In a typical step-growth polymerization, the Anderson–Schulz–Flory distribution is the probability mass function that describes the number fraction or weight fraction of polymers of length $x$ at a given extent of reaction $p$. The extent of reaction, $p$, is a value between 0 and 1, and $x$ is the degree of polymerization for any given oligomer or polymer present in the system.
$$\begin{align} \text{mass fraction} &= p^{x-1} \cdot (p-1)^2 \\ \text{weight fraction} &= x \cdot p^{x-1} \cdot (p-1)^2 \end{align}$$
A nice derivation of these equations can be found in this video. Figure 1 illustrates a plot of the number fraction distribution as a function of the degree of polymerization for several extent of reactions. Figure 2 illustrates the plot of the weight fraction distribution as a function of the degree of polymerization for several extent of reactions.
Note that high extents of reaction in step-growth polymerizations result in very broad distributions, and are also required for high molecular weight polymers. Carothers' equation describes how stoichiometric imbalances necessarily limit the possible degree of polymerization:
$$x_\text{average} = \frac{1+r}{1 + r - 2rp}$$
where $x_\text{average}$ is the average degree of polymerization, $p$ is the extent of reaction, and $r$ is the mole fraction of the reacting functional groups. $r$ is always a number between 0 and 1. The excess functional group is always taken to be in the denominator. When $r$ is equal to 1, stoichiometric balance exists, and the equation reduces to:
$$x_\text{average} = \frac{1}{1-p}$$
For example, if I react a diol and a diacid under step-growth conditions:
My goal is to purposely limit the degree of polymerization of a chemical reaction by introducing a stoichiometric imbalance in order to form short chain oligomers. I'd like to predict/calculate the statistical distribution of molecular weights for a given extent of reaction, p
and stoichiometric imbalance, r
.
I'm having trouble convincing myself that the Flory–Schulz distribution is appropriate for modeling step-growth systems that contain stoichiometric imbalances. Is there a straightforward way to incorporate stoichiometric imbalance into these distribution functions? Any help would be appreciated.