Martin
Dahlqvist
* and
Johanna
Rosen
*
Thin Film Physics, Department of Physics, Chemistry and Biology (IFM), Linköping University, SE-581 83 Linköping, Sweden. E-mail: martin.dahlqvist@liu.se; johanna.rosen@liu.se
First published on 4th December 2019
In this work we systematically explore a class of atomically laminated materials, Mn+1AXn (MAX) phases upon alloying between two transition metals, M′ and M′′, from groups III to VI (Sc, Y, Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W). The materials investigated focus on so called o-MAX phases with out-of-plane chemical ordering of M′ and M′′, and their disordered counterparts, for A = Al and X = C. Through use of predictive phase stability calculations, we confirm all experimentally known phases to date, and also suggest a range of stable ordered and disordered hypothetical elemental combinations. Ordered o-MAX is favoured when (i) M′ next to the Al-layer does not form a corresponding binary rock-salt MC structure, (ii) the size difference between M′ and M′′ is small, and (iii) the difference in electronegativity between M′ and Al is large. Preference for chemical disorder is favoured when the size and electronegativity of M′ and M′′ is similar, in combination with a minor difference in electronegativity of M′ and Al. We also propose guidelines to use in the search for novel o-MAX; to combine M′ from group 6 (Cr, Mo, W) with M′′ from groups 3 to 5 (Sc only for 312, Ti, Zr, Hf, V, Nb, Ta). Correspondingly, we suggest formation of disordered MAX phases by combing M′ and M′′ within groups 3 to 5 (Sc, Ti, Zr, Hf, V, Nb, Ta). The addition of novel elemental combinations in MAX phases, and in turn in their potential two-dimensional MXene derivatives, allow for property tuning of functional materials.
A common route for property tuning is through alloying, which for MAX phases can be realized with a fourth element on either M, A, or X site, as demonstrated to change e.g. hardness,9 to tune properties for isotropic thermal expansion,10 and to introduce magnetic characteristics,11 Traditionally, MAX phase alloys have to a large extent been disordered solid solutions with challenges in attaining an exact a priori composition.12 However, in 2014 Liu et al. synthesized a chemically ordered 312 MAX phase, Cr2TiAlC2,13 defined by out-of-plane ordering through alternating M-layers based on one M-element only, later coined o-MAX. This was soon followed by other out-of-plane ordered phases, Mo2TiAlC2,14,15 Mo2ScAlC2,16 Cr1.5V1.5AlC2,17 and corresponding 413 in the form of Mo2Ti2AlC3 and Cr2V2AlC3.15,17 Common for o-MAX phases are the two crystallographic different M-sites, with Wyckoff positions 4f (M′) and 2a (M′′) for 312 and 4f (M′) and 4e (M′′) for 413. This is opposed to the most recent discovery of a family of in-plane ordered 211 MAX phase related materials, coined i-MAX, where the two metals have a 2:1 ratio within each metal layer.18–23
Chemically ordered o-MAX and i-MAX phases are highly interesting as they allow for incorporation of non-traditional MAX phase elements, e.g., Sc,16,18,23 Y,19,21,23 and W,22 but also for the improved control of the alloy composition. However, there has been no extensive systematic study in the quest for yet hypothetical ordered MAX phases other than for Ti-based o-MAX.24 In this work we focus on alloying between two metals, M′ and M′′, in quaternary 312 and 413 o-MAX phases, to systematically explore chemically ordered (out-of-plane) and disordered distributions of M′ and M′′ with M elements from groups 3 to 6 (Sc, Y, Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, and W) while keeping A and X equal to Al and C, respectively. The A = Al choice is motivated by Al being prone to chemical etching for MXene derivation, as shown for the o-MAX phases Cr2TiAlC2, Mo2TiAlC2, Mo2ScAlC2, and Mo2Ti2AlC3.16,25
This study on o-MAX phases is structured into two parts, starting with predictive stability calculations for chemically ordered and disordered distributions of M′ and M′′. We confirm the stability of the quaternary phases reported to date, and also suggest several ordered as well as disordered novel alloys yet to be experimentally discovered. In the second part, we investigate the impact of choice of element M′ and M′′, and effect of size and electronegativity of M′, M′′ and Al, for the formation of chemically ordered (o-MAX) or disordered MAX phases. We also propose guidelines for which elemental combinations to use in search for novel o-MAX phases, by combining M′ and M′′ from specific groups, and how to promote chemical order vs. disorder.
In this work we assume the o-MAX chemical order with the notation of M′ facing the Al-layer, while M′′ is being sandwiched between carbon layers in the 312 and 413 MAX phase structures. For 312 this corresponds to a 2:1:1:2 composition of M′:M′′:Al:C with M′ at 4f and M′′ at 2a Wyckoff positions, and for the 413 this corresponds to a 2:2:1:3 composition of M′:M′′:Al:C with M′ at 4f and M′′ at 4e Wyckoff positions. In addition, for the 312 MAX phase, five semi-ordered structures have been modelled, with a 2:1:1:2 composition of M′:M′′:Al:C, as shown in Fig. 1a. For the 413 MAX phase, a total of 22 unique ordered structures have been modelled within the unit cell, with a 2:2:1:3 composition of M′:M′′:Al:C, as shown in Fig. S1.† Four selected configurations are shown in Fig. 1b. Note that the o-MAX structure has previously been shown to be dynamically stable.33,34 In the present work we considered M′ and M′′ from groups 3 to 6; Sc, Y, Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, and W. To model chemical disorder of M′ and M′′ on the M sublattices we used the special quasi-random structure (SQS) method35 with supercell sizes of 4 × 4 × 1 unit cells, i.e. 192 and 256 atoms for the 312 and 413 MAX phases, respectively. Convergency tests show that the supercells used give a qualitatively accurate representation and a quantitative convergency in terms of calculated formation enthalpies, equilibrium volumes, and lattice parameters.
Fig. 1 Schematic illustration of considered chemical order of M′ and M′′ for (a) a 312 MAX phase structure with a 2:1:1:2 composition of M′:M′′:Al:C and (b) 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. In (b), only selected structures are shown for the 413 MAX phase, for further details see Fig. S1.† |
The thermodynamic stability of quaternary MAX phases is investigated at 0 K with respect to decomposition into any combination of competing phases. The most competing set of competing phases, denoted equilibrium simplex, is identified using a linear optimization procedure36,37 which have been proven successful to confirm already experimentally known MAX phases as well as predicting the existence of new ones.11,37–41 The stability of the quaternary MAX phase is quantified in terms of formation enthalpy ΔHcp by comparing its energy to the energy of the equilibrium simplex according to
ΔHcp = E(MAX) − E(equilibrium simplex). | (1) |
A phase is concluded stable when ΔHcp < 0. Here E(MAX) represent the chemically ordered or disordered MAX phase structure of lowest energy. However, when T ≠ 0 K, the contribution from configurational entropy due to disorder of M′ and M′′ on the M sublattices will decrease the Gibbs free energy ΔGdisordercp as approximated by
ΔGdisordercp[T] = ΔHdisordercp − TΔS, | (2) |
ΔS = −2kB[zln(z) + (1 − z)ln(1 − z)], | (3) |
(4) |
The trends in thermodynamic stability for both M3AlC2 and M4AlC3 is similar to the trends found for the binary MC phase in its rock salt structure (see Fig. 2c), where M from groups 4 (Ti, Zr, Hf) and 5 (V, Nb, Ta) is found to be stable or close to stable while M from groups 3 (Sc, Y) and 6 (Cr, Mo, W) is far from stable (ΔHcp > +200 meV per atom). The comparable trends can be related to the stacking of M and C in the Mn+1Cn layers of the MAX phases which is also found along the 111 direction of rock salt MC.
Fig. 3 Calculated formation enthalpy ΔHcp of the 312 MAX phase structures with a 2:1:1:2 composition of M′:M′′:Al:C, for six chemically ordered (A to F, as shown in Fig. 1a) and one chemically disordered (SQS) distribution of M′ and M′′. The grey area represents 0 ≤ ΔHcp ≤ 25 and includes those phases which are close to stable. |
The long list in Fig. 3 of ΔHcp for 110 compositions represented with different configurations of chemical order does not provide a clear picture of whether chemical order or disorder are to be preferred at typical synthesis conditions. In Fig. 4, we therefore visualize the trends in thermodynamic stability using a heatmap, where M′ and M′′ are sorted according to the periodic group in which the metal lies. The background colour represents the calculated thermodynamic stability for chemical order of lowest energy, with a blue region representing stable phases (ΔHcp or ΔGcp < 0). We also use a symbol representation to denote the type of chemical order of lowest energy. In addition, experimentally known MAX phases are marked by a square, where the colour represent the type of reported chemical order. Fig. 4a depicts the calculated formation enthalpy at 0 K, showing that the vast majority of M′ and M′′ combinations are chemically ordered, with 55 o-MAX (filled squares) and 45 of order B to F (filled triangles). Reported o-MAX phases to date are identified as stable; Cr2TiAlC2,13 Mo2ScAl2,16 and Mo2TiAlC2.14 However, the result obtained at 0 K do not capture the MAX phases reported with disorder between M′ and M′′; (Zr0.67Ti0.33)3AlC2,42 (V0.67Ti0.33)3AlC2,43 and (Nb0.67Sc0.33)3AlC2.44
Since typical bulk synthesis of MAX phase are performed around 1773 K we need to consider the contribution from configurational entropy to the Gibbs free energy, using eqn (2) and (3) for structures with disordered distribution of M′ and M′′ (SQS). Fig. 4b depicts either the calculated formation enthalpy ΔHcp (for ordered phases) or Gibbs free energy of formation ΔGcp at 1773 K (for disordered phases), depending on which order is of lowest energy. In such a representation, 79 out of 110 M′ and M′′ combinations are identified as disordered, and out of these 28 are stable, including the MAX phases reported to date with disorder on M-sublattices. Furthermore, 31 phases are identified as o-MAX, with 7 being stable, including those reported experimentally; Cr2TiAlC2,13 Mo2ScAl2,16 and Mo2TiAlC2.14 For Cr2VAlC2, we find the o-MAX to be of lowest energy, with ΔHcp = 6 meV per atom, and it has indeed been synthesized, albeit with some intermixing of V on the Cr site (M′), (Cr0.75V0.25)2VAlC2.17 For M′ = Ti and M′′ = Zr, both Ti2ZrAlC2 o-MAX and disordered (Ti0.67Zr0.33)3AlC2 have been reported.42,45 Our theoretical results indicate that disorder is preferred, though a more in-depth discussion of these discrepancies will be presented below.
Beyond the phases reported to date, we predict four stable o-MAX phases; Sc2NbAlC2, Sc2TaAlC2, Sc2WAlC2, and W2TiAlC2. An additional six o-MAX phases are close to stable; Mo2M′′AlC2 (M′′ = Zr, Hf, V, Nb, Ta) and W2ScAlC2. We also identify 26 systems which are stable, though with a preference for disorder at T < 1773 K, with most of these phases combining M′ and M′′ from groups 4 and 5.
Fig. 5 Calculated formation enthalpy ΔHcp of the 413 MAX phase structures with a 2:2:1:3 composition of M′:M′′:Al:C, for chemically ordered o-MAX (A and B), order of type I and V (shown in Fig. 1b), additional (other) ordered configurations, and chemically disordered (SQS) configurations of M′ and M′′. |
Again, we use a heatmap representation to visualize the thermodynamic stability and to determine whether chemical order or disorder is preferred at 0 K and at typical synthesis conditions (1773 K). Fig. 6a depicts ΔHcp at 0 K showing that a majority, 41 of 55, of the M′ and M′′ combinations are ordered in the o-MAX structure Reported o-MAX phases to date are identified as stable, Mo2Ti2AlC3,15 or close to stable for Cr2Ti2AlC3 and Cr2V2AlC3.17,46 11 combinations show preference for order of type I and only 3 show preference for disorder. Again, the reported disordered MAX phases are not captured by the 0 K results, and we therefore consider the contribution from the configurational entropy to the Gibbs free energy, using eqn (2) and (3), at a typical bulk synthesis temperature of 1773 K. Fig. 6b depicts the calculated ΔHcp (for o-MAX phases) or ΔGcp at 1773 K (for disordered MAX phases), depending on which ordering is of lowest energy. Now 32 out of 55 M′ and M′′ combinations are identified as disordered, and out of these 13 are predicted stable, including the MAX phases reported to date with disorder; (Ti0.5Nb0.5)4AlC3,47 (V0.5Cr0.5)4AlC3,48 and (Nb0.5V0.5)4AlC3.43 (Nb0.5Sc0.5)4AlC3 is one of the concluded stable disordered phases, and a chemically related phase, (Nb0.67Sc0.33)4AlC3, has recently been synthesized, albeit with a diverging stoichiometry compared to herein.44 23 phases are identified as o-MAX, with 4 being predicted stable, including the experimentally reported Mo2Ti2AlC3.15
Beyond the reported o-MAX we here predict three phases to be stable with preference for o-MAX ordering; Nb2Hf2AlC3, Mo2Ta2AlC3, and W2Ti2AlC3. An additional four systems are close to stable with preference for o-MAX ordering; Cr2Ta2AlC3, Mo2Zr2AlC3, Mo2Hf2AlC3 and Mo2Nb2AlC3. We also identify 12 systems which are stable and with preference for disorder at T = 1773 K, with a majority combining M′ and M′′ from groups 4 and 5.
Table 1 summarizes selected key results of the thermodynamic stability evaluation, where stable and close to stable 312 and 413 MAX phases with their preference for order or disorder are given. Comparing the experimentally reported phases and their calculated stability, the results are consistent. In addition, we also list hypothetical phases that are stable or close to stable, and provide information on whether or not chemical order or disorder of M′ and M′′ is to be expected.
Phase | Stability criteria | Experimentally reported and here predicted | Predicted |
---|---|---|---|
a Reported with intermixing of V on Cr-site (Cr0.75V0.25)2VAlC2. b Reported with slight off-stoichiometric Cr2.5Ti1.5AlC3. c Reported with some intermixing between M sublattices (Cr0.7V0.3)2(V0.8Cr0.2)2AlC3. d Have also been reported as Ti2ZrAlC2 o-MAX.45,49 e Reported stoichiometry (Nb0.67Sc0.33)4AlC3. | |||
o-MAX stable | ΔHcp < 0 | Cr2TiAlC213 | Sc2M′′AlC2, (M′′ = Nb, Ta, W) |
T disorder > 1773 K | Mo2ScAl216 | W2TiAlC2 | |
Mo2TiAlC214 | Nb2Hf2AlC3 | ||
Mo2Ti2AlC315 | Mo2Ta2AlC3 | ||
W2Ti2AlC3 | |||
o-MAX close to stable | 0 ≤ ΔHcp < +50 | Cr2VAlC217a | Mo2M′′AlC2 (M′′ = Zr, Hf, V, Nb, Ta) |
T disorder > 1773 K | Cr2Ti2AlC346b | W2ScAlC2 | |
Cr2V2AlC317c | Cr2Ta2AlC3 | ||
Mo2M′′2AlC3 (M′′ = Zr, Hf, Nb) | |||
Disordered MAX | ΔGcp[1773 K] ≤ 0 | (Zr0.67Ti0.33)3AlC242 | (Ti0.67M′′0.33)3AlC2 (M′′ = Hf, V, Nb, Ta, Mo, W) |
T disorder < 1773 K | (Ti0.67Zr0.33)3AlC242d | (Zr0.67M′′0.33)3AlC2 (M′′ = Hf, Nb, Ta) | |
(V0.67Ti0.33)3AlC243 | (Hf0.67M′′0.33)3AlC2 (M′′ = Ti, Zr, Nb, Ta) | ||
(Nb0.67Sc0.33)3AlC244 | (V0.67M′′0.33)3AlC2 (M′′ = Ta, Cr) | ||
(Ti0.5Nb0.5)4AlC347 | (Nb0.67M′′0.33)3AlC2 (M′′ = Ti, Zr, Hf, Ta) | ||
(V0.5Cr0.5)4AlC348 | (Ta0.67M′′0.33)3AlC2 (M′′ = Sc, Ti, Zr, Hf, V, Nb) | ||
(Nb0.5V0.5)4AlC343 | (Sc0.5M′′0.5)4AlC3 (M′′ = Ta, Mo, W) | ||
(Nb0.5Sc0.5)4AlC344e | (Ti0.5M′′0.5)4AlC3 (M′′ = V, Ta) | ||
(Zr0.5M′′0.5)4AlC3 (M′′ = Hf, Nb, Ta) | |||
(Hf0.5M′′0.5)4AlC3 (M′′ = Nb, Ta) | |||
(Ta0.5M′′0.5)4AlC3 (M′′ = V, Nb) |
To increase the understanding of the formation of ordered and disordered MAX phase configurations, and to test the previously suggested hypothesis for the origin of o-MAX, we choose to investigate trends in type of metal M′ next to the Al layer (site 4f for 312 and 413), type of metal M′′ sandwiched between C layers (site 2a for 312 and 4e for 413), and the metallic radius (r) and electronegativity (ρ) of M′, M′′ and A. For this analysis, only chemical order of type A (o-MAX) and chemical disorder is considered.
We start by looking at whether the metals M′ and M′′ in o-MAX also prefer formation of the binary MC in its rock salt structure. To be consistent, we herein use ΔHcp in the analysis instead of the isostructural formation enthalpy, ΔHiso, since the latter is biased from whether M prefer an fcc arrangement with C or not. This is most pronounced for the 413 system as seen in Fig. S2.† In Fig. 7, the calculated ΔHcp for o-MAX is shown as a function of the disorder temperature Tdisorder, where the colouring represents if M′ and/or M′′ in the o-MAX structure also form the binary rock salt MC. Note that for Tdisorder < 0, the disorderd MAX is lower in energy than o-MAX, whereas for Tdisorder > 0, the o-MAX is lower in energy. In addition, experimentally reported phases have been marked according to their reported order to aid in the interpretation. For both 312 and 413 o-MAX phases qualitatively similar results are found;
• When both M′ and M′′ in o-MAX also form MC (red), this results in Tdisorder close to zero, i.e., no or small energy difference between order and disorder, indicating preference for disorder at typical synthesis temperatures.
• When only M′ in o-MAX also form MC (light red), this results in Tdisorder < 0, i.e., disorder is lower in energy than order, clearly indicating preference for disorder.
• When only M′′ in o-MAX also form MC (light grey), this generally results in Tdisorder > 0, i.e., order is lower in energy than disorder at 0 K. When Tdisorder > 1773 K, preference for o-MAX is expected.
• When none of M′ and M′′ in o-MAX form MC (dark grey) we find −5000 K < Tdisorder < +5000 K. Hence, no general rule can be identified for this case.
The presented results partially support the previous statement that having a M′′ that also form a rock-salt MC is beneficial for o-MAX formation (Cr2TiAlC2 and Mo2Ti2AlC3), but this do not explain the formation of, e.g., Mo2ScAlC2.
Trends in ΔHcp for o-MAX as a function of the disorder temperature Tdisorder is shown in Fig. 8 for M′2M′′AlC2 (left) and M′2M′′2AlC3 (right). The colouring in the top panels represent the difference in size between M′ and M′′, where red represents rM′ > rM′′ and blue rM′ < rM′′. The mid panels display the difference in electronegativity between M′ and M′′, where red represents ρM′ > ρM′′ and blue ρM′ < ρM′′, while the bottom panels display the difference in electronegativity between M′ and Al, where red represents ρM′ > ρAl and blue ρM′ < ρAl. The values used for r and ρ are found in Table S5.†
Focusing on the region which indicates a preference for chemical order, i.e., Tdisorder > 1773 K, suggests that for a majority of phases M′ should be more electronegative than M′′ and with a size difference of M′ and M′′ being at most 0.2 Å. We also find that M′ should be more electronegative than Al. The latter is in line with previous work suggesting that having a M′ with larger electronegativity than Al results in fewer electrons available for populating antibonding Al–Al orbitals.24
Within the region where disorder is preferred, i.e., Tdisorder < 1773 K, and with 0 < ΔHcp < +50 meV per atom, we find hypothetical as well as experimentally reported disordered phases, with a difference in electronegativity of M′ and M′′ within 0.4, while the size difference of M′ and M′′ does not exceed 0.2 Å. The results also indicate that the difference in electronegativity of M′ and Al should be small.
The Hume-Rothery rules states that if there is a large enough difference in size and electronegativity between two atomic species, chemical order will be preferred, whereas for no or small differences disorder is favoured.50 Our results indicate that an explanation for order/disorder formation upon on alloying between M′ and M′′ within the 312 and 413 MAX phase structure is only consistent with the part of the Hume-Rothery rules related to electronegativity. When it comes to the size criteria for ordering, we find that a large size difference of M′ and M′′ within order of type A (o-MAX) is detrimental, with Tdisorder < 0 in general and ΔHcp ≫ 0. This is most pronounced for Y-based phases due to its large metallic radius (1.80 Å), which results in compressive stress in the Y-layers while the layers with smaller atoms will be under tensile stress. This is opposite to the in-plane ordered i-MAX phases,18,19,21–23 related to the 211 MAX phases, where a significant size difference between M′ and M′′ is required for the ordered formation of M′ and M′′ in a 2:1 ratio within each M-layer.
Based on herein presented result we propose guidelines for which M combination to use when targeting either chemical order (out-of-plane) or disorder in 312 and 413 MAX phases. Stable phases with preference for chemical order at typical bulk synthesis temperatures have mainly been identified for systems where the central M′′, sandwiched by C, also form rock-salt MC and with M′ = Cr and Mo next to the Al-layer. This is most pronounced for the 413 system but is also valid for the 312 systems. For the 312 systems we have also identified ordered phases with M′ = Sc next to the Al-layer. o-MAX is mainly preferred when M′ from group 6 (Cr, Mo, W) is combined with M′′ from groups 4 and 5 (Ti, Zr, Hf, V, Nb, Ta). Disorder of M′and M′′ is preferred when alloying metals from groups 3 to 5 (Sc, Ti, Zr, Hf, V, Nb, Ta).
A comparison of herein presented results with experimentally reported phases with order and/or disorder illustrates that ordered o-MAX phases can all be found within the region where we suggest that order is preferred, i.e., Tdisorder > 1773 K, while disordered MAX is found for Tdisorder < 1773 K. Some M′ and M′′ combinations have been reported both as ordered and disordered, e.g., Cr2V2AlC3 and (Cr0.5V0.5)4AlC3 (choice of notation indicating order/disorder), which have been reported as o-MAX with some degree of intermixing between the two M-sites and as disordered.17,48 We have previously shown that o-MAX phases can be stable both upon limited intermixing of M′ and M′′, and at off-stoichiometries.15,24 Ti2ZrAlC2 or (Ti0.67Zr0.33)3AlC2 is another example where both order45,49 and disorder42,51 have been claimed. Our results indicate that Ti and Zr prefer disorder or possibly semi-order, based on the moderate Tdisorder = 357 K combined with a similar electronegativity of Ti and Al. It should be noted that within this study, only out-of-plane order have been considered, where each M-layer only consists of one elemental type only. Off-stoichiometries for o-MAX have been demonstrated both experimentally and theoretically, and shows that deviation from the ideal o-MAX composition is possible while still retaining the overall chemical order.15,17,24 Impact from off-stoichiometries, intermixing between M′ and M′′ layers, and vacancies at M, Al or C sites have previously been demonstrated for a few M′ and M′′ combinations to have minor influence on the qualitative results.24 The possibility of in-plane order of M′ and M′′, similar to i-MAX phases,18,19 in higher order MAX phases, 312 and 413, is beyond the scope of this work and a topic for future studies.
There are already MXenes realized from o-MAX phases16,25 and the range of materials predicted in the present work suggest that the number of chemically ordered as well as disordered MXenes, from o-MAX or solid solution 312 and 413 MAX phases, respectively, can be increased. This allows tuning of the MXene chemistry through one or two metals in the outmost surface layers, and alteration of fundamental properties through choice of metal embedded within in the MXene.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9nr08675g |
This journal is © The Royal Society of Chemistry 2020 |