Generating a Master Curve Using Dynamic Mechanical Analysis (DMA)

By: Sarah Ackermann, MSc

Imagine the following situation: You’re a polymer scientist working on a new material for a structural application, where it’s critical to understand its long-term behavior under load. This is often referred to as the creep behavior of the material. However, testing the creep behavior of a material can take a very long time, especially if we’re interested in the behavior over a period of years or decades. This is impractical for getting materials to market. What if there was a way to accelerate the creep studies and accurately predict long-term behavior from short-term testing? Dynamic Mechanical Analysis (DMA) is a powerful technique used to study the mechanical properties of materials as a function of temperature, frequency, and time. It provides valuable insights into viscoelastic behavior, glass transitions, and relaxation processes. One of the key applications is to use time-temperature superposition (TTS) in generating a master curve using dynamic mechanical analysis, which allows us to predict material behavior across a wide range of conditions.

Let’s talk about how this works.

1. The Superposition Principle

Boltzmann’s principle of superposition is a fundamental concept in the study of viscoelastic materials. It states that for linear systems, the result of complex inputs is the sum of the individual input components. Applied to viscoelastic materials – this means that the expected strain (deformation) of a material in the linear region of behavior is the sum of the strains observed as a result of all of the component stresses a material is subjected to.

In other words – the expected result of, for example, deformation due to compression and a temperature change at the same time is the sum of the deformation associated with the compression and the deformation associated with the temperature change due to thermal expansion or contraction. Figure 1 illustrates this principle.

1.1 The Superposition Principle In DMA

In the case of DMA, we apply an oscillatory force (stress, σ) or deformation (strain, γ) at a defined frequency (ω) to a material and measure its mechanical response. Stress and strain can be expressed as functions of frequency and time (t) according to the following equations:

From this, using Euler’s identity, we can find the material’s response, termed the complex modulus (E^*), according to the following equation:

The superposition principle and Euler’s identity together allow us to separate the complex modulus into two key components:

1. Storage Modulus (E’ or G’): This represents the material’s elastic behavior. It quantifies how much energy the material can store and release during each cycle of deformation. Mathematically, it is defined as the ratio of stress (σ) to strain (ε) amplitude multiplied by the cosine of the phase angle (δ):

The storage modulus is frequency-dependent and typically increases with increasing frequency.

2. Loss Modulus (E’’ or G’’): This characterizes the material’s viscous behavior. It accounts for energy dissipation (loss) during each deformation cycle. The loss modulus is also frequency-dependent and is related to the damping properties of the material. It is defined according to the following equation:

Using the relation between phase angle, loss modulus, and storage modulus, we can also relate storage and loss modulus to the tangent of the phase angle:

This means that by combining the directly observed complex modulus and phase angle, we can determine both the storage and loss modulus from a single DMA experiment.

To convert the equations above from strain case to shear case, substitute G for E and γ for ε in the above equations.

1.2 DMA Experiments and Superposition

1. Frequency Sweep: In a DMA experiment, we subject the material to oscillatory stress or strain at various frequencies. For each frequency, we measure the storage modulus and loss modulus.
2. Decomposition of the signal: We can extract the individual contributions of elastic and viscous behavior using the superposition principle. At low frequencies, the storage modulus dominates, indicating a more elastic response. At high frequencies, the loss modulus becomes significant, reflecting greater energy dissipation.
3. Temperature Effects: The material will be stiffer at lower temperatures, resulting in a higher degree of elastic behavior. The material will soften at higher temperatures and become rubbery, allowing more viscous behavior.
4. Time–Temperature Equivalence: It is a foundation of polymer science that, in many cases, high frequency mimics low temperature and vice versa. This principle of time-temperature equivalence can be used to extrapolate to higher frequencies by decreasing temperatures and lower frequencies by increasing temperatures (and vice versa).
5. Master Curve Construction: To create a master curve, we plot the storage and loss modulus at different temperatures as frequency functions on a log-log scale. We obtain a comprehensive picture of the material’s viscoelastic behavior by combining data from different frequencies and temperatures. We can then predict the material’s behavior at temperatures and frequencies outside of the measurement range using the principle of time-temperature equivalence.

1.3 Limitations of the Superposition Principle

The superposition principle assumes linearity, which holds only under certain conditions. Notable exceptions include:

• If there is a distribution of activation energies in the system
  • Certain polymer blends and block copolymers have different activation energies and do not behave linearly in relaxation processes.[i]
• If there is more than a single dominant relaxation process. For example:
  • In creep testing, strain relaxation is dominant unless you cross the T_g. For this reason, TTS is usually used to predict creep above or below a Tg, but not through it.[ii]
• If the material is not thermodynamically or physically stable in the temperature range modeled.[iii] In this case, the changing physical or chemical composition with time may cause non-linear behavior. For example:
  • Materials that change chemically as a function of time
    • Foams may outgas for a period of years following their formation.
    • Another example would be polymers at elevated temperatures, which may experience oxidation with time.
    • Light-sensitive materials exposed to light may experience degradation with time
  • Materials with phase changes in the temperature range of interest
  • Materials exhibiting mechanical fatigue
  • In highly viscous, soft materials, slow relaxation times can mean a structural dependence on time, so the material will not show a linear response to strain.
• If the conditions tested or modeled are ones that induce other sources of non-linear behavior from the material, for example:
  • Stresses or strains beyond the linear region of a stress-strain curve
  • Materials at temperatures beyond their dilatometric softening point

1.4 Shifting Data for Extrapolation

A series of curves, known as a multiplex, collected at different temperatures and frequencies, is challenging to use on its own. To rationalize the data and make it more useful for engineers and researchers, the raw data is shifted relative to a reference curve. The shift factor is calculated according to a model appropriate to the material, its state, and whether the behavior range of interest passes through any major transitions.

2. Time-Temperature Superposition (TTS)

TTS is a powerful concept in DMA. It allows us to shift data from different test frequencies (or timescales) to a common reference temperature and vice versa. Here’s how it works:

1. Collect Data: Perform DMA experiments at various frequencies (e.g., 1 Hz, 10 Hz, 100 Hz) over a temperature range.
2. Calculate Shift Factors: Determine the shift factors (α) using the appropriate model. These factors relate the relaxation time at a reference temperature (usually the Tg) to the relaxation time at the test temperature.
3. Shift Data: Apply the shift factors to the relaxation times at each frequency. This aligns the data to a master curve at the reference temperature.
4. Construct the Master Curve: Plot the shifted data on a log-log scale. The resulting master curve represents the material’s viscoelastic behavior across different frequencies and temperatures.

3. Common Models for Shifting Test Data

3.1 Williams-Landel-Ferry Model

The most common model used in Time-Temperature Superposition experiments is the Williams-Landel-Ferry (WLF) model. The WLF Equation is an empirical model which derives from Dolittle’s Mean Free Volume equation and Arrhenius kinetics. It is valid in the temperature range of T_g – (T_g + 100K),[i] although some researchers report a more limited temperature range of validity.[ii] The equation for WLF shift factors is below:

Where:

• a_T is the shift factor
• η is the viscosity
• η_r is the viscosity at a reference temperature, T_r
• C₁ and C₂ are material-specific constants.
• T is temperature

3.2 Arrhenius Model

The Arrhenius Model and its derivatives are applicable below the glass transition temperature.[i] The following equation gives the Arrhenius model:

Where:

• E_A is the activation energy
• R is the ideal gas constant
• Other variables are as described above

3.3 Brostow Model

The Brostow model is a semi-empirical expansion of the WLF model which is less limited by temperature[i]^,[ii] and has demonstrated good performance at temperatures both slightly below T_g and over 100K above T_g. In this model, a factor is added to account for changes in density with temperature, and in pre-drawn materials, a factor may be added to account for the draw ratio. The simplest version of the model is given below:

Where:

• A is a constant
• B comes from the Doolittle viscosity relation
• v is the reduced volume.

4. Benefits of the Master Curve

• Predictive Power: With a master curve, we can predict material behavior at any temperature or frequency, even beyond the experimental range.
• Accelerated Testing: TTS allows us to simulate long-term behavior by testing at higher frequencies or temperatures.
• Material Design: Engineers use master curves to optimize material formulations for specific applications.

5. Case Studies

5.1 Anti-Vibration Mounts

Anti-vibration mounts are used to isolate sensitive equipment from vibrations in their environment. As they’re subject to vibrations and stresses during their service life, it’s important to understand the cumulative effects of creep and vibration with time.

In this case, the sample is mounted in tension jaws and a series of frequency scans at different temperatures were performed. The storage modulus is plotted as a function of temperature, above. The test parameters were as follows:

• Geometry
  • Diameter: 14 mm
  • Height: 15 mm
• Test Configuration: Tension Compression
• Dynamic Strain: 1E-3
• Analysis Temperature Range: -80°C to 60°C
• Analysis Frequency Range: 1 Hz to 100 Hz

The tan δ of the material can be analyzed on the same data to give a good understanding of the material’s damping performance as a function of temperature. This is shown in the plot below:

By transforming the behavior according to both Arrhenius model and the WLF model, a master curve can be developed below:

In the plot above, the WLF model has been used to predict storage modulus and tan delta at a range of frequencies – note that the frequencies outside of the tested range predicted by the WLF model by using the time-temperature equivalence principle.

5.2 Nitrile Butadiene Rubber (NBR)

Nitrile butadiene rubber (NBR) is an elastomeric material with a wide variety of applications. It is used in laboratory and industry personal protective equipment (PPE), where elasticity and toughness are important. It’s also used in the petroleum and chemical industries as a chemically-resistant material for hoses, gaskets, and fuel tanks. It can also be used in a variety of molded and consumer goods, like shoes, floor mats, and synthetic leather.

NBR’s wide operational temperature range, coupled with its good chemical resistance, are what make is so popular as a material of choice for so many applications. A double frequency/temperature sweep can generate a plot of the storage modulus as a function of both frequency and temperature:

This data can be shifted with the WLF law to generate the master curve:

The master curve allows engineers and scientists to predict the behavior of the material at frequencies well outside of the tested range.

6. Conclusion

In summary, DMA and the master curve provide a comprehensive understanding of material behavior. Whether you’re designing polymers, rubbers, or composites, mastering DMA can unlock new possibilities in material science.

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References

[1] Van Gurp, M. and Palmen, J., 1998. Time-temperature superposition for polymeric blends. Rheol. Bull, 67(1), pp.5-8.

[2] Menard, K. P. Dynamic Mechanical Analysis: A Practical Introduction. 1999. CRC Press LLC, Boca Raton, Florida. Ch 7, pages 160-192

[3] Van Gurp, M. and Palmen, J., 1998. Time-temperature superposition for polymeric blends. Rheol. Bull, 67(1), pp.5-8.

[4] Menard, K. P. Dynamic Mechanical Analysis: A Practical Introduction. 1999. CRC Press LLC, Boca Raton, Florida. Ch 6.6 and 7.8, pages 150-154 and 179-184.

[5] Brostow, W. (2000). “Time-stress correspondence in viscoelastic materials: an equation for the stress and temperature shift factor.” Materials Research Innovations, 3(6), 347-351.

[6] Krauklis et al. Polymers 2019, 11(11), 1848;

[7] Menard, K. P. Dynamic Mechanical Analysis: A Practical Introduction. 1999. CRC Press LLC, Boca Raton, Florida. Ch 7.8, pages 179-184. [1] Yu.M. Boiko, Witold Brostow, Anatoly Ya. Goldman, A.C. Ramamurthy, (1995) “Tensile, stress relaxation and dynamic mechanical behaviour of polyethylene crystallized from highly deformed melts,” Polymer, 36(7), 1383-1392,