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Simulation of crack propagation in alumina with ab initio based ...

084707-3 **Simulation** **of** **crack** **propagation** **in** alum**in**a J. Chem. Phys. 136, 084707 (2012) total force cartesian components f 0 m,n (n = 1,2,...,3N m)on N m atoms. The function is m**in**imized, where Z e = 3 Z s = 1 2 Z f = Z = w e Z e + w s Z s + Z f (10) M∑ ( N m em − em) 0 2, m=1 M∑ m=1 l=1 M∑ ∑3N m m=1 n=1 6∑ ( N m sm,l − sm,l) 0 2, (11) ( fm,n − fm,n 0 ) 2, and e m , s m,l , and f m,n are the correspond**in**g values calculated **with** the parametrized force field. w e and w s are weights to balance the different amount **of** avail**ab**le data for each quantity. They are def**in**ed **in** such a way that each weight can be taken as the percentual amount **of** the concern**in**g force data. In the follow**in**g, we assume M reference structures that all consist **of** the same number **of** particles (N m = N), but **in** pr**in**ciple, potfit can handle any different number **of** particles for each reference structure. The root mean square (rms) errors, √ F e = Ze 3MN , F s = √ 2Zs MN , and F f = √ Zf MN , (12) are first **in**dicators **of** the quality **of** the generated force field. Their magnitudes are **in**dependent **of** weight**in**g factors, number, and sizes **of** reference structures. For the m**in**imization **of** the force field parameters, a comb**in**ation **of** a stochastic simulated anneal**in**g algorithm 26 and a conjugate-gradient-like determ**in**istic algorithm 27 is used. For more **in**formation **ab**out the implementation **of** long-range electrostatic **in**teractions **in**clud**in**g the TS force field **with** Wolf summation, see Ref. 16. D. Parametrization We first prepared a set **of** 67 α-alum**in**a crystal structures **with** 360 atoms, **in** total 24 120 atoms. This reference dat**ab**ase is composed **of** three k**in**ds **of** structures: (i) crystals stra**in**ed up to 20% **in** [0001], [2110], and [0110] directions at 0 K, (ii) structures **with** free (0001), (2110), and (0110) surfaces at 0 K, and (iii) equilibrated snapshots taken out **of** an **ab** **in**itio MD trajectory, where an ideal crystal is heated from 0 K up to 2000 K. It has to be mentioned that no **in**itial ad hoc potential as used **in** Refs. 14 and 16 is required to generate a reference dat**ab**ase. This is only necessary **in** the case **of** liquid structures. The present work, however, deals **with** crystall**in**e structures where a dat**ab**ase can be generated **with**out need **of** classical MD trajectories. The 67 structures are applied as **in**put configurations for the first-pr**in**ciples plane wave code VASP. 24, 25 We used ultras**of**t pseudopotentials 28 and the Perdew- Wang 91 (Ref. 29) generalized gradient approximation to the exchange-correlation functional. The latter is recommended **in** Ref. 16. A plane wave cut**of**f **of** 396 eV was used. TABLE I. Force field parameters, given **in** IMD units set eV, Å, and amu (hence charges are multiples **of** the elementary charge). q Al q O α O b Al−O c Al−O 1.122 608 −0.748 406 0.026 576 18.984 286 −5.571 329 D γ r 0 Al–Al 0.000 890 12.737 442 5.405 175 Al–O 1.000 058 8.077 778 1.851 806 O–O 0.005 307 12.081 851 3.994 815 We adopt a gamma centered k mesh **of** 2 × 2 × 2 k po**in**ts. The weights **in** potfit were chosen to w e = 0.03 and w s = 0.28. Sett**in**g κ = 0.1 Å −1 , a cut**of**f radius **of** 10 Å was found to be sufficient. The f**in**al set **of** parameters is shown **in** T**ab**le I.The rmserrorsareF e = 0.0492 eV, F s = 0.0273 eV Å −3 , and F f = 0.3507 eV Å −1 . III. VALIDATION First, we determ**in**e basic properties **of** crystall**in**e alum**in**a such as lattice constants, cohesive energies, and vibrational properties. Additionally and relevant for fracture studies, our validation simulations focus on surface relaxations, surface energies, and stresses **of** stra**in**ed configurations. F**in**ally, we probe the new potential beyond its optimization range by **in**vestigat**in**g two basal tw**in**s. Lattice constants and cohesive energies are obta**in**ed by pressure relaxation: Besides energy m**in**imization, the pressure tensor **of** the sample is calculated at each step, and the size **of** the simulation box is changed by a small amount **in** order to lower that pressure. After **in**sert**in**g surfaces or **in**terfaces **in**to the relaxed sample, a further relaxation is performed which reveals surface or **in**terface energies. The **ab** **in**itio calculations **of** these properties are performed **with** VASP, as described **in** Sec. II D. As can be seen from T**ab**le II, the lattice constants and the cohesive energy **of** crystall**in**e α-alum**in**a obta**in**ed **with** the new potential agree well **with** our **ab** **in**itio calculations and previous studies. The partial vibrational density **of** states (VDOS) for G Al (E) and G O (E) was obta**in**ed by comput**in**g the Fourier transform **of** the time-dependent velocity-velocity autocorrelation function from a 100 ps MD trajectory **with** the s**of**tware package nMoldyn. 37 The generalized VDOS G(E) is then calculated by G(E) = ∑ μ=Al,O σ μ m μ G μ (E) (13) **with** the scatter**in**g cross section σ μ and atomic mass m μ **of** atom μ. Figure 1 shows the partial and generalized VDOS. The key features **of** the curves obta**in**ed **with** the new potential and an **ab** **in**itio study from Ref. 38 co**in**cide. For the partial VDOS **of** alum**in**um, both studies show a broad band between 41 and 83 meV. The **ab** **in**itio results show less states **in** the low-energy band between 15 and 30 meV. There are two sharp peaks between 87 and 100 meV **in** the **ab** **in**ito curve, whereas the new potential shows one broader peak **with** a shoulder, that **in**dicated the second peak. The curves for the partial VDOS **of** oxygen are **in** good agreement. Both simulation and Downloaded 29 Feb 2012 to 129.69.47.171. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/**ab**out/rights_and_permissions

084707-4 Hocker et al. J. Chem. Phys. 136, 084707 (2012) TABLE II. Lattice constants, cohesive energies, and **in**terface energies **of** basal tw**in**s obta**in**ed **with** the new potential compared to **ab** **in**itio results and literature data. a c E coh E {0001} E {1120} E {1010} E m tw**in** E r tw**in** (Å) (Å) (eV) (J/m 2 ) (J/m 2 ) (J/m 2 ) (J/m 2 ) (J/m 2 ) New potential 4.79 12.97 31.85 1.59 1.65 1.89 1.32 0.57 Ab **in**itio 4.78 13.05 32.31 1.55 1.83 1.98 2.12 0.76 Lit. (experiment) 4.76 (Ref. 30) 13.00 (Ref. 30) 31.8 (Ref. 31) Lit. (**ab** **in**itio) 33.0 (Ref. 32) 1.85(Ref.33) 2.44(Ref.33) 2.39(Ref.33) 1.99(Ref.4) 0.73(Ref.4) 2.03 (Ref. 34) 2.23(Ref.34) 2.50(Ref.34) 1.76 (Ref. 35) 1.54 (Ref. 36) **ab** **in**itio calculation show the ma**in** band **of** states between 12 and 85 meV **with** similar curve progression and three local maxima at around 35, 47, and 62 meV. These characteristics are also reflected **in** the generalized VDOS. In addition, the generalized VDOS obta**in**ed **with** neutron scatter**in**g 1 is depicted, which shows the same ma**in** characteristics **of** the curve on a qualitative level. The literature surface energies 33–36 given **in** T**ab**le II differ among each other which orig**in**s from the different methods used **in** the **ab** **in**itio approaches. Our calculation **of** the (0001) surface energy agrees **with** the value obta**in**ed **in** Ref. 36, where also the VASP code **with** the same pseudopotential and exchange-correlation approximation was used. For all **in**vestigated surfaces, the energies obta**in**ed **with** the new potential and **with** **ab** **in**itio methods agree. Both results reveal that the (0001) surface has the lowest surface energy and the (0110) surface has the highest one. The studies **of** Refs. 33–35 yield slightly higher energies for all **in**vestigated surfaces. The surface structure after relaxation is shown **in** Fig. 2.It reveals that the Al atoms **of** the outermost Al-layer are moved closer to the outermost O-layer at the (0001) Al-term**in**ated VDOS (arb. units) New potential Ab-**in**itio Experiment 10 20 30 40 50 60 70 80 90 100 E (meV) FIG. 1. VDOS calculated **with** the new potential compared **with** an **ab** **in**itio study 38 and experiment. 1 (a) Partial VDOS for alum**in**um atoms, (b) partial VDOS for oxygen atoms, and (c) generalized VDOS. (a) (b) (c) surface, which is known to be the most st**ab**le (0001) surface term**in**ation. The atomic adjustment perpendicular to the surface obta**in**ed **with** the new potential agrees very well **with** our **ab** **in**itio results. The distance **of** an Al- and an O-layer is 0.83 Å, whereas this value decreases to 0.15 Å (MD) 0.14 Å (**ab** **in**itio), respectively, at the surface. A relaxation **of** the oxygen atoms can be seen at the (0110) surface. Both MD and **ab** **in**itio study show that the three oxygen atoms per unit cell – which are **in**itially **in** a row along the direction orthogonal to the plane **of** Fig. 2 – relax to different distances from the **in**itial surface. Furthermore, a relaxation towards the neighbor**in**g Al-atoms **in** the first layer occurs. The results obta**in**ed **with** the new potential aga**in** co**in**cide **with** the **ab** **in**itio calculation. One difference, however, can be seen at the second layer: Every second Al atom **in** each row orthogonal to the figure plane is slightly displaced towards the surface **in** the **ab** **in**itio relaxation. This effect is not observed **in** the MD relaxation simulation. Neither the MD nor the **ab** **in**ito relaxation study **of** the (2110) surface yields significant atomic movements. As described **in** Sec. II D, stra**in**ed configurations **with** stra**in** directions [0001], [2110], and [0110] were added **in** the reference dat**ab**ase for the potential optimization approach. To clarify, whether the new potential can reproduce the stresses **of** stra**in**ed configurations, these stresses are calculated us**in**g MD **with** the new potential. Figure 3 shows a comparison **of** stresses obta**in**ed **in** simulations to the stresses **of** the underly**in**g **ab** **in**itio reference configurations, each **with** 360 atoms. They are stra**in**ed up to 15% **in** [0001], [2110], and [0110] directions, respectively. The difference between MD and **ab** **in**itio results is small at lower stra**in**s. With **in**creas**in**g stra**in**, the difference **in**creases up to **ab**out 10 GPa at 15% stra**in**. The new potential underestimates the stress **in** all cases. However, the directions, **in** which the stress **in**creases, can be reproduced correctly: The highest stresses are observed for stra**in**s **in** [0001] direction, the lowest stresses are found for configurations stra**in**ed **in** [0110] direction. F**in**ally, we probe the potential by simulat**in**g basal tw**in**s. These are systems which were not **in**cluded **in** the optimization process. This demonstrates the applic**ab**ility **of** the new force field beyond the range for which it was optimized. It is well known that the basal tw**in** is a frequently observed Al 2 O 3 **in**terface. 39 We consider two types **of** basal tw**in**s which differ by their **in**terface structure: The mirror tw**in** consists **of** Downloaded 29 Feb 2012 to 129.69.47.171. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/**ab**out/rights_and_permissions

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