Strength, transformation toughening and fracture dynamics of rocksalt-structure Ti1-xAlxN (0<= x<= 0.75) alloys

Ab initio-calculated ideal strength and toughness describe the upper limits for mechanical properties attainable in real systems and can, therefore, be used in selection criteria for materials design. We employ density-functional ab initio molecular dynamics (AIMD) to investigate the mechanical properties of defect-free rocksalt-structure (B1) TiN and B1 Ti1-xAlxN (x = 0.25, 0.5, 0.75) solid solutions subject to [001], [110], and [111] tensile deformation at room temperature. We determine the alloys' ideal strength and toughness, elastic responses, and ability to plastically deform up to fracture as a function of the Al content. Overall, TiN exhibits greater ideal moduli of resilience and tensile strengths than TiAlN solid solutions. Nevertheless, AIMD modelingshows that, irrespective of the strain direction, the binary compound systematically fractures by brittle cleavage at its yield point. The simulations also indicate that Ti0.5Al0.5N and Ti0.25Al0.75N solid solutions are inherently more resistant to fracture and possess much greater toughness than TiN, due to the activation of local structural transformations (primarily of B1 ->wurtzite type) beyond the elastic-response regime. In sharp contrast, TiAlN alloys with 25% Al exhibit similar brittleness as TiN. The results of this work are examples of the limitations of elasticity-based criteria for prediction of strength, brittleness, ductility, and toughness in materials able to undergo phase transitions with loading. Furthermore, comparing present and previous findings, we suggest a general principle for design of hard ceramic solid solutions that are thermodynamically inclined to dissipate extreme mechanical stresses via transformation toughening mechanisms.

While single-phase materials generally become softer with temperature [9,10], alloys as (Ti,Al)N are of considerable technological importance due to the spinodally-induced age hardening effect. Over the past decades, several studies [8,[11][12][13][14][15][16][17][18] focused on understanding the surface reactivity and thermodynamics of phase segregation in order to design (Ti,Al)N-based coatings with superior thermal stability and hinder B1→B4 AlN-domain transformations [19][20][21]. In contrast, the toughness and resistance to fracture of (Ti,Al)N and (Ti,Al)N-based solid solutions have not been investigated as extensively, with a few studies available in the literature [22][23][24][25]. Recent experiments suggest that, although detrimental for the alloy hardness, the nucleation of wurtzite phases in B1 AlN-rich regions does not affect, or is even beneficial for, the coating toughness by inhibiting crack formation and/or propagation [26,27]. Nonetheless, the presence of grain boundaries and voids, which act as weakest links [28] in polycrystalline samples, ultimately controls the resistance to fracture of (Ti,Al)N, thus preventing the possibility of describing the alloy mechanical response as a function of metal composition. Moreover, the fact that (Ti,Al)N ceramics are typically synthesized in the form of thin films complicates the experimental evaluation of their strength and toughness. These problems render first-principles approaches an indispensable tool for the investigation of the mechanical properties of single-crystal B1 (Ti,Al)N solid solutions.
As a first step toward understanding the intrinsic ability of defect-free B1 Ti1-xAlxN to withstand loading and plastically deform, we employ ab initio molecular dynamics (AIMD) simulations at 300 Ktemperature at which refractory ceramics are typically brittle [29][30][31] to investigate the effects induced by an increasing Al content (x=0, 0.25, 0.5, 0.75) on the alloys' responses to [001], [110], and [111] tensile deformation [32]. The simulations allow us to observe the dynamics of brittle cleavage vs. lattice-transformation-induced toughening as a function of the metal composition.

II. Computational methods
AIMD [33] simulations are performed using VASP [34][35][36] implemented with the projector augmented wave method [37]. The electronic exchange and correlation energies are parameterized according to the generalized gradient approximation of Perdew, Burke, and Ernzerhof [38]. All AIMD simulations employ Γ-point sampling of the reciprocal space and planewave cutoff energies of 300 eV. The nuclear equations of motion are integrated at 1 fs timesteps, using an energy convergence criterion of 10-5 eV/supercell for the ionic iterations. Prior to modeling tensile deformation, the supercell structural parameters are evaluated via NPT sampling of the configurational space (Parrinello-Rahman barostat [39] and Langevin thermostat set to 300 K).
Subsequently, AIMD within the NVT ensemble (Nose-Hoover thermostat, with a Nose mass of 40 fs) is used to equilibrate the structures at 300 K during three additional ps, ensuring that the timeaveraged stress components |xx|, |yy|, and |zz| are ≤ 0.3 GPa.
In order to model tensile deformation, as well as shear deformation leading to lattice slip (results presented in a parallel study [40] (Fig. 1). [h k l]-oriented supercells are denoted below as Ti1-xAlxN(h k l), where h, k, and l are Miller indexes. B1 Ti1-xAlxN (0 ≤ x ≤ 0.75) simulation boxes contain 288 metal and 288 nitrogen atoms (576 ideal B1 sites with 24 atomic layers orthogonal to the tensile strain z direction), applying periodic boundary conditions in three dimensions (Fig. 1). Al and Ti atoms are stochastically arranged on the cation sublattice, thus ensuring negligible degrees of shortrange metal ordering. Tensile deformation is carried out by following the scheme detailed in Ref. [32]. Briefly, at each strain step (2% of the supercell length along z), the structures are (i) first rapidly equilibrated by isokinetic velocity-rescaling during 300 fs and (ii) then maintained at the same temperature during additional 2.7 ps using the Nose-Hoover thermostat. At each strain step, tensile zz stresses are determined by averaging zz stresses calculated for the 500 final AIMD configuration. Moduli of ideal tensile resilience URenergy density accumulated during elastic deformation (i.e., up to the yield point)and ideal tensile toughness UTenergy density absorbed up to fractureare calculated by integrating the area underlying stress vs. strain curves up to the yield y and fracture f strains, respectively. The supercell size along the lateral x and y directions is maintained unvaried during tensile deformation. Images and videos are generated using the visual molecular dynamics [41] software. strain curves [42] within the alloy elastic-response up to =4% are used to calculate (see equations 2 and 4 in Ref. [43]) the C11 and C12 elastic constants as a function of x. AIMD results yield C11 elastic stiffnesses which, for x increasing from 0 to 0.75, monotonically decrease from 650±50 GPa to 528±38 GPa ( Table I). Noting that x and y supercell axes are parallel to <110> crystallographic directions ( Fig. 1(a)), the C12 elastic constant can be evaluated via 45° rotation of the stress tensor within the xy plane. The calculated C12 values monotonically increase with the Al content from 128±6 GPa (for x = 0) to 174±8 GPa (for x = 0.75). Accordingly, the bulk moduli B remain approximately constant, or exhibit slight reductions with Al substitutions ( Table I). The uncertainties on the C11 and C12 values arise from the sensitivity of calculated elastic constants on the choice of strain ranges and deformation tensors [44] and the presence of small residual stress components in the relaxed supercell structures. The influence of metal-species arrangements, which produces a large scatter on C11 and C12 values calculated for anharmonic transition-metal nitride alloys [45], is expected to have negligible effects on the elastic response of TiN and (Ti,Al)N solid solutions. The trends in, and absolute values of C11, C12, and B vs. x (Table I) agree, within uncertainty ranges, with those reported by previous ab initio calculations at 0 K [46,47] and AIMD simulations at room temperature [43,48].  Ti0.25Al0.75N(001). Conversely to the trend observed for the C11 elastic constants, which demonstrates a reduction in [001] stiffness for increasing x ( Fig. 2(a) and Table I Table II. To summarize, AIMD simulations demonstrate that the room-temperature Ti1-xAlxN mechanical response to [001] tensile deformation up to yield points, which approximate the limit for the elastic response, is not dramatically affected by Al substitutions. This is consistent with the fact that covalent N (p)metal (d-eg) bonding states remain fully occupied even though the valence electron concentration of B1 Ti1-xAlxN solid solutions decreases from 9 e-/f.u. (for TiN) to 8.25 e-/f.u. (for Ti0.25Al0.75N) [49][50][51]. Nonetheless, simulation results (see below) suggest that an increasing Al content significantly promotes the alloys' ability to plastically deform, thus improving the material's toughness.

III. Results and discussion
In agreement with AIMD results of Ref. [32,52], an extension of TiN(001) beyond its tensile yield point (≈10%) leads to brittle fracture of the material. AIMD modeling reveals that 25% replacement of Ti atoms with Al induces negligible effects on the alloy plastic response to [001] uniaxial deformation; cubic Ti0.75Al0.25N(001) solid solutions remain brittle and undergo sudden cleavage on the (001) plane at strains larger than 10% (see Fig. 2(a), Fig. 3 and Table II). In sharp contrast, Ti1-xAlxN alloys with Al contents x ≥ 0.5 are considerably more resistant to fracture than TiN and Ti0.75Al0.25N. This is due to their ability to undergo local structural changes into wurtzitelike atomic environments when the elongation overcomes their yield points (see Figs. 2(a), 4, and

5).
The modifications in the bonding network that become operative in B1 Ti0.5Al0.5N(001) and Ti0.25Al0.75N(001) solid solutions at high tensile strains can be rationalized on the basis of transformation pathways induced by pressure in wurtzite group-III nitrides, such as AlN (see examples of strain-mediated B4→B1 AlN transitions in Ref. [53]), which is a border (x=1) case for the investigated Ti1-xAlxN system [54,55]. Tetragonal [56] and hexagonal (graphitic-like, boronnitride prototype, Bk) crystal structures are the predicted transition states along the B4 → B1 transformation path of group-III nitrides and other semiconductors (see, e.g., figure 1 in Ref. [57]). Bk plane [55]. It is therefore expected that the inverse (B1 → B4) AlN phase transition should also preferentially occur through the Bk metastable configuration.
The formation of tetragonal (Ti,Al)N domains that precede the appearance of wurtzite-like environments is reminiscent of the solid→solid transformation path predicted for other B4-structure crystals as, e.g., GaN and ZnO [56]. As indicated in Ref. [56], B4→tetragonal→B1 transitions are presumably favored due to the presence of d-electrons (note that B1 Ti1-xAlxN with 0 ≤ x ≤ 0.75 is a conductor with d-states at the Fermi level [58]). The lattice transformation active in Ti0.5Al0.5N and Ti0.25Al0.75N ultimately results in a considerably enhanced resistance to fracture and a substantially increased toughness (area underlying stress/strain curves) during [001] tensile deformation (see Table II).
Given that (001) surfaces in B1-structure ceramics have much lower formation energies than (110) and (111)  For an increasing Al concentration, tensile-strained Ti1-xAlxN(110) and Ti1-xAlxN (111) exhibit a monotonic increase in elastic stiffness (i.e., initial slope in zz vs. strain), accompanied by an overall reduction in UR, (see Fig. 2(b,c) and Table II). These trends are opposite to that calculated for Ti1-xAlxN(001). In contrast, the [110] and [111] tensile strengths of the alloy are not significantly affected by Al substitutions. In fact, for each investigated strain direction, the relative strength variation with x remains within 10% ( Fig. 2 and Table II (37)(38)(39). This is consistent with the trend in surface formation energies Es(111) > Es(110) > Es(001) reported for B1-structure materials [59], that is, the uniaxial strength is related to the energy required to cleave the crystal on a plane normal to the elongation direction.  Fig. 2(b,c) and Table   II). Combined with the results described above for [001]-strained materials, these findings indicate that the room-temperature mechanical properties of B1 Ti1-xAlxN are considerably improved by Al substitutions of ≥ 50%. Consistent with AIMD results reported in a previous study [32,52], TiN(110) undergoes sudden brittle failure when the [110] uniaxial strain reaches ≈18%, (Fig. 2(b)). The mechanical response of Ti0.75Al0.25N solid solutions to [110] elongation is nearly equivalent to that determined for the binary compound ( Fig. 2(b)). AIMD simulation snapshots of Ti0.75Al0.25N(110) at a constant tensile strain of 18% display rapid (within 1.3 ps) bond snapping that causes brittle cleavage of the alloy (Fig. 7). Note that the fractured region follows a zig-zag pattern on (001) crystallographic planes. Conversely, B1 solid solutions that contain 50% and 75% Al undergo, after the yield point, local changes in bonding geometries which prevent sudden mechanical failure (in comparison to TiN(110) and Ti0. 75Al0.25N(110)). The transformation toughening effect induced by Al substitutions in [110]-strained Ti0.5Al0.5N and Ti0.25Al0.75N is illustrated by AIMD snapshots in Fig. 8 and Fig. 9, respectively.
Ti0.75Al0.25N(111) displays a mechanical response to [111] deformation qualitatively similar to that of the binary nitride, that is, brittle fracture occurs within few ps at constant strain of 18%, Fig. 2(c). 11 and 12 demonstrate that Ti0.5Al0.5N and Ti0.25Al0.75N break via a progressive, yet slow, reduction in bond densities induced by an increasing strain. A qualitative comparison with TiN (Fig. 10), reveals that elongations of ≈20% (Fig. 11) and ≈24% (Fig. 12) Fig. 2(c)). This is due to the fact that, while both materials fracture at 20% strain, the binary compound reaches mechanical yielding at a much higher elongation than the ternary alloy. In contrast, Ti0.25Al0.75N(111) solid solutions exhibit equal strength, but higher toughness, than TiN(111) owing to slow bond fraying which delays fracture up to an elongation of 22-24% (Fig. 12).
The results of this work provide fundamental insights of the mechanical properties of B1 (Ti,Al)N solid solution ceramics during use. However, it is important to underline that the macroscopic mechanical behavior and resistance to fracture of polycrystalline B1 (Ti,Al)N coatings are primarily controlled by microstructural features such as grain size, texture, and grain boundary properties. For example, cracks can more easily initiate and propagate at the interfaces between crystallites where the density is lower and voids may be present. Nonetheless, the toughening mechanisms observed in AIMD simulations can operate within (Ti,Al)N grains of sufficiently large size (less affected by grain boundary properties), when tensile stresses build up inside the grain. The elongations at fracture f shown in Fig. 2 and Table II are  It should also be emphasized that our present AIMD simulations pertain the mechanical behavior of (Ti,Al)N solid solutions at 300 K. At this temperature, spinodal decomposition is kinetically blocked, i.e., the temperature is not high enough to activate diffusion of vacancies (vacancy migration in B1 (Ti,Al)N systems requires energies in the range ≈2.5 -4.5 eV [63][64][65][66]). In general, if the operation temperature of (Ti,Al)N coatings remains below ≈1000 Ktypically the onset for decompositionwe find it unlikely that the spinodal decomposition process may occur faster than the strain-mediated lattice transformations seen here.
At a fundamental electronic-structure level, a relatively high Al metal content (≈60%) is expected to maximize the hardness of B1 (Ti,Al)N alloys. The effect stems from the fact that ≈8.4 e-/f.u. in B1-structure transition-metal (carbo)nitride solid solutions fully populate strong p-d metal/N bonding states while leaving shear-sensitive d-d metallic states empty [49]. In contrast, a low occupancy of d states is detrimental for the ability of (Ti,Al)N to form metallic bonds upon shearing. That, in turn, has been suggested as a possible cause of brittleness [51]. Consistent with the analysis of Ref. [51], phenomenological ductility/brittleness predictions based on elastic constant values would also (erroneously) indicate that Al substitutions degrade the (Ti,Al)N resistance to fracture. For example, according to the criterion proposed by Pettifor [67], the decrease in C12 -C44 Cauchy's pressure suggests that B1 Ti1-xAlxN solid solutions become progressively more brittle for increasing x (see figure 1 in Ref. [46]). However, DFT predictions of toughness vs. brittleness in (Ti,Al)N [51], primarily based on the analyses of the alloy elastic deformation, are unsuited to reveal the occurrence of transformation toughening mechanisms in the plastic regime. B1 (Ti,Al)N ceramics are of enormous technological importance due to age-hardening induced by spinodal decomposition at elevated temperatures [68]. However, while the spinodal mechanism is kinetically-blocked at ambient conditions, DFT calculations at 0 K show that an Al metal content larger than ≈0.7 renders Ti1-xAlxN solid solutions energetically more stable in the wurtzite than in the rocksalt structure (see figure 3a in [69]). The results of present AIMD simulations, combined with those of Ref. [69], evidence a correlation between the phase stability of the alloys and their inherent room-temperature toughness vs. brittleness.
Presumably, the energy required to induce cleavage in these two systems is smaller than the one necessary to activate any local lattice transformation during uniaxial strain. In contrast, our results suggest that tuning the Ti1-xAlxN metal composition around the threshold value x≈0.7 [69] can be used to optimize the combination of strength and toughness of B1-structure alloys. Indeed, the relatively small EB1-EB4 energy difference calculated for Ti0.5Al0.5N [69] enables B1→B4-like transformations during [001] tensile deformation (Figs. 3 and 5), thus dissipating accumulated stresses and enhancing the material resistance to fracture. On the other hand, Ti0.25Al0.75N solid solutions (which can be synthesized as B1 single-phase films [6,7]) would favorably crystallize in the B4 polymorph structure at ambient conditions [69]. Accordingly, Ti0.25Al0.75N is thermodynamically more inclined than Ti0.5Al0.5N to activate B1→B4 transformations under load.
That metastability is beneficial to enhance the mechanical performance of ceramics is not a new concept. For example, it has been shown that tuning the electron concentration to values near 9.5 e-/f.u. sets hexagonal and cubic polymorph structures of transition-metal carbonitrides to similar energies. This, in turn, promotes formation of hexagonal stacking faults in cubic alloys, thus increasing hardness by hindering dislocation motion across the faults [70,71]. Similarly, plastic deformation along 111 faults in B1 refractory carbonitrides can be assisted by providing facile deformation paths: the energy barrier of {111}<1-10> slip is reduced by synchro-shear mechanisms in B1 Ti0.5W0.5N solid solutions and B1-TiN/B1-WNx superlattices due to the preference of B1 WN-rich domains to transform in more stable hexagonal WC-structures [72,73]. Analogous to the experimental findings for multilayer films of Yalamanchili et al. [61] and Schlögl et al. [62], in this work we show that alloying (in ideal defect-free structures) transition-metal nitrides with AlN canbeside spinodal age-hardening at elevated temperatures -enable B1→B4 transformation toughening mechanisms at 300 K, i.e., much lower than the typical brittle-to-ductile transition temperatures of refractory ceramics [29,31].       50N(001). The three orthographic views are AIMD snapshots taken at elongations of (from left to right) 12% (which corresponds to the Ti0.50Al0.50N(001) yield point in Fig. 2(a), 16%, and 30%. The dynamics bonds have cutoff lengths of 2.6 Å. Color legend: blue = N, pink = Ti, cyan = Al.  Fig. 6) which proceed via lattice shearing within the (001) xy plane: a N atom (yellow) and an Al atom (red) located on different (-110) layers progressively align on a same direction, normal to the page.