Polymer Micelle

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).

$$ P(q) = N^2\beta^2_s\Phi(qr)^2 + N\beta^2_cP_c(q) + 2N^2\beta_s\beta_cS_{sc}(q) + N(N-1)\beta_c^2S_{cc}(q) \\ \beta_s = V_\text{core}(\rho_\text{core} - \rho_\text{solvent}) \\ \beta_c = V_\text{corona}(\rho_\text{corona} - \rho_\text{solvent})
$$
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 $r$

$$ \Phi(qr)= \frac{\sin(qr) - qr\cos(qr)}{(qr)^3}
$$
whilst for the Gaussian coils

$$ P_c(q) = 2 [\exp(-Z) + Z - 1] / Z^2 \\ Z = (q R_g)^2
$$
The sphere to coil (core to corona) and coil to coil (corona to corona) cross terms are approximated by:

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

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