source: sasmodels/sasmodels/models/star_polymer.py @ 2d81cfe

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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
52Authorship and Verification
53----------------------------
54
55* **Author:** Kieran Campbell **Date:** July 24, 2012
56* **Last Modified by:** Paul Butler **Date:** Auguts 26, 2017
57* **Last Reviewed by:** Ziang Li and Richard Heenan **Date:** May 17, 2017
58"""
59
60import numpy as np
61from numpy import inf
62
63name = "star_polymer"
64title = "Star polymer model with Gaussian statistics"
65description = """
66        Benoit 'Star polymer with Gaussian statistics'
67        with
68        P(q) = 2/{fv^2} * (v - (1-exp(-v)) + {f-1}/2 * (1-exp(-v))^2)
69        where
70        - v = u^2f/(3f-2)
71        - u = <R_g^2>q^2, where <R_g^2> is the ensemble average radius of
72        gyration squared of the entire polymer
73        - f is the number of arms on the star
74        - the radius of gyration of an arm is given b
75        Rg_arm^2 = R_g^2 * f/(3f-2)
76        """
77category = "shape-independent"
78single = False
79# pylint: disable=bad-whitespace, line-too-long
80#             ["name", "units", default, [lower, upper], "type","description"],
81parameters = [["rg_squared", "Ang^2", 100.0, [0.0, inf], "", "Ensemble radius of gyration SQUARED of the full polymer"],
82              ["arms",    "",      3,   [1.0, 6.0], "", "Number of arms in the model"],
83             ]
84# pylint: enable=bad-whitespace, line-too-long
85
86source = ["star_polymer.c"]
87
88def random():
89    pars = dict(
90        #background=0,
91        scale=10**np.random.uniform(1, 4),
92        rg_squared=10**np.random.uniform(1, 8),
93        arms=np.random.uniform(1, 6),
94    )
95    return pars
96
97tests = [[{'rg_squared': 2.0,
98           'arms':    3.3,
99          }, 0.5, 0.851646091108],
100
101         [{'rg_squared':    1.0,
102           'arms':       2.0,
103           'background': 1.8,
104          }, 1.0, 2.53575888234],
105        ]
106# 23Mar2016  RKH edited docs, would this better use rg not rg^2 ? Numerical noise at extremely small q.rg
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