Solidification can occur under many different conditions.  
The experimental situation being considered here is an 
undercooled two component liquid with a simple binary 
eutectic phase diagram 
where the system can be held at constant temperature $T_o$ as 
solidification proceeds.  To simplify the notation, 
$NH_4Cl$ will be referred to as A, and $H_2O$ will be 
referred to as B in the remainder of this chapter 
(Fig. \ref{fig:gen_ph_diag}).  
\begin{figure}[thp]
        \includegraphics[angle=0,scale=.90]{mickey.eps}
        \caption{The generalized phase diagram, to which will be referred 
	in this chapter.  A can be thought of as representing $NH_4Cl$ 
	and B can be thought of as representing $H_2O$.}
        \label{fig:gen_ph_diag}
\end{figure}
The system's initial concentration 
of B (impurity) is $C_o$.  When solid forms, it consists entirely of A, 
requiring that all of the B is rejected into the liquid.  
This raises the concentration of B in the liquid above $C_o$.  
Additionally, although it is initially thermodynamically 
favorable throughout the entire system for solid A to 
precipitate from the liquid (rejecting B into solution as 
it grows), the initial solid must be nucleated.  Further growth 
then takes place off the prexistent solid.  
These conditions were all very nearly satisfied in the 
experimental system studied in chapter X, and were strictly 
enforced in the cellular automata model of chapter Y
resulting in extremely good qualitative agreement with the experiments.  
The experiment was constructed to be as two-dimensional as 
possible to facilitate the rapid extraction of any generated 
heat.  Insights from these 2D systems have led to the PDE's described here, 
and the following derivation will thus also be in 2D, 
although it can easily be treated in 1D or in 3D.  
In the latter case, one needs to consider the temperature 
field and potentially 
any gravity-driven convection in the melt \cite{Poirier:94, Chiareli:94}. 

Before beginning the PDE derivation, it is a good idea to have a picture of 
the dynamics.  When the solid is growing, there appears 
to be a moving envelope, behind which there is an array 
of dendrites, and ahead of which is liquid (Figures 
A and B).
It is important to realize that the impurity B can diffuse 
through the liquid parts but not through the solid parts 
(or at best, only very slowly through the solid).  The 
solid therefore acts as a diffusional barrier to the impurity.  
In fact, the structure/morphology of the growing array 
of dendrites, in the absence of significant fluid flow, 
is expected to dictate the speed and shape of this envelope.  
Within the dendritic array itself, there are two distinct states of matter: 
the actual dendrites made up of solid A; and the liquid portion 
in between these dendrites.  Since the impurity B is rejected 
from the solid into the liquid portion, one expects this 
interdendritic liquid to be higher in concentration of the 
impurity B than the original liquid before any solidification 
began.  In fact, the cellular automata model of 
chapter X shows that the higher the 
concentration of impurity B in this interdendritic liquid, 
the faster the structure propagates into the unsolidified 
liquid.  This means that the more effectively the needle 
structure can entrain/contain the impurity B, the less this 
impurity is able to escape the solid network and get ahead 
of the interface.  With a lower concentration of B  
ahead of the front, the driving force for solidification is 
higher and solid will grow more quickly in the form of a denser 
dendritic network.  
With this in mind, the discussion of the PDE derivation can begin.