Time-temperature superposition rheological measurements

The maximum strain rate (e < Is1) for either extensional rheometer is often very slow compared with those of fabrication. Fortunately, time-temperature superposition approaches work well for SAN copolymers, and permit the elevation of the reduced strain rates kaj to those comparable to fabrication. Typical extensional rheology data for a SAN copolymer (h>an = 0.264, Mw = 7 kg/mol,Mw/Mn = 2.8) are illustrated in Figure 13.5 after time-temperature superposition to a reference temperature of 170°C [63]. The tensile stress growth coefficient rj (k, t) was measured at discrete times t during the startup of uniaxial extensional flow. Data points are marked with individual symbols (o) and terminate at the tensile break point at longest time t. Isothermal data points are connected by solid curves. Data were collected at selected k between 0.0167 and 0.0840 s-1 and at temperatures between 130 and 180 °C. Also illustrated in Figure 13.5 (dashed line) is a shear flow curve from a dynamic experiment displayed in a special format (3 versus or1) as suggested by Trouton [64]. The superposition of the low-strain rate data from two types (shear and extensional flow) of rheometers is an important validation of the reliability of both data sets. 

The time-temperature superposition principle, t-T, has been a cornerstone of viscoelastometry. It has been invariably used to determine the viscoelastic properties of materials over the required 10 to 15 decades of reduced frequency, COaj, [Ferry, 1980]. Measuring the rheological properties at several levels of temperature, T, over the experimentally accessible frequency range (usually two to four decades wide), then using the t-T shifting, made it possible to constmct the complete isothermal function. 

Experimentally, one can use the time-temperature superposition (TTS) principle to extend the frequency range of data. It is often observed that rheological response measured at different temperatures is equivalent to one at the reference temperature To if one shifts the time (or frequency) appropriately. Sometimes, the stress has also to be shifted. For example, the complex relaxation modulus of theologically simple polymers defined as G (co) = G (co) +tG"(co), measured at different temperatures, obe3ts... 

Rheological measurements were performed in a stress rheometer fixture with a 2-cm cone and plate having a 1° cone angle and gap of 27 pm. Dynamic shear moduli were measured at 0.5% strain between 0.1 and 100 rad/s. Creep compliance was measured with a constant applied stress in the range of 0.1 to 5 kPa. Both measurements were performed over a series of temperatures to obtain data for time-temperature superposition. 

Ferry went to Harvard University in 1937 and worked there in a variety of posts, including as a Junior Fellow, until he joined the University of Wisconsin in 1946. He was promoted to Full Professor in 1947 His extensive measurements of the temperature dependence of the dynamic mechanical properties of polymers led to the concept of reduced variables in rheology. His demonstration that time-temperature superposition applied to many systems is the basis for the rational description of polymer rheology. He measured the dynamic response over a very wide range of frequency. One of the fruits of this work is the Williams-Landel-Ferry (WLF) equation for time-temperature shift factors. 

In dealing with the behavior of polymers, one gets very accustomed to working with distribution functions. Fundamentally this arises because there are many molecular modes of motion which contribute to the relaxations observed in polymers and these various modes can have enormously different relaxation times. This potentially complicated situation is often simplified by the fact that the distribution functions which measure contributions of these various motions to different types of behavior shift more or less homogeneously when the sample is subjected to change. For example, unless one looks very carefully, (20) time temperature superposition or thermo-rheological simplicity (21) are good approximations for most systems. 

Chapter 4 introduces the subject of linear viscoelasticity for readers somewhat new to rheology and also defines a number of terms that are used in the remainder of the book. The relaxation spectrum is introduced as well as methods for its measurement. Also, time-temperature superposition and its applications are explained. 

---