[7ed702f] | 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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[ca04add] | 24 | I(q) = scale \times P(q) + background \\ |
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[7ed702f] | 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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[ca04add] | 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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[7ed702f] | 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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[9418d75] | 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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[7ed702f] | 42 | |
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| 43 | |
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[95441ff] | 44 | References |
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| 45 | ---------- |
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[7ed702f] | 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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[168052c] | 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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[7ed702f] | 51 | """ |
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| 52 | |
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[2d81cfe] | 53 | import numpy as np |
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[7ed702f] | 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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[a807206] | 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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[30fbe2e] | 67 | rg_primary = rg |
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[7ed702f] | 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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[168052c] | 75 | # pylint: disable=bad-whitespace, line-too-long |
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[6d96b66] | 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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[168052c] | 83 | # pylint: enable=bad-whitespace, line-too-long |
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[7ed702f] | 84 | |
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[9c461c7] | 85 | source = ["mass_surface_fractal.c"] |
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[7ed702f] | 86 | |
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[404ebbd] | 87 | def random(): |
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[232bb12] | 88 | fractal_dim = np.random.uniform(0, 6) |
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[404ebbd] | 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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[7ed702f] | 106 | demo = dict(scale=1, background=0, |
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[a807206] | 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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[30fbe2e] | 110 | rg_primary=4000.0) |
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[7ed702f] | 111 | |
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[07a6700] | 112 | tests = [ |
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[168052c] | 113 | |
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| 114 | # Accuracy tests based on content in test/utest_other_models.py |
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[a807206] | 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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[30fbe2e] | 118 | 'rg_primary': 4000.0, |
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[6dd90c1] | 119 | 'background': 0.0, |
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[168052c] | 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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[a807206] | 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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[30fbe2e] | 126 | 'rg_primary': 4000.0, |
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[6dd90c1] | 127 | }, 0.001, 0.18562699016], |
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[168052c] | 128 | |
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[a807206] | 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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[30fbe2e] | 132 | 'rg_primary': 2000.0, |
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[168052c] | 133 | 'background': 0.8, |
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| 134 | }, 0.001, 1.16539753641], |
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| 135 | |
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[a807206] | 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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[30fbe2e] | 139 | 'rg_primary': 1000.0, |
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[168052c] | 140 | 'scale': 10.0, |
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[6dd90c1] | 141 | 'background': 0.0, |
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[168052c] | 142 | }, 0.051, 0.000169548800377], |
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| 143 | ] |
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