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Tutorial 4 — Time–temperature superposition

Corporate Research Analytical Laboratory, 3M

For thermorheologically simple materials, a change in temperature rescales all relaxation times by the same factor aTa_T. Spectra measured at different temperatures therefore superpose on a single master curve at a reference temperature TrefT_\mathrm{ref} after a horizontal shift along the frequency axis,

G∗(ω;T)=G∗(aT ω;Tref),G^{*}(\omega; T) = G^{*}(a_T\,\omega; T_\mathrm{ref}),
(1)

which extends the accessible frequency window by several decades.

Automated superposition

AutomatedMasterCurve estimates the shifts with the method of Lennon et al. (2023). Each data set is interpolated by a Gaussian process that includes an estimate of interpolation uncertainty, and the shift between neighbouring data sets is the one that maximizes the posterior probability of their overlap. The pairwise shifts are accumulated relative to the reference state, and each aTa_T is returned with its uncertainty. No functional form of aT(T)a_T(T) is assumed.

Shift-factor models

The temperature dependence of aTa_T is described afterwards. Above the glass transition, the Williams–Landel–Ferry (WLF) equation (Williams et al., 1955)

log⁡10aT=−C1 (T−Tref)C2+T−Tref\log_{10} a_T = -\frac{C_1\,(T - T_\mathrm{ref})}{C_2 + T - T_\mathrm{ref}}
(2)

usually applies, while the Arrhenius equation

ln⁡aT=EaR(1T−1Tref),\ln a_T = \frac{E_a}{R} \left(\frac{1}{T} - \frac{1}{T_\mathrm{ref}}\right),
(3)

with TT in kelvin and activation energy EaE_a, suits melts well above TgT_g and thermally activated processes. Both are available in ShiftFactorModel.

The same pressure sensitive adhesive (PSA) was measured with a conventional frequency sweep and with OWChirp (Geri et al., 2018; Perego et al., 2025); OWChirpAnalysis first converts the chirp signals to G′G', G′′G'' and ∣G∗∣|G^{*}| (Tutorial 3).

import warnings
from pathlib import Path

import matplotlib.pyplot as plt

from hermes_rheo import read_file
from hermes_rheo.file_readers.trios_rheo_txt import TriosRheoReader
from hermes_rheo.transforms.automated_mastercurve import AutomatedMasterCurve, CompositeMasterCurve
from hermes_rheo.transforms.owchirp_analysis import OWChirpAnalysis
from hermes_rheo.transforms.shift_factor_models import ShiftFactorModel
from hermes_rheo.visualization import axis_label, plot_master_curve, plot_shift_factors, use_science_style

use_science_style()
warnings.filterwarnings("ignore")  # scikit-learn / SciPy convergence notices
DATA = Path("tutorial_data")
T_REF = 20  # reference temperature, °C

Master curves

Both data sets are superposed on the storage modulus G′(ω)G'(\omega). state_mode="average" assigns each frequency sweep or chirp the mean of its recorded temperature, which absorbs small drifts during the measurement. change_ref moves the reference to the state closest to TrefT_\mathrm{ref} and rescales all shift factors so that aT(Tref)=1a_T(T_\mathrm{ref}) = 1.

settings = dict(state="temperature", state_mode="average",
                x="angular frequency", y="storage modulus")

sweep = read_file(DATA / "aresg2_trios_frequency_sweep_tts.txt",
                  file_reader=TriosRheoReader, create_composite_datasets=True)
chirp = OWChirpAnalysis()(read_file(DATA / "aresg2_trios_owchirp_tts.txt",
                                    file_reader=TriosRheoReader, create_composite_datasets=True))

mc_sweep = AutomatedMasterCurve(**settings)(sweep)
mc_chirp = AutomatedMasterCurve(**settings)(chirp)
for mc in (mc_sweep, mc_chirp):
    mc.change_ref(min(mc.states, key=lambda T: abs(T - T_REF)))
fig, axes = plot_master_curve(mc_sweep)
<Figure size 1040x300 with 4 Axes>

plot_master_curve shows the frequency-sweep data before shifting, the Gaussian-process interpolants used to find the shifts (mean ± one standard deviation), and the master curve against the reduced frequency aT ωa_T\,\omega. The colour bar gives the temperature range. Twenty isotherms between -40 and 150 °C, each covering little more than two decades, combine into a master curve spanning about fourteen decades, from the glassy plateau at high aT ωa_T\,\omega to the low-modulus region at low aT ωa_T\,\omega.

Shift factors

ShiftFactorModel(model="WLF") fits C1C_1 and C2C_2 to the shift factors of each master curve; use model="Arrhenius" for the activation-energy form. Agreement of the two fits confirms that OWChirp recovers the same temperature dependence as the frequency sweep.

wlf_sweep = ShiftFactorModel(model="WLF")(mc_sweep)
wlf_chirp = ShiftFactorModel(model="WLF")(mc_chirp)
wlf_sweep.print_parameters()
wlf_chirp.print_parameters()

fig, ax = plot_shift_factors(wlf_sweep, data_label="frequency sweep", figsize=(4.5, 3.4),
                             color="tab:red", data_color="tab:red")
plot_shift_factors(wlf_chirp, ax=ax, data_label="OWChirp", linestyle="--",
                   color="tab:blue", data_color="tab:blue");
WLF: C1 = 9.203, C2 = 128.8, Tref = 20.01
WLF: C1 = 8.625, C2 = 123.6, Tref = 19.99
<Figure size 450x340 with 1 Axes>

Frequency sweep and OWChirp

The two master curves are compared directly. The units are read from the data: TRIOS exports the frequency-sweep moduli in MPa, while OWChirpAnalysis returns Pa. piblin’s change_dependent_variable_data_unit converts the frequency-sweep curve before plotting.

TO_PA = {"Pa": 1.0, "kPa": 1e3, "MPa": 1e6} # Conversion factors to Pascal (Pa)

fig, ax = plt.subplots(figsize=(4.5, 3.4))
for mc, color, label in ((mc_sweep, "tab:red", "frequency sweep"), (mc_chirp, "tab:blue", "OWChirp")):
    curve = mc.datasets[0]
    curve.change_dependent_variable_data_unit(TO_PA[curve.y_unit], "Pa")
    ax.loglog(curve.x_values, curve.y_values, "o", ms=3, color=color, label=label)
ax.set(xlabel=axis_label(curve, curve.x_name, text=r"$a_T\,\omega$"),
       ylabel=axis_label(curve, curve.y_name, text="$G'$"))
ax.legend(loc="upper left", fontsize=12);
<Figure size 450x340 with 1 Axes>

G′G', G′′G'' and tan⁡δ\tan\delta from the frequency sweep

CompositeMasterCurve builds one master curve per quantity, each with its own Gaussian-process shift. Since tan⁡δ=G′′/G′\tan\delta = G''/G' is independent of any vertical (modulus) shift, its master curve is a useful check of thermorheological simplicity.

composite = CompositeMasterCurve(n_curves=3, y_quantities=["storage modulus", "loss modulus", "tan(delta)"],
                                 state="temperature", x="angular frequency")(sweep)
for mc in composite.measurements:
    mc.change_ref(min(mc.states, key=lambda T: abs(T - T_REF)))
gp, gpp, tan_delta = (mc.datasets[0] for mc in composite.measurements)

fig, ax = plt.subplots(figsize=(5, 3.6))
ax_tan = ax.twinx()
ax.loglog(gp.x_values, gp.y_values, "o", ms=3, color="tab:blue", label="$G'$")
ax.loglog(gpp.x_values, gpp.y_values, "o", ms=3, color="tab:green", label="$G''$")
ax_tan.semilogx(tan_delta.x_values, tan_delta.y_values, "o", ms=3, color="tab:red", label=r"$\tan\delta$")

ax.set(xlabel=axis_label(gp, gp.x_name, text=r"$a_T\,\omega$"),
       ylabel=axis_label(gp, gp.y_name, text="$G'$, $G''$"))
ax.tick_params(which="both", right=False)
ax_tan.set_ylabel(r"$\tan\delta$", color="tab:red")
ax_tan.tick_params(axis="y", which="both", colors="tab:red")
handles = ax.get_legend_handles_labels()[0] + ax_tan.get_legend_handles_labels()[0]
ax.legend(handles=handles, loc="upper left");
<Figure size 500x360 with 2 Axes>

The OWChirp data were decimated by a factor of 4 (sampling rate from 500 pt/s to 125 pt/s, every 4th point kept) to reduce file size; the full-resolution files are available on request from Alessandro Perego (aperego@mmm.com).

References

References
  1. Lennon, K. R., McKinley, G. H., & Swan, J. W. (2023). A data-driven method for automated data superposition with applications in soft matter science. Data-Centric Engineering, 4. 10.1017/dce.2023.3
  2. Geri, M., Keshavarz, B., Divoux, T., Clasen, C., Curtis, D. J., & McKinley, G. H. (2018). Time-Resolved Mechanical Spectroscopy of Soft Materials via Optimally Windowed Chirps. Physical Review X, 8(4). 10.1103/physrevx.8.041042
  3. Perego, A., Vadillo, D. C., Mills, M. J. L., Das, M., & McKinley FRS, G. H. (2025). Evaluation of optimally windowed chirp signals in industrial rheological measurements: method development and data processing. Rheologica Acta, 64(8), 391–406. 10.1007/s00397-025-01511-0
  4. Williams, M. L., Landel, R. F., & Ferry, J. D. (1955). The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-forming Liquids. Journal of the American Chemical Society, 77(14), 3701–3707. 10.1021/ja01619a008
hermes-rheo — Tutorials
Tutorial 3 — Analysing OWChirp experiments
hermes-rheo — Tutorials
Tutorial 5 — Time–cure superposition