1 | r""" |
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2 | The Benoit model for a simple star polymer, with Gaussian coils arms from |
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3 | a common point. |
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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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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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14 | |
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15 | where |
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16 | |
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17 | .. math:: |
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18 | |
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19 | v=\frac{u^2f}{(3f-2)} |
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20 | |
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21 | and |
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22 | |
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23 | .. math:: |
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24 | |
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25 | u = \left\langle R_{g}^2\right\rangle q^2 |
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26 | |
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27 | contains the square of the ensemble average radius-of-gyration of an arm. Note that |
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28 | when there is only one arm, $f$ = 1, the Debye Gaussian coil equation is recovered. |
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29 | Star polymers in solutions tend to have strong interparticle and osmotic effects, so |
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30 | the Benoit equation may not work well. At small q the Guinier term and hence I(q=0) |
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31 | is the same as for $f$ arms of radius of gyration $R_g$, as described for the :ref:`mono_gauss_coil <mono-gauss-coil>`. |
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32 | |
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33 | |
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34 | References |
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35 | ---------- |
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36 | |
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37 | H Benoit *J. Polymer Science*, 11, 596-599 (1953) |
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38 | |
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39 | |
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40 | """ |
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41 | |
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42 | from numpy import inf |
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43 | |
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44 | name = "star_polymer" |
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45 | title = "Star polymer model with Gaussian statistics" |
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46 | description = """ |
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47 | Benoit 'Star polymer with Gaussian statistics' |
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48 | with |
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49 | P(q) = 2/{fv^2} * (v - (1-exp(-v)) + {f-1}/2 * (1-exp(-v))^2) |
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50 | where |
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51 | - v = u^2f/(3f-2) |
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52 | - u = <R_g^2>q^2, where <R_g^2> is the ensemble average radius of |
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53 | gyration squared of an arm |
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54 | - f is the number of arms on the star |
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55 | """ |
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56 | category = "shape-independent" |
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57 | single = False |
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58 | # pylint: disable=bad-whitespace, line-too-long |
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59 | # ["name", "units", default, [lower, upper], "type","description"], |
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60 | parameters = [["rg_squared", "Ang^2", 100.0, [0.0, inf], "", "Ensemble radius of gyration SQUARED of an arm"], |
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61 | ["arms", "", 3, [1.0, 6.0], "", "Number of arms in the model"], |
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62 | ] |
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63 | # pylint: enable=bad-whitespace, line-too-long |
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64 | |
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65 | source = ["star_polymer.c"] |
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66 | |
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67 | demo = dict(scale=1, background=0, |
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68 | rg_squared=100.0, |
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69 | arms=3.0) |
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70 | |
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71 | tests = [[{'rg_squared': 2.0, |
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72 | 'arms': 3.3, |
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73 | }, 0.5, 0.851646091108], |
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74 | |
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75 | [{'rg_squared': 1.0, |
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76 | 'arms': 2.0, |
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77 | 'background': 1.8, |
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78 | }, 1.0, 2.53575888234], |
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79 | ] |
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80 | # 23Mar2016 RKH edited docs, would this better use rg not rg^2 ? Numerical noise at extremely small q.rg |
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