Soft Glassy Rheology (SGR) Models ================================= This section documents the Soft Glassy Rheology (SGR) family of models for disordered soft materials exhibiting glassy dynamics. .. include:: /_includes/glass_transition_physics.rst Quick Reference --------------- .. list-table:: :widths: 25 15 60 :header-rows: 1 * - Model - Parameters - Use Case * - :doc:`sgr_conventional` - 3 (:math:`x`, :math:`G_0`, :math:`\tau_0`) - Soft glassy materials, aging, yield stress fluids * - :doc:`sgr_generic` - 3 (:math:`x`, :math:`G_0`, :math:`\tau_0`) - Thermodynamically consistent extension (GENERIC framework) Overview -------- The Soft Glassy Rheology (SGR) model is a mesoscopic constitutive framework for **soft glassy materials (SGMs)**—systems exhibiting structural disorder and metastability similar to glasses but with interaction energies of order :math:`k_B T`: - **Foams** (shaving cream, bread dough) - **Dense emulsions** (mayonnaise, salad cream) - **Pastes** (toothpaste, hair gel) - **Colloidal glasses** (paints, ceramic slips) - **Polymer gels** (physical gels, block copolymers) **Key physics captured:** - **Noise-activated hopping**: Elements escape energy traps via effective noise temperature :math:`x` - **Glass transition**: Phase transition at :math:`x = 1` (fluid :math:`\leftrightarrow` glass) - **Aging**: Time-dependent evolution of trapped state distribution - **Power-law rheology**: :math:`G' \sim G'' \sim \omega^{x-1}` in fluid regime - **Yield stress**: Emerges in glass regime (:math:`x < 1`) The model was developed by Sollich, Lequeux, Hébraud, and Cates based on Bouchaud's trap model for structural glasses. Model Hierarchy --------------- :: SGR Family │ ├── SGR Conventional (Sollich 1998) │ └── Trap model with Arrhenius hopping │ └── Exponential trap depth distribution ρ(E) = e^(-E) │ └── Strain-warped time Z(t,t') for flow coupling │ └── 3 core parameters: x, G_0, τ_0 │ └── SGR GENERIC (Fuereder & Ilg 2013) └── Thermodynamically consistent extension └── GENERIC framework (reversible + irreversible) └── Proper dissipation and entropy production └── Improved nonlinear response predictions When to Use Which Model ----------------------- .. list-table:: :widths: 35 30 35 :header-rows: 1 * - Feature / Use Case - SGR Conventional - SGR GENERIC * - Linear oscillatory (SAOS) - ✓ Standard choice - ✓ Equivalent * - Aging and rejuvenation - ✓ Full support - ✓ Full support * - Large amplitude (LAOS) - Qualitative - ✓ Better nonlinear * - Thermodynamic consistency - ~ - ✓ Guaranteed * - Steady flow curves - ✓ Good - ✓ Better at high rates * - Computational cost - 1× (faster) - 2-3× (more expensive) * - Simple interpretation - ✓ Standard - More complex **Decision Guide:** - **Start with SGR Conventional** for standard characterization (SAOS, flow curves) - **Use SGR GENERIC** when thermodynamic consistency matters (nonlinear, LAOS) or when conventional model shows systematic deviations SGR Phase Diagram ----------------- The SGR model exhibits a genuine phase transition controlled by the effective noise temperature :math:`x`: :: x (noise temperature) │ │ x > 2 Newtonian Fluid │ G' ~ ω^2, G'' ~ ω │ Classical liquid behavior │ │ 1 < x < 2 Power-Law Fluid │ G' ~ G'' ~ ω^(x-1) │ Flat loss tangent: tan δ = tan(πx/2) │ Broad relaxation spectrum │ │ x = 1 Glass Transition (Critical Point) │ Logarithmic aging, critical slowing │ │ x < 1 Soft Glass │ Yield stress emerges │ G' >> G'', weak frequency dependence │ Aging without equilibration │ └───────────────────────────────────────────── **Physical interpretation of** :math:`x` **:** - :math:`x` **represents the ratio of "noise energy" to typical trap depth** - High :math:`x`: Frequent hopping, equilibrium attained, liquid-like - Low :math:`x`: Rare hopping, aging dominates, solid-like - :math:`x \approx 1`: Marginal stability, critical dynamics Key Parameters -------------- .. list-table:: :widths: 15 10 15 60 :header-rows: 1 * - Parameter - Symbol - Typical Range - Physical Meaning * - Noise temperature - :math:`x` - 0.5–3 - Controls phase: :math:`x < 1` (glass), :math:`x > 1` (fluid) * - Modulus scale - :math:`G_0` - :math:`10\text{--}10^4` Pa - Sets magnitude of :math:`G'`, :math:`G''` * - Attempt time - :math:`\tau_0` - :math:`10^{-6}`–:math:`10^{-2}` s - Microscopic timescale for trap escape Quick Start ----------- **SGR Conventional model:** .. code-block:: python from rheojax.models import SGRConventional import numpy as np # Create model model = SGRConventional() # Set parameters for a soft glassy material model.parameters.set_value('x', 1.3) # Power-law fluid regime model.parameters.set_value('G0', 1000.0) # Pa model.parameters.set_value('tau0', 1e-4) # s # Fit to oscillatory data omega = np.logspace(-2, 2, 50) model.fit(omega, G_star_data, test_mode='oscillation') # Check if material is in glass or fluid regime x = model.parameters.get_value('x') if x < 1: print(f"Glass regime (x = {x:.2f}): Yield stress expected") else: print(f"Fluid regime (x = {x:.2f}): Power-law G' ~ G'' ~ ω^{x-1:.2f}") **Bayesian inference:** .. code-block:: python # Bayesian with NLSQ warm-start result = model.fit_bayesian( omega, G_star_data, test_mode='oscillation', num_warmup=1000, num_samples=2000, num_chains=4, seed=42 ) # Credible interval for noise temperature intervals = model.get_credible_intervals(result.posterior_samples) print(f"x: [{intervals['x'][0]:.2f}, {intervals['x'][1]:.2f}]") **GENERIC formulation:** .. code-block:: python from rheojax.models import SGRGeneric # Thermodynamically consistent version model = SGRGeneric() model.fit(omega, G_star_data, test_mode='oscillation') Model Documentation ------------------- .. toctree:: :maxdepth: 1 sgr_conventional sgr_generic See Also -------- - :doc:`/models/hl/index` — Hébraud-Lequeux: mean-field limit of trap dynamics - :doc:`/models/epm/index` — EPM: spatially-resolved plasticity - :doc:`/models/stz/index` — STZ: shear transformation zones - :doc:`/models/fluidity/index` — Fluidity models for yield stress fluids - :doc:`/transforms/srfs` — Strain-rate frequency superposition (SGR analog of TTS) - :doc:`/user_guide/soft_glassy_materials` — Introduction to soft glassy rheology References ---------- 1. Sollich, P., Lequeux, F., Hébraud, P., & Cates, M.E. (1997). "Rheology of soft glassy materials." *Phys. Rev. Lett.*, 78, 2020–2023. https://doi.org/10.1103/PhysRevLett.78.2020 2. Sollich, P. (1998). "Rheological constitutive equation for a model of soft glassy materials." *Phys. Rev. E*, 58, 738–759. https://doi.org/10.1103/PhysRevE.58.738 3. Fielding, S.M., Sollich, P., & Cates, M.E. (2000). "Aging and rheology in soft materials." *J. Rheol.*, 44, 323–369. https://doi.org/10.1122/1.551088 4. Fuereder, I. & Ilg, P. (2013). "Nonequilibrium thermodynamics of the soft glassy rheology model." *Phys. Rev. E*, 88, 042134. DOI: `10.1103/PhysRevE.88.042134 `_ :download:`PDF <../../../reference/fuereder_ilg_2013_sgr_thermodynamics.pdf>` 5. Sollich, P. & Cates, M.E. (2012). "Thermodynamic interpretation of soft glassy rheology models." *Phys. Rev. E*, 85, 031127. DOI: `10.1103/PhysRevE.85.031127 `_ :download:`PDF <../../../reference/sollich_cates_2012_sgr_thermo.pdf>` 6. Cates, M.E. & Sollich, P. (2004). "Tensorial constitutive models for disordered foams, dense emulsions, and other soft nonergodic materials." *J. Rheol.*, 48, 193–207. https://doi.org/10.1122/1.1634985 7. Bouchaud, J.P. (1992). "Weak ergodicity breaking and aging in disordered systems." *J. Phys. I France*, 2, 1705–1713. https://doi.org/10.1051/jp1:1992238