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14 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2026

Zentropy Theory for Transformative Functionalities of Magnetic and Superconducting Materials

The proposed research developed the zentropy theory through applications to complex magnetic materials and superconductors under the hypothesis that the emergent properties of complex magnetic materials and superconductors can be predicted by statistical mechanics of ergodic microstates with their partition functions computed from DFT-predicted free energies. The key objective is to develop approaches to systematically determine the types and number of microstates and the supercell size in DFT-based calculations through convergency of macroscopic functionalities, with the incorporation of our mixed-space approach accounting for the interactions between periodic supercells. In addition to use scientific intuitions to guide the design of important microstates, the key innovation of the proposed research is to integrate the domain knowledge and the material-property-descriptor database (MPDD) with 4 million microstates, which is supported by our deep neural network machine learning models (SIPFENN: structure-informed prediction of formation energy using neural networks) and integrated with our high throughput DFT Tool Kit (DFTTK). For complex magnetic materials, one of the objectives is to develop approaches to calculate short-range ordering from the statistical distribution of each microstate. For superconductors, the divergency of quasiparticle effective mass at a quantum critical point will be investigated, and the superconducting and non-superconducting microstates will be delineated through analysis of electronic band structure, density of states, charge density, and Fermi surface.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Zentropy theory for accurate prediction of free energy, volume, and thermal expansion without fitting parameters

Based on statistical mechanics, a macroscopically homogeneous system, i.e., a single phase in the present context, is composed of many independent configurations that the system embraces. The macroscopical properties of the system are determined by the properties and statistical probabilities of those configurations with respect to external conditions. The volume of a single phase is thus the weighted sum of the volumes of all configurations. Consequently, the derivative of the volume to temperature of a single phase depends on both the derivatives of the volumes of every configuration to temperature and the derivatives of their statistical probabilities to temperature, with the latter introducing nonlinear emergent behaviors. It is shown that the derivative of the volume to the temperature of the single phase can be negative, i.e., negative thermal expansion, due to the symmetry-breaking non-ground-state configurations with smaller volumes than that of the ground-state configuration and the rapid increase of the statistical probabilities of the former, and negative thermal expansion can be predicted without fitting parameters from the zentropy theory that combines quantum mechanics and statistical mechanics with the free energy of each configuration predicted from quantum mechanics and the partition function of each configuration calculated from its free energy.

36 MATERIALS SCIENCE↗
DOE OSTI2026

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. Furthermore, 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.

42 ENGINEERING↗
DOE OSTI2024

Quantitative Predictive Theories through Integrating Quantum, Statistical, Equilibrium, and Nonequilibrium Thermodynamics

Today's thermodynamics is largely based on the combined law for equilibrium systems and statistical mechanics derived by Gibbs in 1873 and 1901, respectively, while irreversible thermodynamics for nonequilibrium systems resides essentially on the Onsager Theorem as a separate branch of thermodynamics developed in 1930s. Between them, quantum mechanics was invented and was quantitatively solved in terms of density functional theory (DFT) in 1960s. Furthermore, these three scientific domains operate based on different principles and are very much separated from each other. In analogy to the parable of the blind men and the elephant articulated by Perdew, they individually represent different portions of a complex system and thus are incomplete by themselves alone, resulting in the lack of quantitative agreement between their predictions and experimental observations. Over the last two decades, the author's group has developed a multiscale entropy approach (recently termed as zentropy theory) that integrates DFT-based quantum mechanics and Gibbs statistical mechanics and is capable of accurately predicting entropy and free energy of complex systems. Furthermore, in combination with the combined law for nonequilibrium systems developed by Hillert, the author developed the theory of cross phenomena beyond the phenomenological Onsager Theorem. The zentropy theory and theory of cross phenomena jointly provide quantitative predictive theories for systems from electronic to any observable scales as reviewed in the present work.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

Building materials genome from ground‐state configuration to engineering advance

Individual phases are commonly considered as the building blocks of materials. However, the accurate theoretical prediction of properties of individual phases remains elusive. The top-down approach by decoding genomic building blocks of individual phases from experimental observations is nonunique. The density functional theory (DFT), as a state-of-the-art solution of quantum mechanics, prescribes the existence of a ground-state configuration at 0 K for a given system. It is self-evident that the ground-state configuration alone is insufficient to describe a phase at finite temperatures as symmetry-breaking non-ground-state configurations are excited statistically at temperatures above 0 K. Our multiscale entropy approach (recently terms as Zentropy theory) postulates that the entropy of a phase is composed of the sum of the entropy of each configuration weighted by its probability plus the configurational entropy among all configurations. Consequently, the partition function of each configuration in statistical mechanics needs to be evaluated by its free energy rather than total energy. The combination of the ground-state and symmetry-breaking non-ground-state configurations represents the building blocks of materials and can be used to quantitatively predict free energy of individual phases with the free energy of each configuration predicted from DFT as well as all properties derived from free energy of individual phases.

CALPHAD↗
DOE OSTI2023

Parameter-free prediction of phase transition in PbTiO 3 through combination of quantum mechanics and statistical mechanics

Thermodynamics of ferroelectric materials and their ferroelectric to paraelectric (FE-PE) transitions is commonly described by the phenomenological Landau theory and more recently by effective Hamiltonian and various potentials, all with model parameters fitted to experimental or theoretical data. Here, in this work, we show that the zentropy theory, which considers the total entropy of a system as a weighted sum of entropies of configurations that the system may experience and the statistical entropy among the configurations, can predict the FE-PE transition without fitting parameters. For PbTiO 3 , the configurations are identified as the FE configurations with 90- or 180° domain walls in addition to the ground state FE configuration without domain wall. With the domain wall energies predicted from first-principles based on density functional theory in the literature as the only inputs, the FE-PE transition for PbTiO 3 is predicted showing remarkable agreement with experiments, unveiling the microscopic fundamentals of the transition.

36 MATERIALS SCIENCE↗
DOE OSTI2026

ZENN: A thermodynamics-inspired computational framework for heterogeneous data–driven modeling

Traditional entropy-based methods—such as cross-entropy loss in classification problems—have long been essential tools for representing the information uncertainty and physical disorder in data and for developing artificial intelligence algorithms. However, the rapid growth of data across various domains has introduced new challenges, particularly the integration of heterogeneous datasets with intrinsic disparities. To address this, we introduce a zentropy-enhanced neural network (ZENN), extending zentropy theory into the data science domain via intrinsic entropy, enabling more effective learning from heterogeneous data sources. ZENN simultaneously learns both energy and intrinsic entropy components, capturing the underlying structure of multisource data. To support this, we redesign the neural network architecture to better reflect the intrinsic properties and variability inherent in diverse datasets. We demonstrate the effectiveness of ZENN on classification tasks and energy landscape reconstructions, showing its superior generalization capabilities and robustness-particularly in predicting high-order derivatives. In image and text classification tasks, ZENN demonstrates superior generalization by introducing a learnable temperature variable that models latent multisource heterogeneity, allowing it to surpass state-of-the-art models on CIFAR-10/100, BBC News, and AG News. As a practical application in materials science, we employ ZENN to reconstruct the Helmholtz energy landscape of Fe3Pt using data generated from density functional theory and capture key material behaviors, including negative thermal expansion and the critical point in the temperature–pressure space. Overall, this work presents a zentropy-grounded framework for data-driven machine learning, positioning ZENN as a versatile and robust approach for scientific problems involving complex, heterogeneous datasets.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Ab initio studies on structural and thermodynamic properties of magnetic Fe

The present work systematically investigates the total energy, phonon spectra, and thermodynamic properties of different polymorphs of pure Fe, i.e., FCC, BCC, and HCP, with the ab initio approach, considering various magnetic configurations. In general, the calculated energy vs. volume curves and phonon spectra agree well with previous calculations and the experimental data. In addition, their thermodynamic properties are estimated by the quasiharmonic approximation (QHA). Specifically, a superposition approach based on the latest Zentropy theory was utilized to predict magnetic transition temperatures and thermodynamic properties of pure Fe. With the ensemble of the partition function considering the multiplicity of each magnetic microstate, the current work successfully reproduced the Curie/Néel temperature and the Schottky anomaly of heat capacity in FCC, BCC, and HCP Fe purely based on the ab initio input, which exhibits good agreement with the experimental data and CALPHAD modeling.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Ab Initio Modeling on The Thermodynamic and Temperature-Dependent Elastic Properties of Subsystems of The FCC FeNiCoCr Medium Entropy Alloys (MEAs)

The stability, phonon spectra, thermodynamic, and temperature-dependent elasticity of subsystems of the FCC FeNiCoCr MEAs are systematically studied by the ab initio approach. Especially, the quasi-harmonic approximation (QHA) and the innovative Zentropy theory were utilized to predict the thermodynamic properties and elastic properties of FeNi, NiCo, FeNiCo, and FeNiCoCr MEAs with the consideration of magnetic transition. With the ensemble of the partition function based on the multiplicity of each magnetic configuration, the current work successfully reproduced the Curie temperature and the Schottky anomaly of heat capacity of these four MEAs purely based on the ab initio input. Meanwhile, the elastic properties of these alloys at finite temperatures are also successfully predicted with the consideration of magnetic transition. The overall results agree well with the available experimental data and CALPHAD prediction.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Fundamentals on dependence of volume on pressure and temperature

The common wisdom that volume decreases with pressure and increases with temperature is analyzed in terms of Hillert nonequilibrium thermodynamics in the present work. It is shown that the derivative of volume to pressure in a stable system is always negative, i.e., volume decreases with the increase of pressure, when all other natural variables of the system are kept constant. This originates from the stability requirement that the conjugate variables, such as volume and negative pressure, must change in the same direction in a stable system. Consequently, since volume and temperature are not conjugate variables, they do not have to change in the same direction and thus do change in opposite directions in both natural and man-made systems. It is shown that the decrease of volume with the increase of temperature, commonly referred as negative thermal expansion (NTE) in the literature, originates from the statistical competitions of configurations in the system when the volumes of metastable configurations are smaller than that of the ground-state configuration. As a result, it is demonstrated that the zentropy theory can concisely explain and accurately predict NTE based on the density functional theory without fitting parameters.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Quantifying the degree of disorder and associated phenomena in materials through zentropy: Illustrated with Invar Fe 3 Pt

Disorder exists in all materials at finite temperatures, implicating that the degree of disorder (ƒ DoD ) is a key parameter to tailor macroscopic functionalities of materials. Here, in this work, we propose to quantify ƒ DoD as a function of temperature using properties such as configurational entropy predicted by zentropy – a theory to represent total entropy of a system via a nested formula through the integration of quantum mechanics and statistical mechanics. Taking Invar Fe 3 Pt as an example through first-principles based calculations from 0 K to finite temperatures, we demonstrate the capability of the present approach in predicting ƒ DoD and associated phenomena in Fe 3 Pt such as Curie temperature and negative thermal expansion, showing a good agreement with experimental data.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Revising the theory of cross phenomena

In the present paper, the theory of cross phenomena (TCP) proposed by the present author is revised by starting from the first law of thermodynamics instead of the combined law of thermodynamics in the previous publication because the potentials in the latter are not independent of each other. TCP aims to derive the flux equations with respect to independent driving forces from the fundamental laws of thermodynamics rather than the phenomenological Onsager flux equations, which are flawed as shown by the revised TCP.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Predicting phase transitions in PbTi⁢O 3 using zentropy through quasiharmonic phonon calculations

According to x-ray diffraction (XRD) measurements, PbTi⁢O 3 undergoes a phase transition from a tetragonal ferroelectric (FE) phase to a cubic paraelectric phase at 763 K. However, x-ray absorption fine-structure (XAFS) measurements indicate that PbTi⁢O 3 is locally tetragonal even after the phase transition. The difference in these results is because XAFS measurements can probe local features of a structure, while XRD averages over such local features. For both measurements to be consistent, PbTi⁢O 3 is macroscopically cubic but locally tetragonal after the phase transition. Despite this, most models, such as the Landau-Ginsburg-Devonshire theory and effective Hamiltonians, are still unable to explain this phenomenon. Moreover, these methods involve model parameters fitted to experimental or theoretical data and do not consider other tetragonal configurations, such as domain walls, to predict the phase transition. In our previous study, we used our zentropy approach to predict the phase transition by considering the tetragonal FE ground-state configuration and the tetragonal 90° and 180° domain wall configurations with their total energies at 0 K. Here, in this paper, the Helmholtz energies of the three configurations are obtained from density functional theory calculations through energy-volume curves and phonon calculations. The predicted phase transition temperature using the meta-GGA 𝑟 2⁢ SCAN and revised multiplicities of configurations is 716 K, showing good agreement with the experimental value of 763 K.

36 MATERIALS SCIENCE↗