# Star Polymer

## Description:

Definition

Calcuates the scattering from a simple star polymer with f equal Gaussian coil arms. A star being defined as a branched polymer with all the branches emanating from a common central (in the case of this model) point. It is derived as a special case of on the Benoit model for general branched polymers as also used by Richter *et al.*

For a star with $f$ arms the scattering intensity $I(q)$ is calculated as

$$I(q) = \frac{2}{fv^2}\left[ v-1+\exp(-v)+\frac{f-1}{2} \left[ 1-\exp(-v)\right]^2\right]$$
where

$$v=\frac{uf}{(3f-2)}$$
and

$$u = \left\langle R_{g}^2\right\rangle q^2$$
contains the square of the ensemble average radius-of-gyration of the full polymer while v contains the radius of gyration of a single arm $R_{arm}$. The two are related as:

$$R_{arm}^2 = \frac{f}{3f-2} R_{g}^2$$
Note that when there is only one arm, $f = 1$, the Debye Gaussian coil equation is recovered.

.. note:: Star polymers in solutions tend to have strong interparticle and osmotic effects. Thus the Benoit equation may not work well for many real cases. A newer model for star polymer incorporating excluded volume has been developed by Li et al in arXiv:1404.6269 [physics.chem-ph]. Also, at small $q$ the scattering, i.e. the Guinier term, is not sensitive to the number of arms, and hence 'scale' here is simply $I(q=0)$ as described for the `mono-gauss-coil` model, using volume fraction $\phi$ and volume V for the whole star polymer.

References

H Benoit *J. Polymer Science*, 11, 507-510 (1953)
D Richter, B. Farago, J. S. Huang, L. J. Fetters,
B Ewen *Macromolecules*, 22, 468-472 (1989)

Authorship and Verification

**Author:** Kieran Campbell **Date:** July 24, 2012