Approximations for Vibrational Thermodynamics of Disordered Alloys: Effective Supercells and the Quasiharmonic Method
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ABSTRACT Recent work has suggested that vibrational effects can play a significant role in determining alloy phase equilibria. In order to better understand these effects and the methods used in their calculation, we investigate the vibrational properties of disordered Ni3A1 using the Embedded-Atom Method. We examine the effectiveness of the Special Quasirandom Structure (SQS) approximation, and find that an SQS-8 can accurately represent the vibrational thermodynamics of the disordered state. By the use of Monte Carlo (MC) techniques, we also find that the quasiharmonic approximation becomes less accurate as we approach the melting temperature, but that the accuracy may be extended to higher temperatures by resorting to the MC equation of state giving the specific volume as a function of temperature. INTRODUCTION In the past, the difference of vibrational entropy between ordered and disordered phases at the same composition was often neglected when performing ab initio calculations of alloy free energies, since such differences were considered to be small compared to the corresponding configurational entropy difference. It came therefore as somewhat of a surprise when Fultz and co-workers at Caltech [1, 2] found experimental evidence for vibrational entropy differences of roughly the same order as the configurational. A well-studied example is the technologically important Ni 3Al compound, a superstructure of fcc, generally denoted by the symbol L12. The corresponding fcc disordered phase does not exist at equilibrium, but has been prepared experimentally by laser quenching [3], vapor deposition [1] onto cold substrates, and by mechanical alloying [2, 4]. Specific heat and neutron diffraction measurements indicated large entropy differences as summarized in Table I. Such large experimental values pose several problems of a theoretical and practical nature: are these effects real, or are they artifacts of the method of preparation of the disordered state; if real, what are they due to physically, and how might one take vibrational free energy into account in ab initio calculations? In an attempt to answer these questions, we have calculated the vibrational density of states (DOS) and corresponding thermodynamic functions of the ordered and disordered states of NiAl in the quasiharmonic approximation [6], employing the Voter-Chen Embedded-Atom Method (EAM) potential [7]. In a paper actually preceding some of the experimental work of the Caltech group Ackland [5] performed 175 Mat. Res. Soc. Symp. Proc. Vol. 481 © 1998 Materials Research Society
Table I: Values of AS in units of kB. AS 0.3 ± 0.1 [1] 0.2 ± 0.1 [2] 0.29 (5] 0.27 [6]
Method Experiment: Experiment: Calculation: Calculation:
specific heat, etc. Neutron diffraction second moment potential EAM potential
5.0
4.0 S3.0 E 0
.
2.0 1.0 0.0
0.0
1000.0
500.0
1500.0
T (K) Figure 1: Specific heat at constant pressure (isobaric specific heat) for L1 2 Ni3 AI. The solid line is the present calculation, the symbols are experimental data from Ref. [9]. simil
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