.. _transform-owchirp: OWChirp ======= Overview -------- :class:`rheojax.transforms.OWChirp` designs, executes, and analyzes orthogonal windowed chirp experiments for broadband LAOS. A single chirp sweeps logarithmically from ``chirp_span[0]`` to ``chirp_span[1]`` Hz, enabling simultaneous extraction of linear and nonlinear moduli (:math:`G_1'`, :math:`G_1''`, :math:`G_3'`, etc.). Equations --------- OWChirp supports linear and logarithmic sweeps with instantaneous frequency .. math:: f(t) = f_{\mathrm{start}} \exp\left( \frac{\ln(f_{\mathrm{end}} / f_{\mathrm{start}})}{T} t \right) and phase :math:`\phi(t) = 2\pi \int_0^t f(\tau)\,d\tau`. Orthogonal window segments (Planck or Tukey tapers) are applied to sub-bands so harmonics remain separable even when multiple chirps are concatenated. Complete OWCh Waveform Equation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The full OWCh waveform applies a Tukey (cosine-tapered) window to an exponential chirp, incorporating an optional waiting time :math:`t_w` at the signal start. The complete piecewise definition is (Perego et al. 2025, Eq. 5): .. math:: x(t) = x_0 \begin{cases} \cos^2\left[\frac{\pi}{r}\left(\frac{t - t_w}{T_{\mathrm{owc}}} - \frac{r}{2}\right)\right] \sin\left\{\frac{\omega_1 T_{\mathrm{owc}}}{\ln(\omega_2/\omega_1)} \left[\exp\left(\frac{\ln(\omega_2/\omega_1)}{T_{\mathrm{owc}}}(t - t_w)\right) - 1\right]\right\}, & \frac{t - t_w}{T_{\mathrm{owc}}} \le \frac{r}{2} \\[1em] \sin\left\{\frac{\omega_1 T_{\mathrm{owc}}}{\ln(\omega_2/\omega_1)} \left[\exp\left(\frac{\ln(\omega_2/\omega_1)}{T_{\mathrm{owc}}}(t - t_w)\right) - 1\right]\right\}, & \frac{r}{2} \le \frac{t - t_w}{T_{\mathrm{owc}}} \le 1 - \frac{r}{2} \\[1em] \cos^2\left[\frac{\pi}{r}\left(\frac{t - t_w}{T_{\mathrm{owc}}} - 1 + \frac{r}{2}\right)\right] \sin\left\{\frac{\omega_1 T_{\mathrm{owc}}}{\ln(\omega_2/\omega_1)} \left[\exp\left(\frac{\ln(\omega_2/\omega_1)}{T_{\mathrm{owc}}}(t - t_w)\right) - 1\right]\right\}, & \frac{t - t_w}{T_{\mathrm{owc}}} \ge 1 - \frac{r}{2} \end{cases} where: - :math:`x_0` is the stress or strain amplitude - :math:`\omega_1, \omega_2` are the lower and upper angular frequency bounds (rad/s) - :math:`T_{\mathrm{owc}}` is the total chirp duration - :math:`t_w` is the waiting time before the chirp begins - :math:`r` is the Tukey window tapering coefficient (cosine fraction, typically 0.05–0.15) **Tapering coefficient selection:** - **r = 0.05:** Minimal tapering, maximizes signal energy, suitable for high-SNR systems - **r = 0.10 (recommended):** Balanced trade-off between spectral leakage and signal duration - **r = 0.15:** Aggressive tapering, superior sidelobe suppression for precision measurements Duration Selection Guidelines ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The theoretical minimum chirp duration is set by the lowest frequency: .. math:: T_{\mathrm{owc}} \ge \frac{2\pi}{\omega_1} For improved throughput, Perego et al. (2025) adopt a practical **2/3 scaling factor**: .. math:: T_{\mathrm{owc}} \ge \frac{2}{3} \cdot \frac{2\pi}{\omega_1} This reduces acquisition time by approximately 33% while only modestly shifting the lowest reliably probed frequency (e.g., from 0.3 to ~0.45 rad/s). The trade-off is acceptable for most industrial applications where throughput is prioritized. **Example:** For :math:`\omega_1 = 0.3` rad/s: - Theoretical minimum: :math:`T_{\mathrm{owc}} \ge 21` s - Practical (2/3 scaling): :math:`T_{\mathrm{owc}} \ge 14` s Waveform Design Details ----------------------- Planck Taper Window ~~~~~~~~~~~~~~~~~~~ The Planck taper provides smooth onset and offset transitions that minimize spectral leakage while maintaining signal energy. For a signal of duration :math:`T` with taper fraction :math:`\varepsilon`, the window function is: .. math:: w(t) = \begin{cases} \left[1 + \exp\left(\frac{\varepsilon T}{t} + \frac{\varepsilon T}{t - \varepsilon T}\right)\right]^{-1} & 0 < t < \varepsilon T \\ 1 & \varepsilon T \le t \le (1-\varepsilon)T \\ \left[1 + \exp\left(\frac{\varepsilon T}{T-t} + \frac{\varepsilon T}{T-t - \varepsilon T}\right)\right]^{-1} & (1-\varepsilon)T < t < T \end{cases} **Taper parameter selection:** - :math:`\varepsilon` **= 0.10:** Minimal tapering, maximizes flat-top duration, risk of spectral leakage - :math:`\varepsilon` **= 0.15 (default):** Balanced trade-off, recommended for most applications - :math:`\varepsilon` **= 0.20:** Aggressive tapering, reduced leakage but shorter effective duration **Comparison with Tukey taper:** The Tukey (cosine-tapered) window uses a raised cosine transition, which is computationally simpler but produces slightly more spectral leakage than the Planck taper. The Planck taper's exponential rolloff provides superior sidelobe suppression (−60 dB vs −40 dB for Tukey). Mutation Number Validation ~~~~~~~~~~~~~~~~~~~~~~~~~~ Before analyzing chirp data, verify that the material remains quasi-steady during the sweep. The **mutation number** :math:`\delta` quantifies structural evolution during the measurement: .. math:: \delta = \frac{1}{\pi} \int_0^T \left| \frac{d \ln G'(t)}{dt} \right| dt **Quasi-steady criterion:** :math:`\delta < 0.2` indicates <20% structural change during the chirp, validating the assumption of time-invariant material properties. For rapidly mutating materials (gels, curing systems), use shorter chirps or the Time-Curing Superposition approach described in :ref:`owchirp-advanced-applications`. See :doc:`mutation_number` for detailed mutation number theory and calculation. Time-Frequency Trade-Offs ------------------------- The time-bandwidth product :math:`TB = T (f_{\mathrm{end}} - f_{\mathrm{start}})` governs spectral resolution. Values above ~50 yield <2% amplitude error, whereas shorter chirps ( low TB) cover the spectrum faster but broaden frequency bins. OWChirp reports ``tb_product`` and warns when resolution is compromised. **Recommended TB products:** .. list-table:: Time-bandwidth product guidelines :header-rows: 1 :widths: 25 25 50 * - TB Product - Regime - Guidance * - :math:`TB < 30` - Low resolution - Significant spectral broadening (>10% error). Use only for rapid screening. * - :math:`30 \le TB < 66` - Moderate resolution - Acceptable for exploratory measurements (2-10% error). * - :math:`TB \approx 66` - Empirically optimal - Best trade-off between speed and accuracy for most materials. * - :math:`TB \ge 100` - High resolution - Near-discrete frequency accuracy (<2% error). Use for publication-quality data. **Empirical guideline:** A TB product of approximately 66 has been found empirically optimal for balancing measurement speed with spectral resolution in most rheological applications. Harmonic Extraction ------------------- Given recorded stress :math:`\sigma(t)` and measured strain :math:`\gamma(t)`, the transform projects onto harmonic basis functions tied to :math:`\phi(t)`: .. math:: G_n'(\phi) = \frac{2}{T} \int_0^T \sigma(t) \cos(n\phi(t))\,dt, \qquad G_n''(\phi) = \frac{2}{T} \int_0^T \sigma(t) \sin(n\phi(t))\,dt. The resulting moduli are reported versus instantaneous frequency. Nonlinear intensity ratios (:math:`I_{3/1}`) are computed automatically. Deconvolution and Windowing --------------------------- Because chirps excite a continuum of frequencies, OWChirp performs Wiener deconvolution in the joint time-frequency domain: .. math:: H^{-1}(\omega) = \frac{H^*(\omega)}{|H(\omega)|^2 + \lambda}, where :math:`H` is the chirp kernel and :math:`\lambda` is a regularization parameter estimated from the noise floor. Planck or Tukey tapers applied at the start/end of the chirp limit spectral leakage. Parameters ---------- .. list-table:: OWChirp parameters :header-rows: 1 :widths: 25 18 39 18 * - Parameter - Type - Description - Default * - ``chirp_span`` - tuple(float, float) - Frequency range (Hz) for the sweep. - ``(0.1, 30.0)`` * - ``amplitude`` - float - Target strain or stress amplitude. - ``0.05`` * - ``duration`` - float - Chirp length (s); influences time-bandwidth product. - ``30.0`` * - ``taper`` - str - Edge window (``"planck(0.15)"``, ``"tukey(0.2)"`` ...). - ``"planck(0.15)"`` * - ``n_harmonics`` - int - Number of harmonics to extract (odd orders). - ``5`` Industrial Implementation ------------------------- Instrument Integration ~~~~~~~~~~~~~~~~~~~~~~ OWCh experiments can be implemented on commercial rheometers via the **Arbitrary Waveform** method. On TA Instruments ARES-G2 rheometers using TRIOS software, users supply up to four equations describing the OWCh waveform. Key constraints include: - **Memory limit:** :math:`N_{\mathrm{max}} = 2^{15}` points on ARES-G2 - **Sampling rate:** Must satisfy both Nyquist and memory constraints (see Eq. 8) - **Amplitude control:** Manual regulation required; trial-and-error or interpolation for multi-temperature experiments **Recommended workflow:** 1. Define frequency bounds (:math:`\omega_1`, :math:`\omega_2`) and duration (:math:`T_{\mathrm{owc}}`) 2. Calculate valid sampling range using the normalized Nyquist criterion 3. Verify :math:`TB \ge 66` for adequate spectral resolution 4. For temperature sweeps, conduct preliminary OWCh trials at thermal extremes to establish amplitude bounds; interpolate for intermediate temperatures Data Processing with hermes-rheo ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The ``hermes-rheo`` Python package (Perego et al. 2024) provides automated OWCh analysis integrated with the ``piblin`` data pipeline framework. Both packages are MIT-licensed and available on PyPI. **Installation:** .. code-block:: bash pip install piblin hermes-rheo **Key features:** - **OWChGeneration transform:** Automatically generates all waveform parameters for TRIOS Arbitrary Waveform method - **Automated bias correction:** Three filtering methods applied systematically: 1. Subtract average from waiting-time interval only 2. Subtract average from OWCh segment only 3. Subtract average from entire signal duration The algorithm selects the method producing signals symmetric around zero with minimal endpoint deviations. - **Cloud integration:** Modular design enables incorporation into automated, cloud-based data analysis pipelines **Example usage:** .. code-block:: python from hermes_rheo.transforms import OWChGeneration, OWChAnalysis # Generate waveform parameters for TRIOS gen = OWChGeneration( omega1=0.3, # rad/s omega2=60.0, # rad/s towc=14.0, # seconds tw=1.0, # waiting time r=0.1, # tapering coefficient ) waveform_params = gen.generate() # Analyze recorded data analysis = OWChAnalysis(fs=1000.0) # sampling frequency result = analysis.transform(strain_data, stress_data) For detailed tutorials, see the `hermes-rheo documentation `_ and `tutorial notebooks `_. Sampling and Constraints ------------------------ Nyquist Number Analysis ~~~~~~~~~~~~~~~~~~~~~~~ The **Nyquist number** :math:`N_y` quantifies sampling adequacy relative to the highest frequency component: .. math:: N_y = \frac{f_{\mathrm{max}}}{f_{\mathrm{Nyquist}}} = \frac{2 f_{\mathrm{max}}}{f_s} where :math:`f_s` is the sampling rate. For alias-free reconstruction: .. math:: N_y < 1 \quad \Rightarrow \quad f_s > 2 f_{\mathrm{max}} Normalized Nyquist Number ^^^^^^^^^^^^^^^^^^^^^^^^^ For OWCh waveform design, the **normalized Nyquist number** captures the interplay between chirp duration, waiting time, upper frequency, and available sampling points (Perego et al. 2025, Eq. 6): .. math:: N_y = \frac{(T_{\mathrm{owc}} + t_w) \omega_2}{2\pi N} where :math:`N` is the total number of discrete sampling points. The Nyquist–Shannon sampling theorem requires :math:`N_y \le 0.5`. However, in practice, OWCh signals sampled at this rate may exhibit aliasing artifacts, particularly when :math:`\omega_2 \ge 60` rad/s. Hudson-Kershaw et al. (2024) recommend a more stringent constraint: .. math:: N_y < 0.1 \quad \text{(practical guideline)} This translates to a **valid sampling range** (Perego et al. 2025, Eq. 8): .. math:: \frac{10\omega_2}{2\pi} \le f_s \le \frac{N_{\mathrm{max}}}{T_{\mathrm{owc}} + t_w} where :math:`N_{\mathrm{max}}` is the maximum number of recordable points allowed by the rheometer's memory. **Worked example:** For :math:`\omega_1 = 0.3` rad/s, :math:`\omega_2 = 125` rad/s, :math:`T_{\mathrm{owc}} = 14` s, :math:`t_w = 1` s, and :math:`N_{\mathrm{max}} = 2^{15}` (TA ARES-G2): .. math:: 200\ \text{pts/s} \le f_s \le 2184\ \text{pts/s} **Modified sampling condition for harmonics:** When extracting the :math:`n`-th harmonic, the effective maximum frequency is :math:`n \times f_{\mathrm{end}}`. The sampling rate must satisfy: .. math:: f_s > 2 n_{\mathrm{max}} f_{\mathrm{end}} For example, extracting up to the 5th harmonic from a chirp ending at 30 Hz requires :math:`f_s > 300` Hz (typically use 500 Hz with safety margin). **Practical sampling guidelines:** .. list-table:: Sampling rate recommendations :header-rows: 1 :widths: 30 30 40 * - Chirp Range - Max Harmonic - Minimum :math:`f_s` * - 0.1–10 Hz - 5th - 100 Hz (use 200 Hz) * - 0.1–30 Hz - 5th - 300 Hz (use 500 Hz) * - 0.1–100 Hz - 7th - 1400 Hz (use 2000 Hz) Data Density Comparison with DFS ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ OWCh provides significantly higher spectral resolution than discrete frequency sweeps (DFS). The number of frequency-domain data points from an OWCh signal is (Perego et al. 2025, Eq. 11): .. math:: N_f = \frac{2^n (\omega_2 - \omega_1)}{2\pi f_s} where :math:`2^n` is the number of time-domain points (rounded to the nearest power of two for efficient FFT processing). For comparison, the total duration of a DFS is (Perego et al. 2025, Eq. 3): .. math:: T_{\mathrm{DFS}} \ge \sum_{i=1}^{m} \frac{2\pi}{\omega_i} where :math:`m` is the number of tested frequencies. .. list-table:: Data density comparison: OWCh vs DFS :header-rows: 1 :widths: 25 25 25 25 * - Method - Frequency Range - Duration - Data Points * - OWCh - 0.3–30 rad/s - 15 s - ~78 points * - DFS (7 pts/decade) - 0.3–30 rad/s - ~180 s - ~14 points **Key result:** OWCh delivers an **order-of-magnitude increase** in spectral resolution while simultaneously reducing overall testing duration by 10× or more. Wait Time Between Chirps ~~~~~~~~~~~~~~~~~~~~~~~~ When concatenating multiple chirps or performing repeated measurements, allow sufficient **wait time** :math:`t_w` for the material to recover from previous deformation: .. math:: t_w \ge 5 \tau_{\mathrm{relax}} where :math:`\tau_{\mathrm{relax}}` is the material's characteristic relaxation time. For unknown materials, estimate :math:`\tau_{\mathrm{relax}}` from the inverse of the crossover frequency (:math:`\omega` where :math:`G' = G''`). **Wait time guidelines:** - **Purely elastic materials:** :math:`t_w \approx 1` s (fast recovery) - **Viscoelastic fluids:** :math:`t_w \approx 5/\omega_{\mathrm{crossover}}` - **Yield stress materials:** :math:`t_w \approx 30` s or longer (thixotropic recovery) Input / Output Specifications ----------------------------- - **Design input**: sampling rate ``fs`` (Hz), control mode (strain or stress), optional actuator limits. ``OWChirp.design`` returns waveform samples, instantaneous frequency, and metadata for instrument playback. - **Analysis input**: recorded strain ``gamma(t)`` (dimensionless) and stress ``sigma(t)`` (Pa) as :class:`RheoData` objects or arrays with timestamps. - **Outputs**: dict with - ``waveform`` (for design), - ``moduli`` mapping harmonic order to arrays of :math:`G_n'`, :math:`G_n''`, - ``frequency`` grid (Hz) per harmonic, - ``diagnostics`` (time-bandwidth product, crest factor, leakage, Wiener :math:`\lambda`). Usage ----- .. code-block:: python from rheojax.transforms import OWChirp ow = OWChirp(chirp_span=(0.2, 40.0), amplitude=0.1, duration=25.0, taper="tukey(0.2)") plan = ow.design(mode="strain", fs=500.0) # Send plan["waveform"] to the rheometer, then record response traces... result = ow.transform(response_gamma, response_sigma, fs=500.0) G1 = result["moduli"][1] G3 = result["moduli"][3] I31 = G3["G_double_prime"] / G1["G_double_prime"] .. _owchirp-advanced-applications: Advanced Applications --------------------- Time-Curing Superposition (tCS) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ For **mutating materials** (curing polymers, aging gels, crystallizing melts), standard chirp analysis assumes quasi-steady behavior. When the mutation number :math:`Mu > 0.15`, this assumption breaks down. **Time-Curing Superposition (tCS)** extends chirp rheometry to evolving materials by: 1. **Segmenting** the chirp into short sub-windows where :math:`Mu_{\mathrm{local}} < 0.15` 2. **Shifting** each segment's moduli using a time-dependent shift factor :math:`a_t(t_{\mathrm{cure}})` 3. **Constructing** a master curve that tracks the evolving material state **When to use tCS:** - Monitoring gelation kinetics (epoxy, alginate, fibrin) - Tracking crystallization (polymers, fats) - Characterizing aging in colloidal glasses - UV-curable acrylate crosslinking Mutation Number from Consecutive Chirps ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For consecutive OWCh measurements on an evolving material, the mutation number can be computed by tracking changes in a viscoelastic metric :math:`g` (typically :math:`|G^*|`) across chirps (Perego et al. 2025, Eq. 15): .. math:: Mu(t_i, \omega_j) \approx \frac{T_{\mathrm{owc}} \ln\left(\frac{g(t_i, \omega_j)}{g(t_{i-1}, \omega_{j-1})}\right)}{t_i - t_{i-1}} where :math:`i` indexes the chirp number and :math:`j` indexes the frequency. The frequency-averaged mutation number :math:`\overline{Mu}` provides a single metric for each chirp. **Critical threshold:** :math:`Mu_{\mathrm{crit}} = 0.15` (Rathinaraj et al. 2022) When :math:`\overline{Mu} > Mu_{\mathrm{crit}}`, measurement artifacts appear, such as artificial upticks in the complex viscosity master curve. Frequency Selection for Rapid Mutation ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ To keep the mutation number below :math:`Mu_{\mathrm{crit}}`, the minimum angular frequency :math:`\omega_1` must be increased to shorten the OWCh duration. An approximate guideline is (Perego et al. 2025, Eq. 16): .. math:: \omega_1 > \frac{4\pi}{3 \, Mu_{\mathrm{crit}} \, \tau_{\mathrm{max}}} where :math:`\tau_{\mathrm{max}}` is the characteristic time at :math:`\max(\overline{Mu})`. **Trade-off:** Increasing :math:`\omega_1` reduces the time-bandwidth product :math:`TB`, potentially below the optimal :math:`TB \ge 66`. Priority should be given to satisfying :math:`Mu < 0.15` over maintaining high :math:`TB`. **Example:** For a UV-curable acrylate with rapid early-stage kinetics: - Original: :math:`\omega_1 = 0.6` rad/s, :math:`T_{\mathrm{owc}} = 7` s → :math:`TB = 66` - Adjusted: :math:`\omega_1 = 1.5` rad/s, :math:`T_{\mathrm{owc}} = 2.8` s → :math:`TB = 26` The adjusted parameters eliminate the artifact while sacrificing some spectral resolution. Complex Viscosity Master Curves ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For tCS analysis, the reduced complex viscosity provides diagnostic insight: .. math:: |\eta^*_r| = \frac{|G^*_r|}{\omega_r} = \frac{\sqrt{(G'_r)^2 + (G''_r)^2}}{\omega_r} An **artificial uptick** in :math:`|\eta^*_r|` at intermediate reduced frequencies signals that the material transformation rate exceeded the measurement timescale during that phase of curing. This artifact may be masked in :math:`|G^*|` plots due to the linear frequency dependence, but becomes apparent when scaled by :math:`1/\omega_r`. **Diagnostic approach:** 1. Compute :math:`\overline{Mu}` for each chirp 2. Identify time intervals where :math:`\overline{Mu} > Mu_{\mathrm{crit}}` 3. If artifacts present, repeat experiment with higher :math:`\omega_1` to shorten :math:`T_{\mathrm{owc}}` **Practical workflow:** .. code-block:: python from rheojax.transforms import OWChirp, MutationNumber # Check mutation number during cure mn = MutationNumber(tau_c=10.0) delta = mn.calculate(rheo_data_during_cure) if delta > 0.15: # Use short chirps and apply tCS ow_short = OWChirp(duration=5.0) # Short segments # Process each segment independently... else: # Standard analysis ow = OWChirp(duration=30.0) # Monitor mutation number across consecutive chirps mu_values = [] for i in range(1, len(chirp_results)): g_current = chirp_results[i]['G_star_magnitude'] g_previous = chirp_results[i-1]['G_star_magnitude'] dt = chirp_results[i]['time'] - chirp_results[i-1]['time'] mu = (T_owc * np.log(g_current / g_previous)) / dt mu_values.append(np.mean(mu)) # Frequency-averaged # Check for violations if any(mu > 0.15 for mu in mu_values): print("Warning: Mu > 0.15 detected; consider increasing omega1") Accelerated Time-Temperature Superposition ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Chirp rheometry can **accelerate mastercurve construction** by up to 40% compared to traditional discrete-frequency sweeps. Benefits include: **Efficiency gains:** - **Single experiment** covers 3-4 decades of frequency vs multiple isothermal sweeps - **40% reduction** in total measurement time for equivalent frequency coverage - **Continuous data** provides higher frequency density than discrete points **Quantified benefits (Perego et al. 2025):** .. list-table:: OWCh vs DFS for tTS master curves :header-rows: 1 :widths: 35 30 35 * - Metric - DFS - OWCh * - Data points in master curve - ~290 - ~1300 * - Total acquisition time - 90 min - 53 min * - Data density improvement - baseline - **4.5× higher** * - Time reduction - baseline - **40% faster** **Data density advantages:** Traditional frequency sweeps sample 5-10 points per decade. Chirp measurements provide effectively continuous coverage, improving: - Fit quality for multi-mode Maxwell/Prony series - Detection of subtle relaxation features (shoulder peaks, plateaus) - Identification of time-temperature superposition failures Williams-Landel-Ferry (WLF) Equation ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The horizontal shift factors :math:`a_T` are typically described by the WLF equation (Ferry 1980): .. math:: \log_{10}(a_T) = -\frac{C_1 (T - T_r)}{C_2 + (T - T_r)} where :math:`T_r` is the reference temperature and :math:`C_1`, :math:`C_2` are material-specific empirical constants. **Typical WLF parameters for pressure-sensitive adhesives:** .. list-table:: :header-rows: 1 :widths: 40 30 30 * - Method - :math:`C_1` - :math:`C_2` (°C) * - DFS with GPR - 9.19 - 128 * - OWCh with GPR - 8.63 - 124 The close agreement between OWCh and DFS shift factors validates OWCh as an effective alternative for tTS protocols. Automated Superposition with Gaussian Process Regression ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The ``hermes-rheo`` package implements automated master curve construction using Gaussian process regression (GPR) with maximum *a posteriori* estimation, following the methodology of Lennon et al. (2023). This data-driven approach: - Automatically determines optimal shift factors - Provides uncertainty bounds on the master curve - Is robust to elevated noise levels (e.g., near compliance limits) **Integration with mastercurve transform:** .. code-block:: python from rheojax.transforms import OWChirp, Mastercurve # Collect chirp data at multiple temperatures chirp_results = {} for T in [20, 40, 60, 80, 100]: # °C ow = OWChirp(chirp_span=(0.01, 100), duration=30.0) chirp_results[T] = ow.transform(data_at_T, fs=500.0) # Construct mastercurve with automatic shift factors mc = Mastercurve(reference_temp=60, auto_shift=True) master, shifts = mc.transform(chirp_results) Temperature Calibration Best Practices ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ For multi-temperature OWCh experiments: 1. **Amplitude selection:** Conduct preliminary OWCh trials at thermal extremes (e.g., :math:`-40\,^\circ\text{C}` and :math:`150\,^\circ\text{C}`) to establish amplitude bounds 2. **Interpolation:** For intermediate temperatures, interpolate amplitude values; refine near major transitions (e.g., :math:`T_g`) 3. **Equilibration:** Allow 3 min equilibration at each temperature before measurement 4. **Transducer range:** Adjust as needed for complex rheological profiles Instrument Compliance Considerations ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ At low temperatures and high frequencies, material stiffness may approach instrument compliance limits. Perego et al. (2025) recommend working in **compliance space**: .. math:: J' = \frac{G'}{G'^2 + G''^2}, \quad J'' = \frac{G''}{G'^2 + G''^2} The instrument compliance threshold is (Perego et al. 2025, Eq. 13): .. math:: J'_{\mathrm{inst}}(\omega) = \frac{1}{K_\theta(\omega)} \cdot \frac{C_\gamma}{C_\sigma} where: - :math:`K_\theta(\omega)` is the frequency-dependent torsional stiffness - :math:`C_\gamma`, :math:`C_\sigma` are strain and stress conversion coefficients **Warning:** When :math:`J' \lesssim J'_{\mathrm{inst}}`, compliance-related artifacts appear as increased noise at high frequencies. Real-time compliance corrections available in standard oscillatory tests are **not available** for arbitrary waveform protocols, making careful parameter selection essential. Validation Summary ------------------ Perego et al. (2025) validated OWCh against discrete frequency sweeps (DFS) and multi-wave superposition over a temperature range of :math:`-40\,^\circ\text{C}` to :math:`150\,^\circ\text{C}`: - **Excellent agreement** among OWCh, DFS, and multi-wave methods at all temperatures - **WLF parameters** from OWCh within 6% of DFS values - **Master curve overlay** shows negligible differences between methods - **Low-temperature deviations** traceable to instrument compliance limits, not method error These results confirm OWCh as a reliable alternative to traditional frequency sweeps for thermorheologically simple materials. Troubleshooting --------------- **Waveform Design Issues:** - **Spectral holes** — Increase ``duration`` or reduce taper aggressiveness so each octave receives sufficient dwell time. - **TB product warning** — Increase ``duration`` or narrow ``chirp_span`` to achieve :math:`TB \ge 66`. - **Aliasing** — Verify ``fs`` exceeds :math:`10\omega_2/2\pi` for :math:`N_y < 0.1`. **Signal Quality Issues:** - **Weak higher harmonics** — Raise ``amplitude`` (within instrument limits) or average repeated chirps to boost SNR before deconvolution. - **Peak overlap** — Ensure ``n_harmonics`` is not larger than the resolvable bandwidth; use orthogonal window segments when concatenating chirps. - **Bias/drift artifacts** — Use waiting time :math:`t_w > 0` and apply automated bias correction (subtracting signal mean from waiting interval, OWCh segment, or full duration). **Compliance-Related Issues:** - **High-frequency noise at low temperatures** — Material stiffness approaching instrument compliance limits. Check if :math:`J' \lesssim J'_{\mathrm{inst}}` and consider narrowing frequency range or using stiffer geometry. - **Transducer saturation** — Compare programmed vs measured strain amplitude; significant deviation indicates compliance or acceleration limits reached. - **No compliance correction available** — Real-time compliance corrections are not implemented for arbitrary waveform protocols. Must estimate and correct offline. **Mutating Material Issues:** - **High mutation number** — Material evolving during chirp (:math:`Mu > 0.15`). Increase :math:`\omega_1` to shorten :math:`T_{\mathrm{owc}}`, accepting lower :math:`TB`. - **Uptick in** :math:`|\eta^*_r|` — Rapid mutation phase exceeded measurement timescale. Rerun with shorter chirps and verify :math:`\overline{Mu} < Mu_{\mathrm{crit}}` throughout. - **Master curve doesn't collapse** — Check for thermorheological complexity or mutation artifacts; try vertical shift factors :math:`b_T \ne 1` if warranted. References ---------- **Primary OWCh References:** - Perego, A., Vadillo, D.C., Mills, M.J.L., Das, M., & McKinley, G.H. "Evaluation of optimally windowed chirp signals in industrial rheological measurements: method development and data processing." *Rheol. Acta* 64, 391–406 (2025). DOI: `10.1007/s00397-025-01511-0 `_ :download:`PDF <../../reference/perego_2025_owchirp.pdf>` - Geri, M., Keshavarz, B., Divoux, T., Clasen, C., Curtis, D.J., & McKinley, G.H. "Time-resolved mechanical spectroscopy of soft materials via optimally windowed chirps." *Phys. Rev. X* 8, 041042 (2018). DOI: `10.1103/PhysRevX.8.041042 `_ **Extensions and Variants:** - Hudson-Kershaw, R.E., Das, M., McKinley, G.H., & Curtis, D.J. ":math:`\sigma`-OWCh: optimally windowed chirp rheometry using combined motor transducer/single head rheometers." *J. Non-Newtonian Fluid Mech.* 333 (2024). DOI: `10.1016/j.jnnfm.2024.105307 `_ - Athanasiou, T., Geri, M., Roose, P., McKinley, G.H., & Petekidis, G. "High-frequency optimally windowed chirp rheometry for rapidly evolving viscoelastic materials: application to a crosslinking thermoset." *J. Rheol.* 68(3), 445–462 (2024). DOI: `10.1122/8.0000793 `_ - Rathinaraj, J.D.J., Hendricks, J., McKinley, G.H., & Clasen, C. "Orthochirp: a fast spectro-mechanical probe for monitoring transient microstructural evolution of complex fluids during shear." *J. Non-Newtonian Fluid Mech.* 301, 104744 (2022). DOI: `10.1016/j.jnnfm.2022.104744 `_ **Instrument Compliance and Data Processing:** - Hossain, M.T., Macosko, C.W., McKinley, G.H., & Ewoldt, R.H. "Instrument stiffness artifacts: avoiding bad data with operational limit lines of G_max and E_max." *Rheol. Acta* 64, 67–79 (2025). DOI: `10.1007/s00397-024-01481-9 `_ - Lennon, K.R., McKinley, G.H., & Swan, J.W. "A data-driven method for automated data superposition with applications in soft matter science." *Data-Centric Engineering* 4, e13 (2023). DOI: `10.1017/dce.2023.3 `_ **Foundational Works:** - Ghiringhelli, E., Roux, D., Bleses, D., Galliard, H., & Caton, F. "Optimal Fourier rheometry: application to the gelation of an alginate." *Rheol. Acta* 51(5), 413–420 (2012). - Mours, M. & Winter, H.H. "Time-resolved rheometry." *Rheol. Acta* 33, 385–397 (1994). - Ferry, J.D. *Viscoelastic Properties of Polymers*, 3rd ed. Wiley, New York (1980). **Software:** - Mills, M.J.L., et al. ``piblin``: Pipeline data framework. MIT License. `GitHub `_ | `PyPI `_ - Perego, A., Mills, M.J.L., & Vadillo, D.C. ``hermes-rheo``: High-throughput rheological data transformations. MIT License. `GitHub `_ | `Docs `_ See also -------- - :doc:`fft` — OWChirp relies on windowed FFTs for modulus extraction. - :doc:`../models/fractional/fractional_maxwell_model` — broadband chirps enable fitting of multi-order fractional models. - :doc:`../models/flow/herschel_bulkley` — LAOS chirps are often paired with yield-stress model identification. - :doc:`mutation_number` — evaluate whether chirp segments remain quasi-steady. - :doc:`../../examples/transforms/04-owchirp-laos` — notebook demonstrating chirp design, playback, and analysis.