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2. Shape-Independent
3. Gel Fit
4. gel_fit.py

Gel Fit - gel_fit.py

    r"""
*This model was implemented by an interested user!*

Unlike a concentrated polymer solution, the fine-scale polymer distribution
in a gel involves at least two characteristic length scales,
a shorter correlation length ( $a1$ ) to describe the rapid fluctuations
in the position of the polymer chains that ensure thermodynamic equilibrium,
and a longer distance (denoted here as $a2$ ) needed to account for the static
accumulations of polymer pinned down by junction points or clusters of such
points. The latter is derived from a simple Guinier function. Compare also the
gauss_lorentz_gel model.


Definition
----------

The scattered intensity $I(q)$ is calculated as

.. math::

    I(Q) = I(0)_L \frac{1}{\left( 1+\left[ ((D+1/3)Q^2a_{1}^2
    \right]\right)^{D/2}} + I(0)_G exp\left( -Q^2a_{2}^2\right) + B

where

.. math::

    a_{2}^2 \approx \frac{R_{g}^2}{3}

Note that the first term reduces to the Ornstein-Zernicke equation
when $D = 2$; ie, when the Flory exponent is 0.5 (theta conditions).
In gels with significant hydrogen bonding $D$ has been reported to be
~2.6 to 2.8.


References
----------

.. [#] Mitsuhiro Shibayama, Toyoichi Tanaka, Charles C Han, *J. Chem. Phys.* 1992, 97 (9), 6829-6841

.. [#] Simon Mallam, Ferenc Horkay, Anne-Marie Hecht, Adrian R Rennie, Erik Geissler, *Macromolecules* 1991, 24, 543-548

Authorship and Verification
----------------------------

* **Author:**
* **Last Modified by:**
* **Last Reviewed by:**
"""

import numpy as np
from numpy import inf

name = "gel_fit"
title = "Fitting using fine-scale polymer distribution in a gel."
description = """\
    Structure factor for interacting particles:

    Shibayama-Geissler Two-Length Scale Fit for Gels (GelFit)

    Shibayama; Tanaka; Han J Chem Phys (1992), 97(9), 6829-6841
    Mallam; Horkay; Hecht; Rennie; Geissler, Macromol (1991), 24, 543
"""
category = "shape-independent"

# pylint: disable=bad-whitespace, line-too-long
#             ["name", "units", default, [lower, upper], "type","description"],
parameters = [["guinier_scale",    "cm^-1",   1.7, [-inf, inf], "", "Guinier length scale"],
              ["lorentz_scale", "cm^-1",   3.5, [-inf, inf], "", "Lorentzian length scale"],
              ["rg",  "Ang",     104.0, [2, inf],    "", "Radius of gyration"],
              ["fractal_dim",      "",          2.0, [0, inf],    "", "Fractal exponent"],
              ["cor_length",       "Ang",      16.0, [0, inf],    "", "Correlation length"]
             ]
# pylint: enable=bad-whitespace, line-too-long

source = ["gel_fit.c"]

def random():
    """Return a random parameter set for the model."""
    guinier_scale = 10**np.random.uniform(1, 3)
    lorentz_scale = 10**np.random.uniform(1, 3)
    rg = 10**np.random.uniform(1, 5)
    fractal_dim = np.random.uniform(0, 6)
    cor_length = 10**np.random.uniform(0, 3)
    pars = dict(
        #background=0,
        scale=1,
        guinier_scale=guinier_scale,
        lorentz_scale=lorentz_scale,
        rg=rg,
        fractal_dim=fractal_dim,
        cor_length=cor_length
    )
    return pars

demo = dict(background=0.01,
            guinier_scale=1.7,
            lorentz_scale=3.5,
            rg=104,
            fractal_dim=2.0,
            cor_length=16.0)

tests = [[{'guinier_scale': 1.0,
           'lorentz_scale': 1.0,
           'rg': 10.0,
           'fractal_dim': 10.0,
           'cor_length': 20.0,
           'background': 0.0,
          }, 0.1, 0.716532],

         [{'guinier_scale': 4.0,
           'lorentz_scale': 10.0,
           'rg': 500.0,
           'fractal_dim': 1.0,
           'cor_length': 20.0,
           'background': 20.0,
          }, 5.0, 20.1224653026],
        ]

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