S1

Supplementary Information for

Predictive theoretical screening of phase stability for chemical order

and disorder in quaternary 312 and 413 MAX phases

Martin Dahlqvista* and Johanna Rosena*

a Thin Film Physics, Department of Physics, Chemistry and Biology (IFM), Linköping

University, SE-581 83 Linköping, Sweden

* martin.dahlqvist@liu.se, johanna.rosen@liu.se

Electronic Supplementary Material (ESI) for Nanoscale.This journal is © The Royal Society of Chemistry 2019

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Figure S1. Schematic illustration of considered chemical order of M´ and M´´ for a 413 MAX phase structure with a 2:2:1:3 composition of M´:M´´:Al:C, where M´, M´´, Al, and C atoms are represented in red, blue, grey, and black, respectively.

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Table S1. Calculated formation enthalpy ΔHcp (in meV/atom) for M3AlC2 phases. M ∆Hcp (meV/atom) Equilibrium simplex

Sc 163 Sc3AlC, Sc2Al2C3, Sc3C4 Y 214 Y3AlC, Y4C5, YAl2 Ti -6 Ti5Al2C3, Ti7Al2C5 Zr -8 Zr4AlC3, Zr2Al3, Zr4Al3 Hf -12 Hf2AlC, Hf4AlC3 V 5 V12Al3C8, VAl3, Al4C3 Nb -1 Nb2AlC, Nb6C5, NbAl3 Ta -13 Ta2AlC, Ta4AlC3 Cr 81 Cr2AlC, Cr3C2, C Mo 141 C, Mo3Al W 281 WC, W, WAl5

Table S2. Calculated formation enthalpy ΔHcp (in meV/atom) for M4AlC3 phases. M ∆Hcp (meV/atom) Equilibrium simplex

Sc 195 Sc3AlC, Sc3C4, Sc2Al2C3 Y 245 Y3AlC, Y4C5, YAl2 Ti -0.01 Ti3AlC2, TiC Zr -0.3 Zr3AlC2, ZrC Hf -0.2 Hf3AlC2, HfC V 12 V12Al3C8, V6C5, Al4C3 Nb 10 Nb6C5, NbAl3, Nb12Al3C8 Ta 0.2 Ta6AlC5, Ta12Al3C8, TaAl3 Cr 108 Cr2AlC, Cr3C2, C Mo 171 C, Mo3Al, Mo2C W 317 WC, W, WAl5

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Table S3. Calculated formation enthalpy ΔHcp (in meV/atom) for chemical distribution if lowest energy, disorder temperature Tdisorder or T0 (in K), and identified equilibrium simplex for given M´2M´´AlC2 composition.

M´ M´´ Lowest energy

ΔHcp Tdisorder or

T0 Comment Equilibrium simplex

Sc Y SQS 171 6247 not stable Sc3AlC, Y4C5, YAl2, Sc2Al2C3 Sc Ti A 103 4581 not stable TiC, Sc3AlC, Sc2Al2C3, ScAl3 Sc Zr A 86 4580 not stable ZrC, Sc3AlC, Sc2Al2C3 Sc Hf A 80 5195 not stable HfC, Sc3AlC, Sc2Al2C3, ScAl2 Sc V A 91 3532 not stable Sc3AlC, Sc2Al2C3, V6C5, (V2/3Sc1/3)2AlC Sc Nb A -1 1933 order, stable Nb2Sc2AlC3, Sc3AlC, Sc2Al2C3, Nb2ScAlC2 Sc Ta A -14 3075 order, stable Ta2Sc2AlC3, Sc3AlC, Sc2Al2C3, ScAl3 Sc Cr SQS 151 5495 not stable (Cr2/3Sc1/3)2AlC, Sc2CrC3, Sc3AlC, Sc2Al2C3 Sc Mo A 28 2065 not stable (Mo2/3Sc1/3)2AlC, Sc3AlC, Sc3C4 Sc W A -13 2240 order, stable (Sc2/3W1/3)2AlC, WC, Sc3AlC, Sc3C4 Y Sc SQS 169 6159 not stable ScAl3C3, Y4C5, YAl2, YAl3C3 Y Ti SQS 202 7349 not stable TiC, Y3AlC, YAl2, Y4C5 Y Zr A 141 5370 not stable ZrC, Y3AlC, YAl2, Y4C5 Y Hf A 154 6485 not stable HfC, Y3AlC, YAl2, Y4C5 Y V SQS 194 7064 not stable Y3AlC, YAl2, Y4C5, V6C5 Y Nb A 91 3550 not stable Y3AlC, YAl2, Y4C5, Nb6C5 Y Ta A 56 3344 not stable TaC, Y3AlC, YAl2, Y4C5 Y Cr D 237 8631 not stable Y2Cr2C3, YAl2, Y4C5, Y3AlC Y Mo SQS 145 5293 not stable (Mo2/3Y1/3)2AlC, Y3AlC, Y4C5, YAl2 Y W SQS 128 4683 not stable YWC2, Y3AlC, YAl2, (W2/3Y1/3)2AlC Ti Sc D 64 2482 not stable (Ti2/3Sc1/3)2AlC, Sc3AlC, TiC, Sc2Al2C3 Ti Y SQS 183 6667 not stable TiC, (Ti2/3Y1/3)2AlC, YAl2, Y3AlC Ti Zr A 39 1782 not stable (Ti2/3Zr1/3)2AlC, ZrC, Ti3AlC2 Ti Hf A 18 1403 disorder, stabilized by temperature Ti2AlC, HfC Ti V C 3 389 disorder, stabilized by temperature TiV2AlC2, V2AlC Ti Nb D -5 184 disorder, stable Nb2TiAlC2, Ti3AlC2 Ti Ta A -30 529 disorder, stable Ta2TiAlC2, Ti3AlC2 Ti Cr C 19 2091 order, close to stable (Cr2/3Ti1/3)2AlC, TiC, TiAl3, Ti4AlC3 Ti Mo C 9 1139 disorder, stabilized by temperature TiC, Mo3Al, Mo3Al8 Ti W C 6 340 disorder, stabilized by temperature (Ti2/3W1/3)2AlC, TiC, Ti2W2AlC3 Zr Sc F 72 2853 not stable ZrC, ZrAl2, Sc3AlC Zr Y SQS 120 4370 not stable ZrC, YAl2, Y2Al Zr Ti C 41 1708 disorder, stabilized by temperature (Ti2/3Zr1/3)2AlC, ZrC, Zr4AlC3 Zr Hf A -6 401 disorder, stable HfC, Zr4AlC3, Zr2Al3, Zr4Al3 Zr V C 113 4491 not stable (V2/3Zr1/3)2AlC, Zr4AlC3 Zr Nb C 16 684 disorder, stabilized by temperature Nb2Zr, ZrC, Zr4AlC3 Zr Ta A 5 435 disorder, stabilized by temperature ZrC, ZrAl2, Ta2C Zr Cr C 221 10505 not stable (Cr2/3Zr1/3)2AlC, ZrC, ZrAl2, ZrAl3 Zr Mo C 94 4339 not stable ZrC, Mo3Al, Mo3Al8 Zr W C 85 3537 not stable ZrC, (W2/3Zr1/3)2AlC, W, ZrC Hf Sc D 73 3132 not stable HfC, ScAl Hf Y D 159 6134 not stable HfC, Y2Al, YAl2 Hf Ti C 22 1131 disorder, stabilized by temperature (Ti2/3Hf1/3)2AlC, HfC, Hf4AlC3 Hf Zr C 9 576 disorder, stabilized by temperature HfC, Zr4Al3, Zr2Al3 Hf V C 88 4007 not stable (V2/3Hf1/3)2AlC, HfC, Hf4AlC3 Hf Nb D 16 1239 disorder, stabilized by temperature HfC, Nb2Al, NbAl3 Hf Ta C 10 456 disorder, stabilized by temperature (Hf2/3Ta1/3)2AlC, HfC, Hf4AlC3 Hf Cr D 140 7917 not stable HfC, (Cr2/3Hf1/3)2AlC, Cr2Al, HfAl3 Hf Mo C 115 5536 not stable Hf2Al3C4, Hf3Al2, Mo3Al8, C Hf W C 333 13132 not stable (Hf2/3W1/3)2AlC, HfC, WC V Sc C 48 2851 not stable (V2/3Sc1/3)2AlC, V6C5, Sc3AlC, Sc2Al2C3 V Y C 205 8367 not stable YAl2, V6C5, Y3AlC, Y4C5 V Ti A -8 873 disorder, stable Ti2V2AlC3, V2AlC V Zr A 101 4542 not stable ZrC, (V2/3Zr1/3)2AlC, V12Al3C8, ZrAl3 V Hf A 88 4441 not stable V2AlC, HfC V Nb A 47 2011 not stable V12Al3C8, NbAl3, Nb6C5, Nb12Al3C8 V Ta A 15 1582 disorder, stabilized by temperature V2AlC, Ta2V2AlC3 V Cr D 13 1513 disorder, stabilized by temperature Cr2AlC, V12Al3C8, V6C5, Al4C3 V Mo C 38 2534 not stable V6C5, C, Mo3Al, Mo3Al8 V W C 41 2334 not stable WC, V12Al3C8, W, WAl5

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Nb Sc C -21 909 disorder, stable (Nb2/3Sc1/3)2AlC, Nb2Sc2AlC3, Nb12Al3C8, Sc2Al2C3 Nb Y C 74 3446 not stable YAl2, Nb6C5, Y3AlC, Y4C5 Nb Ti A -13 627 disorder, stable Ti2Nb2AlC3, (Nb2/3Ti1/3)2AlC, Nb12Al3C8, NbAl3 Nb Zr A -6 975 disorder, stable ZrC, (Nb2/3Zr1/3)2AlC, Nb12Al3C8, NbAl3 Nb Hf A -3 1483 disorder, stable Hf2Nb2AlC3, Nb2AlC Nb V F 40 1938 not stable NbAl3, Nb12Al3C8, Nb6C5, V12Al3C8 Nb Ta A -11 1055 disorder, stable Nb2Ta2AlC3, Ta2C, NbAl3, Nb12Al3C8 Nb Cr D 123 7033 not stable Nb2CrAlC2, Nb12Al3C8, NbAl3, Nb6C5 Nb Mo F 36 2560 not stable Nb6C5, C, Mo3Al, Mo3Al8 Nb W F 68 3350 not stable WC, W, NbAl3, Nb12Al3C8 Ta Sc C -14 1034 disorder, stable Ta2Sc2AlC3, (Ta2/3Sc1/3)2AlC, Ta12Al3C8, ScAl3 Ta Y C 66 3501 not stable YAl2, Nb6C5, Y3AlC, Y4C5 Ta Ti A -7 10 disorder, stable Ti2TaAlC2, Ta12Al3C8, (Ti0.5Ta0.5)C, TaAl3 Ta Zr A 2 540 disorder, stabilized by temperature ZrC, Ta2Zr, Ta12Al3C8, ZrAl3 Ta Hf A -9 972 disorder, stable Hf2Ta2AlC3, Ta2AlC Ta V F 14 1349 disorder, stabilized by temperature Ta2V2AlC3, V2AlC, Ta12Al3C8, TaAl3 Ta Nb C 4 675 disorder, stabilized by temperature Nb2TaAlC2, Nb2Ta2AlC3, Ta12Al3C8, NbAl3 Ta Cr D 35 4102 not stable (Cr2/3Ta1/3)2AlC, Ta6AlC5 Ta Mo F 39 3519 not stable TaC, Mo3Al, Mo3Al8 Ta W F 45 3231 not stable WC, Ta12Al3C8, W, WAl5 Cr Sc A 103 6407 not stable (Cr2/3Sc1/3)2AlC, ScCrC2, Cr3C2, C Cr Y A 247 10465 not stable (Cr2/3Y1/3)2AlC, C, Y2Cr2C3, Cr3C2 Cr Ti A -2 3757 order, stable Cr2AlC, TiC Cr Zr A 137 8357 not stable (Cr2/3Zr1/3)2AlC, ZrC, Cr3C2, C Cr Hf A 115 8003 not stable Cr2AlC, HfC Cr V A 6 2971 order, close to stable Cr2AlC, V6C5, C Cr Nb A 53 5244 not stable (Cr2/3Nb1/3)2AlC, C, Cr3C2, Nb6C5 Cr Ta A 35 5256 not stable Cr2AlC, TaC Cr Mo A 99 4612 not stable Cr3C2, C, Mo3Al, Mo3Al8 Cr W A 131 6427 not stable Cr2AlC, WC Mo Sc A -20 3128 order, stable (Mo2/3Sc1/3)2AlC, Mo2C, C, Sc3C4 Mo Y A 134 7196 not stable (Mo2/3Y1/3)2AlC, YMoC2, Mo2C, C Mo Ti A -18 4392 order, stable Ti2Mo2AlC3, Mo3Al, Mo3Al8, C Mo Zr A 16 4957 order, close to stable (Mo2/3Zr1/3)2AlC, Mo2C, ZrC, C Mo Hf A 3 5361 order, close to stable HfC, Mo3Al, Mo3Al8 Mo V A 28 3620 not stable C, Mo3Al, V6C5, Mo3Al8 Mo Nb A 24 4160 order, close to stable C, Mo3Al, Nb6C5, Mo3Al8 Mo Ta A 10 4310 order, close to stable Ta2Mo2AlC3, C, Mo3Al, Mo3Al8 Mo Cr C 109 4989 not stable C, Mo3Al8, Mo3Al, Cr3C2 Mo W A 186 7454 not stable WC, C, Mo3Al, Mo3Al8 W Sc A 1 4500 order, close to stable WC, (W2/3Sc1/3)2AlC, (Sc2/3W1/3)2AlC W Y A 158 8548 not stable (W2/3Y1/3)2AlC, YWC2, WC W Ti A -3 4072 order, stable Ti2W2AlC3, WC, W, WAl5 W Zr A 64 5709 not stable (W2/3Zr1/3)2AlC, WC, ZrC W Hf A 220 12257 not stable WC, (Hf2/3W1/3)2AlC, C, WAl5 W V A 98 5561 not stable WC, W, WAl5, V12Al3C8 W Nb A 125 6743 not stable WC, W, NbAl3, Nb12Al3C8 W Ta B 161 6356 not stable WC, W, WAl5, Ta12Al3C8 W Cr C 178 7884 not stable WC, Cr2AlC, W, WAl5 W Mo F 273 10075 not stable WC, Mo3Al, Mo3Al8

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Table S4. Calculated formation enthalpy ΔHcp (in meV/atom) for chemical distribution of lowest energy, disorder temperature Tdisorder or T0 (in K), and identified equilibrium simplex for given M´2M´´2AlC3 composition.

M´ M´´ Lowest energy

ΔHcp Tdisorder

or T0 Comment Equilibrium simplex

Sc Y SQS 237 7920 not stable Sc3AlC, Y4C5, YAl2, Sc2Al2C3 Sc Ti A 79 4249 not stable TiC, Sc3AlC, Sc2Al2C3, ScAl3 Sc Zr A 60 3981 not stable ZrC, Sc3AlC, Sc2Al2C3, ScAl3 Sc Hf A 56 4337 not stable HfC, Sc3AlC, Sc2Al2C3, ScAl3 Sc V I 72 3417 not stable Sc3AlC, V6C5, Sc2Al2C3, Sc3C4 Sc Nb I -22 310 disorder, stable Sc2NbAlC2, Nb6C5, Sc2Al2C3, Sc3C4 Sc Ta I -26 935 disorder, stable Sc2TaAlC2, TaC Sc Cr I 162 7117 not stable (Cr2/3Sc1/3)2AlC, Sc2CrC3 Sc Mo I 15 1526 disorder, stabilized by temperature Mo2ScAlC2, Sc4C3, Sc3C4 Sc W I 0 1304 disorder, stabilized by temperature Sc2WAlC2, WC Y Sc SQS 237 7920 not stable Sc3AlC, Y4C5, YAl2, Sc2Al2C3 Y Ti A 212 7985 not stable TiC, Y3AlC, YAl2, Y4C5 Y Zr A 106 5286 not stable ZrC, Y3AlC, YAl2, Y4C5 Y Hf A 121 6406 not stable HfC, Y3AlC, YAl2, Y4C5 Y V I 253 8649 not stable YAl2, V6C5, Y3AlC, Y4C5 Y Nb I 79 3476 not stable YAl2, Nb6C5, Y3AlC, Y4C5 Y Ta A 41 3067 not stable TaC, Y3AlC, YAl2, Y4C5 Y Cr SQS 304 10177 not stable Y2Cr2C3, (Cr2/3Y1/3)2AlC, YAl2, YAl3C3 Y Mo I 169 6405 not stable (Mo2/3Y1/3)2AlC, Y4C5, Mo2C Y W I 155 6584 not stable YWC2, (W2/3Y1/3)2AlC, YAl2, Y3AlC Ti Sc B 79 4249 not stable TiC, Sc3AlC, Sc2Al2C3, ScAl3 Ti Y B 212 7985 not stable TiC, Y3AlC, YAl2, Y4C5 Ti Zr A 44 2023 not stable ZrC, Ti3AlC2, Zr4AlC3 Ti Hf A 27 1847 not stable HfC, Ti2AlC Ti V B 2 1235 disorder, stabilized by temperature V2TiAlC2, TiC Ti Nb B -5 -1290 disorder, stable Nb2TiAlC2, TiC Ti Ta SQS 4 132 disorder, stabilized by temperature (Ti0.50Ta0.50)C, Ti2TaAlC2, Ta2TiAlC2 Ti Cr B 4 3392 order, close to stable TiC, Cr2AlC Ti Mo B -17 -3406 order, stable Mo2TiAlC2, TiC Ti W B -15 -3143 order, stable W2TiAlC2, TiC Zr Sc B 61 3981 not stable ZrC, Sc3AlC, Sc2Al2C3, ScAl3 Zr Y B 106 5286 not stable ZrC, Y3AlC, YAl2, Y4C5 Zr Ti B 45 2023 not stable ZrC, Ti3AlC2, Zr4AlC3 Zr Hf A 2 543 disorder, stabilized by temperature Zr2HfAlC2, HfC Zr V B 120 5680 not stable ZrC, (V2/3Zr1/3)2AlC, V12Al3C8, Zr2Al3 Zr Nb B 6 1315 disorder, stabilized by temperature Nb2ZrAlC2, ZrC Zr Ta I 7 570 disorder, stabilized by temperature ZrC, (Ta2/3Zr1/3)2AlC, ZrAl3, Ta12Al3C8 Zr Cr B 138 8952 not stable ZrC, (Cr2/3Zr1/3)2AlC, Cr3C2, C Zr Mo B 24 3823 not stable ZrC, (Mo2/3Zr1/3)2AlC, Mo2C, C Zr W B 66 4105 not stable ZrC, (W2/3Zr1/3)2AlC, WC Hf Sc B 56 4337 not stable HfC, Sc3AlC, Sc2Al2C3, ScAl3 Hf Y B 121 6406 not stable HfC, Y3AlC, YAl2, Y4C5 Hf Ti B 27 1847 not stable HfC, Ti2AlC Hf Zr B 2 543 disorder, stabilized by temperature Zr2HfAlC2, HfC Hf V B 99 5601 not stable HfC, V2AlC Hf Nb B -6 -2238 order, stable HfC, Nb2AlC Hf Ta B 3 955 disorder, stabilized by temperature Ta2HfAlC2, HfC Hf Cr B 114 8655 not stable HfC, Cr2AlC Hf Mo B 7 4161 not stable HfC, C, Mo3Al, Mo3Al8 Hf W B 219 10197 not stable WC, (Hf2/3W1/3)2AlC, HfC V Sc I 72 3417 not stable Sc3AlC, V6C5, Sc2Al2C3, Sc3C4 V Y I 253 8649 not stable YAl2, V6C5, Y3AlC, Y4C5 V Ti A 2 1235 disorder, stabilized by temperature V2TiAlC2, TiC V Zr A 120 5680 not stable ZrC, (V2/3Zr1/3)2AlC, V12Al3C8, Zr2Al3 V Hf A 99 5601 not stable HfC, V2AlC V Nb A 37 2076 not stable NbAl3, Nb6C5, V6C5, V12Al3C8 V Ta A -13 1600 disorder, stable Ta6AlC5, V12Al3C8, TaAl3, Al4C3 V Cr B 8 2675 order, close to stable Cr2AlC, V6C5, C V Mo B 41 3208 not stable C, Mo3Al, V6C5, Mo3Al8 V W B 104 4313 not stable WC, V12Al3C8, W, WAl5

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Nb Sc I -22 -3236 disorder, stable Sc2NbAlC2, Nb6C5, Sc2Al2C3, Sc3C4 Nb Y I 79 3476 not stable YAl2, Nb6C5, Y3AlC, Y4C5 Nb Ti A -5 730 disorder, stable Nb2TaAlC2, TiC Nb Zr A 5 1315 disorder, stabilized by temperature Nb2ZrAlC2, ZrC Nb Hf A -7 1870 order, stable HfC, Nb2AlC Nb V B 38 2076 not stable NbAl3, Nb6C5, V6C5, V12Al3C8 Nb Ta A -12 1030 disorder, stable Nb2TaAlC2, TaC Nb Cr B 41 5196 not stable (Cr2/3Nb1/3)2AlC, C, Nb6C5, Cr3C2 Nb Mo B 19 3377 order, close to stable C, Mo3Al, Nb6C5, Mo3Al8 Nb W B 103 5312 not stable WC, W, NbAl3, Nb12Al3C8 Ta Sc I -26 -4066 disorder, stable Sc2TaAlC2, TaC Ta Y B 42 3067 not stable TaC, Y3AlC, YAl2, Y4C5 Ta Ti SQS 4 132 disorder, stabilized by temperature (Ti0.50Ta0.50)C, Ti2TaAlC2, Ta2TiAlC2 Ta Zr I 7 570 disorder, stabilized by temperature ZrC, (Ta2/3Zr1/3)2AlC, ZrAl3, Ta12Al3C8 Ta Hf A 3 955 disorder, stabilized by temperature Ta2HfAlC2, HfC Ta V B -13 -1662 disorder, stable TA6AlC5, V12Al3C8, TaAl3, Al4C3 Ta Nb B -11 -1099 disorder, stable Nb2TaAlC2, TaC Ta Cr B 9 4904 order, close to stable TaC, Cr2AlC Ta Mo B -5 -3632 order, stable TaC, C, Mo3Al, Mo3Al8 Ta W B 60 5069 not stable WC, W, Ta12Al3C8, WAl5 Cr Sc I 162 7117 not stable (Cr2/3Sc1/3)2AlC, Sc2Cr2C3 Cr Y SQS 304 10177 not stable Y2Cr2C3, (Cr2/3Y1/3)2AlC, YAl2, YAl3C3 Cr Ti A 3 3392 order, close to stable TiC, Cr2AlC Cr Zr A 137 8952 not stable ZrC, (Cr2/3Zr1/3)2AlC, Cr3C2, C Cr Hf A 114 8655 not stable HfC, Cr2AlC Cr V A 8 2675 order, close to stable Cr2AlC, V6C5, C Cr Nb A 40 5196 not stable (Cr2/3Nb1/3)2AlC, C, Nb6C5, Cr3C2 Cr Ta A 9 4904 order, close to stable TaC, Cr2AlC Cr Mo A 121 5547 not stable C, Cr3C2, Mo3Al, Mo3Al8 Cr W A 181 7877 not stable WC, Cr2AlC Mo Sc I 15 1526 disorder, stabilized by temperature Mo2ScAlC2, Sc4C3, Sc3C4 Mo Y I 169 6405 not stable (Mo2/3Y1/3)2AlC, Y4C5, Mo2C Mo Ti A -17 2940 order, stable Mo2TiAlC2, TiC Mo Zr A 24 3823 order, close to stable ZrC, (Mo2/3Zr1/3)2AlC, Mo2C, C Mo Hf A 7 4161 order, close to stable HfC, C, Mo3Al, Mo3Al8 Mo V A 40 3208 not stable C, Mo3Al, V6C5, Mo3Al8 Mo Nb A 19 3377 order, close to stable C, Mo3Al, Nb6C5, Mo3Al8 Mo Ta A -5 3785 order, stable TaC, C, Mo3Al, Mo3Al8 Mo Cr B 122 5547 not stable C, Cr3C2, Mo3Al, Mo3Al8 Mo W A 251 9010 not stable WC, C, Mo3Al, Mo3Al8 W Sc I 0 1304 disorder, stabilized by temperature Sc2WAlC2, WC W Y I 155 6584 not stable YWC2, (W2/3Y1/3)2AlC, YAl2, Y3AlC W Ti A -16 1870 order, stable W2TiAlC2, TiC W Zr A 66 4105 not stable ZrC, (W2/3Zr1/3)2AlC, WC W Hf A 219 10197 not stable WC, (Hf2/3W1/3)2AlC, HfC W V A 103 4313 not stable WC, V12Al3C8, W, WAl5 W Nb A 102 5312 not stable WC, W, NbAl3, Nb12Al3C8 W Ta A 60 5069 not stable WC, W, Ta12Al3C8, WAl5 W Cr B 181 7877 not stable WC, Cr2AlC W Mo B 259 9010 not stable WC, C, Mo3Al, Mo3Al8

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Figure S2. Calculated isostructural formation enthalpy ΔHiso as function of the disorder temperature Tdisorder for o-MAX phases with (a) n = 2 and (b) n = 3. The colours represent if the M´ and M´´ in o-MAX also forms rock salt MC. Experimentally reported phases are marked according to their reported order; green squares (o-MAX), black circles (disorder MAX), and orange diamonds (o-MAX, semi-order, or disorder). The vertical dashed line indicates the typical bulk synthesis temperature 1773 K.

Table S5. Covalent radius and electronegativity for M and Al considered in this work.

M Metallic radius r (Å)a Electronegativity �,

(Pauling scale)b Sc 1.62 1.36 Y 1.80 1.22 Ti 1.47 1.54 Zr 1.60 1.33 Hf 1.59 1.30 V 1.34 1.63

Nb 1.46 1.60 Ta 1.46 1.50 Cr 1.28 1.66 Mo 1.39 2.16 W 1.39 2.36 Al 1.43 1.61

a Ref. [1] b Ref. [2]

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References

[1] N. N. Greenwood and A. Earnshaw, Chemistry of the Elements (Butterworth-Heinemann, 1997), 2nd edn.

[2] J. E. Huheey, E. A. Keiter, and R. L. Keiter, Inorganic Chemistry : Principles of Structure

and Reactivity (HarperCollins, New York, USA, 1993), 4th edn.