Polymer RheologyL01 · Non-Newtonian Rheology: Power-Law & Carreau-Yasuda Models for Polymer Melts
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Processing & ManufacturingLesson 0119 PPE Syllabus Aligned

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

Polymer melt rheology, shear-thinning pseudoplasticity, Ostwald-de Waele Power-Law model, Carreau-Yasuda model, zero-shear viscosity, and power-law index n.

~35 min technical deep-dive·Standard Indian Curricula (CIPET / Anna Univ / ICT)

01 · Why This Matters in Industry & GATE XE5

Applied directly across petrochemical refining, compounding plants, mold-flow simulations, and automotive part manufacturing (e.g., Reliance Industries, Supreme Petrochem, IOCL, CIPET testing protocols).

1

Molecular Mechanism: Master conformational physics, transition temperatures, and reaction kinetics.

2

Process & Quality: Predict viscosity behavior, solve moulding defects, and apply ASTM/ISO testing standards.

02 · Technical Theory & Governing Equations

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

1. Why This Topic Matters

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.

2. Learning Objectives

  • 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. Core Theory

Technical Schematic & Mechanism Blueprint: Non-Newtonian Rheology: Power-Law & Carreau-Yasuda Models for Polymer Melts
Technical Schematic & Mechanism Blueprint: Non-Newtonian Rheology: Power-Law & Carreau-Yasuda Models for Polymer Melts

Non-Newtonian Rheology: Power-Law & Carreau-Yasuda Models for Polymer Melts — Mechanism & Physics Diagram
SHEAR RATE (γ̇, s⁻¹) →VISCOSITY (η, Pa·s) →Zero-Shear Viscosity (η_0)Power-Law Region (η = K γ̇¹⁻&supn;)Infinite Shear Limit (η_∞)
Ostwald-de Waele Shear Thinning: Chain disentanglement under high shear rates (γ̇ > 100 s⁻¹) reduces apparent viscosity η, critical for high-speed injection gate filling and extrusion dies.

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:

η=K⋅γ˙n−1\eta = K \cdot \dot{\gamma}^{n-1}

Or equivalently:

τ=K⋅γ˙n\tau = K \cdot \dot{\gamma}^n
ParameterSymbolUnitsPhysical Meaning
Flow consistency indexKPa·sⁿViscosity at unit shear rate
Power-law flow indexndimensionlessn<1: shear-thinning; n=1: Newtonian; n>1: shear-thickening
Apparent viscosityηPa·sShear-rate dependent viscosity

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:

η(γ˙)=η∞+(η0−η∞)[1+(λγ˙)a](n−1)/a\eta(\dot{\gamma}) = \eta_\infty + (\eta_0 - \eta_\infty)\left[1 + (\lambda \dot{\gamma})^a\right]^{(n-1)/a}

For polymers, η_∞ ≈ 0, so:

η(γ˙)=η0[1+(λγ˙)a](n−1)/a\eta(\dot{\gamma}) = \eta_0 \left[1 + (\lambda \dot{\gamma})^a\right]^{(n-1)/a}
ParameterPhysical MeaningTypical Value (HDPE 200°C)
η₀Zero-shear viscosity (Newtonian plateau)50,000–200,000 Pa·s
λRelaxation time — onset of shear-thinning0.01–1 s
nPower-law slope in shear-thinning region0.3–0.5
aTransition breadth (Yasuda parameter)0.5–2

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

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

ΔP=2KLR(3n+14n)n(QπR3)n\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.

4. Worked Example

Non-Newtonian Rheology: Power-Law & Carreau-Yasuda Models for Polymer Melts — Industrial Process Flow Diagram (PFD)

ISO 10628 Unit Operations & Thermodynamic Control Sequence

ISO 10628 PFD Standard
180°C - 100°C
Raw Material Feed & Metering

Drying to < 0.02% moisture and gravimetric dosing

Output:Conditioned Feed
2220°C - 260°C
Reaction & Polymer Melt Processing

Controlled thermal shear dissipation and phase transition

Output:Homogenous Melt
360 - 120 bar
Shaping & Solidification

Die extrusion or mold cavity pressure packing

Output:Formed Component
4QA Passed
Quality Inspection & Testing

ASTM D638 / ISO 527 tensile & MFI compliance check

Output:Finished Grade
Recycle Loop Stream:Closed-loop sprues and edge-trim regrind re-fed into main hopper at maximum 15% blend ratio.
Standard Indian Industrial Practice (CIPET / Reliance / Supreme Petrochem Protocol)
Engineering Note & Problem Analysis

Problem Statement:

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:

γ˙app=4QπR3=4×5×10−7π×(1.5×10−3)3=2×10−61.061×10−8=188 s−1\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:

γ˙true=3n+14n×γ˙app=3(0.35)+14(0.35)×188=2.051.4×188=275 s−1\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:

τw=K⋅γ˙truen=8000×(275)0.35=8000×7.83=62,640 Pa\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:

ΔP=2LτwR=2×0.05×62,6401.5×10−3=4.18 MPa\Delta P = \frac{2 L \tau_w}{R} = \frac{2 \times 0.05 \times 62,640}{1.5 \times 10^{-3}} = \mathbf{4.18 \text{ MPa}}

5. Indian Industry Context

Non-Newtonian Rheology: Power-Law & Carreau-Yasuda Models for Polymer Melts — CAD Tooling & Machine Geometry Blueprint

DWG NO: DWG-POLY-2026-932SCALE: 1:1 (FULL)MATERIAL: P20 Tool Steel / Hardened Alloy (HRC 48-52)
CL MOULD MOUNTINGCAVITY 1CAVITY 2SPRUE BUSHING (Ø6.5mm)WATER INLET (Ø10mm)160.00 ± 0.02 mm
Critical Geometric Dimensions & Tolerances (GD&T)
  • Primary Processing Tolerance:± 0.02 mm (ISO 20457 Class M1)
  • Thermal Expansion Allowance:1.5% (Volumetric Shrinkage)
  • Draft Angle Minimum:1.0° - 1.5° (Core/Cavity Ejection)
Tooling Fabrication Notes
  • • Fabricated according to ISO 20457 plastic molding tolerance standards.
  • • EDM finish SPI C-1 applied across cavity optical contact faces.
  • • Conformal cooling lines pressure tested at 12 bar water flow.

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.

6. Key Takeaways & Glossary

  • 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.

7. Standards Reference

  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

8. GATE / University Practice Questions

  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?

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

Material Synthesis · Processing Hardware · Commercial Application

ASTM / ISO Aligned
1. MaterialResin / Chemistry

Standard Engineering Thermoplastic Resin

—[Monomer Backbone]ₙ— (Calibrated Molecular Weight & PDI)

Specific Gravity:1.05–1.42 g/cm³
Glass Transition (Tg):100–160 °C
Tensile Yield Strength:45–85 MPa
Melt Flow Index:5–25 g/10min
Morphology: Engineered Polymer Morphology (Amorphous / Semi-crystalline Matrix)
2. Machine & MouldShop Floor

Industrial Polymer Processing & Tooling System

Computer-Controlled Extrusion / Injection Moulding Hardware

Thermal Zones:180–280 °C (PID Controlled)
Injection / Melt Pressure:60–140 MPa
Cycle Time:15–45 seconds
Tooling Temperature:40–90 °C (Chiller Regulated)
Tooling: Hardened Tool Steel (H13/P20) Precision Cavity & Runner Layout
3. Real ProductApplication

Commercial Engineering Parts & Quality-Inspected Components

Automotive, Electrical, Medical & Packaging Applications

Standard:ASTM D3641 / ISO 294 / BIS Standard Compliance
Resin Grades: Reliance, SABIC, BASF, Covestro Standard Engineering Resins
Section 05 · Knowledge Check

Test Your Conceptual Understanding

In polymer science and processing thermodynamics, which factor most directly controls the critical transition temperature?

Select the correct option to verifyTake Complete Topic Assessment →
Summary Cheat Sheet & GATE Takeaways
  • Always evaluate molecular weight distribution (MWD) alongside zero-shear viscosity when calculating mold shear rates.
  • Differential Scanning Calorimetry (DSC) provides $T_g$, $T_c$, and $T_m$ to define optimal processing temperatures.
  • Comply with ASTM D638 / ISO 527 tensile specimen sizing to prevent premature necking artifacts.
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