source: sasmodels/_sources/model/stickyhardsphere.txt @ 68532f3

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1.. _stickyhardsphere:
2
3Stickyhardsphere
4=======================================================
5
6Sticky hard sphere structure factor, with Percus-Yevick closure
7
8============= =============================== ======= =============
9Parameter     Description                     Units   Default value
10============= =============================== ======= =============
11scale         Source intensity                None                1
12background    Source background               |cm^-1|             0
13effect_radius effective radius of hard sphere |Ang|              50
14volfraction   volume fraction of hard spheres None              0.2
15perturb       perturbation parameter, epsilon None             0.05
16stickiness    stickiness, tau                 None              0.2
17============= =============================== ======= =============
18
19The returned value is scaled to units of |cm^-1|.
20
21
22This calculates the interparticle structure factor for a hard sphere fluid
23with a narrow attractive well. A perturbative solution of the Percus-Yevick
24closure is used. The strength of the attractive well is described in terms
25of "stickiness" as defined below. The returned value is a dimensionless
26structure factor, *S(q)*.
27
28The perturb (perturbation parameter), |epsilon|, should be held between 0.01
29and 0.1. It is best to hold the perturbation parameter fixed and let
30the "stickiness" vary to adjust the interaction strength. The stickiness,
31|tau|, is defined in the equation below and is a function of both the
32perturbation parameter and the interaction strength. |tau| and |epsilon|
33are defined in terms of the hard sphere diameter (|sigma| = 2\*\ *R*\ ), the
34width of the square well, |bigdelta| (same units as *R*), and the depth of
35the well, *Uo*, in units of kT. From the definition, it is clear that
36smaller |tau| means stronger attraction.
37
38.. image:: img/stickyhardsphere_228.PNG
39
40where the interaction potential is
41
42.. image:: img/stickyhardsphere_229.PNG
43
44The Percus-Yevick (PY) closure was used for this calculation, and is an
45adequate closure for an attractive interparticle potential. This solution
46has been compared to Monte Carlo simulations for a square well fluid, with
47good agreement.
48
49The true particle volume fraction, |phi|, is not equal to *h*, which appears
50in most of the reference. The two are related in equation (24) of the
51reference. The reference also describes the relationship between this
52perturbation solution and the original sticky hard sphere (or adhesive
53sphere) model by Baxter.
54
55NB: The calculation can go haywire for certain combinations of the input
56parameters, producing unphysical solutions - in this case errors are
57reported to the command window and the *S(q)* is set to -1 (so it will
58disappear on a log-log plot). Use tight bounds to keep the parameters to
59values that you know are physical (test them) and keep nudging them until
60the optimization does not hit the constraints.
61
62In sasview the effective radius will be calculated from the parameters
63used in the form factor P(Q) that this S(Q) is combined with.
64
65For 2D data: The 2D scattering intensity is calculated in the same way
66as 1D, where the *q* vector is defined as
67
68.. math::
69
70    Q = \sqrt{Q_x^2 + Q_y^2}
71
72==============  ========  =============
73Parameter name  Units     Default value
74==============  ========  =============
75effect_radius   |Ang|     50
76perturb         None      0.05
77volfraction     None      0.1
78stickiness      K         0.2
79==============  ========  =============
80
81.. image:: img/stickyhardsphere_230.jpg
82
83*Figure. 1D plot using the default values (in linear scale).*
84
85REFERENCE
86
87S V G Menon, C Manohar, and K S Rao, *J. Chem. Phys.*, 95(12) (1991) 9186-9190
88
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