In [ ]:
import escape as esc
esc.require('0.9.1')
import numpy as np
from escape.utils import show

SAXS. Modeling of 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.

In [ ]:
# Q variable
q = esc.var("Q")

# Radii of gyration
RG = esc.par("RG", 150, units="nm")

#excluded volume
VM = esc.par("VM", 0.6, units="arbitr")

#scaling factor
C = esc.par("C", 1, units="arbitr", fixed=True, userlim=[1, 2])


f = 2*VM+C
Ub = q*q*RG*RG*f*(f+1)/6
norm_inv = 0.5*(C*C+C)

#we need normalized lower gamma function
gamma = esc.gamma_p
pow = esc.pow

#intensity
P = (gamma(0.5*C/VM, Ub)/(VM*pow(Ub, 0.5*C/VM))-gamma(0.5*(C+1)/VM, Ub)/(VM*pow(Ub, 0.5*(C+1)/VM)))*norm_inv;
In [ ]:
show(P, coordinates=np.linspace(0, 1, 1000), ylog=True)
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