Tg Predictor

How this works

The Fox equation estimates the glass transition temperature of a random copolymer or a miscible polymer blend from the weight fraction and homopolymer T_g of each component. It assumes the components mix freely at the segmental level with no specific interactions (hydrogen bonding, ionic association) and no leftover monomer or solvent acting as a plasticizer.

This works reasonably well for statistical copolymers and truly miscible blends. Alternating copolymers behave the same way, showing a single intermediate T_g rather than two, so the one-transition behavior is not unique to statistical copolymers and miscible blends. It does not apply to block copolymers or phase separated blends, which typically show two distinct glass transitions instead of one intermediate value, since each block or phase relaxes on its own.

Tg calculator

A worked example

Take a 70:30 styrene / ethyl acrylate statistical copolymer. Polystyrene has a T_g of 100 °C and poly(ethyl acrylate) −24 °C, so in Kelvin the two homopolymer values are 373.15 K and 249.15 K.

The number worth noticing is the one the Fox equation does not give you. A straight weight-average of the two glass transitions – the rule of mixtures, which is the intuitive thing to reach for – predicts 0.70 × 100 + 0.30 × (−24) = 62.8 °C. That is 11 degrees higher than the Fox result, and the gap is not a rounding difference: because the reciprocals are averaged rather than the temperatures, the low-T_g component always pulls harder than its weight fraction suggests. A small amount of a soft comonomer softens a copolymer more than you would guess, which is exactly why the Fox equation is the one used in practice.

The gap widens as the two components get further apart. Run the same 70:30 ratio with a soft acrylate at −54 °C instead and Fox gives about 35 °C against a linear estimate of 54 °C – a 19-degree error, enough to put a coating on the wrong side of room temperature.

When the prediction comes out wrong

A measured T_g well below the calculated one almost always means something is plasticising the sample, and the usual culprits are things the calculation cannot know about:

• Residual monomer. A few percent of unreacted monomer is an efficient plasticiser – small molecules lower T_g far more effectively per unit weight than a soft comonomer does. If conversion was incomplete, expect the measured value to undershoot.
• Retained solvent. A film cast from solution and dried at room temperature can hold solvent for a long time. Dry above the expected T_g, or under vacuum, before you trust the number.
• Absorbed water. Anything with amide, hydroxyl, or acid groups picks up water from the air, and water is the most effective plasticiser of all. This is why the reference table above marks several values “dry”; a conditioned sample will read tens of degrees lower.
• Low molecular weight. Chain ends have more free volume than chain middles, so short chains have a lower T_g. The effect is negligible above roughly 20,000 g/mol but real below it – an oligomer will not match a value tabulated for a high polymer.

A measured T_g above the calculation, or two transitions where you expected one, points the other way: the components are interacting or are not actually mixed. Strong hydrogen bonding between unlike repeat units raises T_g above the Fox line, which is why acid- or amide-containing copolymers often read high. Two separate transitions at, or near, the two homopolymer values mean the blend has phase separated and the calculation does not apply at all – see thermal analysis for reading that off a DSC trace.

One structural caveat the equation cannot capture: it takes composition as the only variable, so it cannot distinguish a statistical copolymer from a block copolymer of identical composition. Those are different materials – the first has one intermediate transition, the second has two – and only you know which one is in the flask.

Reference homopolymer Tg values

Typical literature values for the atactic or common commercial form. Real Tg depends on tacticity, molecular weight, and measurement method (DSC heating rate, for example), so treat these as starting points. The glass transition is partly a kinetic effect, so a faster heating or cooling rate gives a higher measured T_g; a value quoted without a rate is only approximate. It is also not a sharp point but a second-order transition spread over a range of temperatures, read by extrapolating the baselines before and after it, which is part of why tabulated values differ by a few degrees between sources.