Soft Glassy Rheology (SGR) Models¶

This section documents the Soft Glassy Rheology (SGR) family of models for disordered soft materials exhibiting glassy dynamics.

Glass Transition Physics

Common Physical Framework

Models in this category describe materials near or below the glass transition—where thermal fluctuations become insufficient for structural relaxation on experimental timescales. These materials exhibit:

Characteristic Signatures:

Cage effect: Particles trapped by neighbors, requiring cooperative rearrangements
Aging: Properties evolve with waiting time (time since preparation)
Yield stress: Finite stress required for macroscopic flow
Power-law rheology: \(G'(\omega) \sim G''(\omega) \sim \omega^n\) with weak frequency dependence
Structural relaxation: \(\alpha\)-relaxation timescale diverges at glass transition

Key Control Parameters:

Model	Parameter	Physical meaning
SGR	\(x\) (noise temperature)	Ratio of activation energy to trap depth
ITT-MCT	\(\varepsilon\) (separation parameter)	Distance from ideal glass transition
STZ	\(\chi\) (effective temperature)	Configurational disorder
EPM	\(\sigma/\sigma_y\) (stress ratio)	Proximity to yield

Glass Transition Regimes:

Liquid regime (above \(T_g\) or critical point): Equilibrium relaxation, aging absent
Glass regime (below \(T_g\)): Frozen structure, aging, yield stress emerges
Critical point: Power-law divergences, scale-free avalanches

Related Concepts:

/user_guide/soft_glassy_materials — Introduction to SGMs
Mastercurve (Time-Temperature Superposition) — Time-temperature superposition near \(T_g\)
Soft Glassy Rheology (SGR) Models — SGR model family
ITT-MCT Models — Mode-coupling theory approach

Quick Reference¶

Model	Parameters	Use Case
SGR Conventional (Soft Glassy Rheology) — Handbook	3 (\(x\), \(G_0\), \(\tau_0\))	Soft glassy materials, aging, yield stress fluids
SGR GENERIC (Thermodynamically Consistent)	3 (\(x\), \(G_0\), \(\tau_0\))	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 \(k_B T\):

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 \(x\)
Glass transition: Phase transition at \(x = 1\) (fluid \(\leftrightarrow\) glass)
Aging: Time-dependent evolution of trapped state distribution
Power-law rheology: \(G' \sim G'' \sim \omega^{x-1}\) in fluid regime
Yield stress: Emerges in glass regime (\(x < 1\))

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¶

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 \(x\):

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 \(x\) :

\(x\) represents the ratio of “noise energy” to typical trap depth
High \(x\): Frequent hopping, equilibrium attained, liquid-like
Low \(x\): Rare hopping, aging dominates, solid-like
\(x \approx 1\): Marginal stability, critical dynamics

Key Parameters¶

Parameter	Symbol	Typical Range	Physical Meaning
Noise temperature	\(x\)	0.5–3	Controls phase: \(x < 1\) (glass), \(x > 1\) (fluid)
Modulus scale	\(G_0\)	\(10\text{--}10^4\) Pa	Sets magnitude of \(G'\), \(G''\)
Attempt time	\(\tau_0\)	\(10^{-6}\)–\(10^{-2}\) s	Microscopic timescale for trap escape

Quick Start¶

SGR Conventional model:

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:

# 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:

from rheojax.models import SGRGeneric

# Thermodynamically consistent version
model = SGRGeneric()
model.fit(omega, G_star_data, test_mode='oscillation')

Model Documentation¶

References¶

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
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
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
Fuereder, I. & Ilg, P. (2013). “Nonequilibrium thermodynamics of the soft glassy rheology model.” Phys. Rev. E, 88, 042134. DOI: 10.1103/PhysRevE.88.042134 PDF
Sollich, P. & Cates, M.E. (2012). “Thermodynamic interpretation of soft glassy rheology models.” Phys. Rev. E, 85, 031127. DOI: 10.1103/PhysRevE.85.031127 PDF
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
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