Models Summary & Selection Guide¶

This page serves as a comprehensive quick-reference guide for all 53 rheological models in RheoJAX. Use the comparison matrices and decision flowcharts below to select the appropriate model for your experimental data and material system.

Complete Model Comparison Matrix¶

The table below provides a comprehensive overview of all models across key characteristics for rapid model selection.

Comprehensive Model Comparison¶
Model	Family	Params	Test Modes	Material Type	Equilibrium Modulus	Complexity	\(\alpha\) Range	Best For
Maxwell	Classical	2	R, C, O, Rot	Liquid	No (\(G_\infty = 0\))	★☆☆☆☆	N/A	Simple viscoelastic liquids, polymer melts with single relaxation
Zener	Classical	3	R, C, O	Solid	Yes (\(G_e > 0\))	★★☆☆☆	N/A	Soft solids, elastomers with exponential relaxation
SpringPot	Fractional	2	R, O	Gel	No	★★☆☆☆	0-1	Power-law gels, critical gels (Scott-Blair element)
Generalized Maxwell	Multi-Mode	2N+1	R, C, O	Variable	Configurable	★★★★☆	N/A	Prony series, broadband fitting, industrial master curves
Fractional Maxwell Gel	Fractional	3	R, C, O	Gel	No	★★★☆☆	0-1	Gels with elastic plateau + power-law tail
Fractional Maxwell Liquid	Fractional	3	R, C, O	Liquid	No (flows)	★★★☆☆	0-1	Liquid-like materials with fractional memory effects
Fractional Maxwell Model	Fractional	4	R, O	Variable	Configurable	★★★★☆	0-1 (two)	Wideband fitting, materials with multiple fractional processes
Fractional Kelvin-Voigt	Fractional	3-4	C, O	Solid	Yes	★★★☆☆	0-1	Solid-like with slow fractional relaxation, creep-dominated
Fractional Zener SL	Fractional	4	R, C, O	Solid	Yes (\(G_s > 0\))	★★★★☆	0-1	Solid with fractional liquid leg, intermediate behavior
Fractional Zener SS	Fractional	4	R, C, O	Solid	Yes (\(G_e > 0\))	★★★★☆	0-1	Dual elastic plateaus with fractional transition (most common)
Fractional Zener LL	Fractional	4	R, C, O	Liquid	No	★★★★☆	0-1	Liquid-biased Zener, complex liquids with memory
Fractional KV Zener	Fractional	4	C, O	Solid	Yes	★★★★☆	0-1	Fractional KV block in series with spring, creep applications
Fractional Burgers	Fractional	5	R, C, O	Solid/Liquid	Configurable	★★★★★	0-1	Captures creep AND relaxation simultaneously, versatile
Fractional Poynting-Thomson	Fractional	5	R, C, O	Solid	Yes	★★★★★	0-1	Multi-plateau solids, alternate formulation for complex behavior
Fractional Jeffreys	Fractional	4	R, C, O	Liquid	No	★★★★☆	0-1	Liquid-like with fractional damping, two dashpots + springpot
Power Law	Flow	2	Rot	Fluid	N/A	★☆☆☆☆	N/A	Shear-thinning/thickening fluids (Ostwald-de Waele)
Carreau	Flow	4	Rot	Fluid	N/A	★★★☆☆	N/A	Polymer solutions with Newtonian → power-law transition
Carreau-Yasuda	Flow	5	Rot	Fluid	N/A	★★★★☆	N/A	Adjustable transition sharpness, concentrated polymers
Cross	Flow	4	Rot	Fluid	N/A	★★★☆☆	N/A	Alternative interpolation for polymer solutions
Herschel-Bulkley	Flow	3	Rot	Viscoplastic	N/A	★★☆☆☆	N/A	Yield stress fluids with power-law post-yield (gels, slurries)
Bingham	Flow	2	Rot	Viscoplastic	N/A	★★☆☆☆	N/A	Linear viscoplastic (yield stress + constant viscosity)
Giesekus Single Mode	Giesekus	4	R, C, O, Flow, Startup, LAOS	Polymer	No	★★★★☆	\(\alpha\): 0-0.5	Nonlinear viscoelastic with shear thinning, \(N_1, N_2\) predictions
Giesekus Multi Mode	Giesekus	4N	O, Flow, Startup	Polymer	No	★★★★★	\(\alpha_i\): 0-0.5	Multi-mode nonlinear viscoelastic, broadband spectra with normal stresses
SGR Conventional	SGR	3	R, C, O	Soft Glass	No (flows)	★★★★☆	x: 0.5-3	Foams, emulsions, pastes, colloidal suspensions (Sollich 1998)
SGR GENERIC	SGR	3	R, C, O, Flow, Startup, LAOS	Soft Glass	No (flows)	★★★★★	x: 0.5-3	Thermodynamically consistent SGR (Fuereder & Ilg 2013)
Fluidity Local	Fluidity	2-3	O, Flow	Cooperative	No	★★★☆☆	N/A	Local fluidity dynamics, simple cooperative flow
Fluidity Nonlocal	Fluidity	3-4	O, Flow	Cooperative	No	★★★★☆	N/A	Nonlocal fluidity with cooperativity length
Fluidity-Saramito Local	Saramito EVP	10-12	Flow, Startup, Creep, R, O, LAOS	EVP Thixotropic	Configurable	★★★★★	N/A	Tensorial EVP with fluidity coupling, \(N_1\) predictions
Fluidity-Saramito Nonlocal	Saramito EVP	11-13	Flow, Startup, Creep, R, O, LAOS	EVP Thixotropic	Configurable	★★★★★	N/A	Nonlocal EVP for shear banding, cooperativity length
Lattice EPM	EPM	4+	R, C, Startup, Flow	Elasto-plastic	Configurable	★★★★★	N/A	Lattice elasto-plastic model, plastic rearrangements
Tensorial EPM	EPM	4+	R, C, Startup, Flow	Elasto-plastic	Configurable	★★★★★	N/A	Full tensorial EPM for complex loading
MIKH	IKH	4-5	R, C, O	Thixotropic	Configurable	★★★★☆	N/A	Modified IKH for thixotropic materials
MLIKH	IKH	4+	R, C, O	Thixotropic	Configurable	★★★★★	N/A	ML-enhanced IKH with neural network augmentation
FIKH	FIKH	5-6	R, C, O, Flow, Startup, LAOS	Thixotropic	Configurable	★★★★★	\(\alpha\): 0-1	Fractional IKH with Caputo structure kinetics
FMLIKH	FIKH	6+	R, C, O, Flow, Startup, LAOS	Thixotropic	Configurable	★★★★★	\(\alpha\): 0-1	Fractional multi-layer IKH, multiple yield surfaces
DMT Local	DMT	5-7	R, C, O, Flow, Startup, LAOS	Thixotropic	Configurable	★★★★☆	N/A	Structural kinetics with exponential or HB viscosity closure
DMT Nonlocal	DMT	6-8	R, C, O, Flow, Startup, LAOS	Thixotropic	Configurable	★★★★★	N/A	Spatially-resolved thixotropy with structure diffusion, shear banding
Hébraud-Lequeux	HL	3-4	R, C, O, Flow, Startup, LAOS	Soft matter	No	★★★★☆	N/A	Mean-field model for soft glassy materials
STZ Conventional	STZ	4+	R, O, Flow, Startup	Amorphous	No	★★★★★	N/A	Shear transformation zone model (Falk-Langer)
ITT-MCT Schematic	ITT-MCT	6	R, C, O, Flow, Startup, LAOS	Colloidal Glass	Configurable	★★★★★	\(\varepsilon\): -0.5 to 0.5	Dense colloidal suspensions, glass transition (\(F_{12}\) schematic)
ITT-MCT Isotropic	ITT-MCT	5+	R, C, O, Flow, Startup, LAOS	Colloidal Glass	Configurable	★★★★★	\(\phi\): 0.1 to 0.64	Hard-sphere colloids with S(k), full MCT physics
SPP Yield Stress	SPP	3+	LAOS	Yield stress	Yes	★★★★☆	N/A	LAOS-based yield stress analysis (Rogers et al.)
TNT Tanaka-Edwards	TNT	3	R, C, O, Flow, Startup, LAOS	Assoc. Polymer	No	★★☆☆☆	N/A	Baseline transient network (Maxwell via conformation tensor)
TNT Bell	TNT	4	R, C, O, Flow, Startup, LAOS	Assoc. Polymer	No	★★★☆☆	\(\nu\): 0.01-20	Force-dependent bond breakage, shear-thinning networks
TNT FENE-P	TNT	4	R, C, O, Flow, Startup, LAOS	Assoc. Polymer	No	★★★☆☆	\(L_{max}\): 2-100	Finite extensibility, strain hardening at large deformations
TNT Non-Affine	TNT	4	R, C, O, Flow, Startup, LAOS	Assoc. Polymer	No	★★★☆☆	\(\xi\): 0-1	Non-affine chain slip, non-zero \(N_2\)
TNT Stretch-Creation	TNT	4	R, C, O, Flow, Startup, LAOS	Assoc. Polymer	No	★★★☆☆	\(\kappa\): 0-5	Flow-enhanced bond formation, shear thickening
TNT Loop-Bridge	TNT	6	R, C, O, Flow, Startup, LAOS	Telechelic	No	★★★★☆	N/A	Two-species kinetics (loops + bridges), telechelic polymers
TNT Sticky Rouse	TNT	4-6	R, C, O, Flow, Startup, LAOS	Multi-sticker	No	★★★★☆	N/A	Multi-mode sticker dynamics, broad relaxation spectrum
TNT Cates	TNT	4	R, C, O, Flow, Startup, LAOS	Micelles	No	★★★☆☆	N/A	Living polymers, wormlike micelles (\(\tau_d = \sqrt{\tau_{rep} \cdot \tau_{break}}\))
TNT Multi-Species	TNT	2N+1	R, C, O, Flow, Startup, LAOS	Mixed Network	No	★★★★☆	N/A	Heterogeneous networks with multiple bond types
VLB Local	VLB	2	R, C, O, Flow, Startup, LAOS	Assoc. Polymer	No	★☆☆☆☆	N/A	Single transient network (Maxwell via distribution tensor)
VLB Multi-Network	VLB	2N+1	R, C, O, Flow, Startup, LAOS	Assoc. Polymer	Configurable	★★★☆☆	N/A	Multi-network generalized Maxwell with molecular basis
VLB Variant	VLB	2-6	R, C, O, Flow, Startup, LAOS	Assoc. Polymer	No	★★★☆☆	N/A	Bell shear thinning, FENE bounded extension, Arrhenius temperature
VLB Nonlocal	VLB	4-6	Flow, Startup, Creep	Assoc. Polymer	No	★★★★☆	N/A	Spatially-resolved shear banding with tensor diffusion
HVM Local	HVM	6-10	R, C, O, Flow, Startup, LAOS	Vitrimer	Yes (\(G_P\))	★★★★★	N/A	Hybrid vitrimer: permanent + exchangeable (BER/TST) + dissociative networks
HVNM Local	HVNM	13-25	R, C, O, Flow, Startup, LAOS	Filled Vitrimer	Yes (\(G_P X\))	★★★★★	N/A	NP-filled vitrimer: 4 subnetworks, dual TST, Guth-Gold amplification

Legend:

Test Modes: R = Relaxation, C = Creep, O = Oscillation, Rot = Rotation (steady shear), Flow = Flow curve, Startup = Startup shear, LAOS = Large-amplitude oscillatory
Complexity: ★☆☆☆☆ = Simplest, ★★★★★ = Most complex
\(\alpha\) Range: Fractional order range for fractional models; for ITT-MCT: \(\varepsilon\) = separation parameter (glass transition), \(\phi\) = volume fraction; N/A for non-fractional models
Equilibrium Modulus: Whether model predicts finite \(G_\infty\) at long times (solid-like)

Model Selection Decision Flowchart¶

For a comprehensive decision flowchart based on your experimental data, see: /user_guide/model_selection.

Quick Selection Guide:

Quick Model Selection by Data Type¶
Data Type	Data Characteristics	Recommended Models
Oscillation (\(G'\), \(G''\))	Two plateaus visible	FZSS ★★★★☆ (most common)
	One plateau (low-\(\omega\))	FML ★★★☆☆
	Power-law (no plateaus)	FMG ★★★☆☆, SpringPot ★★☆☆☆
Relaxation (G(t))	Exponential decay → 0	Maxwell ★☆☆☆☆
	Exponential decay → plateau	Zener ★★☆☆☆
	Power-law decay	FZSS ★★★★☆, FMG ★★★☆☆
Creep (J(t))	Bounded compliance	Zener ★★☆☆☆, FZSS ★★★★☆
	Unbounded compliance	Maxwell ★☆☆☆☆, FML ★★★☆☆
Flow ( \(\eta vs \dot{\gamma}\) )	Yield stress + linear	Bingham ★★☆☆☆
	Yield stress + power-law	Herschel-Bulkley ★★☆☆☆
	Shear thinning (no yield)	Power Law ★☆☆☆☆, Carreau ★★★☆☆

Model Families Overview¶

Classical Models (3 models)¶

When to use: Exponential decay/recovery, simple viscoelastic behavior, single relaxation time.

Advantages:

Fewest parameters (2-3)
Fast fitting and physically interpretable
Well-established theory and validation
Good for teaching and simple materials

Limitations:

Cannot capture power-law behavior
Single relaxation time unrealistic for most polymers
Poor fit for broad relaxation spectra

Upgrade path to fractional:

Maxwell → Fractional Maxwell Liquid (add fractional memory)
Zener → Fractional Zener SS (add fractional relaxation)

Models:

Maxwell (2 params): Simplest liquid, single relaxation
Zener (3 params): Solid with equilibrium modulus
SpringPot (2 params): Pure power-law element (bridge to fractional)

Fractional Models (11 models)¶

When to use: Power-law relaxation, broad relaxation spectra, non-exponential behavior, soft matter.

Advantages:

Capture power-law dynamics naturally
Fewer parameters than multi-mode Maxwell
Physical interpretation via fractional order \(\alpha\)
Excellent for polymers, gels, biological materials

Fractional order ( \(\alpha\) ) interpretation:

Fractional Order Interpretation Guide¶
\(\alpha\) Value	Physical Meaning	Material Examples
\(\alpha \to 0\)	Elastic-dominated	Stiff gels, crosslinked elastomers (spring-like)
\(\alpha \approx 0.3\text{--}0.5\)	Balanced viscoelasticity	Soft gels, entangled polymers, biological tissues
\(\alpha \approx 0.5\)	Critical gel	Gel point, sol-gel transition
\(\alpha \to 1\)	Viscous-dominated	Polymer melts, concentrated solutions (dashpot-like)

Typical \(\alpha\) ranges by material:

Soft gels: \(\alpha\) = 0.2 - 0.4
Polymer melts: \(\alpha\) = 0.6 - 0.9
Biological tissues: \(\alpha\) = 0.3 - 0.5
Emulsions: \(\alpha\) = 0.4 - 0.7

Model selection within fractional family:

Most common starting point: Fractional Zener SS (FZSS) - dual plateaus, versatile
For liquids: Fractional Maxwell Liquid (FML) or Fractional Zener LL
For gels: Fractional Maxwell Gel (FMG) or SpringPot
For creep: Fractional Kelvin-Voigt (FKV) or Fractional Burgers
For complex materials: Fractional Burgers (5 params) or Fractional Maxwell Model (4 params)

Generalized Maxwell (Multi-Mode) (1 model)¶

When to use: Prony-series fitting of broadband relaxation or oscillatory data, industrial master curve analysis, when no single relaxation time captures the spectrum.

Advantages:

Systematically covers broad relaxation spectra via N Maxwell modes
Automatic mode reduction via optimization_factor — starts from N modes and prunes unnecessary ones
Directly connects to Prony series widely used in industry
Supports relaxation, creep, and oscillation protocols
JIT-compiled element search for fast multi-start optimization

Model selection:

GeneralizedMaxwell (N=2–3): Quick broadband fit, moderate complexity
GeneralizedMaxwell (N=5–10): Publication-quality master curve decomposition
GeneralizedMaxwell (optimization_factor=1.5): Auto-reduce from N=10 to optimal mode count

Key physics:

Parallel Maxwell elements: \(G(t) = G_e + \sum_{i=1}^N G_i \exp(-t/\tau_i)\)
Oscillation: \(G'(\omega) = G_e + \sum G_i \frac{\omega^2 \tau_i^2}{1 + \omega^2 \tau_i^2}\)
Element search warm-starts from N+1, re-uses JIT compilation (2-5x speedup)

Typical applications: Polymer master curves, broadband industrial QC, relaxation spectra decomposition, viscoelastic material databases.

Flow Models (6 models)¶

When to use: Steady shear flow, viscosity vs shear rate, non-Newtonian fluids, process design.

Giesekus Models (2 models)¶

When to use: Polymer melts and solutions exhibiting shear thinning, nonlinear normal stress differences, stress overshoot in startup, and LAOS response. Ideal when both \(N_1\) and \(N_2\) predictions are required.

Advantages:

Quadratic stress term gives physically motivated shear thinning
Predicts both \(N_1 > 0\) and \(N_2 < 0\) with fixed ratio \(N_2/N_1 = -\alpha/2\)
Mobility factor \(\alpha\) directly measurable from normal stress ratio
ODE-based: full support for flow curve, SAOS, startup, relaxation, creep, LAOS
Multi-mode variant for broadband spectra with mode-dependent \(\alpha_i\)

Mobility factor ( \(\alpha\) ) interpretation:

Giesekus Mobility Factor Guide¶
\(\alpha\) Value	Physical Meaning	Material Examples
\(\alpha = 0\)	UCM limit (no shear thinning)	Dilute polymer solutions, Boger fluids
\(\alpha \approx 0.1\text{--}0.3\)	Moderate shear thinning	Polymer melts, semidilute solutions
\(\alpha \approx 0.5\)	Maximum anisotropy	Strongly shear-thinning polymer melts

Model selection within Giesekus family:

GiesekusSingleMode: 4 params (\(\eta_p, \lambda, \alpha, \eta_s\)), single relaxation time, all 6 protocols
GiesekusMultiMode: N modes with independent \(\alpha_i\), broadband spectra, flow curve + SAOS + startup

Key physics:

Constitutive equation: \(\boldsymbol{\tau} + \lambda \overset{\nabla}{\boldsymbol{\tau}} + \frac{\alpha \lambda}{\eta_p} \boldsymbol{\tau} \cdot \boldsymbol{\tau} = 2\eta_p \mathbf{D}\)
Conformation tensor: \(\mathbf{c} = \mathbf{I} + (\lambda/\eta_p)\boldsymbol{\tau}\), quadratic term drives anisotropic relaxation
Analytical flow curve: \(\eta(\dot{\gamma})\) from cubic equation at steady state
Cox-Merz rule: \(\eta(\dot{\gamma}) \approx |\eta^*(\omega)|\) for moderate \(\alpha\)

Typical applications: Polymer melts (PE, PP, PS), concentrated solutions, wormlike micelles, liquid crystals, any system needing \(N_1, N_2\) predictions.

Fluidity Models (2 models)¶

When to use: Thixotropic yield-stress fluids, materials with time-dependent viscosity, fluidity-based structure kinetics, shear banding via cooperative diffusion.

Advantages:

Scalar fluidity parameter \(f\) tracks microstructural state
Coupled aging–rejuvenation kinetics for thixotropy
Simple yet effective: connects naturally to soft glassy rheology
Nonlocal variant adds cooperativity length for shear banding resolution
Supports flow curve, startup, creep, and LAOS protocols

Model selection within Fluidity family:

FluidityLocal: Homogeneous flow, scalar fluidity evolution, fast fitting
FluidityNonlocal: PDE-based spatially resolved flow, banding detection, cooperativity length \(\xi\)

Key physics:

Fluidity evolution: \(df/dt = (f_{eq} - f)/\tau_f + D_f \nabla^2 f\) (nonlocal)
Flow rule: \(\sigma = \eta(f) \dot{\gamma}\) with \(\eta = 1/f\)
Cooperativity length \(\xi\) sets minimum shear band width

Typical applications: Colloidal gels, bentonite suspensions, Laponite, Carbopol, foams, soft glassy materials.

Fluidity-Saramito EVP Models (2 models)¶

When to use: Yield-stress fluids with combined elastic, viscous, and plastic behavior; thixotropic materials requiring stress overshoot prediction; systems needing normal stress difference (\(N_1\)) predictions; shear banding analysis.

Advantages:

Full tensorial stress state: [\(\tau_{xx}, \tau_{yy}, \tau_{xy}\)] for normal stress predictions
Von Mises yield criterion with Herschel-Bulkley plastic flow
Thixotropic fluidity evolution (aging + rejuvenation)
Predicts stress overshoot in startup shear (key thixotropic signature)
Supports 6 protocols: flow curve, startup, creep, relaxation, oscillation, LAOS
Nonlocal variant captures shear banding via cooperativity length

Model selection within Saramito family:

FluiditySaramitoLocal (minimal): Simplest, \(\lambda\) = 1/f only, homogeneous flow
FluiditySaramitoLocal (full): \(\tau_y(f)\) coupling, aging yield stress
FluiditySaramitoNonlocal (minimal): Shear banding capable with \(D_f \nabla^2 f\)
FluiditySaramitoNonlocal (full): Full thixotropic banding

Key physics:

Upper-convected Maxwell viscoelasticity: \(\lambda(d\tau/dt - \mathbf{L} \cdot \tau - \tau \cdot \mathbf{L}^T) + \alpha(\tau)\tau = 2\eta_p \mathbf{D}\)
Plasticity parameter: \(\alpha = \max(0, 1 - \tau_y / |\tau|)\) (Von Mises)
Fluidity evolution: \(df/dt = (f_{\text{age}} - f)/t_a + b|\dot{\gamma}|^n(f_{\text{flow}} - f)\)

Typical applications: Carbopol gels, cement pastes, drilling muds, mayonnaise, blood, cosmetic creams.

Soft Glassy Rheology Models (2 models)¶

When to use: Soft glassy materials (foams, emulsions, pastes, colloidal suspensions), aging systems, power-law fluids near glass transition.

Advantages:

Statistical mechanics foundation (trap model)
Single noise temperature parameter x captures material state
Natural aging dynamics for \(x < 1\)
Power-law rheology emerges from microscopic physics
Bayesian inference support for uncertainty quantification

Noise temperature ( \(x\) ) interpretation:

Noise Temperature Interpretation Guide¶
x Value	Physical Meaning	Material Examples
\(x < 1\)	Glass (aging)	Aged colloidal suspensions, dense pastes (non-ergodic)
\(x \approx 1\)	Glass transition	Critical point, rheological singularity
\(1 < x < 2\)	Power-law fluid	Foams, emulsions, soft gels (SGM regime)
\(x \geq 2\)	Newtonian liquid	Dilute suspensions, simple fluids

Model selection within SGR family:

SGR Conventional (Sollich 1998): Standard trap model, simpler formulation
SGR GENERIC (Fuereder & Ilg 2013): Thermodynamically consistent, better stability near \(x \to 1\)

Connection to SRFS Transform:

The noise temperature \(x\) from SGR models directly relates to SRFS shift factors: \(a(\dot{\gamma}) \sim \dot{\gamma}^{(2-x)}\), enabling complementary analysis of oscillatory and flow data.

ITT-MCT Models (2 models)¶

When to use: Dense colloidal suspensions near the glass transition, hard-sphere systems, microscopic rheological theory, yielding and flow of glassy materials.

Advantages:

Microscopic theory based on Mode-Coupling Theory
Quantitative predictions for hard-sphere colloids
Captures glass transition physics (cage effect)
Full nonlinear rheology including LAOS harmonics
Two-time correlators for non-equilibrium response
Strain decorrelation naturally emerges from advection

Separation parameter ( \(\varepsilon\) ) interpretation:

Glass Transition Parameter Guide¶
\(\varepsilon\) Value	Physical Meaning	Material Examples
\(\varepsilon\) < 0	Glass state	Dense suspensions below \(\phi_c\), kinetically arrested
\(\varepsilon \approx 0\)	Glass transition	Critical point, diverging relaxation time
\(\varepsilon\) > 0	Fluid state	Mobile suspensions, ergodic dynamics

Model selection within ITT-MCT family:

ITTMCTSchematic ( \(F_{12}\) ): Simplified scalar correlator, 6 parameters, fast fitting
ITTMCTIsotropic (ISM): Full k-resolved correlators with S(k) input, quantitative predictions

Key physics:

Memory kernel: \(m(\Phi) = v_1 \Phi + v_2 \Phi^2\) (schematic) or k-integral (isotropic)
Glass transition criterion: \(v_{2,c} = 4\) (for \(v_1 = 0\))
Strain decorrelation: \(h(\gamma) = \exp(-(\gamma/\gamma_c)^2)\)
Integration through transients (ITT) for nonlinear flow

Typical applications: PMMA hard-sphere colloids, silica suspensions, concentrated emulsions, microgel pastes.

Comparison with SGR:

SGR: Phenomenological trap model, noise temperature x, simpler physics
ITT-MCT: Microscopic derivation, volume fraction \(\phi\), full correlator dynamics
Both capture yielding, but ITT-MCT provides quantitative predictions from structure

DMT Thixotropic Models (2 models)¶

When to use: Thixotropic materials with time-dependent rheology, stress overshoot in startup, delayed yielding, materials with structural buildup at rest.

Advantages:

Scalar structure parameter \(\lambda \in [0, 1]\) tracks microstructure
Clear separation of buildup (aging) and breakdown (shear) kinetics
Two viscosity closures: exponential (smooth) or Herschel-Bulkley (yield stress)
Optional Maxwell backbone for stress overshoot and SAOS
Nonlocal variant captures shear banding via structure diffusion

Structure parameter ( \(\lambda\) ) interpretation:

Structure Parameter Guide¶
\(\lambda\) Value	Physical Meaning	Material State
\(\lambda\) = 1	Fully structured	At rest (aged), maximum viscosity, colloidal network intact
0 < \(\lambda\) < 1	Partially broken	Under shear, intermediate microstructure
\(\lambda\) = 0	Fully broken	High shear (rejuvenated), minimum viscosity, network destroyed

Model selection within DMT family:

DMTLocal (exponential): Smooth viscosity transition, no yield stress, simple
DMTLocal (herschel_bulkley): Explicit yield stress, structure-dependent \(\tau_y\) and K
DMTLocal + elasticity: Maxwell backbone for stress overshoot and SAOS
DMTNonlocal: Shear banding via structure diffusion (\(D_{\lambda} \nabla^2 \lambda\))

Key physics:

Structure kinetics: \(d\lambda/dt = (1-\lambda)/t_eq - a\lambda|\dot{\gamma}|^c/t_eq\)
Equilibrium structure: \(\lambda_{eq} = 1/(1 + a|\dot{\gamma}|^c)\)
Exponential viscosity: \(\eta(\lambda) = \eta_{\infty}(\eta_0/\eta_{\infty})^{\lambda}\)
Maxwell stress: \(d\sigma/dt = G\dot{\gamma} - \sigma/\theta(\lambda)\)

Typical applications: Drilling muds, waxy crude oils, cement pastes, mayonnaise, ketchup, paints, concentrated suspensions.

Isotropic Kinematic Hardening Models (2 models)¶

When to use: Materials with yield-stress evolution under deformation history, cyclic loading with Bauschinger effect, isotropic + kinematic hardening, metal-like rheology in complex fluids.

Advantages:

Combined isotropic and kinematic hardening captures evolving yield surfaces
Multi-layer variant (MLIKH) for progressive yielding
ODE-based: startup, creep, relaxation, oscillation, LAOS
Strain-rate-dependent yield for soft materials

Model selection within IKH family:

MIKH: Modified IKH with single yield surface — simpler, 6-8 parameters
MLIKH: Multi-layer IKH with N yield surfaces — progressive yielding, N×3 + base parameters

Key physics:

Yield function: \(f = |\sigma - \alpha| - (\sigma_y + R)\) (kinematic + isotropic)
Back-stress evolution: \(\dot{\alpha} = C \dot{\varepsilon}^p - \gamma_k \alpha |\dot{\varepsilon}^p|\)
Isotropic hardening: \(\dot{R} = b(Q - R) |\dot{\varepsilon}^p|\)

Typical applications: Structured fluids under cyclic loading, waxy crude oils, soft solids, gel fracture.

Fractional IKH Models (2 models)¶

When to use: Same as IKH but with power-law memory effects; materials requiring fractional-order structure kinetics, long-time memory in yielding behavior.

Advantages:

Caputo fractional derivative in structure kinetics — bridges IKH and fractional viscoelasticity
Order \(\alpha \in (0, 1]\) interpolates between integer (IKH) and maximally non-local memory
Inherits all IKH protocols plus fractional relaxation spectra
Multi-layer fractional variant (FMLIKH) for progressive yielding with memory

Model selection within FIKH family:

FIKH: Fractional IKH with single yield surface + Caputo memory, 5-6 parameters
FMLIKH: Fractional multi-layer IKH — N yield surfaces with fractional kinetics

Key physics:

Fractional structure kinetics: \({}^C D_t^{\alpha} \lambda = \text{aging} - \text{shear breakdown}\)
Caputo derivative \({}^C D_t^{\alpha}\) provides long-range temporal memory
Reduces to integer IKH when \(\alpha \to 1\)

Typical applications: Materials with long-time memory effects, thixotropic systems with power-law recovery, structured fluids under complex loading histories.

Hébraud-Lequeux Model (1 model)¶

When to use: Dense amorphous materials (emulsions, foams, granular media) where mesoscopic rearrangement events (T1 events) control rheology; mean-field fluidity approach for amorphous solids.

Advantages:

Mean-field kinetic model for mesoscopic stress redistribution
Predicts flow curves, creep, and oscillatory response from microscopic rearrangements
PDE-based stress probability distribution — captures heterogeneity
Connects to SGR at the mesoscale but with explicit stress redistribution

Key physics:

Stress probability distribution \(P(\sigma, t)\) evolves via advection + diffusion + rearrangement
Rearrangement rate: \(\Gamma = \Gamma_0 \Theta(|\sigma| - \sigma_c)\) (above critical stress)
Mean-field coupling: rearrangement events redistribute stress to neighbors
Diffusion coefficient \(D_\sigma \propto \alpha \Gamma\) from collective rearrangements

Typical applications: Concentrated emulsions, wet foams, colloidal glasses, granular media near jamming.

STZ Model (1 model)¶

When to use: Amorphous solids undergoing plastic deformation via shear transformation zones, metallic glasses, bulk metallic glass forming liquids, granular materials.

Advantages:

Physical basis in localized shear transformation zones
Temperature-dependent transition rates (Arrhenius activated)
Captures strain rate sensitivity and rate-dependent yield stress
ODE-based: 8 parameters, all physically interpretable
Supports flow curve, startup, creep, and relaxation

Key physics:

STZ creation/annihilation: \(\dot{\Lambda} = R_0 [e^{-\Delta F / k_B T} \cosh(\Omega \sigma / k_B T)]\)
Effective disorder temperature \(\chi\) evolves with plastic work
Strain rate: \(\dot{\varepsilon}^{pl} = 2 \epsilon_0 \Lambda e^{-\Delta F / k_B T} \sinh(\Omega \sigma / k_B T)\)
Steady-state flow stress is rate- and temperature-dependent

Typical applications: Metallic glasses, amorphous polymers below \(T_g\), granular shear, simulation benchmarks for amorphous plasticity.

Elasto-Plastic Models (2 models)¶

When to use: Yield-stress materials modeled as ensembles of mesoscopic elastoplastic elements; lattice-based models for heterogeneous yielding; full tensorial stress for anisotropic plasticity.

Advantages:

Mesoscale ensemble approach: many elements sample the stress distribution
Lattice variant adds spatial correlations (Eshelby-like stress propagation)
Tensorial variant for full 3D stress state and anisotropic yield surfaces
SAOS from element-level Maxwell response with yield threshold
Flow curve from element statistics with configurable disorder

Model selection within EPM family:

LatticeEPM: Lattice-based, L×L grid, Eshelby kernel, spatial correlations
TensorialEPM: Full tensor, 3D stress state, anisotropic yield, off-lattice

Key physics:

Element mechanics: \(\sigma_i = G(\gamma - \gamma_i^{pl})\) with local yield \(\sigma_c\)
Yield criterion: \(|\sigma_i| > \sigma_c\) triggers plastic rearrangement
Stress redistribution: Eshelby kernel (lattice) or mean-field (tensorial)
Disorder: \(\sigma_c\) drawn from configurable distribution (Gaussian, Weibull)

Typical applications: Soft glasses, amorphous solids, yield stress fluids with heterogeneous microstructure, earthquake fault mechanics analogues.

Transient Network Theory Models (9 variants across 5 classes)¶

When to use: Associating polymers, physical gels, telechelic networks, wormlike micelles, living polymers, bio-networks with reversible crosslinks, any material with bond-mediated viscoelasticity.

Advantages:

Molecular-level physics: conformation tensor tracks chain stretch and orientation
Composable variants: Bell + FENE + slip can be combined in a single model
Full protocol support: all 6 test modes (flow curve, SAOS, startup, relaxation, creep, LAOS)
GPU-accelerated ODE integration via Diffrax with JAX JIT compilation
Complete Bayesian inference pipeline (NLSQ → NUTS)

Key physics:

Conformation tensor \(\mathbf{S}\) evolves via upper-convected derivative + breakage
Stress: \(\boldsymbol{\sigma} = G \cdot f(\mathbf{S}) + 2\eta_s \mathbf{D}\)
Bond lifetime \(\tau_b\) can be constant (Tanaka-Edwards) or force-dependent (Bell)
Single mode recovers Maxwell behavior; multi-mode gives broad spectra

Model selection within TNT family:

Start here: TNTSingleMode (constant breakage) — 3 params, Maxwell-like baseline
Shear thinning: TNTSingleMode(breakage=”bell”) — force-dependent breakage
Strain hardening: TNTSingleMode(stress_type=”fene”) — finite extensibility
Telechelic networks: TNTLoopBridge — loop-bridge population kinetics
Multi-sticker polymers: TNTStickyRouse — hierarchical Rouse + sticker relaxation
Wormlike micelles: TNTCates — living polymer scission/recombination
Heterogeneous networks: TNTMultiSpecies — discrete relaxation spectrum

Typical applications: HEUR telechelics, PEG-PEO associating polymers, fibrin and collagen bio-networks, CTAB/CPCl wormlike micelles, PVA-borax gels, supramolecular polymer networks, vitrimers.

VLB Transient Network Models (4 models)¶

When to use: Associating polymers, physical gels, hydrogels, vitrimers, self-healing polymers, any material with reversible cross-links where a molecular-statistical foundation is desired.

Advantages:

Molecular-statistical foundation via distribution tensor \(\boldsymbol{\mu}\)
All-analytical single-network predictions (2 parameters, Maxwell behavior)
Multi-network extension for broad relaxation spectra
Uniaxial extension predictions (Trouton ratio, extensional viscosity)
Bell breakage for shear thinning, stress overshoot, nonlinear LAOS
FENE-P for bounded extensional viscosity and strain hardening
Arrhenius temperature dependence
Nonlocal PDE for shear banding with tensor diffusion
Full Bayesian inference pipeline (NLSQ → NUTS)

Key physics:

Distribution tensor \(\boldsymbol{\mu} = \langle \mathbf{r}\mathbf{r} \rangle / \langle r_0^2 \rangle\) from chain statistics
Stress: \(\boldsymbol{\sigma} = G_0(\boldsymbol{\mu} - \mathbf{I})\)
Bond kinetics: \(\dot{\boldsymbol{\mu}} = k_d(\mathbf{I} - \boldsymbol{\mu}) + \mathbf{L} \cdot \boldsymbol{\mu} + \boldsymbol{\mu} \cdot \mathbf{L}^T\)
Single network recovers Maxwell; multi-network gives generalized Maxwell
Bell breakage: \(k_d(\mu) = k_d^0 \exp(\nu(\lambda_c - 1))\)
FENE-P: \(\sigma = G_0 f(\text{tr}(\mu))(\mu - I)\) with bounded extensibility
Nonlocal PDE: \(+ D_\mu \nabla^2 \mu\) for cooperative rearrangements

Model selection within VLB family:

Start here: VLBLocal — 2 params (\(G_0, k_d\)), analytical everywhere
Broad spectrum: VLBMultiNetwork — N modes + optional permanent network + solvent
Nonlinear: VLBVariant — Bell shear thinning, FENE bounded extension, temperature
Shear banding: VLBNonlocal — spatially-resolved PDE with banding detection

Typical applications: PVA-borax hydrogels, boronate ester gels, vitrimers, telechelic polymers, supramolecular networks, shear-banding wormlike micelles.

Comparison with TNT:

Mathematically equivalent to TNT at constant \(k_d\) (both give Maxwell)
VLB now has Bell + FENE-P variants (matching TNT’s nonlinear extensions)
VLB preferred for molecular extensions (Langevin, entropic \(k_d\))
TNT additionally offers non-affine and loop-bridge variants

Hybrid Vitrimer Model (1 model)¶

When to use: Vitrimers (covalent adaptable networks) with associative bond exchange, materials with permanent + exchangeable crosslinks, temperature-dependent topology rearrangement.

Advantages:

3-subnetwork architecture: permanent (P) + exchangeable vitrimer (E) + dissociative physical (D)
Evolving natural-state tensor \(\mu^E_{nat}\) — the vitrimer hallmark (BER rearranges topology)
TST kinetics: stress- or stretch-activated bond exchange rates
Arrhenius temperature dependence with topology freezing transition \(T_v\)
Factory methods for 5 limiting cases (neo-Hookean, Maxwell, Zener, pure/partial vitrimer)
Full protocol support: flow curve, SAOS, startup, relaxation, creep, LAOS

Key physics:

Bond exchange reaction: \(k_{BER} = \nu_0 \exp(-E_a/RT) \cosh(V_{act} \sigma_{VM}/RT)\)
Factor-of-2: \(\tau_E^{eff} = 1/(2 k_{BER,0})\) — both \(\mu^E\) and \(\mu^E_{nat}\) relax toward each other
Stress \(\sigma_E \to 0\) at steady state (natural state fully tracks deformation)
11-component ODE state integrated via Diffrax Tsit5

Typical applications: Epoxy vitrimers, polyester CANs, silicone vitrimers, polyurethane vitrimers, self-healing networks.

Hybrid Vitrimer Nanocomposite Model (1 model)¶

When to use: Nanoparticle-filled vitrimers, nanocomposites with interfacial bond exchange, materials where filler–matrix interphase contributes distinct relaxation.

Advantages:

4-subnetwork architecture: P + E + D + interphase (I) around nanoparticles
Guth-Gold strain amplification: \(X(\phi) = 1 + 2.5\phi + 14.1\phi^2\)
Dual TST kinetics: independent matrix (\(k_{BER}^{mat}\)) and interphase (\(k_{BER}^{int}\)) exchange
\(\phi = 0\) recovers HVM exactly (verified to machine precision)
Factory methods for 5 configurations: unfilled vitrimer, filled elastomer, partial NC, etc.

Key physics:

Interphase reinforcement: \(G_I = \beta_I \cdot G_E\) scales with NP surface area
Separate Arrhenius activation for matrix and interphase exchange
Feature flags for interfacial damage, diffusion, and degradation
17-18 component ODE state depending on configuration

Typical applications: Silica-epoxy vitrimer nanocomposites, CNT-vitrimer networks, graphene-polymer CANs, functional nanocomposites with adaptable bonds.

SPP LAOS Model (1 model)¶

When to use: Large amplitude oscillatory shear (LAOS) analysis, yield stress extraction from oscillatory data, intracycle nonlinear characterization, model validation against SPP trajectories.

Advantages:

Instantaneous moduli \(G'_t, G''_t\) resolve intracycle viscoelastic transitions
Cole-Cole trajectory reveals sequence of physical processes during nonlinear deformation
Robust yield stress determination from trajectory features
Model-experiment comparison via trajectory mismatch metric
Complementary to Fourier-based LAOS (FT-Rheology)

Key physics:

Instantaneous storage: \(G'_t = \dot{\sigma}/\dot{\gamma}\) (elastic contribution)
Instantaneous loss: \(G''_t = (1/\omega)(d\sigma/d\gamma)|_{\dot{\gamma}=\text{const}}\) (viscous contribution)
Cole-Cole trajectory: \(G'_t\) vs \(G''_t\) traces physical process sequence
Yield identification: kink/cusp (Type I) or smooth maximum (Type II)

Typical applications: Yield stress fluids (Carbopol, cement), soft glasses, colloidal gels, biological hydrogels, any material requiring intracycle LAOS analysis.

Non-Newtonian classification:

Shear-thinning (pseudoplastic): Viscosity decreases with shear rate
- Most common: polymer solutions, paints, food products
- Models: Power Law (n<1), Carreau, Cross, Herschel-Bulkley (n<1)
Shear-thickening (dilatant): Viscosity increases with shear rate
- Less common: concentrated suspensions, cornstarch
- Models: Power Law (n>1), Herschel-Bulkley (n>1)
Viscoplastic (yield stress): Requires minimum stress to flow
- Examples: toothpaste, gels, slurries, drilling muds
- Models: Bingham, Herschel-Bulkley

Industrial applications:

Flow Models by Industry¶
Industry	Common Models	Typical Materials
Polymer Processing	Carreau, Cross, Power Law	Polymer melts, concentrated solutions
Food & Cosmetics	Herschel-Bulkley, Bingham	Ketchup, toothpaste, yogurt, creams
Oil & Gas	Herschel-Bulkley, Power Law	Drilling muds, crude oil
Coatings & Paints	Carreau, Herschel-Bulkley	Paints, inks, adhesives
Pharmaceuticals	Bingham, Carreau-Yasuda	Suspensions, gels, ointments

Quick Selection Guide¶

By Material Type¶

Material-to-Model Quick Reference¶
Material Type	Recommended Models	Notes
Polymer Melts	Giesekus, FML, FZSS, Carreau (flow)	Giesekus for \(N_1, N_2\) and startup; \(\alpha\) typically 0.6-0.9 for fractional
Soft Gels	FZSS, FMG, SpringPot	\(\alpha\) typically 0.2-0.4; check for yield stress
Elastomers	FZSS, Zener	Two plateaus common; classical may suffice
Biological Tissues	FZSS, FML, Fractional Burgers	\(\alpha\) typically 0.3-0.5; complex behavior common
Emulsions/Suspensions	FZSS (oscillation), Herschel-Bulkley (flow)	Check for yield stress in flow
Critical Gels	SpringPot, FMG	\(\alpha \approx 0.5\); power-law across all frequencies
Polymer Solutions	Giesekus, Carreau, Cross (flow); FML (oscillation)	Giesekus for nonlinear + \(N_1\); Carreau/Cross for viscosity only
Viscoplastic Materials	Bingham, Herschel-Bulkley	Yield stress present; toothpaste, gels, slurries
Foams/Emulsions	SGR Conventional, SGR GENERIC	Soft glassy materials; x parameter captures state
Colloidal Suspensions	SGR Conventional, ITTMCTSchematic, FZSS	Aging systems (\(x < 1\)), hard-sphere (MCT), or power-law fluids
Hard-Sphere Colloids	ITTMCTSchematic, ITTMCTIsotropic	Near glass transition; use ISM for quantitative S(k) predictions
Pastes/Dense Suspensions	SGR GENERIC, Herschel-Bulkley	Near glass transition; use GENERIC for \(x \to 1\)
Thixotropic Yield Stress	FluiditySaramitoLocal, Herschel-Bulkley	Stress overshoot, aging; use Saramito for \(N_1\)
Shear Banding Materials	FluiditySaramitoNonlocal, FluidityNonlocal	Spatially resolved flow, cooperativity length
Associating Polymers	TNTSingleMode, TNTStickyRouse	Reversible crosslinks; Bell variant for shear thinning
Wormlike Micelles	TNTCates, TNTSingleMode(bell)	Living polymers; \(\tau_d = \sqrt{\tau_{rep} \cdot \tau_{break}}\)
Telechelic Networks	TNTLoopBridge, TNTSingleMode	Loop-bridge kinetics; end-functionalized polymers
Self-Healing Gels	VLBLocal, VLBMultiNetwork	Molecular-statistical foundation; 2 params for Maxwell-like networks
Vitrimers/CANs	HVMLocal, VLBMultiNetwork	Evolving natural state, BER/TST kinetics, Arrhenius \(k_{BER}\)
NP-Filled Vitrimers	HVNMLocal, HVMLocal (unfilled)	Dual TST kinetics, Guth-Gold amplification, Payne effect
DMTA/DMA Specimens	FZSS, GeneralizedMaxwell, Zener	Set `deformation_mode='tension'`; auto E↔G conversion

By Application¶

Application-Based Model Selection¶
Application	Primary Goal	Recommended Models	Complexity
Research	Physical insight, publication	Fractional models (FZSS, FML, Burgers)	★★★★☆
Industrial QC	Fast screening, reproducibility	Maxwell, Zener, Power Law, Bingham	★★☆☆☆
Process Design	Predict flow behavior	Carreau, Herschel-Bulkley, Cross	★★★☆☆
Material Development	Structure-property relationships	Fractional models, multi-technique	★★★★★
Teaching	Conceptual understanding	Maxwell, Zener, Power Law	★☆☆☆☆

By Data Quality¶

Data Quality Considerations¶
Data Characteristics	Model Recommendation	Rationale
Limited data (<20 points)	2-3 parameter models (Maxwell, Zener, Power Law)	Avoid overfitting with simpler models
Moderate data (20-50 points)	3-4 parameter models (FZSS, FML, Carreau)	Balanced complexity and fit quality
Extensive data (>50 points)	Complex models (Burgers, Carreau-Yasuda, FMM)	Sufficient data to constrain 5+ parameters
High noise	Classical models first	Fractional models sensitive to noise; pre-smooth data
Narrow frequency range	Avoid multi-parameter models	Limited information → simpler models
Multi-technique data	Advanced fractional models	Combined relaxation + oscillation → Burgers, FZSS

Parameter Count Comparison¶

2-Parameter Models (Simplest):

Maxwell: \(G_0, \eta\) - Liquid with single relaxation
PowerLaw: K, n - Shear-thinning/thickening
Bingham: \(\tau_0, \eta_{pl}\) - Linear viscoplastic
SpringPot: V, \(\alpha\) - Pure power-law element

3-Parameter Models:

Zener: Gs, Gp, \(\eta_p\) - Classical solid with plateau
FML: V, \(\alpha, \eta\) - Fractional liquid
FMG: Gs, V, \(\alpha\) - Fractional gel
Herschel-Bulkley: \(\tau_0\), K, n - Yield + power-law

4-Parameter Models:

FZSS: Ge, Gm, \(\alpha, \tau\alpha\) - Most common fractional solid
FZSL: Gs, \(\eta_s, V, \alpha\) - Fractional solid-liquid Zener
FZLL: \(\eta_s, \eta_p, V, \alpha\) - Fractional liquid-liquid Zener
FKV: Gp, V, \(\alpha\), (\(\eta_p\) optional) - Fractional Kelvin-Voigt
Carreau: \(\eta_0, \eta_{\infty}, \lambda\), n - Flow with plateaus
Cross: K, m, \(\eta_0, \eta_{\infty}\) - Alternative flow interpolation
Fractional Maxwell Model: \(V_1, V_2, \alpha_1, \alpha_2\) - Dual springpots
Fractional Jeffreys: Two dashpots + springpot parameters

5-Parameter Models (Most Complex):

Fractional Burgers: Maxwell + FKV (5 params) - Creep + relaxation
Fractional Poynting-Thomson: Multi-plateau solid (5 params)
Carreau-Yasuda: \(\eta_0, \eta_{\infty}, \lambda\), n, a - Adjustable transition

Bayesian Inference Support¶

All 53 models support complete Bayesian workflows via NumPyro NUTS sampling:

.fit() - Fast NLSQ point estimation
.fit_bayesian() - Full posterior sampling with MCMC
.sample_prior() - Prior predictive checks
.get_credible_intervals() - Uncertainty quantification

Recommended workflow: NLSQ → NUTS warm-start for 2-5x faster convergence.

See /user_guide/bayesian_inference for comprehensive Bayesian analysis guide.

DMTA / DMA Support¶

All 49 oscillation-capable models support DMTA data through automatic \(E^* \leftrightarrow G^*\) conversion at the BaseModel boundary:

Tensile modulus conversion: \(E^* = 2(1 + \nu) G^*\) applied automatically when deformation_mode='tension'
Poisson ratio presets: rubber (0.5), glassy polymer (0.35), semicrystalline (0.40)
Transparent workflow: Model parameters stay in shear space; conversion at fit/predict boundary
CSV auto-detection: Columns named E', E'', or E* automatically set deformation_mode='tension'

from rheojax.models import FractionalZenerSolidSolid

model = FractionalZenerSolidSolid()
model.fit(omega, E_star, test_mode='oscillation',
          deformation_mode='tension', poisson_ratio=0.5)
E_pred = model.predict(omega)  # Returns E* automatically

See DMTA / DMA Analysis for DMTA theory, model compatibility, and workflow guides.

Next Steps¶

Detailed model documentation: See Models Handbook for individual model handbooks
Multi-technique fitting: /user_guide/multi_technique_fitting
Model selection workflow: /user_guide/model_selection
Compatibility checking: /user_guide/core_concepts (automatic detection of model-data mismatches)
Giesekus models: Giesekus Model — Handbook for nonlinear viscoelastic polymer melts and solutions
SGR models: SGR Conventional (Soft Glassy Rheology) — Handbook and SGR GENERIC (Thermodynamically Consistent)
ITT-MCT models: ITT-MCT Schematic (F_1_2) and ITT-MCT Isotropic (ISM) for colloidal glasses
TNT models: Transient Network Theory (TNT) for transient network theory (associating polymers, micelles)
VLB models: VLB Transient Network Models for VLB transient networks (hydrogels, vitrimers, self-healing polymers)
HVM models: HVM (Hybrid Vitrimer Model) for hybrid vitrimer constitutive models
HVNM models: HVNM (Hybrid Vitrimer Nanocomposite Model) for vitrimer nanocomposite models
DMTA support: DMTA / DMA Analysis for tensile modulus conversion and DMA workflows
SRFS transform: Strain-Rate Frequency Superposition (SRFS) for strain-rate frequency superposition
Example notebooks: 246 examples across 20+ directories in examples/

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