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{{underlinked|date=May 2013}}
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{{refimprove|date=May 2013}}
The '''Williams–Landel–[[John D. Ferry|Ferry]] Equation''' (or '''WLF Equation''') is an empirical equation associated with [[time–temperature superposition]].<ref name="wlf">M.L. Williams, R.F. Landel, J.D. Ferry, J. Amer. Chem. Soc., '''77''':3701, 1955.</ref>
 
The WLF equation has the form
: <math> \log(a_T) = \frac{-C_1(T-T_\mathrm{r})}{C_2 + (T-T_\mathrm{r})} </math>
where ''T'' is the temperature, ''T''<sub>r</sub> is a reference temperature chosen to construct the [[compliance master curve]]  and ''C''<sub>1</sub>, ''C''<sub>2</sub> are empirical constants adjusted to fit the values of the superposition parameter ''a''<sub>T</sub>.
 
The equation can be used to fit (regress) discrete values of the shift factor a<sub>T</sub> vs. temperature. Here, Values of shift factor a<sub>T</sub> are obtained by horizontal shift log(a<sub>T</sub>) of [[Creep (deformation)|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<sub>r</sub>. A minimum of three values of a<sub>T</sub> are needed to obtain C<sub>1</sub>, C<sub>2</sub>, 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| glass transition temperature]] (T<sub>g</sub>), the WLF equation is applicable to temperatures at or above T<sub>g</sub> only; the constants are positive and represent [[Arrhenius equation|Arrhenius]] behavior. Extrapolation to temperatures below T<sub>g</sub> is erroneous.<ref>J. Sullivan, Creep and physical aging of composites, Composites Science and Technology '''39'''(3) (1990) 207-32.</ref> When the constants are obtained with data at temperatures below T<sub>g</sub>, negative values of C<sub>1</sub>, C<sub>2</sub> are obtained, which are not applicable above T<sub>g</sub> and do not represent Arrhenius behavior. Therefore, the constants obtained above T<sub>g</sub> are not useful for predicting the response of the polymer for structural applications, which necessarily must operate at temperatures below T<sub>g</sub>.
 
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 analysis|dynamic mechanical analyzer (DMA)]].
 
While the time span of a TTSP master curve is broad, according to Struik,<ref name=Struik>L. C. E. Struik, Physical aging in amorphous polymers and other materials,  Elsevier Scientific Pub. Co. ; New York, 1978.</ref> 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.<ref name=Struik/> needs to be used to obtain useful prediction for long term time.<ref>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.</ref>
 
Having data above T<sub>g</sub>, it is possible to predict the behavior (compliance, [[Dynamic modulus|storage modulus]], etc.) of viscoelastic materials for temperatures T>T<sub>g</sub>, 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 <math>10^{-2}</math> to <math>10^2</math> Hz), thus extrapolating the results of multi-frequency analysis to a broader range, out of measurement range of machine.
 
== Further reading ==
*[[Temperature dependence of liquid viscosity#Williams-Landel-Ferry model|Williams-Landel-Ferry model]]
*[[Time–temperature superposition]]
*[[Viscoelasticity]]
 
== References ==
<references />
 
{{DEFAULTSORT:Williams-Landel-Ferry Equation}}
[[Category:Polymers]]

Latest revision as of 06:24, 4 December 2014

Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. Thus, it is important to keep pace with this highly advanced age and have a regular interaction with your audience to keep a strong hold in the business market. The effect is to promote older posts by moving them back onto the front page and into the rss feed. If you're using Wordpress and want to make your blog a "dofollow" blog, meaning that links from your blog pass on the benefits of Google pagerank, you can install one of the many dofollow plugins available. provided by Word - Press Automatic Upgrade, so whenever you need to update the new version does not, it automatically creates no webmaster.

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