.. _polymer-micelle:

polymer_micelle
=======================================================

Polymer micelle model

============= ================================================== ============ =============
Parameter     Description                                        Units        Default value
============= ================================================== ============ =============
scale         Source intensity                                   None                     1
background    Source background                                  |cm^-1|              0.001
ndensity      Number density of micelles                         |1e15cm^3|            8.94
v_core        Core volume                                        |Ang^3|              62624
v_corona      Corona volume                                      |Ang^3|              61940
sld_solvent   Solvent scattering length density                  |1e-6Ang^-2|           6.4
sld_core      Core scattering length density                     |1e-6Ang^-2|          0.34
sld_corona    Corona scattering length density                   |1e-6Ang^-2|           0.8
radius_core   Radius of core ( must be >> rg )                   |Ang|                   45
rg            Radius of gyration of chains in corona             |Ang|                   20
d_penetration Factor to mimic non-penetration of Gaussian chains None                     1
n_aggreg      Aggregation number of the micelle                  None                     6
============= ================================================== ============ =============

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



This model provides the form factor, $P(q)$, for a micelle with a spherical
core and Gaussian polymer chains attached to the surface, thus may be applied
to block copolymer micelles. To work well the Gaussian chains must be much
smaller than the core, which is often not the case.  Please study the
reference carefully.

**Definition**

The 1D scattering intensity for this model is calculated according to
the equations given by Pedersen (Pedersen, 2000), summarised briefly here.

The micelle core is imagined as $N\_aggreg$ polymer heads, each of volume $v\_core$,
which then defines a micelle core of $radius\_core$, which is a separate parameter
even though it could be directly determined.
The Gaussian random coil tails, of gyration radius $rg$, are imagined uniformly
distributed around the spherical core, centred at a distance $radius\_core + d\_penetration.rg$
from the micelle centre, where $d\_penetration$ is of order unity.
A volume $v\_corona$ is defined for each coil.
The model in detail seems to separately parametrise the terms for the shape of I(Q) and the
relative intensity of each term, so use with caution and check parameters for consistency.
The spherical core is monodisperse, so it's intensity and the cross terms may have sharp
oscillations (use q resolution smearing if needs be to help remove them).

.. math::
    P(q) = N^2\beta^2_s\Phi(qR)^2+N\beta^2_cP_c(q)+2N^2\beta_s\beta_cS_{sc}s_c(q)+N(N-1)\beta_c^2S_{cc}(q) \\
    \beta_s = v\_core(sld\_core - sld\_solvent) \\
    \beta_c = v\_corona(sld\_corona - sld\_solvent)

where $N = n\_aggreg$, and for the spherical core of radius $R$

.. math::
   \Phi(qR)= \frac{\sin(qr) - qr\cos(qr)}{(qr)^3}

whilst for the Gaussian coils

.. math::

   P_c(q) &= 2 [\exp(-Z) + Z - 1] / Z^2 \\
   Z &= (q R_g)^2

The sphere to coil ( core to corona) and coil to coil (corona to corona) cross terms are
approximated by:

.. math::

   S_{sc}(q)=\Phi(qR)\psi(Z)\frac{sin(q(R+d.R_g))}{q(R+d.R_g)} \\
   S_{cc}(q)=\psi(Z)^2\left[\frac{sin(q(R+d.R_g))}{q(R+d.R_g)} \right ]^2 \\
   \psi(Z)=\frac{[1-exp^{-Z}]}{Z}

**Validation**

$P(q)$ above is multiplied by $ndensity$, and a units conversion of 10^{-13}, so $scale$
is likely 1.0 if the scattering data is in absolute units. This model has not yet been
independently validated.



.. figure:: img/polymer_micelle_autogenfig.png

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

**References**

J Pedersen, *J. Appl. Cryst.*, 33 (2000) 637-640


