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This simple set-up is adequate for illustrative purposes, but in modern laboratories a Thermomechanical Analyzer, or TMA, is used for these types of investigations. This instrument measures the dimensional change as either the force on the sample or the temperature is varied. A schematic diagram of a TMA is shown in Figure 4a. The most important components are the furnace with associated temperature control circuitry and coolant reservoir for subamblent work, the probe, and the linear variable differential transformer (LVDT). In the TMA shown, the lower part of the probe is connected to a metal rod, which fits into the force coil assembly at the bottom. When current is passed through the coil, the probe applies a force to the sample. The metal rod also passes through the core of the LVDT. The voltage output of the LVDT depends on the vertical position of the metal rod, and this is determined by the elongation of the sample.
For use in the study of elastomer elongation the TMA must be equipped with a tension probe (Figure 4b) which allows the rubber band to be stretched by a controlled force. A typical instrument is capable of three general modes of operation: isothermal, in which the dimensional change is measured as the force is changed; isostress, with measurements of dimensional change at constant force as the temperature is varied; and isostrain, in which temperature is varied and the force needed to maintain a constant elongation is measured.
The data obtained in an isostrain experiment may be used to prepare a plot of
force versus temperature, and the slope is equal to 
Equation 9 is
used to calculate 
from the slope and the initial
length of the sample
between the probe clamps. The initial length may be measured with a calipers
or a ruler, but this value is commonly obtained directly from the instrument.
If so, the TMA's initial length measurement should be performed under minimal
force load.
By suitable mathematical manipulations
may be evaluated using the
other TMA operational modes.
The total differential, df, is given by
(32)
For a constant value of f, df is equal to zero, and 
is given by
(33)
The first of the two contributing terms, the derivative of the force with
respect to the relative elongation, may be obtained from the slope of a plot of
f versus 
for an isothermal experiment.
Because of the nonlinear relationship
of f and , the slope must be evaluated for the line
tangent to the curve at a specified elongation.
To evaluate ,
must be multiplied by
the negative derivative of the relative elongation with respect to temperature
at constant force. An isostress TMA experiment provides this derivative as the
slope of a plot of relative elongation versus temperature. The value may also
be evaluated from f vs plots at a series of constant
temperatures. The
corresponding to a specific force is obtained for each set of measurements;
is the slope of a plot of these
values versus temperature.
In this experiment both the TMA and the "weights and ruler" apparatus
(hence-forth called the manual method) will be used to evaluate
for a
common rubber band.
(A 1/16" or 1/8" band seems to work adequately.) The data obtained from the isothermal measurements will also be used to calculate Young's modulus and to verify the predictions of the statistical mechanical model.
For these measurements, the TMA offers the obvious advantages of automated
data acquisition and an on-line computer for the computations, as well as the
capability of performing the isostress and isostrain experiments described
above. However, the instrumentation may tend to be a black-box approach, and
the simple apparatus helps students appreciate the measurements being made.
Use of the manual method also has scientific merits. TMA's are designed for
highly sensitive measurements of dimensional changes, and as a result, they
have small overall elongation limits. Thus, the thermodynamic relationships
cannot be observed over a wide range of values.
By measuring the length with
a ruler the students should be able to test the validity of equation 29 for
relative elongations up to about 4, with curvature observed at higher
's due
to the formation of crystallites in the polymer.
The materials for the manual set-up should be readily available in any laboratory. A bent wire gauze is an adequate pan, and a brass rod can be cut into short lengths to serve as weights (about 15 to 20 , five gram pieces for the rubber bands specified above). Fluid from a circulating constant temperature bath can be used to provide several isothermal sample temperatures. However, if a TMA is available, manual measurements at the ambient temperature should be sufficient for students to understand the fundamental concepts, and the TMA can be used for the remainder of the experiment.
Considering now the TMA measurements, isothermal force/elongation data are collected for several accessible temperatures (e.g., 10, 25, and 50 oC). The range of forces to be applied depends on the extension limits of the TMA and the cross-sectional area of the rubber band. In general, the force is varied to produce relative elongations from about 1.1 (10% elongation) to the maximum allowed by the instrument and/or sample. The temperature ramps for the isostress and isostrain experiments should cover at least the range of temperatures used for the isothermal work (i.e., 10 to 50 oC for the above example). For the later two modes of operation the band is initially strained to a relative elongation approximately in the middle of the elongation range observed in the isothermal sets. For the isostress work the force is fixed at the value needed to produce the desired strain at 25oC or a mid-range temperature. Then the sample is equilibrated at one end of the desired temperature range, prior to starting the ramp itself. The isostrain experiment is conducted in the same manner, but the elongation is held constant and the force is measured as the temperature is ramped
To the inexperienced user of Thermomechanical Analysis, the design of the various acquisition cycles may seem formidable, especially since the operating manuals are commonly written for engineering and materials applications. The author has designed specific instructions for use with the TMA shown in Figure 4, and and these may be found at the end of this document as an addendum
The cross-sectional area of the rubber band may be calculated from the width and thickness of the unstressed sample measured with a calipers. The dimensions should be measured at several locations on the band, and care must be taken not to close the jaws of the calipers too tightly. (These values should theoretically be obtained at each of the specified temperatures, but the effects of thermal expansion/contraction are expected to be very small over this temperature range). The unstressed length of the sample should be obtained prior to and after each experiment, if possible. The values will show some random fluctuation, but should not increase systematically with time.
Although the elongations of the rubber band itself are sizable, the variations
with temperature of the length (isostress mode) or the force (isostrain mode)
are very small, especially over the temperature range suggested in the previous
paragraph. The
TMA is capable of detecting these small effects, but there may still be
considerable uncertainty in the final results. This is especially true of the
isothermal data, for which a plot of
versus T at a constant force must be prepared as part of the calculation of
Probably the most important source of experimental error is failure to allow the system to reach equilibrium. Typically an experiment is designed to equilibrate the rubber band at the designated initial conditions, and then to start the temperature or force ramp. It is essential that the rubber band be allowed to sit at the initial conditions for a sufficient length of time, because some relaxation will occur. The ramps themselves should be as slow as student laboratory time will allow (suggested rates are 1 to 2 oC/minute for temperature; 0.05 newton/minute for force). If possible, runs should be repeated and/or data should be collected using both ascending and descending ramps. For temperature ramps Flory has suggested that the sample be equilibrated initially at the highest temperature, with the descending changes first, followed by an ascending ramp back to the initial temperature. The author has also found that this procedure gives the most reproducible data.
Another major source of uncertainty is the rubber band itself. Students usually relate well to an experiment involving a consumer product, but, if possible, a crosslinked sample without fillers or other additives should be used.
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