# Calculation of Thermal Expansion Coefficient of Rare Earth Zirconate System at High Temperature by First Principles

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

_{∞}) was proposed by combining Grüneisen’s equation and the Debye heat capacity model. Using α

_{∞}model, the thermal expansion property of different compounds can be easily figured out by first principles. Firstly, α

_{∞}of ZrO

_{2}, HfO

_{2}, were calculated, and results are in good agreement with the experimental data from the literature. Moreover, α

_{∞}of La

_{2}Zr

_{2}O

_{7}, Pr

_{2}Zr

_{2}O

_{7}, Gd

_{2}Zr

_{2}O

_{7}, and Dy

_{2}Zr

_{2}O

_{7}were calculated, and results demonstrated that the model of α

_{∞}is a useful tool to predict the thermal expansion coefficient at high temperature. Finally, Gd

_{2}Zr

_{2}O

_{7}with 4 different Yb dopant concentrations (Gd

_{1}

_{-x}Yb

_{x})

_{2}Zr

_{2}O

_{7}(x = 0, 0.125, 0.3125, 0.5) were calculated. Comparing with the experimental data from the literature, the calculation results showed the same tendency with the increasing of Yb concentration.

## 1. Introduction

_{2}Zr

_{2}O

_{7}, as the representative of rare earth zirconates, is evidently one of the most promising candidates for the application of next generation TBCs due to its lower thermal conductivity and higher phase stability [5,6]. However, it still suffers from the problem that mechanical properties are not high enough and thermal cycling performance is poor [7]. The thermal expansion properties play the key role. Comparatively, the thermal expansion coefficient of rare earth zirconates is about 9–10 × 10

^{−6}K

^{−1}(1073 K) [8], which is much lower than that of 8YSZ, about 11 × 10

^{−6}K

^{−1}(1073 K) [3]. The thermal expansion coefficient of NiCoCrAlY bonding layer, is about 17.5 × 10

^{−6}K

^{−1}(1273 K) [5]. The mismatch of the thermal expansion between the ceramic top layer and the bonding layer causes thermal stresses during thermal cycling, which can lead to cracks and failure of the TBCs system [9,10].

_{2}(Zr

_{1-x}Ce

_{x})

_{2}O

_{7}, (Nd

_{1-x}Gd

_{x})

_{2}Zr

_{2}O

_{7}, Gd

_{2}(Zr

_{1-x}Ti

_{x})

_{2}O

_{7}and (Sm

_{1-x}Gd

_{x})Zr

_{2}O

_{7}[6,11,12,13], etc.

_{2}Zr

_{2}O

_{7}pyrochlores by first principles. Atsushi Togo [16] calculated the thermal expansion properties of Ti

_{3}SiC

_{2}, Ti

_{3}AlC

_{2}, and Ti

_{3}GeC

_{2}by the first principles combing with quasi-harmonic approximation (QHA). Feng [17] investigated the thermal expansion properties of rare earth zirconates (Ln

_{2}Zr

_{2}O

_{7}, Ln = La, Nd, Sm and Gd) of pyrochlore structures by using first principles. Meanwhile, the calculation of properties of doped compounds by first principles is very difficult especially for the calculation of the thermal expansion property. For instance, the conventional cell of Gd

_{2}Zr

_{2}O

_{7}contains 88 atoms, and cell expansion must be performed to obtain solid solution structures with different doping concentrations, which makes the calculation of phonons extremely difficult.

_{V}). Further, C

_{V}is the function of Debye temperature and temperature. When the temperature is much greater than Debye temperature, C

_{V}can be considered as a constant. In this case, the calculation of α is simplified to the calculation of elastic properties, which makes the calculation much easier and faster. Coincidentally, the TBCs for aero engines operate at a high temperature, which is much higher than Debye temperature. Moreover, investigations showed that the coefficient of thermal expansion gradually increases with temperature increasing at high temperature, but the increasing rate gradually decreases. Reasonably, the thermal expansion coefficient at super high temperature(α

_{∞}) can be used as a comparison factor to characterize the thermal expansion property of different dopant/concentration compound materials.

_{∞}was developed, by which the calculation of a series of rare earth zirconates were implemented by the first principles.

## 2. Methodology

_{2}Zr

_{2}O

_{7}was used for calculation. The structures of Yb doped Gd

_{2}Zr

_{2}O

_{7}were formed by replacing Gd atoms with different amounts of Yb atoms. The structural models for (Gd

_{1-x}Yb

_{x})

_{2}Zr

_{2}O

_{7}were built using the cluster expansion approach by calculating the lowest forming energy [19,20,21]. Further, the structures were optimized by the Birch–Murnaghan equation of state [22]. The elastic constants of the material were calculated by the stress–strain method [23].

^{2}2p

^{4}, Zr 5s

^{1}4d

^{3}, Hf 5d

^{3}6s

^{1}, Gd 6s

^{2}5p

^{6}5d

^{1}, Yb 6s

^{2}5p

^{6}.

## 3. Results

#### 3.1. Yb Doped Gd_{2}Zr_{2}O_{7} Structure

_{2}Zr

_{2}O

_{7}contains 88 atoms, including 16 Gd atoms, 16 Zr atoms and 56 oxygen atoms. A total of 2, 5 and 8 Yb atoms were used to replace the Gd atoms to obtain three different concentrations: (Gd

_{0.875}Yb

_{0.125})

_{2}Zr

_{2}O

_{7}, (Gd

_{0.875}Yb

_{0.3125})

_{2}Zr

_{2}O

_{7}, and (Gd

_{0.5}Yb

_{0.5})

_{2}Zr

_{2}O

_{7}, respectively. Correspondingly, the possible numbers of (Gd

_{1-x}Yb

_{x})

_{2}Zr

_{2}O

_{7}doped structures were ${C}_{16}^{2}$, ${C}_{16}^{5}$, and ${C}_{16}^{8}$. Excluding the equivalent structures, the number of unequal possible doped structures are 3, 35, and 97, separately. The forming energy E of each structure was calculated using the cluster expansion approach, according to Equation (1) [21], and the final doped structure was identified by the structure with the lowest forming energy.

_{0}is the energy of the doped structure; E

_{1}and E

_{2}are the energy of the single cell of Gd

_{2}Zr

_{2}O

_{7}and Yb

_{2}Zr

_{2}O

_{7}, and n

_{1}and n

_{2}are the number of dopant atoms, respectively. The calculation results were shown in Figure 1. According to the lowest forming energy, the geometrical configurations of three different doping structures were elaborated in Figure 2.

#### 3.2. Lattice Constant and Elastic Modulus

_{2}Zr

_{2}O

_{7}are 176.6 Gpa and 91.9 GPa, respectively, which meets agreement with the data measured by experiments [26]. Generally, the bulk modulus and shear modulus of (Gd

_{1}

_{-x}Yb

_{x})

_{2}Zr

_{2}O

_{7}decrease with the increasing of Yb content, which are possibly caused by structure change. Subramanian M A [27] pointed out that doping of Yb atoms reduce the average cation radius ratio r(A

^{3+})/r(B

^{4+}) and change the crystal structure of Gd

_{2}Zr

_{2}O

_{7}from pyrochlore to disorders in the structures. For the cubic phase, there are three independent elastic constants, C

_{11}, C

_{12}, and C

_{44}[28]. The calculated elastic constants are all positive, satisfying the generalized elastic stability criterion, namely, C

_{11}+ 2C

_{12}> 0; C

_{44}> 0; C

_{11}− C

_{12}> 0, indicating that all studied structures are mechanically stable [29]. According to Pugh’s theory [30], when G/B < 0.5, the material is ductile; otherwise, the material is brittle. The G/B value of all materials calculated is greater than 0.5, indicating that they are brittle materials.

#### 3.3. Thermal Expansion of Rare Earth Zirconates System

_{V}is the heat capacity, B is the bulk modulus, and V is the molar volume.

_{V}can be calculated by the Debye heat capacity model as the following Equation (4) [32]:

_{A}, N

_{A}is Avogadro’s constant, and n is the number of atoms in the molecular formula; k

_{B}is the Boltzmann constant; T

_{D}is the Debye temperature. Meanwhile, T

_{D}can be calculated by the following Equation (5) [33].

_{m}is Planck’s constant, theoretical density, relative molecular mass, and speed of sound, respectively. υ

_{m}is defined as Equation (6).

_{L}, υ

_{S}is the longitudinal and transverse sound velocity and can be expressed as Equations (7) and (8), separately.

_{1}

_{-x}Yb

_{x})

_{2}Zr

_{2}O

_{7}vary within 4% difference, which can be considered as constant. Thus, it is indicated from Equation (10) that α is proportional to T

_{D}

^{−3}at the same temperature, which is compliant with Ruffa’s equation [34].

_{∞}representatives the linear thermal expansion coefficient at super high temperature. Actually, the TBCs are working under the temperature (e.g., the temperature of combustion chamber in F135 turbine engine can be up to 2253 K [35]) much higher than T

_{D}of rare earth zirconate (about 500 K) [36]. Another one, α of rare earth zirconate, increases with the increase in temperature, meanwhile the increasing rate slows down more and more [37,38,39]. Therefore, α

_{∞}can be likely used to compare the difference of thermal expansion property of different dopant/concentration for the same compound.

#### 3.4. The Validity of α_{∞} Model

_{2}and HfO

_{2}are of typical fluorite structure which is the same as that of rare earth zirconates [40]. Firstly, the lattice parameter and elastic properties were calculated by first principles. Secondly utilizing the α

_{∞}model, the thermal expansion properties of cubic ZrO

_{2}and HfO

_{2}were calculated, and both results were listed in Table 2. Comparing to the data from the material project database, calculation results of lattice parameter and elastic properties are very close to the same level. Hong [41] and Irshad, K.A. [42] measured the linear thermal expansion coefficient (α) of cubic ZrO

_{2}and HfO

_{2}by in situ high-temperature X-ray diffraction, and it is (12 ± 3) × 10

^{−}

^{6}K

^{−}

^{1}and 8.80 × 10

^{−6}K

^{−}

^{1}, respectively. Comparably, the calculated results of α

_{∞}are 9.72 × 10

^{−}

^{6}K

^{−}

^{1}and 9.05 × 10

^{−6}K

^{−}

^{1}. It is revealed that both are a good match.

_{∞}model, α

_{∞}of serial rare earth zirconates, including La

_{2}Zr

_{2}O

_{7}, Pr

_{2}Zr

_{2}O

_{7}, Gd

_{2}Zr

_{2}O

_{7}, and Dy

_{2}Zr

_{2}O

_{7}were calculated. The data of elastic property was cited from the literature [36], and both were shown in Table 3. α

_{∞}and α were plotted in Figure 3. Compared to α measured by experiment [8], α

_{∞}are clearly higher because α were measured at 800 °C which is not too much higher than Debye temperature. Henry Lehmann [37] measured the thermal expansion coefficients of Gd

_{2}Zr

_{2}O

_{7}and La

_{2}Zr

_{2}O

_{7}, which were 10.652 × 10

^{−6}K

^{−1}(1473 K) and 9.09 × 10

^{−6}K

^{−1}(1373 K), respectively. The results are very close to α

_{∞}calculated. It is further demonstrated that α

_{∞}can be a useful tool to predict the thermal expansion coefficient at high temperature.

#### 3.5. The Effect of Yb Doping of Gd_{2}Zr_{2}O_{7} on α_{∞}

_{∞}of Gd

_{2}Zr

_{2}O

_{7}with 4 different Yb doping contents Gd

_{2}Zr

_{2}O

_{7}, (Gd

_{0.875}Yb

_{0.125})

_{2}Zr

_{2}O

_{7}, (Gd

_{0.875}Yb

_{0.3125})

_{2}Zr

_{2}O

_{7}, and (Gd

_{0.5}Yb

_{0.5})

_{2}Zr

_{2}O

_{7}were calculated. Table 4 shows the calculation result of theoretical density, sound velocity, and Debye temperature.

_{D}of Gd

_{2}Zr

_{2}O

_{7}was calculated to be 511 K, which is in good agreement with that measured by Toshiaki Kawano [8]. T

_{D}is decreases with the increase of Yb dopant, which is dominated by the decrease of average velocity of sound (υ

_{m}).

_{D}decreases with Yb doping, as shown in Table 4. However, α

_{∞}is a bit lower than α. Feng [17] also discussed and explained the problem. Actually, the first principles were developed based on the material being from an ideally perfect crystal. However, for real bulk materials, various defects (e.g., vacancies and dislocations) and pores ineluctably existed. In general, the total energy of a crystal with defects is higher than that of an ideally perfect crystal, and the anharmonic effect may be affected by various defects and pores in the structure. In addition, the density of the tested ceramic coupon is certainly lower than that of theoretical density. That is the reason why the thermal expansion coefficient measured by experiment is higher than that calculated.

_{∞}and α increase with an increase in Yb content. The increase of thermal coefficient is more remarkable with lower Yb doping concentration. With higher Yb doping concentration, the growth rate slows down. Both the measured and calculated results have the same tendency.

_{1}

_{-x}Yb

_{x})

_{2}Zr

_{2}O

_{7}crystals are shown in Figure 5. In terms of bonding, Gd/Yb-5d, Zr-4d, and O-2p overlap, which means that the electronic state around the Fermi level is primarily determined by the relatively weak p-d bond between O-2p and Zr-4d (or Gd/Yb-5d). The main change from doping is the relative position changes in Gd/Yb- 5d, which may be caused by the difference in the valence electron layers between Gd and Yb. With the increase of Yb content, the total state density curve moves slightly to the lower energy level. Low crystal energy means high coefficient of thermal expansion [45], and the rare earth element Yb influences the O-2p states due to hybridization [44]. The p-d bond strength decreased with the increase in Yb content. According to the value of the PDOS ordinate, the PDOS of Zr-4d is the largest, so the p-d bond strength of Zr-O is higher than Yb-O and Gd-O.

## 4. Conclusions

_{∞}to characterize the thermal expansion coefficient at super high temperature was established. Firstly, using the α

_{∞}model, the high temperature thermal expansion coefficients of cubic ZrO

_{2}and cubic HfO

_{2}were calculated to be 9.72 × 10

^{−6}K

^{−1}and 9.05 × 10

^{−6}K

^{−1}, respectively, which are in agreement with those shown in the literature. Secondly, α

_{∞}of serial rare earth zirconates, including La

_{2}Zr

_{2}O

_{7}, Pr

_{2}Zr

_{2}O

_{7}, Gd

_{2}Zr

_{2}O

_{7}, and Dy

_{2}Zr

_{2}O

_{7}were calculated, and results demonstrated that α

_{∞}can be a useful tool to predict the thermal expansion coefficient at high temperature. Lastly, α

_{∞}of (Gd

_{1}

_{-x}Yb

_{x})

_{2}Zr

_{2}O

_{7}with four different doping contents were calculated, and results showed the same tendency as that measured by experiments. Generally, by characterizing the thermal expansion coefficient at high temperature through the elastic properties and Debye temperature of the material, the complicated calculation of phonon spectrum can be avoided. Thus, the model of α

_{∞}has the broad application prospect to predict the thermal expansion property at high temperature for other rare earth zirconates.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Geometrical configurations of (Gd

_{1}

_{-x}Yb

_{x})

_{2}Zr

_{2}O

_{7}, (

**a**) Gd

_{2}Zr

_{2}O

_{7}; (

**b**) (Gd

_{0.875}Yb

_{0.125})

_{2}Zr

_{2}O

_{7}; (

**c**) (Gd

_{0.6875}Yb

_{0.3125})

_{2}Zr

_{2}O

_{7}; (

**d**) (Gd

_{0.5}Yb

_{0.5})

_{2}Zr

_{2}O

_{7}.

**Figure 3.**α

_{∞}and α of La

_{2}Zr

_{2}O

_{7}, Pr

_{2}Zr

_{2}O

_{7}, Gd

_{2}Zr

_{2}O

_{7}, and Dy

_{2}Zr

_{2}O

_{7}.

**Figure 5.**Partial density of states of (Gd

_{1}

_{-x}Yb

_{x})

_{2}Zr

_{2}O

_{7}: (

**a**) x = 0; (

**b**) x = 0.125; (

**c**) x = 0.3125; (

**d**) x = 0.5.

**Table 1.**Lattice constant, the elastic constants (C

_{11}, C

_{12}, and C

_{44}), bulk modulus (B), shear modulus (G), and Poisson’s ratio (μ) of rare earth zirconates.

a_{0}/(nm) | C_{11}/(GPa) | C_{12}/(GPa) | C_{44}/(GPa) | B/(GPa) | G/(GPa) | G/B | μ | |
---|---|---|---|---|---|---|---|---|

Gd_{2}Zr_{2}O_{7}, cal. | 1.056 | 316.4 | 106.7 | 84.2 | 176.6 | 91.9 | 0.52 | 0.278 |

Gd_{2}Zr_{2}O_{7}, exp. [26] | 1.054 | 174 | 93 | |||||

(Gd_{0.875}Yb_{0.125})_{2}Zr_{2}O_{7} | 1.055 | 312.3 | 100.2 | 83.4 | 170.6 | 91.7 | 0.54 | 0.272 |

(Gd_{0.6875}Yb_{0.3125})_{2}Zr_{2}O_{7} | 1.052 | 308.4 | 96.7 | 83 | 167.2 | 91.6 | 0.55 | 0.269 |

(Gd_{0.5}Yb_{0.5})_{2}Zr_{2}O_{7} | 1.050 | 310.4 | 96 | 82.5 | 167.7 | 91.7 | 0.55 | 0.269 |

**Table 2.**Lattice constant, bulk modulus (B), shear modulus (G), Poisson’s ratio (μ), and α

_{∞}of ZrO

_{2}and HfO

_{2}.

a_{0}/(nm) | B/(GPa) | G/(GPa) | μ | α_{∞} (K^{−1}) | |
---|---|---|---|---|---|

ZrO_{2}, cal. | 0.512 | 238.5 | 100.6 | 0.316 | 9.72 × 10^{−6} |

ZrO_{2} [a] | 0.515 | 235 | 103 | 0.31 | |

HfO_{2}, cal. | 0.508 | 253.8 | 112.6 | 0.307 | 9.05 × 10^{−6} |

HfO_{2} [b] | 0.508 | 248 | 115 | 0.3 |

_{2}(SG:225) by Materials Project. ID:mp-1565. [b] Materials data on HfO

_{2}(SG:225) by Materials Project. ID:mp-550893.

**Table 3.**Lattice constant a

_{0}, bulk modulus (B), shear modulus (G), Poisson’s ratio (μ) of rare earth zirconates system [36].

a_{0}/(nm) | B/(GPa) | G/(GPa) | μ | Thermal Expansion Coefficient/(10^{−6}K^{−1}) | ||
---|---|---|---|---|---|---|

α_{∞} | α/(1073 K) [8] | |||||

La_{2}Zr_{2}O_{7} | 1.081 | 176 | 87 | 0.302 | 9.755 | 8.883 |

Pr_{2}Zr_{2}O_{7} | 1.072 | 155 | 103 | 0.26 | 9.857 | 9.415 |

Gd_{2}Zr_{2}O_{7} | 1.052 | 165 | 63 | 0.284 | 10.61 | 10.094 |

Dy_{2}Zr_{2}O_{7} | 1.054 | 164 | 90 | 0.268 | 10.057 | 9.166 |

**Table 4.**Density (ρ), longitudinal wave velocity (υ

_{L}), shear wave velocity of sound (υ

_{S}), average velocity of sound (υ

_{m}) and Debye temperature (T

_{D}) of rare earth zirconates.

ρ/(kg·m^{−3}) | υ_{L}/(m·s^{−1}) | υ_{S}/(m·s^{−1}) | υ_{m}/(m·s^{−1}) | T_{D}/(K) | |
---|---|---|---|---|---|

Gd_{2}Zr_{2}O_{7} | 6868 | 6600 | 3659 | 4075 | 511 |

(Gd_{0.875}Yb_{0.125})_{2}Zr_{2}O_{7} | 6944 | 6496 | 3635 | 4046 | 508 |

(Gd_{0.6875}Yb_{0.3125})_{2}Zr_{2}O_{7} | 7059 | 6402 | 3602 | 4007 | 504 |

(Gd_{0.5}Yb_{0.5})_{2}Zr_{2}O_{7} | 7176 | 6357 | 3574 | 3977 | 502 |

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**MDPI and ACS Style**

Wang, X.; Bai, X.; Xiao, W.; Liu, Y.; Li, X.; Wang, J.; Peng, C.; Wang, L.; Wang, X. Calculation of Thermal Expansion Coefficient of Rare Earth Zirconate System at High Temperature by First Principles. *Materials* **2022**, *15*, 2264.
https://doi.org/10.3390/ma15062264

**AMA Style**

Wang X, Bai X, Xiao W, Liu Y, Li X, Wang J, Peng C, Wang L, Wang X. Calculation of Thermal Expansion Coefficient of Rare Earth Zirconate System at High Temperature by First Principles. *Materials*. 2022; 15(6):2264.
https://doi.org/10.3390/ma15062264

**Chicago/Turabian Style**

Wang, Xingqi, Xue Bai, Wei Xiao, Yuyang Liu, Xiaoning Li, Jianwei Wang, Cheng Peng, Lijun Wang, and Xingming Wang. 2022. "Calculation of Thermal Expansion Coefficient of Rare Earth Zirconate System at High Temperature by First Principles" *Materials* 15, no. 6: 2264.
https://doi.org/10.3390/ma15062264