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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:
- first-principles /
- Pd-base alloy /
- mixing enthalpy /
- elastic modulus
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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.
图 2 Pd31X晶体结构示意图
Figure 2. Crystal structure of Pd31X.
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).
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.
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.
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.
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.
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.
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 V/(Å3·unit cell–1) ΔH/(J·mol–1) 0L V/(Å3·unit cell–1) ΔH/(J·mol–1) 0L 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
EB/G υ Al 207 151 65 47 170 128 3.643 0.374 Si 197 160 61 38 172 106 4.548 0.398 Sc 208 147 68 49 167 135 3.383 0.365 Ti 212 151 71 51 171 138 3.381 0.365 V 212 153 73 50 173 138 3.432 0.367 Cr 211 153 74 51 172 139 3.375 0.365 Mn 213 155 70 49 174 135 3.544 0.371 Fe 212 154 66 47 173 130 3.655 0.375 Co 212 152 64 47 172 129 3.659 0.375 Ni 211 151 63 47 171 129 3.654 0.375 Cu 207 151 66 47 169 129 3.611 0.373 Zn 203 152 64 44 169 122 3.814 0.379 Ga 205 152 63 45 169 123 3.791 0.379 Y 202 145 66 47 164 129 3.471 0.369 Zr 210 150 70 50 170 136 3.418 0.367 Nb 211 153 74 51 172 139 3.375 0.365 Mo 212 155 76 51 174 140 3.393 0.366 Tc 215 158 75 51 177 140 3.463 0.368 Ru 213 155 69 49 174 134 3.550 0.371 Rh 213 153 64 47 173 130 3.674 0.375 Ag 200 151 64 43 168 120 3.859 0.381 Cd 199 150 64 44 167 120 3.820 0.380 La 193 143 61 42 160 116 3.784 0.379 Ce 199 147 64 44 165 122 3.720 0.377 Hf 210 150 70 50 170 137 3.405 0.366 Ta 213 154 74 51 174 140 3.403 0.366 W 212 156 77 51 175 140 3.409 0.366 Re 213 155 75 50 174 140 3.402 0.366 Os 214 157 74 51 176 138 3.486 0.369 Ir 213 156 67 48 175 131 3.684 0.376 Pt 214 153 63 47 174 129 3.689 0.376 Au 207 151 65 46 170 127 3.666 0.375 Th 198 148 64 44 165 121 3.742 0.377
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