How this works
When two monomers copolymerize, the growing chain end usually doesn't add each monomer at the same rate it appears in the feed. The Mayo–Lewis equation predicts the instantaneous mole fraction of monomer 1 in the copolymer, F1, from the feed mole fraction f1 and the two reactivity ratios r1 and r2 (how much each chain end prefers its own monomer over the other one).
r1 > 1 means a monomer 1 chain end prefers adding another monomer 1 (the copolymer runs rich in monomer 1 relative to the feed); r1 < 1 means it prefers crossing over to monomer 2. When both ratios are below 1, the system tends toward alternation; when both are above 1, it tends toward blockiness. When one ratio is far above 1 and the other far below (styrene/vinyl acetate, at 55 and 0.01, is the classic case), both chain ends add the same monomer, so it homopolymerizes almost to exhaustion first and the other only afterward; the product runs close to a blend of the two homopolymers rather than a random or alternating copolymer. If an azeotropic composition exists (F1 = f1), the copolymer composition stays constant as conversion proceeds even though the composition would otherwise drift as the faster comonomer is consumed first. That crossover exists only when both ratios sit on the same side of 1: both r1 < 1 and r2 < 1 (the common, stable case) or both above 1 (rare). If one ratio is above 1 and the other below, the curve never crosses the diagonal and the composition always drifts, which is what the calculator reports as "None in range." (The one exception is r1 = r2 = 1, where the whole curve lies on the diagonal, so composition never drifts even though there is no single azeotrope point.)
Composition calculator
Drag or click anywhere on the chart to move the feed point – the composition updates live. The dashed diagonal is where copolymer composition equals feed composition. The solid curve is F1 predicted across the whole feed range; the accent dot marks your current feed. If the curve crosses the diagonal, that crossing (marked in red) is the azeotrope.
The microstructure above is what the ratios imply for a conventional radical copolymerization. It is an instantaneous, statistical picture: away from the azeotrope the composition drifts as the faster monomer is consumed, so a batch taken to high conversion is really a gradient with a homopolymer-rich tail. Feeding the fast monomer continuously (starved feed) holds it constant, and controlled/living methods (RAFT, ATRP) let you build true blocks or deliberate gradients that the reactivity ratios alone would not give. In those living systems the reactivity ratios, and this composition equation, are essentially the same as in conventional radical copolymerization; what changes is that all chains start near t = 0 and grow together, so every chain carries the same composition and the same composition gradient instead of the chain-to-chain spread of a conventional batch. The blocks and controlled gradients come from that uniformity plus deliberate feed control, not from altered r values.
The word “instantaneous” is doing the work
The Mayo–Lewis equation describes the copolymer being formed right now, from the feed as it stands right now. If the copolymer is richer in one monomer than the feed is, then that monomer is being removed faster than the other, the feed shifts toward the slower monomer, and the next chains formed are different from the first. Composition drift is not a side effect – it is the direct consequence of r1 and r2 not both being 1.
Take methyl methacrylate with n-butyl acrylate, r1 = 2.2 and r2 = 0.37, a combination made industrially by the tonne. Charge an equimolar batch and the first chains are 70 mol % MMA, not 50. Follow it through the reaction:
| Conversion | Feed f1 | Chains forming now (F1) | Everything made so far |
|---|---|---|---|
| 0 % | 0.500 | 0.700 | — |
| 10 % | 0.479 | 0.683 | 0.692 |
| 25 % | 0.441 | 0.651 | 0.677 |
| 50 % | 0.355 | 0.569 | 0.645 |
| 75 % | 0.212 | 0.401 | 0.596 |
| 90 % | 0.077 | 0.178 | 0.547 |
By 90 % conversion the chains coming off are 18 mol % MMA – a different material from the 70 % chains made at the start. The product is not a copolymer of one composition but a distribution of compositions, and the last column shows the trap: the bulk average ends up near 55 % and, taken to full conversion, must equal the feed exactly. An elemental analysis or an NMR of the isolated product will report that average and look entirely reasonable, while the material actually contains chains ranging from MMA-rich to acrylate-rich. Two Tgs in the DSC, or a hazy film where you expected a clear one, is often this and not an impurity.
Holding the composition constant
Three options, in ascending order of effort:
- Stop early. Drift is mild at low conversion – the MMA/BA case above moves only two points in feed composition by 10 % conversion. Quenching early costs yield but gives a genuinely uniform product, and is the simplest answer when you need a well-defined material rather than a lot of it.
- Run at the azeotrope. When both reactivity ratios are below 1 (or both above), there is one feed composition at which F1 = f1 and nothing drifts at all, at f1 = (1 − r2) ÷ (2 − r1 − r2). For styrene / methyl methacrylate that is f1 = 0.529, and a batch run there stays on composition to complete conversion. The limitation is obvious: the azeotrope is one composition, and it is whatever it is. Note also that MMA / n-butyl acrylate has none – with r1 above 1 and r2 below it, the formula returns a negative fraction, which is how it tells you no azeotrope exists.
- Starve-feed the faster monomer. The general solution. Charge the slow monomer and meter the fast one in at the rate it is consumed, keeping the instantaneous feed pinned wherever you want it. This decouples composition from conversion entirely and is how uniform copolymers are made at scale – at the cost of running a semi-batch process, with the pump rather than the kinetics setting the outcome.
One caution on the reactivity ratios themselves: they are fitted parameters with real uncertainty, and the pairs where drift matters most are often the ones whose values scatter most between sources. Treat a predicted composition as a design starting point and confirm the real one by NMR on the isolated polymer – and see monitoring conversion for taking the points that let you track both at once.
Reference reactivity ratios
Representative conventional free-radical values near 60 °C, each cross-checked against a second source (Polymer Handbook, Odian, or primary literature). Reactivity ratios are sensitive to temperature, solvent, and the fitting method, so treat these as starting points and verify against a primary source for real formulation work. Of those, temperature is the weakest lever for radical systems (styrene / methyl methacrylate shifts only from 0.52/0.46 at 60 °C to 0.59/0.54 at 131 °C); solvent, pH for ionizable monomers, and the fitting method move them more, so do not over-attribute the scatter to temperature. Pick a pair to load its ratios above; the Tendency column is computed from r₁·r₂ (hover for the note). Acrylic acid / acrylamide in particular shifts strongly with pH.
| Monomer 1 | Monomer 2 | r₁ | r₂ | Tendency |
|---|