# source:sasmodels/sasmodels/models/raspberry.py@40a87fa

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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 self-
13correlation of the large droplet, the self-correlation of the particles, the
14correlation terms between different particles and the cross terms between large
15droplet and small particles all need to be calculated.
16
17Consider two infinitely thin shells of radii R1 and R2 separated by distance r.
18The general structure of the equation is then the form factor of the two shells
19multiplied by the phase factor that accounts for the separation of their
20centers.
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 **on:** 2008
105
106**Modified by:** Andrew Jackson **on:** March 20, 2016
107
108**Reviewed by:** Andrew Jackson **on:** March 20, 2016
109"""
110
111from numpy import inf
112
113name = "raspberry"
114title = "Calculates the form factor, *P(q)*, for a 'Raspberry-like' structure \
115where there are smaller spheres at the surface of a larger sphere, such as the \
116structure of a Pickering emulsion."
117description = """
118                RaspBerryModel:
119                volfraction_lg = volume fraction large spheres
121                sld_lg = sld large sphere (A-2)
122                volfraction_sm = volume fraction small spheres
124                surface_fraction = fraction of small spheres at surface
125                sld_sm = sld small sphere
126                penetration = small sphere penetration (A)
127                sld_solvent   = sld solvent
128                background = background (cm-1)
129            Ref: J. coll. inter. sci. (2010) vol. 343 (1) pp. 36-41."""
130category = "shape:sphere"
131#single = False
132
133#             [ "name", "units", default, [lower, upper], "type", "description"],
134parameters = [["sld_lg", "1e-6/Ang^2", -0.4, [-inf, inf], "sld",
135               "large particle scattering length density"],
136              ["sld_sm", "1e-6/Ang^2", 3.5, [-inf, inf], "sld",
137               "small particle scattering length density"],
138              ["sld_solvent", "1e-6/Ang^2", 6.36, [-inf, inf], "sld",
139               "solvent scattering length density"],
140              ["volfraction_lg", "", 0.05, [-inf, inf], "",
141               "volume fraction of large spheres"],
142              ["volfraction_sm", "", 0.005, [-inf, inf], "",
143               "volume fraction of small spheres"],
144              ["surface_fraction", "", 0.4, [-inf, inf], "",
145               "fraction of small spheres at surface"],
146              ["radius_lg", "Ang", 5000, [0, inf], "volume",
148              ["radius_sm", "Ang", 100, [0, inf], "",
150              ["penetration", "Ang", 0, [-1, 1], "",
151               "fractional penetration depth of small spheres into large sphere"],
152             ]
153
154source = ["lib/sph_j1c.c", "raspberry.c"]
155
156# parameters for demo
157demo = dict(scale=1, background=0.001,
158            sld_lg=-0.4, sld_sm=3.5, sld_solvent=6.36,
159            volfraction_lg=0.05, volfraction_sm=0.005, surface_fraction=0.4,
162
163# TODO: update tests so the parameters correspond to SasView parameters
164# The model was re-parameterized so the results have changed.
165# NOTE: test results taken from values returned by SasView 3.1.2, with
166# 0.001 added for a non-zero default background.
167#tests = [[{}, 0.0412755102041, 0.286669115234],
168#         [{}, 0.5, 0.00103818393658],
169#        ]
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