.. _star-polymer:

star_polymer
=======================================================

Star polymer model with Gaussian statistics

========== ======================================================= ======= =============
Parameter  Description                                             Units   Default value
========== ======================================================= ======= =============
scale      Scale factor or Volume fraction                         None                1
background Source background                                       |cm^-1|         0.001
rg_squared Ensemble radius of gyration SQUARED of the full polymer |Ang^2|           100
arms       Number of arms in the model                             None                3
========== ======================================================= ======= =============

The returned value is scaled to units of |cm^-1| |sr^-1|, absolute scale.


**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\ [#Benoit1953]_ as also used by Richter *et al.*\ [#Richter1989]_

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

.. math::

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

where

.. math:: v=\frac{uf}{(3f-2)}

and

.. math:: 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:

.. math:: 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
   :ref:`mono-gauss-coil` model, using volume fraction $\phi$ and volume V
   for the whole star polymer.


.. figure:: img/star_polymer_autogenfig.png

    1D plot corresponding to the default parameters of the model.


**Source**

:download:`star_polymer.py <src/star_polymer.py>`
$\ \star\ $ :download:`star_polymer.c <src/star_polymer.c>`

**References**

.. [#Benoit1953] H Benoit *J. Polymer Science*, 11, 507-510 (1953)
.. [#Richter1989] 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
* **Last Modified by:** Paul Butler **Date:** Auguts 26, 2017
* **Last Reviewed by:** Ziang Li and Richard Heenan **Date:** May 17, 2017