1 | r""" |
---|
2 | The Benoit model for a simple star polymer, with Gaussian coils arms from |
---|
3 | a common point. |
---|
4 | |
---|
5 | Definition |
---|
6 | ---------- |
---|
7 | |
---|
8 | For a star with $f$ arms the scattering intensity $I(q)$ is calculated as |
---|
9 | |
---|
10 | .. math:: |
---|
11 | |
---|
12 | I(q) = \frac{2}{fv^2}\left[ v-1+\exp(-v)+\frac{f-1}{2} |
---|
13 | \left[ 1-\exp(-v)\right]^2\right] |
---|
14 | |
---|
15 | where |
---|
16 | |
---|
17 | .. math:: v=\frac{u^2f}{(3f-2)} |
---|
18 | |
---|
19 | and |
---|
20 | |
---|
21 | .. math:: u = \left\langle R_{g}^2\right\rangle q^2 |
---|
22 | |
---|
23 | contains the square of the ensemble average radius-of-gyration of an arm. |
---|
24 | Note that when there is only one arm, $f = 1$, the Debye Gaussian coil |
---|
25 | equation is recovered. Star polymers in solutions tend to have strong |
---|
26 | interparticle and osmotic effects, so the Benoit equation may not work well. |
---|
27 | At small $q$ the Guinier term and hence $I(q=0)$ is the same as for $f$ arms |
---|
28 | of radius of gyration $R_g$, as described for the :ref:`mono-gauss-coil` model. |
---|
29 | |
---|
30 | References |
---|
31 | ---------- |
---|
32 | |
---|
33 | H Benoit *J. Polymer Science*, 11, 596-599 (1953) |
---|
34 | """ |
---|
35 | |
---|
36 | from numpy import inf |
---|
37 | |
---|
38 | name = "star_polymer" |
---|
39 | title = "Star polymer model with Gaussian statistics" |
---|
40 | description = """ |
---|
41 | Benoit 'Star polymer with Gaussian statistics' |
---|
42 | with |
---|
43 | P(q) = 2/{fv^2} * (v - (1-exp(-v)) + {f-1}/2 * (1-exp(-v))^2) |
---|
44 | where |
---|
45 | - v = u^2f/(3f-2) |
---|
46 | - u = <R_g^2>q^2, where <R_g^2> is the ensemble average radius of |
---|
47 | gyration squared of an arm |
---|
48 | - f is the number of arms on the star |
---|
49 | """ |
---|
50 | category = "shape-independent" |
---|
51 | single = False |
---|
52 | # pylint: disable=bad-whitespace, line-too-long |
---|
53 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
54 | parameters = [["rg_squared", "Ang^2", 100.0, [0.0, inf], "", "Ensemble radius of gyration SQUARED of an arm"], |
---|
55 | ["arms", "", 3, [1.0, 6.0], "", "Number of arms in the model"], |
---|
56 | ] |
---|
57 | # pylint: enable=bad-whitespace, line-too-long |
---|
58 | |
---|
59 | source = ["star_polymer.c"] |
---|
60 | |
---|
61 | demo = dict(scale=1, background=0, |
---|
62 | rg_squared=100.0, |
---|
63 | arms=3.0) |
---|
64 | |
---|
65 | tests = [[{'rg_squared': 2.0, |
---|
66 | 'arms': 3.3, |
---|
67 | }, 0.5, 0.851646091108], |
---|
68 | |
---|
69 | [{'rg_squared': 1.0, |
---|
70 | 'arms': 2.0, |
---|
71 | 'background': 1.8, |
---|
72 | }, 1.0, 2.53575888234], |
---|
73 | ] |
---|
74 | # 23Mar2016 RKH edited docs, would this better use rg not rg^2 ? Numerical noise at extremely small q.rg |
---|