Structure and reactivity of ruthenium nanoparticles

We present a method for obtaining detailed structural information of ruthenium nanoparticles in at least the diameter range from $1.5\text{to}5\mathrm{nm}$. The method is based on an ensemble approach where a large number of low-energy structures are collected in an ensemble, from which average properties can be extracted using Boltzmann averaging. The method is used to obtain the number of catalytic active step sites present on the surface of the ruthenium particles. We find that the presence of highly catalytic active step sites does not depend significantly on the temperature within a relevant temperature range; the presence of step sites is mainly a function of the lowest energy shape of the cluster, i.e., a function of the number of atoms. By combining the structural information with estimations of the single site activities in the ammonia synthesis, we find that the optimal particle diameter is approximately $3\mathrm{nm}$. The single site activities are estimated by using density functional theory to calculate the barrier of the rate limiting step, the dissociation of a nitrogen molecule.

Figure 1

Some of the surfaces for which energies have been calculated using DFT. The calculations are used to determine the energy of the gray atoms. Panel (a) shows the close-packed surface (0001). Panel (b) shows the $\left(10\overline{1}1\right)$ surface. The surfaces on panels (c) and (d) are repeated edges between the (0001) surface and the $\left(10\overline{1}1\right)$ surface with and without a step, respectively.

Figure 2

(Color) A comparison between the energy obtained using four different methods. The estimated EMT, the estimated DFT, and the simple bond breaking model are all compared to the full EMT energy. Each point corresponds to a certain configuration of a 10 046 atom cluster. Points on black line indicate perfect agreement between the full EMT energy and the method in question. Energies are relative to the energy of the found ground state configuration. From this, we conclude that the estimated DFT is the preferred method in our situation (see text).

Figure 3

Flowchart of the algorithm that collects an ensemble of relevant structures for a given cluster.

Figure 4

The definition of the 16 parameters $\{l_0-l_3,h_{0,u}-h_{5,u},h_{0,d}-h_{5,d}\}$ describing the cluster shape. $\{l_0-l_3\}$ are the dimensions of the basal plane, whereas $\{h_{0,u}-h_{5,u}\}$ and $\{h_{0,d}-h_{5,d}\}$ are the number of layers up and down to the edges between the (0001) surfaces and $\left(10\overline{1}1\right)$ surfaces.

Figure 5

Some different ways, one could remove nine atoms (gray) from a cluster. (a)–(c) show examples where all removed atoms are in one group. (d) shows an example where the atoms are removed in two groups (one of five and one of four atoms). The configuration shown in (d) is the found ground state configuration for a 1670 atom cluster.

Figure 6

Only configurations with a energy below a limit value are saved and used in the further calculations. The graph shows the found number of sites divided by the found number of sites with a limit energy of $3\mathrm{eV}$ above the ground state level for a 4000 atoms cluster as a function of the limit energy. Each line corresponds to a certain site type. The inset shows the same thing where the found number of sites are divided by the found number of sites with a limit energy of $2\mathrm{eV}$ instead of $3\mathrm{eV}$. The asymptotic behavior when approaching $3\mathrm{eV}$ shows that $3\mathrm{eV}$ is a sufficiently high limit, whereas the inset shows that a limit of $2\mathrm{eV}$ is insufficient.

Figure 7

An estimation of the statistical uncertainty in the calculations. The abundance of 22 different sites has been estimated in three completely independent calculations on a 4000 atoms cluster. For each site, the three estimations on the number of sites have been used to calculate the relative standard deviation. The graph shows the average of these relative standard deviations taken over the 22 different sites as a function of temperature. The low values (below $2\times {10}^{-3}$ show that the statistical error is negligible).

Figure 8

Definitions of four different edges. Completed edges are seen to the left and stepped edges to the right. The color differences indicate the definitions of the sites. A completed edge site is defined by a dark gray atom having exactly the nearest neighbors sitting as the light gray atoms does. A stepped edge site is defined by a free position having exactly the nearest neighbors sitting as the light gray atoms. Edges A and B connect the (0001) surfaces and $\left(10\overline{1}1\right)$ surfaces. Edge C connects two $\left(1010A\right)$ surfaces. Edge D connects two $\left(10\overline{1}1\right)$ surfaces. Step site B is known as the $B_5$ site.

Figure 9

(Color online) The number of step sites as a function of the number of atoms in the cluster. The four step sites are defined in Fig. 8. The black curve is at a temperature of $300\mathrm{K}$, whereas the gray (light blue) is at $1200\mathrm{K}$. 400 atoms corresponds to a cluster with approximately $1.8\mathrm{nm}$ diameter.

Figure 10

(Color online) The number of step sites as a function of the number of atoms in the cluster. The four step sites are defined in Fig. 8. The black curve is at a temperature of $300\mathrm{K}$, whereas the gray (light blue) is at $1200\mathrm{K}$. 1000 atoms corresponds to a cluster with approximately $2.7\mathrm{nm}$ diameter.

Figure 11

(Color online) The number of step sites per volume as a function of the cluster diameter. The four step sites are defined in Fig. 8. The black curve is at a temperature of $300\mathrm{K}$, whereas the gray (light blue) is at $1200\mathrm{K}$. Each point is an average of approximately 60 different clusters. The uncertainty in the diameter indicates the spread in cluster sizes, whereas the uncertainty in the number of sites is the average standard deviation of the number of sites obtained from the 60 different clusters. The unit in the y axis is ${\mathrm{nm}}^{-3}$.

Figure 12

(Color online) The splitting of a nitrogen molecule on the four different step sites defined in Fig. 8. The initial states are just local minima and not the lowest-energy absorption states. However, in all cases, the barrier between the lowest-energy absorption states and the initial state is lower than the barrier between the initial state and the final state, i.e., the energies of the transition states are the true barrier energies for the dissociation.

Figure 13

(Color online) The energy as a function of the nitrogen bond length for the dissociation of a nitrogen molecule on the four different sites defined in Fig. 8. All energies are relative to the gas phase energy. The lines are cubic splines connecting the actual calculation points.

Figure 14

(Color online) The total catalytic activity of ruthenium clusters in the ammonia synthesis as a function of the cluster diameter. The different colors indicate how much the different step sites contribute to the total activity. The activity peaks at a cluster size of $3\mathrm{nm}$.