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\begin{document}


\subsection{Construct for conserved fields}

The evolution of the concentration field is governed by diffusion, and is
coupled to the non-conserved phase field. \ The diffusion equation is
written as\cite{Loginova-wid}

\begin{equation}
\frac{\partial u_{c}}{\partial t}\frac{1}{V_{m}}=-\nabla \cdot
J_{c}\rightarrow mol/(m^{3}s)  \label{diffEqn}
\end{equation}

The flux of carbon $J_{c}$ is given by

\begin{equation}
J_{c}=-\frac{1}{V_{m}}D_{c}\nabla u_{c}-L^{\prime \prime }\nabla \frac{%
\partial ^{2}G_{m}}{\partial u_{c}\partial \phi }\nabla \phi \rightarrow
mol/(m^{2}s)  \label{Carbon Flux}
\end{equation}

where%
\begin{equation}
D_{c}=V_{m}L^{\prime \prime }\frac{\partial ^{2}G_{m}}{\partial u_{c}^{2}}%
\rightarrow m^{2}/s  \label{CarbonDiff}
\end{equation}

and 
\begin{equation}
L^{\prime \prime }=\frac{u_{c}}{V_{m}}y_{va}M_{c}\rightarrow mol^{2}/(J\cdot
s\cdot m)  \label{Lprime2}
\end{equation}

where $y_{va}$ is the fraction of vacant intersitial sites in $\gamma $, $%
1-u_{c}$\ and in $\alpha $, $1-u_{c}/3$.

\bigskip The second derivative $\frac{\partial ^{2}G_{m}}{\partial u_{c}^{2}}
$ is determined by differentiating equation \ref{PFC - GmPhi} with respect
to $u_{c}$

\begin{equation}
\frac{\partial ^{2}G_{m}}{\partial u_{c}^{2}}=p(\phi )\left( \frac{\partial
^{2}G_{m}^{\alpha }}{\partial u_{c}^{2}}-\frac{\partial ^{2}G_{m}^{\gamma }}{%
\partial u_{c}^{2}}\right) +\frac{\partial ^{2}G_{m}^{\gamma }}{\partial
u_{c}^{2}}\rightarrow J/mol  \label{d2Gdu2}
\end{equation}

\bigskip\ The second derivatives for each phase shown above can obtained
directly from ThermoCalc, or by taking the second derivative of a polynomial
representation of the free energy. \ The second derivative $\frac{\partial
^{2}G_{m}}{\partial u_{c}\partial \phi }$ is determined by differentiating %
\ref{PFC - GmPhi} first with respect to $u_{c}$ and then with respect to $%
\phi $.

\bigskip 
\begin{equation}
\frac{\partial ^{2}G_{m}}{\partial u_{c}\partial \phi }=p^{\prime }(\phi
)\left( \frac{\partial G_{m}^{\alpha }}{\partial u_{c}}-\frac{\partial
G_{m}^{\gamma }}{\partial u_{c}}\right) \rightarrow J/mol  \label{d2Gdudphi}
\end{equation}

As with the non-conserved field, the interpolation function $p(\phi )$ is
used to determine the properties across the entire domain, and in particular
across the diffuse interface region. \ Two postulates are required. \ Using
the postulates forwarded by Loginova \textit{et. al.}, \cite{Loginova-wid}
the first postulate relates to the site fraction of interstitials, $y_{va}$
and the second to the diffusion mobility of carbon, $M_{c}$.

\begin{equation}
u_{c}y_{va}=p(\phi )u_{c}(1-u_{c}/3)+(1-p(\phi ))u_{c}(1-u_{c})\rightarrow
unitless  \label{UcPostulate}
\end{equation}

\begin{equation}
M_{c}=\left( M_{c}^{\gamma }\right) ^{(1-p(\phi ))}\cdot \left(
M_{c}^{\alpha }\right) ^{p(\phi )}\rightarrow m^{2}mol/(J\cdot s)
\label{McPostulate}
\end{equation}

Equations \ref{Fe-C diff mob in ferr} and \ref{Fe-C diff mob in gam} are
used to determine $M_{c}^{\alpha }$ and $M_{c}^{\gamma }$ respectively. \
Substituting equations \ref{CarbonDiff} and \ref{Lprime2} in to \ref{Carbon
Flux},

\begin{equation}
J_{c}=-\frac{1}{V_{m}}u_{c}y_{va}M_{c}\frac{\partial ^{2}G_{m}}{\partial
u_{c}^{2}}\nabla u_{c}-\frac{u_{c}}{V_{m}}y_{va}M_{c}\frac{\partial ^{2}G_{m}%
}{\partial u_{c}\partial \phi }\nabla \phi \rightarrow mol/(m^{2}s)
\label{CarbonFluxExpanded}
\end{equation}

and, substituting \ref{UcPostulate} and \ref{McPostulate} in to \ref%
{CarbonFluxExpanded}

\begin{eqnarray}
J_{c} &=&-\frac{1}{V_{m}}\left[ p(\phi )u_{c}(1-u_{c}/3)+(1-p(\phi
))u_{c}(1-u_{c})\right]  \label{CarbonFluxSuperExpanded} \\
&&\left[ \left( M_{c}^{\gamma }\right) ^{(1-p(\phi ))}\cdot \left(
M_{c}^{\alpha }\right) ^{p(\phi )}\right] \frac{\partial ^{2}G_{m}}{\partial
u_{c}^{2}}\nabla u_{c} \\
&&-\frac{1}{V_{m}}\left[ p(\phi )u_{c}(1-u_{c}/3)+(1-p(\phi ))u_{c}(1-u_{c})%
\right]  \nonumber \\
&&\left[ \left( M_{c}^{\gamma }\right) ^{(1-p(\phi ))}\cdot \left(
M_{c}^{\alpha }\right) ^{p(\phi )}\right] \frac{\partial ^{2}G_{m}}{\partial
u_{c}\partial \phi }\nabla \phi
\end{eqnarray}

and finally substituting \ref{CarbonFluxSuperExpanded} in to \ref{diffEqn}.
\ Here the term $\frac{1}{V_{m}}$ is re-introduced to scale the derivatives
from molar quantities to volume quantities. \ 

\begin{equation}
\frac{\partial u_{c}}{\partial t}=\nabla \left[ 
\begin{array}{c}
\left( M_{c}^{\gamma }\right) ^{(1-p(\phi ))}\cdot \left( M_{c}^{\alpha
}\right) ^{p(\phi )} \\ 
\left[ p(\phi )u_{c}(1-u_{c}/3)+(1-p(\phi ))u_{c}(1-u_{c})\right] \\ 
\left( \frac{\partial ^{2}G_{m}}{\partial u_{c}^{2}}\nabla u_{c}+\frac{%
\partial ^{2}G_{m}}{\partial u_{c}\partial \phi }\nabla \phi \right)%
\end{array}%
\right]  \label{ConservedEqn}
\end{equation}

or

\begin{equation}
\frac{\partial u_{c}}{\partial t}=\nabla \left[ M_{c}\cdot u_{c}y_{va}\cdot
\left( \frac{\partial ^{2}G_{m}}{\partial u_{c}^{2}}\nabla u_{c}+\frac{%
\partial ^{2}G_{m}}{\partial u_{c}\partial \phi }\nabla \phi \right) \right]
\rightarrow s^{-1}  \label{conservedEqn2}
\end{equation}

\subsection{\protect\bigskip FiPy solution for conserved fields}

FiPy solves the conserved field equation using compent diffusion and
component convection terms. \ The convection term ($\chi $) is expressed in
terms of the gradient in $\phi $, while the diffusion term ($D$)is expressed
in terms of the gradient in $u_{c}$. \ Seperating these terms out of the
above equation

\begin{equation}
\chi =\frac{1}{u_{c}}M_{c}\cdot u_{c}y_{va}\cdot \frac{\partial ^{2}G_{m}}{%
\partial u_{c}\partial \phi }\nabla \phi  \label{FiPyConv}
\end{equation}

The term $\frac{1}{u_{c}}$ is required by FiPy in order to solve the
convection equation. \ \textsc{This according to earlier statements by dan
wheeler. \ I dont understand this, but would like to.}

\begin{equation}
D=M_{c}\cdot u_{c}y_{va}\cdot \frac{\partial ^{2}G_{m}}{\partial u_{c}^{2}}%
\nabla u_{c}  \label{FiPyDiff}
\end{equation}

\end{document}
