| Equation of state (EOS) is important for understanding the mechanical and ther-modynamic behaviors of materials, particularly the dynamic responses under high-pressures, and always being the center for experimental and theoretical researches.Along with the advances of the capability of modern computers and first-principlesmethods based on the density functional theory, it has become prevalent to calculateEOS for materials at quantum level for better understanding of their properties. How-ever, for partial and complete disordered alloys, it is very difficult to obtain their EOSdirectly with ab initio calculations, which cumbers this purpose by and large.With cluster variation method (CVM) and cluster expansion method (CEM), wepropose an EOS model which employs CEM for accurate energy and CVM for theexplicit configurational entropy contribution of alloys. Taking substitutional Ni-Al sys-tem as example and making use of plane waves plus pseudo-potentials calculations, itis shown clearly that at temperature T = 0 K much better results are reproduced withthe proposed EOS model than the widely used mixture model. Variations of phasestability of Ni-Al alloys with increasing hydrostatic pressure is also calculated, reveal-ing that the stability of alloys is always enhanced by pressure, and a structural transi-tion from FCC to BCC for elemental Al above 260 GPa. At finite temperatures, bothorder-disorder transition temperatures Tc of L10 NiAl and L12 Ni3Al satisfy the Simonmelting equation. The calculated results of shocked Ni3Al also shows that Tc increaseswith pressure, but has a smaller increment by comparison with melting temperatureTm. This implies that the phase boundaries among ordered, disordered and liquid stateshave been changed by pressure, and it becomes possible to measure the high pressureTc directly excluding in?uences due to melting process to evaluate the modern DFT-based alloys theory fairly. Two phase diagrams are calculated with spin-polarized andnon-polarized effective cluster interactions (ECI) by CVM, with which we clarify theconfusion about the measured formation enthalpies at different temperatures.The adiabatic compression of Ni3Al, namely, its Hugoniot, is calculated withCEM+CVM method. A shock induced order-disorder transition is observed at 205 GPa,corresponding to 3750 K in temperature. Surprisingly, though the magnitude of config-urational contribution is always smaller than that of lattice vibrations, it is much largerthan electronic excitations contribution within a large range of temperature, especiallythe sudden increase of pressure about 8 GPa at shocked Tc. In general, the electronicexcitation free energy of partial disordered states almost cannot be computed with abinitio approaches directly. A set of effective cluster density of state (ECDOS) is de-veloped for this purpose. It is pointed out that the CEM on (free) energy is actuallya CEM on the corresponding electronic and phonon DOS, and all needed quantitiesfor EOS theory can be generated from these ECDOS. Furthermore, it is shown thatthe convergence of CEM on these thermodynamic quantities is much better than EC-DOS itself. As to alloys, it is important to study EOS related quantities as functions ofcomposition for understanding its behavior deeply. We discover that not only the heatcapacity, but also the heat expansion coefficient, Gru¨neisen parameter and the pressurecoefficient have the abnormal "W-shape" around stoichiometric concentrations. Thisstructure originates from the antisite defects of atoms driven by configurational entropyand related to the elements of the inverse of Hessian matrix corresponding to correla-tion functions. Though there is no "W-shape" occurs on compressibility and bulk soundvelocity curves as functions of composition, they are also affected by short and long-range order by and large. This kind of strong dependence of EOS and related quantitiesof alloys on order parameters has not been received much attention before and believedis helpful for understanding subtle thermodynamic properties of order-disordered sys-tems more deeply, e.g., the heating stress in alloys and the propagation of seismic wavesin mineral crystals.Finally, the proposed CEM+CVM EOS is effective and accurate for alloys andcompounds. Some novel understanding has been achieved. However, consideringthe convergence difficulty of CVM iteration algorithms when complicated ECI are in-volved, we recommend CEM+Monte Carlo simulation as a substitute. To improveCEM, a simple model for local lattice distortions is suggested with two kinds ofschemes. The Ising model is generalized to single particle case and shows it is pos-sible to describe quantum motions with discrete space-time as the underlying lattice.We expect these understanding can motivate some further progress more or less.
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