root/trunk/matml/src/ternary/ternary.tex.in

\documentclass{article}
\usepackage{fullpage,lmodern,cxref,url}
\usepackage[T1]{fontenc}
\begin{document}
\title{{\tt Ternary}: Ternary Thermodynamics Visualization}
\author{Adam C. Powell, IV}
\date{Version @VERSION@ released @RELEASE_DATE@}
\maketitle

\begin{abstract}
  {\tt Ternary} is a little code written by Adam Powell back in 2003 to
  visualize ternary free energy functions.  Today it can also calculate the
  two- and three-phase regions in the phase diagram.  Soon it will do a lot
  more.
\end{abstract}

\section{Introduction}

In 2003, Adam Powell wrote the core of this little program to visualize the
free energy function of a polymer-solvent-nonsolvent system as used by one of
his grad students.  In 2008, he expanded it considerably:
\begin{itemize}
\item Removed it from the {\tt RheoPlast} package and made it a stand-alone
  program, with considerably expanded documentation.
\item Craig Carter used a convex hull construct to find and illustrate the
  lowest-energy manifold of a binary free energy function or set of
  functions.\footnote{Though others have noted the relationship between the
    convex hull and the minimum enengy surface before ({\em e.g.}
    \cite{Var1990,Aic2001,Aic2003,Zho2004}), Carter's Mathematica scripts are
    the inspiration for this work.}  Based on that concept, ternary uses {\tt
    qhull} \cite{BDH1997} to calculate that hull, and visualizes the edges of the resulting
  triangles---which will clearly show the one- and two-phase regions, complete
  with two-phase tie lines.
\item Added visualization and convex hull calculation for multiple phases with
  separate free energy functions, {\em e.g.} solid and liquid.
\end{itemize}

And {\tt Ternary} is not stopping there.  In the near future, the following
features are planned:
\begin{itemize}
\item Refine the edges of the binodal region calculation using {\tt
    qh\_addpoint}.
\item Show the edges of the spinodal region.
\item Based on these, build and neatly visualize how free energy functions
  create a ternary phase diagram.
\item Performance enhancements:
  \begin{itemize}
  \item This should use geomview library function calls instead of a pipe to
    the geomview binary.
  \item It's probably faster to inline or vectorize the free energy function.
  \end{itemize}
  Then again, the slowest part of the calculation will likely be qhull, which
  is already quite thoroughly optimized.
\end{itemize}

\section{How it works}

The file {\tt ternary.c} (documentation in appendix \ref{file_ternary.c})
contains {\tt main()}, which calls functions in {\tt book.c} (appendix
\ref{file_book.c}) to create a triangular array of points and set of triangles
connecting them.  It calls {\tt free\_energy()} in {\tt freenergy.c} (appendix
\ref{file_freenergy.c}) to calculate the free energy function on those
triangles.  It then displays the free energy using the functions in {\tt
  geomview.c} (appendix \ref{file_geomview.c}), which forks and controls a
Geomview process.

Next it uses Qhull calls in {\tt qhull.c} (appendix \ref{file_qhull.c}) to
calculate the convex hull of the free energy function.  It refines the
multi-phase triangles using Newton iteration to find the lowest-energy point
starting at each corner.  And it optionally refines the one-phase triangles at
their centroids.

It then strips out the triangles which are in the single-phase region, leaving
two- and three-phase regions with tie lines.  It displays the remaining facets
using transparent triangles with black edges, which clearly show the tie lines
in the two-phase regions.

For more than one free energy function, it creates a double-size array with two
sets of points, and two sets of triangle indices connecting them.  The rest of
this procedure is pretty much the same.

\section{Version History}

This is a condensed summary of new user-visible features in {\tt Ternary}.  A
more detailed description of new features in each release is in the {\tt
  ChangeLog} file.

\subsection{{\tt Ternary} 0.1.0}

First standalone release of ternary (extracted from {\tt RheoPlast}), which
plots a ternary free energy function.  New to this release is the use of the
Geomview {\tt COFF} format, which creates a much smoother pseudocolor plot.

\subsection{{\tt Ternary} 0.2.0}

This release links with the {\tt qhull} library to calculate the convex hull of
the energy function, and displays 2- and 3-phase regions in addition to the
free energy function.

\subsection{{\tt Ternary} 0.3.0}

This version introduced multiple free energy functions which intersect to
produce complex multi-phase regions on the convex hull.  Internally, much of
the code in {\tt main()} moved out to a new file {\tt book.c}, which will make
it a lot easier to wrap these functions using SIDL or SWIG.

\subsection{{\tt Ternary} 0.4.0}

The new Newton refining code in {\tt qhull.c}, which uses new free energy
derivative functions in {\tt freenergy.c}, gives {\tt Ternary} the ability to
accurately predict binodal boundaries.  Another new function in {\tt
  freenergy.c} calculates the shape of the spinodal curve.

\bibliographystyle{unsrturl}
\bibliography{ternary}

\appendix

% Begin-Of-Source-Files

% End-Of-Source-Files
\end{document}