Experimental investigation of solidification can be very 
costly and time-intensive.  Since such investigations are 
required to optimize the solidification processes and their 
final products, good analytical and numerical methods are 
desired to help reduce the time and cost of this optimization.  
In Chapter X of this dissertation, a cellular automata 
model for solidification is discussed, and in chapter Y 
some PDE's are developed for the growth of a network of dendrites.  
There are, of course, many more analytical and numerical tools 
available, and a short list of references is given in this section.  

The varied approaches to modelling solidification have a 
rich history, and the main processes which are taken into account are 
diffusion as well as any reactions which may be taking place in the 
bulk phases or at the interfaces separating the phases.  Even 
relatively simple systems can exhibit surprising behaviour.  One 
of the earliest reported such phenomena is that of Liesegang 
rings \cite{Liesegang:1897}, in which concentric rings of a new phase 
form in an originally homogeneous system as a result of reactions 
which take place as various materials diffuse through the system.  
The rings are centered about a droplet of material added to the 
originally homogeneous system (e.g. a plate of agar gel, or a dish 
of quiescent liquid) and develop as the droplet and any reaction 
products diffuse outwards.  An extensive treatment of similar 
phenomena in such ``reactive-diffusive'' systems is given by 
Kirkaldy et al. \cite{Kirkaldy:87}, encompassing a vast array of 
patterns formed in solidifying and already solid systems.  These 
types of considerations should be recalled with the appearance of 
any regular patterns/oscillations in reactive-diffusive systems\footnote{In 
addition to their occurrence in solidification (as seen in this work, 
\cite{Coriell:83}, and \cite{Trivedi:}), a 
wealth of such phenomena arises in chemical and biological systems.}.

One of the primary points of interest is the motion of the 
solid-liquid interface during solidification.  For simple 
geometries, a thorough review of methods to treat the shape, 
speed, and stability of sharp interfaces is given by 
Langer \cite{Langer:80}.  This work is extended for arrays of 
dendrites in Warren's dissertation \cite{Warren:92}.  One 
attempt to efficiently compute the motion of such interfaces 
has been to stipulate that diffusion occur only within a thin 
boundary layer ahead of the interface.  This method is called 
the ``Boundary Layer Method'' (BLM) \cite{Ben-Jacob:84}, and 
its approximation becomes better justified with increasing 
interface speed.  Another approach, called the ``level set 
approach'' \cite{Merriman:94}, follows a continuous field variable 
(i.e. a field, such as concentration, which takes on real 
values as opposed to discrete values) and defines the 
``interface'' as the set of all points having a certain value 
of this variable (i.e. the ``level set'' with this certain 
value).  Many times, approximate rules governing the local behaviour 
of the level sets can be derived, and then applied to 
yield the evolution of the entire system.  Such continuous 
phase variables have also been employed in studying interfacial 
attachment kinetics \cite{Cahn:64}.

Another method, which defines the interface as a certain 
level set of a real-valued field, is called the ``Phase-field 
Method'' \cite{Wheeler:92, Harrowell:86}.  This field describes 
the phase, in which the system finds itself at a given point 
in space, and is of course called the ``phase-field''.  It 
typically takes on values between 0.0 and 1.0.  In this method, 
the system's total energy is written as the integral of an 
energy functional over the entire spatial domain.  This 
energy functional approximates the contribution to the total 
energy of an infinitesimal element of the system, and is often 
called a Ginzburg-Landau functional.  The local behaviour of 
any fields, which enter into the calculation of the total 
system energy\footnote{For example, the phase-field contributes 
to the total system energy, in that the different phases have 
different thermodynamic energies, and also the interface 
between two phases has an interfacial energy associated with 
it.}, is obtained by assuming that the fields will change 
locally to minimize their contribution to this total energy.  
Such considerations yield a (coupled) set of partial 
differential equations governing the evolution of the 
different fields, which can then be solved numerically.  
A similar approach has been taken by Almgren \cite{Almgren:92}, 
in which the system is propagated in such a way  that, of all 
the configurations possible at the next timestep, the one 
configuration is chosen, which minimizes the total energy of the system.  

To investigate the interface at a molecular level, two 
methods stand out.  The first is a Monte Carlo approach (Gilmer 
and Jackson in \cite{Kaldis:76}), which evolves a very simple model 
many times over, and relies on the statistics of large ensembles 
to converge to values describing the properties of 
interest.  The simple model is 
usually a block lattice, similar to the one described in section 
Intro Basics.  The second method is that of molecular 
dynamics, in which atomic and molecular interactions are governed 
by approximate potential functions.  At each timestep, a given 
particle moves according to the net force exerted on it by the 
rest of the system.  Such models do not restrict the 
particle positions to fixed lattice sites.  As a result, 
the importance of both strain and bond distortion at 
the interface becomes apparent.