Predict Đ for your reaction
In a reversible-deactivation (living) radical polymerization, dispersity is set by two things: how long the chains are, and how often each growing chain swaps between its active and dormant states. Pick a mechanism, set the target degree of polymerization and conversion, and choose the exchange chemistry (the RAFT agent for your monomer, or the level of catalyst/deactivator for ATRP and NMP). The predicted Đ, its molecular-weight distribution, and how Đ evolves with conversion update as you go.
Ideal molecular-weight distribution for this Đ, drawn as an SEC trace (weight fraction vs log M) from a log-normal fit. Mn, peak Mp, and Mw are marked.
Predicted Đ across conversion at this target DP. Dispersity falls as the reaction proceeds, the kinetic signature of a reversible-deactivation process. The dot marks your current conversion.
How the prediction works
RAFT, ATRP, and NMP all narrow the distribution by the same trick: every chain is repeatedly parked in a dormant state and released, so all chains grow in small, near-equal increments instead of a few chains running away. Moment analysis of that exchange (Litvinenko and Müller; Matyjaszewski and Fukuda) gives one equation for all of them:
The 1/DPn term is the chain-length (Poisson) floor: longer chains give narrower distributions, and it is all that remains for a perfect living system. It is the large-chain approximation of the exact living-chain result Đ = 1 + DPn/(DPn + 1)2; either way a Poisson-distributed polymer of DPn = 50 reaches only Đ = 1.02, about as tight as any chain-growth process gets. The C · (2/p − 1) term is the cost of finite exchange. The factor (2/p − 1) is identical to (2 − p)/p; it runs to infinity as conversion p approaches zero and falls to exactly 1 at full conversion, which is why Đ is high early and settles as the reaction proceeds. The whole difference between mechanisms is the exchange coefficient C:
| Mechanism | Exchange coefficient C | What it means |
|---|---|---|
| RAFT | C = 1 / Cex | Cex = ktr/kp is the transfer constant of the polymeric RAFT agent. A fast, well-matched agent (large Cex) makes C small. |
| ATRP | C = kp[R-X]0 / (kdeact[X-CuII/L]) | More deactivator (CuII) or a faster-deactivating ligand shortens each active period, making C small. [R-X]0 is the initial dormant-chain concentration, not the monomer concentration. |
| NMP | C = kp[alkoxyamine]0 / (kc[nitroxide]) | Same structure as ATRP; the dormant chains are alkoxyamines, not alkyl halides, so [alkoxyamine]0 is the NMP analog of [R-X]0. The deactivator is free nitroxide (TEMPO, SG1) and kc is the coupling rate constant. |
| Ideal living | C = 0 | Infinitely fast exchange. Đ = 1 + 1/DPn, the Poisson floor and the best any chain-growth process can reach. |
This is the cumulative dispersity of the whole sample, the quantity a GPC actually measures. It assumes a fixed number of chains, uniform and instant initiation, and a constant C, so it predicts the best case. Real Đ sits above it (see the caveats below).
RAFT: the transfer constant sets the floor
For RAFT the knob is Cex, the exchange (chain-transfer) constant of the polymeric RAFT agent for your monomer. At full conversion the equation collapses to Đ → 1 + 1/Cex for long chains, so Cex above roughly 10 gives Đ near 1.1, above ~20 gives Đ near 1.05, and above ~100 gives Đ near 1.01. The match between the RAFT agent's Z group and the monomer class decides Cex: more-activated monomers (styrene, (meth)acrylates, acrylamides) need dithiobenzoates or trithiocarbonates; less-activated monomers (vinyl acetate, N-vinylpyrrolidone) need xanthates. Cross a monomer with the wrong agent and Cex drops below 1, which the equation turns into a broad distribution.
Representative Cex values. These vary widely with method, temperature, and solvent, so read them as order-of-magnitude, not precise. For the most active dithiobenzoates a single Cex is genuinely ill-defined and spans a wide range.
| RAFT agent (Z group) | Monomer family | Cex | Control |
|---|---|---|---|
| Trithiocarbonate (DDMAT, CDTPA) | Acrylates | ~10–100 | Workhorse; low retardation |
| Trithiocarbonate | Styrene | ~9–20 | Good |
| Trithiocarbonate | Acrylamides (DMA, NIPAM) | ~1–10 | Good |
| Dithiobenzoate (cumyl) | Styrene | ~26–>1000 | Very narrow; can retard |
| Dithiobenzoate (cyano-R, CPDB) | MMA | ~10–40 | Good (needs tertiary R) |
| Xanthate / MADIX | Vinyl acetate | ~10–26 | Correct match for a LAM |
| Xanthate | N-vinylpyrrolidone | ~1–10 | Workable (Đ ~1.2–1.4) |
| Xanthate on an acrylate | Acrylates | ~0.6–3.8 | Mismatched; broad |
| Xanthate on styrene / MMA | Styrene, MMA | <1 | Mismatched; little control |
| Benzyl dithiobenzoate on MMA | MMA | <1 | Poor R leaving group |
One failure mode the equation cannot show: a highly active dithiobenzoate on a less-activated monomer such as vinyl acetate. Addition to the RAFT agent is fast, but the intermediate radical is so stabilized that it fragments too slowly, which retards or stops the reaction outright. Control is lost through slow fragmentation, not through a low Cex, so no single number describes it. Slow fragmentation is not the only retardation channel: the stabilized intermediate radical can also couple with a propagating radical, which both slows the rate and, because it consumes chains, makes stars and other complex architectures hard to build cleanly by RAFT.
ATRP and NMP: the deactivator sets the floor
For ATRP and NMP the exchange coefficient is C = kp[R-X]0 / (kdeact[deactivator]). More deactivator, or a ligand/nitroxide that deactivates faster, shortens each active burst and narrows the distribution. That is the tension behind low-catalyst methods: ARGET and ICAR ATRP run on parts-per-million copper, which means very little CuII deactivator, a larger C, and a broader product unless something else compensates. The scenarios below give order-of-magnitude values of C for common cases (at ~90% conversion, the exchange term is C × 1.22).
| Scenario | Deactivator level | Exchange C | Đ at ~90% |
|---|---|---|---|
| Well-controlled normal ATRP (styrene) | Ample CuII (~10-2 M) | ~1×10-4 | ~1.05–1.10 (DP-limited) |
| Fast acrylate, modest deactivator | CuII ~10-3 M | ~0.03–0.1 | ~1.05–1.15 |
| Low-ppm ARGET / ICAR ATRP (MMA) | Scarce CuII (~10-5 M) | ~0.05–0.2 | ~1.1–1.3 |
NMP behaves the same way with free nitroxide as the deactivator: plenty of persistent nitroxide gives a small C and a narrow product; too little lets chains run long between traps. The predictor exposes C directly for ATRP and NMP so you can dial it between these regimes or enter a value computed from your own rate constants. NMP also covers a narrower set of monomers than ATRP or RAFT: TEMPO gives good control essentially only for styrene and 4-vinylpyridine (Đ below ~1.1–1.2 is hard for other monomers), and it has never worked for methacrylates, whose propagating radical is degraded by β-hydrogen abstraction. Second-generation nitroxides such as SG1 widen the range to acrylates, acrylamides, and dienes.
One reason the ideal curve flatters ATRP and NMP early on: the deactivator is self-regulating. Every irreversible termination event leaves behind an excess of the persistent species (CuII, or free nitroxide in NMP), so its concentration builds over roughly the first 5% of conversion until it is high enough to shut normal termination down. This persistent radical effect is why real Đ starts broad and why the model overstates the control at very low conversion.
Why dispersity falls with conversion
The conversion curve above runs the opposite way to a conventional free-radical polymerization, where Đ is set from the first chains and only broadens. Early in a controlled polymerization the chains are short and have been through few exchange cycles, so both terms in the equation are large and Đ is high. As monomer converts, chains lengthen (1/DPn shrinks) and each has cycled many more times ((2/p − 1) shrinks toward 1), so Đ drops and levels off. Watching Đ decrease as conversion climbs is one of the cleaner experimental tests that a system is genuinely living rather than a fast conventional radical process wearing a control agent. The conversion monitoring guide covers how to get the conversion axis from aliquots.
What this model leaves out
This is an ideal lower bound. Measured Đ is almost always higher. The equation is termination-free and assumes uniform, instant initiation and a fixed number of chains, so it cannot see the effects that broaden real samples:
- Radical termination. Dead chains from combination or disproportionation add high- and low-mass shoulders the ideal form ignores. This bites hardest at high target molecular weight: well-defined ATRP tops out around Mn 150,000–200,000, and above that (especially at high conversion) bimolecular termination claims a growing share of chains and control is lost, so the ideal lower bound is least trustworthy there.
- Slow or incomplete initiation. A RAFT agent with a low initial transfer constant, or an ATRP initiator that starts unevenly, spreads chain births over a range of conversions and can give early bimodality that the steady-state equation never captures. In copper ATRP, pre-loading a little deactivator (roughly 10 mol% CuBr2 relative to CuBr) at the start suppresses that initial burst of termination and removes most of the low-conversion broadening.
- Chain transfer and backbiting. Significant for acrylates at high temperature and high conversion, both broaden the product.
- Loss of chain-end fidelity. At very high conversion, chain ends die faster than they propagate, so pushing to quantitative conversion trades Đ and end-group integrity for yield. For a block copolymer this sets a deadline: add the second monomer no later than about 90–95% conversion, not at 100%, or too many first-block chains will have lost the active end needed to grow the second block.
- A drifting C. Monomer depletes and deactivator builds over a run, so the real coefficient is not the single constant used here.
Treat the predicted number as the narrowest Đ the chemistry allows, and a target to design toward, rather than a value you will read off the GPC. When the measured Đ lands well above the prediction, the gap points at which of the effects above is in play.
Where this fits
To see the whole distribution this number summarizes – and to tell a slow-exchange problem from a termination problem, which give the same Đ and different shapes – use the distribution simulator. Set your feed ratios and predicted Mn on the calculator, track the reaction with the conversion monitoring guide, then read the real distribution against the GPC peak interpretation guide and convert a polystyrene-equivalent result with the GPC calibration converter. The mechanisms guide covers the activation/deactivation and addition/fragmentation steps behind the exchange coefficient, and the glossary defines Đ, DP, Cex, and the rest.