Stability of Zener order in martensite: an atomistic evidencePhilippe Maugisa,<, Damien Connétableb and Paul Eyméouda

aAix Marseille Univ, CNRS, IM2NP, Marseille, FrancebCIRIMAT, UMR 5085, CNRS INP UPS, ENSIACET 4, allée Émile Monso, BP 44362, F-31030 Toulouse Cedex 4, France

A R T I C L E I N F O

Keywords:SteelsLong-range orderingElastic behaviourDensity functional theory (DFT)Mean-field modelling

A B S T R A C T

Martensite is a supersaturated solid solution of carbon in body-centered iron wherein interstitial car-bon atoms preferentially occupy a single octahedral sublattice. Despite a century of research, themechanism of this long-range ordering is still a subject of debate. Recently, Zener’s theory of or-dering was challenged both experimentally and theoretically. In an attempt to settle the controversy,we investigated by density functional theory the ground states of Fe-C configurations having variousdegrees of order. We conclude that the fully Zener-ordered configurations are always the most sta-ble, thus confirming Zener’s theory. Comparison with mean-field elasticity and Ising-type modellingsupports the elastic origin of Zener ordering.

Fink and Campbell in 1926 [1] established that ↵®-martensiteis a supersaturated solid solution of carbon in body-centerediron. The carbon atoms preferentially occupy the octahe-dral interstitial positions between nearest iron atoms in oneof the three Í001Î crystal directions. This preferential occu-pation causes a tetragonal distortion of the matrix, quanti-fied by the lattice parameter ratio c_a > 1. Based on Bain’stransformation, Kurdjumov and Sachs [2] suggested that car-bon distribution in martensite lattice inherits from the di�u-sionless face-centered to body-centered structural transfor-mation of the parent austenite phase. Assuming that long-range carbon–carbon interactions are predominantly elastic,Zener [3] stated that the ordered distribution of carbon atomsis energetically more favorable than disorder. He predictedan order-disorder transition between cubic ferrite and tetrag-onal martensite (Fig. 1). Zener’s mean-field theory has beenchallenged ever since, partly because the so-called Zener or-dering transition was never evidenced directly. Kurdjumov

Figure 1: The Zener order-disorder transition transforms bcc-

ferrite (left) into bct-martensite (right) and vice-versa. Fe

atoms are in gray, preferred carbon sites are coloured according

to their sublattice (RGB code).

et al. [4] provided indirect proof of Zener ordering: theymeasured an increase in tetragonality of quenched marten-site during ageing at room temperature, interpreted as pro-

<Corresponding authorphilippe.maugis@univ-amu.fr (P. Maugis)www.im2np.fr (P. Maugis)

ORCID(s): 0000-0001-9283-0471 (P. Maugis)

gressive ordering via carbon jumps into the favoured octa-hedral sites. Recently, an in situ X-ray di�raction study ofmartensite formation during quench evidenced the set up oftetragonality as soon as martensite forms, confirming Bain’sscheme [5]. In another study, in situ neutron di�raction re-vealed a decrease in axial ratio towards c_a = 1 during age-ing after quench [6], i.e. a case of Zener disordering. Finally,cases were reported of high-carbon austenite isothermallytransformed into cubic martensite, thus challenging Zener’smodel [7, 8]. On the other hand, room-temperature mea-surements of the lattice parameters of Fe-C steels of variouscarbon contents all show an abrupt transition between low-carbon cubic ferrite and high-carbon tetragonal martensite[9, 10, 11], which is coherent with Zener’s theory.

On the theoretical side, investigations of carbon orderingin iron involved various techniques: thermodynamic mean-field modelling [12, 13, 14, 15, 16], the microscopic elas-ticity theory [4, 10, 17, 18, 19], the CALPHAD formalism[20, 21], molecular dynamics [22, 23, 14, 24] and combinedab initio–Monte Carlo [25, 26, 16, 27]. Most of these ap-proaches, although based on di�erent hypotheses and ap-proximations, provide results in favour of Zener ordering.One exception is the work of Ruban [25, 26], which predictsan "anti-Zener" ordering consisting of carbon atoms occu-pying specific sites on two preferred octahedral sublattices,associated with a ratio c_a < 1. The relative stability of thisconfiguration, as compared to Zener order, is explained bythe asserted dominating e�ect of carbon–carbon short-rangeinteractions.

It is worth noting that mechanical external conditions af-fect the phase stabilities (see e.g. [28, 29]) and modify theequilibrium order. In particular, coherency stress or appliedstress can turn the first-order Zener transition into a con-tinuous transition [12, 30]. When compressive, the stressmay also stabilise "beyond-Zener" order, i.e. an orthorhom-bic structure where the three octahedral sublattices are un-evenly occupied by carbon atoms [31, 32, 30, 33].

This literature survey leads to the conclusion that Zenerordering may or may not occur, depending on yet unclearreasons. Moreover, there is no theoretical consensus on whether

P. Maugis et al.: Preprint submitted to Elsevier Page 1 of 6

Stability of Zener order in martensite: an atomistic evidence

Zener order is the most stable state of order in supersaturatediron or not. The subject needs further prospection, given itsimportance in understanding and predicting the mechanismsof martensite transformation and the resulting martensiticmicrostructures.

In this study, we investigated from first principles the en-ergetic stability of various states of carbon order. Specialquasirandom structures (SQS) were used to simulate randomdistributions of carbon atoms over each sublattice. Densityfunctional theory (DFT) computations were performed un-der prescribed applied stress. The results were comparedto the mean-field elastochemical model [33] supplementedby short-range carbon–carbon interactions. The most stablestructures were identified as function of the applied stress.

In body-centered iron, interstitial carbon atoms are dis-tributed over the 3 sublattices of octahedral sites. Carbonfraction on sublattice i is noted c

iand total carbon fraction

is c = ≥ci. To study the e�ect of carbon distribution over

the sublattices at constant carbon fraction c, it is convenientto introduce two long-range order parameters ⌘ and ⇣ de-fined by c⌘ = c3 *

12 (c1 + c2) and c⇣ = c2 * c1. Parameter ⌘

quantifies the degree of Zener order along crystal direction[001], while ⇣ quantifies the unequal occupancy in directions[100] and [010]. In the mean-field approach, a state of orderis characterised by the pair of parameters (⌘, ⇣). The case(1, 0) represents full Zener order along direction [001], i.e.all carbon atoms sitting on sublattice 3. This configurationwill be referred to as the Z3 orientational variant. The com-binations (*0.5,*1) and (*0.5, 1) describe the Z1 and Z2fully ordered variants along directions [100] and [010], re-spectively. Intermediate values of (⌘, ⇣) represent partiallyordered structures. Among them, (*0.5, 0) is the inverse-Zener order: it has empty sublattice 3 and equal occupancyof sublattices 1 and 2. Finally, (0, 0) is the fully disorderedstructure with equal occupancy of the three sublattices.

The thermodynamical stability of a given state of orderunder applied stress tensor � is characterised by the enthalpyof ordering �H(⌘, ⇣), where symbol � expresses that thedisordered state is taken as reference. �H can be calculatedfrom the energy of ordering �U and the strain of ordering�" via the relationship

�H = �U * V � � �", (1)

where V is the volume of the stress-free carbon-free crystal.DFT calculations are known to render correctly the short-

range interactions between solute atoms, irrespective of theirorigin, "chemical" or "elastic". In this study, by allowing thesupercell to relax, we also captured the long-range carbon–strain interaction and the resulting homogeneous strain.

In order to represent carbon–vacancy disorder on octa-hedral interstitial sublattices, we used the SQS approach.Supercells were built with mcsqs [34] modulus of ATATcode [35], following the procedure described in [36]: com-putation of the correlations functions on clusters contain-ing C–C pairs equal to or shorter than

˘3a0 (a0 is the lat-

tice parameter of iron). The computed tetragonality and en-ergy somewhat depend on the relative positions of the car-

bon atoms in the supercell [37, 38]), this is why we averagedeach case over a set 13 SQS structures. To extract energeticand structural properties of these SQS using DFT, we em-ployed an in-house version of VASP code [39, 40] allowingto impose stresses on cells. We used the following parame-ters: generalized gradient approximation [41] with projectoraugmented wave method [42] and Perdew-Burke-Ernzerhofexchange-correlation functional [43, 44], 400 eV energy cut-o� (as used in Refs. [45, 36, 46]), spin-polarized approxima-tion, 10*5 eV energy convergence criterion of the electronicself-consistency, 4ù4ù4 Monkhorst-Pack k-points grid [47]for 128 iron atom supercells and Methfessel-Paxton smear-ing [48] for reciprocal space integration. To relax the su-percells we used a conjugate-gradient algorithm with ionicrelaxation criterion of 10*4 eV in energy for zero-stress cal-culations and 0.01 eV/Å in force for stress-imposed calcula-tions. Calculations were performed on Fe128C12 supercellsof composition Fe-8.6at%C and varying states of order.

In a first series of calculations, the degree of Zener or-der along axis 3 was varied (⌘ variable and ⇣ fixed to zero).Profiles of the averaged ordering enthalpy �H and latticeparameters a and c are drawn in Fig. 2. The energetic stabil-ity of full Zener order with respect to partial order or disor-der is evidenced by the minimum in enthalpy positioned at⌘ = 1. Correlatively, the lattice parameters a and c vary lin-early with the order parameter. They reach the experimentalvalues of Cheng et al. [49] at ⌘ = 1, letting assume full Zenerordering for the experimental data collected in this study.

A second series of calculations aimed at investigatingthe e�ect of tension (+2 GPa) and compression (*2 GPa)along axis 3 on the relative stability of selected states of or-der, namely (1, 0), (*0.5, 0) and (*0.5, 1). The comparativeplotting of Fig. 3 gives the following results: (i) tension sta-bilises tetragonal Zener order along the stress axis (variantZ3); (ii) compression stabilises orthorhombic Zener orderin a direction transverse to the stress axis (variant Z1 or Z2);and (iii) disorder and inverse-Zener order are disfavoured inall cases.

Additional calculations on variants (0, 0) and (1, 0) re-vealed that carbon distribution and applied stress have negli-gible impact on the magnetic moments (less than 0.02�B/atomicspecies). This negligible role of Zener order on magnetismsuggests that an empirical potential without magnetism wouldbe able to capture the discussed e�ects on Zener ordering.

Here, the main advantages of our first-principles approachare: (i) it correctly represents interstitial disorder thanks tothe use of SQS; (ii) it does not raise an issue of empirical po-tential representativeness, as it is the case for the embedded-atom method [50] and pairwise interactions models; and (iii)it considers both homogeneous elasticity by lattice deforma-tion and C–C chemical interactions. However, this approachdoes not easily allow discriminating the long-range elasticcontribution from the short-range interaction contribution.To investigate this point, we used the mean-field elastochem-ical model of Maugis [16, 33] together with pairwise C–Cinteractions. The theory is summarised below.

In the framework of the continuum elasticity theory of

P. Maugis et al.: Preprint submitted to Elsevier Page 2 of 6

Stability of Zener order in martensite: an atomistic evidence

Figure 2: Ordering enthalpy (top) and lattice parameters (bot-

tom) as function of Zener-order parameter ⌘ (⇣ = 0) in stress-

free Fe-8.6at%C. SQS-DFT is compared to elasticity and pair-

wise interactions and to experimental data from Cheng [49].

Error bars indicate the standard deviations.

point defects, long-range carbon–strain interactions are ac-counted for via the dipole moment tensor P of interstitialcarbon [51, 52]. Carbon atoms and axial stress � induce arelaxed strain tensor ", whose normal components write asfunctions of ⌘ and ⇣ :

hnnnlnnnj

"11 =VC3V0

c +V⌃3V0

c

⇠*⌘ * 3

2⇣⇡+ S12�

"22 =VC3V0

c +V⌃3V0

c

⇠*⌘ + 3

2⇣⇡+ S12�

"33 =VC3V0

c +2V⌃3V0

c⌘ + S11�

(2)

Constants V0, VC and V⌃ are respectively the atomic volumeof the lattice, the relaxation volume and the tetragonal dis-tortion due to an interstitial carbon. The latter are related tothe elastic properties of the defect and the host matrix [16]:

VC = (S11 + 2S12)(Pc + 2Pa),V⌃ = (S11 * S12)(Pc * Pa).

(3)

Pa and Pc are the two components of the dipole moment ten-sor and S

ijare the components of the elastic compliance

Figure 3: Ordering enthalpy of various states of order under

tensile or compressive stress in Fe-8.6at%C and schematics of

the relaxed cell shape (arrows represent the state of order).

Fully Zener-ordered configurations are the most stable ones in

all cases.

tensor of the host matrix. It is clear from Eqs. 2 that crys-tal symmetry is governed by both applied stress and orderparameters. Indeed, the Bravais lattice may be either cubic(⌘ = ⇣ = 0, � = 0; full disorder), tetragonal prolate (e.g.⌘ > 0, ⇣ = 0, � > 0; Zener order), tetragonal oblate (e.g.⌘ < 0, ⇣ = 0, � < 0; inverse-Zener order [27, 26]) or or-thorhombic (e.g. ⌘ ë 0, ⇣ ë 0; beyond-Zener order [30, 4]).

The homogeneous strain contributes to the enthalpy ofordering by the amount �Hel written as

�Hel = *h⌃c2⇠⌘2 + 3

4⇣2⇡* 2

3V⌃�c⌘ (4)

where h⌃ is the strain-energy parameter:

h⌃ = 13V0

(S11 * S12)(Pc * Pa)2. (5)

The first term in the r.h.s. of Eq. 4 is the stress-independentordering energy �U el. The second term is the elastic workof the applied stress when the configuration is changed fromdisorder (⌘ = 0) to partial order (⌘ ë 0). Eq. 4 shows thatZener ordering along direction 3 (⌘ > 0) is favoured when atensile stress is applied along that direction (�⌘ > 0, �Hel

decreases); it is disfavoured when a compressive stress is ap-plied along the same direction (�⌘ < 0, �Hel increases).

From Eqs. 3 and 5 we notice that the equilibrium stateof order, obtained from minimising function �Hel(⌘, ⇣) atgiven c and �, does not depend on the elastic sti�ness ofthe host matrix, but only on the deviatoric part of the dipolemoment tensor, via the ratio (Pc * Pa)_V0.

All material parameters entering the elasticity model wereextracted from high-accuracy DFT calculations. Their val-ues are gathered in Table 1. The dipole moment tensor wascomputed using the residual stress method [53]: a supercellcontaining one carbon atom in a bcc-Fe matrix was relaxed,while maintaining the shape and volume of the cell identicalto that of carbon-free iron. On account of the carbon inser-tion, a residual stress � builds up at the cell boundary. The

P. Maugis et al.: Preprint submitted to Elsevier Page 3 of 6

Stability of Zener order in martensite: an atomistic evidence

Table 1Lattice parameter a0 (in nm), elastic compliances S

ij(in GPa*1) and dipole moments Pa,c

(in eV) computed by DFT. Volumes V0, VC, V⌃ (in eV/GPa) and strain-energy parameter

h⌃ (in eV) used in the elasticity calculations.

a0 S11 S12 Pa Pc V0 VC V⌃ h⌃

0.2834 5.55 ù 10*3 *1.91 ù 10*3 9.60 18.88 7.10 ù 10*2 6.56 ù 10*2 6.93 ù 10*2 3.02

dipole moment tensor P is related to the residual stress byP = *V �, where V is the volume of the supercell. Correc-tion of the image forces due to the periodic boundary condi-tions was applied by varying the supercell size and perform-ing a parabolic fit of the residual stress as function of 1_V .

Contribution of the short-range carbon–carbon interac-tions to the enthalpy was written as a pairwise Ising-typemodel: H

nn = 12≥

Vij

, where Vij

is the interaction en-ergy between two carbon atoms indexed i and j. The V

ij’s

were computed by DFT up to the 19th coordination shell[36]. To evaluateHnn(⌘, ⇣), 250,000 carbon atoms were ran-domly distributed over the 3 octahedral sublattices accord-ing to the specified order parameters and the summation wasperformed numerically.

The ordering enthalpy computed with the elasticity modelcoincides well with the decreasing parabolic profile obtainedby SQS-DFT (Fig. 2 top), and even better when the C–C pair-wise contribution is added. The coincidence remains satis-factory in case of applied stress (Fig. 3): although a slightshift is observed, the influence of the order state is well ren-dered by the elasticity and pairwise interactions model. Weconclude that elasticity captures Ì85% of the ordering en-thalpy, while pairwise interactions have a minority contri-bution of Ì15%. The latter moderate e�ect is qualitativelyin line with the findings of Yan et al. [26]. On the otherhand, the influence of ordering on the lattice parameters isperfectly rendered by the elasticity model (Fig. 2 bottom).As expected from elasticity theory, short-range interactionsplay a negligible role on the carbon-induced homogeneousstrain [54].

To generalise our investigation to all possible states of or-der, surface plots of the reduced ordering enthalpy�Hel_h⌃c2were computed with the elasticity model for selected val-ues of the reduced applied stress �V⌃_h⌃c (Fig. 4). In thestress-free condition (� = 0), �Hel exhibits three degener-ate stable states characterised by the (⌘, ⇣) pairs (*0.5,*1),(*0.5, 1) and (1, 0). These pairs represent respectively thefully Zener-ordered variants Z1, Z2 and Z3. We see thatwhen a tensile stress is applied (� > 0), the degeneracy islifted and the stable state is the Zener-ordered variant Z3. Incase of compressive stress (� < 0) variants Z1 and Z2 arethe two degenerate stable states. These variants are orientedtransversaly to the applied compressive stress. Owing to thedownward concavity of the enthalpy surfaces, all partiallyordered states are unstable, including the disordered struc-ture (0, 0) and the inverse-Zener structure (*0.5, 0), and thisregardless the applied stress.

We have demonstrated that full Zener order is alwaysenergetically more favorable than any other state of order,while disorder and inverse-Zener order are unstable. Thisconclusion is true as long as the carbon atoms are randomlydistributed in each sublattice. We did not investigate sec-ondary ordering, i.e. ordering in one or more sublattices,possibly occurring at some specific carbon stoichiometries.Such phenomenon was evidenced experimentally [55] andtheoretically [22, 25, 26, 38]. However, secondary orderingis expected to a�ect only marginally the ordering enthalpies,especially at low-carbon content, thus not invalidating ourgeneral conclusions.

Chen et al. reported the experimental evidence of “cu-bic” or “nontetragonal martensite” formed at room tempera-ture in plastically deformed quenched martensite [7], as wellas in deformation-induced nanograined martensite [8]. Theseresults apparently contradict our finding that Zener order isexpected in martensite, whatever the applied stress. How-ever, the present paper is limited to investigating the 0K statesof a defect-free crystal. At finite temperature, martensite or-ders on the condition that its carbon content be higher thana critical value. According to literature, this value lies inthe range of 0.8 to 2.9 at% at room temperature. Chen etal.’s material may not reach the criterion. Indeed, althoughtheir nominal carbon is high, carbon atoms are likely to havemigrated to the deformation-induced dislocations and/or tothe nanograin boundaries during room-temperature ageingor sample preparation, thus decreasing solute carbon downto a subcritical value.

Our first-principles investigation leads to the conclusionthat Zener-ordered configurations of body-centered Fe-C arealways energetically favoured compared to disordered or par-tially ordered ones. This finding supports Zener’s theory ofordering. Comparison with mean-field elasticity and Ising-type modelling confirms that Zener ordering is mostly drivenby the long-range carbon–carbon strain interaction, whileshort-range carbon–carbon interactions play a minor role.These conclusions remain true when tensile or compressivestress is applied.

Supplementary materialDFT-relaxed SQS structures and values of the correla-

tion functions are provided as supplementary material.

Declaration of Competing InterestThe authors declare that they have no known competing

financial interests or personal relationships that could have

P. Maugis et al.: Preprint submitted to Elsevier Page 4 of 6

Stability of Zener order in martensite: an atomistic evidence

Figure 4: Ordering enthalpy (in units of h⌃c2) as function of the order parameters ⌘ and

⇣ . From left to right: � = *1, 0 and 1 (in units of h⌃c_V⌃). The gray triangle drawn at

ordinate -1 represents all accessible states of order. Z1, Z2 and Z3 fully-ordered variants

(coloured points) are the stable states when no stress is applied (centre). Tension along

axis 3 stabilises longitudinal variant Z3 (right), while compression stabilises both transverse

variants Z1 and Z2 (left).

appeared to influence the work reported in this paper.

AcknowledgementsThis work was supported by the Agence Nationale de la

Recherche (contract C-TRAM ANR-18-CE92-0021). Cen-tre de Calcul Intensif d’Aix-Marseille is acknowledged forgranting access to its high performance computing resources.The authors thank C. Arnold for his technical assistance incompiling the VASP code.

References[1] W. L. Fink and E. D. Campbell, Trans. Am. Soc. Steel Treat. 9, 717

(1926).[2] G. Kurdjumow and G. Sachs, Zeitschrift für Phys. 64, 325 (1930).[3] C. Zener, Phys. Rev. 74, 639 (1948).[4] G. Kurdjumov and A. Khachaturyan, Acta Metall. 23, 1077 (1975).[5] J. Epp, T. Hirsch, and C. Curfs, Metall. Mater. Trans. A 43, 2210

(2012).[6] Y. Wang, Y. Tomota, T. Ohmura, S. Morooka, W. Gong, and S. Harjo,

Acta Mater. 184, 30 (2020).[7] Y. Chen, W. Xiao, K. Jiao, D. Ping, H. Xu, X. Zhao, and Y. Wang,

Phys. Rev. Mater. 2, 050601 (2018).[8] Y. Chen, Q. Liu, W. Xiao, D. Ping, Y. Wang, and X. Zhao, Mater.

Lett. 227, 213 (2018).[9] S. Nagakura, Y. Hirotsu, M. Kusunoki, T. Suzuki, and Y. Nakamura,

Metall. Trans. A 14A, 1025 (1983).[10] Z. Fan, L. Xiao, Z. Jinxiu, K. Mokuang, and G. Zhenqi, Phys. Rev. B

52, 9979 (1995).[11] O. D. Sherby, J. Wadsworth, D. Lesuer, and C. Syn, Mater. Sci. Forum

539-543, 215 (2007).[12] A. Khachaturyan and G. Shatalov, Fiz. Met. Met. 32, 5 (1971).[13] K. Taylor and M. Cohen, Prog. Mater. Sci. 36, 225 (1992).[14] P. Chirkov, A. Mirzoev, and D. Mirzaev, Phys. Met. Metallogr. 117,

34 (2016).[15] P. Maugis, F. Danoix, H. Zapolsky, S. Cazottes, and M. Gouné, Phys.

Rev. B 96, 214104 (2017).[16] P. Maugis, Acta Mater. 158, 454 (2018).[17] A. Udyansky, J. Von Pezold, V. N. Bugaev, M. Friák, and J. Neuge-

bauer, Phys. Rev. B - Condens. Matter Mater. Phys. 79, 224112(2009).

[18] A. Udyansky, J. von Pezold, A. Dick, and J. Neugebauer, Phys. Rev.B 83, 184112 (2011).

[19] X. Zhang, H. Wang, T. Hickel, J. Rogal, Y. Li, and J. Neugebauer,Nat. Mater. (2020).

[20] R. Naraghi and M. Selleby, in 1st World Congr. Integr. Comput. Mater.Eng., edited by P. C. John Allison and G. Spanos (TMS, 2011), pp.235–240.

[21] R. Naraghi, M. Selleby, and J. Ågren, Calphad Comput. CouplingPhase Diagrams Thermochem. 46, 148 (2014).

[22] C. W. Sinclair, M. Perez, R. G. A. Veiga, and A. Weck, Phys. Rev. B81, 224204 (2010).

[23] P. Chirkov, A. Mirzoev, and D. Mirzaev, Mater. Today Proc. 2S, 553(2015).

[24] O. Waseda, J. Morthomas, F. Ribeiro, P. Chantrenne, C. W. Sinclair,and M. Perez, Model. Simul. Mater. Sci. Eng. 27, 015005 (2019).

[25] A. V. Ruban, Phys. Rev. B 90, 144106 (2014).[26] J. Y. Yan and A. V. Ruban, Comput. Mater. Sci. 147, 293 (2018).[27] P. Maugis, S. Chentouf, and D. Connétable, J. Alloys Compd. 769,

1121 (2018).[28] O. Shchyglo, T. Hammerschmidt, M. �ak, R. Drautz, and I. Steinbach,

Materials 9, 669 (2016).[29] A. Artemev, Y. Jin, and A. Khachaturyan, Acta Mater. 49, 1165

(2001).[30] P. Maugis, J. Phase Equilibria Di�us. 41, 269 (2020).[31] M. Shtremel and F. Satdarova, Fiz. Met. Met. 34, 699 (1972).[32] P. Chirkov, A. Mirzoev, and D. Mirzaev, Phys. Status Solidi 255,

1700665 (2018).[33] P. Maugis, Comput. Mater. Sci. 159, 460 (2019).[34] A. Van de Walle, P. Tiwary, M. De Jong, D. L. Olmsted, M. Asta,

A. Dick, D. Shin, Y. Wang, L. Q. Chen, and Z. K. Liu, Calphad 42,13 (2013).

[35] A. Van de Walle, M. Asta, and G. Ceder, Calphad 26, 539 (2002).[36] S. Chentouf, S. Cazottes, F. Danoix, M. Gouné, H. Zapolsky, and

P. Maugis, Intermetallics 89, 92 (2017).[37] H. Ohtsuka, V. A. Dinh, T. Ohno, K. Tsuzaki, K. Tsuchiya, R. Sahara,

H. Kitazawa, and T. Nakamura, ISIJ Int. 55, 2483 (2015).[38] D. Kandaskalov and P. Maugis, Comput. Mater. Sci. 150, 524 (2018).[39] G. Kresse and J. Furthmüller, Comput. Mater. Sci. 6, 15 (1996).[40] G. Kresse and J. Furthmuller, Phys. Rev. B 54, 11169 (1996).[41] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).[42] P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).[43] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865

(1996).[44] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 78, 1396

(1997).[45] L. Ventelon, B. Lüthi, E. Clouet, L. Proville, B. Legrand, D. Rodney,

and F. Willaime, Phys. Rev. B 91, 220102 (2015).

P. Maugis et al.: Preprint submitted to Elsevier Page 5 of 6

Stability of Zener order in martensite: an atomistic evidence

[46] D. Kandaskalov and P. Maugis, Comput. Mater. Sci. 128, 278 (2017).[47] H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976).[48] M. Methfessel and A. T. Paxton, Phys. Rev. B 40, 3616 (1989).[49] L. Cheng, A. Bottger, T. H. de Keijser, and E. Mittemeijer, Scr. Metall.

Mater. 24, 509 (1990).[50] M. S. Daw and M. I. Baskes, Phys. Rev. B 29, 6443 (1984).[51] D. Bacon, D. Barnett, and R. Scattergood, Prog. Mater. Sci. 23, 51

(1980).[52] R. W. Ballu�, Introduction to elasticity theory for crystal defects

(Cambridge University Press, 2012).[53] C. Varvenne and E. Clouet, Phys. Rev. B 96, 224103 (2017).[54] A. G. Khachaturyan, Theory of structural transformations in solids

(Dover Publications, 2008).[55] K. A. Taylor, L. Chang, G. B. Olson, G. D. W. Smith, M. Cohen, and

J. B. V. Sande, Metall. Trans. A 20, 2717 (1989).

P. Maugis et al.: Preprint submitted to Elsevier Page 6 of 6