1 | # Note: model title and parameter table are inserted automatically |
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2 | r""" |
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3 | This calculates the interparticle structure factor for a hard sphere fluid |
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4 | with a narrow attractive well. A perturbative solution of the Percus-Yevick |
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5 | closure is used. The strength of the attractive well is described in terms |
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6 | of "stickiness" as defined below. |
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7 | |
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8 | The perturb (perturbation parameter), $\epsilon$, should be held between 0.01 |
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9 | and 0.1. It is best to hold the perturbation parameter fixed and let |
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10 | the "stickiness" vary to adjust the interaction strength. The stickiness, |
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11 | $\tau$, is defined in the equation below and is a function of both the |
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12 | perturbation parameter and the interaction strength. $\tau$ and $\epsilon$ |
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13 | are defined in terms of the hard sphere diameter $(\sigma = 2 R)$, the |
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14 | width of the square well, $\Delta$ (same units as $R$\ ), and the depth of |
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15 | the well, $U_o$, in units of $kT$. From the definition, it is clear that |
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16 | smaller $\tau$ means stronger attraction. |
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17 | |
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18 | .. math:: |
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19 | |
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20 | \tau &= \frac{1}{12\epsilon} \exp(u_o / kT) \\ |
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21 | \epsilon &= \Delta / (\sigma + \Delta) |
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22 | |
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23 | where the interaction potential is |
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24 | |
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25 | .. math:: |
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26 | |
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27 | U(r) = \begin{cases} |
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28 | \infty & r < \sigma \\ |
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29 | -U_o & \sigma \leq r \leq \sigma + \Delta \\ |
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30 | 0 & r > \sigma + \Delta |
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31 | \end{cases} |
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32 | |
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33 | The Percus-Yevick (PY) closure was used for this calculation, and is an |
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34 | adequate closure for an attractive interparticle potential. This solution |
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35 | has been compared to Monte Carlo simulations for a square well fluid, with |
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36 | good agreement. |
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37 | |
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38 | The true particle volume fraction, $\phi$, is not equal to $h$, which appears |
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39 | in most of the reference. The two are related in equation (24) of the |
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40 | reference. The reference also describes the relationship between this |
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41 | perturbation solution and the original sticky hard sphere (or adhesive |
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42 | sphere) model by Baxter. |
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43 | |
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44 | **NB**: The calculation can go haywire for certain combinations of the input |
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45 | parameters, producing unphysical solutions - in this case errors are |
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46 | reported to the command window and the $S(q)$ is set to -1 (so it will |
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47 | disappear on a log-log plot). Use tight bounds to keep the parameters to |
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48 | values that you know are physical (test them) and keep nudging them until |
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49 | the optimization does not hit the constraints. |
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50 | |
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51 | In sasview the effective radius will be calculated from the parameters |
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52 | used in the form factor $P(q)$ that this $S(q)$ is combined with. |
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53 | |
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54 | For 2D data the scattering intensity is calculated in the same way |
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55 | as 1D, where the $q$ vector is defined as |
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56 | |
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57 | .. math:: |
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58 | |
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59 | q = \sqrt{q_x^2 + q_y^2} |
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60 | |
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61 | .. figure:: img/stickyhardsphere_1d.jpg |
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62 | |
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63 | 1D plot using the default values (in linear scale). |
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64 | |
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65 | References |
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66 | ---------- |
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67 | |
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68 | S V G Menon, C Manohar, and K S Rao, *J. Chem. Phys.*, 95(12) (1991) 9186-9190 |
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69 | """ |
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70 | |
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71 | # TODO: refactor so that we pull in the old sansmodels.c_extensions |
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72 | |
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73 | from numpy import inf |
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74 | |
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75 | name = "stickyhardsphere" |
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76 | title = "Sticky hard sphere structure factor, with Percus-Yevick closure" |
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77 | description = """\ |
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78 | [Sticky hard sphere structure factor, with Percus-Yevick closure] |
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79 | Interparticle structure factor S(Q)for a hard sphere fluid with |
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80 | a narrow attractive well. Fits are prone to deliver non-physical |
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81 | parameters, use with care and read the references in the full manual. |
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82 | In sasview the effective radius will be calculated from the |
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83 | parameters used in P(Q). |
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84 | """ |
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85 | category = "structure-factor" |
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86 | |
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87 | # ["name", "units", default, [lower, upper], "type","description"], |
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88 | parameters = [ |
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89 | # [ "name", "units", default, [lower, upper], "type", |
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90 | # "description" ], |
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91 | ["effect_radius", "Ang", 50.0, [0, inf], "volume", |
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92 | "effective radius of hard sphere"], |
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93 | ["volfraction", "", 0.2, [0, 0.74], "", |
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94 | "volume fraction of hard spheres"], |
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95 | ["perturb", "", 0.05, [0.01, 0.1], "", |
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96 | "perturbation parameter, epsilon"], |
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97 | ["stickiness", "", 0.20, [-inf, inf], "", |
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98 | "stickiness, tau"], |
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99 | ] |
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100 | |
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101 | # No volume normalization despite having a volume parameter |
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102 | # This should perhaps be volume normalized? |
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103 | form_volume = """ |
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104 | return 1.0; |
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105 | """ |
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106 | |
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107 | Iq = """ |
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108 | double onemineps,eta; |
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109 | double sig,aa,etam1,etam1sq,qa,qb,qc,radic; |
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110 | double lam,lam2,test,mu,alpha,beta; |
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111 | double kk,k2,k3,ds,dc,aq1,aq2,aq3,aq,bq1,bq2,bq3,bq,sq; |
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112 | |
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113 | onemineps = 1.0-perturb; |
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114 | eta = volfraction/onemineps/onemineps/onemineps; |
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115 | |
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116 | sig = 2.0 * effect_radius; |
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117 | aa = sig/onemineps; |
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118 | etam1 = 1.0 - eta; |
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119 | etam1sq=etam1*etam1; |
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120 | //C |
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121 | //C SOLVE QUADRATIC FOR LAMBDA |
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122 | //C |
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123 | qa = eta/12.0; |
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124 | qb = -1.0*(stickiness + eta/etam1); |
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125 | qc = (1.0 + eta/2.0)/etam1sq; |
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126 | radic = qb*qb - 4.0*qa*qc; |
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127 | if(radic<0) { |
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128 | //if(x>0.01 && x<0.015) |
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129 | // Print "Lambda unphysical - both roots imaginary" |
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130 | //endif |
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131 | return(-1.0); |
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132 | } |
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133 | //C KEEP THE SMALLER ROOT, THE LARGER ONE IS UNPHYSICAL |
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134 | lam = (-1.0*qb-sqrt(radic))/(2.0*qa); |
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135 | lam2 = (-1.0*qb+sqrt(radic))/(2.0*qa); |
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136 | if(lam2<lam) { |
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137 | lam = lam2; |
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138 | } |
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139 | test = 1.0 + 2.0*eta; |
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140 | mu = lam*eta*etam1; |
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141 | if(mu>test) { |
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142 | //if(x>0.01 && x<0.015) |
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143 | // Print "Lambda unphysical mu>test" |
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144 | //endif |
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145 | return(-1.0); |
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146 | } |
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147 | alpha = (1.0 + 2.0*eta - mu)/etam1sq; |
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148 | beta = (mu - 3.0*eta)/(2.0*etam1sq); |
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149 | //C |
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150 | //C CALCULATE THE STRUCTURE FACTOR |
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151 | //C |
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152 | kk = q*aa; |
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153 | k2 = kk*kk; |
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154 | k3 = kk*k2; |
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155 | SINCOS(kk,ds,dc); |
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156 | //ds = sin(kk); |
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157 | //dc = cos(kk); |
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158 | aq1 = ((ds - kk*dc)*alpha)/k3; |
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159 | aq2 = (beta*(1.0-dc))/k2; |
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160 | aq3 = (lam*ds)/(12.0*kk); |
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161 | aq = 1.0 + 12.0*eta*(aq1+aq2-aq3); |
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162 | // |
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163 | bq1 = alpha*(0.5/kk - ds/k2 + (1.0 - dc)/k3); |
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164 | bq2 = beta*(1.0/kk - ds/k2); |
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165 | bq3 = (lam/12.0)*((1.0 - dc)/kk); |
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166 | bq = 12.0*eta*(bq1+bq2-bq3); |
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167 | // |
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168 | sq = 1.0/(aq*aq +bq*bq); |
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169 | |
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170 | return(sq); |
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171 | """ |
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172 | |
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173 | Iqxy = """ |
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174 | return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS); |
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175 | """ |
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176 | |
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177 | # ER defaults to 0.0 |
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178 | # VR defaults to 1.0 |
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179 | |
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180 | oldname = 'StickyHSStructure' |
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181 | oldpars = dict() |
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182 | demo = dict(effect_radius=200, volfraction=0.2, perturb=0.05, |
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183 | stickiness=0.2, effect_radius_pd=0.1, effect_radius_pd_n=40) |
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184 | # |
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185 | tests = [ |
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186 | [ {'scale': 1.0, 'background' : 0.0, 'effect_radius' : 50.0, 'perturb' : 0.05, 'stickiness' : 0.2, 'volfraction' : 0.1, |
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187 | 'effect_radius_pd' : 0}, [0.001], [1.09718]] |
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188 | ] |
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189 | |
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190 | |
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