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Phase Diagrams 
& Computational Thermodynamics

Simple Solidification Calculations

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Lever Rule
Back Diffusion

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) 20-28) 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) 229-251).  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 Ni-Al-Ta alloy and Sn-Bi-Pb alloys are shown.

Microstructure Simulations

Simulations of the microstructure have been carried out by the CTCMS Phase Field Modeling Tools Working Group.

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Last modified: 15 August 2017