Polymer RheologyL01 · First Normal Stress Difference, Recoverable Strain & Polymer Melt Elasticity
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Processing & ManufacturingLesson 0119 PPE Syllabus Aligned

First Normal Stress Difference, Recoverable Strain & Polymer Melt Elasticity

Melt elasticity, First Normal Stress Difference N1, Weissenberg effect (rod climbing), extrudate die swell, and recoverable shear strain.

~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

First Normal Stress Difference, Recoverable Strain & Polymer Melt Elasticity

1. Why This Topic Matters

Polymer melts are viscoelastic — they simultaneously flow (viscous) and store elastic energy. The First Normal Stress Difference (N₁) is the most important measure of melt elasticity and controls die swell, sharkskin melt fracture, extrudate distortion, and draw resonance in film blowing. Process engineers at RIL (Jamnagar blown film lines), Cosmo Films (BOPP), and UFlex cannot design stable film blowing or profile extrusion without understanding N₁, the Weissenberg number (Wi), and the Deborah number (De).

2. Learning Objectives

  • Define and calculate the First Normal Stress Difference N₁ from cone-and-plate rheometer data.
  • Relate N₁ to die swell and extrudate distortion in practical extrusion.
  • Calculate the Weissenberg number (Wi) and interpret its physical meaning.
  • Apply the Lodge Rubber-Like Liquid (RLL) model to predict N₁ from viscosity data.
  • Identify ISO 6721-10 (oscillatory rheology) and ASTM D4440 standards for melt elasticity.

3. Core Theory

Technical Schematic & Mechanism Blueprint: First Normal Stress Difference, Recoverable Strain & Polymer Melt Elasticity
Technical Schematic & Mechanism Blueprint: First Normal Stress Difference, Recoverable Strain & Polymer Melt Elasticity

First Normal Stress Difference, Recoverable Strain & Polymer Melt Elasticity — 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 Normal Stress Differences in Shear Flow

In simple shear flow of a polymer melt, the stress tensor has both shear and normal components:

Stress ComponentSymbolDefinition
Shear stressτ₁₂Stress driving flow
First normal stress differenceN₁ = σ₁₁ − σ₂₂Flow direction − gradient direction
Second normal stress differenceN₂ = σ₂₂ − σ₃₃Gradient − neutral direction

For polymer melts: N₁ >> N₂ (N₁ > 0, N₂ < 0, |N₂| ≈ 0.1–0.3 N₁)

Measurement: N₁ is measured by cone-and-plate rheometer via the normal force pushing the plates apart:

N1=2FπR2N_1 = \frac{2F}{\pi R^2}

Where: F = normal force (N), R = cone radius (m).

3.2 First Normal Stress Coefficient Ψ₁

N1=Ψ1(γ˙)⋅γ˙2N_1 = \Psi_1(\dot{\gamma}) \cdot \dot{\gamma}^2

Where Ψ₁ = First normal stress coefficient (Pa·s²). Like viscosity η, Ψ₁ decreases with shear rate (shear-thinning).

For Lodge Rubber-Like Liquid:

N1=2ηλγ˙2N_1 = 2 \eta \lambda \dot{\gamma}^2

Where: η = viscosity (Pa·s), λ = longest relaxation time (s), ṁ = shear rate (s⁻¹).

3.3 Die Swell — The Physical Consequence of N₁

When a polymer melt exits a die, elastic energy stored in the flow (characterised by N₁) causes the extrudate to swell — die swell (Barus effect):

B=DextrudateDdie≈1+(N14τwall)2B = \frac{D_{extrudate}}{D_{die}} \approx 1 + \left(\frac{N_1}{4\tau_{wall}}\right)^2

Where: B = die swell ratio, τ_wall = wall shear stress.

Practical die swell values:

PolymerDie swell ratio B
LLDPE (narrow MWD)1.1–1.2
LDPE (broad MWD, high N₁)1.5–2.5
HDPE1.2–1.6
PP1.1–1.3

3.4 Weissenberg Number (Wi) and Deborah Number (De)

Wi=λγ˙(shear flow)De=λtprocessWi = \lambda \dot{\gamma} \quad \text{(shear flow)} \qquad De = \frac{\lambda}{t_{process}}
NumberValuePhysical Regime
Wi < 1Flow dominates elasticityNear-Newtonian, small die swell
Wi > 1Elasticity dominatesLarge N₁, die swell, melt fracture risk
De > 1Elastic deformation faster than process timeSolid-like response — cracking risk

4. Worked Example

First Normal Stress Difference, Recoverable Strain & Polymer Melt Elasticity — 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: A LDPE melt has η = 4000 Pa·s and longest relaxation time λ = 0.8 s at die wall shear rate = 200 s⁻¹. Calculate N₁ and die swell ratio B.

Step 1: N₁ from Lodge RLL model:

N1=2ηλγ˙2=2×4000×0.8×(200)2=2×4000×0.8×40000=256 MPaN_1 = 2 \eta \lambda \dot{\gamma}^2 = 2 \times 4000 \times 0.8 \times (200)^2 = 2 \times 4000 \times 0.8 \times 40000 = \mathbf{256 \text{ MPa}}

Step 2: Wall shear stress:

τwall=ηγ˙=4000×200=800,000 Pa=0.8 MPa\tau_{wall} = \eta \dot{\gamma} = 4000 \times 200 = 800,000 \text{ Pa} = 0.8 \text{ MPa}

Step 3: Die swell:

B=1+(N14τwall)2=1+(2564×0.8)2B = 1 + \left(\frac{N_1}{4 \tau_{wall}}\right)^2 = 1 + \left(\frac{256}{4 \times 0.8}\right)^2 =1+(80)2≈very large — model approximation breaks down at high Wi= 1 + (80)^2 \approx \text{very large — model approximation breaks down at high Wi}

Wi check: Wi = λṁ = 0.8 × 200 = 160 >> 1 — highly elastic regime. This confirms LDPE at high shear rates experiences very large die swell (experimentally B = 2.0–2.5 for LDPE), consistent with LDPE's known high N₁ and broad MWD.

5. Indian Industry Context

First Normal Stress Difference, Recoverable Strain & Polymer Melt Elasticity — CAD Tooling & Machine Geometry Blueprint

DWG NO: DWG-POLY-2026-240SCALE: 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.

Reliance Industries Ltd (Jamnagar) blown film lines for LLDPE and LDPE film are engineered to manage die swell precisely. LLDPE (narrow MWD, low N₁, low die swell) requires neck-in management in MDO; LDPE (broad MWD, high N₁, high die swell) stabilises the bubble naturally. Blending 10–20% LDPE into LLDPE is a common formulation approach used by RIL film processors to improve bubble stability via controlled N₁ enhancement.

Cosmo Films (Aurangabad) extruder die designers account for BOPP melt die swell (B ≈ 1.1–1.15 for PP) when calculating die gap width vs. final film width in their sequential biaxial orientation lines.

6. Key Takeaways & Glossary

  • N₁ (First Normal Stress Difference): Measure of melt elasticity; drives die swell and melt fracture.
  • Ψ₁ (First Normal Stress Coefficient): N₁/ṁ² — analogue of viscosity for elastic component.
  • Die swell (Barus effect): Elastic recovery of stored energy as melt exits die — B = D_extrudate/D_die.
  • Wi (Weissenberg number): λṁ — ratio of elastic to viscous forces; Wi > 1 → elastic-dominant.
  • Lodge RLL model: N₁ = 2ηλṁ² — simplest constitutive model relating viscosity and relaxation time to N₁.
  • Melt fracture: Flow instability at Wi >> 1 — sharkskin, slip-stick, wavy distortion.

7. Standards Reference

  1. ISO 6721-10 — Complex shear viscosity using parallel-plate oscillatory rheometer
  2. ASTM D4440 — Plastics — Determination of dynamic mechanical properties using oscillatory shear
  3. ISO 11443:2021 — Capillary rheometry of polymer melts
  4. ASTM D5422 — Measurement of extrudate swell (die swell)

8. Practice Questions

  1. Explain why LDPE has higher die swell than LLDPE despite similar MFI. What molecular structural feature causes this?
  2. A melt has λ = 0.5 s and experiences die wall shear rate = 150 s⁻¹. Calculate Wi and state whether elastic effects will dominate.
  3. How does broadening the molecular weight distribution of a polymer affect N₁ and die swell?

First Normal Stress Difference, Recoverable Strain & Polymer Melt Elasticity · 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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