APPLICATION OF PHASE DIAGRAMS IN SOLIDIFICATION AND CASTING PROBLEMS
William J. Boettinger
The analysis of problems involving the casting of alloys requires a
complex blend of fluid mechanics, heat flow, chemical diffusion and solid
mechanics. However, the phase diagram, especially when applied to the character
of a material at a given position and time (local equilibrium), provides
the basic constitutive relation regarding the physical state of the alloy.
The information from a phase diagram must be placed upon a framework of
mass, momentum and energy balance equations that describe the kinetics
of a given situation. Since 1950, great progress has been made in the analysis
of solidification problems using this concept of local equilibrium.
A major use of phase diagrams comes about in the prediction of the degree
of microsegregation and inclusion (or second-phase) formation. Modern approaches
to the microsegregation problem accept the inevitable difficulty of solid
state chemical diffusion compared to the liquid state and replace mass
balance accompanied by complete diffusion, i.e., the "lever rule" with
mass balance accompanied by incomplete diffusion (the Scheil equation).
This equation when coupled with the partition coefficients of the various
solutes can be used to estimate the liquid composition (and solid composition)
as a function of fraction solid or, equivalently, the fraction solid as
a function of temperature. Partition coefficients can be read directly
from the phase diagram. Additionally, the solidification path can be calculated
and the prediction made of the fraction solid when the liquid can begin
formation of second-phase particles, inclusions or eutectics. Such liquid
compositions are directly read from phase diagrams. This analysis yields
an estimate for the volume fraction of all the phases that will occur in
the cast structure, as well as the magnitude of concentration gradients
which will exist in the cast solid. In the prediction of macrosegregation,
the mass balance is augmented by fluid flow terms to predict local increases
or decreases in the fraction of second-phase particles and concomitant
shifts in the average composition in that location. Such analysis has been
successfully applied to describe chill zone, centerline, cross section
change and "V" segregation. Data for this require parameters read directly
from phase diagrams.
The prediction of fraction solid as a function of temperature described
above, when coupled with a heat flow analysis of castings, yields the thickness
of the mushy zone or liquid-solid region in castings. The thickness of
the mushy zone is very important for the prediction and control of microporosity
and gas porosity, as well as hot tearing characteristics of castings. The
former are generally caused by the difficulty of fluid flow through the
mushy zone to feed solidification shrinkage, whereas the latter is caused
by the formation of large thermal contraction stresses acting on an insufficiently
strong solid in a mushy zone. Gas porosity analysis also requires data
regarding the solubility of gases in metals. Mushy-zone thickness has also
been associated with grain multiplication and the formation of equiaxed
zone of ingot castings.
Theories of dendrite coarsening, which lead to predictions of dendrite
arm spacings in castings as a function of local solidification time, rely
on the partition coefficient and liquidus slope from the phase diagram
as parameters. Dendrite arm spacings are correlated to the strength of
castings as well as to homogenization times required for subsequent heat
treatment of castings.
The analysis of cast structures produced by rapid solidification has
presented a new challenge to the usefulness of phase diagrams. Here the
phase diagram is best viewed as a representation of the thermodynamically
possible states of alloys. It will, in general, not be able to describe
the exact state of the system, but it will place bounds on the temperatures
and compositions at which certain transformations can occur. In this situation,
the phase diagram is most useful when the underlying free-energy functions
for the phases are known or are estimated from the existing measured phase-diagram
features. These functions can then be used to construct metastable extensions
of phase boundaries and construct new phase diagram boundaries such as
a To curve, which place a bound on partitionless crystallization.
Both of these are useful in the analysis of enhanced solubility during
rapid solidification. Such constructions must be done carefully and compared
to experiments, for at best they are interpolations and at work extrapolations.
They do provide, however, an extremely useful starting point for the analysis
of rapid solidification problems.
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