute temperature, T, which occurs at t for an original temperature, Tq, such that ... 

For the tensile strength of a rubber to follow the time-temperature equivalence principle of linear viscoelasticity it is necessary that the extension at break also follow it. This is most easily verified by use of Equation (23), i.e., with the simplifying assumption of strain-time factorization. In an experiment conducted at fixed rate of strain, i = constant, the stress at any temperature and strain may be shown to be (200) ... 

An increase in strain rate (or crosshead velocity, Vc) leads to effects similar to those just described, with a possible time-temperature equivalence. 

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