# Stress Concentration in the Bulk Cr2O3: Effects of Temperature and Point Defects

Authors: Mazharul M. Islam; Thierry Couvant; Philippe Marcus; Boubakar Diawara
Journal: Journal of Chemistry
 (2017)

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## Abstract

Modeling the growth and failure of passive oxide films formed on stainless steels is of general interest for the use of stainless steel as structural material and of special interest in the context of life time extension of light water reactors in nuclear power plants. Using theDFT+U approach, a theoretical investigation on the resistance to failure of the chromium-rich inner oxide layer formed at the surface of chromium-containing austenitic alloys (stainless steel and nickel based alloys) has been performed. The investigations were done for periodic bulk models. The data at the atomic scale were extrapolated by using the Universal Binding Energy Relationships (UBERs) model in order to estimate the mechanical behavior of a 10 μm thick oxide scale. The calculated stress values are in good agreement with experiments. Tensile stress for the bulk chromia was observed. The effects of temperature and structural defects on cracking were investigated. The possibility of cracking intensifies at high temperature compared to 0 K investigations. Higher susceptibility to cracking was observed in presence of defects compared to nondefective oxide, in agreement with experimental observation.


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## Body

## 1. Introduction

Chromium-containing iron or nickel based alloys such as stainless steel are chemically protected by surface oxide film (passive film). These passive films are generally rich in chromium oxide [1, 2]. The surface oxide growth and properties play a major role in stress corrosion cracking mechanisms of austenitic alloys (austenitic stainless steels and nickel based alloys) exposed to primary water of pressurized water reactor (PWR). Under operating conditions (aqueous environment, high temperature, residual stress, and eventually neutron irradiation), a complex synergy between the microstructure, the oxidation, and the mechanical load can lead to local cracking [3, 4]. Investigating the growth kinetics of passive layers on stainless steels and their corrosion resistance under different physicochemical and mechanical load is a challenge for improving the lifetime of PWR components [5].Bulk Cr2O3 is an antiferromagnetic insulator which crystallizes in the hexagonal corundum structure (space groupR3c) [6]. The unit cell contains six formula units where two-thirds of the octahedral sites are occupied by Cr and one-third remains vacant as shown in Figure 1. Both experimental [7–12] and theoretical [13, 14] studies have been performed for the investigation of mechanical properties of Cr2O3. Experimental investigations [7, 8] suggest that tensile stresses appear during oxide scale growth controlled by the predominant cationic diffusion, whereas compressive stresses are expected due to oxygen diffusion at grain boundaries. Experimental evaluations indicate a tensile strength within the range 178 to 400 MPa [9–11] and a compressive strength within the range −1.6 GPa to −2.6 GPa [7, 8, 12], both depending on the substrate and oxide layer thicknesses. Very recent DFT+U studies [13, 14] have also evaluated the tensile strength in the bulk chromia ranging from 67.0 to 94.6 GPa along the [0001] direction. It is evident that such calculated values seem highly overestimated compared to the experimental values. One of the main reasons is that the quantum chemical DFT calculations were performed at the atomic or nanoscopic scale, whereas the mechanical properties were experimentally measured at a micrometer scale. By applying UBER’s model, the authors [13, 14] have demonstrated that the tensile strength for a 10 μm thick chromia sample is in the range 380–500 MPa. However, in those studies, other factors affecting the mechanical properties such as temperature and presence of defects were not considered.Figure 1
Bulk Cr2O3 showing the octahedral sites. Red and gray spheres represent oxygen and chromium atoms, respectively.The aim of the present study is to compute the resistance to failure of a Cr-rich oxide layer assumed to be composed of chromia (Cr2O3). We have performed a theoretical investigation of the mechanical properties associated with the fracture of chromium oxide by adopting the techniques described in a previous investigation [13]. The present study focuses on the DFT calculations of stresses in bulk Cr2O3 taking into account the effects of point defects and temperature. The techniques will be utilized for the stress/strain localization in grain boundaries (GBs) of Cr2O3 in future study.


## 2. Methodology

### 2.1. Tensile Properties

The maximum tensile stress that a material can withstand before failure is known as the tensile strengthσc. In experimental studies [9–11], tensile properties measurements of oxide layers are performed on macroscopic samples via bending tests where the tensile strength values are estimated from the flexural strength. Typically, this leads to a wide range of values due to various microstructures and presence of defects in the material. The failure of materials is generally initiated through atomic level processes such as bond breaking and dislocation nucleation. DFT calculations provide atomic level insight into these basic processes responsible for the materials’ failure.Total energy calculations were performed within the density functional theory (DFT +U) based on the generalized gradient approximation (GGA) of Perdew and Wang [15]. The Kohn-Sham equations were solved by the Vienna ab initio simulation package (VASP) [16, 17]. VASP code performs an iterative diagonalization of the Kohn-Sham Hamiltonian via unconstrained band-by-band minimization of the norm of the residual vector to each eigenstate and via optimized charge density mixing routines. The (GGA +U) was used as in [18] in order to describe the strong correlation effects of the oxide. For Cr in Cr2O3, an effective on-site Coulomb interaction parameter U = 5 eV, was used as defined in our previous investigations [19, 20]. The eigenstates of the electron wave functions are expanded on a plane-wave basis set using pseudopotentials to describe the electron-ion interactions within the projector augmented waves (PAW) approach [21].The optimization of the atomic geometry at 0 K is performed by determining the exact Hellmann-Feynman forces acting on the ions for each optimization step and by using a conjugate gradient algorithm until the geometric convergence criterion on the energy (0.1 meV/cell or less) is reached. The optimized bulk lattice vectors were calculated to bea=5.07 Å and c=13.85 Å, very close to experimental values, a=4.95 Å and c=13.57 Å [22]. Molecular dynamics calculations at DFT +U level were performed at 300 K.It is to be noted that the first principles methods perform the tensile strength calculation in nanoscopic samples that yield much overestimated values [13]. The establishment of the Universal Binding Energy Relationships (UBERs) model [23–26] provides an alternative means of extrapolating the results of atomic scale calculations to the longer length scales. Therefore, UBERs model is applied in order to estimate the mechanical behavior of a 10 μm thick oxide scale.


### 2.2. Calculation of Stress

The mechanical properties of chromia are estimated in two ways. The first approach is the “ideal tensile test” which considers uniform elongation of the model (Figure2(a)) up to failure providing the theoretical tensile strength of the material. This approach is illustrated in Figure 2(b) where the tensile strain ℇ is operating via a uniaxial and uniform extension of the system with an elongation distance δ (Å). The second approach considers the formation of a single crack across a specific interatomic plane through the application of tensile load as depicted in Figure 2(c). Combining the two approaches, the resistance to failure of a 10 μm thick model can be computed by using the UBERs model as presented in recent theoretical studies [13, 14].Figure 2
Schematic diagrams: (a) the original system showing the lattice parameterc (Å) along which the load is applied, (b) uniform expansion by a distance δ (Å) along the direction of the applied load, and (c) insertion of single crack of width δ (Å).
(a)
(b)
(c)


## 2.1. Tensile Properties

The maximum tensile stress that a material can withstand before failure is known as the tensile strengthσc. In experimental studies [9–11], tensile properties measurements of oxide layers are performed on macroscopic samples via bending tests where the tensile strength values are estimated from the flexural strength. Typically, this leads to a wide range of values due to various microstructures and presence of defects in the material. The failure of materials is generally initiated through atomic level processes such as bond breaking and dislocation nucleation. DFT calculations provide atomic level insight into these basic processes responsible for the materials’ failure.Total energy calculations were performed within the density functional theory (DFT +U) based on the generalized gradient approximation (GGA) of Perdew and Wang [15]. The Kohn-Sham equations were solved by the Vienna ab initio simulation package (VASP) [16, 17]. VASP code performs an iterative diagonalization of the Kohn-Sham Hamiltonian via unconstrained band-by-band minimization of the norm of the residual vector to each eigenstate and via optimized charge density mixing routines. The (GGA +U) was used as in [18] in order to describe the strong correlation effects of the oxide. For Cr in Cr2O3, an effective on-site Coulomb interaction parameter U = 5 eV, was used as defined in our previous investigations [19, 20]. The eigenstates of the electron wave functions are expanded on a plane-wave basis set using pseudopotentials to describe the electron-ion interactions within the projector augmented waves (PAW) approach [21].The optimization of the atomic geometry at 0 K is performed by determining the exact Hellmann-Feynman forces acting on the ions for each optimization step and by using a conjugate gradient algorithm until the geometric convergence criterion on the energy (0.1 meV/cell or less) is reached. The optimized bulk lattice vectors were calculated to bea=5.07 Å and c=13.85 Å, very close to experimental values, a=4.95 Å and c=13.57 Å [22]. Molecular dynamics calculations at DFT +U level were performed at 300 K.It is to be noted that the first principles methods perform the tensile strength calculation in nanoscopic samples that yield much overestimated values [13]. The establishment of the Universal Binding Energy Relationships (UBERs) model [23–26] provides an alternative means of extrapolating the results of atomic scale calculations to the longer length scales. Therefore, UBERs model is applied in order to estimate the mechanical behavior of a 10 μm thick oxide scale.


## 2.2. Calculation of Stress

The mechanical properties of chromia are estimated in two ways. The first approach is the “ideal tensile test” which considers uniform elongation of the model (Figure2(a)) up to failure providing the theoretical tensile strength of the material. This approach is illustrated in Figure 2(b) where the tensile strain ℇ is operating via a uniaxial and uniform extension of the system with an elongation distance δ (Å). The second approach considers the formation of a single crack across a specific interatomic plane through the application of tensile load as depicted in Figure 2(c). Combining the two approaches, the resistance to failure of a 10 μm thick model can be computed by using the UBERs model as presented in recent theoretical studies [13, 14].Figure 2
Schematic diagrams: (a) the original system showing the lattice parameterc (Å) along which the load is applied, (b) uniform expansion by a distance δ (Å) along the direction of the applied load, and (c) insertion of single crack of width δ (Å).
(a)
(b)
(c)


## 3. Results and Discussion

### 3.1. Ideal Tensile Test of Bulk Cr2O3

In the simulated tensile test, tensile strain is applied by elongating the system along the direction of the applied load as shown in Figure2(b). The applied load is directed along the c crystallographic axis ([0001] direction). Thus the axial stress is evaluated for successive elongations. The resulting stress-strain curve (Figure 3(a)) provides a maximum indicating the limit of the resistance to failure of the material. The value of the stress to failure corresponds to the maximum tensile stress (σc) that can be experienced by the simulated system. Figure 3(a) shows that the calculated σc with the DFT +U method is 57.0 GPa. The maximum strain (δc) before fracture is 3.3 Å. The fracture always occurs perpendicular to the direction of the applied load (as shown in Figure 3(b)). The fracture appears at the middle of the system, at the upper part of the Cr–O octahedron breaking the three longest bonds (2.048 Å in the bulk), while the three shorter bonds (1.935 Å in the bulk) remain. It splits the Cr bilayer in its middle as shown in Figure 3(c).Figure 3
(a) Stressσ (GPa) versus strain δ (Å) curve that provides the ideal tensile test. σc is the maximum tensile strength. (b) Crystal structure of Cr2O3 bulk, showing position where the fracture happens; (c) Cr2O3 bulk system with fracture in the middle. The blue atom corresponds to the specific Cr atom and green atoms are the O forming the octahedron.
(a)
(b)
(c)It appears clearly that the calculatedσc value is highly overestimated compared to the experimental value (0.18–0.40 GPa [9–11], see Table 1). Although the ideal tensile test is similar to the experimental tensile test conceptually, there is a large discrepancy between the calculated and experimentally measured values of σc. We assume that, due to the absence of thermal fluctuations and structural defects, the calculated σc is overestimated.Table 1
Comparison of calculatedδc (Å) and σc (GPa) values with the experiment for bulk Cr2O3 (the values from the insertion crack tests are in the bracket).
Nondefective
Cr-defective
Experiment [9–11]
Method
DFT
+
U
UBER
DFT
+
U
Temperature
0 K
300 K
0 K
Supercell
1 × 1 × 1
1 × 1 × 2
1 × 1 × 1
δ
c
3.3 (1.8)
3.35
1.4
165.70
2.5
σ
c
57.0 (39.9)
54.40
21.5
0.53
22.70
0.18–0.40The effect of unit cell size on the calculated properties has been investigated by doubling the unit cell size along thec direction (the direction of applied load). The stress-strain curve is shown in Figure 4(a). The fracture occurs in two planes lying normal to the direction of applied load (Figure 4(b)). The calculated σc is 54.40 GPa which is smaller than a (1 × 1 × 1) supercell (57.00 GPa, Table 1). This gives an indication that simulation of a very large supercell representing the real system would lower σc to a value close to the experimental one. However, due to the limited computer resources, investigation with a larger supercell is out of the scope of the present study. We have also investigated the effects of temperature and defects on the calculated σc, which will be discussed later.Figure 4
Results obtained for a 1 × 1 × 2 supercell of Cr2O3. (a) Stress σ (GPa) versus strain δ (Å) curve that provides the ideal tensile test. σc is the maximum tensile strength. (b) Crystal structure showing the fracture at two points.
(a)
(b)


### 3.2. Strain Localization in Bulk Cr2O3 Structure

Strain localization in Cr2O3 structure was investigated by the analysis of the structures obtained when elongating the system uniformly along the direction of applied load. The variation of the lengths of the Cr–O bonds around a single Cr atom was determined as a function of the applied tensile strain. As shown in Figure 5(a), there exist three equivalent Cr–O bonds (green) above the Cr atom and another three equivalent Cr–O bonds (blue) below this atom, which is indicated as Cr∗. For comparison, we have adopted same notations as discussed in [13]. The variation of the bond length versus tensile strength curves for Cr–O bonds is plotted in Figure 5(b) where the green curve corresponds to the upper Cr–O bonds and the blue curve corresponds to the lower Cr–O bonds.Figure 5
(a) Three equivalent Cr–O bonds above and below the Cr atom designated asCr∗. (b) Bond strain versus tensile strain curves for the Cr–O bonds above (green curve) and below (blue curve) the Cr∗ atom.
(a)
(b)It is observed that the Cr–O bond length increases roughly linearly with the applied elongation. The rate of change of bond length is nearly identical for the Cr–O bonds above and below the designated Cr atom until the system fractures. Note that all Cr atoms in Cr2O3 are structurally equivalent. Our results suggest that all of the Cr–O bonds in the system accommodate equivalently the tensile load applied along the [0001] crystal direction. After fracture, there is huge increase in the lower Cr–O bond (as shown in the blue curve of Figure 5(b)), whereas the upper Cr–O bond decreases (as shown in green curve in the Figure 5(b)). These are in good agreement with a previous theoretical study [13].


### 3.3. UBER Models

Two different limits in which a material can accommodate the applied load can be considered as already depicted in Figure2. Firstly, the system can accommodate the applied strain (ℇ) by uniform expansion with a distance δ (Å) along the direction of the applied load. Secondly, the system can accommodate the applied strain (ℇ) by forming a single crack of width δ (Å).In both cases, several full optimization calculations were performed for variousδ values and the energy Φ of the system is then evaluated for each calculation. The Φ versus δ curves are plotted for both kinds of limit, as shown in Figure 6(a).Figure 6
Φ(eV) versus δ (Å) curves: (a) the pink curve corresponds to the uniformly expanded system and the blue curve corresponds to the system with inserted crack, (b) combination of the two series of results from two different limits, and (c) plot of Φ∗ versus δ∗ providing the UBER curve.
(a)
(b)
(c)For the expanded system, the fracture occurred forδ=3.3 Å (pink curve in Figure 6(a)) whereas, for the crack-inserted system, the failure started to operate at lower δ value (below 1.5 Å, blue curve in Figure 6(a)). For an intermediate value of δ=1.8 Å, the energy of the uniformly expanded system was equal to that of the crack-inserted system. This elongation (δc) corresponds to the maximum value of δ that the system can experience prior to failure. The combination of the pink and the blue curves leads to Figure 6(b), where δc can be evaluated as 1.8 Å. Evaluating the stress at this point provides an estimate of σc.The renormalization ofΦ and δ into the dimensionless quantities can be performed according to following equations: (1)Φ∗=Φ2γ,where γ is the surface energy.(2)δ∗=δc2Nγ/C,where N is the number of stoichiometric units and C is the uniaxial elastic constant given by the slope of the plot of 2NΦ versus δ2.The curve (Figure6(c)) obtained by plotting Φ∗ versus δ∗ is called the UBER; its universality implies that all materials should obey this relationship. More details can be obtained from similar study on Cr2O3 [13].For a 10μm thick model, σc = 0.531 GPa. δc is calculated by (2), where δ∗ is 1 (maximum of Φ∗ versus δ∗ curve, Figure 6(c)). The calculated value of δc for the 10 μm thick sample is 166 Å.The calculatedδc (Å) and σc (GPa) values are compared with experimental values in Table 1. The experimental σc values range from 0.18 to 0.40 GPa. As discussed before, the calculated DFT +U values are much overestimated compared to the experimental value, 57.00 GPa for the uniformly expanded system and 39.90 GPa for the system with inserted crack (the values in the bracket). As indicated by the values of δc, the transition between the uniformly expanded and cracked regimes occurs at 1.80 (Å) when the load is applied along [0001] direction. This can be visualized in the Φ versus δ curve (Figure 6(b)). The transition between the uniformly expanded and cracked regimes is smooth and the curve exhibits the anticipated shape: Φ exhibits an approximately harmonic dependence on δ for small values of δ and asymptotically approaches a constant value as δ tends to infinity. The σc value of 0.53 GPa obtained for a 10 μm thick sample using the UBERs model (see Table 1) is in fair agreement with experiments. This shows the ability of the UBER to transform the results of the atomic scale calculations into realistic values of mechanical properties on much longer length scales.


### 3.4. Effect of Temperature

In reality, experimental measurements of tensile properties are performed at room temperature. In order to simulate the real situation, further investigation was performed at 300 K with the MD simulation. The tensile stress is applied by elongating the system along the direction of the applied load as explained in Figure2(b). Several MD calculations at 300 K were performed for each of the applied loads. The maximum strain (δc) before the fracture is 1.4 Å for Cr2O3 at 300 K which is much smaller than that at 0 K (3.3 Å). The structures before and after fracture are shown in Figure 7. As at 0 K, the fracture occurs at the midpoint of the Cr bilayer and the resulting Cr half layers move toward the adjacent O layers. The value of stress at this point along the stress-strain curve (the maximum tensile stress σc) is 21.50 GPa. Compared to the experimental value of σc, it is observed that the maximum tensile strength is less overestimated at 300 K than that at 0 K (Table 1).Figure 7
Stress calculation in Cr2O3 at 300 K. Indication of the position of crack and the structure after cracking.


### 3.5. Cr-Defective Cr2O3

For the Cr-defective Cr2O3, the tensile strain is applied by elongating the system along the direction of the applied load as explained before (see Figure 2(b)). This is done for a series of strains δ and then the stress σ along the direction of the applied load is evaluated for each strain. It is observed that the fracture occurs perpendicular to the applied load axis and the material breaks down near the defect as shown in Figures 8(a) and 8(b).Figure 8
Stress calculation in the Cr-defective Cr2O3. (a) The arrow indicates the position of the Cr vacancy (Cr V), (b) fracture initiates at the position of the vacancy (c) stress σ versus strain δ plot for the Cr-defective system. σc is the maximum tensile strength.
(a)
(b)
(c)The resulting stress versus strain curve (Figure8(c)) provides a maximum indicating the mechanical failure of the material. The value of stress at this point along the stress-strain curve corresponds to the maximum tensile stress (σc) that can be sustained by the simulated system. In Figure 8(c), it is observed that the calculated σc with the DFT +U method is 22.70 GPa. The maximum strain (δc) before fracture is 2.5 Å. Compared to the experimental value of σc, it is observed that the maximum tensile strength is less overestimated in the Cr-defective system than that in the nondefective system (Table 1). This is reasonable as the presence of defect reduces the strain required for creation of fracture or mechanical failure in the system. As all the Cr atoms in Cr2O3 are structurally equivalent, the variation of location of defect will not vary the maximum tensile stress and strain of the system. The calculated σc and σc values will reduce with the reduced concentration of Cr defects in the structure and converge to the experimental values for a very small vacancy concentration. It must be noted that, due to the limited size of the model, one vacancy per unit cell corresponds to a very high vacancy concentration (3.33%).


## 3.1. Ideal Tensile Test of Bulk Cr2O3

In the simulated tensile test, tensile strain is applied by elongating the system along the direction of the applied load as shown in Figure2(b). The applied load is directed along the c crystallographic axis ([0001] direction). Thus the axial stress is evaluated for successive elongations. The resulting stress-strain curve (Figure 3(a)) provides a maximum indicating the limit of the resistance to failure of the material. The value of the stress to failure corresponds to the maximum tensile stress (σc) that can be experienced by the simulated system. Figure 3(a) shows that the calculated σc with the DFT +U method is 57.0 GPa. The maximum strain (δc) before fracture is 3.3 Å. The fracture always occurs perpendicular to the direction of the applied load (as shown in Figure 3(b)). The fracture appears at the middle of the system, at the upper part of the Cr–O octahedron breaking the three longest bonds (2.048 Å in the bulk), while the three shorter bonds (1.935 Å in the bulk) remain. It splits the Cr bilayer in its middle as shown in Figure 3(c).Figure 3
(a) Stressσ (GPa) versus strain δ (Å) curve that provides the ideal tensile test. σc is the maximum tensile strength. (b) Crystal structure of Cr2O3 bulk, showing position where the fracture happens; (c) Cr2O3 bulk system with fracture in the middle. The blue atom corresponds to the specific Cr atom and green atoms are the O forming the octahedron.
(a)
(b)
(c)It appears clearly that the calculatedσc value is highly overestimated compared to the experimental value (0.18–0.40 GPa [9–11], see Table 1). Although the ideal tensile test is similar to the experimental tensile test conceptually, there is a large discrepancy between the calculated and experimentally measured values of σc. We assume that, due to the absence of thermal fluctuations and structural defects, the calculated σc is overestimated.Table 1
Comparison of calculatedδc (Å) and σc (GPa) values with the experiment for bulk Cr2O3 (the values from the insertion crack tests are in the bracket).
Nondefective
Cr-defective
Experiment [9–11]
Method
DFT
+
U
UBER
DFT
+
U
Temperature
0 K
300 K
0 K
Supercell
1 × 1 × 1
1 × 1 × 2
1 × 1 × 1
δ
c
3.3 (1.8)
3.35
1.4
165.70
2.5
σ
c
57.0 (39.9)
54.40
21.5
0.53
22.70
0.18–0.40The effect of unit cell size on the calculated properties has been investigated by doubling the unit cell size along thec direction (the direction of applied load). The stress-strain curve is shown in Figure 4(a). The fracture occurs in two planes lying normal to the direction of applied load (Figure 4(b)). The calculated σc is 54.40 GPa which is smaller than a (1 × 1 × 1) supercell (57.00 GPa, Table 1). This gives an indication that simulation of a very large supercell representing the real system would lower σc to a value close to the experimental one. However, due to the limited computer resources, investigation with a larger supercell is out of the scope of the present study. We have also investigated the effects of temperature and defects on the calculated σc, which will be discussed later.Figure 4
Results obtained for a 1 × 1 × 2 supercell of Cr2O3. (a) Stress σ (GPa) versus strain δ (Å) curve that provides the ideal tensile test. σc is the maximum tensile strength. (b) Crystal structure showing the fracture at two points.
(a)
(b)


## 3.2. Strain Localization in Bulk Cr2O3 Structure

Strain localization in Cr2O3 structure was investigated by the analysis of the structures obtained when elongating the system uniformly along the direction of applied load. The variation of the lengths of the Cr–O bonds around a single Cr atom was determined as a function of the applied tensile strain. As shown in Figure 5(a), there exist three equivalent Cr–O bonds (green) above the Cr atom and another three equivalent Cr–O bonds (blue) below this atom, which is indicated as Cr∗. For comparison, we have adopted same notations as discussed in [13]. The variation of the bond length versus tensile strength curves for Cr–O bonds is plotted in Figure 5(b) where the green curve corresponds to the upper Cr–O bonds and the blue curve corresponds to the lower Cr–O bonds.Figure 5
(a) Three equivalent Cr–O bonds above and below the Cr atom designated asCr∗. (b) Bond strain versus tensile strain curves for the Cr–O bonds above (green curve) and below (blue curve) the Cr∗ atom.
(a)
(b)It is observed that the Cr–O bond length increases roughly linearly with the applied elongation. The rate of change of bond length is nearly identical for the Cr–O bonds above and below the designated Cr atom until the system fractures. Note that all Cr atoms in Cr2O3 are structurally equivalent. Our results suggest that all of the Cr–O bonds in the system accommodate equivalently the tensile load applied along the [0001] crystal direction. After fracture, there is huge increase in the lower Cr–O bond (as shown in the blue curve of Figure 5(b)), whereas the upper Cr–O bond decreases (as shown in green curve in the Figure 5(b)). These are in good agreement with a previous theoretical study [13].


## 3.3. UBER Models

Two different limits in which a material can accommodate the applied load can be considered as already depicted in Figure2. Firstly, the system can accommodate the applied strain (ℇ) by uniform expansion with a distance δ (Å) along the direction of the applied load. Secondly, the system can accommodate the applied strain (ℇ) by forming a single crack of width δ (Å).In both cases, several full optimization calculations were performed for variousδ values and the energy Φ of the system is then evaluated for each calculation. The Φ versus δ curves are plotted for both kinds of limit, as shown in Figure 6(a).Figure 6
Φ(eV) versus δ (Å) curves: (a) the pink curve corresponds to the uniformly expanded system and the blue curve corresponds to the system with inserted crack, (b) combination of the two series of results from two different limits, and (c) plot of Φ∗ versus δ∗ providing the UBER curve.
(a)
(b)
(c)For the expanded system, the fracture occurred forδ=3.3 Å (pink curve in Figure 6(a)) whereas, for the crack-inserted system, the failure started to operate at lower δ value (below 1.5 Å, blue curve in Figure 6(a)). For an intermediate value of δ=1.8 Å, the energy of the uniformly expanded system was equal to that of the crack-inserted system. This elongation (δc) corresponds to the maximum value of δ that the system can experience prior to failure. The combination of the pink and the blue curves leads to Figure 6(b), where δc can be evaluated as 1.8 Å. Evaluating the stress at this point provides an estimate of σc.The renormalization ofΦ and δ into the dimensionless quantities can be performed according to following equations: (1)Φ∗=Φ2γ,where γ is the surface energy.(2)δ∗=δc2Nγ/C,where N is the number of stoichiometric units and C is the uniaxial elastic constant given by the slope of the plot of 2NΦ versus δ2.The curve (Figure6(c)) obtained by plotting Φ∗ versus δ∗ is called the UBER; its universality implies that all materials should obey this relationship. More details can be obtained from similar study on Cr2O3 [13].For a 10μm thick model, σc = 0.531 GPa. δc is calculated by (2), where δ∗ is 1 (maximum of Φ∗ versus δ∗ curve, Figure 6(c)). The calculated value of δc for the 10 μm thick sample is 166 Å.The calculatedδc (Å) and σc (GPa) values are compared with experimental values in Table 1. The experimental σc values range from 0.18 to 0.40 GPa. As discussed before, the calculated DFT +U values are much overestimated compared to the experimental value, 57.00 GPa for the uniformly expanded system and 39.90 GPa for the system with inserted crack (the values in the bracket). As indicated by the values of δc, the transition between the uniformly expanded and cracked regimes occurs at 1.80 (Å) when the load is applied along [0001] direction. This can be visualized in the Φ versus δ curve (Figure 6(b)). The transition between the uniformly expanded and cracked regimes is smooth and the curve exhibits the anticipated shape: Φ exhibits an approximately harmonic dependence on δ for small values of δ and asymptotically approaches a constant value as δ tends to infinity. The σc value of 0.53 GPa obtained for a 10 μm thick sample using the UBERs model (see Table 1) is in fair agreement with experiments. This shows the ability of the UBER to transform the results of the atomic scale calculations into realistic values of mechanical properties on much longer length scales.


## 3.4. Effect of Temperature

In reality, experimental measurements of tensile properties are performed at room temperature. In order to simulate the real situation, further investigation was performed at 300 K with the MD simulation. The tensile stress is applied by elongating the system along the direction of the applied load as explained in Figure2(b). Several MD calculations at 300 K were performed for each of the applied loads. The maximum strain (δc) before the fracture is 1.4 Å for Cr2O3 at 300 K which is much smaller than that at 0 K (3.3 Å). The structures before and after fracture are shown in Figure 7. As at 0 K, the fracture occurs at the midpoint of the Cr bilayer and the resulting Cr half layers move toward the adjacent O layers. The value of stress at this point along the stress-strain curve (the maximum tensile stress σc) is 21.50 GPa. Compared to the experimental value of σc, it is observed that the maximum tensile strength is less overestimated at 300 K than that at 0 K (Table 1).Figure 7
Stress calculation in Cr2O3 at 300 K. Indication of the position of crack and the structure after cracking.


## 3.5. Cr-Defective Cr2O3

For the Cr-defective Cr2O3, the tensile strain is applied by elongating the system along the direction of the applied load as explained before (see Figure 2(b)). This is done for a series of strains δ and then the stress σ along the direction of the applied load is evaluated for each strain. It is observed that the fracture occurs perpendicular to the applied load axis and the material breaks down near the defect as shown in Figures 8(a) and 8(b).Figure 8
Stress calculation in the Cr-defective Cr2O3. (a) The arrow indicates the position of the Cr vacancy (Cr V), (b) fracture initiates at the position of the vacancy (c) stress σ versus strain δ plot for the Cr-defective system. σc is the maximum tensile strength.
(a)
(b)
(c)The resulting stress versus strain curve (Figure8(c)) provides a maximum indicating the mechanical failure of the material. The value of stress at this point along the stress-strain curve corresponds to the maximum tensile stress (σc) that can be sustained by the simulated system. In Figure 8(c), it is observed that the calculated σc with the DFT +U method is 22.70 GPa. The maximum strain (δc) before fracture is 2.5 Å. Compared to the experimental value of σc, it is observed that the maximum tensile strength is less overestimated in the Cr-defective system than that in the nondefective system (Table 1). This is reasonable as the presence of defect reduces the strain required for creation of fracture or mechanical failure in the system. As all the Cr atoms in Cr2O3 are structurally equivalent, the variation of location of defect will not vary the maximum tensile stress and strain of the system. The calculated σc and σc values will reduce with the reduced concentration of Cr defects in the structure and converge to the experimental values for a very small vacancy concentration. It must be noted that, due to the limited size of the model, one vacancy per unit cell corresponds to a very high vacancy concentration (3.33%).


## 4. Conclusions

The mechanical properties associated with the fracture of chromia (Cr2O3) were evaluated theoretically. The estimation of stress was done by two ways. The first approach considers the effect of subjecting the material to increasing tensile loads until fracture occurs and provides the theoretical tensile strength of the material. The second approach considers the formation of a single crack across a specific interatomic plane through the application of tensile load.The calculations of the stress related properties were done by using the DFT +U approach applying periodic models. Then the atomic level results were extrapolated by using the UBERs model in order to obtain a realistic estimate for a 10 μm thick sample. Our investigation shows also that the temperature and the presence of point defects play important role for the mechanical properties associated with fracture. It was observed that stress for the bulk chromia is tensile. These are in good agreement with experimental observations. Investigations are ongoing to understand the stress/strain localization in the grain boundaries of chromium oxides by applying the above-mentioned techniques and methods.



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*Source: 7039436-2017-04-24.xml*