First Normal Stress Difference, Recoverable Strain & Polymer Melt Elasticity

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).

• 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.

3.1 Normal Stress Differences in Shear Flow

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

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:

$$N_1 = \frac{2F}{\pi R^2}$$

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

3.2 First Normal Stress Coefficient Ψ₁

$$N_1 = \Psi_1(\dot{\gamma}) \cdot \dot{\gamma}^2$$

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

For Lodge Rubber-Like Liquid:

$$N_1 = 2 \eta \lambda \dot{\gamma}^2$$

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):

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

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

Practical die swell values:

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

$$Wi = \lambda \dot{\gamma} \quad \text{(shear flow)} \qquad De = \frac{\lambda}{t_{process}}$$ <div className="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:

$$N_1 = 2 \eta \lambda \dot{\gamma}^2 = 2 \times 4000 \times 0.8 \times (200)^2 = 2 \times 4000 \times 0.8 \times 40000 = \textbf{256 \text{ MPa}}$$

Step 2: Wall shear stress:

$$\tau_{wall} = \eta \dot{\gamma} = 4000 \times 200 = 800,000 \text{ Pa} = 0.8 \text{ MPa}$$

Step 3: Die swell:

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

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.

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.

• 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.

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)

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?

Q1. N₁ is measured by: B) Cone-and-plate rheometer — normal force Q2. Die swell (Barus effect) is caused by: C) Elastic recovery of stored normal stress energy on exiting the die Q3. Weissenberg number Wi > 1 indicates: B) Elastic effects dominate over viscous flow Q4. Which polymer has the highest die swell ratio? B) LDPE (broad MWD, high long-chain branching) Q5. First normal stress coefficient Ψ₁ is defined as: A) N₁ / ṁ²