Bragg-Williams Model Research Articles

Prediction of the ordering behaviours of the orthorhombic phase based on Ti 2AlNb alloys by combining thermodynamic model with ab initio calculation

The order–disorder transformation of intermetallics is of fundamental and technical importance. The ordering behaviours of the O phase based on Ti 2AlNb alloys are predicted by combining thermodynamic model with ab initio calculation. The site occupying tendencies of the constituent elements are studied for the first time theoretically without referring experimental data as input. The predicted results show that Al atoms always tend to occupy the γ (4 c1) sublattice, Ti atoms tend to occupy the α (8 g) sublattice and Nb atoms the β (4 c2) sublattice. The ordering tendencies of Ti and Nb atoms decrease with the increase of temperature, while the site occupation of Al atoms is weakly dependent on the temperature. The order–disorder transformation belongs to a second-order transition with a continuous character. It is also predicted that for the nonstoichiometric O phase with Al contents higher than 25 at.%, the site occupancies of the excess Al atoms prefer the β sublattice. The predicted site occupancy fractions and order parameters agree well with the reliable experimental data. The prediction has been improved compared with the Gorsky–Bragg–Williams model, as well as our early LMTO-ASA calculations.

Ordering phenomena and modelling of pair interactions in Ni 3Ga compound

In the last years, intermetallics with L1 2-structure were of particular interest, both from the point of view of their fundamental properties and their practical application as materials with excellent chemical, mechanical and physical properties. In the present investigation, we have chosen Ni 3Ga as a model L1 2 compound to study (1) the equilibrium degree of order at various temperatures and (2) the kinetics of ordering and disordering. In this paper we report the method of estimating the long-range order parameter from the resistivity data and how to derive the effective pair interaction (EPI) energy. It is then compared with those evaluated from thermodynamic activity measurements and also with those converted from Compound Energy Formalism (CEF) and CALPHAD modelling. Such adequately evaluated EPI energy values can be conveniently employed for calculating point defect concentrations at various temperatures and compositions using common statistical-thermodynamic methods, as a Bragg–Williams model.

Bragg–Williams Model of CsCl-Type Ordering of the FeCo System in Strong Magnetic Fields

A modified Bragg–Williams (B–W) model of α and α′ FeCo is extended to estimate the effect of strong magnetic fields on the critical ordering temperature (TORDER) taking into account long-range chemical and magnetic ordering. The model discussed here is generalized from our previous work in which only the larger average exchange per atom in the chemically ordered state was taken into account. A positive shift of critical temperatures for the higher order α→α′ order-disorder phase transformation has been predicted in the presence of a strong field. In this work, the experimentally observed dependence of the average magnetic moment of Fe atoms on the degree of chemical order has been accounted for explicitly. The estimated shift in the critical ordering temperature (ΔTORDER) is larger when the dependence of the Fe moment on the degree of chemical order is taken into account, particularly in the case of Fe-rich compositions (e.g., ΔTORDER ∼ 13 K vs ΔTORDER ∼ 10 K for H ∼ 50 T at equiatomic composition). For most compositions, however, the contribution to ΔTORDER associated with the larger average exchange per atom in the chemically ordered state accounts for the majority of the shift. The estimated effect remains quite small and is only expected to be experimentally observable in static fields larger than currently available in most laboratories (ΔTORDER is only predicted to be larger than ∼2 to 3 K for H > ∼10 T).

Bragg-Williams model of Fe-Co order-disorder phase transformations in a strong magnetic field

A modified Bragg-Williams statistical model with nearest-neighbor magnetic exchange and chemical interactions and an additional Zeeman energy term has been used to predict an increase in the critical ordering temperature (Torder) of Fe-Co alloys in the presence of a large magnetic field (ΔTorder∼20K near equiatomic composition for H=100T). The magnitude of the predicted ΔTorder increases with increasing field and is only predicted to be significant (ΔTorder≳2−3K near equiatomic) for H≳10T. The predicted shift in the ordering temperature is due to the higher average exchange interaction per atom in the ordered phase resulting in a larger theoretical Curie temperature as compared to the disordered phase. The dependence of the average magnetic moment of Fe on the degree of order was neglected in this work, however it is expected to contribute to an additional shift in Torder. Theoretical magnetization versus temperature curves are also presented for equiatomic FeCo subject to uniform applied magnetic fields.

On the stability of chemical order in small ordered-alloy particles

The reduction in the chemical order of small particles of long-range ordered alloys has recently been discussed by Fukami and Tanaka (Phil. Mag. Lett. 84 33 (2004)) in the framework of the Bragg–Williams model, in apparent accordance with experiment. We re-examine the discussion in the same framework with a closer inspection of the quantitative details. While it reproduces fairly well the lowering of the order–disorder transition temperature of Cu3Au particles of about 10 nm in size, the model is unable to explain the extended stability of the disordered state observed for smaller particles at room temperature; the order–disorder transition temperature does not fall below half of the transition temperature of bulk materials.

Lattice parameter dependence on long-range ordered degree during order–disorder transformation

The order–disorder transformations of stoichiometrical Ni 3Al, NiAl, Cu 3Au, and TiAl are studied by using the Bragg–Williams model based on embedded atom method potentials. The calculated relationship between the lattice parameter and the long-range order (LRO) parameter should be linear for the completely first order transition such as Ni 3Al, non-linear for the second order transition such as NiAl, and a mixture of non-linear (in the range of 1> σ≥ σ 0) and linear (in σ 0> σ≥0) for common first order transition such as Cu 3Au. The calculated linear dependence of lattice parameter on LRO for Ni 3Al is consistent with the existing experimental results. The driving force of the lattice parameter change is the free energy change due to changing of LRO parameter. The relationship between the lattice parameter of a stable ordered structure with a given LRO parameter and its free energy is linear. The alternate distribution of pure Ti and pure Al atomic planes in TiAl leads to strong directional d-bonding between nearest neighbour Ti atoms in the pure Ti plane and the polarization of p-electrons at Al sites aligning with the [001] direction, resulting in an enhancement of the Ti–Al bonding. When LRO=0 the structure becomes cubic fcc, and the directional bonding is eliminated. As LRO increases, the lattice parameter a and the volume of the unit cell are reduced while the lattice parameter c is increased. The calculated change of lattice parameter proportionally corresponds to the change of the free energy of the ordered phase for all four alloys.

Coupled relaxation of concentration and order fields in the linear regime

The relaxation rates of concentration and long-range-order fields in substitutional alloys are known to be coupled not only by the driving forces, as they derive from a single free-energy functional, but also by kinetic coefficients, since the very same atomic jumps drive both relaxations. For the simplest kinetic lattice model (the kinetic counterpart of the Bragg-Williams model), the detailed expressions of the coefficients of the appropriate mobility matrix that had been derived previously [G. Martin, Phys. Rev. B 50, 12 362 (1994)] contain two errors, which are corrected in the present paper. Then, using these expressions, we discuss the conditions for the existence of coupling of the relaxation rates of composition and long-range order. In the most general case, where relaxation takes place around an equilibrium state that is inhomogeneous and ordered, the presence of this kinetic coupling is confirmed. In the case where relaxation takes place around a disordered state (homogeneous or inhomogeneous), we show that the mobility matrix reduces to a diagonal form, i.e., that there is no kinetic coupling. In the case of an ordered homogeneous equilibrium state, however, the mobility matrix is found to be symmetrical with off-diagonal terms which do not vanish, in general, although they may vanish for some specific crystallographic structures, or along specific crystallographic directions. It is shown that, near the homogeneous equilibrium state, even in the presence of kinetic coupling, the relaxation rates are such that the free energy of the system decreases monotonically with time, as expected from the second law of thermodynamics. The model is applied to a few ordered structures with cubic or tetragonal symmetry.

Correlations in the arrangement of antistructure point defects in intermetallic phases with B2 (CsCl) structure

Abstract Based on the Bragg-Williams approximation, a statistical-thermodynamic approach is presented for non-stoichiometric binary B2 phases with a pure anti-structure defect mechanism. Whereas in the original Bragg-Williams model only the state of long-range order is considered and no correlations are allowed in the arrangement of the point defects, short-range order is introduced as an equilibrium property of the system in the present work. This new approach is applicable not only both to highly ordered intermetallic compounds and to completely disordered alloys but also in the region of the order-disorder transition. The obtained phase boundary which describes the order-disorder transition is in good agreement with the results predicted by Monte Carlo calculations.

Correlations in the arrangement of antistructure point defects in intermetallic phases with B2 (CsCl) structure

Abstract Based on the Bragg-Williams approximation, a statistical-thermodynamic approach is presented for non-stoichiometric binary B2 phases with a pure anti-structure defect mechanism. Whereas in the original Bragg-Williams model only the state of long-range order is considered and no correlations are allowed in the arrangement of the point defects, short-range order is introduced as an equilibrium property of the system in the present work. This new approach is applicable not only both to highly ordered intermetallic compounds and to completely disordered alloys but also in the region of the order-disorder transition. The obtained phase boundary which describes the order-disorder transition is in good agreement with the results predicted by Monte Carlo calculations.

The ordering behaviour of the O phase in Ti2AlNb-based alloys

Based on the investigation of Muraleedharan et al. [Intermetallics 3 (1995) 187] concerning the order–disorder transition of O phase in Ti2AlNb-based alloy, the site occupations of the alloying elements are clarified and the ordering behaviour of the O phase in Ti-yAl–zNb (y⩾25 at.%) alloys is investigated with a Bragg–Williams model. As a result, the relationship among the long-range order parameter (s), temperature (T), and the composition of the O phase (y, z) is established. In the temperature and composition range where the O phase exists, there is a continuously ordering character in the ordering behaviour of the O phase. Also the first-order transition character is very ‘weak’.

Thermodynamics of B2 intermetallic compounds with triple defects: a Bragg-Williams model for (Ni,Co)Al

Applying the Bragg-Williams approximation, a pair interaction model has been developed for describing the composition and temperature dependence of thermodynamic properties of ternary intermetallic B2 phases. Taking the activities and the integral enthalpies of formation as input data, values for the enthalpy and the entropy of bonds between atoms have been obtained. An analytical expression has been derived for the dependence of the concentration of point defects (vacancies and antistructure atoms) on composition and temperature. The model has been applied successfully to the available experimental data for the Ni-Al, Co-Al, and Ni-Co-Al systems. It has been shown that the values calculated accordingly for the effective enthalpy (and entropy) of formation of vacancies agree well with those from experimental observations and theoretical ab initio calculations.

Short- and long-range ordering in the ilmenite-hematite solid solution

Magnetic and cation ordering in the ilmenite–hematite solid solution (FeTiO3–Fe2O3) has been studied using in-situ high-temperature time-of-flight neutron powder diffraction. Synthetic samples containing 60 and 65% FeTiO3 (ilm60 and ilm65, respectively) were heated under vacuum up to 1000 °C and their magnetic structure, crystal structure and cation distribution were determined via Rietveld refinement. The quenched starting materials display diffuse superlattice reflections, indicative of short-range cation order. The short-range ordered structure is interpreted with the aid of statistical simulations to be a fine-scale alternation of ordered and antiordered ilmenite-like twin domains, separated by hematite-like twin-domain boundaries (TDBs). Peak width analysis demonstrates that the twin domains have a pronounced shape anisotropy, with average lengths of 20 ± 1 and 60 ± 2 A along the c-axis, and 100 ± 9 and 100 ± 4 A along [0&1macr;1]* in ilm60 and ilm65, respectively. Long-range order increases initially by a process of domain coarsening as the quenched samples are heated below the cation order-disorder temperature, Tod. The degree of order then decreases as they are heated through the transition. This leads to a kinetic relaxation behaviour, in which the observed rate of ordering is determined by the balance between the rate of coarsening and the rate of disordering within the domains. A phenomenological kinetic model is developed, which provides an excellent description of the observed behaviour in both samples. Once long-range order has been established, the equilibrium degree of order as a function of temperature is well described by a modified Bragg-Williams model, yielding values of Tod=830 ± 20 °C and 911 ± 20 °C for ilm60 and ilm65, respectively. Analysis of the magnetic scattering and spontaneous strain demonstrates that short-range magnetic order remains at temperatures well above the bulk Curie temperatures (Tc=178 °C and 143 °C in ilm60 and ilm65, respectively). This indicates significant magnetic heterogeneity in the samples, which may be related to the presence of a high density of Fe-enriched TDBs in the quenched material.

The modified quasi-chemical model: Part II. Multicomponent solutions

Further improvements to the modified quasi-chemical model in the pair approximation for shortrange ordering (SRO) in liquids are extended to multicomponent solutions. The energy of pair formation may be expanded in terms of the pair fractions or in terms of the component fractions, and coordination numbers are permitted to vary with composition. The model permits complete freedom of choice to treat any ternary subsystem with a symmetric or an asymmetric model. An improved general functional form for “ternary terms” in the excess Gibbs energy expression is proposed. These terms are related to the effect of a third component upon the binary pair interaction energies. It is shown how binary subsystems that have been optimized with the quasi-chemical model can be combined in the same multicomponent Gibbs energy equation with binary subsystems that have been optimized with a random-mixing Bragg-Williams model and a polynomial expression for the excess Gibbs energy. This is of much practical importance in the development of large databases for multicomponent solutions. The model also applies to SRO in solid solutions as a special case, when the number of lattice sites and coordination numbers are constant.

First principles study of site substitution of ternary elements in NiAl

Site substitution of ternary elements in ordered compounds influences the electronic structure and hence the properties of compounds at the continuous level. The electronic structure and binding energy of a number of NiAl- X alloy systems ( X=Ti, V, Cr, Mn, Fe, Co, Zr, Nb, Mo, Hf, Ta, W, Si, Ga, or Ge) were calculated using the discrete variational cluster method based on the local density approximation of the density functional theory. The site preference of the ternary additions to NiAl was investigated by employing the Bragg–Williams model to analyse the calculated binding energy. The results show that all the considered ternary elements possess stronger preference to the Al sublattice sites than a Ni atom does. A new method of identifying sublattice substitution of ternary additions in NiAl was proposed by comparison of the binding energies per atom of the ternary and the binary clusters involving the fourth nearest neighbours. The analysis suggests that Fe and Co atoms occupy the Ni sublattice sites, whereas Si, Ga and Ti atoms occupy the Al sublattice sites. The remaining elements may substitute for both sublattices: Mn is most likely to go for the Ni sublattice; V, Cr, Zr, Nb, Mo, Hf, Ta, W and Ge have a larger preference for the Al sublattice, but Cr and W do not show significant preference to any sublattice. The densities of states involving alloying additions of Co, Si and Cr were further investigated to clarify the site preference of the alloying additions.

A first principles study of the influence of alloying elements on TiAl: site preference

The electronic structure and binding energy of a number of TiAl- X alloy systems ( X=V, Cr, Mn, Fe, Co, Ni, Y, Zr, Nb, Mo, Hf, Ta, W, Ga, Ge, In or Sb) were calculated using the discrete variational cluster method, based on the local density approximation of the density functional theory. The site preference of the ternary additions to TiAl was investigated by means of the binding energy data and the Bragg–Williams model. The results showed that Y, Zr, Nb, Mo and Sb preferentially occupy the Ti sublattice sites, Ga and In occupy the Al sublattice sites, while V, Mn, Cr, Co and Ge may occupy either site, depending on the Ti/Al ratio. Investigation of the total and local densities of states for representative elements showed that the substitution behaviour of ternary additions in TiAl is determined by the electronic structure of the systems. The present predictions of the site preference of alloying elements in TiAl show good agreement with the most recent experimental findings.

The effect of Ti/Al ratio on the site occupancies of alloying elements in γ-TiAl

The site occupancies of V, Cr, Mn, Fe, Ni, Zr, Nb, Ta, Mo, Ga and Sn (1∼5 at%) in TiAl alloys with different nominal Ti/Al ratios were measured by the atom location channelling enhanced microanalysis (ALCHEMI) method. The results showed that Zr, Nb and Ta invariably occupy Ti sublattice sites, whereas Fe, Ni, Ga and Sn occupy Al sublattice sites, the alloy composition having no significant influence on their site preference. By contrast, the site preference of V, Cr, and Mn changes significantly with alloy composition (the Ti/Al ratio in particular), the probability of these elements occupying Ti sites decreasing in the above order. In general, with increasing atomic number, elements in the same period show increasing tendency to substitute for Al, so is the tendency to substitute for Ti for elements in the same group of the periodic table. A discussion is made in terms of the Bragg–Williams model by using published bond order data as input, allowing qualitative interpretation of the experimental findings.

MIXED MICELLES OF SDS AND SODIUM CHOLATE. A NUCLEAR MAGNETIC RESONANCE DIFFUSION AND RELAXATION STUDY

A nuclear magnetic resonance diffusion and relaxation investigation of the mixed micellar system sodium dodecylsulfate (SDS)/sodium cholate (SC) is presented. The concentration of SDS is kept constant at 15 mM and the concentration of SC is varied up to a total SC concentration of 100 mM. The diffusion coefficients of SDS and SC and the 2H relaxation rates of specifically 2H labeled SDS are measured. These quantities are used in the description of the distribution of the two surfactants in the various environments (as monomers in solution and associated into micelles). A theoretical framework based on the Bragg-Williams model is used to qualitatively describe the data.

Surface segregation of interacting atoms: analytical approach using a thermodynamic model

Binary atomic interactions are included in the thermodynamic Bragg–Williams model to study the cosegregation of different species of atoms in a multi-component alloy. The kinetics of segregating atoms is described by a set of mutually connected diffusion equations with boundary condition determined by instantaneously equalizing chemical potentials. This model allows analytical solution if t→0 and t→∞. For specific values of parameters our solution shows the approach to equilibrium through a quasi-equilibrium state, that is, the phenomenon of delayed segregation which has been recently analyzed by Jäger in the framework of the quasi-chemical approximation. We also predict a phenomenon of the nonmonotonic segregation.

Short-Range Order in Ni-Rich Ni–Ti Studied by Diffuse Neutron Scattering

Diffuse wide-angle neutron scattering has been measured for 58 Ni-5.8 at% Ti at room temperature after quenching from 777 K, and for 5 8N i-9.6 at% Ti at 1103 K. The fitted sets of short-range order and static displacement parameters are similar to those found in a previous investigation of short-range ordered Ni-8.9 at% Al. Applying a Bragg-Williams model, the coherent solvus of the γ phase is well reproduced. Neither the displacement parameters nor the phonon dispersior branches measured along (ξξξ) for the sample containing 9.6 at% Ti show any signs of instability of the short-range ordered face centered cubic structure.

Explanation for the configurational heat capacity of ordered phases

Recently, an intriguing ``W shape'' has been reported for the curve of the configurational heat capacity versus composition. This feature is calculated in ordered phases with various methods. A detailed explanation has not been published, so a brief derivation and explanation of its origin is in order. Within the context of a Bragg-Williams model expressions for the heat capacity are derived. These expressions indicate that the W shape originates from the composition dependence of the order parameter.