# source:sasmodels/sasmodels/models/raspberry.py@ef07e95

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
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1r"""
2Definition
3----------
4
5The figure below shows a schematic of a large droplet surrounded by several
6smaller particles forming a structure similar to that of Pickering emulsions.
7
8.. figure:: img/raspberry_geometry.jpg
9
10    Schematic of the raspberry model
11
12In order to calculate the form factor of the entire complex, the
13self-correlation of the large droplet, the self-correlation of the particles,
14the correlation terms between different particles and the cross terms between
15large droplet and small particles all need to be calculated.
16
17Consider two infinitely thin shells of radii $R_1$ and $R_2$ separated by
18distance $r$. The general structure of the equation is then the form factor
19of the two shells multiplied by the phase factor that accounts for the
20separation of their centers.
21
22.. math::
23
24    S(q) = \frac{\sin(qR_1)}{qR_1}\frac{\sin(qR_2)}{qR_2}\frac{\sin(qr)}{qr}
25
26In this case, the large droplet and small particles are solid spheres rather
27than thin shells. Thus the two terms must be integrated over $R_L$ and $R_S$
28respectively using the weighting function of a sphere. We then obtain the
29functions for the form of the two spheres:
30
31.. math::
32
33    \Psi_L = \int_0^{R_L}(4\pi R^2_L)\frac{\sin(qR_L)}{qR_L}dR_L =
34    \frac{3[\sin(qR_L)-qR_L\cos(qR_L)]}{(qR_L)^2}
35
36.. math::
37
38    \Psi_S = \int_0^{R_S}(4\pi R^2_S)\frac{\sin(qR_S)}{qR_S}dR_S =
39    \frac{3[\sin(qR_S)-qR_L\cos(qR_S)]}{(qR_S)^2}
40
41The cross term between the large droplet and small particles is given by:
42
43.. math::
44    S_{LS} = \Psi_L\Psi_S\frac{\sin(q(R_L+\delta R_S))}{q(R_L+\delta\ R_S)}
45
46and the self term between small particles is given by:
47
48.. math::
49    S_{SS} = \Psi_S^2\biggl[\frac{\sin(q(R_L+\delta R_S))}{q(R_L+\delta\ R_S)}
50    \biggr]^2
51
52The number of small particles per large droplet, $N_p$, is given by:
53
54.. math::
55
56    N_p = \frac{\phi_S\phi_\text{surface}V_L}{\phi_L V_S}
57
58where $\phi_S$ is the volume fraction of small particles in the sample,
59$\phi_\text{surface}$ is the fraction of the small particles that are adsorbed
60to the large droplets, $\phi_L$ is the volume fraction of large droplets in the
61sample, and $V_S$ and $V_L$ are the volumes of individual small particles and
62large droplets respectively.
63
64The form factor of the entire complex can now be calculated including the excess
65scattering length densities of the components $\Delta\rho_L$ and $\Delta\rho_S$,
66where $\Delta\rho_x = \left|\rho_x-\rho_\text{solvent}\right|$ :
67
68.. math::
69
70    P_{LS} = \frac{1}{M^2}\bigl[(\Delta\rho_L)^2V_L^2\Psi_L^2
71                +N_p(\Delta\rho_S)^2V_S^2\Psi_S^2
72                + N_p(1-N_p)(\Delta\rho_S)^2V_S^2S_{SS}
73                + 2N_p\Delta\rho_L\Delta\rho_SV_LV_SS_{LS}\bigr]
74
75where M is the total scattering length of the whole complex :
76
77.. math::
78    M = \Delta\rho_LV_L + N_p\Delta\rho_SV_S
79
80In a real system, there will ususally be an excess of small particles such that
81some fraction remain unbound. Therefore the overall scattering intensity is
82given by:
83
84.. math::
85    I(Q) = I_{LS}(Q) + I_S(Q) = (\phi_L(\Delta\rho_L)^2V_L +
86            \phi_S\phi_\text{surface}N_p(\Delta\rho_S)^2V_S)P_{LS}
87            + \phi_S(1-\phi_\text{surface})(\Delta\rho_S)^2V_S\Psi_S^2
88
89A useful parameter to extract is the fraction of the surface area of the large
90droplets that is covered by small particles. This can be calculated from the
91model parameters as:
92
93.. math::
94    \chi = \frac{4\phi_L\phi_\text{surface}(R_L+\delta R_S)}{\phi_LR_S}
95
96
97References
98----------
99
100K Larson-Smith, A Jackson, and D C Pozzo, *Small angle scattering model for
101Pickering emulsions and raspberry particles*, *Journal of Colloid and Interface
102Science*, 343(1) (2010) 36-41
103
104* **Author:** Andrew Jackson **Date:** 2008
105* **Modified by:** Andrew Jackson **Date:** March 20, 2016
106* **Reviewed by:** Andrew Jackson **Date:** March 20, 2016
107"""
108
109import numpy as np
110from numpy import inf
111
112name = "raspberry"
113title = "Calculates the form factor, *P(q)*, for a 'Raspberry-like' structure \
114where there are smaller spheres at the surface of a larger sphere, such as the \
115structure of a Pickering emulsion."
116description = """
117                RaspBerryModel:
118                volfraction_lg = volume fraction large spheres
120                sld_lg = sld large sphere (A-2)
121                volfraction_sm = volume fraction small spheres
123                surface_fraction = fraction of small spheres at surface
124                sld_sm = sld small sphere
125                penetration = small sphere penetration (A)
126                sld_solvent   = sld solvent
127                background = background (cm-1)
128            Ref: J. coll. inter. sci. (2010) vol. 343 (1) pp. 36-41."""
129category = "shape:sphere"
130
131
132#             [ "name", "units", default, [lower, upper], "type", "description"],
133parameters = [["sld_lg", "1e-6/Ang^2", -0.4, [-inf, inf], "sld",
134               "large particle scattering length density"],
135              ["sld_sm", "1e-6/Ang^2", 3.5, [-inf, inf], "sld",
136               "small particle scattering length density"],
137              ["sld_solvent", "1e-6/Ang^2", 6.36, [-inf, inf], "sld",
138               "solvent scattering length density"],
139              ["volfraction_lg", "", 0.05, [-inf, inf], "",
140               "volume fraction of large spheres"],
141              ["volfraction_sm", "", 0.005, [-inf, inf], "",
142               "volume fraction of small spheres"],
143              ["surface_fraction", "", 0.4, [-inf, inf], "",
144               "fraction of small spheres at surface"],
145              ["radius_lg", "Ang", 5000, [0, inf], "volume",
147              ["radius_sm", "Ang", 100, [0, inf], "",
149              ["penetration", "Ang", 0, [-1, 1], "",
150               "fractional penetration depth of small spheres into large sphere"],
151             ]
152
153source = ["lib/sas_3j1x_x.c", "raspberry.c"]
154
155def random():
156    # Limit volume fraction to 20% each
157    volfraction_lg = 10**np.random.uniform(-3, -0.3)
158    volfraction_sm = 10**np.random.uniform(-3, -0.3)
159    # Prefer most particles attached (peak near 60%), but not all or none
160    surface_fraction = np.random.beta(6, 4)
161    radius_lg = 10**np.random.uniform(1.7, 4.7)  # 500 - 50000 A
163    penetration = np.random.beta(1, 10) # up to 20% pen. for 90% of examples
164    pars = dict(
165        volfraction_lg=volfraction_lg,
166        volfraction_sm=volfraction_sm,
167        surface_fraction=surface_fraction,
170        penetration=penetration,
171    )
172    return pars
173
174# parameters for demo
175demo = dict(scale=1, background=0.001,
176            sld_lg=-0.4, sld_sm=3.5, sld_solvent=6.36,
177            volfraction_lg=0.05, volfraction_sm=0.005, surface_fraction=0.4,