Williams-Landel-Ferry equation

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)^[1]

$log(a_T) = \frac{-C_1(T-T_r)}{[C_2 %2B (T-T_r)]}$

can be used to fit (regress) discrete values of the shift factor a_T vs. temperature. Here, T_r is the reference temperature chosen to construct the compliance master curve and C₁, C₂ are empirical constants adjusted to fit the values of a_T. Values of shift factor a_T are obtained by horizontal shift log(a_T) 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 T_r. A minimum of three values of a_T 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 (T_g), the WLF equation is applicable to temperatures at or above T_g only; the constants are positive and represent Arrhenius behavior. Extrapolation to temperatures below T_g is erroneous^[2]. When the constants are obtained with data at temperatures below T_g, negative values of C₁, C₂ are obtained, which are not applicable above T_g and do not represent Arrhenius behavior. Therefore, the constants obtained above T_g are not useful for predicting the response of the polymer for structural applications, which necessarily must operate at temperatures below T_g.

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^[3], 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^[3]. needs to be used to obtain useful prediction for long term time^[4].

Having data above T_g, it is possible to predict the behavior (compliance, storage modulus, etc.) of viscoelastic materials for temperatures T>T_g, 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 $10^{-2}$ to $10^2$ Hz), thus extrapolating the results of multi-frequency analysis to a broader range, out of measurement range of machine.

References

^ M.L. Williams, R.F. Landel, J.D. Ferry, J. Amer. Chem. Soc., 77:3701, 1955.
^ J. Sullivan, Creep and physical aging of composites, Composites Science and Technology 39(3) (1990) 207-32.
^ ^a ^b L. C. E. Struik, Physical aging in amorphous polymers and other materials, Elsevier Scientific Pub. Co. ; New York, 1978.
^ E. J. Barbero, Time–temperature–age superposition principle for predicting long-term response of linear viscoelastic materials, chapter 2 in Creep and fatigue in polymer matrix composites, R. M. Guedes, editor, Woodhead Pub. Co., UK, 2010.