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Focus of this work is the simulation of microstructure formation for a ternary eutectic alloy. We apply a general multiphase/multicomponent phase-field model and calculate the microstructure evolution. The required thermophysical data are extracted from a database for the eutectic alloy AlAgCu. Directional solidification for an alloy with eutectic composition is simulated in 2D and 3D. For lamellar solidification the stacking sequence of the three solid phases is a further degree of freedom which does not occurs in binary eutectics. Starting with varying initial sequences of solid phase nuclei, stable lamellar growth with different solid/liquid interface undercoolings as well as tilted lamellae can be obtained in 2D. The relationship between calculated interface undercooling and the wave length of the lamellar structure is not unique. Different stacking sequences of the same wavelength can show different interfacial undercooling. 3D simulations reveals a fibrous like growth structure similar to experimental findings for AlAgCu. In 3D structures quadruple points with a four phase equilibrium appear which are a further qualitative difference compared with lamellar morphologies.