Non-Newtonian Rheology: Power-Law & Carreau-Yasuda Models for Polymer Melts

Polymer melts are non-Newtonian — their viscosity drops dramatically with shear rate (shear-thinning). This behaviour controls how melts flow through extruder dies, injection mould runners, and film blowing dies. Process engineers at Supreme Industries, Reliance Industries (RIL), and Berry Global use power-law and Carreau models to simulate melt flow in Moldflow, Moldex3D, and Ansys Polyflow. Selecting the wrong viscosity model leads to short shots, burn marks, excessive pressure drops, and warpage.

• Explain shear-thinning behaviour and its molecular origin in entangled polymer chains.
• Apply the Power-Law (Ostwald–de Waele) model to calculate apparent viscosity and pressure drop.
• Use the Carreau-Yasuda model and identify its parameters (η₀, λ, n, a).
• Calculate the Power-Law flow consistency index K and flow index n from capillary rheometer data.
• Identify ISO 11443 (capillary rheometry) as the primary standard for polymer melt viscosity.

3.1 Shear-Thinning — The Molecular Origin

At rest, entangled polymer chains form a random coil network with high viscosity. Under shear:

1. Chains align in the flow direction (disentanglement).
2. Coil dimensions collapse (reduced drag)
3. Chain entanglement density drops — viscosity decreases.

This is shear-thinning (pseudoplastic) behaviour — the dominant rheological characteristic of all high-MW thermoplastic melts.

3.2 Power-Law (Ostwald–de Waele) Model

The simplest non-Newtonian viscosity model:

$$\eta = K \cdot \dot{\gamma}^{n-1}$$

Or equivalently:

$$\tau = K \cdot \dot{\gamma}^n$$

For a typical HDPE: K ≈ 18,000 Pa·sⁿ, n ≈ 0.38 at 200°C For PP: K ≈ 8,000 Pa·sⁿ, n ≈ 0.35 at 230°C

Limitation of Power-Law: Predicts infinite viscosity at zero shear rate (unphysical). Cannot capture the Newtonian plateau at low shear rates.

3.3 Carreau-Yasuda Model

A more complete model capturing both Newtonian plateaus:

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

For polymers, η_∞ ≈ 0, so:

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

3.4 Pressure Drop in a Tube (Power-Law Fluid)

For a power-law fluid in a cylindrical die (Hagen-Poiseuille modified):

$$\Delta P = \frac{2KL}{R} \left(\frac{3n+1}{4n}\right)^n \left(\frac{Q}{\pi R^3}\right)^n$$

Where: L = die length, R = die radius, Q = volumetric flow rate.

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Problem: PP melt at 230°C has power-law parameters K = 8000 Pa·sⁿ, n = 0.35. A capillary die has L = 50 mm, R = 1.5 mm. Volumetric flow rate Q = 5 × 10⁻⁷ m³/s. Calculate the pressure drop.

Step 1: Apparent shear rate at die wall:

$$\dot{\gamma}_{app} = \frac{4Q}{\pi R^3} = \frac{4 \times 5 \times 10^{-7}}{\pi \times (1.5 \times 10^{-3})^3} = \frac{2 \times 10^{-6}}{1.061 \times 10^{-8}} = 188 \text{ s}^{-1}$$

Step 2: Corrected (Rabinowitsch) shear rate:

$$\dot{\gamma}_{true} = \frac{3n+1}{4n} \times \dot{\gamma}_{app} = \frac{3(0.35)+1}{4(0.35)} \times 188 = \frac{2.05}{1.4} \times 188 = 275 \text{ s}^{-1}$$

Step 3: Wall shear stress:

$$\tau_w = K \cdot \dot{\gamma}_{true}^n = 8000 \times (275)^{0.35} = 8000 \times 7.83 = 62,640 \text{ Pa}$$

Step 4: Pressure drop:

$$\Delta P = \frac{2 L \tau_w}{R} = \frac{2 \times 0.05 \times 62,640}{1.5 \times 10^{-3}} = \textbf{4.18 \text{ MPa}}$$

Supreme Industries (Mumbai) runs twin-screw compounding lines at 350–450 kg/h throughput. Their process engineers use Power-Law models to estimate screw pressure profiles and motor torque requirements for 30% talc-filled PP compounds — ensuring the gearbox is not overloaded at production line speeds.

Reliance Industries Jamnagar uses Carreau-Yasuda viscosity models in their proprietary die design software for HDPE blown film dies — the low-shear Newtonian plateau (η₀) is critical for predicting bubble stability at the frost line.

• Shear-thinning (pseudoplastic): Viscosity decreases with increasing shear rate — universal for high-MW polymers.
• Power-law index n: n < 1 = shear-thinning; n = 1 = Newtonian; n > 1 = shear-thickening (dilatant).
• K (consistency index): Viscosity magnitude at unit shear rate — higher K = more viscous melt.
• η₀ (zero-shear viscosity): Newtonian plateau — sensitive to molecular weight (η₀ ∝ MW³·⁴).
• Rabinowitsch correction: Converts apparent capillary shear rate to true wall shear rate for non-Newtonian fluids.
• ISO 11443: Standard for capillary rheometry of polymer melts.

1. ISO 11443:2021 — Determination of the fluidity of plastics using capillary and slit-die rheometers
2. ASTM D3835 — Determination of properties of polymeric materials by means of a capillary rheometer
3. ISO 6721-10 — Complex shear viscosity using a parallel-plate oscillatory rheometer
4. ASTM D4440 — Plastics — Determination of dynamic mechanical properties using oscillatory shear

1. HDPE at 220°C: K = 15,000 Pa·sⁿ, n = 0.40. Calculate apparent viscosity at shear rate 500 s⁻¹.
2. Explain why the Power-Law model is inadequate at very low shear rates for polymer melts.
3. What does a higher power-law index n (closer to 1.0) indicate about the degree of shear-thinning?

Q1. For a power-law fluid with n < 1, the material is:

• A) Shear-thinning (pseudoplastic)

Q2. The power-law consistency index K has units of:

• B) Pa·sⁿ

Q3. The Carreau-Yasuda model improves on the power-law by:

• C) Capturing both Newtonian plateaus at low and high shear rates

Q4. Zero-shear viscosity η₀ is particularly sensitive to:

• B) Molecular weight (η₀ ∝ MW³·⁴)

Q5. ISO 11443 governs:

• A) Capillary rheometry of polymer melts