1 | r""" |
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2 | This model calculates the form factor for a flexible cylinder with an |
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3 | elliptical cross section and a uniform scattering length density. |
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4 | The non-negligible diameter of the cylinder is included by accounting |
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5 | for excluded volume interactions within the walk of a single cylinder. |
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6 | The form factor is normalized by the particle volume such that |
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7 | |
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8 | .. math:: |
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9 | |
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10 | P(q) = \text{scale} \left<F^2\right>/V + \text{background} |
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11 | |
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12 | where the averaging $\left<\ldots\right>$ is over all possible orientations |
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13 | of the flexible cylinder. |
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14 | |
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15 | The 2D scattering intensity is the same as 1D, regardless of the orientation |
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16 | of the q vector which is defined as |
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17 | |
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18 | .. math:: |
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19 | |
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20 | q = \sqrt{q_x^2 + q_y^2} |
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21 | |
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22 | |
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23 | Definitions |
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24 | ----------- |
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25 | |
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26 | The function calculated in a similar way to that for the flexible_cylinder model |
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27 | from the reference given below using the author's "Method 3 With Excluded Volume". |
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28 | The model is a parameterization of simulations of a discrete representation of |
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29 | the worm-like chain model of Kratky and Porod applied in the pseudo-continuous |
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30 | limit. See equations (13, 26-27) in the original reference for the details. |
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31 | |
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32 | .. note:: |
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33 | |
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34 | There are several typos in the original reference that have been corrected |
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35 | by WRC. Details of the corrections are in the reference below. Most notably |
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36 | |
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37 | - Equation (13): the term $(1 - w(QR))$ should swap position with $w(QR)$ |
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38 | |
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39 | - Equations (23) and (24) are incorrect; WRC has entered these into |
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40 | Mathematica and solved analytically. The results were then converted to |
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41 | code. |
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42 | |
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43 | - Equation (27) should be $q0 = max(a3/sqrt(RgSquare),3)$ instead of |
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44 | $max(a3*b/sqrt(RgSquare),3)$ |
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45 | |
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46 | - The scattering function is negative for a range of parameter values and |
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47 | q-values that are experimentally accessible. A correction function has been |
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48 | added to give the proper behavior. |
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49 | |
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50 | .. figure:: img/flexible_cylinder_ex_geometry.jpg |
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51 | |
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52 | |
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53 | The chain of contour length, $L$, (the total length) can be described as a chain |
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54 | of some number of locally stiff segments of length $l_p$, the persistence length |
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55 | (the length along the cylinder over which the flexible cylinder can be considered |
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56 | a rigid rod). |
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57 | The Kuhn length $(b = 2*l_p)$ is also used to describe the stiffness of a chain. |
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58 | |
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59 | The cross section of the cylinder is elliptical, with minor radius $a$ . |
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60 | The major radius is larger, so of course, **the axis ratio (parameter 5) must be |
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61 | greater than one.** Simple constraints should be applied during curve fitting to |
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62 | maintain this inequality. |
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63 | |
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64 | The returned value is in units of $cm^{-1}$, on absolute scale. |
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65 | |
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66 | In the parameters, the $sld$ and $sld\_solvent$ represent the SLD of the |
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67 | chain/cylinder and solvent respectively. The *scale*, and the contrast are both |
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68 | multiplicative factors in the model and are perfectly correlated. One or both of |
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69 | these parameters must be held fixed during model fitting. |
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70 | |
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71 | **No inter-cylinder interference effects are included in this calculation.** |
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72 | |
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73 | References |
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74 | ---------- |
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75 | |
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76 | J S Pedersen and P Schurtenberger. *Scattering functions of semiflexible polymers |
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77 | with and without excluded volume effects.* Macromolecules, 29 (1996) 7602-7612 |
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78 | |
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79 | Correction of the formula can be found in |
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80 | |
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81 | W R Chen, P D Butler and L J Magid, *Incorporating Intermicellar Interactions in |
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82 | the Fitting of SANS Data from Cationic Wormlike Micelles.* Langmuir, |
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83 | 22(15) 2006 6539-6548 |
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84 | """ |
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85 | |
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86 | import numpy as np |
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87 | from numpy import inf |
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88 | |
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89 | name = "flexible_cylinder_elliptical" |
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90 | title = "Flexible cylinder wth an elliptical cross section and a uniform " \ |
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91 | "scattering length density." |
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92 | description = """Note : scale and contrast=sldCyl-sldSolv are both multiplicative |
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93 | factors in the |
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94 | model and are perfectly correlated. One or |
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95 | both of these parameters must be held fixed |
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96 | during model fitting. |
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97 | """ |
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98 | single = False |
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99 | |
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100 | category = "shape:cylinder" |
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101 | # pylint: disable=bad-whitespace, line-too-long |
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102 | # ["name", "units", default, [lower, upper], "type", "description"], |
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103 | parameters = [ |
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104 | ["length", "Ang", 1000.0, [0, inf], "volume", "Length of the flexible cylinder"], |
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105 | ["kuhn_length", "Ang", 100.0, [0, inf], "volume", "Kuhn length of the flexible cylinder"], |
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106 | ["radius", "Ang", 20.0, [1, inf], "volume", "Radius of the flexible cylinder"], |
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107 | ["axis_ratio", "", 1.5, [0, inf], "", "Axis_ratio (major_radius/minor_radius"], |
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108 | ["sld", "1e-6/Ang^2", 1.0, [-inf, inf], "sld", "Cylinder scattering length density"], |
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109 | ["sld_solvent", "1e-6/Ang^2", 6.3, [-inf, inf], "sld", "Solvent scattering length density"], |
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110 | ] |
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111 | # pylint: enable=bad-whitespace, line-too-long |
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112 | |
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113 | source = ["lib/polevl.c", "lib/sas_J1.c", "lib/gauss76.c", "lib/wrc_cyl.c", |
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114 | "flexible_cylinder_elliptical.c"] |
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115 | |
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116 | def random(): |
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117 | """Return a random parameter set for the model.""" |
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118 | length = 10**np.random.uniform(2, 6) |
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119 | radius = 10**np.random.uniform(1, 3) |
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120 | axis_ratio = 10**np.random.uniform(-1, 1) |
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121 | kuhn_length = 10**np.random.uniform(-2, -0.7)*length # at least 10 segments |
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122 | pars = dict( |
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123 | length=length, |
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124 | radius=radius, |
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125 | axis_ratio=axis_ratio, |
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126 | kuhn_length=kuhn_length, |
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127 | ) |
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128 | return pars |
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129 | |
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130 | tests = [ |
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131 | # Accuracy tests based on content in test/utest_other_models.py |
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132 | # Currently fails in OCL |
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133 | # [{'length': 1000.0, |
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134 | # 'kuhn_length': 100.0, |
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135 | # 'radius': 20.0, |
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136 | # 'axis_ratio': 1.5, |
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137 | # 'sld': 1.0, |
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138 | # 'sld_solvent': 6.3, |
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139 | # 'background': 0.0001, |
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140 | # }, 0.001, 3509.2187], |
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141 | |
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142 | # Additional tests with larger range of parameters |
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143 | [{'length': 1000.0, |
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144 | 'kuhn_length': 100.0, |
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145 | 'radius': 20.0, |
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146 | 'axis_ratio': 1.5, |
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147 | 'sld': 1.0, |
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148 | 'sld_solvent': 6.3, |
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149 | 'background': 0.0001, |
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150 | }, 1.0, 0.00223819], |
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151 | [{'length': 10.0, |
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152 | 'kuhn_length': 800.0, |
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153 | 'radius': 2.0, |
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154 | 'axis_ratio': 0.5, |
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155 | 'sld': 6.0, |
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156 | 'sld_solvent': 12.3, |
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157 | 'background': 0.001, |
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158 | }, 0.1, 0.390281], |
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159 | [{'length': 100.0, |
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160 | 'kuhn_length': 800.0, |
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161 | 'radius': 50.0, |
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162 | 'axis_ratio': 4.5, |
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163 | 'sld': 0.1, |
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164 | 'sld_solvent': 5.1, |
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165 | 'background': 0.0, |
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166 | }, 1.0, 0.0016338264790] |
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167 | ] |
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