高熵合金藉由多元混雜增加混合熵以形成穩定固溶體，展現出優異的物理和機械性質。而擴散遲緩效應是高熵合金的四大效應之一，對其高溫強度和抗輻射損傷等性質具有重要影響。然而，擴散遲緩效應在高熵合金備受爭議，其機制也不明確，此外，空孔運動行為在受局部複雜環境影響尚未有系統性分析。因此本研究利用蒙地卡羅方法模擬多元系統中空孔的擴散行為，分析局部環境複雜性對空孔運動的影響。在自擴散模擬中，發現空孔傾向與低能障元素交換，形成局部區域快速打轉的現象，且能障大小和能障差距對空孔運動行為的影響大於元素數目，而高溫則會減少局部環境的影響。而在交互擴散模擬中，發現空孔運動受局部環境和濃度梯度影響，且隨著局部環境複雜性的增加，空孔運動相關性變明顯。雖然多元系統中相近的交換機率降低了運動相關性，但若存在特別低能障元素，空孔運動可能會受到限制。最後，本研究提出空孔運動行為與擴散遲緩效應之間的關聯，應與其在低能障元素影響下的局部快速打轉有關，並受跳躍頻率與運動相關性的競爭機制影響。這些發現為高熵合金擴散遲緩效應提供新的理論基礎，並有助於未來在高溫應用中的材料設計。

High-entropy alloys (HEAs), composed of multiple primary elements, tend to form solid solutions through increased entropy contribution, exhibiting excellent physical and mechanical properties. Sluggish diffusion, as one of the four core effects in HEAs, significantly enhances their high-temperature strength, radiation damage tolerance, and other kinetic-related properties. However, sluggish diffusion in HEAs is controversial, and its mechanism remains unclear. Additionally, the behavior of vacancy movement under the influence of local atomic environment complexity has not been systematically analyzed. In this study, we utilize the Monte Carlo method to simulate vacancy diffusion in multicomponent systems, analyzing the impact of local atomic environment complexity on vacancy movement. In the self-diffusion simulations, vacancies tend to exchange with elements that have low migration energy, resulting in rapid but localized motion. The influence of migration barriers and differences between elements on vacancy behavior is more significant than the number of elements. High temperatures reduce the impact of local atomic environment complexity. In interdiffusion simulations, vacancy movement is affected by local atomic environments and concentration gradients, with increased local atomic environment complexity enhancing correlation. Similar exchange rates in multicomponent systems reduce movement correlation, but we suggest the presence of exceptionally low energy barrier elements in HEA can restrict vacancy movement. Finally, this study establishes a link between vacancy behavior and sluggish diffusion, suggesting that rapid but localized motion under the influence of low migration energy, along with the competition between jump frequency and movement correlation, plays a critical role in determining whether sluggish diffusion occurs in HEAs. These findings provide new theoretical insights into the sluggish diffusion effect in HEAs and assist in the future design of materials for high-temperature applications.

誌謝....I
摘要....II
Abstract....III
目錄....V
圖目錄....XI
表目錄....XXI
第一章 前言....1
第二章 文獻回顧....2
2.1 高熵合金與擴散遲緩....2
2.1.1 高熵合金發展....2
2.1.2 高熵合金四大效應....5
2.1.3 擴散遲緩效應的提出....9
2.1.4 擴散遲緩效應的影響....11
2.2 金屬巨觀擴散行為....17
2.2.1 擴散現象....17
2.2.2 擴散係數....19
2.2.3 擴散係數關係式與其物理意義....25
2.2.4 擴散活化能....26
2.3 高熵合金巨觀擴散行為....27
2.3.1 擴散係數實驗量測....27
2.3.2 擴散遲緩與混合熵關聯....31
2.3.3 高熵合金擴散活化能....34
2.3.4 影響高熵合金擴散行為因素....38
2.3.5 數據分析方式....41
2.4 原子尺度擴散行為....43
2.4.1 熱活化原子跳躍與隨機漫步....43
2.4.2 擴散機制....46
2.4.3 隨機漫步與擴散係數....47
2.4.4 傳統金屬原子尺度擴散行為....49
2.5 高熵合金原子尺度擴散行為....50
2.5.1 局部環境複雜性....50
2.5.2 晶格位能差異幅度....51
2.5.3 局部複雜環境對空孔能量之影響....53
2.5.4 局部複雜環境對擴散行為之影響....55
2.6 粗糙位能景觀....56
2.6.1 位能景觀圖....56
2.6.2 粗糙位能景觀....58
2.6.3 粗糙位能景觀模型....60
2.6.4 影響粗糙位能景觀的因素....61
2.6.5 模擬粗糙位能景觀的方法....63
2.6.6 位能景觀對擴散造成的影響....66
2.6.7 粗糙位能景觀對擴散造成的影響....68
2.7 擴散遲緩機制....72
2.7.1 低能原子位置與陷阱....72
2.7.2 空孔運動行為....73
2.7.3 滲透效應與偏好擴散....76
2.7.4 短程有序....79
2.8 文獻回顧總結與研究目的....82
第三章 研究方法....84
3.1 蒙地卡羅與動態蒙地卡羅方法....84
3.1.1 背景介紹....84
3.1.2 模擬方法流程....85
3.1.3 模擬編譯程式與執行工具....86
3.2 研究架構與規劃....87
3.3 自擴散模擬方法....89
3.3.1 設計目的....89
3.3.2 動態蒙地卡羅流程....89
3.3.3 模擬流程....90
3.3.4 系統參數....91
3.3.5 變數設定....92
3.4 交互擴散模擬方法....95
3.4.1 設計目的....95
3.4.2 蒙地卡羅流程....95
3.4.3 模擬流程....95
3.4.4 系統參數....98
3.4.5 變數設定....99
3.5 分析方法....101
3.5.1 統計檢定....101
3.5.2 運動行為....101
3.5.3 擴散行為....102
3.5.4 交互擴散分析....104
3.5.5 統計檢定結果....105
第四章 自擴散模擬結果與討論....106
4.1 假設系統結果....106
4.1.1 移動軌跡視覺化....106
4.1.2 擴散行為....108
4.1.3 空孔運動行為....110
4.1.4 擴散係數與擴散速度....116
4.1.5 溫度效應....118
4.1.6 阿瑞尼斯關係圖....120
4.1.7 有效遷移能障....122
4.1.8 近鄰原子組成....124
4.2 真實系統結果....126
4.2.1 移動軌跡視覺化....126
4.2.2 擴散行為....128
4.2.3 空孔運動行為....130
4.2.4 擴散係數與擴散速度....134
4.2.5 溫度效應....136
4.2.6 阿瑞尼斯關係圖....138
4.2.7 有效能障....140
4.2.8 近鄰原子組成....141
4.3 自擴散統整與討論....143
第五章 交互擴散模擬結果與討論....147
5.1 Mono-V....147
5.1.1 濃度分布曲線....147
5.1.2 運動行為....151
5.1.3 討論與小結....152
5.2 Serial-V....154
5.2.1 濃度分布曲線....154
5.2.2 運動行為....158
5.2.3 濃度臨界值影響....160
5.2.4 討論與小結....162
5.3 Combined-V....163
5.3.1 Erf 交互擴散行為....163
5.3.2 Gradient 交互擴散行為....165
5.3.3 討論與小結....167
5.4 交換機率影響....168
5.4.1 Serial-V....169
5.4.2 Combined-V....172
5.4.3 討論與小結....176
5.5 交互擴散統整與討論....177
第六章 結論....179
參考文獻 181

[1] J.-W. Yeh, S.-K. Chen, S.-J. Lin, J.-Y. Gan, T.-S. Chin, T.-T. Shun, C.-H. Tsau,
and S.-Y. Chang, “Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes,” Advanced engineering materials, vol. 6, no. 5, pp. 299–303, 2004.
[2] B. Cantor, I. Chang, P. Knight, and A. Vincent, “Microstructural development
in equiatomic multicomponent alloys,” Materials Science and Engineering: A,
vol. 375, pp. 213–218, 2004.
[3] E. P. George, D. Raabe, and R. O. Ritchie, “High-entropy alloys,” Nature reviews
materials, vol. 4, no. 8, pp. 515–534, 2019.
[4] O. N. Senkov, G. B. Wilks, J. M. Scott, and D. B. Miracle, “Mechanical properties of nb25mo25ta25w25 and v20nb20mo20ta20w20 refractory high entropy
alloys,” Intermetallics, vol. 19, no. 5, pp. 698–706, 2011.
[5] O. Senkov, J. Scott, S. Senkova, D. Miracle, and C. Woodward, “Microstructure
and room temperature properties of a high-entropy tanbhfzrti alloy,” Journal of
alloys and compounds, vol. 509, no. 20, pp. 6043–6048, 2011.
[6] E. P. George, W. A. Curtin, and C. C. Tasan, “High entropy alloys: A focused
review of mechanical properties and deformation mechanisms,” Acta Materialia,
vol. 188, pp. 435–474, 2020.
[7] F. Otto, A. Dlouhỳ, C. Somsen, H. Bei, G. Eggeler, and E. P. George, “The influences of temperature and microstructure on the tensile properties of a cocrfemnni
high-entropy alloy,” Acta Materialia, vol. 61, no. 15, pp. 5743–5755, 2013.
[8] S. Praveen and H. S. Kim, “High-entropy alloys: potential candidates for
high-temperature applications–an overview,” Advanced Engineering Materials,
vol. 20, no. 1, p. 1700645, 2018.
[9] B. Gludovatz, A. Hohenwarter, D. Catoor, E. H. Chang, E. P. George, and R. O.
Ritchie, “A fracture-resistant high-entropy alloy for cryogenic applications,” Science, vol. 345, no. 6201, pp. 1153–1158, 2014.
[10] N. K. Kumar, C. Li, K. Leonard, H. Bei, and S. Zinkle, “Microstructural stability
and mechanical behavior of fenimncr high entropy alloy under ion irradiation,”
Acta Materialia, vol. 113, pp. 230–244, 2016.
181
[11] Z. Li, K. G. Pradeep, Y. Deng, D. Raabe, and C. C. Tasan, “Metastable highentropy dual-phase alloys overcome the strength–ductility trade-off,” Nature,
vol. 534, no. 7606, pp. 227–230, 2016.
[12] E. Ma and T. Zhu, “Towards strength–ductility synergy through the design of heterogeneous nanostructures in metals,” Materials Today, vol. 20, no. 6, pp. 323–
331, 2017.
[13] T. Yang, Y. Zhao, Y. Tong, Z. Jiao, J. Wei, J. Cai, X. Han, D. Chen, A. Hu,
J. Kai, et al., “Multicomponent intermetallic nanoparticles and superb mechanical
behaviors of complex alloys,” Science, vol. 362, no. 6417, pp. 933–937, 2018.
[14] T. Yang, Y. Zhao, W. Li, C. Yu, J. Luan, D. Lin, L. Fan, Z. Jiao, W. Liu, X. Liu,
et al., “Ultrahigh-strength and ductile superlattice alloys with nanoscale disordered interfaces,” Science, vol. 369, no. 6502, pp. 427–432, 2020.
[15] X. Lim, “Mixed-up metals make for stronger, tougher, stretchier alloys,” Nature,
vol. 533, no. 7603, pp. 306–307, 2016.
[16] D. B. Miracle and O. N. Senkov, “A critical review of high entropy alloys and
related concepts,” Acta Materialia, vol. 122, pp. 448–511, 2017.
[17] W. Li, D. Xie, D. Li, Y. Zhang, Y. Gao, and P. K. Liaw, “Mechanical behavior of
high-entropy alloys,” Progress in Materials Science, vol. 118, p. 100777, 2021.
[18] Y. Jien-Wei, “Recent progress in high entropy alloys,” Ann. Chim. Sci. Mat,
vol. 31, no. 6, pp. 633–648, 2006.
[19] M.-H. Tsai and J.-W. Yeh, “High-entropy alloys: a critical review,” Materials
Research Letters, vol. 2, no. 3, pp. 107–123, 2014.
[20] W.-L. Hsu, C.-W. Tsai, A.-C. Yeh, and J.-W. Yeh, “Clarifying the four core effects
of high-entropy materials,” Nature Reviews Chemistry, pp. 1–15, 2024.
[21] C. Lee, G. Song, M. C. Gao, R. Feng, P. Chen, J. Brechtl, Y. Chen, K. An, W. Guo,
J. D. Poplawsky, et al., “Lattice distortion in a strong and ductile refractory highentropy alloy,” Acta Materialia, vol. 160, pp. 158–172, 2018.
[22] R. K. Nutor, Q. Cao, X. Wang, D. Zhang, Y. Fang, Y. Zhang, and J.-Z. Jiang,
“Phase selection, lattice distortions, and mechanical properties in high-entropy
alloys,” Advanced Engineering Materials, vol. 22, no. 11, p. 2000466, 2020.
[23] H. Wang, Q. He, X. Gao, Y. Shang, W. Zhu, W. Zhao, Z. Chen, H. Gong, and
Y. Yang, “Multifunctional high entropy alloys enabled by severe lattice distortion,” Advanced Materials, p. 2305453, 2023.
182
[24] K.-Y. Tsai, M.-H. Tsai, and J.-W. Yeh, “Sluggish diffusion in co–cr–fe–mn–ni
high-entropy alloys,” Acta Materialia, vol. 61, no. 13, pp. 4887–4897, 2013.
[25] X. Liu, Y. Duan, X. Yang, L. Huang, M. Gao, and T. Wang, “Enhancement of
magnetic properties in feconicr0. 4cux high entropy alloys through the cocktail
effect for megahertz electromagnetic wave absorption,” Journal of Alloys and
Compounds, vol. 872, p. 159602, 2021.
[26] L. Wang, L. Zhang, X. Lu, F. Wu, X. Sun, H. Zhao, and Q. Li, “Surprising cocktail effect in high entropy alloys on catalyzing magnesium hydride for solid-state
hydrogen storage,” Chemical Engineering Journal, vol. 465, p. 142766, 2023.
[27] Z. Wang, Q. Fang, J. Li, B. Liu, and Y. Liu, “Effect of lattice distortion on solid
solution strengthening of bcc high-entropy alloys,” Journal of Materials Science
& Technology, vol. 34, no. 2, pp. 349–354, 2018.
[28] R. Wang, W. Chen, J. Zhong, and L. Zhang, “Experimental and numerical studies on the sluggish diffusion in face centered cubic co-cr-cu-fe-ni high-entropy
alloys,” Journal of materials science & technology, vol. 34, no. 10, pp. 1791–
1798, 2018.
[29] B. Cao, C. Wang, T. Yang, and C. Liu, “Cocktail effects in understanding the
stability and properties of face-centered-cubic high-entropy alloys at ambient and
cryogenic temperatures,” Scripta materialia, vol. 187, pp. 250–255, 2020.
[30] C.-J. Tong, Y.-L. Chen, J.-W. Yeh, S.-J. Lin, S.-K. Chen, T.-T. Shun, C.-H. Tsau,
and S.-Y. Chang, “Microstructure characterization of al x cocrcufeni high-entropy
alloy system with multiprincipal elements,” Metallurgical and Materials Transactions A, vol. 36, pp. 881–893, 2005.
[31] C. Ng, S. Guo, J. Luan, S. Shi, and C. T. Liu, “Entropy-driven phase stability and
slow diffusion kinetics in an al0. 5cocrcufeni high entropy alloy,” Intermetallics,
vol. 31, pp. 165–172, 2012.
[32] T.-T. Shun, C.-H. Hung, and C.-F. Lee, “Formation of ordered/ disordered
nanoparticles in fcc high entropy alloys,” Journal of Alloys and Compounds,
vol. 493, no. 1-2, pp. 105–109, 2010.
[33] W. Liu, Y. Wu, J. He, T. Nieh, and Z. Lu, “Grain growth and the hall–petch relationship in a high-entropy fecrnicomn alloy,” Scripta Materialia, vol. 68, no. 7,
pp. 526–529, 2013.
[34] Y. Zhao, H. Chen, Z. Lu, and T. Nieh, “Thermal stability and coarsening of coherent particles in a precipitation-hardened (nicofecr) 94ti2al4 high-entropy alloy,”
Acta Materialia, vol. 147, pp. 184–194, 2018.
183
[35] B. Xiao, J. Luan, S. Zhao, L. Zhang, S. Chen, Y. Zhao, L. Xu, C. Liu, J.-J. Kai,
and T. Yang, “Achieving thermally stable nanoparticles in chemically complex alloys via controllable sluggish lattice diffusion,” Nature Communications, vol. 13,
no. 1, p. 4870, 2022.
[36] J. Chen, X. Zhou, W. Wang, B. Liu, Y. Lv, W. Yang, D. Xu, and Y. Liu, “A
review on fundamental of high entropy alloys with promising high–temperature
properties,” Journal of Alloys and Compounds, vol. 760, pp. 15–30, 2018.
[37] Y. Zhang, G. M. Stocks, K. Jin, C. Lu, H. Bei, B. C. Sales, L. Wang, L. K. Béland,
R. E. Stoller, G. D. Samolyuk, et al., “Influence of chemical disorder on energy
dissipation and defect evolution in concentrated solid solution alloys,” Nature
communications, vol. 6, no. 1, p. 8736, 2015.
[38] C. Lu, L. Niu, N. Chen, K. Jin, T. Yang, P. Xiu, Y. Zhang, F. Gao, H. Bei, S. Shi,
et al., “Enhancing radiation tolerance by controlling defect mobility and migration pathways in multicomponent single-phase alloys,” Nature communications,
vol. 7, no. 1, p. 13564, 2016.
[39] S.-q. Xia, W. Zhen, T.-f. Yang, and Y. Zhang, “Irradiation behavior in high entropy
alloys,” Journal of iron and steel research, international, vol. 22, no. 10, pp. 879–
884, 2015.
[40] T.-K. Tsao, A.-C. Yeh, C.-M. Kuo, K. Kakehi, H. Murakami, J.-W. Yeh, and
S.-R. Jian, “The high temperature tensile and creep behaviors of high entropy
superalloy,” Scientific reports, vol. 7, no. 1, p. 12658, 2017.
[41] H. Mehrer, Diffusion in solids: fundamentals, methods, materials, diffusioncontrolled processes, vol. 155. Springer Science & Business Media, 2007.
[42] C. Matano, “On the relation between the diffusion-coefficients and concentrations
of solid metals,” Japan. J. Phys., vol. 8, p. 109, 1933.
[43] J. Zhong, L. Chen, and L. Zhang, “High-throughput determination of high-quality
interdiffusion coefficients in metallic solids: a review,” Journal of Materials Science, vol. 55, pp. 10303–10338, 2020.
[44] A. Yelon and B. Movaghar, “Microscopic explanation of the compensation
(meyer-neldel) rule,” Physical review letters, vol. 65, no. 5, p. 618, 1990.
[45] G. Neumann and C. Tuijn, Self-diffusion and impurity diffusion in pure metals:
handbook of experimental data. Elsevier, 2011.
[46] N. Esakkiraja, K. Pandey, A. Dash, and A. Paul, “Pseudo-binary and pseudoternary diffusion couple methods for estimation of the diffusion coefficients
184
in multicomponent systems and high entropy alloys,” Philosophical Magazine,
2019.
[47] A. Dash, N. Esakkiraja, and A. Paul, “Solving the issues of multicomponent diffusion in an equiatomic nicofecr medium entropy alloy,” Acta Materialia, vol. 193,
pp. 163–171, 2020.
[48] V. Verma, C. H. Belcher, D. Apelian, and E. J. Lavernia, “Diffusion in high entropy alloy systems–a review,” Progress in Materials Science, p. 101245, 2024.
[49] J. Dąbrowa and M. Danielewski, “State-of-the-art diffusion studies in the high
entropy alloys,” Metals, vol. 10, no. 3, p. 347, 2020.
[50] D. Gaertner, K. Abrahams, J. Kottke, V. A. Esin, I. Steinbach, G. Wilde, and
S. V. Divinski, “Concentration-dependent atomic mobilities in fcc cocrfemnni
high-entropy alloys,” Acta Materialia, vol. 166, pp. 357–370, 2019.
[51] S. V. Divinski, A. V. Pokoev, N. Esakkiraja, and A. Paul, “A mystery of” sluggish
diffusion” in high-entropy alloys: the truth or a myth?,” Diffusion foundations,
vol. 17, pp. 69–104, 2018.
[52] M. Vaidya, K. Pradeep, B. Murty, G. Wilde, and S. Divinski, “Bulk tracer diffusion in cocrfeni and cocrfemnni high entropy alloys,” Acta Materialia, vol. 146,
pp. 211–224, 2018.
[53] J. Dąbrowa, M. Zajusz, W. Kucza, G. Cieślak, K. Berent, T. Czeppe, T. Kulik,
and M. Danielewski, “Demystifying the sluggish diffusion effect in high entropy
alloys,” Journal of Alloys and Compounds, vol. 783, pp. 193–207, 2019.
[54] A. Mehta and Y. Sohn, “Investigation of sluggish diffusion in fcc al0. 25cocrfeni
high-entropy alloy,” Materials Research Letters, vol. 9, no. 5, pp. 239–246, 2021.
[55] J. Zhang, C. Gadelmeier, S. Sen, R. Wang, X. Zhang, Y. Zhong, U. Glatzel,
B. Grabowski, G. Wilde, and S. V. Divinski, “Zr diffusion in bcc refractory high
entropy alloys: A case of ‘non-sluggish’diffusion behavior,” Acta Materialia,
vol. 233, p. 117970, 2022.
[56] M. Zajusz, M. Jawańska, J. Dąbrowa, K. Berent, G. Cieślak, T. Kulik, and
K. Mroczka, “Evaluation of phase stability and diffusion kinetics in novel bccstructured high entropy alloys,” Materials Research Letters, vol. 10, no. 8,
pp. 556–565, 2022.
[57] S. Sen, X. Zhang, L. Rogal, J. Schell, G. Wilde, B. Grabowski, and S. V. Divinski,
“Sc diffusion in hcp high entropy alloys,” Scripta Materialia, vol. 242, p. 115917,
2024.
185
[58] S. Sen, X. Zhang, L. Rogal, G. Wilde, B. Grabowski, and S. V. Divinski, “Does
zn mimic diffusion of al in the hcp al-sc-hf-ti-zr high entropy alloys?,” Scripta
Materialia, vol. 229, p. 115376, 2023.
[59] G. M. Muralikrishna, S. Sen, S. K. Ayyappan, S. Sankaran, K. Guruvidyathri,
J. Schell, L. Rogal, X. Zhang, J. Mayer, B. Grabowski, et al., “Microstructure
stability and self-diffusion in the equiatomic hfsctizr hcp multi-principal element
alloy,” Journal of Alloys and Compounds, vol. 976, p. 173196, 2024.
[60] M. Vaidya, K. Pradeep, B. Murty, G. Wilde, and S. Divinski, “Radioactive isotopes reveal a non sluggish kinetics of grain boundary diffusion in high entropy
alloys,” Scientific reports, vol. 7, no. 1, p. 12293, 2017.
[61] M. Glienke, M. Vaidya, K. Gururaj, L. Daum, B. Tas, L. Rogal, K. Pradeep, S. V.
Divinski, and G. Wilde, “Grain boundary diffusion in cocrfemnni high entropy
alloy: Kinetic hints towards a phase decomposition,” Acta Materialia, vol. 195,
pp. 304–316, 2020.
[62] R. Li, B. Bian, G. Wilde, Y. Zhang, and S. V. Divinski, “Bulk and grain boundary
tracer diffusion in multiphase alcocrfeniti0. 2 compositionally complex alloy,”
Acta Materialia, vol. 261, p. 119352, 2023.
[63] S. Sen, M. Glienke, B. Yadav, M. Vaidya, K. Gururaj, K. Pradeep, L. Daum,
B. Tas, L. Rogal, G. Wilde, et al., “Grain boundary self-and mn impurity diffusion
in equiatomic cocrfeni multi-principal element alloy,” Acta Materialia, vol. 264,
p. 119588, 2024.
[64] A. Dash, A. Paul, S. Sen, S. Divinski, J. Kundin, I. Steinbach, B. Grabowski, and
X. Zhang, “Recent advances in understanding diffusion in multiprincipal element
systems,” Annual Review of Materials Research, vol. 52, pp. 383–409, 2022.
[65] S. Sen, X. Zhang, L. Rogal, G. Wilde, B. Grabowski, and S. V. Divinski, “‘antisluggish’ti diffusion in hcp high-entropy alloys: Chemical complexity vs. lattice
distortions,” Scripta Materialia, vol. 224, p. 115117, 2023.
[66] M. Vaidya, S. Trubel, B. Murty, G. Wilde, and S. V. Divinski, “Ni tracer diffusion in cocrfeni and cocrfemnni high entropy alloys,” Journal of Alloys and
Compounds, vol. 688, pp. 994–1001, 2016.
[67] C. Zhang, F. Zhang, K. Jin, H. Bei, S. Chen, W. Cao, J. Zhu, and D. Lv, “Understanding of the elemental diffusion behavior in concentrated solid solution
alloys,” Journal of Phase Equilibria and Diffusion, vol. 38, pp. 434–444, 2017.
[68] K. Jin, C. Zhang, F. Zhang, and H. Bei, “Influence of compositional complexity
on interdiffusion in ni-containing concentrated solid-solution alloys,” Materials
Research Letters, vol. 6, no. 5, pp. 293–299, 2018.
186
[69] E.-W. Huang, H.-S. Chou, K. Tu, W.-S. Hung, T.-N. Lam, C.-W. Tsai, C.-Y. Chiang, B.-H. Lin, A.-C. Yeh, S.-H. Chang, et al., “Element effects on high-entropy
alloy vacancy and heterogeneous lattice distortion subjected to quasi-equilibrium
heating,” Scientific reports, vol. 9, no. 1, p. 14788, 2019.
[70] K. Sugita, N. Matsuoka, M. Mizuno, and H. Araki, “Vacancy formation enthalpy
in cocrfemnni high-entropy alloy,” Scripta Materialia, vol. 176, pp. 32–35, 2020.
[71] Z. Wang, C. Liu, and P. Dou, “Thermodynamics of vacancies and clusters in highentropy alloys,” Physical Review Materials, vol. 1, no. 4, p. 043601, 2017.
[72] K. Sugita, R. Ogawa, M. Mizuno, H. Araki, and A. Yabuuchi, “Vacancy migration energies in crmnfeconi, crfeconi, and crfeni alloys and their effect on atomic
diffusion,” Scripta Materialia, vol. 208, p. 114339, 2022.
[73] S. Rothman, L. Nowicki, and G. Murch, “Self-diffusion in austenitic fe-cr-ni alloys,” Journal of Physics F: Metal Physics, vol. 10, no. 3, p. 383, 1980.
[74] J. Kottke, D. Utt, M. Laurent-Brocq, A. Fareed, D. Gaertner, L. Perriere, Ł. Rogal,
A. Stukowski, K. Albe, S. V. Divinski, et al., “Experimental and theoretical study
of tracer diffusion in a series of (cocrfemn) 100- xnix alloys,” Acta Materialia,
vol. 194, pp. 236–248, 2020.
[75] H. Mehrer, “Diffusion in intermetallics,” Materials Transactions, JIM, vol. 37,
no. 6, pp. 1259–1280, 1996.
[76] W. Kucza, J. Dąbrowa, G. Cieślak, K. Berent, T. Kulik, and M. Danielewski,
“Studies of “sluggish diffusion”effect in co-cr-fe-mn-ni, co-cr-fe-ni and co-femn-ni high entropy alloys; determination of tracer diffusivities by combinatorial
approach,” Journal of Alloys and Compounds, vol. 731, pp. 920–928, 2018.
[77] J. Kottke, M. Laurent-Brocq, A. Fareed, D. Gaertner, L. Perrière, Ł. Rogal, S. V.
Divinski, and G. Wilde, “Tracer diffusion in the ni–cocrfemn system: Transition
from a dilute solid solution to a high entropy alloy,” Scripta Materialia, vol. 159,
pp. 94–98, 2019.
[78] M. S. Daw and M. Chandross, “Sluggish diffusion in random equimolar fcc alloys,” Physical Review Materials, vol. 5, no. 4, p. 043603, 2021.
[79] D. Beke and G. Erdélyi, “On the diffusion in high-entropy alloys,” Materials
Letters, vol. 164, pp. 111–113, 2016.
[80] R. W. Balluffi, S. M. Allen, and W. C. Carter, Kinetics of materials. John Wiley
& Sons, 2005.
187
[81] Z. Zhang, Z. Su, B. Zhang, Q. Yu, J. Ding, T. Shi, C. Lu, R. O. Ritchie, and E. Ma,
“Effect of local chemical order on the irradiation-induced defect evolution in crconi medium-entropy alloy,” Proceedings of the National Academy of Sciences,
vol. 120, no. 15, p. e2218673120, 2023.
[82] F. Gao and J. Qu, “Calculating the diffusivity of cu and sn in cu3sn intermetallic
by molecular dynamics simulations,” Materials Letters, vol. 73, pp. 92–94, 2012.
[83] S.-L. Shang, B.-C. Zhou, W. Y. Wang, A. J. Ross, X. L. Liu, Y.-J. Hu, H.-Z.
Fang, Y. Wang, and Z.-K. Liu, “A comprehensive first-principles study of pure
elements: vacancy formation and migration energies and self-diffusion coefficients,” Acta Materialia, vol. 109, pp. 128–141, 2016.
[84] Y.-C. Yang, C. Liu, C.-Y. Lin, and Z. Xia, “Core effect of local atomic configuration and design principles in alxcocrfeni high-entropy alloys,” Scripta Materialia,
vol. 178, pp. 181–186, 2020.
[85] Y.-C. Yang, C. Liu, C.-Y. Lin, and Z. Xia, “The effect of local atomic configuration in high-entropy alloys on the dislocation behaviors and mechanical properties,” Materials Science and Engineering: A, vol. 815, p. 141253, 2021.
[86] Q.-J. Li, H. Sheng, and E. Ma, “Strengthening in multi-principal element alloys
with local-chemical-order roughened dislocation pathways,” Nature communications, vol. 10, no. 1, p. 3563, 2019.
[87] E. Antillon, C. Woodward, S. Rao, B. Akdim, and T. Parthasarathy, “Chemical
short range order strengthening in a model fcc high entropy alloy,” Acta Materialia, vol. 190, pp. 29–42, 2020.
[88] S. Wang, “Atomic structure modeling of multi-principal-element alloys by the
principle of maximum entropy,” Entropy, vol. 15, no. 12, pp. 5536–5548, 2013.
[89] C.-Y. Cheng, Y.-C. Yang, Y.-Z. Zhong, Y.-Y. Chen, T. Hsu, and J.-W. Yeh, “Physical metallurgy of concentrated solid solutions from low-entropy to high-entropy
alloys,” Current Opinion in Solid State and Materials Science, vol. 21, no. 6,
pp. 299–311, 2017.
[90] S. Zhao, G. M. Stocks, and Y. Zhang, “Defect energetics of concentrated solidsolution alloys from ab initio calculations: Ni 0.5 co 0.5, ni 0.5 fe 0.5, ni 0.8
fe 0.2 and ni 0.8 cr 0.2,” Physical Chemistry Chemical Physics, vol. 18, no. 34,
pp. 24043–24056, 2016.
[91] S. Zhao, T. Egami, G. M. Stocks, and Y. Zhang, “Effect of d electrons on defect
properties in equiatomic nicocr and nicofecr concentrated solid solution alloys,”
Physical Review Materials, vol. 2, no. 1, p. 013602, 2018.
188
[92] Y.-Z. Wang and Y.-J. Wang, “Disentangling diffusion heterogeneity in highentropy alloys,” Acta Materialia, vol. 224, p. 117527, 2022.
[93] S. Zhao, “Defect properties in a vtacrw equiatomic high entropy alloy (hea) with
the body centered cubic (bcc) structure,” Journal of Materials Science & Technology, vol. 44, pp. 133–139, 2020.
[94] S. Zhao, Y. Xiong, S. Ma, J. Zhang, B. Xu, and J.-J. Kai, “Defect accumulation and evolution in refractory multi-principal element alloys,” Acta Materialia,
vol. 219, p. 117233, 2021.
[95] A. Roy, P. Singh, G. Balasubramanian, and D. D. Johnson, “Vacancy formation
energies and migration barriers in multi-principal element alloys,” Acta Materialia, vol. 226, p. 117611, 2022.
[96] Y.-K. Dou, Y.-P. Zhao, X.-F. He, J. Gao, J.-l. Cao, and W. Yang, “First-principles
study of vacancy defects in tivta and tivtanb concentrated solid-solution alloys,”
Journal of Nuclear Materials, vol. 573, p. 154096, 2023.
[97] Q. Wang, X. Kong, Y. Yu, W. Zhang, C. Teng, R. Pan, T. Xin, and L. Wu, “Effect
of local chemical environment on the point defects in alnbtizr refractory high
entropy alloys,” Journal of Nuclear Materials, vol. 581, p. 154451, 2023.
[98] D. S. Aidhy, “Chemical randomness, lattice distortion and the wide distributions
in the atomic level properties in high entropy alloys,” Computational Materials
Science, vol. 237, p. 112912, 2024.
[99] H. Guan, S. Huang, J. Ding, F. Tian, Q. Xu, and J. Zhao, “Chemical environment
and magnetic moment effects on point defect formations in cocrni-based concentrated solid-solution alloys,” Acta Materialia, vol. 187, pp. 122–134, 2020.
[100] W.-M. Choi, Y. H. Jo, S. S. Sohn, S. Lee, and B.-J. Lee, “Understanding the
physical metallurgy of the cocrfemnni high-entropy alloy: an atomistic simulation study,” Npj Computational Materials, vol. 4, no. 1, p. 1, 2018.
[101] B. Xing, W. Zou, T. J. Rupert, and P. Cao, “Vacancy diffusion barrier spectrum
and diffusion correlation in multicomponent alloys,” Acta Materialia, vol. 266,
p. 119653, 2024.
[102] D. Utt, S. Lee, Y. Xing, H. Jeong, A. Stukowski, S. H. Oh, G. Dehm, and K. Albe,
“The origin of jerky dislocation motion in high-entropy alloys,” Nature communications, vol. 13, no. 1, p. 4777, 2022.
[103] Y. Zhang, Y. N. Osetsky, and W. J. Weber, “Tunable chemical disorder in concentrated alloys: defect physics and radiation performance,” Chemical Reviews,
vol. 122, no. 1, pp. 789–829, 2021.
189
[104] B. Xu, J. Zhang, Y. Xiong, S. Ma, Y. Osetsky, and S. Zhao, “Mechanism of sluggish diffusion under rough energy landscape,” Cell Reports Physical Science,
vol. 4, no. 4, 2023.
[105] M. Jin, P. Cao, and M. P. Short, “Thermodynamic mixing energy and heterogeneous diffusion uncover the mechanisms of radiation damage reduction in singlephase ni-fe alloys,” Acta Materialia, vol. 147, pp. 16–23, 2018.
[106] B. Xing, X. Wang, W. J. Bowman, and P. Cao, “Short-range order localizing diffusion in multi-principal element alloys,” Scripta Materialia, vol. 210, p. 114450,
2022.
[107] S. L. Thomas and S. Patala, “Vacancy diffusion in multi-principal element alloys:
The role of chemical disorder in the ordered lattice,” Acta Materialia, vol. 196,
pp. 144–153, 2020.
[108] B. Xu, J. Zhang, S. Ma, Y. Xiong, S. Huang, J. Kai, and S. Zhao, “Revealing the
crucial role of rough energy landscape on self-diffusion in high-entropy alloys
based on machine learning and kinetic monte carlo,” Acta Materialia, vol. 234,
p. 118051, 2022.
[109] H. S. Oh, S. J. Kim, K. Odbadrakh, W. H. Ryu, K. N. Yoon, S. Mu, F. Körmann,
Y. Ikeda, C. C. Tasan, D. Raabe, et al., “Engineering atomic-level complexity in
high-entropy and complex concentrated alloys,” Nature communications, vol. 10,
no. 1, p. 2090, 2019.
[110] Y. Zhang, S. Zhao, W. J. Weber, K. Nordlund, F. Granberg, and F. Djurabekova,
“Atomic-level heterogeneity and defect dynamics in concentrated solid-solution
alloys,” Current Opinion in Solid State and Materials Science, vol. 21, no. 5,
pp. 221–237, 2017.
[111] Y. Osetsky, A. V. Barashev, L. K. Béland, Z. Yao, K. Ferasat, and Y. Zhang,
“Tunable chemical complexity to control atomic diffusion in alloys,” npj Computational Materials, vol. 6, no. 1, p. 38, 2020.
[112] T. Shi, S. Lyu, Z. Su, Y. Wang, X. Qiu, D. Sun, Y. Xin, W. Li, J. Cao, Q. Peng,
et al., “Spatial inhomogeneity of point defect properties in refractory multiprincipal element alloy with short-range order: A first-principles study,” Journal
of Applied Physics, vol. 133, no. 7, 2023.
[113] A. Seoane, D. Farkas, and X.-M. Bai, “Influence of compositional complexity
on species diffusion behavior in high-entropy solid-solution alloys,” Journal of
Materials Research, vol. 37, no. 7, pp. 1403–1415, 2022.
190
[114] Y. N. Osetsky, L. K. Béland, A. V. Barashev, and Y. Zhang, “On the existence
and origin of sluggish diffusion in chemically disordered concentrated alloys,”
Current Opinion in Solid State and Materials Science, vol. 22, no. 3, pp. 65–74,
2018.
[115] Z. Fan, B. Xing, and P. Cao, “Predicting path-dependent diffusion barrier spectra
in vast compositional space of multi-principal element alloys via convolutional
neural networks,” Acta Materialia, vol. 237, p. 118159, 2022.
[116] W. Huang and X.-M. Bai, “Machine learning based on-the-fly kinetic monte carlo
simulations of sluggish diffusion in ni-fe concentrated alloys,” Journal of Alloys
and Compounds, vol. 937, p. 168457, 2023.
[117] B. Xu, S. Ma, Y. Xiong, J. Zhang, S. Huang, J.-J. Kai, and S. Zhao, “Exploring the
influence of percolation on vacancy-mediated diffusion in cocrni multi-principal
element alloys,” Materials & Design, vol. 223, p. 111238, 2022.
[118] W. Huang, D. Farkas, and X.-M. Bai, “High-throughput machine learning-kinetic
monte carlo framework for diffusion studies in equiatomic and non-equiatomic
fenicrcocu high-entropy alloys,” Materialia, vol. 32, p. 101966, 2023.
[119] A. Roy, J. Munshi, and G. Balasubramanian, “Low energy atomic traps sluggardize the diffusion in compositionally complex refractory alloys,” Intermetallics, vol. 131, p. 107106, 2021.
[120] C.-H. Chan, Q. Huo, A. Kumar, Y. Shi, H. Hong, Y. Du, S. Ren, K.-P. Wong,
and C.-T. Yip, “Heterogeneity and memory effect in the sluggish dynamics of
vacancy defects in colloidal disordered crystals and their implications to highentropy alloys,” Advanced Science, vol. 9, no. 36, p. 2205522, 2022.
[121] X. Xu, J. Wang, J.-P. Lv, and Y. Deng, “Simultaneous analysis of threedimensional percolation models,” Frontiers of Physics, vol. 9, pp. 113–119, 2014.
[122] Y. N. Osetsky, L. K. Béland, and R. E. Stoller, “Specific features of defect and
mass transport in concentrated fcc alloys,” Acta Materialia, vol. 115, pp. 364–
371, 2016.
[123] Y. Osetsky, A. V. Barashev, and Y. Zhang, “Sluggish, chemical bias and percolation phenomena in atomic transport by vacancy and interstitial diffusion in
nife alloys,” Current Opinion in Solid State and Materials Science, vol. 25, no. 6,
p. 100961, 2021.
[124] R. Zhang, S. Zhao, J. Ding, Y. Chong, T. Jia, C. Ophus, M. Asta, R. O. Ritchie,
and A. M. Minor, “Short-range order and its impact on the crconi medium-entropy
alloy,” Nature, vol. 581, no. 7808, pp. 283–287, 2020.
191
[125] X. Chen, Q. Wang, Z. Cheng, M. Zhu, H. Zhou, P. Jiang, L. Zhou, Q. Xue, F. Yuan,
J. Zhu, et al., “Direct observation of chemical short-range order in a mediumentropy alloy,” Nature, vol. 592, no. 7856, pp. 712–716, 2021.
[126] B. Xu, S. Ma, S. Huang, J. Zhang, Y. Xiong, H. Fu, X. Xiang, and S. Zhao, “Influence of short-range order on diffusion in multiprincipal element alloys from longtime atomistic simulations,” Physical Review Materials, vol. 7, no. 3, p. 033605,
2023.
[127] A. Manzoor and Y. Zhang, “Influence of defect thermodynamics on self-diffusion
in complex concentrated alloys with chemical ordering,” JOM, vol. 74, no. 11,
pp. 4107–4120, 2022.
[128] A. Manzoor and Y. Zhang, “Interplay between thermal vacancy and shortrange order in complex concentrated alloys,” Journal of Alloys and Compounds,
p. 173788, 2024.
[129] Y. Li and W. Qiang, “Defect properties of a body-centered cubic equiatomic
tivzrta high-entropy alloy from atomistic simulations,” Journal of Physics: Condensed Matter, vol. 35, no. 34, p. 345701, 2023.
[130] S. Zhao, “Role of chemical disorder and local ordering on defect evolution in
high-entropy alloys,” Physical Review Materials, vol. 5, no. 10, p. 103604, 2021.
[131] S.-C. Dai, Y. Yang, and Y.-J. Wang, “Role of chemical disorder in slowing down
diffusion in complex concentrated alloys,” Physical Review Materials, vol. 8,
no. 3, p. 033607, 2024.
[132] A. F. Voter, “Introduction to the kinetic monte carlo method,” in Radiation effects
in solids, pp. 1–23, Springer, 2007.
[133] M. Andersen, C. Panosetti, and K. Reuter, “A practical guide to surface kinetic
monte carlo simulations,” Frontiers in chemistry, vol. 7, p. 202, 2019.
[134] C. R. Harris, K. J. Millman, S. J. Van Der Walt, R. Gommers, P. Virtanen,
D. Cournapeau, E. Wieser, J. Taylor, S. Berg, N. J. Smith, et al., “Array programming with numpy,” Nature, vol. 585, no. 7825, pp. 357–362, 2020.
[135] T. pandas development team, “pandas-dev/pandas: Pandas,” Feb. 2020.
[136] J. D. Hunter, “Matplotlib: A 2d graphics environment,” Computing in Science &
Engineering, vol. 9, no. 3, pp. 90–95, 2007.
[137] J. M. Chambers, Graphical methods for data analysis. Chapman and Hall/CRC,
2018.
192
[138] A. Sauvé-Lacoursière, S. Gelin, G. Adjanor, C. Domain, and N. Mousseau, “Unexpected role of prefactors in defects diffusion: The case of vacancies in the
55fe-28ni-17cr concentrated solid-solution alloys,” Acta Materialia, vol. 237,
p. 118153, 2022.
[139] J. Lefèvre López, N. Mousseau, G. Adjanor, and C. Domain, “Harmonic transition state theory applied to vacancy diffusion pre-exponential factors in a concentrated solid-solution alloy,” Physical Review Materials, vol. 8, no. 1, p. 013609,
2024