.. _stickyhardsphere:

stickyhardsphere
=======================================================

Sticky hard sphere structure factor, with Percus-Yevick closure

================ =============================== ======= =============
Parameter        Description                     Units   Default value
================ =============================== ======= =============
scale            Source intensity                None                1
background       Source background               |cm^-1|         0.001
radius_effective effective radius of hard sphere |Ang|              50
volfraction      volume fraction of hard spheres None              0.2
perturb          perturbation parameter, epsilon None             0.05
stickiness       stickiness, tau                 None              0.2
================ =============================== ======= =============

The returned value is a dimensionless structure factor, $S(q)$.


This calculates the interparticle structure factor for a hard sphere fluid
with a narrow attractive well. A perturbative solution of the Percus-Yevick
closure is used. The strength of the attractive well is described in terms
of "stickiness" as defined below.

The perturb (perturbation parameter), $\epsilon$, should be held between 0.01
and 0.1. It is best to hold the perturbation parameter fixed and let
the "stickiness" vary to adjust the interaction strength. The stickiness,
$\tau$, is defined in the equation below and is a function of both the
perturbation parameter and the interaction strength. $\tau$ and $\epsilon$
are defined in terms of the hard sphere diameter $(\sigma = 2 R)$, the
width of the square well, $\Delta$ (same units as $R$\ ), and the depth of
the well, $U_o$, in units of $kT$. From the definition, it is clear that
smaller $\tau$ means stronger attraction.

.. math::

    \tau     &= \frac{1}{12\epsilon} \exp(u_o / kT) \\
    \epsilon &= \Delta / (\sigma + \Delta)

where the interaction potential is

.. math::

    U(r) = \begin{cases}
        \infty & r < \sigma \\
        -U_o   & \sigma \leq r \leq \sigma + \Delta \\
        0      & r > \sigma + \Delta
        \end{cases}

The Percus-Yevick (PY) closure was used for this calculation, and is an
adequate closure for an attractive interparticle potential. This solution
has been compared to Monte Carlo simulations for a square well fluid, with
good agreement.

The true particle volume fraction, $\phi$, is not equal to $h$, which appears
in most of the reference. The two are related in equation (24) of the
reference. The reference also describes the relationship between this
perturbation solution and the original sticky hard sphere (or adhesive
sphere) model by Baxter.

**NB**: The calculation can go haywire for certain combinations of the input
parameters, producing unphysical solutions - in this case errors are
reported to the command window and the $S(q)$ is set to -1 (so it will
disappear on a log-log plot). Use tight bounds to keep the parameters to
values that you know are physical (test them) and keep nudging them until
the optimization does not hit the constraints.

In sasview the effective radius may be calculated from the parameters
used in the form factor $P(q)$ that this $S(q)$ is combined with.

For 2D data the scattering intensity is calculated in the same way
as 1D, where the $q$ vector is defined as

.. math::

    q = \sqrt{q_x^2 + q_y^2}



.. figure:: img/stickyhardsphere_autogenfig.png

    1D plot corresponding to the default parameters of the model.

**References**

S V G Menon, C Manohar, and K S Rao, *J. Chem. Phys.*, 95(12) (1991) 9186-9190

