.. _model-fractional-maxwell-gel:

Fractional Maxwell Gel (Fractional)
===================================

Quick Reference
---------------

- **Use when:** Critical gels, power-law viscoelasticity transitioning to terminal flow
- **Parameters:** 3 (:math:`c_\alpha`, :math:`\alpha`, :math:`\eta`)
- **Key equation:** :math:`G(t) = c_\alpha t^{-\alpha} E_{1-\alpha,1-\alpha}(-t^{1-\alpha}/\tau)` where :math:`\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

.. include:: /_includes/fractional_seealso.rst

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
--------------

.. list-table::
   :widths: 15 85
   :header-rows: 1

   * - Symbol
     - Meaning
   * - :math:`c_\alpha`
     - SpringPot quasi-property (Pa·s\ :sup:`\alpha`). Controls the elastic stiffness scale.
   * - :math:`\alpha`
     - Fractional order (0 < :math:`\alpha` < 1). Controls the relaxation slope (0=solid, 1=liquid).
   * - :math:`\eta`
     - Dashpot viscosity (Pa·s). Controls terminal flow at long times.
   * - :math:`\tau`
     - Characteristic relaxation time (s), :math:`\tau = \eta / c_\alpha^{1/(1-\alpha)}`.
   * - :math:`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** (:math:`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:**

.. code-block:: text

   [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:**

.. math::

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

where:

- :math:`E_{\alpha,\beta}(z)` = two-parameter Mittag-Leffler function
- :math:`\tau = \eta / c_\alpha^{1/(1-\alpha)}` = characteristic relaxation time (s)
- :math:`c_\alpha` = SpringPot quasi-property (Pa·s\ :sup:`\alpha`)
- :math:`\alpha` = fractional order in (0, 1)
- :math:`\eta` = dashpot viscosity (Pa·s)

**Complex Modulus (Oscillatory):**

.. math::

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

Decomposed into storage and loss moduli:

.. math::

   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]

**Creep Compliance:**

.. math::

   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** :math:`E_{\alpha,\beta}(z)` is defined by:

.. math::

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

**Special Cases:**

- :math:`E_{1,1}(z) = e^z` → exponential (classical Maxwell)
- :math:`E_{\alpha,1}(z)` → one-parameter Mittag-Leffler
- :math:`E_{2,1}(-z^2) = \cos(z)` → oscillatory behavior

**Asymptotic Behavior:**

- **Small argument** (:math:`|z| \ll 1`): :math:`E_{\alpha,\beta}(z) \approx 1/\Gamma(\beta) + z/\Gamma(\alpha + \beta)`
- **Large argument** (:math:`|z| \gg 1, z < 0`): :math:`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
----------

.. list-table:: Parameters
   :header-rows: 1
   :widths: 18 12 12 18 40

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

Physical Meaning of :math:`\alpha`
-----------------------------------

The fractional order :math:`\alpha` characterizes the **viscoelastic character**:

- :math:`\alpha < 0.5`: Solid-like (:math:`G' > G''` at intermediate frequencies)
- :math:`\alpha = 0.5`: Critical gel signature (:math:`G' \sim G'' \propto \omega^{0.5}`)
- :math:`\alpha > 0.5`: Liquid-like (:math:`G'' > G'` at low frequencies)

**Material Ranges:**

- **Polymer gels**: :math:`\alpha \approx 0.3-0.6`
- **Wormlike micelles**: :math:`\alpha \approx 0.4-0.7`
- **Weak networks**: :math:`\alpha \approx 0.2-0.5`
- **Colloidal gels**: :math:`\alpha \approx 0.3-0.5`

Regimes and Behavior
--------------------

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

SpringPot dominates, yielding power-law behavior:

.. math::

   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** (:math:`t \gg \tau` or :math:`\omega \ll 1/\tau`):

Dashpot controls the response, leading to **terminal viscous flow**:

.. math::

   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 :math:`\eta`.

**Intermediate Regime** (:math:`t \sim \tau`):

Mittag-Leffler function provides smooth crossover between power-law and viscous regimes. The characteristic time :math:`\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 < :math:`\alpha` < 0.95 for numerical stability
- **Liquid-like behavior**: Zero equilibrium modulus (material flows under stress)
- **Terminal flow**: Dashpot ensures :math:`G(t \to \infty) \to 0` and unbounded creep

Material Examples
-----------------

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

- **Polyacrylamide solutions** (5-10 wt%)
- **PEO (polyethylene oxide)** in water
- **Xanthan gum** solutions

**Physical Gels** (:math:`c_\alpha \approx 10^3-10^5`, :math:`\alpha \approx 0.3-0.5`, :math:`\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** (:math:`\alpha \approx 0.5-0.7`, :math:`\eta \approx 1-100` Pa·s):

- **CTAB** (cetyltrimethylammonium bromide) micelles
- **CPyCl/NaSal** (cetylpyridinium chloride/sodium salicylate)

**Colloidal Gels** (:math:`\alpha \approx 0.3-0.5`, :math:`\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 :math:`\omega`: Power-law with slope :math:`\alpha`
   - Low :math:`\omega`: Terminal flow (:math:`G'' \sim \omega`, :math:`G' \sim \omega^2`)
4. **Crossover frequency**: :math:`\omega_c \approx 1/\tau` where regime transition occurs

**Stress Relaxation:**

1. **Step strain**: :math:`\gamma_0 = 1-5\%` within LVR
2. **Time span**: Cover 4-5 decades (e.g., 0.01-:math:`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 (> :math:`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** :math:`c_\alpha` from high-frequency plateau
2. **Estimate** :math:`\alpha` from power-law slope in intermediate regime
3. **Estimate** :math:`\eta` from low-frequency terminal behavior (:math:`G'' \sim \omega\eta`)
4. **Estimate** :math:`\tau = 1/\omega_c` from crossover frequency

**Manual Initialization:**

.. code-block:: python

   # 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 :math:`\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 (:math:`\alpha = 1`)
- **FMG**: Power-law relaxation (:math:`0 < \alpha < 1`, broad spectrum)
- FMG reduces to Maxwell as :math:`\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
-----------------

- :math:`\alpha \to 1`: Approaches classical Maxwell (:math:`G^*(\omega) \sim i\omega\eta`)
- :math:`\alpha \to 0`: Approaches elastic spring in series with dashpot
- :math:`\eta \to \infty`: Reduces to pure SpringPot (:math:`G^*(\omega) = c_\alpha (i\omega)^\alpha`)
- :math:`\eta \to 0`: Non-physical (no dissipation mechanism)
- :math:`c_\alpha \to 0`: Pure dashpot (:math:`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 (** :math:`\alpha` **)**:
   The fractional order reveals the breadth of the relaxation spectrum and
   proximity to the gel point:

   - :math:`\alpha` **< 0.3**: Very broad spectrum, highly heterogeneous network. Common in
     dense colloidal gels or materials with strong polydispersity.

   - **0.3 <** :math:`\alpha` **< 0.5**: Intermediate behavior. Typical for physical gels with
     moderate cross-link density or entangled polymer solutions.

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

   - **0.5 <** :math:`\alpha` **< 0.7**: Liquid-dominant behavior. Typical for wormlike micelles
     and weakly associated polymers where flow dominates.

   - :math:`\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,
   :math:`\alpha = d_f / (d_f + 2)` where :math:`d_f` is the fractal dimension.
   This connects rheology to network structure.

   *For practitioners*: Target :math:`\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 (** :math:`c_{\alpha}` **)**:
   The quasi-property sets the modulus scale:

   - **Low** :math:`c_{\alpha}` **(< 100 Pa·s^** :math:`\alpha` **)**: Weak network. Soft, easily deformable gel.
   - **Moderate** :math:`c_\alpha` **(100–10⁴ Pa·s^** :math:`\alpha` **)**: Typical gel strength for most applications.
   - **High** :math:`c_\alpha` **(> 10⁴ Pa·s^** :math:`\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, :math:`c_\alpha \propto \nu k_B T` where
   :math:`\nu` is network strand density.

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

**Terminal Viscosity (** :math:`\eta` **)**:
   The dashpot viscosity controls long-time flow:

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

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

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

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

Material Classification
~~~~~~~~~~~~~~~~~~~~~~~

.. list-table:: FMG Material Classification
   :header-rows: 1
   :widths: 20 20 30 30

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

Diagnostic Indicators
~~~~~~~~~~~~~~~~~~~~~

Warning signs in fitted parameters:

- :math:`\alpha \to 0` **or** :math:`\to 1`: Model may be inappropriate. Check if SpringPot-only or
  classical Maxwell fits better.

- **Large uncertainty in** :math:`\alpha`: Data don't span sufficient frequency range.
  Extend measurements to capture both regimes.

- :math:`\eta` **poorly constrained**: Low-frequency data insufficient. Extend to lower
  frequencies or use creep tests to capture terminal flow.

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

Application Examples
~~~~~~~~~~~~~~~~~~~~

**Gel Formulation Development**:
   Track :math:`\alpha` as crosslinker is added. Approach to :math:`\alpha \approx 0.5` indicates proximity to
   gel point. For stable gels, target :math:`\alpha < 0.45` with sufficient margin from
   the transition.

**Quality Control**:
   Monitor :math:`c_{\alpha}` batch-to-batch. A ±20% specification catches network degradation
   while allowing normal variation.

**Process Design**:
   Calculate :math:`\tau` to determine when material transitions from gel-like to flowable.
   For coating applications, ensure :math:`\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+):**

.. code-block:: text

   # 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 :math:`\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 :math:`\eta` accurately
- **Missing power-law regime**: Need broader frequency coverage
- :math:`\alpha \approx 1`: Use classical Maxwell for simpler interpretation

Usage
-----

.. code-block:: python

   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 :doc:`API reference </api/models>`.

Troubleshooting
---------------

.. list-table:: Common Fitting Issues
   :widths: 25 35 40
   :header-rows: 1

   * - Symptom
     - Possible Cause
     - Solution
   * - **Poor fit in terminal regime**
     - Insufficient low-frequency data
     - Extend frequency sweep to lower :math:`\omega` or use longer relaxation test.
   * - :math:`\alpha \to 1`
     - Material is nearly Maxwellian
     - Use classical **Maxwell** model instead (narrow spectrum).
   * - **Oscillatory residuals at high** :math:`\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 :math:`\log(G')`, :math:`\log(G'')` vs :math:`\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
~~~~~~~~~~~~~~

- :doc:`fractional_maxwell_liquid` — complementary model with spring instead of dashpot (solid-like equilibrium)
- :doc:`fractional_maxwell_model` — generalized two-order formulation with independent :math:`\alpha` and :math:`\beta`
- :doc:`fractional_burgers` — adds Kelvin branch for delayed elasticity and creep
- :doc:`../classical/maxwell` — classical limit (:math:`\alpha` → 1, exponential relaxation)
- :doc:`../classical/springpot` — fundamental SpringPot element theory

Transforms
~~~~~~~~~~

- :doc:`../../transforms/owchirp` — broadband LAOS sweeps to estimate fractional slopes
- :doc:`../../transforms/fft` — convert relaxation data to frequency domain for fitting
- :doc:`../../transforms/mutation_number` — monitor gel-to-sol transitions in curing/aging systems
- :doc:`../../transforms/mastercurve` — time-temperature superposition for temperature-dependent :math:`\tau`

Examples
~~~~~~~~

- :doc:`../../examples/advanced/04-fractional-models-deep-dive` — tutorial comparing Fractional Maxwell family
- :doc:`../../examples/bayesian/02-fractional-gel-uncertainty` — uncertainty quantification for FMG
- :doc:`../../examples/fitting/01-smart-initialization` — demonstration of automatic initialization (v0.2.0)