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

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https://doi.org/10.1016/j.calphad.2023.102580Get rights and content

Abstract

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 2^nd 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.

Introduction

This paper is based on the author's presentation given at the CALPHAD Global 2021 virtual conference on the perspectives of CALPHAD modeling in next 50 years in view of its success in last 50 years. The future is hard to predict due to its intrinsic uncertainty in terms of probability of many possible events. Nevertheless, it is important to have perspectives on how the future may look like in both short- and long-terms based on past knowledge and anticipated trajectories. In 2020, in celebrating the 50^th anniversary of “The Bridge” published by National Academy of Engineering, Sinnott and Liu (the present author) attempted such an effort on “Predicted Advances in the Design of New Materials” [1]. With the prehistory and protohistory of humanity divided into three ages in terms of materials of stone, bronze and iron followed by the three industry revolutions in terms of steam power, electricity, and computerization, we are now entering the 4th industry revolution, i.e., Industry 4.0. Industry 4.0 is commonly thought of as the integration of cyber-physical systems, where the physical, digital, and biological worlds are seamlessly unified to form a system with many autonomous subsystems enabled by advanced materials. In other words, digitization of materials and their manufacturing into functional devices in terms of Materials 4.0 and Manufacturing 4.0 [2]. Such digitization will demand increasingly more efficient development and deployment of materials with emergent properties to meet the performance requirements under extreme conditions. Sinnott and Liu [1] concluded that when this integrated system is fully implemented, the residuals from the design, manufacturing, service, and recycling of materials can be drastically reduced, thus lessening the impact of materials usage on the environment.

In the last century, digitization of materials knowledge progressed significantly, including the digitization of the Schrödinger equation in quantum mechanics [3,4] by the density functional theory (DFT) [5,6], resulting in massive digital databases of material properties predicted using high-performance computers, and thermodynamics by the CALculation of PHAse Diagrams (CALPHAD) method [[7], [8], [9], [10]], resulting in CALPHAD databases widely used in academia and industry for education and design of technologically important materials. Those data together with models and mechanistic correlations are enabling the development of artificial intelligence (AI) to connect the data through machine learning (ML) algorithms and deep neural networks (DNNs) [11]. While DFT-based calculations have provided important input data for CALPHAD modeling [12], it is currently still necessary to refine the CALPHAD model parameters using experimental data in order to accurately reproduce experimental observations, particularly phase transitions [13,14]. The need of such refinements significantly hinders the computational discovery and design of materials. To fully understand the differences between DFT-based calculations and CALPHAD modeling, it is necessary to dive deep into their fundamentals and build connections so that in the future the CALPHAD model parameters can be evaluated solely from the DFT-based calculations with experiments as the validation of predictions.

Thermodynamics is a science concerning the state of a system, whether it is stable, metastable, or unstable, when interacting with its surroundings. The interactions can involve exchanges of any combinations of heat, work, and mass between the system and the surroundings, defined by the boundary conditions. The typical work includes contributions from the external mechanic, electric and magnetic fields. Thermodynamics is commonly divided into four branches, i.e., classical Gibbs thermodynamics, statistical thermodynamics, quantum thermodynamics, and irreversible thermodynamics. In a recent overview article [15], the author discussed fundamental thermodynamics, thermodynamic modeling, and the applications of computational thermodynamics. In another recent perspective article [16], the author focused on irreversible thermodynamics as part of a more comprehensive framework of thermodynamics.

In the present paper, the fundamentals of thermodynamics will be reviewed through the derivation of the combined law of thermodynamics with the entropy production due to internal processes and thus without the constraint of equilibrium. The four branches of thermodynamics will then be discussed individually along with their contributions to the combined law of thermodynamics and their integration into a holistic view of thermodynamics. At the end, the author's perspectives on the CALPHAD modeling in the next 50 years will be discussed.

Section snippets

Review of the fundamentals of thermodynamics

The fundamentals of thermodynamics are centered on the first and second laws of thermodynamics and their combination into the combined law of thermodynamics. Since the first and second laws of thermodynamics are represented by an equality and an inequality, respectively, they had remained separately until Gibbs combined them to create the combined law of thermodynamics [[17], [18], [19]]. The first law of thermodynamics describes the interactions between the system and its surroundings and

Combined law of thermodynamics for equilibrium systems

Gibbs [17,18] first considered closed equilibrium systems under hydrostatic pressure (

), i.e., and , and Eq. (10) and Eq. (11) are thus reduced to the following equations as shown in all textbookswhere the negative sign in front of pressure is because is the work added to the system with and changing in opposite directions in a stable system as discussed below. The internal energy is thus a function of entropy and volume, i.e.,

, which is the model

Gibbs and quantum statistical thermodynamics

Statistical mechanics was introduced by Gibbs in 1901 [25] based on the foundations established by Clausius, Boltzmann, and Maxwell. Gibbs considered “a great number of independent systems (states) of the same nature (of a system), but differing in the configurations and velocities which they have at a given instant, and differing not merely infinitesimally, but it may be so as to embrace every conceivable combination of configuration and velocities” [25]. He thus broadened the early

Quantum thermodynamics in the framework of density functional theory

Quantum mechanics provides a description of the physical properties of nature at the scale of atoms and subatomic particles. A solid can be thought of as a collection of interacting positively charged nuclei and negatively charged electrons and can theoretically be treated by solving the many-body Schrödinger quantum mechanics equation involving both the nuclei and the electrons [4]. However, it is extremely difficult to solve the equation due to its many-body nature with too many electrons. On

Classical and extended irreversible thermodynamics

Typical approaches to irreversible thermodynamics in the literature start from the Gibbs thermodynamics, i.e., Eq. (13) or Eq. (14) for closed systems with hydrostatic pressure or Eq. (11) in general. For example, Onsager used the microscopic reversibility that requires that if

and be two quantities which depend only on the configuration of molecules and atoms, the event followed some seconds later by will occur as often as the event , followed later by

. It also requires

Overview of zentropy theory

It is evident from the above discussions that all theories require accurate free energy as a function of both internal and external variables. The key challenge in theoretical predictions of free energy of a phase is because only one or a few configurations are typically considered in computational approaches, whereas experimental measurements stem from sampling all possible configurations at all scales simultaneously. This challenge becomes acute for systems with phase transitions, which is

Perspectives on future of thermodynamic modeling

In the CALPHAD method, the Gibbs energies of individual phases are modeled as a function of temperature, pressure, and composition which are controlled from the surroundings and dictate the ground-state configuration of the system, and additional internal variables that represent the non-ground-state configurations. The Gibbs energy builds from pure element to binary and ternary systems and extrapolates to multicomponent systems. In this special issue, Spencer presented overviews of its

Summary

Thermodynamics is at the core of science and nature, and thermodynamic modeling based on the CALPHAD method has enabled the community to quantitatively go beyond the equilibrium applications of thermodynamics, understand and improve existing materials, and design new materials. The present overview paper further discussed the zentropy theory and the theory of cross phenomena for better prediction of data for CALPHAD modeling. Based on the integration of quantum mechanics and statistical

Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: He is the owner of Materials Genome, Inc. and one of two founders of nonprofit Materials Genome Foundation.

Acknowledgements

The present review article covers research outcomes supported by multiple funding agencies over multiple years with the most recent ones including the Endowed Dorothy Pate Enright Professorship at the Pennsylvania State University, U.S. Department of Energy (DOE) Grant No. DE-SC0023185, DE-NE0008945, and DE-NE0009288, and U.S. National Science Foundation (NSF) Grant No. NSF-2229690. The author would like to thank J.P. Perdew for explaining their recent works on symmetry breaking due to

References (247)