# 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 3 62624
v_corona Corona volume 3 61940
sld_solvent Solvent scattering length density 10-6-2 6.4
sld_core Core scattering length density 10-6-2 0.34
sld_corona Corona scattering length density 10-6-2 0.8
rg Radius of gyration of chains in corona 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.

References

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