An assessment of interatomic potentials for yittria-stablized zirconia

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An assessment of interatomic potentials for yittria-stablized zirconia ... crystal structure, and the cubic phase exists up to the melting point of 2680°C. The addition of certain ..... [31] Liu D W, Perry C H, Feinberg A A, Currat R 1987 Phys. Rev.
Applied Mechanics and Materials Vol. 492 (2014) pp 239-247 © (2014) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.492.239

An assessment of interatomic potentials for yittria-stablized zirconia Chao Yin,1,a Fei Ye,1,2,b Chun Yu Yin,1 Ding Rong Ou,3 Toshiyuki Mori4 1

School of Materials Science and Engineering, Dalian University of Technology, 2 Linggong Road, Dalian, Liaoning 116024, China.

2

Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Dalian University of Technology), Ministry of Education, Dalian, Liaoning 116024, China.

3

Laboratory of Fuel Cells, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, 457 Zhongshan Road, Dalian, Liaoning 116023, China. 4

Fuel Cell Materials Group, Battery Materials Unit, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan. a

[email protected], [email protected]

Keywords: Yttria-stablized zirconia; Simulation; Interatomic potential; Property

Abstract. Six interatomic potentials based on Buckingham potential form for yttria-stablized zirconia have been critically assessed by predicting lattice constants, dielectric constants, and elastic properties using the mean-field approach. The content of Y2O3 is set to the range from 8 to 24 mol%. It has been found out that no potential can reproduce all the fundamental properties. Taking all the simulation and comparison results into consideration, the potential of Butler (1981) displays the highest fidelity, and the potential of Lewis (1985) shows the widest range of applicability. Introduction Stabilized zirconia (ZrO2) has been widely studied and used as electrolyte in solid-oxide fuel cells (SOFCs) because it possesses an adequate level of oxygen-ion conductivity and desirable stability in both oxidizing and reducing atmospheres [1]. Pure zirconia at room temperature has a monoclinic crystal structure, and the cubic phase exists up to the melting point of 2680°C. The addition of certain aliovalent oxides stabilizes the cubic fluorite structure from room temperature to its melting point. This stabilization is accomplished by substitution of divalent or trivalent cations, and the substitution creates a large concentration of oxygen vacancies by charge compensation. For instance, the oxygen vacancies in yttria-stablized zirconia (YSZ) are produced by the following equation in Kröger-Vink notation: (1) The high oxygen vacancy concentration gives rise to a higher conductivity because oxygen-ion conduction takes place in YSZ by the movement of oxygen ions via vacancies. To meet the requirement of the development in SOFCs, the conductivity of the electrolyte should be further increased. According to equation (1), the conductivity of YSZ is closely related to the production and migration of the point defects. However, the individual point defect and its migration are difficult to be observed experimentally. Therefore, computer simulation at atomic level, such as

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molecular dynamics, first-principle and Monte-Carlo methods is an ideal tool to study the interaction and migration of point defects [2-5]. The most accurate theoretical method of investigating the material properties at atomic scale would be first-principle simulation. However, due to the limitation of size or number of atoms, it is often insufficient to investigate material behaviors using only first-principle calculations. Another approach is to use (semi-) empirical interatomic potentials that can deal with more than a million atoms. Therefore, atomistic simulations of molecular dynamics, or Monte Carlo method based on empirical interatomic potentials can be a useful tool for analyzing and predicting defect properties, phase transformation mechanisms, kinetics and resultant structural evolutions, and dynamic behaviors of materials [4,5]. The accuracy of simulation significantly depends on the interatomic potential formalisms and the potential parameters. There are a number of interatomic potentials for YSZ in literatures [4-12]. The properties of YSZ, such as mechanics properties, thermodynamic properties, and defect properties, have been extensively studied using these potentials. Besides the potentials, the accuracy of simulation also depends on the detailed simulation approach. To study the properties of solid solution as YSZ, there are two practical approaches. One approach is to construct a supercell, in which ions and vacancies are distributed randomly. The simulation results may depend on the detailed arrangements of the ions and vacancies, and the size of the supercell. Another approach is an approximate approach named mean-field approach [13]. In this approach, all atoms are assigned an occupancy factor, and all interactions are then scaled by the product of the relevant occupancies. The simulation of some physical properties by the mean-field method can be performed using only one unit cell. Obviously, this method cannot be used to simulate the behavior of the individual atom, such as atom migration. It is clear that the former method can simulate the properties of solid solution more accurately if the supercell is large enough and a suitable potential is used, while its efficiency is much lower than the later approach. However, the fidelity of the potentials reported in previous literatures simulated using the mean-field approach has not been studied. To facilitate the further study of YSZ by computer simulation, a survey of empirical interatomic potentials in Buckingham form is performed. These potentials are assessed by simulating the properties of YSZ using the mean-field approach. Simulation methodology A Born-like description of the lattice [14], which describes the atoms as spherical and formally charged, are used to generate the lattice structure. The procedures are based upon a description of lattice forces in terms of effective potentials. The total energy of the crystal is minimized by allowing the ions in the unit cell and the lattice vectors to relax to zero strain. The constituent ions of the solid are treated as classical charged particles interacting with each other through long-range and short-range pair potentials. The long-rang interactions are described by a simple Coulombic interaction which is given in the following form: .

(2)

where N is the total number of ions in the system, qi and qj are the charge on ions i and j, respectively, rij is the distance between ions i and j. The summations are carried out using the Ewald method [15].

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The short-range interactions are represented by suitable analytical functions of the interatomic separation. For the ionic case, a repulsive potential is usually adequate, with the most common choices being either a positive term which varies inversely with distance, or an exponential form. These lead to the Lennard-Jones potential, and Buckingham potential, and so on. The Lennard-Jones potential is more robust since the repulsion increases faster with decreasing distance than the attractive dispersion term. The Buckingham potential is easier to justify from a theoretical perspective since the repulsion between overlapping electron densities, due to the Pauli principle, takes an exponential form at reasonable distances. In addition, some many-body interactions like the embedded atom method or bond order potential are of significance, and are also widely used, but the computational cost is greater. Therefore, in this work, the most common functional form for YSZ, the Buckingham form, is employed:

Buck(rij) = Ae(−rij/ρ) – C/rij6.

(3) where A, ρ, and C are adjustable parameters, which are chosen to reproduce pertinent physical properties of the real material. In order to reduce the computational time, the short-range interactions are set to zero beyond 20 Å. The composition of the YSZ studied in this work varies from 8 to 24 mol% Y2O3. As mentioned above, the different compositions are simulated with the mean-field approach, in which the solid solution Zr1-xYxO2-x/2 is modeled as an AB2 system of the same fluorite structure in which A and B are a hybrid cation and anion, respectively. The properties of the hybrid species are averages of those of the pure species according to the occupancies. The interaction potential between any two crystallographic sites becomes scaled by the fractional occupancies. For example, the cation-anion short-range potential is

A-B = (1 − x)Zr-O + xY-O.

(4) Besides the potentials, the ions are treated as polarizable in part of the previous literatures, which are described by the shell model [19]. The shell model of an ion consists of a massive core with charge X|e| and a massless shell with charge Y|e|. The overall charge state of each ion is equal to (X + Y)|e|. The core and shell are connected by an isotropic harmonic spring of force constant k. A static lattice calculation is performed at 0 K in this work, and carried out using the GULP (General Utility Lattice Program) [16,17], in which the mean-field method is encoded. Potential evaluation Six different potential parameters of Buckingham potential form are studied in this work, denoted as Butler, Grimes, Lewis, Dwivedi, Schelling, and Kilo, as listed in Table 1. The shell parameters which have been reported in the corresponding literatures are also listed in the table. Among these potentials, the potential denoted as Grimes was derived by Grimes’ group and reported in a series of papers. The O2-−O2- interaction derived by Catlow [7] was used in the potentials of Lewis [12], Dwivedi [6], and Kilo [4]. The Y3+−O2- potential parameters used in the Dwivedi potential were taken from the Lewis potential [12]. The Schelling potential [5] was derived by modifying the Grimes potential. The Kilo potential [4] was derived by modifying the Dwivedi potential and the Lewis potential.

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Table 1 Parameters of the six Buckingham potentials of YSZ Species

O2−

Butler −O2− Grimes

Lewis

Dwivedi

Schelling

Kilo

A[eV]

Zr

4+

3+

Y −O2− O2− −O2− 4+ Zr −O2− Y3+ −O2− O2− −O2− 4+ Zr −O2− Y3+ −O2− O2− −O2− 4+ Zr −O2− Y3+ −O2− O2− −O2− 4+ Zr −O2− Y3+ −O2− O2− −O2− 4+ Zr −O2− Y3+ −O2−

227 106 64.3 826. 7.0 954 744 150 7.960 176 2.11 227 6.40 145 64.3 134 3.8 227 5.1 985. 64.3 134 869 954 5.1 150 7.960 136 2.110 227 6.350 102 64.300 132 4.6

ρ[Å]

0.149 0.376 0.356 4 0.219 0.347 16 0.338 7 0.149 49 0.350

C[eV∙Å6]

112.0 0

2.0

5.1 27.89

K [eV Å]

Ref.

O2−

−3.0

55.0

[8]

6.0

105.0

[8]

Zr

4+

[8] O

2−

−2.0

6.3

[9]

4+

−0.0 4

189.7

[10]

5 −2.0

27.29

77

0

Zr O

2−

0.0 0.0 27.89

0.349

0.0

0.224 1 0.345

32.0

0.348

19.60

0.346

Y∣e∣

19.43

0.149 1 0.376

0.376

Ion

0.0

32.0

0.349

0.149

Shell−Model Parameters

0.0

O

[11]

0

0.0 0.0

[12] [12]

2−

−2.0

27.29

[7]

4+

1.350 77

169.6 0

[10]

Zr

17

5.1 27.89

[7]

[12] [5] [5] [5]

O

2−

−2.0

27.29

[7]

4+

1.350 77

169.6 0

[4]

Zr

17

[4]

To evaluate the of these potentials, the lattice constant, dielectric properties, and elastic −O2−quality 5.6 1 properties are calculated, and the simulation results are compared with the experimental results which are shown in the figures. In the case of the bulk modulus (B), the shear modulus (G), and the Young’s modulus (E), because the experimental results are seldom reported directly, most of them are calculated using C11, C12, and C44. B and G can be calculated using Voigt-Reuss-Hill (VRH) approximations [18], which are the best estimate of the theoretical polycrystalline elastic modulus. In these approximations, B and G are given by BV = (C11 + 2C12)/3,

(5)

BR = 1/(3S11 + 6S12),

(6)

BH = (BV + BR)/2,

(7)

GV = (C11 − C12 + 3C44)/5,

(8)

GR = 5/(4S11 − 4S12 + 3S44),

(9)

GH = (GV + GR)/2, (10) in which the subscripts V, R and H indicate the Voigt, Reuss and VRH averages, and S indicate the elastic compliance, and the compliance of the cubic crystals are given by

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S11 = −(C11 + C12)/(C11 + 2C12)(C12 − C11),

(11)

S12 = C12/(C11 + 2C12)(C12 − C11),

(12)

S44 = 1/C44. (13) The VRH averages BH and GH are taken as the bulk modulus B, and the shear modulus G. Then, the polycrystalline Young’s modulus (E) can be given by E = 9BHGH/(3BH + GH). (14) Lattice constant. Fig. 1 shows the lattice constant of YSZ as a function of Y2O3 content. It can be seen that the lattice constant obtained from both the experimental and most of the simulation results increases with Y2O3 content, while the lattice constant calculated using the Butler potential decreases with Y2O3 content. Moreover, except the results simulated using the Butler potential, the lattice constant calculated using the other potentials are in excellent agreement with the experimental results since the potentials were obtained by fitting to the lattice constant and some other fundamental properties. The difference between the simulation and the experimental results is less than 0.1 Å. However, the discrepancy between the simulation results using the Butler potential and the experimental results is much larger, though the relative error is no more than 0.1%. This is possibly because the Butler potential was obtained by fitting to a low temperature monoclinic phase, while the YSZ with fluorite structure is used to evaluate the potentials in the present work. Static dielectric constant. The simulation results of static dielectric constant are compared with the experimental results in Fig. 2. It can be seen that the Y2O3 content has no significant effect on the static dielectric constants simulated by most potentials, while the results simulated by the Grimes potential and the Dwivedi potential decrease obviously with Y2O3 content. The static dielectric constants simulated by both the Butler and Kilo potential also decrease slightly with Y2O3 content, while those simulated by the Lewis potential and the Schelling potential slightly increase. The trend of the decrease in the dielectric constant with Y2O3 content is same with that of the experimental results. However, all the potentials underestimate the dielectric constant. It is interesting that the dielectric constants yielded by the Butler potential are relatively close to the experimental results though the Butler potential was obtained by fitting to the monoclinic structure. Elastic properties. The simulation and experimental results of the elastic properties including elastic constant C11, C12, C44, bulk modulus, shear modulus, and Young’s modulus are then compared. The elastic constant C11, C12, and C44 obtained from the simulation and experimental results are shown in Fig. 3. It can be seen that most of the simulation results of C11, C12, and C44 decrease with increasing the Y2O3 content, except the simulation results of C44 obtained by the Kilo potential. The same trend can be seen in the experimental results of C11 whose values are between 372 and 415 GPa. However, the experimental results of C12 are disperse and trendless with Y2O3 content, and the dispertion range is from 70 to 140 GPa. Moreover, the experimental results of C44 even increase

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Fig. 1 Lattice constant as a function of Y2O3 content. The experimental results are from ref. [20.21]

Fig. 2 Static dielectric constant as a function of Y2O3 content. The experimental results are from ref. [22.23]

slightly with the increase in the Y2O3 content, which is the same with the trend of C44 simulated by the Dwivedi potential. Besides the variation of the C11, C12, and C44 with the Y2O3 content, the detailed values of the simulation and the experimental results are then compared. For C11 (Fig. 3(a)), it can be found out that the results simulated by the Butler potential overestimate C11 by about 15%, and those simulated by the Lewis, Dwivedi, and Kilo potentials overestimate C11 by about 35%-40%. Those simulated by the Grimes, and Schelling potentials are much larger, which is more than 50%. For C12 (Fig. 3(b)), it can be seen that the values obtained by the Dwivedi, Kilo, Lewis, Schelling, and Grimes potentials are close to the upper limit of the experimental span, but the values simulated by the Butler potential are much larger than the experimental results. For C44 (Fig. 3(c)), the simulation results obtained from the Butler potential is in good agreement with the experimental values though the trend is contrary to the experimental results, while all the other simulation results overestimate C44 by about 40% to 100%. As shown in Fig. 4, the simulation results of bulk modulus, shear modulus, and Young’s modulus decrease with increasing the Y2O3 content. However, the Y2O3 content has no prominent effect on the experimental results of bulk modulus and Young’s modulus, and most of them are around 200 and 220 GPa, respectively. Moreover, the experimental results of shear modulus increase with the Y2O3 content and most of them lay in the range from 80 to 100 GPa. In case of the detailed values of bulk modulus (Fig. 4(a)), the simulation results of the bulk modulus obtained by various potentials are larger than the experimental results. The Kilo potential overestimates the bulk modulus within 20%, which is the closest to the experimental results. For the shear modulus (Fig. 4(b)), the simulation results obtained by the Butler potential is in good agreement with the experimental values though the trend is contrary to the experimental results, while all the other simulation results overestimate the shear modulus. For the Young’s modulus (Fig. 4(c)), the values simulated by all of the potentials lay much higher than the experimental values. The Butler potential overestimates the Young’s modulus by about 50%. The Kilo potential, the Lewis potential, and the Dwivedi potential overestimate the Young’s modulus by almost 100%. The results simulated by the Grimes potential and the Butler potential are even much larger.

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Fig. 3 Elastic constant C11, C12, and C44 as a function of Y2O3 content. [20,24,25,27,29,31-33]

245

Fig. 4 Bulk modulus, shear modulus, and Young’s modulus as a function of Y2O3 content. [20,24-33]

Discussion Applicability of the potentials. In the above comparisons, it can be seen that no single potential can reproduce all these fundamental properties. Therefore, the selection of the appropriate potential for specific application is critical. The applicability of these potentials is summarized in Table 2. According to the relative error between the simulation and the experimental results, the applicability of these potentials is divided into three levels, including good quality (within 20%, denoted by ‘+’), medium quality (from 20% to 50%, denote by ‘+/−’), and low quality (above 50%, denoted by ‘−’), respectively. It can be seen that in general, the Butler potential has the highest fidelity, since more symbol ‘+’ are presented in this column in the table, and the Lewis has the widest range of applicability, since less symbol ‘−’ are presented.

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Table 2 The applicability of the potentials for each property Butler Grimes Lewis Dwivedi Schelling Kilo Lattice Constant Dielectric Constant C11 C12 C44 Bulk Modulus Shear Modulus Young’s Modulus

+ + + − + +/− + −

+ +/− − +/− − +/− − −

+ − +/− +/− +/− +/− +/− −

+ +/− +/− +/− − +/− − −

+ +/− +/− − +/− − +/− −

+ − +/− +/− − + +/− −

Summary The interatomic potentials for YSZ based on Buckingham potential form have been assessed by simulating lattice constants, dielectric constants, and elastic properties using the mean-field approach. According to the relative error between the simulation and the experimental results, the applicability of these potentials is compared. No single potential can reproduce all fundamental properties. Therefore, the selection of the appropriate potential for specific application is critical. In general, the Butler potential has the highest fidelity, and the Lewis potential has the widest range of applicability. Acknowledgement Support from Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry is gratefully acknowledged. This work is also partly supported by the supported by the Fundamental Research Funds for the Central Universities (No. DUT13LAB20). Reference [1] Minh N Q 1993 J. Am. Ceram. Soc 76 563. [2] Giuseppe F, Luciano C, Giovanni Z 2009 Phys. Rev. B 79 214102. [3] Bieberle A, Gauckler L J 2002 Solid State Ionics 146 23. [4] Kilo M, Argirusis C, Borchardt G, Jackson R A 2003 Phys. Chem 5 2219. [5] Schelling P K, Phillpot S R, Wolf D 2001 J. Am. Ceram. Soc 84 1609. [6] Dwivedi A, Cormack A N 1990 Philos. Mag. A 61 1. [7] Catlow C R A 1997 Proc. R. Soc. A. 353 533. [8] Butler V, Catlow C R A, Fender B E F 1981 Solid State Ionics 5 539. [9] Binks D J, Grimes R W 1993 J. Am. Ceram. Soc 76 2370. [10] Grimes L, Grimes R W, SickafusK E 2000 J. Am. Ceram. Soc 83 1873. [11] Grimes R W, Busker G, McCoy M A, Chroneos A,. Kilner J A, Bunsen-Ges B 1997 Phys. Chem 101 1204. [12] Lewis G V, Catlow C R A 1985 J. Phys. C 18 1149. [13] Kaupp M, Reviakine R, Malkina O L, Arbuznikov A, Schimmelpfennig B, Malkin V G 2002 J. Phys. Chem 23 794.

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