[3330bb4] | 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 | r""" |
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| 5 | This empirical model describes the scattering from *polydisperse* polymer |
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| 6 | chains in theta solvents or polymer melts, assuming a Schulz-Zimm type |
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| 7 | molecular weight distribution. |
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| 8 | |
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| 9 | To describe the scattering from *monodisperse* polymer chains, see the |
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| 10 | :ref:`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 | .. math:: |
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| 16 | |
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| 17 | I(q) = \text{scale} \cdot I_0 \cdot P(q) + \text{background} |
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| 18 | |
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| 19 | where |
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| 20 | |
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| 21 | .. math:: |
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| 22 | |
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[ca04add] | 23 | I_0 &= \phi_\text{poly} \cdot V \cdot (\rho_\text{poly}-\rho_\text{solv})^2 \\ |
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| 24 | P(q) &= 2 [(1 + UZ)^{-1/U} + Z - 1] / [(1 + U) Z^2] \\ |
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| 25 | Z &= [(q R_g)^2] / (1 + 2U) \\ |
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| 26 | U &= (Mw / Mn) - 1 = \text{polydispersity ratio} - 1 \\ |
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[3330bb4] | 27 | V &= M / (N_A \delta) |
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| 28 | |
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| 29 | Here, $\phi_\text{poly}$, is the volume fraction of polymer, $V$ is the |
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| 30 | volume of a polymer coil, $M$ is the molecular weight of the polymer, |
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| 31 | $N_A$ is Avogadro's Number, $\delta$ is the bulk density of the polymer, |
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| 32 | $\rho_\text{poly}$ is the sld of the polymer, $\rho_\text{solv}$ is the |
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| 33 | sld of the solvent, and $R_g$ is the radius of gyration of the polymer coil. |
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| 34 | |
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| 35 | The 2D scattering intensity is calculated in the same way as the 1D, |
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| 36 | but where the $q$ vector is redefined as |
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| 37 | |
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| 38 | .. math:: |
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| 39 | |
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| 40 | q = \sqrt{q_x^2 + q_y^2} |
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| 41 | |
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| 42 | References |
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| 43 | ---------- |
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| 44 | |
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| 45 | O Glatter and O Kratky (editors), *Small Angle X-ray Scattering*, |
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| 46 | Academic Press, (1982) Page 404. |
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| 47 | |
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| 48 | J S Higgins, H C Benoit, *Polymers and Neutron Scattering*, |
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| 49 | Oxford Science Publications, (1996). |
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| 50 | |
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| 51 | S M King, *Small Angle Neutron Scattering* in *Modern Techniques for |
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| 52 | Polymer Characterisation*, Wiley, (1999). |
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| 53 | |
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| 54 | http://www.ncnr.nist.gov/staff/hammouda/distance_learning/chapter_28.pdf |
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| 55 | """ |
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| 56 | |
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| 57 | import numpy as np |
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| 58 | from numpy import inf, expm1, power |
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| 59 | |
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| 60 | name = "poly_gauss_coil" |
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| 61 | title = "Scattering from polydisperse polymer coils" |
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| 62 | |
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| 63 | description = """ |
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[404ebbd] | 64 | Evaluates the scattering from |
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[3330bb4] | 65 | polydisperse polymer chains. |
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| 66 | """ |
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| 67 | category = "shape-independent" |
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| 68 | |
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| 69 | # pylint: disable=bad-whitespace, line-too-long |
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| 70 | # ["name", "units", default, [lower, upper], "type", "description"], |
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| 71 | parameters = [ |
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| 72 | ["i_zero", "1/cm", 70.0, [0.0, inf], "", "Intensity at q=0"], |
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| 73 | ["rg", "Ang", 75.0, [0.0, inf], "", "Radius of gyration"], |
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| 74 | ["polydispersity", "None", 2.0, [1.0, inf], "", "Polymer Mw/Mn"], |
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| 75 | ] |
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| 76 | # pylint: enable=bad-whitespace, line-too-long |
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| 77 | |
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| 78 | # NB: Scale and Background are implicit parameters on every model |
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| 79 | def Iq(q, i_zero, rg, polydispersity): |
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| 80 | # pylint: disable = missing-docstring |
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| 81 | u = polydispersity - 1.0 |
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| 82 | z = q**2 * (rg**2 / (1.0 + 2.0*u)) |
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| 83 | |
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| 84 | # need to trap the case of the polydispersity being 1 (ie, monodisperse!) |
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| 85 | if polydispersity == 1.0: |
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| 86 | result = 2.0 * (expm1(-z) + z) |
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| 87 | index = q != 0. |
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| 88 | result[index] /= z[index]**2 |
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| 89 | result[~index] = 1.0 |
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| 90 | else: |
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| 91 | # Taylor series around z=0 of (2*(1+uz)^(-1/u) + z - 1) / (z^2(u+1)) |
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| 92 | p = [ |
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| 93 | #(-1 - 20*u - 155*u**2 - 580*u**3 - 1044*u**4 - 720*u**5) / 2520., |
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[404ebbd] | 94 | #(+1 + 14*u + 71*u**2 + 154*u**3 + 120*u**4) / 360., |
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[3330bb4] | 95 | #(-1 - 9*u - 26*u**2 - 24*u**3) / 60., |
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[404ebbd] | 96 | (+1 + 5*u + 6*u**2) / 12., |
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[3330bb4] | 97 | (-1 - 2*u) / 3., |
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[404ebbd] | 98 | (+1), |
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[3330bb4] | 99 | ] |
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| 100 | result = 2.0 * (power(1.0 + u*z, -1.0/u) + z - 1.0) / (1.0 + u) |
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| 101 | index = z > 1e-4 |
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| 102 | result[index] /= z[index]**2 |
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| 103 | result[~index] = np.polyval(p, z[~index]) |
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| 104 | return i_zero * result |
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| 105 | Iq.vectorized = True # Iq accepts an array of q values |
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| 106 | |
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[404ebbd] | 107 | def random(): |
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| 108 | rg = 10**np.random.uniform(0, 4) |
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| 109 | #rg = 1e3 |
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| 110 | polydispersity = 10**np.random.uniform(0, 3) |
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| 111 | pars = dict( |
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| 112 | #scale=1, background=0, |
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| 113 | i_zero=1e7, # i_zero is a simple scale |
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| 114 | rg=rg, |
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| 115 | polydispersity=polydispersity, |
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| 116 | ) |
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| 117 | return pars |
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| 118 | |
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[3330bb4] | 119 | demo = dict(scale=1.0, |
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| 120 | i_zero=70.0, |
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| 121 | rg=75.0, |
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| 122 | polydispersity=2.0, |
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| 123 | background=0.0) |
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| 124 | |
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| 125 | # these unit test values taken from SasView 3.1.2 |
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| 126 | tests = [ |
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| 127 | [{'scale': 1.0, 'i_zero': 70.0, 'rg': 75.0, |
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| 128 | 'polydispersity': 2.0, 'background': 0.0}, |
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| 129 | [0.0106939, 0.469418], [57.6405, 0.169016]], |
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| 130 | ] |
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