Cross Model

Quick Reference

• Use when: Well-characterized high-rate plateaus, tunable transition sharpness, suspensions and emulsions

• Parameters: 4 (\(\eta_0\), \(\eta_\infty\), \(\lambda\), \(m\))

• Key equation: \(\eta = \eta_{\infty} + \frac{\eta_0 - \eta_{\infty}}{1 + (\lambda\dot{\gamma})^m}\)

• Test modes: Flow (steady shear, rotation)

• Material examples: Polymer melts, colloidal suspensions, emulsions, paints, inks, lubricants

Notation Guide

Symbol	Meaning
\(\eta\)	Apparent (shear) viscosity (Pa·s)
\(\eta_0\)	Zero-shear viscosity (Pa·s); low-shear Newtonian plateau
\(\eta_{\infty}\)	Infinite-shear viscosity (Pa·s); high-shear Newtonian plateau
\(\lambda\)	Time constant (s); reciprocal of critical shear rate
\(m\)	Cross rate constant (dimensionless); controls transition sharpness
\(\dot{\gamma}\)	Shear rate (1/s)

Overview

The Cross model is a four-parameter generalized Newtonian fluid equation that describes the smooth transition between two Newtonian plateaus. It was developed by Malcolm M. Cross in 1965 [1] specifically for polymer solutions and colloidal suspensions, predating the Carreau model by seven years.

The key distinguishing feature is the tunable transition exponent \(m\). While Carreau fixes the transition shape via a square-law term \([1 + (\lambda\dot{\gamma})^2]\), Cross uses a general exponent \(m\) that can be fitted to match experimental data more precisely.

Historical Context

Cross developed the model while working on the rheology of pseudoplastic systems at ICI (Imperial Chemical Industries). His motivation was to create a flow equation that:

1. Predicts finite viscosity at zero shear rate (unlike power law)

2. Allows for a high-shear Newtonian plateau (observed in many real fluids)

3. Has tunable transition sharpness to match diverse materials

The Cross equation became particularly popular for:

• Colloidal suspensions (where both plateaus are experimentally accessible)

• Polymer solutions (especially at low concentrations)

• Paints, inks, and coatings (quality control applications)

• Biomedical fluids (blood, synovial fluid)

Relation to Carreau Model

The Carreau and Cross models are related:

• Carreau: \(\eta = \eta_{\infty} + (\eta_0 - \eta_{\infty})[1 + (\lambda\dot{\gamma})^2]^{(n-1)/2}\)

• Cross: \(\eta = \eta_{\infty} + (\eta_0 - \eta_{\infty})[1 + (\lambda\dot{\gamma})^m]^{-1}\)

When \(m = 2\) and \(n = 0\) (extreme shear-thinning), the models become equivalent in the power-law region. The choice between them often depends on:

• Historical preference in the application area

• Which functional form better fits the specific data

• Whether the transition region or asymptotic behavior is more important

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Physical Foundations

Microstructural Interpretation

The Cross model captures flow behavior arising from shear-induced structural changes:

At low shear rates (\(\eta \approx \eta_0\)):

• Suspended particles or polymer chains are randomly oriented

• Brownian motion maintains isotropic microstructure

• Viscous resistance is maximum due to random collisions/entanglements

• Flow timescale (\(1/\dot{\gamma}\)) exceeds structural relaxation time

At intermediate shear rates (power-law region):

• Shear flow begins to orient particles/chains

• Aggregates or entanglements break up

• Layers of particles slide past each other more easily

• Viscosity decreases following \(\eta \propto \dot{\gamma}^{-m/(1+m\cdot\text{const})}\) approximately

At high shear rates (\(\eta \approx \eta_{\infty}\)):

• Particles/chains are fully aligned with flow

• Minimum structural resistance achieved

• Only hydrodynamic interactions remain

• For suspensions: \(\eta_{\infty}\) approaches solvent viscosity with particle contribution

Physical Meaning of Parameters

Time constant \(\lambda\):

Represents the characteristic time for structural rearrangement. The critical shear rate \(\dot{\gamma}_c = 1/\lambda\) marks where viscosity has dropped halfway from \(\eta_0\) toward \(\eta_{\infty}\).

• For suspensions: Related to particle diffusion time \(\lambda \sim a^2/D_0\) where \(a\) is particle radius

• For polymers: Related to longest relaxation time \(\lambda \sim \tau_d\)

Rate constant \(m\):

Controls how sharply viscosity transitions between plateaus:

• Small \(m\) (0.2-0.5): Gradual, smooth transition over many decades

• Moderate \(m\) (0.5-1.5): Typical for most polymer solutions and suspensions

• Large \(m\) (>1.5): Sharp, switch-like transition (step-function as \(m \to \infty\))

Material Examples with Typical Parameters

Representative Cross parameters

Material	\(\eta_0\) (Pa·s)	\(\eta_{\infty}\) (Pa·s)	\(\lambda\) (s)	\(m\)	T (°C)	Ref
Silicone oil suspension	15.2	0.35	0.08	0.85	25	[2]
Polyisobutylene solution	12.8	0.52	0.15	1.2	25	[1]
Latex paint	8.5	0.15	0.5	0.95	25	[3]
Synovial fluid	2.5	0.005	1.2	0.75	37	[4]
Ink (offset printing)	45.0	1.2	0.02	1.1	30	[5]

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Governing Equations

Constitutive Equation

The Cross viscosity function is:

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

Equivalently, defining the reduced viscosity \(\eta_r = (\eta - \eta_{\infty})/(\eta_0 - \eta_{\infty})\):

\[\eta_r = \frac{1}{1 + (\lambda\dot{\gamma})^m}\]

Shear Stress Relation

The shear stress is:

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

This is monotonically increasing for all \(m > 0\), ensuring flow stability.

Limiting Cases

Asymptotic behavior

Regime	Condition	\(\eta(\dot{\gamma})\)	Physical interpretation
Low shear	\(\lambda\dot{\gamma} \ll 1\)	\(\approx \eta_0\)	First Newtonian plateau
Critical	\(\lambda\dot{\gamma} = 1\)	\((\eta_0 + \eta_{\infty})/2\)	Transition midpoint
Power-law	\(\lambda\dot{\gamma} \gg 1\)	\(\approx \eta_0 (\lambda\dot{\gamma})^{-m}\) + \(\eta_{\infty}\)	Shear-thinning
High shear	\(\lambda\dot{\gamma} \to \infty\)	\(\to \eta_{\infty}\)	Second Newtonian plateau

Power-Law Approximation

In the power-law region (\(\lambda\dot{\gamma} \gg 1\)), ignoring \(\eta_{\infty}\):

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

where \(K = \eta_0 \lambda^{-m}\) and \(n = 1 - m\). This connects Cross parameter \(m\) to power-law index.

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Parameters

Parameters

Name	Symbol	Units	Bounds	Notes
eta0	\(\eta_0\)	Pa·s	\(10^{-3} - 10^{12}\)	Zero-shear viscosity; first Newtonian plateau
eta_inf	\(\eta_{\infty}\)	Pa·s	\(10^{-6} - 10^{6}\)	Infinite-shear viscosity; often solvent viscosity
lambda_	\(\lambda\)	s	\(10^{-6} - 10^{6}\)	Time constant; \(1/\lambda\) is transition shear rate
m	\(m\)	—	\(0.1 - 2.0\)	Rate constant; controls transition sharpness

Parameter Interpretation

eta0 (Zero-Shear Viscosity):

• Physical meaning: Viscosity of the undisturbed structure

• For suspensions: Depends on volume fraction \(\phi\) via Krieger-Dougherty

• For polymers: Related to molecular weight via \(\eta_0 \sim M^{3.4}\)

eta_inf (Infinite-Shear Viscosity):

• Physical meaning: Residual viscosity after complete structure breakdown

• For suspensions: Hydrodynamic contribution only; approaches \(\eta_s (1 - \phi/\phi_m)^{-[\eta]\phi_m}\)

• For solutions: Approximately the solvent viscosity

lambda (Time Constant):

• Physical meaning: Characteristic structural relaxation time

• Interpretation: Faster relaxation (small \(\lambda\)) → early transition to thinning

• Relation: \(\dot{\gamma}_{1/2} = 1/\lambda\) where \(\eta = (\eta_0 + \eta_{\infty})/2\)

m (Rate Constant):

• Physical meaning: Steepness of the viscosity drop in transition region

• Connection to power law: Approximately \(n = 1 - m\) in mid-rate region

• Typical values: 0.5-1.5 for most fluids

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Validity and Assumptions

Model Assumptions

1. Generalized Newtonian: No memory effects, stress depends only on current \(\dot{\gamma}\)

2. Isothermal: Constant temperature (combine with Arrhenius for T-dependence)

3. Simple shear: Steady unidirectional flow

4. Inelastic: No normal stress differences predicted

Data Requirements

• Required: Flow curve \(\eta(\dot{\gamma})\) spanning at least 3 decades

• Ideal: Data capturing both plateaus (may require wide \(\dot{\gamma}\) range)

• For accurate \(m\): Transition region well-resolved (5+ points)

Limitations

No viscoelasticity:

Cannot predict \(G'(\omega)\), \(G''(\omega)\), or stress relaxation. Use Maxwell/Oldroyd-B for elastic effects.

No yield stress:

Material always flows; \(\sigma \to 0\) as \(\dot{\gamma} \to 0\). Use Herschel-Bulkley for yield stress fluids.

No thixotropy:

Instantaneous response assumed; no time-dependent structure changes. Use DMT or fluidity models for thixotropy.

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What You Can Learn

This section explains how to translate fitted Cross parameters into material insights and actionable knowledge.

Parameter Interpretation

eta0 (Zero-Shear Viscosity):

The zero-shear viscosity indicates the structural state at rest:

• High \(\eta_0\) (>100 Pa·s): Strong particle aggregation, high molecular weight polymer, or concentrated system with extensive network formation

• Moderate \(\eta_0\) (1-100 Pa·s): Typical for polymer solutions, emulsions, and moderately concentrated suspensions

• Low \(\eta_0\) (<1 Pa·s): Dilute solution, weak interparticle attractions, or low molecular weight

For graduate students: For suspensions, the Krieger-Dougherty equation relates \(\eta_0\) to volume fraction: \(\eta_0 / \eta_s = (1 - \phi/\phi_m)^{-[\eta]\phi_m}\) where \(\eta_s\) is solvent viscosity, \(\phi\) is volume fraction, and \(\phi_m\) is maximum packing. This enables volume fraction estimation from viscosity measurements.

For practitioners: \(\eta_0\) controls critical processing behaviors—settling/sedimentation rates in storage, coating thickness during low-shear application, and leveling behavior after deposition. Target higher \(\eta_0\) for shelf stability and sag prevention.

eta_inf (Infinite-Shear Viscosity):

The high-shear plateau reveals the fully disrupted microstructure:

• High ratio \(\eta_{\infty}/\eta_0\) (>10%): Significant irreducible structure remains; strong hydrodynamic interactions even when fully aligned

• Low ratio \(\eta_{\infty}/\eta_0\) (<1%): Nearly complete structural breakdown under flow; approaches solvent-like behavior

For graduate students: For suspensions, \(\eta_{\infty}\) approaches the Einstein limit \(\eta_s(1 + 2.5\phi)\) when particles are fully dispersed and aligned. Deviations indicate residual aggregation or non-spherical particle effects.

For practitioners: \(\eta_{\infty}\) determines high-rate processing capability—spray atomization quality, high-speed coating uniformity, and pumping energy requirements at production rates. Lower values enable faster processing.

lambda (Time Constant):

The relaxation time marks the transition between regimes:

• Critical shear rate: \(\dot{\gamma}_c = 1/\lambda\) identifies where viscosity drops to halfway between plateaus

• Short \(\lambda\) (<0.1 s): Fast structural response, suitable for high-speed operations

• Long \(\lambda\) (>10 s): Slow structural relaxation, memory effects important

For graduate students: For Brownian particles, \(\lambda \sim a^2/D_0\) where \(a\) is particle radius and \(D_0\) is diffusion coefficient. For polymers, \(\lambda\) scales with the longest relaxation time from chain dynamics.

For practitioners: Compare \(\lambda\) to process timescales. Operating at \(\dot{\gamma} \gg 1/\lambda\) ensures material is in the thinned state; \(\dot{\gamma} \ll 1/\lambda\) keeps material at rest viscosity. Design mixing speeds accordingly.

m (Rate Constant):

The transition sharpness parameter characterizes structural breakdown:

• Low \(m\) (0.3-0.6): Gradual, smooth transition over many decades—indicates broad distribution of relaxation times or multiple structural elements breaking down at different rates

• Moderate \(m\) (0.6-1.2): Typical for most polymer solutions and suspensions with moderate polydispersity

• High \(m\) (1.2-2.0): Sharp, switch-like transition—indicates narrow relaxation spectrum or cooperative structural breakdown

For graduate students: The parameter \(m\) relates to polydispersity and relaxation time distribution breadth. Compare with Cole-Cole analysis of oscillatory data: broad distributions give low \(m\), narrow distributions give high \(m\).

For practitioners: High \(m\) materials have excellent “smart fluid” behavior—thick when still, thin when worked. This is ideal for coatings (sag-resistant yet sprayable). Low \(m\) gives smoother processing with less abrupt rheology changes.

Material Classification

Material Classification from Cross Parameters

Parameter Pattern	Material Behavior	Typical Materials	Processing Implications
Large \(\eta_0/\eta_{\infty}\), high \(m\)	Strong cooperative structure	Concentrated latex paints, thick emulsions	Excellent sag resistance with spray-ability
Large \(\eta_0/\eta_{\infty}\), low \(m\)	Broad relaxation spectrum	Polydisperse suspensions, polymer blends	Smooth processing window, forgiving
Moderate \(\eta_0/\eta_{\infty}\), moderate \(m\)	Standard structured fluid	Typical coatings, food emulsions	Balanced processing characteristics
Small \(\eta_0/\eta_{\infty}\) (<10)	Weak or minimal structure	Dilute polymer solutions	Limited shear-thinning, consider simpler model

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Experimental Design

When to Use Cross Model

Use Cross when:

• Both Newtonian plateaus are experimentally accessible

• Transition sharpness needs to be a fitted parameter

• Suspension/emulsion with well-defined microstructure

Use Carreau instead when:

• High-shear plateau is not reached

• Polymer melt with standard transition behavior

• Compatibility with existing CFD codes required

Recommended Test Protocol

Steady Shear Flow Curve

Step 1: Sample equilibration

• Load sample, equilibrate at test temperature for 10 min

• Pre-shear at moderate rate (10-100 s\(^{-1}\)) for 60 s, then rest 5 min

Step 2: Flow curve measurement

• Shear rate sweep: \(10^{-3}\) to \(10^{3}\) s\(^{-1}\)

• Log spacing: 5 points per decade minimum

• Equilibration: Wait for steady stress (auto or fixed time)

Step 3: Ascending vs descending

• Ascending sweep preferred for non-thixotropic materials

• Compare ascending/descending to detect time effects

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Fitting Guidance

Parameter Initialization

Step 1: Estimate \(\eta_0\) from lowest shear rates

\(\eta_0 \approx\) average of \(\eta\) at \(\dot{\gamma} < 0.01/\lambda\)

Step 2: Estimate \(\eta_{\infty}\) from highest shear rates

\(\eta_{\infty} \approx\) average of \(\eta\) at \(\dot{\gamma} > 100/\lambda\)

Step 3: Find \(\lambda\) from midpoint

Where \(\eta = (\eta_0 + \eta_{\infty})/2\), \(\lambda = 1/\dot{\gamma}_{1/2}\)

Step 4: Estimate \(m\) from log-log slope

In power-law region: slope \(\approx -m\)

Optimization

RheoJAX default: NLSQ (GPU-accelerated)

• Fast convergence for 4-parameter Cross model

• Bounds recommended to prevent unphysical values

Bounds:

• \(\eta_0\): [1e-2, 1e10] Pa·s

• \(\eta_{\infty}\): [0, 0.9 × \(\eta_0\)] Pa·s

• \(\lambda\): [1e-6, 1e4] s

• \(m\): [0.2, 2.0]

Troubleshooting

Fitting diagnostics

Problem	Diagnostic	Solution
\(m\) hits bounds	Transition shape doesn’t match	Check for artifacts; try Carreau-Yasuda
\(\eta_{\infty}\) negative	Bound violation	Constrain \(\eta_{\infty} \geq 0\); check high-rate data
Poor fit at transition	Functional form mismatch	Try Carreau or Carreau-Yasuda
Correlated \(\lambda\) and \(m\)	Under-resolved transition	More data points in transition region

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Usage

Basic Example

import numpy as np
from rheojax.models import Cross

# Shear rate data
gamma_dot = np.logspace(-3, 4, 100)
eta_data = experimental_viscosity(gamma_dot)

# Create and fit model
model = Cross()
model.fit(gamma_dot, eta_data, test_mode='rotation')

# Extract parameters
eta0 = model.parameters.get_value('eta0')
eta_inf = model.parameters.get_value('eta_inf')
lambda_ = model.parameters.get_value('lambda_')
m = model.parameters.get_value('m')

print(f"Zero-shear viscosity: {eta0:.2f} Pa·s")
print(f"Infinite-shear viscosity: {eta_inf:.4f} Pa·s")
print(f"Time constant: {lambda_:.4f} s")
print(f"Rate constant m: {m:.3f}")

Comparison with Carreau

from rheojax.models import Carreau, Cross

# Fit both models
carreau = Carreau()
carreau.fit(gamma_dot, eta_data, test_mode='rotation')

cross = Cross()
cross.fit(gamma_dot, eta_data, test_mode='rotation')

# Compare fit quality
print(f"Carreau R²: {carreau.score(gamma_dot, eta_data):.4f}")
print(f"Cross R²: {cross.score(gamma_dot, eta_data):.4f}")

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See Also

• Carreau Model — uses square-law exponent; choose based on transition shape

• Carreau–Yasuda Model — adds Yasuda exponent for even more flexibility

• Power-Law (Ostwald–de Waele) — approximates Cross mid-rate slope when plateaus unavailable

• Herschel-Bulkley Model — for yield stress fluids

• SmoothDerivative — differentiate flow curves to estimate \(m\)

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API References

• Module: rheojax.models

• Class: rheojax.models.Cross

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References

[1] (1,2)

Cross, M. M. “Rheology of non-Newtonian fluids: A new flow equation for pseudoplastic systems.” Journal of Colloid Science, 20, 417-437 (1965). https://doi.org/10.1016/0095-8522(65)90022-X

[2]

Barnes, H. A., Hutton, J. F. & Walters, K. An Introduction to Rheology. Elsevier, Amsterdam (1989).

[3]

Patton, T. C. Paint Flow and Pigment Dispersion, 2nd Edition. Wiley-Interscience (1979).

[4]

Fung, Y. C. Biomechanics: Mechanical Properties of Living Tissues, 2nd Edition. Springer (1993).

[5]

Tanner, R. I. & Walters, K. Rheology: An Historical Perspective. Elsevier (1998).

[6]

Larson, R. G. The Structure and Rheology of Complex Fluids. Oxford University Press (1999).

[7]

Macosko, C. W. Rheology: Principles, Measurements, and Applications. Wiley-VCH (1994).

[8]

Mewis, J. & Wagner, N. J. Colloidal Suspension Rheology. Cambridge University Press (2012).

[9]

Krieger, I. M. & Dougherty, T. J. “A mechanism for non-Newtonian flow in suspensions of rigid spheres.” Transactions of the Society of Rheology, 3, 137-152 (1959).

[10]

Morrison, F. A. Understanding Rheology. Oxford University Press (2001).

Further Reading

• Bird, R. B., Armstrong, R. C. & Hassager, O. Dynamics of Polymeric Liquids, Vol. 1. Wiley (1987). [Comprehensive treatment of generalized Newtonian models]