Time-temperature superposition

The Time-temperature superposition principle is a concept in polymer physics. [Hiemenz, Paul C., and Timothy P. Lodge. Polymer Chemistry. 2nd ed. Florida: Taylor & Francis Group, 2007. 486-491.] [Li, Rongzhi. "Time-Temperature Superposition Method for Glass Transition Temperature of Plastic Materials." Science Direct. 27 Mar. 2000. University of Sydney. 7 Dec. 2007 ] [Van Gurp, Marnix, and Jo Palmen. "Time-Temperature Superposition for Polymeric Blends." Society of Rheology. DSM Research.7 Dec. 2007 ]

Glasses, metals, as well as polymers often go through a change of phase which is known as the glass transition. Polymers have a certain temperature known as the glass transition temperature in which it transforms from a non-crystalline material that has a soft, rubbery consistency to a material that is hard, elastic and glassy. The glass transition temperature is dependent on the chemical structure and molecular mobility of the polymer. There aren’t really any efficient means to determine the glass transition temperature of polymers across a wide swath of temperatures and times; however the time-temperature superposition method allows for an accurate portrayal of the glass transition temperature. The theory behind this method is that a measurement of Tg at a certain temperature and time will be equal and behave the same as a polymer measured at a lower temperature and a longer time. And a key assumption is that all relaxation times have identical temperature dependence.

Time-Temperature Superposition

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 shifts to overlap with adjacent curves.

hift Factor Calculation

The distance of the shift is referred to as the time-temperature superposition shift factor a_T which is calculated by the following equations:

a_T= t_T/t₀

Where t_T 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.

a_T=f₀/f_T

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, aTf) = E(T0,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.

Conclusion

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.

References