atom

micro

macro

A

B

C
MC

MC: Monte Carlo simulation
Definition:A simulation method for discrete atom movement based on jump probabilities derived from thermodynamics
Explanation:In general MC simulations, configurations of complex interacting entities are rated by a criterion defining their deviation from an ideal or final state. In discrete steps (a.k.a. MC-steps), the configuration is randomly altered (usually by a small amount) and reevaluated. During the evalutioan, the proabability of the configuration is calculated and the configuration change is determined accordingly. Then, the next MC-step with a new random change is tested repetitively in a virtual infinite loop, so that the system approaches its ideal state.

As an example of MC simulations in materials science, atomic configurations are tested in order to find an ideal spatial distribution for a mixture of chemical elements. The tested alterations are atom exchanges or vacancy/interstitial jumps. The optimisation criterion is the minimization of the free energy or enthalpy. In thermodynamic MC simulations, the probability for the configuration change is calculated as a Boltzmann term considering the energy change involved. In kinetic MC (→kMC) simulations , the configuration change is considered to be thermally activated, and accordingly the activation energy for the change is used to define the transition probability.

In order to get physically sound results, it is important to describe the energy change between the configurations accurately. In simple simulations, only nearest (or second nearest) neighbour interactions between atoms are taken into account, and discrete atom positions in a fixed crystal lattice are prescribed. However, such simple approaches do not account for long-range interactions like elasticity (see e.g. [Jannot et al.]) or ferromagnetism. For an elaborate simulation, such effects must be considered in addition to the local changes. An alternative are Monte-Carlo-Molicular Statics (→MC-MS) simulations, which use a fully consistent energetic description from molecular dynamics (→MD) potentials or, ideally, from ab initio calculations.
Picture /
Figure /
Diagram:

MC-simulated distribution of the pipe diffusion coefficient of Cu around a dissociated edge dislocation in Al at 300 K (left) and 900 K (right).
SFB-Link:In more specialized MC-MS approach, Monte Carlo simulations are tested for applicability in deriving grain boundary properties from MD potentials, with the perspective of using ab initio calculated data. As indicated by the example of pipe diffusion above, any kind of atomistic process can be considered.
References:A. D. Rollett, P. Manohar: The Monte-Carlo-Method. In: Continuum Scale Simulation of Engineering Materials (Eds.: D. Raabe, L.-Q. Chen, F. Barlat, F. Roters), Wiley-VCH 2004, 77-114.
E. Jannot, V. Mohles, B. Thijsse: Calculation of the elastic energy of a solid solution. Phil. Mag. 90 (2010), 963–975