The Steady State of Heterogeneous Catalysis, Studied by First-Principles Statistical Mechanics

The turnover frequency of the catalytic oxidation of CO at $\mathrm{R}\mathrm{u}{\mathrm{O}}_2(110)$ was calculated as a function of temperature and partial pressures using ab initio statistical mechanics. The underlying energetics of the gas-phase molecules, dissociation, adsorption, surface diffusion, surface chemical reactions, and desorption were obtained by all-electron density-functional theory. The resulting $\mathrm{C}{\mathrm{O}}_{\mathrm{2}}$ formation rate [in the full ($T,p_{\mathrm{C}\mathrm{O}},p_{{\mathrm{O}}_{\mathrm{2}}}$) space], the movies displaying the atomic motion and reactions over times scales from picoseconds to seconds, and the statistical analyses provide insight into the concerted actions ruling heterogeneous catalysis and open thermodynamic systems in general.

Figure 1

(a) Perspective view of the $\mathrm{R}\mathrm{u}{\mathrm{O}}_{\mathrm{2}}\mathrm{(}\mathrm{110}\mathrm{)}$ surface, illustrating the two prominent adsorption sites in the rectangular surface unit cell. These sites are labeled as the br (bridge) and the cus (coordinatively unsaturated) site, and both are occupied in the example with oxygen atoms. This is the “high $p_{{\mathrm{O}}_{\mathrm{2}}}$ termination” [2, 3]. (b) Time evolution of the site-occupation numbers from the thermodynamic surface termination shown in (a) to the steady state of catalysis [cf. snapshot (c)]. The temperature is $T=600\mathrm{K}$ and the CO and ${\mathrm{O}}_{\mathrm{2}}$ partial pressures are that of the optimum reaction conditions: $p_{\mathrm{C}\mathrm{O}}=20\mathrm{a}\mathrm{t}\mathrm{m}$; $p_{{\mathrm{O}}_{\mathrm{2}}}=1\mathrm{a}\mathrm{t}\mathrm{m}$. (c) Snapshot (top view of the $\mathrm{R}\mathrm{u}{\mathrm{O}}_2(110)$ surface) from movies [19] displaying the dynamics of the surface under realistic catalytic conditions. Though the full atomic structure is considered in the calculations, for clarity we have marked here the substrate bridge sites only by gray stripes and the cus sites by white stripes. Oxygen adatoms are drawn as (light) red circles and adsorbed CO molecules as (dark) blue circles. Movies are also available for various pressure conditions for runtimes of 500 ns, as well as 1 s [19].

Figure 2

(a) Steady-state surface structures obtained by ab initio kMC calculations at $T=600\mathrm{K}$ and various CO and ${\mathrm{O}}_{\mathrm{2}}$ partial pressures. In all nonwhite areas, the average site occupation is dominated ($>90\%$) by one species, i.e., either O, CO, or empty sites ($-$). (b) Map of the corresponding turnover frequencies (TOFs) in ${\mathrm{c}\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$: white areas have a $\mathrm{T}\mathrm{O}\mathrm{F}<{10}^{11}{\mathrm{c}\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$, and each increasing gray level represents 1 order of magnitude higher activity. Thus, the darkest region corresponds to a TOF higher than ${10}^{17}{\mathrm{c}\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$.

Figure 3

Rate of $\mathrm{C}{\mathrm{O}}_{\mathrm{2}}$ formation at $T=350\mathrm{K}$. The experimental steady-state results of Wang et al. [15] are presented as dotted lines, and the theoretical results are shown as solid lines. Rates given as function of $p_{{\mathrm{O}}_{\mathrm{2}}}$ at $p_{\mathrm{C}\mathrm{O}}={10}^{-10}\mathrm{a}\mathrm{t}\mathrm{m}$ (left), and as function of $p_{\mathrm{C}\mathrm{O}}$ at $p_{{\mathrm{O}}_{\mathrm{2}}}={10}^{-10}\mathrm{a}\mathrm{t}\mathrm{m}$ (right).