Generalized Quasicontinuum Approach to Atomistic-Continuum Modeling of Complex Oxides
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Generalized Quasicontinuum Approach to Atomistic-Continuum Modeling of Complex Oxides Anter El-Azab and Harold Trease Fundamental Science Directorate, Pacific Northwest National Laboratory, Mail Stop: K8-93, Box 999, Richland, WA 99352, USA. ABSTRACT
A formalism of the quasicontinuum method suitable for atomistic-continuum modeling of oxide crystals is presented. Multiple interacting quasicontinua, one per sublattice, which overlap in the physical crystal space, are used to model the oxide crystals. The method is implemented with the shell model for atomic interactions in ionic crystals, along with the Wolf’s method for treating the long-range forces. Results are presented for the structural relaxation of strained and unstrained Fe2 O3 crystal under periodic boundary conditions. INTRODUCTION
The quasicontinuum (QC) method offers a robust approach to the problem of elastic and inelastic response of large atomic systems under various loading conditions. The method has been applied successfully to crystalline materials [1-3]. In this method, the atomic structure problem is cast in the form of an optimization problem for the energy functional of the atomic system with respect to the coordinates of a reduced set of atoms (the representative atoms), which are defined by the vertices of a Delaunay triangulation of the underlying crystal, refined down to the atomic scale around surfaces and defects. The method offers a coarse graining methodology of large atomic system, and is thus an efficient alternative of the lattice statics (LS) method. The latest formalism of the QC method by Knap and Ortiz [4] is particularly interesting. Mathematically, this formalism is stated as follows: let x be the set of atomic coordinates of a finite crystal consisting of N atoms, and assume that the total energy functional of the crystal Ψ (x) is given by the sum of two parts, the interaction energy of the atoms, Ψ◦ (x), and Ψext (x), which is associated with a system of external forces. The equilibrium structure of the crystal is found by minimizing Ψ (x) with respect to a subset of xh of the atomic coordinates x. This can be achieved by requiring that the coordinates of all atoms be found in terms of xh, that is, x = x(xh ). The minimization problem is then stated as follows: min Ψ (x(xh )) , xh
xh ⊆ x.
(1)
The solution to this problem, xh, renders null the force vector Fh given by Fh (xh ) =
X
∇x Ψ (x) · ∇xh x (xh ) =
X
F(x) ϕ(x|xh ).
(2)
In the above, the reduced coordinate set xh includes the coordinates of the representative atoms, Fh (xh ) are coarse-grained forces at these sites, F (x) are the forces on lattice atoms, and ϕ(x|xh ) are the shape functions associated with the crystal triangulation. The implementation of the method relies on one further step, approximating the lattice summation
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