1 | r""" |
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2 | This model describes the scattering from polymer chains subject to excluded |
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3 | volume effects and has been used as a template for describing mass fractals. |
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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 form factor was originally presented in the following integral form |
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9 | (Benoit, 1957) |
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10 | |
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11 | .. math:: |
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12 | |
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13 | P(Q)=2\int_0^{1}dx(1-x)exp\left[-\frac{Q^2a^2}{6}n^{2v}x^{2v}\right] |
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14 | |
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15 | where $\nu$ is the excluded volume parameter |
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16 | (which is related to the Porod exponent $m$ as $\nu=1/m$ ), |
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17 | $a$ is the statistical segment length of the polymer chain, |
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18 | and $n$ is the degree of polymerization. |
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19 | |
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20 | This integral was put into an almost analytical form as follows |
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21 | (Hammouda, 1993) |
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22 | |
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23 | .. math:: |
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24 | |
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25 | P(Q)=\frac{1}{\nu U^{1/2\nu}} |
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26 | \left\{ |
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27 | \gamma\left(\frac{1}{2\nu},U\right) - |
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28 | \frac{1}{U^{1/2\nu}}\gamma\left(\frac{1}{\nu},U\right) |
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29 | \right\} |
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30 | |
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31 | and later recast as (for example, Hore, 2013; Hammouda & Kim, 2017) |
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32 | |
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33 | .. math:: |
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34 | |
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35 | P(Q)=\frac{1}{\nu U^{1/2\nu}}\gamma\left(\frac{1}{2\nu},U\right) - |
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36 | \frac{1}{\nu U^{1/\nu}}\gamma\left(\frac{1}{\nu},U\right) |
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37 | |
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38 | where $\gamma(x,U)$ is the incomplete gamma function |
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39 | |
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40 | .. math:: |
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41 | |
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42 | \gamma(x,U)=\int_0^{U}dt\ \exp(-t)t^{x-1} |
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43 | |
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44 | and the variable $U$ is given in terms of the scattering vector $Q$ as |
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45 | |
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46 | .. math:: |
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47 | |
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48 | U=\frac{Q^2a^2n^{2\nu}}{6} = \frac{Q^2R_{g}^2(2\nu+1)(2\nu+2)}{6} |
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49 | |
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50 | The two analytic forms are equivalent. In the 1993 paper |
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51 | |
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52 | .. math:: |
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53 | |
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54 | \frac{1}{\nu U^{1/2\nu}} |
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55 | |
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56 | has been factored out. |
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57 | |
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58 | **SasView implements the 1993 expression**. |
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59 | |
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60 | The square of the radius-of-gyration is defined as |
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61 | |
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62 | .. math:: |
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63 | |
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64 | R_{g}^2 = \frac{a^2n^{2\nu}}{(2\nu+1)(2\nu+2)} |
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65 | |
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66 | .. note:: |
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67 | This model applies only in the mass fractal range (ie, $5/3<=m<=3$ ) |
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68 | and **does not apply** to surface fractals ( $3<m<=4$ ). |
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69 | It also does not reproduce the rigid rod limit (m=1) because it assumes chain |
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70 | flexibility from the outset. It may cover a portion of the semi-flexible chain |
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71 | range ( $1<m<5/3$ ). |
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72 | |
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73 | A low-Q expansion yields the Guinier form and a high-Q expansion yields the |
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74 | Porod form which is given by |
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75 | |
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76 | .. math:: |
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77 | |
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78 | P(Q\rightarrow \infty) = \frac{1}{\nu U^{1/2\nu}}\Gamma\left( |
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79 | \frac{1}{2\nu}\right) - \frac{1}{\nu U^{1/\nu}}\Gamma\left( |
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80 | \frac{1}{\nu}\right) |
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81 | |
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82 | Here $\Gamma(x) = \gamma(x,\infty)$ is the gamma function. |
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83 | |
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84 | The asymptotic limit is dominated by the first term |
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85 | |
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86 | .. math:: |
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87 | |
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88 | P(Q\rightarrow \infty) \sim \frac{1}{\nu U^{1/2\nu}}\Gamma\left(\frac{1}{2\nu}\right) = |
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89 | \frac{m}{\left(QR_{g}\right)^m}\left[\frac{6}{(2\nu +1)(2\nu +2)} \right]^{m/2} |
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90 | \Gamma (m/2) |
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91 | |
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92 | The special case when $\nu=0.5$ (or $m=1/\nu=2$ ) corresponds to Gaussian chains for |
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93 | which the form factor is given by the familiar Debye function. |
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94 | |
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95 | .. math:: |
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96 | |
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97 | P(Q) = \frac{2}{Q^4R_{g}^4} \left[\exp(-Q^2R_{g}^2) - 1 + Q^2R_{g}^2 \right] |
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98 | |
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99 | For 2D data: The 2D scattering intensity is calculated in the same way as 1D, |
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100 | where the $q$ vector is defined as |
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101 | |
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102 | .. math:: |
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103 | |
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104 | q = \sqrt{q_x^2 + q_y^2} |
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105 | |
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106 | |
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107 | References |
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108 | ---------- |
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109 | |
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110 | H Benoit, *Comptes Rendus*, 245 (1957) 2244-2247 |
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111 | |
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112 | B Hammouda, *SANS from Homogeneous Polymer Mixtures - A Unified Overview, |
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113 | Advances in Polym. Sci.* 106 (1993) 87-133 |
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114 | |
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115 | M Hore et al, *Co-Nonsolvency of Poly(n-isopropylacrylamide) in Deuterated |
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116 | Water/Ethanol Mixtures* 46 (2013) 7894-7901 |
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117 | |
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118 | B Hammouda & M-H Kim, *The empirical core-chain model* 247 (2017) 434-440 |
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119 | """ |
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120 | |
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121 | import numpy as np |
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122 | from numpy import inf, power, errstate |
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123 | from scipy.special import gammainc, gamma |
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124 | |
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125 | name = "polymer_excl_volume" |
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126 | title = "Polymer Excluded Volume model" |
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127 | description = """Compute the scattering intensity from polymers with excluded |
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128 | volume effects. |
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129 | rg: radius of gyration |
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130 | porod_exp: Porod exponent |
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131 | """ |
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132 | category = "shape-independent" |
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133 | |
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134 | # pylint: disable=bad-whitespace, line-too-long |
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135 | # ["name", "units", default, [lower, upper], "type", "description"], |
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136 | parameters = [ |
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137 | ["rg", "Ang", 60.0, [0, inf], "", "Radius of Gyration"], |
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138 | ["porod_exp", "", 3.0, [0, inf], "", "Porod exponent"], |
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139 | ] |
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140 | # pylint: enable=bad-whitespace, line-too-long |
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141 | |
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142 | |
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143 | def Iq(q, rg=60.0, porod_exp=3.0): |
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144 | """ |
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145 | :param q: Input q-value (float or [float, float]) |
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146 | :param rg: Radius of gyration |
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147 | :param porod_exp: Porod exponent |
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148 | :return: Calculated intensity |
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149 | """ |
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150 | usub = (q*rg)**2 * (2.0/porod_exp + 1.0) * (2.0/porod_exp + 2.0)/6.0 |
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151 | with errstate(divide='ignore', invalid='ignore'): |
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152 | upow = power(usub, -0.5*porod_exp) |
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153 | result = (porod_exp*upow * |
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154 | (gamma(0.5*porod_exp)*gammainc(0.5*porod_exp, usub) - |
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155 | upow*gamma(porod_exp)*gammainc(porod_exp, usub))) |
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156 | result[q <= 0] = 1.0 |
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157 | |
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158 | return result |
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159 | |
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160 | Iq.vectorized = True # Iq accepts an array of q values |
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161 | |
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162 | def random(): |
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163 | rg = 10**np.random.uniform(0, 4) |
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164 | porod_exp = np.random.uniform(1e-3, 6) |
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165 | scale = 10**np.random.uniform(1, 5) |
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166 | pars = dict( |
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167 | #background=0, |
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168 | scale=scale, |
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169 | rg=rg, |
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170 | porod_exp=porod_exp, |
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171 | ) |
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172 | return pars |
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173 | |
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174 | tests = [ |
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175 | # Accuracy tests based on content in test/polyexclvol_default_igor.txt |
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176 | [{'rg': 60, 'porod_exp': 3.0}, 0.001, 0.999801], |
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177 | [{'rg': 60, 'porod_exp': 3.0}, 0.105363, 0.0172751], |
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178 | [{'rg': 60, 'porod_exp': 3.0, 'background': 0.0}, 0.665075, 6.56261e-05], |
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179 | |
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180 | # Additional tests with larger range of parameters |
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181 | [{'rg': 10, 'porod_exp': 4.0}, 0.1, 0.724436675809], |
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182 | [{'rg': 2.2, 'porod_exp': 22.0, 'background': 100.0}, 5.0, 100.0], |
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183 | [{'rg': 1.1, 'porod_exp': 1, 'background': 10.0, 'scale': 1.25}, |
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184 | 20000., 10.0000712097] |
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185 | ] |
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