Below shows an example of how the surface energy of |LiTMO2| is calculated for the both non-polar and polar surface facets. The brief overview of how the surface energy 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 ============== .. Substitutions .. |LiTMO2| replace:: LiTMO\ :sub:`2` .. |O2| replace:: O\ :sub:`2` .. Content 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: .. math:: \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}}} The `surface free energy` is the reaction free energy normalized by the surface area `A` and is given by: .. math:: \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 :math:`n_{i}^{\text{slab}}` and :math:`n_{i}^{\text{bulk}}` are the number of atoms of species :math:`i` (O and Li) in the slab and bulk models, respectively. :math:`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 :math:`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, :math:`G_\text{slab}` and :math:`G_\text{bulk}` can be obtained from DFT calculations, representing the energies of the slab model and the bulk structure (one |LiTMO2| formula unit). Reference energies ================== Reference energies are user-provided scientific inputs. The package does not select them from a functional name or contain a supported-method list. Each phase-diagram file declares the Li energy in eV per atom, the raw O2 energy and explicit correction in eV per molecule, and the bulk LiTMO2 energy in eV per formula unit. The method description is retained as free-text provenance but is not interpreted by the code. For example, the committed Ni data used in the reference publications records: ================= ============= ================== ==================== Reference PBE+U SCAN+rVV10+U r2SCAN+rVV10+U ================= ============= ================== ==================== BCC Li -1.89965 -2.33333 -2.32338 isolated O2 -8.50018 -12.00701 -11.54833 LiNiO2 -19.92283375 -36.8133525 unavailable LiCoO2 -22.69242 -37.2001966667 -32.5698933333 LiMnO2 -26.319605 unavailable -36.4717 ================= ============= ================== ==================== These are example-data values, not package defaults. The PBE+U O2 value includes the original +1.36 eV correction to the raw -9.86018 eV isolated-molecule energy. The historic LiMnO2 ``SCAN+rVV10+U`` table entry was zero and is treated as a placeholder, not a physical reference. LiNiO2 ``r2SCAN+rVV10+U`` was not supplied. Users must supply references appropriate to their own calculation settings rather than relying on these historic values. Programmatic callers use :class:`surface_pd.plot.ReferenceEnergies`. Its corrected O2 property is the sum of the separately retained raw energy and correction. Input files use the equivalent metadata keys documented in :doc:`tutorials-surface-plot`. .. For the fully lithiated |LiTMO2| bulk composition, the number of Li and TM atoms is identical, and we identify :math:`n_{\text{Li}}^{\text{bulk}} = n_{\text{TM}}^{\text{bulk}}`, and :math:`n_{\text{O}}^{\text{bulk}} = 2n_{\text{TM}}^{\text{bulk}}`. In equilibrium, the chemical potential of Li can be calculated as: .. math:: \mu_{\text{Li}}=E(\text{Li}_{\text{bcc}})- F\, V where :math:`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. .. math:: \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 :math:`\frac{1}{2}` is a normalization factor accounting for the two oxygen atoms in each |O2| molecule, and :math:`\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. In the implementation, :math:`\mu_{\text{O}_2}^{0\text{K}}` is expressed in eV per O2 molecule, whereas the thermal correction is calculated in kJ/mol of O2. The thermal correction is therefore converted to eV before the terms are added: .. math:: \mu_{\text{O}}(T) = \frac{1}{2}\left\{ E_{\text{O}_2}^{\text{DFT}} + \frac{\Delta\mu_{\text{O}_2}^{\text{thermal}}(T)} {96.487\ \text{kJ mol}^{-1}\text{eV}^{-1}} \right\}. The reference temperature is :math:`T_0 = 298.15` K and pressure dependence is neglected. .. :math:`\mu_{\text{Li}}`, is equal to the sum of the Li ion and electron chemical potentials, :math:`{\mu_{\text{Li}} = \mu_{\text{Li}^+} + \mu_{\text{e}^-}}`. Using the Li metal electrode :math:`\text{Li} \leftrightarrow \text{Li}^+ + e^-` as reference, we introduce the reference chemical potentials :math:`\mu_{\text{Li}^+}^{\circ}` and :math:`\mu_{\text{e}^-}^{\circ}` with :math:`\mu_{\text{Li}^+}^{\circ} + \mu_{\text{e}^-}^{\circ} = G (\text{Li}_{\text{bcc}})`, The difference of the :math:`\text{Li}^+` and :math:`\text{e}^-` chemical potentials from the reference Li metal electrode determines the cell potential `V`: :math:`\Delta\mu_{\text{Li}^+} + \Delta\mu_{\text{e}^-}= - F\, V`, where `F` is Faraday's constant. .. The difference of the enthalpy at standard conditions from its 0~\ Kelvin value, :math:`\Delta{}H^{\circ}`, and the standard entropy :math:`S^{\circ}` were taken from the NIST-JANAF Thermochemical Tables. :math:`\Delta H(T) = C_p(T - T_{\text{0}})` and :math:`\Delta S(T) = C_p\ln{(T/T_{\textup{0}})}`, where the heat capacity :math:`C_{p} = 3.5k_B` was taken to be the value for an ideal gas of diatomic molecules at :math:`T \geq298`. Finally, taken together, the surface energy of any |LiTMO2| surface (ideal stoichiometric or defected/reconstructed) can be approximated as: .. math:: \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\}