Time-temperature equivalence and superposition

Ferry and co-workers [7], on the basis of the molecular theory of viscoelasticity, proposed that superposition should incorporate a small vertical shift factor Topo/Tp, where p is the density at the experimental temperature T and po relates to the reference temperature Tq. Further corrections have been suggested by McCrum and Morris [8] to deal with the changes in unrelaxed and relaxed compliances with temperature. 

The relaxations that lead to viscoelastic behaviour are the result of various types of molecular motions, some of which are described in section 5.7, and they occur more rapidly at higher temperature. The compliance /(/), for instance, is therefore a function of temperature, T, which means that it should really be written as J t, T). Suppose that the effect of a rise in temperature from some chosen standard temperature To is to speed up every stage in a relaxation process by a constant factor that depends on the new temperature T. This is equivalent to saying that the interval of time required for any small change in strain to take place is divided by a factor uj- that depends on T and has the value 1 when T = T . This means that, if measured values of J(t, T) are plotted against ta-r, curves for all temperatures should be superposed. Similarly, if values of J co, T) are plotted against curves for all temperatures should be superposed on the curve for T.  

The curve obtained after the shifts have been applied is called a master curve. The important thing about this curve is that it effectively gives the response of the polymer over a very much wider range of frequencies than any one piece of apparatus is able to provide. It is important to realise, however, that a master curve is obtained only if there is just one important relaxation process in the effective range of frequencies studied. This is 

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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 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. 

In spite of these complications, the viscoelastic response of an amorphous polymer to small stresses turns out to be a relatively simple subject because of two helpful features (1) the behavior is linear in the stress, which permits the application of the powerful superposition principle and (2) the behavior often follows a time-temperature equivalence principle, which permits the rapid viscoelastic response at high temperatures and the slow response at low temperatures to be condensed in a single master curve. 

By use of the time-temperature equivalence principle, the viscoelastic response of a given polymeric material over a wide temperature range can be accommodated in a single master curve. By use the superposition principle, this master curve can be used to estimate the time-dependent response to time-dependent stresses in simple tensile or shear specimens or to nonhomogeneous time-dependent stresses arising in stressed objects and structures. 

In the following sections we discuss the two superposition principles that are important in the theory of viscoelasticity. The first is the Boltzmann superposition principle, which is concerned with linear viscoelasticity, and the second is time-temperature superposition, which deals with the time-temperature equivalence. 

Fortunately for linear amorphous polymers, modulus is a function of time and temperature only (not of load history). Modulus-time and modulus-temperature curves for these polymers have identieal shapes they show the same regions of viscoelastic behavior, and in each region the modulus values vary only within an order of magnitude. Thus, it is reasonable to assume from such similarity in behavior that time and temperature have an equivalent effect on modulus. Such indeed has been found to be the case. Viscoelastic properties of linear amorphous polymers show time-temperature equivalence. This constitutes the basis for the time-temperature superposition principle. The equivalence of time and temperature permits the extrapolation of short-term test data to several decades of time by carrying out experiments at different temperatures. 

Because the time or frequency scales over which the compliance changes are very large, it is usual to use logarithmic scales for t or co. When this is done it is necessary to shift the curves for different temperatures by a constant amount log a or — log aj along the t or co axes, respectively, in order to get superposition with the curve for T. The quantity aj is therefore called the shift factor. Figure 7.14 shows data for a set of measurements before and after the shifts have been applied. The idea that the same effect can be produced on the compliances or moduli either by a change of temperature or by a change of time-scale is called time-temperature equivalence. 

While time-temperature superposition is very useful, it will not work in all cases. Predictions based on TT conform well to the observed behavior of many polymers, but others exhibit behavior inconsistent with TTS. A number of assumptions inherent in the principle of time-temperature equivalence (Ferry 1980) are incorrect for many polymers. For example, implicit in TTS is the assumption that the effect of temperature on the relaxation time spectrum, is consistent for the entire spectrum, but this is frequently in error (Dealy and Wissbrun 1990). 

In addition to the Boltzmann superposition principle, the second consequence of linear viscoelasticity is the time-temperature equivalence, which will be described in greater detail later on. This equivalence implies that functions such as a=/(s), but also moduli, behave at constant temperature and various exten-sional rates similarly to analogues that are measured at constant extensional rates and various temperatures. Time- and temperature-dependent variables such as the tensile and shear moduli (E, G) and the tensile and shear compliance (D, J) can be transformed from E =f(t) into E =f(T) and vice versa, in the limit of small deformations and homogeneous, isotropic, and amorphous samples. These principles are indeed not valid when the sample is anisotropic or is largely strained. 

The WLF equation can be widely applied, and demonstrates the equivalence of time and temperature, the so-called time-temperature superposition principle, on the mechanical relaxations of an amorphous polymer. The equation holds up to about 100° above the glass transition temperature, but after that begins to break down. 

Above Tg the stress relaxation and the creep behaviour of amorphous polymers obey the "time-temperature superposition (or equivalence) principle". 

In the preceding sections, we have looked at the various types of relaxation processes that occur in polymers, focusing predominantly on properties like stress relaxation and creep compliance in amorphous polymers. We have also seen that there is an equivalence between time (or frequency) and temperature behavior. In fact this relationship can be expressed formally in terms of a superposition principle. In the next few paragraphs we will consider this in more detail. First, keep in mind that there are a number of relaxation processes in polymers whose temperature dependence we should explore. These include ... 

The Time-Temperature Superposition Principle. For viscoelastic materials, the time-temperature superposition principle states that time and temperature are equivalent to the extent that data at one temperature can be superimposed upon data at another temperature by shifting the curves horizontally along the log time or log frequency axis. This is illustrated in Figure 8. While the relaxation modulus is illustrated (Young s modulus determined in the relaxation mode), any modulus or compliance measure may be substituted. 

In some epoxy systems ( 1, ), it has been shown that, as expected, creep and stress relaxation depend on the stoichiometry and degree of cure. The time-temperature superposition principle ( 3) has been applied successfully to creep and relaxation behavior in some epoxies (4-6)as well as to other mechanical properties (5-7). More recently, Kitoh and Suzuki ( ) showed that the Williams-Landel-Ferry (WLF) equation (3 ) was applicable to networks (with equivalence of functional groups) based on nineteen-carbon aliphatic segments between crosslinks but not to tighter networks such as those based on bisphenol-A-type prepolymers cured with m-phenylene diamine. Relaxation in the latter resin followed an Arrhenius-type equation. 

For some materials, the principle of time-temperature superposition has been a convenient and useful concept. It Implies that increasing the temperature is equivalent in its effects on properties to increasing the time (decreasing the loading rate). 

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