Gibbs Thermodynamics Research Articles

Thermodynamics and its prediction and CALPHAD modeling: Review, state of the art, and perspectives

Thermodynamics is a science concerning the state of a system, whether it is stable, metastable, or unstable, when interacting with its surroundings. The combined law of thermodynamics derived by Gibbs about 150 years ago laid the foundation of thermodynamics. In Gibbs combined law, the entropy production due to internal processes was not included, and the 2nd law was thus practically removed from the Gibbs combined law, so it is only applicable to systems under equilibrium, thus commonly termed as equilibrium or Gibbs thermodynamics. Gibbs further derived the classical statistical thermodynamics in terms of the probability of configurations in a system in the later 1800's and early 1900's. With the quantum mechanics (QM) developed in 1920's, the QM-based statistical thermodynamics was established and connected to classical statistical thermodynamics at the classical limit as shown by Landau in the 1940's. In 1960's the development of density functional theory (DFT) by Kohn and co-workers enabled the QM prediction of properties of the ground state of a system. On the other hand, the entropy production due to internal processes in non-equilibrium systems was studied separately by Onsager in 1930's and Prigogine and co-workers in the 1950's. In 1960's to 1970's the digitization of thermodynamics was developed by Kaufman in the framework of the CALculation of PHAse Diagrams (CALPHAD) modeling of individual phases with internal degrees of freedom. CALPHAD modeling of thermodynamics and atomic transport properties has enabled computational design of complex materials in the last 50 years. Our recently termed zentropy theory integrates DFT and statistical mechanics through the replacement of the internal energy of each individual configuration by its DFT-predicted free energy. The zentropy theory is capable of accurately predicting the free energy of individual phases, transition temperatures and properties of magnetic and ferroelectric materials with free energies of individual configurations solely from DFT-based calculations and without fitting parameters, and is being tested for other phenomena including superconductivity, quantum criticality, and black holes. Those predictions include the singularity at critical points with divergence of physical properties, negative thermal expansion, and the strongly correlated physics. Those individual configurations may thus be considered as the genomic building blocks of individual phases in the spirit of the materials genome®. This has the potential to shift the paradigm of CALPHAD modeling from being heavily dependent on experimental inputs to becoming fully predictive with inputs solely from DFT-based calculations and machine learning models built on those calculations and existing experimental data through newly developed and future open-source tools. Furthermore, through the combined law of thermodynamics including the internal entropy production, it is shown that the kinetic coefficient matrix of independent internal processes is diagonal with respect to the conjugate potentials in the combined law, and the cross phenomena that the phenomenological Onsager flux and reciprocal relationships are due to the dependence of the conjugate potential of a molar quantity on nonconjugate molar quantities and other potentials, which can be predicted by the zentropy theory and CALPHAD modeling.

Gibbs thermodynamics and surface properties at the nanoscale.

Gibbs's classical thermodynamic framework approximates systems as infinitely large phases separated by infinitely thin surfaces. The range of validity of this classical framework naturally comes under scrutiny as we become interested in the properties of ever smaller systems. This Communication clarifies that while Gibbs's original framework of bulk phase thermodynamics did require modifications to describe the properties of very small (i.e., non-additive) phases, his classical framework remains fundamentally valid to describe the thermodynamic properties of surfaces. We explain why classical surface laws are applicable at the nanoscale, as suggested by simulations and confirmed by experiments. We also show that a generalized Gibbs-Tolman-Koenig-Buff equation and the resulting Tolman's law for surface tension are obtained from a classical thermodynamic analysis in the Tolman region, a region of interaction between the system and the environment.

Thermodynamics of interfaces extended to nanoscales by introducing integral and differential surface tensions

As a system shrinks down in size, more and more molecules are found in its surface region, so surface contribution becomes a large or even a dominant part of its thermodynamic potentials. Surface tension is a venerable scientific concept; Gibbs defined it as the excess of grand potential of an inhomogeneous system with respect to its bulk value per interface area [J. W. Gibbs, "The Collected Works" in Thermodynamics (1928), Vol. 1]. The mechanical definition expresses it in terms of pressure tensor. So far, it has been believed the two definitions always give the same result. We show that the equivalence can break down for fluids confined in narrow pores. New concepts of integral and differential surface tensions, along with integral and differential adsorptions, need to be introduced for extending Gibbs thermodynamics of interfaces. We derived two generalized Gibbs adsorption equations. These concepts are indispensable for an adequate description of nanoscale systems. We also find a relation between integral surface tension and Derjaguin's disjoining pressure. This lays down the basis for measuring integral and differential surface tensions from disjoining pressure by using an atomic force microscope.

Derivation of Kelvin Equation for the Phase Equilibrium between the Curved Interface Based on Gibbs Thermodynamics

Derivation of Kelvin Equation for the Phase Equilibrium between the Curved Interface Based on Gibbs Thermodynamics

Thermodynamics of the formation of composite material structures. A review

Composite material structures at nano-, micro-, and mesolevels have been examined. The application of the Gibbs and Hill thermodynamics has allowed researchers to make a physical interpretation of the Laplace pressure for composite materials; to derive the thermodynamic functions describing the particles consolidation; establish the new phenomenon, namely, metal melt imbibition (MMI), and define its moving forces, substantiate a criterion that makes it possible to determine the direction of a liquid phase migration in a composite body, as well as to predict the final structure of composite materials. For the description of a composite material structure, which forms under extreme conditions, it has been necessary to use the nonequilibrium thermodynamics. For these conditions the Prigogine concept of local equilibrium is promising.

Hydrogen Adsorption on C-Supported Pt and Pt-Co Shell-Core Nano-Catalysts in Proton Exchange Membrane Fuel Cells: Entropic Effects and Catalytic Activity

The Gibbs thermodynamics of hydrogen adsorption for C-supported Pt and Pt-Co shell-core nanoparticles was calculated for the first time on practical electrodes in proton exchange membrane fuel cells (PEMFCs). The variations of the cyclic voltammetry with temperature were used to determine the Gibbs free energy ΔGads(HUPD), enthalpy ΔHads(HUPD), entropy ΔSads(HUPD) and lateral interaction parameter ω of hydrogen underpotential deposition (HUPD). The bond energy DPt − H was further calculated, neglecting the slight change of a double layer over the potential range of H adsorption. The results were compared to those obtained for single crystal Pt surfaces. It is established that the hydrogen adsorption and the catalytic activity are driven by entropic effects. The bond enthalpies of Pt-H are generally found to be unaffected by the structure of the Pt catalysts while entropic effects (order-disorder) are important in determining the catalytic activity. It is shown that the exchange current densities on various single crystal surfaces increase with order. For C-supported Pt-Co shell-core nano-catalyst, the presence of Co core beneath the Pt shell increases the order and hence exhibits better catalytic activity than Pt alone.

Surface Energy, Surface Debye Temperature and Specific Heat of Nanocrystals

The present study provides insight into the role of surface free energy and surface Debye temperatures in the specific heat of nanocrystals. In the case of small phases, consisted of comparable number of bulk and surface atoms, the variation of heat capacity with respect to that of infinitely large phase is significant. This effect, not accounted for in the Debye model is due to the substantial impact of interface energy of clusters. Here, in the framework of classical Gibbs thermodynamics, we introduce surface heat capacity that accounts for surface free energy contribution. Defined as a difference, between surface and bulk terms, the excess heat capacity reveals temperature gap and size limits of nanoclusters where the contribution of surface energy is essential. Considering the role of surface Debye temperatures, we evaluate the deviation of specific heat for nanoclusters having different surface atomic density and surface orientation. Being in agreement with recent experimental findings, the present model opens up a way to predict basic, size-dependent thermodynamic properties of nanocomposite systems.

Gibbs citations and interpretations in the theory of surface tension of solids: A conceptual approach

Josiah Willard Gibbs is definitely the most respectable and citable author in the surface tension of solids, however there are many incorrect citations of concepts and notions ascribed to Gibbs, but completely alien to his scientific heritage. Unfortunately, they became decidedly important for development of the contemporary theory of the surface tension of solids. For example, the erroneous notion of the “reversible cleavage”, as the conceptual background of this theory, appeared half a century after the most recent Gibbs’ works. Here we discuss not just formal linguistic, but conceptual matters using the advanced statistical method to identify wrong concepts ascribed to Gibbs. They existed in the theory of surface tension of solids for long decades and were accepted and used by most experts in this field. Today, we should formally recognize the high status of the current theory of surface tension of solids since it is supported by the scientific majority. It is based on the Shuttleworth equation, the vague concept of “reversible cleavage” and incorrect citations and interpretations of Gibbs thermodynamics. Up till now, neither this equation nor the theory as a whole had any experimental confirmations; moreover, they were shown to contradict all available experimental data. In connection with incorrect citations and interpretations of Gibbs works, we underline some pseudo-concepts and terms that Gibbs never used (like “reversible cleavage” or just “cleavage”), and contrarily, the important notions that Gibbs introduced for the first time, but almost nobody heard about them, for example -”actual and possible components” and the principal absence of “pure phases” in equilibrium of multiphase systems. We have also considered in detail the key difference between citation (as a possible free and often incorrect interpretation of the original text) and quotation (as an exact part of the original). Semantically citations and quotations are usually regarded as synonyms, but in practice it is not always correct and the sequences of their proposed equivalence might be critical for development of new concepts and theories.

Enthalpic and Entropic Effects on Hydrogen and OH Adsorption on Pt(111), Pt(100), and Pt(110) Electrodes As Evaluated by Gibbs Thermodynamics

Gibbs thermodynamics have been used to characterize hydrogen and OH adsorption on Pt(111), Pt(100), and Pt(110) in 0.1 M HClO4. In this way, the Gibbs energy, enthalpy, and entropy of formation of the interface have been evaluated. The results demonstrate that hydrogen adsorption on the three platinum basal planes is mainly driven by entropic effects, while enthalpic effects are the driving force for OH adsorption. Remarkably, this thermodynamic analysis overcomes the limitations inherent to the more common treatment of data, employing a generalized isotherm. Nevertheless, the present work demonstrates that both approaches are adequate for the systems under study, and they provide complementary information. The relationships between the thermodynamic properties, obtained from both approaches, have been derived. Finally, the results are compared with experimental and theoretical data for hydrogen and OH adsorption on Pt(111), Pt(100), and Pt(110) under UHV conditions. It is found that the Pt−H bond energy is similar under electrochemical and UHV conditions, for the three basal planes, although the H−H lateral interactions seem to be more dependent on the environmental conditions. In the case of OH adsorption, it is concluded that coadsorbed water molecules play an important role on both the interaction of OH species with the platinum surface, and the OH−OH lateral interactions. This is related to the formation of hydrogen-bonds between neighboring water and OH species, and these bonds seem to be stronger for Pt(111) than for Pt(100).

Thermodynamic potentials and thermodynamic relations in nonextensive thermodynamics

The generalized Gibbs free energy and enthalpy are derived in the framework of nonextensive thermodynamics by using the concept of the physical temperature and the physical pressure. Some relations of the thermodynamical potentials are changed due to the difference between the physical temperature and the inverse of Lagrange multiplier. We derive the thermodynamical relation between the heat capacities at a constant volume and at a constant pressure using the generalized thermodynamical potential. We find that it has a different form from the traditional one in Gibbs thermodynamics. But, the expressions of the heat capacities in this framework using the generalized thermodynamical potentials are still the same as the traditional one.

Cooper–Frye formula and non-extensive coalescence at RHIC energy

Transverse spectra are calculated for various types of hadrons stemming from AuAu collisions at s=200 GeV. We utilize a quark recombination model based on generalized Boltzmann–Gibbs thermodynamics for local hadron production at various break-up scenarios.

Analysis of temperature effects on hydrogen and OH adsorption on Pt(1 1 1), Pt(1 0 0) and Pt(1 1 0) by means of Gibbs thermodynamics

Gibbs thermodynamics in the presence of charge-transfer processes have been applied to characterize temperature effects on Pt(1 1 1), Pt(1 0 0) and Pt(1 1 0) electrodes in 0.1 M HClO 4 solutions. The present analysis allows the evaluation of the entropy of formation of the interphase for the three platinum basal planes, which is found to be structure-sensitive. In addition, these results have been used to test the suitability of the more common approach to characterize temperature effects, based on the application of a generalized isotherm. Besides, the comparison of both approaches allows the characterization of the different components involved in the entropy of hydrogen and OH adsorption, namely, the standard entropy of adsorption, the entropy due to lateral interactions, and the configurational entropy. Finally, the comparison with statistical mechanical calculations demonstrates that adsorbed hydrogen on Pt(1 1 1) and Pt(1 0 0) is rather mobile, while adsorbed OH is rather immobile. Adsorbed hydrogen on Pt(1 1 0) exhibits an unexpected behavior, since it appears to be rather immobile at low coverages, but it becomes very mobile at near saturation coverages.

Concept of reversible cleavage in surface tension of solids

“Wholly plastic (as in cleavage)” surface area changing is the widely accepted concept formally allowing the application of Gibbs thermodynamics to surface tension of solids considered as a reversible process of partly plastic and partly elastic deformation as recommended by the IUPAC. However, this concept contains some internal contradictions that are serious enough to raise doubts about its correctness and, consequently in validity of the thermodynamic analysis of surface tension of solid electrodes based on this concept. This analysis leads to the generalized Lippmann equation and eventually approves using the classical Lippman equation for solids. The latter is shown to be erroneous since it is based on the misleading concept of “reversible cleavage” and contradicts all available in situ experimental data.

Protein–water electrostatics and principles of bioenergetics

Despite its diversity, life universally relies on a simple basic mechanism of energy transfer in its energy chains-hopping electron transport between centers of electron localization on hydrated proteins and redox cofactors. Since many such hops connect the point of energy input with a catalytic site where energy is stored in chemical bonds, the question of energy losses in (nearly activationless) electron hops, i.e., energetic efficiency, becomes central for the understanding of the energetics of life. We show here that standard considerations based on rules of Gibbs thermodynamics are not sufficient, and the dynamics of the protein and the protein-water interface need to be involved. The rate of electronic transitions is primarily sensitive to the electrostatic potential at the center of electron localization. Numerical simulations show that the statistics of the electrostatic potential produced by hydration water are strongly non-Gaussian, with the breadth of the electrostatic noise far exceeding the expectations of the linear response. This phenomenon, which dramatically alters the energetic balance of a charge-transfer chain, is attributed to the formation of ferroelectric domains in the protein's hydration shell. These dynamically emerging and dissipating domains make the shell enveloping the protein highly polar, as gauged by the variance of the shell dipole which correlates with the variance of the protein dipole. The Stokes-shift dynamics of redox-active proteins are dominated by a slow component with the relaxation time of 100-500 ps. This slow relaxation mode is frozen on the time-scale of fast reactions, such as bacterial charge separation, resulting in a dramatically reduced reorganization free energy of fast electronic transitions. The electron transfer activation barrier becomes a function of the corresponding rate, self-consistently calculated from a non-ergodic version of the transition-state theory. The peculiar structure of the protein-water interface thus provides natural systems with two "non's"-non-Gaussian statistics and non-ergodic kinetics-to tune the efficiency of the redox energy transfer. Both act to reduce the amount of free energy released as heat in electronic transitions. These mechanisms are shown to increase the energetic efficiency of protein electron transfer by up to an order of magnitude compared to the "standard picture" based on canonical free energies and the linear response approximation. In other words, the protein-water tandem allows both the formation of a ferroelectric mesophase in the hydration shell and an efficient control of the energetics by manipulating the relaxation times.

Thermodynamically consistent, maximum dissipation, time-dependent models for non-equilibrium behavior

A thermodynamically consistent construction yields evolution equations for the non-equilibrium behavior of a body. Given 2 n thermodynamic variables divided into control variables y 1 , … , y n and state variables x 1 , … , x n and a thermodynamic function of the controls φ ˆ ( y 1 , … , y n ) such that x i = ∂ φ ˆ / ∂ y i for i = 1 , … , n define the distinguished states of the system, a generalized thermodynamic function is defined such that the distinguished states are determined by a zero-gradient condition. The remaining states are the non-equilibrium states. The construction of objective, time-dependent, non-equilibrium evolution equations for the thermodynamic state variables is based on the Lie time derivative and on a novel maximum dissipation criterion that supplements the second law of thermodynamics. If φ ˆ is the free energy, the evolution tends to the long-term states distinguished by φ ˆ and represents viscoelastic or viscoplastic behavior. If φ ˆ is the entropy production, then the construction gives the non-steady evolution of thermodynamic fluxes to steady states distinguished by φ ˆ and produces physically realistic finite velocity thermal and mass transport. The combination of these two sets of objective evolution equations and the balance laws is the constitutive model that defines the behavior of a non-equilibrium process. The construction is inspired by Gibbs thermodynamics rather than continuum thermodynamics, but the Clausius–Duhem inequality is deduced from a Gibbs one-form that defines admissible processes. The construction is not restricted to processes near equilibrium and reproduces, as validation, several well-known constitutive models as maximum dissipation processes in the sense used here.

TOWARDS POSSIBLE NON-EXTENSIVE THERMODYNAMICS OF ALGORITHMIC PROCESSING — STATISTICAL MECHANICS OF INSERTION SORT ALGORITHM

Tsallis entropy introduced in 1988 is considered to have obtained new possibilities to construct generalized thermodynamical basis for statistical physics expanding classical Boltzmann–Gibbs thermodynamics for nonequilibrium states. During the last two decades this q-generalized theory has been successfully applied to considerable amount of physically interesting complex phenomena. The authors would like to present a new view on the problem of algorithms computational complexity analysis by the example of the possible thermodynamical basis of the sorting process and its dynamical behavior. A classical approach to the analysis of the amount of resources needed for algorithmic computation is based on the assumption that the contact between the algorithm and the input data stream is a simple system, because only the worst-case time complexity is considered to minimize the dependency on specific instances. Meanwhile the article shows that this process can be governed by long-range dependencies with thermodynamical basis expressed by the specific shapes of probability distributions. The classical approach does not allow to describe all properties of processes (especially the dynamical behavior of algorithms) that can appear during the computer algorithmic processing even if one takes into account the average case analysis in computational complexity. The importance of this problem is still neglected especially if one realizes two important things. The first one: nowadays computer systems work also in an interactive mode and for better understanding of its possible behavior one needs a proper thermodynamical basis. The second one: computers from mathematical point of view are Turing machines but in reality they have physical implementations that need energy for processing and the problem of entropy production appears. That is why the thermodynamical analysis of the possible behavior of the simple insertion sort algorithm will be given here.

Towards possible [formula omitted]-generalizations of the Malthus and Verhulst growth models

Tsallis entropy introduced in 1988 based on the equation S q = 1 − ∑ i p i q q − 1 has been considered to obtain new possibilities for constructing a generalized thermodynamical basis for statistical physics, expanding classical Boltzmann–Gibbs thermodynamics for non-equilibrium states. During the last two decades this q -generalized theory has been successfully applied to a considerable amount of physically interesting complex phenomena. The authors wish to present a mathematical analysis of the possible q -generalizations of the Malthus and the Verhulst growth models based on Tsallis definitions of the q -logarithm and the q -exponential.

Kinetic thermodynamics as a physicochemical foundation for fabricating materials under conditions of self-assembly

A thermodynamic approach that allows creating materials with improved combination of properties is developed. The new theory is based on Gibbs thermodynamics, which is a very strict physical theory. Foundations of the phenomenological hierarchical kinetic quasi-equilibrium thermodynamics of quasi-closed systems are presented. It is assumed in the theory of kinetic thermodynamics that at any moment of time the state functions of the systems studied have real physical sense within a justified approximation. Hierarchical thermodynamics is applied successfully in studies of the processes of creation of materials under weakly nonequilibrium conditions and of the evolution of living systems.

Eikonal Representation of Thermodynamics

Potentials are integrals governed by a minimum principle. Taking into account the principle of least time the dependence of the position from time arising therefrom and called the geodesic slope excludes the position as a variable of potentials. Instead of using time und position as variables Hilbert's Independent Theorem presents an integral with respect to time whose integrand contains the geodesic slope. The Gibbs thermodynamics suspending the dependence of time is based on differential theorems. From the fact, however, that the Erdmann-WeierstraB comer conditions arising from Hilbert's theorem are in agreement with the relevant thermodynamic conditions of Gibbs we conclude that Hilbert's Independent Integral presents the action of thermodynamics. The thermodynamic action integral is dependent on the lower limit, the least action, and the higher limit presenting the actual state. Because of the reversibility every state between both limits is available. With the reduction of the higher limit into the past we return to the history of all events that happened in the course of milliards of years. The split of this recording into characteristic components is possible with the help of suitable analytical methods. These reveals the history and the origin of thermodynamic compounds. The genetic fingerprint is a well known example.

On the oxidation state and volatile behavior in multicomponent gas–melt equilibria

The equilibrium between a 4-component H 2O–CO 2–SO 2–H 2S gas phase and a 13-component silicate liquid made of 10 major oxides plus dissolved H 2O, CO 2, and S, is investigated by means of calculations involving homogeneous reactions in the gas phase and heterogeneous gas–liquid saturation modeling based on classical Gibbs thermodynamics and Toop–Samis polymeric approach. Sulfur is assumed to be present in two different oxidation states in the gas (sulfur dioxide and hydrogen sulfide) and liquid (sulfide and sulfate ions) phase, implying a dependence of the equilibrium conditions on the redox state of the system. Sulfur-bearing solid phases and Fe–O–S immiscible liquid are not accounted for in the modeling. The thermodynamic model is an extension of the one presented in Moretti et al. [Moretti R., Papale P. and Ottonello, G., 2003. A model for the saturation of C–H–O–S fluids in silicate melts. In: Oppenheimer C., Pyle D.M., Barclay J. (eds.) Volcanic Degassing, Geol. Soc. London Spec. Publ., 213, 81–101.] to account for iron speciation at high pressure and dissolved water contents. The consequences on the equilibrium conditions of different assumptions concerning the effective redox buffer in magma are examined through calculations made on two different liquids of shoshonitic and rhyolitic composition, determining the equilibrium conditions on the basis of (i) constant ferric to ferrous mass ratio, (ii) constant hydrogen sulfide to sulfur dioxide fugacity ratio, and (iii) constant oxygen fugacity relative to a solid–gas buffer (ΔNNO±0.5). Following Giggenbach [Giggenbach, W.F., 1996. Chemical composition of volcanic gases. In: Scarpa R., Tilling R.I. (eds.) Monitoring and Mitigation of Volcano Hazards, Springer-Berlin, 202–226.], the first two buffers are expected to be effective in basaltic and rhyolitic magmas, respectively, according to the most abundant reservoir of redox couples represented by iron in basalts, and sulfur in rhyolite. The model results show strongly nonlinear dependence of the equilibrium compositions in the gas and liquid phases, as well as of the oxidation state of the system, on the assumed redox buffer. Furthermore, for each assumed redox buffer, the pressure dependence of phase composition and oxidation state of the system also shows strongly nonlinear trends. The largest compositional differences are shown by sulfur species; however, the concentrations of water and carbon dioxide in the two phases at equilibrium also show nonnegligible dependence on the redox conditions. For each assumed redox buffer, sulfur dioxide in the gas phase, and sulfate ions in the liquid phase, are found to be present in appreciable quantities or represent the dominating sulfur species even at the largest employed pressures approaching 500 MPa. The more reliable redox buffers represented by constant ferric to ferrous mass ratio for shoshonite, and constant hydrogen sulfide to sulfur dioxide fugacity ratio for rhyolite, show that oxygen fugacity paths during magma depressurization strongly deviate from those parallel to NNO. Therefore, the characterization of the oxidation state in depressurizing magmas on the basis of deviations from solid buffers (usually NNO or QFM) may not be appropriate.