This simulation shows how the rate of initiation, relative to the rate of propagation, affects the chain-length distribution and the polydispersity index (PDI) of a polymer made by living anionic polymerization in a batch reactor. Because there is no termination or chain transfer, fast initiation produces a narrow Poisson distribution with PDI ≈ 1 + 1/Xn. When initiation is slow, chains start at different times, some initiator is still unreacted when the monomer is consumed, and the distribution broadens.
Set the ratio of initial monomer to initiator concentrations [M]0/[I]0 and the ratio of initiation to propagation rate constants ki/kp (log scale). The fast initiation and slow initiation buttons set ki/kp to 100 and 0.01. Drag the monomer conversion slider, or press Play to follow the polymerization from low conversion to complete conversion. Select number fraction or weight fraction for the distribution plot.
The left plot compares the chain-length distribution (red) with the Poisson distribution for instantaneous initiation at the same monomer conversion (black, dashed). The right plot shows the polydispersity index versus monomer conversion for the same two cases.
Initiation and propagation, with no termination or transfer:
The batch mole balances are
Because every step is first order in [M], a dimensionless time that absorbs the monomer concentration makes the equations linear:
The moments of the chain-length distribution, , have analytical solutions. With
Each monomer that reacts adds one unit to a chain, so the monomer conversion is , where . For a chosen conversion, τ is found from this equation, and then
The full distribution (the Gold distribution) is a superposition of Poisson distributions for chains that started at different times s:
The number fraction is and the weight fraction is .
For instantaneous initiation (κ → ∞) all chains start at τ = 0, the distribution is Poisson with mean number of added monomers ν = τ = Xn − 1, and
When κ < 1, chains initiated early grow long while late ones stay short, and part of the initiator never reacts, so Xn exceeds [M]0/[I]0. Even for very slow initiation, the PDI of a living polymerization remains well below the values of about 2 typical of step-growth polymerization.
Symbols:
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.