#
Semi-Empirical Force-Field Model for the Ti_{1−x}Al_{x}N (0 ≤ x ≤ 1) System

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## Abstract

**:**

_{1−x}Al

_{x}N (0 ≤ x ≤ 1) alloy system. The MEAM parameters, determined via an adaptive simulated-annealing (ASA) minimization scheme, optimize the model’s predictions with respect to 0 K equilibrium volumes, elastic constants, cohesive energies, enthalpies of mixing, and point-defect formation energies, for a set of ≈40 elemental, binary, and ternary Ti-Al-N structures and configurations. Subsequently, the reliability of the model is thoroughly verified against known finite-temperature thermodynamic and kinetic properties of key binary Ti-N and Al-N phases, as well as properties of Ti

_{1−x}Al

_{x}N (0 < x < 1) alloys. The successful outcome of the validation underscores the transferability of our model, opening the way for large-scale molecular dynamics simulations of, e.g., phase evolution, interfacial processes, and mechanical response in Ti-Al-N-based alloys, superlattices, and nanostructures.

## 1. Introduction

_{1−x}Al

_{x}N (0 ≤ x ≤ 1) alloys are widely used for wear and oxidation protection of metal cutting and forming tools, aerospace components, and automotive parts [1,2]. The parent binary phases B1-TiN (space group Fm–3m) and B4-AlN (space group P6

_{3}mc) are immiscible at room temperature and at thermodynamic equilibrium [3]. However, far-from-equilibrium conditions, which prevail during vapor-based thin-film deposition [4], enable synthesis of metastable B1-Ti

_{1−x}Al

_{x}N solid solutions for x ≤ 0.7 [5]. These metastable alloys undergo, in the temperature range ≈1000 to ≈1200 K, spinodal decomposition into strained isostructural Al-rich and Ti-rich B1-Ti

_{1−x}Al

_{x}N domains, which in turn, results in an age-hardened material with superior high-temperature mechanical and oxidation performance [6].

_{1−x}Al

_{x}N (0 ≤ x ≤ 1) alloys, during vapor-based thin-film synthesis and high-temperature operation. Despite the knowledge generated from these investigations, atomic-scale processes that drive phase transformations (including spinodal decomposition) and elastic/plastic response in B1-Ti

_{1−x}Al

_{x}N alloys (0 ≤ x ≤ 1) are poorly understood. This is because the time and length scales at which these processes occur often lie beyond the temporal and spatial resolution of experimental techniques, while their study using first-principle theoretical methods is currently computationally unfeasible.

^{6}–10

^{7}atoms during times reaching 10

^{−6}–10

^{−5}s, which cannot be achieved using first-principle methods. Examples of such phenomena include crystal growth, mass transport, plastic deformation, interfacial reactions, phase separation, and structural transformations.

_{1−x}Al

_{x}N alloys [20]. The second-neighbor modified embedded atom method (MEAM) interatomic potential [21] is an empirical model that has reliably reproduced physical attributes of elemental phases, as well as compounds and alloys in crystalline [22,23,24,25,26,27,28,29] and amorphous [30] form, including B1-TiN [23,31], fcc-Al [26,32], hcp- [33] and bcc-Ti [34], and N

_{2}dimers [35,36]. However, the MEAM models available in the literature are not suitable for describing kinetic and thermodynamic properties of the Ti

_{1−x}Al

_{x}N system over the entire composition range (0 ≤ x ≤ 1), and for different phases and configurations. Hence, a complete re-parametrization of all elemental, pair, and triplet Ti-Al-N interactions is required for developing a MEAM Ti

_{1−x}Al

_{x}N (0 ≤ x ≤ 1) interatomic potential.

_{1−x}Al

_{x}N (0 < x < 1) alloys. We find that our potential reproduces the lattice and elastic constants, phonon-spectra, defect migration energies, phase diagrams, and solid–solid phase transitions induced by temperature and/or pressure changes within the Ti

_{1−x}Al

_{x}N (0 ≤ x ≤ 1) system remarkably well. This opens the way for large-scale molecular dynamics simulations of Ti-Al-N phase transformations and elastic/plastic responses, which may provide critical insights for designing Ti-Al-N-based superlattices and nanostructures with superior thermal stability and mechanical performance.

_{1−x}Al

_{x}N system and beyond.

## 2. Potential Parametrization and Validation Methodology

#### 2.1. Potential Parametrization Methodology

^{2}MEAM parameters (see Table S1 in Supplementary Materials). We address this formidable challenge by using a stochastic algorithm based on an adaptive simulated annealing (ASA) scheme [37], which is explained in detail in Section S2 and illustrated in Figure S3 in Supplementary Materials. This algorithm allows us to determine parameter values that optimize the potential’s predictions relative to experimentally- and theoretically-determined physical properties of phases and configurations of Al, Ti, and N and their binary and ternary combinations, as explained later in the present section. This is a necessary, yet not sufficient, condition for developing a MEAM potential which is fully transferable among different compositions with varying x values (0 ≤ x ≤ 1) and metal-sublattice atomic arrangements within the Ti

_{1−x}Al

_{x}N alloy system. To reduce the size of the multi-dimensional parameter-space that needs to be scanned during the ASA optimization procedure, we constrained the range of key MEAM parameters, i.e., interatomic distance r

_{e}, cohesive energy E

_{c}, bulk modulus B (E

_{c}, B, and r

_{e}determine the MEAM parameter α), and attractive d

_{+}and repulsive d

_{–}interaction, to be close to values that best fit density functional theory (DFT) total energies versus equilibrium volumes (i.e., equation of states), for all reference elemental and binary phases. More details and results on the above-mentioned DFT calculations can be found in Section S1 and in Figures S1 and S2 in Supplementary Materials.

_{3}/mmc), cubic (bcc) β-Ti (space group Im–3m), hexagonal ω-Ti (space group P6/mmm), cubic (fcc) γ-Al (space group Fm–3m), cubic γ-Al

_{98}Ti

_{2}random alloys, cubic L1

_{2}Al

_{3}Ti (space group Pm–3m), tetragonal D0

_{22}Al

_{3}Ti (space group I4/mmm), tetragonal L1

_{0}AlTi (space group P4/mmm), hexagonal Ti

_{3}Al (space group P6

_{3}/mmc), and hexagonal α-Ti

_{98}Al

_{2}random alloys. (ii) Nitrogen-based systems, i.e., N

_{2}dimer, linear N

_{3}trimer, triangular N

_{3}molecule, cubic B1-TiN (rocksalt structure), tetragonal rutile ε-Ti

_{2}N (space group P4

_{2}/mnm), B1-AlN, B3-AlN (cubic zincblende structure, space group F–43m), B4-AlN (hexagonal wurtzite structure), and nineteen ternary random solid solutions of B1-Ti

_{1−x}Al

_{x}N with equidistant compositions with x in the range 0.05 to 0.95.

_{c}); (ii) lattice parameters (a, c), and elastic constants (bulk moduli B and C

_{ij}) for all phases; (iii) mixing enthalpies (ΔH

_{mix}) for B1-Ti

_{1−x}Al

_{x}N random solid solutions calculated with respect to the energies of B1-TiN and B1-AlN parent binary phases (ΔH

_{mix}= E

_{c,B1-Ti1−xAlxN}− (1–x)·E

_{c,B1-TiN}− x·E

_{c,B1-AlN}); and (iv) point-defect (vacancies V and interstitials I) formation energies E

^{f}

_{V/I}for B1-TiN and B1-AlN.

_{ij}were calculated using least-square fitting of energy versus strain curves for structures at their equilibrium volumes. E

^{f}

_{V/I}values were computed with respect to the chemical potential μ of Ti, Al, and N in hcp-Ti, fcc-Al, and N

_{2}molecules, respectively. Specifically, E

^{f}

_{V/I}= E

_{D}± μ – E

_{0}, where E

_{D}is the energy of a crystal containing one point-defect, E

_{0}is the energy of the defect-free crystal, while the sign preceding μ is positive for vacancies and negative for interstitials. All physical properties mentioned above were determined via MEAM conjugate–gradient energy minimization calculations at 0 K using lattices consisting of 250 to 450 atoms. Exception to the latter are the B1-Ti

_{1−x}Al

_{x}N systems, where all properties were evaluated by averaging over the results obtained for 10 different random cation–sublattice configurations using supercells formed by 1000 atoms. For reference, we also calculated N and Ti vacancy formation energies in B1-TiN with respect to the chemical potential of isolated Ti and N atoms by means of both MEAM and DFT. The latter was done using the VASP code [38], the Perdew–Burke–Ernzerhof (PBE) approximation [39] for the exchange and correlation energy, the projector augmented wave (PAW) method to describe electron-core interactions [40], and supercells formed of 215 atoms + 1 vacancy. The total energy of the relaxed TiN system was evaluated to an accuracy of 10

^{–5}eV/supercell, using 3 × 3 × 3 k-point grids and 400 eV cutoff-energy for planewaves. The energy of isolated N and Ti atoms was computed accounting for spin relaxation.

_{m}), and phase transitions induced in AlN upon changing temperature T and/or pressure P. This was accomplished by means of 5 to 30 ps long CMD simulations at finite T and P values for cell sizes of ≈1000 atoms. Structural stability was tested by verifying that the simulated systems exhibit no unphysical behavior during 30 ps and that total energies are conserved during microcanonical NVE sampling at temperatures of ≈0.5 T

_{m}. Melting points were rapidly estimated as the temperature values for which the derivative of equilibrium volumes versus T becomes sharply discontinuous by performing 5 ps CMD isobaric–isothermal NPT simulations at different temperatures near experimental T

_{m}values. The relative stability of competing phases was assessed by calculating the free energies of such phases on a grid of T and/or P values via thermodynamic integration (TI) [41]. The reference energy for TI is the free energy of an Einstein crystal, i.e., a system of non-interacting harmonic oscillators with frequencies determined from the mean square displacements (MSD) of the material system under consideration [42]. Using the assessment process described above, along with the ASA optimization scheme, we defined the best set of MEAM parameters, which is validated as explained in Section 2.2.

#### 2.2. Potential Validation Methodology

_{1−x}Al

_{x}N (0 ≤ x ≤ 1); (ii) finite-temperature phonon spectra of B1-TiN, B1-AlN, and B4-AlN; (iii) nitrogen- and metal-vacancy migration energies for B1-TiN, B1-AlN, and B4-AlN, as well as diffusion energetics of Al interstitials in B1-TiN; (iv) lattice and elastic constants of sub-stoichiometric B1-TiN

_{y}(0.7 ≤ y ≤ 1), B2-TiN (space group Pm–3m), body-centered tetragonal (bct) Ti

_{2}N (space group I4

_{1}/amd); (v) free energies of B1-AlN and B4-AlN and dynamics of B4- to B1-AlN phase transformation; and (vi) B1-Ti

_{1−x}Al

_{x}N alloy mixing free energies with respect to B1-TiN and B1-AlN.

_{1−x}Al

_{x}N random solid solutions, we imposed a symmetric force constant matrix in the framework of TDEP. This was implemented by using reference ideal B1 primitive cells with the metal sublattice occupied using a single elemental species Q of mass M

_{Q}= (1–x)·M

_{Ti}+ x·M

_{Al}, as proposed in Ref. [51]. The configurational entropy S = –k

_{B}[x·ln(x) + (1–x)·ln(1–x)] per formula unit of B1-Ti

_{1−x}Al

_{x}N alloys was evaluated via the mean-field theory approximation. Binodal and spinodal compositions x were determined as a function of temperature. Binodal points, which separate stable from metastable alloy regions in the composition space, were obtained from the tangents to the B1-Ti

_{1−x}Al

_{x}N mixing free energy curves [52]. Spinodal points, which separate metastable from unstable B1-Ti

_{1−x}Al

_{x}N regions, were determined from the change in sign in the second derivative of alloy mixing free energies versus x [52].

_{y}(0.7 ≤ y ≤ 1) were determined by averaging over the values calculated for ≈100 different configurations (supercells containing 512 lattice sites) using the scheme described in Ref. [55]. The dynamics of B4- to B1-AlN transformation was studied via NPT CMD simulations at 300 K using a timestep of 0.1 fs. First, a supercell containing ≈20000 B4 AlN atoms was thermally equilibrated. Then, a pressure ramp with an average rate ≈7 GPa·ps

^{–1}was used during CMD/NPT simulations for a total duration of ≈0.2 ns. A video of the AlN transformation, produced with the Visual Molecular Dynamics [56] software, can be found in the Supplemental Material.

_{1−x}Al

_{x}N (0 ≤ x ≤ 1) potential.

## 3. Potential Parametrization Results

_{2}and N

_{3}molecules can also be found (Table S2) (see Supplemental Materials).

_{2}N, B1-AlN, B3-AlN, and B4-AlN along with their respective experimental (in brackets) and DFT (in parentheses) reference literature values [36,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89] are listed in Table 1. MEAM results for lattice parameters are in excellent agreement with DFT and experimental data. In addition, elastic properties calculated using our MEAM potential are, in general, in very good agreement with the reference values. Furthermore, our potential reproduces the established trend qualitatively with respect to the energetics (i.e., E

_{c}) of AlN polymorph structures; B4-AlN is more stable than B3-AlN, which is, in turn, more stable relative to B1-AlN. From a quantitative point of view, the cohesive energy difference E

_{c,B3-AlN}– E

_{c,B4-AlN}of 30 meV/atom calculated from MEAM is within the range of DFT values (≈20–40 meV/atom), while the MEAM E

_{c,B1-AlN}– E

_{c,B4-AlN}value of 68 meV/atom is lower than DFT predictions (≈150–200 meV/atom). Moreover, we find that the MEAM parameters yield an asymmetric B1-Ti

_{1−x}Al

_{x}N ΔH

_{mix}versus x curve that exhibits a downward concavity and a maximum of ≈230 meV/f.u at x = 0.6. This is within the range of DFT predictions by Alling et al. [3,90,91,92], Wang et al. [93], and Mayrhofer et al. [94], which found ΔH

_{mix}maxima in the interval ≈210–280 meV/f.u. for x ≈ 0.62.

^{f}

_{V}values for N and Ti vacancies (N

_{V}and Ti

_{V}, respectively), obtained using a N atom in a N

_{2}molecule and of a Ti atom in hcp-Ti lattice as reference chemical potentials, are E

^{f}

_{NV}= 3.53 eV and E

^{f}

_{TiV}= 4.76 eV. The corresponding DFT-based values, found in the literature, lie in the ranges 2.41–2.53 eV (N

_{V}) [95,96] and 2.86–3.11 eV (Ti

_{V}) [96,97]. The discrepancy between MEAM and DFT results becomes less than 10% when using the chemical potential of isolated N and Ti for computing E

^{f}

_{V}. In this case, MEAM yields E

^{f}

_{NV}= 8.41 eV and E

^{f}

_{TiV}= 9.81 eV, while the corresponding DFT values are E

^{f}

_{NV}= 7.52 eV and E

^{f}

_{TiV}= 9.71 eV.

_{2}dimer and of Al atom in fcc Al, are E

^{f}

_{NV}= 2.15 eV and E

^{f}

_{AlV}= 5.25 eV for B1-AlN, and E

^{f}

_{NV}= 3.50 eV and E

^{f}

_{AlV}= 6.66 eV for B4-AlN. DFT results [98,99] are in the range between –3 and 4 eV for N

_{V}and –2 and 10 eV for Al

_{V}, i.e., in the same order of magnitude with our MEAM values. It is important to note that the formation energy of point defects in semiconductors is largely determined by their respective charge state. Since the MEAM formalism does not account for charge states, a strict quantitative comparison between classical and ab initio data for vacancy formation energies in AlN is not meaningful.

_{1−x}Al

_{x}N solid solutions versus x calculated from MEAM at 0 K (see Figure 1) are in excellent agreement with DFT (see figure 2 in Ref. [100] and figure 6 in Ref. [92]) and experimental (see figure 6 in Ref. [92]) results. The elastic constants and Zener’s elastic anisotropy factor A = 2·C

_{44}/(C

_{11}–C

_{12}) obtained from MEAM calculations for B1-Ti

_{1−x}Al

_{x}N are plotted as a function of x in Figure 2a. The bulk moduli B and C

_{11}elastic constantly decrease, while C

_{44}, C

_{12}, and the factor A increase monotonically with increasing x. This evolution is in excellent agreement with DFT data [101]. Moreover, our potential yields the same as in DFT calculations [101], stoichiometry (x = 0.28) at which B1-Ti

_{1−x}Al

_{x}N solid solutions are elastically isotropic (A = 1). The latter is a strong indication that our MEAM parameters reproduce, in a reliable fashion, interatomic forces and energetics of Ti-Al-N systems.

_{12}/C

_{11}versus C

_{44}/C

_{11}ratios calculated from our MEAM potential. This representation is known as Blackman’s diagram [102] and entails information on the elastic anisotropy and bonding characteristics of the crystal. The black solid line in Figure 2b represents the condition C

_{12}= C

_{44}. The regions above (below) the C

_{12}= C

_{44}line indicate positive (negative) Cauchy’s pressures C

_{12}− C

_{44}. Based on phenomenological observations, Pettifor suggested that the Cauchy pressure can be used to assess the bonding character in cubic systems: positive values indicate metallic bonds, while negative values indicate more directional and covalent-like bonds [103]. We see that, for B1-Ti

_{1−x}Al

_{x}N alloys, C

_{12}− C

_{44}become progressively more negative for increasing AlN contents (blue curve in Figure 2b). This is due to the increasing directionality of the bonds caused by the presence of Al, and it is in agreement with DFT results [101].

_{m}of all investigated elementary systems, estimated from CMD simulations, are in reasonable agreement with their reference literature values (see Supplemental Material). In the case of AlN, we estimated a T

_{m}value of ≈3200 K. To our knowledge, the melting point of AlN has not been experimentally determined, and previous ab initio predictions indicate that AlN melts at approximately 3000 K [60]. For B1-TiN, our MEAM parameters yield a T

_{m}of approximately 6000 K, which is considerably higher than the experimental value (≈3250 K) [104]. It should be emphasized that CMD simulations using NPT sampling in defect-free crystals may lead to overestimations of melting points relative to values predicted using CMD modelling of solid–liquid interfaces at equilibrium [105]. It is reasonable to expect that such overestimations are greater for TiN versus AlN since the absence of polymorphs competing with the B1-TiN at atmospheric pressure reduces the possibility of nucleating heterogeneous sites that can facilitate melting.

## 4. Potential Validation Results

_{1−x}Al

_{x}N (0 ≤ x ≤ 1) alloys. Hence, in this section we present a thorough validation of the proposed set of Ti-Al-N MEAM parameters focusing on finite-temperature properties, which are difficult to implement within the ASA scheme. We study lattice thermal expansion and dynamics, defect migration energetics, and structural stability and transformations in key binary Ti-N and Al-N phases. These properties, in combination with studies of phase energetics in a multitude of Ti

_{1−x}Al

_{x}N (0 < x < 1) alloy compositions, provide a solid foundation for assessing the robustness and the reliability of the potential beyond the phases and configurations used in the ASA procedure.

#### 4.1. Lattice Thermal Expansion

_{1−x}Al

_{x}N (x = 0, 0.25, 0.5, 0.75, and 1) lattice parameters obtained via CMD simulations is represented in Figure 3. The CMD results (red solid lines in Figure 3a–e) are in good agreement with previous experimental (blue dots in Figure 3a–c) and AIMD simulation data (green solid lines in Figure 3a–e) and black solid line in Figure 3e) [50,71].

_{B1-TiN}(0 K) = 4.252 Å (see Table 1), which is within the range of 0 K DFT calculations (4.188–4.256 Å) based on different electronic exchange/correlation approximations [36]. The relative increase in lattice parameter values calculated for T = 2000 K via CMD simulations is equal to 1.2%, which is comparable to AIMD predictions (2.0%) from Ref. [59] (see solid green line in Figure 3a). The mean linear thermal expansion coefficient of B1-TiN is thus calculated to monotonically increase from 6.7 × 10

^{–6}K

^{–1}at 300 K to 8.1 × 10

^{–6}K

^{–1}at 3000 K. The MEAM estimates are slightly lower than the corresponding experimental values, which lay in the range (7–10) × 10

^{–6}K

^{–1}[104,107,108,109]. Even though our potential underestimates the thermal expansion of B1-TiN, the discrepancy between MEAM and experimental a

_{B1-TiN}is within ≈1% from room temperature up to 3000 K.

_{B1-TiAlN}becomes more pronounced with increasing AlN content x. This is consistent with the results of DFT calculations based on the quasi-harmonic approximation, AIMD simulations [59], and synchrotron X-ray diffraction [109]. For instance, at T = 2000 K, ∆a

_{B1-TiAlN}(T) values determined using CMD are found to increase from 1.2 to 3.3% for x increasing from 0 (B1-TiN) to 1 (B1-AlN). For reference, AIMD simulation results [71] showed that ∆a

_{B1-TiAlN}(2000 K) increases from 2.0 to 2.5% for the same x range.

^{–6}K

^{–1}. For the same temperature range, MEAM results, Figure 3e, yield a thermal expansion coefficient of ≈13 × 10

^{–6}K

^{–1}.

#### 4.2. Lattice Dynamics

_{0.98}compounds. Vibrational frequencies measured via neutron scattering [111] are shown as green circles (transversal modes) and red squares (longitudinal modes). The 300 K CMD dispersion curves (Figure 5a) are in good qualitative agreement with both experimental data (Figure 5a) and AIMD results of Figure 5c. Moreover, our CMD simulations show that an increase of temperature from 300 (Figure 5a) to 1200 K (Figure 5b) does not significantly affect the vibrational frequencies in B1-TiN, in qualitative agreement with the corresponding AIMD data in Figure 5c,d. On a quantitative level, we observe that CMD predicted optical phonon bandwidths of ≈6 THz (Figure 5a,b), which is larger than the corresponding values from experiments (≈4 THz, Figure 5a) [100] and AIMD (≈2 THz, Figure 5c,d). We also found that CMD simulations cannot not reproduce the phonon softening observed in experimental and AIMD data (Kohn anomalies on acoustic modes at L zone boundaries and on the Γ→X and Γ→K paths), since the screening function employed in the MEAM formalism [21] removes long-range interactions.

#### 4.3. Point-Defect Migration Energies

_{1−x}Al

_{x}N, is believed to be kinetically controlled by diffusion of lattice vacancies [6,113].

_{V}and Al

_{V}) and N (N

_{V}) vacancy migration in B1-TiN (Figure 7a), B1-AlN (Figure 7b), and B4-AlN (Figure 7c,d). We found that the migration energies for Ti

_{V}and N

_{V}in B1-TiN along <110> directions are 4.13 and 4.24 eV, respectively, with saddle-point transition states located halfway between initial and final atomic positions (Figure 7a). The MEAM predictions are in good agreement with DFT-NEB results, which yield energy barriers of ≈3.8 eV for N

_{V}[114] and 4.26 eV for Ti

_{V}[97]. Experimental values for vacancy migration energies in B1-TiN are scarce in the literature. Kodambaka et al. [115] determined a global (i.e., both Ti

_{V}and N

_{V}) value of 4.5 ± 0.2 eV for vacancy diffusion energies in bulk B1-TiN. Hultman et al. [116] estimated activation energies for metal interdiffusion at TiN/ZrN superlattice interfaces that range between 2.6 and 4.5 eV, while Wood and Paasche [117] and Anglezio-Abautret et al. [118] reported that activation barriers for N diffusion in B1-TiN lie in the range 1.8 to 5.5 eV.

_{V}and N

_{V}, we also studied formation and diffusion of Al interstitials (Al

_{I}) in B1-TiN. Using the energy of an Al atom in fcc-Al as a chemical potential, we calculated that the energy required for forming an Al

_{I}at tetrahedral sites is 4.92 eV, which is in reasonable agreement with the DFT value of 3.81 by Mei et al. [96]. Then, the corresponding Al

_{I}migration energy across tetrahedral B1-TiN sites was found to be equal to 2.42 eV, which matches perfectly the DFT value from Ref. [96].

_{V}diffusion within and across the B4-AlN (0001) plane yield activation energies of 1.97 and 2.19 eV, respectively (see Figure 7c). These values are within the experimental uncertainty range of O and N interdiffusion activation energies at Al

_{2}O

_{3}/AlN interfaces (2.49 ± 0.42 eV) [119]. For Al

_{V}, we found that migration across the (0001) B4-AlN lattice planes requires an activation energy of 2.29 eV, which is lower than the value of 2.72 eV for in-plane diffusion (Figure 7d). Moreover, our potential predicts that Al

_{V}transport in B1-AlN occurs with an activation energy of 2.47 eV, which is similar to that required in the B4-AlN polymorph. Concurrently, N

_{V}migration in B1-AlN requires significantly higher activation energy (4.00 eV) than that in the B4 structure (Figure 7b). We also note that no experimental and/or theoretical data on diffusion of point defects in B1- and B4-AlN are available in the literature to compare with our MEAM results.

_{1−x}Al

_{x}N samples by Mayrhofer et al. [6] and Norrby et al. [120] attributed activation energies of ≈3.3 eV (in Ti

_{0.36}Al

_{0.64}N) and ≈3.6 eV (in Ti

_{0.55}Al

_{0.45}N) to spinodal decomposition and B1-to-B4 transformations within AlN-rich domains, which in turn, is primarily attributed to diffusion of metal and nitrogen vacancies. The experimental estimates of atomic migration energies during spinodal decomposition in Ti-Al-N (3.3–3.6 eV) are consistent with the range of values that we obtain for cation and anion diffusion in binary Ti-N and Al-N.

_{1−x}Al

_{x}N alloys containing defects, i.e., monovacancies, divacancies, interstitials, and interstitial pairs at different temperatures. We found that point defect migration and point-defect/point-defect interactions do not cause unphysical structural transformations within the alloys during the investigated time scales, which are of the order of one nanosecond. This, together with the results presented above, lends confidence that our model potential is suitable to investigate phase transformation phenomena in Ti-Al-N solid solutions.

#### 4.4. Equilibrium Volumes and Elastic Properties of B1-TiN_{y}, B2-TiN, and bct-Ti_{2}N

_{2}N. To verify transferability of our model to a variety of Ti-N lattice configurations and bonding geometries, which may be encountered during simulations of dynamic processes in Ti-Al-N alloys, we present here the elastic and structural properties calculated for nitrogen-deficient B1-TiN

_{y}(0.69 ≤ y < 1) as well as high-pressure B2-TiN and bct-Ti

_{2}N phases.

_{y}, understoichiometry (y < 1) is primarily accommodated by anion vacancies [123]. In addition, control of the N content during synthesis can be used to tune the TiN

_{y}optical [124], electrical [106], and mechanical [84,125] properties. In Figure 8, we plot the lattice parameter a

_{TiNy}(Figure 8a), Young’s modulus E (Figure 8b), and elastic constants C

_{11}and C

_{44}(Figure 8c,d, respectively) as predicted using 0 K calculations with our MEAM potential along with experimental and DFT data for comparison. We observe (Figure 8a) that a

_{TiNy}increase monotonically with increasing the N content y, in excellent agreement (maximum discrepancy ≈0.5%) with experimental measurements [84]. Remarkably, our MEAM potential reproduces the experimental a

_{TiNy}versus y trend better than DFT [126]. Other results (not included in Figure 8a) show that, depending on the choice of the electronic exchange-correlation approximation, DFT overestimates or underestimates the lattice parameter of stoichiometric TiN by up to 1.3% [36]. DFT calculations [126,127] and experiments (acoustic wave velocities and nanoindentation tests [84,85,128]) have demonstrated that E, C

_{11}, and C

_{44}, in B1-TiN

_{y}increase monotonically as function of y, as shown in Figure 8b–d. Our MEAM results reproduce the above-described trend and specific values of the elastic constants well.

_{2}N phases for comparison with experimental and ab initio results. For bct-Ti

_{2}N, our model yields lattice constants a = 4.152 Å and c = 8.930 Å, in excellent agreement with experimentally-determined ranges of a = 4.140–4.198 Å and c = 8.591–8.805 Å [73], and DFT values of a = 4.151 Å and c = 8.880 Å [129]. The elastic properties determined by MEAM for the bct-Ti

_{2}N phase were consistent with DFT values (given in the parenthesis) as explained in the following: B = 177 (179) GPa, C

_{11}= 456 (372) GPa, C

_{33}= 207 (287) GPa, C

_{12}= 205 (126) GPa, C

_{13}= 120 (87) GPa, C

_{44}= 45 (70) GPa, and C

_{66}= 125 (109) GPa. For the cubic B2 TiN-phase, MEAM yields a lattice parameter of 2.717 Å and a bulk modulus of 256 GPa versus a = 2.638 Å and B = 249 GPa calculated via DFT (present work) employing hard (i.e., optimized to model high-pressure properties) PBE exchange-correlation functionals.

#### 4.5. Phase Stability and Transitions

#### 4.5.1. Ti-N

_{2}N phases, which are the ground-state configurations for the TiN and Ti

_{2}N systems, were reproduced via the ASA parametrization (see Section 3 and Table 1). Post-parametrization calculations were carried out to verify that our model predicted correct trends in the energetics for high-pressure metastable B2-TiN and bct-Ti

_{2}N polymorph structures. The MEAM potential predicts that the B1-TiN structure was ≈1.6 eV/atom more stable than the B2-TiN phase, in fair agreement with present DFT calculations which yield an energy difference of ≈0.9 eV/atom between the two phases. Moreover, we find that ε-Ti

_{2}N is 98 meV/atom more stable than the bct-Ti

_{2}N polymorph, which is qualitatively consistent with DFT results that have shown an energy difference of 16 meV/atom in favor of the ε-Ti

_{2}N structure [129].

#### 4.5.2. Al-N

_{m}≈ 3200 K). This is consistent with the result that MEAM underestimates the energy difference E

_{c,B4-AlN}– E

_{c,B1-AlN}(see Table 1).

^{–1}, at 300 K (see more details in Section 2.2). After a monotonic increase up to a value of 110 ± 10 GPa, the stress accumulated in B4-AlN is partially relieved. As we discuss in detail below, this stress relief is due to the formation of small B1-AlN grains.

^{–10}s), compression experiments are conducted at close-to-equilibrium conditions (≥10

^{2}s) and on polycrystalline B4-AlN samples. Hence, pressures much larger than those predicted by thermodynamics (phase diagrams) are necessary to quickly activate the B4-to-B1 AlN transition during the timescales accessible via molecular dynamics simulations. The fact that our CMD results (Figure 9) match AIMD predictions [133] suggests that the kinetic free-energy barrier that separates the B4- and B1-AlN phases is accurately reproduced by MEAM.

#### 4.5.3. Ti-Al-N

_{1−x}Al

_{x}N solid solutions with high (x > 0.5) AlN contents are known to decompose via the spinodal route, at temperatures near 1200 K [7,8], leading to formation of coherent Ti- and Al-rich B1-Ti

_{1−x}Al

_{x}N domains. This hinders dislocation motion across strained domain interfaces and results in the age-hardening effect, which is of extreme importance for metal cutting at elevated temperatures [134,135]. A further increase in temperature causes B1-to-B4 structural transformation within Al-rich domains, which is detrimental for hardness and coating performance [7,8,134,135]. To date, the atomistic mechanisms that control spinodal decomposition and subsequent formation of B4-AlN in Ti-Al-N coatings are unclear. These processes can be elucidated by means of large-scale CMD simulations using our MEAM potential. The suitability of the potential for such simulations is explored in this section by discussing the B1-Ti

_{1−x}Al

_{x}N phase diagram predicted using MEAM, in comparison to first-principle and experimental results.

_{1−x}Al

_{x}N binodal and spinodal curves calculated on the basis of CMD free energies of mixing are presented in Figure 11. It is important to note that our phase diagram ends at the AlN melting point (T

_{m}≈ 3200 K); the temperature beyond which the free-energy of pure B1-AlN (necessary to compute the alloy free energy of mixing) is not defined. This, however, does not imply that B1-AlN-rich regions may not form within B1-Ti

_{1−x}Al

_{x}N solid solutions during CMD simulations at T > 3200 K. Our estimated binodal and spinodal regions, Figure 11, are in very good agreement with recent thermodynamic assessments (see figure 2 in Ref. [136]), as well as ab initio calculations accounting for vibrational effects on alloy free energies (see figure 3 in Ref. [93]). Our CMD results also match reasonably well with the Ti-Al-N phase diagram determined via AIMD simulations (see figure 2 in Ref. [137]).

_{c}in the range ≈7500–9000 K. Subsequent first-principles investigations by Wang et al. [93] showed that accounting for vibrational entropies S

_{vib}in Ti-Al-N free energies of mixing via the Debye–Grüneisen approximation significantly lowered the calculated T

_{c}(≈3800 K). More recently, AIMD results by Shulumba et al. demonstrated that the explicit inclusion of anharmonic effects in S

_{vib}further reduces the predicted T

_{c}to 2900 K [137]. Consistent with the observations of Refs. [93,137], our CMD-based evaluations also show that the implicit inclusion of S

_{vib}yields narrower miscibility gaps at temperatures above ≈1500 K, as compared to those close to room temperature.

_{1-x}Al

_{x}N phase diagram which is skewed toward the TiN end and with the lowest T

_{c}(≈2100–2200 K) calculated so far (see figure 1 in Ref. [138]).

_{1-x}Al

_{x}N solid solutions. These are included for comparison with our results in the inset of Figure 11, where black stars denote stable or metastable cubic alloy compositions, and red stars mark spinodally-decomposed Ti-Al-N samples. A first apparent inconsistency between theoretical and experimental assessments of miscibility gaps can be seen for T ≤ 1000 K; the experimental observations (black stars) indicate stability (or metastability) of B1-Ti

_{1−x}Al

_{x}N alloys for AlN concentrations well within the spinodal compositional range (0.25 < x < 1) predicted by the CMD phase diagram. Refractory metal-nitride compounds exhibit relatively low atomic mobilities [139], which may shift the onset of structural transformation to temperatures larger than those predicted in theoretical models. In order to explore possible effects of low point-defect mobilities on the outcome of annealing experiments conducted on Ti-Al-N, we consider the case of the parent compound TiN, for which nitrogen diffusivities D(T) are available in the literature [36] (in TiN, Ti migration is even slower than N migration [97]). We consider the model system B1-TiN

_{1–y}with nitrogen vacancy concentrations at the dilute limit (y ≈ 10

^{–5}) and diffusivities D(1100 K) ≈ 10

^{–20±2}cm

^{2}·s

^{–1}and D(800 K) ≈ 10

^{–33±2}cm

^{2}·s

^{–1}, as shown in figure 11 of Ref. [36]. Annealing B1-TiN

_{1−y}at a temperature T = 1100 K during one day (t ≈ 10

^{5}s) would induce nitrogen transport over distances d ≈ [6·t·D(1000 K)]½ of a few nm. A modestly lower annealing temperature (800 K) would require considerably longer times (years) to observe diffusion over comparable length scales. Given the similarity in chemical bonds and crystal structures, it is reasonable to assume that nitrogen and metal atoms in Ti-Al-N migrate at rates comparable to those discussed above for TiN, thus providing a possible explanation for the retained stability of B1-Ti

_{1−x}Al

_{x}N solid solutions at T ≤ 1000 K (inset in Figure 11).

## 5. Summary and Outlook

_{1−x}Al

_{x}N (0 ≤ x ≤ 1) alloy system. This was achieved by developing an adaptive simulated annealing (ASA) algorithm for scanning the MEAM multi-dimensional parameter space and identifying a set of parameters that optimize the potential’s predictions relative to 0 K experimentally- and ab-initio-determined equilibrium volumes, cohesive energies, elastic constants, defect formation energies, and mixing enthalpies of elemental, binary, and ternary phases and configurations within the Ti-Al-N system. We then validated the transferability of the parameters selected via the ASA procedure by establishing the ability of the potential to reproduce known finite-temperature kinetic and thermodynamic properties, not explicitly included in the ASA optimization, for B1-Ti

_{1−x}Al

_{x}N (0 ≤ x ≤ 1), B4-AlN, and various Ti-N phases. We found that, overall, the potential reproduces the following well: (i) temperature-dependence of equilibrium volumes in B1-Ti

_{1−x}Al

_{x}N and B4-AlN; (ii) phonon dispersion curves of B1-TiN, B1-AlN, and B4-AlN; (iii) point-defect migration energies in B1-TiN, B1-AlN, and B4-AlN; (iv) lattice and elastic constants of sub-stoichiometric B1-TiN

_{y}(0.7 ≤ y ≤ 1), B2-TiN, and bct-Ti

_{2}N; (v) free energies of B1-AlN and B4-AlN, and dynamics of B4- to B1-AlN pressure-induced phase transformation; and (vi) B1-Ti

_{1−x}Al

_{x}N alloy mixing free energies and binodal/spinodal phase boundaries with respect to B1-TiN and B1-AlN.

_{1−x}Al

_{x}N solid solutions, multilayers, and self-organized nanostructures and can provide critical insights onto the atomistic pathways that drive segregation via the spinodal route and phase transformation via nucleation and growth. Our MEAM potential can also be used to simulate elastic and plastic responses of B1-Ti

_{1−x}Al

_{x}N alloys, which together with thermal stability simulations can guide the design of protective coatings with superior mechanical and oxidation performance. Other areas where our MEAM set of parameters can be used include diffusion of Al in B1-Ti

_{1−x}Al

_{x}N used as barrier layer and contact material in microelectronic devices [140,141,142]. Moreover, the present set of MEAM parameters can be used as starting point for developing potentials that can simulate vapor-based growth of Ti-Al-N films. Beyond Ti-Al-N, the methodologies presented in this work can serve as the foundation for developing potentials for quaternary systems, e.g., Ti-Al-Si-N, Ti-Al-Ta-N, and Ti-Al-Nb-N, where complex metal-sublattice configurations are used to enhance surface wear and oxidation resistance, and inhibit wurtzite-phase formation subsequent to spinodal decomposition, which allows extending the age hardening effect to temperatures above 1200 K [143,144,145,146,147,148,149,150].

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**(

**a**) Elastic constants B (cyan circles), C

_{11}(black squares), C

_{12}(green triangles), and C

_{44}(blue inversed triangles), as well as Zener’s elastic anisotropy A = 2·C

_{44}/(C

_{11}–C

_{12}) (red diamonds), as a function of x within the Ti

_{1−x}Al

_{x}N system. The red circle and arrow are used to indicate that the elastic anisotropy data point refers to the right-axis scale. (

**b**) Blackman’s diagram constructed using data from (a). The 45° solid black line represents the zero Cauchy pressure, C

_{12}= C

_{44}. The red solid line represents the conditions for elastic isotropy A = 1.

**Figure 3.**Variation of B1-Ti

_{1−x}Al

_{x}N lattice parameter a as a function of temperature T {∆a(T) = [a(T) – a(300 K)]/a(300 K)} for (

**a**) x = 0, (

**b**) x = 0.25, (

**c**) x = 0.5, (

**d**) x = 0.75, and (

**e**) x = 1. AIMD (solid green and black curves) and experimental (blue dots) results are adapted from figure 1 in Ref. [71]. AIMD curves (green lines) shown in the present figure were obtained by fitting the data in the temperature range 0 to 2000 K of Ref. [71] with a second order polynomial. The AIMD curve (solid black line) in (

**e**) was obtained using the data presented for B1-AlN equilibrium volume versus T in figure 2a in Ref. [50].

**Figure 4.**(

**a**) Temperature dependence of B4-AlN and B1-AlN equilibrium volumes as predicted using CMD and AIMD simulations [50] and experiments [86]. The experimental curve (blue) was obtained from the a and c/a ratio variation versus T determined in Ref. [86]. (

**b**) Temperature dependence of a, c, and c/a ratio of B4-AlN lattice parameters as predicted using CMD (red solid line) and measured using X-ray powder diffractometry (blue solid line) in the temperature range 300 to 1400 K from Ref. [86].

**Figure 5.**B1-TiN phonon dispersion curves calculated via: (

**a**) CMD for 300 K, (

**b**) CMD for 1200 K, (

**c**) AIMD for 300 K, and (

**d**) AIMD for 1200 K. Experimental data (green dots for transversal and red squares for longitudinal modes) obtained using neutron scattering measurements on B1-TiN

_{0.98}bulk samples from Ref. [111] are included for comparison.

**Figure 6.**B1- and B4-AlN CMD phonon spectra and phonon densities of states (PDOS) calculated at 300 and 2400 K.

**Figure 7.**Minimum energy pathways for migration of N (N

_{V}) and metal (Ti

_{V}, Al

_{V}) vacancies among nearest-neighbor sites in the anion (for N

_{V}) and cation (for Ti

_{V}and Al

_{V}) sublattices in (

**a**) B1-TiN, (

**b**) B1-AlN, and (

**c**,

**d**) B4-AlN. For B4-AlN structures, we studied vacancy migration both within, as well as across, the (0001) plane.

**Figure 9.**Stress as a function of time/pressure in the B4-AlN simulation box during the CMD runs. The abrupt drop in stress at ≈17 s (≈110 GPa) corresponding to the formation of small B1-AlN grains, which is the first step for the B4- to B1-AlN phase transformation.

**Figure 10.**CMD simulation snapshots of the B4- to B1-AlN phase transformation induced by pressure. B4-AlN is seen from the [0001] direction. See text for details and explanation of the features observed in the figure.

**Figure 11.**Spinodal and binodal curves for B1-Ti

_{1−x}Al

_{x}N solid solutions calculated using CMD free energies of mixing with respect to B1-TiN and B1-AlN. The inset compares our results to experimental data (black and red stars mark B1-Ti

_{1−x}Al

_{x}N and spinodally-decomposed solid solutions, respectively) collected in Ref. [138].

**Table 1.**MEAM-predicted cohesive energies (E

_{c}), lattice constants (a, c), and elastic constants (B, C

_{ij}) for B1-TiN, ε-Ti

_{2}N, B1-AlN, B3-AlN, and B4-AlN. Experimental and DFT values are listed in brackets and parenthesis, respectively.

B1-TiN | ε-Ti_{2}N | B1-AlN | B3-AlN | B4-AlN | ||
---|---|---|---|---|---|---|

E_{c} | 6.613 | 6.180 | 5.690 | 5.728 | 5.758 | |

(eV/at.) | [6.69 ± 0.07^{c1}] (6.8^{d1}, 8.708^{c1}) | (5.597^{e1}) | (6.621^{e}, 5.681^{e}, 5.44^{f}) | [5.76^{g}, 5.76^{o}] (5.701^{e}, 5.779^{e1}, 6.643^{e}, 5.055^{r}, 5.545^{r}) | ||

Energy above hull(meV/at.) | 68 | 30 | 0 | |||

- | - | (147^{q}, 204^{a}, 172^{c}, 182^{e1}) | (43^{q}, 23^{a}, 21^{b}, 21^{e}, 22^{e}, 41^{f}) | |||

a | 4.252 | 4.939 | 4.090 | 4.366 | 3.112 | |

(Å) | [4.240^{z}](4.188–4.254 ^{s}) | [4.938–4.946^{l}] (4.955*, 4.960^{j}) | [4.046^{u}, 4.064^{w}](4.014–4.070 ^{v}, 4.06^{a}, 4.069^{c}, 4.071*) | [4.37^{d}, 4.38^{p}] (4,349^{y}, 4.39^{a}, 4.320^{x}, 4.401^{b}, 4.310^{e}, 4.394^{e}, 4.374^{f}) | [3.111^{d}, 3.111^{b1}, 3.110–3.113^{f}] (3.12^{a}, 3.06^{x}, 3.100^{f}, 3.113^{e}, 3.057^{e}, 3.129^{k}, 3.101^{r}, 3.117^{r}) | |

c/a | 0.616 | 1.600 | ||||

- | [0.613–0.614^{l}] (0.612^{j}, 0.613*) | - | - | [1.601^{d}, 1.600^{b1}, 1.602^{h}, 1.600–1.602^{f}] (1.596^{q}, 1.603^{a}, 1.60^{x}, 1.619^{e}, 1.617^{e}, 1.609^{f}, 1.603^{k}, 1.598^{r}, 1.604^{r}) | ||

B | 298 | 208 | 277 | 237 | 236 | |

(GPa) | [298–324^{a1}](277 ^{n}, 303^{t}, 290–350^{s}) | (204^{j}, 214*) | [221 ± 5^{m}, 295 ± 17^{u}, 319 ± 8^{w}](253–277 ^{v}, 270^{q}, 207^{n}, 265^{t}, 255^{c}, 261*) | [202^{p}](213 ^{y}, 216^{q}, 195^{b}, 206^{e}, 209^{x}, 191^{e}, 218^{f}, 228^{r}) | [208 ± 6^{h}, 211^{b1}, 185 ± 5^{m}, 185–212^{f}, 185–237^{o}, 303 ± 4^{w}] (205^{q}, 202^{x}, 209^{e}, 192^{e}, 194^{k}, 228–243^{r}) | |

C_{11} | 613 | 309 | 480 | 289 | 432 | |

(GPa) | [626^{z}, 605–649^{a1}](590 ^{n}, 610^{t}, 640–710^{s}) | (429^{j}, 434*) | (340^{n}, 425^{t}, 428^{c}, 432*) | [328^{p}] (309^{y}, 284^{b}, 298^{x}, 348^{r}) | [395^{p}, 411^{b1}, 410 ± 10^{i}, 345–411^{o}] (458^{x}, 376^{k}, 389–464^{r}) | |

C_{12} | 140 | 153 | 175 | 213 | 203 | |

(GPa) | [145–165^{a1}](120 ^{n}, 150^{t}, 115–125^{s}) | (105^{j}, 127*) | (140^{n}, 185^{t}, 168^{c}, 175*) | [139^{p}] (164^{y}, 150^{b}, 164^{x}, 168^{r}) | [125^{p}, 149^{b1}, 148 ± 10^{i}, 125–149^{o}] (154^{x}, 129^{k}, 149–158^{r}) | |

C_{44} | 165 | 130 | 271 | 100 | 70 | |

(GPa) | [156^{z}, 162–171^{a1}](160 ^{n}, 165^{t}, 159–169^{s}) | (151^{j},169*) | (260^{n}, 298^{t}, 307^{c}, 296*) | [133^{p}] (78^{y}, 179^{b}, 187^{x},135^{r}) | [118^{p}, 125^{b1}, 125 ± 5^{i}, 118–125^{o}] (85^{x}, 113^{k}) | |

C_{13} | 163 | 153 | ||||

(GPa) | - | (194^{j}, 205*) | - | - | [120^{p}, 99^{b1}, 99 ± 4^{i}, 95–120^{o}] (84^{x}, 98^{k}, 116–138^{r}) | |

C_{33} | 296 | 337 | ||||

(GPa) | - | (300^{j}, 337*) | - | - | [345^{p}, 389^{b1}, 388 ± 10^{i}, 394–402^{o}] (388^{x}, 353^{k}, 408–409^{r}) | |

C_{66} | 121 | 115 | ||||

(GPa) | - | (138^{j}, 136*) | - | - | [135^{p}, 131^{b1}, 131 ± 10^{i}, 130–131^{o}] (152^{x}, 124^{k}, 115–157^{r}) |

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Almyras, G.A.; Sangiovanni, D.G.; Sarakinos, K. Semi-Empirical Force-Field Model for the Ti_{1−x}Al_{x}N (0 ≤ x ≤ 1) System. *Materials* **2019**, *12*, 215.
https://doi.org/10.3390/ma12020215

**AMA Style**

Almyras GA, Sangiovanni DG, Sarakinos K. Semi-Empirical Force-Field Model for the Ti_{1−x}Al_{x}N (0 ≤ x ≤ 1) System. *Materials*. 2019; 12(2):215.
https://doi.org/10.3390/ma12020215

**Chicago/Turabian Style**

Almyras, G. A., D. G. Sangiovanni, and K. Sarakinos. 2019. "Semi-Empirical Force-Field Model for the Ti_{1−x}Al_{x}N (0 ≤ x ≤ 1) System" *Materials* 12, no. 2: 215.
https://doi.org/10.3390/ma12020215