The Time–temperature superposition principle is a concept in polymer physics and in the physics of glass-forming liquids. [1] [2] [3]
Some materials, polymers in particular, show a strong dependence of viscoelastic properties on the temperature at which they are measured. If you plot the elastic modulus of a noncrystallizing crosslinked polymer against the temperature at which you measured it, you will get a curve which can be divided up into distinct regions of physical behavior. At very low temperatures, the polymer will behave like a glass and exhibit a high modulus. As you increase the temperature, the polymer will undergo a transition from a hard “glassy” state to a soft “rubbery” state in which the modulus can be several orders of magnitude lower than it was in the glassy state. The transition from glassy to rubbery behavior is continuous and the transition zone is often referred to as the leathery zone. The onset temperature of the transition zone, moving from glassy to rubbery, is known as the glass transition temperature, or Tg.
In the 1940s Andrews and Tobolsky [4] showed that there was a simple relationship between temperature and time for the mechanical response of a polymer. Modulus measurements are made by stretching or compressing a sample at a prescribed rate of deformation. For polymers, changing the rate of deformation will cause the curve described above to be shifted along the temperature axis. Increasing the rate of deformation will shift the curve to higher temperatures so that the transition from a glassy to a rubbery state will happen at higher temperatures.
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It has been shown experimentally that the elastic modulus (E) of a polymer is influenced by the load and the response time. Time–temperature superposition implies that the response time function of the elastic modulus at a certain temperature resembles the shape of the same functions of adjacent temperatures. Curves of E vs. log(response time) at one temperature can be shifted to overlap with adjacent curves, as long as the data sets did not suffer from ageing effects during the test time (see Williams-Landel-Ferry equation).
The distance of the shift is referred to as the time–temperature superposition shift factor aT which is calculated by the following equations:
aT= tT/t0
Where tT is the time that is required to give a specified response at a certain temperature, and t0 is the time required to give an identical response at the reference temperature.
aT=f0/fT
Where fT is the rate at which the material achieves a particular response at a given temperature and f0 is the rate at which the material achieves the same response time as the reference temperature. Where the curves overlap each other their elastic moduli will be equal. Therefore,
E(T, aT*f) = E(To, f)
The time–temperature superposition shift factor follows the Williams-Landel-Ferry equation where C1 and C2 are constants:
Log10(aT) = (-C1*(T-To))/(C2+(T-To))
The shift factors are then utilized to draw the master curve and plotted against temperature.
The time–temperature shift factor can also be described in terms of the activation energy (Ea). By plotting the shift factor aT versus the reciprocal of temperature (in K), the slope of the curve can be interpreted as Ea/k, where k is the Boltzmann constant = 8.64x10−5 eV/K and the activation energy is expressed in terms of eV.
Time–temperature superposition is a procedure that has become important in the field of polymers to observe the dependence upon temperature on the change of viscosity of a polymeric fluid. Rheology or viscosity can often be a strong indicator of the molecular structure and molecular mobility. Time–temperature superposition avoids the inefficiency of measuring a polymers behavior over long periods of time at a specified temperature by utilizing the fact that at higher temperatures and shorter time the polymer will behave the same.