Hébraud-Lequeux (HL) Models¶

This section documents the Hébraud-Lequeux model for soft glassy materials—a mean-field kinetic theory for yield stress fluids with noise-activated plasticity.

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
Hébraud–Lequeux (HL) Model — Handbook	3 (\(\alpha\), \(\sigma_c\), \(\tau\))	Mean-field plasticity, noise-activated flow, soft glasses

Overview¶

The Hébraud-Lequeux (HL) model is a mesoscopic constitutive theory for soft glassy materials that captures the interplay between elastic loading, plastic yielding, and noise-activated structural relaxation. Originally developed to explain the rheology of concentrated emulsions, it provides a physically-motivated framework for yield stress fluids.

Key physics:

Mean-field approach: Material represented as ensemble of mesoscopic elements
Elastic loading: Elements store stress until yield threshold
Plastic yielding: Stress released when local stress exceeds \(\sigma_c\)
Noise activation: Plastic events occur with rate proportional to noise amplitude
Mechanical noise: Yielding events generate noise that activates neighbors

Connection to other models:

SGR: HL can be viewed as a mean-field limit of SGR dynamics
EPM: HL lacks spatial resolution but captures similar physics
Fluidity models: HL’s noise parameter relates to fluidity evolution

The HL model bridges the gap between phenomenological yield stress models (Bingham, Herschel-Bulkley) and microscopic theories (mode-coupling), providing mechanistic insight while remaining computationally tractable.

Physical Framework¶

Mesoscopic Elements:

The material is coarse-grained into identical mesoscopic elements, each characterized by local stress \(\sigma_{el}\). Elements:

Load elastically: \(d\sigma_{el}/dt = G \cdot \dot{\gamma}\) under macroscopic shear
Yield plastically: Reset to \(\sigma_{el} = 0\) when \(|\sigma_{el}| > \sigma_c\)
Relax via noise: Activated hopping with rate \(\sim \exp(-U/D)\) where \(D\) is noise

Stress Distribution:

The probability distribution \(P(\sigma_{el}, t)\) of local stresses evolves according to a Fokker-Planck equation with:

Convective flux from elastic loading
Diffusive spreading from mechanical noise
Boundary conditions from plastic yielding

Macroscopic Stress:

\[\sigma = \int_{-\sigma_c}^{\sigma_c} \sigma_{el} \, P(\sigma_{el}, t) \, d\sigma_{el}\]

Key Parameters¶

Parameter	Symbol	Units	Physical Meaning
Noise coupling	\(\alpha\)	—	Rate of plastic events generating noise
Yield threshold	\(\sigma_c\)	Pa	Local stress for plastic yielding
Relaxation time	\(\tau\)	s	Microscopic relaxation timescale

Model Predictions¶

Flow Curve:

The HL model predicts a yield stress with continuous transition:

\[\sigma(\dot{\gamma}) = \sigma_y + \eta_{eff}\dot{\gamma}^n\]

where \(\sigma_y\) depends on \(\alpha\) and \(\sigma_c\).

Oscillatory Response:

Low frequency: \(G'\) plateau, \(G''\) peak near yield
High frequency: Classical Maxwell-like behavior
Strain amplitude: Smooth transition from linear to nonlinear

Transient Response:

Startup flow: Stress overshoot for high shear rates
Creep: Delayed yielding with characteristic waiting time
Relaxation: Non-exponential decay with stretched dynamics

Quick Start¶

Hébraud-Lequeux model:

from rheojax.models import HebraudLequeux
import numpy as np

# Create model
model = HebraudLequeux()

# Set parameters
model.parameters.set_value('alpha', 0.3)      # Noise coupling (< 0.5 = glass)
model.parameters.set_value('sigma_c', 50.0)   # Pa
model.parameters.set_value('tau', 1.0)         # s

# Fit to flow curve
gamma_dot = np.logspace(-2, 1, 30)
model.fit(gamma_dot, stress_data, test_mode='steady_shear')

# Extract yield stress
sigma_y = model.get_yield_stress()
print(f"Yield stress: {sigma_y:.1f} Pa")

Bayesian inference:

# Bayesian with NLSQ warm-start
result = model.fit_bayesian(
    gamma_dot, stress_data,
    test_mode='steady_shear',
    num_warmup=1000,
    num_samples=2000,
    num_chains=4,
    seed=42
)

# Parameter uncertainties
intervals = model.get_credible_intervals(result.posterior_samples)
print(f"σ_c: [{intervals['sigma_c'][0]:.1f}, {intervals['sigma_c'][1]:.1f}] Pa")

Model Documentation¶

Hébraud–Lequeux (HL) Model — Handbook

References¶

Hébraud, P. & Lequeux, F. (1998). “Mode-coupling theory for the pasty rheology of soft glassy materials.” Phys. Rev. Lett., 81, 2934–2937. https://doi.org/10.1103/PhysRevLett.81.2934
Hébraud, P., Lequeux, F., Munch, J.P., & Pine, D.J. (1997). “Yielding and rearrangements in disordered emulsions.” Phys. Rev. Lett., 78, 4657–4660. https://doi.org/10.1103/PhysRevLett.78.4657
Picard, G., Ajdari, A., Lequeux, F., & Bocquet, L. (2005). “Slow flows of yield stress fluids: Complex spatiotemporal behavior within a simple elastoplastic model.” Phys. Rev. E, 71, 010501. https://doi.org/10.1103/PhysRevE.71.010501
Derec, C., Ajdari, A., & Lequeux, F. (2001). “Rheology and aging: A simple approach.” Eur. Phys. J. E, 4, 355–361. https://doi.org/10.1007/s101890170118
Coussot, P., Nguyen, Q.D., Huynh, H.T., & Bonn, D. (2002). “Avalanche behavior in yield stress fluids.” Phys. Rev. Lett., 88, 175501. https://doi.org/10.1103/PhysRevLett.88.175501