This simulation shows how a polymer network (a gel) forms during step-growth polymerization when some of the monomers have more than two reactive groups. Two-functional monomers A–A and B–B make linear chains, but a branch monomer with f A groups (f ≥ 3) links chains together. At a critical conversion, the Flory–Stockmayer gel point, the weight-average molecular weight becomes infinite and an insoluble network appears. Beyond the gel point, the polymer is divided into a soluble fraction (sol) and the network (gel).
Set the functionality f, the branching ratio ρ (fraction of all A groups that are on the f-functional monomer), and the stoichiometric ratio r (A groups/B groups). Then move the conversion slider, or press start to increase conversion automatically and reset to return conversion to zero.
The left plot shows the weight-average (Mw) and number-average (Mn) molecular weights on a log scale. The dots mark the values at the current conversion. Mw diverges at the Flory–Stockmayer gel point (vertical dashed line), while Mn stays finite. Beyond the gel point (shaded) an infinite network (the gel) coexists with the sol (the finite molecules), and the curves are averages only for the sol.
Molecules grow by joining together. Each branch unit adds extra unreacted ends, so the larger a molecule gets, the more ways it has to keep growing. At the gel point, the largest clusters link into one network that spans the whole sample. That network is the gel, a single molecule of effectively infinite molecular weight. Everything not connected to it is the sol. Beyond the gel point, the gel acts as a sink.
The right plot shows the weight fractions of sol and gel. When the gel point conversion would be greater than 1, the system cannot gel.
Why the sol molecular weight decreases after the gel point: the gel has a large number of unreacted groups, so most new bonds connect a sol molecule to the gel. A sol molecule joins the gel when any one of its unreacted ends reacts with it, so the largest, most branched sol molecules are captured first. The sol that remains is mostly small, nearly linear molecules and unreacted monomer, so its Mw and Mn decrease. Mass moves only from sol to gel, so the gel fraction wg keeps increasing; when r = 1, all the material is in the gel at complete conversion.
The monomers are A–A (100 g/mol), Af (150 g/mol), and B–B (100 g/mol). Only A–B reactions occur, all groups are equally reactive, and no intramolecular (cyclization) reactions occur before the gel point. The branching ratio is the fraction of A groups on the branch monomer:
Flory’s branching coefficient
The gel point occurs when the expected number of new branches reached from a branch is one, so that
Molecular weights are calculated with the Macosko–Miller recursive method. Let
Before the gel point,
where
and the weight-average molecular weight of the sol is
Solving the linear equations for
Before the gel point this reduces to the Carothers result, total mass/(monomers − bonds formed). Because cyclization is neglected, real systems gel at somewhat higher conversion than predicted.
Symbols:
| reactive groups (A reacts only with B) | |
| branch monomer with | |
| functionality of the branch monomer | |
| mass of monomer | |
| molecular weights of the monomers (g/mol) | |
| number-average molecular weight of the sol (g/mol) | |
| weight-average molecular weight of the sol (g/mol) | |
| moles of A groups and B groups | |
| moles of each monomer | |
| fractional conversions of A groups and B groups | |
| conversion of A groups at the gel point | |
| stoichiometric ratio, A groups/B groups | |
| probabilities that looking out from an A or a B group leads to a finite chain | |
| expected mass attached when looking out from an A or a B group in the sol (g/mol) | |
| weight fractions of gel and sol | |
| branching coefficient | |
| branching coefficient at the gel point | |
| branching ratio, fraction of A groups on the branch monomer |
This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.