MCEN90014 · Materials Engineering
Polymers: Structure, MW, Tg & Viscoelasticity
Polymers are the Weeks 8–9 core of MCEN90014 Materials Engineering at the University of Melbourne, where a material stops being a single molecular weight or a single phase and becomes a distribution of chains that is part-crystalline and time-dependent. This chapter sits on the subject's Process→Structure→Property spine and supplies the polymers half of the final exam — molecular weight, tacticity and crystallinity, the glass transition, and viscoelasticity. Get these four levers straight and the storage-modulus, relaxation and blend questions become routine.
What this chapter covers
- 01Average a chain-length distribution two ways: number-average M̄n = ΣxᵢMᵢ and weight-average M̄w = ΣwᵢMᵢ (g/mol)
- 02Compute degree of polymerisation DP = M̄n/m and polydispersity index PDI = M̄w/M̄n ≥ 1
- 03Classify thermoplastics, thermosets and elastomers by chain architecture and behaviour on heating
- 04Link tacticity (isotactic / syndiotactic / atactic) to how well chains pack and crystallise
- 05Read percent crystallinity from density and its effect on stiffness, density and clarity
- 06Locate the glass transition Tg on a modulus–temperature curve and say what raises or lowers it
- 07Model viscoelasticity with a spring and dashpot: Maxwell stress relaxation vs Kelvin–Voigt creep
- 08Interpret DMA outputs — storage E′, loss E″ and loss tangent tan δ = E″/E′ — and the WLF / time–temperature superposition idea
- 09Test polymer-blend miscibility with the Flory–Huggins free energy ΔG_mix = ΔH_mix − TΔS_mix and χ vs χ_crit
Molecular weight, DP and PDI of a PVC sample
- +2Number-average: M̄n = ΣxᵢMᵢ = 2000 + 10000 + 18000 + 17600 + 13000 = 60,600 g/mol.
- +1Weight fractions: wᵢ = xᵢMᵢ / M̄n = 0.0330, 0.1650, 0.2970, 0.2904, 0.2145 (they sum to 1.00 — a useful check).
- +2Weight-average: M̄w = ΣwᵢMᵢ = 660 + 6600 + 17,820 + 23,230 + 21,450 ≈ 69,770 g/mol (6.98×10⁴ g/mol).
- +1Degree of polymerisation: DP = M̄n / m = 60,600 / 62.5 = 969.6 ≈ 970 repeat units (divide by the repeat-unit mass, not the whole chain).
- +2Polydispersity index: PDI = M̄w / M̄n = 69,770 / 60,600 = 1.15 (greater than 1, so the sample is polydisperse).
Key terms
- Number-average molecular weight (M̄n)
- The distribution averaged by the number of chains, M̄n = ΣxᵢMᵢ (g/mol), where xᵢ is the number fraction in size range i. It is the average a colligative measurement 'sees'.
- Weight-average molecular weight (M̄w)
- The distribution averaged by mass, M̄w = ΣwᵢMᵢ (g/mol), with wᵢ the weight fraction. It gives more weight to long chains, so M̄w ≥ M̄n always.
- Degree of polymerisation (DP)
- The number of repeat units in an average chain, DP = M̄n/m, where m is the repeat-unit (monomer) mass in g/mol. Dimensionless.
- Polydispersity index (PDI)
- The breadth of the distribution, PDI = M̄w/M̄n ≥ 1. PDI = 1 is monodisperse (all chains identical); real polymers give PDI > 1.
- Tacticity
- The stereo-regularity of the pendant side group along the backbone — isotactic (all one side), syndiotactic (strictly alternating) or atactic (irregular). Regular chains pack into crystals; atactic chains stay amorphous.
- Glass transition (Tg)
- The temperature at which the amorphous regions change from a rigid glass to a rubbery solid, driven by free volume. It is a gradual transition of the amorphous phase — distinct from the sharp crystalline melting point Tm (with Tg < Tm).
- Viscoelasticity
- Combined elastic (spring, σ = Eε) and viscous (dashpot, σ = ηε̇) response. The Maxwell model (series) gives stress relaxation σ = σ₀e^(−t/τ); the Kelvin–Voigt model (parallel) gives creep ε = (σ₀/E)(1 − e^(−t/τ)); both share τ = η/E in seconds.
- Loss tangent (tan δ)
- From dynamic mechanical analysis, tan δ = E″/E′, the ratio of loss (viscous) to storage (elastic) modulus. Its peak locates Tg and measures a material's damping.
Polymers: Structure, MW, Tg & Viscoelasticity FAQ
What is the difference between M̄n and M̄w, and why is M̄w always the larger?
M̄n = ΣxᵢMᵢ averages the chain lengths by their NUMBER (each chain counts once), while M̄w = ΣwᵢMᵢ averages by MASS, so the long, heavy chains count for more. Because heavy chains carry extra weight in M̄w, it is always greater than or equal to M̄n, with equality only for a perfectly uniform (monodisperse) sample. Their ratio, the polydispersity index M̄w/M̄n, is therefore always at least 1 and tells you how broad the distribution is.
How is Tg different from melting, and what shifts it?
Tg is the glass transition of the AMORPHOUS regions — a gradual softening from glassy to rubbery driven by free volume — whereas melting (Tm) is a sharp, fixed-temperature change of the CRYSTALLINE regions. A semicrystalline polymer shows both, with Tg below Tm. Plasticisers (small added molecules) increase free volume and LOWER Tg, which is why flexible PVC is soft; a stiffer or bulkier backbone, polar side groups, cross-linking and higher molecular weight all RAISE Tg.
Can AI help me with polymer structure and viscoelasticity in MCEN90014?
Yes — Sia is an AI tutor that can explain the concepts step by step: it can walk you through building the two molecular-weight averages from a distribution, help you tell a Maxwell relaxation problem from a Kelvin–Voigt creep problem, or check the units in a τ = η/E calculation. It is a study aid for understanding the method, not a source of ready-made answers, and it will not sit an assessment or guarantee a grade or a pass. Always follow the University of Melbourne's academic-integrity and generative-AI rules for MCEN90014 and confirm what is permitted on Canvas.
Studying with AI? Sia — free AI mechanical engineering tutor works through MCEN90014 step by step.
Exam move
Organise this chapter around the four levers and one drawing. Levers: chain length (M̄n, M̄w, DP, PDI), stereo-order and crystallinity, the glass transition Tg, and time-dependence (viscoelasticity). The drawing is the log-modulus-versus-temperature curve — sketch the glassy plateau, the step down at Tg and the rubbery plateau, then note that a DMA tan δ peak marks the same Tg. Practise the molecular-weight arithmetic until the M̄w ≥ M̄n check is automatic, and drill the direction words: a plasticiser LOWERS Tg, cross-links RAISE it, Maxwell stress DECAYS at constant strain while Kelvin–Voigt strain GROWS at constant stress. Because a formula sheet is provided in the final exam, spend your time choosing the right relation and substituting cleanly with SI units rather than memorising equations. The final exam is 10 questions of 10 marks each (100 marks, all compulsory) worth 50% of the subject with an exam hurdle, so treat every question as equally weighted — budget about a tenth of your time each — and confirm the exam duration on the timetable in Canvas.
