CASTEP
castep (CAmbridge Serial Total Energy Package) is a first principles code used to cal- culate the properties of materials. It implements DFT and can model a wide range of material properties including total energies, structure, electronic and magnetic proper- ties. castep uses a plane wave basis set to expand the Kohn-Sham wavefunction [206]. VASP
The Vienna Ab initio Simulation Package, also known as vasp, is a package for perform- ing first principles calculations using the projector augmented wave method with a plane wave basis set [207]. It implements DFT, but also contains many post-DFT corrections such as hybrid functionals and many-body perturbation theory (GW).
LAMMPS
The Large-scale Atomic/Molecular Massively Parallel Simulator (lammps) code [208] is a classical molecular dynamics code that can model a range of systems using classical interatomic potential methods and different boundary conditions. In this thesis the code was mainly used to model metals using the EAM formalism. lammps requires that interatomic potentials are supplied in a tabulated format so a simple code written in c++ which tabulates each potential from its parameters was developed. Many pre- existing tabulated potentials can be found on the National Institute for Standards and Technology (NIST) website [209].
VESTA
Visualisation for Electronic STructure Analysis (vesta) is a powerful tool used to visu- alise crystals and molecules [210]. vesta has the ability to visualise most file formats including both castep (.cell) and vasp (POSCAR). However for lammps a new code was written to convert from the .lammpstrj file to a vesta appropriate input. vesta has the ability to show iso-surfaces, draw bonds and offers a significant amount of cus- tomisation.
VMD
Visual Molecular Dynamics (vmd) is a useful tool to visualise and analyse crystallo- graphic systems. It may be used to view more general molecules, such as lammps trajectory formats and display the structure. vmd provides a wide variety of methods for rendering and colouring a molecule: simple points and lines, CPK spheres and cylin- ders, liquorice bonds, backbone tubes and ribbons, cartoon drawings, and others. vmd can be used to animate and analyse the trajectory of a molecular dynamics (MD) simula- tion. In particular, vmd can act as a graphical front end for an external MD program by displaying and animating a molecule undergoing simulation on a remote computer [211]. GULP
General Utility Lattice Program (gulp) is a useful code which allows classical simulations of atoms. It has many useful features such as the ability to automatically calculate the elastic constants [136]. A variety of force fields can be used within gulp spanning the shell model for ionic materials, molecular mechanics for organic systems and the embedded atom model for metals.
Bader analysis
It is possible using techniques developed by Richard Bader to divide the electronic density from clusters of atoms such as those in GBs into individual atom centred charge densities. Bader's method defines an atom based on the total electronic charge density and uses a zero flux surface to divide the atoms. A zero flux surface is a 2-D surface on which the charge density is a minimum perpendicular to the surface. Dr Henkelman's group in Texas have developed a program to perform Bader analysis from the output files of vasp calculations. This offers a convenient way to analyse the results from vasp to show the charges present on atoms in ionic structures [212].
Chapter 4
Excess volume due to grain
boundaries in metals
4.1
Introduction
GBs in metals play an extremely important role in determining their properties and functionality. As discussed in Sec. 2.1.3 the excess volume at GBs influences phenom- ena such as segregation, diffusion and embrittlement in metals. However until recent experiments by Steyskal and Oberdorfer it has been challenging to probe the excess volume in real materials directly [42, 43]. In these investigations a difference of 0.14 ˚A between the average excess volumes of Cu and Ni has been detected. The difference in the lattice constants is insufficient to explain the differences in the excess volume and an atomistic explanation of this observation is currently missing [88]. Understanding the factors which influence the excess volume could have many important benefits not least for materials design relevant to many technological applications such as fusion reactors. In this chapter a detailed theoretical investigation into GB excess volume in the poly- crystalline metals Fe, Cu and Ni is presented. These materials are chosen to allow com- parison with previous theoretical and experimental studies, and due to their numerous applications in areas such as spintronics, fusion, fission, power generation and cataly- sis [213–216]. The focus is on symmetrical tilt GBs over a wide range GB orientations to draw out trends across the three materials. By employing an automated computational approach based on an EAM description of interatomic interactions [85, 217] the stable structures of over 400 distinct symmetrical tilt GBs for Fe, Cu and Ni are determined. The validity of the approach is demonstrated by comparison to first principles calcula-
tions of GB properties using DFT. The results recover a systematic difference in excess volume of between 0.1 and 0.2 ˚A between Cu and Ni which is in very good agreement with experimental data. By analysing the strain at the atomic level it is demonstrated that the excess volume difference is localised in a region of 5 - 10 ˚A around the GB plane. A semi-quantitative explanation for the origin of the difference in terms of the differing bulk moduli of Cu and Ni (138 GPa and 186 GPa respectively) is provided. While Cu and Ni are fcc, Fe is bcc and therefore GBs have different geometric structures. The localisation of GB cusps1 in Fe is also different to Cu and Ni. The range of excess volumes in Fe is comparable to Ni.
The rest of this chapter is structured in the following way. In Secs. 4.2.1 & 4.2.2 the approach is validated by looking at several examples of GBs in Fe, Cu and Ni. From Sec. 4.2.3 the results are presented. In Sec. 4.3 the results are discussed and in Sec. 4.4 the chapter is concluded.