This simulation compares the molecular weight distribution (MWD) of polymer leaving an isothermal batch reactor, a plug flow reactor (PFR) and a continuous stirred-tank reactor (CSTR).

Choose step-growth or free-radical polymerization. Choose whether the reactors are compared at the same monomer conversion (the CSTR then needs a longer residence time) or at the same residence time (the CSTR then reaches a lower conversion).

The left plot shows the weight distribution on a logarithmic chain-length axis, the way a GPC trace is normally plotted. The batch and PFR curves are identical because each fluid element in an ideal PFR is a small batch reactor, so they are shown as one curve labeled batch/PFR. The right plot shows polydispersity (PDI) versus conversion; dots mark the current operating points.

The y-axis of the left plot is dW/d(log₁₀ n) = ln(10) n w_n, the weight fraction of polymer per decade of chain length, where w_n is the weight fraction of chains of length n. With this choice, the area under each curve on the logarithmic axis equals 1, so equal areas anywhere on the plot represent equal masses of polymer. A broad, low curve (high PDI) therefore contains the same total mass as a tall, narrow one. The values can exceed 1 because they are a density per decade, not a fraction. For a most-probable distribution, the peak is near the weight-average chain length, x̄_w.

For free-radical polymerization, set the initial kinetic chain length, the initiator decomposition rate, and the termination mode.

Step-growth (A–B monomer, equal reactivity)

Second-order kinetics in functional groups, with conversion $p$ and Damköhler number $\mathrm{Da} = k C_0 t$ (batch/PFR) or $k C_0 \tau$ (CSTR).

Batch reactor or PFR:

$$p = \frac{\mathrm{Da}}{1+\mathrm{Da}}$$ $$w_n = n(1-p)^2 p^{\,n-1}, \qquad \bar{x}_n = \frac{1}{1-p}, \qquad \bar{x}_w = \frac{1+p}{1-p}, \qquad \mathrm{PDI} = 1+p$$

CSTR (perfectly micromixed, steady state). The population balance for $n$-mers is

$$0 = \frac{\delta_{n1}C_0 - P_n}{\tau} + k\sum_{i=1}^{n-1}P_iP_{n-i} - 2k\lambda_0 P_n$$

where $\lambda_0 = \sum P_n = C_0(1-p)$. The conversion satisfies $p = \mathrm{Da}(1-p)^2$. Solving with a generating function gives

$$\frac{P_n}{C_0} = \frac{a}{2\,\mathrm{Da}}\;\frac{(2n-2)!}{2^{2n-1}\,n!\,(n-1)!}\;z^n, \qquad a = \frac{1+p}{1-p}, \quad z = \frac{4p}{(1+p)^2}$$ $$w_n = n\,\frac{P_n}{C_0}, \qquad \bar{x}_w = 1+2\,\mathrm{Da}, \qquad \mathrm{PDI} = \frac{1+p^2}{1-p}$$

In the CSTR, a molecule's age is distributed exponentially, and old (large) molecules react with new (small) ones. This back-mixing makes the distribution far broader than in a batch reactor: at $p = 0.99$, PDI = 1.99 for batch but 199 for the CSTR.

Free-radical (initiator decomposition, propagation, termination)

With the quasi-steady-state assumption for radicals, $[R\cdot] = \left(2fk_d[I]/k_t\right)^{1/2}$ and the instantaneous kinetic chain length is

$$\nu = \frac{k_p[M]}{2\left(f k_d k_t [I]\right)^{1/2}} = \nu_0\,\frac{[M]}{[M]_0}\left(\frac{[I]_0}{[I]}\right)^{1/2}$$

Dead polymer formed at any instant has a Schulz–Flory distribution (continuous form):

$$\text{disproportionation: } w(n) = \frac{n}{\nu^2}e^{-n/\nu}, \;\; \mathrm{PDI}=2 \qquad \text{combination: } w(n) = \frac{n^2}{2\nu^3}e^{-n/\nu}, \;\; \mathrm{PDI}=1.5$$

Dimensionless time is $\theta = k_p[R\cdot]_0\,t$ and the initiator decay ratio is $\beta = k_d/(k_p[R\cdot]_0)$.

Batch reactor or PFR: $[I]/[I]_0 = e^{-\beta\theta}$ and

$$X = 1 - \exp\!\left[-\frac{2}{\beta}\left(1-e^{-\beta\theta/2}\right)\right], \qquad \nu = \frac{\nu_0(1-X)}{1-\beta L/2}, \quad L = -\ln(1-X)$$

Because $\nu$ drifts as monomer and initiator are consumed, the product is a superposition of instantaneous distributions:

$$W(n) = \frac{1}{X}\int_0^X w\big(n;\nu(X')\big)\,dX'$$

CSTR: $[I]/[I]_0 = 1/(1+\beta\theta)$ and $X/(1-X) = \theta/\sqrt{1+\beta\theta}$. Every radical grows and dies at the same uniform $[M]$ and $[I]$, and a radical's lifetime (seconds) is much shorter than $\tau$, so the residence time distribution does not broaden the product. The CSTR product is the single instantaneous distribution, PDI = 2 (disproportionation) or 1.5 (combination), which is narrower than the batch product.

Chain transfer, gel effect, and temperature changes are not included. Chain lengths are in monomer repeat units; multiply by the repeat-unit molecular weight to obtain $M_n$ and $M_w$.

Definitions

$a$	parameter in the CSTR step-growth solution, $(1+p)/(1-p)$
$C_0$	initial (feed) concentration of functional groups (mol/L)
$\mathrm{Da}$	Damköhler number for step growth, $kC_0t$ (batch/PFR) or $kC_0\tau$ (CSTR)
$dW/d(\log_{10}n)$	weight fraction of polymer per decade of chain length, $\ln(10)\,n\,w_n$
$f$	initiator efficiency
$[I],\ [I]_0$	initiator concentration; feed or initial value (mol/L)
$k$	step-growth rate constant (L/(mol·s))
$k_d$	initiator decomposition rate constant (1/s)
$k_p$	propagation rate constant (L/(mol·s))
$k_t$	termination rate constant (L/(mol·s))
$L$	$-\ln(1-X)$
$[M],\ [M]_0$	monomer concentration; feed or initial value (mol/L)
$M_n,\ M_w$	number- and weight-average molecular weights (g/mol)
$n$	chain length (number of repeat units in a molecule)
$p$	conversion of functional groups (step growth)
$P_n$	concentration of chains of length $n$ (mol/L)
$\mathrm{PDI}$	polydispersity, $\bar{x}_w/\bar{x}_n$
$[R\cdot],\ [R\cdot]_0$	total radical concentration; initial batch value (mol/L)
$t$	reaction time in the batch reactor, or residence time in the PFR (s)
$w_n,\ w(n)$	weight fraction of chains of length $n$ (per unit $n$)
$W(n)$	weight distribution of the accumulated batch/PFR product
$X$	monomer conversion (free radical)
$\bar{x}_n,\ \bar{x}_w$	number- and weight-average chain lengths
$z$	parameter in the CSTR step-growth solution, $4p/(1+p)^2$
$\beta$	initiator decay ratio, $k_d/(k_p[R\cdot]_0)$
$\delta_{n1}$	Kronecker delta (1 if $n=1$, otherwise 0)
$\theta$	dimensionless time, $k_p[R\cdot]_0t$ (batch/PFR) or $k_p[R\cdot]_0\tau$ (CSTR)
$\lambda_0$	total concentration of molecules (zeroth moment, mol/L)
$\nu,\ \nu_0$	instantaneous kinetic chain length; value at feed conditions
$\tau$	CSTR residence time (s)

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.