**5. Migration enthalpy in the local environment**

Although several investigations have shown the formation of local defect structures in doped ceria systems, only a few models exist which describe the doping fraction at which they start appearing and directly influencing the conductivity. Also, in several experimental investigations, the observed trend in macroscopic migration enthalpy is attributed to the microscopic level association between dopant cation and oxygen vacancies without a thorough understanding of underlying atomistic level mechanisms. In earlier studies, the local defect structures are considered as one of the chemical species whose concentration is described using the equilibrium thermodynamics [1, 39]. However, this approach does not take into account the interactions between the local defect structures, especially at higher doping concentrations (>10 mol.% A2 O3 ).

Multiple atomistic simulations studies have been performed to comprehend the relationships between the macroscopic conductivity and the migration in various local environments. In one of the earlier theoretical studies, Murray et al. [50] used Kinetic Monte Carlo (KMC) simulations to consider the migration enthalpy in the local environment to estimate the oxygen-ion conductivity as a function of doping level. It was reported that at higher doping concentration, oxygen vacancies tend to reside next to dopants, and thus do not contribute toward ionic conduction.

In a similar investigation, Nakayama et al. [51] used *ab-initio* density functional theory calculations to identify two key relationships that govern the oxygen-ion migration in rare-earth cation-doped ceria. First, the lowering of migration energy barrier by doping with a smaller trivalent cation would be accompanied by trapping of an oxygen vacancy at the nearest neighboring sites of the dopant. Second, doping with a larger trivalent cation increases the energy barrier but decreases the trapping effect of oxygen vacancies. Thus, the relative magnitude of these two effects is dependent on the size of dopant cation which in turn decides the magnitude of oxygen-ion conductivity.

In recent work by Koettgen et al. [6], the oxygen-ion conductivity was calculated as a function of the doping amount by combining *ab-initio* density functional theory and Kinetic Monte Carlo simulations. A model was developed that can estimate migration energies for all the possible jump configurations present in rare-earth-doped CeO<sup>2</sup> system. Migration energies were analyzed for energy contributions that are either energetically symmetric for both migration directions, that is, forward and backward jumps or energetically asymmetric for the forward and backward directions. If the presence of dopant changes the migration energy identically in both the backward and forward directions, the energy contribution, in this case, is referred to as blocking. On the other hand, if the migration energy is different for forward and backward jumps, the energy contribution is referred to as trapping.

Thus, this study clarifies that the optimal doping concentration to achieve the maximum ionic conductivity cannot be predicted only by trapping effect (association between dopant cation

**Figure 10.** The energy of the system as a function of the reaction coordinate for the configuration change that is shown below in rare-earth-doped ceria. In trapping effect, migration enthalpy increases if the oxygen-ion jump weakens the association between the oxygen vacancies and the dopants. In case of blocking effect, the migration energy increases for an increasing number of large dopants at the migration edge: cerium ions (green), rare-earth ions (blue), oxygen ions

Doped Ceria for Solid Oxide Fuel Cells http://dx.doi.org/10.5772/intechopen.79170 55

This chapter reviews some of the highlights of investigations performed on lower-valent cationdoped ceria materials which have been considered strong candidates for the electrolyte application in SOFCs operating at the intermediate-temperature range. Some of the basic characteristics of doped ceria relating to its high oxygen-ion conductivity are discussed. The maximum in conductivity observed by adding a large amount of lower-valent dopant cation is explained on the basis of formation of local defect structures. The extent of formation of these local defect structures cannot be lowered by just minimizing the elastic lattice distortion. It has been emphasized

and oxygen vacancies) which is commonly assumed in the literature.

**6. Summary and conclusion**

(red spheres), and oxygen vacancies (red boxes) [6].

**Figure 10** shows the schematic of trapping and blocking effects. The energy of the system as a function of a reaction coordinate for the corresponding configuration is also shown. It was demonstrated that both these effects have an impact on the observed ionic conductivity. While blocking effect determines the doping fraction at which the maximum in conductivity is observed, it is the trapping effect which limits the maximum ionic conductivity value.

**Figure 10.** The energy of the system as a function of the reaction coordinate for the configuration change that is shown below in rare-earth-doped ceria. In trapping effect, migration enthalpy increases if the oxygen-ion jump weakens the association between the oxygen vacancies and the dopants. In case of blocking effect, the migration energy increases for an increasing number of large dopants at the migration edge: cerium ions (green), rare-earth ions (blue), oxygen ions (red spheres), and oxygen vacancies (red boxes) [6].

Thus, this study clarifies that the optimal doping concentration to achieve the maximum ionic conductivity cannot be predicted only by trapping effect (association between dopant cation and oxygen vacancies) which is commonly assumed in the literature.
