Department of Chemical
Engineering
Michigan Technological University
Houghton, MI 49931-1295
It is convenient and instructive to shift
rheological
data to a fixed reference temperature [Morrison, Dealy and
Plazek]. The actual shifting is often carried out with software
packages, but for the beginner it is instructive to shift the data by
hand. We discuss two types of data, G' and G" versus freqency and
steady shear viscosity versus shear
rate.
To shift the data and form a master curve, follow the following steps:
1. Take data over a fixed
frequency range (for example 0.01-100 rad/s) for a variety of
temperatures.
2. Plot the data using log-log coordinates. Use a different
sheet of paper (single side only) for each temperature. Use the
same sized axes for all graphs and use at least two decades on each
axis. For linear-viscoelastic data, plot both G' and G" as a
function of frequency on each graph. For steady shear data, plot
viscosity multiplied by shear rate as a function of shear rate on each
graph.
3. Position the lowest-temperature graph on a light box.
4. Superpose the next hightest temperature graph, lining up the
axes.
5. Shift the superposed graph to the left until the graph in the
bottom layer and the graph on the top layer superpose.
6. Note the location of the left edge of the bottom graph by
making a mark along the bottom edge of the superposed graph. The
distance between the left edge of the graph and the mark is an
indication of the log aT value for this data set.
7. With a ruler, measure the horizontal distance between the mark
you made above and the next tic mark to the left on the frequency (or
shear rate) axis. This is length A. Now measure the width
of one
decade of frequency (or shear rate) on the graph; this is length
B. If the
shifting was less than one decade, the ratio of these two numbers (A/B)
is equal to log aT. If the shifting was more than one
decade, add the number of decades to A/B to get log aT.
8. Repeat these steps with each temperature in succession.
Note that when reporting log aT we are always interested in
the shifting with respect to the very lowest temperature, not the
relative shifting between temperatures.
9. If desired, perform these shifts with the estimated aT
factors in a spreadsheet program by multiplying the shift factors with
the
frequency (or shear rate) values. You may use a program like Excel
Solver to optimize the values of aT by setting
Solver to minmize the difference between the shifted curves and the
master curve.
For some practice data from Morrison (2001) based on Gruver and Kraus'
data in Figure 6.46, go to this Excel file.
References:
Morrison, Faith A., Understanding Rheology (Oxford, New
York: 2001).
Dealy, John M. and D. Plazek, "Time-Temperature Superposition
- a Users
Guide," Rheology Bulletin,
78(2), 16-21,30-31 (2009).
