# Example:Polymer clustering

Polymer solutions frequently exhibit some degree of clustering of the polymer chains. A more extreme case are hydrogels, where the polymer chains may be strongly-associated, or even crosslinked, into a network or mesh.

Hammouda et al. proposed the following functional form to describe scattering intensity from such systems:

$I(q) = \frac{A}{q^n} + \frac{C}{1 + (q \xi)^m} + B$

where B is a constant background. The first term represents the Porod scattering from clusters, while the second term is a Lorentzian function ascribed to the scattering of the polymer chains themselves. In the context of a gel, $\scriptstyle \xi$ represents the average mesh size. The parameters A, C, n, and m may be used as fitting parameters.

## Kratky plot

In a Kratky plot ($\scriptstyle q^2I(q)$ vs. $\scriptstyle q$), the equation becomes:

$q^2 I(q) = \frac{A}{q^{n-2}} + \frac{C q^2}{1 + (q \xi)^m} + B q^2$