.. _raspberry:

raspberry
=======================================================

Calculates the form factor, *P(q)*, for a 'Raspberry-like' structure where there are smaller spheres at the surface of a larger sphere, such as the structure of a Pickering emulsion.

================ =============================================================== ============ =============
Parameter        Description                                                     Units        Default value
================ =============================================================== ============ =============
scale            Scale factor or Volume fraction                                 None                     1
background       Source background                                               |cm^-1|              0.001
sld_lg           large particle scattering length density                        |1e-6Ang^-2|          -0.4
sld_sm           small particle scattering length density                        |1e-6Ang^-2|           3.5
sld_solvent      solvent scattering length density                               |1e-6Ang^-2|          6.36
volfraction_lg   volume fraction of large spheres                                None                  0.05
volfraction_sm   volume fraction of small spheres                                None                 0.005
surface_fraction fraction of small spheres at surface                            None                   0.4
radius_lg        radius of large spheres                                         |Ang|                 5000
radius_sm        radius of small spheres                                         |Ang|                  100
penetration      fractional penetration depth of small spheres into large sphere |Ang|                    0
================ =============================================================== ============ =============

The returned value is scaled to units of |cm^-1| |sr^-1|, absolute scale.


**Definition**

The figure below shows a schematic of a large droplet surrounded by several
smaller particles forming a structure similar to that of Pickering emulsions.

.. figure:: img/raspberry_geometry.jpg

    Schematic of the raspberry model

In order to calculate the form factor of the entire complex, the
self-correlation of the large droplet, the self-correlation of the particles,
the correlation terms between different particles and the cross terms between
large droplet and small particles all need to be calculated.

Consider two infinitely thin shells of radii $R_1$ and $R_2$ separated by
distance $r$. The general structure of the equation is then the form factor
of the two shells multiplied by the phase factor that accounts for the
separation of their centers.

.. math::

    S(q) = \frac{\sin(qR_1)}{qR_1}\frac{\sin(qR_2)}{qR_2}\frac{\sin(qr)}{qr}

In this case, the large droplet and small particles are solid spheres rather
than thin shells. Thus the two terms must be integrated over $R_L$ and $R_S$
respectively using the weighting function of a sphere. We then obtain the
functions for the form of the two spheres:

.. math::

    \Psi_L = \int_0^{R_L}(4\pi R^2_L)\frac{\sin(qR_L)}{qR_L}dR_L =
    \frac{3[\sin(qR_L)-qR_L\cos(qR_L)]}{(qR_L)^2}

.. math::

    \Psi_S = \int_0^{R_S}(4\pi R^2_S)\frac{\sin(qR_S)}{qR_S}dR_S =
    \frac{3[\sin(qR_S)-qR_L\cos(qR_S)]}{(qR_S)^2}

The cross term between the large droplet and small particles is given by:

.. math::
    S_{LS} = \Psi_L\Psi_S\frac{\sin(q(R_L+\delta R_S))}{q(R_L+\delta\ R_S)}

and the self term between small particles is given by:

.. math::
    S_{SS} = \Psi_S^2\biggl[\frac{\sin(q(R_L+\delta R_S))}{q(R_L+\delta\ R_S)}
    \biggr]^2

The number of small particles per large droplet, $N_p$, is given by:

.. math::

    N_p = \frac{\phi_S\phi_\text{surface}V_L}{\phi_L V_S}

where $\phi_S$ is the volume fraction of small particles in the sample,
$\phi_\text{surface}$ is the fraction of the small particles that are adsorbed
to the large droplets, $\phi_L$ is the volume fraction of large droplets in the
sample, and $V_S$ and $V_L$ are the volumes of individual small particles and
large droplets respectively.

The form factor of the entire complex can now be calculated including the excess
scattering length densities of the components $\Delta\rho_L$ and $\Delta\rho_S$,
where $\Delta\rho_x = \left|\rho_x-\rho_\text{solvent}\right|$ :

.. math::

    P_{LS} = \frac{1}{M^2}\bigl[(\Delta\rho_L)^2V_L^2\Psi_L^2
                +N_p(\Delta\rho_S)^2V_S^2\Psi_S^2
                + N_p(1-N_p)(\Delta\rho_S)^2V_S^2S_{SS}
                + 2N_p\Delta\rho_L\Delta\rho_SV_LV_SS_{LS}\bigr]

where M is the total scattering length of the whole complex :

.. math::
    M = \Delta\rho_LV_L + N_p\Delta\rho_SV_S

In a real system, there will ususally be an excess of small particles such that
some fraction remain unbound. Therefore the overall scattering intensity is
given by:

.. math::
    I(Q) = I_{LS}(Q) + I_S(Q) = (\phi_L(\Delta\rho_L)^2V_L +
            \phi_S\phi_\text{surface}N_p(\Delta\rho_S)^2V_S)P_{LS}
            + \phi_S(1-\phi_\text{surface})(\Delta\rho_S)^2V_S\Psi_S^2

A useful parameter to extract is the fraction of the surface area of the large
droplets that is covered by small particles. This can be calculated from the
model parameters as:

.. math::
    \chi = \frac{4\phi_L\phi_\text{surface}(R_L+\delta R_S)}{\phi_LR_S}



.. figure:: img/raspberry_autogenfig.png

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


**Source**

:download:`raspberry.py <src/raspberry.py>`
$\ \star\ $ :download:`raspberry.c <src/raspberry.c>`
$\ \star\ $ :download:`sas_3j1x_x.c <src/sas_3j1x_x.c>`

**References**

#. K Larson-Smith, A Jackson, and D C Pozzo, *Small angle scattering model
   for Pickering emulsions and raspberry particles*,
   *Journal of Colloid and Interface Science*, 343(1) (2010) 36-41

**Authorship and Verification**

* **Author:** Andrew Jackson **Date:** 2008
* **Last Modified by:** Andrew Jackson **Date:** March 20, 2016
* **Last Reviewed by:** Andrew Jackson **Date:** March 20, 2016

