Time–Temperature Shift Law

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Time–temperature shift law or time–temperature superposition principle

General

The time–temperature shift law is also referred to in the literature [1] as the time–temperature superposition principle. In addition to their pronounced time dependence, viscoelastic materials also exhibit strong temperature dependence in their properties. The reason for this lies in the molecular motion and rearrangement processes that determine the relaxation and retardation spectrum of the material.

The time-temperature equivalence

As thermally activated processes, these molecular processes occur at increasing velocity as the temperature rises. This shifts the relaxation and retardation time spectrum to shorter times. If only the speed of the molecular processes changes with temperature, but not their type and number, the shape of the relaxation or retardation spectrum and thus also the shape of the viscoelastic characteristic functions along the logarithmic time axis remain unchanged. However, their temporal position changes according to the temperature. One consequence of this behaviour, referred to as ‘thermorheologically simple’, is time-temperature equivalence, the application of which in the form of the time-temperature superposition principle has become very important in practice for predicting long-term behaviour (see: tensile creep test). If the curve of a viscoelastic parameter, for example the modulus E (log t), is known for a specific time interval at different temperatures, the individual curves can be aligned by horizontal displacement with the curve E₀ (log t) determined at the reference temperature T₀, as shown schematically in Fig. 1.

Fig. 1:

Schematic diagram of master curve construction using time–temperature superposition

This produces a master curve that depicts the material behaviour over a wide time range. The displacement function log a_T = log t – log t₀ is temperature-dependent. In many cases, it can be described on the basis of an ARRHENIUS approach:

$log\ a_{T}=log\left({\frac {t}{t_{0}}}\right)={\frac {\Delta H}{2.3k}}\left({\frac {1}{T}}-{\frac {1}{T_{0}}}\right)$

(1)

In the glass transition range (see also: glass transition temperature), however, it often follows the WILLIAMS, LANDEL and FERRY (WLF) equation:

$log\ a_{T}=log\left({\frac {t}{t_{0}}}\right)={\frac {C_{1}(T-T_{0})}{C_{2}+T-T_{0}}}$

(2)

with the universal constants C₁ and C₂ [2].

See also

• BOLTZMANN's superposition principle
• Correspondence principle
• Thermostability PVC
• Stepped isothermal method, tensile stress
• Relaxation behaviour determination

References

[1]	Lüpke, T.: Fundamental Principles of Mechanical Behavior. In: Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022), 3rd Edition, pp. 83/84 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see AMK-Library under A 22)
[2]	Williams, M. L., Landel, R. F., Ferry, J. D.: The Temperature Dependence of Relaxation Mechanism in Amorphous Polymers and other Glass-forming Liquids. J. Amer. Chem. Soc. 77 (1955) 3701−3707; https://doi.org/10.1021/ja01619a008