## Materials and Methods
### Physicochemical Model
The rate of Li\({}^{+}\) and Co\({}^{2+}\) production and H\({}^{+}\) consumption in the extracting liquid, assumed with a constant volume of \(V_{L}\) (m\({}^{3}\)), is given by: \[V_{L}\frac{dC_{\text{Li}^{+}}}{dt}=r_{\text{Li}^{+}}\] (10) \[V_{L}\frac{dC_{\text{Co}^{2+}}}{dt}=\frac{1}{2}r_{\text{Li}^{+}}\] (11) \[V_{L}\frac{dC_{\text{H}^{+}}}{dt}=-2r_{\text{Li}^{+}}\] (12)
5. The outer crust of Co\({}_{3}\)O\({}_{4}\) (\(s\)), which forms during the process, is supposed to be porous. As it forms, the size of the particle remains constant and equal to the initial. Diffusion of the reactant, namely, the H\({}^{+}\), through the layer of Co\({}_{3}\)O\({}_{4}\)(s) from the bulk liquid to the surface of the unreacted core is described as: \[V_{L}\frac{dC_{\text{H}^{+}}}{dt}=-\frac{D_{\text{H}^{+}}^{\text{eff}}\left( C_{\text{H}^{+}}^{bulk}-C_{\text{H}^{+}}^{core}\right)}{\left(R_{p}-r_{c} \right)}n_{p}\Big{(}4\pi R_{p}r_{c}\Big{)}\] (13) where \(C_{\text{H}^{+}}^{bulk}\) is the concentration of protons in the bulk liquid, and \(D_{\text{H}^{+}}^{\text{eff}}\) (m\({}^{2}\) s\({}^{-1}\)) is the effective diffusion coefficient of the proton through the layer of Co\({}_{3}\)O\({}_{4}\). The effective diffusion coefficient is considered to change as the crust increases, according to the following expression: \[\text{LiCoO}_{2}(\text{s})+\text{H}^{+}\rightleftarrows\text{Li}^{+}+\frac{1} {3}\text{Co}_{3}\text{O}_{4}(\text{s})+\frac{1}{2}\text{H}_{2}\text{O}+\frac{1 }{12}\text{O}_{2}(\text{g}). \tag{7}\]0 (14) \(D_{\text{H}^{+}}\) (m\({}^{2}\) s\({}^{-1}\)) is the diffusion coefficient at infinite dilution (\(9.311\times 10^{-9}\) m\({}^{2}\) s\({}^{-1}\)[27]), \(\epsilon\) (\(\rightharpoonup\)) is the porosity, \(X_{\text{Co}}\) (\(\rightharpoonup\)) is the percentage of cobalt extracted, and \(k_{D}\) (m\({}^{-1}\)) is a parameter that measures the increase of the resistance to the mass transport as the crust increases as a consequence of changes in porosity or tortuosity, including pore closing due to counter diffusion of the O\({}_{2}\) gas formed in the reaction. \(X_{\text{Co}}\) (\(\rightharpoonup\)) is included in the equation to take into account the formation of O\({}_{2}\) (g) and accumulation within the porous structure. 6. As the reaction proceeds and the outer Co\({}_{3}\)O\({}_{4}\)(s) crust increases, resistance to the diffusion transport increases according to Equation (14). At a certain thickness of the crust, the process reaches a point in which the rate of proton diffusion is lower than the chemical kinetic consumption. At this point, as soon as protons reach the reactive surface, they are consumed by the reaction. The second Damkohler number, Da\({}_{II}\) (\(\rightharpoonup\)), defined in Equation (15) as the ratio of the chemical reaction rate to the mass transfer rate, is used to identify the controlling mechanisms. \[\text{LiCoO}_{2}(\text{s})+\text{H}^{+}\rightleftarrows\text{Li}^{+}+\frac{1} {3}\text{Co}_{3}\text{O}_{4}(\text{s})+\frac{1}{2}\text{H}_{2}\text{O}+\frac{1 }{12}\text{O}_{2}(\text{g}). \tag{7}\]1 (15)
7. Combining Equations (9), (12), and (13), it can stated that: \[\text{LiCoO}_{2}(\text{s})+\text{H}^{+}\rightleftarrows\text{Li}^{+}+\frac{1} {3}\text{Co}_{3}\text{O}_{4}(\text{s})+\frac{1}{2}\text{H}_{2}\text{O}+\frac{1 }{12}\text{O}_{2}(\text{g}). \tag{7}\]2 (16) where, if Da\({}_{II}\leq 1\), then \(C_{\text{H}^{+}}^{core}=C_{\text{H}^{+}}^{bulk}\), and the process rate is defined by the chemical kinetics law. However, whenever Da\({}_{II}>1\), then \(C_{\text{H}^{+}}^{core}=0\), and the process rate is controlled by the diffusion. The change of control mechanism is schematized in \[\text{LiCoO}_{2}(\text{s})+\text{H}^{+}\rightleftarrows\text{Li}^{+}+\frac{1} {3}\text{Co}_{3}\text{O}_{4}(\text{s})+\frac{1}{2}\text{H}_{2}\text{O}+\frac{1 }{12}\text{O}_{2}(\text{g}). \tag{7}\]3
8. The Diffusion of Li\({}^{+}\) and Co\({}^{2+}\) through the Co\({}_{3}\)O\({}_{4}\)(s) crust layer to the bulk solution is considered to have no effect on the process rate.