Dynamic thermomechanical analysis (or dynamic mechanical analysis) is a technique that measures the dynamic modulus and mechanical loss of a sample in relation to temperature or frequency under programmed temperature control and alternating stress. The instrument used to measure these parameters is called a dynamic mechanical analyzer (DMA).
The structure and key components of a DMA instrument are shown in the figure:
Schematic diagram of DMA structure (left: structure of general DMA; right: structure of improved DMA)
1. Base; 2. Height adjustment device; 3. Drive motor; 4. Drive shaft; 5. (Shearing) specimen; 6. (Shearing) specimen clamp; 7. Furnace body; 8. Displacement sensor (Linear Differential Transformer LVDT); 9. Force sensor
The core components of DMA include a drive motor, sample holder, furnace body, displacement sensor, and force sensor.
Drive motor — drives the drive shaft at a set frequency, force, or displacement.
Depending on the selected fixture, the DMA can employ different measurement modes as shown in the figure:
DMA measurement mode
1. Shearing; 2. Three-point bending; 3. Double cantilever; 4. Single cantilever; 5. Tension or compression
Furnace body—controls the sample to conform to the set temperature program
Displacement sensor—measures the amplitude and phase of sinusoidal displacement.
Force sensor—measures the amplitude and phase of a sinusoidal force. Typical DMAs do not have force sensors; force and phase are determined by the alternating current transmitted to the drive motor.
Concepts of stiffness, stress, strain, modulus, and geometric factors:
The ratio of force to displacement is called stiffness. Stiffness is related to the geometry of the specimen.
The force normalized to the surface area A is called mechanical stress or stress σ (force per unit area), and the displacement normalized to the original length L0 is called relative deformation or strain ε. The ratio of stress to strain is called the modulus, which has physical importance and is independent of the geometry of the specimen.
The modulus measured in tensile, compression, and bending tests is Young's modulus, also known as the elastic modulus, while the modulus obtained in shear tests is the shear modulus.
In dynamic mechanical analysis, the composite modulus is calculated using the force amplitude FA and the displacement amplitude LA. For practical purposes, the calculation of both stiffness and modulus is standardized using the so-called geometric factor g.
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FA/LA represents stiffness. Therefore, the final equation for determining the elastic modulus is:
The modulus is obtained by multiplying the stiffness by the geometric factor.
The calculation formulas for various dynamic thermomechanical measurement modes and geometric factors are shown in the table below:
Table 1. DMA Measurement Modes and Calculation Formulas for Sample Geometric Factors
Note: In the table, b is the thickness, w is the width, and l is the length.
The basic principle of DMA testing:
A specimen subjected to periodically (sinusoidally) varying mechanical vibration stress will exhibit corresponding vibrational strain. The measured strain often lags behind the applied stress, unless the specimen is perfectly elastic. This lag is called the phase difference, or phase angle δ difference. DMA instruments measure the amplitude of the stress, the amplitude of the strain, and the phase difference between stress and strain in the specimen.
The stress applied to the specimen during the test must be within the linear range defined by Hooke's Law, that is, the linear range at the beginning of the stress-strain curve.
DMA testing can be performed under a pre-set force amplitude or a pre-set displacement amplitude. The former is called a force-controlled experiment, and the latter is called a displacement-controlled experiment. Generally, DMA can only perform experiments using one control mode. Improved DMA can automatically switch between force control and displacement control modes during the experiment, ensuring that the force and displacement changes of the sample do not exceed the program-set range.
The relationship between composite modulus, storage modulus, loss modulus and loss angle:
The result of DMA analysis is the composite modulus M* of the sample. The composite modulus consists of the in-phase component M' (or denoted by G', called the storage modulus) and the out-of-phase (phase difference π/2) component M'' (or denoted by G'', called the loss modulus). The ratio of the loss modulus to the storage modulus, M''/M' = tanδ, is called the loss factor (or damping factor).
When polymers are subjected to alternating forces, they exhibit hysteresis. After being subjected to external force, the deformation occurs and before the force can be removed, the next stress is applied, resulting in some elastic energy remaining unreleased. This cycle continues, and the unreleased elastic energy is consumed by the system's self-friction and converted into heat.
The relationship between the composite modulus M*, storage modulus M', loss modulus M'', and loss angle δ can be represented by the triangle shown in the figure below:
The storage modulus M' is directly proportional to the mechanical energy stored in the specimen during stress. Conversely, the loss modulus represents the energy dissipated (as heat) by the specimen during stress. A large loss modulus indicates high viscosity and therefore strong damping. The loss factor tanδ is equal to the ratio of viscosity to elasticity, so a high value indicates a high degree of energy dissipation and a high degree of viscous deformation. It is a measure of the energy dissipated as heat in each deformation cycle. The loss factor is independent of geometric factors, therefore it can be accurately measured even if the specimen geometry is poor.
The reciprocal of the modulus is called the compliance. Corresponding to the modulus, there are composite compliance, energy storage compliance, and loss compliance. For the description of the mechanical properties of materials, composite modulus and composite compliance are equivalent.
Three different types of sample behavior can generally be distinguished:
Pure elasticity means that stress and strain are in phase, i.e., the phase angle δ is 0. There is no energy loss when a pure elastic specimen vibrates.
Pure viscous—stress and strain phase, i.e., phase angle δ is π/2. The deformation energy of a pure viscous specimen is completely converted into heat.
Viscoelastic deformation exhibits a certain hysteresis in its stress response, with the phase angle δ ranging from 0 to π/2. The larger the phase angle, the stronger the vibration damping.
The physical quantities for DMA analysis are listed in the table below:
Table 2 Summary of DMA Physical Quantities
Under constant load, molecules slowly rearrange to minimize stress, causing the material to deform over time. If vibrational stress is applied, the strain decreases with increasing frequency because the time available for rearrangement is reduced. Therefore, materials are stiffer at high frequencies than at low frequencies, meaning the modulus increases with frequency. As temperature increases, molecules rearrange more quickly, increasing displacement amplitude, which is equivalent to a decrease in modulus. The modulus measured at room temperature at a given frequency is equal to the modulus measured at a higher temperature and higher frequency. This means that frequency and temperature influence material properties in a complementary way; this is the temperature-frequency equivalence principle. Because lower frequency corresponds to longer time (and vice versa), temperature-frequency equivalence is also called time-temperature superposition (TTS).
By applying the temperature-frequency equivalence principle, modulus information at frequencies that cannot be directly obtained experimentally can be obtained. For example, the damping behavior of rubber blends at several kilohertz at room temperature cannot be directly measured experimentally because the highest frequency of the DMA (Dielectric-Distributed Damping) is insufficient. In this case, the temperature-frequency equivalence principle can be used to extrapolate the loss factor at room temperature to several kilohertz using tests conducted at low temperatures and within a measurable frequency range.
Typical DMA measurement curve:
There are two main types of DMA measurement curves: dynamic temperature program measurement curves and isothermal frequency scan measurement curves.
The dynamic temperature program measurement curve is obtained by testing under alternating stress at a fixed frequency at a certain heating rate (usually low, preferably 1~3K/min, due to the large size of the sample). The resulting graph is a curve with temperature on the x-axis and modulus on the y-axis. The graph shows the changes in storage modulus G', loss modulus G'', and loss factor tanδ with temperature, reflecting the secondary relaxation, glass transition, cold crystallization, melting, and other processes of the sample.
Isothermal frequency scanning measurement curves are obtained by scanning tests under isothermal conditions with different vibration frequencies of stress. The resulting graph is frequency on the x-axis and modulus on the y-axis, showing the changes in storage modulus G', loss modulus G'', and loss factor tanδ with frequency. The relationship between the mechanical relaxation behavior and frequency in isothermal testing is also known as the mechanical relaxation spectrum. Based on the temperature-frequency equivalence principle, the mechanical relaxation spectrum under different temperature conditions can be shifted laterally along a frequency window to obtain the modulus values corresponding to different temperatures.
