TiAl composite

Journal of Alloys and Compounds 745 (2018) 63e74

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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom

Development and application of a ternary Ti-Al-N interatomic potential for Ti2AlN/TiAl composite Pei Liu, Xiuli Han*, Dongli Sun**, Qing Wang School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, People's Republic of China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 November 2017 Received in revised form 12 February 2018 Accepted 13 February 2018

Ti2AlN/TiAl composite shows significant promise for using in high temperature structural material due to its excellent mechanical properties. However, an atomic scale understanding of the deformation and failure mechanisms under high temperature is still not well understand, this is mainly due to the lack of interatomic potential that accurately describe the interactions between Ti, Al and N atoms. To address this challenge, an interatomic potential of the Ti-Al-N ternary system has been developed on the basis of second-nearest-neighbor modified embedded atom method (2NN MEAM) formalism. Our newly developed 2NN MEAM potential accurately reproduces the structure, elastic, thermodynamics and surface properties of TiAl and Ti2AlN compounds. Through molecular dynamics simulations using the developed potential, the atomic scale mechanisms underlying uniaxial tensile fracture of TiAl, Ti2AlN and Ti2AlN/TiAl composite are also reproduced in good agreement with experiments. These simulations indicate that both single crystal TiAl and Ti2AlN undergoes brittle fracture at low temperature and shows ductile fracture at elevated temperature. When the tensile temperature is 300 K, the fracture behavior of Ti2AlN/TiAl composite is only appeared in the TiAl side, the Ti2AlN side and the interface region remain stable. But when the tensile temperature is 1200 K, the coherent interface becomes unstable and could act as the site for the dislocations nucleation. The interatomic potential for Ti-Al-N ternary system developed in this work could be utilized to further investigate atomic scale mechanisms underlying the response of Ti2AlN/TiAl composite to the other external stimuli under high temperature, such as shear, compression and wear etc. © 2018 Elsevier B.V. All rights reserved.

Keywords: Ti-Al-N ternary system 2NN MEAM Molecular dynamics simulation Uniaxial tensile fracture

1. Introduction TiAl matrix composites have achieved enormous interest globally in recent years due to their lightweight, high specific stiffness and strength, good high-temperature strength and oxidation resistance [1e4]. Among various suitable reinforcements for TiAl matrix composite, Ti2AlN as a layered ternary compounds shows a significant role in reinforcing and toughening of TiAl matrix composites. Due to the combination of strong covalent Ti-N bonds and

* Corresponding author. School of Materials Science and Engineering, Harbin Institute of Technology, 92 West Dazhi Street, Nan Gang District, Harbin 150001, People's Republic of China. ** Corresponding author.School of Materials Science and Engineering, Harbin Institute of Technology, 92 West Dazhi Street, Nan Gang District, Harbin 150001, People's Republic of China. E-mail addresses: [email protected] (X. Han), [email protected] (D. Sun). https://doi.org/10.1016/j.jallcom.2018.02.168 0925-8388/© 2018 Elsevier B.V. All rights reserved.

weak metallic Ti-Al bonds, Ti2AlN phase exhibits interesting combinations of ceramic and metallic properties. Moreover, its layered nature suggests Ti2AlN may has good solid-lubricant qualities [5e8]. Therefore, the Ti2AlN/TiAl composite is expected to acquire excellent mechanical properties, wear-resisting and antifrictional performance, and is very promising to be a new type of structural material served under the harsh conditions, such as high temperature and high wear in fields of aerospace and automotive application. In recent years, many experimental studies [9e14] have been performed on Ti2AlN/TiAl composite with the objective of providing the fundamental knowledge essential to the development of this new composite. However, an atomic level comprehension of the deformation and failure behavior of Ti2AlN/TiAl composite under various service conditions is still lacking, as well as the mechanisms responsible for the deformation and failure process, which is important to prevent the composite degradation.

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P. Liu et al. / Journal of Alloys and Compounds 745 (2018) 63e74

With the exponential growth in computing power, atomic simulations, such as molecular dynamics (MD) simulations, are now emerging as a viable approach to predict the mechanical properties of materials. Particularly for the investigation of failure mechanisms, MD simulations have significant advantages due to the capability of providing the atomic level structural information and dynamic atomic behaviors. The reliability and accuracy of MD calculations, however, relies strongly on the quality of the interatomic potentials employed. The interatomic potentials should reproduce various fundamental physical properties of relevant materials systems correctly. An atomistic simulation to investigate the deformation and failure behavior of Ti2AlN/TiAl composite would need an interatomic potential which can simultaneously reproduce the structural, thermodynamic, elastic and surface properties of TiAl and Ti2AlN, as it is closely relate to the deformation and failure mechanism of Ti2AlN/TiAl composite [15,16]. So far, many sets of interatomic potentials are already available in literature for the Ti-Al binary system [17e19], however, to the knowledge of the present authors, a ternary Ti-Al-N interatomic potential that can simultaneously reproduce the properties of TiAl and Ti2AlN is not available up to now. This is mainly because the Ti2AlN/TiAl composite is a multi-component system and that the constituent elements Ti and Al are very different from constituent element N, development of a (semi-)empirical interatomic potential that can deal with all the different elements, Ti, Al and N, and alloy systems composed of those elements using a common mathematical formalism is extremely challenging. From this point of view, the second nearest neighbor (2NN) MEAM interatomic potential [20] may be said to be highly applicable to multi-component systems, because it can describe a wide range of elements (fcc, bcc, hcp, diamondstructured and even gaseous elements) using a common mathematical formalism. The 2NN MEAM was created by modifying the MEAM [21] to further consider partially second-nearest neighbor atom interactions and to remove some critical shortcomings in the original MEAM. As a part of a long-term project to achieve the ultimate goal of elaborately designing Ti2AlN/TiAl composite and effectively controlling the materials behavior of this composite under harsh condition, the purpose of the present work is to develop a 2NN MEAM potential that successfully captures structure, elastic, surface and thermodynamics properties of TiAl and Ti2AlN. In addition, uniaxial tension were also carried out to confirm that the mechanical properties of single crystal TiAl, single crystal Ti2AlN and Ti2AlN/TiAl composite can be reliably predicted through molecular dynamics (MD) simulation using the developed potential. This paper is organized as follows: Section 2 describes the 2NN MEAM functional forms as well as the construction and simulation of target properties. In Section 3 the optimal values for all the needed interatomic parameters are reported, and the reliability of the developed potentials are examined. Finally, all results are summarized in Section 4. 2. Calculation method

2 3 X X   1 4Fi ðri Þ þ E¼ S f R 5 2 jsi ij ij ij i

(1)

Fi is the embedding function, ri is the background electron density at site i, Sij is the screening function, and fij ðRij Þ is the pair interaction between atom i and j separated by a distance Rij. The embedding function is given as:

!

r r ln 0 r0 r

FðrÞ ¼ AEc

! (2)

where A is an adjustable parameter, Ec is the cohesion energy, and

r0 is the background electron density for a reference structure. The background electron density ri is composed of a spherically symð0Þ

metric partial electron density ri

rið2Þ

ð1Þ

and angular contributions ri ,

rð3Þ , i

and representing the contributions of s, p, d and f atomic electron densities. The partial electron densities can be combined in the following form:

G¼

3 X

" ðhÞ

ti

h¼1

riðhÞ

#2

rið0Þ

(3)

ðhÞ

where ti are adjustable parameters. The total electron density at site i is evaluated as:

ri ¼ rið0Þ GðGÞ GðGÞ ¼

2 1 þ eG

(4)

(5)

In the 2NN MEAM no specific functional expression is given directly to fij ðRij Þ. Instead, the atomic energy (total energy per atom) is evaluated by some means as a function of nearestneighbor distance. Then the value of fij ðRij Þ can be computed from the known values of the total energy and the embedding energy. The total energy per atom is obtained from the universal state equation as a function of R:

  * Еu ðRÞ ¼ Ec 1 þ a* þ da*3 ea

(6)

where d is an adjustable parameter, a* ¼ aðR=re  1Þ, re is the first neighbor distance, a ¼ ð9BUEc Þ1=2 , B is the bulk modulus and U is the equilibrium atomic volume. In the 2NN MEAM, the second nearest-neighbor interactions are considered and the total energy per atom in a reference structure can be written as follows:

2.1. 2NN MEAM interatomic potential model

  Z Z S Еu ðRÞ ¼ F r0 ðRÞ þ 1 fðRÞ þ 2 fðaRÞ 2 2

The 2NN MEAM is a semi-empirical interatomic potential that incorporates angular dependency of electron density into the embedded-atom method and consider partially second-nearest neighbor atom interactions. The complete set of mathematical formalisms of the 2NN MEAM are fully described in literature [20]. In this section, only the main aspects of the model are described. In the 2NN MEAM, the total energy of a multicomponent system is approximated as:

where Z1 and Z2 are the number of first and second nearestneighbor atoms, respectively. a is the ratio between the second and first nearest-neighbor distances. S is the screening factor for second nearest-neighbor interactions (the screening factor for first nearest-neighbor interactions is 1). The screening factor S represents the influence of the neighbor atoms k in the interaction between i and j. It is possible to calculate a C factor of each neighbor atom k:

(7)

P. Liu et al. / Journal of Alloys and Compounds 745 (2018) 63e74

R2ij 1 x þ y2 ¼ c 4 2

(8)

where x, y are the coordinates of k with respect to the ellipse defined by the positions i, j, k. The screening of the k atom varies gradually in the range Cmin < C < Cmax. If C < Cmin the screening is total (S ¼ 0), while if C > Cmax the interaction is independent of k (S ¼ 1). The 2NN MEAM potential formalism gives 14 independent parameters for pure element: four of these parameters [the cohesive energy (Ec), the equilibrium nearest-neighbor distance (re), the exponential decay factor (a) of the reference structure and the adjustable parameter (d)] are related to the universal equation of state. seven of these parameters [ the decay length (b(0), b(1), b(2), b(3)) and the weight factors (t(1), t(2), t(3))] are related to the electron density, one parameter (A) for the embedding function, and two parameters (Cmin, Cmax) are responsible for the many-body screening. To extend the 2NN MEAM formalism to binary system, it is necessary to determine the interactions between pairs of dissimilar elements. For this, it is necessary to define a binary reference structure, and use a similar method to that used to compute pair interaction in a unary system. Thirteen independent model parameters are necessary to describe a binary system using 2NN MEAM potential formalism, four of these parameters (Ec, re, B and d) are related to the universal equation of state of reference structure, eight of these parameters (four Cmin and four Cmax) are responsible for many-body screening, and the ratio between atomic electron density scaling factor (r0 ) belongs to the electron density. The 2NN MEAM potential parameter set for a ternary system is obtained by combining all sub-unary and binary parameters. In addition, the 2NN MEAM potential formalism requires six more parameters: Cmin (iekej), Cmin (iejek), Cmin (jeiek), Cmax (iekej), Cmax (iejek), and Cmax (jeiek). Those parameters represent the degree of screening by a third element atom (k) to the interaction between two neighboring atoms (i and j). 2.2. Target properties and MD simulation In order to fit the 2NN MEAM potential parameters and test the reliability of potential parameters, it is imperative to construct a database of various properties related to L10-TiAl and Ti2AlN. As previously mentioned, the task of this paper is to develop a 2NN MEAM potential for the Ti-Al-N ternary system which will be suitable for furthering the understanding of the deformation and failure of Ti2AlN/TiAl composite under extreme condition. In this case, the accurate reproductions of the structural, elastic, thermodynamic and surface properties of L10-TiAl and Ti2AlN are most critical. Thus, corresponding physical target properties related to L10-TiAl and Ti2AlN are the lattice constants, heat of formation, elastic constants, surface energies and thermal expansion coefficients. The lattice constants, heat of formation and elastic constants are used to fit the potential parameters, while the surface energies and thermal expansion coefficients are used to confirm the transferability of the potential parameters. All the target properties related to L10-TiAl and Ti2AlN are DFT calculation values or experimental values which obtained by the previous research. The MD simulation was performed by using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) MD code [22]. A 6  6  6 periodic cell is used for MD simulations to obtain the bulk properties of the material using the developed potential. The cohesive energy and lattice structure for a given set of 2NN MEAM parameters were obtained by energy minimization using the conjugate gradient algorithm. To calculate the elastic

65

constants, we performed zero static internal and external stress approximations, in which the size and shape of the system varied during the iterations such that the final configuration reached a local potential energy minimum. In terms of surface energy, the supercell model was built along the normal direction of the selected surface and the total energy of supercell was obtained and marked as Ebulk, then the supercell was stretched along the Z axis to create the free surface, the total energy of supercell contained free surface is obtained and marked as Eslab, then the surface energy can be calculated as follows:

Esurf ¼

Eslab  Ebulk 2A

(9)

Thermal expansion was predicted using MD simulation in the NPT (constant number of atoms, pressure and temperature) ensemble with a time step of 1 fs. Pressure and temperature control were maintained using a Nose-Hoover barostat/thermostat with damping constants of 10 ps and 0.1 ps, respectively. The pressure was held 1 atm and the temperature was varied between 300 and 1300 K. The system was allowed to equilibrate for 2 million timesteps at each temperature, with properties averaged over the last 500000 timesteps. Then the linear thermal expansion coefficients of the simulation samples were calculated by the following formula:

a¼

1 dL L0 dT

(10)

where a is the linear thermal expansion coefficient. L0 is the average lattice parameter of the materials at 300 K (initial temdL is the slope of the change in average lattice parameter perature). dT with temperature.

3. Results 3.1. Determination of 2NN MEAM parameters As described in Section 2.1, in order to complete the ternary TiAl-N potential parameter set, fourteen independent parameters for each pure element (Ti, Al, N) are required, thirteen independent parameters are necessary to describe each binary system (Ti-Al, TiN, Al-N), as well as six more ternary potential parameters for the ternary system. In the present work, the 2NN MEAM parameters for pure Ti and Al were taken from Chen et al. [18] and pure N from Lee et al. [23] without any modification, as shown in Table 1. Therefore, a total of forty-five parameters (13 binary 2NN MEAM parameters each for Ti-Al, Ti-N, Al-N binary system; and 6 ternary 2NN MEAM parameters for Ti-Al-N ternary system) were optimized in this study. We first optimize the 2NN MEAM parameters for the Ti-Al, Ti-N, Al-N binary systems. TiAl (BCC_B2), TiN (FCC_B1) and AlN (FCC_B1) were selected as the reference structures. In the 2NN MEAM formalism for binary system, there are certain parameters for the atomic interactions, namely, Ec, re, and a that hold direct relationship with the physical properties of the reference structure, as explained in the previous section. We derived the values of these parameters directly from density functional theory calculations of material properties of the reference structures, that is, cohesive energy, equilibrium nearest-neighbor distance, and bulk modulus. Then the remaining 2NN MEAM parameters for the binary systems and those for the Ti-Al-N ternary system are optimized by fitting the properties containing (a) lattice parameters, (b) heat of formation, (c) elastic constants of L10-TiAl and Ti2AlN. All the target properties related to TiAl and Ti2AlN are DFT calculation values or

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P. Liu et al. / Journal of Alloys and Compounds 745 (2018) 63e74

Table 1 2NN MEAM potential parameter sets for pure Ti, Al and N. The units of the cohesive energy Ec, equilibrium nearest-neighbor distance re are eV and Å, respectively. All other parameters are dimensionless. The reference structure for Ti, Al and N is hcp, fcc and dimer, respectively. Element

Ec

re

a

A

b(0)

b(1)

b(2)

b(3)

t(1)

t(2)

t(3)

Cmin

Cmax

d

Ti [18] Al [18] N [23]

4.87 3.36 4.88

2.92 2.86 1.10

4.63 4.61 5.96

1.17 1.10 1.80

1.32 1.26 2.75

0.0 4.35 4.0

1.95 7.00 4.0

5.0 2.20 4.0

5.3 0.34 0.05

14.1 1.69 1

5.0 8.30 0.00

1 0.49 2

1.44 2.8 2.8

0 0.05 0

experimental values which obtained by the previous research. The finally selected potential parameters are listed in Table 2. It is worth noting that the 2NN MEAM formalism includes up to the second nearest-neighbor interactions, and thus the radial cutoff distance during atomistic simulations should be at least larger than the second nearest-neighbor distance in the structures under consideration. After meeting the above condition, a relatively small radial cutoff distance should be applied to reduce the calculation time. In the present study, a value of 4.2 Å was chosen as the radial cutoff distance, which is larger than the second nearest-neighbor distance and less than the third nearest-neighbor distance of L10TiAl and Ti2AlN. 3.2. Performance of newly developed 2NN MEAM To obtain reasonable insight from the results of atomistic simulations, the interatomic potential should be accurate and wellverified for the system under consideration. We calculate the fundamental physical properties (structural, elastic, surface, and thermodynamics) of the TiAl and Ti2AlN using the developed potentials and compare them with relevant experimental data, DFT calculations and/or other (semi-)empirical calculations to evaluate the quality of the present developed 2NN MEAM potential parameters. Table 3 compares the 2NN MEAM predicted lattice parameters and heat of formation (DHf) for L10-TiAl and Ti2AlN with the values obtained from DFT calculations, experimental data and other (semi)empirical calculations. It is seen that the present 2NN MEAM potential parameters overestimate slightly the lattice parameters of L10-TiAl and Ti2AlN compared to the DFT calculations, experimental data and other (semi-)empirical calculations. The present potential predicted lattice parameters for L10-TiAl is within ~3% of both DFT values, experiments as well as other (semi-)empirical calculations.

Table 2 2NN MEAM potential parameter sets for Ti-Al, Ti-N, Al-N pairs and Ti-Al-N triplet. i-j pair

Ti-Al

Ti-N

Al-N

Reference state Ec (eV) re (Å)

BCC_B2 4.40 2.80 4.577 0.025 0.30 0.01 0.46 0.72 2.80 2.17 1.44 2.80 0.66 Ti-Al-N 0.13 0.96 0.96 2.80 2.06 2.06

FCC_B1 6.615 2.12 5.092 0 0.01 0.01 0.81 1.46 2.80 2.80 1.44 2.80 18

FCC_B1 7.27 2.04 4.043 0.025 0.1 0.1 1.12 1.12 2.80 2.80 2.80 2.80 27.27

a

d Cmin (ieiej) Cmin (jejei) Cmin (iejei) Cmin (iejej) Cmax (ieiej) Cmax (jejei) Cmax (iejei) Cmax (iejej) r0(j)/r0(i) iejek triplet Cmin (iejek) Cmin (iekej) Cmin (jekei) Cmax (iejek) Cmax (iekej) Cmax (jekei)

The present potential predicted lattice parameters for Ti2AlN is within ~0.5% of both DFT values as well as experiments. Similarly, the DHf for L10-TiAl and Ti2AlN obtained from the present potential is in near-perfect agreement with DFT calculations and experiments. In general, the present 2NN MEAM potential parameters could exactly reproduce the lattice parameter and heat of formation for L10-TiAl and Ti2AlN. The internal atomic coordinates in the crystal structure are important crystallographic data for the solid phases, and thus the ability to accurately describe the internal atomic coordinates is a major task for an empirical potential model. In this work, according to the previous DFT calculations [31,32], L10-TiAl has an ordered face-centered tetragonal structure in which Ti and Al atoms alternatively occupy the (002) planes. The space group of L10-TiAl is P4/ mmm (123), in which the Wyckoff positions 1(a) [(0, 0, 0)] are occupied by Al1 atoms, 1(c) [(1/2, 1/2, 0)] are occupied by Al2 atoms and 2(c) [(0, 1/2, 1/2)] are occupied by Ti atoms. Ti2AlN possesses a hexagonal layered structure and belongs to space group P63/mmc (194). In unit cell the atoms occupy the Wyckoff positions 2(a) [(0, 0, 0)] for N, 2(d) [(1/3, 2/3, 3/4)] for Al, and 4(f) [(1/3, 2/3, 0.088)] for Ti. Table 4 compares the optimal values of internal atomic coordinates for L10-TiAl and Ti2AlN using the present 2NN MEAM based on total energy minimization, in comparison with the values obtained from DFT calculations. It can be seen clearly from Table 4 that the difference between the 2NN MEAM and DFT values of internal atomic coordinates for L10-TiAl and Ti2AlN is negligibly small, which indicates that the predictions of internal atomic coordinates for L10-TiAl and Ti2AlN using the present 2NN MEAM potentials are generally satisfactory. Table 5 shows a comparison of elastic constants of L10-TiAl and Ti2AlN predicted by the 2NN MEAM developed in this work with DFT calculations, experiments and other (semi-)empirical calculations (when available). It is clearly seen that the elastic stiffness values for L10-TiAl and Ti2AlN predicted by our newly developed 2NN MEAM potential are in excellent agreement with DFT calculations and experiments. It should be noticed that such a close match is particularly difficult to achieve with interatomic potentials when compared with other (semi-)empirical calculations. One set of EAM (Embedded atom method) potential parameters [17] and MEAM (Modified embedded atom method) potential parameters [19] have been reported previously for L10-TiAl, but there are a common drawback for these two potentials, the two negative Cauchy pressure, (C12 e C66) and (C13 e C44) for L10-TiAl, which are closely related to the anisotropic shear deformation, can not be reproduced correctly by these two potentials. However, this drawback is well offset by the present 2NN MEAM potential. This clearly demonstrates the power of the present potential to investigate the deformation process of L10-TiAl and Ti2AlN. One property considered to confirm the transferability of the present 2NN MEAM potential was the low-index surface energies of the L10-TiAl and Ti2AlN. Table 6 compares the surface energies for several low index surfaces of L10-TiAl and Ti2AlN predicted by the 2NN MEAM developed in this work with DFT calculations and other (semi-)empirical calculations (when available). It is clearly that the present 2NN MEAM parameters are able to give a comparatively

P. Liu et al. / Journal of Alloys and Compounds 745 (2018) 63e74

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Table 3 Comparison of lattice parameters and heat of formation (DHf) of L10-TiAl and Ti2AlN using 2NN MEAM developed in this work with DFT calculations, experimental data and other (semi-)empirical calculations. Structure

Property

Present work

DFT calculations

Experiments

MEAM [19]

L10-TiAl

a (Å) c (Å) c/a ratio DHf(eV/atom) a (Å) c (Å) c/a ratio DHf(eV/atom)

4.03 4.15 1.03 0.41 3.01 13.67 4.53 1.34

3.96 [24], 3.977 [25] 4.04 [24], 4.08 [25] 1.02 [24], 1.026 [25] 0.41 [25], 0.40 [26] 2.995 [27], 3.0 [28] 13.646 [27], 13.68 [28] 4.56 [27], 4.56 [28] 1.32 [30]

3.997 [17] 4.08 [17] 1.02 [17] 0.39 [17] 2.986 [29] 13.60 [29] 4.55 [29]

4.02 4.10 1.02 0.39

Ti2AlN

Table 4 Comparison of internal atomic coordinates for L10-TiAl and Ti2AlN using 2NN MEAM developed in this work with DFT calculations. Structure

Space group

Atoms

Wyckoff notation

(x, y, z) in present work

(x, y, z) in DFT calculations

L10-TiAl

P4/mmm (123)

Ti2AlN

P63/mmc (194)

Al1 Al2 Ti N Al Ti

1(a) 1(c) 2(c) 2(a) 2(d) 4(f)

(0, 0, 0) (0.498, 0.501, 0) (0, 0.497, 0.499) (0, 0, 0) (0.338, 0.662, 0.747) (0.338, 0.662, 0.086)

(0, 0, 0) [31] (1/2, 1/2, 0) [31] (0, 1/2, 1/2) [31] (0, 0, 0) [32] (1/3, 2/3, 3/4) [32] (1/3, 2/3, 0.088) [32]

Table 5 Comparison of elastic constants of L10-TiAl and Ti2AlN predicted by the 2NN MEAM developed in this work with experimental data, DFT calculations and other (semi-) empirical calculations. The unit of elastic constants is GPa. Structure

Property

Present work

Experiments

DFT calculations

EAM [17]

MEAM [19]

L10-TiAl

C11 C12 C13 C33 C44 C66 C11 C12 C13 C33 C44

165.4 70.7 75.3 184.8 94.2 84.6 317.4 80.9 96.3 309.1 125.1

183 [33], 186 [34] 74.1 [33], 72 [34] 74.4 [33], 74 [34] 178 [33], 176 [34] 105 [33], 101 [34] 78.4 [33], 77 [34]

170 [24], 190 [35] 88 [24], 105 [35] 84 [24], 90 [35] 179 [24], 185 [35] 115 [24], 120 [35] 72 [24], 50 [35] 312 [36], 311 [37] 76 [36], 71 [37] 93 [36], 102 [37] 286 [36], 298 [37] 125 [36], 133 [37]

195 107 113 213 92 84

181 76.3 133.7 234 86 61.6

Ti2AlN

Table 6 Comparison of low-index surface energies of L10-TiAl and Ti2AlN predicted by the 2NN MEAM developed in this work with DFT calculations. The unit of the surface energies is J/m2. Structure

Surface

Present work

DFT

L10-TiAl

(001)-Ti terminated (001)-Al terminated (110)-Ti terminated (110)-Al terminated (100) (111) (0001)-Ti terminated (0001)-Al terminated (0001)-N terminated (1010)-TiAl terminated

2.85 1.50 2.66 1.42 1.48 1.53 3.08 1.73 2.45 1.49

2.589 1.620 2.452 1.492 1.618 1.691 3.726 1.835 2.294 1.267

(1010)-N terminated

2.51

2.766 [9]

(1120)

2.27

2.195 [9]

(1013)

1.85

2.017 [9]

Ti2AlN

MEAM [19] [25] [25] [25] [25] [25] [25] [9] [9] [9] [9]

2.138 2.138 1.865 1.865 1.922 1.489

accurate reproduction of low-index surface energies of the L10-TiAl and Ti2AlN as compared to DFT calculations. The same hierarchy of surface energy was predicted by 2NN MEAM and DFT calculations for both L10-TiAl and Ti2AlN. The values of the surface energies for both L10-TiAl and Ti2AlN obtained from 2NN MEAM show an error of ~10% with respect to their DFT values. It should be noticed that such a close match is particularly difficult to achieve with

interatomic potentials when compared with other (semi-)empirical calculations. As shown in Table 6, compared with the values of lowindex surface energies for L10-TiAl determined by other MEAM potential parameters [19], the present 2NN MEAM can more accurately describe the low-index surface energies for L10-TiAl. This clearly demonstrates the power of the present potential to reproduce the interface property of Ti2AlN/TiAl composite due to the interface in the composite is a special area where the surface of Ti2AlN particle interact with that of TiAl matrix. Another property considered to confirm the transferability of the present 2NN MEAM potential parameters was the thermal properties of the L10-TiAl and Ti2AlN. Table 7 shows the lattice constants, thermal expansion of L10-TiAl and Ti2AlN determined from the present 2NN MEAM as a function of temperature. It can be seen from Table 7 that Ti2AlN shows obvious anisotropic thermal expansion behavior and the expansion along the a-axis is higher than along the c-axis. The anisotropic thermal expansion is a common phenomenon in MAX phases due to their crystal structure anisotropy [38,39]. In the crystal structure of Ti2AlN, the Ti-Al bonds and Ti-N bonds along the c-axis are relatively strong, and thus the Ti2AlN is easier to expand along the a-axis than the c-axis. This conclusion is in accord with the previous researches about the thermal expansion anisotropy of Ti2AlN [38,39]. The comparisons of the linear thermal expansion coefficients of L10-TiAl and Ti2AlN from 300 K to 1300 K predicted by the 2NN MEAM developed in this work with available literature data are shown in Table 8. It can

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P. Liu et al. / Journal of Alloys and Compounds 745 (2018) 63e74

Table 7 Lattice constants, thermal expansion of L10-TiAl and Ti2AlN determined from the present 2NN MEAM as a function of temperature. Temperature (K)

300 500 700 900 1100 1300

L10-TiAl

Ti2AlN

a (Å)

(a  a0)/a0  103

a (Å)

c (Å)

(a  a0)/a0  103

(c  c0)/c0  103

4.038 4.047 4.058 4.067 4.076 4.086

0 2.23 4.95 7.18 9.41 11.89

3.014 3.019 3.025 3.031 3.036 3.042

13.687 13.708 13.728 13.749 13.771 13.792

0 1.66 3.65 5.64 7.30 9.29

0 1.53 2.99 4.53 6.14 7.67

Table 8 Calculated linear thermal expansion coefficients of L10-TiAl and Ti2AlN using the present 2NN MEAM potential, in comparison with available literature data. Structure

linear thermal expansion coefficient

Present work

Literature values

L10-TiAl Ti2AlN

a (106 K1) aa (106 K1) ac (106 K1) ac/aa

11.5 9.04 7.61 0.84

12 [13] 10.6 [38], 8.6 [39] 9.75 [38], 7.0 [39] 0.92 [38], 0.81 [39]

be seen that the simulation values are well consistent with the literature data, which indicates that the thermal properties of L10TiAl and Ti2AlN can be well reproduced by the present 2NN MEAM potential. 3.3. Deformation and fracture behavior under uniaxial tension In order to demonstrate the ability of our newly developed 2NN MEAM potential to capture atomic scale dynamical phenomena in Ti2AlN/TiAl composite under various service conditions, we investigated the response of single crystal L10-TiAl, single crystal Ti2AlN and Ti2AlN/TiAl composite to uniaxial tension at 300 K, 600 K, 900 K, 1200 K and/or 1500 K (when required) using molecular dynamics (MD) simulations. MD simulations are performed with the open-source code Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [22]. The single crystal L10-TiAl has a dimension of 6, 6 and 12.2 nm in X, Y and Z directions respectively, and contains 27,000 atoms with crystallography configuration of X [100], Y [010], and Z [001]. The single crystal Ti2AlN simulation model was generated by multiplication of a unit cell along X [1210], Y [1010] and Z [0001] directions, the sample dimension is 6.2 nm (X)  6.3 nm (Y)  12.3 nm (Z), with a total of 36,288 atoms. The simulation configuration of Ti2AlN/TiAl composite was constructed by Ti2AlN and TiAl with the crystallographic orientation relation-

relaxation, these un-deformed models are subjected to a uniaxial loading process along the direction of Z-axis. During the tension process, free boundary condition is employed in the direction of Z and periodic boundary conditions are adopted to the direction of X and Y. The atoms are set to be Newtonian atoms except for the two 7 Å thick regions at the top and bottom, which are set to be rigid atoms to realize the tensile process. The velocity is imposed on the rigid atoms at the top region with a strain rate of 108 s1, and the rigid atoms at the bottom region is fixed. To avoid the atoms to be shocked by an abrupt change in the kinematic quantities, a uniform stretching is imposed to the Newtonian atoms. The simulation is carried out with canonical NVT ensemble. The center symmetry parameters (CSP) method [40] is used to observe the disorder phenomenon and atom positions of local lattice, which can be visualized by Open Visualization Tool (OVITO) [41]. 3.3.1. Tensile deformation process of single-crystal TiAl Fig. 1 shows tensile stress (s)-strain (ε) curves of single-crystal L10-TiAl at different temperatures. It is seen that all the stressstrain curves are almost linear in the region of 0ε  0.05, which indicates that the single-crystal L10-TiAl undergoes an elastic deformation at the initial stage of uniaxial tensile deformation at different temperatures. The Young's modulus of single-crystal L10TiAl can be obtained by the slopes of linear segments, the Young's modulus are 183.1, 176.7, 166.2 and 157.3 GPa for temperatures of 300, 600, 900, and 1200 K, respectively. The Young's modulus of single-crystal L10-TiAl at 300 K predicted in the present interatomic potential is well consistent of the experiment value [34]. In addition, it is obvious that the higher the temperature is, the lower the Young's modulus is. This is because the higher the temperature is, the more intensive the atomic thermal motion is and the smaller the bonding energy between atoms is. Therefore, the lattice is more

ship of [1120]Ti2AlN//[101]TiAl and (0001)Ti2AlN//(111)TiAl. This orientation relationship is defined according to our previous TEM observation [9] and the Ti2AlN(0001)/TiAl(111) interface possesses good matching and is a typical coherent interface. For the Ti2AlN slab, the [1010], [1210] and [0001] directions are parallel to X, Y and Z axis, respectively. For the TiAl slab, the [112], [110] and [111] directions are parallel to X, Y and Z axis, respectively. The sample dimension is 5.8 nm (X)  6.1 nm (Y)  12.0 nm (Z), with a total of 28,800 atoms. Before starting the MD simulations of uniaxial loading, the initial configurations are first optimized using the conjugate gradient (CG) algorithm to achieve a stable configuration with minimum equilibrium energy. Then, the system was thermally equilibrated to 300 K, 600 K, 900 K, 1200 K and/or 1500 K (when required) by running 50,000 steps with a time step of 1 fs. During the relaxation process, the temperature kept constant with NoseHoover thermostat and the initial system stress along the loading direction was adjusted to zero by Nose-Hoover barostat. After

Fig. 1. Stress-strain curves for single-crystal L10-TiAl at four different temperatures.

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easily deformed and the Young's modulus is lower. When the strain surpasses 0.05, the stress increases nonlinearly with the increasing strain until ultimate tensile strength (maximum value of the curve) is obtained, which indicates that the single-crystal L10-TiAl enters the yield stage. With the increase of temperature, the yield phenomenon is becoming more and more obvious. In addition, the tensile strength are decreased with the increase of temperature. After the ultimate tensile strength (maximum value of the curve) is obtained, the stress-strain curves at 300 K and 600 K shows a sudden drop. This sharp transition is suggestive of the typical brittle behavior of TiAl bulk at 300 K and 600 K. This trend is clearly different from that observed in the 900 K deformation, which shows a slow downward trend during the rise in strain, in stark contrast to the sudden drop witnessed at 300 K and 600 K. This effect becomes even more prominent when the deformation is carried out at a higher temperature of 1200 K. These indicate that the single-crystal L10-TiAl shows ductile fracture behavior at 900 K and 1200 K. These simulation results are consistent with the previous research [42], which has shown that the brittle-ductile transition temperature of single crystal g-TiAl is in the range of 873e973 K. Figs. 2 and 3 elaborate the snapshots of single-crystal L10-TiAl under tension at 300 and 1200 K, respectively. The atoms in the snapshots are color-coded according to CSP results. When the CSP value of an atom increases from 0 to 25, it indicates that this atom is in perfect lattices, at point defects, at dislocations, and on free surfaces. Fig. 2(a) shows the initial configuration of single-crystal L10-TiAl at a temperature of 300 K and with strain (ε) ¼ 0, all the TiAl atoms are at site of perfect lattice (blue atoms) except the upper and lower surface atoms (red atoms). When the ε increases to 0.06, as shown in Fig. 2(b), some point defects (light blue atoms) start to appear in the single-crystal L10-TiAl. With the further increasing of ε value, as shown in Fig. 2(c), the dislocations (yellow atoms) start to nuclear, then propagate and accumulate along (001) plane. It is worth noting that only the dislocations along (001) plane are observed during the uniaxial tension of single-crystal L10-TiAl at 300 K, this is because only the ‘ordinary’ dislocations (b ¼ a/2 < 110]) along the TiAl (001) plane is easy to start at room temperature [43]. At ε ¼ 0.129, as shown in Fig. 2(d), a micro-crack (red atoms) nucleates along the (001) plane due to the accumulation of dislocations leads to stress concentration, and micro-crack easily initiates at here.

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Then it can be seen from Fig. 2(e) that the single-crystal L10-TiAl presents a complete fracture. No necking is observed indicative of the brittle nature of the single-crystal L10-TiAl at room temperature. As shown in Fig. 3, the stress-strain behavior of single-crystal L10-TiAl at a temperature of 1200 K is fundamentally different from that at 300 K. It can be seen from Fig. 3(a) that there are some point defects in the initial state of the single-crystal L10-TiAl at a temperature of 1200 K due to the quantity of point defects increases at elevated temperature. With the further increasing of ε value, as shown in Fig. 3(b) and (c), the dislocations (yellow atoms) start to nuclear and propagate. Compared with the tension at 300 K shown in Fig. 2, it is quite obvious that the dislocations in the single-crystal L10-TiAl deformed at 1200 K is no only along the (001) plane but also along other planes, which means that more kinds of dislocations can be activated during the tension at 1200 K. With a further increasing of the strain, as shown in Fig. 3(d), some micro-crack nucleates, but unlike the micro-crack rapidly expand along (001) plane during the tension at 300 K, it can be seen from the Fig. 3(d) that there exists a competition between dislocations and micro-crack, the expansion of the micro-crack is prevented by the high density of dislocations, which results the two TiAl retain contact after fracture via the formation of a gradually thinning neck, as shown in Fig. 3(e). From an overall perspective, the fracture of single-crystal L10-TiAl deformed at 1200 K is of ductile nature. These simulation results are consist with the previous research [42], which has shown that the brittle-ductile transition mechanism of single crystal g-TiAl is controlled by the evolution behaviors of dislocations at elevated temperature. 3.3.2. Tensile deformation process of Ti2AlN The effect of temperature on the deformation mechanisms of single crystal Ti2AlN is studied by applying tensile stress along [0001] direction at temperatures ranging from 300 K to 1500 K. Stress-strain responses at five different temperatures are presented in Fig. 4. The Young's modulus of single crystal Ti2AlN can be obtained by the slopes of linear segments, the Young's modulus are 303.1, 298.6, 285.3, 279.6 and 263.3 GPa for temperatures of 300, 600, 900, 1200 K and 1500 K, respectively. The Young's modulus of single-crystal Ti2AlN at 300 K predicted in the present interatomic potential is well consistent of the DFT calculation values [6,36,37]. When the temperature is in the range of 300 Ke1200 K, the plots

Fig. 2. Atomistic representation of single-crystal L10-TiAl under uniaxial tensile at 300 K. Atoms are color-coded according to CSP results. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 3. Atomistic representation of single-crystal L10-TiAl under uniaxial tensile at 1200 K. Atoms are color-coded according to CSP results. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

exhibit similar trends: Firstly, the stress increases linearly with increasing strain, and to a certain content, the slope of the curve gradually decreases. Finally, the stress shows a sudden drop at the limit strain. These observations indicated single crystal Ti2AlN first shows elastic behavior, then undergoes quasiplastic stress relaxations, and eventually demonstrates typical brittle behavior at the temperatures range of 300 Ke1200 K. It is worth noting that the yield region seen beyond the elastic limit but before brittle fracture has been previously reported for ceramics, particularly at high strain rates [15,44]. When the deformation is carried out at 1500 K, the tensile stress (s)-strain (ε) curve is obvious different. As shown in Fig. 4, when the strain is around 0.086, an obvious yield phenomenon is observed, which indicates that the single-crystal Ti2AlN shows ductile fracture behavior at 1500 K. This simulation result is consistent with the previous research [38], which has shown that the brittle-ductile transition temperature of Ti2AlN is in the range of 1373e1473 K. Figs. 5 and 6 show the snapshots of single-crystal Ti2AlN under tensile testing at temperatures of 300 and 1500 K, respectively. The

Fig. 4. Stress-strain curves for single crystal Ti2AlN at five different temperatures.

atoms in the snapshots are color-coded according to CSP results. It is worth noting that the Ti2AlN possesses a complex hexagonal layered structure, and the nearest neighbors of each kind of atom is different, and thus it is impossible to color the Ti2AlN atoms with one color according to CSP results. As shown in Fig. 5(a), green atoms represent the Ti and Al atoms that are at site of perfect lattice, while blue atoms represent the N atoms that in the perfect lattice. When the deformation temperature is 300 K, as shown in the Fig. 5(b), strain localization forms randomly after the applied tensile stress increased a critical value. With the strain further increased, micro-crack nuclei formed (Fig. 5(c)), which expanded rapidly along (0001) plane and formed cracks as deformation proceeds, eventually resulting in the fracture of the sample (Fig. 5(d)). These simulation results are in line with the previous experimental research [45] on the Ti3SiC2 ceramic whose lattice structure and mechanical properties are similar to Ti2AlN ceramic, which has shown that micro-cracks and their linkage play a dominant role in their brittle response. When the deformation temperature rise to 1500 K, as shown in the Fig. 6(b), pyramidal plane dislocations appeared after the applied tensile strain increased to about 0.086, which corresponding to the first peak in the stress-strain curve. This simulation result indicates that the yield phenomenon of single crystal Ti2AlN at elevated temperature is resulted from the activation of pyramidal plane dislocations. Although some other experimental research [46] has reported that nucleation and propagation of basal plane dislocations is available in MAX phases, the basal plane dislocations were not observed in the present work. This is not strange, because the preferred slip systems are not only determined by the Schmid factor but also the loading conditions [47]. In the present study, the tensile direction is perpendicular to the basal plane of single crystal Ti2AlN, and thus the basal plane dislocations are not easy to be activated but instead of the pyramidal plane dislocations. This simulation result is in accordance with other MD simulation [48], which has shown that the deformation mechanism on the hexagonal structure with the deformed direction normal to the basal plane is controlled by the activation of pyramidal plane dislocations. 3.3.3. Tensile deformation process of Ti2AlN/TiAl composite The tensile deformation mechanisms of Ti2AlN/TiAl composite with a coherent interface was studied by applying tensile stress

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Fig. 5. Atomistic representation of single-crystal Ti2AlN under uniaxial tensile at 300 K. Atoms are color-coded according to CSP results. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. Atomistic representation of single-crystal Ti2AlN under uniaxial tensile at 1500 K. Atoms are color-coded according to CSP results. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

along [0001]Ti2AlN//[111]TiAl direction. Fig. 7 shows the variation of longitudinal stress versus strain for the Ti2AlN/TiAl composite at various temperatures during tensile deformation. In this figure, the yield strength corresponds to the ultimate strength. All plots exhibit similar trends: Firstly, the stress increases linearly with increasing strain, and to a certain content, the slope of the curve gradually decreases. Finally, the stress shows a sudden drop at the limit strain, demonstrating typical brittle behavior. These simulation results are well consistent with the previous experimental research [13], which has shown that the Ti2AlN/TiAl composite exhibit characteristics of brittle fracture both at 300 K and 1073 K. In order to obtain a deeper understanding of the tensile deformation mechanism of Ti2AlN/TiAl composite at both room and elevated temperature, we examined the atomistic structures of Ti2AlN/TiAl composite under tensile testing at temperatures of 300 and 1200 K, respectively. As shown in Figs. 8 and 9, the upper half portion of the sample represents the TiAl slab, and the lower half portion of the sample represents the Ti2AlN slab. The atoms in the snapshots are color-coded according to CSP results. For a Ti2AlN/

TiAl composite sample which all the atoms are at site of perfect lattice, as shown in Fig. 8(a), the Ti and Al atoms in the TiAl slab is colored by blue, the Ti and Al atoms in the Ti2AlN slab is colored by green, and the N atoms in the Ti2AlN slab is colored by blue. When the tensile temperature is 300 K, as shown in Fig. 8, with the applied tensile strain increased continuously, point defects were randomly formed in the TiAl side first (Fig. 8(b)), then the dislocation formed and propagated (Fig. 8(c)), later, micro-crack nuclei appeared and expanded rapidly to form a crack (Fig. 8(d)), and eventually resulting in the fracture of the sample (Fig. 8(e)). It is conspicuous that the deformation mechanism of Ti2AlN/TiAl composite is a typical brittle fracture at room temperature. In addition, it is interesting that the fracture behavior is only appeared in the TiAl side, while the Ti2AlN side and the interface region remain stable during the whole tensile process. These results indicate that the bonding strength of the coherent interface between Ti2AlN and TiAl is so high that dislocations can only nucleate in the softer TiAl side during the tensile process. This results are well consistent with the previous theoretical study [49,50], which has shown that

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low temperature. When the tensile temperature is 1200 K, as shown in Fig. 9, the Ti2AlN/TiAl composite also shows a brittle behavior. However, unlike the tensile process of Ti2AlN/TiAl composite at 300 K, the coherent interface becomes unstable and is the dominant factor of the fracture of Ti2AlN/TiAl composite at 1200 K. As shown in the Fig. 9(b), when the applied strain increased to a certain content, the dislocations nucleated in the interface region between Ti2AlN and TiAl, then the dislocations propagated into the TiAl side and eventually formed the crack resulting in the rupture of composite. These simulation results agreed with previous study [51] showing that the temperature has greater influence on the mechanical properties of the models with coherent interfaces due to the fact that the initial coherency stress level in the model with coherent interfaces may be greatly affected by the variations in temperature. 4. Conclusions Fig. 7. Stress-strain curves for Ti2AlN/TiAl composite at four different temperatures.

nucleation of dislocations is difficult in a perfect coherent interface, and the rupture is tend to occur in the softer matrix of composite at

In summary, we have developed a 2NN MEAM interatomic potential for Ti-Al-N ternary system by fitting to the lattice parameters, heat of formation and elastic constants of TiAl and Ti2AlN. Our newly developed 2NN MEAM potential accurately reproduces the structure, elastic, thermodynamics and surface properties of TiAl

Fig. 8. Atomistic representation of Ti2AlN/TiAl composite under uniaxial tensile at 300 K. Atoms are color-coded according to CSP results. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 9. Atomistic representation of Ti2AlN/TiAl composite under uniaxial tensile at 1200 K. Atoms are color-coded according to CSP results. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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and Ti2AlN compounds. Using the ternary Ti-Al-N 2NN MEAM potential developed in this work, we investigated the deformation and fracture behavior of single crystal L10-TiAl, single crystal Ti2AlN and Ti2AlN/TiAl composite at atomic scale using MD simulations. These simulations indicated that both single crystal L10-TiAl and Ti2AlN undergoes brittle fracture at low temperature, while shows ductile fracture at elevated temperature. When the tensile temperature is 300 K, the fracture behavior of Ti2AlN/TiAl composite is only appeared in the TiAl side, the Ti2AlN side and the interface region remain stable. But when the tensile temperature is 1200 K, the coherent interface becomes unstable and could act as the site for the dislocations nucleation. These MD simulations demonstrate the capability of the 2NN MEAM interatomic potential parameters developed in this work to capture dynamical atomic-scale events in Ti-Al-N systems that occur in response to external stimuli, for example, temperature and applied pressure. In addition, the ternary Ti-Al-N potential developed here can provide a reference to develop empirical potential for other multicomponent systems to investigate interesting atomic-scale phenomena. Acknowledgments The authors acknowledge the financial support from National Natural Science Foundation of China (grant nos. 51471058, 51201046). References [1] W. Li, Y. Yang, J. Liu, Y. Zhou, M. Li, S.F. Wen, Q.S. Wei, C.Z. Yan, Y.S. Shi, Enhanced nanohardness and new insights into texture evolution and phase transformation of TiAl/TiB2 in-situ metal matrix composites prepared via selective laser melting, Acta Mater. 136 (2017) 90e104. [2] Z.L. Lu, J.W. Cao, S.Z. Bai, M.Y. Wang, D.C. Li, Microstructure and mechanical properties of TiAl-based composites prepared by Stereolithography and gelcasting technologies, J. Alloys Comp. 633 (2015) 280e287. 
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