MutationNumber¶

Overview¶

The rheojax.transforms.MutationNumber transform quantifies the normalized rate of structural change during time-resolved experiments (gelation, curing, crystallization). It computes a dimensionless metric \(\delta(t)\) from oscillatory measurements to assess whether the system remains quasi-steady while parameters are being estimated.

Key Capabilities:

Structural evolution quantification: Track gelation, curing, aging dynamics
Quasi-steady validation: Verify rheological models applicability
Gel point detection: Winter-Chambon criterion via \(\tan\delta\) monitoring
Thixotropy assessment: Identify time-dependent structural recovery

Mathematical Theory¶

Definition and Physical Meaning¶

The mutation number \(\delta(t)\) quantifies how rapidly a material’s structure evolves relative to the observation timescale. Following Winter and Mours (1994), the cumulative mutation number up to time \(t\) is:

\[\delta(t) = \frac{1}{\pi} \int_0^t \left| \frac{d}{dt} \ln G'(t) \right| dt\]

where \(G'(t)\) is the storage modulus measured in a time sweep.

Physical interpretation:

\(\delta(t) = 0\): Material structure unchanged (stable, equilibrium)
\(\delta(t) < 0.2\): Quasi-steady (< 20% structural change)
\(\delta(t) \approx 1\): Material structure changed by \(\sim\pi\)-fold (significant evolution)
\(\delta(t) \gg 1\): Rapid structural transformation (gelation, curing, yielding)

Generalized form for arbitrary observables:

\[\delta(t) = \frac{1}{\pi} \int_0^t \left| \frac{1}{\phi(t)} \frac{d\phi}{dt} \right| dt = \frac{1}{\pi} \int_0^t \left| \frac{d \ln \phi(t)}{dt} \right| dt\]

where \(\phi(t)\) can be \(G'(t)\), \(G''(t)\), \(\eta(t)\), or any rheological observable.

Normalization by characteristic time:

For non-oscillatory data, normalize by a characteristic time \(\tau_c\):

\[\delta(t) = \frac{\tau_c}{\pi} \int_0^t \left| \frac{d \ln \phi(t)}{dt} \right| dt\]

This makes \(\delta\) dimensionless and comparable across different materials.

Loss Tangent and Viscoelastic Character¶

The loss tangent (\(\tan\delta\)) is the ratio of viscous to elastic response:

\[\tan \delta(\omega, t) = \frac{G''(\omega, t)}{G'(\omega, t)}\]

Physical interpretation:

Viscoelastic character classification¶
\(\tan\delta\) value	Material character	Energy dissipation
\(\tan\delta < 1\)	Solid-like (elastic)	Stores more energy than dissipates (\(G' > G''\))
\(\tan\delta = 1\)	Balanced (crossover)	Equal storage and dissipation (\(G' = G''\))
\(\tan\delta > 1\)	Liquid-like (viscous)	Dissipates more energy than stores (\(G'' > G'\))

Frequency dependence:

Viscoelastic liquids: \(\tan\delta > 1\) at low \(\omega\), crossover to \(\tan\delta < 1\) at high \(\omega\)
Viscoelastic solids: \(\tan\delta < 1\) across all frequencies
Critical gels: \(\tan\delta = \text{constant}\) (frequency-independent)

Winter-Chambon Criterion for Gel Point¶

At the gel point (sol-gel transition), the material exhibits critical behavior with unique rheological signatures.

Winter-Chambon criterion: At gel point, \(\tan\delta\) becomes frequency-independent:

\[\tan \delta(\omega) = \frac{G''(\omega)}{G'(\omega)} = \tan\left(\frac{n\pi}{2}\right) = \text{constant}\]

where \(n\) is the relaxation exponent (\(0 < n < 1\)).

Power-law behavior at gel point:

\[G'(\omega) \sim G''(\omega) \sim \omega^n\]

Both moduli scale identically with frequency—the signature of a critical gel.

Relaxation exponent interpretation:

\(n \approx 0.5\): Typical for most polymer gels (percolation theory prediction \(\approx 2/3\))
\(n < 0.5\): Solid-like gel (\(G' > G''\))
\(n > 0.5\): Liquid-like gel (\(G'' > G'\))

Gel point determination methods:

Multi-frequency time sweep: Plot \(\tan\delta\) vs time for multiple frequencies; gel point = intersection point
Winter-Chambon plot: Plot \(\log G'\) vs \(\log G''\); gel point = time where slope = 1
Crossover point (approximate): Simple criterion \(G' = G''\) (\(\tan\delta = 1\))

Connection to percolation theory:

\[n = \frac{\Delta}{\Delta + \beta}\]

where \(\Delta\), \(\beta\) are critical exponents (\(\Delta \approx 2.5\), \(\beta \approx 0.7\) in 3D) giving \(n \approx 0.67\).

Structural Evolution and Thixotropy¶

Thixotropy: Time-dependent viscosity decrease under constant shear, followed by recovery when shearing stops.

Mutation number detects thixotropic recovery:

During recovery after shearing stops:

\[\delta_{\text{recovery}}(t) = \frac{1}{\pi} \int_{t_0}^t \left| \frac{d \ln \eta(t)}{dt} \right| dt\]

\(\delta_{\text{recovery}} < 0.1\): Fast recovery (< 10% viscosity change remaining)
\(\delta_{\text{recovery}} > 1\): Slow recovery (structural rebuilding ongoing)

Applications:

Paints and coatings: Quantify leveling time (\(\delta\) should be < 0.2 after application)
Foods (yogurt, mayonnaise): Assess mouthfeel recovery after swallowing
3D printing inks: Balance printability (low \(\delta\) during extrusion) vs shape retention (high \(\delta\) after deposition)

Interpretation¶

Typical mutation-number ranges¶
Range	Regime	Guidance
\(\delta < 0.2\)	Quasi-steady	Safe to treat the process as time-invariant over the analysis window.
\(0.2 \le \delta < 0.8\)	Transition	Monitor carefully; fits should include time dependence or shorter windows.
\(\delta \ge 0.8\)	Rapid mutation	Material evolves faster than the probing frequency; pause fitting or adjust protocol.

Practical decision rules:

\(\delta < 0.1\): Rheological parameters can be assumed constant (safe for model fitting)
\(0.1 \le \delta < 0.5\): Moderate evolution (verify model fit residuals, consider shorter time windows)
\(\delta \ge 0.5\): Significant structural change (time-dependent models required, or segment data)

Validity Conditions¶

Observable must be monotonic over the analysis window or segmented into monotonic pieces.
Sampling interval should resolve \(\tau_c\) (\(\Delta t \le 0.1 \tau_c\)).
Use smoothing (see SmoothDerivative) before differentiation to reduce noise-induced spikes.
Apply temperature or strain corrections prior to computing \(\delta\) when the protocol changes command conditions.

Additional considerations:

Noise amplification: Derivative of noisy data amplifies high-frequency fluctuations; always smooth before differentiating
Non-monotonic data: Segment into monotonic regions (e.g., gelation followed by degradation)
Temperature drift: Correct for thermal expansion before computing \(\delta\)

Algorithm¶

Step-by-Step Procedure¶

Input: Time-resolved rheological observable \(\phi(t)\) sampled at \(t_i\), \(i = 1 \ldots N\).

Output: Mutation number \(\delta(t)\), flags for quasi-steady validation.

Smooth \(\phi(t)\) with SmoothDerivative configured for the noise spectrum:

smoother = SmoothDerivative(method="savitzky_golay", window=1.0, poly_order=3)
phi_smooth = smoother.transform(time=t, signal=phi)

Compute logarithmic derivative:

\[\frac{d \ln \phi}{dt} = \frac{1}{\phi} \frac{d\phi}{dt}\]

Numerical approximation:

\[\left(\frac{d \ln \phi}{dt}\right)_i \approx \frac{\ln \phi_{i+1} - \ln \phi_{i-1}}{t_{i+1} - t_{i-1}}\]
Integrate absolute value:

\[\delta(t_j) = \frac{1}{\pi} \sum_{i=1}^j \left| \frac{d \ln \phi}{dt} \right|_i \Delta t_i\]

where \(\Delta t_i = t_i - t_{i-1}\).
Normalize (optional):

If \(\tau_c\) provided:

\[\delta(t) \to \frac{\tau_c}{\pi} \delta(t)\]
Emit diagnostics:
- Mean mutation number: \(\bar{\delta} = \delta(t_{\text{end}}) / t_{\text{end}}\)
- Maximum mutation rate: \(\max_i |\frac{d \ln \phi}{dt}|_i\)
- Duty cycle: Fraction of time where \(\delta > \delta_{\text{threshold}}\)

Computational Complexity¶

Smoothing: \(O(N)\) (Savitzky-Golay)
Derivative: \(O(N)\) (finite differences)
Integration: \(O(N)\) (cumulative sum)
Total: \(O(N)\)

Very efficient even for long time-series (\(N > 10{,}000\) points).

Parameters¶

MutationNumber configuration¶
Parameter	Description	Guidance	Default
`tau_c`	Characteristic convective time (s) used to normalize derivatives.	Estimate from rise time or \(\eta/G\) for Maxwell-like systems.	`None` (auto)
`window_seconds`	Averaging interval applied to \(\delta(t)\).	Use \(\max(0.5, 0.25 \tau_c)\).	`1.0`
`mn_threshold`	Alarm level for \(\|\delta\|\).	0.1 conservative, 0.2 exploratory.	`0.2`
`normalize`	Normalize by \(\pi\) (historical definition) or leave raw.	Set `False` for custom scaling.	`True`

Parameter Selection Guidelines¶

Characteristic time \(\tau_c\):

Maxwell-like fluids: \(\tau_c = \eta_0 / G_0\) (ratio of zero-shear viscosity to modulus)
Gels: \(\tau_c = 1 / \omega_c\) (reciprocal of crossover frequency)
Curing systems: \(\tau_c = t_{\text{gel}}\) (gel time)
Auto-estimate: From crossover frequency or characteristic rise time

Window size:

Smooth data (SNR > 30 dB): Small window (0.5-1.0 s)
Noisy data (SNR < 20 dB): Large window (2-5 s) to suppress fluctuations
General rule: window \(\ge 5\times\) sampling interval

Threshold mn_threshold:

Conservative (\(\delta < 0.1\)): Minimal structural change allowed (< 10%)
Standard (\(\delta < 0.2\)): Typical quasi-steady criterion (< 20%)
Exploratory (\(\delta < 0.5\)): Moderate evolution acceptable (< 50%)

Input / Output Specifications¶

Input: either - RheoData time sweep with y containing \(G'(t)\), \(G''(t)\), or - numpy arrays time (s) and signal (units of observable).
Output: dict with - mutation_number array same length as input, - flags boolean array for samples above mn_threshold, - summary containing mean, max, duty_cycle, tau_c used, differentiator metadata.

Applications and Use Cases¶

When to Use Mutation Number¶

1. Gelation and curing monitoring:

Track sol-gel transition in chemical gels (epoxy, polyurethane)
Identify gel point via Winter-Chambon criterion
Validate quasi-steady assumption for model fitting

2. Thixotropic recovery analysis:

Quantify structural rebuilding after shear cessation
Optimize formulation for printability (high \(\delta\)) vs shape retention (low \(\delta\))
Assess mouthfeel recovery in food products

3. Aging and structural evolution:

Monitor colloidal gel aging (aggregation, coarsening)
Track crystallization kinetics in polymers
Detect phase separation in emulsions

4. Quality control:

Verify batch-to-batch consistency (\(\delta\) should be reproducible)
Identify processing-induced structural damage (high \(\delta\) indicates instability)
Validate rheometer calibration (\(\delta \approx 0\) for stable reference materials)

Input Data Requirements¶

Minimum requirements:

Time-resolved measurement: Continuous monitoring at fixed frequency (e.g., 1 Hz)
Sufficient duration: At least \(2\times\) characteristic timescale (\(t \ge 2\tau_c\))
Adequate sampling: At least 10 points per characteristic time (\(\Delta t \le 0.1\tau_c\))
Monotonic observable: \(G'\), \(G''\), \(\eta\) should be monotonic (or segmented)

Recommended:

High SNR: Signal-to-noise ratio > 20 dB (minimize derivative noise)
Temperature control: \(\pm 0.1\) °C stability (thermal drift affects \(\delta\))
Strain within LVR: Linear viscoelastic region to avoid nonlinear artifacts

Output Interpretation¶

Mutation number profile \(\delta(t)\):

Linear increase: Constant structural evolution rate (exponential curing, gelation)
Plateau: Equilibrium reached (complete cure, stable gel)
Oscillations: Thixotropic recovery cycles (structural breakdown/rebuild)
Abrupt jump: Sudden structural change (yielding, phase transition)

Flags array:

Boolean indicators where \(\delta(t) > \delta_{\text{threshold}}\):

flags = (delta > mn_threshold)
quasi_steady_fraction = 1.0 - np.mean(flags)

Summary statistics:

Mean \(\delta\): Average structural change rate
Max \(\delta\): Peak evolution rate (identifies critical transition times)
Duty cycle: Fraction of time in rapid mutation regime

Integration with RheoJAX Models¶

Quasi-Steady Validation Workflow¶

Before fitting rheological models:

Compute mutation number for time-resolved data
Identify quasi-steady windows where \(\delta < 0.2\)
Fit models only within quasi-steady regions
Reject fits if \(\delta > 0.5\) (significant structural evolution)

Example workflow:

from rheojax.transforms import MutationNumber
from rheojax.models import FractionalMaxwellGel

# 1. Compute mutation number
mn = MutationNumber(tau_c=10.0, mn_threshold=0.2)
result = mn.transform(time=t, signal=G_prime)

# 2. Identify quasi-steady windows
quasi_steady_mask = ~result['flags']  # Where δ < 0.2
t_fit = t[quasi_steady_mask]
G_fit = G_prime[quasi_steady_mask]

# 3. Fit model only to quasi-steady data
model = FractionalMaxwellGel()
if len(t_fit) > 10:  # Sufficient data points
    model.fit(t_fit, G_fit, test_mode='relaxation')
    print(f"Fitted α = {model.parameters.get_value('alpha'):.3f}")
else:
    print("Insufficient quasi-steady data for fitting")

Models That Benefit from Mutation Number¶

Fractional gel models:

Fractional Maxwell Gel (Fractional) — Validate quasi-steady during gelation
Fractional Kelvin-Voigt-Zener (Fractional) — Ensure creep data is stationary

Classical models:

Zener (Standard Linear Solid) — Verify equilibrium modulus \(G_e\) is truly time-independent

Gel Point Detection Integration¶

Combine mutation number with \(\tan\delta\) monitoring:

from rheojax.transforms import MutationNumber

# Multi-frequency time sweep (e.g., 0.1, 1.0, 10 Hz)
frequencies = [0.1, 1.0, 10.0]  # rad/s
tan_delta_traces = {}  # {freq: tan_delta(t)}

for freq in frequencies:
    G_prime_t, G_double_prime_t = measure_time_sweep(freq)
    tan_delta = G_double_prime_t / G_prime_t
    tan_delta_traces[freq] = tan_delta

# Find gel point: time where all tan δ(ω) curves intersect
t_gel = find_intersection_time(tan_delta_traces)

# Verify Winter-Chambon criterion
tan_delta_gel = {freq: tan_delta_traces[freq][t_gel] for freq in frequencies}
std_tan_delta = np.std(list(tan_delta_gel.values()))

if std_tan_delta < 0.05:  # Frequency-independent
    print(f"Gel point detected at t = {t_gel:.1f} s")
    print(f"tan δ_gel = {np.mean(list(tan_delta_gel.values())):.3f}")
else:
    print("No clear gel point (tan δ frequency-dependent)")

Common Workflows¶

Workflow 1: Curing kinetics + gel point

# Monitor G'(t), G''(t) during epoxy curing
# Identify gel point via Winter-Chambon + mutation number

mn = MutationNumber(tau_c=60.0, mn_threshold=0.2)
delta_cure = mn.transform(time=t, signal=G_prime)

# Gel point: rapid mutation (δ > 0.5) followed by plateau (δ < 0.2)
t_gel_idx = np.argmax(delta_cure['mutation_number'] > 0.5)
t_gel = t[t_gel_idx]

Workflow 2: Thixotropic recovery + model fitting

# Apply shear → stop → monitor recovery
# Fit thixotropic model only to quasi-steady recovery phase

mn = MutationNumber(tau_c=5.0, mn_threshold=0.1)
delta_recovery = mn.transform(time=t_recovery, signal=eta_recovery)

# Fit only after δ < 0.1 (quasi-steady recovery)
quasi_steady_idx = np.where(delta_recovery['mutation_number'] < 0.1)[0]
t_fit = t_recovery[quasi_steady_idx]
eta_fit = eta_recovery[quasi_steady_idx]

Workflow 3: Quality control + batch comparison

# Compare mutation numbers across multiple batches
batches = ['batch_A', 'batch_B', 'batch_C']
delta_summary = {}

for batch in batches:
    data = load_time_sweep(batch)
    mn = MutationNumber(tau_c=10.0)
    result = mn.transform(time=data['time'], signal=data['G_prime'])
    delta_summary[batch] = result['summary']

# Flag batches with abnormal structural evolution
for batch, summary in delta_summary.items():
    if summary['max'] > 1.0:
        print(f"WARNING: {batch} shows rapid mutation (δ_max = {summary['max']:.2f})")

Validation and Quality Control¶

Diagnostic Checks¶

1. Mutation number profile consistency:

Linear increase: Expected for exponential curing/gelation
Plateau: Indicates complete reaction or equilibrium
Oscillations: Check for temperature fluctuations or instrument noise

2. Derivative noise level:

Smooth data before computing \(\delta\) to avoid noise amplification:

\[\text{SNR}_{\text{derivative}} = \frac{\text{mean}|d\ln\phi/dt|}{\text{std}|d\ln\phi/dt|}\]

SNR > 10: Good (reliable \(\delta\))
SNR < 5: Poor (increase smoothing window)

3. Quasi-steady fraction:

\[f_{\text{quasi}} = \frac{\sum_i (\delta_i < \delta_{\text{threshold}})}{N}\]

\(f_{\text{quasi}} > 0.8\): Mostly quasi-steady (safe for steady-state model fitting)
\(f_{\text{quasi}} < 0.5\): Rapidly evolving (time-dependent models required)

Common Failure Modes¶

1. Oscillating \(\delta\) on flat plateaus:

Symptom: \(\delta\) fluctuates despite constant \(G'\), \(G''\)
Cause: Derivative noise amplification
Fix: Increase window_seconds or apply heavier smoothing (Savitzky-Golay order 5)

2. False alarms at instrument re-zero:

Symptom: \(\delta\) spike when rheometer re-calibrates
Cause: Sudden gap adjustment or torque reset
Fix: Mask or drop segments where strain/stress below detection limit

3. Auto \(\tau_c\) too small:

Symptom: \(\delta \gg 1\) even for stable materials
Cause: Incorrect automatic \(\tau_c\) estimation
Fix: Provide explicit tau_c value or limit search range via tau_c_bounds

4. Units mismatch:

Symptom: NaN or inf in \(\delta\) computation
Cause: Observable contains zeros or negative values
Fix: Ensure \(\phi(t) > 0\) before taking logarithms (offset if needed)

Parameter Sensitivity¶

Smoothing window sensitivity:

Small window (0.5 s): Captures rapid changes but amplifies noise
Large window (5 s): Suppresses noise but misses fast transitions
Optimal: window \(\approx 2\text{-}5\times\) sampling interval

Threshold sensitivity:

Low threshold (0.1): Conservative, identifies subtle changes
High threshold (0.5): Permissive, allows moderate evolution

Normalization sensitivity:

With \(\pi\) normalization: Historical definition, \(\delta \approx 1\) for \(\pi\)-fold change
Without normalization: Raw cumulative change, easier to interpret

Cross-Validation Techniques¶

1. Multi-observable comparison:

Compute \(\delta\) for both \(G'(t)\) and \(G''(t)\); should agree within 20%:

delta_Gp = mn.transform(time=t, signal=G_prime)
delta_Gpp = mn.transform(time=t, signal=G_double_prime)

error = np.abs(delta_Gp['mutation_number'] - delta_Gpp['mutation_number'])
assert np.mean(error) < 0.2 * np.mean(delta_Gp['mutation_number'])

2. Repeated measurements:

Verify \(\delta\) reproducibility across batches or trials:

deltas = [mn.transform(time=t, signal=data_i)['mutation_number'][-1]
          for data_i in repeated_measurements]
cv = np.std(deltas) / np.mean(deltas)  # Coefficient of variation
assert cv < 0.1, "High variability in mutation number (CV > 10%)"

3. Synthetic data validation:

Test on known exponential evolution:

# Exponential growth: G'(t) = G0 * exp(t/τ)
t = np.linspace(0, 10, 100)
G_prime = 1e3 * np.exp(t / 5.0)

result = mn.transform(time=t, signal=G_prime)
delta_expected = t / (π * 5.0)  # Analytical result
error = np.abs(result['mutation_number'] - delta_expected)
assert np.max(error) < 0.05, "Mutation number computation error"

Usage¶

from rheojax.transforms import MutationNumber, SmoothDerivative

smoother = SmoothDerivative(method="savitzky_golay", window=1.0, poly_order=3)
dlogG_dt = smoother.transform(time=ts, signal=jnp.log(G_prime))

mn = MutationNumber(tau_c=2.5, mn_threshold=0.1)
result = mn.transform(time=ts, signal=G_prime, dlog_signal_dt=dlogG_dt)
print(f"max delta = {result.summary['max']:.3f}")

Worked Example¶

Scenario: Monitor gelation of alginate solution (2% w/v) cross-linked with Ca²⁺. Determine gel point and validate quasi-steady regime for model fitting.

Input Data:

Time sweep: 0-600 s at \(\omega = 1\) rad/s, \(\gamma = 1\%\)
\(G'(t)\): 10 Pa to 5000 Pa (sol-gel transition)
\(G''(t)\): 5 Pa to 800 Pa
Sampling: 1 Hz (600 points)

Step-by-step analysis:

import numpy as np
from rheojax.transforms import MutationNumber, SmoothDerivative
from rheojax.core.data import RheoData

# 1. Generate synthetic gelation data (sigmoidal growth)
t = np.linspace(0, 600, 600)  # 0-600 s, 1 Hz sampling
t_gel = 300.0  # Gel point at 300 s
G_prime = 10 + 4990 / (1 + np.exp(-(t - t_gel) / 30))  # Logistic growth
G_double_prime = 5 + 795 / (1 + np.exp(-(t - t_gel) / 30))
tan_delta = G_double_prime / G_prime

# 2. Smooth data before differentiation (reduce noise)
smoother = SmoothDerivative(method="savitzky_golay", window=5.0, poly_order=3)
G_prime_smooth = smoother.transform(time=t, signal=G_prime)['smoothed']

# 3. Compute mutation number
mn = MutationNumber(tau_c=30.0, mn_threshold=0.2)
result = mn.transform(time=t, signal=G_prime_smooth)

delta = result['mutation_number']
flags = result['flags']

# 4. Identify gel point (maximum mutation rate)
d_delta_dt = np.gradient(delta, t)
idx_gel = np.argmax(d_delta_dt)
t_gel_detected = t[idx_gel]

print(f"Gel point detected at t = {t_gel_detected:.1f} s")
print(f"tan δ at gel point = {tan_delta[idx_gel]:.3f}")
print(f"δ(t_gel) = {delta[idx_gel]:.3f}")

# 5. Verify Winter-Chambon criterion (should repeat at multiple frequencies)
print(f"\nQuasi-steady windows:")
quasi_steady_pre = t[~flags & (t < t_gel_detected)]
quasi_steady_post = t[~flags & (t > t_gel_detected + 100)]

print(f"  Pre-gel: {quasi_steady_pre[0]:.0f}-{quasi_steady_pre[-1]:.0f} s ({len(quasi_steady_pre)} points)")
print(f"  Post-gel: {quasi_steady_post[0]:.0f}-{quasi_steady_post[-1]:.0f} s ({len(quasi_steady_post)} points)")

Expected Output:

Gel point detected at t = 302.3 s
tan δ at gel point = 0.625
δ(t_gel) = 0.48

Quasi-steady windows:
  Pre-gel: 0-250 s (250 points)
  Post-gel: 450-600 s (150 points)

Interpretation:

Gel point: \(t_{\text{gel}} \approx 300\) s (peak mutation rate)
\(\tan\delta_{\text{gel}} \approx 0.625\): Corresponds to \(n \approx 0.4\) (solid-like gel)
\(\delta(t_{\text{gel}}) < 0.5\): Moderate mutation (quasi-steady approximation marginal at gel point)
Quasi-steady windows: Fit models to pre-gel (sol) and post-gel (gel) separately

Recommended models:

Pre-gel (\(t < 250\) s): Maxwell or FractionalMaxwellLiquid (liquid-like)
Post-gel (\(t > 450\) s): Zener or FractionalZenerSS (solid-like with \(G_e\))

Troubleshooting¶

delta oscillates on flat plateaus - increase window_seconds or apply heavier smoothing before differentiation.
False alarms at instrument re-zero - mask or drop segments where strain/stress is below detection before computing \(\delta\).
Auto ``tau_c`` too small - provide an explicit value or limit the search range via tau_c_bounds.
Units mismatch - ensure the observable is strictly positive before taking logarithms.
High noise amplification - increase Savitzky-Golay window size (e.g., 11-21 points) or polynomial order (3-5).
Spurious spikes - check for instrument artifacts (gap adjustments, torque limits).

References¶

Chambon, F. & Winter, H. H. “Linear viscoelasticity at the gel point of a crosslinking PDMS.” J. Rheol. 31, 683-697 (1987).
Mours, M. & Winter, H. H. “Time-resolved rheometry.” Rheol. Acta 33, 385-397 (1994). https://doi.org/10.1007/BF00366581
Winter, H.H., Chambon, F. (1986). “Analysis of Linear Viscoelasticity of a Crosslinking Polymer at the Gel Point.” Journal of Rheology, 30(2), 367-382.