Joint density functional theory of the electrode-electrolyte interface: Application to fixed electrode potentials, interfacial capacitances, and potentials of zero charge

This work explores the use of joint density functional theory, an extension of density functional theory for the ab initio description of electronic systems in thermodynamic equilibrium with a liquid environment, to describe electrochemical systems. After reviewing the physics of the underlying fundamental electrochemical concepts, we identify the mapping between commonly measured electrochemical observables and microscopically computable quantities within an, in principle, exact theoretical framework. We then introduce a simple, computationally efficient approximate functional which we find to be quite successful in capturing a priori basic electrochemical phenomena, including the capacitive Stern and diffusive Gouy-Chapman regions in the electrochemical double layer, quantitative values for interfacial capacitance, and electrochemical potentials of zero charge for a series of metals. We explore surface charging with applied potential and are able to place our ab initio results directly on the scale associated with the standard hydrogen electrode (SHE). Finally, we provide explicit details for implementation within standard density functional theory software packages at negligible computational cost over standard calculations carried out within vacuum environments.

Figure 1

(a) Schematic of an electrochemical cell. The working electrode is explicitly modeled while the reference electrode is fixed at zero. (b) Relationship between the microscopic electron potential $\langle \phi \left(z\right)\rangle$ (averaged over the directions parallel to the surface), electrochemical potential, and applied potential for a Pt (111) surface. The large variations in potential to the left of $z_{\mathrm{Pt}}$ correspond to the electrons and ionic cores comprising the metal while the decay into the fluid region is visible to the right of $z_{\mathrm{Pt}}$.

Figure 2

Microscopic and model quantities for Pt(111) surface in equilibrium with electrolyte: (a) Pt atoms (white), electron density $n\left(r\right)$ (green), and fluid density $N_{lq}\left(r\right)$ (blue) in a slice passing from surface (left) into the fluid (right) with $z_{\mathrm{Pt}}=5.95$ Å indicating the end of the metal; (b) dielectric constant $\langle \epsilon \left(z\right)\rangle$ and screening length $\langle {\kappa }^{-1}\left(z\right)\rangle$ (averaged over the planes parallel to the surface) for ionic concentrations of 1.0 and 0.1 M along a line passing from surface into the fluid. Position $z-z_{\mathrm{Pt}}$ measures distance from the end of the metal slab. (See Secs. 5 and 6.)

Figure 3

Microscopic electron potential energies $\langle \phi \left(z\right)\rangle$ and macroscopic electrostatic potentials $\langle \Phi \left(z\right)\rangle$ averaged in planes for the Pt (111) surface as a function of distance $z-z_{\mathrm{Pt}}$ from the end of the metal surface: (a) $\langle \phi \left(z\right)\rangle$ for surface with applied voltage $\mathcal{E}=-1.09\mathrm{V}$ vs PZC in vacuum (green dashed) and in monovalent electrolytes of $c=1.0\mathrm{M}$ (red) and $c=0.1\mathrm{M}$ (blue) where the dotted lines represent calculations with an explicit counterelectrode and the solid lines are JDFT calculations; (b) close-up view of $\langle \Phi \left(z\right)\rangle$ for JDFT calculations with $c=1.0\mathrm{M}$ (red) and $c=0.1\mathrm{M}$ (blue) and applied voltage $\mathcal{E}=-1.09\mathrm{V}$ vs PZC (almost indistinguishable in the previous plot); (c) variation of $\langle \Phi \left(z\right)\rangle$ in JDFT monovalent electrolyte of $c=1.0\mathrm{M}$ with $\mathcal{E}=\{-1.09,-0.55,0.0,0.55,1.09\}\mathrm{V}$ vs PZC.

Figure 4

(a) Surface charge $\sigma$ as a function of applied voltage $\mathcal{E}$ for a series of transition-metal surfaces in an electrolyte of monovalent ionic strength $c=1.0\mathrm{M}$. (b) Inverse dielectric constant ${\epsilon }^{-1}$ as a function of distance from a Pt(111) surface for multiple values of applied voltage. (c) Inverse gap capacitance ${\mathcal{C}}_{\Delta }^{-1}$ as a function of the distance from the metal surface at which the fluid begins $\Delta$. The solid line indicates the best fit to the data with slope constrained to ${\epsilon }_0^{-1}$.

Figure 5

Comparisons of ab initio predictions and experimental data (Ref. 43) for potentials of zero free charge (PZC's) and vacuum work functions: (a) ab initio LDA predictions versus experimental PZC relative to SHE; (b) ab initio GGA predictions versus experimental PZC relative to SHE; (c) ab initio GGA vacuum work functions (solid line with squares) and PZC's (solid line with circles), experimental work functions (dotted line), and PZC's (dashed line) versus vacuum for the same series of surfaces. Best linear fits with unit slope [dark diagonal solid lines in (a) and (b)].