Flexible Cylinder Elliptical - flexible_cylinder_elliptical.py
r"""
This model calculates the form factor for a flexible cylinder with an
elliptical cross section and a uniform scattering length density.
The non-negligible diameter of the cylinder is included by accounting
for excluded volume interactions within the walk of a single cylinder.
**Inter-cylinder interactions are NOT provided for.**
The form factor is normalized by the particle volume such that
.. math::
P(q) = \text{scale} \left<F^2\right>/V + \text{background}
where the averaging $\left<\ldots\right>$ is over all possible orientations
of the flexible cylinder.
The 2D scattering intensity is the same as 1D, regardless of the orientation
of the q vector which is defined as
.. math::
q = \sqrt{q_x^2 + q_y^2}
Definitions
-----------
The function is calculated in a similar way to that for the
:ref:`flexible-cylinder` model in reference [1] below using the author's
"Method 3 With Excluded Volume".
The model is a parameterization of simulations of a discrete representation of
the worm-like chain model of Kratky and Porod applied in the pseudo-continuous
limit. See equations (13, 26-27) in the original reference for the details.
.. note::
There are several typos in the original reference that have been corrected
by WRC [2]. Details of the corrections are in the reference below. Most notably
- Equation (13): the term $(1 - w(QR))$ should swap position with $w(QR)$
- Equations (23) and (24) are incorrect; WRC has entered these into
Mathematica and solved analytically. The results were then converted to
code.
- Equation (27) should be $q0 = max(a3/(Rg^2)^{1/2},3)$ instead of
$max(a3*b(Rg^2)^{1/2},3)$
- The scattering function is negative for a range of parameter values and
q-values that are experimentally accessible. A correction function has been
added to give the proper behavior.
.. figure:: img/flexible_cylinder_ex_geometry.jpg
The chain of contour length, $L$, (the total length) can be described as a chain
of some number of locally stiff segments of length $l_p$, the persistence length
(the length along the cylinder over which the flexible cylinder can be considered
a rigid rod).
The Kuhn length $(b = 2*l_p)$ is also used to describe the stiffness of a chain.
The cross section of the cylinder is elliptical, with minor radius $a$ .
The major radius is larger, so of course, **the axis_ratio must be
greater than one.** Simple constraints should be applied during curve fitting to
maintain this inequality.
In the parameters, the $sld$ and $sld\_solvent$ represent the SLD of the
chain/cylinder and solvent respectively. The *scale*, and the contrast are both
multiplicative factors in the model and are perfectly correlated. One or both of
these parameters must be held fixed during model fitting.
**This is a model with complex behaviour depending on the ratio of** $L/b$ **and the
reader is strongly encouraged to read reference [1] before use.**
References
----------
.. [#] J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers with and without excluded volume effects.* Macromolecules, 29 (1996) 7602-7612
Correction of the formula can be found in
.. [#] W R Chen, P D Butler and L J Magid, *Incorporating Intermicellar Interactions in the Fitting of SANS Data from Cationic Wormlike Micelles.* Langmuir, 22(15) 2006 6539-6548
Authorship and Verification
----------------------------
* **Author:**
* **Last Modified by:** Richard Heenan **Date:** December, 2016
* **Last Reviewed by:** Steve King **Date:** March 26, 2019
"""
import numpy as np
from numpy import inf
name = "flexible_cylinder_elliptical"
title = "Flexible cylinder wth an elliptical cross section and a uniform " \
"scattering length density."
description = """Note : scale and contrast=sldCyl-sldSolv are both multiplicative
factors in the
model and are perfectly correlated. One or
both of these parameters must be held fixed
during model fitting.
"""
single = False
category = "shape:cylinder"
# pylint: disable=bad-whitespace, line-too-long
# ["name", "units", default, [lower, upper], "type", "description"],
parameters = [
["length", "Ang", 1000.0, [0, inf], "volume", "Length of the flexible cylinder"],
["kuhn_length", "Ang", 100.0, [0, inf], "volume", "Kuhn length of the flexible cylinder"],
["radius", "Ang", 20.0, [1, inf], "volume", "Radius of the flexible cylinder"],
["axis_ratio", "", 1.5, [0, inf], "", "Axis_ratio (major_radius/minor_radius"],
["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Cylinder scattering length density"],
["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Solvent scattering length density"],
]
# pylint: enable=bad-whitespace, line-too-long
source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "lib/wrc_cyl.c",
"flexible_cylinder_elliptical.c"]
def random():
"""Return a random parameter set for the model."""
length = 10**np.random.uniform(2, 6)
radius = 10**np.random.uniform(1, 3)
axis_ratio = 10**np.random.uniform(-1, 1)
kuhn_length = 10**np.random.uniform(-2, -0.7)*length # at least 10 segments
pars = dict(
length=length,
radius=radius,
axis_ratio=axis_ratio,
kuhn_length=kuhn_length,
)
return pars
tests = [
# Accuracy tests based on content in test/utest_other_models.py
# Currently fails in OCL
# [{'length': 1000.0,
# 'kuhn_length': 100.0,
# 'radius': 20.0,
# 'axis_ratio': 1.5,
# 'sld': 1.0,
# 'sld_solvent': 6.3,
# 'background': 0.0001,
# }, 0.001, 3509.2187],
# Additional tests with larger range of parameters
[{'length': 1000.0,
'kuhn_length': 100.0,
'radius': 20.0,
'axis_ratio': 1.5,
'sld': 1.0,
'sld_solvent': 6.3,
'background': 0.0001,
}, 1.0, 0.00223819],
[{'length': 10.0,
'kuhn_length': 800.0,
'radius': 2.0,
'axis_ratio': 0.5,
'sld': 6.0,
'sld_solvent': 12.3,
'background': 0.001,
}, 0.1, 0.390281],
[{'length': 100.0,
'kuhn_length': 800.0,
'radius': 50.0,
'axis_ratio': 4.5,
'sld': 0.1,
'sld_solvent': 5.1,
'background': 0.0,
}, 1.0, 0.0016338264790]
]
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