First-principles study of stability of the bcc and ω phases of a low Al concentration Nb 1 − x Al x alloy (original) (raw)
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J Phys Condens Matter, 2011
The phase stability and site occupancy of bcc (body centered cubic) Nb 5 Al and slightly rearranged atomic structures have been examined by means of first-principles calculations. In order to use first-principles methods, a periodic cell is required and we used ordered Nb 5 Al compounds as a tractable example of a low Al concentration Nb 1−x Al x alloy (in this case, for about 17 at.% Al). The instability against an ω-structure atomic displacement was also studied, since this structure is detrimental to ductility. Mulliken population analysis was used to provide an understanding of the hybridization between the atoms and the electronic origin of the site occupancy and instability of the underlying bcc structures. By making calculations for several different configurations of the Nb-Al system we estimated the strengths of the Nb-Nb and Nb-Al bonds. It is shown that the stability of the underlying bcc phases is directly related to Nb-Nb and Nb-Al first-nearest-neighbor interactions. The first-principles calculations were extended to finite temperature by including various contributions to the free energy. In particular, the vibrational free energy was calculated within the quasiharmonic approximation, and it is shown that the contribution of the low energy modes to the lattice entropy helps to stabilize ordered bcc phases against ω-type phase transformations. Semi-quasi-random structures were employed to study the stability of the ordered and disordered bcc phases. Our study showed, in agreement with experiment, that the ω, ordered, and disordered phases can coexist in a nonequilibrium state at finite temperature.
First Principles Theory of Disordered Alloys and Alloy Phase Stability
NATO ASI Series, 1994
Carlomethods or the CVM. The difficulty with such an approacnistnatcomplex electronically mediatedinteractions aremapped ontoaneffective classical Hamiltonian. Unfortunately, thereisno apriori guarantee thatsucha procedure iseither uniqueor " rapidly convergent. In addition, since theparameters areextracted from calculations on smallunitcell systems, thereispossible thattheinteractions contain contributions (e.g. fromtheMadelnng energy) thatwill excessively favor suchstructures withrespect tothe disordered phase. Inthese lecture noteswe shall reviewtheLDA-KKR-CPA method fortreating the electronic structure and energetics ofrandom alloys and theMF-CF and GPM theories ofordering and phasestability thathavebeenbuilt on theLDA-KKR-CPA description ofthedisordered phase.Thus,we takethepoint ofviewthatmuch can be learned about metallic alloys by first studying theelectronic structure and energetics ofideal random solid solutions, which,forentropic reasons, arethenatural hightemperature solid state phasesand thento investigate their instabilities to theeither phaseseparation or to theformation ofspecific orderedphases. We shall stress thata direct connection can oftenbe made betweenspecific features intheelectronic structure associated withthe random solid solution and thedriving mechanismsbehindspecific ordering phenomena. Consequently, our understanding ofphasestability willbe underpinned by the same electronic structure thatisresponsible fordetermining theresidual resistivity and other properties ofthedisordered phaseand thatcan be experimentally verified usingoptical spectroscopies, positron annihilation and otherprobes. These lecture notesare structured as follows. In section 2 we layout the basic LDA-KKR-CPA theory oftheelectronic structure and energetics ofrandom alloys and some examplesof itsapplications to theelectronic structure and energie_ ofrandom alloys arepresented. In section 3 we reviewtheprogress thathas beenmade overthe last few yearsin understanding the mechanismsbehindspecific ordering phenomena observed inbinarysolid solutions basedon theMF-CF and GPM theories ofordering and phasestability. We will giveexamplesofa variety ofordering mechanisms:Fermi surface nesting, band filling, off diagonal randomness, charge transfer, size difference or local strain fluctuations, and magnetic effects. Ineachcasewe will trytomake thelink betweenthespecific ordering phenomenon and the underlying electronic structure of thedisordered phase.Insection 4 we will review theresults ofsome recent calculations on theelectronic structure of_-phaseNicAl1_c alloys usinga version oftheLDA-KKR-CPA codes that has been generalized to systems having complex lattices. In section 5 " we provide a few concluding remarks. 2 Theory of Random Substitutional Alloys 2.1 LDA-KKR-CPA The LDA-KKR-CPA method for calculating the energy and other properties of random solid solution alloys rests on three theoretical developments: the local density approximation to density functional theory, multiple scattering theory for solving the effective single particle SchrSdinger equation that is at the heart of the LDA-DFT self-consistent field equations, and the coherent potential approximation for treating the effects of disorder on the electronic structure i.e. for accomplishing the task of configuradonal averaging inherent in the calculation of observables. 2.1.1 Local Density Approximation and Random Alloys Density functional theory (DFT) is, in principle, an exact method for calculating the energetics of an electron system in the field of the atomic nuclei [4],[5], [21],[22],[6]. The. central result of DFT is that the total ground state energy, ELo], of a system of electrons in the presence of the external field provided by the nuclei is a unique functional ELo] = TIp] + U[p] + E,c[p] of the electron density, p(r-'),where Tip], U[p] and E,c[p] are the kinetic, potential and exchange correlation energies respectively. Furthermore, E[p] takes on its minimum value for the correct ground state p(r-').This minimum principle taken together with the constraint foo dar p(r) = N, the total number of electrons in the system leads to a set of self-consistent field equations whose solution yield the ground state charge density and hence the ground state energy. These basic equations of DFT are made into a practical computational method by making the local density approximation (LDA) in which the unknown, but exact, exchange correlation functional for the inhomogeneous interacting electron gas appropriate to the solid is approximated, at each point in space, r, by the exchange correlation functional, E_A[p], appropriate to an interacting but homogeneous electron gas having the density found at that point. Given the specification of a solid in terms of a set of atomic positions, {R/}, and corresponding nuclear charges, {Zi}, of the atoms occupying these sites, the practical applications the LDA involves the solution of a set of Hartree like, Kohn-Sham selfconsistent field equations that take the form [-V 2-I-v,s! (F;p(e; {P_}; {Zi}))] tb, Cr-') = _&,C r-') (1) J where the crystal potential ve!t(F; p(F; { R/); { Zi })) takes the form @ _2Z _ dFp(_') , [_-R'i[ + 2 IF-JI + v.=(r;'°p(r-')) (2) and where p(F; {R_}; {Zi})is given in terms of the eigen-solutions of eq. 1 as I¢,.(r-')12f(e.-p) (3)
Theory of phase transitions in disordered crystal solids
2009
Solid-state amorphization of a crystalline solid to an amorphous phase is extensively studied as a first order phase transition at low temperature for almost thirty years. Many similarities between heat-induced melting and solid-state amorphization have been recognized and a generalized Lindemann melting criterion has been built by focusing on the total mean-square atomic displacement as a generic measure of crystalline disorder in metastable solid solutions. In this dissertation, we report the recent progress on phenomenological models employed for thermodynamic description of macroscopic systems and fluctuations and nucleation of mesoscopic inhomogeneous systems in binary solid solutions under polymorphic constraints with no long-range diffusion involved. Based on our understanding on atomic picture of solid-state amorphization in binary solid solutions, we propose a Landau free energy to describe amorphization as the first order phase transition. The order parameter is defined wh...
The Kinetic Monte Carlo simulation method is used to study the temperature induced order-disorder phase transition in a two dimensional binary alloy (AB). Using the equivalence between the two dimensional Ising spin model and a binary alloy, the second order characteristic of this transition is verified, both with the short and the long range order parameters. The transition is clearly shown for an alloy with 50% of each kind of atoms. It was established that the initial configuration affects the evolution of these parameters at low temperature. The specific case of an alloy with 40% of non interacting B-atoms was investigated. It is shown that for a fixed interaction energy between A-and B-atoms, the order-disorder phase transition is always pointed out and that the critical temperature increases linearly with the pair interaction energy between A-atoms.
First-principles study of ω-phase formation in the Ti 3 Al 2 V system
Journal of Physics: Condensed Matter, 2007
Using first-principles methods, the phase stability of the underlying bodycentered-cubic (bcc) structure of Ti 3 Al 2 V and slightly rearranged atomic structures are investigated. The calculated ground-state energies show an instability in the ternary Ti 3 Al 2 V alloy with respect to the ω structure-type atomic displacement. A Mulliken population analysis shows strong bonding between the transition metals and Al. It is shown that Ti-Al is the strongest bond and that ω-type displacements increase the population overlap for this bond and reduce the energy of the system. The first-principles calculations are extended to finite temperature and various contributions to the free energy are calculated within the quasiharmonic approximation. It is shown that, at high temperatures, the bcc structure is stabilized by the contribution of the low-energy modes to lattice entropy. In agreement with experiment and in contrast to the Ti-Al-Nb system, we find that the metastable B8 2 structure cannot form in this alloy.
Journal of Alloys and Compounds, 2008
Within the framework of the self-consistent field approximation and the static concentration waves approach, a statistical-thermodynamic description of D0 19 -type superstructure in Ti-Al alloy is developed. A model of order-disorder phase transformation is applied for the non-stoichiometric intermetallic Ti 3 Al phase. Interatomic-interaction parameters are estimated for both approximations. One model supposes temperature-independent interatomic-interaction parameters, while the other includes the temperature dependence of mixing energies. The partial phase diagrams (equilibrium compositions for the coexistent ordered ␣ 2 -phase and disordered ␣-phase) are evaluated for both cases.
Physical Review B, 2007
The phase stability of B2 Ti 3 Al 2 X ͑X =Nb or V͒ and slightly rearranged atomic structures is examined by first-principles calculations. The ground-state energy calculations show instability in some of the Ti 3 Al 2 X configurations against the structure type of atomic displacement. We use electronic density of states and Mulliken population analysis to understand the hybridization between the atoms and the electronic origin of the stability or instability of each system. In order to estimate the strength of each bond, the heats of formation for several compounds are calculated. We find that the strength of the transition metal-Al bond increases from V to Nb to Ti, with Ti-V and Ti-Nb being weakly unstable. By examining several atomic configurations, it is shown that the stability of each structure is directly related to the number of Ti-Al bonds in each configuration. It is confirmed that the formation of the phase in Ti 3 Al 2 X is a combined displacive-replacive transformation. The crystal structure parameters, such as lattice constants and bulk modulus, are calculated and compared with available experimental data.
A Unique Framework to Describe Short Range Order and Properties of the Ordered State in Alloys
Le Journal de Physique IV, 1996
Elastic diffuse scattering of neutrons was performed in situ on two different alloys, Ni3V and Pt3V. Effective pair interactions of an Ising model were deduced from the measured short-range order parameters in the disordered phase. For this two examples, we obtain potentials that don't depend on temperature. Thus we can predict properties of the ordered state. In Ni3V, we describe correctly the temperature variation of the dissociation width of superdislocations and we compare the stability of different phases, L12 and D022, In the case of Pt3V, we will show how we can relate the measured interactions with the existence of long period structures in the ordered state.