Monte Carlo Simulations of Phase Transformations in Fe-Cr Alloys
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MONTE CARLO SIMULATIONS OF PHASE TRANSFORMATIONS IN Fe-Cr ALLOYS L. Reinhard and P. E. A. Turchi Lawrence Livermore National Laboratory, Condensed Matter Division (L-268), P. 0. Box 808, Livermore, CA 94550 ABSTRACT Phase stability properties of bcc based Fe-Cr alloys are examined in the framework of the first-principles KKR-CPA-GPM formalism and Monte Carlo simulations. For Fe-rich alloys, ordered configurations are found stable with respect to the random state of the alloy, but unstable with respect to the pure Fe and Cr metals. The results are compared with the ones obtained by using energy parameters extracted from experimental diffuse scattering data. INTRODUCTION The Fe-Cr alloy system, prototypical for highly alloyed stainless steels, displays a complete solubility at elevated temperature with a bcc crystalline structure and an overall tendency towards phase separation which leads to a miscibility gap at lower temparatures [1]. However, diffuse neutron scattering experiments [2] on ferromagnetic, Fe-rich polycrystalline alloys show that the system exhibits a tendency towards ordering for Cr concentrations below -.. 9 at.%. In the following we present results of first-principles calculations of phase stability in bcc Fe-Cr. We have chosen the Monte Carlo technique [3] rather than the cluster variation method (CVM) [4] as a computational tool in calculating bcc-based Fe-Cr phase equilibria because it readily allows the incorporation of more distant interactions which would require clusters of considerable sizes within the CVM. The energetic parameters employed in the Monte Carlo simulations were obtained using the generalized perturbation method (GPM) [5, 6] implemented within the first-principles multiple scattering formalism of the Korringa-Kohn-Rostoker coherent-potential-approximation (KKRCPA) [7]. In the GPM, a perturbation expansion of the band energy is performed with respect to concentration fluctuations around the completely random state, as specified by the CPA. The energy of formation AEfor,, of any configuration of the alloy can then be expressed as the sum of two terms: (1) the energy of mixing AEm,;. which is associated with the creation of the random CPA medium from the pure elements and (2) the ordering energy AEod which represents the energy difference between a particular configuration and the random configuration. Any given configuration can be uniquely specified by a set of occupation numbers pf, where p, = 1 if the lattice site n is occupied by a species i and p*= 0 otherwise. For a binary A 1-,B, alloy one then obtains (with p, = pB = 1 - p): A Efor,,({pn}) =
A E,,i(c) + AEo,d({p.})
(1)
where
AEmz(c)
= EtPA (c) -(1--c)EA
-
cEj,
(2)
is given as the difference between the total energies of the CPA medium and the pure elements. The expansion of the ordering energy is written as follows: AEOd({P
= -EVm(pmc)(p 1) mn
c)+-
Z
V1mn(p, - c)(pm - c)(p - c) +..
IMn 3
Mat. Res. Soc. Symp. Proc. Vol. 291. @1993 Materials Research Society
(3)
408
The concentration-dependent expansion
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