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## 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 | 1015cm3 | 8.94
v_core | Core volume | Å3 | 62624
v_corona | Corona volume | Å3 | 61940
sld_solvent | Solvent scattering length density | 10-6Å-2 | 6.4
sld_core | Core scattering length density | 10-6Å-2 | 0.34
sld_corona | Corona scattering length density | 10-6Å-2 | 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. 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).

\[\begin{split}P(q) &= N^2\beta^2_s\Phi(qr)^2 + N\beta^2_cP_c(q) + 2N^2\beta_s\beta_cS_{sc}(q) + N(N-1)\beta_c^2S_{cc}(q) \\ \beta_s &= V_\text{core}(\rho_\text{core} - \rho_\text{solvent}) \\ \beta_c &= V_\text{corona}(\rho_\text{corona} - \rho_\text{solvent})\end{split}\]

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

\[\begin{split}P_c(q) &= 2 [\exp(-Z) + Z - 1] / Z^2 \\ Z &= (q R_g)^2\end{split}\]

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

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

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.

Fig. 101 Figure 1: 1D plot corresponding to the default parameters of the model.

Source

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

References

1. J Pedersen and M Gerstenberg, Macromolecules, 29 (1996) 1363–1365

2. 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