.. _adsorbed-layer:

adsorbed_layer
=======================================================

Scattering from an adsorbed layer on particles

=============== ===================================== ============ =============
Parameter       Description                           Units        Default value
=============== ===================================== ============ =============
scale           Scale factor or Volume fraction       None                     1
background      Source background                     |cm^-1|              0.001
second_moment   Second moment of polymer distribution |Ang|                   23
adsorbed_amount Adsorbed amount of polymer            |mg/m^2|               1.9
density_shell   Bulk density of polymer in the shell  |g/cm^3|               0.7
radius          Core particle radius                  |Ang|                  500
volfraction     Core particle volume fraction         None                  0.14
sld_shell       Polymer shell SLD                     |1e-6Ang^-2|           1.5
sld_solvent     Solvent SLD                           |1e-6Ang^-2|           6.3
=============== ===================================== ============ =============

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


**Definition**

This model describes the scattering from a layer of surfactant or polymer
adsorbed on large, smooth, notionally spherical particles under the conditions
that (i) the particles (cores) are contrast-matched to the dispersion medium,
(ii) $S(Q) \sim 1$ (ie, the particle volume fraction is dilute), (iii) the
particle radius is >> layer thickness (ie, the interface is locally flat),
and (iv) scattering from excess unadsorbed adsorbate in the bulk medium is
absent or has been corrected for.

Unlike many other core-shell models, this model does not assume any form
for the density distribution of the adsorbed species normal to the interface
(cf, a core-shell model normally assumes the density distribution to be a
homogeneous step-function). For comparison, if the thickness of a (traditional
core-shell like) step function distribution is $t$, the second moment about
the mean of the density distribution (ie, the distance of the centre-of-mass
of the distribution from the interface), $\sigma = \sqrt{t^2/12}$.

.. math::

     I(q) = \text{scale} \cdot (\rho_\text{poly}-\rho_\text{solvent})^2
         \left[
             \frac{6\pi\phi_\text{core}}{Q^2}
             \frac{\Gamma^2}{\delta_\text{poly}^2R_\text{core}}
             \exp(-Q^2\sigma^2)
         \right] + \text{background}

where *scale* is a scale factor, $\rho_\text{poly}$ is the sld of the
polymer (or surfactant) layer, $\rho_\text{solv}$ is the sld of the
solvent/medium and cores, $\phi_\text{core}$ is the volume fraction of
the core particles, $\delta_\text{poly}$ is the bulk density of the
polymer, $\Gamma$ is the adsorbed amount, and $\sigma$ is the second
moment of the thickness distribution.

Note that all parameters except $\sigma$ are correlated so fitting more
than one of these parameters will generally fail. Also note that unlike
other shape models, no volume normalization is applied to this model (the
calculation is exact).

The code for this model is based originally on a a fortran implementation by
Steve King at ISIS in the SANDRA package c. 1990 [#King2002]_.


.. figure:: img/adsorbed_layer_autogenfig.png

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


**Source**

:download:`adsorbed_layer.py <src/adsorbed_layer.py>`

**References**

.. [#King2002] S King, P Griffiths, J Hone, and T Cosgrove, *SANS from
   Adsorbed Polymer Layers*, *Macromol. Symp.*, 190 (2002) 33-42.

**Authorship and Verification**

* **Author:** Jae-Hi Cho **Date:** pre 2010
* **Last Modified by:** Paul Kienzle **Date:** April 14, 2016
* **Last Reviewed by:** Steve King **Date:** March 18, 2016