1 | r""" |
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2 | Definition |
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3 | ---------- |
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4 | |
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5 | The figure below shows a schematic of a large droplet surrounded by several |
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6 | smaller particles forming a structure similar to that of Pickering emulsions. |
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7 | |
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8 | .. figure:: img/raspberry_geometry.jpg |
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9 | |
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10 | Schematic of the raspberry model |
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11 | |
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12 | In order to calculate the form factor of the entire complex, the |
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13 | self-correlation of the large droplet, the self-correlation of the particles, |
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14 | the correlation terms between different particles and the cross terms between |
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15 | large droplet and small particles all need to be calculated. |
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16 | |
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17 | Consider two infinitely thin shells of radii $R_1$ and $R_2$ separated by |
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18 | distance $r$. The general structure of the equation is then the form factor |
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19 | of the two shells multiplied by the phase factor that accounts for the |
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20 | separation of their centers. |
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21 | |
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22 | .. math:: |
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23 | |
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24 | S(q) = \frac{\sin(qR_1)}{qR_1}\frac{\sin(qR_2)}{qR_2}\frac{\sin(qr)}{qr} |
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25 | |
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26 | In this case, the large droplet and small particles are solid spheres rather |
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27 | than thin shells. Thus the two terms must be integrated over $R_L$ and $R_S$ |
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28 | respectively using the weighting function of a sphere. We then obtain the |
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29 | functions for the form of the two spheres: |
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30 | |
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31 | .. math:: |
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32 | |
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33 | \Psi_L = \int_0^{R_L}(4\pi R^2_L)\frac{\sin(qR_L)}{qR_L}dR_L = |
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34 | \frac{3[\sin(qR_L)-qR_L\cos(qR_L)]}{(qR_L)^2} |
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35 | |
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36 | .. math:: |
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37 | |
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38 | \Psi_S = \int_0^{R_S}(4\pi R^2_S)\frac{\sin(qR_S)}{qR_S}dR_S = |
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39 | \frac{3[\sin(qR_S)-qR_L\cos(qR_S)]}{(qR_S)^2} |
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40 | |
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41 | The cross term between the large droplet and small particles is given by: |
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42 | |
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43 | .. math:: |
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44 | S_{LS} = \Psi_L\Psi_S\frac{\sin(q(R_L+\delta R_S))}{q(R_L+\delta\ R_S)} |
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45 | |
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46 | and the self term between small particles is given by: |
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47 | |
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48 | .. math:: |
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49 | S_{SS} = \Psi_S^2\biggl[\frac{\sin(q(R_L+\delta R_S))}{q(R_L+\delta\ R_S)} |
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50 | \biggr]^2 |
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51 | |
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52 | The number of small particles per large droplet, $N_p$, is given by: |
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53 | |
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54 | .. math:: |
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55 | |
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56 | N_p = \frac{\phi_S\phi_\text{surface}V_L}{\phi_L V_S} |
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57 | |
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58 | where $\phi_S$ is the volume fraction of small particles in the sample, |
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59 | $\phi_\text{surface}$ is the fraction of the small particles that are adsorbed |
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60 | to the large droplets, $\phi_L$ is the volume fraction of large droplets in the |
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61 | sample, and $V_S$ and $V_L$ are the volumes of individual small particles and |
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62 | large droplets respectively. |
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63 | |
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64 | The form factor of the entire complex can now be calculated including the excess |
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65 | scattering length densities of the components $\Delta\rho_L$ and $\Delta\rho_S$, |
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66 | where $\Delta\rho_x = \left|\rho_x-\rho_\text{solvent}\right|$ : |
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67 | |
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68 | .. math:: |
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69 | |
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70 | P_{LS} = \frac{1}{M^2}\bigl[(\Delta\rho_L)^2V_L^2\Psi_L^2 |
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71 | +N_p(\Delta\rho_S)^2V_S^2\Psi_S^2 |
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72 | + N_p(1-N_p)(\Delta\rho_S)^2V_S^2S_{SS} |
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73 | + 2N_p\Delta\rho_L\Delta\rho_SV_LV_SS_{LS}\bigr] |
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74 | |
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75 | where M is the total scattering length of the whole complex : |
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76 | |
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77 | .. math:: |
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78 | M = \Delta\rho_LV_L + N_p\Delta\rho_SV_S |
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79 | |
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80 | In a real system, there will ususally be an excess of small particles such that |
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81 | some fraction remain unbound. Therefore the overall scattering intensity is |
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82 | given by: |
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83 | |
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84 | .. math:: |
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85 | I(Q) = I_{LS}(Q) + I_S(Q) = (\phi_L(\Delta\rho_L)^2V_L + |
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86 | \phi_S\phi_\text{surface}N_p(\Delta\rho_S)^2V_S)P_{LS} |
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87 | + \phi_S(1-\phi_\text{surface})(\Delta\rho_S)^2V_S\Psi_S^2 |
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88 | |
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89 | A useful parameter to extract is the fraction of the surface area of the large |
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90 | droplets that is covered by small particles. This can be calculated from the |
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91 | model parameters as: |
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92 | |
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93 | .. math:: |
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94 | \chi = \frac{4\phi_L\phi_\text{surface}(R_L+\delta R_S)}{\phi_LR_S} |
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95 | |
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96 | |
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97 | References |
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98 | ---------- |
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99 | |
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100 | .. [#] K Larson-Smith, A Jackson, and D C Pozzo, *Small angle scattering model for Pickering emulsions and raspberry particles*, *Journal of Colloid and Interface Science*, 343(1) (2010) 36-41 |
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101 | |
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102 | Source |
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103 | ------ |
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104 | |
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105 | `raspberry.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/raspberry.py>`_ |
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106 | |
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107 | `raspberry.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/raspberry.c>`_ |
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108 | |
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109 | Authorship and Verification |
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110 | ---------------------------- |
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111 | |
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112 | * **Author:** Andrew Jackson **Date:** 2008 |
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113 | * **Last Modified by:** Andrew Jackson **Date:** March 20, 2016 |
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114 | * **Last Reviewed by:** Andrew Jackson **Date:** March 20, 2016 |
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115 | * **Source added by :** Steve King **Date:** March 25, 2019 |
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116 | """ |
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117 | |
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118 | import numpy as np |
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119 | from numpy import inf |
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120 | |
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121 | name = "raspberry" |
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122 | title = "Calculates the form factor, *P(q)*, for a 'Raspberry-like' structure \ |
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123 | where there are smaller spheres at the surface of a larger sphere, such as the \ |
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124 | structure of a Pickering emulsion." |
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125 | description = """ |
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126 | RaspBerryModel: |
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127 | volfraction_lg = volume fraction large spheres |
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128 | radius_lg = radius large sphere (A) |
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129 | sld_lg = sld large sphere (A-2) |
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130 | volfraction_sm = volume fraction small spheres |
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131 | radius_sm = radius small sphere (A) |
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132 | surface_fraction = fraction of small spheres at surface |
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133 | sld_sm = sld small sphere |
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134 | penetration = small sphere penetration (A) |
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135 | sld_solvent = sld solvent |
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136 | background = background (cm-1) |
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137 | Ref: J. coll. inter. sci. (2010) vol. 343 (1) pp. 36-41.""" |
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138 | category = "shape:sphere" |
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139 | |
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140 | |
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141 | # [ "name", "units", default, [lower, upper], "type", "description"], |
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142 | parameters = [["sld_lg", "1e-6/Ang^2", -0.4, [-inf, inf], "sld", |
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143 | "large particle scattering length density"], |
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144 | ["sld_sm", "1e-6/Ang^2", 3.5, [-inf, inf], "sld", |
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145 | "small particle scattering length density"], |
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146 | ["sld_solvent", "1e-6/Ang^2", 6.36, [-inf, inf], "sld", |
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147 | "solvent scattering length density"], |
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148 | ["volfraction_lg", "", 0.05, [-inf, inf], "", |
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149 | "volume fraction of large spheres"], |
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150 | ["volfraction_sm", "", 0.005, [-inf, inf], "", |
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151 | "volume fraction of small spheres"], |
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152 | ["surface_fraction", "", 0.4, [-inf, inf], "", |
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153 | "fraction of small spheres at surface"], |
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154 | ["radius_lg", "Ang", 5000, [0, inf], "volume", |
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155 | "radius of large spheres"], |
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156 | ["radius_sm", "Ang", 100, [0, inf], "volume", |
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157 | "radius of small spheres"], |
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158 | ["penetration", "Ang", 0, [-1, 1], "volume", |
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159 | "fractional penetration depth of small spheres into large sphere"], |
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160 | ] |
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161 | |
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162 | source = ["lib/sas_3j1x_x.c", "raspberry.c"] |
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163 | radius_effective_modes = ["radius_large", "radius_outer"] |
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164 | |
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165 | def random(): |
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166 | """Return a random parameter set for the model.""" |
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167 | # Limit volume fraction to 20% each |
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168 | volfraction_lg = 10**np.random.uniform(-3, -0.3) |
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169 | volfraction_sm = 10**np.random.uniform(-3, -0.3) |
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170 | # Prefer most particles attached (peak near 60%), but not all or none |
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171 | surface_fraction = np.random.beta(6, 4) |
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172 | radius_lg = 10**np.random.uniform(1.7, 4.7) # 500 - 50000 A |
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173 | radius_sm = 10**np.random.uniform(-3, -0.3)*radius_lg # 0.1% - 20% |
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174 | penetration = np.random.beta(1, 10) # up to 20% pen. for 90% of examples |
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175 | pars = dict( |
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176 | volfraction_lg=volfraction_lg, |
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177 | volfraction_sm=volfraction_sm, |
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178 | surface_fraction=surface_fraction, |
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179 | radius_lg=radius_lg, |
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180 | radius_sm=radius_sm, |
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181 | penetration=penetration, |
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182 | ) |
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183 | return pars |
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184 | |
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185 | # parameters for demo |
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186 | demo = dict(scale=1, background=0.001, |
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187 | sld_lg=-0.4, sld_sm=3.5, sld_solvent=6.36, |
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188 | volfraction_lg=0.05, volfraction_sm=0.005, surface_fraction=0.4, |
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189 | radius_lg=5000, radius_sm=100, penetration=0.0, |
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190 | radius_lg_pd=.2, radius_lg_pd_n=10) |
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191 | |
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192 | # TODO: update tests so the parameters correspond to SasView parameters |
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193 | # The model was re-parameterized so the results have changed. |
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194 | # NOTE: test results taken from values returned by SasView 3.1.2, with |
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195 | # 0.001 added for a non-zero default background. |
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196 | #tests = [[{}, 0.0412755102041, 0.286669115234], |
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197 | # [{}, 0.5, 0.00103818393658], |
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198 | # ] |
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