What superposition buys you
A rheometer covers three or four decades of frequency. The viscoelastic response of a polymer covers ten or twelve. The gap is bridged by an empirical observation that turns out to hold remarkably well above Tg: raising the temperature and lowering the frequency do the same thing. Every relaxation process in the material speeds up by the same factor, so a curve measured hot and slow lies on top of one measured cold and fast, once you slide it along the frequency axis.
That sliding distance is the shift factor aT. Collect them and you have not just a master curve spanning a dozen decades, but a measurement of how the relaxation time itself depends on temperature – which is the Williams–Landel–Ferry equation:
The thermal analysis guide explains what a DMA sweep tells you. This page does something with one.
Build a master curve
Rheometer exports usually carry G′ and G″ side by side. Superpose each in turn: if they give different shift factors, the material is not thermorheologically simple over this range.
The entropic correction is small next to the horizontal shift and is often left out. Turn it on and see whether it changes the fit.
Shift factors
| T | T − Tref | log aT | WLF fit | residual | points |
|---|
A worked example with a known answer
The data this page loads with is not a measurement. It was generated from a WLF law with C₁ = 12.5 and C₂ = 65.0 K referenced to 373.15 K, sixteen temperatures five kelvin apart, each a four-decade sweep. So the right answer exists before the page computes anything, and what comes out is a measurement of the algorithm rather than a claim about it.
Set the reference to 373.15 K and the fit returns C₁ = 12.49, C₂ = 64.9, with every shift factor within 0.007 decades of the value it was built from. The page opens on 398.15 K instead – the middle sweep – and reports C₁ = 9.10, C₂ = 90.7. Those are not a disagreement. They are the same law stated against a different reference, which is the subject of the next section.
Why five kelvin and four decades. Those are not arbitrary. Superposition needs adjacent sweeps to overlap, and the shift between two temperatures grows fast as you approach Tg: on this WLF law, 10 K steps near the reference demand more than three decades of shift, so consecutive four-decade sweeps would share no frequency range at all and there would be nothing to align. If your own curves refuse to shift cleanly at the cold end, the usual cause is that the temperature steps were too coarse, not that superposition has failed.
Error accumulates away from the reference. Each sweep is aligned onto its neighbour and the shifts are summed outward, so the uncertainty in log aT grows with distance from Tref – in the example above from 0.0003 decades next to the reference to 0.007 at the far end, a factor of twenty. Pick the reference in the middle of your temperature range rather than at one end, and treat the extreme ends of the master curve as the least certain part of it.
C₁ and C₂ mean nothing without their reference
A WLF constant pair is not a property of the material on its own. Move the reference temperature by δ and the same physical law is described by different constants, exactly:
Note that the product C₁C₂ is invariant, and so is the quantity that actually has physical content. Run the demo data at three references and the page reproduces the conversion:
| Reference | C₁ recovered | C₂ recovered | C₁ exact | C₂ exact |
|---|---|---|---|---|
| 373.15 K (as generated) | 12.49 | 64.9 | 12.50 | 65.0 |
| 383.15 K | 10.85 | 75.1 | 10.83 | 75.0 |
| 398.15 K | 9.10 | 90.7 | 9.03 | 90.0 |
Two things to take from that table. Quoting C₁ and C₂ without the reference temperature is not a result, and comparing two literature pairs referenced to different temperatures is comparing nothing – convert first. The "universal" constants C₁ = 17.44, C₂ = 51.6 are specifically referenced to Tg, and applying them at any other temperature without converting is a common and silent error.
And the accuracy degrades as the reference moves off centre: 0.1% at 373.15 K, 0.8% at 398.15 K. That is the shift-factor error accumulating, since a reference near one end of the data leaves the far end many pairwise steps away. Put the reference in the middle of your temperature range when you want the best constants, and move it afterwards with the conversion above rather than by re-shifting.
The activation energy that is not one
It is tempting to fit log aT against 1/T and quote an Arrhenius activation energy. Near Tg that is a mistake, and the WLF fit says exactly why. Differentiating gives an apparent activation energy that is not constant at all:
Put the "universal" constants in – C₁ = 17.44 and C₂ = 51.6 K referenced to Tg, taking Tg = 373 K – and the number at Tg is about 900 kJ/mol. That is not a chemical activation energy; no bond in the polymer is worth 900 kJ/mol. It is the signature of cooperative motion: near Tg a segment cannot move until a whole neighbourhood rearranges with it, so the effective barrier is the sum of many small ones. Fifty kelvin above Tg the same expression gives about 300 kJ/mol – a threefold fall over 50 K, which is not what any Arrhenius process does.
The practical rule that follows: use WLF between Tg and roughly Tg + 100 K, and only switch to an Arrhenius description well above that, where the free volume has stopped changing quickly and the apparent energy really does settle. A single Arrhenius fit spanning both regions will describe neither, and will extrapolate badly in the direction you care about.
When superposition fails, and what that tells you
- A material is only thermorheologically simple if every relaxation shifts by the same factor. That is an assumption, not a law. When it holds, the shifted curves lie on one line; when it does not, they cross or fan out and no shift will fix it. The failure is the measurement: it says you have two processes with different temperature dependences.
- Semicrystalline polymers usually fail it, because the amorphous fraction and the crystal population respond to temperature differently, and because the crystallinity itself changes as you heat. A master curve through a melting range is not a master curve.
- Block copolymers and filled systems fail it near a transition. A diblock approaching its order–disorder transition has a morphology changing underneath the measurement – see the phase behaviour page for where that sits – and a filled rubber has a filler network that yields rather than relaxes.
- Check the overlap, not just the fit. A shift can be numerically optimal while aligning almost nothing, if two sweeps barely touch. This page reports how many points actually overlapped for each shift; anything in single figures should be treated as a guess.
Related tools
- Thermal analysis – what DSC, TGA and DMA each answer, and how to read them together.
- Tg predictor – the transition this whole construction is referenced to.
- Crosslink density – uses the rubbery plateau modulus that sits at one end of the master curve.