Bragg-Williams Model Research Articles

Phase equilibria and thermodynamic modeling of the Sn–S, Ag–S, and Sb–S systems

Critical evaluations and thermodynamic optimizations of the tin-sulfur (Sn–S), silver-sulfur (Ag–S), and antimony-sulfur (Sb–S) binary systems were carried out over the whole composition range. The Modified Quasichemical Model (MQM) in the pair approximation was applied for the liquid phase to consider the short-range-ordering (SRO) in the phase. The binary compounds SnS, Sn2S3, SnS2, Ag2S, and Sb2S3 were treated as stoichiometric compounds. The allotropic transformations were taken into account in the formulation of the Gibbs energy equations. In the Ag–S system, the face-centered cubic (fcc) Ag solid solution was modeled by the Bragg-Williams (BW) model to improve the calculations for the solubility of S as a function of temperature. In the present study, particular focus was placed on the accurate prediction of the measured activities and enthalpies of the liquid phase. Using the MQM, these thermodynamic quantities could be calculated with high accuracy. Published measurements of the phase diagrams, invariant temperatures, and compounds could be reproduced within typical experimental errors.

Bragg–Williams Theory for Particles with a Size-Modulating Internal Degree of Freedom

The field of soft matter teems with molecules and aggregates of molecules that have internal size-modulating degrees of freedom. Proteins, peptides, microgels, polymers, micelles, and even some colloids can exist in multiple—often just two dominating—states with different effective sizes, where size can refer to the volume or to the cross-sectional area for particles residing on surfaces. The size-dependence of their accessible states renders the behavior of these particles pressure-sensitive. The Bragg–Williams model is among the most simple mean-field methods to translate the presence of inter-particle interactions into an approximate phase diagram. Here, we extend the Bragg–Williams model to account for the presence of particles that are immersed in a solvent and exist in two distinct states, one occupying a smaller and the other one a larger size. The basis of the extension is a lattice–sublattice approximation that we use to host the two size-differing states. Our model includes particle–solvent interactions that act as an effective surface tension between particles and solvent and are ignorant of the state in which the particles reside. We analyze how the energetic preference of the particles for one or the other state affects the phase diagrams. The possibility of a single phase-two phases-single phase sequence of phase transitions as a function of increasing temperature is demonstrated.

Thermodynamic assessment of the Fe–V–O system

The Fe–V–O system over the whole composition range was optimized according to the reliable phase equilibria and thermodynamic data. The modified quasichemical model was used to describe the liquid phase from metal alloy to oxide melt. Based upon the Compound Energy Formalism, the FeV2O4–Fe3O4 spinel solution was described by a sublattice model considering the cation distribution between tetrahedral and octahedral sites. Wüstite, corundum and (VO2)s.s. were described using a simple Bragg-Williams model. A set of self-consistent model parameters was obtained and the available phase diagram and thermodynamic data were reproduced well within experimental error limits. The complex phase relations in the Fe–V–O system at various temperatures and oxygen partial pressures were elucidated.

Critical thermodynamic re-evaluation and re-optimization of the CaO–FeO–Fe2O3–SiO2 system

Comprehensive literature review, critical re-assessment and thermodynamic re-optimization of phase diagrams and thermodynamic properties of all phases have been carried out for the CaO–FeO–Fe2O3–SiO2 system. Thermodynamic assessments of this system were previously published, however some of them were incomplete or described only limited range of compositions. In addition, more recent important experimental data have been available after previous assessments. The Modified Quasichemical model is used to describe the Gibbs energy of the liquid slag. The monoxide, wollastonite, α-Ca2SiO4 and α’-Ca2SiO4 solid solutions are described using the random mixing Bragg-Williams model. Spinel, olivine, pyroxene and melilite solid solutions are modelled using sublattice model based on the Compound Energy Formalism. A set of optimized parameters for the thermodynamic models has been obtained which reproduces all available experimental data within the experimental uncertainties from sub-solidus to above the liquidus temperatures at all compositions and atmosphere conditions.

Beyond the Margules equation: a universal thermodynamic equation for solid solutions including coupled substitution, long- and short-range order

Equilibrium calculations in complex chemical systems involving solids require thermodynamic models for solid solutions. This article presents a formalism for solid solutions that assigns an energy contribution for a configuration consisting of a central atom and its neighbors, multiplying it with its probability based on site occupancies and sums the products up to give the total energy of the solid solution. The formulation is a closed-form expression allowing differentiation with respect to composition and hence the direct calculation of the chemical potentials. The formalism is applicable to multicomponent systems as well as to trace elements, handles long- and short-range order and can be applied to coupled substitutions. The necessary parameters can be obtained from force-field , DFT or other calculation methods capable of determining 0 K excess enthalpies and volumes of solid solutions. Because the excess enthalpies are also a function of temperature, methods to calculate excess heat capacities as a function of temperature might be necessary ( e.g . approximation by phonon density of states). For interactions with one neighbor the presented formalism reduces to the regular solution and for two neighbors to the Margules -equation. For a two-site solid solution with interaction to only the neighboring site and the possibility of interchanging atoms between the sites, this formalism reduces to the Bragg-Williams model. As an example, the formalism is applied to the system calcite–dolomite–magnesite and the necessary parameters are derived by computations using force-field methods. Due to the pure computational approach and the limitations of the force-field methods, the phase diagram and the state of order for the respective phases are only reproduced qualitatively.

Thermodynamic Optimization of the Ca-Fe-O System

The present study deals with the thermodynamic optimization of the Ca-Fe-O system. All available phase equilibrium and thermodynamic experimental data are critically assessed to obtain a self-consistent set of model parameters for the Gibbs energies of all stoichiometric and solution phases. Model predictions of the present study are compared with previous assessments. Wustite and lime are described as one monoxide solution with a miscibility gap, using the random mixing Bragg-Williams model. The solubility of CaO in the “Fe3O4” magnetite (spinel) phase is described using the sublattice model based on the Compound Energy Formalism. The effect of CaO on the stability of the spinel phase is evaluated. The liquid CaO-FeO-Fe2O3 slag is modeled using the Modified Quasichemical Formalism. Liquid metal phase is described as a separate solution by an associate model.

Critical Assessment and Thermodynamic Modeling of the Al-Fe-O System

A complete literature review, critical evaluation, and thermodynamic modeling of the phase diagrams and thermodynamic properties of phases in the Al-Fe-O system at 1 atm total pressure are presented. Optimized model equations for the thermodynamic properties of all phases are obtained, which reproduce all available thermodynamic and phase-equilibrium data within experimental error limits from 298.15 K (25 °C) to above the liquidus temperatures at all compositions and oxygen partial pressures from metal saturation to 1 atm. The complex phase relationships in the system have been elucidated, and discrepancies among the data have been resolved. The database of the model parameters can be used along with software for Gibbs-energy minimization in order to calculate all thermodynamic properties and any type of phase diagram section. The modified quasichemical model was used for the liquid oxide phase. A sublattice model, based upon the Compound Energy Formalism, was developed for spinel, which expands from magnetite, Fe3O4, to hercynite, FeAl2O4. The distribution of cations between octahedral and tetrahedral sites and oxygen nonstoichiometry in spinel are taken into account. The model for metallic liquid assumes random mixing of associates: Fe, Al, O, AlO, and Al2O. It describes well the minimum that is observed on the solubility of oxygen in liquid iron as a function of the Al content. The solid solution between hematite and corundum exhibiting a miscibility gap, as well as a small solubility of Al2O3 in wustite are quantitatively described by a simple Bragg-Williams model.

Thermodynamic description of the Ag–(Ca, Li, Zn) and Ca–(In, Li) binary systems

Critical evaluations and thermodynamic modeling of the Ag-(Ca, Li, and Zn) and Ca-(In, Li) binary systems are presented. Thermodynamic optimization of the Ag–Li binary system was carried out in the present work at first. Thermodynamic re-optimizations of the Ag–Ca, Ag–Zn, Ca–In and Ca–Li binary systems were carried out in the present work, which presents improvements in comparison with previous works. The Modified Quasichemical Model in the Pair Approximation (MQMPA) was used for the liquid solution; this model is particularly suited for liquid which exhibits a high degree of short-range order. The intermetallic compounds are modeled with the Compound Energy Formalism (CEF), and terminal solid solutions are modeled with Bragg–Williams model (BWM) with sub-regular solution approximation. All available and reliable experimental data are reproduced within experimental error limits. A self-consistent thermodynamic database was constructed for the Ag–(Ca, Li, and Zn) and Ca–(In, Li) binary systems, which as a part of the thermodynamic database of the Mg–X (X: Ag, Ca, In, Li, Na, Sn, Sr, and Zn) multi-component system shall facilitate the development of Mg alloys for practical industrial purposes.

Assessments of coefficients of linear thermal expansions for magnetic elements Fe, Co and Ni

The coefficients of linear thermal expansions (CLEs) of magnetic elements Fe, Co and Ni were assessed from experimental information using theoretical models combined with MATLAB calculations. Model parameters can be determined accurately, and the assessed data are in good agreement with the experimental results. To facilitate the assessments, theories of thermal expansion were applied to separate CLEs into its nonmagnetic and magnetic components. The calculations of nonmagnetic contribution to CLEs were based on the modified Gruneisen–Debye model, in which the Debye temperature was regarded as an undetermined constant. In order to put the prediction of CLEs at the magnetic transition region on a sound physical basis, two kinds of theoretical models were innovatively used to calculate the magnetic contribution to CLEs, i.e., the Bragg–Williams model and the Fermi–Dirac distribution function. Model parameters were evaluated from experimental data using least square method. Detailed comparisons were made with the published experimental data and the calculated total CLEs. A satisfactory agreement is reached.

CoF2: a model system for magnetoelastic coupling and elastic softening mechanisms associated with paramagnetic ↔ antiferromagnetic phase transitions

Resonant ultrasound spectroscopy has been used to monitor variations in the elastic and anelastic behaviour of polycrystalline CoF2 through the temperature interval 10–290 K and in the frequency range ∼0.4–2 MHz. Marked softening, particularly of the shear modulus, and a peak in attenuation occur as the Néel point (TN = 39 K) is approached from both high and low temperatures. Although the effective thermodynamic behaviour can be represented semiquantitatively with a Bragg–Williams model for a system with spin 1/2, the magnetoelastic coupling follows a pattern which is closely analogous to that of a Landau tricritical transition which is co-elastic in character. Analysis of lattice parameter data from the literature confirms that linear spontaneous strains scale with the square of the magnetic order parameter and combine to give effective shear and volume strains on the order of 1‰. Softening of the shear modulus at T > TN is attributed to coupling of acoustic modes with dynamical local ordering of spins and can be represented by a Vogel–Fulcher expression. At T < TN the coupling of strains with the antiferromagnetic order parameter leads to softening of the shear modulus by up to ∼2%, but this is accompanied by a small and frequency-dependent acoustic loss. The loss mechanism is attributed to spin–lattice relaxations under the influence of externally applied dynamic shear stress. CoF2 provides a reference or end-member behaviour against which the likely antiferromagnetic component of magnetoelastic behaviour in more complex multiferroic materials, with additional displacive instabilities, Jahn–Teller effects and ferroelastic microstructures, can be compared.

ChemInform Abstract: Critical Assessment and Thermodynamic Modeling of the Cu—Fe—O System

AbstractReview: 99 refs.

Critical assessment and thermodynamic modeling of the Cu–Fe–O system

Critical evaluation, thermodynamic modeling and optimization of the Cu–Fe–O system are presented. The model for the liquid phase is developed within the framework of the Quasichemical Formalism. It describes liquid over the whole composition range from metal alloy to oxide melt. The model for spinel is developed within the framework of the Compound Energy Formalism. The spinel solution expands from magnetite, Fe3O4, to cupric ferrite, CuFe2O4. A small solubility of copper in wüstite is described by a simple Bragg–Williams model. Parameters of thermodynamic models have been optimized to reproduce all available thermodynamic and phase equilibrium data within experimental error limits. The thermodynamic optimization of the Cu–Fe–O system performed in the present study lays the groundwork for modeling the solubility of oxygen in sulfide liquid (matte) and in liquid metal alloys, which is of particular importance for simulation of copper smelting and converting.

The temperature and compositional dependence of disordering in Fe-bearing dolomites

Dolomite occurs in a wide range of rock compositions, from peridotites to mafic eclogites and metasediments, up to mantle depths of more than 200 km. At low-temperatures dolomite is ordered ( R 3), but transforms with increasing temperature into a disordered higher symmetry structure ( R 3 c ). To understand the thermodynamics of dolomite, we have investigated temperature, pressure, kinetics, and compositional dependence of the disordering process in Fe-bearing dolomites. To avoid quench effects, in situ X-ray powder diffraction experiments were performed at 300–1350 K and 2.6–4.2 GPa. The long-range order parameter s , quantifying the degree of ordering, has been determined using structural parameters from Rietveld refinement and the normalized peak area variation of superstructure Bragg peaks characterizing structural ordering/disordering. Time-series experiments show that disordering occurs in 20–30 min at 858 K and in a few minutes at temperatures ≥999 K. The order parameter decreases with increasing temperature and X Fe. Complete disorder is attained in dolomite at ~1240 K, 100–220 K lower than previously thought, and in an ankeritic-dolomites.s. with an X Fe of 0.43 at temperatures as low as ~900 K. The temperature-composition dependence of the disorder process was fitted with a phenomenological approach intermediate between the Landau theory and the Bragg-Williams model and predicts complete disorder in pure ankerite to occur already at ~470 K. The relatively low-temperature experiments of this study also constrain the breakdown of dolomite to aragonite+Fe-bearing magnesite at 4.2 GPa to temperature lower than ~800 K favoring an almost straight Clapeyron-slope for this disputed reaction.

Ab Initio Prediction of Adsorption Isotherms for Small Molecules in Metal–Organic Frameworks: The Effect of Lateral Interactions for Methane/CPO-27-Mg

A hybrid method that combines density functional theory for periodic structures with wave function-based electron correlation methods for finite-size models of adsorption sites is employed to calculate energies for adsorption of CH(4) onto different sites in the metal-organic framework (MOF) CPO-27-Mg (Mg-MOF-74) with chemical accuracy. The adsorption energies for the Mg(2+), linker, second layer sites are -27.8, -18.3, and -15.1 kJ/mol. Adsorbate-adsorbate interactions increase the average CH(4) adsorption energy by about 10% (2.4 kJ/mol). The free rotor-harmonic oscillator-ideal gas model is applied to calculate free energies/equilibrium constants for adsorption on the individual sites. This information is used in a multisite Langmuir model, augmented with a Bragg-Williams model for lateral interactions, to calculate adsorption isotherms. This ab initio approach yields the contributions of the individual sites to the final isotherms and also of the lateral interactions that contribute about 15% to the maximum excess adsorption capacity. Isotherms are calculated for both absolute amounts, for calculation of isosteric heats of adsorption as function of coverage, and excess amounts, for comparison with measured isotherms. Agreement with observed excess isotherms is reached if the experimentally determined limited accessibility of adsorption sites (78%) is taken into account.

THE EFFECT OF ORIENTATIONAL DISORDER OF THE ANION IN CALCITE ON THE ARAGONITE (Pmcn) CALCITE (R3c) EQUILIBRIUMvr

A thermodynamic analysis of the order–disorder transition in calcite, based on a modified Bragg–Williams model, has been extended to include the first-order transition from aragonite to calcite, with a view to developing a quantitative understanding of the relative stability of calcite ( R 3 c ) and aragonite and, by extension, a mathematical description of the aragonite–calcite coexistence curve in the phase diagram for CaCO 3 . The thermodynamic model necessarily relies on some estimated values of thermal and calorimetric properties of the two structures. The predictions of our model depend largely on thermodynamically consistent extrapolations of known properties of calcite and (metastable) aragonite at 1 bar pressure and an estimated value for the temperature coefficient of the difference in standard molar heat capacities. Agreement with experimental results is good up to T c = 1240 K, the critical temperature for orientational disorder of the anion in calcite, which we assume to be independent of pressure. The effect of CO 3 orientational disorder is shown as an expansion of the stability field of calcite. The R 3 c → R 3 m second-order transition in calcite is also analyzed in light of this thermodynamic model. Assuming the existence of new calcite phases, or invoking CO 3 disorder, is not necessary to account for the observed curvature of the coexistence line between 600 and 800 K.

Generalized Bragg–Williams model for the size-dependent order–disorder transition of bimetallic nanoparticles

Considering the different effects of exterior atoms (face, edge and corner atoms), the Bragg–Williams model is generalized to account for the size, shape and composition-dependent order–disorder transition of bimetallic nanoparticles (NPs) with B2, L10 and L12 ordered structures. The results show that the order–disorder temperatures TC,p are different for different shapes even in the identical particle size. The order of order–disorder temperatures of different shapes varies for different sizes. The long-range order parameter decreases with the increase in temperature in all size ranges and decreases smoothly in large sizes, but drops dramatically in small sizes. Moreover, it is also found that the order–disorder temperature of bimetallic NPs rises with increasing particle sizes and decreases with a deviation from the ideal compositions. The present predictions are consistent with the available literature results, indicating its capability in predicting other order–disorder transition phenomena of bimetallic NPs.

Role of thermal vibrations in alloy phase transitions

We resolve the fundamental contradiction that has existed from the earliest studies on order–disorder transitions between theoretical models that ignore the effect of thermal vibrations on the chemical interaction parameter, J(0), and the analysis of diffraction data that always incorporates the role of vibrations. New analysis of diffraction data shows that the temperature dependence of the order parameter, the central feature of order–disorder transitions, predicted by existing models is scientifically invalid. All models are constrained by the diffraction data to represent the interaction parameter as J(T) and the ordering energy as temperature dependent. The discrepancy between the experimental and theoretical ordering energies in Ni3V and Pd3V is direct evidence of their temperature dependence. Thermal vibrations influence order–disorder transitions through both the configurational and vibrational terms and not just the vibrational terms as hitherto believed. A modified Bragg–Williams model is proposed, which is the simplest model whose predicted order parameter can be compared with experiments.

Interplay of ordering and spinodal decomposition in the formation of ordered precipitates in binary fcc alloys: Role of second nearest-neighbor interactions

In this paper, the possible interaction of ordering and phase separation tendencies in the formation of an ordered precipitate phase (A3B/L12) within a binary supersaturated fcc solid solution is investigated using computational thermodynamics based on a generalized Bragg–Williams model incorporating first and second nearest-neighbor interactions. The formulation synthesizes and expands upon previous works and incorporates a strong pedagogical approach to elucidate the essential elements of the problem. The diffusional pathways governing microstructural development are predicted to be more complex, allowing for a multiplicity of decomposition mechanisms when second nearest-neighbor interactions are incorporated into the solution energetics, even in this mean field approximation. These higher order interactions markedly influence phase equilibria and phase stability. Ordering and clustering tendencies are not mutually exclusive but can occur synergistically, e.g. a conditional spinodal decomposition is predicted contingent on prior ordering of initially non-stoichiometric, disordered solid solutions. The role of second nearest-neighbor interactions on thermodynamic stability is discussed explicitly and compared to the classic treatments limited to first nearest-neighbor interactions only.

The $$R{\overline{3}} c \to R{\overline{3}} m$$ transition in nitratine, NaNO3, and implications for calcite, CaCO3

The temperature dependences of the crystal structure and superstructure intensities in sodium nitrate, mineral name nitratine, NaNO3, were studied using Rietveld structure refinements based on synchrotron powder X-ray diffraction. Nitratine transforms from $$R{\overline{3}} c\;\hbox{to}\;R{\overline{3}} m$$ at T c = 552(1) K. A NO3 group occupies, statistically, two positions with equal frequency in the disordered $$R{\overline{3}} m$$ phase, but with unequal frequency in the partially ordered $$R{\overline{3}} c$$ phase. One position for the NO3 group is rotated by 60° or 180° with respect to the other. The occupancy of the two orientations in the $$R{\overline{3}} c$$ phase is obtained from the occupancy factor, x, for the O1 site and gives rise to the order parameter, S = 2x − 1, where S is 0 at T c and 1 at 0 K. The NO3 groups rotate in a rapid process from about 541 to T c, where the a axis contracts. Using a modified Bragg–Williams model, a good fit was obtained for the normalized intensities (that is, normalized, NI1/2) for the (113) and (211) reflections in $$R{\overline{3}} c\hbox {\,NaNO}_{3},$$ and indicates a second-order transition. Using the same model, a reasonable fit was obtained for the order parameter, S, and also supports a second-order transition.

In situ study of the $$ R\overline{3} c \to R\overline{3} m $$ orientational disorder in calcite

The temperature dependences of the crystal structure and intensities of the (113) and (211) reflections in calcite, CaCO3, were studied using Rietveld structure refinements based on synchrotron powder X-ray diffraction data. Calcite transforms from $$ R\overline{3} c $$ to $$ R\overline{3} m $$ at about T c = 1,240 K. A CO3 group occupies, statistically, two positions with equal frequency in the disordered $$ R\overline{3} m $$ phase, but with unequal frequency in the partially ordered $$ R\overline{3} c $$ phase. One position for the CO3 group is rotated by 180° with respect to the other. The unequal occupancy of the two orientations in the partially ordered $$ R\overline{3} c $$ phase is obtained directly from the occupancy factor, x, for the O1 site and gives rise to the order parameter, S = 2x − 1. The a cell parameter shows a negative thermal expansion at low T, followed by a plateau region at higher T, then a steeper contraction towards T c, where the CO3 groups disorder in a rapid process. Using a modified Bragg–Williams model, fits were obtained for the order parameter S, and for the intensities of the (113) and (211) reflections.