Why a shape and not a number
The dispersity predictor tells you what Đ a controlled polymerization should reach. That is the right number to design against, but it is a summary, and two very different reactions can summarize to the same one. A living population at Đ = 1.24 and a tight population at Đ = 1.05 carrying a fifth of its chains dead both read 1.24 on a report and look nothing alike on a trace. One is a slow-exchange problem you fix with a better RAFT agent; the other is a termination problem you fix with lower radical flux. The single number cannot tell you which you have. The shape can.
This page builds the distribution from three pieces, each with a closed form behind it, and then shows you what the instrument would do to it.
- The living chains as a Schulz–Zimm distribution of shape k = 1/(Đ − 1), which by construction has exactly the dispersity the predictor's equation gives. k = 1 is the Flory most-probable distribution and Đ = 2; large k is nearly monodisperse.
- The dead chains. A chain that terminated at conversion x stopped growing there, so it carries M = Mtarget · x. Termination runs at a roughly steady rate through the reaction, so the dead population is that family swept across every conversion – a broad low-molecular-weight shoulder rather than a second peak.
- The instrument. Every column bank smears each slice, which in log molecular weight is a Gaussian convolution. You do not have to guess its width: run a narrow standard and read the Đ it gives you.
Simulate a distribution
C in the equation is 1/Cex. Above ~20 gives Đ near 1.05; below 1 means the agent is mismatched to the monomer.
Monomer added per activation cycle, divided by the deactivation rate. Lower is better controlled.
As a fraction of all chains, not of mass. Dead chains are short, so they weigh far less than they count.
This is how you measure your own band broadening. A standard that is genuinely 1.00 would read 1.00 on a perfect instrument.
This converts the simulated distribution into the two-column chromatogram a GPC would have exported, using log₁₀M = 10.5 − 0.70·V. Paste it into the trace analyzer to see what integrating it would have reported – which is the round trip: a distribution you know, turned into a trace, integrated back into numbers that no longer quite match.
Three things the shape tells you that Đ does not
Termination hides from Mw. Take a RAFT polymerization sitting at Đ = 1.05 and kill a tenth of its chains. Đ goes to 1.131 and Mₙ falls by 8.1% – but Mw moves only 1.0%, because the dead chains are short and weigh almost nothing. If you are tracking a reaction by Mw, a termination problem is close to invisible; it shows up in Mₙ and in Đ, which is a good reason to report all three rather than the one the instrument prints largest.
Band broadening multiplies Đ. Convolving with a Gaussian in log molecular weight is exact for a log-normal distribution, and it gives a closed form: Đobserved = Đtrue · eσ², where σ is the broadening width in ln M. The factor does not depend on the sample at all, which is why you can measure it once with a standard and apply it to everything. What it does to your conclusions, though, depends entirely on how narrow the sample is:
| True Đ | Observed Đ | True Đ − 1 | Observed Đ − 1 | Excess inflated by |
|---|---|---|---|---|
| 1.02 | 1.062 | 0.020 | 0.062 | 208% |
| 1.05 | 1.093 | 0.050 | 0.093 | 86% |
| 1.10 | 1.145 | 0.100 | 0.145 | 45% |
| 1.50 | 1.561 | 0.500 | 0.561 | 12% |
| 2.00 | 2.082 | 1.000 | 1.082 | 8% |
That table uses σ = 0.20 in ln M, a middling bank. The dispersity itself is inflated by the same 4.1% in every row – but the quantity you reason about is the excess over 1, and that is inflated by 208% for a genuinely narrow polymer and 8% for a broad one. Band broadening is a narrow-sample problem. It is also why a commercial Đ = 1.02 standard is not really telling you the column is that good, and why chasing Đ below about 1.05 by conventional calibration is chasing your own instrument.
A dead-chain tail and a broad living population are not the same distribution. Set the dispersity to 1.24 two ways – once by dropping Cex to about 4, once by killing 20% of the chains at Cex = 20 – and the summary statistic is the same while the curves separate clearly: the first is a single symmetric peak that got wider, the second is a tight peak with a shoulder running down to low molecular weight. On a real trace that shoulder is the thing to look for, and it is the reason to keep the low-molecular-weight end inside your integration limits rather than clipping it off, which the trace analyzer shows the cost of.
What this model does not include
- Slow initiation. Chains that start late are short at any given moment. This model starts every living chain at the same time, so it will under-predict the low-molecular-weight side for a system whose initiator is sluggish relative to propagation – a common problem with mismatched RAFT agents and one more reason a real trace can be broader than the equation says.
- Chain transfer to monomer, solvent or polymer. Transfer to small molecules makes a separate low-molecular-weight population; transfer to polymer makes branches, which elute early and are covered on the calibration page.
- Coupling. Termination is treated as chain-stopping. If your chains terminate by combination instead, the dead population sits at roughly twice the molecular weight and appears as a high-side shoulder, which is the classic diagnostic for combination in a controlled system.
- Any claim about the absolute Đ. The living-chain dispersity comes from the same equation as the dispersity predictor, with the same assumptions and the same caveats. This page changes what you can see, not how much the underlying model knows.
Related tools
- Dispersity predictor – the equation behind the living-chain population, and what sets C for each mechanism.
- GPC trace analyzer – integrate the chromatogram this page exports, and see what the limits cost.
- Calculator – set the target DP and the recipe that produces it.