First-principles Study Of Phase Stability Research Articles

Thermodynamic and Kinetic Properties of the Lithium-Silver System.

A carbon-silver anode has recently been shown to suppress dendrite formation in all-solid-state lithium-ion batteries. The role that silver plays in enabling the reversible deposition and stripping of lithium remains unknown. Furthermore, very little is known about the thermodynamic and kinetic properties of Li x Ag1-x alloys. Here, we report on an in-depth first-principles study of phase stability and diffusion mechanisms in the Li-Ag alloy system. We identify two new intermetallic phases that are predicted to be stable in Li-rich Li x Ag1-x alloys with stoichiometries of Li3Ag and Li11Ag2. Our calculations show that the peculiar and highly anharmonic energy surface of pure Li along the Bain and Burgers paths persists upon the addition of Ag to BCC Li. This has important implications for room-temperature phase stability and mechanical properties. We have also performed a systematic study of diffusion mechanisms in the Li x Ag1-x alloy system as a function of alloy concentration x. Diffusion in alloys and intermetallics is mediated by vacancies. High vacancy formation energies are predicted in the Li x Ag1-x alloy, especially in Ag-rich FCC solid solutions. Complex diffusion mechanisms are identified in the B2 and γ-brass intermetallic phases that include two-atom hops and second-nearest neighbor hops. The migration barriers are found to decrease with increasing Li concentration, with predictions of exceptionally low migration barriers of 0.1 eV in the D03 Li3Ag phase.

First-principles study of phase stability, mechanical properties and electronic structure of B2-FeAl intermetallic alloying with La and Ce

The structure stability, mechanical properties and electronic structures of B2-FeAl intermetallics and the influence of La/Ce doping were investigated by first-principles density functional theory calculations. Several models were established. The total energies, cohesive energies, lattice constants, elastic constants, electronic density of states of Fe8Al8 and Fe8Al7X (X = La, Ce) were analyzed. The stability of alloy systems were determined according to the cohesive energy results. The calculated lattice constants of Fe8Al7X (X = La, Ce) were found to be related to the atomic radii of the alloy elements. The calculation and analysis of the elastic constants showed that ductility of B2-FeAl can be improved by the addition of La or Ce, the order of the ductility was as follows: Fe8Al7Ce > Fe8Al7La > Fe8Al8. The results of electronic structures showed that the brittleness of B2-FeAl was mainly due to the orbital hybridization of the d orbital of Fe and p orbital of Al, showing typical characteristics of a valent bond. Specifically, the mechanism for improving the ductility of B2-FeAl is that d orbital of La/Ce is mainly involved in the hybridization of B2-FeAl.

First-principles study of phase stability, elastic and thermodynamic properties of AlCrFeNi medium-entropy alloys

We have applied the first-principles method to predict the phase stability, elastic and thermodynamic properties of ternary (AlCrFe, AlCrNi, CrFeNi and AlFeNi) and quaternary (AlCrFeNi) medium-entropy alloys (MEAs). Both body-centered cubic (BCC) and face-centered cubic (FCC) disordered structures are described using the special quasi-random structures (SQSs) technique. AlCrFe, AlCrNi and AlCrFeNi are favorable in single BCC structures, while CrFeNi is likely to form a single FCC structure. Addition of Ni help stabilizes AlCrFeNi quaternary MEAs. Al and Cr addition are in favor of the formation of BCC AlCrFeNi. Addition of Al, Cr and Ni reduce the resistance to volume deformation for quaternary AlCrFeNi due to the effect of the average number of [Formula: see text]-electrons. The ternary MEAs have better resistance to shear deformation and elastic stiffness than quaternary AlCrFeNi. In addition, all the considered MEAs embody elastic anisotropy and AlCrFeNi are predicted to be ductile behavior. Finally, volumetric thermal expansion coefficient, constant volume heat capacity, vibrational and electronic entropy, and Helmholtz free energies of stable BCC AlCrFeNi, BCC AlCrFe, BCC AlCrNi and FCC CrFeNi are calculated using the Debye–Grüneisen model in temperature ranging from 0 to 1200 K to elucidate the relationships between thermodynamic parameters and temperature.

First-principles study of phase stability and temperature-dependent mechanical properties of (Cr, M)23C6 (M = Fe, Mo) phases

Although Cr-rich (Cr, M)23C6 (M = Fe, Mo) carbides as ineluctable precipitates dramatically affect the mechanical performances of advanced materials such as refractory steels and superalloys, their high-temperature material properties are difficult to be determined in experiments. In the present work, the first-principles method is applied to comprehensively investigate the phase stability and thermodynamic and temperature-dependent mechanical properties of Cr23-xMxC6 (M = Fe, Mo; x = 0–23). The results indicate that Cr23-xFexC6 presents thermodynamic phase stability under a broad occupying concentration of Fe (i.e., x = 1, 3, 9, 12, 13, 14, 15). Amongst which, the Cr8Fe15C6 with 4a, 8c and 48 h sites co-occupied by Fe is the most stable. Whereas Cr21Mo2C6 is the only stable phase among Cr23-xMoxC6. The mechanical properties of (Cr, M)23C6 are found to exhibit a strong doping composition-position dependence and a negative temperature dependence. Doping a low concentration of M leads to an increase of the elasticity of Cr23C6. The occupation of 4a or 8c sites alone plays an effective role in enhancing mechanical properties. The present work provides necessary information for understanding the heterogeneous deformation process and the fatigue crack initiation mechanism for alloys with the precipitate of M23C6 inclusions.

Potassium Ordering and Structural Phase Stability in Layered KxCoO2

The recent surge of interest in K-ion batteries necessitates a fundamental understanding of phase stability and K ordering tendencies in common electrode materials. We report on a first-principles study of phase stability in layered KxCoO2 (0 ≤ x ≤ 1) in the O1/P3/O3 family of host structures and identify K ordering preferences within each host. We find that the P3 host is stable at intermediate K concentrations and exhibits a multitude of hierarchical orderings characterized by well-ordered domains separated by antiphase boundaries. We also predict the stability of a new family of layered structures at high K concentrations that are highly distorted and host both octahedrally and prismatically coordinated K within each intercalation layer.

First-principles study of phase stability, electronic and mechanical properties of plutonium sub-oxides.

The formation energies (ΔHf) of fluorite PuO2, α-Pu2O3 and sub-oxides PuO2-x (0.0 < x < 0.5) are determined from atomic scale simulations based on density functional theory (DFT) employing the generalised gradient approximation (GGA) corrected with an effective Hubbard parameter (Ueff). The variation of structural and electronic properties of PuO2 and α-Pu2O3 is determined while ramping up Ueff from 0 eV to 5 eV (Ueff-ramping method) to treat the presence of metastable magnetic states and to determine the most suitable Ueff value matching the experiments. The GGA+U calculated lattice parameter variation as a function of stoichiometry (a(x)) for PuO2-x shows a positive volume of relaxation and an almost linear variation presented by the relation a(x) = a0- 0.522738x, where a0 is the equilibrium lattice parameter of PuO2. The GGA+U calculated ΔHf values of PuO2-x lie above the tie line connecting the ΔHf of PuO2 and Pu2O3, and with decreasing O/Pu ratio, the stability of the sub-oxides increases. The crystal and electronic structure analysis of the oxygen vacancy in PuO2 shows outward anisotropic relaxation of four Pu atoms around the vacancy site. The electronic charges within the Wigner-Seitz sphere around these Pu atoms show an overall gain of only (0.12-0.22)e per Pu atom, signifying an incomplete localization of charges. Finally, the GGA+U calculated single crystal elastic constant values decrease continuously with decreasing O/Pu ratio from 2.0 to 1.5. The rate of decrease of the average C11 is almost 11-15 times higher compared to the rate of decrease of C12 and C44.

First-principles study of phase stability and elastic properties in metastable Ti-Mo alloys with cluster structure

ABSTRACTThis paper provided a novel approach for evaluating phase stability and elastic properties in metastable Ti–Mo alloys with low Mo content by first-principles combined with cluster structure. In 54-atom body-centered-cubic supercell by substituting Ti atoms with 2–7 Mo atoms (7.1–23.0 wt% Mo), individual cluster structure of β-phase was constructed by ‘-Mo-Ti-Mo-’ cluster unit having the lowest cohesive energy. The distorted supercell was more stable than undistorted one at a low Mo content. With increasing Mo content, the density of state at Fermi level decreased, and bonding electron number increased, indicating β-phase stability was gradually promoted. Tetragonal shear elastic constant (C′ = (C11 – C12)/2), shear modulus (G111) and anisotropy factor (A = C44/C′) exhibited a fluctuation with Mo addition, while the change trend of A was opposite to C′ and G111. Calculated Young’s modulus exhibited similar changing trend to the C′, implying that the softening of C′ resulted in low Young’s modulus of β-phase. Measured Young’s modulus exhibited significant difference from calculated one, which was mainly caused by formation of α″-martensite and ω-phase. The values of C′, G111 and A were considered to associate with not only elastic properties of β-phase itself but also transition from β-phase to α″-martensite and/or ω-phase.

First-principles study of phase stability and solubility in Al-(Cu, Zr) alloys

ABSTRACTThe solubility limit of elements plays a crucial role in alloy designing, as well as in the understanding of kinetic processes in alloys. Using first-principles calculations, we calculated the ground states of Al-Cu and Al-Zr compounds, and predicted the stable and metastable structures of these two binaries. The results indicate that the stable structures for Al-Cu compounds are Al2Cu(cF12), AlCu(mC20), Al4Cu9(cP52) and AlCu3(tI8), and those for Al-Zr binary are Al3Zr(tI16), Al2Zr(hP12), Al3Zr4(hP7) and AlZr3(cP4). Based on the statistical-thermodynamic theory of dilute lattice gas, we presented a systematic series of calculations of the solubility limits of Cu and Zr in Al as a function of temperature. It is found that the solubility of Cu is much higher than Zr over the whole investigated temperature range.

First-Principles Study of Phase Stability and Solubility in Fe-RE (Y, La, Ce) Alloys

The solubility limit of elements plays a crucial role in alloy designing, as well as in the understanding of kinetic processes in alloys. Using first-principles calculations, we calculated the ground states of Fe-RE compound, and predicted the stable and metastable structures of Fe-Y and Fe-Ce. The results indicate that the stable structures for Fe-Y compounds are Fe 12 Y.tI26, Fe 17 Y 2 .hP38 and Fe 2 Y.cF24, and those for Fe-Ce binary are Fe 17 Ce 2 .hP38, Fe 19 Ce 5 .hR24 and Fe 2 Ce.cF24. Based on the statistical-thermodynamic theory of dilute lattice gas, we presented a systematic series of calculations of the solubility limits of RE in α -Fe as a function of temperature. It is found that the solubilities are sorted in the following order: S La > S Ce > S Y , corresponding to the trend in the solubility formation enthalpies: H sol (La)< H sol (Ce)< H sol (Y).

Phase and structural stability in Ni-Al systems from first principles

We report on a comprehensive first-principles study of phase stability in the Ni-Al binary, both at zero Kelvin and at finite temperature. First-principles density functional theory calculations of the energies of enumerated orderings on fcc and the sublattices of B2 not only predict the stability of known phases, but also reveal the stability of a family of ordered phases that combine features of $\mathrm{L}{1}_{2}$ and $\mathrm{L}{1}_{0}$ in different ratios to adjust their overall composition. The calculations also confirm the stability of vacancy ordered B2 derivatives that are stable in the Al-rich half of the phase diagram. We introduce strain order parameters to systematically analyze instabilities with respect to the Bain path connecting the fcc and bcc lattices. Many unstable orderings on both fcc and bcc are predicted around compositions of ${x}_{\text{Ni}}=0.625$, where a martensitic phase transformation is known to occur. Cluster expansion techniques together with Monte Carlo simulations were used to calculate a finite-temperature-composition phase diagram of the Ni-Al binary. The calculated phase diagram together with an analysis of Bain instabilities reveals the importance of anharmonicity in determining the phase bounds between the B2 based $\ensuremath{\beta}$ phase and the $\mathrm{L}{1}_{2}$ based ${\ensuremath{\gamma}}^{\ensuremath{'}}$ phase, as well as properties related to martensitic transformations that are observed upon quenching Ni-rich $\ensuremath{\beta}$.

First-principles study of phase stability and elastic properties of binary Ti-xTM (TM = V,Cr,Nb,Mo) and ternary Ti-15TM-yAl alloys

The phase stability and elastic property of binary Ti-xTM (TM=V, Cr, Nb, Mo) and ternary Ti-15TM-yAl alloys are investigated systematically by using a first-principles method. The coherent potential approximation is employed to describe the random distribution of the alloying atoms in the alloys. We show that the transition metal (TM) elements V, Cr, Nb, Mo increase the elastic stability of the β phase but decreases that of the α phase. The polycrystalline bulk modulus of the α phase increases with the concentration of the TM alloying elements whereas the Young's modulus and shear modulus are weakened. For the β phase, all the polycrystalline elastic moduli hardens with the addition of the TM alloying elements. The influence of the simple metal (SM) element Al on the elastic properties is much weaker than the TM ones. The TM alloying elements increase the stability of the β relative to the α and ω phases. Al stabilizes significantly the β phase over the ω phase. Our calculations suggest that the d electron density dominates the elastic modulus but not the relative phase stability. The interaction between the TM and SM alloying elements in titanium alloys matters to the relative stability.

First-principles studies of phase stability and crystal structures in Li-Zn mixed-metal borohydrides

We address the problem of finding mixed-metal borohydrides with favorable thermodynamics and illustrate the approach using the example of ${\mathrm{LiZn}}_{2}{({\mathrm{BH}}_{4})}_{5}$. Using density functional theory (DFT), along with the grand-canonical linear programming method (GCLP), we examine the experimentally and computationally proposed crystal structures and the finite-temperature thermodynamics of dehydrogenation for the quaternary hydride ${\mathrm{LiZn}}_{2}{({\mathrm{BH}}_{4})}_{5}$. We find the following: (i) For ${\mathrm{LiZn}}_{2}{({\mathrm{BH}}_{4})}_{5}$, DFT calculations of the experimental crystal structures reveal that the structure from the neutron diffraction experiments of Ravnsb\ae{}k et al. is more stable [by 24 kJ/(mol f.u.)] than that based on a previous x-ray study. (ii) Our DFT calculations show that when using the neutron-diffraction structure of ${\mathrm{LiZn}}_{2}{({\mathrm{BH}}_{4})}_{5}$, the recently theoretically predicted $\mathrm{LiZn}{({\mathrm{BH}}_{4})}_{3}$ compound is unstable with respect to the decomposition into ${\mathrm{LiZn}}_{2}{({\mathrm{BH}}_{4})}_{5}+{\mathrm{LiBH}}_{4}$. (iii) GCLP calculations show that even though ${\mathrm{LiZn}}_{2}{({\mathrm{BH}}_{4})}_{5}$ is a combination of weakly [$\mathrm{Zn}{({\mathrm{BH}}_{4})}_{2}$] and strongly (${\mathrm{LiBH}}_{4}$) bound borohydrides, its decomposition is not intermediate between the two individual borohydrides. Rather, we find that the decomposition of ${\mathrm{LiZn}}_{2}{({\mathrm{BH}}_{4})}_{5}$ is divided into a weakly exothermic step [${\mathrm{LiZn}}_{2}{({\mathrm{BH}}_{4})}_{5}\ensuremath{\rightarrow}2\mathrm{Zn}+\frac{1}{5}{\mathrm{LiBH}}_{4}+\frac{2}{5}{\mathrm{Li}}_{2}{\mathrm{B}}_{12}{\mathrm{H}}_{12}+\frac{36}{5}{\mathrm{H}}_{2}$] and three strong endothermic steps ($12{\mathrm{LiBH}}_{4}\ensuremath{\rightarrow}10\mathrm{LiH}+{\mathrm{Li}}_{2}{\mathrm{B}}_{12}{\mathrm{H}}_{12}+13{\mathrm{H}}_{2}$; $\mathrm{Zn}+\mathrm{LiH}\ensuremath{\rightarrow}\mathrm{LiZn}+\frac{1}{2}{\mathrm{H}}_{2}$; $2\mathrm{Zn}+{\mathrm{Li}}_{2}{\mathrm{B}}_{12}{\mathrm{H}}_{12}\ensuremath{\rightarrow}2\mathrm{LiZn}+12\mathrm{B}+6{\mathrm{H}}_{2}$). DFT-calculated $\ensuremath{\Delta}{H}_{\text{ZPE}}^{T=0\phantom{\rule{0.28em}{0ex}}\text{K}}$ values for the first three ${\mathrm{LiZn}}_{2}{({\mathrm{BH}}_{4})}_{5}$ decomposition steps are $\ensuremath{-}19$, $+$37, $+$74 kJ/(mol ${\mathrm{H}}_{2}$), respectively. The behavior of ${\mathrm{LiZn}}_{2}{({\mathrm{BH}}_{4})}_{5}$ shows that mixed-metal borohydrides formed by mixing borohydrides of high and low thermodynamics stabilities do not necessarily have an intermediate decomposition tendency. Our results suggest the correct strategy to find intermediate decomposition in mixed-metal borohydrides is to search for stable mixed-metal products such as ternary metal borides.

First-principles study of phase stability of Ti2N under pressure

A first-principles study of phase stability of various phases of Ti${}_{2}$N under normal conditions and as a function of pressure was carried out. Among the \ensuremath{\epsilon} and \ensuremath{\delta}\ensuremath{'}phases of Ti${}_{2}$N that are observed experimentally, \ensuremath{\epsilon}-Ti${}_{2}$N is the most stable. The \ensuremath{\delta}\ensuremath{'} phase can only exist at high temperature due to the soft acoustic modes at the $X$ point. The origin of the tetragonal structure of both the \ensuremath{\epsilon} and \ensuremath{\delta}\ensuremath{'} phases is supposed to be caused by the tetragonal local lattice distortion around a nitrogen vacancy. Based on the results of the total-energy and phonon-spectrum calculations at zero temperature, the following sequence of phase transformation in Ti${}_{2}$N under pressure is predicted: \ensuremath{\epsilon}-Ti${}_{2}$N (space group P4/mnm), $P$ $=$ 77.5 GPa \ensuremath{\rightarrow} Au${}_{2}$Te type (space group C2/m), $P$ $=$ 86.7 GPa \ensuremath{\rightarrow} Al${}_{2}$Cu type (space group I4/mcm). The present study shows that, to correctly predict relative phase stability, the peculiarities of the phonon spectra of the materials under investigation have to be properly accounted for.

First-principles study of phase stability of stoichiometric vanadium nitrides

First-principles pseudopotential total-energy and phonon-spectrum calculations of various phases of stoichiometric vanadium nitrides with a small number of vanadium and nitrogen vacancies, as well as with small carbon and oxygen admixtures, were carried out. It was found that the stoichiometric hexagonal and tetragonal vanadium nitrides (VN) were dynamically stable. The NaCl-type VN exhibited lattice instability. The total energy for the computed ground-state phases of VN increased in the sequence hexagonal (WC type, $P\overline{6}m2$)--hexagonal (AsNi type, $P{6}_{3}/mmc$)--tetragonal ($P{4}_{2}/mcm$)--cubic (NaCl type, $Fm\overline{3}m$)--cubic (ZnS type, $F\overline{4}3m$)--cubic (CsCl type, $Pm\overline{3}m$). At low temperatures, the vacancies in both sublattices stabilize the triclinic phases of V${}_{x}$N${}_{x}$ for $x&lt;0.94$, which is consistent with tetragonal symmetry. At high temperatures, the stability of the NaCl- and WC-type VN structures was estimated using an approach based on band-energy smearing. The results obtained indicate that the cubic structure of VN with the fully occupied sublattices will be stable only at high temperatures. The small carbon and oxygen admixtures do not stabilize the cubic phase compared to the hexagonal one. The phase stability of the cubic and hexagonal structures of V${}_{x}$N${}_{x}$ was explained on the basis of the peculiarities of their electronic densities of states near the Fermi level. The predicted structural transformations in stoichiometric vanadium nitrides are compatible with those observed experimentally.

First-principles study of phase stability and electronic properties of RhZr

First-principles calculations were carried out to investigate the structural stabilities and electronic properties of RhZr. The plane wave based pseudopotential method was used, in which both the local density approximation (LDA) and the generalized gradient approximation (GGA) implanted in the CASTEP code were employed. The internal positions of atoms in the unit cell were optimized and the ground state properties such as lattice parameter, density of state, cohesive energies and enthalpies of formation of ortho-RhZr and cubic-RhZr were calculated. The calculation results indicate that ortho-RhZr can form more easily than cubic-RhZr and the ortho-RhZr is more stable than cubic-RhZr. The density of states (DOS) reveals that the strong bonding in the Rh-Zr and Rh-Rh or Zr-Zr interaction chains accounts for the structural stability of ortho-RhZr and the hybridization between Rh-4d states and Zr-4d states is strong.

First-principles study of phase stability of Gd-doped EuO and EuS

Phase diagrams of isoelectronic Eu${}_{1\ensuremath{-}x}$Gd${}_{x}$O and Eu${}_{1\ensuremath{-}x}$Gd${}_{x}$S quasibinary alloy systems are constructed using first-principles calculations combined with a standard cluster-expansion approach and Monte Carlo simulations. The oxide system has a wide miscibility gap on the Gd-rich side but forms ordered compounds on the Eu-rich side, exhibiting a deep asymmetric convex hull in the formation enthalpy diagram. The sulfide system has no stable compounds. The large difference in the formation enthalpies of the oxide and sulfide compounds is attributed to the contribution of local lattice relaxation, which is sensitive to the anion size. The solubility of Gd in both EuO and EuS is in the range of 10%--20% at room temperature and quickly increases at higher temperatures, indicating that highly doped disordered solid solutions can be produced without the precipitation of secondary phases. We also predict that rocksalt GdO can be stabilized under appropriate experimental conditions.

Low-temperature formation of cubic β-PbF2: precursor-based synthesis and first-principles phase stability study

A precursor-based approach to the cubic β-phase of PbF(2) was developed and allowed the preparation of this high-temperature phase well below the temperature for transition from the orthorhombic α- to the cubic β-phase. The formation of β-PbF(2) from the molecular precursors Pb[Se(C(6)H(2)(CF(3))(3))](2) and Pb(C(6)H(2)(CF(3))(3))(2) is facilitated by the presence of several short PbF contacts in these molecules. The cubic form of PbF(2) was obtained as macroscopic crystals as well as nanoparticulate powder. Its formation at relatively low temperature suggested a theoretical re-investigation of the phase stabilities of the two polymorphs. The theoretical results from the Kohn-Sham density functional theory indicate that the energy content for the β-phase is slightly lower than the one for the α-phase, by 0.5-1.7 kJ mol(-1) depending on the density functional used (zero-point vibrational energy correction included).

First-principles Studies of Phase Stability and the Neutral Atomic Vacancies in LiNbO3, NaNbO3 and KNbO3

AbstractWe overall evaluated the enthalpies of formation and the formation energies of neutral vacancies in ANbO3 (A = Li, Na, K) using a plane-wave pseudopotential method within a density functional formalism. The LiNbO3 phase with the LiNbO3-type structure was confirmed to have lower enthalpy of formation than that with perovskite- or ilmenite-type structure. The NaNbO3 (R3c) and KNbO3 (Bmm2 and R3m) phases with the lowest symmetry were found to have the lowest enthalpy of formation. The formation energy of a A vacancy was found to be the lowest under an oxidizing atmosphere and that of an O vacancy was found to be the lowest under a reducing atmosphere. The formation energy of a Nb vacancy was the highest under both oxygen-rich and -poor conditions. These results are in agreement with the empirical rule that B site defects in perovskite-type oxide do not exist.

First-principles study of phase stability of BN under pressure

The structural and energetic properties of three phases of boron nitride, the graphite-like hexagonal phase ( h-BN), the cubic phase ( c-BN), and the wurtzite phase ( w-BN) have been studied using first-principles pseudopotential calculations. Calculated cohesive energies reveal that c-BN is lower in energy than h-BN by 0.08 eV/atom, in good agreement with the experimental value of 0.17 eV/atom. w-BN is predicted to become a denser phase than c-BN at 16 GPa, in good agreement with experimental observation of a transition from h-BN to w-BN at P>12.5 GPa. The calculated energetic results together with empirical rules on crystal growth provide a good understanding for phase stability and transformation of boron nitride under pressure.

First-principles study of phase stability of Ti–Al intermetallic compounds

Thermodynamic and structural properties of fcc- and hcp-based Ti–Al alloys are calculated from first-principles and are used to perform an ab initio study of phase stability for the intermetallic compounds in this system. The full potential linear muffin tin orbital method is used to determine heats of formation and other zero-temperature properties of 9 fcc- and 7 hcp-based intermetallic compounds, as well as of elemental fcc and hcp Ti and Al. From the results of these calculations, sets of effective cluster interactions are derived and are used in a cluster variation method calculation of the solid-state portion of the composition-temperature phase diagram for fcc- and hcp-based alloy phases. The results of our calculations are compared with those of experimental studies of stable and metastable phases in the Ti–Al system.