Williams-Landel-Ferry equation
Encyclopedia
The Williams–Landel–Ferry Equation(or WLF Equation) is an empirical equation associated to time–temperature superposition.
The empirical Williams–Landel–Ferry equation (WLF equation)
can be used to fit (regress) discrete values of the shift factor aT vs. temperature. Here, Tr is the reference temperature chosen to construct the compliance master curve and C1, C2 are empirical constants adjusted to fit the values of aT. Values of shift factor aT are obtained by horizontal shift log(aT) of creep
compliance data plotted vs. time or frequency in double logarithmic scale so that a data set obtained experimentally at temperature T superposes with the data set at temperature Tr. A minimum of three values of aT are needed to obtain C1, C2, and typically more than three are used.
Once constructed, the WLF equation allows for the estimation of the temperature shift factor for temperatures other than those for which the material was tested. In this way, the master curve can be applied to other temperatures. However, when the constants are obtained with data at temperatures above the glass transition temperature
(Tg), the WLF equation is applicable to temperatures at or above Tg only; the constants are positive and represent Arrhenius
behavior. Extrapolation to temperatures below Tg is erroneous. When the constants are obtained with data at temperatures below Tg, negative values of C1, C2 are obtained, which are not applicable above Tg and do not represent Arrhenius behavior. Therefore, the constants obtained above Tg are not useful for predicting the response of the polymer for structural applications, which necessarily must operate at temperatures below Tg.
The WLF equation is a consequence of time–temperature superposition (TTSP), which mathematically is an application of
Boltzmann's superposition principle. It is TTSP, not WLF, that allows the assembly of a compliance master curve that spans more time, or frequency, than afforded by the time available for experimentation or the frequency range of the instrumentation, such as dynamic mechanical analyzer (DMA)
.
While the time span of a TTSP master curve is broad, according to Struik, it is valid only if the data sets did not suffer from ageing effects during the test time. Even then, the master curve represents a hypothetical material that does not age. Effective Time Theory. needs to be used to obtain useful prediction for long term time.
Having data above Tg, it is possible to predict the behavior (compliance, storage modulus
, etc.) of viscoelastic materials for temperatures T>Tg, and/or for times/frequencies longer/slower than the time available for experimentation. With the master curve and associated WLF equation it is possible to predict the mechanical properties of the polymer out of time scale of the machine (typically
to 
Hz), thus extrapolating the results of multi-frequency analysis to a broader range, out of measurement range of machine.
The empirical Williams–Landel–Ferry equation (WLF equation)
can be used to fit (regress) discrete values of the shift factor aT vs. temperature. Here, Tr is the reference temperature chosen to construct the compliance master curve and C1, C2 are empirical constants adjusted to fit the values of aT. Values of shift factor aT are obtained by horizontal shift log(aT) of creep
Creep (deformation)
In materials science, creep is the tendency of a solid material to slowly move or deform permanently under the influence of stresses. It occurs as a result of long term exposure to high levels of stress that are below the yield strength of the material....
compliance data plotted vs. time or frequency in double logarithmic scale so that a data set obtained experimentally at temperature T superposes with the data set at temperature Tr. A minimum of three values of aT are needed to obtain C1, C2, and typically more than three are used.
Once constructed, the WLF equation allows for the estimation of the temperature shift factor for temperatures other than those for which the material was tested. In this way, the master curve can be applied to other temperatures. However, when the constants are obtained with data at temperatures above the glass transition temperature
Glass transition
The liquid-glass transition is the reversible transition in amorphous materials from a hard and relatively brittle state into a molten or rubber-like state. An amorphous solid that exhibits a glass transition is called a glass...
(Tg), the WLF equation is applicable to temperatures at or above Tg only; the constants are positive and represent Arrhenius
Arrhenius equation
The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of the reaction rate constant, and therefore, rate of a chemical reaction. The equation was first proposed by the Dutch chemist J. H. van 't Hoff in 1884; five years later in 1889, the Swedish...
behavior. Extrapolation to temperatures below Tg is erroneous. When the constants are obtained with data at temperatures below Tg, negative values of C1, C2 are obtained, which are not applicable above Tg and do not represent Arrhenius behavior. Therefore, the constants obtained above Tg are not useful for predicting the response of the polymer for structural applications, which necessarily must operate at temperatures below Tg.
The WLF equation is a consequence of time–temperature superposition (TTSP), which mathematically is an application of
Boltzmann's superposition principle. It is TTSP, not WLF, that allows the assembly of a compliance master curve that spans more time, or frequency, than afforded by the time available for experimentation or the frequency range of the instrumentation, such as dynamic mechanical analyzer (DMA)
Dynamic mechanical analysis
Dynamic mechanical analysis is a technique used to study and characterize materials. It is most useful for studying the viscoelastic behavior of polymers. A sinusoidal stress is applied and the strain in the material is measured, allowing one to determine the complex modulus...
.
While the time span of a TTSP master curve is broad, according to Struik, it is valid only if the data sets did not suffer from ageing effects during the test time. Even then, the master curve represents a hypothetical material that does not age. Effective Time Theory. needs to be used to obtain useful prediction for long term time.
Having data above Tg, it is possible to predict the behavior (compliance, storage modulus
Dynamic modulus
Dynamic modulus is the ratio of stress to strain under vibratory conditions...
, etc.) of viscoelastic materials for temperatures T>Tg, and/or for times/frequencies longer/slower than the time available for experimentation. With the master curve and associated WLF equation it is possible to predict the mechanical properties of the polymer out of time scale of the machine (typically
to Hz), thus extrapolating the results of multi-frequency analysis to a broader range, out of measurement range of machine.