The Time-Temperature Superposition Principle

The extent of molecular motion depends on the free volume. In the glassy state, the free volume depends on the thermal history of the polymer. When a sample is cooled from the melt to some temperature below Tg and held at constant temperature, its volume will decrease (see Rgure 8.18). Because of the lower free volume, the rate of stress relaxation, creep, and related properties will decrease (29-33). This phenomenon is sometimes called physical aging (29,32), although the sample ages in the sense not of degradation or oxidation but rather of an approach to the equilibrium state in the glass. 

The problem of determining the theoretical behavior of polymers in the glassy rates is treated by Curro et al. (34,35). The time dependence of the volume in the glassy state is accounted for by allowing the fraction of unoccupied volume sites to depend on time. This permits the application of the Doolittle equation to predict the shift in viscoelastic relaxation times. 

As indicated above, relaxation and creep occur by molecular diffusional motions which become more rapid as the temperature is increased. Tempera- 

In multicomponent polymeric systems such as polymer blends or blocks, each phase stress relaxes independently (39-41). Thus each phase will show a glass-rubber transition relaxation. While each phase follows the simple superposition rules illustrated above, combining them in a single equation must take into account the continuity of each phase in space. Attempts to do so have been made using the Takayanagi models (41), but the results are not simple. 

The reduced frequency nomograph (42,43) is constructed as follows. First, the storage and loss modulus (or tan S) are plotted versus reduced frequency. 

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The time-temperature superpositioning principle was applied f to the maximum in dielectric loss factors measured on poly(vinyl acetate). Data collected at different temperatures were shifted to match at Tg = 28 C. The shift factors for the frequency (in hertz) at the maximum were found to obey the WLF equation in the following form log co + 6.9 = [ 19.6(T -28)]/[42 (T - 28)]. Estimate the fractional free volume at Tg and a. for the free volume from these data. Recalling from Chap. 3 that the loss factor for the mechanical properties occurs at cor = 1, estimate the relaxation time for poly(vinyl acetate) at 40 and 28.5 C. 

Fig. 49. Illustration of the time—temperature superposition principle as based on stress—relaxation data for polyisobutylene (299,300). To convert Pa to...

Another important characteristic aspect of systems near the glass transition is the time-temperature superposition principle [23,34,45,46]. This simply means that suitably scaled data should all fall on one common curve independent of temperature, chain length, and time. Such generahzed functions which are, for example, known as generalized spin autocorrelation functions from spin glasses can also be defined from computer simulation of polymers. Typical quantities for instance are the autocorrelation function of the end-to-end distance or radius of gyration Rq of a polymer chain in a suitably normalized manner ... 

Since we are interested in this chapter in analyzing the T- and P-dependences of polymer viscoelasticity, our emphasis is on dielectric relaxation results. We focus on the means to extrapolate data measured at low strain rates and ambient pressures to higher rates and pressures. The usual practice is to invoke the time-temperature superposition principle with a similar approach for extrapolation to elevated pressures [22]. The limitations of conventional t-T superpositioning will be discussed. A newly developed thermodynamic scaling procedure, based on consideration of the intermolecular repulsive potential, is presented. Applications and limitations of this scaling procedure are described. 

There are two superposition principles that are important in the theory of Viscoelasticity. The first of these is the Boltzmann superposition principle, which describes the response of a material to different loading histories (22). The second is the time-temperature superposition principle or WLF (Williams, Landel, and Ferry) equation, which describes the effect of temperature on the time scale of the response. 

Time is the major (actor in determining the mechanical properties of a polymer. This is seen directly in creep and stress-relaxation experiments. These tests cover long periods of time, so that they are sensitive to the types of molecular motions that require long times. Tfrey give little direct information on the types of molecular motion that take place at short times. However, by using the time-temperature superposition principle and the WLF equations, access to these short times can be achieved even though they may not easily be attainable by direct experimentation. 

Because of equipment limitations in measuring stress and strain in polymers, the time-temperature superposition principle is used to develop the viscoelastic response curve for real polymers. For example, the time-dependent stress relaxation modulus as a function of time and temperature for a PMMA resin is shown in... 

Moreover, we note that recently in reconstructing relaxation times via the time-temperature superposition principle using double quantum nuclear magnetic resonance (DQ-NMR) the and power laws were invoked without giving the spatial information of NSE [75]. 

PPG (at higher temperatures) behaves like a typical pseudoplastic non-Newtonian fluid. The activation energy of the viscosity in dependence of shear rate (284-2846 Hz) and Mn was detected using a capillary rheometer in the temperature range of 150-180°C at 3.0-5.5 kJ/mol (28,900 Da) and 12-13 kJ/mol (117,700 Da) [15]. The temperature-dependent viscosity for a PPG of 46 kDa between 70 and 170°G was also determined by DMA (torsion mode). A master curve was constructed using the time-temperature superposition principle [62] at a reference temperature of 150°G (Fig. 5) (Borchardt and Luinstra, unpublished data). A plateau for G was not observed for this molecular weight. The temperature-dependent shift factors ax were used to determine the Arrhenius activation energy of about 25 kJ/mol (Borchardt and Luinstra, unpublished data). 

The time-temperature superposition principle has practical applications. Stress relaxation experiments are practical on a time scale of 10 to 10 seconds (10 to 10 hours), but stress relaxation data over much larger time periods, including fractions of a second for impacts and decades for creep, are necessary. Temperature is easily varied in stress relaxation experiments and, when used to shift experimental data over shorter time intervals, can provide a master curve over relatively large time intervals, as shown in Figure 5.65. The master curves for several crystalline and amorphous polymers are shown in Figure 5.66. 

The effects of strain rate and temperature are correlated, and can be modeled (Kinloch and Young, 1983, Kinloch, 1985). For different temperatures and strain rates, GIc and the time to failure, tf, were measured. Using the time-temperature superposition principle, shift factors (aT) applicable to the time to failure tf, were determine. Shift factors plotted against (T — Tg) are independent of the type of test used (Fig. 12.14). The construction of a typical master curve GIc versus tf/aT is shown in Fig. 12.15 (Hunston et al., 1984). The value of GIc may be predicted for any strain rate/temperature combination. This model can also be applied to rubber-modified epoxies (See chapter 13). 

The peak of the dielectric loss of this process reflects its viscoelastic nature by obeying the time-temperature superposition principle, wherein the peak is shifted to higher temperatures for shorter times (higher frequencies) and vice versa. This process has been described by the Havriliak-Negami empirical formula [106, 108]... 

In an earlier section, we have shown that the viscoelastic behavior of homogeneous block copolymers can be treated by the modified Rouse-Bueche-Zimm model. In addition, the Time-Temperature Superposition Principle has also been found to be valid for these systems. However, if the block copolymer shows microphase separation, these conclusions no longer apply. The basic tenet of the Time-Temperature Superposition Principle is valid only if all of the relaxation mechanisms are affected by temperature in the same manner. Materials obeying this Principle are said to be thermorheologically simple. In other words, relaxation times at one temperature are related to the corresponding relaxation times at a reference temperature by a constant ratio (the shift factor). For... 

As an example of the concentration dependence of viscoelastic properties in Fig. 16.11 the shear creep compliance of poly(vinyl acetate) is plotted vs. time for solutions of poly(vinyl acetate) in diethyl phthalate with indicated volume fractions of polymer, reduced to 40 °C with the aid of the time temperature superposition principle (Oyanagi and Ferry, 1966). From this figure it becomes clear that the curves are parallel. We may conclude that the various may be shifted over the time axis to one curve, e.g. to the curve for pure polymer. In general it appears that viscoelastic properties measured at various concentrations may be reduced to one single curve at one concentration with the aid of a time-concentration superposition principle, which resembles the time-temperature superposition principle (see, e.g. Ferry, General references, 1980, Chap. 17). The Doolittle equation reads for this reduction ... 

This is none other than the time-temperature superposition principle. However, the exact shape of the function F(t) is not a generic feature of the theory. A good approximation is provided by the Kohlrausch stretched exponential function. In the frequency domain, Eqs. (23) define the shape of the susceptibility minimum usually observed in the gigahertz range, and an interpolation formula follows ... 

We expect that the modification creates the free volume (Vf) in wood substance from the similarity of the effect of and n on viscoelasticity. The discussion for wood, however, is impossible on the basis of a concept of the free volume, although the flexibility of molecular motion for synthetic amorphous polymers is discussed. Unfortunately, we can not directly know the created free volume because the time-temperature superposition principle is not valid for wood [19]. The principle is related to WLF equation by which the free volume is calculated. The free volume, however, relates to volumetric swelling as follows. 

The WLF equation applies to amorphous polymers in the temperature range of Tg to about Tg + lOO C. In this equation J is the reference temperature, these days taken to be the T, while and C2 are constants, initially thought to be universal (with Cx = 17.44 and C2 = 51.6), but now known to vary somewhat from polymer to polymer. These experimental observations bring up a number of interesting questions. What is the molecular basis of the time-temperature superposition principle What is the significance of the log scale and what does the superposition principle tell us about the temperature dependence of relaxation behavior And what about the temperature dependence of a7 at temperatures well below 2 ... 

A. Briefly explain the time-temperature superposition principle and how it can be used to predict creep properties. 

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. 

It must be noted that changes in density, as well as changes in temperature for higher temperatures, result in vertical shifts. Usually, these are modest in size, however, compared to the horizonal shifts. It should also be noted that the WLF equation, above, is a corollary of the time-temperature superposition principle. 

Important viscoelastic principles include the time-temperature superposition principle and its resultant WLF equation. These can be applied to understand the relationship between literature values of the glass transition temperature and actual needs. Thus, by using the growing amount of science now available in the field of damping, one can select that polymeric material which will damp most effectively. 

Almost always the data from the apparatus above is analyzed by using the time-temperature superposition principle to form a master curve over a wide frequency range at a selected reference temperature. The basis for this procedure is that for thermorheologically simple materials the effect of a change in temperature on... 

The inherent difficulty in the measurement of the complex dynamic moduli of viscoelastic materials is emphasized by the results of this paper. The agreement among the shifted modulus data as measured by different systems is limited by several difficulties (1) measurement inaccuracies of the instruments, (2) differences in the data reduction techniques used to apply the time-temperature superposition principle and propagation of shift curve errors and, (3) nonuniformity of the test samples. 

From dynamic experiments and applying the time temperature superposition principle, the complex shear modulus is measured over about five decades and the Rouse model can be checked extensively [37]. 

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