1 | r""" |
---|
2 | |
---|
3 | A number of natural and commercial processes form high-surface area materials |
---|
4 | as a result of the vapour-phase aggregation of primary particles. |
---|
5 | Examples of such materials include soots, aerosols, and fume or pyrogenic |
---|
6 | silicas. These are all characterised by cluster mass distributions (sometimes |
---|
7 | also cluster size distributions) and internal surfaces that are fractal in |
---|
8 | nature. The scattering from such materials displays two distinct breaks in |
---|
9 | log-log representation, corresponding to the radius-of-gyration of the primary |
---|
10 | particles, $rg$, and the radius-of-gyration of the clusters (aggregates), |
---|
11 | $Rg$. Between these boundaries the scattering follows a power law related to |
---|
12 | the mass fractal dimension, $Dm$, whilst above the high-Q boundary the |
---|
13 | scattering follows a power law related to the surface fractal dimension of |
---|
14 | the primary particles, $Ds$. |
---|
15 | |
---|
16 | Definition |
---|
17 | ---------- |
---|
18 | |
---|
19 | The scattered intensity I(q) is calculated using a modified |
---|
20 | Ornstein-Zernicke equation |
---|
21 | |
---|
22 | .. math:: |
---|
23 | |
---|
24 | I(q) = scale \times P(q) + background \\ |
---|
25 | P(q) = \left\{ \left[ 1+(q^2a)\right]^{D_m/2} \times |
---|
26 | \left[ 1+(q^2b)\right]^{(6-D_s-D_m)/2} |
---|
27 | \right\}^{-1} \\ |
---|
28 | a = R_{g}^2/(3D_m/2) \\ |
---|
29 | b = r_{g}^2/[-3(D_s+D_m-6)/2] \\ |
---|
30 | scale = scale\_factor \times NV^2 (\rho_{particle} - \rho_{solvent})^2 |
---|
31 | |
---|
32 | where $R_g$ is the size of the cluster, $r_g$ is the size of the primary |
---|
33 | particle, $D_s$ is the surface fractal dimension, $D_m$ is the mass fractal |
---|
34 | dimension, $\rho_{solvent}$ is the scattering length density of the solvent, |
---|
35 | and $\rho_{particle}$ is the scattering length density of particles. |
---|
36 | |
---|
37 | .. note:: |
---|
38 | |
---|
39 | The surface ( $D_s$ ) and mass ( $D_m$ ) fractal dimensions are only |
---|
40 | valid if $0 < surface\_dim < 6$ , $0 < mass\_dim < 6$ , and |
---|
41 | $(surface\_dim + mass\_dim ) < 6$ . |
---|
42 | Older versions of sasview may have the default primary particle radius |
---|
43 | larger than the cluster radius, this was an error, also present in the |
---|
44 | Schmidt review paper below. The primary particle should be the smaller |
---|
45 | as described in the original Hurd et.al. who also point out that |
---|
46 | polydispersity in the primary particle sizes may affect their |
---|
47 | apparent surface fractal dimension. |
---|
48 | |
---|
49 | |
---|
50 | References |
---|
51 | ---------- |
---|
52 | |
---|
53 | .. [#] P Schmidt, *J Appl. Cryst.*, 24 (1991) 414-435 Equation(19) |
---|
54 | .. [#] A J Hurd, D W Schaefer, J E Martin, *Phys. Rev. A*, |
---|
55 | 35 (1987) 2361-2364 Equation(2) |
---|
56 | |
---|
57 | Authorship and Verification |
---|
58 | ---------------------------- |
---|
59 | |
---|
60 | * **Converted to sasmodels by:** Piotr Rozyczko **Date:** Jan 20, 2016 |
---|
61 | * **Last Reviewed by:** Richard Heenan **Date:** May 30, 2018 |
---|
62 | """ |
---|
63 | |
---|
64 | import numpy as np |
---|
65 | from numpy import inf |
---|
66 | |
---|
67 | name = "mass_surface_fractal" |
---|
68 | title = "Mass Surface Fractal model" |
---|
69 | description = """ |
---|
70 | The scattering intensity I(x) = scale*P(x)*S(x) + background, where |
---|
71 | p(x)= {[1+(x^2*a)]^(Dm/2) * [1+(x^2*b)]^(6-Ds-Dm)/2}^(-1) |
---|
72 | a = Rg^2/(3*Dm/2) |
---|
73 | b = rg^2/(3*(6-Ds-Dm)/2) |
---|
74 | scale = scale factor * N*Volume^2*contrast^2 |
---|
75 | fractal_dim_mass = Dm (mass fractal dimension) |
---|
76 | fractal_dim_surf = Ds |
---|
77 | rg_cluster = Rg |
---|
78 | rg_primary = rg |
---|
79 | background = background |
---|
80 | Hurd, Schaefer, Martin, Phys Rev A, eq(2),(1987),35, 2361-2364 |
---|
81 | Note that 0 < Ds< 6 and 0 < Dm < 6. |
---|
82 | """ |
---|
83 | category = "shape-independent" |
---|
84 | |
---|
85 | # pylint: disable=bad-whitespace, line-too-long |
---|
86 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
87 | parameters = [ |
---|
88 | ["fractal_dim_mass", "", 1.8, [0.0, 6.0], "", "Mass fractal dimension"], |
---|
89 | ["fractal_dim_surf", "", 2.3, [0.0, 6.0], "", "Surface fractal dimension"], |
---|
90 | ["rg_cluster", "Ang", 4000., [0.0, inf], "", "Cluster radius of gyration"], |
---|
91 | ["rg_primary", "Ang", 86.7, [0.0, inf], "", "Primary particle radius of gyration"], |
---|
92 | ] |
---|
93 | # pylint: enable=bad-whitespace, line-too-long |
---|
94 | |
---|
95 | source = ["mass_surface_fractal.c"] |
---|
96 | |
---|
97 | def random(): |
---|
98 | fractal_dim = np.random.uniform(0, 6) |
---|
99 | surface_portion = np.random.uniform(0, 1) |
---|
100 | fractal_dim_surf = fractal_dim*surface_portion |
---|
101 | fractal_dim_mass = fractal_dim - fractal_dim_surf |
---|
102 | rg_cluster = 10**np.random.uniform(1, 5) |
---|
103 | rg_primary = rg_cluster*10**np.random.uniform(-4, -1) |
---|
104 | scale = 10**np.random.uniform(2, 5) |
---|
105 | pars = dict( |
---|
106 | #background=0, |
---|
107 | scale=scale, |
---|
108 | fractal_dim_mass=fractal_dim_mass, |
---|
109 | fractal_dim_surf=fractal_dim_surf, |
---|
110 | rg_cluster=rg_cluster, |
---|
111 | rg_primary=rg_primary, |
---|
112 | ) |
---|
113 | return pars |
---|
114 | |
---|
115 | |
---|
116 | demo = dict(scale=1, background=0, |
---|
117 | fractal_dim_mass=1.8, |
---|
118 | fractal_dim_surf=2.3, |
---|
119 | rg_cluster=4000.0, |
---|
120 | rg_primary=86.7) |
---|
121 | |
---|
122 | tests = [ |
---|
123 | |
---|
124 | # Accuracy tests based on content in test/utest_other_models.py All except first, changed so rg_cluster is the larger, RKH 30 May 2018 |
---|
125 | [{'fractal_dim_mass': 1.8, |
---|
126 | 'fractal_dim_surf': 2.3, |
---|
127 | 'rg_cluster': 86.7, |
---|
128 | 'rg_primary': 4000.0, |
---|
129 | 'background': 0.0, |
---|
130 | }, 0.05, 1.77537e-05], |
---|
131 | |
---|
132 | # Additional tests with larger range of parameters |
---|
133 | [{'fractal_dim_mass': 3.3, |
---|
134 | 'fractal_dim_surf': 1.0, |
---|
135 | 'rg_cluster': 4000.0, |
---|
136 | 'rg_primary': 90.0, |
---|
137 | }, 0.001, 0.0932516614456], |
---|
138 | |
---|
139 | [{'fractal_dim_mass': 1.3, |
---|
140 | 'fractal_dim_surf': 2.0, |
---|
141 | 'rg_cluster': 2000.0, |
---|
142 | 'rg_primary': 90.0, |
---|
143 | 'background': 0.8, |
---|
144 | }, 0.001, 1.28296431786], |
---|
145 | |
---|
146 | [{'fractal_dim_mass': 2.3, |
---|
147 | 'fractal_dim_surf': 3.1, |
---|
148 | 'rg_cluster': 1000.0, |
---|
149 | 'rg_primary': 30.0, |
---|
150 | 'scale': 10.0, |
---|
151 | 'background': 0.0, |
---|
152 | }, 0.051, 0.00333804044899], |
---|
153 | ] |
---|