Polymer rheology

In short

Polymer rheology is the study of how polymer melts, solutions and rubber compounds flow and deform — they are neither simple liquids nor elastic solids but viscoelastic, so their resistance depends on shear rate, temperature, time and molecular weight. It decides whether a grade can be injection moulded, extruded, blow moulded or calendered, and it explains defects such as short shots, die swell, sharkskin and warpage. It is examined in Polymer Science, Polymer Processing and Rheology papers, and forms a scoring block in GATE (XE-Polymer Science and Engineering, section on polymer rheology and melt flow). Quality labs at Reliance, Supreme, Finolex, Nilkamal, SRF, Apollo Tyres, MRF, Haldia and SABIC run melt flow index, capillary and rotational rheometry daily, so interviewers for QC, process and application-development roles ask it constantly.

Explained

Basic definitions and flow classification

• Shear stress τ = F/A (Pa); shear rate γ̇ = dv/dy (s⁻¹); apparent viscosity η = τ/γ̇ (Pa·s; 1 Pa·s = 10 poise).
• Newtonian fluid: η independent of γ̇ (water, glycerol, very dilute polymer solutions, some polymer melts at very low γ̇).
• Non-Newtonian, time-independent: pseudoplastic (shear-thinning) — η falls with γ̇; almost all polymer melts and solutions. Dilatant (shear-thickening) — η rises with γ̇; PVC plastisols, highly filled slurries. Bingham plastic — needs a yield stress τ₀ before flow; filled rubber compounds, PVC pastes, toothpaste-like systems. Herschel–Bulkley = yield stress + power law.
• Time-dependent: thixotropic — η falls with time of shearing and recovers on rest (paints, PVC pastes); rheopectic — η rises with time of shearing (rare).
• Viscoelastic: stores part of the deformation energy elastically and dissipates the rest; shows normal stresses in shear, which cause die swell and the Weissenberg rod-climbing effect.
• Why melts shear-thin: at rest, chains are random coils with many entanglements; shear orients and disentangles them along the flow, lowering resistance. Beyond a critical stress the structure is fully oriented and η tends to a second (infinite-shear) plateau.

Viscosity curve, molecular weight and temperature

• A log η vs log γ̇ curve has three zones: Newtonian plateau η₀ (zero-shear viscosity) at low γ̇ → power-law region (slope n−1) → upper plateau η∞.
• Molecular weight: η₀ ∝ M_w for M below the critical entanglement molecular weight M_c, and η₀ ∝ M_w^3.4 above it. Hence a small rise in M_w sharply raises melt viscosity and lowers MFI.
• MWD: broad distribution shear-thins earlier and more (easier extrusion, poorer impact consistency); narrow distribution (metallocene grades) stays Newtonian to higher γ̇.
• Branching: long-chain branching (LDPE) raises melt strength and shear sensitivity; linear LLDPE of the same MFI is harder to extrude but tougher in film.
• Temperature: viscosity falls with T. Use Arrhenius form above Tg + 100 °C and WLF form near Tg (Tg to Tg + 100 °C).
• Pressure: viscosity rises with pressure — significant in injection moulding (hundreds of bar) and often ignored in simple calculations.
• Time–temperature superposition (TTS): data at several temperatures shift along log-time/log-frequency by a factor a_T to form one master curve at a reference temperature; a_T follows WLF. Valid for thermorheologically simple (usually amorphous, unfilled) polymers.

Models and constitutive equations

• Power law (Ostwald–de Waele): τ = K γ̇ⁿ, so η = K γ̇ⁿ⁻¹. n = 1 Newtonian, n < 1 pseudoplastic, n > 1 dilatant. Typical n: LDPE 0.35–0.45, HDPE 0.4–0.6, PP 0.3–0.4, PS 0.25–0.35, nylon-6 0.6–0.8, unplasticised PVC 0.2–0.3 (values are grade- and temperature-dependent).
• Limitation: power law fails at very low γ̇ (predicts infinite η). Carreau–Yasuda and Cross models add the η₀ plateau and are used in Moldflow-type simulation.
• Ellis / Sisko / Bingham models suit yield-stress and intermediate systems.
• Linear viscoelastic mechanical models: Maxwell (spring + dashpot in series) models stress relaxation, relaxation time λ = η/E; Kelvin–Voigt (parallel) models creep/retarded elasticity but not relaxation or permanent flow; standard linear solid (Zener) and generalised Maxwell (Wiechert) with a spectrum of relaxation times describe real polymers.
• Boltzmann superposition principle: the response to a sequence of loads is the sum of the responses to each — basis of creep and relaxation prediction in the linear region (small strains, typically < 1–2%).
• Deborah number De = λ/t_obs: De ≫ 1 behaves solid-like (elastic), De ≪ 1 liquid-like (viscous). Weissenberg number Wi = λ·γ̇ measures elasticity in flow.

Dynamic (oscillatory) rheology

• Apply strain γ = γ₀ sin ωt; stress leads strain by phase angle δ: 0° for a perfect elastic solid, 90° for a Newtonian liquid, in between for a viscoelastic polymer.
• Storage modulus G′ = (τ₀/γ₀)cos δ — elastic, recoverable energy; loss modulus G″ = (τ₀/γ₀)sin δ — viscous dissipation; tan δ = G″/G′ = damping or loss factor.
• Complex viscosity η* = η′ − iη″ with η′ = G″/ω. Cox–Merz rule: |η*(ω)| ≈ η(γ̇) when ω (rad/s) = γ̇ (s⁻¹) — lets small-strain oscillatory data substitute for capillary data.
• Crossover point (G′ = G″) correlates with M_w and MWD; G′ plateau at low ω indicates a yield structure, gel or good nanofiller dispersion.
• Amplitude sweep gives the linear viscoelastic limit; frequency sweep gives moduli vs timescale; temperature sweep (DMA) gives Tg as the tan δ peak (also G″ peak and G′ drop) and locates secondary (β) transitions.
• Rubber testing uses Mooney viscosity (ML 1+4 at 100 °C, a rotational disc test) and a rotorless cure meter / MDR for scorch time ts2, cure time t90 and torque rise — the industry's working rheology.

Measurement techniques

1. Melt flow index (MFI/MFR): grams of melt extruded in 10 min through a standard die under a standard load and temperature (e.g. PE 190 °C/2.16 kg, PP 230 °C/2.16 kg). Single-point, low shear rate (≈10 s⁻¹), inversely related to M_w. Standards: ASTM D1238 / ISO 1133; in India, IS 2530 (Methods of test for polyethylene moulding materials and polyethylene compounds) carries the melt flow rate method for polyethylene.
2. MFR ratio (FRR): MFI at 21.6 kg ÷ MFI at 2.16 kg — a cheap index of shear sensitivity and MWD breadth.
3. Capillary (extrusion) rheometer: piston forces melt through a capillary of radius R, length L; gives τ_w and γ̇_w over 10–10⁵ s⁻¹ — the processing range. Needs Bagley correction (entrance/exit pressure loss, from a plot of ΔP vs L/D at constant γ̇, extrapolated to L/D = 0) and Rabinowitsch correction (for non-parabolic velocity profile).
4. Rotational rheometers: cone-and-plate (uniform shear rate, small samples, low γ̇), parallel plate (oscillatory, films), concentric cylinder (solutions, latex).
5. Dilute-solution viscometry: Ubbelohde capillary viscometer → relative, specific, reduced and inherent viscosity → intrinsic viscosity [η] by Huggins extrapolation → Mark–Houwink gives viscosity-average molecular weight.
6. Others: torque rheometer/Brabender plastograph (processing stability, PVC fusion), melt elasticity and melt strength (Rheotens), extensional rheometry (SER fixture) for film blowing and thermoforming.
7. Indian test houses operate under NABL accreditation, with instrument calibration traceable to NPL India; test reports in tenders typically require ISO/ASTM method citation with the accredited lab's scope.

Elastic effects, flow defects and processing links

• Die swell (extrudate swell, Barus effect): elastic recovery of oriented chains on leaving the die; B = D_extrudate/D_die, typically 1.2–2 for PE. Reduced by longer land length, higher temperature, lower γ̇.
• Melt fracture / sharkskin: above a critical wall shear stress (~0.1–0.14 MPa for many polyolefins) the extrudate becomes matt, ridged or grossly distorted; cured by lowering output, raising die temperature, streamlining/polishing the die, polymer processing aids (fluoroelastomer PPA).
• Melt strength / sag: governs parison in blow moulding and bubble stability in film blowing — branched, high-M_w grades are better.
• Entrance (converging) flow creates extensional stress and vortices; stagnation causes degradation and black specks.
• Frozen-in orientation and residual stress cause shrinkage, warpage and anisotropy; relieved by annealing and balanced gating.
• Grade selection rule of thumb: injection moulding needs high MFI/low viscosity (thin sections, γ̇ 10³–10⁵ s⁻¹); extrusion and blow moulding need low MFI/high melt strength (γ̇ 10–10³ s⁻¹); rotomoulding uses very high MFI powders with no shear at all.
• Fillers and additives: carbon black, talc and glass raise η₀ and introduce a yield stress; plasticisers, lubricants and processing aids lower it; fibre orientation makes the melt anisotropic.

Key points

• Viscoelasticity — simultaneous viscous flow and elastic recovery; the defining feature of polymer melts and rubbers.
• Shear thinning — viscosity falls with shear rate due to chain disentanglement and orientation; nearly universal in melts.
• Zero-shear viscosity η₀ — low-shear plateau value; the most molecular-weight-sensitive rheological quantity.
• 3.4 power law — η₀ ∝ M_w^3.4 above the critical entanglement molecular weight M_c.
• Power law index n — slope-derived measure of shear sensitivity; lower n means stronger shear thinning.
• Consistency index K — viscosity scale in the power law; falls steeply with temperature.
• Arrhenius / WLF — temperature dependence far above and near Tg respectively.
• TTS and master curve — shifting data with factor a_T extends the accessible time/frequency range.
• G′, G″, tan δ — elastic modulus, viscous modulus and damping from oscillatory tests; tan δ peak marks Tg.
• Cox–Merz rule — oscillatory |η*| at ω matches steady-shear η at the same numerical γ̇.
• Deborah number — ratio of material relaxation time to observation time; decides solid-like vs liquid-like response.
• MFI — single-point, low-shear inverse viscosity index (ASTM D1238/ISO 1133); never a full flow curve.
• FRR — MFI(21.6 kg)/MFI(2.16 kg); indicates MWD breadth and shear sensitivity.
• Bagley and Rabinowitsch corrections — remove entrance pressure loss and correct wall shear rate in capillary rheometry.
• Die swell — elastic recoil at die exit, ratio 1.2–2 typical; corrected by die design, not by compensating downstream alone.
• Melt fracture — surface or gross extrudate distortion above a critical wall shear stress; fixed by temperature, die geometry or PPA.
• Mooney viscosity and MDR — the standard rheological tests for raw rubber and cure behaviour.
• Processing window — injection γ̇ 10³–10⁵ s⁻¹, extrusion 10–10³ s⁻¹, compression/rotomoulding < 10 s⁻¹.

Formulas and examples

Example 1 — Power-law constants from two viscosity readings. A PP melt at 230 °C shows η = 1,800 Pa·s at γ̇ = 10 s⁻¹ and η = 180 Pa·s at γ̇ = 1,000 s⁻¹.

• η = K γ̇ⁿ⁻¹ → η₂/η₁ = (γ̇₂/γ̇₁)ⁿ⁻¹ → 180/1800 = 0.1 = 100ⁿ⁻¹.
• log 0.1 = (n−1) log 100 → −1 = (n−1)(2) → n − 1 = −0.5 → n = 0.5.
• K = η/γ̇ⁿ⁻¹ = 1800/10^−0.5 = 1800 × 3.1623 = 5,692 Pa·sⁿ.
• Check at 1,000 s⁻¹: η = 5692 × 1000^−0.5 = 5692/31.62 = 180 Pa·s ✓.
• Shear stress at 1,000 s⁻¹: τ = ηγ̇ = 180 × 1000 = 1.8 × 10⁵ Pa = 0.18 MPa — above the usual melt-fracture threshold, so surface defects are likely.

Example 2 — Capillary rheometer with both corrections. Capillary R = 0.5 mm, L = 20 mm; Q = 1.0 × 10⁻⁸ m³/s; measured ΔP = 8.0 MPa; Bagley end correction e = 2.0; n = 0.4.

• Uncorrected τ_w = ΔP·R/(2L) = (8.0 × 10⁶ × 0.0005)/(2 × 0.02) = 4,000/0.04 = 1.0 × 10⁵ Pa.
• With Bagley: L/R = 0.02/0.0005 = 40; τ_w = 8.0 × 10⁶ / [2(40 + 2)] = 8.0 × 10⁶/84 = 9.52 × 10⁴ Pa (4.8% lower).
• γ̇_app = 4Q/(πR³) = (4 × 1.0 × 10⁻⁸)/(π × 1.25 × 10⁻¹⁰) = 4 × 10⁻⁸/3.927 × 10⁻¹⁰ = 101.9 s⁻¹.
• Rabinowitsch: (3n+1)/(4n) = (1.2 + 1)/(1.6) = 2.2/1.6 = 1.375 → γ̇_true = 101.9 × 1.375 = 140.1 s⁻¹.
• True viscosity η = 9.52 × 10⁴/140.1 = 679 Pa·s (against an uncorrected 1.0 × 10⁵/101.9 = 982 Pa·s — a 31% error if corrections are skipped).

Example 3 — Temperature sensitivity (Arrhenius). An HDPE melt has η = 2,000 Pa·s at 190 °C; E_a = 27 kJ/mol. Find η at 230 °C.

• T₁ = 463.15 K, T₂ = 503.15 K; E_a/R = 27,000/8.314 = 3,247.5 K.
• 1/T₁ − 1/T₂ = 1/463.15 − 1/503.15 = 0.0021591 − 0.0019875 = 1.716 × 10⁻⁴ K⁻¹.
• ln(η₁/η₂) = 3247.5 × 1.716 × 10⁻⁴ = 0.5573 → η₁/η₂ = e^0.5573 = 1.746.
• η₂ = 2000/1.746 = 1,146 Pa·s — a 40 °C rise nearly halves the viscosity.

Example 4 — Molecular weight effect on viscosity and MFI. Grade A: M_w = 150,000, η₀ = 10,000 Pa·s. Grade B: M_w = 180,000. Both above M_c.

• η₀B/η₀A = (180/150)^3.4 = (1.2)^3.4. ln 1.2 = 0.18232 → × 3.4 = 0.61990 → e^0.6199 = 1.859.
• η₀B = 10,000 × 1.859 = 18,590 Pa·s — a 20% rise in M_w raises viscosity by 86%.
• MFI varies roughly inversely with η₀, so if grade A has MFI 8 g/10 min, grade B ≈ 8/1.859 ≈ 4.3 g/10 min (an indicative estimate: MFI is a single low-shear point, not a true η₀ measurement).

Example 5 — Shear rate in an extrusion die and defect check. A sheet die of width w = 600 mm and gap h = 1.5 mm runs at output 180 kg/h of PS; melt density 950 kg/m³.

• Q = 180/(950 × 3600) = 180/3,420,000 = 5.263 × 10⁻⁵ m³/s.
• γ̇_app = 6Q/(wh²) = (6 × 5.263 × 10⁻⁵)/(0.6 × 2.25 × 10⁻⁶) = 3.158 × 10⁻⁴/1.35 × 10⁻⁶ = 234 s⁻¹.
• With K = 12,000 Pa·sⁿ and n = 0.3: η = 12,000 × 234^−0.7. ln 234 = 5.455 → × 0.7 = 3.8185 → e^3.8185 = 45.5 → η = 12,000/45.5 = 264 Pa·s.
• τ_w = 264 × 234 = 6.18 × 10⁴ Pa = 0.062 MPa — below the ~0.1 MPa melt-fracture threshold, so the surface should be glossy.

Where it is used

• Grade selection and QC: MFI and FRR testing at polyolefin producers and compounders (Reliance, GAIL, Haldia Petrochemicals, OPaL, Supreme Petrochem) to accept resin lots and match a grade to injection, film or pipe duty.
• Pipe, film and profile extrusion: Finolex, Astral, Supreme, Jain Irrigation use flow-curve data for die land design, output limits and sharkskin control; PVC pipe work also needs torque-rheometer fusion data.
• Injection moulding and simulation: Carreau/Cross viscosity data feed Moldflow and Moldex3D for fill time, gate location, pressure and warpage prediction at tier-1 auto suppliers and appliance moulders (Nilkamal, Supreme, Sintex, Motherson).
• Tyre and rubber industry: Mooney viscosity and MDR curves at MRF, Apollo, CEAT, JK Tyre, TVS Srichakra control mixing, extrusion of treads, calendering and cure cycles.
• Paints, adhesives, inks and personal care: thixotropy, yield stress and viscosity profiles control sag, levelling, brushability and pumpability (Asian Paints, Berger, Pidilite, HUL).
• Job roles: polymer QC chemist, extrusion/moulding process engineer, application development engineer, rheology lab technician, R&D scientist at CIPET centres (under the Department of Chemicals and Petrochemicals), CSIR-NCL and NABL-accredited testing labs.

Interview questions

• Q: Why do polymer melts shear-thin? A: Shear orients and disentangles the random coils along the flow direction, reducing intermolecular drag, so apparent viscosity falls as shear rate rises.
• Q: What does MFI tell you and what does it not? A: MFI is a single-point, low-shear inverse measure of melt viscosity useful for grade identity, but it gives no flow curve, no elasticity and no information on MWD — two grades of equal MFI can process very differently.
• Q: How can two resins have the same MFI but different extrusion behaviour? A: They can differ in MWD and branching, so they shear-thin differently at process shear rates and differ in melt strength and die swell.
• Q: State the molecular weight dependence of zero-shear viscosity. A: η₀ ∝ M_w below the critical entanglement molecular weight and η₀ ∝ M_w^3.4 above it, which is why a modest M_w rise sharply raises viscosity.
• Q: What are the Bagley and Rabinowitsch corrections for? A: Bagley removes the entrance/exit pressure loss from the measured ΔP so wall shear stress is true; Rabinowitsch corrects apparent wall shear rate for the non-parabolic velocity profile of a shear-thinning fluid.
• Q: What is die swell and how do you reduce it? A: Elastic recovery of oriented chains at the die exit, giving an extrudate larger than the die; reduced by a longer die land, higher melt temperature, lower output and lower-elasticity grades.
• Q: What does tan δ mean physically? A: The ratio of dissipated to stored energy per cycle — high tan δ means damping, and its peak in a temperature sweep locates the glass transition.
• Q: Explain the Deborah number. A: De = material relaxation time / observation time; De ≫ 1 gives solid-like elastic response, De ≪ 1 gives liquid-like viscous flow.
• Q: When would you use WLF rather than Arrhenius? A: WLF near the glass transition (Tg to about Tg + 100 °C) where free volume controls mobility; Arrhenius well above that range.
• Q: What is the Cox–Merz rule used for? A: It lets small-amplitude oscillatory |η*(ω)| stand in for steady-shear η(γ̇) at the same numerical value, so delicate or degradation-prone melts can be characterised without steady shear.
• Q: A moulded part shows short shots. What rheological fixes would you try? A: Raise melt and mould temperature, raise injection speed (to exploit shear thinning), enlarge gates/runners, or move to a higher-MFI grade — in that order of cost.
• Q: Why does an extrudate suddenly go matt above a certain output? A: Wall shear stress has exceeded the critical value (~0.1 MPa for many polyolefins), causing sharkskin/melt fracture; reduce output, raise die temperature, polish or redesign the die, or add a processing aid.

Common mistakes

• Treating MFI as a viscosity: it is an inverse, single-point index at low shear, not η at process shear rates, and its unit is g/10 min.
• Using the power-law model at very low shear rates, where it wrongly predicts infinite viscosity instead of the η₀ plateau.
• Forgetting the Rabinowitsch factor (3n+1)/(4n) and the Bagley correction, which together can distort viscosity by 30% or more.
• Mixing units — poise with Pa·s, mm with m, kg/h with m³/s — and dropping the 10³ factors in ΔP and γ̇ calculations.
• Confusing shear-thinning (rate-dependent, instantaneous) with thixotropy (time-dependent, recovers on rest).
• Swapping G′ and G″, or calling tan δ a modulus; also assuming the G′–G″ crossover is always the gel point regardless of test type.
• Explaining die swell as thermal expansion or as a pressure effect rather than elastic recovery of oriented chains.
• Running oscillatory tests outside the linear viscoelastic region (no amplitude sweep first), so G′ and G″ are strain-dependent and meaningless.

Related topics

Polymer physics: molecular weight, crystallinity and glass transition · Polymer processing: extrusion, injection and blow moulding · Rubber compounding and vulcanisation · Polymer testing and characterisation · Fluid mechanics of non-Newtonian flow · Polymer composites and filler reinforcement