The brief overview of how the surface energy is calculated for the both non-polar and polar surface facets is described here. For a detailed description of how all the equations listed below are derived, please refer to the paper.

Understanding the Onset of Surface Degradation in LiNiO2 Cathodes

Surface energy

In thermodynamic equilibrium, the LiTMO₂ (TM represents transition metal species) layered surface forms the reconstruction with the lowest Gibbs free energy for the given conditions. The grand-canonical surface phase diagram is thus determined by the surface free energies of all possible surface structures with different arrangements of lithium and oxygen vacancies, subject to the chemical potentials of lithium and oxygen.

To derive an expression of the surface free energy, we consider the formal truncation of the extended LiTMO₂ crystal structure along a lattice plane. In thermodynamic equilibrium with oxygen and lithium reservoirs (e.g., the reactants during the synthesis of the material), the surfaces may absorb or release Li and O atoms, so that the surface stoichiometry can differ from the stoichiometry of the LiTMO₂ bulk. A surface slab model of any surface, whether ideal stoichiometric or reconstructed, can thus be thought of as the result of the formal formation reaction:

\[\begin{split}\frac{n_\text{TM}^\text{slab}}{n_\text{TM}^\text{bulk}}\text{LiTMO}_2^{\text {bulk}} + \Bigg(n_\text{Li}^\text{slab} - \frac{n_{\text{TM}}^{\text{slab}}}{n_{\text{TM}}^{\text{bulk}}}n_ {\text{Li}}^{\text{bulk}} \Bigg)\text{Li} + \\ \frac{1}{2} \Bigg(n_\text{O}^\text{slab} - \frac{n_\text{TM}^\text{slab}}{n_\text{TM}^\text{bulk}}n_{\text{O }}^{\text{bulk}} \Bigg)\text{O}_2 \rightarrow \text{Li}_{n_{\text{Li}}^{\text{slab}}}\text{TM}_{n_{\text{TM}}^{\text{slab }}}O_{n_{\text{O}}^{\text{slab}}}\end{split}\]

The surface free energy is the reaction free energy normalized by the surface area A and is given by:

\[\gamma = \frac{1}{2\text{A}}\Bigg[ G_{\text{slab}} - \frac{n_{\text{TM}}^{\text{slab}}}{n_{\text{TM}}^{\text {bulk}}} G_{\text{bulk}} - \sum_{{i}}^{\text{Li}, \text{O}}(n_{{i}}^{\text{slab}} - \frac{n_{\text{TM}}^{\text{slab}}}{n_{\text{TM}}^{\text{bulk}}} n_{{i}}^{\text{bulk}}) \mu_{\text{i}} \Bigg]\]

where \(n_{i}^{\text{slab}}\) and \(n_{i}^{\text{bulk}}\) are the number of atoms of species \(i\) (O and Li) in the slab and bulk models, respectively. \(G_{\text{slab}} = G (\text{Li}_{n_{\text{Li}}^{\text{slab}}}\text{TM}_{n_{\text{TM}}^{\text{slab }}}O_{n_{\text{O}}^{\text{slab}}})\) and \(G_{\text{bulk}}=G(\text{LiTMO}_2^{\text{bulk}})\) are the Gibbs free energy of the slab and bulk models, respectively, and the TM content is assumed to be constant. Neglecting the temperature dependence of the solids, \(G_\text{slab}\) and \(G_\text{bulk}\) can be obtained from DFT calculations, representing the energies of the slab model and the bulk structure (one LiTMO₂ formula unit).

In equilibrium, the chemical potential of Li can be calculated as:

\[\mu_{\text{Li}}=E(\text{Li}_{\text{bcc}})- F\, V\]

where \(E(\text{Li}_{\text{bcc}})\) is the free energy of Li metal in the body-centered cubic structure, which is also approximated with the zero-Kelvin DFT energy. F and V are Faraday’s constant and cell potential, respectively.

The chemical potential of oxygen depends, in principle, on the temperature and pressure, though the pressure dependence is negligible compared to the temperature dependence. Thus, we ignore the impact of pressure.

\[\mu_{\text{O}}(T) = \frac{1}{2}\Bigg\{ \mu_{\text{O}_2}^{0\text{K}} + \Bigg[ \Delta H^{\circ} + \Delta H(T) \Bigg] - T\Bigg[ S^{\circ} + \Delta S(T) \Bigg] \Bigg\}\]

where \(\frac{1}{2}\) is a normalization factor accounting for the two oxygen atoms in each O₂ molecule, and \(\mu_{\text{O}_2}^{0\text{K}}\) is the oxygen chemical potential at 0 Kelvin (equal to the enthalpy), which was obtained from DFT calculations as described above. The remaining terms are the enthalpy and entropy contributions to the relative oxygen chemical potential.

Finally, taken together, the surface energy of any LiTMO₂ surface (ideal stoichiometric or defected/reconstructed) can be approximated as:

\[\begin{split}\gamma(T, V) \approx \frac{1}{2A} \Bigg\{ E_{\text{slab}} + (n_{\text{TM}} - n_{\text{Li}}) \Bigg[ E(\text{Li}_{\text{bcc}}) - F\, V \Bigg] \\ + \frac{1}{2} (2n_{\text{TM}} - n_{\text{O}}) \mu_{\text{O}_2}(T) - n_{\text{TM}} E(\text{LiTMO}_2) \Bigg\}\end{split}\]