Carreau–Yasuda Model

Quick Reference

• Use when: Abrupt viscosity transitions, sharp changes between plateaus

• Parameters: 5 (\(\eta_0\), \(\eta_\infty\), \(\lambda\), n, a)

• Key equation: \(\eta = \eta_{\infty} + (\eta_0 - \eta_{\infty})[1 + (\lambda\dot{\gamma})^{a}]^{(n-1)/a}\)

• Test modes: Flow (steady shear)

• Material examples: Wormlike micelles, highly filled polymers, materials with sharp transitions

Overview

The Carreau-Yasuda model extends the classical Carreau Model model by introducing an additional parameter \(a\) that controls the sharpness of the transition between the zero-shear plateau and the power-law region. This generalization was introduced by Yasuda, Armstrong, and Cohen (1981) while studying concentrated polymer solutions.

The model is particularly valuable when:

1. The transition region is sharper or broader than the standard Carreau model predicts

2. Materials exhibit abrupt viscosity drops (e.g., wormlike micelles, associative polymers)

3. High-fidelity modeling of the transition curvature is required for process simulation

Setting \(a = 2\) recovers the original Carreau model, while \(a < 2\) produces sharper transitions and \(a > 2\) produces more gradual ones.

Notation Guide

Symbol	Meaning
\(\eta\)	Apparent viscosity (Pa·s)
\(\eta_0\)	Zero-shear viscosity (Pa·s). Low-rate Newtonian plateau.
\(\eta_\infty\)	Infinite-shear viscosity (Pa·s). High-rate solvent contribution.
\(\dot{\gamma}\)	Shear rate (s-1)
\(\lambda\)	Relaxation time (s). Inverse of critical shear rate.
\(n\)	Power-law index (dimensionless). High-shear slope.
\(a\)	Yasuda exponent (dimensionless). Transition sharpness.

Physical Foundations

Microstructural Interpretation

The Yasuda exponent \(a\) captures the breadth of relaxation time distribution:

• Sharp transition (a < 2): Indicates a relatively narrow distribution of relaxation times. The material transitions rapidly from Newtonian to power-law behavior because most structural elements respond at similar time scales.

• Gradual transition (a > 2): Suggests a broad distribution of relaxation times. Different structural elements (e.g., polymer chains of different lengths, aggregates of different sizes) begin thinning at different shear rates.

• Carreau case (a = 2): The empirical default that works for many polymer solutions.

Connection to Polymer Physics

For polymer solutions, the Yasuda exponent relates to molecular architecture:

Linear Polymers (Narrow MWD):

Monodisperse or narrow-MWD linear polymers typically show \(a \approx 2\), consistent with the Carreau model. The single dominant relaxation mode creates a smooth but definite transition.

Branched Polymers:

Long-chain branching introduces additional relaxation modes at longer times, often producing \(a > 2\) due to the broadened spectrum.

Wormlike Micelles:

These self-assembled structures can break and reform under flow, creating sharp transitions with \(a < 2\). The sudden onset of flow alignment produces an abrupt viscosity drop.

Associative Polymers:

Polymers with sticky groups form transient networks. At critical shear rates, network disruption can cause sharp drops (\(a \approx 1-1.5\)).

Governing Equations

Constitutive Equation

\[\eta(\dot{\gamma}) = \eta_{\infty} + (\eta_0 - \eta_{\infty}) \left[1 + (\lambda \dot{\gamma})^{a}\right]^{\frac{n-1}{a}}\]

This form ensures:

• At \(\dot{\gamma} \to 0\): \(\eta \to \eta_0\) (zero-shear plateau)

• At \(\dot{\gamma} \to \infty\): \(\eta \to \eta_\infty\) (infinite-shear plateau)

• In the power-law region: \(\eta \propto \dot{\gamma}^{n-1}\)

Limiting Cases

Low shear rate (\(\lambda \dot{\gamma} \ll 1\)):

\[\eta \approx \eta_0 - (\eta_0 - \eta_\infty) \frac{(n-1)}{a} (\lambda \dot{\gamma})^a\]

High shear rate (\(\lambda \dot{\gamma} \gg 1\)):

\[\eta \approx \eta_\infty + (\eta_0 - \eta_\infty) (\lambda \dot{\gamma})^{n-1}\]

Power-law approximation (mid-range, \(\eta_\infty \approx 0\)):

\[\eta \approx \eta_0 (\lambda)^{n-1} \dot{\gamma}^{n-1} = K \dot{\gamma}^{n-1}\]

where the effective consistency index is \(K = \eta_0 \lambda^{n-1}\).

Relation to Carreau Model

Setting \(a = 2\):

\[\eta = \eta_\infty + (\eta_0 - \eta_\infty) \left[1 + (\lambda \dot{\gamma})^2\right]^{\frac{n-1}{2}}\]

This is the standard Carreau form.

Parameters

Parameter Summary

Name	Symbol	Units	Description / Constraints
eta0	\(\eta_0\)	Pa·s	Zero-shear viscosity. Must be > 0 and typically ≥ eta_inf.
eta_inf	\(\eta_\infty\)	Pa·s	Infinite-shear viscosity. Must be ≥ 0; often set to 0 when unmeasurable.
lambda_	\(\lambda\)	s	Relaxation time. Inverse of critical shear rate where thinning begins.
n	\(n\)	–	Power-law index. < 1 for thinning, = 1 for Newtonian, > 1 for thickening.
a	\(a\)	–	Yasuda exponent. Controls transition sharpness; = 2 gives Carreau model.

Parameter Bounds

Default Bounds

Parameter	Bounds	Physical Rationale
\(\eta_0\)	(1e-3, 1e12)	Must exceed solvent viscosity
\(\eta_\infty\)	(1e-6, 1e6)	Cannot exceed \(\eta_0\); often ~solvent viscosity
\(\lambda\)	(1e-6, 1e6)	Must capture transition in measured range
\(n\)	(0.01, 1.0)	<0.01 unphysical; >1 rare (shear thickening)
\(a\)	(0.1, 2.0)	<0.1 too sharp (numerical issues); >2 nearly Newtonian transition

Material Behavior Guide

Typical Parameter Ranges

Material Class	\(\eta_0\) (Pa·s)	\(\eta_\infty\) (Pa·s)	n	a	Notes
Wormlike Micelles	10–1000	0.001–0.1	0.1–0.4	0.8–1.5	Sharp transition from network breakup
Associative Polymers	1–100	0.01–1	0.2–0.5	1.0–1.8	HEUR, HASE thickeners
Concentrated Polymer Solutions	100–10000	0.1–10	0.3–0.6	1.5–2.5	Narrow MWD: a ≈ 2
Branched Polymers	1000–100000	1–100	0.4–0.7	2.0–3.5	Long-chain branching broadens transition
Highly Filled Systems	10–1000	0.1–10	0.2–0.5	1.5–2.5	Particle alignment under shear
Blood/Biofluids	0.01–0.1	0.003–0.005	0.3–0.5	2.0–2.5	RBC aggregation/deformation

Validity and Assumptions

When Carreau-Yasuda is Appropriate

Use the Carreau-Yasuda model when:

1. Both plateaus are accessible: Data span from zero-shear to near-infinite-shear plateaus, or at least show clear approach to both.

2. Transition sharpness matters: The standard Carreau model (\(a=2\)) provides poor fits to the transition region.

3. No yield stress: The material flows freely at all stresses (no intercept at \(\dot{\gamma}=0\)).

4. Steady-state flow: Time-independent response (no thixotropy or viscoelastic overshoot).

When to Use Alternatives

Model Selection Guide

Observation	Issue	Better Model
Transition fits well with \(a \approx 2\)	Carreau-Yasuda overparameterized	Carreau Model (4 parameters)
No visible zero-shear plateau	\(\eta_0\) unconstrained	Power-Law (Ostwald–de Waele) or Cross Model
Stress intercept at zero rate	Material has yield stress	Herschel-Bulkley Model
Fitted \(a < 0.5\)	Approaching step-function (unphysical)	Check data; consider yield stress model
Strong parameter correlations	Data don’t resolve all 5 parameters	Carreau Model or reduce \(a\) to fixed value

What You Can Learn

This section explains how to translate fitted Carreau-Yasuda parameters into material insights and actionable knowledge.

Parameter Interpretation

Yasuda Exponent (a):

The Yasuda exponent reveals the breadth of relaxation time distribution:

• a < 1.5: Sharp transition indicating a narrow relaxation spectrum. Common in wormlike micelles and associative polymers where cooperative structural breakdown occurs at a critical shear rate.

• a ≈ 2.0: Standard Carreau behavior. Typical for well-characterized polymer solutions with moderate polydispersity.

• a > 2.5: Broad transition suggesting wide relaxation time distribution. Common in branched polymers and materials with multiple structural components.

For graduate students: The Yasuda exponent connects to the Cole-Davidson parameter in dielectric relaxation and the stretched exponential \(\beta\) in KWW relaxation. Lower \(a\) corresponds to more exponential (single-mode) relaxation; higher \(a\) corresponds to stretched relaxation.

For practitioners: Sharp transitions (low \(a\)) can cause processing instabilities. If \(a < 1.5\), consider whether sudden viscosity drops might cause flow instabilities or poor coating uniformity.

Relaxation Time ( \(\lambda\) ):

The relaxation time identifies the critical shear rate for structural response:

• Critical shear rate: \(\dot{\gamma}_c = 1/\lambda\) marks where viscosity begins significant departure from \(\eta_0\).

• Weissenberg number: At \(Wi = \lambda \dot{\gamma} = 1\), elastic and viscous timescales balance.

For graduate students: For entangled polymers, \(\lambda\) scales with the terminal relaxation time \(\tau_d\), which in turn scales as \(\tau_d \propto M_w^{3.4}/c^{1.5}\) (reptation theory).

For practitioners: Compare \(\lambda\) to process timescales. Coating at 100 s\(^{-1}\) with \(\lambda = 0.1\) s gives \(Wi = 10\)—firmly in the power-law regime with good leveling.

Viscosity Ratio ( \(\eta_0/\eta_\infty\) ):

The ratio of plateau viscosities quantifies total thinning capacity:

• Small ratio (< 10): Mild thinning; limited shear-rate sensitivity

• Large ratio (> 1000): Strong thinning; dramatic viscosity reduction

Material Classification

Material Classification from Carreau-Yasuda Parameters

Parameter Pattern	Material Behavior	Typical Materials	Processing Implications
Low a, high \(\eta_0/\eta_\infty\)	Sharp transition, breakable network	Wormlike micelles, associative gels	Shear-banding risk, flow instabilities
a ≈ 2, moderate ratio	Standard polymer behavior	Linear polymer solutions, melts	Predictable processing, stable flow
High a, high \(\eta_0\)	Broad relaxation spectrum	Branched polymers, blends	Wide processing window, forgiving
n close to 1, any a	Weak shear-thinning	Dilute solutions, low MW	Near-Newtonian behavior, consider simpler model

Comparing Carreau vs Carreau-Yasuda Fits

Fit both models to your data and compare:

1. AIC/BIC comparison: If Carreau-Yasuda doesn’t significantly improve fit statistics, use simpler Carreau model.

2. Residual analysis: Systematic residuals in transition region favor Carreau-Yasuda.

3. Parameter uncertainty: If \(a\) has uncertainty >50%, data don’t constrain it—use fixed \(a = 2\) (Carreau).

Diagnostic Indicators

Warning signs in fitted parameters:

• a approaching bounds: If \(a < 0.5\) or \(a > 4\), the model may be compensating for other issues (yield stress, data artifacts).

• \(\lambda\) at measurement bounds: If \(\lambda\) equals 1/(max shear rate) or 1/(min shear rate), the transition is outside your measurement window.

• Strong a- \(\lambda\) correlation: These parameters are inherently correlated. Consider fixing one based on literature or prior measurements.

• \(\eta_\infty > \eta_0\): Physically impossible. Check data for slip or inertia at high rates.

Experimental Design

Recommended Protocol

1. Wide shear rate range: Span at least 4 decades, ideally 6 (0.01–10,000 s\(^{-1}\)).

2. Logarithmic spacing: Use 10 points per decade for good resolution.

3. Steady-state verification: Allow 30–120 s equilibration per point, longer at low rates. Confirm by checking that viscosity doesn’t drift.

4. Bidirectional sweeps: Run both up-ramp and down-ramp. Hysteresis indicates thixotropy or structure evolution.

5. Temperature control: ±0.1°C stability; viscosity changes ~3%/°C for polymers.

Geometry Selection

Recommended Geometries

Shear Rate Range	Geometry	Notes
0.001–100 s\(^{-1}\)	Cone-plate (1–2°)	Uniform shear rate; best for low rates
0.1–1000 s\(^{-1}\)	Parallel plate	Adjustable gap; good for moderate rates
10–10,000 s\(^{-1}\)	Capillary	Best for high rates; requires Rabinowitsch correction
Full range	Combine geometries	Stitch data from multiple tests

Fitting Guidance

Initialization Strategy

Smart initialization dramatically improves convergence:

1. From plateaus: - \(\eta_0\) = average viscosity at lowest 3 shear rates - \(\eta_\infty\) = average viscosity at highest 3 shear rates (or 0)

2. From transition: - Find \(\dot{\gamma}_{1/2}\) where \(\eta = (\eta_0 + \eta_\infty)/2\) - Initialize \(\lambda = 1/\dot{\gamma}_{1/2}\)

3. From slope: - \(n\) = 1 + slope of log-log plot at high rates

4. Default for a: - Start with \(a = 2\) (Carreau default)

Optimization Strategy

Two-stage fitting often works best:

Stage 1: Fix \(a = 2\) and fit other 4 parameters (Carreau fit).

Stage 2: Release \(a\) and refine all 5 parameters from Stage 1 solution.

from rheojax.models import CarreauYasuda

model = CarreauYasuda()

# Stage 1: Carreau fit (fixed a=2)
model.parameters.set_value('a', 2.0)
model.parameters.get('a').vary = False
model.fit(gamma_dot, eta, test_mode='flow_curve')

# Stage 2: Release a and refine
model.parameters.get('a').vary = True
model.fit(gamma_dot, eta, test_mode='flow_curve')

Troubleshooting

Common Issues

Symptom	Possible Cause	Solution
a → 0 (lower bound)	Near-yield behavior	Try Herschel-Bulkley; check for yield stress
a → upper bound	Effectively Newtonian	Use Carreau or simpler model
\(\lambda\) poorly constrained	Transition outside data range	Extend shear rate range
\(\eta_\infty\) negative	Optimization artifact	Constrain \(\eta_{\infty}\) ≥ 0; check high-rate data
Strong a-\(\lambda\) correlation	Insufficient transition data	Fix a = 2 or increase mid-range points

Usage

Basic Fitting

from rheojax.core.jax_config import safe_import_jax
jax, jnp = safe_import_jax()

from rheojax.models import CarreauYasuda
from rheojax.core.data import RheoData

# Flow curve data
gamma_dot = jnp.logspace(-3, 4, 50)  # s^-1
eta = jnp.array([...])  # Pa·s

# Fit with default bounds
model = CarreauYasuda()
model.fit(gamma_dot, eta, test_mode='flow_curve')

# Extract parameters
print(f"eta0 = {model.parameters.get_value('eta0'):.1f} Pa·s")
print(f"eta_inf = {model.parameters.get_value('eta_inf'):.3f} Pa·s")
print(f"lambda = {model.parameters.get_value('lambda_'):.4f} s")
print(f"n = {model.parameters.get_value('n'):.3f}")
print(f"a = {model.parameters.get_value('a'):.2f}")

Two-Stage Fitting

from rheojax.models import CarreauYasuda

model = CarreauYasuda()

# Stage 1: Carreau fit (fix a=2)
model.parameters.set_value('a', 2.0)
model.parameters.get('a').vary = False
model.fit(gamma_dot, eta, test_mode='flow_curve')

# Check if Carreau is sufficient
r2_carreau = model.r_squared
print(f"Carreau R² = {r2_carreau:.5f}")

# Stage 2: Full Carreau-Yasuda (release a)
model.parameters.get('a').vary = True
model.fit(gamma_dot, eta, test_mode='flow_curve')
r2_cy = model.r_squared

# Compare improvement
delta_r2 = r2_cy - r2_carreau
print(f"Carreau-Yasuda R² = {r2_cy:.5f}")
print(f"Improvement: {delta_r2:.6f}")

# If improvement < 0.001, Carreau is sufficient
if delta_r2 < 0.001:
    print("Carreau model is adequate")

Bayesian Inference

from rheojax.models import CarreauYasuda

model = CarreauYasuda()
model.fit(gamma_dot, eta, test_mode='flow_curve')  # NLSQ warm-start

result = model.fit_bayesian(
    gamma_dot, eta,
    test_mode='flow_curve',
    num_warmup=1000,
    num_samples=2000,
    num_chains=4
)

intervals = model.get_credible_intervals(result.posterior_samples)
for param in ['eta0', 'eta_inf', 'lambda_', 'n', 'a']:
    ci = intervals[param]
    print(f"{param}: {ci['mean']:.3f} [{ci['hdi_2.5%']:.3f}, {ci['hdi_97.5%']:.3f}]")

Computational Implementation

JAX Vectorization

The model is fully JIT-compiled:

from functools import partial
from rheojax.core.jax_config import safe_import_jax
jax, jnp = safe_import_jax()

@partial(jax.jit, static_argnums=())
def carreau_yasuda(gamma_dot, eta_0, eta_inf, lambda_, n, a):
    return eta_inf + (eta_0 - eta_inf) * (
        1 + (lambda_ * gamma_dot) ** a
    ) ** ((n - 1) / a)

Numerical Stability

1. Exponent limiting: The term \((n-1)/a\) is bounded to prevent overflow when \(a\) is very small.

2. Log-space fitting: Internal optimization uses \(\log(\eta_0)\), \(\log(\eta_\infty)\), \(\log(\lambda)\) for numerical stability.

3. Gradient clipping: JAX gradients are clipped to prevent NaN propagation

See Also

Related Flow Models

• Carreau Model — Special case with \(a = 2\); preferred when simpler model suffices

• Cross Model — Alternative sigmoidal model with denominator exponent

• Power-Law (Ostwald–de Waele) — Simple power-law for mid-range rates only

• Herschel-Bulkley Model — For materials with yield stress

Transforms

• Mastercurve (Time-Temperature Superposition) — Time-temperature superposition

• Strain-Rate Frequency Superposition (SRFS) — Strain-rate frequency superposition

API Reference

• rheojax.models.CarreauYasuda

• rheojax.models.Carreau

References

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[2]

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[3]

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[8]

Berret, J.-F. “Rheology of wormlike micelles: Equilibrium properties and shear banding transitions.” In Molecular Gels, edited by R. G. Weiss and P. Terech, 667–720. Springer, Dordrecht (2006). https://doi.org/10.1007/1-4020-3689-2_20

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[10]

Mewis, J., and Wagner, N. J. Colloidal Suspension Rheology. Cambridge University Press (2012). ISBN: 978-0521515993

[11]

Dealy, J. M., and Larson, R. G. Structure and Rheology of Molten Polymers: From Structure to Flow Behavior and Back Again. Hanser (2006). https://doi.org/10.3139/9783446412811