Configuration Dependence of the Vibrational Free Energy in Substitutional Alloys and Its Effects on Phase Stability
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of the lattice vibrations and the trends with size mismatch and chemical affinity. In this paper, we study model systems for which the vibrational free energy can be computed exactly within the harmonic approximation. This allows us to derive conclusions about the size and origin of the vibrational effect on alloy phase stability. THE VIBRATIONAL FREE ENERGY The effect of the lattice vibrations on the phase stability of substitutional alloys can be studied with an Ising-like lattice model [8, 12]. The only requirement for this transformation is that every microstate of the alloy be uniquely mapped to a configuration of the atoms on a fixed lattice. The difference in time scales between lattice vibrations and substitutions implies that for every substitutional arrangement, there is a well defined vibrational free energy. As our focus is on the vibrational free energy, we ignore the electronic excitations. The lattice Hamiltonian contains the "chemical" energy of the relaxed ground state structure and its vibrational free energy. Therefore, the partition function of the system defined by this lattice Hamiltonian includes both configurational and vibrational degrees of freedom. In the harmonic approximation and for temperatures larger than the characteristic Debye temperatures of the system, the lattice Hamiltonian is [10]: 4 (U4) (6) H(8, T) =Eo(8) + (log(w)) (6)kB T + h224kBT (w2) (6) h2880k0T 3 +
""'
(1)
where 8 labels the configuration of the A and B atoms on the lattice, () symbolizes averages (per atom) over the Brillouin zone, E 0 is the fully relaxed ground state energy, w is the vibrational frequency of a phonon mode, and kB and h are the Boltzmann and Planck constants respectively. The terms that are not dependent on 8 have been left out of the lattice Hamiltonian, as they do not contribute to the configurational thermodynamics. The leading term, (log(w)) kBT, in this high temperature expansion is enough to accurately represent the vibrational free energy for temperatures of the order of, or higher than the Debye temperatures of the system [10]. Since most phase transformations occur in this temperature range, in the rest of the paper we will concentrate on this term. For systems described with classical potentials, the dynamical matrix is easily obtained from the second derivates of the total energy [13]. By diagonalizing this matrix, the values of the frequencies of the normal modes of the system can be obtained and the averages in Eq. (1) are easily computed for a given configuration 8' of A and B atoms. To study the thermodynamic properties of the alloy system, the actual dependence of H on the configuration 8 is parametrized using a cluster expansion technique [14]. The occupancy of site i is labeled by a spinlike variable ai [oi = +1(-1) when a B (A) atom is on site i]. The substitutional state (or configuration) of an N-site alloy, 8, is then an N-dimensional vector of l's and -l's. The alloy Hamiltonian is written as H(8,T) = Z- V0 (T)ao(6),
(2)
where the sum is over al clusters a of lattice po
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