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 ($\xi$) to describe the rapid fluctuations in the position
of the polymer chains that ensure thermodynamic equilibrium (based on an
Ornstein-Zernicke, or Lorentz, model), and a longer distance (denoted here
as $R_g$) needed to account for the static accumulations of polymer pinned
down by junction points or clusters of such points (based on a simple
Guinier model). The relative contributions of these two contributions,
$I_L(0)$ and $I_G(0)$, are parameterised as *lorentz_scale* and
*guinier_scale*, respectively.

See also the lorentz model and the gauss_lorentz_gel model.


Definition
----------

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

.. math::

    I(Q) \approx \frac{I_L(0)}{\left(1+\left[(D+1)/3\right]Q^2\xi^2
    \right)^{D/2}} + I_G(0) \cdot \exp\left( -Q^2R_{g}^2/3\right) + B

Note that the first term reduces to the Ornstein-Zernicke equation
when the fractal dimension $D = 2$; 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. DOI: 10.1063/1.463637

#. Simon Mallam, Ferenc Horkay, Anne-Marie Hecht, Adrian R Rennie,
   Erik Geissler, *Macromolecules* 1991, 24, 543-548. DOI: 10.1021/MA00002A031

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

* **Author:**
* **Last Modified by:** Steve King **Date:** November 22, 2022
* **Last Reviewed by:** Paul Kienzle **Date:** November 21, 2022
"""

import numpy as np
from numpy import inf

name = "gel_fit"
title = "Fitting using fine-scale polymer distribution in a gel."
description = """\
    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 term scale"],
              ["lorentz_scale", "cm^-1",   3.5, [-inf, inf], "", "Lorentz term 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

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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