Journal of Inorganic Materials ›› 2026, Vol. 41 ›› Issue (4): 445-454.DOI: 10.15541/jim20250272
• RESEARCH ARTICLE • Previous Articles Next Articles
WANG Yuhe1,2(
), LUO Yixiu1(
), GUO Huiming3, ZHANG Guangheng4, ZHANG Siyan4, SUN Luchao1, WANG Jiemin1, WANG Jingyang1(
)
Received:2025-06-27
Revised:2025-08-22
Published:2026-04-20
Online:2025-08-26
Contact:
LUO Yixiu, professor. E-mail: yxluo13s@imr.ac.cn;About author:WANG Yuhe (2001-), male, Master candidate. E-mail: yhwang23s@imr.ac.cn
Supported by:CLC Number:
WANG Yuhe, LUO Yixiu, GUO Huiming, ZHANG Guangheng, ZHANG Siyan, SUN Luchao, WANG Jiemin, WANG Jingyang. First-principles Investigation of Elastic and Thermophysical Properties of High-entropy Rare-earth Oxide Thermal Barrier Coating Materials[J]. Journal of Inorganic Materials, 2026, 41(4): 445-454.
Fig. 1 Crystal structures of high-entropy rare-earth oxide materials (a) C-RE2O3; (b) C-(REI1/4REII1/4REIII1/4REIV1/4)2O3; (c) C-(REI3/16REII3/16REIII3/16REIV3/16Hf1/4)2O3
| Compound | Method | a/Å | b/Å | c/Å | α/(°) | β/(°) | γ/(°) | ρ/(g•cm-3) |
|---|---|---|---|---|---|---|---|---|
| (Y1/4Ho1/4Er1/4Yb1/4)2O3 | Calc | 9.13 | 9.14 | 9.15 | 109.49 | 109.52 | 109.46 | 7.81 |
| (Er1/4Tm1/4Yb1/4Lu1/4)2O3 | Calc | 9.05 | 9.04 | 9.04 | 109.43 | 109.51 | 109.46 | 9.10 |
| (Y3/16Ho3/16Er3/16Yb3/16Hf1/4)2O3 | Calc | 9.01 | 9.06 | 9.01 | 109.34 | 109.37 | 109.55 | 8.43 |
| (Er3/16Tm3/16Yb3/16Lu3/16Hf1/4)2O3 | Calc | 8.92 | 9.00 | 8.98 | 109.53 | 109.74 | 109.47 | 9.46 |
Table 1 Calculated lattice parameters of four high-entropy rare-earth oxide materials
| Compound | Method | a/Å | b/Å | c/Å | α/(°) | β/(°) | γ/(°) | ρ/(g•cm-3) |
|---|---|---|---|---|---|---|---|---|
| (Y1/4Ho1/4Er1/4Yb1/4)2O3 | Calc | 9.13 | 9.14 | 9.15 | 109.49 | 109.52 | 109.46 | 7.81 |
| (Er1/4Tm1/4Yb1/4Lu1/4)2O3 | Calc | 9.05 | 9.04 | 9.04 | 109.43 | 109.51 | 109.46 | 9.10 |
| (Y3/16Ho3/16Er3/16Yb3/16Hf1/4)2O3 | Calc | 9.01 | 9.06 | 9.01 | 109.34 | 109.37 | 109.55 | 8.43 |
| (Er3/16Tm3/16Yb3/16Lu3/16Hf1/4)2O3 | Calc | 8.92 | 9.00 | 8.98 | 109.53 | 109.74 | 109.47 | 9.46 |
| Compound | c11 | c12 | c13 | c22 | c23 | c33 | c44 | c55 | c66 |
|---|---|---|---|---|---|---|---|---|---|
| (Y1/4Ho1/4Er1/4Yb1/4)2O3 | 244.79 | 95.97 | 95.90 | 240.43 | 101.79 | 239.03 | 69.14 | 62.94 | 63.10 |
| (Er1/4Tm1/4Yb1/4Lu1/4)2O3 | 257.22 | 100.81 | 99.30 | 251.14 | 107.11 | 250.74 | 72.87 | 66.14 | 66.16 |
| (Y3/16Ho3/16Er3/16Yb3/16Hf1/4)2O3 | 250.73 | 98.29 | 98.37 | 248.15 | 97.99 | 241.45 | 70.74 | 69.71 | 69.93 |
| (Er3/16Tm3/16Yb3/16Lu3/16Hf1/4)2O3 | 259.08 | 102.63 | 100.97 | 250.68 | 97.74 | 251.63 | 77.04 | 71.63 | 75.61 |
Table 2 Calculated second-order elastic coefficients of four high-entropy rare-earth oxide materials (GPa)
| Compound | c11 | c12 | c13 | c22 | c23 | c33 | c44 | c55 | c66 |
|---|---|---|---|---|---|---|---|---|---|
| (Y1/4Ho1/4Er1/4Yb1/4)2O3 | 244.79 | 95.97 | 95.90 | 240.43 | 101.79 | 239.03 | 69.14 | 62.94 | 63.10 |
| (Er1/4Tm1/4Yb1/4Lu1/4)2O3 | 257.22 | 100.81 | 99.30 | 251.14 | 107.11 | 250.74 | 72.87 | 66.14 | 66.16 |
| (Y3/16Ho3/16Er3/16Yb3/16Hf1/4)2O3 | 250.73 | 98.29 | 98.37 | 248.15 | 97.99 | 241.45 | 70.74 | 69.71 | 69.93 |
| (Er3/16Tm3/16Yb3/16Lu3/16Hf1/4)2O3 | 259.08 | 102.63 | 100.97 | 250.68 | 97.74 | 251.63 | 77.04 | 71.63 | 75.61 |
| Compound | B/GPa | G/GPa | E/GPa | Emax/GPa | Emin/GPa | Emax/Emin | G/B | ν/GPa |
|---|---|---|---|---|---|---|---|---|
| (Y1/4Ho1/4Er1/4Yb1/4)2O3 | 146 | 67 | 175 | 192 | 151 | 1.2753 | 0.4606 | 0.3004 |
| (Er1/4Tm1/4Yb1/4Lu1/4)2O3 | 153 | 71 | 183 | 201 | 159 | 1.2639 | 0.4624 | 0.2997 |
| (Y3/16Ho3/16Er3/16Yb3/16Hf1/4)2O3 | 148 | 72 | 185 | 197 | 174 | 1.1323 | 0.4852 | 0.2912 |
| (Er3/16Tm3/16Yb3/16Lu3/16Hf1/4)2O3 | 151 | 75 | 194 | 209 | 180 | 1.1605 | 0.4975 | 0.2866 |
Table 3 Calculated elastic modulus of four high-entropy rare-earth oxide materials
| Compound | B/GPa | G/GPa | E/GPa | Emax/GPa | Emin/GPa | Emax/Emin | G/B | ν/GPa |
|---|---|---|---|---|---|---|---|---|
| (Y1/4Ho1/4Er1/4Yb1/4)2O3 | 146 | 67 | 175 | 192 | 151 | 1.2753 | 0.4606 | 0.3004 |
| (Er1/4Tm1/4Yb1/4Lu1/4)2O3 | 153 | 71 | 183 | 201 | 159 | 1.2639 | 0.4624 | 0.2997 |
| (Y3/16Ho3/16Er3/16Yb3/16Hf1/4)2O3 | 148 | 72 | 185 | 197 | 174 | 1.1323 | 0.4852 | 0.2912 |
| (Er3/16Tm3/16Yb3/16Lu3/16Hf1/4)2O3 | 151 | 75 | 194 | 209 | 180 | 1.1605 | 0.4975 | 0.2866 |
| Compound | vs/(m·s-1) | vl/(m·s-1) | vm/(m·s-1) | ΘD/K | γ | αL/(×10-6, K-1) | kmin/(W·m-1·K-1) |
|---|---|---|---|---|---|---|---|
| (Y1/4Ho1/4Er1/4Yb1/4)2O3 | 2932 | 5489 | 3276 | 398 | 1.7749 | 11.4574 | 0.7567 |
| (Er1/4Tm1/4Yb1/4Lu1/4)2O3 | 2785 | 5207 | 3111 | 382 | 1.7707 | 11.2533 | 0.7334 |
| (Y3/16Ho3/16Er3/16Yb3/16Hf1/4)2O3 | 2915 | 5371 | 3253 | 400 | 1.7194 | 11.3375 | 0.7690 |
| (Er3/16Tm3/16Yb3/16Lu3/16Hf1/4)2O3 | 2823 | 5162 | 3148 | 390 | 1.6928 | 11.1577 | 0.7566 |
Table 4 Thermophysical parameters of four high-entropy rare-earth oxide materials
| Compound | vs/(m·s-1) | vl/(m·s-1) | vm/(m·s-1) | ΘD/K | γ | αL/(×10-6, K-1) | kmin/(W·m-1·K-1) |
|---|---|---|---|---|---|---|---|
| (Y1/4Ho1/4Er1/4Yb1/4)2O3 | 2932 | 5489 | 3276 | 398 | 1.7749 | 11.4574 | 0.7567 |
| (Er1/4Tm1/4Yb1/4Lu1/4)2O3 | 2785 | 5207 | 3111 | 382 | 1.7707 | 11.2533 | 0.7334 |
| (Y3/16Ho3/16Er3/16Yb3/16Hf1/4)2O3 | 2915 | 5371 | 3253 | 400 | 1.7194 | 11.3375 | 0.7690 |
| (Er3/16Tm3/16Yb3/16Lu3/16Hf1/4)2O3 | 2823 | 5162 | 3148 | 390 | 1.6928 | 11.1577 | 0.7566 |
| Parameter | 1-YHoErYb | 2-ErTmYbLu | 3-YHoErYbHf | 4-ErTmYbLuHf | Diff(1-3)/% | Diff(2-4)/% | Diff(1-2)/% | Diff(3-4)/% |
|---|---|---|---|---|---|---|---|---|
| kT | 898.98 | 892.94 | 1006.36 | 1050.75 | 12.07 | 17.70 | -0.68 | 4.32 |
| 1.44×10-12 | 1.62×10-12 | 1.49×10-12 | 1.62×10-12 | 3.15 | -0.01 | 11.90 | 8.48 | |
| ΘD3 | 6.31×107 | 5.58×107 | 6.40×107 | 5.95×107 | 1.36 | 6.69 | -11.69 | -7.05 |
| A(γ)/γ2 | 9.85×106 | 9.90×106 | 1.06×107 | 1.09×107 | 7.19 | 10.33 | 0.52 | 3.47 |
Table 5 Quantitative relationship of k~αT-1 and origin of differences in four high-entropy rare-earth oxide materials
| Parameter | 1-YHoErYb | 2-ErTmYbLu | 3-YHoErYbHf | 4-ErTmYbLuHf | Diff(1-3)/% | Diff(2-4)/% | Diff(1-2)/% | Diff(3-4)/% |
|---|---|---|---|---|---|---|---|---|
| kT | 898.98 | 892.94 | 1006.36 | 1050.75 | 12.07 | 17.70 | -0.68 | 4.32 |
| 1.44×10-12 | 1.62×10-12 | 1.49×10-12 | 1.62×10-12 | 3.15 | -0.01 | 11.90 | 8.48 | |
| ΘD3 | 6.31×107 | 5.58×107 | 6.40×107 | 5.95×107 | 1.36 | 6.69 | -11.69 | -7.05 |
| A(γ)/γ2 | 9.85×106 | 9.90×106 | 1.06×107 | 1.09×107 | 7.19 | 10.33 | 0.52 | 3.47 |
| Site | RE1 | RE2 | |
|---|---|---|---|
| No.1 | NS | 4 | 12 |
| XS | XREⅠ=XREⅡ=XREⅢ= XREⅣ=1/4 | XREⅠ=XREⅡ=XREⅢ= XREIV=1/4 |
Table S1 Number of sites on sublattice and composition for C-(REⅠ1/4REⅡ1/4REⅢ1/4REⅣ1/4)2O3
| Site | RE1 | RE2 | |
|---|---|---|---|
| No.1 | NS | 4 | 12 |
| XS | XREⅠ=XREⅡ=XREⅢ= XREⅣ=1/4 | XREⅠ=XREⅡ=XREⅢ= XREIV=1/4 |
| Site | RE1 | RE2 | |
|---|---|---|---|
| No.1 | NS | 4 | 12 |
| XS | XREⅠ=0 XREⅡ=XREⅢ=XREⅣ= XHf =1/4 | XREⅠ=XHf=1/4 XREⅡ=XREⅢ=XREⅣ= 1/6 | |
| No.2 | NS | 4 | 12 |
| XS | XREⅡ=0 XREⅠ=XREⅢ=XREⅣ= XHf=1/4 | XREⅡ=XHf=1/4 XREⅠ=XREⅢ=XREⅣ= 1/6 | |
| No.3 | NS | 4 | 12 |
| XS | XREⅢ=0 XREⅠ=XREⅡ=XREⅣ= XHf=1/4 | XREⅢ=XHf=1/4 XREⅠ=XREⅡ=XREⅣ= 1/6 | |
| No.4 | NS | 4 | 12 |
| XS | XREⅣ=0 XREⅠ=XREⅡ=XREⅢ= XHf=1/4 | XREⅣ=XHf=1/4 XREⅠ=XREⅡ=XREⅢ= 1/6 | |
| No.5 | NS | 4 | 12 |
| XS | XHf=0 XREⅠ=XREⅡ=XREⅢ= XREⅣ=1/4 | XHf=1/3 XREⅠ=XREⅡ=XREⅢ= XREⅣ=1/6 |
Table S2 Number of sites on sublattice and composition for C-(REⅠ3/16REⅡ3/16REⅢ3/16REⅣ3/16Hf1/4)2O3
| Site | RE1 | RE2 | |
|---|---|---|---|
| No.1 | NS | 4 | 12 |
| XS | XREⅠ=0 XREⅡ=XREⅢ=XREⅣ= XHf =1/4 | XREⅠ=XHf=1/4 XREⅡ=XREⅢ=XREⅣ= 1/6 | |
| No.2 | NS | 4 | 12 |
| XS | XREⅡ=0 XREⅠ=XREⅢ=XREⅣ= XHf=1/4 | XREⅡ=XHf=1/4 XREⅠ=XREⅢ=XREⅣ= 1/6 | |
| No.3 | NS | 4 | 12 |
| XS | XREⅢ=0 XREⅠ=XREⅡ=XREⅣ= XHf=1/4 | XREⅢ=XHf=1/4 XREⅠ=XREⅡ=XREⅣ= 1/6 | |
| No.4 | NS | 4 | 12 |
| XS | XREⅣ=0 XREⅠ=XREⅡ=XREⅢ= XHf=1/4 | XREⅣ=XHf=1/4 XREⅠ=XREⅡ=XREⅢ= 1/6 | |
| No.5 | NS | 4 | 12 |
| XS | XHf=0 XREⅠ=XREⅡ=XREⅢ= XREⅣ=1/4 | XHf=1/3 XREⅠ=XREⅡ=XREⅢ= XREⅣ=1/6 |
| Compound | Unit | Average bond length/Å | |
|---|---|---|---|
| (Y1/4Ho1/4Er1/4Yb1/4)2O3 | YO6 | 2.2868 | |
| HoO6 | 2.2794 | ||
| ErO6 | 2.2700 | ||
| YbO6 | 2.2518 | ||
| REO6 | 2.2720 | ||
| (Er1/4Tm1/4Yb1/4Lu1/4)2O3 | ErO6 | 2.2631 | |
| TmO6 | 2.2535 | ||
| YbO6 | 2.2446 | ||
| LuO6 | 2.2356 | ||
| REO6 | 2.2492 | ||
| (Y3/16Ho3/16Er3/16Yb3/16Hf1/4)2O3 | YO6 | 2.2991 | |
| HoO6 | 2.2880 | ||
| ErO6 | 2.2794 | ||
| YbO6 | 2.2451 | ||
| REO6 | 2.2779 | ||
| HfO6 | 2.1893 | ||
| XO6* | 2.2558 | ||
| (Er3/16Tm3/16Yb3/16Lu3/16Hf1/4)2O3 | ErO6 | 2.2684 | |
| TmO6 | 2.2541 | ||
| YbO6 | 2.2437 | ||
| LuO6 | 2.2399 | ||
| REO6 | 2.2515 | ||
| HfO6 | 2.1936 | ||
| XO6* | 2.2370 | ||
Table S3 Average bond length of polyhedral of four high-entropy rare-earth oxide materials
| Compound | Unit | Average bond length/Å | |
|---|---|---|---|
| (Y1/4Ho1/4Er1/4Yb1/4)2O3 | YO6 | 2.2868 | |
| HoO6 | 2.2794 | ||
| ErO6 | 2.2700 | ||
| YbO6 | 2.2518 | ||
| REO6 | 2.2720 | ||
| (Er1/4Tm1/4Yb1/4Lu1/4)2O3 | ErO6 | 2.2631 | |
| TmO6 | 2.2535 | ||
| YbO6 | 2.2446 | ||
| LuO6 | 2.2356 | ||
| REO6 | 2.2492 | ||
| (Y3/16Ho3/16Er3/16Yb3/16Hf1/4)2O3 | YO6 | 2.2991 | |
| HoO6 | 2.2880 | ||
| ErO6 | 2.2794 | ||
| YbO6 | 2.2451 | ||
| REO6 | 2.2779 | ||
| HfO6 | 2.1893 | ||
| XO6* | 2.2558 | ||
| (Er3/16Tm3/16Yb3/16Lu3/16Hf1/4)2O3 | ErO6 | 2.2684 | |
| TmO6 | 2.2541 | ||
| YbO6 | 2.2437 | ||
| LuO6 | 2.2399 | ||
| REO6 | 2.2515 | ||
| HfO6 | 2.1936 | ||
| XO6* | 2.2370 | ||
| Material | 1-YHoErYb | 2-ErTmYbLu | 3-YHoErYbHf | 4-ErTmYbLuHf |
|---|---|---|---|---|
| s11 | 0.005239 | 0.004968 | 0.005147 | 0.005011 |
| s12 | -0.001469 | -0.001396 | -0.001445 | -0.001504 |
| s13 | -0.001491 | -0.001365 | -0.001516 | -0.001426 |
| s14 | -0.000049 | -0.000006 | -0.000043 | -0.000120 |
| s15 | -0.000082 | -0.000019 | 0.000079 | 0.000086 |
| s16 | 0.000013 | 0.000002 | 0.000050 | -0.000122 |
| s22 | 0.005544 | 0.005321 | 0.005213 | 0.005182 |
| s23 | -0.001794 | -0.001739 | -0.001541 | -0.001420 |
| s24 | -0.000058 | 0.000003 | 0.000073 | -0.000020 |
| s25 | 0.000077 | -0.000031 | -0.000081 | -0.000353 |
| s26 | 0.000974 | 0.000913 | 0.000381 | 0.000603 |
| s33 | 0.005584 | 0.005303 | 0.005388 | 0.005103 |
| s34 | 0.000051 | -0.000014 | 0.000040 | 0.000062 |
| s35 | 0.000000 | 0.000049 | 0.000058 | 0.000069 |
| s36 | -0.000990 | -0.000888 | -0.000400 | -0.000397 |
| s44 | 0.014726 | 0.013938 | 0.014169 | 0.012992 |
| s45 | -0.002028 | -0.001789 | -0.000457 | -0.000304 |
| s46 | 0.000017 | -0.000002 | -0.000460 | 0.000002 |
| s55 | 0.016174 | 0.015356 | 0.014366 | 0.014004 |
| s56 | -0.000060 | 0.000077 | -0.000237 | -0.000343 |
| s66 | 0.016120 | 0.015338 | 0.014369 | 0.013311 |
Table S4 Elastic compliance constants of four high-entropy rare-earth oxide materials
| Material | 1-YHoErYb | 2-ErTmYbLu | 3-YHoErYbHf | 4-ErTmYbLuHf |
|---|---|---|---|---|
| s11 | 0.005239 | 0.004968 | 0.005147 | 0.005011 |
| s12 | -0.001469 | -0.001396 | -0.001445 | -0.001504 |
| s13 | -0.001491 | -0.001365 | -0.001516 | -0.001426 |
| s14 | -0.000049 | -0.000006 | -0.000043 | -0.000120 |
| s15 | -0.000082 | -0.000019 | 0.000079 | 0.000086 |
| s16 | 0.000013 | 0.000002 | 0.000050 | -0.000122 |
| s22 | 0.005544 | 0.005321 | 0.005213 | 0.005182 |
| s23 | -0.001794 | -0.001739 | -0.001541 | -0.001420 |
| s24 | -0.000058 | 0.000003 | 0.000073 | -0.000020 |
| s25 | 0.000077 | -0.000031 | -0.000081 | -0.000353 |
| s26 | 0.000974 | 0.000913 | 0.000381 | 0.000603 |
| s33 | 0.005584 | 0.005303 | 0.005388 | 0.005103 |
| s34 | 0.000051 | -0.000014 | 0.000040 | 0.000062 |
| s35 | 0.000000 | 0.000049 | 0.000058 | 0.000069 |
| s36 | -0.000990 | -0.000888 | -0.000400 | -0.000397 |
| s44 | 0.014726 | 0.013938 | 0.014169 | 0.012992 |
| s45 | -0.002028 | -0.001789 | -0.000457 | -0.000304 |
| s46 | 0.000017 | -0.000002 | -0.000460 | 0.000002 |
| s55 | 0.016174 | 0.015356 | 0.014366 | 0.014004 |
| s56 | -0.000060 | 0.000077 | -0.000237 | -0.000343 |
| s66 | 0.016120 | 0.015338 | 0.014369 | 0.013311 |
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