Phase Field Analysis of Eutectic Breakdown
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INTRODUCTION
PHASE field modeling has been used to study both single-phase[1–4] systems (single-phase models have only one solid phase with the liquid and hence require only a scalar phase variable /) such as the growth of dendrites and multiphase systems[5–9] such as eutectic, peritectic, and monotectic solidification. The general concept of all of these models is that rather than a sharp solid-liquid (or solid-solid) boundary, these interfaces between different phases are smooth and continuous. The amount of a particular phase present is given by the phase field variable /. Multiphase systems contain three or more phases and, because of this, require one-phase variable for each phase often written as a vector /, where /i represents the amount of phase i. Eutectic microstructures form in alloys when two or more phases grow simultaneously in a cooperative manner. The majority of industrial casting alloys are eutectic or near eutectic in composition. There is a wide variation in the features of eutectic systems, the number of solid phases present limited to the number of alloy components. Recently, there has been an increased interest in ternary eutectics for lead-free solder. As well as the number of phases present, the amount of each phase and the entropy of fusion of the phase will significantly affect what structures are formed. With roughly equal amounts of each phase preset, lamellarlike structures are most prevalent. The theory behind the growth of these was described by Jackson and Hunt,[10] while low entropies of fusion will tend to produce regular morphologies. If the amounts of each phase vary significantly, then the structures will tend to be more fibrous and higher entropies will tend to make the boundaries between phases more faceted. Eutectics can grow in a similar manner to constrained growth J.R. GREEN, Gratuate Student, P.K. JIMACK, Professor, Institute of Materials Research, and A.M. MULLIS, Professor, School of Computing, are with the University of Leeds, Leeds LS2-9JT, United Kingdom. Contact e-mail: [email protected] This article is based on a presentation made in the symposium entitled ‘‘Solidification Modeling and Microstructure Formation: in Honor of Prof. John Hunt,’’ which occurred March 13–15, 2006 during the TMS Spring Meeting in San Antonio, Texas, under the auspices of the TMS Materials Processing and Manufacturing Division, Solidification Committee. Article published online June 13, 2007. 1426—VOLUME 38A, JULY 2007
dendrites, without the dendritic arms; in this situation, the eutectic equivalent of the primary arm spacing is roughly 1/10 that of the dendritic case. This means that the specific surface area between eutectic phases is significantly greater than for dendritic structures. Additionally, at higher undercooling values, the lamellar eutectic structures have been noted to break down[11] and roughly spherical grains of one phase form within the second phase; furthermore, the growth of eutectics and spherical grains within dendrites has been reported.[12] For a given undercooling, DT
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