Thermomechanical Analysis

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Thermomechanical analysis

Fundamentals of thermal expansion

With increasing operating temperatures, polymeric materials undergo linear expansion. Measuring thermal expansion provides information about the average linear (α) and cubic (β) thermal expansion coefficients of the respective plastic material, as well as important transformation phenomena during heating.

The thermal expansion coefficient α, also known as the thermal expansion factor, describes the change in length L₁ or change in volume V₁ of a body when the temperature increases by 1 K and is expressed in K^-1 [1]. In a limited temperature range, the linear expansion is given by

$L_{1}\,=\,L_{0}\,+\,\alpha \,L_{0}\,\left(T_{1}\,-\,T_{0}\right)$

(1)

and for the volume expansion

$V_{1}\,=\,V_{0}\,+\,\beta \,V_{0}\,\left(T_{1}\,-\,T_{0}\right)$

(2)

whereas for the isotropic body the relationship

$\beta \,=\,3\,\alpha$

(3)

applies.

However, due to the temperature dependence of the coefficients α and β, nonlinear dependencies are to be expected, and the following applies

$\alpha \,=\,{\frac {1}{L_{0}}}\,\left({\frac {\delta \,L}{\delta \,T}}\right)_{p}$ bzw. $\beta \,=\,{\frac {1}{V_{0}}}\,\left({\frac {\delta \,V}{\delta \,T}}\right)_{p}$.

(4)

The nonlinearity is a consequence of the local movement of small molecule groups (secondary relaxation) that begins with rising temperatures and the subsequent cooperative movements of entire molecule parts (primary relaxation). In the transition areas, the expansion coefficients change abruptly.

Length measurement methods

The determination of expansion coefficients is limited to temperature ranges in which the thermal expansion is virtually independent of temperature. This places high demands on the sensitivity of length measurement technology. Exact characteristic values can only be obtained when the plastic material to be characterised is in a solid state, as the result is influenced by a number of significant factors during the determination process. Polymer materials are more or less hygroscopic or contain volatile components that tend to processing shrinkage and dry out when heat is applied externally, counteracting thermal expansion. Therefore, methods should be used that eliminate secondary influences but correspond to the conditions of practical use.
With optical expansion measuring devices, the expansion measurement is carried out visually using a measuring microscope. Tin foil strips are stuck onto the test specimen as measuring marks. The test specimens are heated in hot air using a suitable heating table. A special control loop is used to monitor and suppress the thermal sensors and disruptive setpoint fluctuations. The temperature increase should be in the order of 5 K h^-1. Even the slightest curvature or shrinkage of the test specimen as it approaches the softening range can distort the result. When measuring with a quartz tube dilatometer, the change in length after a temperature T is recorded using a dial gauge or an inductive displacement transducer. This causes a deformation impedance that counteracts the expansion. Heating can be carried out in air or in a liquid bath, whereby an accuracy of 0.2 K should be aimed for in the individual stages.
Displacement dilatometers are essentially pycnometers in which the change in the level of the liquid is read on a calibrated capillary tube by heating a measuring chamber in which the test specimen is located. Mercury and methanol have proven to be suitable measuring liquids. The measuring chamber is heated in a liquid bath, also with an accuracy of 0.2 K. This method is suitable for measuring volume expansion directly, without interruption, down to the liquid range. The temperature range of interest is traversed in small steps.

Experimental implementation of TMA

Thermomechanical analysis (TMA) has proven itself as a method for measuring the linear thermal expansion coefficient of plastics. In contrast to the force-free dilatometer method, TMA measures with a constant, low load. Cylindrical or cuboid test specimens with plane-parallel measuring surfaces are used. A quartz stamp is used to apply the low load (0.1 to 5 g) and, at the same time, an inductive measuring system is used to measure the thermal expansion. The test setup is located in an oven that is heated at a low heating rate. Based on DIN 53752 [2] or ISO 11359 [3], a mean (Eq. 5) or a differential thermal coefficient of linear expansion (Eq. 6) can be determined.

${\bar {\alpha }}\left(T\right)\,=\,{\frac {1}{L_{0}}}\,\cdot \,{\frac {L_{2}\,-\,L_{1}}{T_{2}\,-\,T_{1}}}\,=\,{\frac {1}{L_{0}}}\,\cdot \,{\frac {\Delta L}{\Delta T}}$

(5)

$\alpha \left(T\right)\,=\,{\frac {1}{L_{0}}}\,\cdot \,{\frac {dL}{dT}}$

(6)

The differential thermal coefficient of linear expansion is determined by the increase in the tangent to the dependence ΔL/L₀. It is always ‘0’ at the start of the test. As with differential scanning calorimetry (DSC), the first heating cycle of a TMA always provides information about the thermal and mechanical history. Heating not only allows volatile components to escape, but can also lead to the breakdown of orientations and internal stress and can trigger recrystallisation processes. All of these processes are associated with shrinkage and counteract thermal expansion. In thermosets, post-curing processes have the same effect. In addition, anisotropy effects must be taken into account in injection-moulded and extruded components. This also applies to filled and reinforced plastics.

In semi-crystalline polymers, heating causes a more or less pronounced contraction, and the linear expansion coefficient in the chain direction can take on negative values. The cause lies in the undisturbed rubber-elastic recovery of the tie molecules [4] in the amorphous regions. Since a volume measurement yields positive values, there must be a correspondingly stronger increase in the coefficient of expansion perpendicular to the orientation direction. For polyethylene (abbreviation: PE), α_|| = - 2.4•10^-5 K^-1 and α_⊥ = 19•10^-5 K^-1 were found at room temperature. Conversely, it should therefore be possible to obtain information about the orientation state by measuring the directional dependence of the linear coefficient of expansion [5].

Analogous to semi-crystalline polymers, the coefficient of thermal expansion in amorphous multiphase systems depends, as expected, on the proportion of components and the compatibility of the phases. Above the glass transition temperature of both components, the expansion coefficient usually follows a simple additive law. In the range be-tween the glass transition temperatures of the polymers involved, this is only partially true. In addition, the different expansion behaviour of the phases can lead to the formation of thermally induced stresses (residual stresses), which negatively influence the macro properties of the polymer blend.

Application example

Composite systems made of plastics with inorganic fillers generally exhibit lower thermal expansion depending on the filler content, particle shape and manufacturing-related or-der state, as the matrix material expands more than the fillers. As a result, the expected internal stresses, especially at the polymer/filler interfaces, are also more pronounced. There are limits to the application of the mixing rule for the analytical evaluation of the expansion coefficient of a composite. As long as the determination equations do not take into account the interaction between the matrix and the filler surface, changes in free volume, percolation effects and particle size, only approximate values can be given.

Fig. 1:

Thermal expansion behaviour of a fibre-reinforced polyphenylene sulfide (abbreviation: PPS) plate in the radial, tangential, and thickness directions

Fig. 1 shows the influence of fibre reinforcement on the thermal expansion behaviour of a circular plate. While there are only minor differences in thermal expansion behaviour in the radial and tangential directions, significantly greater thermal expansion can be detected in the thickness direction, which is largely determined by the thermal expansion behaviour of the unreinforced matrix. Conclusions about the fibre orientation can be drawn from the anisotropy of the thermal expansion behaviour.

The thermal expansion behaviour makes a significant contribution to the development of internal stresses. In this context, it should also be noted that the coefficient of thermal expansion (see: thermal expansion coefficient) decreases with increasing elastic modulus. Any obstruction of thermal expansion leads to the build-up of stress in the material, known as thermal stress. This applies both to the case of a force-fit combination of mate-rials with different thermal and elastic properties and to the case of different temperatures within a product. Tensile stresses build up in the material or material areas with the lower coefficient of thermal expansion, while compressive stresses build up in the others. If the causes of the thermal stresses are eliminated, the internal stresses disappear completely, provided that no plastic deformations occur. Otherwise, residual stresses arise.

See also

• Thermal stress analysis
• Thermal strain analysis
• Thermoelastic effect
• Shrinkage test
• Curing

References

[1]	Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser, Munich (2022) 3rd Edition, pp. 296–299 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see AMK-Library under A 22)
[2]	DIN 53752 (1980-12): Testing of Plastics – Determination of the Coefficient of Linear Thermal Expansion (withdrawn; replaced by ISO 11359)
[3]	ISO 11359: Plastics – Thermomechanical Analysis (TMA) Teil 1 (2023-02): General Principles Teil 2 (2021-11): Determination of Coefficient of Linear Thermal Expansion and Glass Transition Temperature
[4]	Michler, G. H.: Kunststoff-Mikromechanik – Morphologie, Deformations- und Bruch-mechanismen. Carl Hanser, Munich Vienna (1992), (ISBN 3-446-17068-5; see AMK-Library under F 4)
[5]	Schmiedel, H. (Ed.): Handbuch der Kunststoffprüfung. Carl Hanser, Munich Vienna (1992), (ISBN 3-446-16336-0; see AMK-Library under A 3)