Calculations of Alloy Phases with a Direct Monte-Carlo Method
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CALCULATIONS OF ALLOY PHASES WITH A DIRECT MONTE-CARLO METHOD
J. S. FAULKNER,* YANG WANG,* EVA A. HORVATH,*and G. M. STOCKS** *Alloy Research Center and Department of Physics, Florida Atlantic University, Boca Raton, FL 33431 **Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831
ABSTRACT A method for calculating the boundaries that describe solid-solid phase transformations in the phase diagrams of alloys is described. The method is first-principles in the sense that the only input is the atomic numbers of the constituents. It proceeds from the observation that the crux of the Monte-Carlo method for obtaining the equilibrium distribution of atoms in an alloy is a calculation of the energy required to replace an A atom on site i with a B atom when the configuration of the atoms on the neighboring sites, K,'is specified, 8HK(A-4B) = EB(K) -EA(K). Normally, this energy difference is obtained by introducing interatomic potentials, vij, into an Ising Hamiltonian, but we calculate it using the embedded cluster method (ECM). In the ECM an A or B atom is placed at the center of a cluster of atoms with the specified configuration K, and the atoms on all the other sites in the alloy are simulated by the effective scattering matrix obtained from the coherent potential approximation. The interchange energy is calculated directly from the electronic structure of the cluster. The table of 8HK(A----B)'s for all configurations K and several alloy concentrations is used in a Monte Carlo calculation that predicts the phase of the alloy at any temperature and concentration. The detailed shape of the miscibility gaps in the palladium-rhodium and copper-nickel alloy systems are shown.
INTRODUCTION Phase diagrams of alloys are extremely useful to materials scientists who wish to design processes to improve the physical properties of materials through control of their micro structure. It has long been a goal of alloy theory to use electronic structure calculations to predict phase stabilities from first principles. Since most of the binary phase diagrams are known, theoretical predictions will be more useful when they can make reliable predictions about ternaries and quatemaries, but the early results that we will discuss here indicate that they can provide a useful feedback for experimental studies of binaries. A number of theories for predicting the structure of an alloy as a function of concentration and temperature are based on the assumption that its energy can be obtained from an Ising model Hamiltonian H
Ej[VApA A + VBBBB 2H= j iP ii i Pij + vAB( A B+ BA ij i Pj PiP.)]
Mat. Res. Soc. Symp. Proc. Vol. 291. Q1993 Materials Research Society
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where p1A is one if the atom at site i is an A atom, and zero if it is not. Theorists have made efforts over the years to write the exact expression for the total energy of an alloy in the form of this Hamiltonian,"- and to calculate the interatomic potentials, V•1, from the electronic structure. The major reason for the popularity of this approach is
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