**6. Summary and conclusion**

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

(>10 mol.% A2

conduction.

O3 ).

54 Cerium Oxide - Applications and Attributes

magnitude of oxygen-ion conductivity.

possible jump configurations present in rare-earth-doped CeO<sup>2</sup>

and backward jumps, the energy contribution is referred to as trapping.

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

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

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

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

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

**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.

system. Migration energies

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 that the concept of critical ionic radius alone cannot explain the maximum oxygen-ion conductivity observed in Pm3+-doped CeO2 as found by the first-principles density functional theory calculations. Particular attention has been given to a more recent atomistic simulations study on rare-earth-doped ceria which calculates the migration energies for all the possible jump configurations that may present in rare-earth-doped CeO2 . This study explains the importance of the shape of migration energy barrier and its impact on the ionic conductivity.

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