Monte Carlo Methodology for Grand Canonical Simulations of Vacancies at Crystalline Defects

The design of new materials and the optimization of the existing ones require more and more knowledge of the elementary processes underlying the macroscopic properties. Computer simulations have become, together with ever finer experimental technics, the modern tools for probing these mechanisms. This paper focuses on the development of Monte Carlo simulations, at the atomic scale, of vacancies in crystals. These defects have been extensively studied, in their isolated state, because they are the vectors of diffusion in solids. Their concentration and dynamics determine the kinetics of most phase transformations and thermal annealings which enter the processes of the production of materials, for example metallic alloys, or surface deposits for microelectronic applications. They can also contribute to the loss of mechanical properties. For example, in irradiated steels, the clustering of vacancies induce the formation of loops which harden the matrix and, at the same time, their diffusion to the grain boundaries lead to all sorts of segregations that sometimes reduce their cohesion. The combined effect of a hard matrix and weak interfaces can lead to the premature formation of cracks. It is therefore not only important to model vacancies in a hole range of temperatures and concentrations in perfect crystals, but also at pre-existing defects, like grain boundaries and dislocations. The methodology presented is inherited from statistical mechanics. Molecular Dynamics (MD) (Allen & Tildesley (1991)) is the method of choice if the details of the trajectories of the particles in the system are needed. The amount of physical time that can be simulated (of the order of the nano second) is often too limited to give enough statistics to measure the property of interest. Kinetic Monte Carlo (Landau & Binder (2000); Soisson et al. (1996); Dai et al. (2005)) is event based. It eliminates all the details of the trajectory and keeps only the jumps from one local minium of the energy to another one. In its simple form, a limited list of the most important events is provided at the beginning of the simulation, together with the list of rates and the particles are constrained to be on a rigid lattice. It can be refined to build the list on the fly (Henkelmann & Jonsson (2001)) or to get the events from MD (Sorensen & Voter (2000)). A last class of methods is the one where MD is accelerated (Voter et al. (2001)), for example, by the use of a bias in the interactions (Wang et al. (2001)) which does not modify the saddles in between the local energy minima, but reduces the waiting time in the basins. Each method has its limitations: KMC, on a rigid lattice, can not treat realistic diffusion mechanisms with collective movements and relaxations (for example, if the system has different components 28