Shear-Thinning and the Power Law Model

Polymer melts are non-Newtonian fluids that display shear-thinning (pseudoplastic) behavior: their viscosity decreases as the shear rate increases. This property is crucial for processing: it allows high-velocity injection moulding and extrusion without requiring high pressures. Sizing runners, dies, and gating layouts requires calculating viscosity drops using mathematical flow models. Extrusion and moulding engineers use these models to optimize throughput and prevent melt degradation.

• Explain the molecular origin of shear-thinning (polymer chain entanglement kinetics).
• Formulate the Ostwald-de Waele Power Law model for shear stress and viscosity.
• Solve shear stress and apparent viscosity calculations for a power-law fluid.
• Distinguish Newtonian, shear-thinning (pseudoplastic), and shear-thickening (dilatant) fluids.
• Reference rheological testing standards (ASTM D3835).

3.1 Molecular Origin of Shear-Thinning

At rest or low shear rates, polymer chains are highly entangled, forming a dense network that resists flow (high zero-shear viscosity $\eta_0$). As the shear rate increases:

1. The rate of shear deformation exceeds the rate of thermal molecular relaxation.
2. Polymer chains untangle and align parallel to the flow direction.
3. The hydrodynamic drag decreases, causing a reduction in apparent viscosity (shear-thinning).

3.2 The Power Law (Ostwald-de Waele) Model

The relationship between shear stress ($\tau$) and shear rate ($\dot{\gamma}$) is:

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

The apparent viscosity ($\eta$) is:

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

Where:

• $K$: Flow consistency index (Pa·s$^n$)
• $n$: Flow behavior index (dimensionless):
  • $n < 1$: Shear-thinning (pseudoplastic) — typical for polymer melts (typically $0.2 - 0.6$).
  • $n = 1$: Newtonian (viscosity is constant, $\eta = K$).
  • $n > 1$: Shear-thickening (dilatant).

3.3 Limitations of the Power Law Model

The power law model predicts that as $\dot{\gamma} \rightarrow 0$, apparent viscosity $\eta \rightarrow \infty$, which is physically incorrect. Actual polymer melts show a constant zero-shear viscosity ($\eta_0$) plateau at low shear rates. The Cross or Carreau models must be used if low-shear ranges are critical.

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Problem: A Polypropylene (PP) melt at 230°C is evaluated on a capillary rheometer. The flow consistency index is $K = 8500$ Pa·s$^n$, and the flow behavior index is $n = 0.35$. Calculate:

1. The apparent viscosity ($\eta_1$) at a low processing shear rate $\dot{\gamma}_1 = 10$ s$^{-1}$ (typical for profile extrusion).
2. The apparent viscosity ($\eta_2$) at a high processing shear rate $\dot{\gamma}_2 = 1000$ s$^{-1}$ (typical for injection moulding).
3. The percentage viscosity reduction achieved by increasing the shear rate.

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Solution:

1. Calculate apparent viscosity $\eta_1$ at $\dot{\gamma}_1 = 10$ s$^{-1}$:

$$\eta_1 = K \cdot \dot{\gamma}^{n-1} = 8500 \times (10)^{0.35 - 1} = 8500 \times (10)^{-0.65}$$ $$\eta_1 = 8500 \times 0.22387 = \textbf{1902.9 Pa·s}$$

1. Calculate apparent viscosity $\eta_2$ at $\dot{\gamma}_2 = 1000$ s$^{-1}$:

$$\eta_2 = 8500 \times (1000)^{-0.65} = 8500 \times 0.01122 = \textbf{95.37 Pa·s}$$

1. Calculate the percentage viscosity reduction:

$$\text{Viscosity Drop (\%)} = \frac{1902.9 - 95.37}{1902.9} \times 100\% = \textbf{94.99\%}$$

Interpretation: Increasing the processing shear rate from 10 s⁻¹ to 1000 s⁻¹ reduces PP apparent viscosity by 95% (from 1903 Pa·s to 95.4 Pa·s). This demonstrates the extreme shear-thinning behavior of PP melts. Injection mould designers take advantage of this by using high injection speeds to lower viscosity, allowing thin walls to fill easily.

Raw material producers like Reliance Industries publish rheological curves for LLDPE film grades (e.g., Relene LLDPE) showing power-law indices to help blown film processors calculate back-pressure and melt temperature build-up in their extruder dies.

• Shear-Thinning: Rheological transition where fluid viscosity decreases with increasing shear rate.
• Flow Behavior Index ($n$): Dimensionless number indicating deviation from Newtonian flow ($n < 1$ for shear-thinning).
• Flow Consistency Index ($K$): Measure of fluid thickness; equals viscosity at a shear rate of 1 s⁻¹.
• Pseudoplastic: Term synonymous with shear-thinning.
• Zero-Shear Viscosity ($\eta_0$): Constant viscosity plateau reached at the limit of zero shear rate.

1. ASTM D3835 — Standard Test Method for Determination of Properties of Polymeric Materials by Means of a Capillary Rheometer
2. ISO 11443 — Capillary and slit-die rheometry guidelines

1. Explain why polymer chains orient under shear. How does the molecular weight distribution (MWD) of a polymer impact the flow behavior index $n$?
2. A polymer melt has $K = 5000$ Pa·s$^n$ and $n = 0.40$. Write the expression for shear stress $\tau$ as a function of shear rate $\dot{\gamma}$, and calculate the shear stress at $\dot{\gamma} = 500$ s$^{-1}$.
3. Describe the physical limitations of the Power Law model at very low and very high shear rates. Why is the Carreau-Yasuda model preferred for full-range simulations?

Q1. A fluid that displays a decrease in apparent viscosity as shear rate increases is classified as:

• B) Shear-thinning (pseudoplastic)

Q2. In the Power Law model, a flow behavior index of $n = 1$ indicates that the fluid is:

• A) Newtonian

Q3. What is the typical flow behavior index ($n$) range for commercial polymer melts?

• B) $0.2$ to $0.6$

Q4. Apparent viscosity of a power-law fluid is mathematically defined as:

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

Q5. The power law model is physically inaccurate in which flow region?

• A) Very low shear rates (where zero-shear viscosity $\eta_0$ is a constant plateau)