Thermodynamic Properties of Coherent Interphase Boundaries in Substitutional Fcc Alloys
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thermodynamic properties of coherent IPBs between phases with FCC-based crystal structures in the Ag-Al and Al-Li alloy systems. Specifically, in Ag-Al we consider IPBs between two disordered phases: the oa-Al (solid solution) and e-Guinier-Preston-zone (GP-zone) phases (see for example [3]). In Al-Li by contrast, calculations have been performed for IPBs between disordered oa-Al and the ordered 8'A13Li phase with an L1 2 crystal structure. The cluster variation method (CVM) [4, 5] and low-temperature-expansion (LTE) [6] statistical-mechanical techniques have been used to calculate interphase energies and equilibrium composition profiles as a function of temperature for IPBs with high-symmetry crystallographic orientations. The energetic parameters required as input for these calculations have been obtained in two ways: in the case of Al-Li they were taken from the phase-diagram-fitting work of Garland and Sanchez [7], whereas for Ag-Al they were derived from the results of first-principles total energy calculations. The results presented below for Ag-Al provide an example demonstrating how ab-initio techniques can be applied to the study of finite-temperature IPB properties in alloys. COMPUTATIONAL APPROACH Let S2(Ap, T) denote the value of the grandpotential [8] in a binary alloy at temperature Tand chemical field (defined here as half the difference between the chemical potentials for the two atomic species) Ayt. S20 and Aluo will denote the values of the grand potential and chemical field corresponding to bulk thermodynamic equilibrium between the two alloy phases of interest. Additionally, Qhkl(AyO, T) will denote the value of the grand potential for an inhomogeneous
system containing separate spatial regions of these two phases together with coherent IPBs 281 Mat. Res. Soc. Symp. Proc. Vol. 398 ©1996 Materials Research Society
between them oriented along (hkl). If one can calculate f2hkl(A/o, T) and ,20 (Apto, T) a value of the interphase energy Yhkl can be derived from the following equation [8]: Q2hkl( AIo0, T )--Q-o( Alto, T )= Yhkl Ahkl
()
where Ahkl is the total cross-sectional area associated with the IPBs. For the purpose of calculating the values of the various thermodynamic potentials entering Eq. (1), the CVM [4, 5] and LTE [6] techniques have been used in the present study. In the CVM variational estimates of grand potentials are determined from minimizations of a free energy functional which explicitly takes into account contributions to the enthalpy and entropy arising from correlation between atoms within some "maximal" cluster of lattice points. The CVM free energy functional can be written in terms of a linearly independent set of correlationfunctions [5] defined as generalized short-range order parameters for multisite clusters. At low temperatures minimization of the CVM free energy functional becomes numerically ill-conditioned due to the fact that the entropy is expressed in terms of logarithms of cluster probabilities, the values of which approach zero or one at low T. At these low t
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