Appendix 1
The Williams Landau Ferry (WLF) equation
and Time Temperature Superposition
The method of reduced variables, or time temperature superposition, has traditionally been
applied to pure polymers, where it is used to make predictive measurements of the material.
It has also found wider application in the foods area with the treatment of sugar rich
hydrocolloid gels and gelatin. A detailed treatment can be found in standard texts such as
Viscoelastic Properties of Polymers by Ferry (1) and some basic information is reproduced
here.
The fundamental WLF equation can be derived from the Dolittle equation which describes
the variation of viscosity with free volume.
ln = ln A + B([V − Vf ]/V)
Where is the tensile viscosity, V and Vf are the total volume and free volume of the system
respectively, and A and B are constants.
⇒ ln = ln A + B(1/f − 1)
If it is assumed that the fractional free volume increases linearly with temperature then
f = fg + f (T − Tg )
where f is the thermal coefficient of expansion, f is the fractional free volume at T, a
temperature above Tg, and fg is the fractional free volume at Tg .
Therefore
ln (T) = ln A + B(1/[fg + f (T − Tg )] − 1) at T > Tg
And ln (T) = ln A + B(1/f − 1) at T = Tg
subtracting ⇒ log (T)/(Tg ) = −B/2.303 fg [T − Tg ]/{(fg /f ) + T − Tg }
where the universal constants C1 = B/2.303 fg and C2 = fg /f
It can be shown that log (T)/(Tg ) = log aT
The method of extracting the universal constants of the WLF equation is to use the method
of reduced variables (time temperature superposition). A number of DMA frequency scans
are recorded at a series of isothermal temperatures, and the resulting data displayed on a
The Williams Landau Ferry (WLF) equation and Time Temperature Superposition 451
screen. A reference temperature is chosen (typically the isothermal temperature from the
middle section of data where properties vary most as a function of frequency, equivalent to
Tg ) and the factors required to shift the responses at the other temperatures calculated so that
one continuous smooth mastercurve is produced. Most commercial analysers provide soft-
ware to create the mastercurve and calculate the shift factors involved. Different approaches
have been used to obtain this type of data including a frequency multiplexing approach
where a low underlying scan rate is used and the frequency continuously varied. However
this means that data collected at each individual frequency will be at a slightly different
temperature, and it is best to ensure that truly isothermal conditions are employed.
The factor for the reference temperature (aT ) is 1 with those for temperatures less than
the reference temperature being less than 1 and those for temperatures greater than the
reference temperature being greater than 1. The mastercurve, in conjunction with either
the modulus temperature curve or the shift factors (aT ) relative to a reference temperature
(Tref ), enables the response of the polymer under any conditions of time and temperature
to be obtained.
There is also a correction to be applied in the vertical direction as well as the shift along
the x-axis. This is related to the absolute value of temperature, on which the moduli depend,
and the change in density as the temperature changes. The entire procedure can be expressed
thus
E(T1 , t)/ (T1 )T1 = E(T2 , t/aT )/ (T2 )T2
This correction should only be made for the rubbery state, ie above Tg . It is invalid for the
glassy state.
By fitting the WLF equation to the shift factors, (y-axis), at a particular temperature,
(x-axis) we can obtain an estimate of the universal constants C1 and C2 and hence obtain an
estimate of T∞ , the underlying true second order transition temperature, where viscosity
becomes infinite. The WLF equation can also be expressed in terms of the shear moduli as
log aT = G(T)/G(Tref ) = −C1 (T − Tref )/(C2 + T − Tref )
and can be rearranged to obtain the universal constants C1 and C2 from a linear graphical
plot.
(T − Tref )/ log aT = −1/C1 (T − Tref ) + (−C2 /C1 )
y = mx + c
If the term [C2 + T − T0 ] went to zero the term log aT would tend to −∞ which would
correspond to a shift factor of 0 and the response of the polymer at the limiting true transition
temperature T∞ .
and T∞ = Tref − C2 = Tg − 50
if Tg is chosen as the reference temperature.
452 Principles and Applications of Thermal Analysis
T∞ is thought to be the true underlying 2nd order transition for these materials and
only accessible for experiments carried out over very long time scales. In practice the 50◦ C,
which is normally added to the T∞ , returns a value for Tg more in keeping with everyday
experience and measured values. This procedure places the glass transition somewhere in
the transition region.
It can therefore be seen that C1 can give information on the free volume at the glass
transition whilst C2 can give information on the thermal expansion as well as the underlying
glass transition.
WLF Kinetics
It is found for some diffusion limited reactions that the temperature dependence of the
rate of the reaction follows a WLF type of equation rather than an Arrhenius equation in
the region of the glass transition. This is also true for rheological properties such as shear
moduli. Plots which linearise the data are of the form of the logarithm of reaction rate
against the reciprocal of terms similar to T − Tg , rather than the Arrhenius form, 1/T. It is
only for reactions limited by diffusion, such as the translational motion of a reactant in a
glass/rubber. Reactions which are limited by some other non diffusion based step are known
as reaction limited and will obey Arrhenius kinetics.
1) Viscoelastic Properties of Polymers, Third edition, Ferry, 1980, Wiley, Pages 280–298
