1 | # Note: model title and parameter table are inserted automatically |
---|
2 | r""" |
---|
3 | Calculates the interparticle structure factor for a hard sphere fluid |
---|
4 | with a narrow, attractive, potential well. Unlike the :ref:`squarewell` |
---|
5 | model, here a perturbative solution of the Percus-Yevick closure |
---|
6 | relationship is used. The strength of the attractive well is described |
---|
7 | in terms of "stickiness" as defined below. |
---|
8 | |
---|
9 | The perturbation parameter (perturb), $\tau$, should be fixed between 0.01 |
---|
10 | and 0.1 and the "stickiness", $\epsilon$, allowed to vary to adjust the |
---|
11 | interaction strength. The "stickiness" is defined in the equation below and is |
---|
12 | a function of both the perturbation parameter and the interaction strength. |
---|
13 | $\epsilon$ and $\tau$ are defined in terms of the hard sphere diameter $(\sigma = 2 R)$, |
---|
14 | the width of the square well, $\Delta$ (having the same units as $R$\ ), |
---|
15 | and the depth of the well, $U_o$, in units of $kT$. From the definition, it |
---|
16 | is clear that smaller $\epsilon$ means a stronger attraction. |
---|
17 | |
---|
18 | .. math:: |
---|
19 | |
---|
20 | \epsilon &= \frac{1}{12\tau} \exp(u_o / kT) \\ |
---|
21 | \tau &= \Delta / (\sigma + \Delta) |
---|
22 | |
---|
23 | where 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 | |
---|
33 | The Percus-Yevick (PY) closure is used for this calculation, and is an |
---|
34 | adequate closure for an attractive interparticle potential. The solution |
---|
35 | has been compared to Monte Carlo simulations for a square well fluid, with |
---|
36 | good agreement. |
---|
37 | |
---|
38 | The true particle volume fraction, $\phi$, is not equal to $h$ which appears |
---|
39 | in most of reference [1]. The two are related in equation (24). Reference |
---|
40 | [1] also describes the relationship between this perturbative solution and |
---|
41 | the original sticky hard sphere (or "adhesive sphere") model of Baxter [2]. |
---|
42 | |
---|
43 | .. note:: |
---|
44 | |
---|
45 | The calculation can go haywire for certain combinations of the input |
---|
46 | parameters, producing unphysical solutions. In this case errors are |
---|
47 | reported to the command window and $S(q)$ is set to -1 (so it will |
---|
48 | disappear on a log-log plot!). |
---|
49 | |
---|
50 | Use tight bounds to keep the parameters to values that you know are |
---|
51 | physical (test them), and keep nudging them until the optimization |
---|
52 | does not hit the constraints. |
---|
53 | |
---|
54 | .. note:: |
---|
55 | |
---|
56 | Earlier versions of SasView did not incorporate the so-called |
---|
57 | $\beta(q)$ ("beta") correction [3] for polydispersity and non-sphericity. |
---|
58 | This is only available in SasView versions 5.0 and higher. |
---|
59 | |
---|
60 | In SasView the effective radius may be calculated from the parameters |
---|
61 | used in the form factor $P(q)$ that this $S(q)$ is combined with. |
---|
62 | |
---|
63 | For 2D data the scattering intensity is calculated in the same way |
---|
64 | as 1D, where the $q$ vector is defined as |
---|
65 | |
---|
66 | .. math:: |
---|
67 | |
---|
68 | q = \sqrt{q_x^2 + q_y^2} |
---|
69 | |
---|
70 | |
---|
71 | References |
---|
72 | ---------- |
---|
73 | |
---|
74 | .. [#] S V G Menon, C Manohar, and K S Rao, *J. Chem. Phys.*, 95(12) (1991) 9186-9190 |
---|
75 | |
---|
76 | .. [#] R J Baxter, *J. Chem. Phys.*, 49 (1968), 2770-2774 |
---|
77 | |
---|
78 | .. [#] M Kotlarchyk and S-H Chen, *J. Chem. Phys.*, 79 (1983) 2461-2469 |
---|
79 | |
---|
80 | Authorship and Verification |
---|
81 | ---------------------------- |
---|
82 | |
---|
83 | * **Author:** |
---|
84 | * **Last Modified by:** |
---|
85 | * **Last Reviewed by:** Steve King **Date:** March 27, 2019 |
---|
86 | """ |
---|
87 | |
---|
88 | # TODO: refactor so that we pull in the old sansmodels.c_extensions |
---|
89 | |
---|
90 | import numpy as np |
---|
91 | from numpy import inf |
---|
92 | |
---|
93 | name = "stickyhardsphere" |
---|
94 | title = "'Sticky' hard sphere structure factor with Percus-Yevick closure" |
---|
95 | description = """\ |
---|
96 | [Sticky hard sphere structure factor, with Percus-Yevick closure] |
---|
97 | Interparticle structure factor S(Q) for a hard sphere fluid |
---|
98 | with a narrow attractive well. Fits are prone to deliver non- |
---|
99 | physical parameters; use with care and read the references in |
---|
100 | the model documentation.The "beta(q)" correction is available |
---|
101 | in versions 4.2.2 and higher. |
---|
102 | """ |
---|
103 | category = "structure-factor" |
---|
104 | structure_factor = True |
---|
105 | |
---|
106 | single = False |
---|
107 | # ["name", "units", default, [lower, upper], "type","description"], |
---|
108 | parameters = [ |
---|
109 | # [ "name", "units", default, [lower, upper], "type", |
---|
110 | # "description" ], |
---|
111 | ["radius_effective", "Ang", 50.0, [0, inf], "volume", |
---|
112 | "effective radius of hard sphere"], |
---|
113 | ["volfraction", "", 0.2, [0, 0.74], "", |
---|
114 | "volume fraction of hard spheres"], |
---|
115 | ["perturb", "", 0.05, [0.01, 0.1], "", |
---|
116 | "perturbation parameter, tau"], |
---|
117 | ["stickiness", "", 0.20, [-inf, inf], "", |
---|
118 | "stickiness, epsilon"], |
---|
119 | ] |
---|
120 | |
---|
121 | def random(): |
---|
122 | """Return a random parameter set for the model.""" |
---|
123 | pars = dict( |
---|
124 | scale=1, background=0, |
---|
125 | radius_effective=10**np.random.uniform(1, 4.7), |
---|
126 | volfraction=np.random.uniform(0.00001, 0.74), |
---|
127 | perturb=10**np.random.uniform(-2, -1), |
---|
128 | stickiness=np.random.uniform(0, 1), |
---|
129 | ) |
---|
130 | return pars |
---|
131 | |
---|
132 | # No volume normalization despite having a volume parameter |
---|
133 | # This should perhaps be volume normalized? |
---|
134 | form_volume = """ |
---|
135 | return 1.0; |
---|
136 | """ |
---|
137 | |
---|
138 | Iq = """ |
---|
139 | double onemineps,eta; |
---|
140 | double sig,aa,etam1,etam1sq,qa,qb,qc,radic; |
---|
141 | double lam,lam2,test,mu,alpha,beta; |
---|
142 | double kk,k2,k3,ds,dc,aq1,aq2,aq3,aq,bq1,bq2,bq3,bq,sq; |
---|
143 | |
---|
144 | onemineps = 1.0-perturb; |
---|
145 | eta = volfraction/onemineps/onemineps/onemineps; |
---|
146 | |
---|
147 | sig = 2.0 * radius_effective; |
---|
148 | aa = sig/onemineps; |
---|
149 | etam1 = 1.0 - eta; |
---|
150 | etam1sq=etam1*etam1; |
---|
151 | //C |
---|
152 | //C SOLVE QUADRATIC FOR LAMBDA |
---|
153 | //C |
---|
154 | qa = eta/6.0; |
---|
155 | qb = stickiness + eta/etam1; |
---|
156 | qc = (1.0 + eta/2.0)/etam1sq; |
---|
157 | radic = qb*qb - 2.0*qa*qc; |
---|
158 | if(radic<0) { |
---|
159 | //if(x>0.01 && x<0.015) |
---|
160 | // Print "Lambda unphysical - both roots imaginary" |
---|
161 | //endif |
---|
162 | return(-1.0); |
---|
163 | } |
---|
164 | //C KEEP THE SMALLER ROOT, THE LARGER ONE IS UNPHYSICAL |
---|
165 | radic = sqrt(radic); |
---|
166 | lam = (qb-radic)/qa; |
---|
167 | lam2 = (qb+radic)/qa; |
---|
168 | if(lam2<lam) { |
---|
169 | lam = lam2; |
---|
170 | } |
---|
171 | test = 1.0 + 2.0*eta; |
---|
172 | mu = lam*eta*etam1; |
---|
173 | if(mu>test) { |
---|
174 | //if(x>0.01 && x<0.015) |
---|
175 | // Print "Lambda unphysical mu>test" |
---|
176 | //endif |
---|
177 | return(-1.0); |
---|
178 | } |
---|
179 | alpha = (1.0 + 2.0*eta - mu)/etam1sq; |
---|
180 | beta = (mu - 3.0*eta)/(2.0*etam1sq); |
---|
181 | //C |
---|
182 | //C CALCULATE THE STRUCTURE FACTOR |
---|
183 | //C |
---|
184 | kk = q*aa; |
---|
185 | k2 = kk*kk; |
---|
186 | k3 = kk*k2; |
---|
187 | SINCOS(kk,ds,dc); |
---|
188 | //ds = sin(kk); |
---|
189 | //dc = cos(kk); |
---|
190 | aq1 = ((ds - kk*dc)*alpha)/k3; |
---|
191 | aq2 = (beta*(1.0-dc))/k2; |
---|
192 | aq3 = (lam*ds)/(12.0*kk); |
---|
193 | aq = 1.0 + 12.0*eta*(aq1+aq2-aq3); |
---|
194 | // |
---|
195 | bq1 = alpha*(0.5/kk - ds/k2 + (1.0 - dc)/k3); |
---|
196 | bq2 = beta*(1.0/kk - ds/k2); |
---|
197 | bq3 = (lam/12.0)*((1.0 - dc)/kk); |
---|
198 | bq = 12.0*eta*(bq1+bq2-bq3); |
---|
199 | // |
---|
200 | sq = 1.0/(aq*aq +bq*bq); |
---|
201 | |
---|
202 | return(sq); |
---|
203 | """ |
---|
204 | |
---|
205 | demo = dict(radius_effective=200, volfraction=0.2, perturb=0.05, |
---|
206 | stickiness=0.2, radius_effective_pd=0.1, radius_effective_pd_n=40) |
---|
207 | # |
---|
208 | tests = [ |
---|
209 | [{'scale': 1.0, 'background': 0.0, 'radius_effective': 50.0, |
---|
210 | 'perturb': 0.05, 'stickiness': 0.2, 'volfraction': 0.1, |
---|
211 | 'radius_effective_pd': 0}, |
---|
212 | [0.001, 0.003], [1.09718, 1.087830]], |
---|
213 | ] |
---|