Diffusion kinetics of silver (Ag) in nuclear grade graphites, IG-110 and ZXF-5Q were investigated from 500 to 900 °C by the thin film diffusion technique. Dynamic secondary ion mass spectroscopy was utilized for depth profiling of concentrations and diffusion coefficients were calculated utilizing solutions to Fick’s second law. Ag was found to diffuse through the bulk of both graphite grades with diffusivities on the order of 10−20 to 10−22 m2/s. For IG-110 graphite, temperature-dependence of Ag diffusion coefficients yielded activation energy and pre-exponential factor of 66.7 kJ/mol and 2.11 x 10−17 m2/s, respectively. For ZXF-5Q graphite, activation energy and pre-exponential factor were determined to be 66.6 kJ/mol and 2.04 x 10−17 m2/s, respectively. These diffusion coefficients were very consistent despite the difference grain size, porosity and filler presence for the two types of graphite.
The phase equilibria of the Dy-Fe-B ternary system at 873 and 1073 K were investigated experimentally in this work using the equilibrated alloy method combined with scanning electron microscopy (SEM) with energy dispersive spectroscopy (EDS) and X-ray powder diffraction (XRD). The experimental analysis identified five stable ternary intermetallic compounds, namely Dy2Fe14B (τ1) with the Nd2Fe14B-type structure and space group P42/mnm, DyFe4B4 (τ3) with the Nd1+εFe4B4-type structure and space group Pccn, Dy5Fe2B6 (τ4) with the Pr5−xCo2+xB6-type structure and space group R 3 m, Dy3FeB7 (τ5) with the Y3ReB7-type structure and space group Cmcm, and DyFeB4 (τ6) with the YCrB4-type structure and space group Pbam, at 873 and 1073 K. It is noteworthy that DyFe2B2 (τ2) reported previously in the literature was not found in the present analyses of the measured equilibrated alloys. Finally, based on the experimentally determined results, two isothermal sections of the Dy-Fe-B ternary system at 873 and 1073 K were constructed, providing key experimental data and a theoretical basis for the thermodynamic calculation of this ternary system and the design of high-performance and low-cost Nd-Dy-Fe-B permanent magnets.
Vapor–liquid equilibria and impurity separation behavior in Pb-Zn, Pb-Bi and Pb-Ag binary systems were investigated using CALPHAD-based activity calculations combined with Gibbs-energy minimization equilibrium simulations performed in HSC Chemistry software. Separation coefficients of Zn, Bi, and Ag were evaluated as a function of temperature and composition to quantify the relative volatility of impurity elements with respect to lead. The results reveal three distinct thermodynamic regimes. The Pb-Zn system exhibits strong positive deviations from ideality and extremely high separation coefficients (βZn ≫ 1), indicating a very high selective volatility of Zn relative to Pb. The Pb-Bi system shows near-ideal solution characteristics with separation coefficients approaching, but remaining below unity (βBi ≈ 0.56-0.93), indicating limited separation efficiency. In contrast, the Pb-Ag system is characterized by very low separation coefficients (βAg ≪ 1), reflecting the thermodynamic stability of Ag in the condensed phase and its negligible tendency to evaporate relative to Pb. Equilibrium composition diagrams demonstrate the combined influence of temperature and total pressure on phase redistribution. Both increasing temperature and decreasing pressure promote vapor-phase enrichment of impurities; however, these conditions simultaneously enhance lead volatilization, thereby reducing separation selectivity under single-stage equilibrium conditions. The theoretical predictions are consistent with experimental data reported in the literature, where efficient impurity removal was achieved through two-stage high-low temperature vacuum distillation of crude lead. The present study provides a quantitative thermodynamic framework for evaluating the feasibility and limitations of vacuum refining of lead alloys.
In first-principles-based calculations of thermodynamic quantities "thermodynamic consistency" has not been guaranteed when applying disparate approximation methods to diverse phases such as, compounds, solid solutions, or liquids, constituting alloy phase diagrams. In this study, we performed tailored calculations for each phase in the Cu-Ni-Si system and verified consistency using phase equilibrium calculations and likelihood analysis. The resulting phase diagrams generally reproduced experimental ones, despite extreme sensitivity to minute free energy differences. Furthermore, likelihood analysis quantitatively demonstrated that free energy residuals fall within a narrow 1-2 kJ/mol range. This indicates that a high degree of thermodynamic consistency is preserved across all phases. Therefore, we newly define the computational process encompassing the first-principles uncertainty revealed by this likelihood analysis as the “theoretical phase diagram calculation method.” By applying this method to the binaries and ternary of the Cu-Ni-Si system, we demonstrated its capability to reproduce experimental phase diagrams with high precision.
A comprehensive thermodynamic assessment of the Ni-Ti-Zr ternary system has been conducted using the CALPHAD (CALculation of PHAse Diagrams) methodology. Thermodynamic models for the constituent binary systems—Ni-Ti and Ni-Zr—were critically evaluated and refined based on recent experimental data and literature assessments. These models, along with an established description for the Ti-Zr system, were extrapolated to the ternary domain, where further optimization was performed using newly available experimental phase equilibria, isothermal sections, and invariant reactions. The resulting thermodynamic description captures the main features of the Ni-Ti-Zr phase diagram, including the liquidus and solidus projections, as well as the isothermal sections and isopleths. This model offers a useful basis for exploring and developing Ni-Ti-Zr-based materials, such as shape memory alloys, metallic glasses, and high-temperature structural alloys.
Diffusion in the ordered V3Au and disordered face-centred cubic solid-solution Au(V)ss phases in the binary V-Au system is studied. Interdiffusion coefficients increase with increase in the Au content in Au(V)ss due to increased pre-exponential factor and reduced migration barrier. Since the thermodynamic factor decreases with increasing Au content in Au(V)ss, interdiffusion may likely be kinetically-driven. Integrated diffusion coefficients of V3Au were determined with an activation energy of 87 ± 19 kJ/mol, indicating the possible dominance of grain boundary diffusion.
Magnetite concentrates from different parts of the Mesabi Range, exhibiting varying tendencies to encapsulate, were reduced using thermogravimetric analysis (TGA) and a high-temperature confocal scanning laser microscope (CSLM). Gas-phase mass-transfer conditions were intentionally varied by employing two reactors with different geometries: the TGA produced significantly slower reduction, whereas the CSLM setup yielded reduction rates up to 20 times faster under mass-transfer-controlled conditions. A preliminary mechanism is proposed to explain the transition from a porous iron product to a dense encapsulating layer. Encapsulation is interpreted as the outcome of competition between pore creation, driven by the reduction rate, and pore elimination, driven by surface diffusion. When pore formation is insufficient to counteract pore coarsening, a dense iron layer develops and restricts further reduction.
Thermodynamics is traditionally viewed as a theory of equilibrium because Gibbs formulated the combined law strictly for equilibrium systems. This historical limitation motivated the emergence of irreversible thermodynamics as a distinct field. Subsequent advances, including Kaufman’s introduction of lattice stability for quantitative treatment of nonequilibrium phases, Hillert’s integration of entropy production into the combined law, and Ågren’s development of the modern theory of atomic mobility, collectively extended Gibbs’s framework to describe real materials under evolving thermodynamic driving forces. The present author further refined Hillert’s nonequilibrium formalism by introducing partial internal energy, partial entropy, and partial volume for each component directly into the first, second, and combined laws. This refinement yields an explicit definition for the chemical potential in terms of these partial quantities, resolving subtle interdependencies among entropy, volume, and composition in open systems. Building on this foundation and Ågren’s mobility theory, the author developed and later revised the theory of cross phenomena, deriving transport equations from the first law and addressing limitations inherent in phenomenological Onsager formulations. In parallel, the author and collaborators established zentropy theory, which unifies quantum mechanics and Gibbs statistical mechanics to predict entropy and Helmholtz energy from a full ensemble of symmetry-broken configurations. This framework enables quantitative prediction of Helmholtz energy landscapes with basins, ridges, and apices across stable, metastable, and unstable states. Together, these developments establish a unified thermodynamic framework spanning equilibrium, nonequilibrium, statistical, and quantum descriptions across all-scales, and provide a thermodynamics-based artificial intelligence (AI) framework exemplified by the zentropy-enhanced neural network (ZENN), yielding physically grounded, interpretable predictions. This new AI framework also incorporates built-in safety mechanisms, including structural containment through configuration partitioning and dynamic containment via zentropy-based regulation of driving forces and system stability.
Refractory multi-principal element alloys (RMPEAs) are promising candidates for replacing Ni-based and Co-based superalloys in high temperature gas turbine engines due to their high melting temperatures. However, these alloys generally exhibit poor oxidation resistance, and will therefore require oxidation-resistant environmental barrier coatings, which historically have provided protection via formation of a dense alumina scale. Unfortunately, current alumina-forming alloys have inadequate temperature capabilities and are thermochemically incompatible with state-of-the-art RMPEAs, motivating the design of new oxidation-resistant coatings. Thermodynamic modeling (Thermo-Calc) was leveraged to calculate the elemental activity and phase constitutions of tens of thousands of alloys in the Nb-Mo-Ti-Al-Hf composition space. Alloys were filtered based upon their Al activity and phase constitution, which gave shape to the remaining data. The remaining alloys were dimensionally reduced using principal component analysis and k-means clustering to identify promising families of alloys for synthesis. The methodology was validated by synthesis and testing. Alloys that oxidize favorably to form a continuous and adherent alumina scale at 1400 °C under isothermal conditions were identified. The methodology offers an efficient means of designing coatings for RMPEAs and is transferrable to coating design for a wide range of substrate classes.
In this paper aspects of the first two papers written by Gibbs on the topic of thermodynamics are discussed. These early papers set the stage for Gibbs’ major work on thermodynamics. His use of graphical methods and geometry are reviewed and display the reasoning behind his mathematical construction of thermodynamics. This paper ends with later extensions of his work to interesting graphical plots as well as an example taken from his later work on Statistical Mechanics.
Phase equilibria and solid-state transformations in the Ni-rich region of the Ti–Zr–Ni system were investigated through a combined application of scanning electron microscopy, electron probe microanalysis, x ray diffraction, and differential thermal analysis. Comprehensive analysis enabled the construction of isothermal sections at 960 °C and 750 °C, revealing substantial differences in phase topology across this temperature interval. At 960 °C, the ternary system exhibits equilibria that differ markedly from the Zr–Ni binary: the intermetallic compounds ZrNi3 and Zr9Ni11 remain stable due to the stabilizing effect of Ti, forming isolated phase fields that are not present in the binary diagram. The formation temperature of ZrNi3 was established as 1000 °C, while Zr9Ni11 decomposes below 750 °C. At 750 °C, the phase relations significantly differ from those at 960 °C as a result of a sequence of solid-state reactions. The most noteworthy of these is the decomposition of the ternary compound τ2, producing Zr8Ni21, Zr7Ni10, and TiNi3, and fundamentally altering the topology of the isothermal section. Intermediate annealing at 900 °C provided insight into the transitional behavior between the two isothermal sections, revealing incomplete transformations and confirming the peritectoid formation of ZrNi3 from Zr2Ni7, τ3, and Zr8Ni21. The resulting reaction scheme for the ZrNi–Ni–TiNi subsystem integrates high-temperature stability, intermediate transformations, and low-temperature decomposition pathways. These findings establish a coherent thermochemical framework for the Ni-rich portion of the Ti–Zr–Ni system and provide essential reference data for thermodynamic modeling and the design of Ti–Zr–Ni intermetallic alloys.
The phase equilibria of the Zn-Cu-Cr ternary system at 450 and 600 °C were studied using equilibrium alloy method. The phase constitution of the alloys were analyzed by means of the scanning electron microscope equipped with energy dispersive x-ray spectroscopy (SEM-EDS), and x-ray diffraction (XRD). The results show that five three-phase regions exist in the isothermal section at 450 °C, and four three-phase regions exist in the isothermal section at 600 °C. No new ternary compound was found in the system. The (Cr) phase is found in equilibrium with all phases. At 450 °C the solubility of Cu in CrZn17 is 0.69 at.
Despite the potential of copper and silver halogen-containing argyrodites as environmentally friendly functional materials, phase equilibria in the corresponding systems remain virtually unstudied. In this study, phase equilibria in the quasi-ternary Ag2Se-AgI-GeSe2 system are investigated using differential thermal and X-ray phase analysis, as well as microhardness measurements. Solid-phase equilibria diagrams at 300 and 750 K, a liquidus surface projection, and several polythermal sections of the phase diagram are constructed. The primary crystallization fields of four phases, including ion-conducting solid solutions of the Ag8−xGeSe6−xIx (δ-phase), were determined. The presented polythermal sections of the phase diagram clearly demonstrate the processes of phase crystallization from the melt. The relatively large primary crystallization area of the δ-phase facilitates the growth of its single crystals with various compositions by directional solution-melt crystallization over a wide range of compositions and temperatures.
Phase equilibria in the Fe-rich corner of the quaternary Fe-Al-Ti-C system upon solidification and at 1000 °C were studied using differential thermal analysis (DTA), powder x-ray diffraction (XRD), microscopic examination and electron probe microanalysis (EPMA). The partial liquidus and solidus projections and the isothermal section at 1000 °C were constructed. The results are presented on the Fe-25Al-Ti-C composition triangle, where the corner Fe-25Al corresponds to the disordered (αFe) solid solution or its ordered derivatives FeAl or Fe3Al, depending on the temperature. The boundaries of the primary crystallization regions of the phases, types and coordinates of mono- and bivariant equilibria are determined. The liquidus surface in the studied region is characterized by the regions of primary crystallization of the (αFe), Fe2Ti, Fe3AlC (κ-carbide), TiC and graphite (C) phases. The solidus surface and isothermal section at 1000 °C are defined by the co-existence of the TiC phase with all phases of the system forming two three-phase regions, which form by transition (U) type four-phase reactions.
Experimental investigation of the phase equilibria in the Ce-Co-Cu ternary system was conducted via the equilibrated alloy technique, utilizing x-ray diffraction (XRD) and scanning electron microscopy coupled with energy dispersive spectroscopy (SEM-EDS). Ten binary intermetallic compounds, CeCo2, CeCo3, Ce2Co7, Ce5Co19, CeCo5, Ce2Co17, CeCu2, CeCu4, CeCu5, and CeCu6, were confirmed, and no ternary intermetallic compounds were detected. It was found that the continuous solid solution phase Ce(Co, Cu)5 was formed between the CeCo5 and CeCu5 binary intermetallic compounds due to having the same crystal structure. The solid solubilities of Cu in CeCo2, CeCo3, Ce2Co7, Ce5Co19, and Ce2Co17 as well as that of Co in CeCu2, CeCu4, and CeCu6 were measured. Furthermore, a thermodynamic assessment of the Ce-Co-Cu ternary system was performed via the calculation of phase diagrams (CALPHAD) approach, incorporating both the experimental data obtained in this study and available literature results. The calculated isothermal sections exhibit good agreement with the experimental observations. Consequently, a set of self-consistent thermodynamic parameters has been acquired, providing a solid foundation for developing a thermodynamic database for Sm-Co-based permanent magnets containing abundant rare-earth and transition metals.
The activities of the constituent metals in the binary solid alloys Cu-Ni at 973 K, Ni-Pt at 1625 K, and Cu-Pt at 1350 K, as well as their excess Gibbs energy and the activities of Cu in the ternary solid alloys Cu-Ni-Pt at 1000 K for three cross-sections, i.e., for different ratios of nickel to platinum, XNi:XPt = 2:1, 1:1, and 1:2, have been computed using the statistical thermodynamic model based on the free volume concept, i.e., the molecular interaction volume model (MIVM). The predicted values have been compared with the available experimental data and analyzed for the binary solid solutions. In addition, the excess Gibbs energy of mixing (ΔGXS) of the ternary Cu-Ni-Pt alloys for all three cross-sections has been calculated using the same model. The agreement between the theoretical and experimental results for the binary solid alloys is found to be satisfactory, confirming the suitability of the model in predicting thermodynamic properties and interaction behavior in ternary and multicomponent metallic solid solutions.
This study introduces an integrated design and optimization framework for AlCoCrFeNi-based composites reinforced with graphene nanoplatelets (GNPs), combining machine learning-driven predictive modelling with experimental validation to achieve uniform reinforcement distribution across a predefined microstructure. A comprehensive dataset was constructed incorporating ten elements (Al, Co, Cr, Fe, Ni, Cu, Mn, Ti, V, Mo) with carbon additions, where carbon specifically represents various reinforcement phases, including graphene nanoplatelets, enabling diverse carbon-based reinforcement strategies. A comparative analysis of five machine learning models was performed using six thermodynamic descriptors: valence electron concentration (VEC), atomic size difference (δ), electronegativity difference (Δχ), mixing enthalpy (ΔHmix), mixing entropy (ΔSmix) and Gibbs energy of mixing (ΔGmix) for the 750-sample dataset. Both random forest (RF) and extreme gradient boosting (XGBoost) demonstrated superior predictive accuracy (test accuracy: 0.94927; receiver-operating characteristic curve - area under the curve (ROC-AUC): 0.99446-0.99627; 10-fold cross-validation: 0.83315-0.85184), consistently identifying ΔHmix as the most critical feature influencing phase stability. X-ray diffraction confirmed dual-phase (FCC + BCC) structures in all samples, with BCC phase volume fraction increasing from 77.25
The brain is a dissipative, far-from-equilibrium system that continuously consumes energy to sustain structured neural activity while producing entropy. How system-level interactions among energy, entropy, structure, and physiological constraints give rise to cognition through distributed computation remains a central challenge in neuroscience. Existing free-energy frameworks have provided valuable conceptual insights, yet many remain largely metaphorical and lack explicit biological grounding, limiting the specificity and falsifiability of their predictions. Here, we propose a biologically grounded, thermodynamically inspired framework in which brain dynamics emerge from a continual balance between stability and flexibility, formalized as minimizing a Gibbs-free-energy-like function (ΔG = ΔH − TΔS). Stable neural configurations correspond to low-enthalpy, ordered states that support reliable function, whereas flexible configurations correspond to higher-entropy states that enable adaptive behavior. We further propose that neural population dynamics can be described as evolving on a hyperbolic statistical landscape shaped by metabolic, biophysical, anatomical, and bodily constraints. Neural activity samples this landscape, and experience gradually reshapes it over time. Metastable brain states occupy local free-energy minima, with transitions governed by Boltzmann-like probabilities. Learning sculpts the landscape via synaptic optimization and pruning, reducing entropy and stabilizing task-relevant dynamics. Importantly, this framework accounts for adaptive reallocation of neural resources under physiological stress, whereby higher-order cognitive functions may be transiently constrained to prioritize survival-essential processes, thereby predicting that cognitive function can remain rescuable in some cases despite structural damage. Together, this quantitative thermodynamic and geometric framework provides a unified foundation for understanding cognition as an emergent, system-level property of whole-brain dynamics.
This paper develops a unified phenomenological description of solidification by using a quaternionic orientational order parameter to represent local atomic rotations in undercooled liquids. The novelty of the work is that the same topological language is used to describe both crystallization and glass formation. In the crystalline pathway, quantized misorientation defects bind, order, and support long-range orientational and translational coherence. In the glassy pathway, those defects proliferate and remain frustrated, producing a rigid but non-periodic solid. The competition is organized by a non-thermal tuning parameter g = U/J that balances orientational stiffness J against localization U, in close analogy with duality ideas known from Josephson-junction arrays. Within this framework, geometrical frustration explains persistent defect skeletons in topologically close-packed phases and the opposite temperature dependences of thermal conductivity in crystals and glasses are linked to the presence or loss of coherent heat-carrying vibrational modes. The paper is intended as a materials-oriented topological phenomenology that complements, rather than replaces, atomistic and first-principles approaches.
High-temperature shape memory alloys (SMAs) are essential for advanced actuation in aerospace and energy systems, yet their continued development is constrained by challenges in both equilibrium and metastable phase prediction, transformation temperature control, and material sustainability. This study presents a data-driven framework for discovery and multi-objective optimization of novel, concentrated, multi-component SMA compositions that integrates high-throughput CALPHAD (calculation of phase diagrams) with machine learning (ML). The framework is used to identify regions of B2 phase which have potential for exhibiting transitions to the B19’ phase upon cooling. An Extra Trees regression model was trained to predict martensitic and austenitic transformation temperatures. Compositions were selected for experimental validation using cost and supply chain risk metrics to promote practical viability. Selected compositions were synthesized via arc melting for experimental validation using differential scanning calorimetry and dilatometry. The integrated approach enables rapid exploration of complex compositional spaces, linking thermodynamic calculations with functional behavior and real-world constraints. Strengths and limitations of ML in the alloy optimization context are exposed and discussed, along with prospects of such a framework as a scalable pathway for accelerating the discovery and optimization of high-performance SMAs with tailored transformation behavior.
