# polymer_micelle

Polymer micelle model

Parameter |
Description |
Units |
Default value |
---|---|---|---|

scale |
Scale factor or Volume fraction |
None |
1 |

background |
Source background |
cm |
0.001 |

ndensity |
Number density of micelles |
10 |
8.94 |

v_core |
Core volume |
Å |
62624 |

v_corona |
Corona volume |
Å |
61940 |

sld_solvent |
Solvent scattering length density |
10 |
6.4 |

sld_core |
Core scattering length density |
10 |
0.34 |

sld_corona |
Corona scattering length density |
10 |
0.8 |

radius_core |
Radius of core ( must be >> rg ) |
Å |
45 |

rg |
Radius of gyration of chains in corona |
Å |
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 \(N\) = *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).

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 \(r\)

whilst for the Gaussian coils

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

**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.

**Source**

`polymer_micelle.py`

\(\ \star\ \) `polymer_micelle.c`

\(\ \star\ \) `sas_3j1x_x.c`

**References**

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:**Steve King**Date:**November 30, 2017