Time–temperature superposition

Consider a viscoelastic body that is subjected to dynamic loading. If the excitation frequency is low enough[9] the viscous behavior is paramount and all polymer chains have the time to respond to the applied load within a time period. In contrast, at higher frequencies, the chains do not have the time to fully respond and the resulting artificial viscosity results in an increase in the macroscopic modulus. Moreover, at constant frequency, an increase in temperature results in a reduction of the modulus due to an increase in free volume and chain movement.

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 polymer's 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, provided there are no phase transitions.

Schematic of the evolution of the instantaneous modulus E(t,T) in a static relaxation test. t is the time and T is the temperature.

Consider the relaxation modulus E at two temperatures T and T₀ such that T > T₀. At constant strain, the stress relaxes faster at the higher temperature. The principle of time-temperature superposition states that the change in temperature from T to T₀ is equivalent to multiplying the time scale by a constant factor a_T which is only a function of the two temperatures T and T₀. In other words,

${\displaystyle E(t,T)=E({\frac {t}{a_{\rm {T}}}},T_{0})\,.}$

The quantity a_T is called the horizontal translation factor or the shift factor and has the properties:

${\displaystyle {\begin{aligned}&T>T_{0}\quad \implies \quad a_{\rm {T}}<1\\&T<T_{0}\quad \implies \quad a_{\rm {T}}>1\\&T=T_{0}\quad \implies \quad a_{\rm {T}}=1\,.\end{aligned}}}$

The superposition principle for complex dynamic moduli (G* = G' + i G'' ) at a fixed frequency ω is obtained similarly:

${\displaystyle {\begin{aligned}G'(\omega ,T)&=G'\left(a_{\rm {T}}\,\omega ,T_{0}\right)\\G''(\omega ,T)&=G''\left(a_{\rm {T}}\,\omega ,T_{0}\right).\end{aligned}}}$

A decrease in temperature increases the time characteristics while frequency characteristics decrease.

For a polymer in solution or "molten" state the following relationship can be used to determine the shift factor:

${\displaystyle a_{\rm {T}}={\frac {\eta _{\rm {T}}}{\eta _{\rm {T0}}}}}$

where η_T0 is the viscosity (non-Newtonian) during continuous flow at temperature T₀ and η_T is the viscosity at temperature T.

The time–temperature shift factor can also be described in terms of the activation energy (E_a). By plotting the shift factor a_T versus the reciprocal of temperature (in K), the slope of the curve can be interpreted as E_a/k, where k is the Boltzmann constant = 8.64x10⁻⁵ eV/K and the activation energy is expressed in terms of eV.

Curve of the variation of a_T as a function of T for a reference temperature T₀.[10]

The empirical relationship of Williams-Landel-Ferry,[11] combined with the principle of time-temperature superposition, can account for variations in the intrinsic viscosity η₀ of amorphous polymers as a function of temperature, for temperatures near the glass transition temperature T_g. The WLF model also expresses the change with the temperature of the shift factor.

Williams, Landel and Ferry proposed the following relationship for a_T in terms of (T-T₀) :

${\displaystyle \log a_{\rm {T}}=-{\frac {C_{1}(T-T_{0})}{C_{2}+(T-T_{0})}}}$

where ${\displaystyle \log }$ is the decadic logarithm and C₁ and C₂ are positive constants that depend on the material and the reference temperature. This relationship holds only in the approximate temperature range [T_g, T_g + 100 °C]. To determine the constants, the factor a_T is calculated for each component M′ and M of the complex measured modulus M*. A good correlation between the two shift factors gives the values of the coefficients C₁ and C₂ that characterize the material.

If T₀ = T_g:

${\displaystyle \log a_{\rm {T}}=-{\frac {C_{1}^{g}(T-T_{g})}{C_{2}^{g}+(T-T_{g})}}=\log \left({\frac {\eta _{\rm {T}}}{\eta _{T_{g}}}}\right)}$

where C^g₁ and C^g₂ are the coefficients of the WLF model when the reference temperature is the glass transition temperature.

The coefficients C₁ and C₂ depend on the reference temperature. If the reference temperature is changed from T₀ to T′₀, the new coefficients are given by

${\displaystyle C'_{1}={\frac {C_{1}\,C_{2}}{C_{2}+(T'_{0}-T_{0})}}\qquad {\rm {and}}\qquad C'_{2}=C_{2}+(T'_{0}-T_{0})\,.}$

In particular, to transform the constants from those obtained at the glass transition temperature to a reference temperature T₀,

${\displaystyle C_{1}^{0}={\frac {C_{1}^{g}\,C_{2}^{g}}{C_{2}^{g}+(T_{0}-T_{g})}}\qquad {\rm {and}}\qquad C_{2}^{0}=C_{2}^{g}+(T_{0}-T_{g})\,.}$

These same authors have proposed the "universal constants" C^g₁ and C^g₂ for a given polymer system be collected in a table. These constants are approximately the same for a large number of polymers and can be written C^g₁ ≈ 15 and C^g₂ ≈ 50 K. Experimentally observed values deviate from the values in the table. These orders of magnitude are useful and are a good indicator of the quality of a relationship that has been computed from experimental data.

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.

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(a_T). In the area of glass transition, a_T 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 .

Principle of construction of a master curve for E′ for a reference temperatureT₀. f=ω is the frequency. The shift factor is computed from data in the frequency range ω₁ = 1 Hz and ω₂ = 1000 Hz.

The shift factor (which depends on the nature of the transition) can be defined, below T_g, using an Arrhenius law:

${\displaystyle \log(a_{\rm {T}})=-{\frac {E_{a}}{2.303R}}\left({\frac {1}{T}}-{\frac {1}{T_{0}}}\right)}$

where E_a is the activation energy, R is the universal gas constant, and T₀ is a reference temperature in kelvins. This Arrhenius law, under this glass transition temperature, applies to secondary transitions (relaxation) called β-transitions.

For the superposition principle to apply, the sample must be homogeneous, isotropic and amorphous. The material must be linear viscoelastic under the deformations of interest, i.e., the deformation must be expressed as a linear function of the stress by applying very small strains, e.g. 0.01%.

To apply the WLF relationship, such a sample should be sought in the approximate temperature range [T_g, T_g + 100 °C], where α-transitions are observed (relaxation). The study to determine a_T and the coefficients C₁ and C₂ requires extensive dynamic testing at a number of scanning frequencies and temperature, which represents at least a hundred measurement points.

• Viscoelasticity
• Temperature dependence of liquid viscosity
• Williams-Landel-Ferry equation