page_title: Time–temperature superposition
section_title: Construction of master curves
hierarchical_section_title: Time–temperature superposition / Construction of master curves
caption_attribution_description: Français&#160;: DMA - Courbes maîtresses du module de conservation E' et du facteur de perte d'un matériau polymère, générées par logiciel, pour une température de référence T0. Ces courbes sont créées en décalant horizontalement les données du module et du facteur de perte sur l'axe des fréquences (échelle log&#160;; f en Hz). Application du principe d'équivalence temps-température. Cela a été réalisé à partir des données d'un test à double balayage fréquence/température sur une grandeur viscoélastique (pour cet exemple, le module de conservation E’ et le facteur de perte (aussi appelé facteur d'amortissement) tan δ&#160;; mesurés à plusieurs fréquences de 1 à 100 Hz). English: DMA - Storage modulus E' and loss factor master curves generated by software on a polymer material using a reference temperature T0. These curves were produced by horizontally shifting the modulus and loss factor data along the frequency axis (log scale; f in Hz). Application of the time–temperature superposition principle. This was done using data obtained from a double frequency/temperature sweep test on a viscoelastic property (for this example storage modulus E' and loss factor tan δ; measured at several frequencies from 1 to 100 Hz).
context_page_description: The time–temperature superposition principle is a concept in polymer physics and in the physics of glass-forming liquids.
  This superposition principle is used to determine temperature-dependent mechanical properties of linear viscoelastic materials from known properties at a reference temperature. The elastic moduli of typical amorphous polymers increase with loading rate but decrease when the temperature is increased. Curves of the instantaneous modulus as a function of time do not change shape as the temperature is changed but appear only to shift left or right. This implies that a master curve at a given temperature can be used as the reference to predict curves at various temperatures by applying a shift operation. The time-temperature superposition principle of linear viscoelasticity is based on the above observation.
  The application of the principle typically involves the following steps:
  experimental determination of frequency-dependent curves of isothermal viscoelastic mechanical properties at several temperatures and for a small range of frequencies
  computation of a translation factor to correlate these properties for the temperature and frequency range
context_section_description: The principle of time-temperature superposition requires the assumption of thermorheologically simple behavior (all curves have the same characteristic time variation law with temperature). From an initial spectral window [ω₁, ω₂] and a series of isotherms in this window, we can calculate the master curves of a material which extends over a broader frequency range. An arbitrary temperature T₀ is taken as a reference for setting the frequency scale (the curve at that temperature undergoes no shift).
  In the frequency range [ω₁, ω₂], if the temperature increases from T₀, the complex modulus E' (ω) decreases.  This amounts to explore a part of the master curve corresponding to frequencies lower than ω₁ while maintaining the temperature at T₀. Conversely, lowering the temperature corresponds to the exploration of the part of the curve corresponding to high frequencies. For a reference temperature T₀, shifts of the modulus curves have the amplitude log(aT).  In the area of glass transition, aT is described by an homographic function of the temperature.
  The viscoelastic behavior is well modeled and allows extrapolation beyond the field of experimental frequencies which typically ranges from 0.01 to 100 Hz .
caption: Master curves for the instantaneous modulus E'  and the loss factor tanδ as a function of frequency.  The data have been fit to a polynomial of degree 7.
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