End Members Of Solid Solution Research Articles

Development of a flexible quasi-harmonic-based approach for fast generation of self-consistent thermodynamic properties used in computational thermochemistry

We present in this work a new formalism for the rapid and accurate evaluation of the thermodynamic properties of compounds and solid solution end-members which are required in the high-throughput computational thermochemistry discovery of new materials. The method is based on the Debye approximation and is fully thermodynamically self-consistent. It considers several energetic contributions, such as the vibrational free energy (including both implicit and explicit anharmonicity), the defect-free energy (via the introduction of thermal vacancies), the energetic effect of thermal excitation of electrons (for metallic solids) as well as other excess terms to precisely reproduce experimental data. Sets of fully self-consistent pressure- and temperature-dependent thermodynamic properties are evaluated from the exact differentiation of this complete free energy formalism. This strategy allows the formalism to be fully integrated in the CALPHAD framework for the development of large thermodynamic databases of both stable and metastable compounds which may form in multicomponent/high-order systems. An emphasis is put in this work on the analysis and precise description of anharmonic vibrational contributions. Finally, sets of self-consistent thermodynamic and thermo-physical properties have been predicted as a function of temperature and compared with both experiments and classical QHA for pure elements (i.e. Ag, Li, V, Al, and C) and stoichiometric compounds of different chemical nature (i.e. MgO, CaO, Al2O3, NaCl, LiFePO4 and InP). The impact of various input parameters of the formalism on the free energy and the derived thermodynamic properties for these solids is analyzed. A better predictive capability and the enhanced flexibility of the proposed formalism are demonstrated.

Quantifying the Effects of Non‐Hydrostatic Stress on Multi‐Component Minerals

AbstractMineral compositions are used to infer pressures, temperatures, and timescales of geological processes. The thermodynamic techniques underlying these inferences assume a uniform, constant pressure. Nonetheless, convergent margins generate significant non‐hydrostatic (unequal) stresses, violating the uniform pressure assumption and creating uncertainty. Materials scientists F. Larché and J. Cahn derived an equation suitable for non‐hydrostatically stressed geologic environments that links stress and equilibrium composition in elastic, multi‐component crystals. However, previous works have shown that for binary solid solutions with ideal mixing behavior, hundreds of MPa to GPa‐level stresses are required to shift mineral compositions by a few hundredths of a mole fraction, limiting the equation's applicability. Here, we apply Larché and Cahn's equation to garnet, clinopyroxene, and plagioclase solid solutions, incorporating for the first time non‐ideal mixing behavior and more than two endmembers. We show that non‐ideal mixing increases predicted stress‐induced composition changes by up to an order of magnitude. Further, incorporating additional solid solution endmembers changes the predicted stress‐induced composition shifts of the other endmembers being considered. Finally, we demonstrate that Larché and Cahn's approach yields positive entropy production, a requirement for any real process to occur. Our findings reveal that stresses between tens and a few hundred MPa can shift mineral compositions by several hundredths of a mole fraction. Consequently, mineral compositions could plausibly be used to infer stress states. We suggest that stress‐composition effects could develop via intracrystalline diffusion in any high‐grade metamorphic setting, but are most likely in hot, dry, and strong rocks such as lower crustal granulites.

Enhanced piezoelectric response in ferroelectric solid solution with rhombohedral end members

High-performance piezoelectric materials are, in general, ferroelectric solid solutions exhibiting morphotropic phase boundaries (MPB) – a composition driven interferroelectric instabilities. The most common approach to engineer MPB in ferroelectric systems is to choose the end members of the solid solution with different polar symmetries and anticipate for the interferroelectric transformation to occur around a certain critical composition. Here I report a non-classical approach of obtaining MPB by choosing end members of solid solutions with similar polar symmetry (rhombohedral). This is demonstrated in the alloy system x(Na0.5Bi0.5)TiO3–(1-x)Pb(Zr0.6Ti0.4)O3 (NBT-PZT) which shows a narrow MPB region (0.07 ≤ .x ≤ 0.10) at which the piezoelectric response increases remarkably by nearly 3-times to its parent end members. This strategy opens the possibility of discovering new MPB ferroelectric systems with an enhanced electromechanical response.

Estimation of standard molar entropy of cement hydrates and clinker minerals

It is not straightforward to experimentally measure the standard molar entropy of cement hydrates or clinker minerals. This is further compounded by the controversies surrounding the entropy values reported in established thermodynamic datasets for cements. The purpose of this study is to assess the reliability of standard entropies compiled in those datasets. To this end, a simple but robust method is used in which the standard entropy of an inorganic solid is correlated to its formula unit volume via a linear equation. The results of this analysis show that the standard entropies and/or molar volumes (and in cases solubility products) of the following phases deserve closer scrutiny: meta-ettringite phases; magnesium/aluminium layered double hydroxide solid solutions; almost all iron-bearing monosulfate and hydrogarnet phases; and several calcium silicate hydrate solid solution end-members. In addition, this study reports the provisional estimates for the standard entropies of minerals ternesite and ye'elimite.

Pressure-induced amorphization in plagioclase feldspars: A time-resolved powder diffraction study during rapid compression

The pressure-induced amorphization of the two endmembers of the plagioclase ((Na1−xCax)Al1+xSi3−xO8) solid-solution, anorthite (CaAl2Si2O8) and albite (NaAlSi3O8), has been studied as a function of compression rate by means of time-resolved powder diffraction. Anorthite and albite were compressed in a diamond anvil cell to 80 GPa at multiple rates from 0.05 GPa/s to 80 GPa/s. The amorphization pressure decreases with increasing compression rate. This negative strain rate sensitivity indicates a change in deformation mechanism in the plagioclase solid-solution endmembers from brittle to ductile with increasing compression rate. The presented data support the previously proposed shear deformation mechanism for the amorphization of plagioclase. Furthermore, amorphization progresses over a wide pressure range suggesting heterogeneous amorphization, similar to observations based on recovered material from shock-compression experiments of plagioclase. Our experiments support the contention that amorphization pressures for plagioclase may occur at lower pressures than usually considered.

Kinetics of Precipitation of Non-ideal Binary Solid-solutions in an Aqueous Medium

We have developed a formalism which accounts for the kinetics of precipitation of binary non-ideal solid-solutions, and includes nucleation, growth and/or re-dissolution processes. It allows prediction of the time evolution of the aqueous ion activities together with all characteristics of the solid phase: crystal size distribution function and composition profiles of particles. We show that different scenarios of precipitation are possible depending on the thermodynamic properties of the solid solution end-members, the degree of non-ideality of the solid solution and the initial conditions of precipitation. We illustrate the various scenarios by simulations performed under a few simplifying assumptions for some carbonate solid solutions of geochemical interest. To our knowledge, this is the first time in the field of geochemistry that these out-of-equilibrium precipitation processes of non-ideal solid solutions are fully described.

GEMSFITS: Code package for optimization of geochemical model parameters and inverse modeling

GEMSFITS is a new code package for fitting internally consistent input parameters of GEM (Gibbs Energy Minimization) geochemical–thermodynamic models against various types of experimental or geochemical data, and for performing inverse modeling tasks. It consists of the gemsfit2 (parameter optimizer) and gfshell2 (graphical user interface) programs both accessing a NoSQL database, all developed with flexibility, generality, efficiency, and user friendliness in mind. The parameter optimizer gemsfit2 includes the GEMS3K chemical speciation solver (http://gems.web.psi.ch/GEMS3K), which features a comprehensive suite of non-ideal activity- and equation-of-state models of solution phases (aqueous electrolyte, gas and fluid mixtures, solid solutions, (ad)sorption. The gemsfit2 code uses the robust open-source NLopt library for parameter fitting, which provides a selection between several nonlinear optimization algorithms (global, local, gradient-based), and supports large-scale parallelization. The gemsfit2 code can also perform comprehensive statistical analysis of the fitted parameters (basic statistics, sensitivity, Monte Carlo confidence intervals), thus supporting the user with powerful tools for evaluating the quality of the fits and the physical significance of the model parameters. The gfshell2 code provides menu-driven setup of optimization options (data selection, properties to fit and their constraints, measured properties to compare with computed counterparts, and statistics). The practical utility, efficiency, and geochemical relevance of GEMSFITS is demonstrated by examples of typical classes of problems that include fitting of parameters of thermodynamic mixing models, optimization of standard state Gibbs energies of aqueous species and solid-solution end-members, thermobarometry, inverse titrations, and optimization problems that combine several parameter- and property types.

Sapphirine-bearing assemblages in the system MgOAl2O3SiO2: a continuing ambiguity

Based on experimental data for reactions involving sapphirine in the system MgO–Al 2 O 3 –SiO 2 (MAS), thermodynamic properties of the sapphirine solid solution end-members have been optimized for both ideal and non-ideal models, and internally consistent P–T grids are proposed. According to the calculated P–T relationships in MAS, the sapphirine + quartz assemblage, which is widely recognized as indicative of ultrahigh-temperature metamorphism, can be stable down to 840 ° C and 0.67 GPa, i.e ., at temperatures up to 150 ° C lower than estimated by Kelsey et al . (2004). The sapphirine + kyanite assemblage has been found stable at temperatures below 860 ° C and 1.13 GPa, whereas the sapphirine + forsterite assemblage may be stable below 800 ° C only under specific conditions of a very low activity of water. Calculations of P–T relationships involving sapphirine using the Thermocalc dataset revealed its inconsistency with both experimental and natural assemblages.

Dual-thermodynamic estimation of stoichiometry and stability of solid solution end members in aqueous–solid solution systems

The dual-thermodynamic (DualTh) approach is shown to provide a useful alternative to other methods in: (i) forward modelling of equilibrium speciation, activities, and element partitioning in a heterogeneous system involving several variable-composition phases, such as the aqueous–solid solution system; (ii) estimation of interaction parameters of a non-ideal mixing model from known bulk compositions of coexisting aqueous and solid-solution phases; and (iii) retrieval of unknown stoichiometries and apparent standard chemical potentials of trace solid-solution end-members. Inverse-modelling tasks (ii) and (iii) can be performed when the solid solution of interest is shown experimentally to co-exist with the aqueous phase either in the equilibrium or at the minimum stoichiometric saturation state. DualTh calculations exploit the ability of Gibbs energy minimisation (GEM) algorithms to find simultaneously two numerical solutions of the isobaric–isothermal chemical equilibrium speciation problem: (1) primal solution x — a vector of amounts of components (species) in phases; and (2) dual solution u — a vector of chemical potentials of stoichiometry units (usually chemical elements and charge). Conversely, the chemical potential of a phase component can be found in two complementary ways: (i) primal via its standard-state potential, concentration and activity coefficient (the latter two are functions of the x vector); and (ii) dual through its formula stoichiometry multiplied by the u vector. The DualTh methods compare primal and dual values of the chemical potential in simple and straightforward equations that can be easily computed in a spreadsheet, or implemented in GEM geochemical modelling codes.

A Two Order Parameter Thermodynamic Model for Pb(Zr1-xTix)O3

A two order parameter thermodynamic model is proposed which accounts for a number of features of the solid solution system Pb(Zr1-xTix)O3 not predicted by the conventional 6th order model. The free energy comprises three sets of terms: two expansions of polarization, representing contributions to the polarization of each of the solid solution end-members, weighted linearly with respect to composition, plus terms representing the coupling between the two contributions. Employing previously published coefficients for PbTiO3 and values extrapolated from Zr-rich PZT compositions for PbZrO3, calculations predict that the composition of the morphotropic phase boundary between the rhombohedral and tetragonal phases lies at x=0.45 and is independent of the strength of coupling between the two order parameters. The model allows for the existence of a monoclinic phase, the composition range of which depends upon the strength of the coupling terms. A satisfactory fit to the lattice parameters of Pb(Zr0.52Ti0.48)O3 is obtained over a wide range of temperature.

Random-Site Cation Ordering and Dielectric Properties of PbMg1/3Nb2/3O3-PbSc1/2Nb1/2O3

(1 − x)PbMg1/3Nb2/3O3-(x)PbSc1/2Nb1/2O3 (PMN-PSN) solid solution crystals have been grown by the flux method in the whole concentration range. X-ray supercell reflections due to B-cation ordering were observed for as-grown crystals from the 0.1 ≤ x ≤ 0.65 compositional range. Though the ordered domains are rather large (∼50 nm) the relaxor-like dielectric behavior is observed for compositions with x < 0.6. The diffusion of the dielectric permittivity maximum in as-grown crystals is the lowest at x = 0.6 and increases towards the end members of solid solution. Such behavior is explained within a Bragg-Williams approach by employing the random layer model. At x ∼ 0.6 the excitation energy determined from the Vogel-Fulcher relation exhibits a jump which we regard to changing the kind of the polar regions from PbMg1/3Nb2/3O3 to PbSc1/2Nb1/2O3 related type.

Cation Ordering and Dielectric Properties of PbMg 1/3 Nb 2/3 O 3 -PbSc 1/2 Nb 1/2 O 3 Solid Solution Crystals

(1 m x )PbMg 1/3 Nb 2/3 O 3 -( x )PbSc 1/2 Nb 1/2 O 3 solid solution crystals have been grown by the flux method. X-ray supercell reflections due to B-cation ordering were observed for crystals in the 0.1 < x < 0.65 compositional range. In as-grown crystals diffuseness of dielectric permittivity maximum is the lowest for x = 0.65 composition and increases toward the end members of solid solution. Such behavior is assumed to be due to the random-site ordering of B-site cations and non-monotonic dependence of the compositional order-disorder phase transition temperature on concentration.

Gibbs energy minimization approach to modeling sorption equilibria at the mineral-water interface: Thermodynamic relations for multi-site-surface complexation

A Gibbs energy minimization (GEM) approach to calculate chemical equilibria involving surface complexation at the mineral-water interfaces is presented, permitting the whole continuum “aqueous speciation - heterogeneous adsorption - surface (co)precipitation - solid solution” to be modeled only in elemental stoichiometry, without additional material balance constraints for the total number of surface sites. The thermodynamic stability of surface-bound species is considered in a way similar to that of the solid-solution endmembers, gases, and aqueous species. This consideration is made possible by introducing: (1) the standard and reference states of surface species involving a single value of reference site density Γ_o; (2) an expression of (electro)chemical potential linking quantities of the surface species with the amount and specific surface area of the sorbent; (3) activity/concentration relationships between aqueous sorbates and surface species, based on the surface activity terms (SAT) as functions of the maximum site densities Γ_max; (4) the surface type area fractions φ_α,t for describing faces or patches on heterogeneous surface of multi-site-surface sorption mineral phases; (5) an elemental stoichiometry and standard partial molal properties of the amphoteric neutral surface functional group ≡OH and derived surface complexes at ambient conditions, comparable between all (hydr)oxide mineral surfaces. For the ;OH group (in 2pK_A surface complexation models), a O_0.5H^o stoichiometry is conventionally defined from a reaction 0.5H₂O_aq = &gt;O_0.5H^o, with logK_n = 1.74436 (at Γ_o = 20 μmol · m⁻²) which is independent of temperature, pressure, pH, fO₂, amount and specific surface area of the sorbent, aqueous speciation, and surface speciation. It follows that all the non-reacted &gt;O_0.5H^o groups have constant activity in the presence of liquid H₂O and are thus macroscopically indistinguishable. The surface heterogeneity is described via standard thermodynamic properties of the surface complexes (reacted groups), such as “surface hydronium” &gt;O_0.5H₂⁺, “surface hydroxyl” &gt;O_0.5⁻, adsorbed cations or anions, as well as through φ_α,t and the common or individual Γ_t,max parameters. Theoretical findings, implemented in the GEM-Selektor code, are illustrated by modeling the literature potentiometric titration and adsorption data for silica and rutile in NaCl solutions with the triple-layer (TLM), the generalized double-layer (DLM), and the non-electrostatic (NEM) surface complexation models (SCM). Some examples were also calculated with the FITEQL3.2 code to check for compatibility. It is shown: (1) how approximate conversions between partial molal properties of surface species and intrinsic adsorption constants can be performed for different total site densities Γ_C ≠ Γ_o; (2) what are the regions of “geometrically ideal” and “non-ideal” adsorption behavior for the chosen SCM and how they are controlled by the SAT = f(Γ_max) on heterogeneous surfaces; and (3) why different sets of TLM parameters are not thermodynamically equivalent, and how a meaningful “optimal” set can be determined from titration data. Due to the separation of Γ_max parameters from Γ_o and, hence, from thermodynamic constants, the thermodynamic SCMs (in GEM implementation) can provide more insight into the relationships between surface types, complexes, bulk mineral sorbent, and aqueous sorbates, therefore permitting a more thorough use of microscopic/spectroscopic data in sorption modeling than has ever been possible.

Layered Li-Co-Mn Oxide as a High-Voltage Positive Electrode Material for Lithium Batteries

A layered Li-Co-Mn oxide was synthesized from a host layered Na-Co-Mn oxide by ion-exchange technique. Its electrode performance showed anomalous high redox potential of ca. 4.5 V vs. for the intercalation and deintercalation of lithium, although the end members of solid solution and did only 4.0 V. The oxidation state of the cobalt was measured by X-ray photoelectron spectroscopy and was trivalent. © 2001 The Electrochemical Society. All rights reserved.

Synthesis and structural properties of clinopyroxenes of the hedenbergite CaFe2+Si2O6 - aegirine NaFe3+Si2O6 solid-solution series

Clinopyroxenes along the solid solution hedenbergite-aegirine M2[Ca2+1-xNa+xM1{Fe2+1-xFe3+x}Si2O6 were synthesized using hydrothermal techniques at 4 kbar. Different temperatures and redox conditions were used to determine optimum synthesis conditions and the stability range of individual compositions in the T - log fO2 field. Synthesized samples were characterized using microprobe analysis, X-ray powder diffraction and Mossbauer spectroscopy at 298 K and 80 K. The structure was refined in the C 2/ c space group by means of the Rietveld method. Along the solid-solution series between hedenbergite (a = 9.8448(6) A, b = 9.0296(6) A, c = 5.2452(4) A, β = 104.813) and aegirine endmembers (a = 9.6547(6) A, b = 8.7941(8) A, c = 5.2944(4) A, β = 107.398), the changes in unit cell dimensions show significant deviations from linearity. Mean and individual M1-O distances decrease linearly from hedenbergite to aegirine; mean M2-O and T-O distances do not change significantly, whereas individual length may vary. While in hedenbergite the coordination of the M2 site is 6+2, it is 4+4 in aegirine. The Mossbauer spectra of the solid-solution endmembers display narrowly split resonance absorption lines with hyperfine parameters typical for Fe2+ (δ = 1.18 mm/s, Δ = 2.25 mm/s at 298 K) and Fe3+ (δ = 0.38 mm/s, Δ = 0.30 mm/s at 298 K). Fe occupies only the M1 site. The Fe2+ resonance absorption is somewhat broadened in the 80 K spectra of the solid solution, which is due to a distribution of quadrupole splittings.

Phase transitions in SrTiO3 quantum paraelectric with impurities

Phase diagrams of SrTiO3-based solid solutions with PbTiO3, BaTiO3, and CaTiO3 have been analyzed using our own experimental data and some data from the literature on dielectric, ultrasonic, and optical properties. At small concentration x of the second end-member of solid solutions, ferroelectric phase transition temperature varies as Tc = A(x-xc )1/2 where the critical concentration xc is almost the same for all the impurities; xc = 0.002. At high x values, the Tc (x) dependences are linear. For the case of PbTiO3 and BaTiO3, the structural and ferroelectric phase transition lines in the (T − x) plane converge at x near 0.04.

Estimation of thermodynamic functions of minerals of variable composition at high pressure using equations of state for end members

Knowledge of the values of excess thermodynamic functions (enthalpy, excess volume) for minerals of variable composition are necessary for phase equilibria calculations under conditions appropriate to the Earth's mantle and core. On the basis of the equation derived by Hietala for the mixing heat of the alkali halides, and expanded by Urusov for compounds with various structures and chemical bonding more complex than purely ionic, and using the equations of state for the end members of solid solutions, the expressions for enthalpies and volumes of mixing at high pressures have been derived. The quantitative results are presented for some alkali halides and oxide systems. General regularities of variation of mixing enthalpy at high pressure have been established.