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34          <span>Home</span></a></h1>
35        <h2 class="heading"><span>2.3.2. Stickyhardsphere</span></h2>
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39        <p>
40        «&#160;&#160;<a href="hardsphere.html">2.3.1. Hardsphere</a>
41        &#160;&#160;::&#160;&#160;
42        <a class="uplink" href="../index.html">Contents</a>
43        &#160;&#160;::&#160;&#160;
44        <a href="../ref/models/custom-models.html">2.4. Custom Models</a>&#160;&#160;»
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51  <div class="section" id="stickyhardsphere">
52<span id="id1"></span><h1>2.3.2. Stickyhardsphere<a class="headerlink" href="#stickyhardsphere" title="Permalink to this headline">¶</a></h1>
53<p>Sticky hard sphere structure factor, with Percus-Yevick closure</p>
54<table border="1" class="docutils">
55<colgroup>
56<col width="20%" />
57<col width="48%" />
58<col width="11%" />
59<col width="20%" />
60</colgroup>
61<thead valign="bottom">
62<tr class="row-odd"><th class="head">Parameter</th>
63<th class="head">Description</th>
64<th class="head">Units</th>
65<th class="head">Default value</th>
66</tr>
67</thead>
68<tbody valign="top">
69<tr class="row-even"><td>scale</td>
70<td>Source intensity</td>
71<td>None</td>
72<td>1</td>
73</tr>
74<tr class="row-odd"><td>background</td>
75<td>Source background</td>
76<td>cm<sup>-1</sup></td>
77<td>0</td>
78</tr>
79<tr class="row-even"><td>effect_radius</td>
80<td>effective radius of hard sphere</td>
81<td>Å</td>
82<td>50</td>
83</tr>
84<tr class="row-odd"><td>volfraction</td>
85<td>volume fraction of hard spheres</td>
86<td>None</td>
87<td>0.2</td>
88</tr>
89<tr class="row-even"><td>perturb</td>
90<td>perturbation parameter, epsilon</td>
91<td>None</td>
92<td>0.05</td>
93</tr>
94<tr class="row-odd"><td>stickiness</td>
95<td>stickiness, tau</td>
96<td>None</td>
97<td>0.2</td>
98</tr>
99</tbody>
100</table>
101<p>The returned value is scaled to units of cm<sup>-1</sup>.</p>
102<p>This calculates the interparticle structure factor for a hard sphere fluid
103with a narrow attractive well. A perturbative solution of the Percus-Yevick
104closure is used. The strength of the attractive well is described in terms
105of &#8220;stickiness&#8221; as defined below. The returned value is a dimensionless
106structure factor, <em>S(q)</em>.</p>
107<p>The perturb (perturbation parameter), ε, should be held between 0.01
108and 0.1. It is best to hold the perturbation parameter fixed and let
109the &#8220;stickiness&#8221; vary to adjust the interaction strength. The stickiness,
110τ, is defined in the equation below and is a function of both the
111perturbation parameter and the interaction strength. τ and ε
112are defined in terms of the hard sphere diameter (σ = 2*<em>R</em>), the
113width of the square well, Δ (same units as <em>R</em>), and the depth of
114the well, <em>Uo</em>, in units of kT. From the definition, it is clear that
115smaller τ means stronger attraction.</p>
116<img alt="../_images/stickyhardsphere_228.PNG" src="../_images/stickyhardsphere_228.PNG" />
117<p>where the interaction potential is</p>
118<img alt="../_images/stickyhardsphere_229.PNG" src="../_images/stickyhardsphere_229.PNG" />
119<p>The Percus-Yevick (PY) closure was used for this calculation, and is an
120adequate closure for an attractive interparticle potential. This solution
121has been compared to Monte Carlo simulations for a square well fluid, with
122good agreement.</p>
123<p>The true particle volume fraction, φ, is not equal to <em>h</em>, which appears
124in most of the reference. The two are related in equation (24) of the
125reference. The reference also describes the relationship between this
126perturbation solution and the original sticky hard sphere (or adhesive
127sphere) model by Baxter.</p>
128<p>NB: The calculation can go haywire for certain combinations of the input
129parameters, producing unphysical solutions - in this case errors are
130reported to the command window and the <em>S(q)</em> is set to -1 (so it will
131disappear on a log-log plot). Use tight bounds to keep the parameters to
132values that you know are physical (test them) and keep nudging them until
133the optimization does not hit the constraints.</p>
134<p>In sasview the effective radius will be calculated from the parameters
135used in the form factor P(Q) that this S(Q) is combined with.</p>
136<p>For 2D data: The 2D scattering intensity is calculated in the same way
137as 1D, where the <em>q</em> vector is defined as</p>
138<div class="math">
139\[Q = \sqrt{Q_x^2 + Q_y^2}\]</div>
140<table border="1" class="docutils">
141<colgroup>
142<col width="40%" />
143<col width="23%" />
144<col width="37%" />
145</colgroup>
146<thead valign="bottom">
147<tr class="row-odd"><th class="head">Parameter name</th>
148<th class="head">Units</th>
149<th class="head">Default value</th>
150</tr>
151</thead>
152<tbody valign="top">
153<tr class="row-even"><td>effect_radius</td>
154<td>Å</td>
155<td>50</td>
156</tr>
157<tr class="row-odd"><td>perturb</td>
158<td>None</td>
159<td>0.05</td>
160</tr>
161<tr class="row-even"><td>volfraction</td>
162<td>None</td>
163<td>0.1</td>
164</tr>
165<tr class="row-odd"><td>stickiness</td>
166<td>K</td>
167<td>0.2</td>
168</tr>
169</tbody>
170</table>
171<img alt="../_images/stickyhardsphere_230.jpg" src="../_images/stickyhardsphere_230.jpg" />
172<p><em>Figure. 1D plot using the default values (in linear scale).</em></p>
173<p>REFERENCE</p>
174<p>S V G Menon, C Manohar, and K S Rao, <em>J. Chem. Phys.</em>, 95(12) (1991) 9186-9190</p>
175</div>
176
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182        «&#160;&#160;<a href="hardsphere.html">2.3.1. Hardsphere</a>
183        &#160;&#160;::&#160;&#160;
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185        &#160;&#160;::&#160;&#160;
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