.. _model-cross: ============= Cross Model ============= Quick Reference --------------- - **Use when:** Well-characterized high-rate plateaus, tunable transition sharpness, suspensions and emulsions - **Parameters:** 4 (:math:`\eta_0`, :math:`\eta_\infty`, :math:`\lambda`, :math:`m`) - **Key equation:** :math:`\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 -------------- .. list-table:: :widths: 15 85 :header-rows: 1 * - Symbol - Meaning * - :math:`\eta` - Apparent (shear) viscosity (Pa·s) * - :math:`\eta_0` - Zero-shear viscosity (Pa·s); low-shear Newtonian plateau * - :math:`\eta_{\infty}` - Infinite-shear viscosity (Pa·s); high-shear Newtonian plateau * - :math:`\lambda` - Time constant (s); reciprocal of critical shear rate * - :math:`m` - Cross rate constant (dimensionless); controls transition sharpness * - :math:`\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** :math:`m`. While Carreau fixes the transition shape via a square-law term :math:`[1 + (\lambda\dot{\gamma})^2]`, Cross uses a general exponent :math:`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**: :math:`\eta = \eta_{\infty} + (\eta_0 - \eta_{\infty})[1 + (\lambda\dot{\gamma})^2]^{(n-1)/2}` - **Cross**: :math:`\eta = \eta_{\infty} + (\eta_0 - \eta_{\infty})[1 + (\lambda\dot{\gamma})^m]^{-1}` When :math:`m = 2` and :math:`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 ---- Physical Foundations -------------------- Microstructural Interpretation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The Cross model captures flow behavior arising from shear-induced structural changes: **At low shear rates** (:math:`\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 (:math:`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 :math:`\eta \propto \dot{\gamma}^{-m/(1+m\cdot\text{const})}` approximately **At high shear rates** (:math:`\eta \approx \eta_{\infty}`): - Particles/chains are fully aligned with flow - Minimum structural resistance achieved - Only hydrodynamic interactions remain - For suspensions: :math:`\eta_{\infty}` approaches solvent viscosity with particle contribution Physical Meaning of Parameters ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ **Time constant** :math:`\lambda`: Represents the characteristic time for structural rearrangement. The critical shear rate :math:`\dot{\gamma}_c = 1/\lambda` marks where viscosity has dropped halfway from :math:`\eta_0` toward :math:`\eta_{\infty}`. - **For suspensions**: Related to particle diffusion time :math:`\lambda \sim a^2/D_0` where :math:`a` is particle radius - **For polymers**: Related to longest relaxation time :math:`\lambda \sim \tau_d` **Rate constant** :math:`m`: Controls how sharply viscosity transitions between plateaus: - **Small** :math:`m` **(0.2-0.5)**: Gradual, smooth transition over many decades - **Moderate** :math:`m` **(0.5-1.5)**: Typical for most polymer solutions and suspensions - **Large** :math:`m` **(>1.5)**: Sharp, switch-like transition (step-function as :math:`m \to \infty`) Material Examples with Typical Parameters ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. list-table:: Representative Cross parameters :header-rows: 1 :widths: 25 15 12 12 10 10 16 * - Material - :math:`\eta_0` (Pa·s) - :math:`\eta_{\infty}` (Pa·s) - :math:`\lambda` (s) - :math:`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]_ ---- Governing Equations ------------------- Constitutive Equation ~~~~~~~~~~~~~~~~~~~~~ The Cross viscosity function is: .. math:: \eta(\dot{\gamma}) = \eta_{\infty} + \frac{\eta_0 - \eta_{\infty}}{1 + (\lambda\dot{\gamma})^m} Equivalently, defining the reduced viscosity :math:`\eta_r = (\eta - \eta_{\infty})/(\eta_0 - \eta_{\infty})`: .. math:: \eta_r = \frac{1}{1 + (\lambda\dot{\gamma})^m} Shear Stress Relation ~~~~~~~~~~~~~~~~~~~~~ The shear stress is: .. math:: \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 :math:`m > 0`, ensuring flow stability. Limiting Cases ~~~~~~~~~~~~~~ .. list-table:: Asymptotic behavior :header-rows: 1 :widths: 25 25 25 25 * - Regime - Condition - :math:`\eta(\dot{\gamma})` - Physical interpretation * - Low shear - :math:`\lambda\dot{\gamma} \ll 1` - :math:`\approx \eta_0` - First Newtonian plateau * - Critical - :math:`\lambda\dot{\gamma} = 1` - :math:`(\eta_0 + \eta_{\infty})/2` - Transition midpoint * - Power-law - :math:`\lambda\dot{\gamma} \gg 1` - :math:`\approx \eta_0 (\lambda\dot{\gamma})^{-m}` + :math:`\eta_{\infty}` - Shear-thinning * - High shear - :math:`\lambda\dot{\gamma} \to \infty` - :math:`\to \eta_{\infty}` - Second Newtonian plateau Power-Law Approximation ~~~~~~~~~~~~~~~~~~~~~~~ In the power-law region (:math:`\lambda\dot{\gamma} \gg 1`), ignoring :math:`\eta_{\infty}`: .. math:: \eta \approx \eta_0 \lambda^{-m} \dot{\gamma}^{-m} = K \dot{\gamma}^{n-1} where :math:`K = \eta_0 \lambda^{-m}` and :math:`n = 1 - m`. This connects Cross parameter :math:`m` to power-law index. ---- Parameters ---------- .. list-table:: Parameters :header-rows: 1 :widths: 15 12 12 18 43 * - Name - Symbol - Units - Bounds - Notes * - ``eta0`` - :math:`\eta_0` - Pa·s - :math:`10^{-3} - 10^{12}` - Zero-shear viscosity; first Newtonian plateau * - ``eta_inf`` - :math:`\eta_{\infty}` - Pa·s - :math:`10^{-6} - 10^{6}` - Infinite-shear viscosity; often solvent viscosity * - ``lambda_`` - :math:`\lambda` - s - :math:`10^{-6} - 10^{6}` - Time constant; :math:`1/\lambda` is transition shear rate * - ``m`` - :math:`m` - — - :math:`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 :math:`\phi` via Krieger-Dougherty - **For polymers**: Related to molecular weight via :math:`\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 :math:`\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 :math:`\lambda`) → early transition to thinning - **Relation**: :math:`\dot{\gamma}_{1/2} = 1/\lambda` where :math:`\eta = (\eta_0 + \eta_{\infty})/2` **m (Rate Constant)**: - **Physical meaning**: Steepness of the viscosity drop in transition region - **Connection to power law**: Approximately :math:`n = 1 - m` in mid-rate region - **Typical values**: 0.5-1.5 for most fluids ---- Validity and Assumptions ------------------------ Model Assumptions ~~~~~~~~~~~~~~~~~ 1. **Generalized Newtonian**: No memory effects, stress depends only on current :math:`\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 :math:`\eta(\dot{\gamma})` spanning at least 3 decades - **Ideal**: Data capturing both plateaus (may require wide :math:`\dot{\gamma}` range) - **For accurate** :math:`m`: Transition region well-resolved (5+ points) Limitations ~~~~~~~~~~~ **No viscoelasticity**: Cannot predict :math:`G'(\omega)`, :math:`G''(\omega)`, or stress relaxation. Use Maxwell/Oldroyd-B for elastic effects. **No yield stress**: Material always flows; :math:`\sigma \to 0` as :math:`\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. ---- 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** :math:`\eta_0` **(>100 Pa·s)**: Strong particle aggregation, high molecular weight polymer, or concentrated system with extensive network formation - **Moderate** :math:`\eta_0` **(1-100 Pa·s)**: Typical for polymer solutions, emulsions, and moderately concentrated suspensions - **Low** :math:`\eta_0` **(<1 Pa·s)**: Dilute solution, weak interparticle attractions, or low molecular weight *For graduate students*: For suspensions, the Krieger-Dougherty equation relates :math:`\eta_0` to volume fraction: :math:`\eta_0 / \eta_s = (1 - \phi/\phi_m)^{-[\eta]\phi_m}` where :math:`\eta_s` is solvent viscosity, :math:`\phi` is volume fraction, and :math:`\phi_m` is maximum packing. This enables volume fraction estimation from viscosity measurements. *For practitioners*: :math:`\eta_0` controls critical processing behaviors—settling/sedimentation rates in storage, coating thickness during low-shear application, and leveling behavior after deposition. Target higher :math:`\eta_0` for shelf stability and sag prevention. **eta_inf (Infinite-Shear Viscosity)**: The high-shear plateau reveals the fully disrupted microstructure: - **High ratio** :math:`\eta_{\infty}/\eta_0` **(>10%)**: Significant irreducible structure remains; strong hydrodynamic interactions even when fully aligned - **Low ratio** :math:`\eta_{\infty}/\eta_0` **(<1%)**: Nearly complete structural breakdown under flow; approaches solvent-like behavior *For graduate students*: For suspensions, :math:`\eta_{\infty}` approaches the Einstein limit :math:`\eta_s(1 + 2.5\phi)` when particles are fully dispersed and aligned. Deviations indicate residual aggregation or non-spherical particle effects. *For practitioners*: :math:`\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**: :math:`\dot{\gamma}_c = 1/\lambda` identifies where viscosity drops to halfway between plateaus - **Short** :math:`\lambda` **(<0.1 s)**: Fast structural response, suitable for high-speed operations - **Long** :math:`\lambda` **(>10 s)**: Slow structural relaxation, memory effects important *For graduate students*: For Brownian particles, :math:`\lambda \sim a^2/D_0` where :math:`a` is particle radius and :math:`D_0` is diffusion coefficient. For polymers, :math:`\lambda` scales with the longest relaxation time from chain dynamics. *For practitioners*: Compare :math:`\lambda` to process timescales. Operating at :math:`\dot{\gamma} \gg 1/\lambda` ensures material is in the thinned state; :math:`\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** :math:`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** :math:`m` **(0.6-1.2)**: Typical for most polymer solutions and suspensions with moderate polydispersity - **High** :math:`m` **(1.2-2.0)**: Sharp, switch-like transition—indicates narrow relaxation spectrum or cooperative structural breakdown *For graduate students*: The parameter :math:`m` relates to polydispersity and relaxation time distribution breadth. Compare with Cole-Cole analysis of oscillatory data: broad distributions give low :math:`m`, narrow distributions give high :math:`m`. *For practitioners*: High :math:`m` materials have excellent "smart fluid" behavior—thick when still, thin when worked. This is ideal for coatings (sag-resistant yet sprayable). Low :math:`m` gives smoother processing with less abrupt rheology changes. Material Classification ~~~~~~~~~~~~~~~~~~~~~~~ .. list-table:: Material Classification from Cross Parameters :header-rows: 1 :widths: 20 20 30 30 * - Parameter Pattern - Material Behavior - Typical Materials - Processing Implications * - Large :math:`\eta_0/\eta_{\infty}`, high :math:`m` - Strong cooperative structure - Concentrated latex paints, thick emulsions - Excellent sag resistance with spray-ability * - Large :math:`\eta_0/\eta_{\infty}`, low :math:`m` - Broad relaxation spectrum - Polydisperse suspensions, polymer blends - Smooth processing window, forgiving * - Moderate :math:`\eta_0/\eta_{\infty}`, moderate :math:`m` - Standard structured fluid - Typical coatings, food emulsions - Balanced processing characteristics * - Small :math:`\eta_0/\eta_{\infty}` (<10) - Weak or minimal structure - Dilute polymer solutions - Limited shear-thinning, consider simpler model ---- 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\ :math:`^{-1}`) for 60 s, then rest 5 min **Step 2**: Flow curve measurement - Shear rate sweep: :math:`10^{-3}` to :math:`10^{3}` s\ :math:`^{-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 ---- Fitting Guidance ---------------- Parameter Initialization ~~~~~~~~~~~~~~~~~~~~~~~~ **Step 1**: Estimate :math:`\eta_0` from lowest shear rates :math:`\eta_0 \approx` average of :math:`\eta` at :math:`\dot{\gamma} < 0.01/\lambda` **Step 2**: Estimate :math:`\eta_{\infty}` from highest shear rates :math:`\eta_{\infty} \approx` average of :math:`\eta` at :math:`\dot{\gamma} > 100/\lambda` **Step 3**: Find :math:`\lambda` from midpoint Where :math:`\eta = (\eta_0 + \eta_{\infty})/2`, :math:`\lambda = 1/\dot{\gamma}_{1/2}` **Step 4**: Estimate :math:`m` from log-log slope In power-law region: slope :math:`\approx -m` Optimization ~~~~~~~~~~~~ **RheoJAX default: NLSQ (GPU-accelerated)** - Fast convergence for 4-parameter Cross model - Bounds recommended to prevent unphysical values **Bounds**: - :math:`\eta_0`: [1e-2, 1e10] Pa·s - :math:`\eta_{\infty}`: [0, 0.9 × :math:`\eta_0`] Pa·s - :math:`\lambda`: [1e-6, 1e4] s - :math:`m`: [0.2, 2.0] Troubleshooting ~~~~~~~~~~~~~~~ .. list-table:: Fitting diagnostics :header-rows: 1 :widths: 25 35 40 * - Problem - Diagnostic - Solution * - :math:`m` hits bounds - Transition shape doesn't match - Check for artifacts; try Carreau-Yasuda * - :math:`\eta_{\infty}` negative - Bound violation - Constrain :math:`\eta_{\infty} \geq 0`; check high-rate data * - Poor fit at transition - Functional form mismatch - Try Carreau or Carreau-Yasuda * - Correlated :math:`\lambda` and :math:`m` - Under-resolved transition - More data points in transition region ---- Usage ----- Basic Example ~~~~~~~~~~~~~ .. code-block:: python 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 ~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: python 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}") ---- See Also -------- - :doc:`carreau` — uses square-law exponent; choose based on transition shape - :doc:`carreau_yasuda` — adds Yasuda exponent for even more flexibility - :doc:`power_law` — approximates Cross mid-rate slope when plateaus unavailable - :doc:`herschel_bulkley` — for yield stress fluids - :doc:`../../transforms/smooth_derivative` — differentiate flow curves to estimate :math:`m` ---- API References -------------- - Module: :mod:`rheojax.models` - Class: :class:`rheojax.models.Cross` ---- References ---------- .. [1] 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]