Accelerating TTS Experiments Using Frequency Chirps

Frequency chirps generate moduli and phase angles from a broad range of frequencies in a short period of time. Recent work has minimized spectral leakage in strain [1] and stress-based [2] chirps, enabling fast frequency data collection on both separate motor/transducer and combined motor/transducer instruments such as the ARES™ System platform and HR series instruments. This note will describe using chirps to accelerate Time-Temperature Superposition (TTS) data on the HR series instruments using Polystyrene and Polycarbonate as examples.

Time-temperature superposition is used to determine the mechanical properties of polymers using the relationship between the relaxation of polymers, their mechanical properties, and the temperature of the measurement. Frequency chirps can be used to accelerate data acquisition, shortening measurements from hours to less than an hour in ideal cases and otherwise offering significant time savings in addition to higher data density.

A traditional TTS experiment uses frequency sweeps done isothermally and increments temperature over a range of interest. The measurements are then shifted by a shift factor αT that multiplies the frequency axis. Some measurements benefit from shifting the moduli (y axis) to account for density changes. A significant overlap of the data is required to successfully determine the shift factor. Once collected, these shift factors can be fit to models such as the WLF and Arrhenius, depending on the temperature range.

A frequency chirp experiment can be used to complete an isothermal frequency sweep in a fraction of the time while maintaining the same isothermal conditions. The time savings for a chirp relative to a frequency sweep can be realized in three ways: less time for the same frequency range, same time for a wider frequency range and less temperature steps, or ramping temperature while chirping to save time on equilibration. These methods can be optimized for the size of the shift factors, e.g. a chirp ramp from the glassy state to rubbery plateau is more efficient than long chirps while long chirps in wide temperature steps are more efficient in the rubbery plateau into the molten state of a polymer.

These strategies of acceleration for TTS experiments with chirps can also be mixed, utilizing a short chirp for ramps in the glass transition region and then long chirps where the shift factors are lower.

A TA Instruments™ Discovery™ HR 30 Rheometer was used for all experiments. Temperature ramps and steps from glass transition to the molten region were done with an 8 mm plate for polystyrene (PS) and polycarbonate (PC) samples.

Frequency chirp ramps and temperature steps can be done with the same step in TRIOS™ Software based on the mode selection. An example of the ramp is shown in Figure 1. Axial force and auto-strain can be set in the conditioning options step. A target axial force of 0 N, minimum torque of 10 µNm, and maximum strain of 10% was set for these experiments.

A torque or stress-based chirp is used because the experiment starts in the stiff glassy region where stress/torque chirps typically offer better performance. An initial amplitude of 1000 µNm was used. A zero-torque equilibration is performed between each chirp.

This equilibration gives time for the material to relax between chirps and adjust the gap to maintain 0 axial force. These parameters can be adjusted in the equilibration drop-down of the step.

The Frequency Chirp ramp can be followed by a much longer isothermal chirp (Figure 2) by selecting the Single radial button for a single chirp or the Step radial button for a set of chirps at pre-determined temperature increments.

The shift factors for TTS between the glass and molten states of a polymer typically follow a WLF form and are much higher in the glass transition region and rubbery plateau than near the melt region. The WLF fit equation used is below [3]:

where:
α_T is the shift factor
C₁ and C₂ are fitting constants
T is temperature and
T_ref is the reference temperature

Using a short chirp while ramping is more effective at lower temperatures in the glass transition where the shift factor is large and changing quickly. Large shift factors mean that data needs to be collected close together in temperature. A stepped experiment would require many small steps with equilibration for each while a ramped chirp experiment collects data quickly and frequently. A longer chirp or set of longer chirps with much higher temperature spacing is more efficient than ramping when approaching the melt region because the shift is much smaller.

The method described in Figure 1 produces a little less than one frequency chirp every 1 °C when accounting for the chirp duration, baseline, and equilibration between chirps. This chirp produces frequency content from 0.1 to 10 Hz. A select few chirps are shown in Figure 3 for each of the key regions of a thermoplastic: near the glass state (100 °C), the transitions state (110 °C, 125 °C), the rubbery plateau (150-170 °C), and the final longer chirp for the molten region (180 °C).

All of the chirps in the experiment, including the select examples in Figure 3, can be shifted automatically in TRIOS Software to produce the curve in Figure 4 along with a similar experiment performed using discrete frequency sweeps (DFS) for comparison. The frequency sweeps take much longer than a chirp and require several more temperature steps above 180 to acquire data to shift into the same frequency range as the chirp data. The high temperatures and long times result in degradation of the polymer as seen where the frequency sweep data deviates from the chirp at the lowest frequencies.

Until the degradation at higher temperatures, the DFS and chirp data are in very good agreement. The overall experimental duration for the chirp is ~3,500 s (< 1h) where the frequency sweep data in this setup take ~13,700 s or nearly four hours. These discrete frequency sweeps were collected using isothermal frequency sweeps with 5-minute temperature soaks at 5 °C increments between 100 and 180 °C, then using 20 °C increments up to 260 °C. The chirp method also provides significantly more data points to produce reliable shifts. The chirp method near the extremes of stiffness (very low and very high) can be noisier as chirps don’t utilize multiple isothermal cycles as a discrete frequency sweep does, but data between the extremes is nearly identical.