1 | r""" |
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2 | This model calculates the structure factor of a polyelectrolyte solution with |
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3 | the RPA expression derived by Borue and Erukhimovich. |
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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 | 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) = K\frac{q^2+k^2}{4\pi L\alpha ^2} |
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13 | \frac{1}{1+r_{0}^2(q^2+k^2)(q^2-12hC_a/b^2)} + background |
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14 | |
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15 | k^2 = 4\pi L(2C_s + \alpha C_a) |
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16 | |
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17 | r_{0}^2 = \frac{1}{\alpha \sqrt{C_a} \left( b/\sqrt{48\pi L_b}\right)} |
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18 | |
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19 | where $K$ is the contrast factor for the polymer, $L_b$ is the Bjerrum length, |
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20 | $h$ is the virial parameter, $b$ is the monomer length, |
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21 | $C_s$ is the concentration of monovalent salt, $\alpha$ is the ionization |
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22 | degree, $C_a$ is the polymer molar concentration, and $background$ is the |
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23 | incoherent background. |
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24 | |
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25 | For 2D data the scattering intensity is calculated in the same way as 1D, |
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26 | where the $q$ vector is defined as |
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27 | |
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28 | .. math:: |
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29 | |
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30 | q = \sqrt{q_x^2 + q_y^2} |
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31 | |
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32 | |
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33 | .. figure:: img/be_polyelectrolyte_1d.jpg |
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34 | |
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35 | 1D plot using the default values (w/500 data point). |
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36 | |
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37 | NB: $1 barn = 10^{-24} cm^2$ |
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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 | V Y Borue, I Y Erukhimovich, *Macromolecules*, 21 (1988) 3240 |
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43 | |
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44 | J F Joanny, L Leibler, *Journal de Physique*, 51 (1990) 545 |
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45 | |
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46 | A Moussaid, F Schosseler, J P Munch, S Candau, |
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47 | *J. Journal de Physique II France*, 3 (1993) 573 |
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48 | |
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49 | E Raphael, J F Joanny, *Europhysics Letters*, 11 (1990) 179 |
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50 | |
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51 | """ |
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52 | |
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53 | from numpy import inf, pi, sqrt |
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54 | |
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55 | name = "be_polyelectrolyte" |
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56 | title = "Polyelectrolyte with the RPA expression derived by Borue and Erukhimovich" |
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57 | description = """ |
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58 | Evaluate |
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59 | F(x) = K 1/(4 pi Lb (alpha)^(2)) (q^(2)+k2)/(1+(r02)^(2)) |
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60 | (q^(2)+k2) (q^(2)-(12 h C/b^(2))) |
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61 | |
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62 | has 3 internal parameters : |
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63 | The inverse Debye Length: K2 = 4 pi Lb (2 Cs+alpha C) |
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64 | r02 =1/alpha/Ca^(0.5) (B/(48 pi Lb)^(0.5)) |
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65 | Ca = 6.022136e-4 C |
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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 | ["contrast_factor", "barns", 10.0, [-inf, inf], "", "Contrast factor of the polymer"], |
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73 | ["bjerrum_length", "Ang", 7.1, [0, inf], "", "Bjerrum length"], |
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74 | ["virial_param", "1/Ang^2", 12.0, [-inf, inf], "", "Virial parameter"], |
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75 | ["monomer_length", "Ang", 10.0, [0, inf], "", "Monomer length"], |
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76 | ["salt_concentration", "mol/L", 0.0, [-inf, inf], "", "Concentration of monovalent salt"], |
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77 | ["ionization_degree", "", 0.05, [0, inf], "", "Degree of ionization"], |
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78 | ["polymer_concentration", "mol/L", 0.7, [0, inf], "", "Polymer molar concentration"], |
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79 | ] |
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80 | # pylint: enable=bad-whitespace, line-too-long |
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81 | |
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82 | |
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83 | def Iq(q, |
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84 | contrast_factor=10.0, |
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85 | bjerrum_length=7.1, |
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86 | virial_param=12.0, |
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87 | monomer_length=10.0, |
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88 | salt_concentration=0.0, |
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89 | ionization_degree=0.05, |
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90 | polymer_concentration=0.7): |
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91 | """ |
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92 | :param q: Input q-value |
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93 | :param contrast_factor: Contrast factor of the polymer |
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94 | :param bjerrum_length: Bjerrum length |
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95 | :param virial_param: Virial parameter |
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96 | :param monomer_length: Monomer length |
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97 | :param salt_concentration: Concentration of monovalent salt |
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98 | :param ionization_degree: Degree of ionization |
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99 | :param polymer_concentration: Polymer molar concentration |
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100 | :return: 1-D intensity |
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101 | """ |
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102 | |
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103 | concentration = polymer_concentration * 6.022136e-4 |
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104 | |
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105 | k_square = 4.0 * pi * bjerrum_length * (2*salt_concentration + |
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106 | ionization_degree * concentration) |
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107 | |
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108 | r0_square = 1.0/ionization_degree/sqrt(concentration) * \ |
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109 | (monomer_length/sqrt((48.0*pi*bjerrum_length))) |
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110 | |
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111 | term1 = contrast_factor/(4.0 * pi * bjerrum_length * |
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112 | ionization_degree**2) * (q**2 + k_square) |
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113 | |
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114 | term2 = 1.0 + r0_square**2 * (q**2 + k_square) * \ |
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115 | (q**2 - (12.0 * virial_param * concentration/(monomer_length**2))) |
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116 | |
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117 | return term1/term2 |
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118 | |
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119 | Iq.vectorized = True # Iq accepts an array of q values |
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120 | |
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121 | |
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122 | def Iqxy(qx, qy, *args): |
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123 | """ |
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124 | :param qx: Input q_x-value |
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125 | :param qy: Input q_y-value |
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126 | :param args: Remaining arguments |
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127 | :return: 2D-Intensity |
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128 | """ |
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129 | intensity = Iq(sqrt(qx**2 + qy**2), *args) |
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130 | return intensity |
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131 | |
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132 | Iqxy.vectorized = True # Iqxy accepts an array of qx, qy values |
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133 | |
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134 | |
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135 | demo = dict(scale=1, background=0.1, |
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136 | contrast_factor=10.0, |
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137 | bjerrum_length=7.1, |
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138 | virial_param=12.0, |
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139 | monomer_length=10.0, |
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140 | salt_concentration=0.0, |
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141 | ionization_degree=0.05, |
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142 | polymer_concentration=0.7) |
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143 | |
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144 | oldname = "BEPolyelectrolyte" |
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145 | |
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146 | oldpars = dict(background='background', |
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147 | contrast_factor='k', |
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148 | bjerrum_length='lb', |
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149 | virial_param='h', |
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150 | monomer_length='b', |
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151 | salt_concentration='cs', |
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152 | ionization_degree='alpha', |
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153 | polymer_concentration='c') |
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154 | |
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155 | tests = [ |
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156 | |
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157 | # Accuracy tests based on content in test/utest_other_models.py |
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158 | [{'contrast_factor': 10.0, |
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159 | 'bjerrum_length': 7.1, |
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160 | 'virial_param': 12.0, |
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161 | 'monomer_length': 10.0, |
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162 | 'salt_concentration': 0.0, |
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163 | 'ionization_degree': 0.05, |
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164 | 'polymer_concentration': 0.7, |
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165 | 'background': 0.001, |
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166 | }, 0.001, 0.0948379], |
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167 | |
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168 | # Additional tests with larger range of parameters |
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169 | [{'contrast_factor': 10.0, |
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170 | 'bjerrum_length': 100.0, |
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171 | 'virial_param': 3.0, |
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172 | 'monomer_length': 1.0, |
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173 | 'salt_concentration': 10.0, |
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174 | 'ionization_degree': 2.0, |
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175 | 'polymer_concentration': 10.0, |
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176 | }, 0.1, -3.75693800588], |
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177 | |
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178 | [{'contrast_factor': 10.0, |
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179 | 'bjerrum_length': 100.0, |
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180 | 'virial_param': 3.0, |
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181 | 'monomer_length': 1.0, |
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182 | 'salt_concentration': 10.0, |
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183 | 'ionization_degree': 2.0, |
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184 | 'polymer_concentration': 10.0, |
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185 | 'background': 100.0 |
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186 | }, 5.0, 100.029142149], |
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187 | |
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188 | [{'contrast_factor': 100.0, |
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189 | 'bjerrum_length': 10.0, |
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190 | 'virial_param': 180.0, |
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191 | 'monomer_length': 1.0, |
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192 | 'salt_concentration': 0.1, |
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193 | 'ionization_degree': 0.5, |
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194 | 'polymer_concentration': 0.1, |
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195 | }, 200., 1.80664667511e-06], |
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196 | ] |
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