The topic of this dissertation is the solidification 
of a binary melt.  The investigation is separated into three portions:  An 
experimental investigation on the $NH_4Cl-H_2O$ system; 
the development of a Cellular Automata code; and the development of a pair 
of coupled partial differential equations governing the 
evolution of an array of dendrites.  Any necessary concepts are reviewed 
in the introduction.

The experimental investigation focusses on the morphological 
transition from ``slow'' $\langle 100 \rangle$ dendrites to 
``fast'' $\langle 111 \rangle$ dendrites.  It is shown how the 
very complicated structures occuring during the transition actually 
have a simple explanation.  The ``slow-to-fast'' transition has 
been previously investigated in the literature, and the 
relationships obtained in those studies can not account for 
the data collected in the present study.  
When ``slow'' dendrites are cooled into the ``fast'' regime, 
a curious stagnation of growth has also been observed.
Additionally, two mechanisms are 
proposed as possible contributions to the order-of-magnitude jump 
in speed at the slow-to-fast transition.  One mechanism is that of a 
``herringbone structure'', and the other is that of a vortical 
fluid flow occuring at the tip of the dendrite.  A relationship 
is also found which further indicates the importance of fluid flow.  

The cellular automata model developed in this dissertation has proven to 
be a valuable tool in gaining insight into the solidification 
process.  The simulated growth is governed predominantly 
by the diffusion of solute and the Gibbs-Thomson effect.  
Solutal diffusion, is accurately treated, diffusing differently through 
liquid than through solid.  The 
interface curvature is approximated using a template method, 
into which crystalline anisotropy has also been introduced.  
Several features were added to explore interface kinetics, 
solute partitioning, and fluid flow due to shrinkage.  
Simulations on a $100 \times 100$ system typically required less than 
a minute on a workstation, and only qualitative agreement 
with the experiments was sought.  

The partial differential equations for the evolution of a 
growing array of dendrites 
are derived taking into account only diffusion.  It is 
explicitly shown how the non-conservative equations conserve 
all of the material in the solidifying system.