First-Principles Investigations of the Structural, Anisotropic Mechanical, Thermodynamic and Electronic Properties of the AlNi₂Ti Compound

Abstract

In this paper, the electronic, mechanical and thermodynamic properties of AlNi₂Ti are studied by first-principles calculations in order to reveal the influence of AlNi₂Ti as an interfacial phase on ZTA (zirconia toughened alumina)/Fe. The results show that AlNi₂Ti has relatively high mechanical properties, which will benefit the impact or wear resistance of the ZTA/Fe composite. The values of bulk, shear and Young’s modulus are 164.2, 63.2 and 168.1 GPa respectively, and the hardness of AlNi₂Ti (4.4 GPa) is comparable to common ferrous materials. The intrinsic ductile nature and strong metallic bonding character of AlNi₂Ti are confirmed by B/G and Poisson’s ratio. AlNi₂Ti shows isotropy bulk modulus and anisotropic elasticity in different crystallographic directions. At room temperature, the linear thermal expansion coefficient (LTEC) of AlNi₂Ti estimated by quasi-harmonic approximation (QHA) based on Debye model is 10.6 × 10⁻⁶ K⁻¹, close to LTECs of zirconia toughened alumina and iron. Therefore, the thermal matching of ZTA/Fe composite with AlNi₂Ti interfacial phase can be improved. Other thermodynamic properties including Debye temperature, sound velocity, thermal conductivity and heat capacity, as well as electronic properties, are also calculated.

1. Introduction

AlNi₂Ti is one of the Heusler compounds first identified by Heusler in 1903 [1]. The Heusler compounds are a group of ternary intermetallic alloys formed at the stoichiometric composition X₂YZ (generally, X and Y are transition metals and Z is a b-subgroup element), with the doubly ordered L2₁ structure based on the ordered B2 [2]. They have been widely reported to have excellent ferromagnetic, paramagnetic, mechanical and thermoelectric properties [3,4]. AlNi₂Ti with L2₁ structure can provide NiAl or Ni-base and Ti-base alloys with good high-temperature mechanical properties by precipitate strengthening [5,6,7,8,9,10,11,12,13,14,15,16]. Especially the mechanical properties of Fe-Ni-Al-Ti ferritic alloys, such as creep resistance and yield strength, can be significantly enhanced due to the AlNi₂Ti precipitates [17,18,19,20,21,22,23,24,25,26]. In addition, AlNi₂Ti precipitate phase has also been observed in microstructures of Al-doped Ni-Ti shape memory alloys that have huge potential to be applied as functional material in many areas such as clinical medicine, biotechnology, automation, energy engineering, electronics industry, aeronautics and astronautics, owing to their pseudoelasticity, superplasticity and shape memory properties [27,28,29,30,31,32].

AlNi₂Ti is also a common reaction product at the interface between phases. Recently, AlNi₂Ti was observed at the interface in zirconia-toughened alumina (ZTA) or Al₂O₃ reinforced iron matrix composite with an Ni-Ti transition layer that is expected to be applied for wear-resistant material [33,34]. The active elements Ni and Ti can promote the atom diffusion and chemical reaction between ZTA and Fe. Thus, Al₂O₃ will decompose and diffuse into the Ni-Ti layer, leading to the formation of interfacial phase AlNi₂Ti. The mechanical and thermal performance of AlNi₂Ti is a key point for improving the interface properties and then wear resistance of composites, especially the thermal expansion coefficient (TEC) that influences the thermal matching of composite interfaces.

By now, a number of research works have been carried out on AlNi₂Ti and other Heusler compounds by first-principles calculations. Wen et al. [35] reported the elastic constants and mechanical moduli under different pressures or temperatures of Ni₂XAl (X = Sc, Ti, V) compounds. The thermodynamic properties including heat capacity and thermal expansion coefficient as a function of temperature and pressure were demonstrated. They claimed that the influence of pressure on moduli and hardness decreases as the sequence of Ni₂ScAl > Ni₂TiAl > Ni₂VAl, while it has an opposite effect on ductility or anisotropy. Reddy and Kanchana [36] studied the effect of pressure and temperature on the Fermi surface and dynamic properties of a series of compounds Ni₂XAl (X = Ti, V, Zr, Nb, Hf and Ta) using density functional theory (DFT). Their calculating results suggested that AlNi₂Ti has lower specific heat and entropy but higher internal energy and free energy among the considered compounds [36]. The superconducting property of Ni₂NbAl has also been studied and the critical superconducting transition temperature Tc calculated to be 3.1 K [36], which agrees well with the experimental result (Tc around 2.1 K) in previous work [37]. Compared with representative high-pressure superconductors, such as H₃S (Tc: 178 K), PH₃ (Tc: 81 K) [38] and PtH (Tc: 12.94~20.01 K) [39], Ni₂NbAl performs at a relatively low value of Tc. Sahariya and Ahuja [40] presented the energy bands and density of states of AlNi₂Ti calculated using the DFT by linear combination of atomic orbitals (LCAO) approach. Zhou et al. [41] investigated the electronic and magnetic properties of AlNi₂Ti affected by swap defects and atomic antisite using first-principles calculations. The Ni_Ti(A) antisite was found to be the most probable defect because of the lowest formation energy. Nevertheless, the properties of AlNi₂Ti are still short of systematic theoretical study, especially the thermal properties and their anisotropies.

In this paper, a comprehensive and complementary study of the mechanical and thermal properties of AlNi₂Ti crystal is conducted by first-principles calculation in order to provide significant theoretical directions for the fabrication and application of composites with interfacial phase AlNi₂Ti and to offer a reference for the future research into AlNi₂Ti or other Heusler alloys.

2. Calculation Methods

The L2₁ cubic crystal structure with the $Fm\overline{3}m$ space group of AlNi₂Ti is shown in Figure 1. The unit cell of AlNi₂Ti has four formulas with a total of 16 atoms.

The first-principles calculations based on DFT were implemented in Cambridge Serial Total Energy Package (CASTEP) code [42,43,44]. The ultra-soft pseudo potentials were employed to represent the interactions between ionic core and valence electrons. The valence electron configurations for Al, Ni and Ti were 3s² 3p¹, 3d⁸ 4s² and 3s² 3p⁶ 3d² 4s², respectively. A special k-point mesh was used for the numerical integrations in the entire Brillouin zone by setting as 8 × 8 × 8 grid [45]. The Broyden–Fletcher–Goldfarb–Shannon (BFGS) minimization scheme was utilized for the optimization of lattice parameters and atomic positions in order to obtain the ground-state crystal structure. For the purpose of providing more abundant and reliable results, two approaches, i.e., the generalized gradient approximation (GGA) of Perdew Burke Ernzerhof (PBE) and Perdew Wang in 1991 (PW91) approaches, were adopted in exchange–correlation energy calculations [46]. It is capable of comparison with other DFT studies on AlNi₂Ti by the same PBE-GGA approach [35,36] and local density approximation (LDA) approach [7,35]. A kinetic energy cut-off value of 500 eV was used for plane wave expansions. The self-consistence convergence of the energy was set to 0.5 × 10⁻⁶ eV/atom. The quasi-harmonic approximation (QHA) Debye model was applied to predict the thermal properties of AlNi₂Ti compounds.

3. Results and Discussion

3.1. Crystal Parameters and Mechanical Properties

The optimized ground-state structural parameters of AlNi₂Ti compounds calculated by the GGA-PBE and GGA-PW91 approaches are shown in

Table 1. The optimized ground-state structural parameters (a, b, c, in Å; V_cell in ų; ρ in g/cm³), atomic site, cohesive energy (E_coh, eV/f.u.) and formation enthalpy (Δ_rH, eV/atom) of AlNi₂Ti intermetallic compound.

Method	Lattice Parameters	Vcell	ρ	Atomic Site			${\Delta }_{\mathit{r}}\mathit{H}$	Ecoh
	a = b = c			Al	Ni	Ti
GGA-PBE	5.908 (5.865 a)	206.242 (201.75 a)	6.193 (6.33 a)	4a (0, 0, 0)	8c (0.25, 0.25, 0.25)	4b (0.5, 0.5, 0.5)	−0.671	−27.602
GGA-PW91	5.903	205.676	6.210				−0.643	−33.173
Other cal. data	5.906 b, 5.90 c, 5.828 d						−0.658 b, −0.662 m
Exp. data	5.906 e, 5.883–5.910 f, 5.819 g, 5.876 i, 5.942 j, 5.889 k, 5.865–5.886 l	203.61–206.40 f					−0.747 e, −0.578 h, −0.579 n
EOS	5.911	206.499	6.186

^a From Inorganic Crystal Structure Database (ICSD) #58063; ^b Calculation data evaluated by GGA in [35]; ^c Cal. data evaluated by Reddy, et al. in [36]; ^d Cal. data evaluated by Monte Carlo simulations in [11]; ^e Exp. data in [47]; ^f Exp. data from [48]; ^g Data calculated from the selected area electron diffraction pattern in [22]; ^h Exp. data from [49]; ⁱ Data from handbook [50]; ^j Exp. data measured by X-ray diffraction in [51]; ^k Exp. data in [52]; ^l Exp. data in [53]; ^m Calculated values from a thermodynamic database in [54]; ⁿ Data measured by high-temperature reaction calorimeter in [54].

. It can be seen that the lattice parameters at 0 K in this study are close to other theoretical and experimental data [22,35,36,47,48,49,50,51,52,53,54]. The average deviation of results in this work to references for lattice parameters is less than 1%. The chemical stability of the AlNi₂Ti compound was evaluated by the cohesive energy E_coh and formation enthalpy Δ_rH, which can be calculated using equations as follows [55].

E_coh(AlNi₂Ti) = E_tot(AlNi₂Ti) − E_iso(Al) − 2E_iso(Ni) − E_iso(Ti)

(1)

Δ_rH(AlNi₂Ti) = E_coh(AlNi₂Ti) − E_coh(Al) − 2E_coh(Ni) − E_coh(Ti)

(2)

where E_tot represents the total energy of material and E_iso represents the total energy of a single atom. From the calculated result in Table 1, the negative values of formation enthalpy and cohesive energy indicate the thermodynamic stability of AlNi₂Ti.

In order to study the mechanical properties of AlNi₂Ti under external stress, the second-order elastic constants c_ij are calculated by stress versus strain without stress controls. The stress–strain method is conducted using strain modes [ε]^T = η[1 0 0 1 0 0], where [ε]^T is the transpose of the strain matrix with Voigt notation and η is the magnitude of strain [56]. The maximum strain amplitude is set as 0.003 and the number of steps for each strain is four. Then elastic constants c_ij can be computed by Hooker’s law, as presented in

Table 2. Calculated independent elastic constants (GPa) compared to experimental and other theoretical values for the single crystalline AlNi₂Ti intermetallic compound. Hereafter the mechanical modulus (GPa), Poisson’s ratio, Vicker’s hardness, shear anisotropic factors, and percent anisotropic index are also provided.

Method	Elastic Constants			B			G			BH/GH	E	v	HV	Shear Anisotropic Factors			Anisotropic Index
	c11	c12	c44	BV	BR	BH	GV	GR	GH					A1	A2	A3	AB	AG	AU
GGA-PBE	208.1	142.2	97.6	164.2	164.2	164.2	71.8	54.7	63.2	2.6	168.1	0.32	4.4	2.96	2.96	2.96	0.00	0.14	1.56
GGA-PW91	209.3	143.4	94.8	165.4	165.4	165.4	70.1	54.2	62.1	2.7	165.6	0.33	4.1	2.88	2.88	2.88	0.00	0.13	1.47
Cal. data a	215.8 a, 223 b	139.4 a, 135 b	98.7 a, 104 b	164.9 a, 164 b			67.5 a, 74 b			2.4 a	178.1 a, 193 b		8.1 a	2.36 b				0.104 a	1.165 a
Exp. data	120 c	97 c	56 c
EOS				162.9

^a Cal. data from [35]; ^b Cal. data from [36]; ^c Data adjusted by comparing with experimental diffraction elastic constants in [18];

. Our results are similar to other calculation data [18,35,36] and the values from the GGA-PW91 method are quite close to those from GGA-PBE. All elastic constants satisfy the Born–Huang stability criterion, so the AlNi₂Ti compound has intrinsic stability [57]. The detailed criterion for cubic structure is that an elastically stable crystal should follow restrictions: c₄₄ > 0, c₁₁ > c₁₂, and c₁₁ + 2c₁₂ > 0 [58,59,60]. The value of c₁₁ is much larger than c₁₂ and c₄₄, indicating that it is stiff against uniaxial strain ε₁. Then the bulk modulus (B) and shear modulus (G) are computed by Voigt–Reuss–Hill approximation [61]. The Young’s modulus E, Poisson’s ratio v and Vicker’s hardness Hv are estimated by well-known relations E = 9BG/(3B + G), v = (3B − 2G)/(6B + 2G) [62,63,64,65] and Hv = 2(k²G)^0.585 − 3, where k = G/B [66]. The results show that the hardness of AlNi₂Ti (4.4 GPa) is comparable to some common ferrous materials like pearlite steel (about 2 GPa), martensite steel (about 5.5 GPa) and white cast iron (about 6 GPa). Compared with Ni-Ti binary compounds, the hardness of AlNi₂Ti is much higher than NiTi (2.7 or 1.1 GPa with B2 or B19’ structure respectively) and NiTi₂ (1.6 GPa), and the mechanical moduli of AlNi₂Ti are also higher than NiTi and NiTi₂, but lower than Ni₃Ti [55]. The Poisson’s ratio of AlNi₂Ti is determined as 0.32. Since the Poisson’s ratio is normally 0.25 for covalent materials and 0.3 for metallic materials, our result indicates that AlNi₂Ti has a strong metallic bonding character. Besides, the value of B/G is a criterion for justifying the ductility of the material; for tough materials, B/G > 1.75, and for brittle materials, B/G < 1.75. The B/G value (2.6) indicates that AlNi₂Ti has relatively higher toughness and ductility than common ceramics, such as WC (B/G = 1.37 [67]), SiO₂ (1.39 [68]), TiC (1.41 [69,70]), Al₂O₃ (1.52 [71]) and ZrO₂ (2.3 [71]). Therefore, the existence of AlNi₂Ti at the ZTA/Fe interface will benefit the mechanical properties of the composite with considerable certainty. Our results are close to other calculated data, except for hardness, which was estimated by another empirical equation in reference [35].

The Helmholtz free energy is calculated by the following equation to investigate the stability of AlNi₂Ti under elevated temperature [45]:

E(V, T) = E_gs(V) + E_vib(V, T) + E_ele(V, T),

(3)

where E_gs refers to the ground state total energy; E_vib refers to vibrational free energy which can be given by QHA approximation based on the Debye model; E_ele refers to the electron thermal excitations at finite temperature. Then the equilibrium volume and bulk modulus at the certain temperature can be obtained from isothermal curves (E(V, T)-V) by fitting the Birch–Murnaghan equation of state as follows [72,73,74]:

$$E\left(V\right)=\frac{B_0{V}_0}{B_0^{\prime }}\left[\frac{1}{B_0^{\prime }-1}{\left(\frac{V}{V_0}\right)}^{\left(1-B_0^{\prime }\right)}+\frac{V}{V_0}+\frac{B_0^{\prime }}{1-B_0^{\prime }}\right]+E_0,$$

(4)

Here E(V) represents the Helmholtz free energy of crystal with a volume of V; E₀, B₀ and V₀ are the energy, bulk modulus and cell volume, respectively, under 0 K; B′₀ refers to the first-order derivative of bulk modulus versus pressure under 0 K. Then the computed lattice parameters and bulk moduli as the function of temperature are shown in Figure 2. It can be seen that with the rise of temperature from 0 to 1500 K, the lattice parameter increases from 5.91 to 6.01 Å while the bulk modulus declines from 162.95 to 138.05 GPa. The result indicates that AlNi₂Ti is not particularly stable under elevated temperatures and there might be a certain amount of volume expansion when the temperature rises.

In addition, the compressibility calculated from the ratio of volume (V) under a certain pressure to original volume (V₀) without pressing is adopted to characterize the stability under high pressure and plasticity. Figure 3 shows the computed results of AlNi₂Ti as well as some ceramic and metal materials for comparison. From Figure 3, under the same pressure, the volume change of AlNi₂Ti is larger than TiO, TiO₂, Fe, α-Al₂O₃ and diamond, so the intermetallic compound AlNi₂Ti performs the best plasticity and deformation ability.

3.2. Anisotropy of Mechanical Moduli

The elastic anisotropy of AlNi₂Ti with cubic structure is investigated by computing the values of bulk and Young’s moduli under different directions, as in the following equations [75]:

$$B=1/\left(s_{11}+2s_{12}\right),$$

(5)

$$E=1/\left[s_{11}-2\left(s_{11}-s_{12}-\frac{1}{2}{s}_{44}\right)\left(l_1^2{l}_2^2+l_2^2{l}_3^2+l_3^2{l}_1^2\right)\right],$$

(6)

Here s_ij represents the elastic compliance matrix, which is the inverse matrix of c_ij. l₁, l₂ and l₃ are the directional cosines. In spherical coordinates, l₁, l₂ and l₃ are computed by l₁ = sinθcosφ, l₂ = sinθsinφ, and l₃ = cosθ respectively. Then the obtained three-dimensional contour plots of elastic moduli are shown in Figure 4, revealing that the bulk modulus of AlNi₂Ti shows spherical morphology while the Young’s modulus shows relatively stronger anisotropy. From the planar projections of bulk and Young’s moduli on [001] and [110] crystallographic planes in Figure 4, the anisotropy of the Young’s modulus on both [001] and [110] planes is apparent. The bulk and Young’s moduli on principal crystallographic directions can be calculated by

B_[100] = 1/(s₁₁ + s₁₂ + s₁₃), B_[010] = 1/(s₁₂ + s₂₂ + s₂₃), B_[001] = 1/(s₁₃ + s₂₃ + s₃₃),

(7)

E_[100] = 1/s₁₁, E_[010] = 1/s₂₂, E_[001] = 1/s₃₃,

(8)

The results indicate that the bulk modulus of AlNi2Ti is B_[100] = B_[010] = 492.4 GPa, which shows obvious isotropy. As for the Young’s modulus, the maximum and minimum values on the [001] plane are E_[110] = 173.4 GPa and E_[100] = 92.7 GPa; the maximum and minimum values on the [110] plane are E_[111] = 244.4 GPa and E_[001] = 92.7 GPa, respectively.

Three shear anisotropic factors (A₁, A₂ and A₃) for cubic crystal are calculated to characterize the elastic anisotropy, which were defined by Ravindran et al. as A₁ = A₂ = A₃ = 2c₄₄/(c₁₁ − c₁₂) [76]. As in Table 2, the computed values of A₁, A₂ and A₃ were 2.96 by the GGA-PBE method and 2.88 by the GGA-PW91 method, demonstrating that the shear modulus of AlNi₂Ti has a strong directional dependence, because the values of A₁, A₂ and A₃ should be one for an isotropic crystal [77]. Besides, Ranganathan and Ostoja-Starzewski proposed another universal anisotropy index A^U determined by [78]:

A^U = 5(G_V/G_R) + (B_V/B_R) − 6,

(9)

The elastic anisotropy can also be evaluated by the percent anisotropies A_B and A_G in compression and shear modes as follows [79]:

A_B = (B_V − B_R)/(B_V + B_R),

(10)

A_G = (G_V − G_R)/(G_V + G_R),

(11)

The results in Table 2 show that our results are close to the data calculated by others. The value of A^U over zero implies that AlNi₂Ti shows the universal anisotropy of mechanical moduli; A_B has zero value while the value of A_G is slightly larger than zero, so AlNi₂Ti has isotropic bulk modulus and anisotropic shear modulus [77], which agrees well with the results from the Voigt and Reuss approximations.

3.3. Sound Velocities and Thermal Conductivity

The sound velocity in a crystal is an important parameter for thermodynamic properties. For cubic structure, the sound velocities of both longitudinal and transverse waves on principal crystallographic directions should be evaluated by the procedure of Braggar using elastic constants, as in the relations below [80,81,82,83]:

$$\mathrm{for} [100]v_l=\sqrt{c_{11}/\rho }, [010]v_{t1}=[001]v_{t2}=\sqrt{c_{44}/\rho },$$

(12)

$$\mathrm{for} [110]v_l=\sqrt{\left(c_{11}+c_{12}+2c_{44}\right)/2\rho }, [1\overline{1}0]v_{t1}=\sqrt{\left(c_{11}-c_{12}\right)/\rho }, [001]v_{t2}=\sqrt{c_{44}/\rho },$$

(13)

$$\mathrm{for} [111]v_l=\sqrt{\left(c_{11}+2c_{12}+4c_{44}\right)/3\rho }, [11\overline{2}]v_{t1}=v_{t2}=\sqrt{\left(c_{11}-c_{12}+c_{44}\right)/3\rho },$$

(14)

Here ρ is the theoretical density of the compound; v_l is the longitudinal sound velocity; v_t1 and v_t2 are the first and the second transverse mode, respectively. Obviously, c₁₁ determines the longitudinal sound velocity along [100] direction, while c₄₄ corresponds to the transverse modes. The sound velocities in the [110] and [111] directions are related to all values of c₁₁, c₁₂ and c₄₄. The anisotropy of sound velocities implies the elastic anisotropy of crystal. The calculated results are tabulated in

Table 3. Calculated anisotropic sound velocities of AlNi₂Ti intermetallic compounds, together with the values of Ni₃Ti. The unit of velocity is m/s.

Species	[100]			[010]			[001]			[110]			[111]			νl	νt	νm
	νl	νt1	νt2	νl	νt1	νt2	νl	νt1	νt2	νl	νt1	νt2	νl	νt1	νt2
AlNi2Ti	5797	3971	3971	5797	3971	3971	5797	3971	3971	6637	3262	3971	6894	2967	2967	6334	3195	3583
Ni3Ti	6565	3277	3625	6565	3277	3625	6565	3277	3277	6565	3277	3625	7243	1172	3513	6427	3550	3955

. For comparison, the sound velocities of the Ni₃Ti intermetallic compound with a hexagonal structure are also calculated. It is indicated that the longitudinal sound velocities along all directions of Ni₃Ti are higher than AlNi₂Ti.

Besides, the total transverse, longitudinal and mean sound velocities (v_t, v_l and v_m) can be calculated from density ρ, bulk and shear moduli (B and G), as following equations [84,85].

$$v_t=\sqrt{\frac{G}{\rho }},$$

(15)

$$v_l=\sqrt{\frac{\left(B+\frac{4}{3}G\right)}{\rho }},$$

(16)

$$v_m={\left[\frac{1}{3}\left(\frac{2}{v_t^3}+\frac{1}{v_l^3}\right)\right]}^{-1/3},$$

(17)

In Table 3, the velocities of AlNi₂Ti are slightly lower than Ni₃Ti. Once the mean sound velocities are obtained, the Debye temperature (Θ_D) can be estimated by [86]:

$${\Theta }_D=\frac{h}{k_B}{\left[\frac{3n}{4\pi }\left(\frac{N_A\rho }{M}\right)\right]}^{1/3}{v}_m,$$

(18)

where h is the Planck’s constant; k_B is the Boltzmann constant; N_A is the Avagadro’s constant; n is the number of atoms per formula; and M is the molecular weight. The calculated Debye temperature values of AlNi₂Ti and Ni₃Ti are shown in

Table 4. Calculated thermodynamic parameters: Debye temperatures (K), and minimum thermal conductivities (k, W/(m·K)) of the AlNi₂Ti intermetallic compound and Ni₃Ti for comparison.

Species	Model	ΘD	$\overline{\mathit{M}}$ (10−26)	P (1028)	[100] kmin	[010] kmin	[001] kmin	[110] kmin	[111] kmin	kmin	k (Exp.)
AlNi2Ti	Clark	455.1 (462 a, 411 b)	7.97		0.85	0.85	0.85	1.16	1.37	1.14	21.4 c
	Cahill			7.76	1.39	1.39	1.39	1.40	1.30	1.29
Ni3Ti	Clark	516.6	9.29		1.36	1.36	1.45	1.36	1.26	1.31
	Cahill			8.43	1.44	1.44	1.40	1.44	1.28	1.45

^a Cal. data from [35]; ^b Exp. data from [87]; ^c Exp. data from [2].

. Our results are in good agreement with both the theoretical values and experimental data [35,87]. The Debye temperature of Ni₃Ti is higher than that of AlNi₂Ti, so the inter-atomic force in Ni₃Ti is correspondingly stronger than AlNi₂Ti. This may be the reason why the moduli of Ni₃Ti are relatively larger than AlNi₂Ti.

Thermal conductivity characterizes the thermal transportation behavior of material. In this study, the minimum thermal conductivity (k_min) of AlNi₂Ti as well as Ni₃Ti for comparison is estimated by Clarke’s model and Cahill’s model as follows [88,89]:

in Clark’s model,

$$k_{\min }=0.87k_B{\overline{M}}^{-2/3}{E}^{1/2}{\rho }^{1/6},$$

(19)

in Cahill’s model,

$$k_{\min }=\frac{k}{2.48}{P}^{2/3}\left(v_l+2v_t\right),$$

(20)

where k_B is the Boltzmann constant; $\overline{M}=M/(m·N_A)$ represents the average mass per atom; E is the Young’s modulus; ρ is the density; P is the density of number of atoms per volume; and v_l and v_t refer to the longitudinal and transverse sound velocities, respectively. The results in Table 4 show that the minimum thermal conductivities of AlNi₂Ti predicted by Clark’s model (1.14 W·m⁻¹ K⁻¹) are slightly smaller than Cahill’s model (1.29 W·m⁻¹ K⁻¹). The minimum thermal conductivities of AlNi₂Ti by both models are close to that of Ni₃Ti, while the values of AlNi₂Ti are relatively smaller, probably due to the higher modulus of Ni₃Ti. The calculated thermal conductivities are smaller than the experimental value [2], because the experimental measurement is conducted using AlNi₂Ti alloy rather than single phase. Besides, the anisotropy of minimum thermal conductivity estimated by Clark’s model is consistent with that by Cahill’s model to some degree. The k_min has the same value in the [100], [010] and [001] directions by both models. However, the largest k_min estimated within Clark’s model is 1.37 W·m⁻¹ K⁻¹ in [111] direction; as for Cahill’s model, the largest k_min (1.40 W·m⁻¹ K⁻¹) is in [110] direction.

3.4. Thermal Expansion Coefficient and Heat Capacity

The Helmholtz free energy E(V, T)-V curves under different temperatures are obtained and presented in Figure 5. Then the equilibrium volumes at each temperature can be computed by fitting the Birch–Murnaghan equation of state. The volume expansions (V/V₀) under various temperatures are shown in Figure 6. It can be seen that the volume expansion of AlNi₂Ti increases with temperature and is greater than that of Ni-Ti binary compounds, which were investigated in reference [55].

Then the volumetric TEC λ can be calculated by λ = (∂V/∂T)/V. Moreover, the linear TEC α of cubic AlNi₂Ti determined as α = λ/3 is near the LTEC of Ni₃Ti, and higher than other Ni-Ti binary compounds, as shown in Figure 7. With the temperature rising, the LTEC of AlNi₂Ti increases rapidly first and then shows nearly linear growth with relatively slow increments above 200 K. At room temperature (300 K approximately), the LTECs of AlNi₂Ti and Ni-Ti binary compounds are AlNi₂Ti: 10.6 × 10⁻⁶ K⁻¹, Ni₃Ti: 8.92 × 10⁻⁶ K⁻¹, B2_NiTi: 7.50 × 10⁻⁶ K⁻¹, NiTi₂: 6.31 × 10⁻⁶ K⁻¹, respectively, so AlNi₂Ti has the largest LTEC. However, the LTEC of Ni₃Ti increases faster and exceeds the LTEC of AlNi₂Ti under elevated temperatures above 1200 K. Since AlNi₂Ti is considered one of the interfacial phases in ZTA-reinforced iron matrix composite, the thermal matching property of the ZTA/Fe interface should be improved as much as possible. In this study, the LTEC of AlNi₂Ti is quite near to ZTA (7.0–9.5 × 10⁻⁶ K⁻¹ [90]) and Fe (9.2–16.9 × 10⁻⁶ K⁻¹ [91]), so the ZTA/Fe composite with AlNi₂Ti at the interface will have excellent thermal shock resistance.

Figure 7 also shows the LTEC of another Heusler compound AlFe₂Ti (14.5 × 10⁻⁶ K⁻¹ [52]), which is higher than our result. In addition, the specific heats at constant pressure (C_p) and constant volume (C_v) are studied and their values follow the equation

C_p − C_v = λ²V(T)TB,

(21)

where λ is the volumetric thermal expansion coefficient; V(T) and B are the cell volume and crystal bulk modulus at temperature T. The calculated results for specific heats are shown in Figure 8, indicating that our results agree well with the measured C_p of AlNi₂Ti in experiment from 102 to 606.1 K [92]. At low temperature, C_p and C_v both increase rapidly with the temperature rising and the increment speed decreases substantially at higher temperatures. The high temperature limit of C_v tends to approaching the classic Dulong–Petit limit, and C_p keeps growing due to the work done by the volumetric expansion.

In order to explore the low-temperature heat capacity of AlNi₂Ti with metallic features, C_p can be estimated from the calculations of the electronic structures and elastic constants, which is given by [67].

$$C_p(T)=\gamma T+\beta T^3,$$

(22)

$$\gamma =\frac{1}{3}{\pi }^2{k}_{\mathrm{B}}^2{D}_{\mathrm{f}},$$

(23)

$$\beta =\frac{12{\pi }^{4}Rn}{5{\Theta }_D^3},$$

(24)

where γ and β are the coefficients of electronic and lattice heat, respectively; D_f is the density of state (DOS) value at the Fermi level; R is the molar gas constant and n is the total number of atoms per formula unit. It is worth mentioning that Θ_D can only describe the temperature-dependence of C_p when the temperature is lower than Θ_D/10 [93]. Therefore, the values of D_f, γ and β are shown in

Table 5. Theoretically calculated low-temperature specific heat of AlNi₂Ti and Ni-Ti binary compounds, including the total density of states at the Fermi level (D_f, states/(eV·atom)), the characteristic parameters of electron (γ, mJ/(K²·mol)) and phonon (β, mJ/(K⁴·mol)) specific heat, and the specific heat (C_p, J/(K·mol)) at 3 K and 30 K for comparison.

Species	Df	γ	β	Cp (3 K)	Cp (30 K)
AlNi2Ti	0.416	0.981	0.0825	0.00517	2.26
Ni3Ti	0.113	0.267	0.0569	0.00234	1.54
B2_NiTi	0.714	1.68	0.0577	0.00661	1.61
B19’_NiTi	0.406	0.957	0.0806	0.00505	2.21
NiTi2	0.763	1.80	0.115	0.00850	3.16

and the C_p-T curve of AlNi₂Ti and Ni-Ti binary compounds for comparison from 0 to 45 K is obtained in Figure 9.

From the illustration at the top left corner of Figure 9, it can be seen that at temperatures up to around 5.25 K, the values of C_p have the following order: NiTi₂ > B2_NiTi > AlNi₂Ti ≈ B19’_NiTi > Ni₃Ti, which is the same as the sequence of γ, indicating that the main contribution to C_p is the excitation of electrons at first. The smallest value of γ is from Ni₃Ti (γ = 0.267 mJ/(K²·mol)), because it has the smallest DOS value at the Fermi level and thus the weakest metallic nature among these compounds. At the temperature from 5.25 to 45 K, and contributions from phonon excitations (β) have to be taken into account, so the various growth rates result in a different order of C_p: NiTi₂ > AlNi₂Ti ≈ B19’_NiTi > B₂_NiTi ≈ Ni₃Ti, the same sequence as β. The calculated results of C_p in Table 5 give the examples below and above 5.25 K, which can demonstrate the distinction clearly. Therefore, it is concluded that the C_p curve is dominated by the heat capacity from electrons at first and when the temperature further rises the heat capacity is mainly determined by phonon excitation.

3.5. Formatting of Mathematical Components

The electronic properties of AlNi₂Ti are investigated by calculating the total density of states (TDOS) and partial density of states (PDOS). As shown in Figure 10, there are two sharp peaks which arise from the Ti-3s and 3p states respectively in the lower energy part of the DOS curve. They are localized and have little effect on chemical bonding. Around the Fermi level, the DOS is mainly composed of three wide peaks. The resonance peak I appearing in the DOS is caused by the hybridization of Al-p and Ni-d orbitals. There exists a hollow between peaks II and III, which reflects the splitting of bonding and anti-bonding states. The bonding states (peak II) mainly consists of Al-p and Ni-d states, while the anti-bonding states above the Fermi level are a conduction band composed of 3d bands of Ti, which form peak III. Obviously, the metallic nature of AlNi₂Ti can be determined by a nonzero DOS value at the Fermi level (3.81 electrons/eV·f.u.), which provides this compound with an excellent ductile character.

4. Conclusions

The mechanical and thermodynamic properties of AlNi₂Ti were investigated by first-principles calculations based on DFT. The mechanical moduli of AlNi₂Ti were computed from the independent elastic constants. The computed values of bulk, shear and Young’s modulus were 164.2, 63.2 and 168.1 GPa, respectively, indicating that it has relatively high mechanical properties. The hardness of AlNi₂Ti (4.4 GPa) is comparable to common ferrous materials, like martensite steel and white cast iron. Compared with Ni-Ti binary compounds, the hardness and moduli of AlNi₂Ti are much higher than NiTi and NiTi₂, but lower than Ni₃Ti. The intrinsic ductile nature and strong metallic bonding character of AlNi₂Ti are confirmed by B/G and Poisson’s ratio. Besides, AlNi₂Ti shows the isotropy bulk modulus and anisotropic elasticity in different crystallographic directions. Therefore, the existence of AlNi₂Ti at the ZTA/Fe interface will probably benefit the mechanical properties of the composite. At room temperature, the linear thermal expansion coefficient (LTEC) of AlNi₂Ti, estimated by quasi-harmonic approximation based on the Debye model, was 10.6 × 10⁻⁶ K⁻¹, quite close to LTECs of ZTA and iron, so the thermal shock resistance of the ZTA/Fe composite with an AlNi₂Ti interfacial phase can be improved. Other thermodynamic properties including Debye temperature, sound velocity, thermal conductivity and heat capacity were also calculated and compared with Ni-Ti binary compounds. The electronic properties of AlNi₂Ti imply its excellent ductile character.