Tutorial 5 — Time–cure superposition - hermes-rheo

During curing, crosslinking progressively lengthens the relaxation times of a material until it gels. If the shape of the relaxation spectrum is preserved while its characteristic time grows, spectra measured at successive cure times $t_\mathrm{c}$ superpose after a horizontal shift $a_t$ along the frequency axis,

G^{*}(\omega; t_\mathrm{c}) = G^{*}(a_t\,\omega; t_\mathrm{ref}),

(1)

the time–cure superposition (tCS) analogue of time–temperature superposition (Tutorial 4). The evolution of $a_t$ with $t_\mathrm{c}$ tracks cure kinetics, and at the gel point the spectrum becomes a power law, $G' \sim G'' \sim \omega^{n}$ (Chambon and Winter, 1987).

Conventional frequency sweeps are too slow for a sample that changes during the measurement. OWChirp records a full spectrum in a few seconds, so a sequence of chirps gives quasi-static snapshots of the curing material (Perego et al., 2025). The shifts are estimated with the automated method of Lennon et al. (2023), using the cure time as state variable.

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
from hermes_rheo.transforms.owchirp_analysis import OWChirpAnalysis
from hermes_rheo.visualization import axis_label, plot_master_curve, use_science_style

use_science_style()
warnings.filterwarnings("ignore")  # scikit-learn / SciPy convergence notices
DATA = Path("tutorial_data")

Master curve¶

The pipeline matches Tutorial 4 with time as the state variable. Each chirp is assigned the time of its first point (state_mode="first point"), since averaging over a short chirp adds no information. The superposition starts from the last, most cured spectrum (reverse_data=True), whose high modulus gives the best-defined Gaussian-process fit. change_ref then moves the reference to the first chirp, so that $a_t = 1$ at the start of the cure.

experiment = read_file(DATA / "aresg2_trios_owchirp_time_sweep.txt",
                       file_reader=TriosRheoReader, create_composite_datasets=True)

mastercurve = (OWChirpAnalysis() + AutomatedMasterCurve(
    state="time", state_mode="first point",
    x="angular frequency", y="complex modulus",
    vertical_shift=False, reverse_data=True,
))(experiment)
mastercurve.change_ref(mastercurve.states[-1])

fig, axes = plot_master_curve(mastercurve, cmap="viridis", lowest_on_top=False)

The panels show the measured spectra, their Gaussian-process fits and the master curve; the colour bar gives the cure time. The shift factors are stored in mastercurve.datasets[1] (cure time against $a_t$ ), with their uncertainties in mastercurve.details["horizontal_shift_uncertainties"].

Horizontal shift factors¶

shifts = mastercurve.datasets[1]
a_t_err = mastercurve.details["horizontal_shift_uncertainties"]

fig, ax = plt.subplots(figsize=(4.5, 3.4))
ax.errorbar(shifts.x_values, shifts.y_values, yerr=a_t_err, fmt="s", ms=4,
            color="black", capsize=2)
ax.set_yscale("log")
ax.set(xlabel=axis_label(shifts, shifts.x_name, text="cure time"),
       ylabel=axis_label(shifts, shifts.y_name, text="$a_t$"));

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

References¶

Chambon, F. and Winter, H. H., Linear Viscoelasticity at the Gel Point of a Crosslinking PDMS with Imbalanced Stoichiometry, Journal of Rheology 31, 683–697 (1987). DOI: Chambon & Winter (1987).
Lennon, K. R., McKinley, G. H. and Swan, J. W., A data-driven method for automated data superposition with applications in soft matter science, Data-Centric Engineering 4, e13 (2023). DOI: Lennon et al. (2023).
Perego, A., Vadillo, D. C., Mills, M. J. L., Das, M. and McKinley, G. H., Evaluation of Optimally Windowed Chirp Signals in Industrial Rheological Measurements: Method Development and Data Processing, Rheologica Acta 64, 391–406 (2025). DOI: Perego et al. (2025).

References¶

Chambon, F., & Winter, H. H. (1987). Linear Viscoelasticity at the Gel Point of a Crosslinking PDMS with Imbalanced Stoichiometry. Journal of Rheology, 31(8), 683–697. 10.1122/1.549955
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
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