.. _polymer-micelle:

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

Polymer micelle model

============= ================================================== ============ =============
Parameter     Description                                        Units        Default value
============= ================================================== ============ =============
scale         Scale factor or Volume fraction                    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. Take special care before use and read both of the
provided references carefully; this model typically only works well when the
Gaussian chains are much smaller than the core, which is often not the case.

**Definition**

The 1D scattering intensity for this model is calculated according to
the equations given by Pedersen (1996, 2000), summarised briefly here.

The micelle core is imagined as $N$ = *n_aggreg* polymer heads, each of volume
$V_\text{core}$, which then defines a micelle core of radius $r$ = *r_core*,
which is a separate parameter even though it could be directly determined.
The Gaussian random coil tails, of gyration radius $R_g$, are imagined
uniformly distributed around the spherical core, centred at a distance
$r + d \cdot R_g$ from the micelle centre, where $d$ = *d_penetration* is
of order unity. A volume $V_\text{corona}$ is defined for each coil. The
model in detail seems to separately parameterize 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}(q) + N(N-1)\beta_c^2S_{cc}(q) \\
    \beta_s &= V_\text{core}(\rho_\text{core} - \rho_\text{solvent}) \\
    \beta_c &= V_\text{corona}(\rho_\text{corona} - \rho_\text{solvent})

where $\rho_\text{core}$, $\rho_\text{corona}$ and $\rho_\text{solvent}$ are
the scattering length densities *sld_core*, *sld_corona* and *sld_solvent*.
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 \cdot R_g))}{q(r+d \cdot R_g)} \\
   S_{cc}(q) &= \psi(Z)^2
       \left[\frac{\sin(q(r+d \cdot R_g))}{q(r+d \cdot 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.


**Source**

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

**References**

#.  J Pedersen and M Gerstenberg, *Macromolecules*, 29 (1996) 1363–1365
#.  J Pedersen, *J. Appl. Cryst.*, 33 (2000) 637-640


**Authorship and Verification**

* **Translated by   :** Richard Heenan **Date:** March 20, 2016
* **Last modified by:** Paul Kienzle **Date:** November 29, 2017
* **Last reviewed by:** Oliver Hammond **Date:** January 13, 2024

