# Mechanical and Thermal Properties of the Hf–Si System: First-Principles Calculations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}, HfSi, Hf

_{5}Si

_{4}, Hf

_{3}Si

_{2}, and Hf

_{2}Si with much higher melting points than that of Si. Among them, HfSi

_{2}has the lowest modulus capable of good modulus matching with SiC substrate. In addition, these Hf-Si compounds have much lower high temperature thermal conductivity with Hf

_{2}Si being the lowest of 0.63 W m

^{−1}K

^{−1}, which is only half of Si, capable of improved heat insulation.

## 1. Introduction

_{3}N

_{4}, SiC, and SiC matrix composites, have great potential for application in gas turbine engines because of their excellent high temperature mechanical properties [1]. However, Si-based ceramics and their composites easily react with water vapor to produce Si(OH)

_{4}in the engine operating environment, leading to its performance decline [2,3]. In order to improve the performance of high temperature resistance, chemical corrosion resistance, and high gas flow resistance, environmental barrier coatings (EBCs) are introduced to protect Si-based ceramics and their composites [4].

^{−6}°C

^{−1}) [5] matching with that of SiC (4.5 × 10

^{−6}°C

^{−1}) [5], low Young’s modulus, and advantageous adhesion property [6,7,8]. However, the relatively low melting point (1414 °C) of Si limits its upper using temperature [9,10]. In addition, Si was oxidized into SiO

_{2}at high temperature, which undergoes a β to α phase transformation at approximately 277 °C during cooling and accompanied by a volume shrinkage of approximately 5% [11,12]. When the high temperature oxidation atmosphere contains water vapor, SiO

_{2}directly reacts with water vapor to form volatilized Si(OH)

_{4}[13], resulting in the coating’s failure. NASA proposed mixing HfO

_{2}into the Si bonding layer [7,14,15,16]. HfO

_{2}not only has a high melting point (2800 °C) [17] and low creep rate at high temperature [18], it also reacts with SiO

_{2}at a high temperature to produce HfSiO

_{4}, which has good phase stability up to 1700 °C, and a better matched CTE (3.3–6.6 × 10

^{−6}°C

^{−1}) [14,16]. Based on our previous work, the property of Si + HfO

_{2}bonding layer can be improved by optimizing the amount and the distribution state of HfO

_{2}inside Si [19]. However, the upper-temperature limit has not yet changed.

_{2}, HfSi, Hf

_{5}Si

_{4}, Hf

_{3}Si

_{2}, and Hf

_{2}Si hafnium silicide are stable phases with much higher melting points (1543 °C, 2142 °C, 2315 °C, 2480 °C and 2360 °C, respectively) than that of Si [21], making them good candidates for a bonding layer used at ahigher temperature. Among them, HfSi

_{2}was also reported as having good antioxidant capacity [22,23]. In addition to the high temperature capability, good mechanical properties (such as relatively low modulus that matched with SiC) and low thermal conductivity are also required for better stress relaxation and strong thermal insulation ability.

## 2. Computation Methods

_{2}, HfSi, Hf

_{5}Si

_{4}, Hf

_{3}Si

_{2}and Hf

_{2}Si, respectively. The spin polarization of the electron was taken into account in all calculations. Finally, structural relaxation used a tolerance of 10

^{−4}eV for the electronic self-consistent calculations, and 10

^{−5}eV for electronic static computing. In order to obtain accurate mechanical and thermal properties, the models of the pure Si and Hf-Si system were fully structurally optimized.

_{V}) and shear modulus (G

_{V}) are bounded by the Voigt approximation as below [31], respectively.

_{ij}are second-order elastic constants. Then, Reuss proposed the approximation of the lower bound of volume (B

_{R}) and shear modulus (G

_{R}) [32]:

_{ij}are the compliance constants [33]:

_{T}and longitudinal v

_{L}components of the speed of sound could be estimated using the values of bulk modulus B, shear modulus G, and density ρ of the crystal.

_{m}is bounded from above by [36]:

_{D}is written as [36]:

_{B}is the Boltzmann constant, N

_{A}is the Avogadro constant, and M is the molecular weight.

^{3}is the average volume of the atom, $\overline{M}$ is the average mass of the atoms in the crystal, and A is a physical constant (A = 3.1 × 10

^{−6}if $k$ is in W m

^{−1}L

^{−1}, and δ in $\dot{A}$). In addition, γ is the high-temperature limit of the acoustic phonon mode Grüneisen parameter, which could be derived from the sound velocity [37]:

_{D}< T < 1.6 Θ

_{D}). At very high temperatures, the thermal conductivity is independent of temperature and tends to be stable (i.e., minimum thermal conductivity), which can be estimated according to Clarke’s model [24]:

_{A}n (n is the number of atoms in a molecule), $\hslash $ is the reduced Planck constant (h/2π).

## 3. Results and Discussion

#### 3.1. Structural Properties

_{2}Si, Hf

_{5}Si

_{3}, Hf

_{3}Si

_{2}, Hf

_{5}Si

_{4}, HfSi, and HfSi

_{2}with Fd3m1, I4/mcm, P6

_{3}/mcm, P4/mbm, P4

_{1}2

_{1}2, Pnma, and Cmcm space groups, their structural parameters were optimized, and all lattice parameters and ionic positions were fully relaxed during the geometry optimization. The calculated lattice parameters, along with the corresponding JCPDS card data are presented in Table 1. It can be clearly seen that the calculated parameters of the Hf-Si system were in agreement with the available JCPDS card data. Relative errors of lattice constants of this Hf-Si system are also shown in Figure 2, from which the accuracy of the calculation can be directly observed. The maximum relative error was 1.505% for Hf

_{2}Si crystal, and the minimum relative error was only 0.062%. This further explained the accuracy of the calculation results.

#### 3.2. Elastic and Mechanical Properties

_{5}Si

_{4}, Hf

_{3}Si

_{2,}and Hf

_{2}Si, which are tetragonal system with six independent elastic constants, and the rest (HfSi

_{2}and HfSi) are orthomorphic system with nine independent elastic constants. Before further calculation, it is necessary to evaluate whether the above system can resist external deformation and restore its own structure during calculation, namely mechanical stability. The formula suitable for determining the mechanical stability of cubic crystals (Si) [38] is:

_{5}Si

_{4}> Hf

_{3}Si

_{2}> Hf

_{2}Si > HfSi

_{2}, and they were much larger than that of Si. Liu et al. once used first principles to calculate the bulk modulus of the Hf-Si system, and found that HfSi had the largest value and HfSi

_{2}had the smallest [41], which is the same as the results in this work. B represents the elasticity of a substance over an elastic range, so among the above materials, HfSi has the strongest incompressible properties. Secondly, G describes a material’s resistance to shape change. The order of G in the Hf-Si system was Hf

_{2}Si > HfSi > Hf

_{3}Si

_{2}> Hf

_{5}Si

_{4}> HfSi

_{2}> Si. For these materials, the G value was obviously smaller than B (Table 3), which had good ductility and machinability. In addition, lower E is preferred as a thermal coating material, because it has good bonding properties and can reduce the influence of thermal stress [42]. Lee et al. calculated the Young’s modulus of Si ‹001› nanowires by using first-principles calculations to be 122.8 GPa [43], which is relatively close to the value calculated here (139 Gpa). This suggests that the calculation result is reliable. The calculated E of Si (139 GPa) was much smaller than that of the Hf-Si system, so Si was better than the Hf-Si system in resistance to thermal stress. In all, the modulus of Si was much smaller than that of the other five materials, which showed that alloying Hf element into Si can increase modulus.

_{5}Si

_{4}and HfSi is closer to 0.26 and their plasticity is relatively better. Liu et al. also found that the brittleness of Hf

_{2}Si, Hf

_{3}Si

_{2}, and HfSi

_{2}is obvious, while the brittleness of HfSi and Hf

_{5}Si

_{4}is not, preferring ductile materials [41]. It also can be seen that the Hf-Si system has a higher value of Poisson’s ratio than that of Si, and thus improved plasticity. The hardness sequence of the Si and Hf-Si system was Hf

_{2}Si > Hf

_{3}Si

_{2}> HfSi = HfSi

_{2}> Hf

_{5}Si

_{4}> Si at zero temperature and zero pressure. Finally, the G/B values of the Hf-Si system were ordered as Si > Hf

_{2}Si > HfSi

_{2}> Hf

_{3}Si

_{2}> HfSi > Hf

_{5}Si

_{4}. A smaller G/B indicates good ductility and damage tolerance, which ensures the integrity of the coating against foreign particles and thermal cycling by avoiding crack formation. In conclusion, these Hf-Si silicides exhibited enhanced plasticity and damage tolerance as compared with Si, which is beneficial to protect SiC composites at high temperatures.

_{ij}is the elastic compliance, and then l

_{1}, l

_{2}, and l

_{3}are the directional cosines of angles with the three principal directions, respectively.

_{1}–e

_{1}. For HfSi

_{2}(Figure 4a,a

_{1}) and HfSi (Figure 4b,b

_{1}), the anisotropy of Young’s modulus on the (100) plane is stronger than that on the other two planes. For Hf

_{5}Si

_{4}(Figure 4c,c

_{1}), Hf

_{3}Si

_{2}(Figure 4d,d

_{1}), and Hf

_{2}Si (Figure 4e,e

_{1}), the anisotropy of (100) and (010) faces was the same. In addition, the minimum Young’s modulus of HfSi

_{2}and Hf

_{2}Si is parallel to the (010) direction, and the minimum ones of HfSi, Hf

_{5}Si

_{4}and Hf

_{3}Si

_{2}are parallel to the (001) direction. Finally, the maximum Young’s modulus of Hf

_{5}Si

_{4}is parallel to the (110) direction. The above analysis shows that the anisotropy of the Young’s modulus of the material is closely related to the crystal symmetry. Mohapatra and Eckhardt [47] believe that the anisotropy of the elastic modulus is mainly affected by the non-diagonal elements of the flexibility matrix. When calculating Young’s modulus of different crystallization directions, if the non-diagonal elements (i.e., S

_{12}, S

_{13}, S

_{23}, in this case) are ignored, the degree of anisotropy of the elastic modulus will be significantly reduced. For simplicity, this procedure was not presented here. This confirms that the anisotropy of the Young’s modulus of Si was the lowest among the above substances. Anyway, Figure 3 and Figure 4 showed the anisotropy of the elastic properties of Hf-Si system. Using this information, the most important directions of mechanical property measurements and applications were defined.

#### 3.3. Thermal Conductivity

_{L}, v

_{T}, v

_{m}and Θ

_{D}. According to the calculated elastic moduli and density of the equilibrium structure, the v

_{L}, v

_{T}, v

_{m}and Θ

_{D}were derived by Equations (10)–(12). The calculated parameters for Hf-Si system are listed in Table 4. First of all, it shows that the sound velocities of Si (values of v

_{L}, v

_{T}and v

_{m}were 4.42, 2.67 and 2.94 m/s, respectively) are significantly lower than those of the Hf-Si system. In addition, Si also has the highest Θ

_{D}(488 K), which is ranked as Si > HfSi

_{2}> HfSi > Hf

_{5}Si

_{4}> Hf

_{3}Si

_{2}> Hf

_{2}Si.

_{p}of the Hf-Si system obtained from Equation (13) is listed in Table 5, and Equation (13) can be written as follows:

_{p}, the lower the thermal conductivity of the substance at the same temperature. From Table 5, the thermal conductivity of Si (1025.82 W m

^{−1}) was much greater than that of Hf-Si system (sorted as Si > HfSi > Hf

_{3}Si

_{2}>HfSi

_{2}> Hf

_{2}Si> Hf

_{5}Si

_{4}). Therefore, according to Slack’s model, the Hf-Si system has reduced thermal conductivity than that of Si. In addition, among the Hf-Si system, the E value of HfSi

_{2}is the closest to Si. So for simplicity, taking Si and HfSi

_{2}as examples, their temperature-dependent thermal conductivity estimated from Slack’s model is shown in Figure 5. With the increase in temperature, the thermal conductivity of Si and HfSi

_{2}declined as k = 1025.82/T and k = 811.96/T, respectively. If the temperature is further increased, the phonon mean-free path decreases to the average atomic distance, and thus the thermal conductivity approaches its minimum [24].

_{min}is evaluated by the modified Clarke’s model, as illustrated by Equation (15). Compared to the perfect crystal in the calculation, there are inevitable defects in the real Si material. In addition, lattice thermal conductivity was only calculated in this paper, ignoring the contribution of electrons to the thermal conductivity. So, the experimental value of the minimum thermal conductivity of single-crystal Si (2.87 W m

^{−1}K

^{−1}) [48] is larger than the calculated value in this paper (1.26 W m

^{−1}K

^{−1}). As can be seen in Table 4, the minimum thermal conductivity of the Hf-Si system ordered as Si > HfSi

_{2}> HfSi > Hf

_{5}Si

_{4}= Hf

_{3}Si

_{2}> Hf

_{2}Si, where the thermal conductivity of Hf

_{2}Si (0.63 W m

^{−1}K

^{−1}) was only half of that of Si (1.26 W m

^{−1}K

^{−1}). Clarke found that mixing ions of different atomic masses reduces the minimum thermal conductivity of the system [24]. This explains the decrease in thermal conductivity caused by the incorporation of Hf elements into Si. According to the above, the Hf-Si system could reduce the thermal conductivity and improve the heat insulation ability.

## 4. Conclusions

- (1)
- The Hf-Si system has improved plasticity and hardness as compared to Si, and reduced G/B value, which benefits in minimizing the thermal stress on the substrate, and increases their thermal shock resistance. In addition, the Young’s modulus of Hf-Si system is higher than that of Si.
- (2)
- The addition of the Hf element to Si forming silicide can increase the sound velocities and reduce the Debye temperature, and thus reduce the thermal conductivity. Compared with Si, the theoretical minimum thermal conductivity of the Hf-Si system was substantially small, which was only 0.63 W m
^{−1}K^{−1}for Hf_{2}Si with improved heat insulation ability than that of Si. - (3)
- The calculation results show that HfSi
_{2}in the Hf-Si system has the lowest Young’s modulus and good plasticity, making it a good candidate as a bond layer for EBCs used at a high temperature.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Crystal structures of the Hf-Si system: (

**a**) Si, (

**b**) HfSi

_{2}, (

**c**) HfSi, (

**d**) Hf

_{3}Si

_{2}, (

**e**) Hf

_{2}Si, (

**f**) Hf

_{5}Si

_{4}(the blue represented Si atoms and the brown represented Hf atoms).

**Figure 3.**(

**a**) Surface contour of direction-dependent Young’s modulus of Si and (

**b**) its planar projections on (100), (010), and (001) crystallographic planes.

**Figure 4.**Surface contour of direction-dependent Young’s modulus of (

**a**) HfSi

_{2}, (

**b**) HfSi, (

**c**) Hf

_{5}Si

_{4}, (

**d**) Hf

_{3}Si

_{2}, (

**e**) Hf

_{2}Si and (

**a**–

_{1}**e**) planar projections on (100), (010), and (001) crystallographic planes.

_{1}**Figure 5.**Temperature dependence of thermal conductivity of Si and HfSi

_{2}. The minimum thermal conductivity (dash line) was also shown.

**Table 1.**The equilibrium lattice parameters (Å) of the Hf-Si system compared with the JCPDS card data.

Materials | a ($\dot{\mathit{A}}$) | b ($\dot{\mathit{A}}$) | c ($\dot{\mathit{A}}$) |
---|---|---|---|

Si | 5.450 | 5.450 | 5.450 |

Si (27-1402) | 5.431 | 5.431 | 5.431 |

HfSi_{2} | 3.656 | 14.640 | 3.670 |

HfSi_{2} (38-1373) | 3.680 | 14.556 | 3.649 |

HfSi | 6.896 | 3.788 | 5.249 |

HfSi (13-0369) | 6.885 | 3.753 | 5.191 |

Hf_{5}Si_{4} | 7.067 | 7.067 | 12.877 |

Hf_{5}Si_{4} (42-1166) | 7.039 | 7.039 | 12.869 |

Hf_{3}Si_{2} | 7.014 | 7.014 | 3.681 |

Hf_{3}Si_{2} (14-0427) | 7.000 | 7.000 | 3.671 |

Hf_{2}Si | 6.579 | 6.579 | 5.180 |

Hf_{2}Si (12-0467) | 6.480 | 6.480 | 5.210 |

**Table 2.**Calculated independent second-order elastic constants C

_{ij}(in GPa) for the Hf-Si system.

Materials | C_{11} | C_{12} | C_{13} | C_{22} | C_{23} | C_{33} | C_{44} | ${\mathit{C}}_{55}$ | C_{66} |
---|---|---|---|---|---|---|---|---|---|

Si | 138 | 52 | 69 | ||||||

HfSi_{2} | 237 | 57 | 113 | 156 | 97 | 258 | 111 | 92 | 104 |

HfSi | 238 | 109 | 83 | 250 | 86 | 316 | 139 | 81 | 92 |

Hf_{5}Si_{4} | 272 | 91 | 88 | 255 | 99 | 81 | |||

Hf_{3}Si_{2} | 294 | 62 | 100 | 185 | 84 | 117 | |||

Hf_{2}Si | 250 | 86 | 80 | 272 | 83 | 114 |

**Table 3.**Calculated bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa), Poisson’s ratio (μ), and Hardness (H) of Hf-Si system.

Materials | B (GPa) | G (GPa) | E (GPa) | μ | H (HV) | G/B |
---|---|---|---|---|---|---|

Si | 80 | 57 | 139 | 0.213 | 11 | 0.722 |

HfSi_{2} | 125 | 83 | 204 | 0.229 | 13 | 0.664 |

HfSi | 150 | 94 | 234 | 0.241 | 13 | 0.627 |

Hf_{5}Si_{4} | 148 | 88 | 219 | 0.253 | 12 | 0.595 |

Hf_{3}Si_{2} | 142 | 93 | 229 | 0.231 | 14 | 0.655 |

Hf_{2}Si | 140 | 96 | 235 | 0.221 | 15 | 0.686 |

**Table 4.**Sound velocities (v

_{L}, v

_{T}, v

_{m}, in km s

^{−1}), Debye temperature Θ

_{D}(in K), and minimum thermal conductivity k

_{min}(in W m

^{−1}K

^{−1}) of Hf-Si system.

Materials | v_{L} (m/s) | v_{T} (m/s) | v_{m} (m/s) | Θ_{D} (K) | k_{min} (w/(m·k)) |
---|---|---|---|---|---|

Si | 4.42 | 2.67 | 2.94 | 488 | 1.26 |

HfSi_{2} | 5.42 | 3.21 | 3.56 | 418 | 0.77 |

HfSi | 5.21 | 3.05 | 3.38 | 391 | 0.71 |

Hf_{5}Si_{4} | 5.02 | 2.88 | 3.20 | 366 | 0.65 |

Hf_{3}Si_{2} | 4.93 | 2.91 | 3.23 | 361 | 0.65 |

Hf_{2}Si | 4.83 | 2.89 | 3.20 | 359 | 0.63 |

**Table 5.**The coefficient k

_{p}(equivalent to $A\frac{\overline{M}{\theta}_{D}^{3}\delta}{{\gamma}^{2}{n}^{\frac{2}{3}}T}$) in Equation (13) of Hf-Si system.

Materials | Si | HfSi_{2} | HfSi | Hf_{5}Si_{4} | Hf_{3}Si_{2} | Hf_{2}Si |
---|---|---|---|---|---|---|

k_{p} | 1025.82 | 811.96 | 996.11 | 771.05 | 878.01 | 792.10 |

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## Share and Cite

**MDPI and ACS Style**

Huang, P.; Han, G.; Liu, H.; Zhang, W.; Peng, K.; Li, J.; Wang, W.; Zhang, J.
Mechanical and Thermal Properties of the Hf–Si System: First-Principles Calculations. *J. Compos. Sci.* **2024**, *8*, 129.
https://doi.org/10.3390/jcs8040129

**AMA Style**

Huang P, Han G, Liu H, Zhang W, Peng K, Li J, Wang W, Zhang J.
Mechanical and Thermal Properties of the Hf–Si System: First-Principles Calculations. *Journal of Composites Science*. 2024; 8(4):129.
https://doi.org/10.3390/jcs8040129

**Chicago/Turabian Style**

Huang, Panxin, Guifang Han, Huan Liu, Weibin Zhang, Kexue Peng, Jianzhang Li, Weili Wang, and Jingde Zhang.
2024. "Mechanical and Thermal Properties of the Hf–Si System: First-Principles Calculations" *Journal of Composites Science* 8, no. 4: 129.
https://doi.org/10.3390/jcs8040129