Schulz–Zimm distribution explained

Schulz–Zimm
Type:	continuous
Pdf Image:	SchulzZimmPDF.pdf
Parameters:	k (shape parameter)
Support:	x\in R _{>0}
Pdf:	(k ^{k} x ^{k} e ^{-kx})/(\Gamma(k))
Mean:	1
Variance:	(1)/(k)

The Schulz–Zimm distribution is a special case of the gamma distribution. It is widely used to model the polydispersity of polymers. In this context it has been introduced in 1939 by Günter Victor Schulz^[1] and in 1948 by Bruno H. Zimm.^[2]

This distribution has only a shape parameter k, the scale being fixed at θ=1/k. Accordingly, the probability density function is$$f(x)=\backslash frac.$$

When applied to polymers, the variable x is the relative mass or chain length

x = M/M _{n} . Accordingly, the mass distribution f(M) is just a gamma distribution with scale parameter \theta = M _{n/k} . This explains why the Schulz–Zimm distribution is unheard of outside its conventional application domain.

The distribution has mean 1 and variance 1/k. The polymer dispersity is

\langle x ^{2\rangle} / \langle x\rangle = 1+1 /k .

For large k the Schulz–Zimm distribution approaches a Gaussian distribution. In algorithms where one needs to draw samples

x\ge 0 , the Schulz–Zimm distribution is to be preferred over a Gaussian because the latter requires an arbitrary cut-off to prevent negative x.

Notes and References

1. G V Schulz (1939), Z. Phys. Chem. 43B, 25-46. - Eq (27a) with -ln(a), k+1 in place of our x,k.
2. B H Zimm (1948), J. Chem. Phys. 16, 1099. - Proposes a two-parameter variant of Eq (13) without derivation and without reference to Schulz or whomsoever. One of the two parameters can be eliminated by the requirement =1.