Anomalous mechanical strengths and shear deformation paths of Al₂O₃ polymorphs with high ionicity

Abstract

Alumina (Al₂O₃) formed by selective oxidization provides an effective way to protect aluminide alloys against corrosion for sustainable applications. Despite a broad interest and investigations on Al₂O₃ polymorphs such as α-Al₂O₃ and θ-Al₂O₃, their intrinsic mechanical strengths and atomic deformation mechanisms are not yet fully understood. In this research, density functional theory is used to show that the calculated shear moduli and mechanical strengths of θ-Al₂O₃ are substantially lower than those of α-Al₂O₃, and this explains why θ-Al₂O₃ is much weaker than α-Al₂O₃. An analysis of shear deformation paths and electronic structure indicates that the longest Al–O ionic bonds are responsible for the lattice instability of both polymorphs during shear, showing they have different anisotropic features. This study gives a novel view on the failure of thermally grown α-Al₂O₃ and θ-Al₂O₃, and it should help to improve the performance of thermal barrier coatings.

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1. Introduction

Alumina (Al₂O₃) polymorphs are important ceramic materials with many desirable properties such as high hardness, chemical inertness, thermal shock resistance and wear resistance, which makes them useful for many technological applications such as protective coatings, abrasives, insulators, and catalysts.^1–4 The most stable form of alumina is α-Al₂O₃, which is described as a quasi-harmonic crystal with negligible intrinsic anharmonic effects.⁵ The α-Al₂O₃, alumina may also adopt a series of metastable transition polymorphs such as γ, η, θ, γ′ and δ structures with different fcc packing of oxygen, and α, κ, κ′ and ι structures based on hcp packing lattice.^6,7 Some other structures include θ′, θ′′, λ, U and χ have also been reported.^6,7 When Al₂O₃ adopts the fcc packing lattice, a phase transformation of alumina may occur by the reactions shown in the following four sequences:⁷

Amorphous (anodic film) → γ → δ → θ → α-Al₂O₃ (iii)

Melt → γ → δ, θ → α-Al₂O₃ (iv)

With the alumina polymorphs and hydroxides as input, Peintinger et al.⁶ systematically investigated their relative stability using hybrid density functional theory (DFT), and then obtained the following energetic order: gibbsite < bayerite < boehmite < akdalaite < α-Al₂O₃ < κ-Al₂O₃ < θ-Al₂O₃ < δ-Al₂O₃ < γ-Al₂O₃ < η-Al₂O₃ < ι-Al₂O₃. Lee et al.⁸ compared the band gap of some aluminas and provided an order of α > κ > θ > γ, which is agreement with the most recent results by Museur et al.⁹

Among these alumina polymorphs, the α-Al₂O₃ and κ-Al₂O₃ are used as protective coatings against wear on tools because of their superior mechanical properties, whereas the metastable γ-Al₂O₃ and θ-Al₂O₃ are used as catalytic supports because of their lower surface energy.¹⁰ It is generally known that the α- and θ-Al₂O₃ play an important role in the performance of the early oxidation of nickel–aluminium alloys in thermal barrier coatings (TBCs), whereas the others are generally regarded as defectively metastable ones.^11–13 However, the stress concentration because of the fast phase transformation of θ-Al₂O₃ to α-Al₂O₃ is an important factor that leads to alumina scale cracking and spallation at Al₂O₃/alloy interface in TBCs.^14,15 To better understand the effect of metastable θ-Al₂O₃ on the performance of TBCs, much effort has been made to evaluate the properties of θ-Al₂O₃ both practically and using theoretical calculations. However, only limited equilibrium data, such as structural stability, electronic structure and elastic constants, have been reported so far. Therefore, in the present study, the former two polymorphs were the ones that were focused on. In the previous theoretical studies, the band gaps of α-Al₂O₃ were calculated to be about 6.72 eV,⁸ 6.1 eV (ref. 16) and 6.24 eV,¹⁷ which underestimate the experimental value of 8.8 eV.^16,18 The indirect band gaps of 5.04 eV (ref. 8) and 4.98 eV (ref. 17) were also reported for θ-Al₂O₃, which are about 1.3–1.7 eV smaller than that of α-Al₂O₃. Although the calculated site- and orbital-resolved partial electronic density of states (EDOS) show profound dependence on local coordination of Al and oxygen (O) atoms, the derived effective charge and bond order are similar for both θ-Al₂O₃ and α-Al₂O₃.¹⁷ These results suggest a fundamental question about whether both polymorphs could possess comparable mechanical properties.

For this purpose, a comparative mechanical test of α-Al₂O₃ and θ-Al₂O₃ was performed by Shang et al.¹⁹ several years ago. They obtained a positive value of c₁₄ for α-Al₂O₃ by means of ab initio DFT calculations, and these results were in good agreement with the results obtained in previous experiments and theoretical studies by Gladden et al.²⁰ and Hovis et al.²¹ With this validation, the derived bulk moduli and shear moduli of α-Al₂O₃ are shown to be much higher than those of θ-Al₂O₃, and this provides guidance for understanding the failure origin of the TBCs. In Shang's later work,²² the calculated temperature-dependent elastic constants enable a more realistic stress analysis at elevated temperatures in thermally grown oxides containing α-Al₂O₃ and θ-Al₂O₃, which provide a deeper understanding of the failure of TBCs in gas turbine engines.

Although numerous investigations have been performed on the structure and elastic properties of Al₂O₃ polymorphs, relatively little work has been devoted to the calculations of ideal strength and deformation paths which are critical for an in-depth understanding of the failure of turbine parts operating under extreme conditions. In general, a higher elastic modulus does not guarantee a higher strength because the former one describes only the reversible response of a crystal to a small strain, whereas the ideal strength gives an upper limit of stress that an ideal crystal can sustain at large strain.^23,24 In the present paper, α-Al₂O₃ and θ-Al₂O₃ polymorphs are used to illustrate that, although both are experimentally accessible and have similar effective charge and ionicity,¹⁷ their mechanical strengths and shear deformation paths are substantially different.

Recent advancements in computational techniques have made it the routine method to calculate the ideal strengths in a reasonably precise way using first principle methods and have provided a method to analyze the deformation paths and electronic instabilities under various loadings.^23,25,26 It is generally recognized that the ideal strength is a fundamental indicator of material strength,²⁷ which is defined as the minimum stress required to plastically deform an ideal crystal and the stress required for the formation of stacking faults.²⁸ A close connection between Peierls–Nabarro stress of dislocations and ideal shear strength has also made it possible to evaluate the mechanical strength of a real material via a comprehensive analysis of two intrinsic mechanical parameters: shear moduli and shear strength.^25,29 In addition, the ideal shear strength has attracted much more attention with the progress of nanotechnology, which provides a way to quantitatively measure the near ideal strength, which can be obtained from modern first principle calculations.^27,30

2. Computational methods

The ab initio DFT calculations^31,32 were performed using the Vienna ab initio simulation package (VASP) with the Perdew–Burke–Ernzerhof (PBE)³³ version of the generalized gradient approximation (GGA) for the exchange–correlation functional. For the EDOS calculations, the screened hybrid functional (HF) of Heyd, Scuseria, and Ernzerhof (HSE06)³⁴ was chosen, where correlation is described in GGA (PBE) and the exchange is a mixture of 25% exact (HF) exchange and 75% PBE exchange. The integration in the Brillouin zone was done on special k points of 9 × 9 × 9 grids for the phases under consideration, with an energy cutoff of 600 eV, which was sufficient to keep the total energy convergence at 10⁻³ meV per atom or less. The atomic positions were relaxed using the conjugate gradient method until the forces on all the atoms were converged to less than 10⁻³ eV Å⁻¹.

As a further check of the validity of the applied projected augmented wave (PAW)-PBE potentials, the elastic constants and moduli were determined by applying a set of small deformations, δ, between −0.02 and +0.02, and then the second order coefficients in a polynomial fit of the total energy as a function of the distortion δ were calculated. Atomic relaxations were allowed when straining the cell. The applied strain configurations and the corresponding strain–energy density variations, ΔE/V₀, for hcp α-Al₂O₃ structure were: ε = (δ, δ, 0, 0, 0, 0) with ΔE/V₀ = (c₁₁ + c₁₂)δ², ε = (0, 0, 0, 0, 0, δ) with ΔE/V₀ = 1/4(c₁₁ − c₁₂)δ², ε = (0, 0, δ, 0, 0, 0) with ΔE/V₀ = 1/2c₃₃δ², ε = (0, 0, 0, δ, 0, 0) with ΔE/V₀ = 1/2c₄₄δ², and ε = (δ, δ, δ, 0, 0, 0) with ΔE/V₀ = (c₁₁ + c₁₂ + 2c₁₃ + 1/2c₃₃)δ². For θ-Al₂O₃, ε = (δ, 0, 0, 0, 0, 0) with ΔE/V₀ = 1/2c₁₁δ², ε = (0, δ, 0, 0, 0, 0) with ΔE/V₀ = 1/2c₂₂δ², ε = (0, 0, δ, 0, 0, 0) with ΔE/V₀ = 1/2c₃₃δ², ε = (0, 0, 0, δ, 0, 0) with ΔE/V₀ = 1/2c₄₄δ², ε = (0, 0, 0, 0, δ, 0) with ΔE/V₀ = 1/2c₅₅δ², ε = (0, 0, 0, 0, 0, δ) with ΔE/V₀ = 1/2c₆₆δ², ε = (δ, δ, 0, 0, 0, 0) with ΔE/V₀ = (1/2c₁₁ + c₁₂ + 1/2c₂₂)δ², ε = (δ, 0, δ, 0, 0, 0) with ΔE/V₀ = (1/2c₁₁ + c₁₃ + 1/2c₃₃)δ².

The ab initio DFT calculations of stress–strain dependences conducted in this research are technically similar to those performed in previous studies by Zhang et al.^35–39 and these should be referred to for further details. The atomic basis vectors of a given unit cell were firstly projected onto the Cartesian coordinate system with one cell axis vector being parallel to the strain direction for the tension loading. For the shear deformation, one cell axis vector was perpendicular to the slip plane and another one was parallel to the slip direction in that plane. Afterwards, the crystal was incrementally deformed by transforming the unstrained cell basis vector matrices to the strained ones using the deformation matrices. In order to keep the crystal under a stress state of uniaxial tension or shear, the strained cell was relaxed for both the cell basis vectors and for the atom position inside the unit cell by keeping the applied strain component fixed and relaxing the other five strain components until their conjugate stress components, i.e., the Hellmann–Feynman stresses reached negligible values.

3. Results and discussion

The α-Al₂O₃ with a space group of R3̄c has the most symmetric ordering of Al atoms, which gives it a higher density compared to other polymorphs. The relaxed topological bond structure of α-Al₂O₃ is shown in Fig. 1(a); the Al cations and the O anions are located at the 12c (0, 0, z) and 18e (x, 0, 0.25) Wyckoff positions, respectively. The obtained values of the x- and z-coordinates are 0.306 and 0.352, respectively, which are in good agreement with those results obtained previously (x = 0.306 and z = 0.347).⁴⁰ The α-Al₂O₃ can also be represented in a hexagonal lattice with six layers of close packed O atoms filled by Al atoms at the octahedral hole sites. Each Al atom is bonded to six O atoms with two non-equivalent bond distances, and each O atom has four neighbor Al atoms [see Fig. 1(b)]. Fig. 1(c) shows the crystal structure and local coordination of anions of θ-Al₂O₃ that belong to the monoclinic system with a space group of C2/m, with all the ions located at 4i (x, 0, z) Wyckoff positions. Our calculated coordinates of O are: (0.160 0 0.109), (0.495 0 0.257), (0.826 0 0.433), and those of Al are: (0.090 0 0.796) (0.342 0 0.683), and these values agree with the previous values of O: (0.163 0 0.123), (0.489 0 0.261), (0.830 0 0.439) and those of Al: (0.101 0 0.794) (0.352 0 0.687).⁴¹ The Al atoms occupy four octahedral and four tetrahedral interstitials of the O sublattice⁴² [see Fig. 1(d) and (e)]. The calculated lattice parameters of α-Al₂O₃ are a = 4.806 Å, c/a = 2.7285 and those of θ-Al₂O₃ are a = 11.922 Å, b = 2.940 Å, c = 5.668 Å, β = 104.0°. They are in agreement with the available theoretical and experimental values (a = 4.8195 Å^a, 5.180 Å^b, c/a = 2.7284^a for α-Al₂O₃ and a = 11.923Å^b, 11.8530^c; b = 2.941 Å^b, 2.904^c; c = 5.671 Å^b, 5.622^c, β = 103.90°^b, 103.8°^c for θ-Al₂O₃) (^a ref. 43, ^b ref. 19, ^c ref. 44). There is only one type of O coordination in α-Al₂O₃, but θ-Al₂O₃ has three different types of O coordination, which produce a more complicated bond anisotropy. Fig. 1(c) shows that O₁ is bonded to one Al_tet and three Al_oct cations, where Al_oct (Al_tet) stands for the Al in octahedral (tetrahedral) coordination. It can be further seen in Fig. 1(c) that O₂ is bonded to two Al_tet and one Al_oct cations, and O₃ is bonded to one Al_tet and two Al_oct cations.¹⁷ The Al–O bond lengths of 1.9901 (1.969^d) Å and 1.8730 (1.857^d) Å (^d ref. 17) for α-Al₂O₃ [see Fig. 1(b)] show some differences from those of θ-Al₂O₃ in which seven equivalent Al–O bond lengths are 1.7701, 1.7702, 1.8134, 1.8881, 1.8883, 1.9566, 2.0208 Å [see Fig. 1(d) and (e)], which are in agreement with the previous values of 1.710, 1.745, 1.896, 1.904, 1.936, 2.025 Å^d, respectively, (^d ref. 17). The difference of local coordination between α-Al₂O₃ and θ-Al₂O₃, especially the bond anisotropy and the longest Al–O bonds are responsible for the different mechanical properties and the deformation paths during shear, as will be shown later in this paper.

Fig. 1 Lattice topology and polyhedra of the (a) hexagonal α-Al₂O₃ and (c) monoclinic θ-Al₂O₃. In each polyhedron, one Al cation is bonded to six O anions for α-Al₂O₃, and four or six O anions are bonded for θ-Al₂O₃ structures. The surrounding bond arrangements and the length of the neighbor to each O atom are shown for (b) α-Al₂O₃ and (d) and (e) θ-Al₂O₃. The large and small spheres represent the Al (cations) and O (anions), respectively. O₁, O₂, O₃ are the O atoms with different local coordinations.

Based on the reaction 4Al + 3O₂ = 2Al₂O₃, the calculated energy difference per formula unit between α-Al₂O₃ and θ-Al₂O₃ is 0.047 eV, which is consistent with results from previous reports where the following values were reported: ΔE(θ − α) = 0.03 eV per formula unit in⁴² and 0.04 eV per formula unit in⁴⁵ thus confirming that α-Al₂O₃ is thermodynamically more stable than θ-Al₂O₃.

The elastic constants were determined by applying an appropriate set of distortions within elastic limits.^46,47 Table 1 shows the calculated and measured values of the elastic constants c_ij for α-Al₂O₃ and θ-Al₂O₃. The elastic constants satisfy the Born stability criteria,^43,48,49 thus confirming that both hexagonal α-Al₂O₃ and monoclinic θ-Al₂O₃ are mechanically stable. Voigt average bulk modulus, B_V, of α-Al₂O₃ (235.6 GPa) is much larger than that of θ-Al₂O₃ (187.4 GPa), i.e., α-Al₂O₃ is more incompressible than θ-Al₂O₃. Additionally, it is found that the anisotropy ratio between c₁₁ and c₃₃ is nearly unity for α-Al₂O₃, whereas it is about 1.5 for θ-Al₂O₃, suggesting a higher elastic anisotropy for θ-Al₂O₃. A well-established consideration of whether a crystalline solid is intrinsically ductile or brittle^50–52 can be characterized approximately by using the ratio of the shear modulus to the bulk modulus, G/B, by considering B as the resistance to fracture and G as the resistance to plastic deformation.^52–54 The critical G/B ratio which separates ductile and brittle materials is around 0.57, i.e., if G/B < 0.57 the material behaves in a ductile manner, otherwise the material is brittle. The relatively high ratio of G/B ≈ 0.63 for α-Al₂O₃ reveals that it is intrinsically brittle, and this in agreement with the results of the experiments. The somewhat lower value of 0.58 for θ-Al₂O₃ is closer to the boundary of brittleness/ductility.

Table 1 The elastic constants (c₁₁, c₁₂, c₁₃, c₃₃, c₄₄, c₆₆), bulk moduli (B_V), Young's moduli (E), shear moduli (G_V, GPa), and Poisson ratio (ν) of α-Al₂O₃ and θ-Al₂O₃ compared with previous calculations and available experimental measurements

Property	α-Al2O3				θ-Al2O3
	This work	GGA	LDA	Experimental	This work	GGA	LDA
a Ref. 43.b Ref. 21.c Ref. 44.d Ref. 17.e Ref. 48.
c11	455.2	437.2^a, 451.5^b	476.8^a, 497.1^b	497.5^c, 497^d, 497.4^e	268.0	269.1^b	283.8^b
c22					389.7	389.3^b	420.4^b
c33	457.1	443.3^a, 454.9^b	476.6^a, 492.5^b	503.3^c, 501^d, 499.1^e	405.8	404.2^b	435.3^b
c44	132.6	125.5^a, 131.7^b	145.5^a, 154.2^b	147.4^c, 147^d, 147.4^e	72.5	74.2^b	86.8^b
c55					102.0	102.2^b	104.3^b
c66	151.6	146.4^a	159.6^a		122.3	126.0^b	124.5^b
c12	152.0	144.3^a, 148.4^b	157.6^a, 164.7^b	162.7^c, 163^d, 164.0^e	112.2	113.2^b	119.3^b
c13	112.2	101.5^a, 107.6^b	119.4^a, 129.5^b	115.5^c, 116^d, 112.2^e	135.5	132.2^b	159.8^b
BV	235.6	223.4^a, 231.7^b	246.9^a, 259.3^b	252.3^e	187.4	186.9^b	207.1^b
GV	149.4	143.0^a, 149.3^b	158.5^a, 165.8^b	166.0^e	109.5	110.7^b	114.9^b
E	370.0	353.6^a	391.6^a	403.0^e	274.9
ν	0.238	0.2362^a	0.2356^a	0.23^e	0.256

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Next the ideal strengths of both Al₂O₃ polymorphs were investigated. For α-Al₂O₃, the ab initio DFT calculations were performed with a unit cell along the 〈0001〉, 〈1̄21̄0〉 and 〈101̄0〉 directions for tension and on the (0001) 〈1̄21̄0〉, (0001) 〈101̄0〉 and (101̄0) 〈1̄21̄0〉 slip systems for shear. The crystallographic directions of 〈010〉 and 〈001〉 of monoclinic θ-Al₂O₃ are parallel to the y and z axes in the Cartesian coordinate system. Fig. 2(a) shows the calculated stress–strain curves for α-Al₂O₃. The ideal tensile strength of α-Al₂O₃ along the 〈0001〉 direction of about 58.3 GPa is much larger than in the two other directions. The anisotropy ratio of tensile strengths for α-Al₂O₃ of σ_〈0001〉 = 58.3 : σ_〈101̄0〉 = 36.5 : σ_{〈1̄21̄0〉} = 26.3 GPa ≈ 2.22 : 1.39 : 1 is slightly larger than those of θ-Al₂O₃ [see Fig. 2(b)]: σ_〈010〉 = 28.9 : σ_{001}normal = 14.8 : σ_〈100〉 = 13.9 GPa ≈ 2.08 : 1.06 : 1. It is interesting to note that the anisotropy of ideal tensile strengths does not follow the previously mentioned elastic anisotropy for both Al₂O₃ polymorphs. The tensile strength of 26.3 GPa along the weakest 〈1̄21̄0〉 direction of α-Al₂O₃ is comparable to the maximum value of 28.9 GPa of θ-Al₂O₃ along the 〈010〉 direction. Because plastic deformation occurs in shear, the calculated anisotropic ideal shear strengths must be compared.^35,36,55,56 The lowest shear strength of 14.4 GPa of α-Al₂O₃ is found in the (0001) 〈1̄21̄0〉 slip system [Fig. 2(a)]. This value is about 67% higher than the lowest shear strength of 4.7 GPa found for θ-Al₂O₃ [Fig. 2(b)], and this provides direct theoretical evidence that θ-Al₂O₃ is intrinsically much weaker than α-Al₂O₃.

Fig. 2 The stress–strain curves calculated by an ab initio DFT method under various tension and shear deformations for (a) α-Al₂O₃ and (c) θ-Al₂O₃ and the top view structures for (b) α-Al₂O₃ and (d) θ-Al₂O₃. Three tensile deformations are along crystallographic directions: 〈0001〉, 〈1̄21̄0〉 and 〈101̄0〉 for α-Al₂O₃ and 〈100〉, 〈010〉 and perpendicular to (001) for θ-Al₂O₃; the shear deformations are along: (0001) 〈1̄21̄0〉, (0001) 〈101̄0〉 and (101̄0) 〈 1̄21̄0〉 for α-Al₂O₃ and (100) 〈010〉, (001) 〈010〉 and (001) 〈100〉 for θ-Al₂O₃.

To further understand the deformation paths of both Al₂O₃ polymorphs in shear, the change of Al–O bond length and valence charge density differences (VCDD) during shear deformation are compared in Fig. 3. The VCDD is defined as the difference between the calculated total valence charge density of the crystal minus the superposition of the valence charge densities of neutral atoms³⁹). A negative value (yellow color in Fig. 3) means a depletion of charge when compared to neutral atoms (bond weakening). To emphasize the variation of charge depletion between Al–O bonds, only the negative isosurface of VCDD is shown in Fig. 3. Note that the expansion of the negative VCDD isosurface between Al–O bonds indicates their weakening during shear deformation. Although both α-Al₂O₃ and θ-Al₂O₃ have similar ionic bonds, their bond deformation paths and electronic instability modes are substantially different as can be seen from the change of bond topology and deformed VCDD isosurfaces shown in Fig. 3. Fig. 3(b) and (c) show the typical Al–O bond length and VCDD at shear strain of γ = 0.1934 (before) and γ = 0.3400 (after instability) for α-Al₂O₃, and Fig. 3(e) and (f) show them at a shear strain of γ = 0.0924 (before) and γ = 0.1366 (after instability) for θ-Al₂O₃. For α-Al₂O₃, the shear instability occurs between Al {0001} and O {0001} planes [Fig. 3(b) and (c)] with an increase of Al–O bond length from 2.33 to 3.04 Å, whereas for θ-Al₂O₃, the shear instability appears between Al/O {001} planes with the Al–O bond lengths increasing from 2.23 to 2.73 Å [Fig. 3(e) and (f)]. In Fig. 3, it can also be seen that the dramatic change of VCDD before and after the instability of α-Al₂O₃ and θ-Al₂O₃ (see the yellow isosurface marked by big blue arrows), which shows that the process of Al–O bond strength becomes too weak for bond cleavage.

Fig. 3 The isosurfaces of VCDD of α-Al₂O₃ (a) equilibrium, (b) before and (c) after instability under shear deformation along the weakest (0001) 〈1̄21̄0〉 slip system, and those of θ-Al₂O₃ (d) equilibrium, (e) before and (f) after instability under shear deformation along the weakest (001) 〈010〉 slip system (i.e., τ_yz). The isosurface maps of the VCDD correspond to −0.014 electrons per Bohr³. A yellow color means a negative value. The crystallographic directions of 〈010〉 and 〈001〉 of monoclinic θ-Al₂O₃ are parallel to the y and z axes in a Cartesian coordinate system, respectively, as represented in (d).

To quantify the electronic origin of distinct mechanical strength in both Al₂O₃, the quantum theory of atoms in molecules proposed by Bader^57,58 is first used to quantitatively determine the change of charge transfer during shearing. The calculated non-equivalent values of Bader charges are: for α-Al₂O₃, at equilibrium: (Al^−2.4751)₂(O^+1.6501)₃, at γ = 0.1934 (before instability): (Al^−2.4732)₂(O^+1.6488)₃, at γ = 0.3400 (after instability): (Al^−2.4640)₂(O^+1.6427)₃; for θ-Al₂O₃, at equilibrium: (Al^{−2.4442/−2.4637})₂(O^{+1.6431/+1.6346/+1.6303})₃, at γ = 0.0924 (before instability): (Al^{−2.4451/−2.4749})₂(O^{+1.6481/+1.6359/+1.6360})₃, at γ = 0.1366 (after instability): (Al^{−2.4479/−2.4733})₂(O^{+1.6340/+1.6442/+1.6431})₃. It is seen that the highly ionicity of α-Al₂O₃ and θ-Al₂O₃ are nearly identical. The minor difference of the Bader charges between α-Al₂O₃ and θ-Al₂O₃ provides an electronic explanation on the significant differences of their ideal tensile and shear strengths. Further consideration of the distributions of non-equivalent, highly ionic Al–O bonds for θ-Al₂O₃ may give an explanation of the choice of deformation paths of both structures during the shear loadings (see the change of bond lengths and VCDD isosurface marked in Fig. (3).

In order to develop a deeper understanding of the mechanical properties of α-Al₂O₃ and θ-Al₂O₃, the EDOS and localized charge accumulation between Al–O bonds was next analyzed. Fig. 4(a) and (b) shows the total EDOS which presents three distinct regions: the lower part of the valence bands is dominated by O 2s states and the upper part by O 2p- and Al 3s-, 3p-like states. Both α-Al₂O₃ [Fig. 4(a)] and θ-Al₂O₃ [Fig. 4(b)] are insulators with band gaps of about 7.6 and 6.2 eV, respectively. The previously calculated band gaps of α-Al₂O₃ are 8.6 eV,⁶ 9.1 eV,⁵⁹ and 8.5 eV,⁶⁰ and that of θ-Al₂O₃ is 6.9 eV,⁶ by utilizing the HSE functional. A higher value of the band gaps of α-Al₂O₃ was reported by Weber et al.,⁵⁹ because the Hartree–Fock mixing parameter in the HSE calculation is 32%, which is higher than the standard HSE06 (25%). Note that the calculated band gap for α-Al₂O₃ using HSE06 pseudopotentials is much more close to the experimental value (∼8.8 eV) when compared to the previous studies.^16,18 The upper valence band width (from −7.5 eV to 0 eV) of α-Al₂O₃, which is dominated by overlapping of O 2p and Al 3p orbitals, shows strong similarity to that of θ-Al₂O₃. To obtain the difference of bond strength between α-Al₂O₃ and θ-Al₂O₃, Fig. 4(c) and (d) show the contour plot of the VCDD cross section for α-Al₂O₃ and θ-Al₂O₃. The circled regions between the longest Al–O bonds in Fig. 4(c) and (d) indicate the magnitude of charge depletion. It can be clearly seen that the longer Al–O bonds for θ-Al₂O₃ corresponds to a higher depletion of valence charge density between them and a significant bond weakening, providing a physical explanation why α-Al₂O₃ is stronger than θ-Al₂O₃.

Fig. 4 The calculated EDOS for (a) α-Al₂O₃ and (b) θ-Al₂O₃ and the cross-sectional plot of the VCDD for (c) α-Al₂O₃ and (d) θ-Al₂O₃ to indicate the VCDD between the longest Al–O bonds. The circled region indicates the charge depletion between the longest Al–O bonds for both polymorphs of Al₂O₃.

4. Conclusions

In conclusion, the structural, thermodynamic, mechanical, electronic properties and shear deformation paths of α-Al₂O₃ and θ-Al₂O₃ have been studied using ab initio DFT calculations. The resulting equilibrium properties of α-Al₂O₃ and θ-Al₂O₃ obtained are consistent with the experimental and theoretical data. The relatively high bulk modulus of α-Al₂O₃ confirms that it is a low compressible material, whereas the relatively low ratio of the shear modulus to bulk modulus shows that it is still intrinsically brittle, and somewhat more brittle than θ-Al₂O₃. The shear strength of θ-Al₂O₃ is much lower than that of α-Al₂O₃ because of the highly ionic Al–O bonds with the longest bond length being more intense for θ-Al₂O₃. The longest Al–O bond lengths determine the deformation paths in both of the Al₂O₃ polymorphs.

Acknowledgements

This research is sponsored by National Basic Research Program (973 Program) of China under Grant No. 2012CB625100, the National High-tech R&D Program (863 project) No. 2012AA03A512, and the Nature Science Foundations of China (NFSC) under Grant No. 51231001 and No. 51425102. R. F. Zhang is supported by the Fundamental Research Funds for the Central Universities, NFSC with No. 51471018, and the National Thousand Young Talents Program of China. D. Legut acknowledges financial and computational support from the IT4 Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070), funded by the European Regional Development Fund and the national budget of the Czech Republic (Large Research, Development and Innovations Infrastructures project, LM2011033). We would like to thank Prof. Dr. Stan Veprek for constructive comments and suggestions, and Prof. G. Kresse for valuable advice for the application of VASP.

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