1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | Calculates the macroscopic scattering intensity for a multi-component |
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6 | homogeneous mixture of polymers using the Random Phase Approximation. |
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7 | This general formalism contains 10 specific cases |
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8 | |
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9 | Case 0: C/D binary mixture of homopolymers |
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10 | |
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11 | Case 1: C-D diblock copolymer |
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12 | |
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13 | Case 2: B/C/D ternary mixture of homopolymers |
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14 | |
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15 | Case 3: C/C-D mixture of a homopolymer B and a diblock copolymer C-D |
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16 | |
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17 | Case 4: B-C-D triblock copolymer |
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18 | |
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19 | Case 5: A/B/C/D quaternary mixture of homopolymers |
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20 | |
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21 | Case 6: A/B/C-D mixture of two homopolymers A/B and a diblock C-D |
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22 | |
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23 | Case 7: A/B-C-D mixture of a homopolymer A and a triblock B-C-D |
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24 | |
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25 | Case 8: A-B/C-D mixture of two diblock copolymers A-B and C-D |
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26 | |
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27 | Case 9: A-B-C-D tetra-block copolymer |
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28 | |
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29 | .. note:: |
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30 | These case numbers are different from those in the NIST SANS package! |
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31 | |
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32 | The models are based on the papers by Akcasu *et al.* and by |
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33 | Hammouda assuming the polymer follows Gaussian statistics such |
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34 | that $R_g^2 = n b^2/6$ where $b$ is the statistical segment length and $n$ is |
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35 | the number of statistical segment lengths. A nice tutorial on how these are |
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36 | constructed and implemented can be found in chapters 28 and 39 of Boualem |
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37 | Hammouda's 'SANS Toolbox'. |
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38 | |
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39 | In brief the macroscopic cross sections are derived from the general forms |
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40 | for homopolymer scattering and the multiblock cross-terms while the inter |
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41 | polymer cross terms are described in the usual way by the $\chi$ parameter. |
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42 | |
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43 | USAGE NOTES: |
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44 | |
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45 | * Only one case can be used at any one time. |
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46 | * The RPA (mean field) formalism only applies only when the multicomponent |
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47 | polymer mixture is in the homogeneous mixed-phase region. |
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48 | * **Component D is assumed to be the "background" component (ie, all contrasts |
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49 | are calculated with respect to component D).** So the scattering contrast |
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50 | for a C/D blend = [SLD(component C) - SLD(component D)]\ :sup:`2`. |
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51 | * Depending on which case is being used, the number of fitting parameters can |
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52 | vary. |
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53 | |
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54 | .. Note:: |
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55 | * In general the degrees of polymerization, the volume |
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56 | fractions, the molar volumes, and the neutron scattering lengths for each |
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57 | component are obtained from other methods and held fixed while The *scale* |
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58 | parameter should be held equal to unity. |
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59 | * The variables are normally the segment lengths ($b_a$, $b_b$, |
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60 | etc.) and $\chi$ parameters ($K_{ab}$, $K_{ac}$, etc). |
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61 | |
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62 | References |
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63 | ---------- |
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64 | |
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65 | A Z Akcasu, R Klein and B Hammouda, *Macromolecules*, 26 (1993) 4136. |
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66 | |
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67 | B. Hammouda, *Advances in Polymer Science* 106 (1993) 87. |
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68 | |
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69 | B. Hammouda, *SANS Toolbox* |
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70 | https://www.ncnr.nist.gov/staff/hammouda/the_sans_toolbox.pdf. |
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71 | |
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72 | Authorship and Verification |
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73 | ---------------------------- |
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74 | |
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75 | * **Author:** Boualem Hammouda - NIST IGOR/DANSE **Date:** pre 2010 |
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76 | * **Converted to sasmodels by:** Paul Kienzle **Date:** July 18, 2016 |
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77 | * **Last Modified by:** Paul Butler **Date:** March 12, 2017 |
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78 | * **Last Reviewed by:** Paul Butler **Date:** March 12, 2017 |
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79 | """ |
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80 | |
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81 | from numpy import inf |
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82 | |
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83 | name = "rpa" |
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84 | title = "Random Phase Approximation" |
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85 | description = """ |
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86 | This formalism applies to multicomponent polymer mixtures in the |
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87 | homogeneous (mixed) phase region only. |
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88 | Case 0: C/D binary mixture of homopolymers |
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89 | Case 1: C-D diblock copolymer |
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90 | Case 2: B/C/D ternary mixture of homopolymers |
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91 | Case 3: B/C-D mixture of homopolymer b and diblock copolymer C-D |
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92 | Case 4: B-C-D triblock copolymer |
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93 | Case 5: A/B/C/D quaternary mixture of homopolymers |
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94 | Case 6: A/B/C-D mixture of two homopolymers A/B and a diblock C-D |
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95 | Case 7: A/B-C-D mixture of a homopolymer A and a triblock B-C-D |
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96 | Case 8: A-B/C-D mixture of two diblock copolymers A-B and C-D |
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97 | Case 9: A-B-C-D four-block copolymer |
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98 | See details in the model function help |
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99 | """ |
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100 | category = "shape-independent" |
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101 | |
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102 | CASES = [ |
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103 | "C+D binary mixture", |
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104 | "C:D diblock copolymer", |
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105 | "B+C+D ternary mixture", |
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106 | "B+C:D binary mixture", |
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107 | "B:C:D triblock copolymer", |
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108 | "A+B+C+D quaternary mixture", |
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109 | "A+B+C:D ternary mixture", |
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110 | "A+B:C:D binary mixture", |
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111 | "A:B+C:D binary mixture", |
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112 | "A:B:C:D quadblock copolymer", |
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113 | ] |
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114 | |
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115 | # ["name", "units", default, [lower, upper], "type","description"], |
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116 | parameters = [ |
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117 | ["case_num", "", 1, [CASES], "", "Component organization"], |
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118 | |
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119 | ["N[4]", "", 1000.0, [1, inf], "", "Degree of polymerization"], |
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120 | ["Phi[4]", "", 0.25, [0, 1], "", "volume fraction"], |
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121 | ["v[4]", "mL/mol", 100.0, [0, inf], "", "molar volume"], |
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122 | ["L[4]", "fm", 10.0, [-inf, inf], "", "scattering length"], |
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123 | ["b[4]", "Ang", 5.0, [0, inf], "", "segment length"], |
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124 | |
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125 | ["K12", "", -0.0004, [-inf, inf], "", "A:B interaction parameter"], |
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126 | ["K13", "", -0.0004, [-inf, inf], "", "A:C interaction parameter"], |
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127 | ["K14", "", -0.0004, [-inf, inf], "", "A:D interaction parameter"], |
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128 | ["K23", "", -0.0004, [-inf, inf], "", "B:C interaction parameter"], |
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129 | ["K24", "", -0.0004, [-inf, inf], "", "B:D interaction parameter"], |
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130 | ["K34", "", -0.0004, [-inf, inf], "", "C:D interaction parameter"], |
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131 | ] |
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132 | |
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133 | |
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134 | source = ["rpa.c"] |
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135 | single = False |
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136 | |
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137 | control = "case_num" |
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138 | HIDE_ALL = set("Phi4".split()) |
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139 | HIDE_A = set("N1 Phi1 v1 L1 b1 K12 K13 K14".split()).union(HIDE_ALL) |
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140 | HIDE_AB = set("N2 Phi2 v2 L2 b2 K23 K24".split()).union(HIDE_A) |
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141 | def hidden(case_num): |
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142 | """ |
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143 | Return a list of parameters to hide depending on the multiplicity parameter. |
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144 | """ |
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145 | case_num = int(case_num+0.5) |
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146 | if case_num < 2: |
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147 | return HIDE_AB |
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148 | elif case_num < 5: |
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149 | return HIDE_A |
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150 | else: |
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151 | return HIDE_ALL |
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152 | |
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153 | # TODO: no random parameters generated for RPA |
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