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. .. list-table:: Comprehensive Model Comparison :header-rows: 1 :widths: 16 10 6 12 12 10 8 8 18 :class: longtable * - Model - Family - Params - Test Modes - Material Type - Equilibrium Modulus - Complexity - :math:`\alpha` Range - Best For * - :doc:`Maxwell ` - Classical - 2 - R, C, O, Rot - Liquid - No (:math:`G_\infty = 0`) - ★☆☆☆☆ - N/A - Simple viscoelastic liquids, polymer melts with single relaxation * - :doc:`Zener ` - Classical - 3 - R, C, O - Solid - Yes (:math:`G_e > 0`) - ★★☆☆☆ - N/A - Soft solids, elastomers with exponential relaxation * - :doc:`SpringPot ` - Fractional - 2 - R, O - Gel - No - ★★☆☆☆ - 0-1 - Power-law gels, critical gels (Scott-Blair element) * - :doc:`Generalized Maxwell ` - Multi-Mode - 2N+1 - R, C, O - Variable - Configurable - ★★★★☆ - N/A - Prony series, broadband fitting, industrial master curves * - :doc:`Fractional Maxwell Gel ` - Fractional - 3 - R, C, O - Gel - No - ★★★☆☆ - 0-1 - Gels with elastic plateau + power-law tail * - :doc:`Fractional Maxwell Liquid ` - Fractional - 3 - R, C, O - Liquid - No (flows) - ★★★☆☆ - 0-1 - Liquid-like materials with fractional memory effects * - :doc:`Fractional Maxwell Model ` - Fractional - 4 - R, O - Variable - Configurable - ★★★★☆ - 0-1 (two) - Wideband fitting, materials with multiple fractional processes * - :doc:`Fractional Kelvin-Voigt ` - Fractional - 3-4 - C, O - Solid - Yes - ★★★☆☆ - 0-1 - Solid-like with slow fractional relaxation, creep-dominated * - :doc:`Fractional Zener SL ` - Fractional - 4 - R, C, O - Solid - Yes (:math:`G_s > 0`) - ★★★★☆ - 0-1 - Solid with fractional liquid leg, intermediate behavior * - :doc:`Fractional Zener SS ` - Fractional - 4 - R, C, O - Solid - Yes (:math:`G_e > 0`) - ★★★★☆ - 0-1 - Dual elastic plateaus with fractional transition (most common) * - :doc:`Fractional Zener LL ` - Fractional - 4 - R, C, O - Liquid - No - ★★★★☆ - 0-1 - Liquid-biased Zener, complex liquids with memory * - :doc:`Fractional KV Zener ` - Fractional - 4 - C, O - Solid - Yes - ★★★★☆ - 0-1 - Fractional KV block in series with spring, creep applications * - :doc:`Fractional Burgers ` - Fractional - 5 - R, C, O - Solid/Liquid - Configurable - ★★★★★ - 0-1 - Captures creep AND relaxation simultaneously, versatile * - :doc:`Fractional Poynting-Thomson ` - Fractional - 5 - R, C, O - Solid - Yes - ★★★★★ - 0-1 - Multi-plateau solids, alternate formulation for complex behavior * - :doc:`Fractional Jeffreys ` - Fractional - 4 - R, C, O - Liquid - No - ★★★★☆ - 0-1 - Liquid-like with fractional damping, two dashpots + springpot * - :doc:`Power Law ` - Flow - 2 - Rot - Fluid - N/A - ★☆☆☆☆ - N/A - Shear-thinning/thickening fluids (Ostwald-de Waele) * - :doc:`Carreau ` - Flow - 4 - Rot - Fluid - N/A - ★★★☆☆ - N/A - Polymer solutions with Newtonian → power-law transition * - :doc:`Carreau-Yasuda ` - Flow - 5 - Rot - Fluid - N/A - ★★★★☆ - N/A - Adjustable transition sharpness, concentrated polymers * - :doc:`Cross ` - Flow - 4 - Rot - Fluid - N/A - ★★★☆☆ - N/A - Alternative interpolation for polymer solutions * - :doc:`Herschel-Bulkley ` - Flow - 3 - Rot - Viscoplastic - N/A - ★★☆☆☆ - N/A - Yield stress fluids with power-law post-yield (gels, slurries) * - :doc:`Bingham ` - Flow - 2 - Rot - Viscoplastic - N/A - ★★☆☆☆ - N/A - Linear viscoplastic (yield stress + constant viscosity) * - :doc:`Giesekus Single Mode ` - Giesekus - 4 - R, C, O, Flow, Startup, LAOS - Polymer - No - ★★★★☆ - :math:`\alpha`: 0-0.5 - Nonlinear viscoelastic with shear thinning, :math:`N_1, N_2` predictions * - :doc:`Giesekus Multi Mode ` - Giesekus - 4N - O, Flow, Startup - Polymer - No - ★★★★★ - :math:`\alpha_i`: 0-0.5 - Multi-mode nonlinear viscoelastic, broadband spectra with normal stresses * - :doc:`SGR Conventional ` - SGR - 3 - R, C, O - Soft Glass - No (flows) - ★★★★☆ - x: 0.5-3 - Foams, emulsions, pastes, colloidal suspensions (Sollich 1998) * - :doc:`SGR GENERIC ` - SGR - 3 - R, C, O, Flow, Startup, LAOS - Soft Glass - No (flows) - ★★★★★ - x: 0.5-3 - Thermodynamically consistent SGR (Fuereder & Ilg 2013) * - :doc:`Fluidity Local ` - Fluidity - 2-3 - O, Flow - Cooperative - No - ★★★☆☆ - N/A - Local fluidity dynamics, simple cooperative flow * - :doc:`Fluidity Nonlocal ` - Fluidity - 3-4 - O, Flow - Cooperative - No - ★★★★☆ - N/A - Nonlocal fluidity with cooperativity length * - :doc:`Fluidity-Saramito Local ` - Saramito EVP - 10-12 - Flow, Startup, Creep, R, O, LAOS - EVP Thixotropic - Configurable - ★★★★★ - N/A - Tensorial EVP with fluidity coupling, :math:`N_1` predictions * - :doc:`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 * - :doc:`Lattice EPM ` - EPM - 4+ - R, C, Startup, Flow - Elasto-plastic - Configurable - ★★★★★ - N/A - Lattice elasto-plastic model, plastic rearrangements * - :doc:`Tensorial EPM ` - EPM - 4+ - R, C, Startup, Flow - Elasto-plastic - Configurable - ★★★★★ - N/A - Full tensorial EPM for complex loading * - :doc:`MIKH ` - IKH - 4-5 - R, C, O - Thixotropic - Configurable - ★★★★☆ - N/A - Modified IKH for thixotropic materials * - :doc:`MLIKH ` - IKH - 4+ - R, C, O - Thixotropic - Configurable - ★★★★★ - N/A - ML-enhanced IKH with neural network augmentation * - :doc:`FIKH ` - FIKH - 5-6 - R, C, O, Flow, Startup, LAOS - Thixotropic - Configurable - ★★★★★ - :math:`\alpha`: 0-1 - Fractional IKH with Caputo structure kinetics * - :doc:`FMLIKH ` - FIKH - 6+ - R, C, O, Flow, Startup, LAOS - Thixotropic - Configurable - ★★★★★ - :math:`\alpha`: 0-1 - Fractional multi-layer IKH, multiple yield surfaces * - :doc:`DMT Local ` - DMT - 5-7 - R, C, O, Flow, Startup, LAOS - Thixotropic - Configurable - ★★★★☆ - N/A - Structural kinetics with exponential or HB viscosity closure * - :doc:`DMT Nonlocal ` - DMT - 6-8 - R, C, O, Flow, Startup, LAOS - Thixotropic - Configurable - ★★★★★ - N/A - Spatially-resolved thixotropy with structure diffusion, shear banding * - :doc:`Hébraud-Lequeux ` - HL - 3-4 - R, C, O, Flow, Startup, LAOS - Soft matter - No - ★★★★☆ - N/A - Mean-field model for soft glassy materials * - :doc:`STZ Conventional ` - STZ - 4+ - R, O, Flow, Startup - Amorphous - No - ★★★★★ - N/A - Shear transformation zone model (Falk-Langer) * - :doc:`ITT-MCT Schematic ` - ITT-MCT - 6 - R, C, O, Flow, Startup, LAOS - Colloidal Glass - Configurable - ★★★★★ - :math:`\varepsilon`: -0.5 to 0.5 - Dense colloidal suspensions, glass transition (:math:`F_{12}` schematic) * - :doc:`ITT-MCT Isotropic ` - ITT-MCT - 5+ - R, C, O, Flow, Startup, LAOS - Colloidal Glass - Configurable - ★★★★★ - :math:`\phi`: 0.1 to 0.64 - Hard-sphere colloids with S(k), full MCT physics * - :doc:`SPP Yield Stress ` - SPP - 3+ - LAOS - Yield stress - Yes - ★★★★☆ - N/A - LAOS-based yield stress analysis (Rogers et al.) * - :doc:`TNT Tanaka-Edwards ` - TNT - 3 - R, C, O, Flow, Startup, LAOS - Assoc. Polymer - No - ★★☆☆☆ - N/A - Baseline transient network (Maxwell via conformation tensor) * - :doc:`TNT Bell ` - TNT - 4 - R, C, O, Flow, Startup, LAOS - Assoc. Polymer - No - ★★★☆☆ - :math:`\nu`: 0.01-20 - Force-dependent bond breakage, shear-thinning networks * - :doc:`TNT FENE-P ` - TNT - 4 - R, C, O, Flow, Startup, LAOS - Assoc. Polymer - No - ★★★☆☆ - :math:`L_{max}`: 2-100 - Finite extensibility, strain hardening at large deformations * - :doc:`TNT Non-Affine ` - TNT - 4 - R, C, O, Flow, Startup, LAOS - Assoc. Polymer - No - ★★★☆☆ - :math:`\xi`: 0-1 - Non-affine chain slip, non-zero :math:`N_2` * - :doc:`TNT Stretch-Creation ` - TNT - 4 - R, C, O, Flow, Startup, LAOS - Assoc. Polymer - No - ★★★☆☆ - :math:`\kappa`: 0-5 - Flow-enhanced bond formation, shear thickening * - :doc:`TNT Loop-Bridge ` - TNT - 6 - R, C, O, Flow, Startup, LAOS - Telechelic - No - ★★★★☆ - N/A - Two-species kinetics (loops + bridges), telechelic polymers * - :doc:`TNT Sticky Rouse ` - TNT - 4-6 - R, C, O, Flow, Startup, LAOS - Multi-sticker - No - ★★★★☆ - N/A - Multi-mode sticker dynamics, broad relaxation spectrum * - :doc:`TNT Cates ` - TNT - 4 - R, C, O, Flow, Startup, LAOS - Micelles - No - ★★★☆☆ - N/A - Living polymers, wormlike micelles (:math:`\tau_d = \sqrt{\tau_{rep} \cdot \tau_{break}}`) * - :doc:`TNT Multi-Species ` - TNT - 2N+1 - R, C, O, Flow, Startup, LAOS - Mixed Network - No - ★★★★☆ - N/A - Heterogeneous networks with multiple bond types * - :doc:`VLB Local ` - VLB - 2 - R, C, O, Flow, Startup, LAOS - Assoc. Polymer - No - ★☆☆☆☆ - N/A - Single transient network (Maxwell via distribution tensor) * - :doc:`VLB Multi-Network ` - VLB - 2N+1 - R, C, O, Flow, Startup, LAOS - Assoc. Polymer - Configurable - ★★★☆☆ - N/A - Multi-network generalized Maxwell with molecular basis * - :doc:`VLB Variant ` - VLB - 2-6 - R, C, O, Flow, Startup, LAOS - Assoc. Polymer - No - ★★★☆☆ - N/A - Bell shear thinning, FENE bounded extension, Arrhenius temperature * - :doc:`VLB Nonlocal ` - VLB - 4-6 - Flow, Startup, Creep - Assoc. Polymer - No - ★★★★☆ - N/A - Spatially-resolved shear banding with tensor diffusion * - :doc:`HVM Local ` - HVM - 6-10 - R, C, O, Flow, Startup, LAOS - Vitrimer - Yes (:math:`G_P`) - ★★★★★ - N/A - Hybrid vitrimer: permanent + exchangeable (BER/TST) + dissociative networks * - :doc:`HVNM Local ` - HVNM - 13-25 - R, C, O, Flow, Startup, LAOS - Filled Vitrimer - Yes (:math:`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 * :math:`\alpha` **Range:** Fractional order range for fractional models; for ITT-MCT: :math:`\varepsilon` = separation parameter (glass transition), :math:`\phi` = volume fraction; N/A for non-fractional models * **Equilibrium Modulus:** Whether model predicts finite :math:`G_\infty` at long times (solid-like) Model Selection Decision Flowchart ----------------------------------- For a comprehensive decision flowchart based on your experimental data, see: :doc:`/user_guide/model_selection`. **Quick Selection Guide:** .. list-table:: Quick Model Selection by Data Type :header-rows: 1 :widths: 25 40 35 * - Data Type - Data Characteristics - Recommended Models * - **Oscillation** (:math:`G'`, :math:`G''`) - Two plateaus visible - FZSS ★★★★☆ (most common) * - - One plateau (low-:math:`\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 (** :math:`\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 :math:`\alpha` * Excellent for polymers, gels, biological materials **Fractional order (** :math:`\alpha` **) interpretation:** .. list-table:: Fractional Order Interpretation Guide :header-rows: 1 :widths: 15 25 60 * - :math:`\alpha` Value - Physical Meaning - Material Examples * - :math:`\alpha \to 0` - Elastic-dominated - Stiff gels, crosslinked elastomers (spring-like) * - :math:`\alpha \approx 0.3\text{--}0.5` - Balanced viscoelasticity - Soft gels, entangled polymers, biological tissues * - :math:`\alpha \approx 0.5` - Critical gel - Gel point, sol-gel transition * - :math:`\alpha \to 1` - Viscous-dominated - Polymer melts, concentrated solutions (dashpot-like) **Typical** :math:`\alpha` **ranges by material:** * **Soft gels:** :math:`\alpha` = 0.2 - 0.4 * **Polymer melts:** :math:`\alpha` = 0.6 - 0.9 * **Biological tissues:** :math:`\alpha` = 0.3 - 0.5 * **Emulsions:** :math:`\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: :math:`G(t) = G_e + \sum_{i=1}^N G_i \exp(-t/\tau_i)` * Oscillation: :math:`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 :math:`N_1` and :math:`N_2` predictions are required. **Advantages:** * Quadratic stress term gives physically motivated shear thinning * Predicts both :math:`N_1 > 0` and :math:`N_2 < 0` with fixed ratio :math:`N_2/N_1 = -\alpha/2` * Mobility factor :math:`\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 :math:`\alpha_i` **Mobility factor (** :math:`\alpha` **) interpretation:** .. list-table:: Giesekus Mobility Factor Guide :header-rows: 1 :widths: 15 25 60 * - :math:`\alpha` Value - Physical Meaning - Material Examples * - :math:`\alpha = 0` - UCM limit (no shear thinning) - Dilute polymer solutions, Boger fluids * - :math:`\alpha \approx 0.1\text{--}0.3` - Moderate shear thinning - Polymer melts, semidilute solutions * - :math:`\alpha \approx 0.5` - Maximum anisotropy - Strongly shear-thinning polymer melts **Model selection within Giesekus family:** * **GiesekusSingleMode**: 4 params (:math:`\eta_p, \lambda, \alpha, \eta_s`), single relaxation time, all 6 protocols * **GiesekusMultiMode**: N modes with independent :math:`\alpha_i`, broadband spectra, flow curve + SAOS + startup **Key physics:** * Constitutive equation: :math:`\boldsymbol{\tau} + \lambda \overset{\nabla}{\boldsymbol{\tau}} + \frac{\alpha \lambda}{\eta_p} \boldsymbol{\tau} \cdot \boldsymbol{\tau} = 2\eta_p \mathbf{D}` * Conformation tensor: :math:`\mathbf{c} = \mathbf{I} + (\lambda/\eta_p)\boldsymbol{\tau}`, quadratic term drives anisotropic relaxation * Analytical flow curve: :math:`\eta(\dot{\gamma})` from cubic equation at steady state * Cox-Merz rule: :math:`\eta(\dot{\gamma}) \approx |\eta^*(\omega)|` for moderate :math:`\alpha` **Typical applications:** Polymer melts (PE, PP, PS), concentrated solutions, wormlike micelles, liquid crystals, any system needing :math:`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 :math:`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 :math:`\xi` **Key physics:** * Fluidity evolution: :math:`df/dt = (f_{eq} - f)/\tau_f + D_f \nabla^2 f` (nonlocal) * Flow rule: :math:`\sigma = \eta(f) \dot{\gamma}` with :math:`\eta = 1/f` * Cooperativity length :math:`\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 (:math:`N_1`) predictions; shear banding analysis. **Advantages:** * Full tensorial stress state: [:math:`\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, :math:`\lambda` = 1/f only, homogeneous flow * **FluiditySaramitoLocal (full)**: :math:`\tau_y(f)` coupling, aging yield stress * **FluiditySaramitoNonlocal (minimal)**: Shear banding capable with :math:`D_f \nabla^2 f` * **FluiditySaramitoNonlocal (full)**: Full thixotropic banding **Key physics:** * Upper-convected Maxwell viscoelasticity: :math:`\lambda(d\tau/dt - \mathbf{L} \cdot \tau - \tau \cdot \mathbf{L}^T) + \alpha(\tau)\tau = 2\eta_p \mathbf{D}` * Plasticity parameter: :math:`\alpha = \max(0, 1 - \tau_y / |\tau|)` (Von Mises) * Fluidity evolution: :math:`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 :math:`x < 1` * Power-law rheology emerges from microscopic physics * Bayesian inference support for uncertainty quantification **Noise temperature (** :math:`x` **) interpretation:** .. list-table:: Noise Temperature Interpretation Guide :header-rows: 1 :widths: 15 25 60 * - x Value - Physical Meaning - Material Examples * - :math:`x < 1` - Glass (aging) - Aged colloidal suspensions, dense pastes (non-ergodic) * - :math:`x \approx 1` - Glass transition - Critical point, rheological singularity * - :math:`1 < x < 2` - Power-law fluid - Foams, emulsions, soft gels (SGM regime) * - :math:`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 :math:`x \to 1` **Connection to SRFS Transform:** The noise temperature :math:`x` from SGR models directly relates to SRFS shift factors: :math:`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 (** :math:`\varepsilon` **) interpretation:** .. list-table:: Glass Transition Parameter Guide :header-rows: 1 :widths: 15 25 60 * - :math:`\varepsilon` Value - Physical Meaning - Material Examples * - :math:`\varepsilon` < 0 - Glass state - Dense suspensions below :math:`\phi_c`, kinetically arrested * - :math:`\varepsilon \approx 0` - Glass transition - Critical point, diverging relaxation time * - :math:`\varepsilon` > 0 - Fluid state - Mobile suspensions, ergodic dynamics **Model selection within ITT-MCT family:** * **ITTMCTSchematic (** :math:`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: :math:`m(\Phi) = v_1 \Phi + v_2 \Phi^2` (schematic) or k-integral (isotropic) * Glass transition criterion: :math:`v_{2,c} = 4` (for :math:`v_1 = 0`) * Strain decorrelation: :math:`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 :math:`\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 :math:`\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 (** :math:`\lambda` **) interpretation:** .. list-table:: Structure Parameter Guide :header-rows: 1 :widths: 15 25 60 * - :math:`\lambda` Value - Physical Meaning - Material State * - :math:`\lambda` = 1 - Fully structured - At rest (aged), maximum viscosity, colloidal network intact * - 0 < :math:`\lambda` < 1 - Partially broken - Under shear, intermediate microstructure * - :math:`\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 :math:`\tau_y` and K * **DMTLocal + elasticity**: Maxwell backbone for stress overshoot and SAOS * **DMTNonlocal**: Shear banding via structure diffusion (:math:`D_{\lambda} \nabla^2 \lambda`) **Key physics:** * Structure kinetics: :math:`d\lambda/dt = (1-\lambda)/t_eq - a\lambda|\dot{\gamma}|^c/t_eq` * Equilibrium structure: :math:`\lambda_{eq} = 1/(1 + a|\dot{\gamma}|^c)` * Exponential viscosity: :math:`\eta(\lambda) = \eta_{\infty}(\eta_0/\eta_{\infty})^{\lambda}` * Maxwell stress: :math:`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: :math:`f = |\sigma - \alpha| - (\sigma_y + R)` (kinematic + isotropic) * Back-stress evolution: :math:`\dot{\alpha} = C \dot{\varepsilon}^p - \gamma_k \alpha |\dot{\varepsilon}^p|` * Isotropic hardening: :math:`\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 :math:`\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: :math:`{}^C D_t^{\alpha} \lambda = \text{aging} - \text{shear breakdown}` * Caputo derivative :math:`{}^C D_t^{\alpha}` provides long-range temporal memory * Reduces to integer IKH when :math:`\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 :math:`P(\sigma, t)` evolves via advection + diffusion + rearrangement * Rearrangement rate: :math:`\Gamma = \Gamma_0 \Theta(|\sigma| - \sigma_c)` (above critical stress) * Mean-field coupling: rearrangement events redistribute stress to neighbors * Diffusion coefficient :math:`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: :math:`\dot{\Lambda} = R_0 [e^{-\Delta F / k_B T} \cosh(\Omega \sigma / k_B T)]` * Effective disorder temperature :math:`\chi` evolves with plastic work * Strain rate: :math:`\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 :math:`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: :math:`\sigma_i = G(\gamma - \gamma_i^{pl})` with local yield :math:`\sigma_c` * Yield criterion: :math:`|\sigma_i| > \sigma_c` triggers plastic rearrangement * Stress redistribution: Eshelby kernel (lattice) or mean-field (tensorial) * Disorder: :math:`\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 :math:`\mathbf{S}` evolves via upper-convected derivative + breakage * Stress: :math:`\boldsymbol{\sigma} = G \cdot f(\mathbf{S}) + 2\eta_s \mathbf{D}` * Bond lifetime :math:`\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 :math:`\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 :math:`\boldsymbol{\mu} = \langle \mathbf{r}\mathbf{r} \rangle / \langle r_0^2 \rangle` from chain statistics * Stress: :math:`\boldsymbol{\sigma} = G_0(\boldsymbol{\mu} - \mathbf{I})` * Bond kinetics: :math:`\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: :math:`k_d(\mu) = k_d^0 \exp(\nu(\lambda_c - 1))` * FENE-P: :math:`\sigma = G_0 f(\text{tr}(\mu))(\mu - I)` with bounded extensibility * Nonlocal PDE: :math:`+ D_\mu \nabla^2 \mu` for cooperative rearrangements **Model selection within VLB family:** * **Start here:** VLBLocal — 2 params (:math:`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 :math:`k_d` (both give Maxwell) * VLB now has Bell + FENE-P variants (matching TNT's nonlinear extensions) * VLB preferred for molecular extensions (Langevin, entropic :math:`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 :math:`\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 :math:`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: :math:`k_{BER} = \nu_0 \exp(-E_a/RT) \cosh(V_{act} \sigma_{VM}/RT)` * Factor-of-2: :math:`\tau_E^{eff} = 1/(2 k_{BER,0})` — both :math:`\mu^E` and :math:`\mu^E_{nat}` relax toward each other * Stress :math:`\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: :math:`X(\phi) = 1 + 2.5\phi + 14.1\phi^2` * Dual TST kinetics: independent matrix (:math:`k_{BER}^{mat}`) and interphase (:math:`k_{BER}^{int}`) exchange * :math:`\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: :math:`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 :math:`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: :math:`G'_t = \dot{\sigma}/\dot{\gamma}` (elastic contribution) * Instantaneous loss: :math:`G''_t = (1/\omega)(d\sigma/d\gamma)|_{\dot{\gamma}=\text{const}}` (viscous contribution) * Cole-Cole trajectory: :math:`G'_t` vs :math:`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:** 1. **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) 2. **Shear-thickening (dilatant):** Viscosity increases with shear rate * Less common: concentrated suspensions, cornstarch * Models: Power Law (n>1), Herschel-Bulkley (n>1) 3. **Viscoplastic (yield stress):** Requires minimum stress to flow * Examples: toothpaste, gels, slurries, drilling muds * Models: Bingham, Herschel-Bulkley **Industrial applications:** .. list-table:: Flow Models by Industry :header-rows: 1 :widths: 25 35 40 * - 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 ~~~~~~~~~~~~~~~~ .. list-table:: Material-to-Model Quick Reference :header-rows: 1 :widths: 25 35 40 * - Material Type - Recommended Models - Notes * - Polymer Melts - Giesekus, FML, FZSS, Carreau (flow) - Giesekus for :math:`N_1, N_2` and startup; :math:`\alpha` typically 0.6-0.9 for fractional * - Soft Gels - FZSS, FMG, SpringPot - :math:`\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 - :math:`\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 - :math:`\alpha \approx 0.5`; power-law across all frequencies * - Polymer Solutions - Giesekus, Carreau, Cross (flow); FML (oscillation) - Giesekus for nonlinear + :math:`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 (:math:`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 :math:`x \to 1` * - Thixotropic Yield Stress - FluiditySaramitoLocal, Herschel-Bulkley - Stress overshoot, aging; use Saramito for :math:`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; :math:`\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 :math:`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 ~~~~~~~~~~~~~~ .. list-table:: Application-Based Model Selection :header-rows: 1 :widths: 20 30 25 25 * - 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 ~~~~~~~~~~~~~~~ .. list-table:: Data Quality Considerations :header-rows: 1 :widths: 20 35 45 * - 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**: :math:`G_0, \eta` - Liquid with single relaxation * **PowerLaw**: K, n - Shear-thinning/thickening * **Bingham**: :math:`\tau_0, \eta_{pl}` - Linear viscoplastic * **SpringPot**: V, :math:`\alpha` - Pure power-law element **3-Parameter Models:** * **Zener**: Gs, Gp, :math:`\eta_p` - Classical solid with plateau * **FML**: V, :math:`\alpha, \eta` - Fractional liquid * **FMG**: Gs, V, :math:`\alpha` - Fractional gel * **Herschel-Bulkley**: :math:`\tau_0`, K, n - Yield + power-law **4-Parameter Models:** * **FZSS**: Ge, Gm, :math:`\alpha, \tau\alpha` - Most common fractional solid * **FZSL**: Gs, :math:`\eta_s, V, \alpha` - Fractional solid-liquid Zener * **FZLL**: :math:`\eta_s, \eta_p, V, \alpha` - Fractional liquid-liquid Zener * **FKV**: Gp, V, :math:`\alpha`, (:math:`\eta_p` optional) - Fractional Kelvin-Voigt * **Carreau**: :math:`\eta_0, \eta_{\infty}, \lambda`, n - Flow with plateaus * **Cross**: K, m, :math:`\eta_0, \eta_{\infty}` - Alternative flow interpolation * **Fractional Maxwell Model**: :math:`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**: :math:`\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 :doc:`/user_guide/bayesian_inference` for comprehensive Bayesian analysis guide. DMTA / DMA Support ------------------- **All 49 oscillation-capable models support DMTA data** through automatic :math:`E^* \leftrightarrow G^*` conversion at the ``BaseModel`` boundary: * **Tensile modulus conversion:** :math:`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'`` .. code-block:: python 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 :doc:`/models/dmta/index` for DMTA theory, model compatibility, and workflow guides. Next Steps ---------- * **Detailed model documentation:** See :doc:`/models/index` for individual model handbooks * **Multi-technique fitting:** :doc:`/user_guide/multi_technique_fitting` * **Model selection workflow:** :doc:`/user_guide/model_selection` * **Compatibility checking:** :doc:`/user_guide/core_concepts` (automatic detection of model-data mismatches) * **Giesekus models:** :doc:`/models/giesekus/giesekus` for nonlinear viscoelastic polymer melts and solutions * **SGR models:** :doc:`/models/sgr/sgr_conventional` and :doc:`/models/sgr/sgr_generic` * **ITT-MCT models:** :doc:`/models/itt_mct/itt_mct_schematic` and :doc:`/models/itt_mct/itt_mct_isotropic` for colloidal glasses * **TNT models:** :doc:`/models/tnt/index` for transient network theory (associating polymers, micelles) * **VLB models:** :doc:`/models/vlb/index` for VLB transient networks (hydrogels, vitrimers, self-healing polymers) * **HVM models:** :doc:`/models/hvm/index` for hybrid vitrimer constitutive models * **HVNM models:** :doc:`/models/hvnm/index` for vitrimer nanocomposite models * **DMTA support:** :doc:`/models/dmta/index` for tensile modulus conversion and DMA workflows * **SRFS transform:** :doc:`/transforms/srfs` for strain-rate frequency superposition * **Example notebooks:** 246 examples across 20+ directories in ``examples/`` **Need a model not listed?** Open an issue via :doc:`/developer/contributing`.