First-principles kinetic Monte Carlo simulations for heterogeneous catalysis: Application to the CO oxidation at $\mathrm{Ru}{\mathrm{O}}_2\left(110\right)$

We describe a first-principles statistical mechanics approach enabling us to simulate the steady-state situation of heterogeneous catalysis. In a first step, density-functional theory together with transition-state theory is employed to obtain the energetics of the relevant elementary processes. Subsequently the statistical mechanics problem is solved by the kinetic Monte Carlo method, which accounts for the correlations, fluctuations, and spatial distributions of the chemicals at the surface of the catalyst under steady-state conditions. Applying this approach to the catalytic oxidation of CO at $\mathrm{Ru}{\mathrm{O}}_2\left(110\right)$, we determine the surface atomic structure and composition in reactive environments ranging from ultra-high vacuum (UHV) to technologically relevant conditions, i.e., up to pressures of several atmospheres and elevated temperatures. We also compute the $\mathrm{C}{\mathrm{O}}_2$ formation rates (turnover frequencies). The results are in quantitative agreement with all existing experimental data. We find that the high catalytic activity of this system is intimately connected with a disordered, dynamic surface “phase” with significant compositional fluctuations. In this active state the catalytic function results from a self-regulating interplay of several elementary processes.

Figure 1

(Color online) Potential-energy surface (schematic) along the minimum energy path connecting the free gas phase and adsorbed state in surface site $\mathit{st}$. Shown is a case where the adsorption process is exothermic $(E_{b,\mathit{st},i}<0)$, but activated $(\Delta E_{\mathit{st},i}^{\mathrm{ad}}>0)$.

Figure 2

Top view of the $\mathrm{Ru}{\mathrm{O}}_2\left(110\right)$ surface showing the two prominent adsorption sites (bridge and cus), which continue the bulk-stacking sequence (central panel). When focusing on these two site types, the surface can be coarse grained to the lattice model shown schematically to the right. Additionally shown are two perspective views to the left, exemplifying what the atomic structure of the surface looks like, if all bridge sites are occupied with oxygen atoms and the cus sites remain empty (${\mathrm{O}}^{\mathrm{br}}∕-$ termination, top left panel), or if oxygen atoms occupy both site types (${\mathrm{O}}^{\mathrm{br}}∕{\mathrm{O}}^{\mathrm{cus}}$ termination, bottom left panel). $\mathrm{Ru}=\text{light}$, large spheres, $\mathrm{O}=\text{dark}$, medium spheres. Atoms lying in deeper layers have been whitened in the topview for clarity.

Figure 3

(Color online) Potential-energy surface (PES) for associative oxygen desorption from two neighboring cus sites into an ${\mathrm{O}}_2$ molecule with orientation parallel to the surface and aligned along the [001] trench (as schematically shown in the inset). The height above the topmost $\left({\mathrm{Ru}}_2{\mathrm{O}}_2\right)$ surface and the distance between the two O atoms are used as coordinates of the plot. For every (height, bondlength) of the parallel ${\mathrm{O}}_2$ molecule, all atomic positions of the oxide substrate are fully relaxed. Actually calculated points are indicated, and the energy zero corresponds to a free ${\mathrm{O}}_2$ molecule far above the surface.

Figure 4

(Color online) Time evolution of the site occupation by O and CO of the two prominent adsorption sites, bridge and cus, cf. Fig. 2. The temperature and pressure conditions chosen ($T=600\mathrm{K}$, $p_{\mathrm{C}\mathrm{O}}=7\mathrm{atm}$, $p_{{\mathrm{O}}_2}=1\mathrm{atm}$) correspond to optimum catalytic performance. Under these conditions kinetics builds up a steady-state surface population in which O and CO compete for either site type at the surface, as reflected by the strong fluctuations in the site occupations within the $(20\times 20)$ simulation cell. Note the extended time scale, also for the “induction period,” until the steady-state populations are reached when starting from a purely oxygen-covered surface.

Figure 5

(Color online) Steady-state surface structures obtained by first-principles kMC simulations, where the surface reaction events are switched off (see text). In all nonwhite areas, the average site occupation is dominated $(>90\%)$ by one species, i.e., either O, CO, or empty sites (−). The labels correspondingly characterize the surface populations by indicating this majority species at the bridge and cus sites. The unlabeled small region just appearing in the upper left edge of the figure corresponds to a fully CO-covered surface, $\mathrm{C}{\mathrm{O}}^{\mathrm{br}}∕\mathrm{C}{\mathrm{O}}^{\mathrm{cus}}$. Note that in the thermodynamic case the stability regions of the various surface phases depend only on the gas phase chemical potentials ${\mu }_i$ ($i=\mathrm{O}$, CO), leading to a simple scaling of the pressure axes at different temperatures, cf. Eq. (8). The only exception to this is the width of the white coexistence regions, which increases with temperature. The one shown here corresponds to $600\mathrm{K}$. Above the dashed line bulk $\mathrm{Ru}{\mathrm{O}}_2$ is thermodynamically unstable against CO-induced decomposition (see text).

Figure 6

(Color online) Left panel: Steady-state surface structures obtained by first-principles kMC simulations at $T=600\mathrm{K}$ (see Fig. 5 for an explanation of the labeling). Right panel: Map of the corresponding turn-over frequencies (TOFs) in ${\text{site}}^{-1}{\mathrm{s}}^{-1}$ $\left({10}^{15}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}\right)$: White areas have a TOF $<{10}^{-4}{\mathrm{site}}^{-1}{\mathrm{s}}^{-1}$ $(<{10}^{11}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1})$, and each increasing gray level represents one order of magnitude higher activity. Thus, the darkest gray region corresponds to TOFs above ${10}^2\mathrm{mol}.{\text{site}}^{-1}{\mathrm{s}}^{-1}$ $\left({10}^{17}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}\right)$, while in a narrow $(p_{\mathrm{C}\mathrm{O}},p_{{\mathrm{O}}_2})$ region the TOFs actually peak over ${10}^4{\text{site}}^{-1}{\mathrm{s}}^{-1}$ $\left({10}^{19}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}\right)$. The plots are based on $\approx 400\mathrm{kMC}$ simulations for different $(p_{\mathrm{C}\mathrm{O}},p_{{\mathrm{O}}_2})$ conditions.

Figure 7

(Color online) Left panel: Steady-state surface structures obtained by first-principles kMC simulations at $T=350\mathrm{K}$ (see Fig. 5 for an explanation of the labeling). Right panel: Map of the corresponding turn-over frequencies (TOFs) in ${\text{site}}^{-1}{\mathrm{s}}^{-1}$ $\left({10}^{15}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}\right)$: White areas have a TOF $<{10}^{-12}{\text{site}}^{-1}{\mathrm{s}}^{-1}$ $(<{10}^3{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1})$, and each increasing gray level represents one order of magnitude higher activity. Thus, the black region corresponds to TOFs above ${10}^{-4}{\text{site}}^{-1}{\mathrm{s}}^{-1}$ $\left({10}^{11}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}\right)$, while in a narrow $(p_{\mathrm{C}\mathrm{O}},p_{{\mathrm{O}}_2})$ region the TOFs actually peak over ${10}^{-3}{\text{site}}^{-1}{\mathrm{s}}^{-1}$ $\left({10}^{12}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}\right)$. The plots are based on $\approx 400\mathrm{kMC}$ simulations for different $(p_{\mathrm{C}\mathrm{O}},p_{{\mathrm{O}}_2})$ conditions.

Figure 8

(Color online) Snapshot of the steady-state surface population under optimum catalytic conditions at $T=600\mathrm{K}$ ($p_{{\mathrm{O}}_2}=1\mathrm{atm}$, $p_{\mathrm{C}\mathrm{O}}=7\mathrm{atm}$). Shown is a schematic top view of the simulation area of $(20\times 20)$ surface sites, where the substrate bridge sites are marked by gray stripes and the cus sites by white stripes. Oxygen adatoms are drawn as light gray (red) circles and adsorbed CO molecules as dark gray (blue) circles. Movies of entire kMC simulations over $500\mathrm{ns}$ and $1\mathrm{s}$ at various pressure and temperature conditions are also available.[53]

Figure 9

(Color online) Snapshot of the steady-state surface population under optimum catalytic conditions at $T=350\mathrm{K}$ ($p_{{\mathrm{O}}_2}={10}^{-10}\mathrm{atm}$, $p_{\mathrm{C}\mathrm{O}}=4\times {10}^{-10}\mathrm{atm}$). Shown is a schematic top view of the simulation area of $(20\times 20)$ surface sites, where the substrate bridge sites are marked by gray stripes and the cus sites by white stripes. Oxygen adatoms are drawn as light gray (red) circles and adsorbed CO molecules as dark gray (blue) circles. Movies of entire kMC simulations over $500\mathrm{ns}$ and $1\mathrm{s}$ at various pressure and temperature conditions are also available.[53]