Past the single gel-point formula
The step-growth calculator gives the Flory–Stockmayer gel point, which answers the question for one functionality at one stoichiometry. Real formulations are rarely that tidy: a triol with a little diol in it, a diisocyanate with a trifunctional impurity, deliberate excess of one component to stop short of gelation, or a chain extender added to soften the network. Each of those changes the answer, and none of them fits the single formula.
The Macosko–Miller recursive method handles all of them with one idea. Stand on a reacted group and look outward: the material beyond is either finite or part of the infinite network. Call the probability that it is finite Pout. Whatever is out there is reached through the monomer on the far side, so Pout can be written in terms of itself:
PBout = (1 − pB) + pB · Σi αi (PAout)fi−1
An unreacted group leads nowhere, which is finite – that is the (1 − p) term. A reacted one leads into a monomer of functionality f, and everything beyond its other f − 1 groups must also be finite. The sums run over the functionalities present, weighted by αi, the fraction of all A groups sitting on monomers of functionality fi. Arbitrary mixtures come for free, because they are just more terms in a sum.
Everything else follows. The gel point is where a solution below 1 first appears, which happens when pApB·E[f−1]·E[g−1] = 1. The sol fraction is the weight of material all of whose arms lead nowhere. And a junction is elastically effective when at least three of its arms reach the infinite network, which turns the same probability into a crosslink density.
Build a network
The other group's conversion follows from the stoichiometry: pA·[A] = pB·[B], because every reaction consumes one of each.
Three things that only show up when you can vary the formulation
Excess of one component can stop gelation entirely. A triol cured with a diisocyanate gels at 70.7% conversion when the groups are balanced. Run the isocyanate in excess – or, equivalently, leave hydroxyl short – and the gel point climbs, because the limiting group cannot reach the conversion the network needs. At a group ratio of exactly 0.5 the gel point lands precisely at full conversion, so below that ratio an A₃ + B₂ system never gels no matter how long you cook it. That is not a numerical curiosity; it is how you formulate a branched prepolymer that stays processable.
| A groups ÷ B groups | Conversion of A at the gel point |
|---|---|
| 1.00 (balanced) | 0.7071 |
| 0.90 | 0.7454 |
| 0.80 | 0.7906 |
| 0.70 | 0.8452 |
| 0.60 | 0.9129 |
| 0.50 | 1.0000 — the limit |
The sol fraction collapses far faster than the conversion moves. Past the gel point a very small further extent of reaction pulls most of the remaining soluble material into the network. For the balanced triol system, going 1% past the gel point – conversion 0.707 to 0.714 – takes the sol fraction from 1 to 0.89; 5% past it, to 0.55; 20% past it, to 0.07. That steepness is why extraction is such a sensitive test of cure, and why a sample that still gives 10% extractables is not nearly cured, it is barely past gelling.
Elastic strands are not the same as crosslinks. A junction only contributes to the modulus if at least three of its arms actually reach the infinite network; one with two attached arms is a chain extender and one with fewer is a dangling end. Near the gel point most junctions fail that test even though the material is technically a gel, which is why modulus rises so much more slowly than gel fraction just past the gel point. The elastic strand density this page reports counts only the junctions that qualify, and the Mc it gives can be compared directly against the value the crosslink density page extracts from swelling or the rubbery plateau. If the formulation predicts a much tighter network than the measurement finds – the usual result – the difference is not dangling ends, which the three-arm rule has already removed. It is intramolecular loops, which the theory assumes never form, or a real conversion lower than the one you entered.
What the theory assumes, and where that bites
- No intramolecular reaction. Every bond is assumed to join two different molecules. Real systems form loops, especially in dilute conditions and with short flexible spacers, and every loop is a bond that consumed two groups without contributing to the network. The consequence is systematic and always in one direction: the real gel point is later than this page predicts, and the real modulus lower. Dilution makes it worse.
- Equal and independent reactivity. All groups of one type are assumed identical and unaffected by whether their neighbours have reacted. Substitution effects are real – the second hydroxyl of a diol often differs from the first, and steric hindrance grows as a junction fills – and they shift the gel point either way.
- One reaction. Side reactions that consume groups without linking (isocyanate with water, for instance) are not modelled. Subtract them from the functionality or the moles before you start, because the page cannot see them.
- A mean-field count, not a structure. This returns how much network there is, not how it is arranged. Two formulations with the same strand density and very different spatial heterogeneity will behave differently, and nothing here will tell you that.
Related tools
- Step-growth and gel point – the single-formula version, and Carothers for the pre-gel molecular weight.
- Crosslink density – Mc measured from swelling or the rubbery plateau, to compare against what the formulation predicts.
- Polyurethane calculator – equivalent weights and index for the most common system this applies to.