Maxwell (Classical)

Quick Reference

  • Use when: Single relaxation time, exponential stress decay, viscoelastic liquids

  • Parameters: 2 (\(G_0\), \(\eta\))

  • Key equation: \(G(t) = G \exp(-t/\tau)\) where \(\tau = \eta/G\)

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

  • Material examples: Polymer melts (PS, PDMS), viscoelastic liquids, dilute solutions

Notation Guide

Symbol

Meaning

\(G\)

Spring modulus (Pa). Controls instantaneous elasticity.

\(\eta\)

Dashpot viscosity (Pa·s). Controls energy dissipation.

\(\tau\)

Relaxation time (s), \(\tau = \eta/G\).

Overview

Two-parameter linear viscoelastic model with a spring (\(G\)) and dashpot (\(\eta\)) in series. Captures single-time-constant stress relaxation with \(G(t) = G\,e^{-t/\tau}\) where \(\tau = \eta/G\).

Physical Foundations

Mechanical Analogue

The Maxwell model consists of a linear spring (Hookean elastic element) connected in series with a Newtonian dashpot (viscous element):

┌────────┐         ┌────────┐
│ Spring │─────────│Dashpot │
│   G    │         │   η    │
└────────┘         └────────┘

Total deformation: γ_total = γ_spring + γ_dashpot
Same stress: σ_spring = σ_dashpot = σ

The series configuration means:

  • Strain is additive: \(\gamma(t) = \gamma_{\text{spring}}(t) + \gamma_{\text{dashpot}}(t)\)

  • Stress is identical: Both elements experience the same stress \(\sigma(t)\)

Microstructural Interpretation

The Maxwell model represents materials where:

Spring (elastic storage):

  • Entropic elasticity from chain stretching in polymer melts

  • Temporary network junctions that store elastic energy

  • Reversible conformational changes

Dashpot (viscous dissipation):

  • Chain reptation through entanglement network

  • Molecular rearrangements that dissipate energy

  • Irreversible flow at long timescales

Physical meaning of relaxation time \(\tau = \eta/G\):

  • Characteristic time for stress to decay to \(1/e \approx 37\%\) of initial value

  • Ratio of viscous resistance to elastic restoring force

  • Related to molecular weight and entanglement density in polymers

Material Examples with Typical Parameters

Representative Maxwell parameters

Material

G (Pa)

\(\eta\) (Pa·s)

\(\tau\) (s)

Ref

Polystyrene melt (170°C)

\(1 \times 10^5\)

\(1 \times 10^6\)

10

[1]

Polyethylene melt (190°C)

\(3 \times 10^4\)

\(3 \times 10^5\)

10

[1]

Bitumen (25°C)

\(1 \times 10^6\)

\(1 \times 10^8\)

100

[2]

Dilute polymer solution

\(1 \times 10^2\)

\(1 \times 10^1\)

0.1

[3]

PDMS (crosslinked)

\(5 \times 10^5\)

\(5 \times 10^4\)

0.1

[4]

Connection to Polymer Physics

For entangled polymer melts, the Maxwell relaxation time relates to molecular parameters via the Rouse-reptation theory (Doi-Edwards):

\[\tau_d \sim \frac{M^3}{\rho RT} \cdot \frac{\zeta N_e^2}{M_e}\]
where:

  • \(M\) = molecular weight

  • \(M_e\) = entanglement molecular weight

  • \(\zeta\) = monomeric friction coefficient

  • \(N_e\) = entanglement strand length

Scaling laws (experimental):

  • \(\eta_0 \sim M^{3.4}\) for \(M > M_c\) (entangled)

  • \(\eta_0 \sim M^{1.0}\) for \(M < M_c\) (Rouse regime)

  • \(G_N^0 \sim \rho RT / M_e\) (plateau modulus)

Governing Equations

Mathematical Derivation

Starting from the mechanical analogue with series connection:

Step 1: Express strain rates

Spring obeys Hooke’s law: \(\sigma = G \gamma_{\text{spring}}\)

Dashpot obeys Newton’s law: \(\sigma = \eta \dot{\gamma}_{\text{dashpot}}\)

Step 2: Differentiate spring equation

\(\dot{\sigma} = G \dot{\gamma}_{\text{spring}}\)

Step 3: Substitute dashpot relation

\(\dot{\gamma}_{\text{dashpot}} = \sigma / \eta\)

Step 4: Total strain rate (series)

\(\dot{\gamma} = \dot{\gamma}_{\text{spring}} + \dot{\gamma}_{\text{dashpot}} = \frac{\dot{\sigma}}{G} + \frac{\sigma}{\eta}\)

Step 5: Rearrange to constitutive form

\(\sigma + \frac{\eta}{G} \dot{\sigma} = \eta \dot{\gamma}\)

Defining \(\tau = \eta/G\):

\[\tau\,\dot{\sigma}(t) + \sigma(t) = \eta\,\dot{\gamma}(t)\]

Constitutive (differential) form:

\[\tau\,\dot{\sigma}(t) + \sigma(t) = \eta\,\dot{\gamma}(t), \qquad \tau = \frac{\eta}{G}\]

Relaxation modulus:

\[G(t) = G\,e^{-t/\tau}\]

Derivation: For stress relaxation (step strain \(\gamma_0\) at \(t=0\)), the governing ODE with \(\dot{\gamma}=0\) for \(t>0\) gives \(\dot{\sigma} = -\sigma/\tau\), yielding \(\sigma(t) = \sigma_0 e^{-t/\tau} = G\gamma_0 e^{-t/\tau}\).

Fourier Transform to Frequency Domain

For oscillatory shear \(\gamma(t) = \gamma_0 e^{i\omega t}\), apply Fourier transform:

Step 1: Substitute into constitutive equation

\(\sigma(t) = \sigma_0 e^{i\omega t}\) (harmonic response)

\(\dot{\sigma} = i\omega \sigma\), \(\dot{\gamma} = i\omega \gamma\)

Step 2: Transform constitutive equation

\(\tau (i\omega \sigma) + \sigma = \eta (i\omega \gamma)\)

\(\sigma (1 + i\omega\tau) = i\omega\eta \gamma\)

Step 3: Define complex modulus
\[G^*(\omega) = \frac{\sigma}{\gamma} = \frac{i\omega\eta}{1 + i\omega\tau} = G \frac{i\omega\tau}{1 + i\omega\tau}\]

Step 4: Separate real and imaginary parts

Oscillatory complex modulus:

\[G^*(\omega) = G'(\omega) + i\,G''(\omega) = G \frac{i\,\omega \tau}{1 + i\,\omega \tau}\]

with storage and loss moduli:

\[G'(\omega) = G\,\frac{(\omega \tau)^2}{1 + (\omega \tau)^2}, \qquad G''(\omega) = G\,\frac{\omega \tau}{1 + (\omega \tau)^2}\]

Mathematical Significance

First-order linear ODE: The Maxwell constitutive equation is the simplest differential equation describing viscoelasticity. The exponential solution is characteristic of all first-order linear systems.

Complex modulus structure: The factor \(i\omega\tau/(1+i\omega\tau)\) is a high-pass filter in signal processing, transitioning from viscous (low \(\omega\)) to elastic (high \(\omega\)) behavior.

Loss tangent:

\[\tan\delta = \frac{G''}{G'} = \frac{1}{\omega\tau}\]

This monotonically decreases with frequency, confirming liquid-like character (no solid plateau).

Parameters

Parameters

Name

Symbol

Units

Bounds

Notes

G0

\(G\)

Pa

\(G > 0\)

Spring modulus

eta

\(\eta\)

Pa·s

\(\eta > 0\)

Dashpot viscosity

(derived)

\(\tau\)

s

\(\tau > 0\)

\(\tau = \eta/G\) (not an independent parameter)

Parameter Interpretation

G (Spring Modulus):

  • Physical meaning: Instantaneous elastic response at short times (or high frequencies)

  • Molecular origin: Entropic resistance to chain deformation

  • Typical ranges:

    • Polymer melts: \(10^4 - 10^6\) Pa

    • Dilute solutions: \(10^1 - 10^3\) Pa

    • Gels: \(10^2 - 10^5\) Pa

  • Scaling: \(G \sim \rho RT / M\) (low MW), \(G \sim \rho RT / M_e\) (entangled)

\(\eta\) (Dashpot Viscosity):

  • Physical meaning: Resistance to flow, energy dissipation rate

  • Molecular origin: Chain friction during reptation or Rouse relaxation

  • Typical ranges:

    • Polymer melts: \(10^3 - 10^7\) Pa·s (strongly temperature-dependent)

    • Bitumen: \(10^6 - 10^{10}\) Pa·s

    • Dilute solutions: \(10^{-2} - 10^2\) Pa·s

  • Scaling: \(\eta \sim M^{3.4}\) (entangled polymers), \(\eta \sim M\) (Rouse)

\(\tau\) (Relaxation Time):

  • Physical meaning: Timescale separating elastic (solid-like) from viscous (liquid-like) behavior

  • Diagnostic: \(\tau^{-1}\) corresponds to frequency where \(G''(\omega)\) peaks

  • Material design: Long \(\tau\) → more elastic character; short \(\tau\) → more viscous

  • Typical ranges: \(10^{-3}\) s (dilute solutions) to \(10^3\) s (bitumen at room T)

Relation to Molecular Properties

For linear polymer melts:

\[\eta_0 = \frac{\pi^2}{12} \rho \frac{RT}{M_e} \tau_e \left(\frac{M}{M_e}\right)^3\]
where:

  • \(M_e \approx 1800\) g/mol (polyethylene), 13000 g/mol (polystyrene)

  • \(\tau_e\) = Rouse time of entanglement strand

  • \(\rho\) = density

This connects measurable rheological parameters to fundamental molecular architecture.

Validity and Assumptions

  • Linear viscoelasticity: yes

  • Small amplitude: yes

  • Isothermal: yes

  • Data/test modes: relaxation, oscillation

  • Additional assumptions: single relaxation time

Limitations

Critical limitation: Predicts unbounded creep

For creep compliance \(J(t) = \gamma(t)/\sigma_0\) under constant stress \(\sigma_0\):

\[J(t) = \frac{1}{G} + \frac{t}{\eta} \quad \to \infty \text{ as } t \to \infty\]
The Maxwell model predicts linear viscous flow with no creep recovery, making it:

  • Inappropriate for viscoelastic solids (rubbers, gels)

  • Appropriate for viscoelastic liquids (polymer melts, dilute solutions)

Single relaxation time:

Real polymers exhibit continuous distributions of relaxation times \(H(\tau)\). The Maxwell model is adequate only when: - Material is nearly monodisperse - One relaxation process dominates experimental window - Data span < 2 decades in time/frequency

No equilibrium modulus:

\(G_e = \lim_{t\to\infty} G(t) = 0\), meaning the material always flows eventually. This fails for crosslinked networks.

Regimes and Behavior

Limiting Cases

Low frequency ( \(\omega\) → 0, terminal region):

\[ \begin{align}\begin{aligned}G'(\omega) \approx G (\omega\tau)^2 \sim \omega^2\\G''(\omega) \approx G \omega\tau \sim \omega\end{aligned}\end{align} \]

Interpretation: Viscous liquid-like behavior dominates. Energy dissipation (\(G''\)) exceeds storage (\(G'\)).

High frequency ( \(\omega \to \infty\) , glassy region):

\[ \begin{align}\begin{aligned}G'(\omega) \to G \quad (\text{plateau})\\G''(\omega) \to 0\end{aligned}\end{align} \]

Interpretation: Elastic solid-like response. Chains don’t have time to relax, behaving as frozen network.

Crossover frequency \(\omega_c = 1/\tau\):

\[G'(\omega_c) = G''(\omega_c) = \frac{G}{2}\]

At this point, elastic and viscous contributions are equal, defining the characteristic relaxation timescale.

Asymptotic Behavior Summary

Frequency-dependent regimes

Regime

\(G'(\omega)\)

\(G''(\omega)\)

Physical interpretation

Low \(\omega \ll 1/\tau\)

\(\sim \omega^2\)

\(\sim \omega\)

Viscous liquid (\(\tan \delta \gg 1\))

\(\omega \approx 1/\tau\)

\(\approx G/2\)

\(\approx G/2\)

Balanced viscoelastic

High \(\omega \gg 1/\tau\)

\(\to G\)

\(\to 0\)

Elastic solid (\(\tan \delta \to 0\))

Diagnostic Signatures

  • Peak in \(G''\) at \(\omega \approx 1/\tau\): Characteristic signature of single relaxation time

  • Slope of log \(G'\) vs log \(\omega\) = 2 at low \(\omega\): Terminal behavior of viscoelastic liquids

  • Loss tangent: \(\tan\delta = 1/(\omega\tau)\) is monotonically decreasing (unlike Zener model with minimum)

What You Can Learn

This section explains how to translate fitted Maxwell parameters into material insights and actionable knowledge for both research and industrial applications.

Parameter Interpretation

G (Spring Modulus):

Fitted \(G\) reveals the instantaneous elastic response:

  • Low values (< \(10^4\) Pa): Dilute solution, low entanglement density, or near-terminal regime

  • Moderate values ( \(10^4-10^6\) Pa): Typical processing-grade polymer melts, well-entangled

  • High values (> \(10^6\) Pa): Very high MW, high entanglement density, or glassy contribution

For graduate students: Compare with plateau modulus \(G_N^0\) from Generalized Maxwell or from \(G_N^0 = \rho RT / M_e\) to estimate entanglement MW. The single Maxwell \(G\) underestimates \(G_N^0\) when multiple modes contribute.

For practitioners: \(G\) indicates die swell magnitude and elastic recoil strength. Higher \(G\) means more pronounced elastic effects in processing.

eta (Dashpot Viscosity):

Fitted \(\eta\) reveals the flow resistance:

  • Low values (< \(10^3\) Pa·s): Low MW, high temperature, or weak entanglement

  • Moderate values ( \(10^3-10^6\) Pa·s): Typical polymer melt processing range

  • High values (> \(10^6\) Pa·s): Very high MW, low temperature, or near \(T_g\)

For graduate students: Use \(\eta \sim M^{3.4}\) scaling (for \(M > 2M_c\)) to estimate molecular weight. Compare with capillary viscometry or GPC data.

For practitioners: \(\eta\) controls pumping power requirements and flow rates in processing. Higher \(\eta\) means slower filling, higher pressures needed.

tau (Relaxation Time):

The derived parameter \(\tau = \eta/G\) is the most important for processing:

  • Short \(\tau\) (<0.1 s): Fast relaxation, minimal melt memory, easy processing

  • Moderate \(\tau\) (0.1-10 s): Typical processing regime, some elastic effects

  • Long \(\tau\) (>10 s): Strong melt memory, stress relaxation issues, orientation effects

For graduate students: Compare \(\tau\) with reptation time \(\tau_d\) from tube model. For monodisperse melts, single Maxwell \(\tau \approx \tau_d\).

For practitioners: Processing Deborah number \(De = \tau \cdot \dot{\gamma}_{process}\) determines whether elastic (\(De > 1\)) or viscous (\(De < 1\)) effects dominate.

Material Classification

Material Classification from Maxwell Parameters

Parameter Pattern

Material Type

Examples

Processing Notes

High \(G\), high \(\eta\)

High-MW entangled melt

UHMWPE, high-MW PS

Strong elastic effects, die swell

Low \(G\), low \(\eta\)

Low-MW oligomer/liquid

Wax, low-MW PDMS

Near-Newtonian, easy processing

High \(G\), low \(\eta\) (short \(\tau\))

Concentrated, fast-relaxing

Branched polymers at high T

Good processability

Low \(G\), high \(\eta\) (long \(\tau\))

Dilute but entangled

Dilute polymer solution

Slow dynamics, low elasticity

Molecular Weight Estimation

From fitted \(\eta\) using the empirical scaling relation:

\[M_w \approx \left( \frac{\eta}{K_\eta} \right)^{1/3.4}\]
where \(K_\eta\) is polymer- and temperature-specific:

  • Polyethylene (190°C): \(K_\eta \approx 3.4 \times 10^{-14}\) (Pa·s)/(g/mol)^3.4

  • Polystyrene (170°C): \(K_\eta \approx 1.1 \times 10^{-14}\) (Pa·s)/(g/mol)^3.4

Process Window Estimation

From \(\tau\), estimate the shear rate range for different flow behaviors:

  • Newtonian regime (De < 0.1): \(\dot{\gamma} < 0.1/\tau\)

  • Transition regime (0.1 < De < 10): \(0.1/\tau < \dot{\gamma} < 10/\tau\)

  • Elastic-dominated (De > 10): \(\dot{\gamma} > 10/\tau\)

For typical processing rates (\(\dot{\gamma} \approx 10^2 - 10^4\) s\(^{-1}\)), target \(\tau < 0.01\) s for minimal elastic effects.

Diagnostic Indicators

Warning signs in fitted parameters:

  • If \(\tau\) outside data frequency range: \(G''\) peak not captured; extend frequency sweep

  • If \(R^2\) < 0.95: Multiple relaxation times present; use Generalized Maxwell

  • If fit residuals show curvature: Single exponential inadequate; try fractional models

  • If \(G\) hits bounds: Data may be in terminal regime only; verify \(G' \sim \omega^2\) slope

Application Examples

Quality Control:

  • Track \(\tau\) across batches to monitor MW consistency

  • Verify \(G\) within specification for grade identification

  • Use \(\eta\) to detect contamination or degradation

Process Troubleshooting:

  • High die swell → \(G\) or \(\tau\) too high → increase temperature or reduce MW

  • Shark skin melt fracture → \(\tau\) too long → blend with lower-MW grade

  • Poor weld line strength → \(\tau\) too short → increase MW or reduce temperature

Material Development:

  • Target \(\tau \approx 0.1-1\) s for balanced processability

  • Increase \(G\) by increasing MW or crosslink density

  • Reduce \(\tau\) via chain branching or blending

Experimental Design

Sample Preparation Considerations

Polymer melts:

  • Compression molding at \(T > T_g + 50°C\) to erase thermal history

  • Annealing at test temperature for 10-30 min before measurement

  • Avoid air bubbles (reduce pressure slowly during molding)

  • Typical geometry: 25 mm parallel plates, 1 mm gap

Polymer solutions:

  • Dissolve at \(T > T_g\) with gentle stirring (avoid degradation)

  • Filter through 0.45 \(\mu\text{m}\) PTFE filter to remove aggregates

  • Equilibrate at test temperature for 30 min

  • Use solvent trap to prevent evaporation

Temperature-dependent materials:

Common Experimental Artifacts

Troubleshooting experimental issues

Artifact

Symptom

Solution

Wall slip

\(G'\), \(G''\) artificially low, non-reproducible

Use serrated plates, reduce gap, check with multiple geometries

Inertia (high \(\omega\))

\(G'\) increases spuriously at high \(\omega\)

Reduce tool inertia, use smaller geometry, limit \(\omega < 100\) rad/s

Edge fracture

Sudden drop in \(G'\) at high strain

Reduce \(\gamma_0\), use cone-plate geometry

Sample degradation

Drift in \(G'\), \(G''\) over time

Reduce temperature, minimize air exposure, use antioxidants

Insufficient relaxation

Non-exponential \(G(t)\) decay

Extend measurement to \(10\tau\), check for multi-mode behavior

Fitting Guidance

Parameter Initialization Strategies

Method 1: From frequency sweep data

Step 1: Estimate \(\tau\) from \(G''(\omega)\) peak

\(\tau \approx 1 / \omega_{\max}\) where \(G''\) is maximum

Step 2: Estimate \(G\) from high-frequency plateau

\(G \approx \lim_{\omega \to \infty} G'(\omega)\)

Step 3: Calculate \(\eta\)

\(\eta = G \tau\)

Method 2: From stress relaxation data

Step 1: Linear regression on \(\log G(t)\) vs \(t\)

Slope = \(-1/\tau\), Intercept = \(\log G\)

Step 3: Extract parameters directly from fit

Method 3: From zero-shear viscosity (flow curve)

If \(\eta_0\) is known from steady shear:

\(\eta = \eta_0\)

Then fit \(G\) from oscillatory data with \(\eta\) fixed.

Optimization Algorithm Selection

RheoJAX default: NLSQ (GPU-accelerated)

  • Recommended for Maxwell model (2 parameters, well-conditioned)

  • 5-270× faster than scipy.optimize

  • Robust to initial guesses if parameters initialized correctly

Alternative: Bayesian inference (NUTS)

  • Use when parameter uncertainty quantification needed

  • Warm-start from NLSQ fit for faster convergence

  • See ../../examples/bayesian/01-bayesian-basics

Bounds:

  • \(G\): [1e2, 1e8] Pa (adjust based on material)

  • \(\eta\): [1e-2, 1e10] Pa·s

  • \(\tau = \eta/G\) not fitted directly (derived parameter)

Troubleshooting Common Fitting Problems

Fitting diagnostics and solutions

Problem

Diagnostic

Solution

Poor fit at low \(\omega\)

\(G'\) underestimated (terminal slope wrong)

Check for multi-mode relaxation, consider Generalized Maxwell

Poor fit at high \(\omega\)

\(G'\) doesn’t plateau

Extend frequency range, check for glass transition effects

\(G''\) peak not captured

\(\tau\) outside data range

Expand frequency window to bracket \(1/\tau\)

Converged but \(R^2 < 0.95\)

Single Maxwell inadequate

Use multi-mode Maxwell or fractional model (FML)

Fitted \(\eta\) unrealistic

Units mismatch or poor initialization

Verify data units (Pa, rad/s), reinitialize from \(G''\) peak

Validation Strategies

1. Residual Analysis

Visual check:

  • Plot residuals \(r_i = \log|G^*_{\text{data}}| - \log|G^*_{\text{fit}}|\) vs \(\omega\)

  • Should be randomly scattered around zero (no trends)

  • Systematic curvature → model inadequacy (try multi-mode or fractional)

Statistical test:

  • \(R^2 > 0.98\) for good fit (oscillatory data typically noisy)

  • RMSE in log-space should be \(< 0.1\) (10% error)

2. Physical Plausibility

\[\text{Check: } \tau_{\text{fitted}} = \eta_{\text{fitted}} / G_{\text{fitted}}\]

Should match \(\tau\) from \(G''\) peak location within 10%.

3. Kramers-Kronig Relations

Verify causality:

\[G'(\omega) = \frac{2}{\pi} \int_0^\infty \frac{x G''(x)}{x^2 - \omega^2} dx\]

For Maxwell model, this is automatically satisfied (analytical model). Use for experimental data validation.

4. Cross-validation with Different Test Modes

  • Fit SAOS data → predict \(G(t)\) via inverse Fourier transform

  • Compare with stress relaxation measurements

  • Discrepancies indicate time-temperature superposition failure or nonlinearity

Worked Example with Numbers

Material: Polystyrene melt at 170°C

Experimental data (SAOS frequency sweep):

  • \(G''\) peak at \(\omega = 0.1\) rad/s

  • High-frequency plateau: \(G' \approx 1.0 \times 10^5\) Pa

Initialization:

  • \(\tau = 1/0.1 = 10\) s

  • \(G = 1.0 \times 10^5\) Pa

  • \(\eta = G\tau = 1.0 \times 10^6\) Pa·s

Optimization (NLSQ with 100 iterations):

  • Fitted: \(G = 9.8 \times 10^4\) Pa, \(\eta = 1.05 \times 10^6\) Pa·s

  • \(R^2 = 0.992\), RMSE = 0.08

  • Validation: \(\tau_{\text{fit}} = 10.7\) s vs \(\tau_{\text{init}} = 10\) s (7% difference, excellent)

Interpretation:

  • Molecular weight: \(M \sim (\eta_0)^{1/3.4} \approx 180\) kg/mol (using \(\eta \sim M^{3.4}\))

  • Entanglement time: \(\tau \sim M^3 / M_e^3 \approx 10\) s consistent with literature

Model Comparison

When to Use Maxwell vs Alternatives

Model selection decision tree

Use Maxwell when…

Use alternative when…

Recommended model

Viscoelastic liquid (flows)

Viscoelastic solid (finite \(G_e\))

Zener (SLS), FZSS

Single relaxation time dominates

Broad relaxation spectrum

Generalized Maxwell, FML

Exponential \(G(t)\) decay

Power-law \(G(t) \sim t^{-\alpha}\) decay

FMG, SpringPot

Stress relaxation analysis

Creep/recovery analysis

Zener, Burgers, FKV

2 parameters sufficient

Higher accuracy needed

Multi-mode Maxwell (Prony)

Educational/conceptual

Quantitative predictions

Fractional or multi-mode

Model Hierarchy (Simpler → More Complex)

Level 1: Maxwell (this model)

  • 2 parameters: \(G\), \(\eta\)

  • Exponential relaxation

  • Viscoelastic liquid only

Level 2: Zener (Standard Linear Solid)

  • 3 parameters: \(G_e\), \(G_m\), \(\eta\)

  • Adds equilibrium modulus → viscoelastic solid

  • Exponential relaxation to finite plateau

  • See Zener (Standard Linear Solid)

Level 3: Generalized Maxwell (Prony series)

  • \(2N\) parameters (N Maxwell elements in parallel)

  • Multiple relaxation times → broader spectra

  • Computationally expensive (\(N = 10-20\) typical)

Level 4: Fractional Maxwell Liquid (FML)
Level 5: Fractional Maxwell Gel (FMG)

Diagnostic Tests to Discriminate Models

Test 1: Plot log \(G(t)\) vs \(t\)

  • Linear → Maxwell (exponential)

  • Curved → Multi-mode or fractional

Test 2: Plot log \(G''\) vs log \(\omega\)

  • Single symmetric peak → Maxwell

  • Broad peak or multiple peaks → Multi-mode

  • Linear slope (\(\alpha < 1\)) → Fractional (SpringPot)

Test 3: Creep-recovery test

  • No recovery → Maxwell (pure liquid)

  • Partial recovery → Zener, Burgers (viscoelastic solid)

  • Power-law creep → Fractional models

Test 4: Tan \(\delta\) behavior

  • Monotonic decrease with \(\omega\) → Maxwell

  • Minimum in tan \(\delta\) → Zener

  • Constant tan \(\delta\) → Critical gel (FMG with \(\alpha \approx 0.5\))

Connection to Advanced Models

Reptation (Doi-Edwards):

The Maxwell model is the single-mode approximation of reptation theory. The disengagement time \(\tau_d\) corresponds to Maxwell \(\tau\), and the plateau modulus \(G_N^0\) corresponds to \(G\).

Cox-Merz rule:

For materials obeying Cox-Merz, steady-shear viscosity \(\eta(\dot{\gamma})\) can be predicted from complex viscosity \(|\eta^*(\omega)|\):

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

Maxwell model: \(|\eta^*| = \eta / \sqrt{1 + (\omega\tau)^2}\)

Winter-Chambon criterion:

Maxwell model does not satisfy gel point criterion (\(\tan \delta \neq \text{constant}\)). Use FMG for critical gels.

API References

  • Module: rheojax.models

  • Class: rheojax.models.Maxwell

Usage

Basic Fitting Example

import numpy as np
from rheojax.models import Maxwell

omega = np.logspace(-2, 2, 100)
maxwell = Maxwell()
maxwell.fit(omega, data)  # replace ``data`` with target complex modulus

G0 = maxwell.parameters.get_value('G0')
eta = maxwell.parameters.get_value('eta')
tau = eta / G0

Gstar = maxwell.predict(omega)

Advanced Usage: Bayesian Inference

from rheojax.models import Maxwell
import numpy as np

# 1. NLSQ point estimation (fast)
model = Maxwell()
model.fit(omega, G_data)

# 2. Bayesian inference with warm-start
result = model.fit_bayesian(
    omega, G_data,
    num_warmup=1000,
    num_samples=2000
)

# 3. Get credible intervals
intervals = model.get_credible_intervals(result.posterior_samples, credibility=0.95)
print(f"G0: [{intervals['G0'][0]:.2e}, {intervals['G0'][1]:.2e}] Pa")
print(f"eta: [{intervals['eta'][0]:.2e}, {intervals['eta'][1]:.2e}] Pa·s")

Examples

Fit to oscillatory data

from rheojax.models import Maxwell
model = Maxwell()
model.fit(omega, G_star)
print(model.score(omega, G_star))

Time-Temperature Superposition

from rheojax.models import Maxwell
from rheojax.transforms import Mastercurve

# Create master curve at reference temperature
mc = Mastercurve(reference_temp=170, method='wlf', C1=17.44, C2=51.6)
master_data, shifts = mc.transform(multi_temp_datasets)

# Fit Maxwell to extended frequency range
model = Maxwell()
model.fit(master_data.omega, master_data.G_star)

See Also

Classical Models:

Fractional Models:

Transforms:

Examples:

  • ../../examples/basic/01-maxwell-fitting — notebook demonstrating parameter estimation and validation

User Guides:

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

References