# 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:

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

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, ${\displaystyle \scriptstyle \xi }$ represents the average mesh size. The parameters A, C, n, and m may be used as fitting parameters.

## Kratky plot

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

${\displaystyle q^{2}I(q)={\frac {A}{q^{n-2}}}+{\frac {Cq^{2}}{1+(q\xi )^{m}}}+Bq^{2}}$