MATS3004 · Polymer Science and Engineering 1
Molecular-Weight Characterization
Week 4 defines the molecular-weight averages — number-average Mn, weight-average Mw and dispersity Đ = Mw/Mn — and the methods that measure them: gel-permeation/size-exclusion chromatography (GPC/SEC), dynamic light scattering (DLS) and MALDI mass spectrometry. The exam asks you to compute averages from distribution data and to reason about which technique gives which average, so the bankable skill is the Mn/Mw/Đ calculation plus knowing what each instrument reports.
What this chapter covers
- 01Number-average Mn = ΣNᵢMᵢ/ΣNᵢ (weights every chain equally; from colligative/end-group methods)
- 02Weight-average Mw = ΣNᵢMᵢ²/ΣNᵢMᵢ (weights by mass; from light scattering)
- 03Dispersity Đ = Mw/Mn ≥ 1 (1 monodisperse; ~2 ideal step-growth or disproportionation; ~1.5 FRP by combination)
- 04Ordering Mn ≤ Mv ≤ Mw ≤ Mz across the averages
- 05GPC/SEC: separates by hydrodynamic volume, large coils elute first; gives the whole distribution and all averages
- 06Universal calibration using the product [η]·M
- 07MALDI-TOF MS: absolute mass of individual chains, best for narrow-Đ samples, resolves repeat-unit spacing and end groups
- 08DLS: diffusion coefficient → hydrodynamic radius via Stokes-Einstein R_h = k_B·T/(6π·η·D)
Molecular-weight averages of a blend of two monodisperse polymers
- +1Take N chains of each type (equal numbers). Number-average weights every chain equally: Mn = ΣNᵢMᵢ/ΣNᵢ = (N·20 + N·60)/(N + N) = 80/2 = 40 kg/mol.
- +1Weight-average weights each chain by its mass: Mw = ΣNᵢMᵢ²/ΣNᵢMᵢ. Numerator N·20² + N·60² = N(400 + 3600) = 4000N; denominator N·20 + N·60 = 80N.
- +1So Mw = 4000N / 80N = 50 kg/mol (the N cancels — only the ratios of chain numbers matter).
- +1Dispersity Đ = Mw/Mn = 50/40 = 1.25, and Mw (50) > Mn (40). Mixing two very different chain lengths creates a spread, so the blend is polydisperse even though each component was monodisperse.
Key terms
- Number-average molar mass (Mn)
- Mn = ΣNᵢMᵢ/ΣNᵢ, the mean molar mass counting every chain equally. It is sensitive to small chains and is measured by colligative methods, end-group analysis and membrane osmometry.
- Weight-average molar mass (Mw)
- Mw = ΣNᵢMᵢ²/ΣNᵢMᵢ, the mean molar mass weighting each chain by its mass. It is sensitive to large chains and is measured by (static) light scattering. Always Mw ≥ Mn.
- Dispersity (Đ)
- The breadth of the distribution, Đ = Mw/Mn ≥ 1. It is 1 for a monodisperse sample, about 2 for ideal step-growth or radical polymer terminated by disproportionation, and about 1.5 for radical polymer terminated by combination.
- GPC / SEC
- Gel-permeation / size-exclusion chromatography separates chains by hydrodynamic volume — large coils are excluded from the pores and elute first, small coils last. With calibration it gives the full distribution and every average.
- MALDI-TOF MS
- Matrix-assisted laser desorption/ionization time-of-flight mass spectrometry measures the absolute mass of individual chains. It works best for narrow-dispersity samples and resolves the repeat-unit spacing and end-group masses.
- Dynamic light scattering (DLS)
- Measures the diffusion coefficient D of coils in solution and converts it to a hydrodynamic radius via the Stokes-Einstein relation R_h = k_B·T/(6π·η·D), giving a size distribution in solution.
Molecular-Weight Characterization FAQ
Why is the weight-average molar mass always larger than the number-average?
Because Mw weights each chain by its mass while Mn weights every chain equally. Squaring the molar mass in the Mw sum gives the heavy chains disproportionate influence, so Mw is pulled upward toward the long chains, whereas Mn is dominated by the numerous short ones. The two are equal only when every chain has the same mass (a monodisperse sample, Đ = 1); for any real distribution Mw > Mn, which is exactly why the ratio Đ = Mw/Mn measures the breadth of the distribution.
Which technique gives which molecular-weight average?
End-group analysis, membrane osmometry and other colligative methods count molecules, so they give the number-average Mn. Static light scattering responds to mass, so it gives the weight-average Mw (and the radius of gyration). GPC/SEC, once calibrated, delivers the whole distribution and hence every average. MALDI-TOF measures the absolute mass of individual chains (best on narrow samples), and DLS gives a hydrodynamic radius rather than a mass average directly. Matching the method to the average it reports is a common exam point.
How does size-exclusion chromatography separate polymer chains?
By hydrodynamic volume, not by mass directly. The column is packed with porous beads; small coils can enter the pores and take a long, winding path, so they elute late, while large coils are excluded from the pores and travel the short route between beads, eluting first. Because separation is really by size, calibration is needed to turn elution time into molar mass — the universal calibration using the product [η]·M makes the calibration transferable between different polymers.
How is molecular-weight characterization examined in MATS3004?
As a calculation of Mn, Mw and Đ from tabulated distribution or blend data, and as short-answer questions matching a technique (GPC/SEC, DLS, MALDI, osmometry, light scattering) to the average or property it measures. The Group Lab Report also draws on characterization thinking. Watch the number-versus-mass weighting in the arithmetic and give answers with units. Confirm the assessed methods and any provided data on the UNSW course outline / Moodle.
Exam move
Make the Mn/Mw/Đ calculation second nature and always start by identifying the weighting — 'equal numbers of chains' fixes Nᵢ, 'equal masses' fixes NᵢMᵢ — because the exam plants that trap. Memorise the two sums (Mn = ΣNᵢMᵢ/ΣNᵢ, Mw = ΣNᵢMᵢ²/ΣNᵢMᵢ) and the sanity check Mw ≥ Mn, plus the benchmark dispersities (1 monodisperse, ~1.5 combination, ~2 step-growth/disproportionation) so you can gut-check an answer. Build a one-line technique table (which method → which average → what it is sensitive to) covering GPC/SEC, osmometry, light scattering, MALDI and DLS. Link back to the previous chapter's coil size, since GPC separates by hydrodynamic volume. Confirm the examinable technique set on the UNSW course outline / Moodle.
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