Mechanical Properties and Thermal Conductivity of Y-Si and Gd-Si Silicides: First-Principles Calculations

Abstract

The traditional Si bonding layer in environmental barrier coatings has a low melting point (1414 °C), which is a significant challenge in meeting the requirements of the next generation higher thrust-to-weight ratio aero-engines. To seek new bonding layer materials with higher melting points, the mechanical properties of Y-Si and Gd-Si silicides were calculated by the first-principles method. Subsequently, empirical formulae were employed to compute the sound velocities, Debye temperatures, and the minimum coefficients of thermal conductivity for the YSi, Y₅Si₄, Y₅Si₃, GdSi, and Gd₅Si₄. The results showed that Y₅Si₄ has the best plasticity and ductility among all these materials. In addition, Gd₅Si₄ has the minimum Debye temperature (267 K) and thermal conductivity (0.43 W m⁻¹ K⁻¹) compared with others. The theoretical calculation results indicate that some silicides in the Y-Si and Gd-Si systems possess potential application value in high-temperature bonding layers for thermal and/or environmental barrier coating.

1. Introduction

The melting point of the traditional Si bond layer, which is 1414 °C, limits its application temperature in environmental barrier coatings (EBCs) [1]. Furthermore, the oxidation product, SiO₂, undergoes phase transformations accompanied by volume changes (~4.3%), which tends to cause the coating to crack [2,3]. Considering that the coefficient of thermal expansion (CTE) of HfSiO₄ (3.6–6.6 × 10⁻⁶ °C⁻¹ [4]) matches well with that of the SiC matrix (4.5 × 10⁻⁶ °C⁻¹ [5]), and given its good phase stability, HfO₂ was incorporated into the Si bond layer to dynamically convert SiO₂ into HfSiO₄ phase [6,7]. However, HfO₂ has a much higher oxygen diffusion rate, which accelerates the oxidation of the Si bond layer [8]. Although the oxidation resistance and the service life were improved by optimizing the content and distribution of HfO₂ [9,10], the service temperature did not improve, posing a challenge in fulfilling the demands for the next generation aero-engines with a higher thrust-to-weight ratio.

Y₂SiO₅ exhibits minimal oxygen permeability across a broad temperature spectrum, reaching a permeability of 10⁻¹⁰ kg/(m·s) at 1973 K [11]. Additionally, it demonstrates a CTE of 5–8 × 10⁻⁶ °C⁻¹, while Y₂Si₂O₇ has a CTE of 3.90 × 10⁻⁶ °C⁻¹ [11], which matches well with the SiC matrix when used in combination [12]. Yttrium silicides (Y_xSi_y) have much higher melting points than that of Si. And their oxidation products, Y₂O₃ and SiO₂, will react with each other to form Y₂SiO₅ and/or Y₂Si₂O₇, which have good environmental barrier properties. This strategy simultaneously increases not only the operating temperature of the bond layer but also its service life. The same applies to the rare earth silicate Gd₂SiO₅, which has a relatively low thermal conductivity and excellent corrosion resistance [13,14], and has been extensively studied as a coating material in recent years [15,16]. Some gadolinium silicides (Gd_xSi_y) also have high melting points and good phase structural stability, making them promising candidates for use as high-temperature bonding layer materials.

When a certain material is used as a bond layer, it is important to consider not only its temperature resistance and oxidation resistance, but also its mechanical properties and thermal conductivity. The coating material should have good ductility and a large damage tolerance to ensure that the coating does not peel or crack under the impact of foreign particles and the influence of thermal cycling [17]. In addition, materials with low thermal conductivity can play a certain role in insulation, thereby reducing the surface temperature of SiC-based composite [18].

Therefore, in this study, the mechanical properties of high-melting-point rare earth silicides YSi, Y₅Si₄, Y₅Si₃, GdSi and Gd₅Si₄ were assessed using first-principles calculations. These materials were selected based on the phase diagrams of Y-Si and Gd-Si binary system [19,20]. The ductility of selected silicides was evaluated according to the ratio of shear modulus to bulk modulus. Subsequently, the models proposed by Clarke [21] and Slack [22] were utilized to forecast the temperature-dependent thermal conductivity and the theoretical minimum values for Y-Si and Gd-Si silicides. The findings revealed that these yttrium and gadolinium silicides exhibit promising characteristics as potential high-temperature bond layers in EBCs applications.

2. Computation Methods

Density functional theory (DFT) calculations were conducted using the projection augmented wave (PAW) method [23,24]. These calculations were executed with the Vienna Ab-initio Simulation Package (VASP) [25]. A plane-wave basis cutoff energy of 520 eV was utilized, and electron spin polarization was incorporated into all calculations. Subsequently, K-point sampling in the Brillouin zone was performed using the Monkhorst–Pack method. For YSi, Y₅Si₄, Y₅Si₃, GdSi and Gd₅Si₄, Brillouin-zone integrations were conducted on grid sizes of 8 × 2 × 8, 5 × 2 × 5, 5 × 5 × 6, 5 × 8 × 7 and 5 × 2 × 5, respectively. During the structural optimization process, which included electron self-consistent calculations, a tolerance of 10⁻⁴ eV was applied. For the computation of electron statics, a tolerance of 10⁻⁵ eV was utilized. To ensure the accuracy of mechanical and thermal properties, all lattices and atoms underwent full relaxation.

The characteristic elastic constants were calculated using the Voigt–Reuss–Hill averaging scheme [26]. Utilizing the Voigt approximation [27], the upper bulk modulus (B_V) and shear modulus (G_V) were determined as follows:

$$B_V=\frac{1}{9}\left(C_{11}+C_{22}+C_{33}\right)+\frac{2}{9}\left(C_{12}+C_{13}+C_{23}\right)$$

(1)

$$G_V=\frac{1}{15}\left(C_{11}+C_{22}+C_{33}-C_{12}-C_{13}-C_{23}\right)+\frac{1}{5}\left(C_{44}+C_{55}+C_{66}\right)$$

(2)

where the C_ij represents the second-order elastic constants. While the Reuss approximation (lower bound) of bulk modulus (B_R) and shear modulus (G_R) were determined as follows [28]:

$$B_R=\frac{1}{\left(S_{11}+S_{22}+S_{33}\right)+2\left(S_{12}+S_{13}+S_{23}\right)}$$

(3)

$$G_R=\frac{15}{4\left(S_{11}+S_{22}+S_{33}\right)-4\left(S_{12}+S_{13}+S_{23}\right)+3\left(S_{44}+S_{55}+S_{66}\right)}$$

(4)

in which the S_ij are the compliance constants [29]:

$$S_{11}+S_{12}=C_{33}/C, S_{11}+S_{12}=1/\left(C_{11}-C_{12}\right),$$

$$S_{13}=-C_{13}/C, S_{33}=\left(C_{11}+C_{12}\right)/C, S_{44}=1/C_{44}, S_{66}=1/C_{66}$$

(5)

where

$$C=C_{33}\left(C_{11}+C_{12}-{2C}_{13}^2\right)$$

(6)

The average values of the bulk modulus (B_V, and B_R) and shear modulus (G_V and G_R) were adopted as the values of the modulus [26].

$$B=\frac{1}{2}\left(B_V+B_R\right),G=\frac{1}{2}\left(G_V+G_R\right)$$

(7)

The average Young’s modulus (E) and Poisson’s ratio (μ) were calculated using the following expression [30]:

$$E=\frac{9BG}{\left(3B+G\right)},u=\frac{3B-2G}{2\left(3B+G\right)}$$

(8)

The Vickers hardness was assessed using the following formula [31]:

$$H=0.92{\left(\frac{G}{B}\right)}^{1.137}{G}^{0.708}$$

(9)

Based on B and G obtained from Equation (7), the mean values of the transverse (v_T) and longitudinal (v_L) sound velocity components were calculated as follows:

$$v_T={\left(\frac{G}{\rho }\right)}^{\frac{1}{2}},v_L={\left(\frac{B+\frac{4}{3}G}{\rho }\right)}^{\frac{1}{2}}$$

(10)

where ρ is the density. Then, the average velocity of sound (v_m) was written as [32]:

$$v_m={\left[\frac{1}{3}\left(\frac{2}{v_T^3}+\frac{1}{v_L^3}\right)\right]}^{\frac{-1}{3}}$$

(11)

Based on this, the Debye temperature (Θ_D) was obtained as [32]:

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

(12)

where n represents the number of atoms in a formula unit, k_B denotes the Boltzmann constant, h signifies the Planck constant, N_A is the Avogadro constant, and M corresponds to the molecular weight.

At lower temperature (0.5 Θ_D < T < 1.6 Θ_D), the thermal conductivity (k) was calculated from Slack’s model [22]:

$$k=A\frac{\overline{M}{\Theta }^3\delta }{{\gamma }^2{n}^{\frac{2}{3}}T}$$

(13)

where δ: cube root of the average volume of the atom, $\overline{M}$: the average mass of the atoms in the crystal, A: a physical constant, γ: Grüneisen’s parameter [33]:

$$\gamma =\frac{9\left(v_L^2-\frac{4}{3}{v}_T^2\right)}{2\left(v_L^2-2v_T^2\right)}=\frac{3\left(1+v_m\right)}{2\left(2-3v_m\right)}$$

(14)

Thermal conductivity decreases with increasing temperature and then saturated to a constant value (k_min), which was determined using the Clarke’s model [21]:

$$k_{min}=0.257k_B^2{\hslash }^{-1}{〈M〉}^{-\frac{1}{3}}{\rho }^{\frac{1}{3}}{\Theta }_D$$

(15)

where <M> is the average atomic mass equal to M/N_An_a (n_a is the number of atoms in a molecule), $\hslash$ represents the reduced Planck constant (h/2π).

3. Results and Discussion

3.1. Structural Properties

For YSi, Y₅Si₄, Y₅Si₃, GdSi, and Gd₅Si₄ with Cmcm, Pnma, P6₃/mcm, Pnma and Pnma space groups, the structural parameters were firstly optimized. Both Y_xSi_y and Gd_xSi_y crystals for structural optimization are single-cell, as shown in Figure 1. The calculated structural parameters and JCPDS data are listed in

Table 1. Calculated equilibrium lattice parameters of Y_xSi_y and Gd_xSi_y compared to their JCPDS card data.

Materials	$\mathit{a}$($\dot{\mathit{A}}$)	$\mathit{b}$($\dot{\mathit{A}}$)	$\mathit{c}$($\dot{\mathit{A}}$)
YSi	4.283	10.541	3.842
YSi (89-2305)	4.251	10.526	3.826
Y5Si4	7.443	14.585	7.701
Y5Si3	8.445	8.445	6.386
Y5Si3 (89-3037)	8.403	8.403	6.303
GdSi	7.980	3.878	5.767
GdSi (80-0705)	7.973	3.858	5.753
Gd5Si4	7.516	14.735	7.774
Gd5Si4 (87-2319)	7.486	14.750	7.751

(the JCPDS card of Y₅Si₄ was not retrieved), and it can be seen that the structural parameters calculated by first principles are consistent with the JCPDS card data.

In addition, to clearly see the accuracy of the calculation, the relative errors of lattice constants for Y_xSi_y and Gd_xSi_y compared to JCPDS data are plotted (Figure 2). The maximum observed relative error is 1.317% for the Y₅Si₃ crystal, while errors for other materials are less than 0.8%. This further illustrated the accuracy of the calculation results.

3.2. Elastic and Mechanical Properties

The elastic constants, as calculated, are listed in

Table 2. Calculated elastic constants C_ij (in GPa) for Y_xSi_y and Gd_xSi_y.

Materials	C11	C12	C13	C22	C23	C33	C44	C55	C66
YSi	161	43	67	203	25	182	48	101	62
Y5Si4	119	37	51	139	50	139	44	31	52
Y5Si3	167	40	32			119	51		62
GdSi	161	43	67	203	25	182	48	101	62
Gd5Si4	107	36	48	134	50	130	38	31	51

. It can be seen that Y₅Si₃, which is a tetragonal structure, has six independent elastic constants; YSi, Y₅Si₄, GdSi and Gd₅Si₄ are orthomorphic structures, each have nine independent elastic constants. To evaluate the mechanical stability of Y_xSi_y and Gd_xSi_y, Born’s requirements for tetragonal structures were given as follows [34,35,36]:

$$\left(C_{11}-C_{12}\right)>0, \left(C_{11}+C_{33}-2C_{13}\right)>0, C_{11}>0, C_{33}>0, C_{44}>0,C_{66}>0, \left(2C_{11}+C_{33}+2C_{12}+4C_{13}\right)>0$$

(16)

And for orthorhombic crystals are [36]:

$$\begin{gathered} \left(C_{11}+C_{22}-2C_{12}\right)>0, \left(C_{11}+C_{33}-2C_{13}\right)>0, \left(C_{22}+C_{33}-2C_{23}\right)>0, C_{11}>0, \\ {C}_{22}>0,C_{33}>0,C_{44}>0,C_{55}>0,C_{66}>0,\left(C_{11}+C_{22}+C_{33}+2C_{12}+2C_{13}+2C_{23}\right)>0 \end{gathered}$$

(17)

Based on these requirements and calculated elastic constants, these five Y_xSi_y and Gd_xSi_y considered in this paper were all mechanically stable.

The calculated mechanical properties of Y_xSi_y and Gd_xSi_y are tabulated in

Table 3. Calculated bulk modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio (μ), and hardness (H) of Y_xSi_y and Gd_xSi_y.

Materials	B (GPa)	G (GPa)	E (GPa)	μ	H (HV)	G/B
YSi	91	67	161	0.205	13	0.736
Y5Si4	75	42	106	0.265	7	0.560
Y5Si3	71	56	133	0.194	12	0.789
GdSi	74	57	136	0.193	12	0.730
Gd5Si4	62	38	94	0.247	7	0.613

. B represents the elasticity of a substance within the elastic range, and for YSi, the B value is the largest among the silicides in Table 3, indicating that YSi has the strongest incompressibility. G describes the resistance to shape change in materials, with a lower value indicating higher ductility. For Y_xSi_y and Gd_xSi_y, the G value is obviously smaller than B, suggesting good ductility and machinability. A lower value of E can reduce the impact of thermal stress and potentially extend the service life of the coating materials [37]. As shown in Table 3, in the Y-Si system, the order of the calculated E is YSi > Y₅Si₃ > Y₅Si₄, and in the Gd-Si system, it is GdSi > Gd₅Si₄. Specifically, the calculated E value for Gd₅Si₄ is only 94 GPa (Table 3), which is significantly lower than that of the traditional Si bond layer (140 ± 2 GPa) [38].

Furthermore, materials that serve as high-temperature coatings must also exhibit plasticity, as the plastic deformation process is crucial for the effective release of thermal stress. The toughness and brittleness of the material can be judged by Equation (8), if μ > 0.26, the material is more plastic than brittle [39]. As shown in Table 3, Poisson’s ratio of Y₅Si₄ was more than 0.26. This showed that Y₅Si₄ is relative plastic, and the rest are more brittle. In addition, the calculation is performed at zero temperature and zero pressure, in which case the hardness order of Y Y_xSi_y and Gd_xSi_y was YSi > Y₅Si₃ = GdSi > Y₅Si₄ = Gd₅Si₄. A small G/B ratio indicates good ductility, processability, and damage tolerance, which helps to maintain the integrity of the coating. This property prevents issues such as foreign particle impact and thermal cycling, as well as avoiding cracks that can be caused by thermal expansion mismatches. After calculation, the order of G/B values is Y₅Si₃ > YSi > GdSi > Gd₅Si₄ > Y₅Si₄. It is predicted that Y₅Si₄ has the best ductility among these materials.

Young’s modulus (E) can be used to evaluate the strength and stiffness of a material. In practical applications, it needs to accurately obtain the 3D Young’s modulus diagram to understand the change in E with crystal orientation. In the case of Y_xSi_y and Gd_xSi_y, the relation between E and direction can be obtained by the following equation [40]:

$$\frac{1}{E}=l_1^4+2l_1^2{l}_2^2{s}_{12}+2l_1^2{l}_3^2{s}_{13}+l_2^4{s}_{22}+2l_2^2{l}_3^2{s}_{23}+l_3^4{s}_{33}+l_2^2{l}_3^2{s}_{44}+l_1^2{l}_3^2{s}_{55}+l_1^2{l}_2^2{s}_{66}$$

(18)

The elastic compliance S_ij is related to the directional cosines l₁, l₂, and l₃ with respect to the three principal directions. The surface contours of E for Y_xSi_y and Gd_xSi_y are depicted in Figure 3a–e, and the planar projections of E for (100), (010), and (001) crystallographic planes are shown in Figure 3(a1–e1). According to [41], crystal directions A and B correspond to different crystallographic orientations on various planes: for the (001) plane, A corresponds to the [100] direction and B to the [010] direction; for the (010) plane, they correspond to the [100] direction and the [001] direction, respectively; and for the (100) plane, they are the [001] direction and the [010] direction.

Figure 3 clearly illustrates the elastic anisotropy of Y_xSi_y and Gd_xSi_y. For YSi, the anisotropy of E on the (001) plane was stronger than on the other two planes, with the minimum E occurring in the <100> direction (Figure 3(a,a1)). For Y₅Si₃, the anisotropy of the (010) and (100) crystal planes are the same, and the minimum and maximum values occur in the <001> and <100> directions, respectively (Figure 3(c,c1)). For Y₅Si₄ (Figure 3(b,b1)), GdSi (Figure 3(d,d1)) and Gd₅Si₄ (Figure 3(e,e1)), the anisotropy of E on the (010) plane is stronger than on the other two planes. Based on the above analysis, the degree of anisotropy of Young’s modulus correlates closely with crystal symmetry. According to Mohapatra and Eckhardt [42], the anisotropy of the elastic modulus is primarily influenced by the non-diagonal elements of the compliance matrix. Therefore, if these non-diagonal elements (i.e., S₁₂, S₁₃, S₂₃, in this case) are ignored when calculating 3D Young’s modulus, the degree of anisotropy will be significantly reduced. This explains why, among the materials mentioned, Y₅Si₃ exhibits the lowest degree of anisotropy in Young’s modulus. The anisotropy data for the elastic properties of Y_xSi_y and Gd_xSi_y depicted in Figure 3 can offer substantial support for the design, selection, and simulation of materials.

3.3. Thermal Conductivity

Coatings with low thermal conductivity possess strong thermal insulation capabilities, effectively mitigating the thermal damage from the environment to the matrix. Therefore, thermal conductivity is a critical factor to consider when selecting suitable EBCs. According to Equations (13) and (15), the estimation of the intrinsic thermal conductivity of Y_xSi_y and Gd_xSi_y depends on the v_L, v_T, v_m, and Θ_D. According to Equations (10)–(12), the calculation results for the aforementioned parameters are listed in

Table 4. Calculated sound velocities (v_L, v_T, v_m), Debye temperature (Θ_D), and minimum thermal conductivity (k_min) of Y_xSi_y and Gd_xSi_y.

Materials	vL (km/s)	vT (km/s)	vm (km/s)	ΘD·(K)	kmin(w/(m·K))
YSi	6.34	3.86	4.27	455	0.76
Y5Si4	5.44	3.08	3.42	356	0.58
Y5Si3	5.75	3.54	3.91	398	0.63
GdSi	4.65	2.87	3.17	335	0.55
Gd5Si4	4.03	2.34	2.59	267	0.43

. The results indicate that the sound velocities of Gd_xSi_y are significantly lower than those of Y_xSi_y. Additionally, the order of Θ_D values is YSi > Y₅Si₃ > Y₅Si₄ > GdSi > Gd₅Si₄. Figure 4 displays a comparison between the Θ_D values calculated in this study and those from the literature. The minimal difference between the calculated and reported values suggests that the results of our calculations are reliable.

The material’s inherent thermal conductivity is determined by how phonons interact and scatter at varying temperature levels [21]. Slack’s model, as presented in Equation (13) [22], offers a suitable approach to temperature-dependent thermal conductivity at a relatively low temperature. The model-estimated temperature-dependent thermal conductivity for Y_xSi_y and Gd_xSi_y is in Figure 5. It can also be intuitively seen from Figure 5 that although the slopes of each curve were different, they all exhibit an inverse proportionality to temperature. Based on Slack’s model, the behavior of thermal transportation Y_xSi_y could be described the following: with the increase in temperature, the thermal conductivity of YSi, Y₅Si₄, and Y₅Si₃ declined as k = 989.11/T, k = 902.99/T and k = 531.61/T. Similarly, the thermal conductivity of GdSi and Gd₅Si₄ in relation to temperature was k = 579.48/T and k = 313.53/T, respectively. Then, if the temperature is higher, the phonon mean free path will reduce to the average atomic distances when the thermal conductivity is close to the minimum [21]. As Slack’s model does not provide a rigorous theory for high-temperature thermal conductivity, this study utilized the modified Clarke’s model (Equation (15)) to assess the minimum thermal conductivity (k_min). Table 4 indicates a calculated thermal conductivity sequence for Y_xSi_y and Gd_xSi_y as follows: Gd₅Si₄ < Y₅Si₃ < GdSi < Y₅Si₄ < YSi, with Gd₅Si₄ exhibiting the lowest thermal conductivity at 0.43 W m⁻¹ K⁻¹. Calculations revealed that the thermal conductivities of Gd_xSi_y compounds are lower than that of silicides in the Y-Si system, suggesting that when used as coating materials, specific compounds such as GdSi and Gd₅Si₄ will exhibit superior thermal insulation ability, potentially mitigating thermal damage to the SiC composites substrate.

4. Conclusions

In this work, the elastic constants of silicides YSi, Y₅Si₄, Y₅Si₃, GdSi, and Gd₅Si₄ in the Y-Si and Gd-Si systems were predicted using first-principles calculations. Subsequently, the volume modulus, shear modulus, Young’s modulus, sound velocity, Debye temperature, and thermal conductivity were calculated by empirical formulas. The optimized structural parameters of Y_xSi_y and Gd_xSi_y showed minimal differences compared to the existing JCPDS card data. The conclusions regarding their mechanical and thermal properties are as follows:

(1)

The results for elastic properties indicated that Y₅Si₄ is a ductile material, and its G/B value is lower than that of the other materials in this study. This characteristic helps to minimize the thermal stress and enhances the thermal shock resistance when used as coating materials. In addition, Young’s moduli of all the calculated materials are anisotropic.

(2)

The calculated thermal conductivity sequence for Y_xSi_y and Gd_xSi_y is as follows: Gd₅Si₄ < Y₅Si₃ < GdSi < Y₅Si₄ < YSi, with Gd₅Si₄ exhibiting the lowest thermal conductivity at 0.43 W m⁻¹ K⁻¹. This study ascertains that they are promising materials for environmental barrier coatings.