Equilibrium Crystal Shape of Ni from First Principles - American

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Equilibrium Crystal Shape of Ni from First Principles Wei-Bing Zhang,* Chuan Chen, and Shun-Ying Zhang School of Physics and Electronic Sciences, Changsha University of Science and Technology, Changsha 410004, People’s Republic of China ABSTRACT: We have determined the equilibrium crystal shape of Ni at 0 K using Wulff construction based on 36 orientation-dependent surface energies, which are calculated from first principles. The (111), (100), (110), (210), (221), (311), and (322) surfaces are found in the predicted equilibrium shape based on DFT data, whereas the commonly used broken-bond model predicts that only (111) and (100) surfaces are present. This significant difference of equilibrium shape can be ascribed to the fact that the calculated surface energies show an orientation-dependent deviation from that predicted by the broken-bond model. Moreover, the (111), (100), (110), and (210) surfaces observed in available experiments have also been identified in the present calculations. This indicates that the broken-bond model is insufficient for obtaining the accurate surface energies and their anisotropy of real systems, and the high Miller index surfaces have to be considered to predict the reliable equilibrium shape of the metal.

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Hong et al.13 investigated the equilibrium shape of pure and carburized Ni systems and suggested that the ECS of pure Ni consisted of (111), (100), (110), (210), and (320) surfaces. Unexpectedly, they found that the surface energy of (210) surface was quite low, which was even lower than the low index (111) and (100) surfaces. Recently, Meltzman et al.14 have studied the crystal shape of Ni particles as a function of the partial pressure of oxygen (P(O2)) and iron content using scanning and transmission electron microscopy. The (111), (100), (110), (531), and (831) surfaces were found to be stabilized at low P(O2), while (210) and (310) surfaces were found at higher P(O2). According to Wulff construction, the equilibrium crystal shape can be determined based on the orientation-dependent surface energies assuming minimal surface free energy. Given the difficulties of a direct measurement of equilibrium crystal shape, accurate and reliable knowledge of the surface energy plays a relevant role. Unfortunately, high accurate experimental surface energies are difficult to obtain. The surface energies of a solid crystal are known to be anisotropic, which depend on the crystallographic orientation. However, most of the available experimental surface energy data were obtained from surface tension measurements in the liquid phase and extrapolated to zero temperature, which are unfortunately average values of an unknown range of orientations. Meanwhile, surface energy also suffers from various uncertainties such as impurities. As an important supplement to experiment, different theoretical approaches including early (semi)empirical studies and firstprinciples calculations have been used to calculate the surface

INTRODUCTION Because of the critical role in a wide variety of applications such as heterogeneous catalysis and solar energy conversion, the study of transition metals has been focused during the last decades. Generally, transition metals are often dispersed as nanoparticles (NPs) in real systems. In order to understand and improve the various performance of transition metal NPs, considerable efforts have been made. Studies have shown that by altering several important physical parameters, such as composition,1,2 size,3 and reactive environments,4 one can tailor and fine-tune the properties of materials. Moreover, recent studies also indicate that the properties of noble metal are strongly dependent on the shape of NPs,5−9 which also intrigues considerable research for controlling and understanding the shape of metal particles. Since many critical parameters are responsible for the final shapes of metal, the formation mechanisms of various shapes are very complicated and still not clear. Clearly, the detailed knowledge of equilibrium crystal shape is the first step to obtain insight into the underlying mechanism. Although equilibrium crystal shapes (ECS) of pure metal have been extensively investigated in recent decades, the discrepancies among different experiments are present widely due to the difficulties in achieving equilibrium shape of the pure metal particles. In the present study, we focus on a typical 3d transition metal Ni, which is widely used as a catalyst or substrate in the growth of many exciting materials such as graphene.10−12 Compared to other metals, the relatively high temperature is required to create a facet on the surface for Ni, which results in lengthier experiments to achieve equilibrated particles. However, ECSs of metals are known to depend on adsorption, very low concentration of impurities, and experimental conditions including temperature and pressure. © 2013 American Chemical Society

Received: May 8, 2013 Revised: September 18, 2013 Published: September 23, 2013 21274

dx.doi.org/10.1021/jp404569m | J. Phys. Chem. C 2013, 117, 21274−21280

The Journal of Physical Chemistry C

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energy of different metals.15−19 However, most of the high accurate first-principle calculations are limited to the low index surfaces. Recently, Yu et al. 15−17 have calculated the orientation-dependent surface energies of 35 low-index and vicinal Pb surface orientations. The (211), (221), and (320) facets are found on the equilibrium crystal shape of Pb constructed using Wulff theory in addition to (111) and (100) surfaces.16 The energetics of vicinal Cu surfaces were also investigated by the all electron full-potential linearized augmented plane-wave (FLAPW) method.18 The surface energy, step energy, and stability with respect to faceting of the low- and high-Miller-index vicinal Cu surfaces were obtained. For Ni crystal focused here, the surface energies of low index surfaces have been studied extensively. Unfortunately, a full consistent surface energy result is still lacking, even the relative stability order of low index Ni surface is found to conflict with each other in different calculations. Most of the calculations including Mittendorfer et al.20 suggested that surface energy order is γ111 < γ100 < γ110. However, Vitos et al.21 has established a database of surface energies for low index surfaces of 60 metals including Ni in the periodic table and found that the surface energy order in units of eV/Å2 is γ111 < γ110 < γ100. Recently, Hong et al.22 studied the surface energies of (111), (100), (110), and (311) surfaces based on densityfunctional calculations and constructed a 2D equilibrium shape with the Wulff construction. Although such a calculation can give some clues in equilibrium crystal shape of Ni, the reliability is restricted clearly to the number of the considered surfaces. In order to clarify the debate present in experiment and theory and give more reliable surface energies and ECS of Ni, we have performed a systematical density functional theory calculation for 36 Ni surfaces. The equilibrium shape of Ni particle is then obtained from the Wulff construction using the calculated surface energies and compared with available experimental and theoretical results.

convergence, the Brillouin zone (BZ) was sampled with a mesh of 12 × 12 × 1 k-points generated by the scheme of Monkhorst Pack for the (1 × 1) Ni(111) surface unit cell. For other surfaces, the k-points were reduced proportionately to keep the similar sampling of the reciprocal spaces. All atoms were optimized using the BFGS method29 implemented in ASE until the maximum absolute forces of all atoms converged to 0.05 eV/Å. Increasing the equivalent k-points to 16 × 16 × 1 for Ni(111) surface and decreasing the grid spacing to 0.18 Å, the change of surface energy is smaller than 10 meV, which does not affect our discussion and conclusions. Surface Energy Calculations. According to Wulff construction,30 the equilibrium crystal shape can be obtained by minimizing the total surface free energy for a fixed crystal volume, in which the length of a vector drawn normal to a crystal face (hkl) will be proportional to its surface energy γhkl. Clearly, the accuracy of surface energies will affect the validity of such a Wulff construction. The standard method for calculating the surface energy is to evaluate the total energy of a slab including N Ni atoms and to subtract from the bulk energy obtained from a separate calculation. γ = lim

n →∞

1 [Eslab − NE bulk ] 2A

(1)

Here, Ebulk, Eslab, and A represent the energy of bulk fcc Ni, total energy of bare slab, and surface area, respectively. A central but often underestimated problem using this equation in calculating surface energy is to choose the reliable bulk energy Ebulk. Boettger et al.31 pointed out that any difference between Ebulk and Eslab calculation will cause the calculated surface energy to diverge linearly with atomic layers. Da Silva et al.32 have discussed this problem and pointed out that the converged surface energies can be obtained in the case that the similar high quality integrations over the surface and bulk BZ are used in the calculation. Meanwhile, the slab and bulk systems need to be treated using the exact same numerical details such as the basis function type and cutoff energies in FLAPW method they used. In the present calculation, we use grid based DFT code in which the pseudo wave functions, pseudo electron densities, and potentials are represented on uniform real-space grids. Assuming that the above scheme is valid, both the real-space grids and k-points sampling of the reciprocal spaces need to be treated using the same high quality for bulk and all surfaces. Since the surfaces involved in the calculations have different geometry and symmetry, it is infeasible to keep grid and k-point sampling in all surfaces with the same accuracy quality. There are also other methods available to obtain the bulk energy in literature. One can use the same supercell for the bulk and slab systems in which the bulk supercell consists of the slab plus the vacuum space between the slabs filled with atoms.33 However, a separate calculation is still needed to get bulk energy in such a filled slab method. In order to avoid the divergence problem, Boettger31 suggested a method using only slab-related quantities, making no reference to separate bulk energies calculations.

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THEORETICAL METHODS Computational Details. The present calculations have been performed using the grid-based real-space DFT code GPAW,23,24 which is an implementation of the projector augmented wave (PAW) method of Blöchl.25 The atomic simulation environment (ASE)26 provides an interface to GPAW. Wave functions, electron densities, and potentials were represented on grids in real space, and a grid spacing of 0.20 Å was used in all directions of the supercell. All calculations were performed with spin-polarization, and general gradient approximations (GGA) in the Perdew−Burke−Ernzerhof (PBE) implementation27 were chosen for the exchange correlation functional. A Fermi smearing of 0.1 eV and Pulay density mixing were used for accelerating electronic convergence. In the present work, the different Ni surfaces are modeled using central symmetric slabs separated by at least 18 Å vacuum space. The surface unit cells are constructed based on the scheme developed by Herman,28 and the obtained surface cell is confirmed by the criterion that the cross of the two surface unit vectors should be parallel to the normal direction of the surface. The theoretical lattice parameter 3.52 Å was used in the surface calculation, corresponding to the experimental result of 3.524 Å, and the calculated magnetic moment of bulk Ni is 0.63 μB. Since the surfaces involved in the calculations have different geometry and symmetry, it is important to ensure the similar numerical accuracy for all surfaces. To ensure enough

γ = lim

n →∞

1 [Eslab − N ΔEN ] 2A

(2)

in which ΔEN is defined as − Clearly, one need calculate repeatedly the total energies for slabs of increasing thickness. Using the same data used in the eq 2, Fiorentini et al.34 suggested an alternative method for calculating the surface ENslab

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EN−1 slab .



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Figure 1. Surface energies of four different surfaces Ni(111), Ni(100), Ni(110), and Ni(311) as a function of the slab thickness evaluated by different methods.

energy (we called it linear fitting method hereafter). They found as N became large and convergence was approaching, the definition of the surface energy in eq 1 implied that N Eslab ≈ 2γ + NE bulk

convergence is reached. The maximum error bar with atomic layer and the maximum atomic layer used in the calculation for each surface have been listed in Table 1.

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RESULTS AND DISCUSSION Surface Energies. The calculated surface energies of 36 high-index Ni surfaces are listed in Table 1. Both results of surface energies in units of eV and J/m2; the surface energies based on the number of broken nearest-neighbor bonds are also included for comparison. The total number of broken nearestneighbor bonds is given by

The bulk crystal energies are obtained directly by making an overall linear fit to the total energy of slab as a function of the thickness and to take its slope. In order to get the reliable surface energies, we first compared the performance of the different schemes. The surface energies of three low index Ni(111) (Figure 1a), Ni(100)(Figure 1b), and Ni(110) (Figure 1c) surfaces, and a high Miller index Ni(311) surface (Figure 1d) as a function of slab layers evaluated by the various schemes described above is shown in Figure 1. One can find clearly that a very fast convergence of surface energies to 5 meV was achieved in the linear fitting method. The approach using a separately calculated bulk energy (i.e., eq 1) evidently suffers from the divergence problem in all surfaces considered here. The divergent behavior of surface energies with the number of layers can be ascribed to the inequivalence convergence between slab and bulk calculations with respect to the number of k-points and real-space uniform grids. The method developed by Boettger31 gives much better convergence, but the error bar is about 50 meV, which is 1 order of magnitude greater than the linear fitting method. We thus adopt the linear fitting method developed by Fiorentini et al.34 to extract surface energies in the present research. For each surface, we have checked the numerical convergence of surface energy with respect to slab layers with a great care. The slabs with three different layers are calculated first, and then the surface energies are extracted using the linear fitting method. If the surface energies of slabs using the fitted bulk energy reference converged with atomic layers within a few meV (typical value is 5 meV), the fitting surface energies are used. Otherwise, more atomic layers will be considered in the slabs until the

⎧ 2h + k h , k , l odd Nb(hkl) = ⎨ ⎩ 4h + 2k otherwise h > k > l

(4)

while the surface energy γNb in units of eV is given by γf111 × Nb/ Nb111. We first focus on the results of low index surfaces. From available DFT calculations in the literature, we can find that there are large discrepancies about surface energy. Mittendorfer et al.’s calculations20 using PW91 gave the value of Ni(111) surface to 1.93 J/m2, while Hong et al.’s GGA22 calculations gave a result of 2.011 J/m2. Fortunately, both calculations predicted that γ111 < γ100 < γ110. A full charge density LMTO calculation by Vitos et al.21 gave 2.02 J/m2 for Ni(111) surface, closed to Hong et al.’s results, but suggested a different relative surface energy order γ111 < γ110 < γ100. As shown in Table 2, (111) surface is found to be the lowest surface in the present calculation. The obtained surface energy of Ni(111) surface is 1.946 J/m2, which is closed to Mittendorfer et al.’s20 result. Kumikov and Khokonov35 suggested that the most reliable experiment value for the average surface energy of Ni was 1.94 J/m2, while recently Meltzman et al.14 gave an estimation of the absolute solid surface energy of Ni 2.05 ± 0.05 J/m2. Considered the uncertainty in experiment, the agreement between experiment and theory should be satisfactory. 21276



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Table 1. Calculated Surface Energies in Units of eV and J/m2 of Relaxed Surfacesa surface

γf (eV)

γr (eV)

area (Å2)

γf (J/m2)

γr (J/m2)

Nb

γNb (eV)

γNb (J/m2)

111 100 110 115 211 331 551 1111 117 119 210 221 310 311 320 322 410 430 510 533 553 755 771 775 332 433 443 540 544 554 650 655 710 910 531 831

0.655 0.883 1.313 2.427 2.194 1.944 3.304 5.176 3.311 4.240 2.165 2.627 2.996 1.550 3.516 3.485 3.956 4.809 4.992 2.857 3.256 4.144 4.735 4.570 3.836 4.875 5.212 6.067 6.156 6.592 7.684 7.271 6.478 8.427 2.840 8.424

0.652 0.869 1.265 2.366 2.159 1.916 3.173 5.008 3.269 4.109 2.100 2.548 2.973 1.494 3.406 3.388 3.911 4.716 4.794 2.769 3.186 4.138 4.412 4.527 3.783 4.582 5.093 5.738 6.095 6.458 7.094 7.119 6.427 8.296 2.766 7.853

5.365 6.195 8.761 16.132 15.175 13.502 22.121 34.354 22.121 28.220 13.853 18.586 19.591 10.274 22.337 25.543 25.543 30.976 31.589 20.312 23.793 30.821 30.821 34.354 29.058 36.124 39.669 39.669 46.773 50.330 48.386 57.452 43.807 56.100 18.326 53.293

1.956 2.284 2.401 2.410 2.316 2.307 2.393 2.414 2.398 2.407 2.504 2.264 2.450 2.418 2.522 2.186 2.481 2.487 2.532 2.254 2.192 2.154 2.461 2.131 2.115 2.162 2.105 2.450 2.109 2.098 2.544 2.028 2.369 2.407 2.483 2.533

1.946 2.248 2.314 2.350 2.279 2.273 2.298 2.336 2.368 2.333 2.428 2.196 2.431 2.330 2.443 2.125 2.453 2.439 2.431 2.184 2.145 2.151 2.293 2.111 2.086 2.032 2.057 2.318 2.088 2.056 2.349 1.985 2.351 2.369 2.418 2.361

3 4 6 11 10 9 15 23 15 19 10 12 14 7 16 16 18 22 22 13 15 19 21 21 18 22 24 28 28 30 34 34 30 38 13 38

0.655 0.873 1.310 2.402 2.183 1.965 3.275 5.022 3.275 4.148 2.183 2.620 3.057 1.528 3.493 3.493 3.930 4.803 4.803 2.838 3.275 4.148 4.585 4.585 3.930 4.803 5.240 6.113 6.113 6.550 7.423 7.423 6.550 8.297 2.838 8.297

1.956 2.259 2.396 2.385 2.305 2.332 2.372 2.342 2.372 2.355 2.525 2.259 2.500 2.383 2.506 2.191 2.465 2.484 2.436 2.239 2.205 2.156 2.383 2.138 2.167 2.130 2.116 2.469 2.094 2.085 2.458 2.070 2.396 2.369 2.482 2.494

Nm

m ErreV(J/m 2 )

19 18 19 19 35 25 30 17 21 25 14 19 27 24 20 29 21 26 34 32 35 25 22 25 26 33 31 29 33 30 31 31 35

0.002(0.007) 0.003(0.008) 0.003(0.006) 0.003(0.003) 0.002 (0.002) 0.002(0.002) 0.008(0.005) 0.002(0.001) 0.001(0.001) 0.002(0.001) 0.006(0.007) 0.001(0.001) 0.005(0.004) 0.001(0.002) 0.009(0.006) 0.004(0.002) 0.002(0.001) 0.019(0.010) 0.004(0.002) 0.003 (0.002) 0.005 (0.004) 0.002 (0.001) 0.002 (0.001) 0.003 (0.002) 0.001 (0.001) 0.012 (0.005) 0.002 (0.001) 0.009 (0.003) 0.002 (0.001) 0.006 (0.002) 0.030(0.010) 0.006 (0.002) 0.000(0.000) 0.012 (0.003) 0.002(0.001) 0.014 (0.004)

a Thirty-six different index surfaces are considered in the calculations. γf (γr) represents the results obtained on fixed (relaxed) surface slab. The surface energies based on the broken-bond model are also given in the table. The maximum atomic layers used in the calculations and the maximum error bar in the fitting for each surface are also listed.

anisotropy ratios. In order to verify this model, we plot the surface energies versus the number of broken bonds in Figure 2. As shown in Figure 2a, we can find an almost linear correlation of surface energies in units of eV both for static and relaxed surfaces with the number of broken surface bonds. The average and maximum deviations from linearity are found to be 1.39(1.21) and 3.71(3.09) % for relaxed (static) surfaces. Similar linear correlation of surface energies with the broken surface bonds is also discussed by Galanakis et al.36,37 and supported by DFT calculations for Cu18 and Pb surfaces.15 However, compared with the broken bonds model, the calculated surface energy anisotropy ratios are always lower than that for the broken bonds model. As shown in the Figure 2b, we can also find that the surface energies of relaxed surfaces are lower than ideal ones and the deviation from linearity depends on different surface orientations. This can be attributed to the fact that surface relaxation and surface charge smoothing are neglected in the ideal broken surface bond model and that many different inequivalent bonds will appear during relaxation. The surface relaxations of different surfaces

Table 2. Surface Energies Available in the Literature for the Low-Miller-Index Ni Surfaces surface

present calculation

ref 20

ref 21

ref 22

111 100 110

1.946 2.248 2.314

1.93 2.19 2.25

2.011 2.426 2.368

2.02 2.23 2.29

However, the present predicted surface energy order is γ111 < γ100 < γ110, which is consistent with Mittendorfer et al.’s20 and Hong et al.’s22 GGA calculations. This order is also in line with the experimental results by Meltzman et al.14 and the broken bond model as shown in Table 1, but conflicts with the experimental result given by Hong et al.13 On the basis of the strict convergence test described above, our results should be reliable under the present exchange correlation approximation, and the reasonable stability order of Ni surfaces should be γ111 < γ100 < γ110. The broken nearest-neighbor bonds rule is a well-known model to understand and predict the surface energies and their 21277



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we can find the (322) and (311) surfaces have same 4-fold symmetry with (100) surface, while (110) and (221) facets are 2-fold symmetric. Hong et al.22 constructed 2D equilibrium shape with the Wulff construction theory based on the surface energies of (111), (100), (110), and (311) surfaces. Most of the high-Miller-index surfaces are missing in this shape since they are not considered in the calculation. However, the symmetries of different facets are the same as the results found here. However, the equilibrium shape is extremely sensitive to surface energies. We have also considered the variation of the equilibrium crystal shape by varying the surface energies of certain facets within the error bars obtained above. By keeping the surface energies of other surfaces fixed, we change the surface energy of one surface to examine whether the error bar affects the equilibrium crystal shape. Especially, the high-Millerindex surfaces, which are observed in the experiments and appeared in present calculations, are focused. The results indicate that increasing/decreasing surface energy by the error bar will change the relative proportion of different facets but does not lead to existence/nonexistence of certain facets except for the (311) surface. By decreasing the surface energy of the (322) surface or increasing surface energy of the (311) surface by the error bar (±0.01 J/m2), the (311) surface will disappear on ECS. However, it is interesting that the (210) surface, which occupied the least area seems to be quite robust. Experimentally, the equilibrium crystal shapes of Ni have been investigated by different groups. Hong et al.13 suggested that the equilibrium crystal shape (ECS) of pure Ni is a polyhedron consisting of (111), (100), (110), and (210) surfaces, and the (210) surface is found to occupy a quite large part of the shape. However, Meltzman et al.14 found that in addition to the (111), (100), and (110) facets, high-index facets were identified such as (531) and (831) at low P(O2), and (210) and (310) at higher P(O2). Although the conflicts between different experiments are present, both experiments show that (111), (100), and (110) are present, which is consistent with the present prediction. The (210) facet observed in both experiments is also evidenced in our DFT results. However, the broken bond model predicts only the low index (111) and (100) surfaces, which conflicts with all experiments. This indicates the broken bond model is not enough for predicting the real equilibrium crystal shape of material. In addition, the present calculation also predicts the (221) and (322) surfaces present in the shape of Ni crystal. The presence of high Miller index on ECS suggests that in order to obtain a reliable description for equilibrium shape of material, the high-Miller index-surface must be considered. It should be noticed that a quantitative comparison between experiment and theory is quite challenging. This can be understood by the difficulties to get an equilibrated shape of Ni particles both from theory and experiment. Experimentally, low concentration impurity, adsorption, and experimental conditions can significantly change the equilibrium crystal shape. Further theoretical modeling about the effect of oxygen environment to the equilibrium crystal shape of Ni is in progress, in which a grand-canonical ensemble is required and reactions on the surface should also be accounted. However, although the convergence of surface energy are checked carefully, the accuracy of surface energy suffers from the approximation of exchange and correlation in DFT. Recently, high level theory such as random phase approximation (RPA)39 have been evidenced to be able to give a quite reasonable surface energy and adsorption energy. However, the calculation

Figure 2. Surface energies of Ni surfaces versus the number of broken nearest-neighbor bonds in the surface (a), and the difference between surface energy in units of J/m2 calculated by DFT and broken nearestneighbor bond model (b). The continuous lines are obtained from a linear fitting of the surface energy DFT results.

can be understood by the broken-bonding rule, which have been discussed in our recent work for Ni(331), Ni(115), and Ni(551) stepped surfaces.38 The detailed surface relaxation and magnetism of high-Miller-index surfaces will be published elsewhere. This difference between calculated and ideal surface energy results in fully different equilibrium shape, which will be discussed below. Equilibrium Shape. On the basis of the calculated surface energies, we obtain the equilibrium shape of Ni crystal by Wulff construction at 0 K. As given in Figure 3a, we can find that the

Figure 3. Equilibrium shape of Ni crystal produced by Wulff construction using surface energies obtained by broken bond model (a) and DFT-PBE calculations (b). The different facets are labeled by different colors in the figure.

equilibrium shape of Ni NP only consists with two main low index (111) and (100) surfaces and that the (111) facet occupies the most area of the shape. In addition, the (111) and (100) facets are found to have 6-fold and 4-fold symmetry. Figure 3b shows the equilibrium shape based on the DFT surface energies. Besides the expected (111) and (100) surface, five surfaces (110), (210), (221), (311), and (322) are found to be present, and the area ratio of (111) surface decreases largely. We can also find that there are substantial differences about the symmetry of different surfaces. The (111) facet becomes 3-fold, while the 4-fold symmetry of (100) facet is kept. In addition, 21278



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is quite involved and infeasible for high-Miller-index surface calculations. Further effort should be undertaken to obtain the more reliable experimental and theoretical results.

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CONCLUSIONS In order to verify the validity of the broken bond model and elucidate the role of high-Miller-index surface in the prediction of equilibrium crystal shape, we have performed systematical density functional theory calculations for 36 different Ni surfaces. The surface energies have been obtained by linear fitting method, in which a sufficient convergence of surface energy was achieved. Consistent with experiment, the surface energy order of three low index surfaces is γ111 < γ110 < γ100. The obtained orientation-dependent surface energies are then used in the Wulff construction to obtain the equilibrium shape of Ni crystal. The results indicated that different from equilibrium shape predicted using broken bond model, in which only two low index surfaces (111) and (100) present, other surfaces including (110), (210), (221), (311), and (322) are also found in the equilibrium shape of Ni. This suggests that the commonly used broken bond model is not enough for predicting the real shape of nanoparticle and that the highMiller-index surfaces also play a very important role.

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AUTHOR INFORMATION

Corresponding Author

*(W.-B.Z.) E-mail: [email protected] or zhangwb@ csust.edu.cn. Phone: +86(0)73185258223. Fax: +86(0) 73185258217. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial supports from the National Natural Science Foundation of China (No. 11004018) and the Hunan Provincial Natural Science Foundation of China (No. 10JJ4002) are gratefully appreciated. Computations were performed partially at Shanghai Supercomputer Center. This work was also supported by the construct program of the key discipline in Hunan Province and aid program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.

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REFERENCES

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Equilibrium Crystal Shape of Ni from First Principles - American

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