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