Voltage-dependent cluster expansion for electrified solid-liquid interfaces: Application to the electrochemical deposition of transition metals

The detailed atomistic modeling of electrochemically deposited metal monolayers is challenging due to the complex structure of the metal-solution interface and the critical effects of surface electrification during electrode polarization. Accurate models of interfacial electrochemical equilibria are further challenged by the need to include entropic effects to obtain accurate surface chemical potentials. We present an embedded quantum-continuum model of the interfacial environment that addresses each of these challenges and study the underpotential deposition of silver on the gold (100) surface. We leverage these results to parametrize a cluster expansion of the electrified interface and show through grand canonical Monte Carlo calculations the crucial need to account for variations in the interfacial dipole when modeling electrodeposited metals under finite-temperature electrochemical conditions.

Figure 1

Equilibrium voltages extracted from the quantum-continuum calculations. Voltages on the absolute scale (left axis) are aligned to the SHE scale (right axis) by recovering the experimental potential of zero charge of the bare surface. Noncompact (dispersed) configurations (lower insets) exhibit larger interfacial dipoles compared to compact (island forming) structures (upper insets).

Figure 2

Effects of the differential capacitance on the voltage-dependent binding energies $\mathcal{F}(\mathbf{\sigma },\mathrm{\Phi })$ for $C_0=0$ and $30\mu \mathrm{F}/{\mathrm{cm}}^2$ (green/blue). The enhanced binding energy of intermediate coverages is driven by their large interfacial dipoles. Predicted energies from the cluster expansion are overlaid as solid circles. Ground state structures are identified with thicker markers.

Figure 3

Clusters identified from the cluster selection process. Clusters with diameters that sample up to fourth nearest neighbors and cluster sizes up to quadruplets were included in the search. Sampled sites are shown in blue (except the empty point cluster shown in white).

Figure 4

Theoretical adsorption isotherms obtained for a bulk solution silver concentration of $\left[{\mathrm{Ag}}^{+}\right]={10}^{-2}\mathrm{M}$. Isotherms were obtained using both the Monte Carlo (MC) and common tangent method (CTM) for differential capacitance values of 0 and $30\mu \mathrm{F}/{\mathrm{cm}}^2$. Coverages were averaged over 20 000 Monte Carlo steps after 5000 Monte Carlo steps of warm up with standard deviations lower than $5\times {10}^{-2}$.