Duhem Relation Research Articles

Nonstoichiometry, defects and thermodynamic properties of YFeO3, YFe2O4 and Y3Fe5O12

Oxygen nonstoichiometry of three ternary oxides, YFeO3−δ, YFe2O4−α and Y3Fe5O12−θ, in the system Y–Fe–O was investigated as a function of oxygen partial pressure by thermogravimetry at high temperature. The defects responsible for nonstoichiometry were identified as oxygen vacancies for YFeO3−δ and YFe2O4−α, although the manner of variation of nonstoichiometric parameter with oxygen partial pressure for these two oxides is quite different. Cation interstitials are the predominant defects in Y3Fe5O12−θ. Gibbs energies of formation of the three nonstoichiometric oxides were determined using solid-state electrochemical cells in the temperature range from 975 to 1475K. YFe2O4−α was found to be stable only above 1391K. Gibbs energies of formation of the three stoichiometric compounds from their component binary oxides were obtained by combining information from solid state cells with results of thermogravimetric analysis using the Gibbs–Duhem relation. The results can be summarized as:(1/2)Y2O3+(1/2)Fe2O3→YFeO3;ΔGf(ox)O(±250)(J/mol)=−17,126−8.263T(1/2)Y2O3+FeO+(1/2)Fe2O3→YFe2O4;ΔGf(ox)O(±260)(J/mol)=−10,352−13.24T(3/2)Y2O3+(5/2)Fe2O3→Y3Fe5O12;ΔGf(ox)O(±780)(J/mol)=−56,647−31.091T.

A cubic formalism for linking dilute and concentrated regions of ternary and multicomponent solutions

A cubic formalism is proposed for representation of thermodynamic properties of ternary and higher order systems that connect data on interaction parameters for solutes in metallic solvents with available thermodynamic data on binary systems. Since most metallic alloys have asymmetric excess Gibbs energies of mixing, a cubic expression in mole fraction is the minimum requirement for representation of binary data. Subregular solution model provides such a minimum framework. To link seamlessly with binary data, activity coefficients of solutes in metallic solvents have to be represented by third-order interaction parameter formalism. Although this requires values for a large number of interaction parameters, it is shown that these parameters are interrelated and many parameters can be obtained from data on the constituent binaries; only a limited number have to determined by measurement on ternary systems. For example, for a ternary 1–2–3 rich in component 1, 18 interaction parameters are required for defining the activity coefficients of both solutes 2 and 3 when third-order formalism is used. However, when interaction parameter formalism is made consistent with the Gibbs–Duhem relation, 11 separate relations are obtained between interaction parameters, reducing the number of independent parameters to seven. Six of these parameters can be obtained from the properties of the constituent binaries and only one parameter need to be determined from measurements in the ternary system. The number of independent parameters required becomes explicit when the excess Gibbs energy of mixing of the ternary system is represented by a subregular type model with an additional ternary parameter. Relations between coefficients of the subregular model and interaction parameters are derived. Thus, the use of the cubic formalism does not necessarily require additional measurements. The advantage is better representation of data at higher concentrations compared to the quadratic formalism of Darken or the unified interaction parameter formalism, which are essentially identical.

The Gibbs–Duhem type relation for ionic/nonionic mixed micelles – An alternative approach to Hall’s method

The micellar Gibbs–Duhem relation for two-component ionic/nonionic mixed micelles is presented in terms of the apparent bound amount to micelles. For simplicity, the ionic surfactant and a salt are both uni-uni valent electrolytes and have a common counterion species. Compared to the theory by D.G. Hall on the basis of a consideration on the Donnan equilibrium [D.G. Hall, J. Chem. Soc. Faraday Trans. 87 (1991) 3529], the present approach is simple and unambiguous. Considering changes at constant temperature and pressure, the two approaches provide nearly identical results.

Constitutive equations for Ca 2+ -alginate gels

Constitutive equations accounting for the coupling between chemical and mechanical phenomena are developed for the Ca 2+ alginate gel from the framework of thermodynamics of irreversible processes with internal variables. The development of the model is based on the Gibbs–Duhem relation and kinetic relations acting on generalized non-equilibrium forces. The constitutive equations are then compared to mechanical data obtained from uniaxial compressive tests at different velocities. A good agreement is observed between model and data. Furthermore, the model is able to predict the evolution of the mechanical response when the initial quantity of crosslinked Ca 2+ ions varies.

Tracking multi-phase boundaries using an integration-based approach

An integration-based method to track multi-phase boundaries for multi-component alloys systems is proposed. The foundation of the method is considering the phase boundary tracking as an initial value problem. It starts with the utilization of Gibbs–Duhem relation to find the slope of a phase boundary (the partial derivatives of the solvus temperature with respect to the equilibrium phase composition) and the partial derivatives of partition coefficients with respect to the equilibrium phase composition. Then, with these derivatives and a pair of known initial points on a phase boundary, the rest of the phase boundary can be calculated using a numerical integration technique such as the Runge–Kutta or Predictor–Corrector method in a stepwise manner. The integration method has been applied to track various types of phase boundaries in binary, ternary and quaternary systems. The calculated ones are in good agreement with the ones obtained by the commercial CALPHAD software Thermo-Calc. It provides a different computational method for the tracking of multi-phase boundaries and might be useful in the calculation of phase diagram when combined with the overlapping of Zero Phase Fraction (ZPF) lines technique and global minimization technique.

Molecular transport in nanopores: a theoretical perspective

Molecular transport in nanopores plays a central role in many emerging nanotechnologies for gas separation and storage, as well as in nanofluidics. Theories of the transport provide an understanding of the mechanisms that influence the transport and their interplay, and can lead to tractable models that can be used to advance these nanotechnologies through process analysis and optimisation. We review some of the most influential theories of fluid transport in small pores and confined spaces. Starting from the century old Knudsen formulation, the dusty gas model and several other related approaches that share a common point of departure in the Maxwell-Stefan diffusion equations are discussed. In particular, the conceptual basis of the models and the validity of the assumptions and simplifications necessary to obtain their final results are analysed. It is shown that the effect of adsorption is frequently either neglected, or treated on an ad hoc basis, such as through the division of the pore flux into gas-phase and surface diffusion contributions. Furthermore, while it is commonplace to assume that cross-sectional pressure is uniform, it is demonstrated that this violates the Gibbs-Duhem relation and that it is the chemical potential that essentially remains constant in the cross-section, as near-equilibrium density profiles are preserved even during transport. The Dusty Gas model and Maxwell-Stefan model for surface diffusion are analysed, and their strengths and weaknesses discussed, illustrating the use of conflicting choices of frames of reference in the former case, and the importance of assigning appropriate values for the binary diffusivity in the latter case. The oscillator model, developed in this laboratory, which is exact in the low density limit under diffuse reflection conditions, is shown to represent an advance on the classical Knudsen formula, although the latter frequently appears as a fundamental part of many transport models. The distributed friction model, also developed in this laboratory for the study of multi-component transport at any Knudsen number is discussed and compared with previous approaches. Finally, the outlook for theory and future research needs are discussed.

Thermodynamics of Binary and Ternary Solutions of Multivalent Electrolytes with Formation of 1:1 and 1:2 Complexes, within the Mean Spherical Approximation

The mean activity (γ±) and osmotic (Φ) coefficients for binary and ternary aqueous solutions of trivalent electrolytes (mainly made up of lanthanide salts) are described in the framework of the primitive model of ionic solutions, using the binding mean spherical approximation (BiMSA). This model, based on the Wertheim formalism, accounts for (chemical or electrostatic) association of ions. In this work, the multivalent cation and the anion are allowed to form 1:1 (pairs) and 1:2 (trimers) complexes. Expressions for γ± and Φ are given which satisfy the Gibbs−Duhem relation. The model involves concentration-dependent cation size and effective relative permittivity, variations that can be interpreted in terms of solvent effects. The theory is applied to aqueous solutions of binary and ternary mixtures at 25 °C with common anion.

The thermodynamics of protein folding: a critique of widely used quasi-thermodynamic interpretations and a restatement based on the Gibbs–Duhem relation and consistent with the Phase Rule

Interpretations of data in the extensive literature on the unfolding of proteins in aqueous solution follow a variety of methods involving assumptions leading to estimates of thermodynamic quantities associated with the unfolding transition. Inconsistencies and thermodynamic errors in these methods are identified. Estimates of standard molar free energies and enthalpies of unfolding using incompletely defined equilibrium constants and the van't Hoff relation are unsound, and typically contradict model-free interpretation of the data. A widely used routine for estimating the change in heat capacity associated with unfolding based on changes in the unfolding temperature and enthalpy co-induced by addition of denaturant or protective additives is thermodynamically incorrect by neglect of the Phase Rule. Many models and simulations predicting thermodynamic measures of unfolding are presently making comparisons with insecure quantities derived by incorrect thermodynamic analyses of experimental data. Analysis of unfolding via the Gibbs-Duhem equation with the correct Phase Rule constraints avoids the assumptions associated with incomplete equilibrium constants and misuse of the van't Hoff relation, and applies equally to positive, negative, sitewise or diffuse solute binding to the protein. The method gives the necessary relations between the thermodynamic parameters for thermal and isothermal unfolding and is developed for the case of two-state unfolding. The differences in binding of denaturants or stabilizers to the folded and unfolded forms of the protein are identified as major determinants of the unfolding process. The Phase Rule requires the temperature and enthalpy of unfolding to depend generally on the protein concentration. The available evidence bears out this expectation for thermal unfolding, indicating that protein-protein interactions influence folding. A parallel dependence of the denaturant concentrations for isothermal unfolding on the protein concentration is anticipated. The degree of unfolding as measured by UV, CD, fluorescence and other non-calorimetric methods may not show the same temperature and concentration ranges for unfolding among themselves or as compared to DSC or isothermal calorimetry. Such disparities indicate distinct stages in unfolding detectable by particular methods.

Thermodynamics of SiO 2–H 2O fluid near the upper critical end point from quartz solubility measurements at 10 kbar

Quartz solubility in SiO 2–H 2O fluid was measured at 10 kbar and 750–1130 °C in a piston-cylinder apparatus. At ≤ 1035 °C, solubility was determined by weight loss of single crystals equilibrated with H 2O-rich fluid; at higher temperature ( T), the fluid phase coexisting with quartz quenched to a glass, and solubility was determined by phase-assemblage bracketing (quartz present or absent with glass in quenched charges). At 700 to 1000 °C, quartz solubility increases at an accelerating rate from 1.2 mol% to 8.9 mol% SiO 2. Above 1000 °C, there is a sharp increase to nearly equimolar fluids in equilibrium with quartz at 1080 °C. At T > 1080 °C, SiO 2 concentration increases less strongly. The data confirm the existence of a critical end point on the hydrous melting curve of quartz, and imply it lies at 1080 °C and 9.5–10 kbar. Two independent approaches to calculating SiO 2 activity at 1080 °C — from a mixing model for aqueous SiO 2 extrapolated from lower temperatures, and from depression of the melting temperature of quartz by H 2O — indicate nearly constant values over the wide compositional range of 20–60 mol% SiO 2. The activity of H 2O ( a h ) at the same conditions, calculated by integration of the Gibbs–Duhem relation, shows a plateau of very high a h (∼ 0.93) in the same broad composition range. At ≥ 60 mol%, there is an abrupt change in H 2O dissolution mechanism, as SiO 2 activity becomes proportional to its mole fraction at high silica concentration. The activity-concentration relations at 1080 °C and 10 kbar were fitted to a subregular solution model, giving interchange energies for SiO 2 and H 2O of respectively 25.3 and 14.3 kJ/mol. Roughly constant SiO 2 activity of the near-critical fluid despite a factor of 3 increase in X H2O can be explained by progressively greater amounts of hydrogen bonding of H 2O molecules to polymerized silica units, so that the effective concentration of these units remains nearly constant as H 2O increases. The sudden onset of critical phenomena in the system SiO 2–H 2O as temperature and pressure increase over narrow intervals approaching 1080 °C and 10 kbar is thus explained as mainly a convergent or cooperative compositional effect, in which polymerization and hydrogen bonding both play important roles.

Dynamics of binary mixtures in inhomogeneous temperatures

A dynamical description for fluid binary mixtures with variable temperature and concentration gradient contributions to entropy and internal energy is given. By using mass, momentum and energy balance equations together with the standard expression for entropy production, a generalized Gibbs–Duhem relation is obtained which takes into account thermal and concentration gradient contributions. Then an expression for the pressure tensor is derived. As examples of applications, interface behavior and phase separation have been numerically studied in two dimensions neglecting the contributions of the velocity field. In the simplest case with a constant thermal gradient, the growth exponent for the averaged size of domains is found to have the usual value z = 1/3 and the domains appear elongated in the direction of the thermal gradient. When the system is quenched by contact with external walls, the evolution of temperature profiles in the system is shown and the domain morphology is characterized by interfaces perpendicular to the thermal gradient.

THERMODYNAMICS OF ROTATING BLACK BRANES IN GAUSS–BONNET–BORN–INFELD GRAVITY

Considering both the Gauss–Bonnet and the Born–Infeld terms, which are on a similar footing with regard to the string corrections on the gravity and electrodynamics sides respectively, we present a new class of rotating solutions in Gauss–Bonnet–Born–Infeld gravity with k rotation parameters. Although these solutions are not real in the whole space–time, they can be made real by a suitable transformation. Depending on the metric parameters, these real solutions may be interpreted as black brane solutions with inner and outer event horizons, an extreme black brane or naked singularity. We calculate the finite action and conserved quantities of the solutions by using the counterterm method, and find that these quantities do not depend on the Gauss–Bonnet parameter. We also compute the temperature, the angular velocities, the electric charge and the electric potential. Then, we calculate the entropy of the black brane through the use of the Gibbs–Duhem relation and show that it obeys the area law of entropy. We obtain a Smarr-type formula for the mass as a function of the entropy, the angular momenta and the charge, and show that the conserved and thermodynamic quantities satisfy the first law of thermodynamics. Finally, we perform a stability analysis in both the canonical and the grand-canonical ensemble and show that the presence of a nonlinear electromagnetic field has no effect on the stability of the black branes, which are stable in the whole phase space.

Accurate and local formulation for thermodynamic properties directly from integral equation method

A local formulation for determination of excess chemical potential is derived out by applying an assumption of linear dependence of correlation function and bridge function on the charging parameter to the Kirkwood charging formula and scaling the bridge function, the scaling parameter is specified by a Gibbs–Duhem relation. The local formulation for the excess chemical potential only requires the correlation function and bridge function of the investigated state as input and is therefore free of an unwieldy thermodynamic integration. A comprehensive comparison between the presently calculated thermodynamic quantities for a Lennard–Jones (LJ) fluid including two key quantities, i.e. the excess chemical potential and excess entropy, corresponding simulation data available in literature, and corresponding calculated results by several other global and local formulations, indicates that the present formulation is the only one capable of predicting locally and excellently all of the thermodynamic properties of the LJ fluid. The GCMC simulation is carried out for a core-softened potential fluid and the LJ fluid near critical state and at subcritical state near the gas–liquid coexistence line to obtain the excess chemical potential which is also in excellent agreement with the theoretical prediction from the present formalism; this indicates that the present formalism is of general interest in fluid statistical mechanics and applicable to parameter space covering over the entire phase diagram.

Correction of the Buckingham–Darcy Law for flow of high strength salts in variably saturated porous media

The Buckingham–Darcy Law is used to describe fluid flow in unsaturated porous media at low Reynolds number. In order to provide a priori corrections to this law, a process thermodynamic approach is utilized to ascertain the functional dependence. Using this knowledge, corrections to the hydraulic conductivity coefficient are proposed and compared with available data. The proposed corrections substantially predict the observed behavior of flow of high concentration (saturated) sodium chloride solutions in porous media. During the derivation, physical principles consistent with the thermodynamics of the system were utilized. A review of these principles and their results provides an alternative form of the generalized Gibbs–Duhem Relation for continuous processes, indicating that the identical equivalence to zero is unlikely to occur for dissipative processes. Further, the postulated Gibbs and Gibbs–Duhem Relations indicate that special differential operators need to be used for continuous processes rather than the usual use of a generic differential.

Mass Variables and Chemical Potentials in the Gibbs and De Donder Formulations of Chemical Equilibrium

The classical thermodynamic treatment of chemical equilibria in closed systems is founded on theoretical formulations by J.W. Gibbs and Th. De Donder. These two theories represent mathematically equivalent energy variation problems, related to each other through a mass variable transformation analogous to coordinate transformations in mechanics. Associated with the change in mass variables, chemical potentials of reactive substances are defined differently in the two theories, and are subject to different sets of constraints. Traditionally, the two sets of constraints have been merged into one, by assuming that the chemical potential represents the same variable in both theories, an assumption that is formally inconsistent with the difference in chemical potential definitions. Merging of constraints is still possible if chemical potentials remain invariant in value as a manifestation of symmetry under a Gibbs – De Donder mass transformation, but such symmetry has not been investigated. Here we demonstrate chemical potential invariance by reformulating De Donder’s theory using Lagrange multipliers, and combining the resulting constraints on chemical potentials with the classical Gibbs constraints, in a phase equilibration thought experiment. A new form of the Gibbs–Duhem (GD) relation is derived in the De Donder mass system. Comparing this relation to the classical GD equation in Gibbs’ component framework, subject to chemical potential invariance, directly yields the linear operator mapping the De Donder mass variables onto the set of Gibbs component masses. Results are illustrated for a simple ion exchange reaction, in the process solving the longstanding problem of determining solid exchanger activities in ion exchanging mixtures. It is pointed out that, even though the Gibbs and De Donder formulations are mathematically equivalent, the formal structure of De Donder’s theory allows a more explicit and systematic treatment of constraints, and also defines chemical masses in a form more naturally adapted to typical experimental measurements.

Theoretical analysis of experimental tracer and interdiffusion data in Cu–Ni–Fe alloys

In this paper, we present strategies to extract fundamental atomistic information from measured diffusion coefficients in a ternary alloy system. The strategies are exemplified with Cu–Ni–Fe alloys at 1271K where recent extensive interdiffusion coefficients and tracer diffusion coefficients for all three components have become available. We develop new defining phenomenological expressions for the vacancy-wind factors in terms of the diffusion coefficients. We show that the measured tracer diffusion coefficients can be processed using the Manning and Moleko, Allnatt and Allnatt random alloy diffusion kinetics formalisms (with and without the assumption of the Gibbs–Duhem relation between the thermodynamic activities) to give jump frequencies, tracer correlation factors, vacancy-wind factors and phenomenological coefficients. It is shown for example that Cu is generally the most correlated component in its diffusion behavior and that the off-diagonal phenomenological coefficients can be as high as 64% of the smallest of the diagonal phenomenological coefficients. It is also shown that the Darken formalism (which ignores off-diagonal phenomenological coefficients) is in fact a reasonable approximation for expressing the diagonal phenomenological coefficients in terms of the tracer diffusion coefficients. It is then shown how the measured interdiffusivities can be processed with these formalisms to give tracer diffusivities, vacancy-wind factors and phenomenological coefficients. Finally, we show how a straightforward strategy starting with the Darken analysis that is then followed by the Manning or Moleko, Allnatt and Allnatt analysis can be used to gain access to the vacancy-wind factors and the off-diagonal phenomenological coefficients.

Scaling relations in equilibrium nonextensive thermostatistics

The Euler and Gibbs–Duhem relations discussed in thermodynamics of ordinary extensive systems are shown to hold in the same forms also for nonextensive systems with long-range interactions with a novel interpretation of the entities appearing therein. In this way, Tsallis' conjecture for the scaling relations in equilibrium nonextensive thermostatistics is proved to be justified.

Integral equations for simple fluids in a general reference functional approach

The integral equations for the correlation functions of an inhomogeneous fluid mixture arederived using a functional Taylor expansion of the free energy around an inhomogeneousequilibrium distribution. The system of equations is closed by the introduction of areference functional for the correlations beyond second order in the density difference fromthe equilibrium distribution. Explicit expressions are obtained for energies required toinsert particles of the fluid mixture into the inhomogeneous system. The approach isillustrated by the determination of the equation of state of a simple, truncatedLennard-Jones fluid and the analysis of the behaviour of this fluid near a hard wall. Thewall–fluid integral equation exhibits complete drying and the corresponding coexistingdensities are in good agreement with those obtained from the standard (Maxwell)construction applied to the bulk fluid. The self-consistency of the approach is examinedby analysing the virial/compressibility routes to the equation of state and theGibbs–Duhem relation for the bulk fluid, and the contact density sum rule andthe Gibbs adsorption equation for the hard-wall problem. For the bulk fluid, wefind good self-consistency for stable states outside the critical region. For thehard-wall problem, the Gibbs adsorption equation is fulfilled very well near phasecoexistence where the adsorption is large. For the contact density sum rule, wefind deviations of up to 20% in the ratio of the contact densities predicted bythe present method and predicted by the sum rule. These deviations are largelydue to a slight disagreement between the coexisting density for the gas phaseobtained from the Maxwell construction and from complete drying at the hard wall.

Analysis of composition profiles to obtain interdiffusion coefficients in binary intermetallic compounds

Intermetallic compounds containing high concentrations of structural vacancies have made the determination of the interdiffusivity from the composition profile from a diffusion couple a confusing issue. In this paper, an exact procedure is formulated from fundamental principles. The only assumption made is the use of the Gibbs–Duhem relation. We show that the standard procedure for analysing composition profiles in diffusion couples slightly overestimates the interdiffusivity. We also show that the procedure put forward by Kim and Chang (Metall. Mater. Trans., 31A (2000) 1519) to analyse the composition profiles in NiAl is incorrect near the stoichiometric composition where it seriously underestimates the interdiffusivity.

Analysis of open system Carnot cycle and state functions

A first principle analysis of an open system thermodynamical Carnot cycle is provided and the results are compared to those proposed by Gibbs for open systems. The Kelvin–Clausius statement concerning heat transfer for reversible cycles is taken as an axiom, from which several rigorous theorems are proven. An equation is derived that resembles a Gibbs–Duhem relation equating convected entropies, from which two distinguishable forms of entropy are proven to exist for such systems; this questions prevailing developments which presume a singular or characteristic entropic form that couple all work and heat flows, such as in Onsager first-order thermodynamics. In particular, a closed path undergone by the system does not return the environment to the initial state for one of these entropic forms. The Biot assertion that the entropy contribution due to diathermal heat transfer does not form a state function is therefore contradicted and a local entropy is shown to exist. Several other new composite state functions for work and heat flow are shown to exist for open systems. From Gibbs’ results, it is suggested that the ensuing chemical potentials used routinely may possibly ignore heat effects. The functions developed here are suitable for application since the functions are proven to exist, rather than presumed to exist.

New insights into water?phospholipid model membrane interactions

Modulating the relative humidity (RH) of the ambient gas phase of a phospholipid/water sample for modifying the activity of phospholipid-sorbed water [humidity-controlled osmotic stress methods, J. Chem. Phys. 92 (1990) 4519 and J. Phys. Chem. 96 (1992) 446] has opened a new field of research of paramount importance. New types of phase transitions, occurring at specific values of this activity, have been then disclosed. Hence, it is become recognized that this activity, like the temperature T , is an intensive parameter of the thermodynamical state of these samples. This state can be therefore changed (phase transition) either, by modulating T at a given water activity (a given hydration level), or, by modulating the water activity, at a given T . The underlying mechanisms of these two types of transition differ, especially when they appear as disorderings of fatty chains. In lyotropic transitions, this disordering follows from two thermodynamical laws. First, acting on the activity (the chemical potential) of water external to a phospholipid/water sample, a transbilayer gradient of water chemical potential is created, leading to a transbilayer flux of water (Fick's law). Second, water molecules present within the hydrocarbon region of this phospholipid bilayer interact with phospholipid molecules through their chemical potential (Gibbs–Duhem relation): the conformational state of fatty chains (the thermodynamical state of the phospholipid molecules) changes. This process is slow, as revealed by osmotic stress time-resolved experiments. In thermal chain-melting transitions, the first rapid step is the disordering of fatty chains of a fraction of phospholipid molecules. It occurs a few degrees before the main transition temperature, T m , during the pretransition and the sub-main transition. The second step, less rapid, is the redistribution of water molecules between the different parts of the sample, as revealed by T-jump time-resolved experiments. Finally, in lyotropic and thermal transitions, hydration and conformation are linked but the order of anteriority of their change, in each case, is probably not the same. In this review, first, the interactions of phospholipid submolecular fragments and water molecules, in the interfacial and hydrocarbon regions of phospholipid/water multibilayer stacks, will be described. Second, the coupling of the conformational states of phospholipid and water molecules, during thermal and lyotropic transitions, will be demonstrated through examples.