Simulation Set-up

During the simulation, the forces onto each atom are evaluated once every femtosecond. This is sufficiently often for atomistic simulations, where typical time scales of crystal vibrations are given by the Debye frequencyνD≈ 10¹³Hz. Therefore, the forces are evaluated ˜100 times during one oscillation cycle due to lattice vibrations. The simulation frequency is equally much faster than typical frequencies for diffusive processes.

After calculating the forces that act onto each atom at a time step t , the atoms are moved during the time interval∆t according to~r^{i +1}=~rⁱ+~vⁱ∆t +¹₂(∆t)²~fⁱ/m. When the time moves forward t^{i +1}= tⁱ+ ∆t , the calculation process is repeated from the beginning by calculating the forces.

Chapter 4. Microscopic Mechanisms Observed in Molecular Dynamics Simulations

The simulations are performed by imposing a finite temperature to the system. Since the atom velocities determine the temperature of the particle ensemble through their kinetic energy, the speed of each particle has to be adapted in order to reach the imposed temperature. At each calculation step, a thermostat removes or adds energy to the system. Using the Berendsen thermostat [Bere84], the temperature of the system is corrected such that the deviation decays exponentially with a time constantτB: ˙T = (T0− T )/τB. The updated positions and velocities are consistent with the microcanonical ensemble, since the atom number N is constant as well as the simulation volume V and the energy is controlled by the thermostat. The Berendsen thermostat is efficient for the calculation of large systems and yields thermodynamically correct results for systems with more than 1000 atoms.

Figure 4.1 – Snapshot during a molecular dynamics simulation. Two different crystal orienta-tions are separated by a GB (horizontal plane).

The simulations have been performed using the software package Lammps [Plim95b,Plim95a], which stands for Large Scale Atomic/Molecular Massively Parallel Simulator. It is an open-source software package written in C++, that can be parallelized easily with the Message passing Interface (MPI) on multiple processors. Further acceleration of the simulation can be achieved by calculating on a 48-cores Nvidia graphics card via the CUDA platform, which works together with Lammps.

The parallelization of the MD simulation using the graphics card is very good, since the crystal can be divided into geometrical chunks, for example horizontal slices, which can be treated on parallel cores. Since the heaviest calculation is the one of the forces and the update of particle positions, the main calculation is done between neighbouring atoms. Thus, most calculations can be performed inside one chunk and the information exchange between chunks is rather small. The speed-up from parallelization is nearly linear for a pure simulation of the atom positions, which means that doubling the number of processors approximately halves the 92

4.1. Molecular Dynamics Simulations

Figure 4.2 – Geometry of a typical MD simulation. The sample is composed of two blocks with different crystal orientations. An upper and a lower slice of 5 Å are hold fixed. After a thermalization step the upper atoms are moved in positive y direction with a velocity of 10 m/s.

runtime. Generating output files like the snapshot in Figure 4.1 costs comparably a lot of computing resources, since the whole simulation data has to be transferred to the CPU, which destroys the advantages from parallelization.

Lammps works with an input file, in which a first part sets up the simulation and a second part contains commands how to run the simulation and what to write as output files. The crystal structure can be created directly via the input file by defining a certain volume in space and by filling it up with atoms in form of an fcc lattice of a defined orientation. Defects like dislocations can be introduced into a perfect crystal by performing displacement operations to each atom dependent on their position from the dislocation line. GBs can be created from two adjacent volumes filled up with crystal lattices of different orientations.

Care must be taken by bringing both lattices into contact. A GB in a bi-crystal has eight degrees of freedom in total: five are macroscopic degrees of freedom (DoF) and can be chosen as three angles to bring the two crystal lattices into superposition and two DoF to define the GB plane normal vector. The remaining three DoF are microscopic that define the relative translation parallel and perpendicular to the GB plane of one crystal part with respect to the other. If the microscopic translations are chosen in a wrong way, the bi-crystal does not relax into the lowest energy state and the simulations cannot be compared with observations in real materials.

To overcome the difficulty of finding the experimentally realized GB structure, a series of simulation with different translations should be performed calculating the energy of the

Chapter 4. Microscopic Mechanisms Observed in Molecular Dynamics Simulations

final GB structure [Alex13], or the relaxed GB structure should be compared to microscopic observations of GBs. The equilibrium GB structure of aΣ5 symmetric tilt boundary has been observed by MD simulations [Trau12,Chen14] and by TEM imaging [Merk87,Wang11]. The main part of the simulations in this chapter is done with aΣ5 boundary so that we can be sure to start with a boundary observed in fcc polycrystals.

All simulated geometries have free surfaces in x, y and z direction, which differs from most MD simulations on bi-crystals, where periodic boundary conditions are applied in the x y-plane to simulate an infinite boundary. We chose another approach with a finite sample, since this geometry does not restrict the simulation to coincidence-lattice-site (CLS)-boundaries. Also, the boundary box does not need an adjustment upon a temperature change to avoid internal stresses due to the periodic boundary condition.

Due to the free surfaces, the outer sample regions will differ from the bulk, but artefacts due to free surfaces are easily identified, since their effect will be limited to a surface region. The analysis of the sample was limited to the inner part, 10-20 Å away from any surface.

Typical sample sizes are 100 x 100 x 60 Å with 20 000-40 000 atoms. A schematic drawing of a bi-crystal is shown in Figure 4.2. In order to apply a shear stress onto the GB, a surface layer of 5 Å thickness at the bottom is fixed and a constant velocity along y is imposed on the atoms situated in a top layer. The simulated times were typically between 0.1 ns and 1 ns, but sporadic long-time simulations up to 5 ns showed the same effects as measured with the shorter simulations.

After creating the geometry and introducing defects like dislocations, all atoms are thermalized using the Berendsen thermostat. The temperature is monitored until an equilibrium is reached.

If there are parts of the geometry, where the atoms are fixed like the upper and the lower slice of Figure 4.2, these atoms are excluded in a second thermalization step and their position is fixed. These atoms are not fixed in the first place, because their lattice parameter would not be the same as for the free atoms. After reaching thermal equilibrium for a second time, the top layer is moved with a constant velocity of 10 m s⁻¹to apply a stress to the GB.