1 | #poly_gauss_coil model |
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2 | #conversion of Poly_GaussCoil.py |
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3 | #converted by Steve King, Mar 2016 |
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4 | |
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5 | |
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6 | |
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7 | r""" |
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8 | This empirical model describes the scattering from *polydisperse* polymer chains in theta solvents or polymer melts, assuming a Schulz-Zimm type molecular weight distribution. |
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9 | |
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10 | To describe the scattering from *monodisperse* polymer chains, see the :ref:`mono_gauss_coil <mono-gauss-coil>` model. |
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11 | |
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12 | Definition |
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13 | ---------- |
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14 | |
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15 | *I(q)* = *scale* |cdot| *I* \ :sub:`0` |cdot| *P(q)* + *background* |
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16 | |
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17 | where |
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18 | |
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19 | *I*\ :sub:`0` = |phi|\ :sub:`poly` |cdot| *V* |cdot| (|rho|\ :sub:`poly` - |rho|\ :sub:`solv`)\ :sup:`2` |
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20 | |
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21 | *P(q)* = 2 [(1 + UZ)\ :sup:`-1/U` + Z - 1] / [(1 + U) Z\ :sup:`2`] |
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22 | |
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23 | *Z* = [(*q R*\ :sub:`g`)\ :sup:`2`] / (1 + 2U) |
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24 | |
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25 | *U* = (Mw / Mn) - 1 = (*polydispersity ratio*) - 1 |
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26 | |
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27 | and |
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28 | |
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29 | *V* = *M* / (*N*\ :sub:`A` |delta|) |
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30 | |
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31 | Here, |phi|\ :sub:`poly`, is the volume fraction of polymer, *V* is the volume of a polymer coil, *M* is the molecular weight of the polymer, *N*\ :sub:`A` is Avogadro's Number, |delta| is the bulk density of the polymer, |rho|\ :sub:`poly` is the sld of the polymer, |rho|\ :sub:`solv` is the sld of the solvent, and *R*\ :sub:`g` is the radius of gyration of the polymer coil. |
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32 | |
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33 | The 2D scattering intensity is calculated in the same way as the 1D, but where the *q* vector is redefined as |
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34 | |
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35 | .. math:: |
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36 | |
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37 | q = \sqrt{q_x^2 + q_y^2} |
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38 | |
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39 | References |
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40 | ---------- |
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41 | |
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42 | O Glatter and O Kratky (editors), *Small Angle X-ray Scattering*, Academic Press, (1982) |
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43 | Page 404. |
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44 | |
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45 | J S Higgins, H C Benoit, *Polymers and Neutron Scattering*, Oxford Science Publications, (1996). |
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46 | |
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47 | S M King, *Small Angle Neutron Scattering* in *Modern Techniques for Polymer Characterisation*, Wiley, (1999). |
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48 | |
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49 | http://www.ncnr.nist.gov/staff/hammouda/distance_learning/chapter_28.pdf |
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50 | """ |
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51 | |
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52 | from numpy import inf, exp, power |
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53 | |
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54 | name = "poly_gauss_coil" |
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55 | title = "Scattering from polydisperse polymer coils" |
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56 | |
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57 | description = """ |
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58 | Evaluates the scattering from |
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59 | polydisperse polymer chains. |
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60 | """ |
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61 | category = "shape-independent" |
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62 | |
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63 | # ["name", "units", default, [lower, upper], "type", "description"], |
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64 | parameters = [["i_zero", "1/cm", 70.0, [0.0, inf], "", "Intensity at q=0"], |
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65 | ["radius_gyration", "Ang", 75.0, [0.0, inf], "", "Radius of gyration"], |
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66 | ["polydispersity", "None", 2.0, [1.0, inf], "", "Polymer Mw/Mn"]] |
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67 | |
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68 | # NB: Scale and Background are implicit parameters on every model |
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69 | def Iq(q, i_zero, radius_gyration, polydispersity): |
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70 | # pylint: disable = missing-docstring |
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71 | u = polydispersity - 1.0 |
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72 | z = (q*radius_gyration)**2 / (1.0 + 2.0*u) |
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73 | # need to trap the case of the polydispersity being 1 (ie, monodispersity!) |
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74 | if polydispersity == 1.0: |
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75 | inten = i_zero * 2.0 * (exp(-z) + z - 1.0) |
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76 | else: |
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77 | inten = i_zero * 2.0 * (power(1.0 + u*z, -1.0/u) + z - 1.0) / (1.0 + u) |
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78 | index = q != 0. |
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79 | inten[~index] = i_zero |
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80 | inten[index] /= z[index]**2 |
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81 | return inten |
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82 | Iq.vectorized = True # Iq accepts an array of q values |
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83 | |
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84 | demo = dict(scale = 1.0, |
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85 | i_zero = 70.0, |
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86 | radius_gyration = 75.0, |
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87 | polydispersity = 2.0, |
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88 | background = 0.0) |
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89 | |
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90 | # these unit test values taken from SasView 3.1.2 |
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91 | tests = [ |
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92 | [{'scale': 1.0, 'i_zero': 70.0, 'radius_gyration': 75.0, 'polydispersity': 2.0, 'background': 0.0}, |
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93 | [0.0106939, 0.469418], [57.6405, 0.169016]], |
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94 | ] |
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