source: sasmodels/sasmodels/models/star_polymer.py @ 0507e09

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Last change on this file since 0507e09 was 0507e09, checked in by smk78, 5 months ago

Added link to source code to each model. Closes #883

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Line 
1r"""
2Definition
3----------
4
5Calcuates the scattering from a simple star polymer with f equal Gaussian coil
6arms. A star being defined as a branched polymer with all the branches
7emanating from a common central (in the case of this model) point.  It is
8derived as a special case of on the Benoit model for general branched
9polymers\ [#CITBenoit]_ as also used by Richter *et al.*\ [#CITRichter]_
10
11For a star with $f$ arms the scattering intensity $I(q)$ is calculated as
12
13.. math::
14
15    I(q) = \frac{2}{fv^2}\left[ v-1+\exp(-v)+\frac{f-1}{2}
16           \left[ 1-\exp(-v)\right]^2\right]
17
18where
19
20.. math:: v=\frac{uf}{(3f-2)}
21
22and
23
24.. math:: u = \left\langle R_{g}^2\right\rangle q^2
25
26contains the square of the ensemble average radius-of-gyration of the full
27polymer while v contains the radius of gyration of a single arm $R_{arm}$.
28The two are related as:
29
30.. math:: R_{arm}^2 = \frac{f}{3f-2} R_{g}^2
31
32Note that when there is only one arm, $f = 1$, the Debye Gaussian coil
33equation is recovered.
34
35.. note::
36   Star polymers in solutions tend to have strong interparticle and osmotic
37   effects. Thus the Benoit equation may not work well for many real cases.
38   A newer model for star polymer incorporating excluded volume has been
39   developed by Li et al in arXiv:1404.6269 [physics.chem-ph].  Also, at small
40   $q$ the scattering, i.e. the Guinier term, is not sensitive to the number of
41   arms, and hence 'scale' here is simply $I(q=0)$ as described for the
42   :ref:`mono-gauss-coil` model, using volume fraction $\phi$ and volume V
43   for the whole star polymer.
44
45References
46----------
47
48.. [#CITBenoit] H Benoit *J. Polymer Science*, 11, 507-510 (1953)
49.. [#CITRichter] D Richter, B. Farago, J. S. Huang, L. J. Fetters,
50   B Ewen *Macromolecules*, 22, 468-472 (1989)
51
52Source
53------
54
55`star_polymer.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/star_polymer.py>`_
56
57`star_polymer.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/star_polymer.c>`_
58
59Authorship and Verification
60----------------------------
61
62* **Author:** Kieran Campbell **Date:** July 24, 2012
63* **Last Modified by:** Paul Butler **Date:** Auguts 26, 2017
64* **Last Reviewed by:** Ziang Li and Richard Heenan **Date:** May 17, 2017
65* **Source added by :** Steve King **Date:** March 25, 2019
66"""
67
68import numpy as np
69from numpy import inf
70
71name = "star_polymer"
72title = "Star polymer model with Gaussian statistics"
73description = """
74        Benoit 'Star polymer with Gaussian statistics'
75        with
76        P(q) = 2/{fv^2} * (v - (1-exp(-v)) + {f-1}/2 * (1-exp(-v))^2)
77        where
78        - v = u^2f/(3f-2)
79        - u = <R_g^2>q^2, where <R_g^2> is the ensemble average radius of
80        gyration squared of the entire polymer
81        - f is the number of arms on the star
82        - the radius of gyration of an arm is given b
83        Rg_arm^2 = R_g^2 * f/(3f-2)
84        """
85category = "shape-independent"
86single = False
87# pylint: disable=bad-whitespace, line-too-long
88#             ["name", "units", default, [lower, upper], "type","description"],
89parameters = [["rg_squared", "Ang^2", 100.0, [0.0, inf], "", "Ensemble radius of gyration SQUARED of the full polymer"],
90              ["arms",    "",      3,   [1.0, 6.0], "", "Number of arms in the model"],
91             ]
92# pylint: enable=bad-whitespace, line-too-long
93
94source = ["star_polymer.c"]
95
96def random():
97    """Return a random parameter set for the model."""
98    pars = dict(
99        #background=0,
100        scale=10**np.random.uniform(1, 4),
101        rg_squared=10**np.random.uniform(1, 8),
102        arms=np.random.uniform(1, 6),
103    )
104    return pars
105
106tests = [[{'rg_squared': 2.0,
107           'arms':    3.3,
108          }, 0.5, 0.851646091108],
109
110         [{'rg_squared':    1.0,
111           'arms':       2.0,
112           'background': 1.8,
113          }, 1.0, 2.53575888234],
114        ]
115# 23Mar2016  RKH edited docs, would this better use rg not rg^2 ? Numerical noise at extremely small q.rg
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