source: sasmodels/sasmodels/models/stickyhardsphere.py @ b297ba9

core_shell_microgelsmagnetic_modelticket-1257-vesicle-productticket_1156ticket_1265_superballticket_822_more_unit_tests
Last change on this file since b297ba9 was b297ba9, checked in by Paul Kienzle <pkienzle@…>, 5 years ago

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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    """Return a random parameter set for the model."""
103    pars = dict(
104        scale=1, background=0,
105        radius_effective=10**np.random.uniform(1, 4.7),
106        volfraction=np.random.uniform(0.00001, 0.74),
107        perturb=10**np.random.uniform(-2, -1),
108        stickiness=np.random.uniform(0, 1),
109    )
110    return pars
111
112# No volume normalization despite having a volume parameter
113# This should perhaps be volume normalized?
114form_volume = """
115    return 1.0;
116    """
117
118Iq = """
119    double onemineps,eta;
120    double sig,aa,etam1,etam1sq,qa,qb,qc,radic;
121    double lam,lam2,test,mu,alpha,beta;
122    double kk,k2,k3,ds,dc,aq1,aq2,aq3,aq,bq1,bq2,bq3,bq,sq;
123
124    onemineps = 1.0-perturb;
125    eta = volfraction/onemineps/onemineps/onemineps;
126
127    sig = 2.0 * radius_effective;
128    aa = sig/onemineps;
129    etam1 = 1.0 - eta;
130    etam1sq=etam1*etam1;
131    //C
132    //C  SOLVE QUADRATIC FOR LAMBDA
133    //C
134    qa = eta/6.0;
135    qb = stickiness + eta/etam1;
136    qc = (1.0 + eta/2.0)/etam1sq;
137    radic = qb*qb - 2.0*qa*qc;
138    if(radic<0) {
139        //if(x>0.01 && x<0.015)
140        //    Print "Lambda unphysical - both roots imaginary"
141        //endif
142        return(-1.0);
143    }
144    //C   KEEP THE SMALLER ROOT, THE LARGER ONE IS UNPHYSICAL
145    radic = sqrt(radic);
146    lam = (qb-radic)/qa;
147    lam2 = (qb+radic)/qa;
148    if(lam2<lam) {
149        lam = lam2;
150    }
151    test = 1.0 + 2.0*eta;
152    mu = lam*eta*etam1;
153    if(mu>test) {
154        //if(x>0.01 && x<0.015)
155        // Print "Lambda unphysical mu>test"
156        //endif
157        return(-1.0);
158    }
159    alpha = (1.0 + 2.0*eta - mu)/etam1sq;
160    beta = (mu - 3.0*eta)/(2.0*etam1sq);
161    //C
162    //C   CALCULATE THE STRUCTURE FACTOR
163    //C
164    kk = q*aa;
165    k2 = kk*kk;
166    k3 = kk*k2;
167    SINCOS(kk,ds,dc);
168    //ds = sin(kk);
169    //dc = cos(kk);
170    aq1 = ((ds - kk*dc)*alpha)/k3;
171    aq2 = (beta*(1.0-dc))/k2;
172    aq3 = (lam*ds)/(12.0*kk);
173    aq = 1.0 + 12.0*eta*(aq1+aq2-aq3);
174    //
175    bq1 = alpha*(0.5/kk - ds/k2 + (1.0 - dc)/k3);
176    bq2 = beta*(1.0/kk - ds/k2);
177    bq3 = (lam/12.0)*((1.0 - dc)/kk);
178    bq = 12.0*eta*(bq1+bq2-bq3);
179    //
180    sq = 1.0/(aq*aq +bq*bq);
181
182    return(sq);
183"""
184
185demo = dict(radius_effective=200, volfraction=0.2, perturb=0.05,
186            stickiness=0.2, radius_effective_pd=0.1, radius_effective_pd_n=40)
187#
188tests = [
189    [{'scale': 1.0, 'background': 0.0, 'radius_effective': 50.0,
190      'perturb': 0.05, 'stickiness': 0.2, 'volfraction': 0.1,
191      'radius_effective_pd': 0},
192     [0.001, 0.003], [1.09718, 1.087830]],
193    ]
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