Influence of alloying elements on the thermodynamic and elastic properties of palladium based alloys and database construction

ZHU Hanyu; CHONG Xiaoyu; GAO Xingyu; WU Haijun; LI Zulai; FENG Jing; SONG Haifeng

doi:10.7498/aps.74.20251058

Abstract

The lower friction coefficient and superior mechanical properties of palladium (Pd) alloys make them potentially advantageous for use in high-precision instruments and devices that require long-term stable performance. However, the high cost of raw materials and experimental expenses result in a lack of fundamental data, which hinders the design of high-performance Pd alloys. Therefore, in this study, first-principles calculations are used to determine the lattice constant and elastic modulus of Pd. A model of dilute solid solutions formed by Pd with 33 alloying elements including Al, Si, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, and others, is established. The mixing enthalpy, elastic constant, and elastic modulus are calculated. The results show that, all other alloying elements except for Mn, Fe, Co, Ni, Ru, Rh, Os, and Ir can form solid solutions with Pd. Alloying elements from both sides of the periodic table enhance the ductility of Pd solid solutions, with La, Ag, and Zn having the most significant effects, while Cu and Hf reduce the ductility of Pd. Differential charge density analysis indicates that the electron cloud formed after doping with Ag is spherically distributed, thereby improving ductility. After doping with Hf, the degree of delocalization around the atoms is maximized, indicating a strong ionic bond between Hf and Pd, which results in a higher hardness of Pd₃₁Hf. The datasets presented in this paper are openly available at https://www.doi.org/10.57760/sciencedb.j00213.00186.

Keywords:

Authors and contacts

1.
Faculty of Materials Science and Engineering, Kunming University of Science and Technology, Kunming 650093, China
2.
Beijing Institute of Applied Physics and Computational Mathematics, Beijing 100098, China
3.
State Key Laboratory of Precious Metal Functional Materials, Kunming 650106, China

Corresponding author: CHONG Xiaoyu, xiaoyuchong@kust.edu.cn ; GAO Xingyu, gao_xingyu@iapcm.ac.cn ;

Funds: Project supported by the Key Research and Development Program of Yunnan Province, China (Grant No. 202403AA080016).

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References

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Cited By

图 1 (a) Pd超胞模型; (b) Pd的能量-体积曲线; (c) Pd的弹性模量与实验值对比

Figure 1. (a) Supercell models of Pd; (b) the energy-volume curves for Pd; (c) comparison of the elastic modulus of Pd with experimental values.

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图 2 Pd₃₁X晶体结构示意图

Figure 2. Crystal structure of Pd₃₁X.

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图 3 Pd₃₁X (X = Al, Si, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Cd, La, Ce, Hf, Ta, W, Re, Os, Ir, Pt, Th)的能量-体积曲线

Figure 3. Energy-volume curves for Pd₃₁X (X = Al, Si, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Cd, La, Ce, Hf, Ta, W, Re, Os, Ir, Pt, Th).

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图 4 原子弛豫(a)和完全弛豫(b)策略下Ir-X二元合金的混合焓的三点拟合

Figure 4. Three-point fitting of the mixing enthalpy of Ir-X binary alloys under atomic relaxation (a) and complete relaxation (b) strategies.

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图 5 钯基稀固溶体合金的弹性特性　(a) 体模量; (b) 剪切模量; (c) 杨氏模量; (d) 泊松比

Figure 5. Elastic properties of Pd-based dilute alloys: (a) Bulk modulus; (b) shear modulus; (c) Young’s modulus; (d) Poisson’s ratio.

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图 6 通过R-K多项式得到的Pd-X二元合金的弹性常数和弹性模量

Figure 6. Elastic constants and elastic modulus of Pd-X binary alloys obtained through R-K polynomials.

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图 7 钯基稀固溶体合金在完全弛豫策略下的泊松比与B/G的关系

Figure 7. Poisson’s ratio and B/G relationship of dilute Pd-based alloys under the complete relaxation strategy in terms of computational materials science.

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图 8 差分电荷密度　(a) (111)面; (b) (100)面

Figure 8. Differential charge density: (a) (111) plane; (b) (100) plane.

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表 1 Pd稀固溶体原子及完全松弛豫略下的体积(V )、混合焓(ΔH )与零阶相互作用参数(⁰L)

Table 1. Volume (V ), mixing enthalpy (ΔH ), and zero-order interaction parameter (⁰L) for atomic and complete relaxation strategies of Pd dilute solid solution.

$ {\text{P}}{{\text{d}}_{31}}X $	Atom relaxing strategy			Full relaxing strategy			Solid solubility
$ {\text{P}}{{\text{d}}_{31}}X $	V/(Å³·unit cell^–1)	ΔH/(J·mol^–1)	⁰L	V/(Å³·unit cell^–1)	ΔH/(J·mol^–1)	⁰L	Solid solubility
Al	487.87	–8564.71	–282911.59	486.32	–8681.80	–286779.56	5%
Si	487.87	–7619.23	–251680.33	484.34	–7644.04	–252499.95	6.00%
Sc	487.87	–11153.07	–368411.25	491.01	–11418.30	–377172.35	10%
Ti	487.87	–10113.64	–334076.21	487.22	–10255.70	–338768.78	6%
V	487.87	–6175.68	–203996.75	485.01	–6249.44	–206433.24	10%
Cr	487.87	–1968.7	–65030.61	485.57	–1897.47	–62677.72	12%
Mn	487.87	1093.49	36120.32	483.34	962.92	31807.49	15%
Fe	487.87	2205.78	72862.04	483.35	2006.79	66288.66	10%
Co	487.87	1678.78	55453.81	483.60	1427.42	47151.00	3%
Ni	487.87	524.51	17325.86	484.15	270.46	8933.84	完全互溶
Cu	487.87	–766.65	–25324.15	485.30	–997.37	–32945.41	20%
Zn	487.87	–4513.42	–149088.54	486.68	–4589.15	–151590.11	7%
Ga	487.87	–6361.88	–210147.41	487.05	–6472.09	–213787.63
Y	487.87	–10057.30	–332215.41	496.61	–10411.01	–343899.13	8%
Zr	487.87	–11826.39	–390652.51	492.53	–12047.20	–397946.22	8%
Nb	487.87	–9494.62	–313628.61	489.21	–9616.53	–317655.76	15%
Mo	487.87	–5023.61	–165941.15	487.39	–5089.07	–168103.51	23%
Tc	487.87	–973.48	–32156.31	486.18	–1072.43	–35424.78	25%—86%
Ru	487.87	1059.92	35011.71	486.21	995.81	32893.73	4%
Rh	487.87	814.70	26911.42	486.61	531.91	17570.24	8%
Ag	487.87	–18.04	–595.92	490.16	–128.24	–4236.11	完全互溶
Cd	487.87	–3068.11	–101346.48	492.11	–3208.36	–105979.25
La	487.87	–8248.78	–272475.91	501.57	–8649.21	–285703.03
Ce	487.87	–11290.17	–372939.69	497.96	–11613.00	–383603.62	17%
Hf	487.87	–12565.04	–415051.67	491.86	–12765.53	–421674.24	12%
Ta	487.87	–10076.76	–332858.26	489.28	–10186.26	–336475.27	4%
W	487.87	–5839.20	–192881.85	487.40	–5887.76	–194485.85	28%
Re	487.87	–1398.74	–46203.54	487.82	–1356.74	–44816.19	18%
Os	487.87	1196.45	39521.60	486.10	1091.13	36042.65	9%
Ir	487.87	1114.98	36830.25	486.88	910.96	30091.00	3%
Pt	487.87	–229.64	–7585.51	488.04	–509.93	–16844.30	完全互溶
Au	487.87	–458.08	–15131.47	490.65	–733.56	–24231.25	完全互溶
Th	487.87	–12313.59	–406745.80	501.50	–12676.97	–418748.79

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表 2 完全驰豫下的钯基稀固溶体合金的计算弹性性能(GPa), 包括弹性常数$ {C_{ij}} $、体模量、剪切模量、杨氏模量、B/G和泊松比

Table 2. Calculated elastic properties (GPa) of Pd-based dilute alloys in full relaxing strategy, including Elastic constants $ {C_{ij}} $, bulk modulus, shear modulus, Young’s modulus, B/G and Poisson’s ratio.

$ {\text{P}}{{\text{d}}_{31}}X $	C₁₁	C₁₂	C₄₄	G	B	E	B/G	υ	$ {\text{P}}{{\text{d}}_{31}}X $	C₁₁	C₁₂	C₄₄	G	B	E	B/G	υ
Al	207	151	65	47	170	128	3.643	0.374	Tc	215	158	75	51	177	140	3.463	0.368
Si	197	160	61	38	172	106	4.548	0.398	Ru	213	155	69	49	174	134	3.550	0.371
Sc	208	147	68	49	167	135	3.383	0.365	Rh	213	153	64	47	173	130	3.674	0.375
Ti	212	151	71	51	171	138	3.381	0.365	Ag	200	151	64	43	168	120	3.859	0.381
V	212	153	73	50	173	138	3.432	0.367	Cd	199	150	64	44	167	120	3.820	0.380
Cr	211	153	74	51	172	139	3.375	0.365	La	193	143	61	42	160	116	3.784	0.379
Mn	213	155	70	49	174	135	3.544	0.371	Ce	199	147	64	44	165	122	3.720	0.377
Fe	212	154	66	47	173	130	3.655	0.375	Hf	210	150	70	50	170	137	3.405	0.366
Co	212	152	64	47	172	129	3.659	0.375	Ta	213	154	74	51	174	140	3.403	0.366
Ni	211	151	63	47	171	129	3.654	0.375	W	212	156	77	51	175	140	3.409	0.366
Cu	207	151	66	47	169	129	3.611	0.373	Re	213	155	75	50	174	140	3.402	0.366
Zn	203	152	64	44	169	122	3.814	0.379	Os	214	157	74	51	176	138	3.486	0.369
Ga	205	152	63	45	169	123	3.791	0.379	Ir	213	156	67	48	175	131	3.684	0.376
Y	202	145	66	47	164	129	3.471	0.369	Pt	214	153	63	47	174	129	3.689	0.376
Zr	210	150	70	50	170	136	3.418	0.367	Au	207	151	65	46	170	127	3.666	0.375
Nb	211	153	74	51	172	139	3.375	0.365	Th	198	148	64	44	165	121	3.742	0.377
Mo	212	155	76	51	174	140	3.393	0.366

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[1]	高书亮, 樊思思, 袁成, 曹军伟 2023 航空兵器 30 11 Google Scholar Gao S L, Fan S S, Yuan C, Cao J W 2023 Aero Weapon. 30 11 Google Scholar
[2]	Xu X, Lv J X, Wang Y, Li M, Wang Z, Wang H 2025 Mater. Genome Eng. Adv. 3 e69 Google Scholar
[3]	焦磊 2018 硕士学位论文 (北京: 北京工业大学) Jiao L 2018 M. S. Thesis ( Beijing: Beijing University of Technology
[4]	马晓东余建军赵涛 2016 山东工业技术 19 2 Google Scholar Ma X D, Yu J J, Zhao T 2016 Shandong Ind. Technol. 19 2 Google Scholar
[5]	宿彦京, 付华栋, 白洋, 姜雪, 谢建新 2020 金属学报 56 1313 Google Scholar Su Y J, Fu H D, Bai Y, Jiang X, Xie J X 2020 Acta Metall. Sin. 56 1313 Google Scholar
[6]	Wang Z R, Qin M Y, Zhang P, Xu Y G, Que S T, Yan F, Xiang X D 2025 Mater. Genome Eng. Adv. 3 e70010 Google Scholar
[7]	汪洪, 向勇, 项晓东, 陈立泉 2015 科技导报 33 13 Google Scholar Wang H, Xiang Y, Xiang X D, Chen L Q 2015 Sci. Technol. Rev. 33 13 Google Scholar
[8]	Guo Z K, Li R, He X F, Guo J, Ju S H 2024 Mater. Genome Eng. Adv. 2 e73 Google Scholar
[9]	Meng H Y, Huang J, Zhao T H, Zhang X Y, Tong Y, Qu J L, Li W D, Jiang L, Meng F C, Chen S Y 2025 Mater. Genome Eng. Adv. 9 e70002 Google Scholar
[10]	Wang W Y, Tang B, Shang S L, Wang J, Li S, Wang Y, Zhu J, Wei S, Wang J, Darling K A, Mathaudhu S N, Wang Y, Ren Y, Hui X D, Kecskes L J, Li J, Liu Z K 2019 Acta Mater. 170 231 Google Scholar
[11]	Xia Y X, He J G, Chen N F, Chen J K 2024 Rare Metals 43 3460 Google Scholar
[12]	Chong X Y, Paz Soldan Palma J, Wang Y, Shang S L, Drymiotis F, Ravi V A, Star K E, Fleurial J P, Liu Z K 2021 Acta Mater. 217 117169 Google Scholar
[13]	Hu Y C, Tian J 2023 J. Mater. Informat. 3 1 Google Scholar
[14]	Wang W Y, Gan B, Lin D, Wang J, Wang Y, Tang B, Kou H C, Shang S L, Wang Y, Gao X Y, Song H F, Hui X D, Kecskes L J, Xia Z H, Dahmen K A, Liaw P K, Li J S, Liu Z K 2020 J. Mater. Sci. Technol. 53 192 Google Scholar
[15]	Zou C, Li J, Wang W Y, Zhang Y, Lin D, Yuan R, Wang X, Tang B, Wang J, Gao X Y, Kou H, Hui X D, Zeng X Q, Ma Q, Song H F, Liu Z K, Xu D S 2021 Acta Mater. 202 211 Google Scholar
[16]	Liao M Q, Liu Y, Min L, Lai Z H, Han T Y, Yang D, Zhu J C 2018 Intermetallics. 101 152 Google Scholar
[17]	Birch F 1947 Phys. Rev. 71 809 Google Scholar
[18]	Shang S L, Wang Y, Kim D, Liu Z K 2010 Comput. Mater. Sci. 47 1040 Google Scholar
[19]	周云轩 2021 博士学位论文 (昆明: 昆明理工大学) Zhou Y X 2021 Ph. D. Dissertation ( Kunming: Kunming University of Science and Technology
[20]	Chong X Y, Wei Y, Liang Y X, Shang S L, Li C, Zhang A M, Wei Y, Gao X Y, Wang Y, Feng J, Chen L, Song H F, Liu Z K 2023 J. Mater. Inf. 3 21 Google Scholar
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[24]	Duan Y H, Sun Y, Peng M J, Zhou S G 2014 J. Alloys Compd. 595 14 Google Scholar
[25]	Tang B Y, Chen P, Li D L, Yi J X, Wen L, Peng L M, Ding W J 2010 J. Alloys Compd. 492 416 Google Scholar
[26]	Wang Y, Liao M Q, Bocklund B J, Gao P, Shang S L, Kim H J, Beese A M, Chen L Q, Liu Z K 2021 Calphad 75 102355 Google Scholar
[27]	Liu Y, Lu Y H, Wang W, Li J, Zhang Y, Yin J, Pan X Q, Chen Y, Li J S, Song H F 2023 J. Mater. Inf. 3 17 Google Scholar
[28]	Chen R T, Li E, Zou Y 2024 J. Mater. Inf. 4 26 Google Scholar
[29]	Perdew J P, Ruzsinszky A, Csonka G I, Vydrov O A, Scuseria G E, Constantin L A, Zhou X, Burke K 2008 Phys. Rev. Lett. 100 136406 Google Scholar
[30]	Vega L, Vines F 2020 J. Comput. Chem. 41 2598 Google Scholar
[31]	Tretiakov K V, Wojciechowski K W 2008 J. Non-Cryst. Solids 354 35 Google Scholar
[32]	Ji H, Yin J, Wei G, Lai W S, Liu B X, Liu J B 2023 Rare Metals 42 1663 Google Scholar
[33]	Li R Y, Duan Y H 2016 Philos. Mag. 96 972 Google Scholar
[34]	Ozisik H B, Deligoz E, Ozisik H, Ateser E 2020 Mater. Res. Express 7 025004 Google Scholar
[35]	Kenneth Barbalace https://EnvironmentalChemistry.com/yogi/periodic/Pd.html [2025-10-12]
[36]	Samsonov G V 1968 Handbook of the Physicochemical Properties of the Elements (New York: Springer New
[37]	彭红建, 谢佑卿, 陶辉锦 2006 中国有色金属学报 16 100 Google Scholar Peng H J, Xie Y Q, Tao H J 2006 Trans. Nonferrous Met. Soc. China 16 100 Google Scholar
[38]	段亚娟, 乔吉超 2022 物理学报 71 086101 Google Scholar Duan Y J, Qiao J C 2022 Acta Phys. Sin. 71 086101 Google Scholar
[39]	Zhang X Z, Wang D T, Nagaumi H, Wang R, Wu Z B, Zhang M H, Gao D S, Chen H, Wang P F, Zhou P F, Zhou Y X, Wang Z X, Li T L 2025 Mater. Genome Eng. Adv. 3 e70008 Google Scholar
[40]	Born M 1939 J. Chem. Phys. 7 591 Google Scholar
[41]	Wang G C, Jiang Y H, Li Z L, Chong X Y, Feng J 2021 Ceram. Int. 47 4758 Google Scholar
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Vol. 74, Issue 24 - 2025-12-20

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