Zentropy Theory: Bridging Materials Gene to Materials Properties

doi:10.11900/0412.1961.2024.00147

Acta Metall Sin

2024, Vol. 60

Issue (10): 1379-1387 DOI: 10.11900/0412.1961.2024.00147

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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

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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.

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Abstract

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.

Key words: zentropy theory thermodynamics materials genome engineering thermal expansion mechanism ferroelectric transition temperature order-disorder transition

Received: 08 May 2024

ZTFLH:

O414

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

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https://www.ams.org.cn/EN/10.11900/0412.1961.2024.00147 OR https://www.ams.org.cn/EN/Y2024/V60/I10/1379

Fig.1 Schematic of entropy over multiscale

Fig.2 Schematic of zentropy theory (k_B—Boltzmann constant; p^k and pⁱ —probabilities of configuration k and its sub-configuration i; S^k and S^ki —entropies of configuration k and its sub-configuration i)

Property	Statistical mechanics	Zentropy framework
Entropy	$S = - k B ∑ k p k l n p k$	$S = ∑ k p k S k - k B ∑ k p k l n p k$
Free energy	$F = ∑ k p k E k + k B T ∑ k p k l n p k$	$F = ∑ k p k F k + k B T ∑ k p k l n p k$
Partition function	$Z = e x p (- F k B T) = ∑ k e x p (- E k k B T)$	$Z = e x p (- F k B T) = ∑ k e x p (- F k k B T)$
Probability	$p k = Z k Z = e x p (- E k - F k B T)$	$p k = Z k Z = e x p (- F k - F k B T)$

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 E_tot 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 (V_N) 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, T_C—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 ( $f D o D S$ ) as a function of temperature in Fe₃Pt via entropy^[46] (ΔE₀ and E_k (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)

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