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

THE JOURNAL OF CHEMICAL PHYSICS 136, 084707 (2012) **Simulation** **of** **crack** **propagation** **in** alum**in**a **with** **ab** **in**itio **based** polariz**ab**le force field Stephen Hocker, 1,a) Philipp Beck, 2 Siegfried Schmauder, 1 Johannes Roth, 2 and Hans-Ra**in**er Treb**in** 2 1 Institut für Materialprüfung, Werkst**of**fkunde und Festigkeitslehre (IMWF), Universität Stuttgart, Pfaffenwaldr**in**g 32, 70550 Stuttgart, Germany 2 Institut für Theoretische und Angewandte Physik (ITAP), Universität Stuttgart, Pfaffenwaldr**in**g 57, 70550 Stuttgart, Germany (Received 23 November 2011; accepted 31 January 2012; published onl**in**e 28 February 2012) We present an effective atomic **in**teraction potential for crystall**in**e α-Al 2 O 3 generated by the program potfit. The Wolf direct, pairwise summation method **with** spherical truncation is used for electrostatic **in**teractions. The polariz**ab**ility **of** oxygen atoms is **in**cluded by use **of** the Tangney-Scandolo **in**teratomic force field approach. The potential is optimized to reproduce the forces, energies, and stresses **in** relaxed and stra**in**ed configurations as well as {0001}, {1010}, and {1120} surfaces **of** Al 2 O 3 .Details **of** the force field generation are given, and its validation is demonstrated. We apply the developed potential to **in**vestigate **crack** **propagation** **in** α-Al 2 O 3 s**in**gle crystals. © 2012 American Institute **of** Physics. [http://dx.doi.org/10.1063/1.3685900] I. INTRODUCTION Alum**in**um oxide is the most commonly used ceramic **in** technological applications. Due to its **in**sulat**in**g properties, it is frequently adopted **in** microelectronic devices, i.e., field effect transistors, **in**tegrated circuits, or superconduct**in**g quantum **in**terference devices. Another important field **of** application is the coat**in**g **of** metallic components. Alum**in**a cover**in**g alum**in**um is known to prevent further oxidation **of** the metal. S**in**ce alum**in**a is characterized by a high melt**in**g po**in**t and also a high degree **of** hardness, applications at high temperatures and high mechanical demands are possible. Mechanical properties such as tensile strengths and fracture processes are **of** high importance regard**in**g these applications. Because surfaces and **in**ternal **in**terfaces play an important part **in** these technologies, numerical modell**in**g is **of**ten focused on such systems. Due to this technological significance, alum**in**a and metal/alum**in**a **in**terfaces were frequently **in**vestigated experimentally 1–3 and by **ab** **in**itio methods. 4–7 Firstpr**in**ciples methods are well est**ab**lished for calculat**in**g the work **of** separation or the surface energy. However, the **in**vestigation **of** dynamic processes such as **crack** **propagation** requires systems **with** significantly more atoms and larger time scales than **ab** **in**itio **based** methods nowadays can deal **with**. Hence, it is an important challenge to develop a suit**ab**le **in**teraction force field for classical molecular dynamics (MD) simulations. There exist **in**teraction potential approaches for alum**in**a, which reach a good accuracy by us**in**g angular dependent terms. 8, 9 Pair potentials are also widely applied **in** simulations **of** alum**in**a 10–13 by reason **of** usually much lower coma) Author to whom correspondence should be addressed. Electronic mail: stephen.hocker@imwf.uni-stuttgart.de. putational demands. Due to the lattice misfit, especially the simulation **of** **in**terface structures has to be performed **with** many atoms. Hence, for the simulation **of** **crack** **propagation**, a pair potential is to be favored concern**in**g the computational effort. Many-body effects can be important for correctly describ**in**g bond angles and bond-bend**in**g vibration frequencies **in** oxides. However, for SiO 2 (Refs. 14–16) and MgO, 16 it was shown that an accurate description can be achieved **with**out angular dependent terms, but **with** polariz**ab**le oxygen atoms. There, the potential model **of** Tangney and Scandolo 14 (TS) is applied, **in** which the dipole moments **of** oxygen atoms are determ**in**ed self-consistently from the local electric field **of** surround**in**g charges and dipoles. In Ref. 16, the computational effort **in** simulations scales l**in**early **in** the number **of** particles, which is due to the Wolf summation. 17 We apply this comb**in**ation **of** the TS model and the Wolf summation method **in** the potential generation as well as **in** simulations. An effective atomic **in**teraction potential is generated **with** the program potfit. 18 The potential parameters are optimized by match**in**g the result**in**g forces, energies, and stresses **with** respective **ab** **in**itio values **with** the force match**in**g method. 19 **Simulation**s are performed **with** the MD code IMD. 20 A detailed description **of** the adopted methods and the potential generation is given **in** Sec. II. MD simulations and **ab**-**in**itio calculations which were performed **in** order to validate the obta**in**ed force field are presented **in** Sec. III. Asa first appplication **of** the new potential, we **in**vestigate **crack** **propagation** **of** alum**in**a s**in**gle crystals. To our knowledge, these are the first MD simulations **of** **crack** **propagation** **in** alum**in**a **with** a potential that takes **in**to account the polariz**ab**ility **of** oxygen atoms. Results **of** these simulations are shown **in** Sec. IV. F**in**ally, conclusions and an outlook are presented **in** Sec. V. 0021-9606/2012/136(8)/084707/9/$30.00 136, 084707-1 © 2012 American Institute **of** Physics 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

084707-2 Hocker et al. J. Chem. Phys. 136, 084707 (2012) II. FORCE FIELD GENERATION A. Tangney-Scandolo force field The TS force field is a sum **of** two contributions: a shortrange pair potential **of** Morse-Stretch (MS) form, and a longrange part, which describes the electrostatic **in**teractions between charges and **in**duced dipoles on the oxygen atoms. The MS **in**teraction between an atom **of** type i and an atom **of** type j has the form ( )] [ ( )]] Uij MS = D ij [exp [γ ij 1 − r ij γ ij rij 0 − 2exp 1 − r ij 2 rij 0 **with** r ij =|r ij |, r ij = r j − r i and the model parameters D ij , γ ij , and rij 0 , which have to be determ**in**ed dur**in**g the force field generation. Because the dipole moments depend on the local electric field **of** the surround**in**g charges and dipoles, a self-consistent iterative solution has to be found. In the TS approach, a dipole moment p n i at position r i **in** iteration step n consists **of** an **in**duced part due to an electric field E(r i ) and a short-range part p SR i due to the short-range **in**teractions between charges q i and q j . Follow**in**g Rowley et al., 21 this contribution is given by ∑ p SR q j r ij i = α i rij 3 f ij (r ij ) (2) **with** f ij (r ij ) = c ij j≠i (1) 4∑ (b ij r ij ) k e −b ij r ij . (3) k! k=0 b ij and c ij are parameters **of** the model. Together **with** the **in**duced part, one obta**in**s p n i = α i E ( r i ; { p n−1 j }j=1,N , {r ) j } j=1,N + p SR i , (4) where α i is the polariz**ab**ility **of** atom i and E(r i ) is the electric field at position r i , which is determ**in**ed by the dipole moments p j **in** the previous iteration step. Tak**in**g **in**to account the **in**teractions between charges U qq , between dipole moments U pp and between a charge and a dipole U pq ,thetotal electrostatic contribution is given by and the total **in**teraction is B. Wolf summation U EL = U qq + U pq + U pp , (5) U tot = U MS + U EL . (6) The electrostatic energies **of** a condensed system are described by functions **with** r −n dependence, n ∈ {1,2,3}.For po**in**t charges (r −1 ), it is common to apply the Ewald method, where the total Coulomb energy **of** a set **of** N ions, E qq = 1 2 N∑ i=1 N∑ j=1 j≠i q i q j r ij , (7) is decomposed **in**to two terms Er qq and E qq k by **in**sert**in**g a unity **of** the form 1 = erfc(κr) + erf(κr) **with** the error function erf(κr) := √ 2 ∫ κr dt e −t 2 . (8) π The splitt**in**g parameter κ controls the distribution **of** energy between the two terms. The short-ranged erfc term is summed up directly, while the smooth erf term is Fourier transformed and evaluated **in** reciprocal space. This restricts the technique to periodic systems. However, the ma**in** disadvantage is the scal**in**g **of** the computational effort **with** the number **of** particles **in** the simulation box, which **in**creases as O(N 3/2 ), 22 even for the optimal choice **of** κ. Wolf et al. 17 designed a method **with** l**in**ear scal**in**g properties O(N) for Coulomb **in**teractions. By tak**in**g **in**to account the physical properties **of** the system, the reciprocal-space term E qq k is disregarded. It can be written as E qq k = 2π V ∑ k≠0 |k|

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