Zentropy Theory: Bridging Materials Gene to Materials Properties
LIAO Mingqing1, WANG William Yi2(), WANG Yi3, SHANG Shun-Li3, LIU Zi-Kui3()
1 School of Materials Science and Engineering, Jiangsu University of Science and Technology, Zhenjiang 212100, China 2 State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi'an 710072, China 3 Department of Materials Science and Engineering, the Pennsylvania State University, University Park, PA, 16802, USA
Cite this article:
LIAO Mingqing, WANG William Yi, WANG Yi, SHANG Shun-Li, LIU Zi-Kui. Zentropy Theory: Bridging Materials Gene to Materials Properties. Acta Metall Sin, 2024, 60(10): 1379-1387.
Entropy is an important concept in science and is ubiquitous from quantum to astronomy. By integrating statistical mechanics, quantum mechanics, and thermodynamics, Professor Zi-Kui Liu proposed the zentropy theory, which stacks entropy over configurations. The zentropy theory takes the configurations in Gibbs' statistical mechanics of a given ensemble as the material gene with the ground state as the basic configuration and additional configurations ergodically derived from its internal degrees of freedom. In the zentropy theory, the total entropy of a system is defined as the weighted average of the entropy of each configuration plus the statistical entropy among all configurations. In this paper, the basic equations and principles of the zentropy theory are introduced, and their typical applications, including magnetic and ferroelectric transformations, thermal expansion mechanisms, and critical phenomenon prediction are outlined. Furthermore, a perspective on the development of this theory, software ecosystems, high-throughput computing, and integration with artificial intelligence is provided in this study.
Fund: Natural Science Foundation of Jiangsu Province(BK20230673);Doctor of Entrepreneurship and Innovation of Jiangsu Province(JSSCBS20221270)
Corresponding Authors:
WANG William Yi, professor, Tel: (029)88460294, E-mail: wywang@nwpu.edu.cn; LIU Zi-Kui, professor, Tel: (814)8651934, E-mail: zxl15@psu.edu
Fig.2 Schematic of zentropy theory (kB—Boltzmann constant; pk and pi —probabilities of configuration k and its sub-configuration i; Sk and Ski —entropies of configuration k and its sub-configuration i)
Property
Statistical mechanics
Zentropy framework
Entropy
Free energy
Partition function
Probability
Table 1 Comparison of thermodynamic terminology between zentropy theory and classic statistical mechanics[41]
Fig.3 Volume-energy curves at 0 K of different configurations[30] (FM, AFM, and NM mean ferromagnetic, antiferromagnetic, and nonmagnetic, respectively; V and Etot mean atomic volume and total energy at 0 K, respectively) (a) Ce (b) Fe3Pt
Fig.4 Temperature-volume phase diagram with isobaric volumes at various pressures[45] (T means temperature; the volume (V) is normalized to their respective equilibrium volume (VN) at atmospheric pressure and room temperature; CPTE and NTE mean colossal positive thermal expansion and negative thermal expansion, respectively; Exp. means experimental results) (a) Ce (b) Fe3Pt
Fig.5 Probability of configurations as a function of temperature in PbTiO3[47] (p—probability, DW—domain wall, FEG—ground state of ferroelectric without DW, TC—critical temperature, the domain wall energies for 90DW and 180DW are 35 and 132 mJ/m2, respectively, which is taken from Ref.[58])
Fig.6 Predicted degree of disorder () as a function of temperature in Fe3Pt via entropy[46] (ΔE0 and Ek (V) are equilibrium energy and static total energy at 0 K, respectively. QHA means quasiharmonic approach, in which the free energy with contributions from vibrations and thermal electrons is evaluated)
1
Gan Y. Research on the innovative development of new materials science and technology in China [J]. Engineering, 2024, 32: 10
2
Xie M, Gan Y, Wang H. Research on new material power strategy by 2035 [J]. Strategic Study CAE, 2020, 22(5): 1
Liu Z K. Perspective on materials genome [J]. Chin. Sci. Bull., 2013, 58: 3618
刘梓葵. 关于材料基因组的基本观点及展望 [J]. 科学通报, 2013, 58: 3618
4
National Science and Technology Council. Materials genome initiative for global competitiveness [EB/OL]. (2011-06-24).
5
National Science and Technology Council. Materials genome initiative strategic plan [EB/OL]. (2014-12-04).
6
Su Y J, Fu H D, Bai Y, et al. Progress in materials genome engineering in China [J]. Acta. Metall. Sin., 2020, 56: 1313
doi: 10.11900/0412.1961.2020.00199
National Research Council. Integrated Computational Materials Engineering: A Transformational Discipline for Improved Competitiveness and National Security [M]. Washington, D.C.: The National Academies Press, 2008: 9
8
Wang W Y, Yin J L, Chai Z X, et al. Big data-assisted digital twins for the smart design and manufacturing of advanced materials: From atoms to products [J]. J. Mater. Inf., 2022, 2: 1
9
Liu Z K, Chen L Q, Raghavan P, et al. An integrated framework for multi-scale materials simulation and design [J]. J. Comput. Aided Mater. Des., 2004, 11: 183
10
Liu X, Furrer D, Kosters J, et al. Vision 2040 : A roadmap for integrated, multiscale modeling and simulation of materials and systems [EB/OL]. (2018-03-22).
11
National Academies of Sciences, Engineering, and Medicine. NSF Efforts to Achieve the Nation's Vision for the Materials Genome Initiative: Designing Materials to Revolutionize and Engineer Our Future (DMREF) [M]. Washington, D.C.: The National Academies Press, 2023: 57
Xie J X, Su Y J, Zhang D W, et al. A vision of materials genome engineering in China [J]. Engineering, 2022, 10: 10
14
Li M X, Zhao S F, Lu Z, et al. High-temperature bulk metallic glasses developed by combinatorial methods [J]. Nature, 2019, 569: 99
15
Lu Z C, Zhang Y B, Li W Y, et al. Materials genome strategy for metallic glasses [J]. J. Mater. Sci. Technol., 2023, 166: 173
doi: 10.1016/j.jmst.2023.04.074
16
Wang G J, Peng L Y, Li K Q, et al. ALKEMIE: An intelligent computational platform for accelerating materials discovery and design [J]. Comput. Mater. Sci., 2021, 186: 110064
17
Wang G J, Li K Q, Peng L Y, et al. High-throughput automatic integrated material calculations and data management intelligent platform and the application in novel alloys [J]. Acta. Metall. Sin., 2022, 58: 75
doi: 10.11900/0412.1961.2021.00041
Chong X Y, Yu W, Liang Y X, et al. Understanding oxidation resistance of Pt-based alloys through computations of Ellingham diagrams with experimental verifications [J]. J. Mater. Inf., 2023, 3: 21
19
Agrawal A, Choudhary A. Perspective: Materials informatics and big data: Realization of the “fourth paradigm” of science in materials science [J]. APL Mater., 2016, 4: 053208
20
Liu Y L, Niu C, Wang Z, et al. Machine learning in materials genome initiative: A review [J]. J. Mater. Sci. Technol., 2020, 57: 113
doi: 10.1016/j.jmst.2020.01.067
21
E W N. AI helps to establish a new paradigm for scientific research [J]. Bull. Chin. Acad. Sci., 2024, 39(1): 10
鄂维南. AI助力打造科学研究新范式 [J]. 中国科学院院刊, 2024, 39(1): 10
22
Li G J. AI4R: The fifth scientific research paradigm [J]. Bull. Chin. Acad. Sci., 2024, 39: 1
李国杰. 智能化科研(AI4R)：第五科研范式 [J]. 中国科学院院刊, 2024, 39: 1
23
Ågren J. The materials genome and CALPHAD [J]. Chin. Sci. Bull., 2013, 58: 3633
Ågren J. 材料基因组与相图计算 [J]. 科学通报, 2013, 58: 3633
24
Wang S Q, Ye H Q. First-principles calculation of crystalline materials genome [J]. Chin. Sci. Bull., 2013, 58: 3623
Liu Z K. Computational thermodynamics and its applications [J]. Acta Mater., 2020, 200: 745
26
Liu Z K. First-principles calculations and CALPHAD modeling of thermodynamics [J]. J. Phase Equilib. Diffus., 2009, 30: 517
27
Olson G B, Liu Z K. Genomic materials design: CALculation of PHAse Dynamics [J]. Calphad, 2023, 82: 102590
28
Liu Z K. Thermodynamics and its prediction and CALPHAD modeling: Review, state of the art, and perspectives [J]. Calphad, 2023, 82: 102580
29
Campbell C E, Kattner U R, Liu Z K. File and data repositories for next generation CALPHAD [J]. Scr. Mater., 2014, 70: 7
30
Liu Z K, Wang Y, Shang S L. Zentropy theory for positive and negative thermal expansion [J]. J. Phase Equilib. Diffus., 2022, 43: 598
31
Liu Z K. Building materials genome from ground‐state configuration to engineering advance [J]. Mater. Genome Eng. Adv., 2023, 1: e15
32
Liu Z K, Hew N L E, Shang S L. Zentropy theory for accurate prediction of free energy, volume, and thermal expansion without fitting parameters [J]. Microstructures, 2024, 4: 2024009
33
Mooraj S, Chen W. A review on high-throughput development of high-entropy alloys by combinatorial methods [J]. J. Mater. Inf., 2023, 3: 4
34
Jiang B B, Yu Y, Cui J, et al. High-entropy-stabilized chalcogenides with high thermoelectric performance [J]. Science, 2021, 371: 830
doi: 10.1126/science.abe1292
pmid: 33602853
35
Liu Y, Lu Y H, Wang W Y, et al. Effects of solutes on thermodynamic properties of (TMZrU)C (TM = Ta, Y) medium-entropy carbides: A first-principles study [J]. J. Mater. Inf., 2023, 3: 17
36
Liao M Q, Gong H S, Qu N, et al. CALPHAD aided mechanical properties screening in full composition space of NbC-TiC-VC-ZrC ultra-high temperature ceramics [J]. Int. J. Refract. Met. Hard Mater., 2023, 113: 106191
37
Wang J, Chong X Y, Lv L, et al. High-entropy ferroelastic (10RE0.1)TaO4 ceramics with oxygen vacancies and improved thermophysical properties [J]. J. Mater. Sci. Technol., 2023, 157: 98
38
Chen L X, Chen Z W, Yao X, et al. High-entropy alloy catalysts: High-throughput and machine learning-driven design [J]. J. Mater. Inf., 2022, 2: 19
39
Ceder G. A derivation of the Ising model for the computation of phase diagrams [J]. Comput. Mater. Sci., 1993, 1: 144
40
Liu Z K. Theory of cross phenomena and their coefficients beyond Onsager theorem [J]. Mater. Res. Lett., 2022, 10: 393
41
Liu Z K. Quantitative predictive theories through integrating quantum, statistical, equilibrium, and nonequilibrium thermodynamics [J]. J. Phys.: Condens. Matter., 2024, 36: 343003
42
Liu Z K. On Gibbs Equilibrium and hillert nonequilibrium thermodynamics [DB/OL]. arXiv: 2402. 14231, 2024
43
Wang Y, Liao M Q, Bocklund B J, et al. DFTTK: Density functional theory toolKit for high-throughput lattice dynamics calculations [J]. Calphad, 2021, 75: 102355
44
Liu Z K, Li B, Lin H. Multiscale entropy and its implications to critical phenomena, emergent behaviors, and information [J]. J. Phase Equilib. Diffus., 2019, 40: 508
45
Liu Z K, Wang Y, Shang S L. Thermal expansion anomaly regulated by entropy [J]. Sci. Rep., 2014, 4: 7043
46
Shang S L, Wang Y, Liu Z K. Quantifying the degree of disorder and associated phenomena in materials through zentropy: Illustrated with Invar Fe3Pt [J]. Scr. Mater., 2023, 225: 115164
47
Liu Z K, Shang S L, Du J L, et al. Parameter-free prediction of phase transition in PbTiO3 through combination of quantum mechanics and statistical mechanics [J]. Scr. Mater., 2023, 232: 115480
48
Liang E J, Sun Q, Yuan H L, et al. Negative thermal expansion: Mechanisms and materials [J]. Front. Phys., 2021, 16: 53302
49
Zhou C, Tang Z Y, Kong X Q, et al. High-performance zero thermal expansion in Al metal matrix composites [J]. Acta Mater., 2024, 275: 120076
50
Liao M Q, Wang Y, Wang F J, et al. Unexpected low thermal expansion coefficients of pentadiamond [J]. Phys. Chem. Chem. Phys., 2022, 24: 23561
51
Wang Y, Hector Jr L G, Zhang H, et al. A thermodynamic framework for a system with itinerant-electron magnetism [J]. J. Phys.: Condens. Matter., 2009, 21: 326003
52
Wang Y, Shang S L, Zhang H, et al. Thermodynamic fluctuations in magnetic states: Fe3Pt as a prototype [J]. Philos. Mag. Lett., 2010, 90: 851
53
Shang S L, Wang Y, Liu Z K. Thermodynamic fluctuations between magnetic states from first-principles phonon calculations: The case of bcc Fe [J]. Phys. Rev., 2010, 82B: 014425
54
Shang S L, Saal J E, Mei Z G, et al. Magnetic thermodynamics of fcc Ni from first-principles partition function approach [J]. J. Appl. Phys., 2010, 108: 123514
55
Liu Z K, Wang Y, Shang S L. Origin of negative thermal expansion phenomenon in solids [J]. Scr. Mater., 2011, 65: 664
56
Wang Y, Hector Jr L G, Zhang H, et al. Thermodynamics of the Ce γ-α transition: Density-functional study [J]. Phys. Rev, 2008, 78B: 104113
57
Li Y L, Hu S Y, Liu Z K, et al. Effect of substrate constraint on the stability and evolution of ferroelectric domain structures in thin films [J]. Acta Mater., 2002, 50: 395
58
Meyer B, Vanderbilt D. Ab initio study of ferroelectric domain walls in PbTiO3 [J]. Phys. Rev., 2002, 65B: 104111
59
Fang H Z, Wang Y, Shang S L, et al. Nature of ferroelectric-paraelectric phase transition and origin of negative thermal expansion in PbTiO3 [J]. Phys. Rev., 2015, 91B: 024104
60
Du J L, Malyi O I, Shang S L, et al. Density functional thermodynamic description of spin, phonon and displacement degrees of freedom in antiferromagnetic-to-paramagnetic phase transition in YNiO3 [J]. Mater. Today Phys., 2022, 27: 100805
61
Du J L, Zhang Z L, Shang S L, et al. Underpinnings behind the magnetic order-to-disorder transition and property anomaly of disproportionated insulating samarium nickelate [J]. Acta Mater., 2024, 268: 119783
62
Liu Z K, Shang S L. Revealing symmetry-broken superconducting configurations by density functional theory [DB/OL]. arXiv: 2404. 00719, 2024
63
Hong Q J, Liu Z K. A generalized approach for rapid entropy calculation of liquids and solids [DB/OL]. arXiv: 2403. 19872, 2024
64
Bocklund B, Otis R, Egorov A, et al. ESPEI for efficient thermodynamic database development, modification, and uncertainty quantification: Application to Cu-Mg [J]. MRS Commun., 2019, 9: 618
doi: 10.1557/mrc.2019.59
65
Otis R, Liu Z K. pycalphad: CALPHAD-based computational thermodynamics in python [J]. J. Open Res. Softw., 2017, 5: 1
66
Paulson N H, Bocklund B J, Otis R A, et al. Quantified uncertainty in thermodynamic modeling for materials design [J]. Acta Mater., 2019, 174: 9
doi: 10.1016/j.actamat.2019.05.017
67
Peng J, Lee S, Williams A, et al. Advanced data science toolkit for non-data scientists—A user guide [J]. Calphad, 2020, 68: 101733
68
Krajewski A M, Siegel J W, Xu J, et al. Extensible structure-informed prediction of formation energy with improved accuracy and usability employing neural networks [J]. Comput. Mater. Sci., 2022, 208: 111254
69
Liu Z K. Ocean of data: Integrating first-principles calculations and CALPHAD modeling with machine learning [J]. J. Phase Equilib. Diffus., 2018, 39: 635
70
Liu Z K. View and comments on the data ecosystem: “Ocean of data” [J]. Engineering, 2020, 6: 604
