1 | r""" |
---|
2 | Definition |
---|
3 | ---------- |
---|
4 | |
---|
5 | Calcuates the scattering from a simple star polymer with f equal Gaussian coil |
---|
6 | arms. A star being defined as a branched polymer with all the branches |
---|
7 | emanating from a common central (in the case of this model) point. It is |
---|
8 | derived as a special case of on the Benoit model for general branched |
---|
9 | polymers\ [#CITBenoit]_ as also used by Richter *et al.*\ [#CITRichter]_ |
---|
10 | |
---|
11 | For 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 | |
---|
18 | where |
---|
19 | |
---|
20 | .. math:: v=\frac{uf}{(3f-2)} |
---|
21 | |
---|
22 | and |
---|
23 | |
---|
24 | .. math:: u = \left\langle R_{g}^2\right\rangle q^2 |
---|
25 | |
---|
26 | contains the square of the ensemble average radius-of-gyration of the full |
---|
27 | polymer while v contains the radius of gyration of a single arm $R_{arm}$. |
---|
28 | The two are related as: |
---|
29 | |
---|
30 | .. math:: R_{arm}^2 = \frac{f}{3f-2} R_{g}^2 |
---|
31 | |
---|
32 | Note that when there is only one arm, $f = 1$, the Debye Gaussian coil |
---|
33 | equation 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 | |
---|
45 | References |
---|
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 | |
---|
52 | Source |
---|
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 | |
---|
59 | Authorship 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 | |
---|
68 | import numpy as np |
---|
69 | from numpy import inf |
---|
70 | |
---|
71 | name = "star_polymer" |
---|
72 | title = "Star polymer model with Gaussian statistics" |
---|
73 | description = """ |
---|
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 | """ |
---|
85 | category = "shape-independent" |
---|
86 | single = False |
---|
87 | # pylint: disable=bad-whitespace, line-too-long |
---|
88 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
89 | parameters = [["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 | |
---|
94 | source = ["star_polymer.c"] |
---|
95 | |
---|
96 | def 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 | |
---|
106 | tests = [[{'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 |
---|