[55b283e8] | 1 | r""" |
---|
| 2 | Definition |
---|
| 3 | ---------- |
---|
| 4 | |
---|
[d439007] | 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 |
---|
[30b60d2] | 9 | polymers\ [#CITBenoit]_ as also used by Richter *et al.*\ [#CITRichter]_ |
---|
[d439007] | 10 | |
---|
[55b283e8] | 11 | For a star with $f$ arms the scattering intensity $I(q)$ is calculated as |
---|
| 12 | |
---|
| 13 | .. math:: |
---|
| 14 | |
---|
[40a87fa] | 15 | I(q) = \frac{2}{fv^2}\left[ v-1+\exp(-v)+\frac{f-1}{2} |
---|
| 16 | \left[ 1-\exp(-v)\right]^2\right] |
---|
[55b283e8] | 17 | |
---|
| 18 | where |
---|
| 19 | |
---|
[d439007] | 20 | .. math:: v=\frac{uf}{(3f-2)} |
---|
[55b283e8] | 21 | |
---|
| 22 | and |
---|
| 23 | |
---|
[40a87fa] | 24 | .. math:: u = \left\langle R_{g}^2\right\rangle q^2 |
---|
[55b283e8] | 25 | |
---|
[d439007] | 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 | |
---|
[40a87fa] | 32 | Note that when there is only one arm, $f = 1$, the Debye Gaussian coil |
---|
[d439007] | 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. |
---|
[5da1ac8] | 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. |
---|
[55b283e8] | 44 | |
---|
[6b4f7f6] | 45 | References |
---|
| 46 | ---------- |
---|
[55b283e8] | 47 | |
---|
[d439007] | 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 | |
---|
[0507e09] | 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 | |
---|
[d439007] | 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 |
---|
[0507e09] | 65 | * **Source added by :** Steve King **Date:** March 25, 2019 |
---|
[55b283e8] | 66 | """ |
---|
| 67 | |
---|
[2d81cfe] | 68 | import numpy as np |
---|
[55b283e8] | 69 | from numpy import inf |
---|
| 70 | |
---|
| 71 | name = "star_polymer" |
---|
| 72 | title = "Star polymer model with Gaussian statistics" |
---|
| 73 | description = """ |
---|
[6b4f7f6] | 74 | Benoit 'Star polymer with Gaussian statistics' |
---|
[55b283e8] | 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 |
---|
[d439007] | 80 | gyration squared of the entire polymer |
---|
[55b283e8] | 81 | - f is the number of arms on the star |
---|
[d439007] | 82 | - the radius of gyration of an arm is given b |
---|
| 83 | Rg_arm^2 = R_g^2 * f/(3f-2) |
---|
[55b283e8] | 84 | """ |
---|
| 85 | category = "shape-independent" |
---|
[13ed84c] | 86 | single = False |
---|
[168052c] | 87 | # pylint: disable=bad-whitespace, line-too-long |
---|
[55b283e8] | 88 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
[d439007] | 89 | parameters = [["rg_squared", "Ang^2", 100.0, [0.0, inf], "", "Ensemble radius of gyration SQUARED of the full polymer"], |
---|
[55b283e8] | 90 | ["arms", "", 3, [1.0, 6.0], "", "Number of arms in the model"], |
---|
[168052c] | 91 | ] |
---|
| 92 | # pylint: enable=bad-whitespace, line-too-long |
---|
[55b283e8] | 93 | |
---|
| 94 | source = ["star_polymer.c"] |
---|
| 95 | |
---|
[48462b0] | 96 | def random(): |
---|
[b297ba9] | 97 | """Return a random parameter set for the model.""" |
---|
[48462b0] | 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 |
---|
[55b283e8] | 105 | |
---|
[6b4f7f6] | 106 | tests = [[{'rg_squared': 2.0, |
---|
[55b283e8] | 107 | 'arms': 3.3, |
---|
[6dd90c1] | 108 | }, 0.5, 0.851646091108], |
---|
[55b283e8] | 109 | |
---|
[6b4f7f6] | 110 | [{'rg_squared': 1.0, |
---|
[55b283e8] | 111 | 'arms': 2.0, |
---|
| 112 | 'background': 1.8, |
---|
[168052c] | 113 | }, 1.0, 2.53575888234], |
---|
| 114 | ] |
---|
[40a87fa] | 115 | # 23Mar2016 RKH edited docs, would this better use rg not rg^2 ? Numerical noise at extremely small q.rg |
---|