Fractional Maxwell Gel (Fractional)

Quick Reference

• Use when: Critical gels, power-law viscoelasticity transitioning to terminal flow

• Parameters: 3 (\(c_\alpha\), \(\alpha\), \(\eta\))

• Key equation: \(G(t) = c_\alpha t^{-\alpha} E_{1-\alpha,1-\alpha}(-t^{1-\alpha}/\tau)\) where \(\tau = \eta / c_\alpha^{1/(1-\alpha)}\)

• Test modes: Oscillation, relaxation, creep

• Material examples: Critical gels, wormlike micelles, weak polymer networks, polymer solutions near gel point

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 Gel (FMG) model consists of a SpringPot element (fractional viscoelastic element) in series with a Newtonian dashpot. This configuration captures the rheological behavior of materials transitioning from power-law viscoelastic response at short times to terminal viscous flow at long times.

Notation Guide

Symbol	Meaning
\(c_\alpha\)	SpringPot quasi-property (Pa·salpha). Controls the elastic stiffness scale.
\(\alpha\)	Fractional order (0 < \(\alpha\) < 1). Controls the relaxation slope (0=solid, 1=liquid).
\(\eta\)	Dashpot viscosity (Pa·s). Controls terminal flow at long times.
\(\tau\)	Characteristic relaxation time (s), \(\tau = \eta / c_\alpha^{1/(1-\alpha)}\).
\(E_{\alpha,\beta}\)	Mittag-Leffler function (generalized exponential).

Overview

The FMG model is particularly effective for describing polymer solutions, physical gels, and soft materials exhibiting gel-like characteristics with eventual viscous dissipation—materials that behave as soft solids at short timescales but flow like liquids over extended durations.

The SpringPot element provides fractional-order power-law viscoelasticity characterized by a broad relaxation spectrum, while the series dashpot ensures terminal flow behavior (\(G(t \to \infty) \to 0\)). This combination makes the FMG model especially suitable for materials that exhibit intermediate behavior between pure elastic solids and Newtonian liquids, such as critical gels evolving toward sol states, wormlike micelle solutions, and weak polymer networks undergoing structural rearrangement.

Physical Foundations

The Fractional Maxwell Gel extends the classical Maxwell model by replacing the spring with a SpringPot:

Mechanical Analogue:

[SpringPot (c_α, α)] ---- series ---- [Dashpot (η)]

The SpringPot provides power-law elasticity while the dashpot guarantees liquid-like behavior at long times.

Microstructural Interpretation:

• SpringPot contribution: Broad distribution of network relaxation modes (chain rearrangements, bond breaking/reformation)

• Dashpot contribution: Irreversible viscous flow from chain reptation or solvent drag

• Combined behavior: Gel-like response at short times transitions to flow at long times

Governing Equations

Relaxation Modulus:

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

where:

• \(E_{\alpha,\beta}(z)\) = two-parameter Mittag-Leffler function

• \(\tau = \eta / c_\alpha^{1/(1-\alpha)}\) = characteristic relaxation time (s)

• \(c_\alpha\) = SpringPot quasi-property (Pa·s^alpha)

• \(\alpha\) = fractional order in (0, 1)

• \(\eta\) = dashpot viscosity (Pa·s)

Complex Modulus (Oscillatory):

\[G^*(\omega) = c_\alpha (i\omega)^\alpha \cdot \frac{i\omega\tau}{1 + i\omega\tau}\]

Decomposed into storage and loss moduli:

\[\begin{split}G'(\omega) &= c_\alpha \omega^{\alpha} \left[\cos\left(\frac{\alpha\pi}{2}\right) \frac{(\omega\tau)^2}{1 + (\omega\tau)^2} + \sin\left(\frac{\alpha\pi}{2}\right) \frac{\omega\tau}{1 + (\omega\tau)^2}\right] \\ G''(\omega) &= c_\alpha \omega^{\alpha} \left[\sin\left(\frac{\alpha\pi}{2}\right) \frac{(\omega\tau)^2}{1 + (\omega\tau)^2} - \cos\left(\frac{\alpha\pi}{2}\right) \frac{\omega\tau}{1 + (\omega\tau)^2}\right]\end{split}\]

Creep Compliance:

\[J(t) = \frac{1}{c_\alpha} t^\alpha E_{1+\alpha,1+\alpha}\left(-\left(\frac{t}{\tau}\right)^{1-\alpha}\right)\]

Shows bounded creep at short times transitioning to unbounded viscous flow at long times.

Mittag-Leffler Function

The two-parameter Mittag-Leffler function \(E_{\alpha,\beta}(z)\) is defined by:

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

Special Cases:

• \(E_{1,1}(z) = e^z\) → exponential (classical Maxwell)

• \(E_{\alpha,1}(z)\) → one-parameter Mittag-Leffler

• \(E_{2,1}(-z^2) = \cos(z)\) → oscillatory behavior

Asymptotic Behavior:

• Small argument (\(|z| \ll 1\)): \(E_{\alpha,\beta}(z) \approx 1/\Gamma(\beta) + z/\Gamma(\alpha + \beta)\)

• Large argument (\(|z| \gg 1, z < 0\)): \(E_{\alpha,\beta}(z) \sim |z|^{-1}/\Gamma(\beta - \alpha)\) → power-law decay

These asymptotics produce the crossover from power-law to viscous behavior in FMG.

Parameters

Parameters

Name	Symbol	Units	Bounds	Notes
c_alpha	\(c_\alpha\)	Pa·salpha	[1e-3, 1e9]	SpringPot material constant (sets modulus scale)
alpha	\(\alpha\)	dimensionless	[0.05, 0.95]	Power-law exponent (0.3-0.7 typical for gels)
eta	\(\eta\)	Pa·s	[1e-6, 1e12]	Dashpot viscosity (controls terminal flow)

Physical Meaning of \(\alpha\)

The fractional order \(\alpha\) characterizes the viscoelastic character:

• \(\alpha < 0.5\): Solid-like (\(G' > G''\) at intermediate frequencies)

• \(\alpha = 0.5\): Critical gel signature (\(G' \sim G'' \propto \omega^{0.5}\))

• \(\alpha > 0.5\): Liquid-like (\(G'' > G'\) at low frequencies)

Material Ranges:

• Polymer gels: \(\alpha \approx 0.3-0.6\)

• Wormlike micelles: \(\alpha \approx 0.4-0.7\)

• Weak networks: \(\alpha \approx 0.2-0.5\)

• Colloidal gels: \(\alpha \approx 0.3-0.5\)

Regimes and Behavior

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

SpringPot dominates, yielding power-law behavior:

\[G(t) \sim c_\alpha t^{-\alpha}, \quad G^*(\omega) \sim c_\alpha (i\omega)^\alpha\]

Material behaves as a fractional gel with broad relaxation spectrum.

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

Dashpot controls the response, leading to terminal viscous flow:

\[G(t) \sim \frac{\eta}{t}, \quad G''(\omega) \sim \omega\eta, \quad G'(\omega) \sim \omega^2\]

Material flows like a Newtonian liquid with viscosity \(\eta\).

Intermediate Regime (\(t \sim \tau\)):

Mittag-Leffler function provides smooth crossover between power-law and viscous regimes. The characteristic time \(\tau\) marks the transition from gel-like to liquid-like behavior.

Validity and Assumptions

• Linear viscoelasticity: Strain amplitudes remain small (< 5-10% typically)

• Isothermal conditions: Temperature constant throughout measurement

• Time-invariant material: No aging, gelation, or structural evolution

• Supported test modes: Oscillation, relaxation, creep

• Fractional order bounds: 0.05 < \(\alpha\) < 0.95 for numerical stability

• Liquid-like behavior: Zero equilibrium modulus (material flows under stress)

• Terminal flow: Dashpot ensures \(G(t \to \infty) \to 0\) and unbounded creep

Material Examples

Polymer Solutions (\(c_\alpha \approx 10^2-10^4\) Pa·s^alpha, \(\alpha \approx 0.4-0.6\), \(\eta \approx 10-10^3\) Pa·s):

• Polyacrylamide solutions (5-10 wt%)

• PEO (polyethylene oxide) in water

• Xanthan gum solutions

Physical Gels (\(c_\alpha \approx 10^3-10^5\), \(\alpha \approx 0.3-0.5\), \(\eta \approx 10^2-10^4\)):

• Gelatin gels near sol-gel transition

• Agar gels at low concentration (< 1%)

• Alginate gels (weak cross-linking)

Wormlike Micelle Solutions (\(\alpha \approx 0.5-0.7\), \(\eta \approx 1-100\) Pa·s):

• CTAB (cetyltrimethylammonium bromide) micelles

• CPyCl/NaSal (cetylpyridinium chloride/sodium salicylate)

Colloidal Gels (\(\alpha \approx 0.3-0.5\), \(\eta \approx 10-10^3\)):

• Carbon black suspensions

• Silica gel networks

Experimental Design

Frequency Sweep (SAOS):

1. Frequency range: 0.01-100 rad/s (minimum 3 decades)

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

3. Identify regimes: - High \(\omega\): Power-law with slope \(\alpha\) - Low \(\omega\): Terminal flow (\(G'' \sim \omega\), \(G' \sim \omega^2\))

4. Crossover frequency: \(\omega_c \approx 1/\tau\) where regime transition occurs

Stress Relaxation:

1. Step strain: \(\gamma_0 = 1-5\%\) within LVR

2. Time span: Cover 4-5 decades (e.g., 0.01-\(10^3\) s)

3. Sampling: Log-spaced to capture both regimes

4. Analysis: Early-time power-law → late-time viscous decay

Creep Test:

1. Constant stress: Within LVR

2. Time span: Long enough to observe viscous flow (> \(10^3\) s)

3. Expected: Bounded creep → unbounded flow

Fitting Strategies

Smart Initialization (v0.2.0):

RheoJAX automatically initializes FMG parameters from oscillation data using frequency-domain analysis:

1. Estimate \(c_\alpha\) from high-frequency plateau

2. Estimate \(\alpha\) from power-law slope in intermediate regime

3. Estimate \(\eta\) from low-frequency terminal behavior (\(G'' \sim \omega\eta\))

4. Estimate \(\tau = 1/\omega_c\) from crossover frequency

Manual Initialization:

# From frequency sweep log-log plot
alpha_init = slope_of_log_Gp_vs_log_omega  # intermediate regime
eta_init = Gpp_low_freq / omega_low        # terminal region
c_alpha_init = Gp_high_freq / (omega_high**alpha * cos(pi*alpha/2))
tau_init = 1 / omega_crossover

Optimization Tips:

• Fit in log-space for better conditioning

• Constrain \(\alpha\) bounds to [0.1, 0.9] to avoid singularities

• Use NLSQ optimizer (5-270x faster than scipy)

• Verify residuals show no systematic trends

Model Comparison

FMG vs FML (Fractional Maxwell Liquid):

• FMG: SpringPot + dashpot → power-law + terminal flow

• FML: SpringPot + spring → power-law + equilibrium plateau

• Use FMG for flowing gels; FML for soft solids

FMG vs Classical Maxwell:

• Maxwell: Exponential relaxation (\(\alpha = 1\))

• FMG: Power-law relaxation (\(0 < \alpha < 1\), broad spectrum)

• FMG reduces to Maxwell as \(\alpha \to 1\)

FMG vs Fractional Burgers:

• FMG: 3 parameters, single relaxation mode

• Burgers: 5 parameters, adds retardation mode (delayed elasticity)

• Use Burgers for complex creep with multiple timescales

Limiting Behavior

• \(\alpha \to 1\): Approaches classical Maxwell (\(G^*(\omega) \sim i\omega\eta\))

• \(\alpha \to 0\): Approaches elastic spring in series with dashpot

• \(\eta \to \infty\): Reduces to pure SpringPot (\(G^*(\omega) = c_\alpha (i\omega)^\alpha\))

• \(\eta \to 0\): Non-physical (no dissipation mechanism)

• \(c_\alpha \to 0\): Pure dashpot (\(G^*(\omega) = i\omega\eta\))

What You Can Learn

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

Parameter Interpretation

Fractional Order ( \(\alpha\) ):

The fractional order reveals the breadth of the relaxation spectrum and proximity to the gel point:

• \(\alpha\) < 0.3: Very broad spectrum, highly heterogeneous network. Common in dense colloidal gels or materials with strong polydispersity.

• 0.3 < \(\alpha\) < 0.5: Intermediate behavior. Typical for physical gels with moderate cross-link density or entangled polymer solutions.

• \(\alpha \approx 0.5\): Critical gel signature (Winter-Chambon criterion). Material is at or near the gel point with \(G' \approx G'' \propto \omega^{0.5}\).

• 0.5 < \(\alpha\) < 0.7: Liquid-dominant behavior. Typical for wormlike micelles and weakly associated polymers where flow dominates.

• \(\alpha\) > 0.7: Nearly Maxwellian. Consider using classical Maxwell model for simpler interpretation.

For graduate students: The fractional order relates to the fractal dimension of the network. For percolating gels at the gel point, \(\alpha = d_f / (d_f + 2)\) where \(d_f\) is the fractal dimension. This connects rheology to network structure.

For practitioners: Target \(\alpha \approx 0.4-0.6\) for stable gel textures. Values approaching 0.5 indicate proximity to sol-gel transition— small formulation changes can dramatically shift behavior.

SpringPot Quasi-Property ( \(c_{\alpha}\) ):

The quasi-property sets the modulus scale:

• Low \(c_{\alpha}\) (< 100 Pa·s^ \(\alpha\) ): Weak network. Soft, easily deformable gel.

• Moderate \(c_\alpha\) (100–10⁴ Pa·s^ \(\alpha\) ): Typical gel strength for most applications.

• High \(c_\alpha\) (> 10⁴ Pa·s^ \(\alpha\) ): Stiff network. Strong gel with high elastic character.

For graduate students: The quasi-property relates to network density and strand stiffness. For polymer gels, \(c_\alpha \propto \nu k_B T\) where \(\nu\) is network strand density.

For practitioners: Use \(c_\alpha\) as a QC metric for gel strength. A 50% drop indicates network degradation or incomplete gelation.

Terminal Viscosity ( \(\eta\) ):

The dashpot viscosity controls long-time flow:

• High \(\eta (> 10^3\) Pa·s): Slow flow at long times. Material maintains shape for extended periods but will eventually sag or level.

• Moderate \(\eta (10-10^3\) Pa·s): Balanced behavior. Typical for controlled- release applications.

• Low \(\eta\) (< 10 Pa·s): Rapid terminal flow. Material levels quickly once network relaxes.

For practitioners: The ratio \(\tau = \eta/c_\alpha^{1/(1-\alpha)}\) is the characteristic time for gel-to-liquid transition. For stability, ensure \(\tau\) exceeds your process timescale.

Material Classification

FMG Material Classification

\(\alpha\) Range	Material State	Typical Materials	Process Implications
\(\alpha\) < 0.4	Strong gel	Dense colloidal gels, stiff hydrogels	Good shape retention, difficult to pump
0.4 < \(\alpha\) < 0.55	Critical gel	Polymer gels near gel point, weak networks	Sensitive to conditions, handle carefully
0.55 < \(\alpha\) < 0.7	Weak gel / sol	Wormlike micelles, associative polymers	Easy flow, may not hold shape
\(\alpha\) > 0.7	Near-Maxwellian	Dilute polymer solutions	Use classical Maxwell model

Diagnostic Indicators

Warning signs in fitted parameters:

• \(\alpha \to 0\) or \(\to 1\): Model may be inappropriate. Check if SpringPot-only or classical Maxwell fits better.

• Large uncertainty in \(\alpha\): Data don’t span sufficient frequency range. Extend measurements to capture both regimes.

• \(\eta\) poorly constrained: Low-frequency data insufficient. Extend to lower frequencies or use creep tests to capture terminal flow.

• \(c_{\alpha}\) and \(\eta\) strongly correlated: The characteristic time \(\tau\) is well- determined but individual parameters are not. Report \(\tau\) instead.

Application Examples

Gel Formulation Development:

Track \(\alpha\) as crosslinker is added. Approach to \(\alpha \approx 0.5\) indicates proximity to gel point. For stable gels, target \(\alpha < 0.45\) with sufficient margin from the transition.

Quality Control:

Monitor \(c_{\alpha}\) batch-to-batch. A ±20% specification catches network degradation while allowing normal variation.

Process Design:

Calculate \(\tau\) to determine when material transitions from gel-like to flowable. For coating applications, ensure \(\tau\) exceeds leveling time to prevent sagging.

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 power-law and terminal flow regimes captured

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

Initialization Strategy (Automatic in RheoJAX v0.2.0+):

# Smart initialization applied automatically when test_mode='oscillation'
# From frequency sweep |G*|(ω):
c_alpha_init = high_freq_plateau  # SpringPot quasi-property
tau_init = 1 / (frequency at crossover to terminal regime)
alpha_init = slope in power-law region
eta_init = G''(ω → 0) / ω  # Low-frequency terminal viscosity

Optimization Tips:

• Use smart initialization (automatic for oscillation mode)

• Fit in log-space for better conditioning

• Constrain \(\alpha\) bounds to [0.1, 0.9] to avoid singularities

• Use NLSQ optimizer (5-270x faster than scipy)

• Verify residuals show no systematic trends

Common Pitfalls:

• Insufficient low-frequency data: Cannot determine \(\eta\) accurately

• Missing power-law regime: Need broader frequency coverage

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

Usage

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

# Create model instance
model = FractionalMaxwellGel()

# Frequency sweep (wormlike micelle solution)
omega = np.logspace(-2, 2, 50)
G_star_exp = load_experimental_data()  # Complex modulus

# Automatic smart initialization + fit (v0.2.0)
model.fit(omega, G_star_exp, test_mode='oscillation')

# Inspect fitted parameters
print(f"c_alpha = {model.parameters.get_value('c_alpha'):.2e} Pa·s^α")
print(f"alpha = {model.parameters.get_value('alpha'):.4f}")
print(f"eta = {model.parameters.get_value('eta'):.2e} Pa·s")
tau = model.parameters.get_value('eta') / model.parameters.get_value('c_alpha')**(1/(1-model.parameters.get_value('alpha')))
print(f"tau = {tau:.2e} s")

# 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, G_star_exp,
    num_warmup=1000,
    num_samples=2000,
    test_mode='oscillation'
)
ci = model.get_credible_intervals(result.posterior_samples, credibility=0.95)

For more details, see API reference.

Troubleshooting

Common Fitting Issues

Symptom	Possible Cause	Solution
Poor fit in terminal regime	Insufficient low-frequency data	Extend frequency sweep to lower \(\omega\) or use longer relaxation test.
\(\alpha \to 1\)	Material is nearly Maxwellian	Use classical Maxwell model instead (narrow spectrum).
Oscillatory residuals at high \(\omega\)	Multiple relaxation modes	Use Fractional Maxwell Model (FMM) which has two fractional orders.
Non-convergence	Poor initial guess or parameter correlation	Use Smart Initialization (automatic in v0.2.0) or warm-start with NLSQ.

Tips & Best Practices

1. Verify regimes: Plot \(\log(G')\), \(\log(G'')\) vs \(\log(\omega)\) to confirm power-law and terminal regions

2. Use smart initialization: Automatic in RheoJAX v0.2.0 for oscillation mode

3. Check Mittag-Leffler implementation: RheoJAX uses optimized JAX-based computation

4. Bayesian inference: Quantify parameter uncertainty with fit_bayesian()

5. Warm-start: Use NLSQ fit to initialize NUTS sampling (2-5x faster convergence)

References

[1]

Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press (2010). https://doi.org/10.1142/p614

[2]

Bagley, R. L., and Torvik, P. J. “On the fractional calculus model of viscoelastic behavior.” Journal of Rheology, 30, 133–155 (1986). https://doi.org/10.1122/1.549887

[3]

Friedrich, C. “Relaxation and retardation functions of the Maxwell model with fractional derivatives.” Rheologica Acta, 30, 151–158 (1991). https://doi.org/10.1007/BF01134604

[4]

Gorenflo, R., Kilbas, A. A., Mainardi, F., and Rogosin, S. V. Mittag-Leffler Functions, Related Topics and Applications. Springer (2014). https://doi.org/10.1007/978-3-662-43930-2

[5]

Hilfer, R. (ed.) Applications of Fractional Calculus in Physics. World Scientific (2000). ISBN: 978-9810234577. https://doi.org/10.1142/3779

[6]

Scott Blair, G. W., Veinoglou, B. C., and Caffyn, J. E. “Limitations of the Newtonian time scale in relation to non-equilibrium rheological states.” Proceedings of the Royal Society A, 189, 69–87 (1947). https://doi.org/10.1098/rspa.1947.0029

[7]

Winter, H. H., and Chambon, F. “Analysis of linear viscoelasticity of a crosslinking polymer at the gel point.” Journal of Rheology, 30, 367–382 (1986). https://doi.org/10.1122/1.549853

[8]

Metzler, R., and Nonnenmacher, T. F. “Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials.” International Journal of Plasticity, 19, 941–959 (2003). https://doi.org/10.1016/S0749-6419(02)00087-6

[9]

Schiessel, H., and Blumen, A. “Hierarchical analogues to fractional relaxation equations.” Journal of Physics A, 26, 5057–5069 (1993). https://doi.org/10.1088/0305-4470/26/19/034

[10]

Jaishankar, A., and McKinley, G. H. “Power-law rheology in the bulk and at the interface.” Proceedings of the Royal Society A, 469, 20120284 (2013). https://doi.org/10.1098/rspa.2012.0284

See Also

Related Models

• Fractional Maxwell Liquid (Fractional) — complementary model with spring instead of dashpot (solid-like equilibrium)

• Generalized Fractional Maxwell (Two-Order) — generalized two-order formulation with independent \(\alpha\) and \(\beta\)

• Fractional Burgers Model (Fractional) — adds Kelvin branch for delayed elasticity and creep

• Maxwell (Classical) — classical limit (\(\alpha\) → 1, exponential relaxation)

• SpringPot (Scott-Blair Element) — fundamental SpringPot element theory

Transforms

• OWChirp — broadband LAOS sweeps to estimate fractional slopes

• FFTAnalysis — convert relaxation data to frequency domain for fitting

• MutationNumber — monitor gel-to-sol transitions in curing/aging systems

• Mastercurve (Time-Temperature Superposition) — time-temperature superposition for temperature-dependent \(\tau\)

Examples

• ../../examples/advanced/04-fractional-models-deep-dive — tutorial comparing Fractional Maxwell family

• ../../examples/bayesian/02-fractional-gel-uncertainty — uncertainty quantification for FMG

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