Electronic structure calculations of liquid-solid interfaces: Combination of density functional theory and modified Poisson-Boltzmann theory

A robust and efficient computational method for electronic structure calculations of liquid-solid interfaces is presented. The theory employs the density functional theory and a modified Poisson-Boltzmann theory, combining them through a smooth dielectric model function. The free energy, including electrostatic and nonelectrostatic interactions between solutes and the solvation medium, is formulated, and its first derivatives with atomic positions are presented. This methodology is applied to two different topics; one is the potential of zero charge (PZC) of Pt(111), and the other is a poisoning of active sites for the oxygen-reduction reaction (ORR) by interfacial water molecules on Pt(111). The results of the first topic show that induced charge redistributions caused by the adsorption of water molecules form a surface dipole moment that dominates the experimentally observed negative shift in the PZC when platinum is immersed in an aqueous electrolyte. The results of the second topic show the possibility of a decrease in the surface coverage of the first reaction precursor to the ORR due to site blocking by the adsorbed water molecules.

Figure 1

Schematic of models: (a) a slab sandwiched by solvation mediums and (b) a slab sandwiched by the solvation medium and the vacuum. $z_0$ and $z_1$ denote the positions of the nucleus in the bottom and top layers, respectively, of the slab.

Figure 2

Distributions of relative permittivity $\epsilon$ for (a) a slab sandwiched by solvation mediums and (b) a slab sandwiched by the solvation medium and the vacuum and (c) the top view of the slab. The distributions are on the cross section shown as a dotted arrow in (c). The line spacing of the contour plot is 10. $z_0$ and $z_1$ denote the positions of the nucleus in the bottom and top layers, respectively, of the slab.

Figure 3

Free energies for nonelectrostatic interactions as functions of the surface area of solutes, $S\left({\text{Å}}^{-2}\right)$, (a) dispersion free energies, $\Delta {\Omega }_{\text{SS},\text{dr}}\left(\text{eV}\right)$, and (b) repulsion free energies, $\Delta {\Omega }_{\text{SS},\text{rep}}\left(\text{eV}\right)$. The circles are the free energies given from GAUSSIAN03. The solid lines are the fitted free energies to the circles.

Figure 4

Flowchart for the SCF procedure. The shaded squares mean additional parts needed for systems including the solvation medium.

Figure 5

(Color online) Top views of the models for the PZC calculations. The coverages for water molecules are (a) 0, (b) 1/4, (c) 1/2, (d) 1/6, (e) 1/3, (f) 1/2, (g) 2/3, and (h) 2/3 ML. The dotted lines show translation unit cells. The big, medium, and small spheres show Pt, O, and H atoms, respectively.

Figure 6

PZC $U_{\text{PZC}}$ (V) (SHE) as a function of the coverage for water molecules and ${\theta }_{\text{H}2\text{O}}$ (ML) from vacuum calculations. The squares are calculated PZCs. The solid line is the experimentally measured PZC in solutions, and the dotted line is the experimentally measured PZC in the vacuum.

Figure 7

Potential drop $\Delta {\varphi }_{\text{dip}}$ (V) caused by the net dipole moment in the $z$ direction of the system as a function of the change in the PZC by adsorptions of water molecules, $U_{\text{PZC}}\left({\theta }_{\text{H}2\text{O}}\right)-U_{\text{PZC}}\left(0\right)$ (V). The circles correspond to calculated data. The solid line is a diagonal line.

Figure 8

Potential drops $\Delta {\varphi }_{\text{dip}}$ (V) caused by the total dipole moment, the dipole moment by water orientations, and the dipole moment due to charge redistributions caused by the adsorption as functions of the coverage of water molecules, ${\theta }_{\text{H}2\text{O}}$ (ML).

Figure 9

PZC $U_{\text{PZC}}$ (V) (SHE), calculated as $U_{\text{PZC}}\left(0\right)-\Delta {\varphi }_{\text{dip},\text{pol}}\left({\theta }_{\text{H}2\text{O}}\right)$, as a function of the coverage of water molecules, ${\theta }_{\text{H}2\text{O}}$ (ML). The solid and dotted lines are the same as those in Fig. 6.

Figure 10

Charge redistributions caused by the adsorption of water molecules for model d. $\Delta \rho$ is the change in the electron density due to adsorption of water.

Figure 11

PZCs $U_{\text{PZC}}$ (V) (SHE) calculated from three different methods. The squares denote the PZCs calculated from vacuum calculations, the triangles denote the PZCs calculated from vacuum calculations with the correction to dipole moments caused by water orientations, and the circles denote the PZCs calculated from solvation calculations. The solid and dotted lines are same as those in Fig. 6.

Figure 12

Charge redistributions $\Delta {\rho }_{\epsilon }$ and $\Delta \rho$ caused by the solvation medium and the adsorption of explicit water molecules for model d.

Figure 13

Schematic of the model for the adsorption of the oxygen molecule on the surface in the solution.

Figure 14

(Color online) Top view of the surface (a) before and (b) after the adsorption of the oxygen molecule.

Figure 15

Total free energies ${\Omega }_{\text{tot}}\left(\text{eV}\right)$ before and after the adsorption of the oxygen molecule as functions of the electrode potential $U$ (V) (SHE). (a) Results by RPBE and (b) results by PBE. The triangles and circles are for systems before and after the adsorption, respectively. The solid lines are free energy curves fitted to the calculated data using least square fittings with quadratic functions.

Figure 16

Free energy changes $\Delta {\Omega }_{\text{tot}}\left(\text{eV}\right)$ caused by the adsorption of oxygen molecule as a function of electrode potential $U$ (V) (SHE).

Figure 17

Coverage of the adsorbed oxygen molecule, ${\theta }_{\text{O}2}$ (ML), as a function of electrode potential $U$ (V) (SHE).