Implicit Solvation Methods for Catalysis at Electrified Interfaces

Abstract

Implicit solvation is an effective, highly coarse-grained approach in atomic-scale simulations to account for a surrounding liquid electrolyte on the level of a continuous polarizable medium. Originating in molecular chemistry with finite solutes, implicit solvation techniques are now increasingly used in the context of first-principles modeling of electrochemistry and electrocatalysis at extended (often metallic) electrodes. The prevalent ansatz to model the latter electrodes and the reactive surface chemistry at them through slabs in periodic boundary condition supercells brings its specific challenges. Foremost this concerns the difficulty of describing the entire double layer forming at the electrified solid-liquid interface (SLI) within supercell sizes tractable by commonly employed density functional theory (DFT). We review liquid solvation methodology from this specific application angle, highlighting in particular its use in the widespread ab initio thermodynamics approach to surface catalysis. Notably, implicit solvation can be employed to mimic a polarization of the electrode's electronic density under the applied potential and the concomitant capacitive charging of the entire double layer beyond the limitations of the employed DFT supercell. Most critical for continuing advances of this effective methodology for the SLI context is the lack of pertinent (experimental or high-level theoretical) reference data needed for parametrization.

Figure 1

Ab initio thermodynamics approach to electrified solid–liquid interfaces as occurring in electrocatalysis. The electrode is here negatively charged, and this surface charge is compensated by the buildup of counter charge in the electrolyte. The formed electric double layer (DL) can be pictured as a localized capacitor at the interface of electrode and a rather rigid layer of ions (inner DL or Helmholtz layer) and a long-range contribution (outer or diffuse DL). This leads to an exponential decay of the electrostatic potential along the surface normal in the diffuse DL which is illustrated by the plot of the electrostatic potential averaged over the electrode surface (xy) plane. As in particular the spatial extent of the diffuse DL challenges efficient first-principles calculations, the ab initio thermodynamics approach considers a grand-canonical ensemble in which a finite supercell computed, e.g., with DFT, is in equilibrium with appropriate reservoirs for the catalyst atoms, solvent species, and electrons. Since the supercell does then generally not comprise the entire DL, it misses part of the compensating charge and does not necessarily have to be overall charge neutral.

Figure 2

Hierarchy of coarse-graining approaches for the liquid phase in the context of electrocatalysis at SLIs. The sketch depicts an aqueous electrolyte with salt ions (blue spheres, described at a varying level of theory) and dissolved CO₂ (red and black molecules) at a crystalline surface (described throughout on a quantum chemical level). Starting from a fully explicit quantum mechanical description (far left, indicated by electron density isosurfaces) one can conceptually coarse-grain away electronic DOFs to arrive at a force field or interatomic potential description (center left). From that one can gradually remove nuclear solvent DOFs to represent solvent molecules, e.g., only through their spatial distributions or correlation functions as in RISM-type models (center right). Finally, replacing even this with simply a polarizable continuum, one arrives at fully implicit models (far right). Note that in the derivation and parametrization of each coarse-grained level one does not necessarily need to follow each step and can, e.g., directly parametrize an implicit model from fully explicit data.

Figure 3

Categorization of different electrostatic solvation models. From the general starting point of a static nonlinear, nonlocal (“nl”), and anisotropic model (top), several approximations can be made to ultimately arrive at the linear, local, and isotropic polarization model most commonly applied in present-day DFT codes. In this figure, we have in addition to our standard notation in this review (bold symbols indicating vectors and tensors) utilized arrows to improve the readibility.

Figure 4

Illustration of different types of dielectric transition between solute and solvent. For the example of an adsorbed CO₂ molecule at a single-crystal surface, (a) shows the solvation cavity resulting from the superposition of atom-centered spheres based on eq 13 and (b) shows the solvation cavity as defined by an isosurface of the electron density.

Figure 5

Schematic representation of various electrolyte models currently used for the description of SLIs. Planar counter charge (PCC) models place rigid ions in a Helmholtz-layer-like arrangement, while Poisson–Boltzmann (PB) models determine the ionic distribution self-consistently in the total electrostatic potential. Various important modifications of PB theory are highlighted and discussed in the text.

Figure 6

Creation of an ion-free Stern layer around a molecular solute. Compared are the solvation environments around the center of mass (COM) of naphthalene in a 2.18 M NaCl solution as obtained from explicit molecular dynamics simulations (dashed lines) and with a Stern layer corrected implicit MPB model (solid lines). Data in red represent the spherically averaged radial distribution function (RDF) of the oxygen atoms in the explicit water solvent (g_H2O) and the corresponding spherically averaged dielectric function ε for the implicit model. Data in black are the spherically averaged RDF for the ions and the corresponding ion-exclusion function α_ion. Both the onset of the solute solvation shell and the radial Stern layer shift of the ionic distribution are rather well reproduced. To better grasp the involved scales, two dashed vertical lines illustrate the radial distance to the molecule COM as shown in the top view in the inset. Adapted with permission from ref (132). Copyright 2017 American Institute of Physics.

Figure 7

Number of solvation energy entries (“training set size”) per nonaqueous solvent in the three largest corresponding experimental databases. The solvents are sorted according to their largest training set size in all the three databases. From our previous work, we estimate even heterogeneous training sets with sizes below 50 to be potentially prone to significant overfitting errors.

Figure 8

Relative trend of PZC values as obtained in experiments and implicit solvent calculations. Experimental PZCs for the low-index surfaces of Ag, Cu, Au, and Pt (gray line) are on the absolute scale and taken as averages of literature data compiled in the SI of ref (26). Calculated PZCs are arbitrarily aligned to the experimental PZC of Au(111) and are taken from refs (26) (red) and (336) (blue).

Figure 9

Surface phase diagram of Pt(111) in water as determined within the CHE approach. Shown are computed potential-dependent surface free energies of bare Pt(111) and various H, OH, and O coverages on it. Within the ab initio thermodynamics framework, surface terminations with the lowest surface free energy are declared as the most stable one at the corresponding potential. This yields the indicated gradual transition from H-covered over bare surface to OH- and O-covered terminations with increasingly positive potential. Reproduced from ref (398). Copyright 2017 American Chemical Society.

Figure 10

Dependence of the interfacial capacitance on implicit solvation model parameters. Shown is the variation of the fixed-composition total grand potential energy around the PZC for a model Li(110) electrode in implicit ethylene carbonate (EC) solvent (ε_r = 89.9). The parabolic variation nicely reflects the Taylor expansion of eq 45 and allows fitting of the interfacial capacitance. (left) Variation as a function of the bulk permittivity employed in the implicit solvation model. (right) Variation as a function of the threshold charge density (called n_c) employed to define the solvation cavity. Reproduced from ref (402). Copyright 2015 American Chemical Society.

Figure 11

Theoretical surface Pourbaix diagram of Cu(100) in implicit water considering H and CO adsorbates. The diagram obtained within the CHE approximation (left) shows only a trivial Nernstian pH dependence, which vanishes on the here-employed RHE scale. In contrast, nontrivial pH dependencies are obtained with constant-potential (FGC) calculations (right). Figure created from data published in ref (457).