A. Fernández Guillermet

Gibbs energy coupling of the phase diagram and thermochemistry in the tantalum-carbon system

Abstract An analysis of the thermochemical properties and the phase diagram of the Taue5f8C system is presented, including the b.c.c., Ta 2 C x , TaC x , graphite, and liquid phases. The analysis was performed using a theoretical approach that involves the evaluation of the Gibbs energy functions by computer coupling phase diagram data and thermochemical measurements. The analysis is based or a combination of experimental and estimated values. The estimates were based on the regular behaviour of the vibrational entropy and other cohesive properties, which have been established in recent studies on transition metal compounds. A calculated phase diagram is reported, and extensive comparisons between calculations and experiments are presented.

Phase stability and fcc/hcp martensitic transformation in Fe–Mn–Si alloys: Part II. Thermodynamic modelling of the driving forces and the MS and AS temperatures

Abstract This paper presents the second part of a study of the fcc/hcp relative phase stability and the fcc↔hcp martensitic transformation (MT) in the Fe–Mn–Si system. In part I, an experimental database was built up using dilatometric measurements, which covers the composition range in which the fcc↔hcp MT is detected. This new information is analysed in the present work using models for the molar Gibbs energy (Gm) of the various phases. Hcp is a metastable phase in the system, but we show that its properties can be inferred from selected pieces of experimental data, using the concept of T0 temperature. The assessed Gm functions of the ternary fcc and hcp phases are used to evaluate the so-called resistance-to-start-the-transformation energy (RSTE). According to our results the RSTE in this system vary smoothly with composition, and we account phenomenologically for that using low-order polynomials in the atomic fractions. Finally, the optimum Gm and RSTE functions are also used to calculate the MS and AS temperatures for arbitrary compositions in the ternary system. The extensive comparisons between calculations and measurements presented in this work show a very good agreement, which adds to the credibility of the present approach.

Structural properties of metastable phases in Zr–Nb alloys: I. Neutron diffraction study and analysis of lattice parameters

Abstract This paper reports the results of a neutron diffraction study of a series of Zr–Nb alloys quenched from high temperatures, where bcc (β) is the stable phase. Upon quenching, two additional structures are formed and retained metastably, viz. the hcp (α′) phase, which forms through a martensitic transformation, and the omega (ω) phase, which forms by a displacive transformation involving the collapse of (111) bcc planes. The lattice parameters (LPs) of the β, α′ and ω phases are determined as functions of the Nb content in the composition interval 4≤at.% Nb≤31, which represents a significant expansion of the experimental database for these various metastable structures. With the present data, reliable trends in the composition dependence of the LPs are established. In general, a smooth, essentially linear dependence upon the Nb content is detected, corresponding to Vegard’s law. However, a striking deviation from this simple behaviour is detected in the a ω parameter for Zr-rich alloys, i.e. in the region where this LP is directly related to the shortest interatomic distance (ID) in the ω phase. The need for a new analysis of the composition dependence of the various IDs in the ω phase is emphasized.

Theoretical study of the structural properties and thermodynamic stability of the omega phase in the 4d-transition series

Abstract Using the full potential-linearised-muffin-tin-orbitals (FP-LMTO) method, we have studied the structural properties and the thermodynamic stability of the AlB2 (C32) type-structure, so-called omega (Ω) phase, of Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd and Ag. The equilibrium volume, lattice parameters, interatomic distances, and density-of-electron-states at the Fermi level of the stable or metastable Ω phase of these elements are reported, and compared with those for the respective bcc phase. The zero kelvin energy difference between Ω and bcc is determined as a function of the number of electrons per atom. The energetics of the bcc→Ω phase transition in Y, Zr and Nb is studied by calculation, and a renewed contact is established with the Ho, Fu and Harmon picture of this transition, which is based on the softening of the 2/3 [1,1,1] longitudinal phonon mode of the bcc phase.

Atomic ordering and systematics of bonding lengths in the Ti–V omega phase: a neutron diffraction study

Abstract A new model describing the structural and bonding properties of the omega (Ω) phase in Zr–Nb alloys has recently been presented [12] . This model, which was aimed at explaining the composition dependence of the bonding lengths, predicts that the Ω phase is ordered, i.e., that some crystallographic sites are preferentially occupied by Zr atoms. Such feature, which should in principle be observed in other, related Ω phases, has not yet been tested against direct measurements. This problem has now been studied in the Ti–V system, which is the analogue of Zr–Nb in the 3d-transition series. Neutron diffraction measurements have been performed in quenched Ti–V alloys with V contents between 14 and 17 at.%. The diffraction spectra have been analysed using the Rietveld method, and a systematic analysis is reported here of the possibility of deviations from the random occupation of the two sublattices which are distinguished in the Ω structure. In addition, these new diffraction data are used in an evaluation of the shortest interatomic distances which are relevant for a comparison with the predictions of the model of Grad et al.

First-principles bond-length correlation for the omega structure of the 4d-transition elements

Abstract Using the full potential-linearised muffin tin orbitals (FP-LMTO) method, we have studied the shortest interatomic distance (SID) of the AlB 2 (C32) type-structure, so-called omega (Ω) phase, of Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd and Ag. It is shown that the SID in this phase correlates remarkably well with the single bond length D (1) given to these elements by Pauling’s bond length (BL) versus bond number relation. The physical relevance of this result, which holds for stable and metastable Ω phases, lies in the identification of the quantity D (1), which for metallic elements has long represented only a useful empirical parameter, to be used in the analysis and prediction of BLs, with a structural property that can be either directly measured, or accurately calculated by current ab initio methods.

Interatomic distances in the stable and metastable bcc and omega structures of the transition metals : analysis of experimental and theoretical trends and correlations with Pauling's bond lengths

Abstract The name omega (Ω) phase refers to a high-pressure modification of the transition metals (TMs) Ti, Zr, and Hf. In alloys of Ti, Zr and Hf with other TMs, the Ω phase can be formed and retained metastably at room temperature by quenching the bcc structure, which is usually the stable high-temperature phase in these alloy systems. As a part of a long-term research program on the structural and bonding properties of TM phases, we present in this paper a detailed analysis of the shortest interatomic distances (SIDs) of the stable and metastable bcc and Ω structures of the 3d, 4d and 5d TMs, as well as the connections between these and the single-bond length D (1) entering into Pauling’s classical bond length vs. bond number relation (BLBNR). The analysis involves an extensive database including experimental data, very recent ab initio calculations, and new predictions based on the BLBNR for the bcc phase of the TMs. Our results lend support to early suggestions by Jamieson of the existence of a correlation between D (1) and the SID of the Ω phase of Ti and Zr. We also find that, for most TMs, D (1) and the SID of the Ω structure differ by less than about 2%, which implies that the Jamieson correlation might be considered as a reasonably accurate first approximation in estimating IDs and lattice parameters for unknown Ω phases of the TMs.

Phenomenological bond length-bond number correlations for Ti, Zr and Hf: assessment and applications to metastable structures

Abstract This paper deals with (i) the establishment of Pauling’s bond length vs. bond number relation (BLBNR) for Ti, Zr and Hf, and (ii) the analysis of the connections between the parameters entering into the BLBNR and the experimental interatomic distances (IDs) in the known solid phases of these elements. Recently assessed ID data on the hcp structure, which is the stable phase at room temperature and atmospheric pressure (RTAP) as well as on the bcc and omega (Ω) structures, which are metastable at RTAP, are treated in terms of the BLBNR, and various approaches are tested for the evaluation of the univalent metallic bond-length parameter, D (1), of Ti, Zr and Hf. Next, a comparison is made with Pauling’s method, which is based on determining an optimum D (1) parameter. With these results, a new interpretation is presented of the so-called Jamieson correlation, which involves the possibility of a connection between D (1) and the shortest ID in the Ω phase. Finally, the BLBNR is applied to the metastable fcc structure of Ti, Zr and Hf, and a new evaluation is performed of the difference between metallic bond lengths for coordination numbers 12 ( D (12)) and 8 ( D (8)), which is compared with early generalisations on D (12)– D (8).

Structural properties of metastable phases in Zr-Nb alloys. III. The athermal Ω phase

Abstract By means of high-resolution neutron diffraction experiments, a new study of the lattice parameters and other structural properties as well as the mass fraction of the Ω phase formed by quenching Zr–Nb alloys, has been made. With the present results and those from the recent neutron diffraction work by Benites and co-workers (J. Alloys Comp. 299 (2000) 183; 302 (2000) 192) reported on papers I and II of this series, an analysis is performed of the composition dependence of various key aspects of the diffusionless (‘athermal’) Ω phase transition in the Zr–Nb system. In particular, we discuss the misfit between the Ω particles and the parent bcc phase, the hexagonal to trigonal symmetry change occurring in Ω and the effect of the latter upon the shortest interatomic distance in this phase. The picture emerging from the present study is also used to investigate the interatomic distance correlation previously reported by Grad et al. (Z. Metallkd. 87 (1996) 726).

Phase diagram and thermochemical properties of the ZrTa system. An assessment based on Gibbs energy modelling

Abstract The thermochemical properties of the Zrue5f8Ta system are not known from direct measurements and there is a lack of information on the Gibbs energy ( G m ) of its various phases, which is necessary in systematic analyses of cohesive and thermal properties of transition metal systems, and in thermodynamic modelling of ternary and higher-order phase diagrams based on Zrue5f8Ta. This kind of information has been obtained by using a theoretical method which relies on the coupling between thermochemistry and phase diagrams. The approach involves the analysis and synthesis of selected pieces of phase equilibrium data using models for G m of bcc, hcp and liquid, and the evaluation of a set of optimum model parameters through a computer optimisation technique. The G m functions arrived at in this way are applied to construct, by calculation, a phase diagram for the Zrue5f8Ta system, which is used in critical comparisons with the experimental data available. In addition, the enthalpy of formation values generated by the present study are compared with estimates based on the Miedema method and other predictive approaches.