source: sasmodels/sasmodels/models/polymer_micelle.py @ 5a99d43

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Last change on this file since 5a99d43 was 5a99d43, checked in by Paul Kienzle <pkienzle@…>, 6 years ago

polymer micelle: correct typo in doc to show the correct equation

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1r"""
2
3This model provides the form factor, $P(q)$, for a micelle with a spherical
4core and Gaussian polymer chains attached to the surface, thus may be applied
5to block copolymer micelles. To work well the Gaussian chains must be much
6smaller than the core, which is often not the case.  Please study the
7reference carefully.
8
9Definition
10----------
11
12The 1D scattering intensity for this model is calculated according to
13the equations given by Pedersen (Pedersen, 2000), summarised briefly here.
14
15The micelle core is imagined as $N$ = *n_aggreg* polymer heads, each of volume
16$V_\text{core}$, which then defines a micelle core of radius $r$ = *r_core*,
17which is a separate parameter even though it could be directly determined.
18The Gaussian random coil tails, of gyration radius $R_g$, are imagined
19uniformly distributed around the spherical core, centred at a distance
20$r + d \cdot R_g$ from the micelle centre, where $d$ = *d_penetration* is
21of order unity. A volume $V_\text{corona}$ is defined for each coil. The
22model in detail seems to separately parametrise the terms for the shape
23of $I(Q)$ and the relative intensity of each term, so use with caution
24and check parameters for consistency. The spherical core is monodisperse,
25so it's intensity and the cross terms may have sharp oscillations (use $q$
26resolution smearing if needs be to help remove them).
27
28.. math::
29    P(q) &= N^2\beta^2_s\Phi(qr)^2 + N\beta^2_cP_c(q)
30            + 2N^2\beta_s\beta_cS_{sc}(q) + N(N-1)\beta_c^2S_{cc}(q) \\
31    \beta_s &= V_\text{core}(\rho_\text{core} - \rho_\text{solvent}) \\
32    \beta_c &= V_\text{corona}(\rho_\text{corona} - \rho_\text{solvent})
33
34where $\rho_\text{core}$, $\rho_\text{corona}$ and $\rho_\text{solvent}$ are
35the scattering length densities *sld_core*, *sld_corona* and *sld_solvent*.
36For the spherical core of radius $r$
37
38.. math::
39   \Phi(qr)= \frac{\sin(qr) - qr\cos(qr)}{(qr)^3}
40
41whilst for the Gaussian coils
42
43.. math::
44
45   P_c(q) &= 2 [\exp(-Z) + Z - 1] / Z^2 \\
46   Z &= (q R_g)^2
47
48The sphere to coil (core to corona) and coil to coil (corona to corona) cross
49terms are approximated by:
50
51.. math::
52
53   S_{sc}(q) &= \Phi(qr)\psi(Z)
54       \frac{\sin(q(r+d \cdot R_g))}{q(r+d \cdot R_g)} \\
55   S_{cc}(q) &= \psi(Z)^2
56       \left[\frac{\sin(q(r+d \cdot R_g))}{q(r+d \cdot R_g)} \right]^2 \\
57   \psi(Z) &= \frac{[1-\exp^{-Z}]}{Z}
58
59Validation
60----------
61
62$P(q)$ above is multiplied by *ndensity*, and a units conversion of $10^{-13}$,
63so *scale* is likely 1.0 if the scattering data is in absolute units. This
64model has not yet been independently validated.
65
66
67References
68----------
69
70J Pedersen, *J. Appl. Cryst.*, 33 (2000) 637-640
71"""
72
73import numpy as np
74from numpy import inf, pi
75
76name = "polymer_micelle"
77title = "Polymer micelle model"
78description = """
79This model provides the form factor, $P(q)$, for a micelle with a spherical
80core and Gaussian polymer chains attached to the surface, thus may be applied
81to block copolymer micelles. To work well the Gaussian chains must be much
82smaller than the core, which is often not the case.  Please study the
83reference to Pedersen and full documentation carefully.
84    """
85
86
87category = "shape:sphere"
88
89# pylint: disable=bad-whitespace, line-too-long
90#   ["name", "units", default, [lower, upper], "type","description"],
91parameters = [
92    ["ndensity",      "1e15/cm^3",  8.94, [0.0, inf], "", "Number density of micelles"],
93    ["v_core",        "Ang^3",  62624.0,  [0.0, inf], "", "Core volume "],
94    ["v_corona",      "Ang^3",  61940.0,  [0.0, inf], "", "Corona volume"],
95    ["sld_solvent",   "1e-6/Ang^2", 6.4,  [0.0, inf], "sld", "Solvent scattering length density"],
96    ["sld_core",      "1e-6/Ang^2", 0.34, [0.0, inf], "sld", "Core scattering length density"],
97    ["sld_corona",    "1e-6/Ang^2", 0.8,  [0.0, inf], "sld", "Corona scattering length density"],
98    ["radius_core",   "Ang",       45.0,  [0.0, inf], "", "Radius of core ( must be >> rg )"],
99    ["rg",    "Ang",       20.0,  [0.0, inf], "", "Radius of gyration of chains in corona"],
100    ["d_penetration", "",           1.0,  [-inf, inf], "", "Factor to mimic non-penetration of Gaussian chains"],
101    ["n_aggreg",      "",           6.0,  [-inf, inf], "", "Aggregation number of the micelle"],
102    ]
103# pylint: enable=bad-whitespace, line-too-long
104
105single = False
106
107source = ["lib/sas_3j1x_x.c", "polymer_micelle.c"]
108
109def random():
110    radius_core = 10**np.random.uniform(1, 3)
111    rg = radius_core * 10**np.random.uniform(-2, -0.3)
112    d_penetration = np.random.randn()*0.05 + 1
113    n_aggreg = np.random.randint(3, 30)
114    # volume of head groups is the core volume over the number of groups,
115    # with a correction for packing fraction of the head groups.
116    v_core = 4*pi/3*radius_core**3/n_aggreg * 0.68
117    # Rg^2 for gaussian coil is a^2n/6 => a^2 = 6 Rg^2/n
118    # a=2r => r = Rg sqrt(3/2n)
119    # v = 4/3 pi r^3 n => v = 4/3 pi Rg^3 (3/2n)^(3/2) n = pi Rg^3 sqrt(6/n)
120    tail_segments = np.random.randint(6, 30)
121    v_corona = pi * rg**3 * np.sqrt(6/tail_segments)
122    V = 4*pi/3*(radius_core + rg)**3
123    pars = dict(
124        background=0,
125        scale=1e7/V,
126        ndensity=8.94,
127        v_core=v_core,
128        v_corona=v_corona,
129        radius_core=radius_core,
130        rg=rg,
131        d_penetration=d_penetration,
132        n_aggreg=n_aggreg,
133    )
134    return pars
135
136tests = [
137    [{}, 0.01, 15.3532],
138    ]
139# RKH 20Mar2016 - need to check whether the core & corona volumes are per
140#                 monomer ??? and how aggregation number works!
141# renamed from micelle_spherical_core to polymer_micelle,
142# moved from shape-independent to spheres section.
143# Ought to be able to add polydisp to core? And add ability to x by S(Q) ?
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