Theory¶
surfinpy is a python module to generate surface phase diagrams from DFT calculation data. Before using this code you will need to generate the relevant data. surfinpy has the capability to generate phase diagrams as a function of chemical potential of two species e.g. water and carbon dioxide. In such an example the user would require calculations with varying concentrations of water, carbon dioxide and water/carbon dioxide on a surface. Assuming that you have generated enough, reliable data then you are ready to use surfinpy.
Surface Energy¶
The physical quantity that is used to define the stability of a surface with a given composition is its surface energy \(\gamma\) (J \(m^{-2}\)). At its core, surfinpy is a code that calculates the surface energy of different slabs at varying chemical potential and uses these surface energies to construct a phase diagram. In this exmplantion of theory we will use the example of water adsorbing onto a surface of \(TiO_2\) containing oxygen vacancies. In such an example there are two variables, water concentration and oxygen vacancy concentration. We are able to calculate the surface energy according to
where A is the surface area, \(E_{TiO_2}^{slab}\) is the DFT energy of the slab, \(nTi_{Slab}\) is the number of cations in the slab, \(nTi_{Bulk}\) is the number of cations in the bulk unit cell and \(E_{TiO_2}^{Bulk}\) is the DFT energy of the bulk unit cell and
where \(nO_{Slab}\) is the number of anions in the slab, \(nO_{Bulk}\) is the number of anions in the bulk and \(nH_2O\) is the number of adsorbing water molecules. \(\Gamma_O\) / \(\Gamma_{H_2O}\) is the excess oxygen / water at the surface and \(\mu_O\) / \(\mu_{H_2O}\) is the oxygen / water chemcial potential. Clearly \(\Gamma\) and \(mu\) will only matter when the surface is non stoichiometric.
Temperature¶
The above phase diagram is at 0K. It is possible to use experimental data from the NIST_JANAF database to make the chemical potential a temperature dependent term and thus generate a phase diagram at a temperature (T). This is done according to
where
\(\mu_O\) (T) (0 K , DFT) is the 0K free energy of an isolated oxygen molecule evaluated with DFT, \(\mu_O\) (T) (0 K , EXP) is the 0 K experimental Gibbs energy for oxygen gas and $Delta$ \(G_{O_2}\) ( \(\Delta\) T, Exp) is the Gibbs energy defined at temperature T as
This will generate a phase diagram at temperature (T)
Pressure¶
Chemical potential can be converted to pressure values according to
where P is the pressure, \(\mu\) is the chemical potential of oxygen, \(k_B\) is the Boltzmnann constant and T is the temperature.
Pressure vs temperature¶
Surfinpy has the functionality to generate phase diagrams as a function of pressure vs temperature based upon the methodology used in Molinari et al according to
where \(\gamma_{adsorbed, T, p}\) is the surface energy of the surface with adsorbed species at temperature (T) and pressure (P), \(\gamma_{bare}\) is the suface energy of the bare surface, C is the coverage of adsorbed species, \(E_{ads}\) is the adsorption energy,
where \(E_{slab, adsorbant}\) is the energy of the surface and the adsorbed species, \(n_{H_2O}\) is he number of adsorbed species,
where \(S_{(T)}\) is the experimental entropy of gaseous water in the standard state.