Temperature dependence of TiN elastic constants from ab initio molecular dynamics simulations

Elastic properties of cubic TiN are studied theoretically in a wide temperature interval. First-principles simulations are based on ab initio molecular dynamics (AIMD). Computational efficiency of the method is greatly enhanced by a careful preparation of the initial state of the simulation cell that minimizes or completely removes a need for equilibration and therefore allows for parallel AIMD calculations. Elastic constants $C_{11}$, $C_{12}$, and $C_{44}$ are calculated. A strong dependence on the temperature is predicted, with $C_{11}$ decreasing by more than 29$\%$ at 1800 K as compared to its value obtained at $T=0$ K. Furthermore, we analyze the effect of temperature on the elastic properties of polycrystalline TiN in terms of the bulk and shear moduli, the Young's modulus and Poisson ratio. We construct sound velocity anisotropy maps, investigate the temperature dependence of elastic anisotropy of TiN, and observe that the material becomes substantially more isotropic at high temperatures. Our results unambiguously demonstrate the importance of taking into account finite temperature effects in theoretical calculations of elastic properties of materials intended for high-temperature applications.

Figure 1

Distribution of stresses obtained by AIMD simulations at each value of distortion $\eta$ and at two temperatures, 300 and 1200 K. Solid lines are histograms of all the instantaneous ${\sigma }_{11}$ stress values. Dashed lines correspond to $C_{11}$ strain-stress curves fitted to the stress data, from which the values of $C_{11}\left(T\right)$ are extracted.

Figure 2

Calculated single-crystal elastic constants for B1 TiN as a function of temperature. Theoretical results obtained from AIMD simulations are shown with filled circles connected with solid lines. Static calculations at $T=0$ K are shown with open circles. The error bars correspond to a standard deviation, calculated from the variance of five independent molecular dynamics simulations.

Figure 3

Single-crystal longitudinal and transverse sound velocities for B1 TiN at temperatures $T=300$ K: (a) $V_p$, (b) $V_{s1}$, and (c) $V_{s2}$ and $T=1500$ K: (d) $V_p$, (e) $V_{s1}$, and (f) $V_{s2}$ as a function of propagation direction. All velocities are given in (km/s). The deformed spheres show the directional dependency of the sound velocities. The radius and color are proportional to the sound velocity in the corresponding direction. The isocontours are drawn at fixed intervals for each polarization, starting from the minimum velocity at each temperature and polarization. Fewer contours correspond to a more isotropic velocity distribution. White and black spheres indicate the direction of the minimum and maximum velocity, respectively.

Figure 4

Calculated polycrystalline elastic constants, bulk modulus $B$ (a), Young's modulus $E$ (b), shear modulus $G$ (c), as well as Poisson ratio (d) for B1 TiN as a function of temperature. The results of the averaging using the Reuss and the Voigt approaches are indicated with circles and squares, respectively. Theoretical results obtained from AIMD simulations are connected with solid lines. Static calculations at $T=0$ K are shown with open symbols. The error bars correspond to a standard deviation, calculated from the standard deviations in $C_{ij}$ assuming the errors are normally distributed.

Figure 5

Calculated Voigt-Reuss-Hill anisotropy $A_{VR}$ and Zenner elastic-shear anisotropy index $A_Z$ for B1 TiN as a function of temperature. Theoretical results obtained from AIMD simulations are connected with solid lines. Static calculations at $T=0$ K are shown with open symbols. The error bars correspond to one standard deviation, calculated from the standard deviations in $C_{ij}$ assuming the errors are normally distributed.