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Cohesive Energy of Clusters Referenced by Wulff Construction H. Li, M. Zhao,* and Q. Jiang* Key Laboratory of Automobile Materials (Jilin UniVersity), Ministry of Education, and Department of Material Science and Engineering, Jilin UniVersity, Changchun 130022, China ReceiVed: January 11, 2009
The geometrical and energetic characteristics of Wulff construction are expressed by variants δ, Ba/Bt, and Ec(N)/Eb0 in an N atom system, where δ ) Ns/N is the surface/volume ratio with Ns being the surface atom number, Ba is the rest bond number, Bt denotes the total bond number without broken bonds, and Ec(N) and Eb0 are the cohesive energy values in an N atom system and in bulk. It is found that these functions of Wulff construction correspond to that of several standard clusters. Thus, Wulff construction as a first-order approximation could estimate the shape and energy of clusters without any structural consideration. 1. Introduction Metallic clusters have unique properties due to their large surface/volume ratio and special structures, which strongly change both chemical and physical properties in comparison with that of the corresponding crystalline bulk.1-3 It is known that the shape and structure of clusters are strong functions of the number of atoms (N) in a system.4 Due to the evident effect of surface energy on the total cohesive energy Ec(N), all clusters, however, have similar spherical shape although they could be in different structures. It has been found that Na4 and Mo5 clusters would have an FCC or more like icosahedron (IH) structures. Co clusters with 4 e N e 60 have an IH structure.6 Similar phenomena are also observed in Au,7 Ni,8,9 Pd,8 Cu,10 Ca,11 and Sr11 clusters while their shapes are all quasispherical with many facets.12 Among these clusters, some possess noncrystalline structures, such as IH and truncated decahedron (DH).13 Although clusters have been widely studied, suitable models describing their cohesive energy and shape in a unified form are absent. The structure uncertainty is the key difficulty to realizing the above object. It is well-known that Wulff construction, which is developed by minimizing surface energy for a given enclosed volume,14 is a standard method for determining the equilibrium shape of bulk crystals. Although the structures of clusters differ from Wulff construction, the energetic requirement of minimizing surface energy is the same for both crystals and clusters. Thus, the shapes and energetics of clusters could be similar to that of Wulff construction. As a result, through developing a sizedependent function of Wulff construction, we could approximately describe shapes and Ec(N) functions of clusters without detailed consideration on structures of clusters while Ec(N) is usually structure and shape dependent. This is the goal of this contribution, which is realized through establishing models and comparing the model predictions and experimental results. The obtained results and comparisons confirm the usefulness of the developed models.
Figure 1. A sketch of a general TO structure with eight hexagonal (111) and six square (100) facets at the surfaces.
with the lowest surface energy is established by truncating symmetrically six vertices of the octahedron with six square (100) and eight hexagonal (111) facets at its surface. Concerning the hexagonal (111) facets, three edges of the hexagon are in common with square (100) facets, each having nsqu atoms, while each of the remaining three edges have nhex atoms, as shown in Figure 1. The geometrical properties of Wulff construction can be studied by introducing a variant Ba/Bt where Ba is the rest bond number or the actual bond number under consideration of the surface bond deficits in a cluster system with N atoms, and Bt is the total bond number without broken bonds in a perfect crystal. This is because the largest possible Ba/Bt value should be the most stable with the smallest surface bond deficits. Note that an alternative is the surface/volume ratio δ, which is expressed as
δ ) Ns ⁄ N
(1)
where Ns denotes number of surface atoms in the system. For Wulff construction, surface atoms are composed of atoms on (111) facets N(111), atoms on (100) faces N(100), atoms at edges N,e and that on vertex Nv. These values can be expressed mathematically according to Figure 1 and its geometry
N(111) ) [4(nsqu + nhex)(nsqu + nhex - 9) + 2nsqunhex + 14]
(2.1) N(100) ) 6(nsqu - 2)2
2. Modeling Owing to the densest consideration on packing, the FCC structure is taken into account as an example. Wulff construction * Author to whom correspondence should be addressed. Fax: +86 431 85095876. E-mail:
[email protected] and
[email protected].
Ne ) 12(2nsqu + nhex - 6)
and
(2.2) Nv ) 24
(2.3)
Adding eqs 2.1-2.3 together, Ns ) N111 + N100 + Ne + Nv. In addition, N as functions of both nsqu and nhex can also be determined by15
10.1021/jp902319z CCC: $40.75 2009 American Chemical Society Published on Web 04/08/2009
Cohesive Energy Referenced by Wulff Construction
J. Phys. Chem. C, Vol. 113, No. 18, 2009 7595 TABLE 1: The Values of γ(100) and γ(111) and the Corresponding Ratios γ(100)/γ(111) for FCC Metallic Elementsa
N ) {2[2(nsqu - 1) + nhex]3 + 2(nsqu - 1) + nhex}/3 2(nsqu - 1)3 - 3(nsqu - 1)2 - (nsqu - 1)
(3)
element
Defining Zs(N) as the average coordination number (CN) of surface atoms, Zs(N) is then obtained by determining the corresponding CN of surface atoms at different sites, namely,
where Z111 ) 9, Z100 ) 8, Ze ) 7, and Zv ) 6. Let Zb ) 12 denote bulk CN, Ba ) [NsZs(N) + (N - Ns)Zb]/2 and Bt ) NZb/ 2, that is,
(5)
Determination of Ba/Bt or δ of Wulff construction of different substances is difficult since two variants of nsqu and nhex in all the above equations are involved. Therefore, the combination of them into one variant will simplify our calculation, which could be realized by considering the formation condition of Wulff construction,15
γ(100) D(100) ) γ(111) D(111)
(6)
where γ(100) and γ(111) are the surface energies at (100) and (111) facets, respectively, and D is the distance from the facets to the center of truncated octahedron (TO). The geometrical shape of Wulff construction thus will rely on substances through the ratio γ(100)/γ(111). It is known that D(100)/D(111) ) d(100)(ξ(100) 1)/d(111)(ξ(111) - 1), where d(100)/d(111) ) 31/2/2 for FCC crystals with d(100) denoting the distance of the two neighboring (100) facets and d(111) that of the two neighboring (111) facets. ξ(100) and ξ(111) show the layer number of (100) and (111) facets respectively within the distances of 2D(100) and 2D(111). Through considering a general TO structure with hexagonal and square facets at surfaces as presented in Figure 1, there is (ξ(100) 1)/(ξ(111) - 1) ) (2nsqu + 2nhex - 4)/(2nsqu + nhex - 3). Thus, in terms of eq 6, the above equation can be rewritten as
2γ(100)
√3γ(111)
)
2nsqu + 2nhex - 4 2nsqu + nhex - 3
(7)
According to eq 7 and the known γ(100)/γ(111) values, the relationship between nhex and nsqu can be determined. Table 1 summaries the γ(100)/γ(111) values of the known FCC elements. As displayed in Table 1, γ(100) values are usually larger than γ(111) values due to the corresponding larger bond deficits, which leads to γ(100)/γ(111) > 1. The ratio nhex/nsqu in eq 7 as the monotonous function of γ(100)/ γ(111) increases with the increasing of γ(100)/γ(111), which implies that a larger γ(100) value requires a smaller surface area of (100) facets in equilibrium. Note that when nhex ) nsqu, γ(100)/γ(111) ) 2/3, which corresponds to a special Wulff construction, i.e., regular truncated octahedron (R-TO) with regular hexagons and squares at surfaces. As a result, nhex > nsqu when γ(100)/γ(111) > 2/3, and nhex < nsqu when γ(100)/γ(111) < 2/3. Note that at the limit case of nsqu ≡ 1 for any nhex, γ(100)/γ(111) ) 3, which shows a complete octahedron (OH) without any truncating although this structure is unstable due to its high energetic state. Note that the most substances, except two rare earth elements, have similar γ(100)/γ(111) values, which are located between 1.15 and 1.19. Namely all γ(100)/γ(111) values shown in Table 1 can be taken as γ(100)/γ(111) ) 2/3 with errors being smaller than 4%. This can be observed in later results.
γ(111) (J m-2)23
γ(100)/γ(111)
1.80 1.40 2.88 2.15 0.5 0.39 2.17 2.98 3.15 3.74 0.64 1.68 1.14 2.36
1.52 1.20 2.44 1.85 0.43 0.33 1.83 2.54 2.7 3.19 0.55 1.45 1.03 1.85
1.184 1.167 1.180 1.162 1.163 1.182 1.186 1.173 1.167 1.172 1.164 1.159 1.107 1.276 2/3 ) 1.155 3 ) 1.732
Au Ag Ni Pd Ca Sr Cu Pt Rh Ir Pb Al Ac Th R-TO OH
Zs(N) ) (Z(111)N(111) + Z(100)N(100) + ZeNe + ZvNv)/Ns (4)
Ba NsZs(N) + (N - Ns)Zb ) Bt NZb
γ(100) (J m-2)23
a The presented γ(100)/γ(111) values respectively for R-TO and OH structures are determined according to Wulff theorem (eq 6) for comparison.
For comparison, the geometrical parameters of IH and cuboctahedron (CO) structures are also adopted in a similar way as4,13
Ns ) 10(ν - 1)2 + 2 N)
(IH and CO)
10 3 11 ν - 5ν2 + ν - 1 3 3
(IH and CO)
(7.1) (7.2)
Ba 20ν3 - 45ν2 + 37ν - 12 ) Bt 20ν3 - 30ν2 + 22ν - 6
(IH)
(8.1)
Ba 10ν3 - 24ν2 + 20ν - 6 ) Bt 10ν3 - 15ν2 + 11ν - 3
(CO)
(8.2)
where ν represents the number of atoms at one edge since all edges of a IH or a CO structure are equal. Also as a TO structure, CO, however, has eight triangular (111) and six square (100) facets at the surface.16 It is known that Ec(N)/Eb0 ) (Ba/Bt)1/2, where Eb0 shows bulk cohesive energy.17 This equation is in agreement with simulation results17 although it only considers attractive forces, i.e., overestimating the relaxation effect. In the literature,18-22 by j ) into account, Z j N ) Ba taking an average CN for clusters (Z j /Zb ) Ba/Bt, which is and ZbN ) Bt. Hence, Ec(N)/Eb0 ) Z successful when the melting points of clusters are used for comparison,18-22 although the surface bond contraction and surface bond strength enhancement are ignored. As a more general formula with a first-order approximation, an algebra sum of them is arbitrarily carried out to make up the deficiency of both:
Ec(N)/Eb0 ) [(Ba/Bt)1/2 +Ba ⁄ Bt]/2
(9)
. 3. Results and Discussion Figure 2 shows the comparison of δ(N) functions of IH, CO, and Wulff structures with different γ(100)/γ(111) values in light of eq 1. δ is the simplest function to describe the shape of clusters and δ values of IH and CO are the same. It is clear that a small difference among them is found, and tends to zero as N increases. Moreover, the good agreement between IH and Wulff construction for δ values implies that IH has a similar
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J. Phys. Chem. C, Vol. 113, No. 18, 2009
Figure 2. δ(N) functions for IH (and CO) (red line) and different Wulff constructions in terms of eq 1, where WulffAc and WulffTh denote the Wulff construction of Ac and Th for comparison. The corresponding γ(100)/γ(111) values for WulffAc (dotted line), WulffTh (dashed line), OH (blue line), and R-TO (black line) are given in Table 1, and the olive line represents Wulff construction with γ(100)/γ(111) ) 1.
Figure 3. Ba/Bt as a function of N, separately for different geometrical structures of IH (red line), CO (pink line), and Wulff structures in terms of eqs 8.2 and 5, respectively, where black, blue, and olive lines denote R-TO, OH, and other Wulff constructions with 1 e γ(100)/γ(111) < 3.
shape of Wulff construction. The relatively large difference of δ between IH and Wulff construction indicates that even if the crystalline structure takes the Wulff construction, its Ns is still larger than that of IH clusters. Thus, the shape of clusters is more spherical. In addition, when N is fixed, δ of Wulff construction relies on the γ(100)/γ(111) ratio. As γ(100)/γ(111) decreases, δ drops, which implies that the larger nsqu value is in favor of forming a spherical shape. This also explains why the δ value of the CO structure is smaller than that of any Wulff construction in the full size range. The smallest and largest δ values for Wulff construction respectively correspond to γ(100)/ γ(111) ) 1 and γ(100)/γ(111) ) 3 (OH), as shown in Figure 2. It is noteworthy that the small difference of δ values between them suggests that the errors induced by γ(100)/γ(111) values among different Wulff constructions are negligibly small. As a result, a standard Wulff construction with γ(100)/γ(111) ) 2/3 could represent the δ function of any clusters with an acceptable error range. Figure 3 presents the comparisons of Ba/Bt functions among Wulff construction, IH and CO structures using eqs 5 and 8, respectively. In light of Figure 3, the Ba/Bt function also depends on structures, which declines with reducing N. The Ba/Bt value of IH is larger than that of Wulff construction and CO in the full size range, which denotes that IH has the largest rest bond number. The Ba/Bt value of Wulff construction is larger than that of the CO structure where CO is in fact also an FCC structure, although CO has a smaller δ value as displayed in Figure 2. In addition, the comparison of Ba/Bt of Wulff construction as a function of γ(100)/γ(111) changed from 1 to 3
Li et al.
Figure 4. By considering real Wulff construction for different elements where the needed parameter γ(100)/γ(111) values are listed in Table 1, the comparison of Ec(N)/Eb0 function based on eq 9 (olive lines) with experimental and simulation results for metallic clusters is made. The symbol 9 displays IH structure for Au,7 Ag,22 Ni,9 Pd,8 Ca,11 and Sr11 clusters. 2 shows DH for Ag24 and Au25 clusters. b represents CO and other TO structures for Au7 and Ag24 clusters. The black lines, according to R-TO structure, represent the model predictions of eq 9 for comparison.
is seen in Figure 3. OH has the smallest Ba/Bt value for Wulff construction. Again the difference of Ba/Bt among different γ(100)/ γ(111) values in the shown size range is indistinguishable. Since Ba/Bt in one side is related to δ, or the surface shape, and in another side is related to Ec(N), the similarity of Ba/Bt among different Wulff constructions, IH and CO implies that Wulff construction could approximately characterize clusters not only in shape, but also in energy. The above consideration is confirmed indeed in Figure 4, where Ec(N)/Eb0 functions of Wulff construction in terms of eq 9 and that of experimental and simulation results of several metallic clusters are plotted. Although the changing rate of the considered metallic elements with N is a little different, the model predictions by the real Wulff construction for metals correspond to experimental and simulation results. As mentioned above, N could be roughly considered as a unique variant of Ec(N)/Eb0 since the effect of γ(100)/γ(111) is weak. Namely, when a structure at a certain N is stable, it has a similar energetic level and takes a quasispherical shape. Thus, Wulff construction with a similar shape and energy of clusters can describe the energetic states, such as that of IH, DH, and CO, and also that of quasicrystals. The easiest way to do the above is using the R-TO structure where nsqu ) nhex, which leads to negligible error as stated above. As a result, even if we do not know the exact structure and shape, using R-TO Wulff construction, their energetics can be roughly estimated. 4. Conclusion In summary, δ, Ba/Bt, and Ec(N) functions of Wulff construction are present, which correspond to experimental and simulation results of clusters. Thus, Wulff construction could be utilized to describe cohesive energy of clusters roughly even when they have different structures. Acknowledgment. The authors acknowledge the financial support of the National Key Basic Research and Development Program of China (Grant No. 2004CB619301). References and Notes (1) Schmid, G.; Baumle, M.; Geerkens, M.; Helm, I.; Osemann, C.; Sawitowski, T. Chem. Soc. ReV. 1999, 28, 179.
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JP902319Z
