**3. Elastic lattice strain**

The amount of local defect structures increases with the incorporation of oxygen vacancies inside the lattice. For achieving high oxygen-ion conductivity, the association of defects needs to be minimized. Kilner and Brook [45] stated that the oxygen-ion conductivity in doped CeO2 could be enhanced if the elastic strain present in the lattice is lowered. Kim [46] partially replaced acceptor cations in fluorite-structured MO<sup>2</sup> (where M is a tetravalent cation) and studied the change in lattice parameter of the host oxides. By performing multiple regression analysis, an empirical model was developed that can predict the elastic strain in doped ceria systems for any given dopant cation. This model takes into consideration the size mismatch and difference in charge valence between the dopant and host cations and the dopant concentration. Further, Kim [46] proposed the concept of critical ionic radius (*rc* ) which was described as the ionic radius of a cation which on doping leads to a negligible distortion in the host fluorite lattice. For the trivalent dopant cation and host ceria, the value of *rc* was estimated to be 1.038 Å. Based on Kilner and Brook's prediction, the cation with the ionic radius of *rc* is the optimal dopant that can lead to the maximum oxygen-ion conductivity in CeO2 [45]. As a result, the oxygen-ion conductivity in doped CeO2 depend upon the ( <sup>|</sup> *<sup>r</sup> <sup>d</sup>* <sup>−</sup> *<sup>r</sup> <sup>c</sup>* <sup>|</sup> ) value, where *rd* is the ionic radius of dopant. The lower the <sup>|</sup> *<sup>r</sup> <sup>d</sup>* <sup>−</sup> *<sup>r</sup> <sup>c</sup>* | value for a particular dopant, the higher the ionic conductivity for that system is expected to be. Kim [46] argued that the highest ionic conductivity is observed around Gd3+ (see **Figure 4**) as *rc* value is close to the ionic radius of Gd3+ (*<sup>r</sup> Gd*,*VIII* 3+ = 1.053 Å).

CeO2

et al. [32].

co-doped CeO2

inferior to that of Gd3+-doped CeO2

a much larger ionic radius than the *rc*

lattice distortion for various doped CeO<sup>2</sup>

higher temperatures. The *r*<sup>c</sup>

between doped and pure CeO<sup>2</sup>

system was found to be lower than that of Gd3+-doped CeO2

**Figure 7.** Elastic strain is plotted against the dopant concentration [23]. Data for Nd*<sup>x</sup>*

.

tice strain. However, it is noteworthy to mention that all the *rc*

On plotting the activation energy for oxygen-ion diffusion in trivalent cation-doped CeO<sup>2</sup>

has been found that the minimum value for activation energy occurs for Sm3+ which possesses

activation energy values do not support the hypothesis given by Kilner and Brook [45] that the highest oxygen-ion conductivity is obtained in the system having a negligible elastic lat-

are derived from the empirical relationships which are based on the lattice expansion data

with the increasing temperature. Omar et al. [15] tested this hypothesis by measuring the

decreases with the increase in temperature and does not follow the conductivity trend. The obtained results clearly indicate that the oxygen-ion conductivity is not solely dependent on elastic strain, and therefore, a structure-conductivity relationship based on the critical ionic

acquired at room temperature. As the oxygen-ion conductivity of doped CeO2

radius concept is insufficient to describe the conduction behavior in doped ceria.

usually measured between 400°C and 700°C, there is a possibility that the *r*<sup>c</sup>

system exhibits minimal elastic strain, its oxygen-ion conductivity remains

value [15]. Thus, both co-doping results and the obtained

Ce1-*<sup>x</sup>* O2-*<sup>δ</sup>*

systems (with 10 mol.% dopant concentration) at

value was then determined, where the lattice parameter mismatch

becomes zero (see **Figure 8**). It was noticed that the *rc*

[23]. Thus, even though the

Doped Ceria for Solid Oxide Fuel Cells http://dx.doi.org/10.5772/intechopen.79170 51

were taken after Stephens

values reported in the literature

, it

materials is

value

value increases

In a similar study, Hong and Virkar [47] calculated the *rc* value to be 1.024 Å using a simple analytical model that can also predict the elastic lattice strain present in doped CeO<sup>2</sup> . In this model, the oxygen vacancy is treated as one of the chemical species whose size remains constant irrespective of the size and concentration of dopant cation.

To examine the validity of minimum strain's hypothesis, Omar et al. [23] have carried out the experimental studies using a novel co-doping strategy in ceria. A co-dopant pair of Lu3+ and Nd3+ were chosen and added in a ratio such that the weighted average dopant ionic radius of co-dopants matches the magnitude of *r*<sup>c</sup> . Using this approach, it is expected that, on average, the positive elastic strain because of larger dopant cation (i.e., Nd3+) is compensated by the negative elastic strain caused by the smaller dopant cation (i.e., Lu3+). This, as a result, prevents any distortion in fluorite lattice that is usually present in singly doped ceria systems. On calculating the lattice parameter, almost negligible elastic lattice strain was seen in these systems (see **Figure 7**). However, the ionic conductivity in Lu3+ and Nd3+ co-doped

of 0.02–0.05 eV [42]. Nowick et al. [43] obtained the association enthalpy by estimating the

enthalpy. The association enthalpy initially decreases and then increases after reaching a minimum value at *x* = 0.03 (shown in **Figure 6**) [32]. The increase in association enthalpy is attributed to the development of deep vacancy association. A similar trend in association enthalpy is

The amount of local defect structures increases with the incorporation of oxygen vacancies inside the lattice. For achieving high oxygen-ion conductivity, the association of defects needs to be minimized. Kilner and Brook [45] stated that the oxygen-ion conductivity in doped

studied the change in lattice parameter of the host oxides. By performing multiple regression analysis, an empirical model was developed that can predict the elastic strain in doped ceria systems for any given dopant cation. This model takes into consideration the size mismatch and difference in charge valence between the dopant and host cations and the dopant

described as the ionic radius of a cation which on doping leads to a negligible distortion in the

is the ionic radius of dopant. The lower the <sup>|</sup> *<sup>r</sup> <sup>d</sup>* <sup>−</sup> *<sup>r</sup> <sup>c</sup>* | value for a particular dopant, the higher the ionic conductivity for that system is expected to be. Kim [46] argued that the highest ionic

model, the oxygen vacancy is treated as one of the chemical species whose size remains con-

To examine the validity of minimum strain's hypothesis, Omar et al. [23] have carried out the experimental studies using a novel co-doping strategy in ceria. A co-dopant pair of Lu3+ and Nd3+ were chosen and added in a ratio such that the weighted average dopant ionic

on average, the positive elastic strain because of larger dopant cation (i.e., Nd3+) is compensated by the negative elastic strain caused by the smaller dopant cation (i.e., Lu3+). This, as a result, prevents any distortion in fluorite lattice that is usually present in singly doped ceria systems. On calculating the lattice parameter, almost negligible elastic lattice strain was seen in these systems (see **Figure 7**). However, the ionic conductivity in Lu3+ and Nd3+ co-doped

to be 1.038 Å. Based on Kilner and Brook's prediction, the cation with the ionic radius of *rc*

concentration. Further, Kim [46] proposed the concept of critical ionic radius (*rc*

host fluorite lattice. For the trivalent dopant cation and host ceria, the value of *rc*

the optimal dopant that can lead to the maximum oxygen-ion conductivity in CeO2

analytical model that can also predict the elastic lattice strain present in doped CeO<sup>2</sup>

could be enhanced if the elastic strain present in the lattice is lowered. Kim [46] partially

also observed for other dopant cations such as Y3+, Gd3+ and Sm3+ and so on [11, 14, 44].

mum value of *EA* measured in the low-temperature regime in Nd*<sup>x</sup>*

replaced acceptor cations in fluorite-structured MO<sup>2</sup>

result, the oxygen-ion conductivity in doped CeO2

3+ = 1.053 Å).

conductivity is observed around Gd3+ (see **Figure 4**) as *rc*

In a similar study, Hong and Virkar [47] calculated the *rc*

radius of co-dopants matches the magnitude of *r*<sup>c</sup>

stant irrespective of the size and concentration of dopant cation.

to be 0.67 eV. Stephen and Kilner [32] have taken the mini-

Ce1-*<sup>x</sup>*

(where M is a tetravalent cation) and

depend upon the ( <sup>|</sup> *<sup>r</sup> <sup>d</sup>* <sup>−</sup> *<sup>r</sup> <sup>c</sup>* <sup>|</sup> ) value, where *rd*

value is close to the ionic radius of

value to be 1.024 Å using a simple

. Using this approach, it is expected that,

O2-*x*/2 as a migration

) which was

was estimated

[45]. As a

. In this

is

migration enthalpy in pure CeO2

50 Cerium Oxide - Applications and Attributes

**3. Elastic lattice strain**

CeO2

Gd3+ (*<sup>r</sup> Gd*,*VIII*

**Figure 7.** Elastic strain is plotted against the dopant concentration [23]. Data for Nd*<sup>x</sup>* Ce1-*<sup>x</sup>* O2-*<sup>δ</sup>* were taken after Stephens et al. [32].

CeO2 system was found to be lower than that of Gd3+-doped CeO2 [23]. Thus, even though the co-doped CeO2 system exhibits minimal elastic strain, its oxygen-ion conductivity remains inferior to that of Gd3+-doped CeO2 .

On plotting the activation energy for oxygen-ion diffusion in trivalent cation-doped CeO<sup>2</sup> , it has been found that the minimum value for activation energy occurs for Sm3+ which possesses a much larger ionic radius than the *rc* value [15]. Thus, both co-doping results and the obtained activation energy values do not support the hypothesis given by Kilner and Brook [45] that the highest oxygen-ion conductivity is obtained in the system having a negligible elastic lattice strain. However, it is noteworthy to mention that all the *rc* values reported in the literature are derived from the empirical relationships which are based on the lattice expansion data acquired at room temperature. As the oxygen-ion conductivity of doped CeO2 materials is usually measured between 400°C and 700°C, there is a possibility that the *r*<sup>c</sup> value increases with the increasing temperature. Omar et al. [15] tested this hypothesis by measuring the lattice distortion for various doped CeO<sup>2</sup> systems (with 10 mol.% dopant concentration) at higher temperatures. The *r*<sup>c</sup> value was then determined, where the lattice parameter mismatch between doped and pure CeO<sup>2</sup> becomes zero (see **Figure 8**). It was noticed that the *rc* value decreases with the increase in temperature and does not follow the conductivity trend. The obtained results clearly indicate that the oxygen-ion conductivity is not solely dependent on elastic strain, and therefore, a structure-conductivity relationship based on the critical ionic radius concept is insufficient to describe the conduction behavior in doped ceria.

**Figure 8.** Lattice parameter mismatch between A0.10Ce0.90O2-*<sup>δ</sup>* and CeO2 is plotted against the ionic radius of dopant cation (A3+) at 500°C. The grain (bulk) oxygen-ion conductivity of A0.10Ce0.90O2-*<sup>δ</sup>* at 500°C is also shown [15].
