[55b283e8] | 1 | r""" |
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| 2 | Definition |
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| 3 | ---------- |
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| 4 | |
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[d439007] | 5 | Calcuates the scattering from a simple star polymer with f equal Gaussian coil |
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| 6 | arms. A star being defined as a branched polymer with all the branches |
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| 7 | emanating from a common central (in the case of this model) point. It is |
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| 8 | derived as a special case of on the Benoit model for general branched |
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[30b60d2] | 9 | polymers\ [#CITBenoit]_ as also used by Richter *et al.*\ [#CITRichter]_ |
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[d439007] | 10 | |
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[55b283e8] | 11 | For a star with $f$ arms the scattering intensity $I(q)$ is calculated as |
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| 12 | |
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| 13 | .. math:: |
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| 14 | |
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[40a87fa] | 15 | I(q) = \frac{2}{fv^2}\left[ v-1+\exp(-v)+\frac{f-1}{2} |
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| 16 | \left[ 1-\exp(-v)\right]^2\right] |
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[55b283e8] | 17 | |
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| 18 | where |
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| 19 | |
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[d439007] | 20 | .. math:: v=\frac{uf}{(3f-2)} |
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[55b283e8] | 21 | |
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| 22 | and |
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| 23 | |
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[40a87fa] | 24 | .. math:: u = \left\langle R_{g}^2\right\rangle q^2 |
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[55b283e8] | 25 | |
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[d439007] | 26 | contains the square of the ensemble average radius-of-gyration of the full |
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| 27 | polymer while v contains the radius of gyration of a single arm $R_{arm}$. |
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| 28 | The two are related as: |
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| 29 | |
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| 30 | .. math:: R_{arm}^2 = \frac{f}{3f-2} R_{g}^2 |
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| 31 | |
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[40a87fa] | 32 | Note that when there is only one arm, $f = 1$, the Debye Gaussian coil |
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[d439007] | 33 | equation is recovered. |
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| 34 | |
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| 35 | .. note:: |
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| 36 | Star polymers in solutions tend to have strong interparticle and osmotic |
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| 37 | effects. Thus the Benoit equation may not work well for many real cases. |
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[5da1ac8] | 38 | A newer model for star polymer incorporating excluded volume has been |
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| 39 | developed by Li et al in arXiv:1404.6269 [physics.chem-ph]. Also, at small |
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| 40 | $q$ the scattering, i.e. the Guinier term, is not sensitive to the number of |
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| 41 | arms, and hence 'scale' here is simply $I(q=0)$ as described for the |
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| 42 | :ref:`mono-gauss-coil` model, using volume fraction $\phi$ and volume V |
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| 43 | for the whole star polymer. |
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[55b283e8] | 44 | |
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[6b4f7f6] | 45 | References |
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| 46 | ---------- |
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[55b283e8] | 47 | |
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[d439007] | 48 | .. [#CITBenoit] H Benoit *J. Polymer Science*, 11, 507-510 (1953) |
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| 49 | .. [#CITRichter] D Richter, B. Farago, J. S. Huang, L. J. Fetters, |
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| 50 | B Ewen *Macromolecules*, 22, 468-472 (1989) |
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| 51 | |
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| 52 | Authorship and Verification |
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| 53 | ---------------------------- |
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| 54 | |
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| 55 | * **Author:** Kieran Campbell **Date:** July 24, 2012 |
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| 56 | * **Last Modified by:** Paul Butler **Date:** Auguts 26, 2017 |
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| 57 | * **Last Reviewed by:** Ziang Li and Richard Heenan **Date:** May 17, 2017 |
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[55b283e8] | 58 | """ |
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| 59 | |
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| 60 | from numpy import inf |
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| 61 | |
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| 62 | name = "star_polymer" |
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| 63 | title = "Star polymer model with Gaussian statistics" |
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| 64 | description = """ |
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[6b4f7f6] | 65 | Benoit 'Star polymer with Gaussian statistics' |
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[55b283e8] | 66 | with |
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| 67 | P(q) = 2/{fv^2} * (v - (1-exp(-v)) + {f-1}/2 * (1-exp(-v))^2) |
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| 68 | where |
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| 69 | - v = u^2f/(3f-2) |
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| 70 | - u = <R_g^2>q^2, where <R_g^2> is the ensemble average radius of |
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[d439007] | 71 | gyration squared of the entire polymer |
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[55b283e8] | 72 | - f is the number of arms on the star |
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[d439007] | 73 | - the radius of gyration of an arm is given b |
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| 74 | Rg_arm^2 = R_g^2 * f/(3f-2) |
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[55b283e8] | 75 | """ |
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| 76 | category = "shape-independent" |
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[13ed84c] | 77 | single = False |
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[168052c] | 78 | # pylint: disable=bad-whitespace, line-too-long |
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[55b283e8] | 79 | # ["name", "units", default, [lower, upper], "type","description"], |
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[d439007] | 80 | parameters = [["rg_squared", "Ang^2", 100.0, [0.0, inf], "", "Ensemble radius of gyration SQUARED of the full polymer"], |
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[55b283e8] | 81 | ["arms", "", 3, [1.0, 6.0], "", "Number of arms in the model"], |
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[168052c] | 82 | ] |
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| 83 | # pylint: enable=bad-whitespace, line-too-long |
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[55b283e8] | 84 | |
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| 85 | source = ["star_polymer.c"] |
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| 86 | |
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[48462b0] | 87 | def random(): |
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| 88 | import numpy as np |
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| 89 | pars = dict( |
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| 90 | #background=0, |
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| 91 | scale=10**np.random.uniform(1, 4), |
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| 92 | rg_squared=10**np.random.uniform(1, 8), |
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| 93 | arms=np.random.uniform(1, 6), |
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| 94 | ) |
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| 95 | return pars |
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[55b283e8] | 96 | |
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[6b4f7f6] | 97 | tests = [[{'rg_squared': 2.0, |
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[55b283e8] | 98 | 'arms': 3.3, |
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[6dd90c1] | 99 | }, 0.5, 0.851646091108], |
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[55b283e8] | 100 | |
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[6b4f7f6] | 101 | [{'rg_squared': 1.0, |
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[55b283e8] | 102 | 'arms': 2.0, |
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| 103 | 'background': 1.8, |
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[168052c] | 104 | }, 1.0, 2.53575888234], |
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| 105 | ] |
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[40a87fa] | 106 | # 23Mar2016 RKH edited docs, would this better use rg not rg^2 ? Numerical noise at extremely small q.rg |
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