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Properties of Ti/TiC Interfaces from Molecular Dynamics Simulations Tao Liang, Michael Ashton, Kamal Choudhary, Difan Zhang, Alexandre F. Fonseca, Benjamin C. Revard, Richard G Hennig, Simon R. Phillpot, and Susan B. Sinnott J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b02763 • Publication Date (Web): 17 May 2016 Downloaded from http://pubs.acs.org on May 19, 2016
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The Journal of Physical Chemistry
Properties of Ti/TiC Interfaces from Molecular Dynamics Simulations Tao Liang (梁涛)1, Michael Ashton2, Kamal Choudhary2, Difan Zhang (张迪凡)2,Alexandre F. Fonseca2,3, Benjamin C. Revard4,2, Richard G. Hennig2, Simon R. Phillpot2, and Susan B. Sinnott1,* 1
Department of Materials Science and Engineering, Pennsylvania State University, University Park, PA, 16801, USA
2
Departmentof Materials Science and Engineering, University of Florida, Gainesville, FL, 32611, USA 3
4
Applied Physics Department, State University of Campinas, Campinas, SP, 13083-970, Brazil
Department of Materials Science and Engineering, Cornell University, Ithaca, New York, 14853, USA
ABSTRACT Titanium carbide is used as a primary component in coating materials, thin films for electronic devices, and composites. Here, the structure of coherent and semicoherent interfaces formed between closepacked TiC (111) and Ti (0001) is investigated in classical molecular dynamics simulations. The forces on the atoms in the simulations are determined using a newly developed TiC potential under the framework of the third-generation charge optimized many-body (COMB3) suite of potentials. The work of adhesion energies for the coherent interfaces are calculated and compared with the predictions of density functional theory calculations. In the case of relaxed semicoherent interfaces, a two-dimensional (2D) misfit dislocation network is predicted to form that separates the interface into different regions in which the positions of atoms are similar to the positions at the corresponding coherent interfaces. After annealing the interface at an elevated temperature, the climb of edge dislocations is activated which modifies the 2D misfit dislocation network and increases the work of adhesion. These findings can be used as inputs for sequential larger simulation models to understand and predict the macroscopic properties of TiC/Ti interfaces. *Corresponding author:
[email protected] and 814-863-3117
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I. INTRODUCTION Titanium carbide (TiC) has a rocksalt structure (space group 3) at room temperature and is a well-known refractory metal compound with high melting point, hardness, and electrical conductivity. In addition, experimental and theoretical studies 1-3 reveal that the chemical bonding in TiC is a mixture of ionic, covalent and metallic bonds. These unusual physical, chemical and mechanical properties of TiC have led to its use in a wide range of applications such as coating materials, thin films for electronic devices, constituent materials for composites, structural components in aerospace and military applications, and as raw materials to produce carbide derived carbon4-10. Surfaces of TiC and TiC/Ti interfaces have attracted a substantial amount of attention due to the crucial roles they play in these applications. Transmission electron microscopy indicates that the primary orientation relationship for the TiC/Ti interface is TiC (111)//Ti (0001) 11-12. As is the case for NaCl structures, stoichiometric TiC (111) is a polar surface and has the highest cleavage energy among the planes with low Miller Indices. The surface energy of non-stoichiometric TiC (111) with different terminations is a function of the chemical potential of the Ti or C atoms. Using density functional theory (DFT) calculations, Li et al
13
found that the Ti-terminated TiC (111) surface has the lowest surface
energy and ‘ideal’ TiC/Ti interfaces, in which the Ti slab atoms occupy hollow sites on C-terminated TiC and center sites on Ti-terminated TiC, have the lowest interfacial energies. Coherent interfaces between TiC and Ti slabs results in elastic deformation of one or both slabs to accommodate lattice mismatch, and dislocations are often observed12 that reduce the strain energy and lead to lower energy semicoherent interfaces. Unfortunately, these semicoherent interfaces are too large to be investigated with DFT. Empirical potentials are a more practical solution, including in combination with classical molecular dynamics (MD) simulations. For example, Shao et al
14
examined spiral patterns of
dislocations at the Cu (111)/Ni (111) semicoherent interface using the embedded atom method (EAM) potential. More recently, Pilania et al 15 employed response-free modified EAM potentials with dynamic charge transfer to investigate semicoherent interfaces of Al/α-Al2O3.
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A modified embedded atom method (MEAM) potential for TiC was developed by Kim et al.16 that predicted reasonable lattice constants, elastic constants, and cleavage surface energies. This potential was extended to include Fe by the same investigators
17
. The resulting Ti-C-Fe MEAM potential was
subsequently applied to investigate problems such as nucleation kinetics of TiC precipitates in steels. However, this potential underestimates the work of adhesion at the interface between TiC (001) and face centered cubic (fcc) Ti (001) by 87%. In addition, this potential lacks charge transfer capabilities, which are vitally important in determining the structure and properties of the interfaces. Here, we report on the development of a reactive, dynamic charge interatomic potential for TiC and Ti within the framework of the third-generation charge optimized many-body (COMB3)
18
potential.
We further demonstrate their capabilities by using them in classical MD simulations to examine the properties of TiC surfaces, in addition to both coherent and semicoherent interfaces between TiC and Ti. The results provide new insights into the generation of misfit dislocations at semicoherent interfaces and can serve as inputs for subsequent large-scale simulation approaches such as dislocation dynamics models.
II. PARAMETERIZATION OF COMB3 The COMB3 potential is a variable charge and reactive interatomic empirical potential. As shown in Eq. 1, the total energy of the system is expressed as a sum of electrostatic energy ( ), chargedependent short-range interactions ( ), van der Waals interactions ( ), and correction terms (
). Here {q} and {r} refer to the charge and coordinates of the atoms in the system, respectively. A complete description of each energy component of the COMB3 formalism can be found elsewhere18. , = , + , + +
(1)
The parameterization of TiC made use of the Parameterization Optimization Software for MATerials (POSMat)
19
to minimize the cost function, which is a sum of weighted squared residuals between the
calculated values and the training database. Wherever possible, the training database was populated with experimental values and first-principles data from the literature. When this data was not available we ACS Paragon Plus Environment
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performed our own DFT calculations to complete the dataset for fitting and testing, which are listed in the DFT column in Table 1 without a reference. All the DFT calculations that we carried out used the Vienna ab initio Program (VASP)
20-21
program with the generalized-gradient approximation (GGA) 22 and the Perdew–Burke–Ernzerhof (PBE) 23
exchange-correlation functional. The energy cutoff is 20% higher than the default energy cutoff of the
pseudopotentials and the Brillouin zones of the simulation cells were sampled using a Monkhorst-Pack mesh with the product of the number of k-points and the lattice parameter corresponding to a value greater than 20 Å. The convergence criteria of geometry optimization was set at 1.0×10-5 eV and 1.0×10-3 eV·Å-1 for energies and forces, respectively. The potential energy, interatomic forces and the stress tensor were corrected to include van der Waals contributions using the DFT-D3
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method with
Becke-Jonson damping. The values of the bulk and surface properties of the TiC rocksalt phase calculated using the COMB3 potential are provided in Table 1. The cleavage energies listed were calculated as
γ =
Eslab - Ebulk
(2)
A
where Eslab is the total energy of the slab with 20 Å vacuum, Ebulk is the energy of the bulk structure, and A is the surface area. The resulting properties are compared to experimental data obtained using the MEAM potential
16
16
, published data
, and the results of our DFT calculations and DFT calculations
from the literature, including data in Ref. 16.
Table 1: Properties of the rocksalt and other phases of TiC from experimental data, DFT calculations,
MEAM, and COMB3. DFT calculation results without references were carried out by the authors. Properties TiC Rocksalt Ecoh (eV/TiC) ∆Hf (eV/atom) a0 (Å) B (GPa) G (GPa)
Expa
DFT
-13.77 -0.78 4.32 242 188
-19.28 -0.92 4.32-4.38a 214-286a
MEAMa
COMB3
-0.78 4.42 242 161
-13.87 -0.82d 4.32 263 190
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C11 (GPa) C12 (GPa) C44 (GPa) qTi (e) Cleavage Energies γ(100) (J/m2) γ(110) (J/m2) γ(111) (J/m2) Other structures ∆Hf (eV/atom) Antifluorite [ZnS] [CsCl] Perovskite Fluorite a : Ref. 16 b : Mulliken charge in Ref. 25 c
513 106 178
480-610a 97-124a 167-230a 0.68b, 2.1c
522 102 129
516 136 190 1.02
3.3-5.5a 7.3-7.6a 6.2-11.3a
5.8 7.5 8.1
6.8 9.2 11.2
-0.36 -0.17 0.26 1.21 1.14
-0.30 -0.18 0.17 1.28 0.80
: Bader charge from our DFT calculation.
: The reference energies that were used to calculate the heats of formation in COMB3 are -4.83 eV/atom 26 for 27-28 for graphite. hexagonal close packed (hcp) Ti and -7.40 eV/atom
d
As indicated in Table 1, COMB3 accurately reproduces the heat of formation, lattice constants, elastic constants, and surface energies for bulk TiC. The atomic charge on Ti atom (qTi) in TiC is 1.02 e, which is 0.34 e higher than the Mulliken charge25 and 1.08 e smaller than Bader charge from our own calculation. As shown in the review papers29-30 the Mulliken charge is normally smaller than the Bader charge. As COMB3 takes Mulliken electronegativity and self-Coulomb for the ionization energy of a single atom18, it is not surprise that the COMB3 charge in TiC is about 50% of Bader charge. The heats of formation of TiC in the [ZnS] and [CsCl] crystal structures, fluorite (TiC2), in addition to the antifluorite (Ti2C) and perovskite (Ti2C3) structures are calculated. Within the crystalline phases in the database, the heats of formation calculated from COMB3 are in good agreement with those calculated with DFT. To further probe the ability of the potential to describe the energetics of phases not explicitly included in the fitting database, genetic algorithm (GA) searches for low-energy compounds were performed in the Ti-C spaces. The structure search employs the Genetic Algorithm for Structure Prediction (GASP) package Simulator (LAMMPS)
33-34
31-32
coupled to the Large-scale Atomic/Molecular Massively Parallel
software for structure optimization and energy evaluation with COMB3. ACS Paragon Plus Environment
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GASP enables grand-canonical searches for structures with different numbers of atoms in the unit cell and different compositions. The initial structures and compositions were chosen randomly. The GASP package was previously applied to test empirical potentials for the Mo, Li-S, and Al-N systems35-37. A more complete description of the algorithms employed in GASP is provided elsewhere31,32. The 0 K phase diagrams, commonly referred to as the convex hulls, are provided in Fig. 1. The Ti-C system is predicted by the COMB3 potential to have one stable binary compound, and in DFT there are three. It is difficult for a parameterized potential to compute phase diagrams without error, and while the COMB3 potential captures much of the relative complexity of the Ti-C system, including many of the experimental phases, it is clear that the agreement with experimental data is imperfect.
Figure 1. The Ti-C phase diagrams at 0 K, generated using a genetic algorithm search with COMB3.
The newly developed parameter sets for TiC are seamlessly coupled with existing COMB3 potentials, including those for Ti
26
, TiO2 26, and TiN
38
, which allow us to simulate a wide range of
heterogeneous material systems with LAMMPS, which is an open source, massively parallel MD ACS Paragon Plus Environment
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software package distributed by Sandia National Laboratory. The parameter set that was used in this paper is provided in Supplementary Materials with the format that is compatible with LAMMPS.
III. TiC (111)/Ti (0001) INTERFACES A.
TiC (111) Surfaces
Cleaving bulk TiC with the rocksalt crystal structure in a direction that is perpendicular to the [001] and [110] directions produces two symmetric and stoichiometric surfaces; the cleavage energies listed in Table 1 equal the surface energies of the corresponding TiC (001) and (110) surfaces multiplied by two. Along the [111] direction TiC is composed of alternatively stacked hexagonal Ti and C layers with an abc stacking sequence to produce a structure that is similar to that in an fcc crystal structure. The stoichiometric TiC (111) slabs are terminated with carbon on one side and titanium on the other. To eliminate spurious surface dipole effects, we also calculated the surface energies of the symmetric but non-stoichiometric TiC (111) slabs that are terminated by either carbon (TiC-C) or titanium (TiC-Ti) on both sides. The surface energies (γs) for the non-stoichiometric surface slabs are thus dependent on the ! chemical potential of the Ti atoms in the extra layer in the case of the TiC-Ti surface slab ( ):
" = #
(/&0 %&'( $%&'( )*+ ,-.+ 1(*+ )*-. ),-.
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5 ,
(3)
where nC and nTi are numbers of C and Ti atoms in the slab and !7 8 6 is the chemical potential of bulk TiC. This !7 8 6 (-13.87 eV/TiC) can be written as: 9 :
!7 8 6 = 6 9 :
where 6
+ : + ∆?@) ,
(4)
, : and ∆?@) represent the energy per atom of C in graphite (-7.40 eV), the
heat of formation of bulk Ti in the hcp crystal structure (-4.83 eV), and the heat of formation of bulk TiC (-1.64 eV/TiC or -0.82 eV/atom), respectively. The TiC compound is stable since its heat of formation is negative, which indicates that 9 :
and 6
!
and 6 ! in the surface slab have to be less than :
. Combining these terms provides a range for the chemical potential of Ti in the surface
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∆
