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Kohlbrecher committed Sep 5, 2019
1 parent 0ed69af commit 0f3b5a826bbb4ee6d35152c26fcaa0806abeb3d4
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@@ -137,7 +137,7 @@ \section{Schulz-Zimm (Flory) distribution}
$k=X_a^2/\sigma^2 > 0$} \label{fig:SchulzZimm}
\end{figure}

A function commonly used to present polymer molecular weight distributions is the Schulz-Zimm function
A function commonly used to present polymer molecular weight distributions is the Schulz-Zimm function \cite{Zimm1948}
\begin{equation}
\textrm{SZ}_n(X,N,X_a,k) = \frac{N}{X_a}
\left(\frac{X}{X_a}\right)^{k-1}
@@ -188,6 +188,7 @@ \section{Schulz-Zimm (Flory) distribution}

\clearpage
\section{Gamma distribution}
\label{sec:SZGammaDistr}

The Gamma distribution is a two parameter continuous distribution with
a scale parameter $\theta$ and a shape parameter $k$.
@@ -103,6 +103,7 @@ \subsection{Ellipsoid of revolution or spheroid}
\end{description}



\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.768\textwidth]{../images/form_factor/Ellipsoid/ellipsoid_ii.png}
@@ -112,6 +113,64 @@ \subsection{Ellipsoid of revolution or spheroid}
\end{figure}



\clearpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Spheroid with a Gamma size distribution}
\label{sec:SpheroidSZdistr}~\\
The Gamma-distribution or Schulz-Zimm (Flory) distribution in section \ref{sec:SZGammaDistr} or \ref{sec:SchulzZimm} has been used since many decades \cite{Schmidt1958,Aragon1976,Schmidt1984,Bartlett1992,Heinemann2000,Wagner2004,Forster2005} to analytically integrate the form factor of a spherical particle or spheroids over a size distribution. The form factor of a homogeneous randomly oriented spheroid with a gamma distribution has been calculated by Schmidt in \cite{Schmidt1958}, where he performed the integration over the size distribution analytically.
\begin{align}\label{eq:spherid_start}
I(q) &= \Delta\eta^2 \int_0^\infty\int_0^1 N(R) V^2 \frac92\pi\frac{\mathrm{J}_{3/2}^2(qtR)}{(qtR)^3} \mathrm{d}y \, \mathrm{d}R\\
&= \Delta\eta^2 \int_0^\infty\int_0^1 \frac{1}{\theta}
\left(\frac{R}{\theta}\right)^{k-1}
\frac{\exp(-R/\theta)}{\Gamma(k)} \left(\frac{4}{3}\pi\nu R^3\right)^2 \frac92\pi\frac{\mathrm{J}_{3/2}^2(qtR)}{(qtR)^3} \mathrm{d}y \, \mathrm{d}R
\end{align}
\begin{multline}
\label{eq:spheroid_GD}
I(q) = \Delta\eta^2 \int_0^1 \frac{8\nu^2}{(qt)^6} \Bigg((A-1) k (k+1) \left(C^{-\frac{k}{2}-1} \cos (B (k+2))+1\right) \\
-2 k qt\theta C^{-\frac{k}{2}-\frac{1}{2}} \sin (B(k+1))-C^{-k/2} \cos (B k)+1\Bigg) \mathrm{d}y
\end{multline}
with
\begin{align}\label{eq:spheroid_tABC}
t &= \sqrt{1+\left(\nu^2-1\right)y^2}\\
\theta &= R_\mathrm{e}/(k-1)\\
\Delta\eta&=\eta_\mathrm{c}-\eta_\mathrm{s}\\
A &= 1+(qt\theta)^2\\
B &= \arctan\left(2qt\theta\right)\\
C &=4A-3
\end{align}
~\\
\underline{Input Parameters for model \texttt{spheroid w. g-size distr}:}
\begin{description}
\item[\texttt{nu}]
ratio between radius of the polar axes $R_\mathrm{p}$ and equatorial axis $R_\mathrm{e}$.
\item[\texttt{R\_equatorial}] length of the equatorial semi-axes $R_\mathrm{e}$
\item[\texttt{dummy}] not used
\item[\texttt{dummy}] not used
\item[\texttt{k}] width parameter of the gamma distribution $(k>1)$.
\item[\texttt{eta\_core}] scattering length density of spheroid $\eta_\mathrm{c}$
\item[\texttt{dummy}] not used
\item[\texttt{eta\_sol}] scattering length density of solvent $\eta_\mathrm{s}$
\end{description}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.768\textwidth]{../images/form_factor/Ellipsoid/spheroid_w_g_SD.png}
\end{center}
\caption{form factor of an spheroid with axis $R_\mathrm{e}$, $R_\mathrm{e}$ and $R_\mathrm{p}\nu
R_\mathrm{e}$.Values of $\nu<1$ describe oblate ellipsoids, a value of $\nu=1$ a
sphere, and $\nu>1$ a prolate ellipsoids. The plot show both the analytically integrated variant and the numerically averaged intensity using the same gamma size distribution.} \label{fig:I_spheroid_w_gSD}
\end{figure}

\newpage

\underline{Note:}
\begin{itemize}
\item the width parameter needs to be larger than 1, $k>1$. The most probable size and the size parameter $\theta$ are related by $R_\mathrm{e} = (k-1) \theta$ and the variance of the distribution is $\sigma^2 = k \theta^2$
\item The eccentricity parameter $\nu$ needs to be a nonzero positive number $\nu>0$ as well as the equatorial axis $R_\mathrm{e}>0$.
\item compared to the numerical integration over the size distribution this analytical one is especially much faster for large $q$-values which is in the numerical variant an integration over an oscillating function and numerically more demanding.
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage
\subsection{Ellipsoid with two equal equatorial semi-axis $R$ and volume $V$}
@@ -233,7 +292,6 @@ \subsection{Ellipsoidal core shell structure}
\caption{Form factor of an ellipsoidal core shell structure including a lognormal size distribution.} \label{fig:I_ellipsoidal_core_shell}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage
\subsection{triaxial ellipsoidal core shell structure}

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