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	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. 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).

\[ \begin{align}\begin{aligned}P(q) = N^2\beta^2_s\Phi(qR)^2+N\beta^2_cP_c(q)+2N^2\beta_s\beta_cS_{sc}s_c(q)+N(N-1)\beta_c^2S_{cc}(q)\\\beta_s = v\_core(sld\_core - sld\_solvent)\\\beta_c = v\_corona(sld\_corona - sld\_solvent)\end{aligned}\end{align} \]

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

\[\Phi(qR)= \frac{\sin(qr) - qr\cos(qr)}{(qr)^3}\]

whilst for the Gaussian coils

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

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

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

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. 73 1D plot corresponding to the default parameters of the model.

References

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