source: sasmodels/sasmodels/models/stickyhardsphere.py @ 99658f6

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since 99658f6 was 304c775, checked in by Paul Kienzle <pkienzle@…>, 5 years ago

provide method for testing Fq results. Refs #1202.

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1# Note: model title and parameter table are inserted automatically
2r"""
3This calculates the interparticle structure factor for a hard sphere fluid
4with a narrow attractive well. A perturbative solution of the Percus-Yevick
5closure is used. The strength of the attractive well is described in terms
6of "stickiness" as defined below.
7
8The perturb (perturbation parameter), $\epsilon$, should be held between 0.01
9and 0.1. It is best to hold the perturbation parameter fixed and let
10the "stickiness" vary to adjust the interaction strength. The stickiness,
11$\tau$, is defined in the equation below and is a function of both the
12perturbation parameter and the interaction strength. $\tau$ and $\epsilon$
13are defined in terms of the hard sphere diameter $(\sigma = 2 R)$, the
14width of the square well, $\Delta$ (same units as $R$\ ), and the depth of
15the well, $U_o$, in units of $kT$. From the definition, it is clear that
16smaller $\tau$ means stronger attraction.
17
18.. math::
19
20    \tau     &= \frac{1}{12\epsilon} \exp(u_o / kT) \\
21    \epsilon &= \Delta / (\sigma + \Delta)
22
23where the interaction potential is
24
25.. math::
26
27    U(r) = \begin{cases}
28        \infty & r < \sigma \\
29        -U_o   & \sigma \leq r \leq \sigma + \Delta \\
30        0      & r > \sigma + \Delta
31        \end{cases}
32
33The Percus-Yevick (PY) closure was used for this calculation, and is an
34adequate closure for an attractive interparticle potential. This solution
35has been compared to Monte Carlo simulations for a square well fluid, with
36good agreement.
37
38The true particle volume fraction, $\phi$, is not equal to $h$, which appears
39in most of the reference. The two are related in equation (24) of the
40reference. The reference also describes the relationship between this
41perturbation solution and the original sticky hard sphere (or adhesive
42sphere) model by Baxter.
43
44**NB**: The calculation can go haywire for certain combinations of the input
45parameters, producing unphysical solutions - in this case errors are
46reported to the command window and the $S(q)$ is set to -1 (so it will
47disappear on a log-log plot). Use tight bounds to keep the parameters to
48values that you know are physical (test them) and keep nudging them until
49the optimization does not hit the constraints.
50
51In sasview the effective radius may be calculated from the parameters
52used in the form factor $P(q)$ that this $S(q)$ is combined with.
53
54For 2D data the scattering intensity is calculated in the same way
55as 1D, where the $q$ vector is defined as
56
57.. math::
58
59    q = \sqrt{q_x^2 + q_y^2}
60
61
62References
63----------
64
65S V G Menon, C Manohar, and K S Rao, *J. Chem. Phys.*, 95(12) (1991) 9186-9190
66"""
67
68# TODO: refactor so that we pull in the old sansmodels.c_extensions
69
70import numpy as np
71from numpy import inf
72
73name = "stickyhardsphere"
74title = "Sticky hard sphere structure factor, with Percus-Yevick closure"
75description = """\
76    [Sticky hard sphere structure factor, with Percus-Yevick closure]
77        Interparticle structure factor S(Q)for a hard sphere fluid with
78        a narrow attractive well. Fits are prone to deliver non-physical
79        parameters, use with care and read the references in the full manual.
80        In sasview the effective radius will be calculated from the
81        parameters used in P(Q).
82"""
83category = "structure-factor"
84structure_factor = True
85
86single = False
87#             ["name", "units", default, [lower, upper], "type","description"],
88parameters = [
89    #   [ "name", "units", default, [lower, upper], "type",
90    #     "description" ],
91    ["radius_effective", "Ang", 50.0, [0, inf], "volume",
92     "effective radius of hard sphere"],
93    ["volfraction", "", 0.2, [0, 0.74], "",
94     "volume fraction of hard spheres"],
95    ["perturb", "", 0.05, [0.01, 0.1], "",
96     "perturbation parameter, epsilon"],
97    ["stickiness", "", 0.20, [-inf, inf], "",
98     "stickiness, tau"],
99    ]
100
101def random():
102    pars = dict(
103        scale=1, background=0,
104        radius_effective=10**np.random.uniform(1, 4.7),
105        volfraction=np.random.uniform(0.00001, 0.74),
106        perturb=10**np.random.uniform(-2, -1),
107        stickiness=np.random.uniform(0, 1),
108    )
109    return pars
110
111# No volume normalization despite having a volume parameter
112# This should perhaps be volume normalized?
113form_volume = """
114    return 1.0;
115    """
116
117Iq = """
118    double onemineps,eta;
119    double sig,aa,etam1,etam1sq,qa,qb,qc,radic;
120    double lam,lam2,test,mu,alpha,beta;
121    double kk,k2,k3,ds,dc,aq1,aq2,aq3,aq,bq1,bq2,bq3,bq,sq;
122
123    onemineps = 1.0-perturb;
124    eta = volfraction/onemineps/onemineps/onemineps;
125
126    sig = 2.0 * radius_effective;
127    aa = sig/onemineps;
128    etam1 = 1.0 - eta;
129    etam1sq=etam1*etam1;
130    //C
131    //C  SOLVE QUADRATIC FOR LAMBDA
132    //C
133    qa = eta/6.0;
134    qb = stickiness + eta/etam1;
135    qc = (1.0 + eta/2.0)/etam1sq;
136    radic = qb*qb - 2.0*qa*qc;
137    if(radic<0) {
138        //if(x>0.01 && x<0.015)
139        //    Print "Lambda unphysical - both roots imaginary"
140        //endif
141        return(-1.0);
142    }
143    //C   KEEP THE SMALLER ROOT, THE LARGER ONE IS UNPHYSICAL
144    radic = sqrt(radic);
145    lam = (qb-radic)/qa;
146    lam2 = (qb+radic)/qa;
147    if(lam2<lam) {
148        lam = lam2;
149    }
150    test = 1.0 + 2.0*eta;
151    mu = lam*eta*etam1;
152    if(mu>test) {
153        //if(x>0.01 && x<0.015)
154        // Print "Lambda unphysical mu>test"
155        //endif
156        return(-1.0);
157    }
158    alpha = (1.0 + 2.0*eta - mu)/etam1sq;
159    beta = (mu - 3.0*eta)/(2.0*etam1sq);
160    //C
161    //C   CALCULATE THE STRUCTURE FACTOR
162    //C
163    kk = q*aa;
164    k2 = kk*kk;
165    k3 = kk*k2;
166    SINCOS(kk,ds,dc);
167    //ds = sin(kk);
168    //dc = cos(kk);
169    aq1 = ((ds - kk*dc)*alpha)/k3;
170    aq2 = (beta*(1.0-dc))/k2;
171    aq3 = (lam*ds)/(12.0*kk);
172    aq = 1.0 + 12.0*eta*(aq1+aq2-aq3);
173    //
174    bq1 = alpha*(0.5/kk - ds/k2 + (1.0 - dc)/k3);
175    bq2 = beta*(1.0/kk - ds/k2);
176    bq3 = (lam/12.0)*((1.0 - dc)/kk);
177    bq = 12.0*eta*(bq1+bq2-bq3);
178    //
179    sq = 1.0/(aq*aq +bq*bq);
180
181    return(sq);
182"""
183
184demo = dict(radius_effective=200, volfraction=0.2, perturb=0.05,
185            stickiness=0.2, radius_effective_pd=0.1, radius_effective_pd_n=40)
186#
187tests = [
188    [{'scale': 1.0, 'background': 0.0, 'radius_effective': 50.0,
189      'perturb': 0.05, 'stickiness': 0.2, 'volfraction': 0.1,
190      'radius_effective_pd': 0},
191     [0.001, 0.003], [1.09718, 1.087830]],
192    ]
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