Schulz-Zimm distribution

The Schulz-Zimm distribution is a widely used model for the polydispersity of polymers.

Probability density function of the Schulz-Zimm distribution for three values of the shape parameter k.

The Schulz-Zimm distribution is a special case of the gamma distribution. The latter has two parameters, shape k, and scale θ. The Schulz-Zimm distribution has only a shape parameter k, the scale being fixed at θ=1/k. Accordingly, the probability density function is

${\displaystyle f(x)={\frac {x^{k-1}e^{-kx}}{\Gamma (k)k^{k}}}.}$

When applied to polymers, the variable x is the relative mass or chain length ${\displaystyle x=N/N_{n}}$.

The distribution has mean 1 and variance 1/k. The polymer dispersity is ${\displaystyle \langle x^{2}\rangle /\langle x\rangle =1+1/k}$.

For large k the Schulz-Zimm distribution approaches a Gaussian. In algorithms where one needs to draw samples ${\displaystyle x\geq 0}$, the Schulz-Zimm distribution is to be preferred over a Gaussian because the latter requires an arbitrary cut-off to prevent negative x.

Schulz-Zimm distribution with k=100, and Gaussian distribution with same mean and variance