Bond order potentials for fracture, wear, and plasticity
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Introduction A quantitative microscopic understanding of the behavior of real materials under realistic conditions still presents a great challenge for atomic-scale modeling. Especially challenging are simulations of processes involving plastic deformation, fracture, wear, thin-film growth, or ion-beam etching, since they require an accurate description of strongly distorted atomic configurations with rapidly changing bonding conditions in large atomic ensembles containing thousands or even millions of atoms. Interatomic potentials suitable for description of such complex processes therefore must be both computationally efficient and reliable in the description of interatomic interactions. One of the most successful families of interatomic potentials that has been able to stand the demanding requirement of simultaneous efficiency and reliability is the family of models based on the quantum-mechanical concept of bond order, introduced by Coulson1 long before the advent of atomistic simulations. In the intuitive language of chemical molecular orbital theory,
the bond order simply characterizes the strength of a chemical bond, since it corresponds to one-half of the difference between the number of electrons in the bonding and anti-bonding states. In the elementary H2 molecule, the bond order is equal to one since the bonding state is fully occupied by two electrons with opposite spins, while the anti-bonding state remains empty. In more complicated molecules or in solids, the bonds are usually not fully saturated due to the influence of the atomic environment, and, as a result, their bond orders are less than unified. A more rigorous, quantitative definition of the bond order requires an introduction of some basic quantum mechanical terminology. The usual way to solve the one-particle Schrödinger equation
Hˆ n = ε(n ) n ,
(1)
where Ĥ is the Hamiltonian operator, |n〉 are the single-particle eigenfunctions, and ε(n) the corresponding eigenvalues, is to
Lars Pastewka, Fraunhofer Institute for Mechanics of Materials IWM and Institute for Applied Materials IAM at Karlsruhe Institute of Technology, Germany; [email protected] Matous Mrovec, Fraunhofer Institute for Mechanics of Materials IWM and Institute for Applied Materials IAM at Karlsruhe Institute of Technology, Germany; [email protected] Michael Moseler, Fraunhofer Institute for Mechanics of Materials IWM and Physics Department of the University of Freiburg, Germany; [email protected] Peter Gumbsch, Fraunhofer Institute for Mechanics of Materials IWM and Institute for Applied Materials IAM at Karlsruhe Institute of Technology, Germany; [email protected] DOI: 10.1557/mrs.2012.94
© 2012 Materials Research Society
MRS BULLETIN • VOLUME 37 • MAY 2012 • www.mrs.org/bulletin
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BOND ORDER POTENTIALS FOR FRACTURE, WEAR, AND PLASTICITY
convert it into a matrix eigenvalue problem. Following the chemically intuitive tight-binding (TB) model,2,3 the eigenfunctions are expanded into a minimal set of localized atomic-like orbitals
n ² = ¦
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