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moved color table into different directory

added form factor functions for azimuthal analysis
added form factor for deformed polymer network according to Warner-Edwards and Heinrich-Straube-Helmis
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Kohlbrecher committed Mar 8, 2019
1 parent 8ba309c commit a0e3d92966378d0277ea9d6e645a182c87121191
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@@ -61,7 +61,7 @@ \subsection{Maier-Saupe azimuthal analysis} ~\\
\begin{center}
\includegraphics[width=0.7\textwidth]{../images/form_factor/azimuthal/maiersaupe.png}
\end{center}
\caption{Azimuthal intensity distribution of the Maiser-Saupe model from Picken \cite{Picken1990}}
\caption{Azimuthal intensity distribution of the Maier-Saupe model from Picken \cite{Picken1990}}
\label{fig:maiersaupe}
\end{figure}

@@ -80,7 +80,7 @@ \subsection{affine shrinkage} ~\\
\underline{Input Parameters for models \texttt{affine shrinkage (deg)} and \texttt{affine shrinkage (rad)}:}\\
\begin{description}
\item[\texttt{I0}] flat background $I_0$
\item[\texttt{A}] Amplitude $A$ of the angle dependent intensity
\item[\texttt{A}] Amplitude $A$ of the angle dependent intensity
\item[\texttt{lambda}] shrinkage factor $\lambda$
\item[\texttt{delta}] direction of the polarisation $\delta$ in degree or radian
\end{description}
@@ -120,3 +120,23 @@ \subsection{Ellipsoidal azimuthal analysis} ~\\
I_\mathrm{rad}(\psi) &= \left(\left(\frac{\cos(\psi-\delta)}{A}\right)^2 + \left(\frac{\sin(\psi-\delta)}{B}\right)^2\right)^{-N/2} +C\\
I_\mathrm{deg}(\psi) &= \left(\left(\frac{\cos\left((\psi-\delta)\frac{\pi}{180}\right)}{A}\right)^2 + \left(\frac{\sin\left((\psi-\delta)\frac{\pi}{180}\right)}{B}\right)^2\right)^{-N/2}+C
\end{align}
\hspace{1pt}\\
\underline{Input Parameters for models \texttt{elliptically averaged (deg)} and \texttt{elliptically averaged (rad)}:}\\
\begin{description}
\item[\texttt{A}] semi-axis along $\psi-\delta=0$ of iso-contours in reciprocal space
\item[\texttt{B}] semi-axis along $\psi-\delta=\pi/2$ of iso-contours in reciprocal space
\item[\texttt{C}] Amplitude $A$ of the angle dependent intensity
\item[\texttt{delta}] direction of the polarisation $\delta$ in degree or radian
\end{description}

\underline{Note:}
The reciprocal value of $A$ and $B$ are related to the corresponding correlation length of the scattering object in that direction.


\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{../images/form_factor/azimuthal/ellipsoidal.png}
\end{center}
\caption{Azimuthal intensity distribution of the elliptically averaged model}
\label{fig:elliptically_averaged}
\end{figure}
@@ -875,7 +875,7 @@ \subsubsection{coiled superhelix} ~\\
\begin{align}
\begin{split}
f(\gamma) &= R_2^2+R_1^2N^2 \left(\cos^2\left(N\gamma\right)+\sin^2\left(N\gamma\right)\cos^2(\alpha_2)\right) \\
&+\frac{P^2}{(2\pi)^2} + 2R_2*R_1\cos\left(N\gamma\right) +2R_1^2N\cos(\alpha_2)
&+\frac{P^2}{(2\pi)^2} + 2R_2R_1\cos\left(N\gamma\right) +2R_1^2N\cos(\alpha_2)
\end{split}
\end{align}
and
@@ -1,5 +1,5 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\clearpage
\section{Sheared and deformed objects}

\subsection{Non-equilibrium static form factor of a reptating chain}
@@ -38,10 +38,10 @@ \subsection{Non-equilibrium static form factor of a reptating chain}
\end{align}

\hspace{1pt}\\
\underline{Input Parameters for models \texttt{CylShell1}, \texttt{CylShell2} and \texttt{LongCylShell}:}\\
\underline{Input Parameters for models \texttt{reptating chain}:}\\
\begin{description}
\item[\texttt{I0}] forward scattering $I_0$
\item[\texttt{Rg}] radius of gyration of unstretched polymer $R_r$
\item[\texttt{Rg}] radius of gyration of unstretched polymer $R_g$
\item[\texttt{lambda}] stretching factor $\lambda$
\item[\texttt{t/tau}] relaxation time in units of Rouse time $t/\tau$
\item[\texttt{theta\_0}] stretching direction in the detector plane $\theta_0$ in degree
@@ -61,6 +61,108 @@ \subsection{Non-equilibrium static form factor of a reptating chain}
\label{fig:IQ2Dstretchedpolymermelt}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\subsection{step-deformed polymer networks}
\label{sect:DeformedPolymerNetwork}
\hspace{1pt}\\
Two slightly different tube models for labeled polymer chains in a deformed network are implemented in this plugin: the original one from Warner-Edwards (WE) \cite{Warner1978} and a refined one from Heinrich-Straube-Helmis (HSH) \cite{Heinrich1988}, which have been extensively used to study uni-axial stretched polymer networks \cite{Straube1995,Mergell2001,Westermann2001,Westermann1999,Westermann1996}.
The form factor of an uniaxial stretched polymer network with a stretching direction in the plane of the detector reads as
\begin{align}
\label{eq:SQlambda}
S(\mathbf{q},\lambda) &= \int_0^1\mathrm{d}\eta \int_0^1 \mathrm{d}\eta' \prod_{\mu} e^{-\left(Q_\mu\lambda_\mu\right)^2\abs{\eta-\eta'}-Q_\mu^2(1-\lambda_\mu^2)\xi\left\{1-\exp\left(-\frac{\abs{\eta-\eta'}}{\xi}\right)\right\}} \\
\xi &= \frac{d_\mathrm{t}^2}{2\sqrt{6}R_g^2} \\
Q_\mu &= q_\mu R_g\\
\mu &= x,y,z \\
\mathbf{q} &= \Vek{q_x}{q_y}{q_z} = \Vek{q \cos(\psi-\delta)}{q\sin(\psi-\delta)}{0}
\end{align}
As the function of the integral kernel in eq.\ \ref{eq:SQlambda} only depends on $\abs{\eta-\eta'}$ one can transform the 2D-integral into a 1D-integral by using the identity
\begin{align}
\int_0^1\mathrm{d}\eta \int_0^1 \mathrm{d}\eta' \: K\left(\abs{\eta-\eta'}\right) &= 2 \int_0^1 (1-x)K(x)\: \mathrm{d}x
\end{align}
The difference between the WE-model and the HSH-model is how the deformed tube diameter $d_\mathrm{t}$ is calculated. In case of the EW-model the projection $d_\mu$ of $d_t$ onto the Cartesian axis are used, whereas the HSH-model uses an effective angle dependent diameter $d_\phi$.
\begin{align}
\mbox{(WE-model): } d_\mathrm{t} &= d_\mu^2 = d_0^2\lambda_\mu^{2\nu} \\
\mbox{(HSH-model): } d_\mathrm{t} &= d_\phi^2 = d_0^2\lambda_\phi^{2\nu}
\end{align}
If we assume an elongation ratio of $\lambda$ in the $x$-direction
\begin{align}
\lambda_x &=\lambda_{\|}=\lambda, \quad \lambda_y=\lambda_z=\lambda_\perp=\frac{1}{\sqrt{\lambda}} \\
\lambda_\phi^2 &= \lambda_{\|}^2 \cos^2(\psi-\delta)+\lambda_\perp^2\sin^2(\psi-\delta)
\end{align}
where $R_g$ the radius of gyration of the un-deformed polymer network $\lambda_\mu$ is the macroscopic stretch ratio in this Cartesian direction and the exponent $\nu$ is predicted to take a value of $1/2$ in the case of networks \cite{Straube1995,Read2004}. Nevertheless $\nu$ is kept as an input parameter in this plug-in function.
\begin{align}
\label{eq:SQ_WE_HSH}
%\begin{split}
S^\mathrm{WE}(\mathbf{q},R_g,\lambda,\nu,d_0) = & 2\int_0^1\mathrm{d}x\: (1-x) \exp\left(\sum_{\mu} -Q_\mu^2\lambda_\mu^2 x- %\\ &\qquad \qquad
(1-\lambda_\mu^2)Q_\mu^2\xi_\mu\left\{1-\exp\left(-\frac{x}{\xi_\mu}\right)\right\}\right) \\
%\end{split} \\
%\begin{split}
S^\mathrm{HSH}(\mathbf{q},R_g,\lambda,\nu,d_0) = & 2\int_0^1\mathrm{d}x\: (1-x) \exp\left(\sum_{\mu} -Q_\mu^2\lambda_\mu^2 x- % \\ &\qquad \qquad
(1-\lambda_\mu^2)Q_\mu^2\xi_\phi\left\{1-\exp\left(-\frac{x}{\xi_\phi}\right)\right\}\right)
%\end{split}
\end{align}
with
\begin{align}
\xi_\mu =& \frac{d_\mu^2}{2\sqrt{6}R_g^2} \\
\xi_\phi =& \frac{d_\phi^2}{2\sqrt{6}R_g^2}
\end{align}
In a work of \cite{Ariane2004} an additional retraction coefficient $\gamma(t)$ has been included so that the final equation for the HSH model reads as
\begin{align}
\label{eq:SQ_HSH_retraction}
\begin{split}
S^\mathrm{HSH}(\mathbf{q},R_g,\lambda,\nu,\gamma,d_0) = & 2\int_0^1\mathrm{d}x\: (1-x) \exp\Bigg(\sum_{\mu} Q_\mu^2\left[-\lambda_\mu^2\frac{x}{\gamma(t)}- \right. \\ &\qquad \qquad \left. \xi_\phi\left\{1-\exp\left(-\frac{x}{\xi_\phi\gamma(t)}\right)\right\} +\lambda_\mu^2\xi_\phi\left\{1-\exp\left(-\frac{x}{\xi_\phi}\right)\right\} \right]\Bigg)
\end{split}
\end{align}
The retraction coefficient is for short times $\gamma(t=0)=1$ and becomes larger for increasing relaxation times $\gamma(t>0)>1$.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{../images/form_factor/deformed_sheared/deformed_polymer_network_I(sector).png}
\end{center}
\caption{sector averaged scattering curves of the Warner-Edwards and Heinrich-Straube-Helmis model of step-deformed polymer networks for tube diameters $d_0$ of 2nm and zero nm}
\label{fig:I(sector)polymernetwork}
\end{figure}

\hspace{1pt}\\
\underline{Input Parameters for models \texttt{WE: deformed polym. netw.}:}\\
\begin{description}
\item[\texttt{I0}] forward scattering $I_0$
\item[\texttt{Rg}] radius of gyration of unstretched polymer $R_g$
\item[\texttt{do}] equilibrium tube diameter $d_0$
\item[\texttt{nu}] exponent ($\nu=1/2$, $\nu=1$:affine deformation)
\item[\texttt{lambda}] deformation ratio along deformation axis $\lambda$
\item[\texttt{dummy}] not used
\item[\texttt{psi}] direction $\psi$ of the scattering vector $\mathbf{q}$ in the detector plane in degree
\item[\texttt{delta}] stretching direction in the detector plane $\delta$ in degree
\end{description}

\hspace{1pt}\\
\underline{Input Parameters for models \texttt{HSH: deformed polym. netw.}:}\\
\begin{description}
\item[\texttt{I0}] forward scattering $I_0$
\item[\texttt{Rg}] radius of gyration of unstretched polymer $R_g$
\item[\texttt{do}] equilibrium tube diameter $d_0$
\item[\texttt{nu}] exponent ($\nu=1/2$, $\nu=1$:affine deformation)
\item[\texttt{lambda}] deformation ratio along deformation axis $\lambda$
\item[\texttt{gamma}] retraction coefficient ($1 < \gamma \lesssim 2$)
\item[\texttt{psi}] direction $\psi$ of the scattering vector $\mathbf{q}$ in the detector plane in degree
\item[\texttt{delta}] stretching direction in the detector plane $\delta$ in degree
\end{description}

\underline{Note:}
\begin{itemize}
\item for large $q$-values the integration routines might fail to converge.
\end{itemize}

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{../images/form_factor/deformed_sheared/deformed_polymer_network.png}
\end{center}
\caption{2D scattering patterns of the Warner-Edwards and Heinrich-Straube-Helmis model of step-deformed polymer networks}
\label{fig:IQ2Dpolymernetwork}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
\subsection{Sheared Cylinder according to Hayter and Penfold}
@@ -223,7 +325,7 @@ \subsection{partly aligned anisotropic objects}
axis $\mathbf{n}$. $L$ is the length of the cylinder, $R$ its
radius, $\Delta\eta$ the scattering length density contrast relative
to the solvent and $J_1(x)$ is the first order Bessel function of
the first kind.
the first kind.

The scattering amplitude of an ellispoid of revolution is given by
\begin{align}
@@ -270,7 +372,7 @@ \subsection{partly aligned anisotropic objects}
\end{align}
For this form factor it is assumed that the orientation distribution is independent of $\phi$,
i.e. $p(\theta,\phi;\kappa)=p(\theta;\kappa)$ and that $p(\theta;\kappa)=p(\pi-\theta;\kappa)$, which means that turning the cylinder by 180$^\circ$ results in the same scattering intensity.
Several orientation distributions have been implemented in a way that their resulting order parameter $S_2$ can have values between -0.5 and 1, which correspond to perfect alignment perpendicular to the $\mathrm{x}$ axis and perfect alignment parallel to it. All probability distributions have been normalized
Several orientation distributions have been implemented in a way that their resulting order parameter $S_2$ can have values between -0.5 and 1, which correspond to perfect alignment perpendicular to the $\mathrm{x}$ axis and perfect alignment parallel to it. All probability distributions have been normalized
\begin{align}
\int_0^\pi \int_0^{2\pi} p(\theta,\phi;\kappa) \sin \theta \, d\theta \, d\phi&=1
\end{align}
@@ -326,14 +428,14 @@ \subsection{partly aligned anisotropic objects}
The order parameter for the Hayter-Penfold orientation distribution has been calculated by performing first a coordination transformation of the polar coordinates with $\mathbf{z}$ being the polar axis to coordination system with a polar axis pointing into the direction of the most probable orientation. The order parameter $S_2(\kappa)$ in eq.\ \ref{eq:S2kappa} is then calculated in this new polar coordinate system.

The form factors additional contain already a size distribution to profit from the speed enhancement by using a specialized multidimensional integration routine. The size distribution is included as
\begin{align}
\begin{align}
I(Q) &= \int_0^\infty \mathrm{LogNorm}(\nu,\sigma,1) I_\mathrm{p.a.CylShell}(Q,R\nu,\Delta R\nu,L\nu) \, \mathrm{d}\nu \\
\end{align}
and
\begin{align}
I(Q) &= \int_0^\infty \mathrm{LogNorm}(\nu,\sigma,1) I_\mathrm{p.a.EllShell}(Q,R_\mathrm{p}\nu,\Delta R\nu,R_\mathrm{e}\nu) \, \mathrm{d}\nu
\end{align}
with
with
\begin{align}
\mathrm{LogNorm}(\nu,\sigma,\mu) &= \frac{1}{\sqrt{2\pi}\sigma}\frac{1}{\nu} \exp\left(-\frac{(\ln(\nu/\mu))^2}{2\sigma^2}\right)
\end{align}
@@ -27,8 +27,8 @@
\caption{#5}\label{#6}
\end{center}\end{figure}}

\newcommand{\vek}[2]{\left(\begin{array}{r}{#1}\\{#2}\end{array}\right)}
\newcommand{\Vek}[3]{\left(\begin{array}{r}#1\\#2\\#3\end{array}\right)}
\newcommand{\vek}[2]{\left(\begin{array}{c}{#1}\\{#2}\end{array}\right)}
\newcommand{\Vek}[3]{\left(\begin{array}{c}#1\\#2\\#3\end{array}\right)}

% \def\spvec#1{\left(\vcenter{\halign{\hfil$##$\hfil\cr \spvecA#1;;}}\right)}
% \def\spvecA#1;{\if;#1;\else #1\cr \expandafter \spvecA \fi}
@@ -118,4 +118,4 @@
\def\corr{\mbox{\small$\,\:{\rlapa{\star}\bigcirc}\,$}}
\def\scorr{\mbox{\footnotesize$\,{\rlapb{\bigcirc}\star}\,\:$}}

\newcommand*{\doi}[1]{\href{http://dx.doi.org/\detokenize{#1}}{doi: \detokenize{#1}}}
\newcommand*{\doi}[1]{\href{http://dx.doi.org/\detokenize{#1}}{doi: \detokenize{#1}}}
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