Time-temperature equivalence polymers

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... 

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. 

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.). 

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). 

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. 

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... 

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... 

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. 

The time-temperature equivalence principle makes it possible to predict the viscoelastic properties of an amorphous polymer at one temperature from measurements made at other temperatures. The major effect of a temperature increase is to increase the rates of the various modes of retarded conformational elastic response, that is, to reduce the retarding viscosity values in the spring-dashpot model. This appears as a shift of the creep function along the log t scale to shorter times. A secondary effect of increasing temperature is to increase the elastic moduli slightly because an equilibrium conformational modulus tends to be proportional to the absolute temperature (13). 

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. 

Howe er, once the polymer is drawn and subsequently redrawn, the degree of orientation of the crystallites becomes more fixed [7], unless or until the structure is subjected to a time -temperature equivalence that exceeds the original energy equilibrium, thus allowing shrinkage tension to develop [8],... 

The time t is called the retardation time and its value depends on the nature of the polymer and on the temperature, T. If T increases, the frequency of molecular rearrangements rises and t falls. This time-temperature equivalence is considered in detail later. 

The procedure was carried out on the measured Jp t) of Epon 1007/DDS in Figure 5.2. The reduced curve extending over 10 decades of time with the choice To of 100.7°C is shown in Figure 5.3. This test of time-temperature equivalence is eminently successful as found in other molecular network polymers (Plazek and Chay, 1991 Plazek et al., 1983 Plazek et al., 1988). 

To of 100.7°C is shown in Rg. 7. This test of time-temperature equivalence is eminently successful as found in other molecular network polymers [14-16]. 

Time-temperature equivalence n. Because an increase in temperature accelerates molecular motions, mechanical behavior of polymers at one temperature can be used to predict those at another by means of a shift factor equal to the ratio of relaxation time at the second temperature to that at the first. The principle can be used to combine measurements made at many temperatures into a single master curve for a reference temperature over many decades of time. 

The correspondence principles accomplish an important goal prediction of long-term behavior from short-term tests. However, some polymer scientists and engineers do not believe that the prediction methods work apparently, because they think that the time-temperature equivalence is the same thing as the so-called WLF equation of 1955 (63) for the shift factor ax- Ferry who co-created WLF warned (61) that the use of that equation is limited to a temperature range... 

Condensed matter physicists calculate many properties of eiystalline solids in terms of a model, due initially to Debye, in which massive point atoms are connected by linear elastic springs. The dynamies of molecular chains can be considered from this starting point. The theories diseussed below, although initially derived for polymer solutions, can be used to predict relaxation spectra and time temperature equivalence for amorphous solid polymers. As a full treatment involves quite advanced mathematics, we shall discuss the theories only in outline. 

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). 

The idea of time-temperature equivalence introduced in Section 14.3.1 is of considerable practical importance because one would often like to predict the longterm response of materials on the basis of experiments carried out on a laboratory time scale. This is to some extent possible in polymers, for which it has been widely verified that viscoelastic functions determined at different T over a fixed range of oj or t, slightly adjusted to take into account the effect of T and density p on the elastic response [through Eq. (30)], superpose if the or t scale is multiplied by a shift factor, flr(T, Tr), where Tr is some convenient reference temperature. A typical master curve obtained in this way is shown in Figure 14.8 for stress relaxation data from polyisobutene, taking Tr = —66.5°C. The effect is to expand the t or scale of the measurement carried out at Tr, revealing the whole of the a transition in this case. 

Physical properties of polymers handbook. Mark JE (ed). Springer, New York, 1996) See Time-Temperature Equivalence and Williams-Landell-Ferry Equation. 

The Time-Temperature Equivalence of the Glass Transition Viscoelastic Behaviour in Amorphous Polymers and the Williams, Landel and Ferry (WLF) Equation... 

We have so far discussed two types of theories, those based on the site model, and thosebased on the WLF equation and its ramifications, which deal with time-temperature equivalence. The site model theories predict constant activation energies and are more applicable to relaxation transitions originating from localised chain motions, whereas the WLF equation theories deal with the glass transition behaviour in amorphous polymers. 

In the introductory section on amorphous polymers (Section 7.1.1), we considered the relaxation spectrum of amorphous polymers and noted that it was quite complex. The normal mode theories, now to be discussed, attempt to predict the relaxation spectrum for amorphous polymers, as well as the time-temperature equivalence. 

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