[68532f3] | 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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