[55b283e8] | 1 | r""" |
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[40a87fa] | 2 | The Benoit model for a simple star polymer, with Gaussian coils arms from |
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[6b4f7f6] | 3 | a common point. |
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[55b283e8] | 4 | |
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| 5 | Definition |
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| 6 | ---------- |
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| 7 | |
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| 8 | For a star with $f$ arms the scattering intensity $I(q)$ is calculated as |
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| 9 | |
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| 10 | .. math:: |
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| 11 | |
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[40a87fa] | 12 | I(q) = \frac{2}{fv^2}\left[ v-1+\exp(-v)+\frac{f-1}{2} |
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| 13 | \left[ 1-\exp(-v)\right]^2\right] |
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[55b283e8] | 14 | |
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| 15 | where |
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| 16 | |
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[40a87fa] | 17 | .. math:: v=\frac{u^2f}{(3f-2)} |
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[55b283e8] | 18 | |
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| 19 | and |
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| 20 | |
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[40a87fa] | 21 | .. math:: u = \left\langle R_{g}^2\right\rangle q^2 |
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[55b283e8] | 22 | |
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[40a87fa] | 23 | contains the square of the ensemble average radius-of-gyration of an arm. |
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| 24 | Note that when there is only one arm, $f = 1$, the Debye Gaussian coil |
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| 25 | equation is recovered. Star polymers in solutions tend to have strong |
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| 26 | interparticle and osmotic effects, so the Benoit equation may not work well. |
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| 27 | At small $q$ the Guinier term and hence $I(q=0)$ is the same as for $f$ arms |
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| 28 | of radius of gyration $R_g$, as described for the :ref:`mono-gauss-coil` model. |
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[55b283e8] | 29 | |
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[6b4f7f6] | 30 | References |
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| 31 | ---------- |
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[55b283e8] | 32 | |
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| 33 | H Benoit *J. Polymer Science*, 11, 596-599 (1953) |
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| 34 | """ |
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| 35 | |
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| 36 | from numpy import inf |
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| 37 | |
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| 38 | name = "star_polymer" |
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| 39 | title = "Star polymer model with Gaussian statistics" |
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| 40 | description = """ |
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[6b4f7f6] | 41 | Benoit 'Star polymer with Gaussian statistics' |
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[55b283e8] | 42 | with |
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| 43 | P(q) = 2/{fv^2} * (v - (1-exp(-v)) + {f-1}/2 * (1-exp(-v))^2) |
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| 44 | where |
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| 45 | - v = u^2f/(3f-2) |
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| 46 | - u = <R_g^2>q^2, where <R_g^2> is the ensemble average radius of |
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[6b4f7f6] | 47 | gyration squared of an arm |
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[55b283e8] | 48 | - f is the number of arms on the star |
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| 49 | """ |
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| 50 | category = "shape-independent" |
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[13ed84c] | 51 | single = False |
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[168052c] | 52 | # pylint: disable=bad-whitespace, line-too-long |
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[55b283e8] | 53 | # ["name", "units", default, [lower, upper], "type","description"], |
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[6b4f7f6] | 54 | parameters = [["rg_squared", "Ang^2", 100.0, [0.0, inf], "", "Ensemble radius of gyration SQUARED of an arm"], |
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[55b283e8] | 55 | ["arms", "", 3, [1.0, 6.0], "", "Number of arms in the model"], |
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[168052c] | 56 | ] |
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| 57 | # pylint: enable=bad-whitespace, line-too-long |
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[55b283e8] | 58 | |
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| 59 | source = ["star_polymer.c"] |
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| 60 | |
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| 61 | demo = dict(scale=1, background=0, |
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[6b4f7f6] | 62 | rg_squared=100.0, |
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[55b283e8] | 63 | arms=3.0) |
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| 64 | |
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[6b4f7f6] | 65 | tests = [[{'rg_squared': 2.0, |
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[55b283e8] | 66 | 'arms': 3.3, |
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[6dd90c1] | 67 | }, 0.5, 0.851646091108], |
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[55b283e8] | 68 | |
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[6b4f7f6] | 69 | [{'rg_squared': 1.0, |
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[55b283e8] | 70 | 'arms': 2.0, |
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| 71 | 'background': 1.8, |
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[168052c] | 72 | }, 1.0, 2.53575888234], |
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| 73 | ] |
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[40a87fa] | 74 | # 23Mar2016 RKH edited docs, would this better use rg not rg^2 ? Numerical noise at extremely small q.rg |
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