coreshellmicrogel (SASfit)

This file has been automatically generated by sasfit_convert and manually edited by Wojciech Potrzebowski, ESS on 2017-12-07.

The model calculates an empirical functional form for SAS data characterized by coreshellmicrogel (as defined in SASfit https://github.com/SASfit/SASfit/)

Definition:
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This model can be used to calculate the scattering from spherical particles with a parabolic "fuzzy" interface

The radial profile is given by
\begin{align}
\rho(r,R,\sigma) &=
\begin{cases}
1 & \mbox{for } r\leq R-\sigma \\
1-\frac{1}{2}\frac{\left((r-R)+\sigma\right)^2}{\sigma^2} & \mbox{for } R-\sigma < r \leq R \\
\frac{1}{2}\frac{\left((R-r)+\sigma\right)^2}{\sigma^2} & \mbox{for } R< r\leq R+\sigma \\
0 & \mbox{for } r > R+\sigma
\end{cases}
\end{align}

where $R=W+\sigma$. For such a radial profile the Fourier-transformation can be calculated analytically as

\begin{multline}
F(Q,R,\sigma) = \mathcal{F}[\rho(r,R,\sigma)] = \\
4 \pi \Bigg(
\left(\frac{R}{\sigma^2}+\frac{1}{\sigma}\right) \frac{\cos (q(R+\sigma))}{q^4}
+ \left(\frac{R}{\sigma^2}-\frac{1}{\sigma}\right) \frac{\cos (q(R-\sigma))}{q^4} \\
- 3 \frac{\sin(q(R+\sigma))}{q^5 \sigma^2}
- 3 \frac{\sin(q(R-\sigma))}{q^5 \sigma^2}
- 6 \frac{\sin(qR)}{q^5 \sigma^2}
- 2 R \frac{\cos(qR)}{q^4 \sigma^2}
\Bigg)
\end{multline}

The last term in the brackets needed to be corrected compared to the referenced papers due to a typo in the original papers.
The radial scattering length density profile of a fuzzy core can be for example obtained by

\begin{multline}
\eta_{core,sh}(r,W_\textrm{core},\sigma_\textrm{core},D,\sigma_\textrm{sh,in},W_\textrm{sh},\sigma_\textrm{sh,out}) =
\eta_\textrm{sol}
+ (\eta_\textrm{shell}-\eta_\textrm{sol}) \rho(r,R_\textrm{out},\sigma_\textrm{out}) \\
+ (\eta_\textrm{shell}-\eta_\textrm{sol}) \rho(r,R_\textrm{sh,in},\sigma_\textrm{sh,in})
+ (\eta_\textrm{core} -\eta_\textrm{sol}) \rho(r,R_\textrm{core},\sigma_\textrm{core})
\end{multline}

with
\begin{align}
R_\textrm{core} &= W_\textrm{core}+\sigma_\textrm{core} \\
R_\textrm{sh,in}&= R_\textrm{core}+D \\
R_\textrm{out} &= R_\textrm{sh,in}+\sigma_\textrm{sh,in}+W_\textrm{sh}+\sigma_\textrm{sh,out}
\end{align}

In the same way also the scattering amplitude $F_\textrm{core,sh}(Q,\cdots)$ and the scattering intensity
$I_\textrm{core,sh}(Q,\cdots)=|F_\textrm{core,sh}(Q,\cdots)|^2$ can be calculated:

\begin{multline}
F_\textrm{core,sh}(Q,W_\textrm{core},\sigma_\textrm{core},D,\sigma_\textrm{sh,in},W_\textrm{sh},\sigma_\textrm{sh,out}) =
(\eta_\textrm{shell}-\eta_\textrm{sol}) F(Q,R_\textrm{out},\sigma_\textrm{out}) \\
+ (\eta_\textrm{shell}-\eta_\textrm{sol}) F(Q,R_\textrm{sh,in},\sigma_\textrm{sh,in})
+ (\eta_\textrm{core} -\eta_\textrm{sol}) F(Q,R_\textrm{core},\sigma_\textrm{core})
\end{multline}

Parameters definition:
$W_{core}$ : radius of center parts of core Wcore with homogeneous scattering length density
$\sigma_{core}$ : interface half width of the core
$W_{sh}$ : width of center parts of shell Wsh with homogeneous scattering length density
$\sigma_{sh,in}$ : half width of the inner interface of shell
$D$ : distance between interface of core and in interface of shell
$\sigma_{out}$ : half width of the outer surface profile
$\eta_{core}$ : scattering length density of homogeneous core part
$\eta_{shell}$ : scattering length density of homogeneous shell part
$\eta_{sol}$ : scattering length density of solvent

References:
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Author(s) of the original file: src/plugins/fuzzysphere/sasfit_ff_coreshellmicrogel.c: Joachim Kohlbrecher (joachim.kohlbrecher@psi.ch)

A paper about SASfit has been published in J. Appl. Cryst. (2015). 48, 1587-1598 doi:10.1107/S1600576715016544

Ingo Berndt, Jan Skov Pedersen, Peter Lindner, and Walter Richtering. Influence of shell thickness and cross-link density on the structure of temperature-sensitive poly-n isopropylacrylamidepoly-n-isopropylmethacrylamide coreshell microgels investigated by small-angle neutron scattering.
Langmuir, 22(1):459-468,2006.PMID: 16378460.

Ingo Berndt, Jan Skov Pedersen, and Walter Richtering. Structure of multiresponsive intelligent? coreshell microgels.
Journal of the American Chemical Society, 127(26):9372-9373, 2005 PMID: 15984856

Ingo Berndt, Jan Skov Pedersen, and Walter Richtering. Temperature-sensitive coreshell microgel particles with dense shell.
Angewandte Chemie, 118(11):1769-1773, 2006.

Example Data: