Boundary conditions for molecular dynamics simulations in crystalline solids: from zero temperature to finite temperature

Xiantao Li
Department of Mathematics
Penn State University

Molecular dynamics (MD), which uses Newton's equation to describe the motion of atoms/molecules, has been an important simulation tool to study crystalline structure and defect dynamics from the atomic scale. The typical time scale of such simulation is at femtoseconds (10^{-15} second), and the inter-atomic spacing is usually at the scale of angstroms (10^{-10} meter).

Because of the small time/length scale, MD simulations have to be conducted on a small system truncated from a much larger sample to make the overall computation tractable. As a result, artificial boundaries are introduced, where boundary conditions have to be imposed. The role of the boundary condition is to take in account the atoms that have been removed from the system and therefore reduce the finite size effect. In particular, the purpose of the imposed boundary conditions is to,

a. suppress the wave reflection at the boundary,
b. maintain the external loading,
c. bring the system to thermal equilibrium.

I will first present a systematic approach for finding boundary conditions at zero temperature. This approach is based on a variational formulation that aims to minimize the total reflection. Local boundary conditions, which are efficient for practical applications, are obtained from this formulation.

I will then discuss how to extend these boundary conditions to finite temperature to simulate a system surround by an infinite heat reservoir at thermal equilibrium. The extension relies on a Generalized Langevin equation (GLE) formulation. The boundary conditions for the zero temperature case will serve as the memory kernel in the GLE, and provide the time correlation of the thermal noise to respect the second fluctuation-dissipation theorem.

Fracture simulations in Iron-alpha and Aluminum will be presented as applications of these boundary conditions.