Nanorod Calculations on Body-Centered Cubic Iron: A Method for

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J. Phys. Chem. C 2009, 113, 644–649

Nanorod Calculations on Body-Centered Cubic Iron: A Method for Estimation of Size-Dependent Surface Energies of Metal Nanocrystals Pieter van Helden,†,‡ Rutger A. van Santen,*,† and Eric van Steen‡ Molecular Heterogeneous Catalysis, Chemical Engineering and Chemistry, EindhoVen UniVersity of Technology, P.O. Box 513, 5600 MB EindhoVen, The Netherlands, and Centre for Catalysis Research, Department of Chemical Engineering, UniVersity of Cape Town, PriVate Bag X3, Rondebosch 7701, South Africa

Downloaded by UNIV OF NEBRASKA-LINCOLN on September 3, 2015 | http://pubs.acs.org Publication Date (Web): December 19, 2008 | doi: 10.1021/jp807426f

ReceiVed: August 19, 2008; ReVised Manuscript ReceiVed: October 30, 2008

Density functional theory calculations on various Fe nanorod morphologies were performed. The resulting excess energies of low coordinated ridge atoms increased with decreasing coordination number. It was found that the square root dependence of excess energy on coordination number is a good first estimate, but that Fe excess energies scale with the coordination number to the power of 0.82 after including surface energy corrections. A simple general model for the size-dependent surface energy of metal nanocrystals has also been derived. Using this model with the obtained parameters for iron showed that the average surface energy of various Fe nanocrystals per surface atom increases as the size of the nanocrystals decreases. On the other hand, the average surface energy normalized with respect to the surface area increased only slightly with a decrease in size. Introduction The properties of low coordination atoms differ from those in the bulk due to the reduced interaction with neighboring atoms. Nanosized crystals have a large portion of low coordination atoms, and the physical properties of nanosized materials (such as the surface energies1 and morphology2) will change as a result of this. The contribution of this reduced interaction of nanosized clusters to the system’s overall energy can change the thermodynamic stability of these small clusters in various atmospheres.3 The development of methods to estimate surface energies of nanosized crystals is therefore of great interest. An example is the thermodynamic model to estimate the cohesive energy as a function of the crystal size developed by Jiang et al.,4 which is based on a size-dependent cohesive energy.5,6 A simpler, but crude approach is the so-called broken bond method.7,8 In this approximation it is assumed that all broken bonds are similar. A model was proposed to get the surface energy of a plane with a specific coordination9

γ ) (1 - CNS⁄CNB) · Ecoh

(1)

where Ecoh is the bulk cohesion energy per bond and CNS and CNB are the coordination numbers of the surface atoms and the bulk atoms, respectively. An improved expression has been proposed for the low index surfaces of 4d transition metals:10-12

( √

γ) 1-

CNS

)

⁄CNB · Ecoh

(2)

Li and Jiang5 pointed out that although both expressions and the corresponding results are different, they clearly show that

γ ) k · Ecoh

(3)

where γ is the surface energy, Ecoh is the bulk cohesion energy, and k < 1 is a function of CN. * To whom correspondence should be addressed. E-mail: R.A.v.Santen@ tue.nl. Tel.: +31 40 247 3082. Fax: +31 40 245 5054. † Eindhoven University of Technology. ‡ University of Cape Town.

Hartog and van Hardeveld13 showed that the number and type of low coordination atoms change according to the size and structure of nanocrystals. In a recent paper,14 nanorod calculations were done on four fcc metals. These calculations show that different kinds of low coordination atoms contribute differently toward the total energy. It was also shown that the excess energy of the various atoms of fcc metals scaled well with the square root model in eq 2. This resulted in an increase in the average surface energy per atom with decreasing size of fcc nanocrystals. Body-centered cubic metals have a different coordination sphere, and the same dependency cannot be assumed. Another factor has been shown to contribute to the consideration of calculated surface energies. In DFT (density functional theory) various exchange-correlation functionals can be used to calculate the electron-electron interactions. It has been shown that upon using DFT calculations we cannot expect to obtain accurate surface energies,15 but that the introduced error due to a specific functional can be corrected for.15,16 In this paper we will consider the excess energies of various coordinated bcc Fe atoms by using our nanorod calculations and the subsequent surface energy corrections. We will also propose a simple general model for the estimation of the average surface energies of transition metal nanocrystals. Methods and Models Calculational Setup. Plane-wave DFT calculations were performed using the code VASP 17-20 to solve the Kohn-Sham equations21 within periodic boundary conditions. The GGA-PAW (generalized gradient approximation-projector augmented wave) potentials22,23 were used for the calculations to represent the electron-ion interactions. The electron-electron interaction was modeled by the GGA with the exchange-correlation interaction PW91 functional.24 The cutoff energy was 400 eV and convergence was checked for all systems. The Monkhorst-Pack scheme25 was used to perform the k-point sampling. The k-points were optimized for all cells. A k-point set of 14 × 14 × 14

10.1021/jp807426f CCC: $40.75  2009 American Chemical Society Published on Web 12/19/2008

Modeling Surface Energies of Nanosized bcc Fe

J. Phys. Chem. C, Vol. 113, No. 2, 2009 645


Figure 1. Cross-sectional view of examples of the three considered bcc nanorod morphologies: an octagonal rod (left), a 100 square rod (middle), and a 110 square rod (right).

yielded converged energies for bulk cells. For the surface slabs a k-point set of 14 × 14 × 1 was used, and the nanorods were calculated by using a k-point set of 1 × 1 × 14. A Gaussian electron smearing with σ ) 0.2 eV was used to improve convergence time. No atomic constraints were used, thereby allowing all atoms to move in the geometry optimization calculations. All calculations were performed using the conjugate gradient method with a 0.02 eV/Å force tolerance. Bulk Calculations. The bulk of bcc Fe was calculated. The measurable properties for the bulk, such as the bulk lattice parameter, bulk modulus, and magnetization, were calculated. A spin-polarized setup was used in order to ensure that the lowest energy states were obtained. Fe yielded a bulk ferromagnetic ground state. To ensure that the energies and the geometries were converged within the accuracy of the model used, the minimum energy bulk lattice parameters were used in all further calculations. Surface Calculations. For the surface calculations, a p(1 × 1) surface slab was used to represent the square bcc (100) and the bcc (110) surfaces. The vacuum separation was converged at 10 Å, at which it has a negligible effect on the estimated surface energy. The convergence was checked for all calculations. Surface energies were calculated for surface slabs with a thickness of up to 13 layers. The surface excess energy (Esurf) per surface atom (Nsurf) was determined. This surface energy is the difference between the total energy of a surface slab (Eslab) and the total number of metal atoms (N) with an energy per atom of Ebulk

Esurf )

Eslab - N · Ebulk Nsurf

(4)

In this equation the excess energy is seen to originate solely from the reduced nearest neighbor interaction of the surface atoms. The surface energy (γ) is usually represented per unit surface area. This value can be obtained by calculating the area per surface atom (A) and the number of surface atoms in the unit cell (Ns) with surface atoms exposed to vacuum on both sides of the slab

γ)

Eslab - N · Ebulk Ns · A

(5)

These values (surface energy per unit area) can be used for comparison to the experimental data found in the literature. Rod Calculations. In our earlier work,14 we applied the method of making a second cut of the bulk structure along another plane, which leads to the formation of a nanorod structure. A variety of nanorods can be created by cutting surfaces along various planes (Figure 1). These cuts will lead to special low coordinated surface atoms on the nanorod ridge. These atoms lie on the interface between two of the created surface planes. Rods can be created containing ridge atoms with

various coordination numbers and various morphologies. In this study, two different rectangular rods, representative of a cubical morphology, were investigated. The first rectangular rod has four bcc (100) surfaces in the x and y directions, and the second has four bcc (110) surfaces in the x and y directions. These rods will be named 100 square and 110 square, respectively. The number of ridge atoms relative to those with higher coordination can be varied by changing the width of the rod. For these nanorod systems, the vacuum spacing between the closest two flat surfaces of the rods was kept at 10 Å. No constraints were placed on the atoms allowing all the atoms to fully relax in order to minimize the forces. Similar to the fcc metals,14 the excess energy of the ridge atoms is given by

Eridge )

Erod - Nsurf · Esurf - NB · EB Nridge

(6)

where Eridge is the excess energy for the ridge atom, Erod is the total calculated rod energy, Nsurf is the number of the exposed flat surface atoms of energy Esurf, Nridge is the number of ridges, and NB is the total number of atoms with a bulk contribution of EB. Octagonal rods can also be created which expose four bcc (100) surfaces and four bcc (110) surfaces. The resulting octagonal rod has eight ridges, which is the surface interface between the bcc (110) and bcc (100) surfaces. For these rods we used the same calculational parameters as for the rectangular rods. The excess energy of the octagonal edge atoms is therefore given by

Eridge )

Erod - N110 · E110 - N100 · E100 - NB · EB Nridge

(7)

where N110 is the number of bcc (110) surface atoms exposed of surface energy E110 and N100 is the number of bcc (100) surface atoms exposed of surface energy E100. Surface Energy Corrections. We also calculated the proposed surface energy corrections for all the structures.16 The applied surface correction amounts to 0.0401 eV/Å2. The area of edge atoms was estimated by using the Wigner-Seitz radius of iron for the rounded edges (1.302 Å). Atomic Energies. The energy of the isolated Fe atom was calculated by placing the atom in the center of a box with dimension 10 Å × 10 Å × 10 Å. The method proposed in ref 14 was used in trying to calculate the electronic ground state. The difference between the atomic energies and the bulk energies is generally known as the cohesive energy (Ecoh), which is the binding energy of the single metal atoms in a periodic bulk structure. The cohesive energy can thus be regarded as having the same energy as the excess energy for an atom with a coordination of 0 with reference to the bulk state. Results and Discussion Body-Centered Cubic Coordination Number. The coordination number (CN) of an atom is a very widely used and useful concept. For fcc and hcp metals this concept is very simple, since it corresponds directly to the number of nearest neighbor atoms. The bulk coordination number (CB) of both these close-packed structures is 12. We use this coordination number concept to create a localized bonding picture of the considered metal, although the electronic band structures of metals are viewed as delocalized. When we consider the bcc metal structures, we find that using only the eight nearest neighbors is not sufficient to describe

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TABLE 1: Summary of the Coordination Numbers (CN), Excess Energies (Eexcess), and Surface Energies (γ) of the Considered Atoms type of Fe atom

NNNa

NSNNb

N′SLSNNc

CN

Eexcessd (eV)

bulk bcc 110 bcc 100 octagonal ridge 110 square ridge 100 square ridge atom

8 6 4 4 4 2 0

6 4 5 4 3 4 0

0 0 1 0.5 0 2 0

10.16 7.44 5.44 5.26 5.08 2.72 0

0.00 0.87 1.29 1.40 1.52 1.90 5.04 (4.29)e

γ (J/m2) 2.46 2.59

Ecorrf (eV) 0.00 1.09 1.61 1.90 2.20 2.46 4.29


a NNN is the number of nearest neighbor atoms. b NSNN is the number of second-nearest neighbor atoms. c N′SLSNN is the number of the missing second-nearest neighbor interactions in the subsurface layer. d Either Esurf or Eridge. e Same value as the bulk cohesion energy (Ecoh). Experimental value in brackets.33 f Either Esurf or Eridge including the surface energy corrections.16

the coordination number. This problem arises since the secondnearest neighbor bond length is only approximately 15% longer. It is clear that the bcc metals have a second coordination sphere with six s-nearest neighbor bonds. This gives us a total of coordination of 14. Since this total coordination cannot be larger than 12 (close-packing), we need to scale the second-nearest neighbor contribution. We can therefore scale the bcc coordination number by CBbcc ) 8 + 6r, where r is the scaling factor. Various values for r have been proposed. Bol’shakov26 has proposed a value of r ) 0.328 based on the energetic contributions of the two coordination spheres. By using purely geometrical terms, O’Keeffe27 arrived at a similar value of r ) 0.36. Since we would like to relate excess energies to a purely geometrical coordination number, we will use the value of r ) 0.36 when calculating the coordination numbers of the atoms. This results in a total bulk coordination number of CBbcc ) 10.16. We will use the same second-nearest neighbor scaling factor upon considering the various types of atoms that we have calculated

CN ) NNN + 0.36NSNN

(8)

where CN is the effective coordination number of the atom, NNN is the number of nearest neighbors, and NSNN is the number of second-nearest neighbors. We also include the number of missing second-nearest neighbor interactions of the subsurface layer (N′SLSNN) by incorporating them in the second-nearest neighbor number (NSNN). The effective coordination of the considered types of atoms can be seen in Table 1. Bulk Calculations. We calculated the properties of bulk Fe, such as the bulk lattice parameter, the bulk modulus (B0), and the magnetic moment (m), using the PAW-PW91 exchangecorrelation functional. Single point calculations for various lattice parameters were used to obtain the bulk lattice parameter corresponding to the minimum energy. Using the BirchMurnaghan equation of state,28,29 we calculated the bulk lattice constant for iron at 2.827 Å with a bulk modulus of B0 ) 193.5 GPa. These are in agreement with the experimental values of 2.86 Å and 168 GPa.30 The magnetic moment per iron atom was calculated at 2.18 µB, which corresponds well to the experimental value of 2.24 µB.30 Surface Calculations. The surface energies for the bcc (100) p(1 × 1) and the bcc (110) p(1 × 1) surfaces were calculated to determine the excess energy of the various types of atoms. No constraints were imposed on these systems, and therefore all of the atoms were allowed to relax in any direction within the unit cell. Figure 2 shows the influence of the slab thickness on the resulting surface energies (as determined using eq 4) for the bcc (100) and bcc (110) surfaces. The surface energy calculations converge rapidly for these surfaces with variations

Figure 2. Surface energies (Esurf) per atom of the bcc (110) and bcc (100) surfaces as a function of the number of layers in the surface slab.

of less than 1% in the surface energy. The surface energy of the relaxed surface does not significantly differ from that of the unrelaxed surface (less than 1% change). Our calculated values agree reasonably well with the experimental estimates for the polycrystalline surface based on the liquid metal surface tension data extrapolated to 0 K (2.41 J/m231 and 2.55 J/m2 32). A summary of the converged surface energies, as well as the corrected surface energy values for the iron bcc (110) and bcc (100) surfaces, can be seen in Table 1. A slight increase in the average magnetic moment per atom was seen upon creation of these two surfaces. Nanorod Calculations. To estimate the energy of the atoms on the nanorod ridges, we calculated a series of rectangular rods for the 100 square rod and 110 square rod systems. We also calculated a series of octagonal rods with increasing rod diameters. These rods expose four bcc (100) and four bcc (110) surfaces. As expressed in eqs 6 and 7, it is assumed that the excess energies of the atoms in the flat surface areas are equal to the converged surface energies of the bcc (100) and bcc (110) surfaces, respectively. Figure 3 shows the estimated excess energy of the ridge atoms of the nanorods as a function of the total number of exposed atoms per unit cell. From this figure it is clear that the excess energy converges very quickly for the relaxed nanorods. The converged excess energies, as well as the corrected surface energy values, are summarized in Table 1. In all cases the obtained excess energy for the ridge atoms is higher than those for the atoms of the flat surfaces. This is due to the lower coordination of the atoms on the ridges. This is similar to the results for fcc nanorods.14 The octagonal rod’s ridge atoms have the lowest excess energy of the three types of



Figure 3. Ridge energies (Eridge) as a function of the total number of exposed atoms in a unit cell for various nanorod morphologies.

nanorods (1.40 eV/atom). The converged energy for the 110 square rod ridge is only slightly higher than that of the octagonal rod at 1.52 eV/atom. At 1.90 eV/atom, the 100 square rod excess ridge energy is the highest of the considered nanorods. From Table 1, it is clear that the increase of the excess ridge energy is accompanied by a decrease in the coordination number. Relaxation of the atomic positions occurs for all of these nanorod systems, resulting in a decrease in the excess energy. The flat surfaces showed only small relaxations (less that 1%), but as the coordination numbers of the nanorod systems decrease, the relaxations increase slightly. The octagonal rod systems show changes in the excess energy between 1% and 2% upon relaxation. The excess energies of the 110 square rod systems showed changes of approximately 1%, and the excess energies of the 100 square rod systems showed changes of between 3% and 5%. In all cases the atoms on the ridge relaxed toward the center of the rod. The rods showed a higher average magnetic moment than the flat surfaces and bulk systems. Cohesion Energy. We calculated the difference in energy of the bulk atoms and of a single atom in a cell. This corresponds to the estimation of the bulk cohesion energy (Ecoh), and it has the same absolute value as the excess energy of a single atom (coordination number 0). We calculated this value as 5.04 eV. This is much larger than the experimental bulk cohesion energy of iron (4.29 eV).33 The calculation of isolated atoms in the gas phase is a known problem of the pseudopotential GGA approach.34 This arises from the fact that the electronic ground state of isolated spin-polarized atoms are calculated incorrectly, although the correct spin state is achieved. This incorrect atomic energy results in an overestimated bulk cohesion energy. Our calculated bulk cohesion energy is similar to the value calculated in ref 34 since the same error applies. Since the calculation of the atomic ground state energy is shown to be far from the experimental values, we will use the experimental bulk cohesion energy in all further considerations. Dependence of Energy on Coordination. In our earlier work,14 we showed that the excess energies of the ridge atoms (ECN) are proportional to (CN/CB) by using nanorod calculations on four fcc metals. Using eq 2 as a guide, we can represent their data as the relation of the excess energy to the cohesion energy (ECN/Ecoh) as a function of 1 - (CN/CB) (see Figure 4). Their values correspond well to the expected linear relation (these values are still uncorrected). If we construct a similar plot for the uncorrected data we obtained for the Fe bcc nanorods (along with the experimental bulk cohesion energy), we see a similar correspondence to the square root dependency (Figure

J. Phys. Chem. C, Vol. 113, No. 2, 2009 647

Figure 4. Excess energy per atom divided by the corresponding atom’s cohesion energy (ECN/Ecoh) as a function of 1 - (CN/CB) for the fcc metals Co, Cu, Rh, and Au.14 The dashed line indicates the ideal relation.

Figure 5. Excess energy per atom divided by the bulk cohesion energy (ECN/Ecoh) as a function of the coordination number (1 - (CN/CB)n) for bcc iron. O indicates values with n ) 0.52; 9 indicates values with n ) 0.82. The dashed line indicates the ideal relation.

5). Guevara et al.35 used bulk vacancy calculations to suggest that the s- and d-bands scale differently with the coordination number. The square root approximation is therefore only fully valid for pure s-type metals. According to them the d-bands scale according to

ECN ∝ (CNS⁄CNB)n

(9)

where n is close to 2/3. This is an extension of the square root model (eq 2) in which n ) 0.5. By fitting our calculated energies to this relation, we find the best correlation with n ) 0.52 (see Figure 5), which is close to the proposed value of n ) 0.5. By fitting the corrected excess energy values, we obtain a much larger value of n ) 0.82. This value is larger than the n ) 0.5 of the square root model. This indicates that the purely s-bands approximation (n ) 0.5) does not hold for the corrected values of Fe and that the d-band scaling contributes significantly to the binding of Fe atoms. Size-Dependent Model. We can now derive a simple general formulation for the size-dependent surface energy of nanocrystals. The following assumptions will be kept in mind: 1. k is the size-dependent factor. 2. Ecoh is the size-independent contribution of the bulk atoms. In this case Ecoh will be the experimental cohesion energy of the bulk metal. This is also the absolute value of the maximum excess energy (that of a completely isolated atom).

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These assumptions make it possible to reconsider the total surface energy of a nanocrystal. The total excess energy (Eexcess) of a nanocrystal structure can be defined as

Eexcess ) Etot - Ntot · Ebulk

(10)

with Etot as the total energy of the nanocrystal structure, Ebulk as the energy contribution of a bulk atom, and Ntot as the total number of atoms in the nanocrystal structure. The total energy of the nanocrystal structure can be defined as the sum of the energy contributions of all the atoms. The atoms can be divided into groups where each group has a specific coordination number. The energy contribution of a group is the product of the energy of an atom with the specific coordination number (Ei) and the amount of atoms with the specific coordination number (Ni) CB


Etot )

∑ Ni · Ei

(11)

i)0

where CB is the coordination number of a bulk atom. Upon inclusion of eq 11 in eq 10, the total excess energy is given by CB

∑ Ni · Ei - Ntot · Ebulk

Eexcess )

(12)

i)0

The total cohesion energy (Ecoh) or heat of atomization of a bulk system is defined as the difference between the energy of the gas atom (Eatom) and that of an atom in the bulk (Ebulk). If we define the atomic energy as our reference state (Eatom ) 0) then we find that

Ecoh ) Eatom - Ebulk ∴

Ebulk ) -Ecoh

(13)

and eq 13 rearranges to CB

Eexcess )

∑ Ni · EiCN + Ntot · Ecoh

(14)

i)0

The relationship between the coordination number (CNi) of an atom and the excess energy of the specific atom (CN i ) is considered to be given by

[ ( )] CNi CB

εiCN ) 1 -

n

· Ecoh

(15)

Since CN is the excess energy of an atom with a specific i coordination number, it is related to the energy of the remaining bonds of an atom with the specific coordination number (ECN i ) by the cohesion energy (Ecoh):

EiCN ) εiCN - Ecoh

γs )

[

( )]

CB Ecoh CNi · Ntot Ni · Nlow CB i)0

∑

n

(19)

We can see that this model also corresponds to the relationship

γs ) k · Ecoh where

k)

[

(20)

( )]

CB CNi 1 · Ntot Ni · Nlow CB i)0

∑

n

(21)

In eq 20 k is the only factor that can be size dependent. In applying this model, Ecoh can be obtained from experimental results whereas Nlow, Ntot, and Ni can be obtained from the Hartog and van Hardeveld13 statistics, which provides the number of different types of surface atoms as a function of the total number of atoms and thus the size of the crystal for various crystal types. As a first approximation a value of n ) 0.5 can be used, but as we have shown for metals like Fe the value of n can be as large as 0.82 if we include the surface energy corrections. Figure 6 shows the size-dependent average surface energy (in eV/atom) for three types of bcc Fe nanocrystals as a function of an effective diameter. The bcc cubic, octahedral, and rhombic dodecahedral crystal structures were considered.13 As input to the model described in eq 19, we used the experimental bulk cohesion energy (4.29 eV 33) and an exponential factor of n ) 0.82, as obtained from our nanorod calculations on Fe. The effective diameter was defined as the diameter of a sphere with a volume equal to that of the considered nanocrystal. In this example we give the surface energy per mole of surface atoms. From the figure it is clear that for all three of these crystals the average surface energy increases with decreasing nanocrystal size. As a nanocrystal becomes smaller, the ratio of very low coordinated atoms to normal surface atoms increases. This is accompanied by a larger contribution to the excess energy by these very low coordinated atoms, which results in an increase in the average surface energy per atom of the nanocrystal. This is similar to the results for fcc cuboctahedrons.14 Thus, we also find that nanosized crystals have a higher surface energy than bulk materials, which seems to correspond to the experimental estimations of Nanda et al.1 Since the surface area of a small crystal cannot be determined unambiguously, we used the effective diameter to estimate the equivalent spherical surface area of the nanocrystals. Figure 7 shows the size-dependent average surface energy in J/m2 for the same three types of bcc Fe nanocrystals as a function of an

(16)

Substitution of eq 15 into eq 16 and rearrangement of the resulting equation leads to

EiCN ) -

( ) CNi CB

n

· Ecoh

(17)

Substitution of eq 17 into eq 14 results in the following expression CB

Eexcess )

∑ i)0

[( ) -

CNi CB

n

]

· Ecoh · Ni + Ntot · Ecoh

(18)

Considering that the total excess energy of the nanocrystal system is the product of the excess energy per atom (γs) and the total number of lower coordination atoms (Nlow), eq 18 can be rearranged to the final simple model for the average surface energy per atom:

Figure 6. Average surface energy per atom as a function of nanocrystal diameter for three types of bcc crystals.


J. Phys. Chem. C, Vol. 113, No. 2, 2009 649 surface energy per atom increased with decreasing nanocrystal size. The surface area of the nanocrystals also increases with decreasing size. The average surface energy normalized with area only slightly increased with a decrease in size. Acknowledgment. We thank the Stichting Nationale Computerfaciliteiten (NCF) for granting us supercomputer time (grant no. SH-025-07). P.v.H. would like to express his appreciation to Sasol Technology and the University of Cape Town for the opportunity to work in the group of R.A.v.S. References and Notes


Figure 7. Average surface energy per unit area as a function of nanocrystal diameter for three types of bcc crystals.

effective diameter. From this figure, it is clear that the size dependence of the metal nanocrystals gives a different trend when the surface energies are normalized to area. As we have shown, the average surface energy per surface atom increases as the size decreases, but the area of the nanocrystal also increases as the size decreases. Since the area-normalized average surface energy value is a ratio of the surface energy to the area, it is not surprising that the trends seen in Figure 7 are relatively flat with only slight increases in the cubic and octahedral structures. We can therefore note that two important parameters need to be considered with regard to the estimation of the surface energy of small nanocrystals, i.e., the surface energy per surface atom and the area that the specific atom contributes to the structure. Conclusions We calculated bulk, surface, and nanorod systems of bcc iron. We used various nanorod morphologies of Fe to obtain the excess energies of various types of low coordinated atoms. We found that the excess energy of these atoms increases with decreasing coordination. Relaxation of the structures slightly reduces the excess energy of the low coordinated atoms. We found the nanorod approach very useful to study the nature of low coordinated metal atoms. We derived a simple general model for the size dependence of metal nanocrystals. The only inputs that are needed for a first estimate of the size-dependent behavior of metal nanocrystals are the crystal geometry and the experimental bulk cohesion energy. Further refinement can be added by optimizing the exponential factor (n) for each considered element. The nanorod method is ideal to find this parameter for a particular metal. By using Fe as an example, we used this model to show that for various crystal morphologies of the bcc structure the average

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JP807426F

Nanorod Calculations on Body-Centered Cubic Iron: A Method for

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