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J. Phys. Chem. C 2007, 111, 4998-5005
Surface Energy Estimation of Catalytically Relevant fcc Transition Metals Using DFT Calculations on Nanorods Jurie C. W. Swart, Pieter van Helden, and Eric van Steen* Centre for Catalysis Research, Department of Chemical Engineering, UniVersity of Cape Town, PriVate Bag X3, Rondebosch 7701, South Africa ReceiVed: December 11, 2006; In Final Form: February 2, 2007
The use of nanorods is presented to investigate the properties of low coordination atoms in a metal with an fcc structure. Calculations on nanorods can be performed in a similar manner to surface calculations keeping the efficiency of the 3D periodic plane wave approach. Nanorods with fcc(100) surfaces exposed have been used to calculate the excess energy of low coordination atoms. It was observed that the excess energy per “broken bond” decreases with an increase in the coordination number. The excess energy is further reduced due to relaxation of the surface atoms. Surface magnetization results in a further reduction of the excess energy for magnetic materials, such as Co, but not for nonmagnetic materials, such as Rh. Furthermore, it is shown that the bond strength of the surface atoms with nearest neighboring atoms in surface and bulk increase with a decrease in the coordination.
Introduction Transition metals such as Co, Cu, Rh, and Au are catalytically active but are rather expensive relative to metals such as Fe. Maximum utilization of the catalytic activity is obtained by dispersing these metals as nanosized crystallites on a support, so that the fraction of catalytically active surface atoms increases significantly. The properties of surface atoms differ from those in the bulk of the structure due to the reduced interaction with neighboring atoms and a lower coordination. This results in a change in the physical properties of nanosized materials, such as magnetization,1 Fermi-level,2,3 surface energies,4 and morphology.5 The change in the intrinsic physical properties of nanosized materials is believed to cause a modified catalytic behavior.6 However, the surface contribution of nanosized clusters to the overall energy of the system can change the thermodynamic stability of these small clusters in various atmospheres.7 Hence, it is of great interest to develop methods to estimate surface energies of nanosized crystals. It has been proposed that the surface energy is related to the number of “broken bonds”,8,9 in which case the size dependent surface energy can be estimated utilizing the statistical models developed by Hartog and van Hardeveld.10 Methfessel et al.11 investigated the surface energy of 4d metals using the fullpotential linear-muffin-tin-orbital (FP LTMO) method. They argue that although the main features of the surface energy can be related to the number of “broken bonds”, the change in the bond strength with changing coordination of the surface atom should be taken into account. Robertson et al.12 tried to obtain insight in the coordination dependent energy of aluminum using so-called “glue models”. They only considered different bulk crystal lattices with different coordination numbers and surface slabs with a two layer thickness but could not obtain a quantitative insight into the coordination dependent energies per atom. Jiang et al.13 developed an thermodynamic model to estimate the cohesive energy as a function of the crystal size. It becomes * To whom correspondence should be addressed. E-mail:
[email protected]. Tel: + 27 21 650 3796. FAX: + 27 21 650 5501.
less negative with decreasing crystallite size. A size-dependent surface energy was derived based on the size dependent cohesive energy.14,15 A counter-intuitive increase in the surface energy with increasing size of nanocrystals was obtained, contradicting experimentally observed high surface energy of nanosized Ag crystals.4 Surface energies are typically obtained for particular surfaces using “slab”-type calculations.5,11 It should however be realized that nanocrystals contain surface atoms on the facets of the crystals as well as surface atoms on the ridges and edges of the nanocrystal. Surface atoms on ridges and edges have a much lower coordination than the surface atoms on the facets. The fraction of these low coordination surface atoms increases strongly with decreasing size of the nanocrystal.10 Surface energy calculations using “slab”-type models will only yield an insight in the energy of the surface atoms in the facets. In this paper, a new method for investigating lower coordinated atoms using nanorods in a 3-D periodic supercell will be investigated to gain insight into the behavior of nanoclusters with a view to determine the size dependent surface energies using density functional theory (DFT). Furthermore, insight will be obtained into the influence of the coordination of surface atoms on the excess surface energy. Methodology Calculation Setup. All calculations are performed using the planewave DFT (density functional theory) code VASP16-19 solving the Kohn-Sham equations20 with 3D periodic boundary conditions. The electron-ion interactions are represented by Vanderbilt Ultrasoft Pseudopotentials21 as given by Kresse and Hafner,22 whereas the electron-electron interaction was modeled by the generalized gradient approximation (US-GGA) with the exchange-correlation interaction function PW91 functional proposed by Perdew et al.23 or the localized density approximation (LDA) with the exchange and correlation interaction proposed by Ceperley and Alder24 and parametrized by Perdew and Zunger.25 The GGA-PAW (projector augmented wave) potentials26 implemented by Kresse and Joubert27 were used for some of the calculations to compare the performance of the
10.1021/jp0684980 CCC: $37.00 © 2007 American Chemical Society Published on Web 03/10/2007
Surface Energy Estimation using Nanorods
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potentials. The cutoff energy was 400 eV ensuring convergence, which was checked for all systems. The k-point sampling was done using the Monkhorst-Pack scheme.28 The k-points were optimized for all cells. For bulk cells, a k-point set of 13 × 13 × 13 yielded converged energies. For the surface cells, slabs, a k-point set of 13 × 13 × 1 was used and for the nanorods a k-point set of 1 × 1 × 13 gave converged energies within 2 meV. The electron smearing over the Fermi surface was used to improve convergence with k-points with σ ) 0.1 eV as implemented the first-order Methfessel-Paxton method.29 The error due to smearing was lower than 1 meV for all calculations when extrapolated to σ ) 0. The extrapolated values (σ ) 0) were used in all cases. The geometry optimization calculations for the surface slabs and square rods were performed using the conjugate gradient method converging with a 0.02 eV/atom force tolerance with no atomic constraints thereby allowing all atoms to move. Bulk Calculations. The measurable properties for the bulk fcc Co, Cu, Rh, and Au, such as bulk lattice parameter, bulk modulus and magnetization (where applicable), were calculated using the spin polarized setup in order to ensure that the lowest energy states are obtained. Only Co yielded a bulk magnetic ground state with a lower energy while Rh, Cu, and Au calculations gave the same ground state energy and lattice parameter using a spin polarized and nonpolarized calculation setup. The minimum energy bulk lattice parameters were used in all further calculations in order to ensure that the energies and geometries calculated are converged within the accuracy of the model used. Surface Calculations. A k-point set of 13 × 13 × 1 was used for all surface slabs, resulting in 21 irreducible k-points for the fcc (111) and 28 irreducible k-points for the fcc (100) surfaces. A cutoff energy of 400 eV was used along with a smearing of σ ) 0.1 eV. For the surface calculations, a p(1 × 1) surface slab was used to represent the square fcc (100) and fcc (111) surfaces. The vacuum separation was converged at five times the interlayer spacing (8.75-10 Å depending on the lattice parameter) as shown by a negligible effect on the estimated surface energy (
