Langmuir 1996, 12, 4605-4609
4605
Molecular Dynamics Simulations of Elastic Response and Tensile Failure of Alumina† F. H. Streitz‡ and J. W. Mintmire* U.S. Naval Research Laboratory, Code 6179, Washington, D.C. 20375-5342 Received October 17, 1995. In Final Form: June 24, 1996X We present the results of molecular dynamics simulations of R-alumina performed using the ES+ potential. We concentrate here on the applicability of these potentials to the study of surface structure and thin film elastic response. The aluminum-terminated (0001) surface of R-alumina is found to undergo dramatic relaxation away from bulk values, driven by the electrostatics of atoms near the surface. We find that the local valence is reduced by 25% for surface and near-surface atoms in this orientation. We demonstrate that the strain variation of the elastic constants in a 25 Å thick film of R-alumina can differ substantially from the expected behavior in a bulk crystal. The thin film was found to yield at a stress of approximately 45 GPa, in agreement with a calculation of the theoretical limit of yield stress.
Introduction Adhesion, friction, and wear have been ubiquitous concerns in the history of technology, with serious scientific study of these topics occurring over the past 3 centuries. These three processes, though not equivalent, are intimately related and are central to the basic mechanical and materials processes of a vast range of technological problems. The mechanical basis of all of these derive from the interaction of two surfaces at an interface, and a fundamental understanding of the dynamics and energetics of the interface is an important component of our understanding of these processes. Progress in understanding the interfacial interactions has historically been limited by the inability to examine what is going on at the atomic scale. As pointed out by several recent reviews,1-3 experimental techniques have been developed in the last few decades that allow resolutions at or near atomic-scale dimensions, with the most common methods being the surface force apparatus (SFA),4 the scanning-tunneling microscope (STM),5 and other proximal probe techniques related to the STM, the atomic-force6 and friction-force7 microscopes (AFM and FFM). Theoretical techniques have also evolved to treat problems in this new area of atomic-scale tribology, “nanotribology,”2,3,8 in this case by extending earlier simulation techniques to larger domains both in time and spatial extent so that realistic simulations * To whom correspondence should be addressed: phone, (202) 767-2026; fax, (202) 767-3321; e-mail,
[email protected]. † Presented at the Workshop on Physical and Chemical Mechanisms in Tribology, held at Bar Harbor, ME, August 27 to September 1, 1995. ‡ Current address: Department of Physics, Auburn University, Auburn, AL 36849-5311. X Abstract published in Advance ACS Abstracts, September 1, 1996. (1) Singer, I. L.; Pollock, H. M. Fundamentals of Friction; Kluwer: Dordrecht, 1992. (2) Bhushan, B.; Israelachvili, J. N.; Landman, U. Nature 1995, 374, 607. (3) Singer, I. J. Vac. Sci. Technol., A 1994, 12, 2605. (4) Tabor, D.; Winterton, R. H. S. Proc. R. Soc. London, Ser. A 1969, 312, 435. Israelachvili, J. N.; Tabor, D. Nature 1973, 241, 148; Wear 1973, 24, 386. (5) Binnig, G.; Rohrer, H.; Gerber, Ch.; Weibel, E. Phys. Rev. Lett. 1982, 49, 57; Phys. Rev. Lett. 1983, 50, 120. (6) Binnig, G.; Quate, C. F.; Gerber, Ch. Phys. Rev. Lett. 1986, 56, 930. (7) Mate, C. M.; McClelland, G. M.; Erlandsson, R.; Chiang, S. Phys. Rev. Lett. 1987, 59, 1942; Meyer, G.; Amer, N. M. Appl. Phys. Lett. 1990, 56, 2100. (8) Yoshizawa, H.; Chen, Y.-L.; Israelachvili, J. J. Phys. Chem. 1993, 97, 4128.
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can be made of the interfacial interactions contributing to adhesion, friction, and wear. At the atomic scale, adhesion and friction are essentially different macroscopic measurements of the same physical interactions at the nanoscale. In general, friction measurements are set up as a measure of a dynamic response of the system: what is the energy loss over time for a moving interface under a load? Conversely, adhesion measurements are commonly viewed as a measurement of a static structural property: what is the work required to break the interface? Nevertheless, these two processes have been shown to be closely related. Yoshizawa et al.9 have demonstrated that the coefficient of friction is correlated not directly to the work of adhesion at an interface but with the adhesion hysteresis. Similarly, they further discuss that the two major causes of wear in sliding are believed to be the breaking of adhesive junctions and plastic deformation of surface asperities. Thus, atomic-scale molecular dynamics (MD) simulations of adhesion at interfaces can be expected to describe the properties of friction at an interface. In addition to sliding friction, which has been treated with MD methods,8 rolling friction should be a prime candidate for study using MD methods. As a concrete example, let us consider a typical macroscopic example: a quarter-inch diameter bearing rolling at 3600 rpm with a contact diameter of 10 µm. The time that a single point on the bearing surface will be in contact with the opposing contact surface will be on the order of 1-10 µs. Molecular dynamics simulations (and the atomic-scale dynamics described by the simulations) typically consider time periods ranging from picoseconds to nanoseconds. The contact time of a bearing with its opposing surface is thus practically infinite as far as the atomic-scale simulations are concerned. If we want to perform a simulation that tells us something useful about the adhesion hysteresis, we will thus want to study the atomic-scale adhesive interactions at the interface and the surrounding interphase region through the contact process and until pull-off. One primary objective we have had in our research effort has been to simulate the atomic-scale dynamics and energetics of technologically important metal/metal oxide interfaces. For an accurate treatment of the energy balance we must describe well both the interface and the structural response of the two surface films, typically denoted as the interphase region. Because of the strong contribution of electrostatic effects, we need an empirical (9) Harrison, J. A.; White, C. T.; Colton, R. J.; Brenner, D. W. Surf. Sci. Lett. 1992, 271, 57; Phys. Rev. B 1992, 46, 9700.
© 1996 American Chemical Society
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method that allows the local cation valence to vary according to the local environment and which includes the Coulombic electrostatic interaction among the anions and cations. An extensive literature of modeling local atomic charge for organic and biological systems based on long-range electrostatic interactions and local atomic electronegativity-based properties has developed over the last several decades. We have investigated the use of such electronegativity-based models for a direct calculation of charge transfer in metal oxide systems and incorporated the resulting electrostatic potential into an overall model potential for the titanium dioxide (rutile) and R-alumina systems.10 We present below a brief outline of our approach for modeling the electrostatic component of the potential energy of ionic systems and how standard empirical potential techniques can be effectively merged with this approach. In other work we have presented atomic-scale simulations of the rupture under tensile stress of an interface between an R-alumina (0001) face and an aluminum(111) face.11 We describe molecular dynamics simulations of the elastic response and yield behavior of R-alumina systems and compare with experimental results where available. Our results indicate that this approach can provide physically realistic empirical potentials for future dynamics simulations of adhesive and tribological importance. Approach One of the dominant interactions in the metal oxides is the Coulombic interaction between anions and cations. Earlier models using empirical potentials included such effects by incorporating fixed atomic charges and polarizability functions into the energetics of metal oxides. Because our ultimate goal in this work is to develop relatively simple empirical potentials for studying the adhesion of metal oxides with metal substrates, any new approach must include an ability to calculate the local atomic charge (or equivalently, the valence) based on the local environment of each atom. The long-range nature of the Coulomb interaction will lead to this “local” environment typically being relatively large-scale, of the dimensions of the screening length in the metal oxides. Conceptually then, what is needed is a description of the total electrostatic energy of an array of atoms as a function of atomic charges (valences) and position. We define an atomic energy term Ei in terms of the local charge qi on atom i.
1 Ei(qi) ) Ei(0) + χi0qi + Ji0qi2 2
(1)
where χi0 and J0 correspond to local atomic properties traditionally denoted as the electronegativity12,13 and hardness.14 The electrostatic energy, Ees, of a set of interacting atoms with total atomic charges qi is then given by the sum of the atomic energies Ei, and the electrostatic interaction energies between all pairs of atoms,
Ees )
∑i
Ei(qi) +
1 2
∑ i*j
Vij(Rij;qi,qj)
(2)
(10) Streitz, F. H.; Mintmire, J. W. J. Adhes. Sci. Technol. 1994, 8, 853; Thin Solid Films 1994, 253, 179; Phys. Rev. B 1994, 50, 11996. (11) Streitz, F. H.; Mintmire, J. W. Compos. Interfaces 1994, 2, 473. (12) Iczkowsky, R. P.; Margrave, J. L. J. Am. Chem. Soc. 1961, 83, 3547. (13) Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J. J. Chem. Phys. 1978, 68, 3801. (14) Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512.
where Rij denotes the vector Ri - Rj. The Coulomb pair interaction Vij(Rij;qi,qj) is given by the classic electrostatic interaction between two charge distributions Fi(r1;qi) and Fj(r2; qj), where Fi(r;qi) is the charge distribution about atom i (including the nuclear point charge) for total charge qi and r12 is the distance between two atoms. The simplest model for Fi(r;qi) is as a point charge of charge qi; this leads to Vij(Rij;qi,qj) ) qiqj/Rij. Rappe and Goddard15 have suggested the use of spherically symmetric exponential functions to generate a Fi(r;qi) linear in qi. For the work herein, we assume an atomic charge density distribution of the form
Fi(r;qi) ) Ziδ(r - Ri) + (qi - Zi)fi(ri)
(3)
where fi is a spherically symmetric function in ri ) |r Ri|. The effective core charge, Zi, should satisfy the condition 0 < Zi < Zi, where Zi is the total nuclear charge of the atom. For simplicity, we model the atomic densities as single Slater 1s orbitals of the form fi(r) ) (ζi3π) exp(-2ζi|r - Ri|). We then choose the values of qi as those that minimize Ees subject to the constraint that the sum of the qi be constant. The electrostatic energy is just one component of the total energy of a metal oxide (or metal) system. Indeed, the electrostatic interaction between cations and anions will be strictly attractive at any internuclear separation, so that in the short-range limit a repulsive potential will be needed to maintain physically reasonable internuclear separations in any empirical potential constructed using the above described electrostatic potential. What is needed for a complete potential is a description of the nonCoulombic part of the interatomic interactions. This remaining interaction could be modeled using any standard empirical potential, such as a sum of pair potentials, an embedded atom method (EAM) approach,16-19 or a many-body potential such as that developed by Abell and others for covalent systems.22-26 We have chosen to merge our above described electrostatic model with an EAM potential.10 We chose an EAM scheme in large part because this approach is a commonly accepted and used technique for metallic bonding, such as in the fcc aluminum. That is, we assume that the total energy is given as the sum of the electrostatic energy Ees defined above and an EAM potential energy of the form
EEAM )
φij(Rij) ∑i Fi[Fi] + ∑ i
