TIME TEMPERATURE PRINCIPLE

With the long term objective of treating the effects of moisture and other plasticizers on the mechanical properties of materials, a new scheme that yields a complete constitutive model of viscoelastic materials has been developed. The time-temperature principle is an integral part of this modeling with a quantitative description of the glass transition behavior of pol3nmers. 

Resonance Techniques. Resonance methods (183-185) are used for measurements below about 100 kHz. The complex modulus (usually Young s or shear) is determined over a limited frequency range at a number of fixed temperatures, usually over one to three decades of frequency and over the useful temperature range of the material. From the measured data, reduced frequency plots at constant temperature are generated by the application of the time-temperature principle (3). In these plots, the real and imaginary parts of the complex modulus are plotted over many decades of frequencies, typically, as many as six or more decades of frequency than were actually measured. 

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 procedure described above is an application of the time-temperature correspondence principle. By shifting a set of plots of modulus (or compliance) versus time (or frequency) at any temperature (subscript 1) along the log t axis, we obtain the value of that mechanical property at another time and temperature (subscript 2). Using the shear modulus as an example, the time-temperature correspondence principle states... 

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 ... 

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 ... 

In the first one, the desorption rates and the corresponding desorbed amounts at a set of particular temperatures are extracted from the output data. These pairs of values are then substituted into the Arrhenius equation, and from their temperature dependence its parameters are estimated. This is the most general treatment, for which a more empirical knowledge of the time-temperature dependence is sufficient, and which in principle does not presume a constancy of the parameters in the Arrhenius equation. It requires, however, a graphical or numerical integration of experimental data and in some cases their differentiation as well, which inherently brings about some loss of information and accuracy, The reliability of the temperature estimate throughout the whole experiment with this... 

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. 

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. 

To get accurate distributions of relaxation or retardation times, the expetimcntal data should cover about 10 or 15 decades of time. It is impossible to get experimental data covering such a great range of times at one temperature from a single type of experiment, such as creep or stress relaxation-t Therefore, master curves (discussed later) have been developed that cover the required time scales by combining data at different temperatures through the use of time-temperature superposition principles. 

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. 

The temperature-time superposition principle is illustrated in Figure 8 by a hypothetical polymer with a TK value of 0°C for the case of stress relaxation. First, experimental stress relaxation curves are obtained at a series of temperatures over as great a time period as is convenient, say from 1 min to 10 min (1 week) in (he example in Figure 8. In making the master curve from the experimental data, the stress relaxation modulus ,(0 must first be multiplied by a small temperature correction factor/(r). Above Tg this correction factor is where Ttrt is the chosen reference... 

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. 

The Fourier transform of the record of annual air temperature versus time shows principle periods at 7.4, 9.8, and 17.5 years ... 

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). 

Apply time-temperature superposition principles to polymer moduli and calculate shift factors. 

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. 

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