.. _model-carreau-yasuda: Carreau–Yasuda Model ==================== Quick Reference --------------- - **Use when:** Abrupt viscosity transitions, sharp changes between plateaus - **Parameters:** 5 (:math:`\eta_0`, :math:`\eta_\infty`, :math:`\lambda`, n, a) - **Key equation:** :math:`\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 :doc:`carreau` model by introducing an additional parameter :math:`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 :math:`a = 2` recovers the original Carreau model, while :math:`a < 2` produces sharper transitions and :math:`a > 2` produces more gradual ones. Notation Guide -------------- .. list-table:: :widths: 15 85 :header-rows: 1 * - Symbol - Meaning * - :math:`\eta` - Apparent viscosity (Pa·s) * - :math:`\eta_0` - Zero-shear viscosity (Pa·s). Low-rate Newtonian plateau. * - :math:`\eta_\infty` - Infinite-shear viscosity (Pa·s). High-rate solvent contribution. * - :math:`\dot{\gamma}` - Shear rate (s\ :sup:`-1`) * - :math:`\lambda` - Relaxation time (s). Inverse of critical shear rate. * - :math:`n` - Power-law index (dimensionless). High-shear slope. * - :math:`a` - Yasuda exponent (dimensionless). Transition sharpness. Physical Foundations -------------------- Microstructural Interpretation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The Yasuda exponent :math:`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 :math:`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 :math:`a > 2` due to the broadened spectrum. **Wormlike Micelles**: These self-assembled structures can break and reform under flow, creating sharp transitions with :math:`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 (:math:`a \approx 1-1.5`). Governing Equations ------------------- Constitutive Equation ~~~~~~~~~~~~~~~~~~~~~ .. math:: \eta(\dot{\gamma}) = \eta_{\infty} + (\eta_0 - \eta_{\infty}) \left[1 + (\lambda \dot{\gamma})^{a}\right]^{\frac{n-1}{a}} This form ensures: - At :math:`\dot{\gamma} \to 0`: :math:`\eta \to \eta_0` (zero-shear plateau) - At :math:`\dot{\gamma} \to \infty`: :math:`\eta \to \eta_\infty` (infinite-shear plateau) - In the power-law region: :math:`\eta \propto \dot{\gamma}^{n-1}` Limiting Cases ~~~~~~~~~~~~~~ **Low shear rate** (:math:`\lambda \dot{\gamma} \ll 1`): .. math:: \eta \approx \eta_0 - (\eta_0 - \eta_\infty) \frac{(n-1)}{a} (\lambda \dot{\gamma})^a **High shear rate** (:math:`\lambda \dot{\gamma} \gg 1`): .. math:: \eta \approx \eta_\infty + (\eta_0 - \eta_\infty) (\lambda \dot{\gamma})^{n-1} **Power-law approximation** (mid-range, :math:`\eta_\infty \approx 0`): .. math:: \eta \approx \eta_0 (\lambda)^{n-1} \dot{\gamma}^{n-1} = K \dot{\gamma}^{n-1} where the effective consistency index is :math:`K = \eta_0 \lambda^{n-1}`. Relation to Carreau Model ~~~~~~~~~~~~~~~~~~~~~~~~~ Setting :math:`a = 2`: .. math:: \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 ---------- .. list-table:: Parameter Summary :header-rows: 1 :widths: 15 15 15 55 * - Name - Symbol - Units - Description / Constraints * - ``eta0`` - :math:`\eta_0` - Pa·s - Zero-shear viscosity. Must be > 0 and typically ≥ ``eta_inf``. * - ``eta_inf`` - :math:`\eta_\infty` - Pa·s - Infinite-shear viscosity. Must be ≥ 0; often set to 0 when unmeasurable. * - ``lambda_`` - :math:`\lambda` - s - Relaxation time. Inverse of critical shear rate where thinning begins. * - ``n`` - :math:`n` - – - Power-law index. < 1 for thinning, = 1 for Newtonian, > 1 for thickening. * - ``a`` - :math:`a` - – - Yasuda exponent. Controls transition sharpness; = 2 gives Carreau model. Parameter Bounds ~~~~~~~~~~~~~~~~ .. list-table:: Default Bounds :header-rows: 1 :widths: 20 30 50 * - Parameter - Bounds - Physical Rationale * - :math:`\eta_0` - (1e-3, 1e12) - Must exceed solvent viscosity * - :math:`\eta_\infty` - (1e-6, 1e6) - Cannot exceed :math:`\eta_0`; often ~solvent viscosity * - :math:`\lambda` - (1e-6, 1e6) - Must capture transition in measured range * - :math:`n` - (0.01, 1.0) - <0.01 unphysical; >1 rare (shear thickening) * - :math:`a` - (0.1, 2.0) - <0.1 too sharp (numerical issues); >2 nearly Newtonian transition Material Behavior Guide ----------------------- .. list-table:: Typical Parameter Ranges :widths: 25 12 12 12 12 27 :header-rows: 1 * - Material Class - :math:`\eta_0` (Pa·s) - :math:`\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 (:math:`a=2`) provides poor fits to the transition region. 3. **No yield stress**: The material flows freely at all stresses (no intercept at :math:`\dot{\gamma}=0`). 4. **Steady-state flow**: Time-independent response (no thixotropy or viscoelastic overshoot). When to Use Alternatives ~~~~~~~~~~~~~~~~~~~~~~~~ .. list-table:: Model Selection Guide :widths: 35 30 35 :header-rows: 1 * - Observation - Issue - Better Model * - Transition fits well with :math:`a \approx 2` - Carreau-Yasuda overparameterized - :doc:`carreau` (4 parameters) * - No visible zero-shear plateau - :math:`\eta_0` unconstrained - :doc:`power_law` or :doc:`cross` * - Stress intercept at zero rate - Material has yield stress - :doc:`herschel_bulkley` * - Fitted :math:`a < 0.5` - Approaching step-function (unphysical) - Check data; consider yield stress model * - Strong parameter correlations - Data don't resolve all 5 parameters - :doc:`carreau` or reduce :math:`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 :math:`\beta` in KWW relaxation. Lower :math:`a` corresponds to more exponential (single-mode) relaxation; higher :math:`a` corresponds to stretched relaxation. *For practitioners*: Sharp transitions (low :math:`a`) can cause processing instabilities. If :math:`a < 1.5`, consider whether sudden viscosity drops might cause flow instabilities or poor coating uniformity. **Relaxation Time (** :math:`\lambda` **)**: The relaxation time identifies the critical shear rate for structural response: - **Critical shear rate**: :math:`\dot{\gamma}_c = 1/\lambda` marks where viscosity begins significant departure from :math:`\eta_0`. - **Weissenberg number**: At :math:`Wi = \lambda \dot{\gamma} = 1`, elastic and viscous timescales balance. *For graduate students*: For entangled polymers, :math:`\lambda` scales with the terminal relaxation time :math:`\tau_d`, which in turn scales as :math:`\tau_d \propto M_w^{3.4}/c^{1.5}` (reptation theory). *For practitioners*: Compare :math:`\lambda` to process timescales. Coating at 100 s\ :math:`^{-1}` with :math:`\lambda = 0.1` s gives :math:`Wi = 10`—firmly in the power-law regime with good leveling. **Viscosity Ratio (** :math:`\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 ~~~~~~~~~~~~~~~~~~~~~~~ .. list-table:: Material Classification from Carreau-Yasuda Parameters :header-rows: 1 :widths: 25 25 25 25 * - Parameter Pattern - Material Behavior - Typical Materials - Processing Implications * - Low a, high :math:`\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 :math:`\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 :math:`a` has uncertainty >50%, data don't constrain it—use fixed :math:`a = 2` (Carreau). Diagnostic Indicators ~~~~~~~~~~~~~~~~~~~~~ Warning signs in fitted parameters: - **a approaching bounds**: If :math:`a < 0.5` or :math:`a > 4`, the model may be compensating for other issues (yield stress, data artifacts). - :math:`\lambda` **at measurement bounds**: If :math:`\lambda` equals 1/(max shear rate) or 1/(min shear rate), the transition is outside your measurement window. - **Strong a-** :math:`\lambda` **correlation**: These parameters are inherently correlated. Consider fixing one based on literature or prior measurements. - :math:`\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\ :math:`^{-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 ~~~~~~~~~~~~~~~~~~ .. list-table:: Recommended Geometries :header-rows: 1 :widths: 25 25 50 * - Shear Rate Range - Geometry - Notes * - 0.001–100 s\ :math:`^{-1}` - Cone-plate (1–2°) - Uniform shear rate; best for low rates * - 0.1–1000 s\ :math:`^{-1}` - Parallel plate - Adjustable gap; good for moderate rates * - 10–10,000 s\ :math:`^{-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**: - :math:`\eta_0` = average viscosity at lowest 3 shear rates - :math:`\eta_\infty` = average viscosity at highest 3 shear rates (or 0) 2. **From transition**: - Find :math:`\dot{\gamma}_{1/2}` where :math:`\eta = (\eta_0 + \eta_\infty)/2` - Initialize :math:`\lambda = 1/\dot{\gamma}_{1/2}` 3. **From slope**: - :math:`n` = 1 + slope of log-log plot at high rates 4. **Default for a**: - Start with :math:`a = 2` (Carreau default) Optimization Strategy ~~~~~~~~~~~~~~~~~~~~~ Two-stage fitting often works best: **Stage 1**: Fix :math:`a = 2` and fit other 4 parameters (Carreau fit). **Stage 2**: Release :math:`a` and refine all 5 parameters from Stage 1 solution. .. code-block:: python 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 ~~~~~~~~~~~~~~~ .. list-table:: Common Issues :widths: 25 35 40 :header-rows: 1 * - 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 * - :math:`\lambda` poorly constrained - Transition outside data range - Extend shear rate range * - :math:`\eta_\infty` negative - Optimization artifact - Constrain :math:`\eta_{\infty}` ≥ 0; check high-rate data * - Strong a-:math:`\lambda` correlation - Insufficient transition data - Fix a = 2 or increase mid-range points Usage ----- Basic Fitting ~~~~~~~~~~~~~ .. code-block:: python 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 ~~~~~~~~~~~~~~~~~ .. code-block:: python 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 ~~~~~~~~~~~~~~~~~~ .. code-block:: python 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: .. code-block:: python 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 :math:`(n-1)/a` is bounded to prevent overflow when :math:`a` is very small. 2. **Log-space fitting**: Internal optimization uses :math:`\log(\eta_0)`, :math:`\log(\eta_\infty)`, :math:`\log(\lambda)` for numerical stability. 3. **Gradient clipping**: JAX gradients are clipped to prevent NaN propagation See Also -------- Related Flow Models ~~~~~~~~~~~~~~~~~~~ - :doc:`carreau` — Special case with :math:`a = 2`; preferred when simpler model suffices - :doc:`cross` — Alternative sigmoidal model with denominator exponent - :doc:`power_law` — Simple power-law for mid-range rates only - :doc:`herschel_bulkley` — For materials with yield stress Transforms ~~~~~~~~~~ - :doc:`../../transforms/mastercurve` — Time-temperature superposition - :doc:`../../transforms/srfs` — Strain-rate frequency superposition API Reference ~~~~~~~~~~~~~ - :class:`rheojax.models.CarreauYasuda` - :class:`rheojax.models.Carreau` References ---------- .. 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