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Lever Rule
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Back Diffusion
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Working Group
Ursula
Kattner
Bill Boettinger
Dilip Banerjee

Simple Solidification Calculations
Two simple models describe the limiting
cases of solidification behavior. For solidification obeying the lever rule at each temperature during cooling,
complete diffusion is assumed in the solid as well as in the liquid,
thus all phases are assumed to be in thermodynamic equilibrium at
each temperature during solidification. In comparison,
solidification following the Scheil path,
where diffusion in the solid
is forbidden, produces the worst case of microsegration with the lrowest
final freezing temperature. Thermodynamic equilibrium exists only
as local equilibrium at the liquid/solid interface. Modeling of
real
solidification behavior requires the incorporation of a kinetic
analysis
of liquid and solid (back) diffusion. In a first approach to the
kinetic analysis of solidification, the assumption of a
completely mixed liquid is retained and back diffusion is considered to occur only in the primary phase.
The necessary phase equilibria information for all solidification path
calculations can be obtained from thermodynamic descriptions of the
alloy system, such as the CALPHAD
method. The calculation gives not only information about the
solid phases formed during solidification but provides also information
about thermodynamic quantities, such as enthalpy.
Computer Programs
Kraft and Chang (JOM 49 (12) (1997) 2028) gave a summary of
solidification modeling efforts, including those that employ
thermodynamic phase equilibria calculations.
For our work we found it convenient to develop simple programs for
lever rule, Scheil and back diffusion calculations based on the PMLFKT code developed
by H.L. Lukas, MPI Stuttgart (Lukas et al., CALPHAD 6 (1982)
229251). This code was used to calculate solidification paths
for a variety of alloys, such as superalloy or solder alloy systems.
Results from these calculations have been published in a number
of articles.
Examples for the lever
rule and Scheil solidification paths of a NiAlTa
alloy and SnBiPb
alloys are shown.
Microstructure Simulations
Simulations of the microstructure have been carried out by the CTCMS Phase Field
Modeling
Tools Working Group.
