Temperature Effects on the Elastic Constants, Stacking Fault Energy and Twinnability of Ni₃Si and Ni₃Ge: A First-Principles Study

Abstract

The volume versus temperature relations for Ni${}_3$Si and Ni${}_3$Ge are obtained by using the first principles calculations combined with the quasiharmonic approach. Based on the equilibrium volumes at temperature T, the temperature dependence of the elastic constants, generalized stacking fault energies and generalized planar fault energies of Ni${}_3$Si and Ni${}_3$Ge are investigated by first principles calculations. The elastic constants, antiphase boundary energies, complex stacking fault energies, superlattice intrinsic stacking fault energies and twinning energy decrease with increasing temperature. The twinnability of Ni${}_3$Si and Ni${}_3$Ge are examined using the twinnability criteria. It is found that their twinnability decrease with increasing temperature. Furthermore, Ni${}_3$Si has better twinnability than Ni${}_3$Ge at different temperatures.

1. Introduction

Ni-based single-crystal (SC) superalloys are widely used for turbine blades and vanes in the most advanced gas turbine engines due to their high melting temperature, adequate high temperature strength, elevated thermal stability, excellent oxidation and corrosion resistance [1,2]. The L1${}_2$-type ordered phases Ni${}_3$Si and Ni${}_3$Ge exhibit unique mechanical properties which makes them attractive for structural applications at elevated temperatures.

A number of reports, particularly on mechanical properties, have been carried on Ni${}_3$Si and Ni${}_3$Ge. For example, Ochial et al. [3] evaluated the direction of solubility lobe, the extent of homogeneity of L1${}_2$ phase and the site preference of ternary additions in Ni${}_3$Ga, Ni${}_3$Si, Ni${}_3$Ge and Ni${}_3$Al through the experiment. Yasuda et al. [4] determined the three elastic stiffness constants of Ni${}_3$X (X = Mn, Fe, Al, Ga, Ge and Si) by the rectangular parallelepiped resonance method. Iotova et al. [5] studied the electronic and mechanical properties of the Ni${}_3$X (X = Mn, Al, Ga, Si, Ge) using first-principles total-energy electronic structure calculations based on the full-potential linear-muffin-tin-orbital method. Chen et al. [6] investigated the structural properties, elastic properties, thermal stability, thermodynamics and electronic structures of Ni${}_3$X (X: Al, Mo, Ti, Pt, Si, Nb, V and Zr) from first-principles calculation. Hou et al. [7] investigated the structural, elastic, thermodynamic and electronic properties of L1${}_2$-ordered intermetallic compounds Ni${}_3$X (X = Al, Ga and Ge) under pressure using first-principles methods based on density functional theory. Tanaka et al. [8] measured the elastic-stiffness constants of Ni${}_3$Ga and Ni${}_3$Ge over the temperature range of 77–1200 K using the rectangular parallelepiped resonance (RPR) method. Prikhodko et al. [9] measured the single-crystal elastic constants of Ni-Ge and Ni-Ga solid solution alloys up to 1100 K. However, the effect of temperature on the dislocation of Ni${}_3$Si and Ni${}_3$Ge have not been reported yet.

It is widely accepted that dislocations play an important role in determining the mechanical properties of the materials [10,11]. Deformation twinning is another deformation mechanism in Ni${}_3$Si and Ni${}_3$Ge. Dislocation-mediated slip and deformation twinning are two major and competing modes of plastic deformation in fcc metals and alloys [12,13]. The generalized stacking fault energy (GSFE) curves play a critical role in revealing the competition between dislocation and deformation twinning. There have been many theoretical studies focused on the qualitative dependence of deformation twinning tendency on the GSFEs in fcc metals and alloys [14,15,16,17,18,19,20,21,22,23,24]. Fu et al. [25] determined the (100) and (111) antiphase boundary (APB) energies and the superlattice intrinsic stacking fault (SISF) energies of Ni${}_3$Si using the first-principle total-energy methods. Mryasov et al. [26] employed the local-density approximation (LDA) and the full-potential linear muffin-tin orbital (FLMTO) method to determine the GSFEs of Ni${}_3$Ge on the {111} plane at 0 K. However, the entire GSFE curves and the twinnability of Ni${}_3$Si and Ni${}_3$Ge at different temperatures have not yet been reported. In this paper, the first principles methods and quasiharmonic approximation are used to systematically examine the GSFEs and twinnabilities of Ni${}_3$Si and Ni${}_3$Ge at different temperatures.

2. Method of Calculation

In terms of first-principles quasiharmonic approach, the Helmholtz free energy F for a crystal can be written in the form of

$$\begin{array}{c}\hfill F(V,T)=E_0\left(V\right)+F_{\mathrm{el}}(V,T)+F_{\mathrm{vib}}(V,T)\end{array}$$

(1)

where $E_0\left(V\right)$ is the static energy at 0 K and $F_{\mathrm{el}}$ is the thermal electronic contribution to free energy from finite temperature. Both the $E_0\left(V\right)$ and $F_{\mathrm{el}}$ can be obtained from first principles calculations. $F_{\mathrm{vib}}$ is the vibrational free energy of the lattice atoms given by

$$\begin{array}{c}\hfill F_{\mathrm{vib}}(V,T)=\sum _{\kappa }\{\frac{1}{2}\hslash {\omega }_{\kappa }\left(V\right)+k_{B}T\ln \left[1-\exp \left(-\frac{\hslash {\omega }_{\kappa }\left(V\right)}{k_{B}T}\right)\right]\}\end{array}$$

(2)

where $k_B$ is the Boltzmann constant, ℏ is the reduced Planck constant, ${\omega }_{\kappa }$ represents an individual phonon frequency.

The isothermal elastic constants $C_{ij}^T$, with respect to the finite strain variables, are defined as [27]

$$\begin{array}{c}\hfill C_{ij}^T=\frac{1}{V}\frac{{\partial }^{2}F}{\partial {\eta }_i{\eta }_j}{|}_{T,{\eta }^{\prime }}\end{array}$$

(3)

where $V=V\left(T\right)$ is the volume at temperature T and ${\eta }^{\prime }$ represents that all other strains are held fixed. For a cubic system, there are three independent elastic constants $C_{11}^T$, $C_{12}^T$ and $C_{44}^T$. The bulk modulus $B_T$ determined from the Vinet equation of state [28] is approximated by the combinations of the elastic moduli

$$\begin{array}{c}\hfill B_T^T=(C_{11}^T+2C_{12}^T)/3\end{array}$$

(4)

A volume-conserving orthorhombic strain is applied to determine the difference between the $C_{11}^T$ and $C_{12}^T$

$$\eta \left(\xi \right)=\left(\begin{array}{ccc}\xi & 0& 0\\ 0& \xi & 0\\ 0& 0& {(1+\xi )}^{-2}-1\end{array}\right)$$

(5)

which leads to the Helmholtz free energy change

$$\begin{array}{c}\hfill F(V,\xi )=F(V,0)+3(C_{11}^T-C_{12}^T)V{\xi }^2+O\left({\xi }^3\right)\end{array}$$

(6)

A volume-conserving tetragonal strain is applied to determine $C_{44}^T$

$$\eta \left(\xi \right)=\left(\begin{array}{ccc}0& \xi & 0\\ \xi & 0& 0\\ 0& 0& {\xi }^2/(1-{\xi }^2)\end{array}\right)$$

(7)

and the corresponding Helmholtz free energy change is

$$\begin{array}{c}\hfill F(V,\xi )=F(V,0)+2C_{44}^{T}V{\xi }^2+O\left({\xi }^4\right)\end{array}$$

(8)

In present work, a first-principles quasistatic approach proposed by Wang et al. [29,30] is employed to determine the temperature dependent elastic constants $C_{ij}\left(T\right)$. In this approach, the change of elastic constants at higher temperature is mainly caused by volume change. The temperature dependence of the elastic constants $C_{ij}\left(T\right)$ can be estimated by the application of the following three-step procedure. In the first step, the equilibrium volume $V\left(T\right)$ at given T is obtained by employing the first-principles quasiharmonic approach. In the second step, the static elastic constants at 0 K as a function of volume are obtained by using the energy-strain relations based on Equations (4)–(8). In the third step, the calculated elastic constants from the second step at the volume are estimated as those at finite temperature. The elastic constants at higher temperature are usually measured by the resonance method. That is, the crystal system is adiabatic because elastic waves travel faster than heat diffuses and the deformation can be viewed as a isoentropic process [31]. Thus, the isothermal elastic constants $C_{ij}^T$ need to be transformed to the adiabatic elastic constants $C_{ij}^S$ by the following relation

$$\begin{array}{c}\hfill C_{11}^S=C_{11}^T+\frac{TV}{C_V}{\alpha }^2{B}_T^2\end{array}$$

(9)

$$\begin{array}{c}\hfill C_{12}^S=C_{12}^T+\frac{TV}{C_V}{\alpha }^2{B}_T^2\end{array}$$

(10)

$$\begin{array}{c}\hfill C_{44}^S=C_{44}^T\end{array}$$

(11)

where $C_V$ is the specific heat at constant volume, and $\alpha$ is the volume thermal expansion coefficient obtained by $\alpha \left(T\right)=\frac{1}{V}\left(\frac{dV\left(T\right)}{dT}\right)$.

In this work, all calculations were performed using the VASP code [32,33,34], which is based on the implementation of density functional theory with the exchange-correlation functional depicted by the generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) [35]. The ion electron interaction is described by the plane-wave basis projector augmented wave (PAW) method [36,37]. For calculation of the GSFEs of Ni${}_3$Si and Ni${}_3$Ge at different temperatures, a periodic supercell consisting of ten atomic planes is employed. Between the periodically repeated supercell a vacuum space with six atomic layers is added to avoid interactions between faults in two neighboring slabs. A plane wave basis set with energy cutoff of 550 eV is used for all calculations. Brillouin zone (BZ) integrations are performed using the Monkhorst–Pack special k-point scheme [38]. The samplings of k meshes are 15 × 15 × 15 for lattice and elastic constants calculations and 15 × 15 × 1 for generalized stacking fault energy calculations. The self-consistent convergence of the total energy is set to 10${}^{-6}$ eV/atom and the maximum force on the atom is 10${}^{-4}$ eV/Å.

Phonon calculations are carried out by the supercell approach within the framework of force-constant method. The chosen supercell size strongly influences on the thermal properties. After a lot of testing, the adequate supercell size consisting of 3 × 3 × 3 unit cells with 108 atoms is good enough for calculating the phonon dispersions. The forces acting on atoms for each perturbed supercell are calculated by VASP with 5 × 5 × 5 for Ni${}_3$Si and Ni${}_3$Ge. Phonon dispersion curves for Ni${}_3$Si and Ni${}_3$Ge along high-symmetry directions in the Brillouin zone are plotted in Figure 1. The Helmholtz free energy curves as a function of unit cell volume with a step of 100 K from 0 to 900 K at 14 volume points are given in Figure 2. At each temperature point, the equilibrium volume V depicted by the red triangle is obtained by minimizing the Helmholtz free energy with respect to V from fitting the integral form of the Vinet equation of state (EOS) [28]. The specific details of calculating the phonon and thermodynamic properties can be found in Refs. [39] and [40].

3. Results and Discussion

The calculated lattice constants and isentropic elastic constants $C_{ij}^S$ at $T=0$ K are listed in

Table 1. Calculated lattice parameter a and elastic constants $C_{11}$, $C_{12}$, $C_{44}$ of Ni${}_3$Si and Ni${}_3$Ge at 0 K compared with the previous theoretical and experimental data.

Materials	Ref.	a	${\mathit{C}}_{11}$	${\mathit{C}}_{12}$	${\mathit{C}}_{44}$
Ni${}_3$Si	This work	3.553	268.5	140.4	107.5
	[6]	3.517	317	162	129
Ni${}_3$Ge	This work	3.629	235.3	130.6	85.2
	[7]	3.584	268.9	148.4	103.4
	[4]		263.0	143.0	103.0

, along with the available experimental and theoretical values. It can be seen that the calculated values in this work are consistent with the theoretical calculations [6,7] at 0 K and experimental values [4] at room temperature. The temperature dependent elastic constants are important for understanding and predicting the mechanical strength, stability and phase transitions of a material [41]. Figure 3 shows the predicted isoentropic elastic constants of Ni${}_3$Si and Ni${}_3$Ge as a function of temperature in the temperature range of 0–900 K. It can be seen that all the values are almost invariant at low temperatures and decrease with increasing temperature at high temperatures. $C_{11}^S$ represents the elasticity in length, whereas $C_{12}^S$ and $C_{44}^S$ are related to the elasticity in shape. Whatever Ni${}_3$Si or Ni${}_3$Ge, it can be seen that $C_{11}^S$ decreases to the largest extant within the temperature range of the study. $C_{11}^S$, $C_{12}^S$ and $C_{44}^S$ for Ni${}_3$Si decrease by 18.3, 9.7 and 10.2 GPa, and those for Ni${}_3$Ge decrease by 18.9, 11.1 and 8.9 GPa, respectively. The elastic constants of Ni${}_3$Si and Ni${}_3$Ge compounds at different temperatures satisfy the mechanical stability criteria in a cubic crystal [42]: $C_{11}$ − $C_{12}>$0, $C_{11}>$ 0 and $C_{44}>$ 0, which indicates that they are mechanically stable in the temperature range of 0–900 K.

The plastic behavior of Ni${}_3$Si and Ni${}_3$Ge is mainly determined by the dislocation motion and deformation twinning. The (001) and (111) planes are the two primary slip planes in L1${}_2$ structure [43]. It is widely accepted that there are three different types of dislocation dissociations in Ni${}_3$Si and Ni${}_3$Ge. In the (111) or (001) plane, a [$\overline{1}$0$\overline{1}$] superdislocation will collinearly dissociated into two 1/2[$\overline{1}$0$\overline{1}$] superpartials bound by an APB as following reaction:

$$\begin{array}{c}\hfill \left[\overline{1}0\overline{1}\right]\to \frac{1}{2}\left[\overline{1}0\overline{1}\right]+\mathrm{APB}+\frac{1}{2}\left[\overline{1}0\overline{1}\right]\end{array}$$

(12)

where [$\overline{1}$0$\overline{1}$] is the Burgers vector of perfect superdislocations in (111) and (001) planes. Furthermore, the [$\overline{1}$0$\overline{1}$]{111} superpartial may further dissociate into two 1/6〈112〉 Schockley partials separated by two complex stacking faults (CSF), which is called as a metastable dissociated dissociation, namely

$$\begin{array}{c}\hfill \left[\overline{1}0\overline{1}\right]\to \frac{1}{6}\left[\overline{1}\overline{1}2\right]+\mathrm{CSF}+\frac{1}{6}\left[\overline{2}11\right]+\mathrm{APB}+\frac{1}{6}\left[\overline{1}\overline{1}2\right]+\mathrm{CSF}+\frac{1}{6}\left[\overline{2}11\right]\end{array}$$

(13)

The $\left[\overline{1}0\overline{1}\right]${111} perfect superdislocations may be also divided into 1/3〈112〉{111} superpartials with a superlattice intrinsic stacking fault (SISF), namely,

$$\begin{array}{c}\hfill \left[\overline{1}0\overline{1}\right]\to \frac{1}{3}\left[\overline{2}11\right]+\mathrm{SISF}+\frac{1}{3}\left[\overline{1}\overline{1}2\right]\end{array}$$

(14)

In this work, on the basis of the temperature dependent lattice constants $a\left(T\right)$, all the calculated stacking fault energies of Ni${}_3$Si and Ni${}_3$Ge as a function of temperature are given in Figure 4 and Figure 5, respectively. The discrete points denote the values of first principles calculations and the solid curves represent the results obtained from polynomial fitting. All the various typical energies (${\gamma }_{\mathrm{us}}$, ${\gamma }_{\mathrm{CSF}}$, ${\gamma }_{\mathrm{SISF}}$, ${\gamma }_{\mathrm{APB}}$) at different temperatures for Ni${}_3$Si and Ni${}_3$Ge are listed in

Table 2. Calculated various fault energies (J/m${}^2$), relative barrier difference ${\delta }_{\mathrm{us}}^{\mathrm{ut}}\equiv {\gamma }_{\mathrm{ut}}-{\gamma }_{\mathrm{us}}$ (J/m${}^2$) and twinnability ${\tau }_a$ at different temperatures of Ni${}_3$Si and Ni${}_3$Ge.

Ni${}_{\mathbf{3}}$Si	$\mathit{T}$ (K)	0	300	600	900
	${\gamma }_{\mathrm{us}1}^{\langle 112\rangle \left\{111\right\}}$	0.367	0.366	0.359	0.346
	${\gamma }_{\mathrm{us}2}^{\langle 112\rangle \left\{111\right\}}$	0.990	0.988	0.973	0.948
	${\gamma }_{\mathrm{us}}^{\langle 110\rangle \left\{111\right\}}$	1.193	1.189	1.164	1.122
	${\gamma }_{\mathrm{CSF}}^{\langle 112\rangle \left\{111\right\}}$	0.610	0.609	0.602	0.588
	${\gamma }_{\mathrm{SISF}}^{\langle 112\rangle \left\{111\right\}}$	0.308	0.307	0.304	0.298
	${\gamma }_{\mathrm{ut}}^{\langle 112\rangle \left\{111\right\}}$	0.951	0.949	0.935	0.911
	2${\gamma }_{\mathrm{tsf}}^{\langle 112\rangle \left\{111\right\}}$	0.944	0.942	0.929	0.908
	${\gamma }_{\mathrm{us}}^{\langle 110\rangle \left\{010\right\}}$	1.984	1.975	1.925	1.840
	${\gamma }_{\mathrm{APB}}^{\left\{111\right\}}$	0.599	0.598	0.592	0.580
	${\gamma }_{\mathrm{APB}}^{\left\{010\right\}}$	0.477	0.476	0.471	0.463
	${\delta }_{\mathrm{us}}^{\mathrm{ut}}$	0.584	0.583	0.576	0.565
	$\tau$	0.550	0.549	0.547	0.541
Ni${}_{\mathbf{3}}$Ge	$\mathit{T}$ (K)	0	300	600	900
	${\gamma }_{\mathrm{us}1}^{\langle 112\rangle \left\{111\right\}}$	0.321	0.320	0.314	0.301
	${\gamma }_{\mathrm{us}2}^{\langle 112\rangle \left\{111\right\}}$	0.953	0.951	0.936	0.908
	${\gamma }_{\mathrm{us}}^{\langle 110\rangle \left\{111\right\}}$	1.047	1.044	1.021	0.980
	${\gamma }_{\mathrm{CSF}}^{\langle 112\rangle \left\{111\right\}}$	0.609	0.608	0.600	0.584
	${\gamma }_{\mathrm{SISF}}^{\langle 112\rangle \left\{111\right\}}$	0.306	0.305	0.303	0.298
	${\gamma }_{\mathrm{ut}}^{\langle 112\rangle \left\{111\right\}}$	0.925	0.923	0.908	0.882
	2${\gamma }_{\mathrm{tsf}}^{\langle 112\rangle \left\{111\right\}}$	0.947	0.944	0.931	0.906
	${\gamma }_{\mathrm{us}}^{\langle 110\rangle \left\{010\right\}}$	1.667	1.661	1.617	1.537
	${\gamma }_{\mathrm{APB}}^{\left\{111\right\}}$	0.622	0.620	0.613	0.598
	${\gamma }_{\mathrm{APB}}^{\left\{010\right\}}$	0.466	0.465	0.462	0.456
	${\delta }_{\mathrm{us}}^{\mathrm{ut}}$	0.604	0.603	0.594	0.581
	$\tau$	0.500	0.499	0.498	0.492

, respectively. The ${\gamma }_{\mathrm{us}}$ is the unstable stacking fault energy, which is an important solid state parameter that demonstrates the resistance to dislocation nucleation in materials [44,45]. The present results (${\gamma }_{\mathrm{APB}}^{\left\{111\right\}}$ = 599 mJ/m${}^2$, ${\gamma }_{\mathrm{SISF}}^{\left\{111\right\}}$ = 308 mJ/m${}^2$ and ${\gamma }_{\mathrm{CSF}}^{\left\{111\right\}}$ = 610 mJ/m${}^2$ for Ni${}_3$Si, ${\gamma }_{\mathrm{APB}}^{\left\{111\right\}}$ = 622 mJ/m${}^2$, ${\gamma }_{\mathrm{SISF}}^{\left\{111\right\}}$ = 306 mJ/m${}^2$ and ${\gamma }_{\mathrm{CSF}}^{\left\{111\right\}}$ = 609 mJ/m${}^2$ for Ni${}_3$Ge) at 0 K agree with the results (${\gamma }_{\mathrm{APB}}^{\left\{111\right\}}$ = 625 mJ/m${}^2$, ${\gamma }_{\mathrm{SISF}}^{\left\{111\right\}}$ = 460 mJ/m${}^2$ and ${\gamma }_{\mathrm{CSF}}^{\left\{111\right\}}$ = 710 mJ/m${}^2$ for Ni${}_3$Si) obtained by Yoo et al. [46] through the first principles total-energy calculations within the framework of the local-density functional theory and the results (${\gamma }_{\mathrm{APB}}^{\left\{111\right\}}$ = 660 mJ/m${}^2$, ${\gamma }_{\mathrm{SISF}}^{\left\{111\right\}}$ = 420 mJ/m${}^2$ and ${\gamma }_{\mathrm{CSF}}^{\left\{111\right\}}$ = 620 mJ/m${}^2$ for Ni${}_3$Ge) obtained by Mryasov et al. [26] through the ab-initio calculations. However, there are no experimental and theoretical values of stacking fault energies at finite temperature for comparison. For Ni${}_3$Si and Ni${}_3$Ge, it can be found that all the stacking fault energies decrease with increasing temperature, since thermal expansion may soften the stacking fault configuration at higher temperatures. From Table 2, it is easily found that the ${\gamma }_{\mathrm{us}}$ along the $\langle 112\rangle \left\{111\right\}$ is the lowest, indicating that the emission of a dislocation is most likely at this slip system.

Generally, there are no deformation twinning in Ni${}_3$Si and Ni${}_3$Ge intermetallic compounds. However, the deformation micro-twinning can be observed in the precipitation phases of the Ni${}_3$Si and Ni${}_3$Ge intermetallic compounds at higher temperature. There are two types of deformation twinning in Ni${}_3$Si and Ni${}_3$Ge. One is pseudo-twinning and the other is anti-twinning. The former is generated by the sequential motions of identical 1/6〈$\overline{2}$11〉 Shockley’s partial dislocations and the latter is formed by sequential shear of 1/3〈$\overline{1}$2$\overline{1}$〉 dislocations on successive (111) planes. The 1/6〈$\overline{2}$11〉 Shockley’s partial dislocation is easy to form, but the formation of the 1/3〈$\overline{1}$2$\overline{1}$〉 dislocation is very difficult. Besides, the dislocations of the disordered solid solution matrix phase of nickel-based superalloys are also the 1/6〈$\overline{2}$11〉 Shockley’s partial dislocations. Therefore, the generalized planar fault energy (GPFE) curves along the 1/6〈$\overline{2}$11〉 pseudo-twinning direction at different temperatures for Ni${}_3$Si and Ni${}_3$Ge are plotted in Figure 6. The first half of each GPFE curve is created by sliding the upper 6–10 atomic layers relative to the lower 1–5 layers along 〈$\overline{2}$11〉 over a distance $a/\sqrt{6}$ and the stacking fault ⋯ABCABCBCABCA⋯ is generated. The second half is then constructed by displacing the upper 7–10 layers in a similar fashion and the two-layer twin ⋯ABCABCBABCAB⋯ is generated. For each GPFE curve, the first maximum value is the unstable stacking fault energy ${\gamma }_{\mathrm{us}1}$ and the second maximum value is the unstable twinning energy ${\gamma }_{\mathrm{ut}}$, the first minimum value is the intrinsic stacking fault energy ${\gamma }_{\mathrm{sf}}$ and the second minimum value is the twin stacking fault energy 2${\gamma }_{\mathrm{tsf}}$. Numerical values extracted from the curves’ extrema are listed in Table 2. All these values decrease with increasing temperature.

Van Swygenhoven et al. [14] proposed that the difference ${\delta }_{\mathrm{us}}^{\mathrm{ut}}\equiv {\gamma }_{\mathrm{ut}}-{\gamma }_{\mathrm{us}}$ can estimate the ease with which an fcc material deforms by twinning relative to deforming by dislocation-mediated slip. Tadmor and Bernstein [15] proposed a more thorough treatment of the balance between dislocation mediated slip and deformation twinning, that is

$$\begin{array}{c}\hfill {\tau }_a=\left[1.136-0.151\frac{{\gamma }_{\mathrm{sf}}}{{\gamma }_{\mathrm{us}}}\right]\sqrt{\frac{{\gamma }_{\mathrm{us}}}{{\gamma }_{\mathrm{ut}}}}\end{array}$$

(15)

where ${\gamma }_{\mathrm{sf}}$ and ${\gamma }_{\mathrm{us}}$ are replaced by ${\gamma }_{\mathrm{CSF}}$ and ${\gamma }_{\mathrm{us}1}$ for the pseudo-twinning in Ni${}_3$Si and Ni${}_3$Ge intermetallic compounds, respectively. ${\tau }_a$ is called as the twinnability parameter of a crystal. ${\tau }_a>1$ indicates that the main deformation mechanism is the deformation twinning, ${\tau }_a<1$ shows that the main deformation mechanism is the dislocation slip and ${\tau }_a=1$ demonstrates that the main deformation mechanism is the full dislocation movement. Besides, a larger ${\tau }_a$ indicates a greater propensity for twinning. All the values of ${\delta }_{\mathrm{us}}^{\mathrm{ut}}$ and ${\tau }_a$ at different temperatures for Ni${}_3$Si and Ni${}_3$Ge are listed in Table 2. Obviously, both ${\delta }_{\mathrm{us}}^{\mathrm{ut}}$ and ${\tau }_a$ decrease with increasing temperature. Whatever Ni${}_3$Si or Ni${}_3$Ge, the values of ${\delta }_{\mathrm{us}}^{\mathrm{ut}}$ are always larger than 0 in the temperature range of 0–900 K. This phenomenon demonstrates that the main deformation mechanism does not change with increasing temperature, but the twinnability of Ni${}_3$Si and Ni${}_3$Ge increase with increasing temperature. From Table 2, it can be seen that the twinnability parameters of Ni${}_3$Si (${\tau }_a$ = 0.550) and Ni${}_3$Ge (${\tau }_a$ = 0.500) at 0 K are much smaller than those of the fcc pure metals Al (${\tau }_a$ = 0.930), Cu (${\tau }_a$ = 1.001) and Ag (${\tau }_a$ = 1.042) [15]. However, the twinnability of Ni${}_3$Si and Ni${}_3$Ge are similar to that of Ni${}_3$Al. Besides, the twinnability parameters of Ni${}_3$Si and Ni${}_3$Ge are smaller than 1 in the whole temperature and their twinnability parameters decrease with increasing temperature, which indicates that it is difficult to form the pseudo-twinning through the direct shearing and the twinnability of Ni${}_3$Si and Ni${}_3$Ge decrease with increasing temperature. That is to say, the twinnability of Ni${}_3$Si and Ni${}_3$Ge with temperature predicted by ${\tau }_a$ and ${\delta }_{\mathrm{us}}^{\mathrm{ut}}$ are different. The twinnability criterion ${\tau }_a$ is dependent on the ${\gamma }_{\mathrm{ut}}$ and ${\gamma }_{\mathrm{us}}$ in addition to ${\gamma }_{\mathrm{sf}}$, but ${\delta }_{\mathrm{us}}^{\mathrm{ut}}$ is based on the ${\gamma }_{\mathrm{ut}}$ and ${\gamma }_{\mathrm{us}}$. Therefore, ${\tau }_a$ can more accurately describe the twinnability of a material in comparison with ${\delta }_{\mathrm{us}}^{\mathrm{ut}}$. Furthermore, the twinnability of Ni${}_3$Si is greater than that of Ni${}_3$Ge at different temperatures according to the two criteria ${\tau }_a$ and ${\delta }_{\mathrm{us}}^{\mathrm{ut}}$. This may be resulted from that the difference between ${\gamma }_{\mathrm{ut}}$ and ${\gamma }_{\mathrm{us}}$ of Ni${}_3$Ge is greater than that of Ni${}_3$Si.

4. Conclusions

In this work, temperature effects on the elastic constants, generalized stacking fault energies and generalized planar fault energies of Ni${}_3$Si and Ni${}_3$Ge are investigated by the first principles calculations combined with a quasiharmonic approach. It is found that the elastic constants, APB energies, CSF energies, SISF energies and twinning energy decrease with increasing temperature. Besides, the twinnability of Ni${}_3$Si and Ni${}_3$Ge with temperature have been investigated through the two criteria ${\tau }_a$ and ${\delta }_{\mathrm{us}}^{\mathrm{ut}}$, but the two criteria give different results. ${\tau }_a$ is dependent on the ${\gamma }_{\mathrm{ut}}$ and ${\gamma }_{\mathrm{us}}$ in addition to ${\gamma }_{\mathrm{sf}}$, but ${\delta }_{\mathrm{us}}^{\mathrm{ut}}$ is based on ${\gamma }_{\mathrm{ut}}$ and ${\gamma }_{\mathrm{us}}$, that is to say, ${\tau }_a$ can more accurately give the twinnability of a material. Furthermore, the twinnability of Ni${}_3$Si is greater than that of Ni${}_3$Ge at different temperatures.