**Appendix A**

*Appendix A.1 Chemical Expansion of Cobalt Ions upon Reduction*

Using a Kröger–Vink-compatible notation, the following description of Co ions can be introduced (Table A1) [35]:

**Table A1.** Cobalt ions notation.


Two reaction equations linking the reduction of cobalt and the formation of oxygen vacancies can be written. One would be the reduction of *Co*4+ to *Co*3+ associated with vacancy formation (Equation (A1)), and the second can be formulated similarly for the reduction of *Co*3+ to *Co*2+ (Equation (A2)).

$$2\operatorname{Co}\_{\text{Co}}^{\frac{1}{2}\bullet} + \operatorname{O}\_{\text{O}}^{X} \rightarrow 2\operatorname{Co}\_{\text{Co}}^{\frac{1}{2}\prime} + v\_{\text{O}}^{\bullet\bullet} + \frac{1}{2}\operatorname{O}\_{2\text{ (g)}}\tag{A1}$$

$$2\operatorname{Co}\_{\text{Co}}^{\frac{1}{2}\text{ }/} + \text{O}\_{\text{O}}^{\text{X}} \rightarrow 2\operatorname{Co}\_{\text{Co}}^{\frac{3}{2}\text{ }/} + v\_{\text{O}}^{\bullet\bullet} + \frac{1}{2}\text{O}\_{2} \text{ (g)}\tag{A2}$$

The regimes can be distinguished with regard to the presence of Co oxidation states in the material: oxidized state for δ < 0.5 and reduced state for δ > 0.5. In the former *Co* 12 • *Co* + *Co* 12 / *Co Co* 32 / *Co* and in the latter *Co* 12 / *Co* + *Co* 32 / *Co Co* 12 • *Co* . The transition value of δ is 0.5, where the average cobalt oxidation state is 3.0.

The chemical expansion of cobalt results from the difference in size between cobalt oxidation states, and it is defined as

$$
\beta\_{\rm Co} = \frac{1}{\Delta \delta} \cdot \frac{\Delta V\_{\rm Co}}{V\_{\rm Co\_0}} \tag{A3}
$$

where Δδ denotes the change of oxygen stoichiometry, which can be detailed as

$$
\beta\_{\rm Co} = \frac{1}{\delta - \delta\_0} \cdot \frac{V\_{\rm Co(\delta)} - V\_{\rm Co\_0}}{V\_{\rm Co\_0}} \tag{A4}
$$

Here, *VCo*(δ) is the average volume occupied by Co at any given δ and *VCo*0 is the average volume occupied by Co at room temperature (δ0).

#### *Appendix A.2 Reduction from Co4*+ *to Co3*+

The average cobalt oxidation state (*CoAVG*) can be calculated from

$$2\text{Co}^{AVG} = 2\left[\text{Co}\_{\text{Co}}^{\frac{2}{2}}\right] + 3\left[\text{Co}\_{\text{Co}}^{\frac{1}{2}}\right] + 4\left[\text{Co}\_{\text{Co}}^{\frac{1}{2}}\right] \tag{A5}$$

In the oxidized regime, this gives

$$\mathbb{C}o^{AVG} = \frac{3}{2} \Big[ \mathbb{C}o^{\frac{1}{2}}\_{\mathbb{C}o} \Big] + 2 \Big[ \mathbb{C}o^{\frac{1}{2}}\_{\mathbb{C}o} \Big] \tag{A6}$$

*CoAVG* can be related to oxygen non-stoichiometry according to

$$
\mathbb{C}\sigma^{AVG} = 3.5 - \delta \tag{A7}
$$

The concentration of cobalt ions in the oxidized regime is given by

$$\left[\text{Co}\_{\text{Co}}^{\frac{1}{2}}\,\!\!\!\!\sigma\right] + \left[\text{Co}\_{\text{Co}}^{\frac{1}{2}}\,\!\!\/'\right] = 2\tag{A8}$$

The combination of Equations (A6)–(A8) gives the relations between the concentration of *Co*3+ and *Co*4+ with oxygen non-stoichiometry (Equation (A9)) in the oxidized regime.

$$\left[\text{Co}\_{\text{Co}}^{\frac{1}{2}}\right] = 1 - 2\delta\tag{A9}$$

$$\left[\text{Co}\_{\text{Co}}^{\frac{1}{2}}\right] = 2\delta + 1\tag{A10}$$

The volume of cobalt can be described as the weighted arithmetic mean of *Co*4+ and *Co*3+ volume, where the concentration of each species is a weight (Equation (A11)).

$$V\_{\odot o} = \frac{\left[\text{Co}\_{\text{Co}}^{\ddagger}\right] \cdot V\_{\text{Co}^{4+}} + \left[\text{Co}\_{\text{Co}}^{\ddagger}\right] V\_{\text{Co}^{3+}}}{2} \tag{A11}$$

The average cobalt volume as a function of δ (Equation (A10)) can be obtained by including the Equations (A9) and (A10) to the Equation (A11).

$$V\_{\mathbb{C}o}(\delta) = \frac{(1 - 2\delta) \cdot V\_{\mathbb{C}o^{4+}} + (2\delta + 1) \cdot V\_{\mathbb{C}o^{3+}}}{2} \tag{A12}$$

Similarly, the average volume occupied by Co at room temperature relates to δ0:

$$V\_{\rm Co\_0} = \frac{(1 - 2\delta\_0) \cdot V\_{\rm Co^{4+}} + (2\delta\_0 + 1) \cdot V\_{\rm Co^{3+}}}{2} \tag{A13}$$

Inserting Equations (A12) and (A13) into Equation (A5) leads to the expression on chemical expansion coefficient of *Co*4+ to *Co*3+ reduction (Equations (A14) and (A15)),

$$\beta\_{\text{Co 4}\to\text{3}} = \frac{1}{\Delta\delta} \cdot \frac{2\left((\delta\_0 - \delta) \cdot V\_{\text{Co}^{4+}} + (\delta - \delta\_0) \cdot V\_{\text{Co}^{3+}}\right)}{(1 - 2\delta\_0) \cdot V\_{\text{Co}^{4+}} + (2\delta + 1) \cdot V\_{\text{Co}^{3+}}} \tag{A14}$$

$$
\beta\_{\text{Co 4}\to\text{3}} = \frac{2 \cdot (V\_{\text{Co}^{3+}} - V\_{\text{Co}^{4+}})}{(1 - 2\delta\_{\text{0}}) \cdot V\_{\text{Co}^{4+}} + (2\delta\_{\text{0}} + 1) \cdot V\_{\text{Co}^{3+}}} \tag{A15}
$$

Cobalt volume can be calculated as sphere volume, leading to the Equation (A16).

$$\beta\_{\text{Co}\ 4\to3} = \frac{2\cdot \left(r\_{\text{Co}^{3+}}^3 - r\_{\text{Co}^{4+}}^3\right)}{(1-2\delta\_0)\cdot r\_{\text{Co}^{4+}}^3 + (2\delta\_0 + 1)\cdot r\_{\text{Co}^{3+}}^3} \tag{A16}$$

*Appendix A.3 Reduction of Co3*+ *to Co2*+

The analogous consideration can be made in the reduced regime. The average cobalt oxidation state is given with Equation (A17) and cobalt ion concentrations in the reduced regime are given by Equation (A18).

$$\text{Co}^{AVG} = \frac{3}{2} \cdot \left[ \text{Co}\_{\text{Co}}^{\frac{1}{2}} \right] + \left[ \text{Co}\_{\text{Co}}^{\frac{3}{2}} \right] \tag{A17}$$

$$\left[\text{Co}\_{\text{Co}}^{\frac{1}{2}}\right] + \left[\text{Co}\_{\text{Co}}^{\frac{3}{2}}\right] = 2\tag{A18}$$

The relation between the concentration of cobalt species with δ is given by Equations (A19) and (A20).

$$\left[\text{Co}\_{\text{Co}}^{\frac{1}{2}}\right] = 3 - 2\delta \tag{A19}$$

$$\left[\text{Co}\_{\text{Co}}^{\frac{3}{2}}\right] = 2\delta - 1\tag{A20}$$

In the reduced state, the cobalt volume is also a weighted mean of *Co*3+ and *Co*2+ volume (Equation (A21)).

$$V\_{Co} = \frac{\left[\text{Co}\_{Co}^{\frac{3}{2}}/\right]V\_{Co^{3+}} + \left[\text{Co}\_{Co}^{\frac{3}{2}}/\right]V\_{Co^{2+}}}{2} \tag{A21}$$

Including Equations (A19) and (A20) to Equations (A21) and (A22) gives the relation between cobalt average volume and oxygen non-stoichiometry.

$$V\_{Co}(\delta) = \frac{(3-2\delta) \cdot V\_{Co^{3+}} + (2\delta - 1) \cdot V\_{Co^{2+}}}{2} \tag{A22}$$

With the Equation (A21) the chemical expansion coefficient of *Co*3+ to *Co*2+ reduction can be calculated:

.

$$
\beta\_{\text{Co 3} \to 2} = \frac{1}{\delta - \delta\_0} \cdot \frac{\Delta V\_{\text{Co}}}{V\_{\text{Co}\_0}} \tag{A23}
$$

In this regime, δ0 = 0.5 and *VCo*0 = *VCo*<sup>3</sup>+

$$\beta\_{\text{Co 3}\to2} = \frac{1}{\delta - 0.5} \cdot \frac{(1 - 2\delta) \cdot V\_{\text{Co}^{3+}} + (2\delta - 1) \cdot V\_{\text{Co}^{2+}}}{2 \cdot V\_{\text{Co}^{3+}}} \tag{A24}$$

$$\beta\_{\text{Co 3}\to2} = \frac{1}{\delta - 0.5} \cdot \frac{(0.5 - \delta) \cdot V\_{\text{Co}^{3+}} + (\delta - 0.5) \cdot V\_{\text{Co}^{2+}}}{V\_{\text{Co}^{3+}}} \tag{A25}$$

$$
\beta\_{\text{Co 3} \to 2} = \frac{V\_{\text{Co2}^{2+}} - V\_{\text{Co3}^{3+}}}{V\_{\text{Co3}^{3+}}} \tag{A26}
$$

Equation (A24) is equivalent to Equation (A4) with δ = 1.5, and where all Co is in oxidation state 2+. The volume of cobalt can be related to the cobalt ionic radius, which leads to Equation (A26).

$$
\beta\_{\text{Co 3} \to 2} = \frac{r\_{\text{Co}^{2+}}^3 - r\_{\text{Co}^{3+}}^3}{r\_{\text{Co}^{3+}}^3} \tag{A27}
$$

#### *Appendix A.4 Chemical Expansion of Oxygen Vacancies Formation*

The expression of the chemical expansion coefficient of oxygen vacancies is the same in both δ ranges and may be defined with the Equation (A28).

$$
\beta\_{v\_O^\*} = \frac{1}{\Delta \delta} \cdot \frac{\Delta V\_O}{V\_{O\_0}} \tag{A28}
$$

As a *VO* is the average volume of oxygen site volume, which can also be calculated as weighted arithmetic means of oxygen ions and oxygen vacancies volume (Equation (A29)).

$$
\Delta V\_O = \frac{\left[\rm O\_O^X\right] \cdot V\_{O\_O^X} + \left[v\_O^{\bullet \bullet}\right] \cdot V\_{V\_O^{\bullet \bullet}}}{6} \tag{A29}
$$

The molar concentration of oxygen vacancies is by definition equal to δ, thus the relation between the volume of oxygen site and δ can be written (Equation (A30)).

$$V\_O = \frac{(6-\delta) \cdot V\_{O\_O^X} + \delta \cdot V\_{V\_O^\bullet}}{6} \tag{A30}$$

Substituting Equation (A27) to Equation (A25), the expression on the chemical expansion coefficient of oxygen vacancies is obtained (Equations (A31) and (A32)). The value of δ0 is now equivalent to δ at RT.

$$
\beta\_{v\_O^{\bullet \bullet}} = \frac{1}{\delta - \delta\_0} \cdot \frac{(\delta\_0 - \delta) \cdot V\_{O\_O^X} + (\delta - \delta\_0) \cdot V\_{V\_O^{\bullet \bullet}}}{(\delta - \delta\_0) \cdot V\_{O\_O^X} + \delta\_0 \cdot V\_{V\_O^{\bullet \bullet}}} \tag{A31}
$$

$$\beta\_{v\_O^{\bullet \bullet}} = \frac{V\_{V\_O^{\bullet \bullet}} - V\_{O\_O^X}}{(6 - \delta\_0) \cdot V\_{O\_O^X} + \delta\_0 \cdot V\_{V\_O^{\bullet \bullet}}} \tag{A32}$$

Subtracting the calculated values of chemical expansion coefficient upon cobalt reduction from the total value, the chemical expansion coefficient of oxygen vacancies formation may be obtained; knowing the ionic radii of the oxygen ion, the size of oxygen vacancy can be determined.
