source: sasmodels/sasmodels/models/stickyhardsphere.py @ 58c3367

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Last change on this file since 58c3367 was 40a87fa, checked in by Paul Kienzle <pkienzle@…>, 8 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
70from numpy import inf
71
72name = "stickyhardsphere"
73title = "Sticky hard sphere structure factor, with Percus-Yevick closure"
74description = """\
75    [Sticky hard sphere structure factor, with Percus-Yevick closure]
76        Interparticle structure factor S(Q)for a hard sphere fluid with
77        a narrow attractive well. Fits are prone to deliver non-physical
78        parameters, use with care and read the references in the full manual.
79        In sasview the effective radius will be calculated from the
80        parameters used in P(Q).
81"""
82category = "structure-factor"
83structure_factor = True
84
85single = False
86#             ["name", "units", default, [lower, upper], "type","description"],
87parameters = [
88    #   [ "name", "units", default, [lower, upper], "type",
89    #     "description" ],
90    ["radius_effective", "Ang", 50.0, [0, inf], "volume",
91     "effective radius of hard sphere"],
92    ["volfraction", "", 0.2, [0, 0.74], "",
93     "volume fraction of hard spheres"],
94    ["perturb", "", 0.05, [0.01, 0.1], "",
95     "perturbation parameter, epsilon"],
96    ["stickiness", "", 0.20, [-inf, inf], "",
97     "stickiness, tau"],
98    ]
99
100# No volume normalization despite having a volume parameter
101# This should perhaps be volume normalized?
102form_volume = """
103    return 1.0;
104    """
105
106Iq = """
107    double onemineps,eta;
108    double sig,aa,etam1,etam1sq,qa,qb,qc,radic;
109    double lam,lam2,test,mu,alpha,beta;
110    double kk,k2,k3,ds,dc,aq1,aq2,aq3,aq,bq1,bq2,bq3,bq,sq;
111
112    onemineps = 1.0-perturb;
113    eta = volfraction/onemineps/onemineps/onemineps;
114
115    sig = 2.0 * radius_effective;
116    aa = sig/onemineps;
117    etam1 = 1.0 - eta;
118    etam1sq=etam1*etam1;
119    //C
120    //C  SOLVE QUADRATIC FOR LAMBDA
121    //C
122    qa = eta/6.0;
123    qb = stickiness + eta/etam1;
124    qc = (1.0 + eta/2.0)/etam1sq;
125    radic = qb*qb - 2.0*qa*qc;
126    if(radic<0) {
127        //if(x>0.01 && x<0.015)
128        //    Print "Lambda unphysical - both roots imaginary"
129        //endif
130        return(-1.0);
131    }
132    //C   KEEP THE SMALLER ROOT, THE LARGER ONE IS UNPHYSICAL
133    radic = sqrt(radic);
134    lam = (qb-radic)/qa;
135    lam2 = (qb+radic)/qa;
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
173# ER defaults to 0.0
174# VR defaults to 1.0
175
176demo = dict(radius_effective=200, volfraction=0.2, perturb=0.05,
177            stickiness=0.2, radius_effective_pd=0.1, radius_effective_pd_n=40)
178#
179tests = [
180    [{'scale': 1.0, 'background': 0.0, 'radius_effective': 50.0,
181      'perturb': 0.05, 'stickiness': 0.2, 'volfraction': 0.1,
182      'radius_effective_pd': 0},
183     [0.001, 0.003], [1.09718, 1.087830]],
184    ]
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