Modeling of small angle scattering from branched polymers

In this notebook we continue to demonstrate the capabilities of ESCAPE to simulate small-angle scattering data with custom model functions.

The following example implements a single polymer form-factor for branched polymers formulated by B. Hammouda et al. ( https://doi.org/10.1002/mats.201100111)

Full article is available here: http://www.sciencetopics.net/5.recent_scientific_publications/4.publications_4/2012_hammouda_macromol_theory_and_simul.pdf

Without going into details of the (mass)fractal model, the form-factor of branched polymer is given by:

$P_B(Q)=\frac{1}{Norm}\left[\frac{1}{\nu U_B^{c/2\nu}}\gamma\left(c/2\nu, U_B\right)-\frac{1}{\nu U_B^{(c+1)/2\nu}}\gamma\left((c+1)/2\nu, U_B\right)\right]$

where $Norm = \frac{2}{c(c+1)}$ - the normalization factor,

$U_B=Q^2R_g^2(2\nu+c)(2\nu+c+1)/6$ - scattering variable which expressed in terms of the radii of gyration.

$\nu$ - excluded volume, $c$ - scaling factor and raddi of gyration are the model fit parameters.

The implementation of this model in terms of ESCAPE entities is straightforward.