Schulz–Zimm distribution

Schulz–Zimm
	Probability density function
Parameters	(shape parameter)
Support
PDF
Mean
Variance

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

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 $x\geq 0$ , 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

References

^ G V Schulz (1939), Z. Phys. Chem. 43B, 25-46. - Eq (27a) with -ln(a), k+1 in place of our x,k.
^ 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 <n>=1.

[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 <n>=1.

[1]

[2]