Soft Glassy Rheology (SGR) Analysis

Statistical Mechanics Framework for Complex Soft Materials

Overview

Soft Glassy Rheology (SGR) is a statistical mechanics framework developed by Sollich and coworkers (1997-1998) that unifies the rheological behavior of diverse complex fluids— foams, emulsions, pastes, colloidal glasses, and other metastable “soft glassy” materials— under a single theoretical picture.

Key Insight

SGR treats soft materials as collections of mesoscopic trapped elements that can yield and rearrange via an activated hopping process. The single dimensionless parameter \(x\) (effective noise temperature) controls whether the material behaves as a glass (\(x < 1\)), a power-law fluid (\(1 < x < 2\)), or a Newtonian liquid (\(x \geq 2\)).

Unlike phenomenological models (Maxwell, Kelvin-Voigt) that describe material response without microscopic interpretation, SGR provides a physical picture connecting macroscopic rheology to mesoscopic rearrangement dynamics.

When to Use SGR

SGR is ideal for:

  • Soft glassy materials: Foams, concentrated emulsions, pastes, slurries, colloidal gels

  • Yield stress fluids: Materials with solid-to-liquid transitions under stress

  • Aging materials: Systems whose properties evolve over time (\(x < 1\))

  • Power-law rheology: Materials showing \(G' \sim G'' \sim \omega^n\) across broad frequency ranges

  • Shear rejuvenation: Systems fluidized by deformation

  • Phase classification: Determining whether a material is a glass, fluid, or near transition

SGR complements (rather than replaces):

  • Fractional models: For mathematical convenience without microscopic picture

  • Herschel-Bulkley: For simpler steady-shear yield stress fitting

  • Multi-mode Maxwell: For discrete relaxation time spectra

Material Classification

The SGR framework classifies materials by their effective noise temperature \(x\):

SGR Phase Diagram

Regime

\(x\) Range

Physical Characteristics

Glass

\(x < 1\)

Aging, yield stress, metastable, non-ergodic

Transition

\(x \approx 1\)

Critical behavior, \(G' \approx G''\), borderline yield

Power-law Fluid

\(1 < x < 2\)

\(G' \sim G'' \sim \omega^{x-1}\), viscoelastic liquid

Newtonian

\(x \geq 2\)

Exponential relaxation, simple viscous flow

The glass transition at \(x = 1\) is not a thermodynamic phase transition but a dynamical arrest: below \(x = 1\), the system cannot equilibrate on any finite timescale.

Theoretical Foundations

The Trap Model Picture

SGR models the material as many mesoscopic elements—local regions containing multiple particles/droplets/bubbles—each characterized by:

  1. Local strain \(l\): The strain stored in that element

  2. Trap depth \(E\): The energy barrier to rearrangement (how “stuck” it is)

  3. Elastic constant \(k\): Local stiffness (typically equal to \(G_0\))

Elements are trapped in local energy minima (“cages”) until activated by noise. The exponential trap distribution \(\rho(E) = \exp(-E)\) means deep traps are exponentially rare—a consequence of entropic arguments.

Activation and Yielding

Elements yield with rate:

\[\Gamma(E, l) = \Gamma_0 \exp\left(-\frac{E - \frac{1}{2}kl^2}{x}\right)\]
where:

  • \(\Gamma_0 = 1/\tau_0\) is the attempt frequency

  • \(x\) is the effective noise temperature

  • \(\frac{1}{2}kl^2\) is the elastic energy (lowers the barrier)

Key physics: Large local strain makes yielding more likely by reducing the effective barrier height.

Effective Noise Temperature

The parameter \(x\) is not thermal temperature (which is essentially fixed at room temperature for soft materials). Instead, \(x\) represents an effective noise level arising from:

  • Mechanical noise: Rearrangements of neighbors creating local fluctuations

  • Shear rejuvenation: Applied deformation increasing \(x\)

  • Aging: Gradual decrease of \(x\) as system settles into deeper traps

  • External perturbations: Vibrations, stirring, etc.

Connecting x to Rheology

The remarkable power of SGR is that \(x\) alone determines rheological character:

\[G'(\omega) \sim G''(\omega) \sim \omega^{x-1} \quad \text{for } 1 < x < 2\]
This gives:

  • Phase angle: \(\delta = \pi(x-1)/2\) = constant (frequency-independent)

  • Flow index: \(n = x - 1\) in Herschel-Bulkley-like steady shear

  • Loss tangent: \(\tan(\delta) = \tan(\pi(x-1)/2)\) = constant

How to estimate \(x\) from data:

  1. Plot \(\log G'\) vs \(\log \omega\)

  2. Fit linear region to get slope \(m\)

  3. \(x \approx m + 1\)

Or from loss tangent:

\[x = 1 + \frac{2}{\pi} \arctan(\tan\delta)\]

SGR Model Variants

RheoJAX implements two SGR variants:

SGRConventional (Sollich 1998)

The standard SGR model for rheological fitting:

from rheojax.models import SGRConventional

model = SGRConventional()
model.fit(omega, G_star, test_mode='oscillation')

x = model.parameters.get_value('x')
print(f"Effective temperature: x = {x:.3f}")
print(f"Phase: {'glass' if x < 1 else 'fluid'}")

Parameters: \(x\) (noise temperature), \(G_0\) (modulus scale), \(\tau_0\) (attempt time)

Use for: Standard fitting, phase classification, steady shear flow curves

SGRGeneric (Fuereder & Ilg 2013)

Thermodynamically consistent version satisfying the GENERIC framework:

from rheojax.models import SGRGeneric

model = SGRGeneric()
model.fit(omega, G_star, test_mode='oscillation')

# Access thermodynamic functions
S_prod = model.entropy_production(gamma_dot=1.0)
print(f"Entropy production: {S_prod:.4f} J/(K·m³·s)")
Additional features:

  • Entropy production calculation

  • Free energy landscape

  • Fluctuation-dissipation verification

  • GENERIC structure validation

Use for: Thermodynamic research, entropy analysis, theoretical validation

Note

Both models give identical rheological predictions. Use SGRConventional for standard fitting and SGRGeneric when you need thermodynamic information.

Practical Implementation

Basic SGR Analysis Workflow

import numpy as np
from rheojax.models import SGRConventional
from rheojax.io.readers import auto_read

# 1. Load oscillatory data
data = auto_read("frequency_sweep.csv")
omega = data.x  # Angular frequency
G_star = data.y  # Complex modulus

# 2. Fit SGR model
model = SGRConventional()
model.fit(omega, G_star, test_mode='oscillation')

# 3. Extract and interpret parameters
x = model.parameters.get_value('x')
G0 = model.parameters.get_value('G0')
tau0 = model.parameters.get_value('tau0')

print(f"Effective temperature x = {x:.3f}")
print(f"Plateau modulus G₀ = {G0:.1f} Pa")
print(f"Attempt time τ₀ = {tau0:.2e} s")

# 4. Phase classification
if x < 0.9:
    print("Material is in GLASS phase (aging, yield stress)")
elif x < 1.1:
    print("Material is near GLASS TRANSITION")
elif x < 2.0:
    print("Material is a POWER-LAW FLUID")
else:
    print("Material is near NEWTONIAN")

# 5. Predict other test modes
t = np.logspace(-3, 3, 100)
G_t = model.predict(t, test_mode='relaxation')  # Stress relaxation

gamma_dot = np.logspace(-3, 2, 50)
sigma = model.predict(gamma_dot, test_mode='steady_shear')  # Flow curve

Bayesian Inference for x

Since \(x\) is the critical parameter determining phase behavior, Bayesian inference is highly recommended to quantify uncertainty:

from rheojax.models import SGRConventional

model = SGRConventional()
model.fit(omega, G_star, test_mode='oscillation')

# Bayesian inference with NLSQ warm-start
result = model.fit_bayesian(
    omega, G_star,
    test_mode='oscillation',
    num_warmup=1000,
    num_samples=2000
)

# Get credible intervals
intervals = model.get_credible_intervals(result.posterior_samples, credibility=0.95)
x_low, x_high = intervals['x']

print(f"x = {model.parameters.get_value('x'):.3f} "
      f"[{x_low:.3f}, {x_high:.3f}] (95% CI)")

# Phase classification with uncertainty
if x_high < 1.0:
    print("Material is DEFINITELY in glass phase (95% CI)")
elif x_low > 1.0:
    print("Material is DEFINITELY in fluid phase (95% CI)")
else:
    print("Material straddles glass transition (uncertain phase)")

SRFS Transform (Flow Curve Analysis)

The Strain-Rate Frequency Superposition (SRFS) transform creates flow curve mastercurves analogous to time-temperature superposition:

from rheojax.transforms import SRFS
import numpy as np

# Flow curves at different shear rates
datasets = [
    {'gamma_dot': gamma_dot_1, 'sigma': sigma_1},
    {'gamma_dot': gamma_dot_2, 'sigma': sigma_2},
    # ...
]

srfs = SRFS(reference_gamma_dot=1.0, auto_shift=True)
master_curve, shift_factors = srfs.transform(datasets)

# Shift factor scaling reveals x
# a(γ̇) ~ γ̇^(2-x) for SGR materials
slope = np.polyfit(np.log(gamma_dots), np.log(shift_factors), 1)[0]
x_from_srfs = 2 - slope
print(f"x from SRFS: {x_from_srfs:.3f}")

Shear Banding Detection

SGR can predict shear banding when the flow curve is non-monotonic:

from rheojax.transforms import SRFS
from rheojax.models import SGRConventional

model = SGRConventional(x=0.8, G0=100.0, tau0=0.01)
srfs = SRFS()

# Check for shear banding (non-monotonic flow curve)
is_banding, critical_rates = srfs.detect_shear_banding(
    model,
    gamma_dot_range=(1e-3, 1e2)
)

if is_banding:
    print("Shear banding predicted!")
    low_rate, high_rate = critical_rates
    print(f"Coexisting bands at γ̇ = {low_rate:.2e} and {high_rate:.2e} s⁻¹")

    # Lever rule for band fractions
    band_info = srfs.compute_shear_band_coexistence(model, applied_rate=1.0)
    print(f"High-rate band fraction: {band_info['high_fraction']:.2f}")

Aging Dynamics (\(x < 1\))

For glassy materials (\(x < 1\)), properties evolve with waiting time:

from rheojax.models import SGRConventional
import numpy as np

# Simulate aging: x approaches 1 from below
waiting_times = [100, 1000, 10000]  # seconds
x_values = [0.7, 0.85, 0.95]  # Effective temperature increases toward 1

for t_w, x in zip(waiting_times, x_values):
    model = SGRConventional(x=x, G0=100.0, tau0=0.01)
    G_star = model.predict(omega, test_mode='oscillation')

    print(f"t_w = {t_w} s: x = {x:.2f}, G'(1 rad/s) = {G_star[25].real:.1f} Pa")

# Effective relaxation time grows as τ_eff ~ t_w^μ with μ = (1-x)/x
x = 0.8
mu = (1 - x) / x
print(f"Aging exponent μ = {mu:.2f}")

Visualization

import matplotlib.pyplot as plt
import numpy as np
from rheojax.models import SGRConventional

model = SGRConventional()
model.fit(omega, G_star_data, test_mode='oscillation')

fig, axes = plt.subplots(2, 2, figsize=(12, 10))

# 1. Frequency sweep with fit
ax = axes[0, 0]
ax.loglog(omega, np.real(G_star_data), 'o', label="G' data")
ax.loglog(omega, np.imag(G_star_data), 's', label="G'' data")
G_fit = model.predict(omega, test_mode='oscillation')
ax.loglog(omega, np.real(G_fit), '-', label="G' fit")
ax.loglog(omega, np.imag(G_fit), '--', label="G'' fit")
ax.set_xlabel('ω (rad/s)')
ax.set_ylabel('G*, G" (Pa)')
ax.legend()
ax.set_title(f'SGR Fit: x = {model.parameters.get_value("x"):.2f}')

# 2. Loss tangent (should be constant for SGR)
ax = axes[0, 1]
tan_delta = np.imag(G_star_data) / np.real(G_star_data)
ax.semilogx(omega, tan_delta, 'o-')
x = model.parameters.get_value('x')
theoretical_tan_delta = np.tan(np.pi * (x - 1) / 2)
ax.axhline(theoretical_tan_delta, color='r', linestyle='--',
           label=f'SGR prediction: tan(δ) = {theoretical_tan_delta:.2f}')
ax.set_xlabel('ω (rad/s)')
ax.set_ylabel('tan(δ)')
ax.legend()
ax.set_title('Loss Tangent (Should Be Constant)')

# 3. Stress relaxation prediction
ax = axes[1, 0]
t = np.logspace(-3, 3, 100)
G_t = model.predict(t, test_mode='relaxation')
ax.loglog(t, np.real(G_t))
ax.set_xlabel('t (s)')
ax.set_ylabel('G(t) (Pa)')
ax.set_title('Predicted Stress Relaxation')

# 4. Flow curve prediction
ax = axes[1, 1]
gamma_dot = np.logspace(-3, 2, 50)
sigma = model.predict(gamma_dot, test_mode='steady_shear')
ax.loglog(gamma_dot, sigma)
ax.set_xlabel('γ̇ (s⁻¹)')
ax.set_ylabel('σ (Pa)')
ax.set_title('Predicted Flow Curve')

plt.tight_layout()
plt.savefig('sgr_analysis.png', dpi=150)

SGR vs Other Models

Model Comparison for Soft Materials

Model

Best For

Limitations

Key Parameters

SGR

Physical interpretation, phase classification, aging

Mean-field (no spatial correlations)

\(x\) (noise temp), \(G_0\), \(\tau_0\)

Fractional Maxwell

Mathematical convenience, broad relaxation

No microscopic picture

\(G\), \(\tau\), \(\alpha\) (fractional order)

Herschel-Bulkley

Simple yield stress fitting

Steady shear only, no dynamics

\(\sigma_y\), \(K\), \(n\)

Multi-mode Maxwell

Discrete relaxation spectrum

Many parameters, no physical basis

\(G_i\), \(\tau_i\) for each mode

When to choose SGR over fractional models:

  1. You need phase classification (glass vs fluid)

  2. Material shows aging behavior

  3. You want microscopic interpretation (trap hopping)

  4. Loss tangent is approximately constant with frequency

When to choose fractional models over SGR:

  1. You need mathematical simplicity

  2. Material doesn’t fit SGR predictions

  3. You don’t need phase classification

  4. You’re doing multi-technique fitting (easier optimization)

Limitations and Caveats

Mean-Field Approximation

SGR neglects spatial correlations between elements. In reality:

  • Yielding events trigger neighbors (avalanches)

  • Shear banding involves spatial heterogeneity

  • Localization effects not captured

These phenomena require spatially-resolved extensions (mode-coupling theory, elastoplastic models, or mesoscale simulations).

Phenomenological Noise Temperature

The origin of \(x\) is not derived from first principles:

  • Must be fitted to data or estimated

  • Physical mechanism of “noise” unclear

  • Cannot predict \(x\) from molecular properties alone

For fundamental understanding, pair SGR with molecular dynamics simulations.

Single Element Size

Real soft glasses have polydisperse element sizes:

  • Affects trap distribution shape

  • May cause deviations from pure power-law behavior

  • Consider polydisperse extensions for better fits

Thixotropy Limitations

Basic SGR treats \(x\) as constant. For strongly thixotropic materials:

  • \(x\) should be time-dependent: \(x(t)\)

  • Consider structural parameter extensions: \(\lambda \in [0, 1]\)

  • See SRFS transform for thixotropy detection

Tutorial Notebooks

RheoJAX provides comprehensive SGR tutorial notebooks:

SGR Soft Glassy Rheology Tutorial

Notebook: examples/advanced/09-sgr-soft-glassy-rheology.ipynb

This tutorial covers:

  • Loading frequency sweep data for soft glassy materials

  • Fitting SGRConventional and SGRGeneric models

  • Phase classification from fitted \(x\) parameter

  • Bayesian uncertainty quantification

  • SRFS transform for flow curve analysis

  • Shear banding detection and analysis

  • Comparison with fractional models

SRFS Flow Curve Analysis

Notebook: examples/transforms/07-srfs-strain-rate-superposition.ipynb

This tutorial demonstrates:

  • Creating flow curve mastercurves

  • Extracting SGR parameters from shift factors

  • Detecting non-monotonic flow curves

  • Shear band coexistence analysis

References

Foundational SGR Papers:

Thermodynamic Extensions:

Shear Banding and Aging:

  • Fielding, S. M., Cates, M. E., & Sollich, P. (2009). “Shear banding, aging and noise dynamics in soft glassy materials.” Soft Matter 5, 2378-2382. https://doi.org/10.1039/B812394M

Reviews:

See Also