1 | r""" |
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2 | |
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3 | A number of natural and commercial processes form high-surface area materials |
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4 | as a result of the vapour-phase aggregation of primary particles. |
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5 | Examples of such materials include soots, aerosols, and fume or pyrogenic |
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6 | silicas. These are all characterised by cluster mass distributions (sometimes |
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7 | also cluster size distributions) and internal surfaces that are fractal in |
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8 | nature. The scattering from such materials displays two distinct breaks in |
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9 | log-log representation, corresponding to the radius-of-gyration of the primary |
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10 | particles, $rg$, and the radius-of-gyration of the clusters (aggregates), |
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11 | $Rg$. Between these boundaries the scattering follows a power law related to |
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12 | the mass fractal dimension, $Dm$, whilst above the high-Q boundary the |
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13 | scattering follows a power law related to the surface fractal dimension of |
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14 | the primary particles, $Ds$. |
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15 | |
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16 | Definition |
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17 | ---------- |
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18 | |
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19 | The scattered intensity I(q) is calculated using a modified |
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20 | Ornstein-Zernicke equation |
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21 | |
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22 | .. math:: |
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23 | |
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24 | I(q) = scale \times P(q) + background \\ |
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25 | P(q) = \left\{ \left[ 1+(q^2a)\right]^{D_m/2} \times |
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26 | \left[ 1+(q^2b)\right]^{(6-D_s-D_m)/2} |
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27 | \right\}^{-1} \\ |
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28 | a = R_{g}^2/(3D_m/2) \\ |
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29 | b = r_{g}^2/[-3(D_s+D_m-6)/2] \\ |
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30 | scale = scale\_factor \times NV^2 (\rho_{particle} - \rho_{solvent})^2 |
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31 | |
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32 | where $R_g$ is the size of the cluster, $r_g$ is the size of the primary |
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33 | particle, $D_s$ is the surface fractal dimension, $D_m$ is the mass fractal |
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34 | dimension, $\rho_{solvent}$ is the scattering length density of the solvent, |
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35 | and $\rho_{particle}$ is the scattering length density of particles. |
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36 | |
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37 | .. note:: |
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38 | |
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39 | The surface ( $D_s$ ) and mass ( $D_m$ ) fractal dimensions are only |
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40 | valid if $0 < surface\_dim < 6$ , $0 < mass\_dim < 6$ , and |
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41 | $(surface\_dim + mass\_dim ) < 6$ . |
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42 | |
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43 | |
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44 | References |
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45 | ---------- |
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46 | |
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47 | P Schmidt, *J Appl. Cryst.*, 24 (1991) 414-435 Equation(19) |
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48 | |
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49 | A J Hurd, D W Schaefer, J E Martin, *Phys. Rev. A*, |
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50 | 35 (1987) 2361-2364 Equation(2) |
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51 | """ |
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52 | |
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53 | import numpy as np |
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54 | from numpy import inf |
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55 | |
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56 | name = "mass_surface_fractal" |
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57 | title = "Mass Surface Fractal model" |
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58 | description = """ |
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59 | The scattering intensity I(x) = scale*P(x)*S(x) + background, where |
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60 | p(x)= {[1+(x^2*a)]^(Dm/2) * [1+(x^2*b)]^(6-Ds-Dm)/2}^(-1) |
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61 | a = Rg^2/(3*Dm/2) |
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62 | b = rg^2/(3*(6-Ds-Dm)/2) |
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63 | scale = scale factor * N*Volume^2*contrast^2 |
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64 | fractal_dim_mass = Dm (mass fractal dimension) |
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65 | fractal_dim_surf = Ds |
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66 | rg_cluster = Rg |
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67 | rg_primary = rg |
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68 | background = background |
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69 | Ref: Schmidt, J Appl Cryst, eq(19), (1991), 24, 414-435 |
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70 | Hurd, Schaefer, Martin, Phys Rev A, eq(2),(1987),35, 2361-2364 |
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71 | Note that 0 < Ds< 6 and 0 < Dm < 6. |
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72 | """ |
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73 | category = "shape-independent" |
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74 | |
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75 | # pylint: disable=bad-whitespace, line-too-long |
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76 | # ["name", "units", default, [lower, upper], "type","description"], |
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77 | parameters = [ |
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78 | ["fractal_dim_mass", "", 1.8, [0.0, 6.0], "", "Mass fractal dimension"], |
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79 | ["fractal_dim_surf", "", 2.3, [0.0, 6.0], "", "Surface fractal dimension"], |
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80 | ["rg_cluster", "Ang", 86.7, [0.0, inf], "", "Cluster radius of gyration"], |
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81 | ["rg_primary", "Ang", 4000., [0.0, inf], "", "Primary particle radius of gyration"], |
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82 | ] |
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83 | # pylint: enable=bad-whitespace, line-too-long |
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84 | |
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85 | source = ["mass_surface_fractal.c"] |
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86 | |
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87 | def random(): |
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88 | fractal_dim = np.random.uniform(0, 6) |
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89 | surface_portion = np.random.uniform(0, 1) |
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90 | fractal_dim_surf = fractal_dim*surface_portion |
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91 | fractal_dim_mass = fractal_dim - fractal_dim_surf |
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92 | rg_cluster = 10**np.random.uniform(1, 5) |
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93 | rg_primary = rg_cluster*10**np.random.uniform(-4, -1) |
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94 | scale = 10**np.random.uniform(2, 5) |
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95 | pars = dict( |
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96 | #background=0, |
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97 | scale=scale, |
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98 | fractal_dim_mass=fractal_dim_mass, |
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99 | fractal_dim_surf=fractal_dim_surf, |
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100 | rg_cluster=rg_cluster, |
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101 | rg_primary=rg_primary, |
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102 | ) |
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103 | return pars |
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104 | |
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105 | |
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106 | demo = dict(scale=1, background=0, |
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107 | fractal_dim_mass=1.8, |
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108 | fractal_dim_surf=2.3, |
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109 | rg_cluster=86.7, |
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110 | rg_primary=4000.0) |
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111 | |
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112 | tests = [ |
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113 | |
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114 | # Accuracy tests based on content in test/utest_other_models.py |
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115 | [{'fractal_dim_mass': 1.8, |
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116 | 'fractal_dim_surf': 2.3, |
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117 | 'rg_cluster': 86.7, |
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118 | 'rg_primary': 4000.0, |
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119 | 'background': 0.0, |
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120 | }, 0.05, 1.77537e-05], |
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121 | |
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122 | # Additional tests with larger range of parameters |
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123 | [{'fractal_dim_mass': 3.3, |
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124 | 'fractal_dim_surf': 1.0, |
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125 | 'rg_cluster': 90.0, |
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126 | 'rg_primary': 4000.0, |
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127 | }, 0.001, 0.18562699016], |
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128 | |
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129 | [{'fractal_dim_mass': 1.3, |
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130 | 'fractal_dim_surf': 1.0, |
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131 | 'rg_cluster': 90.0, |
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132 | 'rg_primary': 2000.0, |
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133 | 'background': 0.8, |
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134 | }, 0.001, 1.16539753641], |
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135 | |
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136 | [{'fractal_dim_mass': 2.3, |
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137 | 'fractal_dim_surf': 1.0, |
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138 | 'rg_cluster': 90.0, |
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139 | 'rg_primary': 1000.0, |
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140 | 'scale': 10.0, |
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141 | 'background': 0.0, |
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142 | }, 0.051, 0.000169548800377], |
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143 | ] |
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