High-throughput estimation of planar fault energies in A3B compounds with L12 structure

Accepted Manuscript High-throughput estimation of planar fault energies in A3B compounds with L12 structure K.V. Vamsi, S. Karthikeyan PII:

S1359-6454(17)30887-X

DOI:

10.1016/j.actamat.2017.10.029

Reference:

AM 14128

To appear in:

Acta Materialia

Received Date: 4 May 2017 Revised Date:

9 September 2017

Accepted Date: 12 October 2017

Please cite this article as: K.V. Vamsi, S. Karthikeyan, High-throughput estimation of planar fault energies in A3B compounds with L12 structure, Acta Materialia (2017), doi: 10.1016/ j.actamat.2017.10.029. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Comparison of time taken for calculations involved in the model proposed in this work versus direct simulation, for a variety of planar faults

RI PT

Comparison of planar fault energies predicted from the model proposed in this work versus direct simulation, for a variety of planar faults

ACCEPTED MANUSCRIPT

M AN U

SC

20

AC C

EP

TE D

10

Data for 9 fcc metals and 38 L12 compounds Select cases (compositions and faults) highlighted

1

ACCEPTED MANUSCRIPT

High-throughput estimation of planar fault energies in A3B compounds with L12 structure K.V. Vamsi and S. Karthikeyan*

Abstract

RI PT

Department of Materials Engineering, Indian Institute of Science, Bangalore, India -560012

Deformation of alloys containing L12-ordered A3B precipitates is strongly influenced by planar

SC

fault energies of the precipitate. However, accurate data on fault energies is not available for many A3B compounds owing to time and cost constraints. In this work, we propose a Diffuse

M AN U

Multi-Layer Fault model that enables high-throughput computational estimation of planar fault energies. The model accounts for the change in stacking and bonding environment of atoms on multiple atomic layers in the vicinity of the fault. The new bonding environment in each layer was compared against that in a library of over 1300 geometrically close packed A3B compounds,

TE D

each with less than 16 atoms in the unit cell, and proximate structures were identified for each atomic layer. Fault energy was expressed in terms of energy of the proximate structures and L12. This work enabled the identification of hitherto unknown proximate structures, viz. ω and χ,

EP

relevant to antiphase boundary and complex stacking fault on {111} planes. Density functional theory was used to estimate the energy of the proximate structures and to predict fault energies

AC C

which were compared with results from direct simulations of faults. It was found that the proposed model predicted energies of different superlattice faults in over 40 A3B compounds with a high degree of accuracy. The model has no fitting parameters, has a fifteen-fold computational advantage over direct simulation, and is extendable to several novel A3B compounds where data is presently lacking.

ACCEPTED MANUSCRIPT

Keywords: Stacking fault energy; Superalloys; Antiphase Boundaries; Ab-initio calculations; High-Throughput Computation.

RI PT

1. Introduction The search for the ‘next Ni-base superalloy’ is driven by the ever-expanding demand for creepresistant materials that can operate at higher temperatures and in more corrosive environments.

SC

In this context, there is significant ongoing research on Co, Pt, Ir, and Rh-based alloy systems [1–4]. The composition and microstructural design of these alloys borrows heavily from

M AN U

prototypical Ni-base superalloys which owe their high temperature strength to a two-phase microstructure of solute-strengthened Ni-rich f.c.c (γ) matrix with a high-volume fraction of ordered precipitates (γ′, γ′′, δ, η) of nominal composition, Ni3X (X=Al, Ti, Ta, Nb, etc) [5]. Shear of these precipitates by matrix dislocations results in the formation of planar faults such as

TE D

anti-phase boundaries (APBs) and other super-lattice stacking faults (SSFs). The penalties, i.e., the Planar Fault Energies (PFEs) involved in the creation of such defects correlate strongly with the order strengthening provided by the precipitate [6,7]. Peak strength in Ni-base superalloys

EP

has been found to scale with the square root of APB energy on the (111) plane when deformation involved cutting of γ′ precipitates (L12 structure) by weakly- or strongly-coupled ½[110]

AC C

dislocation pairs [5]. The absolute and relative magnitudes of various PFEs also affects deformation mechanisms—looping versus shearing [8], slip versus twinning [9], ½[110] slip versus ⅓[112] slip in γ′ [10], etc. Moreover, PFEs affect cross-slip related locking mechanisms which result in phenomena such as yield anomaly [11] and tension-compression asymmetry [12]. Given the centrality of PFEs to high temperature deformation of A3B compounds, assessment of future Co, Pt, Ir, Rh-based systems with (γ + γ′) microstructure requires estimates of PFEs. The

ACCEPTED MANUSCRIPT

nominal composition of the precipitate in these systems is Co3X, Pt3X, Ir3X and Rh3X, (X = Al, W, Ti, Sn, Ta, Zr, etc). Only a subset of these A3B compounds is expected to have the desirable combination of PFEs to deliver the expected mechanical response. However, data on PFEs is not

RI PT

available for the aforementioned compounds; indeed, this is available only for a limited set of compositions and structures such as a few binary A3B compounds with L12 crystal structure and for only some of the many possible SSFs. Even in mature systems such as Ni-based superalloys

SC

[13–19], PFEs are not well characterized because the A3B composition deviates considerably from binary Ni3Al and is better represented by (Ni,Co)3(Al, Ti, Ta, W). If the crystal structure of

M AN U

the A3B precipitate is not L12 (γ′), but D022 (γ′′) or D024 (η) instead, data on PFE is even more meagre since these structures have their own characteristic APBs and SSFs. Clearly, there is a need for a database of PFEs for different A3B compounds such that one could select promising candidates which have the desirable combination of PFEs. However, this database is difficult to

TE D

generate due to the time, effort and cost involved in accurate estimation. Recovery and X-ray methods are indirect experimental techniques requiring additional models and are prone to errors [13,20]. Transmission electron microscopy (TEM) based experiments enable direct estimation of

EP

PFEs [15,21] from the separation of partial dislocations at extended dislocations and dislocation nodes. However, accurate estimation by TEM is non-trivial and time-consuming, particularly if

AC C

the aim is to obtain statistically relevant data. An alternate approach is to compute PFEs via ‘direct’ atomistic simulations of planar faults, either employing semi-empirical interatomic potentials or via electronic structure calculations [22,23]. While the former technique is computationally inexpensive, the non-availability of reliable potentials for ternary systems, and in many cases even binary systems [24] limits the technique’s usefulness. Electronic structure methods avoid this bottleneck since energies are directly derived from quantum-mechanical

ACCEPTED MANUSCRIPT

calculations without a need for interatomic potentials. Several recent studies in ternary Ni3(Al,X) systems are based on such first principles calculations [22,25,26]. However, despite the technique’s accuracy, typical direct supercell simulations involve 50-200 atoms and thus

RI PT

computationally expensive particularly for simulating multi-component compositions.

In this paper, we propose an alternate computational method for rapidly predicting PFEs in A3B

SC

compounds. The aim is to provide a ‘high-throughput’ means of identifying promising A3B compounds with the desired combination of PFEs. The method uses first principles Density

M AN U

Functional Theory (DFT) calculations but avoids computationally expensive direct simulation of faults. Instead, PFEs are estimated from DFT calculations of formation energy of A3B compounds in several simple ordered structures with unit cells of 16 atoms or fewer. These calculations are significantly faster and efficient for shortlisting candidate alloy systems for more detailed experimental and computational studies. In the first part of this paper, theoretical

TE D

background on existing models for predicting PFEs are reviewed and the new high-throughput model is proposed. The DFT methodology for calculation of structural energies and direct simulation of faults are described subsequently. The validity of the model was tested for a variety

EP

of faults in 46 A3B compounds (both real and hypothetical) with L12 structure by comparing

AC C

model predictions to direct simulations and these are presented in the results section. Finally, these results are discussed and the advantages and shortcomings of the model are highlighted. Predictions are also made for a few Co3X, Pt3X, Ir3X and Rh3X compounds for which experimental data is currently unavailable. 2. Theoretical background

ACCEPTED MANUSCRIPT

A planar fault on the (hkl) plane is created by shearing one half of a parent crystal with respect to the other, across adjacent (hkl) planes by an in-plane non-lattice translation. This results in a bonding environment in the vicinity of the fault that is distinct from that in the bulk, thus

RI PT

resulting in an energy penalty. Olson and Cohen [27] proposed a model for estimating Intrinsic Stacking Fault (ISF) energy in fcc alloys wherein the ISF is treated as an  atomic layer thick plate of a Stacking Fault Phase (SFP) with hcp structure inserted into the fcc matrix. If  is the

SC

number of atoms per unit area, the ISF energy ( ) is expressed in terms of the chemical free energy difference (∆ 

), the penalty for creation of two interfaces (2) and the strain

 = ∆ 

) + 2 +  

M AN U

energy (Estrain) due to modulus or lattice parameter misfit between the hcp and fcc phases: (1)

∆ 

from thermodynamic databases has been used to predict  in several fcc alloys

TE D

[28]. The model suffers from several limitations. Interface energy () between hcp and fcc is typically unknown and used as a fitting parameter. For e.g., Curtze et al. [28] estimated ISF energies in several austenitic steels using the Olson-Cohen model by accounting for the

EP

compositional dependence of ∆ 

, but ignoring the compositional dependence of . This is problematic since in many cases the 2 term contributed to >75% of  . Moreover,  

AC C

contribution was ignored. The application of the model to L12 compounds is further complicated because of the diversity of planar faults (Figure 1). The faults on the (111) plane include the Antiphase Boundary (APB), Superlattice Intrinsic Stacking Fault (SISF) and Complex Stacking Fault (CSF) created by shear vectors,  ! *

!!!)

= #1%01', !!!) = #1%21' and )!!!) = ! "

! (

% '. A Superlattice Extrinsic Stacking Fault (SESF) on the (111) plane is formed when the #12%1

crystal is sheared by #1%21' on two adjacent (111) planes. An APB on the (010) plane is created ! (

ACCEPTED MANUSCRIPT

by a shear vector 

+!+)

= #1%01'. Application of Eq. (1) to L12 compounds requires ! "

knowledge of the appropriate SFP for each fault and this is presently lacking for some faults.

RI PT

Possible candidates for the SFP for each fault include one of many geometrically close packed (GCP) A3B compounds. Even in cases where the SFP is known, free energy is usually unknown. An alternate approach for estimating PFEs in fcc alloys [26,29] and ordered compounds [30,31]

SC

is the Axial Next Nearest Neighbor Ising (ANNNI) model [32] where the energy of a faulted or perfect crystal with a specific stacking sequence is expressed as a 1-D Ising expansion: (2)

M AN U

 = ,+ − ,! ∑ / /0! − ," ∑ / /0" − ,( ∑ / /0( − ⋯

where ,+ , ,! , ," , etc., are the respective interaction energy parameters between atoms within each atomic plane, atoms on adjacent planes, atoms on alternate planes, etc. The model uses two-state spin variable / for each plane, and their combinations to quantify the stacking sequence. By

TE D

comparing the energies of variously stacked GCP structures, ,+ , ,! , ," , etc are estimated and applied to the stacking sequence created by a planar fault to estimate its energy. PFE is expressed 2 345 0"2 6345 (2744 

EP

in terms of energy of one or more GCP structures, for eg.,  =

where A is

the fault area. An advantage of the model is that it avoids a description of interface energy ()

AC C

between different phases. Moreover, it is computationally inexpensive to estimate the energies of perfect crystals consisting of 1-4 atoms via DFT techniques, instead of directly simulating faults. This technique has been applied successfully to some faults in L12 compounds. For instance, SISF (111) and SESF (111) energies have been expressed as

"82 9:;< 2 =;> ? 

[31] and

@82 9:>A 2 =;> ? 

(based on [33]). However, the technique cannot be readily extended to all SSFs. For instance, Figures 2a and b show the projection of atoms on the {110} plane, after creation of an SISF and

ACCEPTED MANUSCRIPT

an APB on the {111} plane. There are no 1st or 2nd nearest neighbor (NN) violations due to the creation of the SISF (Fig 2a), though there is a change in stacking of close packed planes (CPPs) to |ABCBCABC|. In contrast for APB(111) (Fig 2b) the stacking sequence is identical to that in

RI PT

L12 (i.e. |ABC|) but first NN bond violations, also called chemical violations, result in the energy penalty. CSF(111) involves both stacking and chemical violations (supplementary material). A modified two-spin Ising model can be used to capture both types of violations observed in an

SC

APB or a CSF. In such a case, the APB (or CSF) energy can be expressed in terms of a set of GCP structures which incorporate such chemical violations. For instance, the appropriate 4-atom

M AN U

structure identified by this methodology is equivalent to incorporating an APB on every CPP. Even for the appropriate 12-atom structure, the high density of APBs−one in every three planes−results in interactions between the ‘faults’. One may have to use appropriate GCP structures with 24 atoms or more to avoid such interactions. However, for GCP structures of this

to be significant.

TE D

size, the computational advantage of the ANNNI method over direct simulations is not expected

EP

In the present work, we propose a generalized model for predicting PFEs in fcc alloys and A3B compounds with L12 structure, and which can be readily extended to other ordered compounds.

AC C

The model borrows elements from the SFP and ANNNI models; bonding environment in the vicinity of the planar fault is approximated to that in an appropriate reference crystal structure and the energy penalty due to the fault is attributed to the energy difference between the reference and parent crystal structures. As seen in Fig 2, the two immediate planes across which the fault is created, are designated B! and B! . Similarly, subsequent planes away from the fault plane are designated B" and B" and so on. Atoms that lie on any of these layers, say B! , have a bonding environment (i.e., number and type of bonds at different distances) that is different from

ACCEPTED MANUSCRIPT

that in the parent structure. We propose there exists a specific reference structure designated /! , also called proximate structure, for B! with the following characteristics: it is GCP, it has the same composition as the parent and it has a bonding environment that is approximately the same

RI PT

as that for atoms on B! . The average potential energy of an atom sitting in B! , in the purview of the NN bond approximation, is the same as the potential energy per atom in /! which is designated /! ). One can identify proximate structures S±1, S±2, etc. for layers L±1, L±2, etc. The

SC

energy of 2m layers of the faulted configuration can be expressed as CDEF =  ∑G HG / ) where / ) is the energy per atom for structure / and  is as defined in Eq (1). If the energy

M AN U

per atom for the parent structure is I), the energy for 2m layers of unfaulted structure is CCDEF = 2JI). PFE (2 ) is the difference in the energies of the faulted and unfaulted structures:

(3)

TE D

G G 2 =  ∑G HG / ) − 2J I ) =  ∑HG# / ) −  I)' ≡  ∑HG8∆L , ?

The current approach is more general than that of Olson-Cohen; the matrix and SFP are not delineated explicitly thus avoiding descriptions of SFP thickness and SFP-matrix interface

EP

energies. In other words, the model of an SFP separated from the matrix by a sharp interface is

AC C

supplanted by a Diffuse Multi-Layer Fault (DMLF). The PFE is the result of bonding violations not only for atoms in the immediate vicinity of the fault but also for those in layers further away from the fault plane. The energy penalty due to the latter violations is expected to be lesser. Henceforth, the present approach will be termed the DMLF model for brevity. The first step (details in section 3.1) in predicting energy of any fault from the DMLF model is the identification of a set of proximate structures (i.e., /G to /G ) which mimic BG to BG in terms of their bonding environment. After identifying / , ∆L , was estimated via DFT

ACCEPTED MANUSCRIPT

calculations (details in section 3.2) for each layer, which in conjunction with Eq. (3) allowed prediction of PFE. The model was applied to ISF in 9 fcc metals, and five SSFs in 46 A3B compounds with L12 structure. The maximum number of atoms in the unit cell of the proximate

RI PT

structures was limited to 16. This constraint was imposed to ensure computational efficiency over direct DFT simulation of faults which typically involve 32-200 atoms in the supercell; computational time in DFT typically scales as " ln , where  is the number of atoms. To

SC

validate the model, direct DFT fault simulations with 32-40 atoms per supercell were also done

M AN U

(details in section 3.2). 3. Methodology

3.1. Structure Proximity (SP) model to identify proximate structures The SP model is used to identify the most proximate structure / of layer B for each fault,

TE D

involved two steps: (i) matching the stacking environment and (ii) matching the chemical environment, of atoms in B . The steps correspond to the energy penalties due to stacking and chemical

violations.

A

library

of

all

possible

(1397

in

total)

GCP

structures

EP

 = P! , " , ⋯ ,  , ⋯ , !(QR , S with unit cell ≤16 atoms and a composition of A3B was

AC C

generated. It was assumed that / ∈ . These structures are either ordered fcc (|ABC| stacking of CPPs), or ordered hcp (|AB| stacking), or ordered dhcp (|ABAC| stacking). For each structure, the number and type of bonds (A-A, A-B, and B-B) at different distances was evaluated. For instance, ,V5

U!

in

 ,V5

, U!

, U!



structure

,V5

,V5

, U"

 ,V5

, U"

, U"

,V5

counts at different distances with U!

,V5

the ,V5

⋯ , U

set  ,V5

, U

, U

of ,V5

)

numbers,

represents

bond

being the number of A-A bonds at distance 1 and so on.

ACCEPTED MANUSCRIPT

Also tabulated (see Table 1) for the structure was the coordination (total bond count) at each V5

V5

V5

V5

,V5

distance given by: WU! , U" , … U Y, where U! = U!

 ,V5

+ U!

+ U!

,V5

, etc.

RI PT

The first task was to identify the subset of structures with stacking sequence similar to that for an atom on B . As seen in Table 1, for structure  the stacking sequence (one of |ABC|, |AB| or |ABAC|) results in specific coordination at specific distances. The stacking sequence can be V5

V5

V5

SC

equivalently represented by a sequence of numbers WU! , U" , … U Y. This set of numbers represents a point  in n dimensional “stacking environment” space where each dimension

M AN U

corresponds to coordination at a certain distance. The stacking sequence for an atom in B can Z

Z

also be can be quantified by a unique sequence of numbers, (U! L , U" L , … ) which corresponds to Z

Z

point B in the same n dimensional space. By comparing (U! L , U" L , … ) against corresponding sequences for all structures in , a subset of structures, [ with a stacking sequence identical to

TE D

that for an atom in B was shortlisted. This was implemented by evaluating Ψ8B ,  ? = ]W∆U! 8B ,  ?, ∆U" 8B ,  ?, ⋯ , ∆U 8B ,  ?Y] = ^∑_H! W∆U_ 8B ,  ?Y Z

where ∆U_ 8B ,  ? =

− U_ L is the difference in coordination at the jth distance between  and B . ΨB ,  ) is the

EP

V5

U_

"

norm of the vector connecting B to _ . The subset [ was thus identified by invoking the

AC C

criterion of Ψ8B ,  ? = 0 which corresponds to atoms in B and  having identical stacking environments.

In the second step, the most proximate structure / with a chemical environment closest to that for an atom on layer B was identified from subset [. The chemical bonding environment for a specific structure [` can be represented in terms of the number of A-A, A-B, and B-B bonds at

ACCEPTED MANUSCRIPT

different distances. This, in turn, can be represented by point [` in a ‘3n’ dimensional bondcount

space

,ab

U!

 ,ab

, U!

, U!

,ab

,ab

, U"

with  ,ab

, U"

, U"

,ab

,ab

⋯ , U

coordinates:  ,ab

, U

, U

,ab

)

The

bonding

RI PT

environment for an atom in B can be quantified by point B in the same 3n dimensional bondcount space. Π8B , [` ? is the norm of the vector connecting points B to [` and given by "

"

"

,ab

∆U_ 8B , [` ? = U_

,ZL

− U_

where

SC

Π8B , [` ? = ^∑_H! dW∆U_ 8B , [` ?Y + W∆U_ 8B , [` ?Y + W∆U_ 8B , [` ?Y e,

. This is a possible measure of proximity of B and [` in

M AN U

bond-count space. While a small value of Π8B , [` ? essentially means that B and [` are proximate, a large Π8B , [` ? does not necessarily imply that they are distant energetically. This is because ∆U8B , [` ? at large NN distances are less important than differences at short NN distances since 1st NN bonds are stronger than 2nd NN bonds, and so on. Π8B , [` ? gives equal

TE D

weightage to ∆U at all distances and thus overestimates ∆U contributions at large distances. This can be corrected by giving 1st NN differences greater weightage than 2nd NN differences, etc. In this work, the weight given to each NN interaction was assumed to scale with typical bond

= h_ . Thus, the true chemical proximity, Π  8B , [` ? between B and [` can

AC C

h_ = h_ = h_

EP

strength at that distance such that h! > h" > ⋯ > h . Further, it was assumed that

be written as:

"

"

"

"

Π 8B , [` ? = ^∑_H!8h_ ? dW∆U_ 8B , [` ?Y + W∆U_ 8B , [` ?Y + W∆U_ 8B , [` ?Y e

(4)

where h_ is the weighting factor at the jth NN distance. The weights used in this work (Table 1) were proportional to the bond energy at specific NN distances derived from the Morse potential

ACCEPTED MANUSCRIPT

of Ni [34]. Note that other weighting schemes including LJ potential for an fcc solid were attempted and the same proximate structures were identified. So as long as 1st NN differences were assigned a greater weightage that 2nd NN differences (and so on), approximate weighting

RI PT

factors suffice. This also justifies the assumption of h_ = h_ = h_ . While Π 8B , [` ? = 0 implies identical bonding environments in B and [` , small values of Π 8B , [` ? suggest that [` has approximately the same chemical environment B . Thus, Π was evaluated between B and

SC

each structure in subset [. The structure with a minimum Π was identified as the proximate

M AN U

structure / for layer B .

The procedure was repeated for layers L±1, L±2, etc of the fault and proximate structures S±1, S±2, etc. were identified from . The two-step procedure was applied to APB(111), SISF(111), SESF(111), APB(010) and CSF(111) in L12. For the ISF in fcc, only step 1 was used to identify

TE D

the proximate structure(s) since there are no chemical violations. Proximate structures identified for each layer of each fault are indicated in Table 2 along with the expressions for evaluating PFE via Eq. (3). Layer nomenclature for each fault is pictorially shown in the supplementary

EP

material. The crystallographic details of the relevant proximate structures are presented in Table 3 and in the supplementary material. The results of this analysis are presented and discussed in

AC C

section 4.

3.2. Validation of DMLF model After identifying / for each layer of each fault, PFE was estimated using the DMLF model (i.e., from ∆L, via expressions shown in Table 2) and via direct simulations. This was done for several pure metals and A3B compounds (Table 4). Vienna Ab-initio Simulation Package (VASP 4.6) was used for DFT calculations at 0 K. [35,36]. The MedeA platform was used for generating

ACCEPTED MANUSCRIPT

atomic configurations, setting up simulation parameters and post-processing [37]. Projected augmented wave method was used with a plane wave cut-off of 450 eV. First Brilluoin zone integration done under Methfessel-Paxton scheme [38] with a smearing width of 0.12 eV and a

RI PT

k-point spacing of 0.167 Å-1. Calculations were terminated when Hellmann-Feynman forces were <0.02 eV/Å. Pseudopotentials based on generalized gradient approximation (GGA-PBE)

SC

were used [39,40].

Two sets of calculations were done to independently evaluate the left and right hand sides of Eq

M AN U

(3). The first set involved direct simulation of the faults to estimate 2 in A3B L12 compounds or fcc metals. The faults simulated are listed in Table 2. Depending on the crystallographic plane of the fault, appropriate supercell orientations were chosen. The fault was created using the tiling method [41] or by creating faulted supercells (for SESF). The number of atoms in the supercell was 32-40 depending on the fault. The supercell dimension perpendicular to fault plane was

estimated as: 2 =

TE D

sufficiently large (>15Å) to avoid interactions between the periodic images of the fault. PFE was 27klmno6 25op7o4n 

, where Efaulted and Eperfect were the energies of the faulted

EP

and perfect supercells obtained from DFT calculations and A is the fault area. To validate the DMLF model, PFEs were evaluated from the supercells where atomic relaxations were not

AC C

allowed, in other words the lattice was rigid. Further details on PFEs estimation from direct simulations may be found elsewhere [41]. In the second set of calculations, the structural energies of the pure metals in fcc and dhcp structures, and A3B compounds in the relevant GCP structures were evaluated. Lattice parameters of the GCP structures were constrained by the equilibrium lattice parameter of A3B compound in L12 structure so that the GCP structure was wholly coherent (without misfit) with L12. Atomic relaxations were permitted. With these calculations, the appropriate ∆L, values on the right hand side of the Eq, (3) were estimated.

ACCEPTED MANUSCRIPT

Cubic elastic constants (C11, C12 and C44) were determined for Ni3X compounds and some A3B other compounds in L12 structure. The reasons for these calculations are explained in the context of the results. Strains ranging from -0.01 to 0.01 in steps of 0.002 were imposed to 4-atom L12

RI PT

unit cells and the symmetry-general least-square extraction method [42] was used to evaluate the elastic constants.

SC

4. Results

Proximate structures corresponding to each layer of the different faults, their key

M AN U

crystallographic details and expressions for evaluating PFEs are provided in Tables 2 and 3. The unit cells of the most important proximate structures are shown in Figure 3(a). Atomic coordinates are provided in the supplementary material. Some of these—D019 (Ni3Sn), D023 (Al3Zr) and D024 (Ni3Ti)—are well-known GCP structures with experimental prototypes (in

TE D

parenthesis). Interestingly, this work has allowed the identification of two hitherto-unknown structures relevant to APB(111) and CSF(111). These structures are hypothetical and without prototypes. Since this is the first known report of these structures, we designate them ω (oP16

EP

and P/mmm) and χ (hP16 and C/2m) respectively. GCP structures of A3B can be visualized as a stacking of CPPs. If the structure is disordered, the CPPs are identical to the (111) plane in fcc.

AC C

However, in ordered GCP structures, there is ordering within the CPP with B atoms occupying specific positions resulting in different motifs. B atoms can arrange themselves in one of three motifs: triangular (T), rectangular (R) or mixed triangular/rectangular (T+R). The GCP structure can be built by stacking CPPs (of different motifs) in different stacking sequences. The L12 structure can be thought to be an |ATBTCT| stacking of CPPs with (T) motif. Similarly D019, D024 and

D023

can

be

represented

by

|ATBT|,

|ATBTATCT|

and

a

twelve-layer

|A(T+R)B(T+R)…K(T+R)L(T+R)|. The newly-identified ω can be visualized as alternating CPPs with

ACCEPTED MANUSCRIPT

(T) and (R) motifs and represented by a six-layer |ATBRCTDRETFR|. χ is similar to D024 with a ! |rts Bs A s C s | stacking where rt is translated with respect its adjacent layers by a 〈112%0〉 type (

RI PT

non-lattice vector. Using expressions in Table 2, PFEs were estimated and these values were compared to those from direct simulations (with no relaxations of supercell dimensions or atomic positions) and

SC

these results are shown in Figure 3(b). Data is for ISF in a variety of fcc metals, and the SSFs in several Ni3X compounds. The unity line to test the validity of the DMLF model is indicated. It is

M AN U

observed that for all faults, the DMLF model predictions match the results from the direct simulation closely with the data falling close to the unity line. It is noted that the results from the model and direct simulations are highly correlated with an R2 ~1 and a linear fit has a slope of ~1 with an intercept of ~0 mJ/m2. The low RMS values of the residuals, (, i.e., z{/ = > ∑ L€;|6Lpo4n,L |9}=~,L )



) also underscore the tight scatter band and predictive capability of the

TE D

^

DMLF model (FE  and ‚Z are the PFEs estimated from direct simulations and the DMLF model). This suggests that the DMLF model predictions not only correlate well with the direct

EP

simulations, but also requires no additional fitting parameters. The model is robust for a variety

AC C

of faults, and for fcc metals and A3B compounds that are either stable or unstable in the parent structure. For unstable metals and compounds, the DMLF model and direct simulations typically predict PFEs that are negative implying that the local structure near the fault is energetically more favorable than the parent. It should be highlighted that SISF(111) has been previously associated with a D019 structure while our model suggests that D024 is a more proximate. This is confirmed by a significantly poorer fit if one were to use D019 as the proximate structure (RMSE = 65 mJ/m2 for D024 versus RMSE = 109 mJ/m2 for D019). Bonding environment around an

ACCEPTED MANUSCRIPT

APB(010) has been be approximated to both D022 and D023 previously [43] while our analysis suggests that D023 is the better approximation (RMSE = 48 mJ/m2 for D023 versus RMSE = 164 mJ/m2 for D022). Similarly, bonding about a SESF(111) has been previously approximated to

RI PT

D024 [33] while our analysis suggests that the local bonding is better represented by a combination of D024 and D019. Structures corresponding to APB (111) and CSF (111) have been identified for the first time in this work, and the model predictions are very good for both these

SC

faults, though not as good as for the other faults. These deviations are discussed in section 5.

M AN U

Given the efficacy of the model for a variety of faults and compounds in the Ni3X family, it was extended to other A3B L12 compounds and results are shown in Figure 4. To minimize clutter, results are separated into three panels. Figure 4(a) compares model predictions to direct simulations for several stable L12 compounds (i.e., positive PFE). Data for different SSFs are collapsed into one plot. This again confirms a close match (RMSE = 88 mJ/m2) between the

TE D

model and direct simulations with data falling in a tight scatter band about the unity line. Fig 4(b) and (c) are companion plots to Figure 3(b) and correspond to experimental and hypothetical Co3X and Al3X type L12 compounds. Again, the match between model and direct simulations is

EP

good (RMSE = 122 mJ/m2 for Co3X, RMSE = 98 mJ/m2 for Al3X). Thus, it can be concluded

AC C

that the DMLF model is able to predict PFEs in a variety of A3B compounds and for a variety of faults with a high degree of accuracy. In the next section, the advantages and shortcomings of the model are discussed. Predictions are also made for novel A3B compounds for which PFE data is not available.

5. Discussion

ACCEPTED MANUSCRIPT

As can be seen from figures 3 and 4, the DMLF model is capable of predicting the energy of a variety of planar faults in several fcc metals and A3B compounds in L12 structure with a high degree of accuracy and with no fitting parameters. The absence of fitting parameters makes the

RI PT

model transferable and reliable for making predictions in novel A3B compounds. It must be reiterated that the primary aim of this work is to provide a high-throughput means of screening potential A3B compounds which can then be taken up for further experimental or computational

SC

studies. In this sense, in addition to accuracy, the model was expected to provide significant computational efficiency. It was found that estimating PFEs using the DMLF model was ~15

M AN U

times faster than via direct simulations (as shown in supplementary material and graphical abstract). The advantage is further heightened since a thermodynamic database (heat of formation, cohesive energy) of various A3B compounds in different structures is also additionally generated. The model thus provides an alternate means of predicting PFEs that is

TE D

both rapid and accurate. It should be noted that it can be easily extended to other high temperature A3B compounds which do not have an L12 structure (such as Ti3Al (D019) and Ni3Nb (D022)). Finally, a significant advantage of the technique is that it can be extended to

EP

estimate high temperature PFEs. Using Hellmann-Feynman forces from DFT calculations one can estimate phonon density of states (P-DOS) and free energy (in the quasi-harmonic

AC C

approximation) of various proximate phases as a function of temperature. These free energies in conjunction with DMLF model can be used to predict high temperature PFEs. While these calculations are more expensive than 0 K single point calculations, they are significantly cheaper than direct simulations involving P-DOS evaluation. The extension to non-L12 A3B compounds and to high temperature PFE will be presented elsewhere [44].

ACCEPTED MANUSCRIPT

It is also important to highlight the limitations of the present model. While model predictions are good for most faults (Fig 3 and 4), the RMSE values are relatively high for APB(111) and CSF(111) (Fig 3). Data suggests that DMLF model underestimates the energies for APB(111)

RI PT

(slope = 1.17) and overestimates that for CSF(111) (slope = 0.86). The causes of these deviations are two-fold. First, while the identification of the most proximate structures for L1 for either fault was accurate, structures with a high degree of match could not be found for the L2 of either fault.

SC

L2 layer was approximated to L12 in APB(111) resulting in no energy penalty, ∆>,Z!> in turn leading to an under-prediction. Likewise, L2 layer was approximated to D024 in CSF(111)

M AN U

resulting in an over-prediction. This relatively poor match for L2 was due to the limited set of structures against which L2 was compared. Had a wider palette of structures been available, a better match for L2 may have been found. However, this would have involved relaxing the ≤16 atom/unit cell constraint leading to a much higher computational cost, with only a marginal

borne in mind.

TE D

improvement in accuracy. So, the tradeoff between accuracy and computational time has to be

EP

The second cause of deviations arises from the central assumption in this work that if the bonding environment of an atom in the vicinity of the fault (in terms of number and types of

AC C

bonds) matches that for an atom in a proximate structure, then both atoms have the same energy. This is a statement of the nth NN bond model where the total potential energy is expressed as a summation of pairwise interactions between atoms bound by a central force potential. While central force potentials describe some bulk properties well, they are found to be simplistic in describing defect formation enthalpies particularly in systems with a high degree of metallic bonding where electron redistribution readily occurs. Atoms in the vicinity of defects like vacancies, grain boundaries and free surfaces, are more strongly bound than atoms in the bulk

ACCEPTED MANUSCRIPT

due to their sparser coordination. Thus, central force potentials significantly over-predict energy of such defects [45]. The NN bond model has been shown to be unsatisfactory even in modeling PFEs in pseudo-binary Ni3(Al,X) [41]. Thus, the degree of bond metallicity in the A3B

RI PT

compound determines the exactness of the NN approximation on which the DMLF model is broadly based. Higher the bond metallicity, greater the electron redistribution and higher the discrepancies between the model and direct simulations. An indirect measure of bond metallicity

SC

in cubic crystals is the ratio, C44/C12 between elastic constants. This ratio is unity in cubic crystals bound by central force potentials (Cauchy’s condition) and significantly <1 in highly

M AN U

metallic crystals (C44/C12 ~ 0.3 in Au, Pt). Figure 5(a) is a bar chart indicating C44/C12 for different Ni3X compounds. Also indicated are the residuals between the DMLF model and direct simulations for APB(111) energies. A strong negative correlation is observed with residuals being significantly greater for compounds with a lower C44/C12, i.e., those with a higher degree

TE D

of metallic bonding such as Ni3Ta. Two extreme cases of Ni3Al (C44/C12 = 0.78, RMSE = 28 mJ/m2) and of Ni3Ta (C44/C12 = -0.03, RMSE = 264 mJ/m2) can be examined in greater detail. Figure 5b shows the line profile of electron density change due the creation of an APB(111) in

EP

Ni3Al and Ni3Ta. In both compounds, fault creation results in electron redistribution, however, the amplitude of the redistribution is higher in Ni3Ta compared to Ni3Al. It is more informative

AC C

to look at the volume-average electron density difference indicated by the bar graphs. The bars for the case of Ni3Al are significantly smaller in magnitude than for Ni3Ta. It is seen that while electron redistribution after fault creation is fairly localized in Ni3Al, this occurs over larger distances in Ni3Ta (enrichment in the volume containing L±1 at the expense of those containing L±2 and L±3). This electron redistribution in Ni3Ta is consistent with the lower C44/C12 and the high degree of metallicity. The enhancement in electron density is expected to cause the Ta-Ta

ACCEPTED MANUSCRIPT

1st NN bonds across the APB to be stronger than what is expected from a NN bond model resulting in a smaller APB energy from direct simulation than that predicted by DMLF model. Figure 3a) confirms that this is so for APB(111) in Ni3Ta and other compounds with low C44/C12.

RI PT

The C44/C12 thus serves as a figure of merit for the reliability of the prediction. In summary, the DMLF model is significantly more computationally efficient in predicting PFEs than direct

SC

simulations, its accuracy is dependent on degree of metallicity of the compound.

Comparisons to this point have been between competing computational techniques for estimating

M AN U

PFEs, with the DMLF model offering significant computational advantage without a major loss in accuracy. The efficacy of the DMLF model in reproducing experimental PFE values was tested for a limited set of compounds and faults (Figure 6a) [15,16,18,46–53]. While the data is still correlated, it is less so than earlier. This is related to errors in prediction and wide scatter in experimental data. Experimental values are usually derived from TEM measurements of

TE D

separation of superpartial dislocations in samples deformed at high temperatures. Accurate estimation of PFEs requires an accounting of elastic anisotropy, imaging conditions, dislocation character, thin film effects, dislocation curvature, stresses due to neighboring dislocations, local

EP

fault composition, etc. Moreover, PFE data is typically derived from dislocation fine structure

AC C

locked in from high temperature, while incorporating low temperature elastic modulus (for lack of data). These result in the wide scatter in experimental PFE values. Despite this, there is a correlation between experiments and the model. Experimentally measured PFE values are typically half that predicted from the DMLF model (slope of the best linear fit is ~0.5). This is for two reasons. First, atomic relaxations can take place in the vicinity of the fault which can result in significant energy reduction, whereas in the DMLF model, lattice parameters of proximate structures were constrained to match the parent L12 lattice parameters. Indeed, as

ACCEPTED MANUSCRIPT

shown in supplementary material, while predictions from DMLF model (involving a rigid lattice) correlate strongly with computational PFE values where atomic relaxations were allowed, the relaxed PFE values were ~25% lower than that of the DMLF model. This suggests that atomic

RI PT

relaxations do contribute to the discrepancy between the DMLF model and experiments. Complete relaxation of proximate structures (lattice parameters and atomic relaxations) and accounting for strain energy may remedy the discrepancy between the model and experiments.

SC

However, the modes of relaxation in the fault and in the proximate structures can be different and an improvement in accuracy is not guaranteed. Moreover, complete relaxations are

M AN U

computationally expensive. Thus, the advantage of high-throughput is lost without a clear advantage in accuracy of prediction. The second cause for experimental values being lower than the model is that PFEs are temperature dependent. For e.g., APB(111) energy in Ni60Fe15Ge25 decreases from 202 mJ/m2 (at 77 K) to 153 mJ/m2 (at 600 K) [16]. Thus, while experimental

TE D

values typically correspond to high temperature PFEs, DMLF model predictions are for 0 K. Thus, if accurate prediction of experimental values is the objective, then high temperature direct simulation of faults with complete relaxations is the computational approach of choice, but this

EP

comes at a significant computational cost. Instead, once could obtain first-order estimates of experimental PFEs within a margin of error at insignificant computational cost by using the

AC C

DMLF model and incorporating the appropriate correction factor. The latter approach was used for high-throughput estimation of experimental PFEs for a series of Co3X, Pt3X, Ir3X and Rh3X based compounds for which experimental PFE data is currently not available. These results are shown in Figure 6b (note that these include a scaling factor of 0.5 and a margin of error). The C44/C12 values are also indicated. The DMLF model predictions are expected to be reasonably accurate for all compounds except Pt3Ga and Pt3Sn (which have a low

ACCEPTED MANUSCRIPT

C44/C12). It is noteworthy that these calculations for 7 compounds and 4 faults took only 9hrs on a 20-core machine (64 GB RAM, processor speed of 3.0 GHz). Experimental evaluation of the same data could have taken several man-years, if not man-decades. Since the aim of this work

RI PT

was not to analyze PFEs in these compounds but to demonstrate the methodology, we do not comment further on these values. It suffices to say that the absolute and relative magnitudes of these initial estimates of PFEs in conjunction with the elastic constants can be used in models for

SC

predicting strength and the propensity for yield anomaly. With these initial estimates, one can then select the most promising candidate materials for more detailed experimental or

M AN U

computational study, thus expediting alloy design and selection significantly. 6. Conclusions

A Diffuse Multi-Layer Fault model was developed that enabled rapid estimation of PFEs in A3B

TE D

compounds with L12 structure. The planar fault was assumed to be a diffuse interface. Changes to stacking and bonding environment of atoms on multiple atomic layers in the vicinity of the fault were assumed to contribute to PFE. Using a two-step SP model, the bonding environment

EP

in each layer in the vicinity of the fault was compared against that in over 1300 GCP A3B compounds with ≤16 atoms per unit cell. By this procedure, proximate structures were identified

AC C

for each atomic layer and PFE was expressed in terms of the energy difference between these proximate structures and L12. Two hitherto-unknown structures, viz. ω and χ, were found to be proximate to APB(111) and CSF(111). DFT calculations were used to estimate the energy of the proximate structures and to predict PFEs which were compared with results from direct simulations. The proposed model predicted energies of a variety of SSFs in over forty A3B compounds with a high degree of accuracy, with no fitting parameters and with a fifteen-fold computational advantage over direct simulations. The model was also extended to predict

ACCEPTED MANUSCRIPT

expected experimental PFEs in a series of Co3X, Pt3X, Ir3X and Rh3X based compounds for which this data is currently not available. The model thus enables high-throughput initial estimates of PFEs which may be used to shortlist promising candidate compounds for detailed

RI PT

subsequent studies. Acknowledgements

SC

The authors acknowledge the Department of Materials Engineering, Indian Institute of Science, for supporting this work. The authors are also grateful for the assistance provided by Nano

AC C

EP

TE D

M AN U

Mission, Department of Science of Technology, Government of India, via project DSTO 1169.

ACCEPTED MANUSCRIPT

References [1] A. Suzuki, H. Inui, T.M. Pollock, L12-Strengthened Cobalt-Base Superalloys, Annu. Rev. Mater. Res. 45 (2015) 345–368.

RI PT

[2] Y. Yamabe-Mitarai, M.-H. Hong, Y. Ro, H. Harada, Temperature dependence of the flow stress in Ir3Nb with the L12 structure, Philos. Mag. Lett. 79 (1999) 673–682. [3] Y. Yamabe-Mitarai, Y. Ro, S. Nakazawa, Temperature dependence of the flow stress of Irbased L12 intermetallics, Intermetallics. 9 (2001) 423–429. [4] M.S. Titus, Y.M. Eggeler, A. Suzuki, T.M. Pollock, Creep-induced planar defects in L12containing Co- and CoNi-base single-crystal superalloys, Acta Mater. 82 (2015) 530–539.

SC

[5] R.C. Reed, The superalloys: fundamentals and applications, Cambridge University Press, 2006.

M AN U

[6] C.T. Liu, D.P. Pope, Ni3Al and its alloys, in: J.H. Westbrook, R.L. Fleischer (Eds.), Intermetallic Compounds – Structural Applications of Intermetallic Compounds, John Wiley & Sons, Ltd, 2002, pp. 55–74. [7] T.M. Pollock, R.D. Field, Chapter 63, Dislocations and high-temperature plastic deformation of superalloy single crystals, in: F.R.N.N. and M.S. Duesbery (Ed.), Dislocations in Solids, Elsevier, 2002, pp. 547–618. [8] E. Nembach, G. Neite, Precipitation hardening of superalloys by ordered γ′-particles, Prog. Mater. Sci. 29 (1985) 177–319.

TE D

[9] G.B. Viswanathan, P.M. Sarosi, M.F. Henry, D.D. Whitis, W.W. Milligan, M.J. Mills, Investigation of creep deformation mechanisms at intermediate temperatures in René 88 DT, Acta Mater. 53 (2005) 3041–3057. [10] V. Paidar, D.P. Pope, M. Yamaguchi, Structural stability and deformation behavior of L12 ordered alloys, Scr. Metall. 15 (1981) 1029–1031.

EP

[11] V. Vitek, D.P. Pope, J.L. Bassani, Chapter 51, Anomalous yield behaviour of compounds with LI2 structure, in: F.R.N. Nabarro, M.S. Duesbery (Eds.), Dislocations in Solids, Elsevier, 1996, pp. 135–185.

AC C

[12] N. Tsuno, R.C. Reed, K. Kakehi, S. Shimabayashi, C.M.F. Rae, Tension/Compression asymmetry in yield and creep strengths of Ni-based Superalloys, in: R.C. Reed, K.A. Green, P. Caron, T. Gabb, M.G. Fahrmann, E.S. Huron, S.A. Woodard (Eds.), Superalloys 2008, Wiley, Warrendale, PA, 2008, pp. 433–442 [13] P.C.J. Gallagher, The influence of alloying, temperature, and related effects on the stacking fault energy, Metall. Trans. 1 (1970) 2429–2461. [14] D.M. Dimiduk, Dislocation structures and anomalous flow in L12 compounds, J. Phys. III. 1 (1991) 1025–1053. [15] T. Kruml, E. Conforto, B. Lo Piccolo, D. Caillard, J.L. Martin, From dislocation cores to strength and work-hardening: a study of binary Ni3Al, Acta Mater. 50 (2002) 5091–5101.

ACCEPTED MANUSCRIPT

[16] T.J. Balk, M. Kumar, K.J. Hemker, Influence of Fe substitutions on the deformation behavior and fault energies of Ni3Ge–Fe3Ge L12 intermetallic alloys, Acta Mater. 49 (2001) 1725–1736. [17] B.L. Lü, G.Q. Chen, S. Qu, H. Su, W.L. Zhou, First-principle calculation of yield stress anomaly of Ni3Al-based alloys, Mater. Sci. Eng. A. 565 (2013) 317–320.

RI PT

[18] E. Conforto, G. Molénat, D. Caillard, Comparison of Ni-based alloys with extreme values of antiphase boundary energies: dislocation mechanisms and mechanical properties, Philos. Mag. 85 (2005) 117–137.

SC

[19] V. Paidar, V. Vitek, Stacking-Fault-Type Interfaces and their Role in Deformation, in: J.H. Westbrook, R.L. Fleischer (Eds.), Intermet. Compd. - Princ. Pract., John Wiley & Sons, Ltd, Chichester, UK, 2002, pp. 437–467. [20] R.P. Reed, R.E. Schramm, Relationship between stacking‐fault energy and x‐ray measurements of stacking‐fault probability and microstrain, J. Appl. Phys. 45 (1974) 4705–4711.

M AN U

[21] H.P. Karnthaler, E.T. Mühlbacher, C. Rentenberger, The influence of the fault energies on the anomalous mechanical behaviour of Ni3Al alloys, Acta Mater. 44 (1996) 547–560. [22] D.J. Crudden, A. Mottura, N. Warnken, B. Raeisinia, R.C. Reed, Modelling of the influence of alloy composition on flow stress in high-strength nickel-based superalloys, Acta Mater. 75 (2014) 356–370. [23] J.M. MacLaren, A. Gonis, G. Schadler, First-principles calculation of stacking-fault energies in substitutionally disordered alloys, Phys. Rev. B. 45 (1992) 14392–14395.

TE D

[24] National Institute of Standards and Technology, U. S. Department of Commerce, Interatomic Potentials Repository Project, http://www.ctcms.nist.gov/potentials/ (accessed April 26, 2017).

EP

[25] K.V. Vamsi, S. Karthikeyan, Effect of Off-Stoichiometry and Ternary Additions on Planar Fault Energies in Ni3Al, in: E.S. Huron, R.C. Reed, M.C. Hardy, M.J. Mills, R.E. Montero, P.D. Portella, J. Telesman (Eds.), The Minerals, Metals and Materials Society (TMS), Warrendale, PA, 2012, pp. 521–530.

AC C

[26] M. Chandran, S.K. Sondhi, First-principle calculation of stacking fault energies in Ni and Ni-Co alloy, J. Appl. Phys. 109 (2011) 103525-103525–6. [27] G.B. Olson, M. Cohen, A general mechanism of martensitic nucleation: Part I. General concepts and the FCC → HCP transformation, Metall. Trans. A. 7 (1976) 1897–1904. [28] S. Curtze, V.-T. Kuokkala, A. Oikari, J. Talonen, H. Hänninen, Thermodynamic modeling of the stacking fault energy of austenitic steels, Acta Mater. 59 (2011) 1068–1076. [29] L. Vitos, J.-O. Nilsson, B. Johansson, Alloying effects on the stacking fault energy in austenitic stainless steels from first-principles theory, Acta Mater. 54 (2006) 3821–3826. [30] C. Amador, J.J. Hoyt, B.C. Chakoumakos, D. de Fontaine, Theoretical and experimental study of relaxations in A13Ti and Al3Zr ordered phases, Phys. Rev. Lett. 74 (1995) 4955– 4958.

ACCEPTED MANUSCRIPT

[31] A. Mottura, A. Janotti, T.M. Pollock, A first-principles study of the effect of Ta on the superlattice intrinsic stacking fault energy of L12-Co3(Al,W), Intermetallics. 28 (2012) 138–143. [32] P.J.H. Denteneer, W. van Haeringen, Stacking-fault energies in semiconductors from firstprinciples calculations, J. Phys. C Solid State Phys. 20 (1987) L883–L887.

RI PT

[33] S. Sandlöbes, M. Friák, S. Zaefferer, A. Dick, S. Yi, D. Letzig, Z. Pei, L.-F. Zhu, J. Neugebauer, D. Raabe, The relation between ductility and stacking fault energies in Mg and Mg–Y alloys, Acta Mater. 60 (2012) 3011–3021. [34] L.A. Girifalco, V.G. Weizer, Application of the Morse potential function to cubic metals, Phys. Rev. 114 (1959) 687–690.

SC

[35] J. Hafner, Materials simulations using VASP—a quantum perspective to materials science, Comput. Phys. Commun. 177 (2007) 6–13.

M AN U

[36] G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B. 54 (1996) 11169–11186. [37] MedeA version 2.6.6, Materials Design Inc, Angel Fire, NM, USA, 2009. [38] M. Methfessel, A.T. Paxton, High-precision sampling for Brillouin-zone integration in metals, Phys. Rev. B. 40 (1989) 3616–3621. [39] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient Approximation made simple, Phys. Rev. Lett. 77 (1996) 3865–3868.

TE D

[40] J J.P. Perdew, K. Burke, M. Ernzerhof, Generalized Gradient Approximation made simple [Phys. Rev. Lett. 77, 3865 (1996)], Phys. Rev. Lett. 78 (1997) 1396–1396. [41] K.V. Vamsi, S. Karthikeyan, Modelling ternary effects on antiphase boundary energy of Ni3Al, MATEC Web Conf. 14 (2014) 11005-11010. [42] Y. Le Page, P. Saxe, Symmetry-general least-squares extraction of elastic data for strained materials from ab initio calculations of stress, Phys. Rev. B. 65 (2002) 104104-104114.

EP

[43] A.T. Paxton, Y.Q. Sun, The role of planar fault energy in the yield anomaly in L12 intermetallics, Philos. Mag. A. 78 (1998) 85–104.

AC C

[44] K.V. Vamsi, Planar Fault Energies in L12 Compounds, PhD Thesis, Indian Institute of Science, Bangalore, India, 2017. [45] J.M. Blakely, Introduction to the Properties of Crystal Surfaces: International Series on Materials Science and Technology, Elsevier, 2013. [46] S.M.L. Sastry, B. Ramaswami, Fault energies in ordered and disordered Cu3Au, Philos. Mag. 33 (1976) 375–380. [47] L.M. Howe, M. Rainville, E.M. Schulson, Transmission electron microscopy investigations of ordered Zr3Al, J. Nucl. Mater. 50 (1974) 139–154. [48] B. Tounsia, P. Beauchamp, Y. Mishima, T. Suzuki, P. Veyssière, The deformation microstructure in Ni3Si polycrystals strained over the range of temperature of flow stress anomaly, in: MRS Proc., Cambridge Univ Press, (1988) 731-736

ACCEPTED MANUSCRIPT

[49] N.L. Okamoto, Y. Hasegawa, W. Hashimoto, H. Inui, Plastic deformation of single crystals of Pt3Al with the L12 structure, Philos. Mag. 93 (2013) 60–81. [50] K. Suzuki, M. Ichihara, S. Takeuchi, Dissociated structure of superlattice dislocations in Ni3Ga with the L12 structure, Acta Metall. 27 (1979) 193–200.

RI PT

[51] J. Douin, A dissociation transition in Zr3Al deformed at room temperature, Philos. Mag. Lett. 63 (1991) 109–116. [52] J. Douin, P. Veyssière, P. Beauchamp, Dislocation line stability in Ni3Al, Philos. Mag. A. 54 (1986) 375–393.

AC C

EP

TE D

M AN U

SC

[53] P. Veyssière, Properties of surface defects in Intermetallics, in: C.T. Liu, R.W. Cahn, G. Sauthoff (Eds.), Ordered Intermetallics — Physical Metallurgy and Mechanical Behaviour, Springer Netherlands, 1992, pp. 165–175.

ACCEPTED MANUSCRIPT

Figure captions Figure 1. (a) Schematic of the L12 crystal structure with the close packed plane indicated in

RI PT

gray. Projections of atoms on (b) the close packed (111) plane, and (c) the cube plane (010), are indicated. Circles, squares and triangles represent atoms on parallel planes. Also indicated are the fault vectors bSISF(111) = ( #1%21%', bCSF(111)= * #12%1', and bAPB(111) = bAPB(010) = " #1%01'. !

!

!

SC

Figure 2. Projection of atoms on the 1%01) plane for (a) an SISF(111) with stacking violations, and (b) an APB(111) with nearest neighbor chemical violations (highlighted). Also indicated is

M AN U

the fault plane, the adjacent atomic layers and their nomenclature. Similar projection for other faults are provided in the supplementary material.

Figure 3. (a) The unit cells of the most important proximate GCP structures corresponding to L1

TE D

layer of each fault, and (b) PFEs predicted by DMLF model compared against results from direct simulations for a variety of fcc metals and L12 compounds. The six panels correspond to ISF(111), APB(010), SISF(111), CSF(111) SESF(111) and APB(111). The fitness of the model

EP

as estimated by R2, slope of best linear fit, intercept of best linear fit (in mJm-2) and RMS of residuals (in mJm-2) are also indicated. Details of data labels can be found in Table 4.

AC C

Figure 4. PFEs predicted by DMLF model compared against results from direct simulations for (a) stable L12 compounds, (b) Co3X compounds and (c) Al3X compounds. Details of data labels can be found in Table 4. The data used in this figure is tabulated in supplementary material.

ACCEPTED MANUSCRIPT

Figure 5. (a) C44/C12 ratio and residual between DMLF model and direct simulation indicated for several Ni3X compounds. (b) Line profile (along the fault plane normal) of the plane-averaged electron density difference (solid curves) between the faulted (APB(111)) and perfect (L12)

RI PT

structures for Ni3Al(black) and Ni3Ta (blue). The electron density difference is averaged over a plane parallel to the fault and at a distance from the fault indicated in terms of layers. The bars represent the volume-averaged electron density difference for Ni3Al (gray) and Ni3Ta (purple).

M AN U

difference in the volume enclosed by L0.5 to L1.5.

SC

For e.g., the magnitude of the bar spanning L0.5 to L1.5 represents the average electron density

Figure 6. (a) A comparison of predictions from DMFL model versus experimental values for a variety of faults in various L12 compounds. References for experimental data provided in text. (b) Expected experimental PFE values obtained from DMLF model (using slope and error band derived from Fig 6a) for Co3Al0.5W0.5, Pt3X, (X=Ga, Sn), Rh3X (Ti, Nb, Ta) and Ir3X (Hf, Zr).

TE D

Also indicated in parenthesis below each compound is the C44/C12 ratio to serve as a measure of

AC C

EP

the reliability of these predictions.

ACCEPTED MANUSCRIPT

Table captions Table 1. A summary of the coordination and relative weights assigned to bonds at different

RI PT

nearest neighbour (NN) distances in various classes of GCP structures Table 2. Proximate structures for different layers of each fault and the relevant expressions for PFEs (i.e., specific forms of Eq (3)). Refer to supplementary material for details on which layers

SC

are present in which fault. Also note that for all faults, beyond the second layer (i.e., Bƒ|"| ), the

M AN U

most proximate structure is the parent phase, L12 or fcc as the case may be.

Table 3. Crystallographic details of the relevant proximate structures identified in this study. Dimensions and axial angles correspond to the simulation supercell (also shown in Figure 3(a)) which may correspond to conventional or primitive unit cells in some cases.

TE D

Table 4. List of metals and compounds for which validity of the DMLF model was tested. Data

AC C

EP

labels used in Figures 3 and 4 are indicated in parenthesis.

ACCEPTED MANUSCRIPT Table 1. A summary of the coordination and relative weights assigned to bonds at different nearest neighbour (NN) distances in various classes of GCP structures Nearest neighbor

Relative

Coordination

distance

weights,

Ordered fcc

Ordered hcp

Ordered dhcp

 at

of ‘a’, lattice

(e.g., L12)

(e.g., D019)

(e.g., D024)

different NN

parameter of L12)

|ABC|

|AB|

|ABAC|

distances

1

1/√2

12

12

12

0.386

2

1

6

6

6

0.263

3

2/√3

0

2

1

0.133

4

3/2

24

18

21

0.095

5

11/6

0

12

6

0.051

6

√2

12

6

9

0.038

7

5/2

24

12

18

0.016

8

17/6

0

12

6

0.010

9

√3

8

6

6

0.008

SC

M AN U

TE D EP AC C

j

RI PT

(distance in terms

ACCEPTED MANUSCRIPT Table 2. Proximate structures for different layers of each fault and the relevant expressions for PFEs (i.e., specific forms of Eq (3)). Refer to supplementary material for details on which layers are present in which fault. Also note that for all faults, beyond the second layer (i.e.,

|| ), the most proximate structure is the parent phase, L12 or fcc as the case may be.



±.

±

±.

±

DMLF model for fault energy

APB(010)

-

D023

-

D023

-

4() ∆( ,! ) "

APB(111)

-

-

ω

-

L12

2() ∆(#,! ) "

CSF(111)

-

-

χ

-

D024

2() ∆($,! ) + ∆(& ,! ) "

SISF(111)

-

-

D024

-

D024

4() ∆(& ,! ) "

SESF(111)

D019

-

L12

-

D024

() 2∆(& ,!) + ∆('( ,! ) "

ISF(111)

-

-

dhcp

-

dhcp

SC

RI PT

Fault

AC C

EP

TE D

M AN U

4() ∆()*+,,-++) "

ACCEPTED MANUSCRIPT Table 3. Crystallographic details of the relevant proximate structures identified in this study. Dimensions and axial angles correspond to the simulation supercell (also shown in Figure 3(a)) which may correspond to conventional or primitive unit cells in some cases.

(., /, 0)

(1, 2, 3)

(in units of lattice parameter of L12)

(in degrees) (90,90, 90)

Space group (45365)

(1,1,1)

D019

(469 /550)

√2, √2, 2/√3"

D023

(:4/555)

(1,1,4)

D024

(469 /550)

√2, √2, 4/√3"

ω

(4/555)

√2, √2, 2"

χ

(;2/5)

√2, √2, 4/√3"

(90,90, 120) (90,90, 90)

(90,90, 120)

AC C

EP

TE D

M AN U

L12

RI PT

Supercell axial angles

SC

Structure

Supercell dimensions

(90,90, 90)

(90,90, 120)

ACCEPTED MANUSCRIPT Table 4. List of metals and compounds for which validity of the DMLF model was tested. Data labels used in Figures 3 and 4 are indicated in parenthesis. Ni3X

Other stable L12

Co3X

Al3X

in Fig

compounds

compounds

compounds

compounds

3a

(Label in Fig 3b-f)

(Label in Fig

(Label in Fig

(Label in Fig

4a)

4b)

4c)

Al

Ni3Al (Al)

Ni3Si (Si)

Cu3Au (1)

Co3V (7)

Al3Li (17)

Ag

Ni3Ga (Ga)

Ni3V (Si)

Pt3Al (2)

Co3Nb (8)

Al3Ti (18)

Au

Ni3Ge (Ge)

Ni3Sn (Sn)

Al3Sc (3)

Co3Mo (9)

Al3Zr (19)

Cu

Ni3Ti (Ti)

Ni3Pt (Pt)

Zr3Al (4)

Co3W (10)

Al3Hf (20)

Co

Ni3Ta (Ta)

Ni3Y (Y)

Co3Ta (5)

Co3Co (11)

Al3Ta (21)

Mg

Ni3Nb (Nb)

Ni3Sc (Sc)

Co3Ti (6)

Co3Ni (12)

Al3Nb (22)

Ni

Ni3W (W)

Ni3Re (Re)

SC

RI PT

Metals

Pt

Ni3Mo (Mo) Ni3Ru (Ru)

M AN U

Co3Al (13)

Co3Hf (14) Co3Zr (15)

Zn

AC C

EP

TE D

Co3Re (16)

ACCEPTED MANUSCRIPT

Figure 3b -200

-100

0

100

200

300

-800

ISF(111) ) 2

Slope = 0.99

Intercept = 7

Intercept = 7

Adj. R

2

Adj. R

= 0.96

S i G e

400

= 0.99 Sc Sn Y

RMSE= 48 Al

100

Al GaTi N P ti

Au Cu

Ru

Ag

0

Mg Zn Co

L =

dhcp

L =

dhcp

1

Mo Re

Ta

2

)

Slope = 0.98

2

Adj. R

S G ie Sn Sc Y Pt

0

Al Ga Ti Ru V

L = D0

W Nb

1

L = D0

Ta

2

AC C = 0.96

400

= 0.98

Al Pt Ga V

Ta Nb

-400

L = L = D0 2

APB(111) SG i e

Slope = 1.17

24

Sn Y Sc

Adj. R

2

800

Ti

Intercept = -91

400

= 0.97

Al Ga

RMSE= 128 Pt

0

Ni V

Ti Ru

-400

N b Ta

Re Mo

L = D0

W

0

Nb

L = L1 1

Ta

L = D0 2

-400

-800

1

V

-800

Ru

24

Sc Y S GA aln Pt

Ru

-800

0

Re

24

S ie G

RMSE= 81

-400

800

W

EP

) 2

Intercept = 41 Adj. R

SnY

Mo

SESF(111) Slope = 1.02

23

Si Sc TiGe

RMSE= 117

TE

Re Mo

800

2

D

-400

2

= D0

1.5

-800 23

Intercept = 42

= 0.98

RMSE= 65

-800

L

Slope = 0.86

Intercept = -11 Adj. R

= D0

CSF(111)

800

M AN U

SISF(111)

L

0.5

Nb W

-200

0

-400

V

-100

400

0

RI PT

Direct simulations (mJ/m

2

Ni

400

800

800

Slope = 1.01

2

Direct simulations (mJ/m

400

Pt

RMSE= 24

Direct simulations (mJ/m

0

APB(010)

300

200

-400

SC

(b)

0

400

Predictions fromDMLF model (mJ/m

Mo 19

L = L = L1 2

24

800 2

-800

)

-800

1

R e W

2

-400

0

400

Predictions fromDMLF model (mJ/m

2

800

2

)

AC C EP TE D

SC

M AN U

RI PT

AC C EP TE D

SC

M AN U

RI PT

AC C

EP

TE

D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Figure 04

AC C

EP

TE

D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

1200

900

(b)

5

2

5

1000

3

800

600

3

6 6

2

400 6

4

4

5

3

2

2 2

3

200

6

4

SISF(111)

5 1

APB(010)

1 1

0

8

900

8 15

600

7

14

14

9

9 8 14

6 6 5

6

300 6

0

14

5 15

10 13 8 7 9 11 12

7

13

11 11 13 10 16

-300

9

SISF(111)

16

APB(010) APB(111)

16

CSF(111)

(c)

3

5

10 7 10

APB(111)

4

1

0

Direct simulation (mJ/m )

2

Direct simulation (mJ/m )

5

Direct simulation (mJ/m )

(a)

2

1200

3

600 19 318

20

300

3

19

20

19

18

20 18 22 22 19 22 21 17 20 17 18 21 17 17

0

-300

21

SISF(111) APB(010)

22 21

APB(111)

CSF(111)

CSF(111)

-600 200

400

600

800

1000

1200 2

Predictions from DMLF model (mJ/m )

-600

-300

0

300

600

900

1200 2

Predictions from DMLF model (mJ/m )

-300

0

300

600

900 2

Predictions from DMLF model (mJ/m )

Figure 05

Electron

AC C

3

density difference (e/A )

EP

TE

D

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

Ni Al 3

Ni Ta

Ni Ta

3

3

L

-5

L

-4

L

-3

L

-2

L

-1

L

1

L

2

L

3

L

4

L

5

Figure 06

ACCEPTED MANUSCRIPT (a)

Experimental values (mJ/m )

SISF(111) 2

APB(010) APB(111)

M AN U

SC

RI PT

CSF(111)

2

D

Predictions from DMLF model (mJ/m )

600

TE

SISF(111)

2

Planar fault energies (mJ/m )

(b)

APB(010)

500

EP

APB(111) CSF(111)

300

200

AC C

400

100

0

p

Com ound

(C /C ) 44

12

Co Al 3

0.5

W

(0.942)

Pt Ga

Pt Sn

Rh Ti

Rh Nb

Rh Ta

Ir Hf

Ir Zr

(0.532)

(0.408)

(0.973)

(1.014)

(0.984)

(0.928)

(0.958)

0.5

3

3

3

3

3

3

3