In order to obtain the primary phase field of a given phase of interest, the first step is to determine all compounds that are in equilibrium with the phase of interest. Next, the multiphase compatibility regions involving this phase are identified. For example, in a ternary system, one would determine all 3-phase compatibility regions involving the ternary phase of interest, or 4-phase compatibility regions in a quaternary system, or five-phase volumes in a five-component system, etc. For each of the compatibility regions, the onset melting temperature is then determined. The set of compositions of the first liquids formed in each of these regions will determine the outline of the primary phase field.
The compositions of the multi-phase volumes were modeled using a computational
geometry technique based on forming a convex hull from the experimentally
determined chemical compositions. This mathematical notion of a the
convex hull, the smallest convex set of points that contains all of the
given data points, has been used in many physical science applications.
In this case, the convex hull represents the extent of the compositional
volume. This numerical procedure results in a well-formed, hyper-volume
in 5-dimensional space. It is important to note the requirement that chemical
compositions sum to unity results in the loss of one degree of freedom.
In general the convex hull is defined by the matrix equation Ax +
b < = 0 where A is a matrix whose rows define
the unit normal vectors to the faces of the convex hull. Each element of
the vector b defines the proximity of the given face to the
origin of the multi-dimensional space. The vector x gives
the coordinates corresponding to a given point. The matrix A
and vector b can be found using a so-called "sweeping algorithm''.
When the procedure is complete, the matrix A has dimension
k × 5 and the vector b has dimension
k where k is the number of faces in the convex
hull.
This equation can be used to determine if a given composition lies
inside the (Bi,Pb)-2223 primary phase field. If Ax-b < 0,
the particular composition lies inside the crystallization volume. If the
value of all components is zero, the composition is on the surface; and
if the value of any one component is positive, it is outside the volume.
Compositions falling outside this volume should be avoided for the processing
of the 2223 phase, as they would be expected to produce large primary crystals
of unwanted phases.