## 2 Diffusion Mechanisms  
Diffusion in materials is a fundamental and complicated phenomenon. The added complexity in oxide materials, as compared to metals, is the anion and cation sublattices. These sublattices can restrict the ionic diffusion to its own sublattice, however, the cation sublattice may have a significant impact upon the anion diffusion. In oxides oxygen self-diffusion is typically orders of magnitude faster than cation self-diffusion [79; 80; 81; 82].  
For net diffusion to occur in a crystalline material atoms have to migrate away from their equilibrium positions towards neighbouring equilibrium positions. For this process to occur point defects (interstitials and/or vacancies) are necessary [34]. For the materials considered is the present review there are mainly three mechanisms: the interstitial, the interstitialcy and the vacancy mechanisms. There are also mechanisms, which are based upon the point defects but also involve cooperative motion of the local polyhedral structures.  
Considering first the vacancy mechanism of diffusion, in which an ion diffuses by migrating to a neighbouring vacancy [81; 83]. The vacancy mechanism necessitates the presence of lattice vacancies, with their concentration in the lattice and migration energy barriers effectively controlling the transport kinetics. The vacancy mechanism is common for oxygen self-diffusion in a number of oxygen hypostoichiometric oxides that are of interest to SOFC (for example fluorite- and perovskite-related systems) [84].  
In the interstitial mechanism, interstitial ions migrate from one interstitial site to a neighbouring interstitial site. Conversely, to the vacancy mechanism in the interstitial mechanism when a jump between interstitial sites is completed there is no net displacement of the other ions. Additionally, the interstitial mechanism does only require the interstitial ions and no other point defects. Finally, the interstitialcy mechanism is distinct from the interstitial mechanism, as the interstitial ion displaces another ion from its equilibrium lattice site. Following this the displaced ion moves to another interstitial site and so on.  
In the vacancy mechanism, the migration enthalpies for the thermally activated vacancy hopping is the critical factor determining how low a temperature the electrolyte and cathode can effectively function at. According to transition state theory, the rate of vacancy hops, \(\
u\), is given by the Boltzmann relationship,  
\[v=v_{0}\exp\left(\frac{-E_{M}}{k_{B}T}\right), \tag{2}\]  
where \(E_{M}\) is the migration barrier and \(\
u_{0}\) is a constant.  
The vacancy diffusivity, \(D_{\
u}\), is related to the mean square displacement \(\left\langle R^{2}\right\rangle\) of the vacancies via the [PERSON] relation,  
\[\left\langle R^{2}\right\rangle=6D_{\
u}t. \tag{3}\]  
It should be noted that It should be noted that \(D_{\
u}\) exponentially depends on the energy barrier for oxygen vacancy migration. The oxygen diffusivity, \(D_{O}\), is connected to \(D_{\
u}\) via the following relation,  
\[D_{O}=\frac{c_{\
u}}{1-c_{\
u}}D_{\
u}, \tag{4}\]  
where \(c_{\
u}\) is the vacancy concentration fraction.  
Figure 1: A schematic of the cubic perovskite structure ABO\({}_{3}\).