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X. J.
Zhao
^{a},
X. L.
Xue
^{a},
Z. X.
Guo
^{b} and
S. F.
Li
*^{a}
^{a}International Laboratory for Quantum Functional Materials of Henan, School of Physics and Engineering, Zhengzhou University, Zhengzhou, Henan 450001, China. E-mail: sflizzu@zzu.edu.cn
^{b}Department of Chemistry, University College London, London WC1H 0AJ, UK

Received
19th January 2016
, Accepted 1st May 2016

First published on 3rd May 2016

When a structure is reduced to a nanometer scale, the proportion of the lowly-coordinated edge atoms increases significantly, which can play a crucial role in determining both their geometric and electronic properties, as demonstrated by the recently established generalized Wulff construction principle [S. F. Li, et al., Phys. Rev. Lett., 2013, 111, 115501]. Consequently, it is of great interest to clarify quantitatively the role of the edge atoms that dominate the motifs of these nanostructures. In principle, establishing an effective method valid for determining the absolute value of the surface energy and particularly the edge energy for a given nanostructure is expected to resolve such a problem. However, hitherto, it is difficult to obtain the absolute edge energy of transition metal clusters, particularly when their sizes approach the nanometer regime. In this paper, taking Ru nanoclusters as a prototypical example, our first-principles calculations introduce the concept of relative edge energy (REE), reflecting the net edge atom effect over the surface (facet) atom effect, which is fairly powerful to quasi-quantitatively estimate the critical size at which the crossover occurs between different configurations of a given motif, such as from an icosahedron to an fcc nanocrystal. By contrast, the bulk effect should be re-considered to rationalize the power of the REE in predicting the relative stability of larger nanostructures between different motifs, such as fcc-like and hcp-like nanocrystals.

Here, we emphasize that with further reduction of the cluster size, the atoms on the edges constructed by the intersection of the adjacent mini-facets become increasingly important in determining the physical and chemical properties of a given polyhedral cluster.^{15–20} On the one hand, the fraction of the edge atoms considerably increases, on the other hand, the coordination numbers of the edge atoms are further reduced; both result in d-type dangling bonds and extra chemical activities, compared with the facet atoms.^{21,22} Recently, we identified that the classic Wulff construction principle (CWCP) should be generalized to emphasize the edge atom effect to identify both the geometric structure and the magic number of TM nanoclusters,^{21,22} as also strongly supported by experiments.^{23} Edge atoms have also been found to play a critically important role in catalysis.^{19,20,24,25} For example, N_{2} dissociation on the Ru(0001) surface is totally dominated by the step edges where the measured adsorption rate is at least nine orders of magnitude higher than that on the terraces at 500 K.^{24} For water-gas-shift catalysis, it is identified that those metallic corner atoms located on the intersection of adjacent step edges of gold clusters act as the dominant active components.^{26} In addition, in Jaramillo's work,^{27} it is shown that the electrocatalytic activity measurements for hydrogen evolution correlate linearly with the number of edge sites of MoS_{2} particles.

Consequently, it is of great interest to identify quantitatively or quasi-quantitatively the role of edge atoms in the configuration and hence the properties of a given TM nanostructure. In principle, establishing an effective method of determining the accurate value of the edge energy for a given nanostructure is expected to resolve such a problem. However, up to date, it is still intractable to quantitatively obtain the absolute edge energies of a given nanostructure in the framework of first-principles calculations, though the surface properties and surface energies^{28–31} of various crystals have been extensively studied and can be accurately determined. Only limited work has been done to estimate quantitatively the edge energy of nanoclusters. For example, by using an empirical potential and continuum approach,^{32} simulations reveal negligible edge energy for some noble TM nanoclusters. In this paper, first of all, choosing a Ru_{55} cluster as a prototypical example, we identify that the previously proposed continuum method^{32} is inadequate for calculating the edge energy of a given TM particle, when the size approaches the nanometer regime, due to the discontinuity of the energy density originating from the localized d-type dangling bonds on the edges. Thus, we establish a concept of REE reflecting the net edge atom effect over the surface (facet) atom to estimate the critical size at which a crossover occurs between different configurations of a given motif (such as fcc-like). Furthermore, to rationalize the predictive power of the REE in determining the stabilities of different larger nanocluster motifs, such as fcc-like relative to hcp-like nanocrystals, the bulk effect of the nanocluster core should also be invoked.

E_{Total} = Al^{3} + Bl^{2} + Cl + D | (1) |

(n − 1)d ≤ l ≤ nd | (2) |

(3) |

Fig. 1 Calculated total energy of I_{h}-Ru_{N} clusters (N = 13, 55, 147, 309, 561) as a function of the edge length for four different definitions of l. The curve labeled “l ≈ (n − 0.5)d” corresponds to the value obtained using eqn (3) in the text. |

One can get that l ≈ (n − 0.5)d.

Then, taking Ru_{N} clusters (N = 13, 55, 147…) as typical examples, we demonstrate that although the edge length can be precisely defined as introduced, this continuum approach is still not applicable for estimating the edge energy of nanoclusters in the framework of first-principles calculations. To address this point, we first calculated the total energies (E_{Total}) of a series of I_{h}-Ru_{N} clusters with N = 13, 55, 147, 309, 561, then we plot the E_{Total} as a function of l, as represented by the curve labeled “l ≈ (n − 0.5)d” in Fig. 1. According to the continuum approach, by fitting the calculated data of E_{Total} as expressed in eqn (1) and Fig. 1, one can get the bulk, surface, edge, and the vertex (corner) energies, which relate directly to the coefficients A, B, C and D, respectively, as shown in Table 1. Unfortunately, though we can obtain a negative value for the coefficient A and positive values for both B and C, the last term D has a totally wrong sign: D = −4.529 eV! For further comparison, we have also considered other three choices for the definition of the edge length l, namely, (n − 1)d, (n − 0.5)d, and nd, and plotted the E_{Total} curves as a function of l with these new definitions. It is found that all these three new definitions result in similar values for the coefficient A to the case with precisely defined edge length, however, none of those can consistently yield reasonable signs for all the four coefficients, A, B, C, and D. Thus we can conclude that the previously established continuum approach is invalid to determine correctly the edge energy of a given polyhedral TM cluster, at least in the nanometer size range investigated.

A | B | C | D | |
---|---|---|---|---|

≈(n − 0.5)d | −1.454 | 1.209 | 3.094 | −4.529 |

=(n − 1)d | −1.522 | −4.917 | −5.904 | −3.983 |

=(n − 0.5)d | −1.522 | 1.261 | −0.957 | −1.227 |

=nd | −1.522 | 7.439 | −12.728 | 6.146 |

We now discuss in more detail the underlying mechanism why the continuum approach is insufficient to estimate the edge energy of the present Ru_{N} nanoclusters. In principle, the continuum theory stresses the gradual quantitative transitions of a variation without abrupt changes or discontinuities. Therefore, in Hamilton's continuum approach^{32} of calculating the edge energy, the energy density is essentially supposed to be close to a constant within the whole “cluster domain” enveloped by the dividing surface, because the empirical potential intrinsically cannot provide such important information reflecting the abrupt changes in the electronic structures on different local sites, such as on the edges as compared to the inner sites or facet sites. Particularly, on the edge sites, the atoms are more lowly-coordinated than those of the inner or the facet atoms, therefore the edge atoms possess significantly different bonding features and stabilities as compared to the latter.

The above statements are further supported by the detailed local electronic structure analysis. As a typical example, the electronic charge density (see the lower panel) projected onto the high-symmetry plane bisecting the I_{h}-Ru_{55} cluster in two representative energy windows (see the upper panel) are presented in Fig. 2. Significantly, on the edge sites of the I_{h}-Ru_{55}, the electronic charge density shows strong features of directional d-type dangling bonds (Fig. 2(II)) around the Fermi level, particularly compared to the charge density in the similar energy window far below the Fermi level, see Fig. 2(I). Such high electronic density of states of the edge atoms by the Fermi level (Fig. 2) results in considerable energy increases (see also ref. 21) and contrast edge properties, which cannot be accounted for by a continuum approach, and thus is the main reason why the above continuum approach is invalid to estimate the edge energy.

Panel (I) of Fig. 3 displays six representative low-energy Ru_{55} structures optimized from various initial configurations (some of these configurations were also reported previously in ref. 21). These structures can be grouped into three types of motif, fcc-like (structures (a), (b), and (d)), I_{h} (f), and the others hcp-like ((c) and (e)) configurations. Due to their similar geometries, we present the fcc-like and I_{h} structures in the same color, and show the hcp-like by differently colored polyhedra, respectively. The three low-symmetry structures in Fig. 3(I-a), (-b) and (-d) are found to be much lower in energy than the high-symmetry I_{h} (see structure (f)), by 3.120, 2.691, and 1.479 eV, respectively. As reported in our previous paper,^{21} these three stable structures can be optimized from an fcc crystal fragment (FCCCF). For more details on the geometric structures, thermal dynamic and kinetic stabilities, see ESI, S1.†

To assess the importance of the edge atom effect in determining the relative stability of the above structures within a given motif, we qualitatively separate the total energy of a given polyhedral cluster of size N into three terms:

E_{Total} = E_{Bulk} + E_{Surf} + E_{Edge} | (4) |

We can now estimate E_{Surf} by the total surface area times the energy per unit area, and E_{Edge} by the total edge length times the energy per unit length. Additionally, the total surface area and total edge length of these four representative structures can be expressed as N_{Surf} × δ and N_{Edge} × λ, respectively. Here, N_{Surf} and N_{Edge} denote the total numbers of triangularly shaped mini-facets defined by the three adjacent atoms on the surfaces and the total numbers of the atomic bonds defined on the edges,^{21} respectively. Correspondingly, the average area δ per triangularly shaped mini-facet and the average length λ per atomic bond on the edges can be taken as constants due to their negligible fluctuation (within ∼2%) between different structures. First, as indicated in Fig. 4(b), each of the three more stable FCCCF structures in panel (I) of Fig. 3 possess a larger total surface area (N_{Surf}) than the I_{h} structure. Furthermore, the I_{h} structure contains 20 fcc(111) mini-facets, while the mini-facets on the three new cluster structures are either also fcc(111)-like, or are bcc(100)-like with higher energy per unit area, showing that each new structure corresponds to a higher E_{Surf} than the I_{h} structure.

Fig. 4 Relative energies and atomic arrangements of four Ru_{55} low energy structures with configurations 1, 2, 3, and 4 corresponding to the structures shown in Fig. 3(I-a), (-b), (-d), and (-f), respectively. The relative energies are measured from that of the I_{h} structure, given by ΔE = E(TM_{N}) − E(TM_{N}(I_{h})). |

Based on these analyses, we must attribute the overall energy reductions associated with the three FCCCF structures to the dramatic reductions in the third term, E_{Edge}. Indeed, as shown in Fig. 4(d), we find that each of the three FCCCF structures has a significantly reduced total edge length (N_{Edge}) as compared to the I_{h} structure. We also observe a close correspondence between the total energies shown in Fig. 4(a) and the total edge lengths in Fig. 4(d), while such a correspondence is absent between Fig. 4(a) and (b) or (c), confirming that the edge atom effect is the dominant factor in determining both the energetic stability and the geometric structure of these nanoclusters.^{21}

Because we have convincingly identified that the total energy differences (ΔE) of these structures are dominated by the edge atom effects, the observation of the close correspondence between the variation of the quantitative values of the total energies and the difference of the total edge length (ΔL) of these structures provides us with a good opportunity to estimate the average edge energy. Namely, the average edge energy per unit length (E_{edge}) can be semi-quantitatively obtained in the lowest boundary approximation as ΔE/ΔL. For example, taking the I_{h} structure as a reference, the E_{edge} of the FCCCF nanocluster series can be specifically defined as:

(5) |

Fig. 5 Relative edge energy (REE). The least square fitting of the REEs of (a) ten (three) fcc-like configurations and the I_{h} cluster as a function of the total edge length for Ru_{55} (Ru_{54}) clusters. The olive circled pentacles correspond to the low-energy structures of Ru_{55} presented in Fig. 3(I-a), (-b), (-d) and (-f), and other low-energy isomers of Ru_{55} and Ru_{54} are provided in ESI, S1 and S2,† respectively; (b) four fcc-like configurations and the I_{h} cluster as a function of the total edge length for Rh_{54} and Rh_{55} clusters. (c) Comparison between the REEs obtained by the present comparative method based on first-principles calculations and that by the empirical potential approach for Ru and other prototypical TM elements. |

Note that I_{h} is a highly strained structure with a volumetric strain contribution to the energy that is not taken into account in −Ne_{0}. This strain energy is mainly due to the strong compression of the central part of the structure.^{40–42} To check this strain effect in the validity of the proposed approach in calculating the REE, we perform additional calculations on the I_{h}-Ru_{54} nanocluster, obtained by the removal of the central atom from the respective I_{h}-Ru_{55}. Correspondingly, the Ru_{54} nanoclusters are now used to estimate the REE with the elimination of the contribution of the strain effect. As presented in Fig. 5(a), for both Ru_{54} and Ru_{55}, by comparing with other representative low-lying FCCCF isomers, we can obtain a clear linear relationship between the relative energy and the total edge length, reassuring again the validity of the present comparative approach to calculate the REE. Specifically, for Ru_{54}, the values of the REE per unit length is estimated to be around 48 meV Å^{−1} (Fig. 5(a)), which is slightly smaller than that of 50 meV Å^{−1} obtained for Ru_{55} without considering the strain effect. This is as expected, as the release of the volumetric strain in the structural reference (I_{h}-Ru_{54}) increases the energy contribution from the bulk and hence reduces the REE of other structures.

To examine this argument further, we perform substantial additional first-principles calculations on the energetics of larger clusters, namely Ru_{147} and Ru_{309} nanoclusters; for the detailed geometric structures, see ESI, S3.† Similarly, as shown in Fig. 6(a), by linear fitting of the data of ΔE/ΔL, we can estimate that the minimum value of the average relative edge energy per unit length is close to 27 meV Å^{−1}, which is already reduced to about half of that calculated from Ru_{55}, mainly due to the reduction of the ratio of the edge atoms relative to the surface or facet atoms. Here, we note again that the strain effect is also further checked for the Ru_{146} cluster series in Fig. 6(a) by using the same method as previously discussed for the case of Ru_{54}.

Fig. 6 The least squares fitting of the relative energies as a function of the total edge length of (a) three FCCCF and the I_{h} cluster for both Ru_{146} and Ru_{147} cluster series; (b) three FCCCF configurations and the I_{h}-Ru_{309} clusters. For both Ru_{147} and Ru_{309}, the numbers of inner atoms in clusters (N_{Inner}) represented by the triangles (in red) versus relative energy are also shown. The geometry configurations of Ru_{147} and Ru_{309} are shown in Fig. 3(II) and (III), respectively. |

Next, we further investigate the most stable geometric candidates for an even larger Ru_{309} cluster. In panel (III) of Fig. 3, the six lowest energy structures are presented, including two HCPCF, I_{h}, and three FCCCF configurations. Interestingly, the HCPCF motif now is much more stable than the FCCCF structures. In addition, for Ru_{309}, the highly symmetric Mackey I_{h} (Fig. 3(III-c)) is more stable than the FCCCF configurations preferred by relatively smaller nanoclusters of Ru_{55} and Ru_{147} based on GWCP. Particularly, the most stable FCCCF eight-layered structure presented in Fig. 3(III-d) is 1.419 eV less stable, and the seven-layered configurations in (e) and (f) possess a much higher energy, by 2.571 and 4.208 eV, respectively. On the other hand, the HCPCF motif possesses much lower energy than the FCCCF one. For example, structure (a) and (b) are 6.320 and 3.922 eV lower in energy than the highly symmetric I_{h} configuration as shown in (c). Taking these FCCCF configurations again as a typical example, we demonstrate that the relative edge energy obtained by fitting the data of ΔE/ΔL in our comparative approach is now negative, −51 meV Å^{−1}. The contrasting signs of the relative edge energies of Ru_{147} and Ru_{309} within the fcc-like motif indicate a structural change of the Ru nanoclusters between size 147 and 309, as directly supported by the calculated stability reversion between FCCCF and HCPCF.

Note that to make the above size effect on the REE convincingly statistically valid, we also considered another two sizes for Ru_{N} nanoclusters, i.e., N = 101 and 231. At these two sizes, highly symmetric I_{h} configurations cannot be formed, however, other low-energy perfect polyhedrons can be obtained, as detailed in the ESI, S4 and S5.† In these two cases, the obtained REEs are 37 meV Å^{−1} for Ru_{101} and −36 meV Å^{−1} for Ru_{231}, respectively, which locate almost exactly in the fitted linear relationship obtained by Ru_{N} with N = 54, 55, 146, 147 and 309.

Fig. 7 REE as a function of nanocluster size N. The REEs are obtained by the fitting method as shown in Fig. 5(a) for Ru_{54} and Ru_{55}, Fig. 6 for Ru_{146}, Ru_{147}, and Ru_{309}, Fig. S4† for Ru_{101} and Fig. S5† for Ru_{231}, respectively. |

Here, we note again that the evolution of the sign of the relative edge energy in Fig. 7 is due to two factors: on the one hand, due to the gradually reduced edge atom effect because of the reduced ratio of the edge atom to the atoms on the facets and core; on the other hand, due to the structural changes in the cluster cores of different FCCCF configurations, as implied by the different number of inner atoms shown in Fig. 6(b); for more detailed discussion, see ESI, S6.†

Overall, we have provided further quantitatively/semi-quantitative analysis of relative edge atom effects on the structures and stabilities of nanoclusters of transition metals. Such analyses are of great significance for nanocatalyst design and for the discovery of effective nanostructures in chemical/photochemical catalysis, nanomagnetism, electronic/photonic clusters, quantum dots and related subject areas.

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## Footnote |

† Electronic supplementary information (ESI) available: Detailed geometric structures of Ru_{54}, Ru_{55}, Ru_{101}, Ru_{147} and Ru_{231} cluster series; REEs of Ru_{101} and Ru_{231} cluster series; evolution of the sign of the REE when cluster size increases. See DOI: 10.1039/C6NR00486E |

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