# source:sasmodels/sasmodels/models/stickyhardsphere.py@7f47777

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Last change on this file since 7f47777 was 7f47777, checked in by richardh, 9 years ago

RKH added simple 1d unit test as per sasview

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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 will 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.. figure:: img/stickyhardsphere_1d.jpg
62
63    1D plot using the default values (in linear scale).
64
65References
66----------
67
68S V G Menon, C Manohar, and K S Rao, *J. Chem. Phys.*, 95(12) (1991) 9186-9190
69"""
70
71# TODO: refactor so that we pull in the old sansmodels.c_extensions
72
73from numpy import inf
74
75name = "stickyhardsphere"
76title = "Sticky hard sphere structure factor, with Percus-Yevick closure"
77description = """\
78    [Sticky hard sphere structure factor, with Percus-Yevick closure]
79        Interparticle structure factor S(Q)for a hard sphere fluid with
80        a narrow attractive well. Fits are prone to deliver non-physical
81        parameters, use with care and read the references in the full manual.
82        In sasview the effective radius will be calculated from the
83        parameters used in P(Q).
84"""
85category = "structure-factor"
86
87#             ["name", "units", default, [lower, upper], "type","description"],
88parameters = [
89    #   [ "name", "units", default, [lower, upper], "type",
90    #     "description" ],
91    ["effect_radius", "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
101# No volume normalization despite having a volume parameter
102# This should perhaps be volume normalized?
103form_volume = """
104    return 1.0;
105    """
106
107Iq = """
108    double onemineps,eta;
110    double lam,lam2,test,mu,alpha,beta;
111    double kk,k2,k3,ds,dc,aq1,aq2,aq3,aq,bq1,bq2,bq3,bq,sq;
112
113    onemineps = 1.0-perturb;
114    eta = volfraction/onemineps/onemineps/onemineps;
115
116    sig = 2.0 * effect_radius;
117    aa = sig/onemineps;
118    etam1 = 1.0 - eta;
119    etam1sq=etam1*etam1;
120    //C
121    //C  SOLVE QUADRATIC FOR LAMBDA
122    //C
123    qa = eta/12.0;
124    qb = -1.0*(stickiness + eta/etam1);
125    qc = (1.0 + eta/2.0)/etam1sq;
126    radic = qb*qb - 4.0*qa*qc;
128        //if(x>0.01 && x<0.015)
129        //    Print "Lambda unphysical - both roots imaginary"
130        //endif
131        return(-1.0);
132    }
133    //C   KEEP THE SMALLER ROOT, THE LARGER ONE IS UNPHYSICAL
136    if(lam2<lam) {
137        lam = lam2;
138    }
139    test = 1.0 + 2.0*eta;
140    mu = lam*eta*etam1;
141    if(mu>test) {
142        //if(x>0.01 && x<0.015)
143        // Print "Lambda unphysical mu>test"
144        //endif
145        return(-1.0);
146    }
147    alpha = (1.0 + 2.0*eta - mu)/etam1sq;
148    beta = (mu - 3.0*eta)/(2.0*etam1sq);
149    //C
150    //C   CALCULATE THE STRUCTURE FACTOR
151    //C
152    kk = q*aa;
153    k2 = kk*kk;
154    k3 = kk*k2;
155    SINCOS(kk,ds,dc);
156    //ds = sin(kk);
157    //dc = cos(kk);
158    aq1 = ((ds - kk*dc)*alpha)/k3;
159    aq2 = (beta*(1.0-dc))/k2;
160    aq3 = (lam*ds)/(12.0*kk);
161    aq = 1.0 + 12.0*eta*(aq1+aq2-aq3);
162    //
163    bq1 = alpha*(0.5/kk - ds/k2 + (1.0 - dc)/k3);
164    bq2 = beta*(1.0/kk - ds/k2);
165    bq3 = (lam/12.0)*((1.0 - dc)/kk);
166    bq = 12.0*eta*(bq1+bq2-bq3);
167    //
168    sq = 1.0/(aq*aq +bq*bq);
169
170    return(sq);
171"""
172
173Iqxy = """
174    return Iq(sqrt(qx*qx+qy*qy), IQ_PARAMETERS);
175    """
176
177# ER defaults to 0.0
178# VR defaults to 1.0
179
180oldname = 'StickyHSStructure'
181oldpars = dict()