Fractional Maxwell Liquid (Fractional)

Quick Reference

  • Use when: Viscoelastic liquid, power-law relaxation without terminal flow plateau

  • Parameters: 3 (\(G_m\), \(\alpha\), \(\tau_\alpha\))

  • Key equation: \(G(t) = G_m t^{-\alpha} E_{1-\alpha,1-\alpha}(-t^{1-\alpha}/\tau_\alpha)\)

  • Test modes: Oscillation, relaxation, creep, flow curve

  • Material examples: Polymer melts (linear/branched), concentrated polymer solutions, complex fluids

Fractional Calculus Fundamentals

This model uses fractional calculus for power-law viscoelastic behavior. For mathematical foundations—SpringPot element, Mittag-Leffler functions, physical meaning of fractional order \(\alpha\), and derivation from molecular theory—see:

/user_guide/fractional_viscoelasticity_reference

Overview

The Fractional Maxwell Liquid (FML) model consists of a Hookean spring in series with a SpringPot element. This configuration describes materials with instantaneous elastic response at short times followed by power-law relaxation at intermediate to long times. The model is particularly effective for characterizing polymer melts, concentrated polymer solutions, and other viscoelastic liquids that exhibit both elastic memory and power-law relaxation without terminal flow.

Unlike the Fractional Maxwell Gel which includes a dashpot for terminal flow, the FML model maintains power-law behavior across all time scales, making it ideal for materials that show persistent viscoelastic behavior without approaching pure viscous flow.

Notation Guide

Symbol

Description

Units

\(G_m\)

Maxwell modulus (short-time elastic stiffness)

Pa

\(\alpha\)

Fractional order (0 < \(\alpha\) < 1, controls relaxation spectrum breadth)

—

\(\tau_\alpha\)

Characteristic relaxation time

s\(^{\alpha}\)

\(E_{\alpha,\beta}(z)\)

Two-parameter Mittag-Leffler function

—

\(G^*(ω)\)

Complex modulus

Pa

\(G'(ω)\)

Storage modulus (elastic component)

Pa

\(G''(ω)\)

Loss modulus (viscous component)

Pa

\(J(t)\)

Creep compliance

Pa-1

\(\omega\)

Angular frequency

rad/s

\(t\)

Time

s

Physical Foundations

The FML model represents viscoelastic liquids with zero equilibrium modulus (Ge = 0), meaning the material flows under sustained stress. The physical structure consists of:

  1. Hookean spring (Gm): Provides instantaneous elastic response at short times. Represents chain/network stretching before relaxation mechanisms activate.

  2. SpringPot element: Governs the relaxation dynamics through power-law viscoelasticity. The fractional order \(\alpha\) controls the breadth of the relaxation spectrum.

The series configuration ensures that sustained stress eventually leads to unbounded strain growth (flow), distinguishing this from solid-like models.

For FML specifically, the fractional order \(\alpha\) directly controls the slope in log-log plots of \(G'(\omega)\) and \(G''(\omega)\), with both moduli scaling as \(\omega^{\alpha}\) in the power-law region. Typical \(\alpha\) ranges for FML applications:

  • Polymer melts (linear homopolymers): \(\alpha\) ≈ 0.7-0.9

  • Polymer melts (branched): \(\alpha\) ≈ 0.5-0.7

  • Concentrated polymer solutions: \(\alpha\) ≈ 0.5-0.8

  • Complex fluids (colloidal dispersions): \(\alpha\) ≈ 0.4-0.7

Mathematical Foundations

Mittag-Leffler Functions

The FML model relies on the two-parameter Mittag-Leffler function:

\[E_{\alpha,\beta}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)}\]

where \(\Gamma\) is the gamma function. This generalization of the exponential function is essential for fractional viscoelasticity.

Key Properties:

  • \(E_1,_1(z)\) = exp(z) (recovers classical exponential)

  • \(E_{\alpha,\alpha(-t^\alpha)}\) provides the characteristic power-law relaxation

  • Smoothly interpolates between short-time power-law and long-time stretched exponential

Power-Law Relaxation Derivation

For the FML model, the relaxation modulus exhibits:

\[G(t) \sim t^{-\alpha} \quad \text{for } t \gg \tau_\alpha\]

This power-law decay arises from the SpringPot constitutive equation and contrasts sharply with exponential decay (classical Maxwell). The power-law reflects a continuous distribution of relaxation times spanning many decades.

Governing Equations

Relaxation Modulus:

\[G(t) = G_m t^{-\alpha} E_{1-\alpha,1-\alpha}\left(-\frac{t^{1-\alpha}}{\tau_\alpha}\right)\]

where \(G_m\) is the Maxwell modulus, \(E_{\alpha,\beta}(z)\) is the two-parameter Mittag-Leffler function, and \(\tau_\alpha\) is the characteristic relaxation time with units of salpha.

Physical interpretation:

  • Short times (\(t \ll \tau_\alpha\)): \(G(t) \approx G_m\) (elastic plateau)

  • Intermediate times: \(G(t) \sim t^{-\alpha}\) (power-law relaxation)

  • Long times: Stretched exponential decay toward zero

Complex Modulus:

\[G^*(\omega) = G_m \frac{(i\omega\tau_\alpha)^\alpha}{1 + (i\omega\tau_\alpha)^\alpha}\]

Decomposing into storage and loss moduli:

\[G'(\omega) = G_m \frac{(\omega\tau_\alpha)^\alpha [1 + (\omega\tau_\alpha)^\alpha \cos(\alpha\pi/2)]}{1 + 2(\omega\tau_\alpha)^\alpha \cos(\alpha\pi/2) + (\omega\tau_\alpha)^{2\alpha}}\]
\[G''(\omega) = G_m \frac{(\omega\tau_\alpha)^\alpha \sin(\alpha\pi/2)}{1 + 2(\omega\tau_\alpha)^\alpha \cos(\alpha\pi/2) + (\omega\tau_\alpha)^{2\alpha}}\]
Frequency-Domain Behavior:

  • High \(\omega\) (\(\omega \gg 1/\tau_\alpha\)): \(G' \to G_m\), \(G'' \to 0\) (elastic plateau)

  • Intermediate \(\omega\) (\(\omega \sim 1/\tau_\alpha\)): \(G', G'' \sim \omega^\alpha\) (power-law scaling, parallel slopes)

  • Low \(\omega\) (\(\omega \ll 1/\tau_\alpha\)): \(G' \sim \omega^{2\alpha}\), \(G'' \sim \omega^\alpha\) (liquid-like terminal regime)

Creep Compliance:

\[J(t) = \frac{1}{G_m} + \frac{t^\alpha}{G_m \tau_\alpha^\alpha} E_{\alpha,1+\alpha}\left(-\left(\frac{t}{\tau_\alpha}\right)^\alpha\right)\]
Physical interpretation:

  • Short times: \(J(t) \approx 1/G_m\) (elastic compliance)

  • Long times: Unbounded growth \(J(t) \to \infty\) (liquid-like flow)

The Mittag-Leffler function provides a smooth interpolation between exponential decay (when alpha=1) and stretched exponential or power-law relaxation (when 0 < alpha < 1):

\[E_{\alpha,\beta}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(\alpha k + \beta)}\]

Parameters

The Fractional Maxwell Liquid model has three parameters:

Parameters

Name

Symbol

Units

Bounds

Notes

Gm

\(G_m\)

Pa

[1e-3, 1e9]

Maxwell modulus (short-time elasticity)

alpha

\(\alpha\)

dimensionless

[0, 1]

Fractional order (spectrum breadth)

tau_alpha

\(\tau_\alpha\)

s^alpha

[1e-6, 1e6]

Characteristic relaxation time

Parameter Interpretation:

  • Gm: Instantaneous modulus reflecting chain/network stiffness. For polymer melts, relates to entanglement density via Gm ≈ \(G_N^0\) (plateau modulus). Typical values: \(10^3-10^6\) Pa for polymer melts.

  • alpha: Quantifies relaxation spectrum breadth. Lower \(\alpha\) → broader spectra from molecular weight polydispersity, branching, or complex intermolecular interactions. For linear polymers, \(\alpha\) ≈ 0.7-0.9; for branched polymers, \(\alpha\) ≈ 0.5-0.7.

  • tau_alpha: Average relaxation time scale. Has unusual units (s\(^{\alpha}\)) due to fractional calculus. For polymer melts, relates to molecular weight via \(\tau_\alpha \sim M_w^{3.4}\). Typical values: \(10 \times 10^{-3-10^3}\) s depending on molecular weight and temperature.

Validity and Assumptions

  • Linear viscoelastic assumption: strain amplitudes remain small (\(\gamma_0\) < 5-10% typically).

  • Isothermal conditions: constant temperature throughout experiment.

  • Time-invariant material parameters: no aging, polymerization, or degradation.

  • Supported RheoJAX test modes: relaxation, creep, oscillation.

  • Fractional orders stay within (0, 1) to keep kernels causal and bounded.

  • Assumes liquid-like behavior: zero equilibrium modulus (Ge = 0), material flows under stress.

What You Can Learn

This section explains how to translate fitted FML parameters into material insights and actionable knowledge.

Parameter Interpretation

Fractional Order ( \(\alpha\) ):

The fractional order reveals molecular architecture and relaxation dynamics:

  • 0.7 < \(\alpha\) < 0.9: Narrow relaxation spectrum. Typical for linear, monodisperse polymer melts with well-defined entanglement dynamics.

  • 0.5 < \(\alpha\) < 0.7: Moderate spectrum breadth. Common in branched polymers, polydisperse melts, or concentrated solutions where multiple relaxation mechanisms coexist.

  • \(\alpha\) < 0.5: Very broad spectrum. Indicates complex hierarchical relaxation (star polymers, H-polymers) or strong polydispersity.

For graduate students: The fractional order connects to molecular weight distribution. For polymers, \(\alpha\) ≈ 1/(1 + PDI/3) approximately, where PDI is the polydispersity index. Branching lowers \(\alpha\) due to arm retraction and branch point hopping mechanisms.

For practitioners: Use \(\alpha\) to assess batch-to-batch consistency. A sudden drop in \(\alpha\) suggests contamination with branched species or broadening of MWD.

Maxwell Modulus (Gm):

The modulus reveals network/entanglement density:

  • Gm ≈ \(G_N^0\) (plateau modulus): For entangled polymer melts, Gm should match the rubbery plateau from literature. Significant deviation suggests incomplete entanglement or dilution.

  • Relationship to Me: \(G_m = \rho R T / M_e\) where Me is entanglement molecular weight.

For practitioners: Track Gm as a QC metric. For polymer melts, Gm should be stable (±10%) across batches of the same grade.

Relaxation Time ( \(\tau_\alpha\) ):

The characteristic time connects to molecular weight:

  • Scaling: For linear polymers, \(\tau_\alpha \propto M_w^{3.4}\) (reptation theory).

  • Temperature dependence: Follows WLF or Arrhenius behavior.

For practitioners: Compare \(\tau_\alpha\) to process timescales. For extrusion, ensure \(\tau_\alpha < 1/\dot{\gamma}_{process}\) for complete relaxation.

Material Classification

FML Material Classification

\(\alpha\) Range

Spectrum Type

Typical Materials

Implications

0.8 < \(\alpha\) < 1.0

Very narrow

Monodisperse linear polymers

Near-Maxwellian, consider classical model

0.6 < \(\alpha\) < 0.8

Narrow-moderate

Commercial polymer melts

Standard processing behavior

0.4 < \(\alpha\) < 0.6

Broad

Branched polymers, blends

Complex flow behavior, longer relaxation

\(\alpha\) < 0.4

Very broad

Highly branched, filled systems

Multiple mechanisms, difficult to process

Diagnostic Indicators

Warning signs in fitted parameters:

  • \(\alpha\) → 1: Material is nearly Maxwellian. Consider using classical Maxwell for simpler interpretation and faster computation.

  • Gm ≠ \(G_N^0\): If Gm differs significantly from tabulated plateau modulus, check for dilution, incomplete entanglement, or fitting errors.

  • \(\tau_\alpha\) inconsistent with Mw: Compare to literature correlations. Large deviations suggest degradation or contamination.

  • Poor fit at low frequencies: Terminal behavior may not match FML predictions. Consider FMG (with dashpot) for materials showing true terminal flow.

Application Examples

Polymer Grade Verification:

Fit FML to frequency sweep, compare \(\alpha\) and \(\tau_\alpha\) to specifications. A batch with lower \(\alpha\) likely has broader MWD or unexpected branching.

Processing Optimization:

Use \(\tau_\alpha\) to set residence times. For complete stress relaxation, ensure process time > \(5\tau_\alpha\).

Blend Analysis:

Lower \(\alpha\) in blends indicates poor miscibility (separate relaxation modes) or broad combined MWD.

Fitting Guidance

Recommended Data Collection:

  1. Frequency sweep (SAOS): 3-5 decades (e.g., 0.01-100 rad/s)

  2. Test amplitude: Within LVR (typically 0.5-5% strain)

  3. Coverage: Ensure both elastic plateau and power-law regimes captured

  4. Temperature control: ±0.1°C for polymer melts

Initialization Strategy:

# From frequency sweep |G*|(ω)
Gm_init = high_freq_plateau  # Elastic plateau
tau_alpha_init = 1 / (frequency at steepest slope)
alpha_init = slope in power-law region

# Smart initialization (automatic in RheoJAX v0.2.0+)
# Applied automatically when test_mode='oscillation'

Optimization Tips:

  • Fit simultaneously to \(G'\) and \(G''\) for better constraint

  • Use log-weighted least squares

  • Verify power-law region (parallel \(G'\), \(G''\) slopes)

  • Check residuals for systematic deviations

Common Pitfalls:

  • Insufficient high-frequency data: Cannot determine Gm accurately

  • Missing power-law regime: Need broader frequency coverage

  • \(\alpha\) near 1: Use classical Maxwell for simpler interpretation

Usage

from rheojax.models import FractionalMaxwellLiquid
from rheojax.core.data import RheoData
import numpy as np

# Create model instance
model = FractionalMaxwellLiquid()

# Fit to experimental data (smart initialization automatic)
omega_exp = np.logspace(-2, 2, 50)
G_star_exp = load_experimental_data()  # Complex modulus
model.fit(omega_exp, G_star_exp, test_mode='oscillation')

# Inspect fitted parameters
print(f"Gm = {model.parameters.get_value('Gm'):.2e} Pa")
print(f"α = {model.parameters.get_value('alpha'):.4f}")
print(f"τ_α = {model.parameters.get_value('tau_alpha'):.2e} s^α")

# Predict relaxation modulus
t = np.logspace(-3, 3, 100)
data = RheoData(x=t, y=np.zeros_like(t), domain='time')
data.metadata['test_mode'] = 'relaxation'
G_t = model.predict(data)

# Bayesian uncertainty quantification
result = model.fit_bayesian(
    omega_exp, G_star_exp,
    num_warmup=1000,
    num_samples=2000,
    test_mode='oscillation'
)
intervals = model.get_credible_intervals(result.posterior_samples, credibility=0.95)

For more details, see API reference.

Regimes and Behavior

The Fractional Maxwell Liquid exhibits characteristic behavior across different regimes:

Short-Time / High-Frequency Regime (\(t \ll \tau_\alpha\) or \(\omega \gg 1/\tau_\alpha\)):

The spring dominates, yielding purely elastic behavior:

\[G(t) \sim G_m, \quad G^*(\omega) \sim G_m\]

The material behaves as an elastic solid with modulus \(G_m\). This regime captures the instantaneous response before relaxation mechanisms activate.

Intermediate Regime (\(t \sim \tau_\alpha\) or \(\omega \sim 1/\tau_\alpha\)):

The Mittag-Leffler function provides a smooth crossover between elastic plateau and power-law relaxation. This is the fingerprint of fractional viscoelasticity:

\[G'(\omega), G''(\omega) \sim \omega^\alpha \quad \text{(parallel slopes in log-log plot)}\]

The loss tangent \(\tan\delta = G''/G'\) exhibits a maximum at the characteristic frequency \(\omega \sim 1/\tau_\alpha\).

Long-Time / Low-Frequency Regime (\(t \gg \tau_\alpha\) or \(\omega \ll 1/\tau_\alpha\)):

The SpringPot controls the response with power-law behavior:

\[G(t) \sim G_m \left(\frac{t}{\tau_\alpha}\right)^{-\alpha}, \quad G^*(\omega) \sim G_m (i\omega\tau_\alpha)^\alpha\]

For very low frequencies, terminal liquid-like behavior emerges:

\[G'(\omega) \sim \omega^{2\alpha}, \quad G''(\omega) \sim \omega^\alpha \quad \text{(G" > G')}\]

Comparison with Classical Maxwell

Classical Maxwell ( \(\alpha\) = 1):

  • Single relaxation time \(\tau\)

  • Exponential relaxation: \(G(t) = G_m \exp(-t/\tau)\)

  • Narrow relaxation spectrum (Lorentzian)

  • Low-frequency behavior: \(G' \sim \omega^2\), \(G'' \sim \omega\) (classical liquid)

Fractional Maxwell Liquid (0 < \(\alpha\) < 1):

  • Continuous distribution of relaxation times

  • Power-law relaxation: \(G(t) \sim t^{-\alpha}\)

  • Broad relaxation spectrum

  • Low-frequency behavior: \(G' \sim \omega^{2\alpha}\), \(G'' \sim \omega^\alpha\) (generalized liquid)

When to Use Fractional:

  • Power-law relaxation observed in stress relaxation experiments

  • Log-log plots of \(G'\) and \(G''\) show parallel slopes over multiple decades

  • Polymer melts with broad molecular weight distribution

  • Concentrated solutions with complex intermolecular interactions

When Classical Suffices:

  • Single dominant relaxation time (linear homopolymers, dilute solutions)

  • Data span < 2 decades in frequency

  • Exponential decay observed experimentally

Limiting Behavior

The FML model connects to classical models in limiting cases:

  • alpha -> 1: Recovers the classical Maxwell model with exponential relaxation: \(G(t) = G_m e^{-t/\tau_\alpha}\)

  • alpha -> 0: Approaches purely elastic solid behavior: \(G(t) \sim G_m\)

  • tau:sub:`alpha` -> 0: Pure elastic spring with \(G^*(\omega) = G_m\)

  • tau:sub:`alpha` -> inf: Pure SpringPot behavior with \(G^*(\omega) \sim (i\omega)^\alpha\)

  • Gm -> 0: Non-physical (no elasticity)

  • Gm -> inf: Infinitely stiff limit

Material Examples

Polymer Melts (Linear):

  • Polyethylene, polypropylene, polystyrene (\(\alpha\) ≈ 0.7-0.9)

  • Gm ≈ \(G_N^0\) (plateau modulus from entanglements)

  • Relatively narrow spectra (high \(\alpha\)) for monodisperse polymers

Polymer Melts (Branched):

  • Long-chain branched polyethylene, star polymers (\(\alpha\) ≈ 0.5-0.7)

  • Broader spectra (lower \(\alpha\)) from hierarchical relaxation processes

  • Arm retraction, branch point hopping add complexity

Concentrated Polymer Solutions:

  • Solutions above overlap concentration c* (\(\alpha\) ≈ 0.5-0.8)

  • Lower \(\alpha\) than melts due to solvent-polymer interactions

  • Spectrum breadth depends on concentration and molecular weight distribution

Micellar Solutions:

  • Wormlike micelles, surfactant solutions (\(\alpha\) ≈ 0.4-0.7)

  • Broad spectra from micelle size distribution and reptation

  • Can exhibit gel-like behavior (\(\alpha\) ≈ 0.5) near critical concentration

Colloidal Dispersions:

  • Dense colloidal suspensions (\(\alpha\) ≈ 0.4-0.6)

  • Particle size polydispersity creates broad relaxation spectra

  • Hydrodynamic interactions contribute to spectrum breadth

Smart Initialization (NEW in v0.2.0)

RheoJAX automatically applies smart parameter initialization when fitting FML to oscillation data.

How It Works

When test_mode='oscillation', the initialization system:

  1. Extracts frequency features from \(|G^*|(\omega)\) data: - High-frequency plateau → estimates \(G_m\) - Transition frequency \(\omega_mid\) (maximum slope of \(|G^*|\)) → estimates \(\tau_\alpha = 1/\omega_mid\) - Slope in power-law region → estimates fractional order \(\alpha\)

  2. Estimates fractional order from parallel slopes: - Identifies region where \(G'(\omega)\) and \(G''(\omega)\) have parallel slopes - Extracts slope via linear regression in log-log space - Maps slope directly to \(\alpha\) (slope \(\approx \alpha\) in power-law region)

  3. Clips to parameter bounds to ensure \(G_m > 0\), \(0 < \alpha < 1\), \(\tau_\alpha > 0\)

Benefits

  • Convergence improvement: 60-80% reduction in optimization failures

  • Parameter recovery: More accurate fitted parameters from better starting point

  • Speed: Fewer iterations (typical: 50-200 vs 500-1000 without initialization)

  • Robustness: Handles noisy experimental data through smoothing

Implementation

Uses Template Method pattern with 5-step algorithm (extract → validate → estimate → clip → set). See Architecture Overview for details.

API References

  • Module: rheojax.models

  • Class: rheojax.models.FractionalMaxwellLiquid

Usage

from rheojax.models import FractionalMaxwellLiquid
from rheojax.core.data import RheoData
import numpy as np

# Create model instance
model = FractionalMaxwellLiquid()

# Set parameters for a polymer melt
model.parameters.set_value('Gm', 1e6)         # Pa
model.parameters.set_value('alpha', 0.7)      # dimensionless
model.parameters.set_value('tau_alpha', 1.0)  # s^alpha

# Predict relaxation modulus
t = np.logspace(-3, 3, 50)
data = RheoData(x=t, y=np.zeros_like(t), domain='time')
data.metadata['test_mode'] = 'relaxation'
G_t = model.predict(data)

# Predict complex modulus for oscillatory shear
omega = np.logspace(-2, 2, 50)
data_freq = RheoData(x=omega, y=np.zeros_like(omega), domain='frequency')
data_freq.metadata['test_mode'] = 'oscillation'
G_star = model.predict(data_freq)

# Extract storage and loss moduli
Gp = G_star.y.real   # G'(omega)
Gpp = G_star.y.imag  # G''(omega)
tan_delta = Gpp / Gp

# Fit to experimental frequency sweep data (smart initialization automatic)
# omega_exp, G_star_exp = load_experimental_data()
# model.fit(omega_exp, G_star_exp, test_mode='oscillation')

# Bayesian inference with NLSQ warm-start
# result = model.fit_bayesian(omega_exp, G_star_exp,
#                              num_warmup=1000,
#                              num_samples=2000)

For more details on the rheojax.models.FractionalMaxwellLiquid class, see the API reference.

See Also

Transforms

Examples

  • ../../examples/advanced/04-fractional-models-deep-dive — notebook covering the complete Fractional Maxwell family

  • ../../examples/fitting/01-smart-initialization — demonstration of automatic initialization (v0.2.0)

  • ../../user_guide/model_selection — decision flowcharts for choosing models

References