Time-Temperature Superposition (TTS)¶

Overview¶

Time-Temperature Superposition (TTS) is a powerful principle in rheology enabling extension of measurement frequency ranges by 4-8 decades through temperature sweeps. Instead of mechanically accessing \(\omega = 10^{-6}\) to \(10^6\) rad/s (requires 12 decades, impossible for most instruments), measure at 5-10 temperatures and shift data horizontally to create a mastercurve at a reference temperature.

Physical basis: The temperature dependence of molecular relaxation processes is separable from their fundamental shape. Increasing temperature accelerates all relaxation processes by the same factor, equivalent to shifting the time scale.

Mathematical formulation:

\[ \begin{align}\begin{aligned}G(t, T) = G(t/a_T, T_{\text{ref}})\\G'(\omega, T) = G'(\omega \cdot a_T, T_{\text{ref}})\end{aligned}\end{align} \]

where \(a_T\) is the shift factor depending on temperature T and reference temperature \(T_{\text{ref}}\).

Applicability:

Valid for: Amorphous polymers above \(T_g\), polymer melts, viscoelastic liquids
Requires: Thermorheologically simple materials (all relaxation modes shift equally)
Fails for: Crystalline polymers, multi-phase systems with different T-dependencies, materials undergoing phase transitions

Manual Shift Methods¶

WLF Equation (Williams-Landel-Ferry)¶

Empirical equation for shift factors of amorphous polymers near glass transition:

\[\log_{10}(a_T) = \frac{-C_1 (T - T_{\text{ref}})}{C2 + (T - T_{\text{ref}})}\]

Parameters:

\(C_1\): Dimensionless constant (universal ≈ 17.44)
\(C_2\): Temperature constant in Kelvin (universal ≈ 51.6 K)
\(T_{\text{ref}}\): Reference temperature (often \(T_g\) + 50 °C)

Universal values (reference at \(T_g\)):

\(C_1 = 17.44\), \(C_2 = 51.6\) K when \(T_{\text{ref}} = T_g\)
Valid for most polymers within \(T_g\) to \(T_g + 100\) °C

Material-specific values: Can be optimized for better accuracy (see optimize_wlf_parameters()).

Physical interpretation:

Free volume theory: Molecular mobility increases with free volume
\(C_1\) relates to fractional free volume at \(T_g\)
\(C_2\) relates to thermal expansion coefficient

Usage example:

from rheojax.transforms.mastercurve import Mastercurve

# Create WLF mastercurve
mc = Mastercurve(
    reference_temp=373.15,  # 100°C in Kelvin
    method='wlf',
    C1=17.44,
    C2=51.6,
    auto_shift=False  # Manual WLF
)

# Transform multi-temperature datasets
mastercurve, shift_factors = mc.transform(datasets)

# Get WLF parameters
params = mc.get_wlf_parameters()
print(params)  # {'C1': 17.44, 'C2': 51.6, 'T_ref': 373.15}

Arrhenius Equation¶

Exponential temperature dependence for high-temperature flows (\(T \gg T_g\)):

\[\log(a_T) = \frac{E_a}{R} \left( \frac{1}{T} - \frac{1}{T_{\text{ref}}} \right)\]

Parameters:

\(E_a\): Activation energy (J/mol)
\(R = 8.314\) J/(mol·K): Gas constant
\(T\): Absolute temperature (Kelvin)

Physical interpretation:

Energy barrier for molecular rearrangement
Higher \(E_a\) → stronger temperature dependence

When to use Arrhenius:

High temperatures (\(T > T_g + 100\) °C)
Polymer melts well above glass transition
Liquids without glass transition (oils, solvents)

Usage example:

from rheojax.transforms.mastercurve import Mastercurve

# Create Arrhenius mastercurve
mc = Mastercurve(
    reference_temp=453.15,  # 180°C in Kelvin
    method='arrhenius',
    E_a=85000,  # J/mol (85 kJ/mol for PS melt)
    auto_shift=False
)

mastercurve, shift_factors = mc.transform(datasets)

# Get Arrhenius parameters
params = mc.get_arrhenius_parameters()
print(params)  # {'E_a': 85000, 'T_ref': 453.15}

When to Use Each Method¶

WLF vs Arrhenius decision guide¶
Criterion	WLF	Arrhenius
Temperature range	\(T_g\) to \(T_g + 100\) °C	\(T > T_g + 100\) °C
Material state	Rubbery plateau	Viscous melt
Physical basis	Free volume theory	Activation energy
Typical materials	Amorphous polymers near \(T_g\)	Polymer melts, oils
Shift factor curvature	Non-linear (curved)	Linear (straight line)
Parameter availability	Universal \(C_1\), \(C_2\) available	Material-specific \(E_a\) needed

Automatic Shift Method (NEW in v0.2.2)¶

Power-Law Intersection Algorithm¶

Problem: What if WLF/Arrhenius parameters are unknown or material is non-standard?

Solution: Automatic shift factor calculation via power-law intersection method (PyVisco algorithm).

Key idea: Each temperature curve locally behaves as a power-law in log-log space:

\[y = a \cdot x^b + e\]

Fit adjacent curves, find their intersection in the overlap or gap region, compute relative shift.

Algorithm steps:

Fit power-law to each temperature curve:
- Model: \(y = a \cdot x^b + e\)
- NLSQ optimization (JAX-accelerated, 5-270× speedup vs scipy)
Detect outliers:
- Try removing first data point
- Keep removal if exponent error improves (lower uncertainty in b)
Compute pairwise shifts:
- For adjacent temperatures \(T_i\) and \(T_{i+1}\)
- Sample 10 points in overlap/gap region
- Compute inverse power-law: \(x = \left(\frac{y - e}{a}\right)^{1/b}\)
- Average log shift: \(\log(a_T) = \text{mean}(\log_{10}(x_{\text{top}} / x_{\text{bot}}))\)
Sequential cumulative shifting:
- Start from reference temperature (shift = 0)
- Move outward (both higher and lower T)
- Accumulate shifts: \(\log(a_{T_i}) = \log(a_{T_{i-1}}) + \Delta \log(a_T)\)

Advantages:

No material parameters needed (WLF \(C_1\), \(C_2\) or Arrhenius \(E_a\))
Handles non-standard materials (blends, composites)
Robust to outliers (first-point removal)
JAX-accelerated (fast NLSQ fitting)

Limitations:

Requires clear overlap or gap between temperatures
Less accurate for sparse data (<10 points per curve)
Assumes local power-law behavior

Usage example:

from rheojax.transforms.mastercurve import Mastercurve
from rheojax.core.data import RheoData
import numpy as np

# Multi-temperature SAOS data
temps = [140, 160, 180, 200, 220]  # °C
datasets = []

for T in temps:
    omega = np.logspace(-2, 2, 50)
    G_prime = ...  # Experimental data
    G_double_prime = ...

    data = RheoData(
        x=omega,
        y=np.column_stack([G_prime, G_double_prime]),
        domain='frequency',
        metadata={'temperature': T + 273.15}  # Kelvin
    )
    datasets.append(data)

# Automatic shift factor calculation
mc_auto = Mastercurve(
    reference_temp=180+273.15,  # 180°C
    auto_shift=True  # Enable automatic shifting
)

mastercurve, shift_factors = mc_auto.transform(datasets)

# Get computed shift factors
temps_arr, log_aT_arr = mc_auto.get_auto_shift_factors()

# Plot shift factors
import matplotlib.pyplot as plt
plt.plot(temps_arr - 273.15, log_aT_arr, 'o-')
plt.xlabel('Temperature (°C)')
plt.ylabel('log(a_T)')
plt.axhline(0, color='k', linestyle='--', label='Reference T')
plt.grid(True)
plt.legend()
plt.show()

Transition Guide: Auto vs Manual¶

Decision Flowchart¶

Do you know WLF or Arrhenius parameters?
   ├─ YES → Are they from literature or trusted source?
   │         ├─ YES → Use manual WLF/Arrhenius
   │         │         Validate with compute_overlap_error()
   │         │
   │         └─ NO  → Use auto-shift, then validate against manual
   │
   └─ NO  → Is the material standard (PS, PMMA, PE)?
             ├─ YES → Try universal WLF (C1=17.44, C2=51.6)
             │         Optimize if needed: optimize_wlf_parameters()
             │
             └─ NO  → Use auto-shift (power-law intersection)

When Auto-Shift is Appropriate¶

Use automatic shift factors when:

Unknown material parameters:
- New polymer blend (no literature WLF)
- Composite with multiple phases
- Modified polymer (plasticizers, fillers)
Non-standard temperature ranges:
- T range outside WLF validity (\(T_g\) to \(T_g + 100\) °C)
- High-temperature melts where WLF breaks down
- Low-temperature regime near \(T_g\) where Arrhenius fails
Quick screening:
- Exploratory measurements
- Quality control (batch-to-batch comparison)
- Preliminary characterization
Validation:
- Check if literature WLF parameters work
- Compare auto-shift vs manual for consistency

When Manual Shift is Better¶

Use manual WLF/Arrhenius when:

Known material with established parameters:
- Commercial polymers (PS, PMMA, PDMS)
- Material matches literature closely
- WLF parameters from supplier datasheet
Extrapolation needed:
- Predict behavior at temperatures outside measurement range
- WLF/Arrhenius have physical basis (can extrapolate)
- Auto-shift only interpolates between measured temperatures
Sparse data:
- Few temperatures (< 4)
- Few points per temperature (< 10)
- Auto-shift unreliable with sparse data
Physical interpretation required:
- Need activation energy \(E_a\) for reaction kinetics
- Compare \(C_1\), \(C_2\) across material batches
- Fundamental understanding of molecular mobility

Hybrid Workflow: Auto → Validate → Refine with Manual¶

Best practice: Use auto-shift for initial exploration, then refine with manual methods.

Workflow:

from rheojax.transforms.mastercurve import Mastercurve
import matplotlib.pyplot as plt

# Step 1: Auto-shift (exploratory)
mc_auto = Mastercurve(reference_temp=180+273.15, auto_shift=True)
mastercurve_auto, shifts_auto = mc_auto.transform(datasets)

# Step 2: Manual WLF (literature parameters)
mc_wlf = Mastercurve(
    reference_temp=180+273.15,
    method='wlf',
    C1=8.86,   # PS universal WLF at T_ref=170°C
    C2=101.6,  # Adjusted C2 for PS
    auto_shift=False
)
mastercurve_wlf, shifts_wlf = mc_wlf.transform(datasets)

# Step 3: Compare shift factors
temps_auto, log_aT_auto = mc_auto.get_auto_shift_factors()
temps_wlf, log_aT_wlf = mc_wlf.get_shift_factors_array()

plt.figure(figsize=(10, 5))

plt.subplot(1, 2, 1)
plt.plot(temps_auto - 273.15, log_aT_auto, 'o-', label='Auto-shift')
plt.plot(temps_wlf - 273.15, log_aT_wlf, 's-', label='WLF (literature)')
plt.xlabel('Temperature (°C)')
plt.ylabel('log(a_T)')
plt.legend()
plt.grid(True)
plt.title('Shift Factor Comparison')

plt.subplot(1, 2, 2)
# Plot mastercurves
plt.loglog(mastercurve_auto.x, mastercurve_auto.y[:, 0], 'o',
           alpha=0.5, label="G' (auto)")
plt.loglog(mastercurve_wlf.x, mastercurve_wlf.y[:, 0], 's',
           alpha=0.5, label="G' (WLF)")
plt.xlabel('Shifted frequency (rad/s)')
plt.ylabel("G' (Pa)")
plt.legend()
plt.grid(True)
plt.title('Mastercurve Comparison')

plt.tight_layout()
plt.show()

# Step 4: Compute overlap errors
error_auto = mc_auto.compute_overlap_error(datasets)
error_wlf = mc_wlf.compute_overlap_error(datasets)

print(f"Overlap error (auto): {error_auto:.2e}")
print(f"Overlap error (WLF):  {error_wlf:.2e}")

# Step 5: Optimize WLF if needed
if error_auto < error_wlf * 0.8:
    # Auto-shift significantly better, optimize WLF
    C1_opt, C2_opt = mc_wlf.optimize_wlf_parameters(
        datasets,
        initial_C1=8.86,
        initial_C2=101.6
    )
    print(f"Optimized WLF: C1={C1_opt:.2f}, C2={C2_opt:.2f}")

DMA Applications (Dynamic Mechanical Analyzer)¶

Temperature Sweep Protocols¶

Goal: Measure viscoelastic properties of solid-like materials (polymers, composites) as function of temperature.

Standard DMA temperature sweep:

Setup:
- Geometry: Tension, compression, or torsion (material-dependent)
- Sample: Rectangular bar (10×5×1 mm typical)
- Fixed frequency: \(\omega = 1\) rad/s (or 1 Hz = 6.28 rad/s)
- Strain amplitude: 0.1-1% (within LVR, verified by amplitude sweep)
Temperature ramp:
- Range: \(T_g - 50\) °C to \(T_g + 100\) °C (e.g., -50 °C to 150 °C for PMMA)
- Ramp rate: 2-5°C/min (slow enough for thermal equilibrium)
- Data points: Every 2-5°C (50-100 points total)
Measurements:
- Storage modulus \(E'(T)\) or \(G'(T)\)
- Loss modulus \(E''(T)\) or \(G''(T)\)
- Loss tangent \(\tan \delta = E'' / E'\)

Glass transition identification:

Peak in \(\tan \delta\): T at maximum loss
Inflection in \(E'\): Rapid drop in modulus
Peak in \(E''\): Energy dissipation maximum

Storage/Loss Modulus Master Curves¶

Frequency sweep mastercurve (isothermal at multiple T):

Isothermal frequency sweeps:
- Temperatures: 5-10 values spanning \(T_g\) to \(T_g + 100\) °C
- Example for PMMA: T = 110°C, 120°C, 130°C, 140°C, 150°C
- Frequency range per T: \(\omega = 10^{-2}\) to \(10^2\) rad/s (4 decades)
- Points per decade: 10 (total 40 points per T)

Time-temperature superposition:

from rheojax.transforms.mastercurve import Mastercurve
from rheojax.core.data import RheoData
import numpy as np

# Prepare datasets
temps = [110, 120, 130, 140, 150]  # °C
datasets = []

for T in temps:
    omega = np.logspace(-2, 2, 40)
    E_prime = ...  # DMA measurements
    E_double_prime = ...

    data = RheoData(
        x=omega,
        y=np.column_stack([E_prime, E_double_prime]),
        domain='frequency',
        metadata={'temperature': T + 273.15}
    )
    datasets.append(data)

# Create mastercurve (auto-shift)
mc = Mastercurve(reference_temp=130+273.15, auto_shift=True)
mastercurve, shift_factors = mc.transform(datasets)

# Result: ω_reduced spans 10⁻⁶ to 10⁶ rad/s (12 decades)

Plotting mastercurve:

import matplotlib.pyplot as plt

omega_master = mastercurve.x
E_prime_master = mastercurve.y[:, 0]
E_double_prime_master = mastercurve.y[:, 1]

plt.figure(figsize=(10, 6))
plt.loglog(omega_master, E_prime_master, 'o', label="E' (mastercurve)")
plt.loglog(omega_master, E_double_prime_master, 's', label='E" (mastercurve)')
plt.xlabel('Reduced frequency ω (rad/s)')
plt.ylabel('Modulus (Pa)')
plt.title(f'PMMA Mastercurve at T_ref = 130°C')
plt.legend()
plt.grid(True)
plt.show()

Expected behavior:

Low \(\omega\): Rubbery plateau (\(E' \approx 10^6\) Pa, nearly constant)
Mid \(\omega\): Transition region (\(E'\) increases, \(E''\) peak)
High \(\omega\): Glassy plateau (\(E' \approx 3 \times 10^9\) Pa)

Rheometer Applications (Fluid Dynamics)¶

Complex Viscosity Master Curves¶

Rotational rheometer for polymer melts and viscoelastic fluids:

from rheojax.transforms.mastercurve import Mastercurve
import numpy as np

# Temperatures for PS melt
temps = [140, 160, 180, 200, 220]  # °C
datasets = []

for T in temps:
    omega = np.logspace(-2, 2, 50)
    G_prime = ...  # Pa
    G_double_prime = ...  # Pa

    data = RheoData(
        x=omega,
        y=np.column_stack([G_prime, G_double_prime]),
        domain='frequency',
        metadata={'temperature': T + 273.15}
    )
    datasets.append(data)

# Automatic shift factors
mc = Mastercurve(reference_temp=180+273.15, auto_shift=True)
mastercurve, shifts = mc.transform(datasets)

# Complex viscosity
omega_master = mastercurve.x
G_star_master = mastercurve.y
G_prime = G_star_master[:, 0]
G_double_prime = G_star_master[:, 1]

eta_star = np.sqrt(G_prime**2 + G_double_prime**2) / omega_master

Cox-Merz Rule Validation¶

Cox-Merz empirical rule: For many polymers,

\[\eta(\dot{\gamma}) \approx |\eta^*(\omega)| \bigg|_{\omega = \dot{\gamma}}\]

Test procedure:

SAOS (oscillatory): Measure \(G'(\omega)\), \(G''(\omega)\) → compute \(|\eta^*| = |G^*| / \omega\)
Steady shear (flow curve): Measure \(\eta(\dot{\gamma})\) vs shear rate

Compare at matched frequencies/rates:

import matplotlib.pyplot as plt

# SAOS complex viscosity (from mastercurve)
omega = ...  # rad/s
eta_star = ...  # Pa·s

# Steady shear viscosity (flow curve)
shear_rate = ...  # 1/s
viscosity_steady = ...  # Pa·s

# Plot Cox-Merz comparison
plt.figure(figsize=(8, 6))
plt.loglog(omega, eta_star, 'o-', label='|η*| (SAOS)')
plt.loglog(shear_rate, viscosity_steady, 's--', label='η (steady shear)')
plt.xlabel('ω or γ̇ (rad/s or 1/s)')
plt.ylabel('Viscosity (Pa·s)')
plt.title('Cox-Merz Rule Validation (PS melt, 180°C)')
plt.legend()
plt.grid(True)
plt.show()

Troubleshooting¶

Common TTS problems and solutions¶
Problem	Symptom	Solution
Poor superposition	Curves don’t overlap	Check thermorheological simplicity (try auto-shift)
Vertical shift needed	Horizontal shift insufficient	Enable vertical_shift=True, or data has density/T effects
Shift factors nonphysical	Negative \(a_T\) or extreme values	Wrong T units (use Kelvin), or material not simple
Auto-shift fails	RuntimeError in power-law fit	Insufficient overlap between curves, increase T range
WLF extrapolation poor	Mastercurve breaks down at edges	T outside \(T_g\) to \(T_g + 100\) °C, use Arrhenius
Sparse data (< 10 pts/curve)	Auto-shift unreliable	Use manual WLF/Arrhenius with literature values

References¶

Ferry, J. D. (1980). Viscoelastic Properties of Polymers, 3rd Edition. Wiley.
Williams, M. L., Landel, R. F., & Ferry, J. D. (1955). “The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers.” JACS, 77(14), 3701-3707.
Springer, M. “pyvisco: A Python library for identifying Prony series parameters of linear viscoelastic materials.” NREL (2026). DOI: 10.5281/zenodo.7672127 PDF

Time-Temperature Superposition (TTS)¶

Overview¶

Manual Shift Methods¶

WLF Equation (Williams-Landel-Ferry)¶

Arrhenius Equation¶

When to Use Each Method¶

Automatic Shift Method (NEW in v0.2.2)¶

Power-Law Intersection Algorithm¶

Transition Guide: Auto vs Manual¶

Decision Flowchart¶

When Auto-Shift is Appropriate¶

When Manual Shift is Better¶

Hybrid Workflow: Auto → Validate → Refine with Manual¶

DMA Applications (Dynamic Mechanical Analyzer)¶

Temperature Sweep Protocols¶

Storage/Loss Modulus Master Curves¶

Rheometer Applications (Fluid Dynamics)¶

Complex Viscosity Master Curves¶

Cox-Merz Rule Validation¶

Troubleshooting¶

See Also¶

References¶