# The Evolution of Solid Oxide Fuel Cell Materials
## 2 Methodology
### Atomistic Simulation  
The most common ways to investigate the defect processes on SOFC materials at the atomistic level is via classical MD and DFT, as has been shown in previous theoretical studies [63; 64; 65; 66; 67; 68; 69; 70]. Classical MD has been a popular way to investigate the atomic scale properties of inorganic materials for over 40 years. The present computational resources allow the MD simulations of many millions of atoms and therefore research can include extended defects such as grain boundaries. This is important as polycrystalline material processes can be dominated or severely impacted at the extended defects. Classical MD is a fairly simple method where the state of the system is predicted by considering the positions and the momenta of the particles using [PERSON]'s equations of motion, which are solved iteratively in order to calculate the evolution of the system. Interparticle interactions are described using the potential energy function. For SOFC, the ionic systems are modelled within the classical Born-like description [71; 72]. Typically, the ionic interactions can be modelled by a Coulombic term (long-range) and by a short-range parameterized pair potential [73; 74]. Although MD calculations in ionic systems can work very well and can be a useful tool when considering dynamical properties such as diffusion, they cannot calculate the electronic structure of the system.  
When considering electronic properties, DFT calculations are advantageous over MD. It is acknowledged that the quantum mechanical description is the most complete depiction of nature. What is also acknowledged is that the analytic solution of the Schrodingerequation for practical systems (i.e., a substantial number of electrons and nuclei) is intractable due to the complexity of many-electron interactions [75; 76; 77]. The DFT method was discovered to address these issues via the description of the exchange-correlation energy of electrons through approximations (typically the local density approximation (LDA), the generalized gradient approximation (GGA) or with hybrid functionals) [75; 76]. A key aspect of DFT is to use a plane-wave basis set with the pseudopotential method, where the core electrons can be modelled by effective potentials and the valence electrons can evolve explicitly [75; 76; 77]. Although DFT is at zero kelvin and diffusion typically occurs at high temperatures, there are ways to gain information for the energetics of diffusion using this approach. For example, formation energies (the energy to form a point defect) of ionic systems can be easily modelled, whereas for the activation energy of migration (i.e., the energy barrier that the ion has to overcome to move from one minimum energy site to the next), methods such as the nudged elastic band method can be employed. An issue with the DFT study of ionic materials is the supercell size, as it needs to be extended enough so that periodic images have little effect upon the calculated energies. This is particularly important for extended and charged defects. Finally, it should be considered that the ab initio MD can overcome many of the problems of classical MD (for example, the inability to calculate electronic properties) and DFT (i.e., it is dynamic and can model high temperatures); however, it is a technique that is very computationally intensive, so it is typically limited to small number of atoms.