Changeset eb69cce in sasmodels for sasmodels/models/stickyhardsphere.py
- Timestamp:
- Nov 30, 2015 9:18:41 PM (8 years ago)
- Branches:
- master, core_shell_microgels, costrafo411, magnetic_model, release_v0.94, release_v0.95, ticket-1257-vesicle-product, ticket_1156, ticket_1265_superball, ticket_822_more_unit_tests
- Children:
- d18f8a8
- Parents:
- d138d43
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- 1 edited
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sasmodels/models/stickyhardsphere.py (modified) (4 diffs)
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sasmodels/models/stickyhardsphere.py
r3e428ec reb69cce 4 4 with a narrow attractive well. A perturbative solution of the Percus-Yevick 5 5 closure is used. The strength of the attractive well is described in terms 6 of "stickiness" as defined below. The returned value is a dimensionless 7 structure factor, *S(q)*. 6 of "stickiness" as defined below. 8 7 9 The perturb (perturbation parameter), |epsilon|, should be held between 0.018 The perturb (perturbation parameter), $\epsilon$, should be held between 0.01 10 9 and 0.1. It is best to hold the perturbation parameter fixed and let 11 10 the "stickiness" vary to adjust the interaction strength. The stickiness, 12 |tau|, is defined in the equation below and is a function of both the13 perturbation parameter and the interaction strength. |tau| and |epsilon|14 are defined in terms of the hard sphere diameter (|sigma| = 2\*\ *R*\ ), the15 width of the square well, |bigdelta| (same units as *R*), and the depth of16 the well, *Uo*, in units of kT. From the definition, it is clear that17 smaller |tau|means stronger attraction.11 $\tau$, is defined in the equation below and is a function of both the 12 perturbation parameter and the interaction strength. $\tau$ and $\epsilon$ 13 are defined in terms of the hard sphere diameter $(\sigma = 2 R)$, the 14 width of the square well, $\Delta$ (same units as $R$\ ), and the depth of 15 the well, $U_o$, in units of $kT$. From the definition, it is clear that 16 smaller $\tau$ means stronger attraction. 18 17 19 .. image:: img/stickyhardsphere_228.PNG 18 .. math:: 19 20 %\begin{align*} % isn't working with pdflatex 21 \begin{array}{rl} 22 \tau &= \frac{1}{12\epsilon} \exp(u_o / kT) \\ 23 \epsilon &= \Delta / (\sigma + \Delta) \\ 24 \end{array} 20 25 21 26 where the interaction potential is 22 27 23 .. image:: img/stickyhardsphere_229.PNG 28 .. math:: 29 30 U(r) = \begin{cases} 31 \infty & r < \sigma \\ 32 -U_o & \sigma \leq r \leq \sigma + \Delta \\ 33 0 & r > \sigma + \Delta 34 \end{cases} 24 35 25 36 The Percus-Yevick (PY) closure was used for this calculation, and is an … … 28 39 good agreement. 29 40 30 The true particle volume fraction, |phi|, is not equal to *h*, which appears41 The true particle volume fraction, $\phi$, is not equal to $h$, which appears 31 42 in most of the reference. The two are related in equation (24) of the 32 43 reference. The reference also describes the relationship between this … … 34 45 sphere) model by Baxter. 35 46 36 NB: The calculation can go haywire for certain combinations of the input47 **NB**: The calculation can go haywire for certain combinations of the input 37 48 parameters, producing unphysical solutions - in this case errors are 38 reported to the command window and the *S(q)*is set to -1 (so it will49 reported to the command window and the $S(q)$ is set to -1 (so it will 39 50 disappear on a log-log plot). Use tight bounds to keep the parameters to 40 51 values that you know are physical (test them) and keep nudging them until … … 42 53 43 54 In sasview the effective radius will be calculated from the parameters 44 used in the form factor P(Q) that this S(Q)is combined with.55 used in the form factor $P(q)$ that this $S(q)$ is combined with. 45 56 46 For 2D data : The 2Dscattering intensity is calculated in the same way47 as 1D, where the *q*vector is defined as57 For 2D data the scattering intensity is calculated in the same way 58 as 1D, where the $q$ vector is defined as 48 59 49 60 .. math:: 50 61 51 Q = \sqrt{Q_x^2 + Q_y^2}62 q = \sqrt{q_x^2 + q_y^2} 52 63 53 ============== ======== ============= 54 Parameter name Units Default value 55 ============== ======== ============= 56 effect_radius |Ang| 50 57 perturb None 0.05 58 volfraction None 0.1 59 stickiness K 0.2 60 ============== ======== ============= 64 .. figure:: img/stickyhardsphere_1d.jpg 61 65 62 .. image:: img/stickyhardsphere_230.jpg 66 1D plot using the default values (in linear scale). 63 67 64 *Figure. 1D plot using the default values (in linear scale).* 65 66 REFERENCE 68 References 69 ---------- 67 70 68 71 S V G Menon, C Manohar, and K S Rao, *J. Chem. Phys.*, 95(12) (1991) 9186-9190
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