2.12.1.3 Case III: Simultaneous presence of oxygen vacancies and metal interstitials

Such a scenario is often found in ceramic oxides like TiO₂, and Nb₂O₅.
Consider a metal oxide (MO₂)  with doubly charged oxygen vacancies and metal ion interstitials. The corresponding defect reaction is

OR

Assuming , the defect equilibrium can be written as

[]

(2.38)

According to the electrical neutrality condition

Two limiting cases can be considered:

When [V₀^••] >>[M_i^••]

= [VO••] =

(2.40)

And

(2.41)

i.e.

[Mi••]

(2.42)

As you can see, under such conditions, [M_i^••] decreases more rapidly with increasing pO₂. This is commonly observed in TiO₂ and Nb₂O₅ where [V₀^••] can be 10¹⁰ times higher than  [M_i^••].

When [M_i^••] >>[V₀^••]

Following similar exercise as above, we can calculate

[Mi••] = =

(2.43)

and

[] =

(2.44)

Here, [V₀^••]  increases with increasing pO₂ while keep decreasing with increasing pO₂ but at a different rate.