Download Modulated Thermomechanical Analysis

Modulated Thermomechanical Analysis – Measuring Expansion and
Contraction Simultaneously
Roger L. Blaine, Ph.D.
TA Instruments, 109 Lukens Drive, New Castle DE 19720, USA
ABSTRACT
In modulated thermal analysis, a sinusoidally varying temperature program is
added to an underlying linearly changing or static temperature program. Fourier
transformation of the oscillatory temperature “forcing function” and the result dependent
variable provides the deconvolution of these signals into reversing and nonreversing
components. The reversing signal is associated with properties dependent upon the
temperature rate of change while the nonreversing signal defines kinetic events (that is,
those associated with both time and temperature). Thus for modulated thermomechanical
analysis (MTMA™), coefficient of thermal expansion is observed in the reversing signal
and stress relaxation, softening and heat shrinking are observed in the nonreversing
signal. The ability of MTMA to measure both expansion and contraction simultaneously
is demonstrated on samples including thermoset composite printed circuit boards and
heat-shrink packaging film.
INTRODUCTION
Reading introduced modulated temperature thermal analysis in 1993 (1) with
what has become known as modulated differential scanning calorimetry (Modulated
DSC®, MDSC®). This was followed a few years later by Blaine who reported on
modulated thermogravimetry (MTGA™) (2) and by Price with modulated
thermomechanical analysis (MTMA™) (3). All of these approaches are commercially
available from TA Instruments (New Castle, DE).
The impact of modulated temperature measurements on thermal analysis can
hardly be underestimated. One leading observer has called modulated DSC “the greatest
advance in DSC since its inception” (4). And nearly half of the articles published in
thermal analysis journals now feature one of the modulated temperature techniques.
In modulated temperature thermal analysis, a sinusoidal temperature “forcing
function” is added to the traditional linear (or isothermal) underlying heating rate. This
forcing function induces in the sample a sinusoidal response that may be deconvoluted to
yield reversing and nonreversing information.
When applied to thermomechanical analysis (TMA), the temperature modulation
produces a sinusoidal change in test specimen length. Figure 1, for example, shows the
modulated temperature at the figure bottom and the length change at the top both as their
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first derivatives. Discrete Fourier transformation is applied in real time to continuously
determine the average value and the amplitude value for each signal.
Deriv. Mod Dimension Change (µm/min)
0
16
-1
8
-2
0
-3
-4
40
60
80
100
Temperature (°C)
120
140
[ – – – – ] Deriv. Modulated Temperature (°C/min)
24
1
-8
160
Figure 1 - Modulated Temperature and Resultant Modulated Length
The governing equation for MTMA is given by (3):
dL/dt = A dT/dt + f (t, T)
(1)
where L is length, T is temperature, t is time, A is expansivity, and f(x) is “some function
of…”. The left hand term of equation 1, known as the total length change rate, is shown
on the right to be composed of two parts; a reversing contribution proportional to the
time rate of change of the independent parameter (temperature) and a nonreversing
contribution from the absolute value of that independent parameter.
Of course in TMA, the dependent parameter measured is length (not the time rate
of change of length) so the governing equation takes on the form of:
Ltotal =Lreversing + Lnonreversing
(2)
where Ltotal is the average length from the Fourier deconvoltion, Lreversing is the
(length amplitude / temperature amplitude) ∫ dT/dt K. The “average length”, “length
amplitude” and “temperature amplitude” are the signals derived from the Fourier
transformation process. ∫ dT/dt is the underlying heating rate averaged over a single
period while K is a calibration constant close to unity. The nonreversing length is
obtained from the difference between the total and reversing contributions.
Lnonreversing = Ltotal – Lreversing
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(3)
Deriv. Mod Dimension Change (µm/min)
4
δ = arcsin a/b = arcsin 0.862/1.57 = 33 °
= 28 s
1.571µm/min
2
0.8621µm/min
0
-2
-4
-15
-10
-5
0
5
Deriv. Modulated Temperature (°C/min)
10
15
Figure 2 - Lissajou Plot to Determine Phase Lag
4
-65
2
-70
0
-75
-2
-80
-4
70
90
110
130
Temperature (°C)
150
170
[ – – – – ] Dimension Change (µm)
Deriv. Mod Dimension Change (µm/min)
There is always a “lag” between the applications of the forcing function and the
resultant response. The use of sinusoidal forcing functions provides for the easy
measurement and interpretation of the magnitude of this lag through the use of a
Lissajous plot such as the shown in Figure 2. Evaluation of the midpoint and extrema
values of the Lissajous figure yields the phase lag. In MTMA, this lag is rather large
-85
Figure 3 - Length Change and Modulated Length Change at the Glass Transition
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compared to other modulated temperature techniques, about 28 s, due to the large
thermal mass and low thermal conductivity of the quartz sample stage and measuring
probe. A practical rule-of-thumb is that the period of the applied forcing function should
be about ten times the time constant. So for MTMA, a period of about 300 seconds is
used. The applied temperature amplitude is selected based upon the value of the
expansivity value A in equation 1, but is typically ± 5 °C. And as with all modulated
temperature approaches, at least 5 cycles is required across a transition in order to have
reliable Fourier deconvolution. This results in the typical underlying heating rate of
2 °C/min or less.
The value of the calibration coefficient is determined, as is the length change
calibration of the TMA (5), from the use of a reference material with a known CTE value.
We used copper for this purpose as it has no nonreversing or kinetic behavior in the
region of interest between 50 and 150 °C.
5.5
4.5
0
3.5
-1
2.5
-2
1.5
-3
-4
0.5
0
50
100
Temperature (°C)
150
[ – – – – ] Deriv. Modulated Temperature (°C/min)
Deriv. Mod Dimension Change (µm/min)
1
-0.5
200
Figure 4 - Glass Transition with Enthalpic Recovery
RESULTS AND DISCUSSION
Figure 3 shows the dimension ( i.e., Ltotal) and modulated dimension signals for
a cured thermoset sample undergoing a change in coefficient of linear thermal expansion.
Where there is no nonreversing phenomenon, the amplitude of the modulated dimension
derivative is proportional to the coefficient of linear thermal expansion. The expansion is
proportional to the amplitude of this signal. As the expansion increases so does the
oscillatory dimension amplitude signal.
A more complicated example is shown in Figure 4 where the oscillatory
temperature signal is shown at the bottom (as its derivative) while the modulated length
signal is shown at the top. In this case both the amplitude and average value for the
length change during the course of the experiment with a “peak” in the derivative curve
appearing near 140 °C. The resultant total, reversing and nonreversing signals for this
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thermal curve are shown in Figure 5. The total dimension, presented in the middle curve
shows the masking effect on the glass transition by the enthalpic recover transition
typical of aged thermoset material. The enthalpic recovery causes the test specimen to
shrink as it gains mobility upon passing through the glass transition. That is, the sample
contracts (due to the enthalpic recovery) and expands (due to the glass transition) at the
same time with the total length change reflecting the sum of these two events. The upper
trace in Figure 5 shows the reversing expansion of the sample as a result of the increase
in the coefficient of linear thermal expansion. The bottom nonreversing curve contains
the shrinkage information resulting from the enthalpic recovery.
15
124.12°C
-15
-45
0
50
100
Temperature (°C)
150
25 20
-5 -10
[ – – – – ] Dimension Change (µm)
50
[ ––– ––– ] Non-Rev Dimension Change (µm)
Rev Dimension Change (µm)
45
-40
200
Figure 5 -Separation of the Glass Transition from the Enthalpic Recovery
Another example is shown in Figure 6 for a thin polymer film examined in
tension. As the sample goes through the glass transition, the expansion of the material
increases. Additionally, the sample softens in the region of the glass transition and the
sample begins to stretch under tension. The expansion is separated out into the reversing
length change while the stretching is resolved into the nonreversing dimension change.
In this case both the thermodynamic and kinetic components are in the same direction but
are different in effect by an order of magnitude.
In TMA, the glass transition temperature is identified by the extrapolation of the
linear portions of the expansion curve before and after the transition to their intersection
(6). This value taken from the reversing length change curve corresponds to 154 °C. In
some cases, the softening temperature, either as the extrapolated onset (6) or at a specific
modulus value (7), is used to estimate the glass transition temperature. Figure 7 shows
that the extrapolated onset from the soften curve estimate the glass transition at 149 °C,
some 5 °C lower that that obtained from the change in linear expansion.
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5000
700
Yield Point
-1000
-3000
-5000
40
154.68°C
80
120
Temperature (°C)
160
2000
500
0
300
-2000
100
-4000
-100
[ – – – – ] Rev Dimension Change (µm)
600
1000
10 X
[ ––– ––– ] Non-Rev Dimension Change (µm)
Dimension Change (µm)
3000
Deriv. Mod Dimension Change (µm/min)
162.97°C
-300
200
Figure 6 - Separation of Film Stretching from Expansivity
SUMMARY
Modulated thermomechanical analysis is shown to be a useful tool for
determining thermodynamic and kinetic, reversing and nonreversing length changes in
materials when these thermally induced events take place simultaneously. This ability to
separate thermodynamic from kinetic events aids in the interpretation of the thermal
curve, measures expansion and contraction taking place at the same time and provides a
more accurate estimation of the glass transition temperature than the softening
temperature.
MTMA may have applications in other measurements where thermodynamic and
kinetic length changes occur simultaneously. Such examples may include the softening
of organic or inorganic glasses, heat set films and fibers, and shape memory alloys.
REFERENCES
1.
2.
3.
M. Reading, “Modulated Differential Scanning Calorimetry – A New Way
Forward in Materials Characterization“, Trends in Polymer Science, 1993, 1, pp.
248-253.
R. L. Blaine and B. K. Hahn, “Obtaining Kinetic Parameters by Modulated
Thermogravimetry“, Journal of Thermal Analysis, 1998, 54, pp. 695-704.
D. M. Price, “Novel Methods of Modulated Temperature Thermal Analysis“,
Thermochimica Acta, 1998, 315, pp. 11-18.
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5.
6.
7.
B. Wunderlich, Y. Jin and A. Bollar, “Mathematical Description of Differential
Scanning Calorimetry Based on Periodic Temperature Modulation”.
Thermochimica Acta, 1994, 238, pp. 277-293.
E 2113, “Length Change Calibration of Thermomechanical Analyzers”, ASTM
International, West Conshohocken, PA
E 1545, “ Assignment of the Glass Transition Temperature by Thermomechanical
Analysis”, ASTM International, West Conshohocken, PA.
E 2092, “Distortion Temperature in Three-Point Bending by Thermomechanical
Analysis”, ASTM International, West Conshohocken, PA
4000
800
3000
Dimension Change (µm)
600
2000
Softening Temperature
1000
400
Glass Transition Temperature
148.86°C
0
200
153.84°C
-1000
-2000
40
60
80
100
120
Temperature (°C)
140
160
[ – – – – ] Rev Dimension Change (µm)
4.
0
180
Figure 7 - Comparison of Softening and Glass Transition Temperatures
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