# source:sasmodels/sasmodels/models/raspberry.py@0507e09

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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
100.. [#] K Larson-Smith, A Jackson, and D C Pozzo, *Small angle scattering model for Pickering emulsions and raspberry particles*, *Journal of Colloid and Interface Science*, 343(1) (2010) 36-41
101
102Source
103------
104
105raspberry.py <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/raspberry.py>_
106
107raspberry.c <https://github.com/SasView/sasmodels/blob/master/sasmodels/models/raspberry.c>_
108
109Authorship and Verification
110----------------------------
111
112* **Author:** Andrew Jackson **Date:** 2008
114* **Last Reviewed by:** Andrew Jackson **Date:** March 20, 2016
115* **Source added by :** Steve King **Date:** March 25, 2019
116"""
117
118import numpy as np
119from numpy import inf
120
121name = "raspberry"
122title = "Calculates the form factor, *P(q)*, for a 'Raspberry-like' structure \
123where there are smaller spheres at the surface of a larger sphere, such as the \
124structure of a Pickering emulsion."
125description = """
126                RaspBerryModel:
127                volfraction_lg = volume fraction large spheres
129                sld_lg = sld large sphere (A-2)
130                volfraction_sm = volume fraction small spheres
132                surface_fraction = fraction of small spheres at surface
133                sld_sm = sld small sphere
134                penetration = small sphere penetration (A)
135                sld_solvent   = sld solvent
136                background = background (cm-1)
137            Ref: J. coll. inter. sci. (2010) vol. 343 (1) pp. 36-41."""
138category = "shape:sphere"
139
140
141#             [ "name", "units", default, [lower, upper], "type", "description"],
142parameters = [["sld_lg", "1e-6/Ang^2", -0.4, [-inf, inf], "sld",
143               "large particle scattering length density"],
144              ["sld_sm", "1e-6/Ang^2", 3.5, [-inf, inf], "sld",
145               "small particle scattering length density"],
146              ["sld_solvent", "1e-6/Ang^2", 6.36, [-inf, inf], "sld",
147               "solvent scattering length density"],
148              ["volfraction_lg", "", 0.05, [-inf, inf], "",
149               "volume fraction of large spheres"],
150              ["volfraction_sm", "", 0.005, [-inf, inf], "",
151               "volume fraction of small spheres"],
152              ["surface_fraction", "", 0.4, [-inf, inf], "",
153               "fraction of small spheres at surface"],
154              ["radius_lg", "Ang", 5000, [0, inf], "volume",
156              ["radius_sm", "Ang", 100, [0, inf], "volume",
158              ["penetration", "Ang", 0, [-1, 1], "volume",
159               "fractional penetration depth of small spheres into large sphere"],
160             ]
161
162source = ["lib/sas_3j1x_x.c", "raspberry.c"]
164
165def random():
166    """Return a random parameter set for the model."""
167    # Limit volume fraction to 20% each
168    volfraction_lg = 10**np.random.uniform(-3, -0.3)
169    volfraction_sm = 10**np.random.uniform(-3, -0.3)
170    # Prefer most particles attached (peak near 60%), but not all or none
171    surface_fraction = np.random.beta(6, 4)
172    radius_lg = 10**np.random.uniform(1.7, 4.7)  # 500 - 50000 A
174    penetration = np.random.beta(1, 10) # up to 20% pen. for 90% of examples
175    pars = dict(
176        volfraction_lg=volfraction_lg,
177        volfraction_sm=volfraction_sm,
178        surface_fraction=surface_fraction,
181        penetration=penetration,
182    )
183    return pars
184
185# parameters for demo
186demo = dict(scale=1, background=0.001,
187            sld_lg=-0.4, sld_sm=3.5, sld_solvent=6.36,
188            volfraction_lg=0.05, volfraction_sm=0.005, surface_fraction=0.4,