Shape, Orientation, and Stability of Twinned Gold Nanorods - The

J. Phys. Chem. C 2008, 112, 1385-1390

1385

Shape, Orientation, and Stability of Twinned Gold Nanorods A. S. Barnard* Department of Materials, UniVersity of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom ReceiVed: September 25, 2007

Although energetically unfavorable, twinning is often observed in many faceted nanomaterials including nanocrystals, nanorods, and nanowires, even following annealing. Unfortunately, twinning is neglected by most analytical models, which consequentially limits theoretical descriptions of nanostructured materials to more idealized (less realistic) approximations. In the present study, a new general analytical model for the investigation of nanomaterials of arbitrary shape and with any configuration of twin planes is used to examine twinning in unsupported gold nanorods of various orientations and aspect ratios. The results point to a close competition between nonconvex pentagonal nanorods, and 〈011〉 oriented face-centered cubic (fcc) nanorods containing multiple longitudinal twin planes, in excellent agreement with experimental studies reported in the literature.

1. Introduction As the field of nanotechnology matures, more attention is being given to identifying and characterizing potential hazards associated with nanomaterials, particularly isolated or “free” (unsupported) nanostructures. Considerable effort is being directed toward managing integration of these nanomaterials into devices and to understanding their stability with respect to their thermal and chemical environment during synthesis, storage, and operation.1 Among the variety of engineered nanoparticles produced today, gold is among the most studied because of a number of interesting physical, chemical, and mechanical properties that show great promise for a range of nanoscale applications.2 Many of the desirable properties are strongly linked with the nanomorphology, including such features as size, geometric shape, aspect ratio, and surface and bulk defects. For example, gold nanorods exhibit strong optical extinction at visible and nearinfrared wavelengths which can be tuned by adjusting the nanorod length and diameter.3-9 This results in the enhancement of fluorescence by a factor of 106 compared with “bulk” gold metal.10 The physical origin of the surface plasmon absorption11 and surface plasmon resonances12 in gold nanoparticles have been reported, with emphasis on the effects of particle size and shape.8 Most of the experimental studies on gold nanorods have used samples synthesized by electrochemical methods,4,12-16 bioreduction methods,17,18 or seed-mediated surfactant-directed synthesis.19-21 In each case, the shape anisotropy has been attributed to a type of template-directed process that physically constrains crystal growth in particular directions.22 These syntheses generally produce nanorods with average aspects between approximately 2-40,6,12,23,24 which may be controlled using precursors,24,25 surfactants,26,27 or temperature.23 In addition to this, gold nanorods often exhibit structural modification such as twinning, even following annealing.2,18,28,29 These include single or multiple (symmetric) contact twins in either a lateral or a longitudinal configuration, as well as fivefold (cyclic) twinning about the principal axis, resulting in pentagonal * Corresponding author. E-mail: [email protected].

nanorods akin to an elongated decahedron. Unfortunately, planar defects such as twinning are neglected by most analytical models, which has (until recently) made a rigorous analytical investigation of the structure and stability of realistic gold nanorods practically impossible. In the present study, a general analytical model designed for investigating nanomaterials of arbitrary shape and with any configuration of twin planes is used to examine twinning in unsupported gold nanorods of different orientations, over a range of aspect ratios. Nanorods with 〈001〉 and 〈011〉 orientations are considered, with various configurations of 〈111〉 twin planes (that are accommodated crystallographically), in addition to the fivefold decahedral nanorods characterized by a pentagonal cross section. In general, the results point to a close competition between nonconvex pentagonal nanorods and 〈011〉 oriented face-centered cubic (fcc) nanorods containing multiple longitudinal twin planes, in excellent agreement with experimental studies reported in the literature. 2. Theoretical Method The shape-dependent thermodynamic model for these nanostructures has already been derived for the case of defect-free, pure nanostructures and reported elsewhere.30 It is based on a summation of the Gibbs free energy Gx(T) of a nanoparticle of material in phase x and includes contributions various (defect free) geometric features. This treatment was recently extended31 to include planar defects such as twin planes or stacking faults, becoming a sum of contributions from the particle bulk, surfaces, edges, corners, and planar defects such that: surface Gx ) Gbulk (T) + Gedge x (T) + Gx x (T) +

(T) + Gpd Gcorner x x (T) + . . . (1) Briefly, Gbulk x (T) is defined as the standard free energy of formation ∆Gx(T) which depends on the temperature T. The (T), is expressed in terms of the molar surface term, Gsurface x mass M and density Fx of the material in a phase x and the surface to volume ratio q:

10.1021/jp077688n CCC: $40.75 © 2008 American Chemical Society Published on Web 01/16/2008

1386 J. Phys. Chem. C, Vol. 112, No. 5, 2008

Gsurface (T) ) x

M q Fx

∑i fiγxi(T)

Barnard

(2)

where γxi(T) is the specific surface free energy of facet i. Similarly, the energy associated with an edge Gedge x (T) is expressed as a sum of the specific edge free energy λxj of the edges j and the edge to volume ratio p:

Gedge x (T) )

M p Fx

∑j gjλxj(T)

(3)

,and the energy associated with a corner Gcorner (T) may be x expressed in terms of the specific corner free energy x of corners k along with the corner to volume ratio w such that:

(T) ) Gcorner x

M w Fx

∑k hkxk(T)

(4)

In each case, the weighting factors are defined so that:

∑i fi ) ∑j gj ) ∑k hk ) 1

(5)

Finally, the free energy for a planar defect is written as:

Gpd x (T) )

∑θ [aθνxθ(T) + ∑φ φ lθφηxφ(T)]

(6)

in terms of the specific free energy of a particular twin plane νxθ(T) in the crystallographic orientation θ, the area of that plane aθ, the re-entrant line tension ηxφ(T) along the direction φ, and the length of the re-entrant edge lθφ forming part of the circumference of the defect plane θ. For consistency, this is in turn written in terms of M and Fx, by introducing the number density of planar defects n ) nt/V (where nt is the total number of planar defects in the nanostructure, and V is the total volume of the nanostructure), so that:

Gx(T) ) ∆Gx(T) + p

(

M Fx

1-

2

∑i ifiσi B0R

+

)

Pex B0

[q

∑i fiγxi(T) +

∑j gjλxj(T) + w∑k hkxk(T) + n∑θ (aθνxθ(T) + ∑φ lθφηxφ(T))] (7)

This provides a general model for arbitrary nanostructures containing nt planar defects that may be solved for any desired size or shape and has already been successfully used to examine the shape and stability of gold nanorods in the absence of planar defects.30 Note that a volume dilation is also included, due to isotropic surface stress σxi of the particular crystallographic surface i, which incorporates the bulk modulus B0 and the contribution from external pressure Pex.32 Although the overall affect of this is small, inclusion of this term is particularly important, since it has been reported experimentally that small nanoparticles typically exhibit a lattice contraction, resulting in lattice parameters smaller than that of their bulk counterpart.33 In this model, the shape-dependence is introduced by ratios q and p and w, as well as the weighted sums of the surface energies and the isotropic surface stresses, corresponding to the surfaces present in the particular morphology under consideration. The size dependence is introduced by the ratios q, p, and w, and by the reduction of the volume dilation as the crystal

grows larger. The model is however, rather cumbersome and requires the input of a relatively large number of terms to parametrize it, especially if realistic polyhedral nanostructures are to be considered. These include γxi(T), λxj(T), xk(T), σxi(T), νxθ(T), and ηxφ(T), which must be calculated explicitly for all i, j, k, θ, and φ of interest. For the purposes of studying nanorods of the sizes of interest here, it is possible to use the truncated version of the model (given in ref 31), where the energetic contribution from edges and corners are ignored. This provides , which is a sum of contributions an approximate value, Gapprox x from the bulk and surfaces and twin planes, such that (at T ) 0):

Gapprox ) ∆Gox + x

M (1 - e)[q Fx

∑i fiγxi + n∑θ (aθνxθ +

∑φ lθφηxφ)]

(8)

Using this truncated version, Gapprox may be expected to x give a good approximation of the total free energy at sizes within the range of applicability between Dc to ∼100 nm. The precise value of Dc depends upon the material, but as a general rule of thumb, Dc ≈ [(Vuc‚6 × 104)/(4πNuc)]1/3, where Nuc is the number of atoms in the unit cell and Vuc is total volume of the unit cell. Therefore, in the present context, Dc ≈ 4 nm. The principle advantage of this version is that it only requires that B0, γxi, σxi, νxθ, and ηxφ must be calculated explicitly for each i, θ, and φ. However, it is also of great importance that the same computational method be used for all of these parameters and to the same convergence criteria, so that energetic differences can reliably be attributed to materials properties and not to numerical inconsistencies. Previously, the surface free energies γxi and isotropic surface stresses σxi of reconstructed {111} and {100} gold surfaces have been calculated, using fully relaxed periodic slabs with 2 × 2 (surface) supercells, and nine and eight atomic layers perpendicular to the desired surface, respectively.34 These values, along with the bulk modulus B0, were calculated using scalar relativistic density functional theory (DFT) within the generalized-gradient approximation (GGA), and the final values of B0 ) 157 ( 1.9 GPa, γAu(111) ) 84 meV/Å2, σAu(111) ) 174 meV/Å2, γAu(100) ) 97 meV/Å2, and σAu(100) ) 188 meV/Å2 have been shown to be in excellent agreement with other theoretical studies and experimental results where available. Similarly, by using the same computational method and convergence as that used in ref 34, the energy of a {111} twin plane has been calculated. The final value of νAu(111) ) 1.7 meV/Å2 is in very good agreement other theoretical and experimental results.31,35-37 The values for re-entrant line tensions have also been previously determined by fitting to the results for the set of decahedrons (Au75, Au101, and Au146) reported in ref 38, which (again) were calculated with the same computational procedure used for all of the values for B0, γ, σ, and ν listed above. Fitting to these particles gave the values of ηAu|(111) ) -0.54 eV/Å and ηAu|(100) ) -0.40 eV/Å. Although it is obvious that twin planes must be accompanied by re-entrant angles where the planes intersect with free surfaces (and that certain configurations of a re-entrant angle will minimize the total energy of the defect), the physical consequences of negative line tensions means that particular configurations will actually stabilize the defect. If the endothermic contribution to the total free energy from the twin boundary plane is compensated by an exothermic contribution from the line tension of the circumference, then

Twinned Gold Nanorods

J. Phys. Chem. C, Vol. 112, No. 5, 2008 1387

(depending on the size of the structure) a twinned nanostructure may be energetically preferred over a pristine (untwinned) structure. It should be pointed out that all of the parameters (above) have been calculated at T ) 0. A number of simple expressions have been used here to describe the temperature dependence, that have previously been shown to be suitable for this type of analysis.34 First, a semiempirical expression for determining of γxi(T) proposed by Guggenheim39,40 has been used,

(

γxi(T) ) γxi(0) 1 -

T Tc

)

a

(9)

where a is an empirical parameter (known to be unity for metals41) and Tc is the critical temperature at which the structure of the surface deteriorates or changes significantly from the structure in the bulk.42 The value of Tc is usually taken as the bulk surface melting temperature, but since this has been shown to be size dependent,43 the expression of Qi and Wang44 has also been employed:

(

Tc ) T m 1 -

)

6ratS D

(10)

where Tm are the macroscopic surface melting temperatures, D is the average diameter of the nanostructure, rat is the atomic radius of gold, and S is a shape dependent factor defined as the ratio of the surface area of the particle divided by the surface area of a sphere of equivalent volume.44 The temperature dependence of other quantities such as σxi has been described in the same way. 3. Discussion of Results A review of the literature on gold nanorods shows that considerable attention has been focused on pentagonally twinned structures, with five twin planes radiating from the center or the nanorod perpendicular to the principal axis. Schematic representations of these model shapes are shown in Figure 1. There are two possible variants which are analogous to the convex (simple) decahedron (Figure 1a), and the nonconvex (Marks) decahedron (Figure 1b), depending upon whether the concave re-entrant angles are present along the (100)/(100) edges. In addition to these structures, different configurations of twins may be observed in fcc gold nanorods, as illustrated by Figure 2. Shown to the far left are “pristine” (untwinned) nanorods oriented in the 〈001〉 direction (Figure 2a), along with twin planes in a “lateral” configurations (right); and 〈011〉 direction (Figure 2b), with twin planes in “radial”, “longitudinal”, and “lateral” configurations (left to right), respectively. By using the truncated version of the model given in eq 8, the approximate free energy of formation has been calculated for gold nanorods with each of the twinned and pristine shapes shown in Figures 1 and 2. In each case, care has been taken in selecting n (where geometrically possible) to ensure that the total twin area is approximately the same, that is, that in the fcc nanorods Σθaθ (longitudinal) ≈ Σθaθ (lateral) ≈ Σθaθ (radial), and that Σθaθ (fcc) ≈ Σθaθ (decahedral). The results are plotted as a function of size in Figure 3a, where a number of interesting results are revealed. The first point that is apparent is that the 〈011〉 orientation is energetically preferred over the 〈001〉 orientation, for the shapes shown in Figure 2. Next, we see that these gold nanorods prefer to contain twins than to be pristine, and in the case of the 〈011〉 nanorods, longitudinal twinning is preferred over radial or lateral twinning. Twinning is favored

Figure 1. Schematic representations of pentagonally twinned gold nanorods, showing (a) the simple convex (Ino) variation, and (b) the nonconvex (Marks) variation. The {100} surfaces are shown in red; the {111} surfaces are shown in green, and the 〈111〉 twin planes are shown in gray.

Figure 2. Schematic representations of pristine (untwinned) and twinned fcc gold nanorods oriented in (a) the 〈001〉 direction and (b) the 〈110〉 direction. The {100} surfaces are shown in red; the {111} surfaces are shown in green, and the 〈111〉 twin planes are shown in gray.

at small sizes due to exothermic ηxφ stabilizing the defect (see ref 31). Among the entire collection of fcc nanorods, the longitudinally twinned 〈011〉 oriented nanorod is the most energetically preferred. In general, the prediction that longitudinal twinning is the preferred configuration for gold nanorods is in excellent agreement with the experimental observations of Canizal et al.,18 who concluded that gold nanorods either exhibit parallel (longitudinal) bands of twin planes when growing along the

1388 J. Phys. Chem. C, Vol. 112, No. 5, 2008

Barnard

Figure 3. Approximate free energy of formation for isomorphic (R ) 5) gold nanorods as illustrated in Figures 1 and 2, as a function of total volume at (a) ambient temperatures, and (b) T ) 100 °C.

〈011〉 direction or exhibit concentric (fivefold) twins forming pentagonal nanorods when growing along the 〈110〉 direction. The present results predict that under ambient conditions the 〈110〉 oriented Marks configuration (Figure 1b) is preferred over the longitudinally twinned 〈011〉 nanorod only at small sizes (NAu j 7500). Over this size, the longitudinal twinned 〈011〉 nanorod is energetically preferred, and at large size J45000, the radial and lateral twinning configuration are also more stable than the Marks configuration. This is consistent with the energetic preference of this shape in quasi-zero dimensional nanocrystals31 but does not explain the repeated observation of this shape in numerous independent studies of higher aspect gold nanorods.18,19,28,29 A common feature of these cited studies is the use of elevated temperatures, usually between 20-100 C. When elevated temperatures are used in eq 8, as shown in Figure 3b, we can see that the stability of the Marks configuration is increased. Note that the temperature (of 373.15 K) used in Figure 3b is at the upper end of the temperature range used during synthesis, but it serves to highlight the temperaturedependence of the calculated results. Comparing Figure 3a,b,

we can clearly see that temperature does play a role, but the stability of the pentagonal Marks configuration cannot be attributed to temperature dependence alone. Another important feature of gold nanorods produced via colloidal synthesis methods is the aspect ratio (R), which may be systematically moderated using surfactants.26,27 The calculations presented in Figure 3 are isomorphic; that is, the shape and aspect ratio are constant at all sizes. In both Figure 3a and Figure 3b, R ) 5; however, it may be expected that the relative stability of these different orientations and twinning configurations could be different for longer nanorods (for example, with R > 15). Therefore, the results of free energy for this collection of gold nanorods has been calculated as a function of R and is shown in Figure 4. These calculations are all volume conserving (equivalent to a total number of atoms NAu ) 10 000), so that both the diameter and the length change as R is varied. Looking to Figure 4, we can see that low aspect gold nanorods (of any orientation or twinning configuration) are metastable with respect to a transition to a quasi-zero dimensional gold nanocrystal, given suitable perturbation. This is consistent with

Twinned Gold Nanorods

J. Phys. Chem. C, Vol. 112, No. 5, 2008 1389 expected to be stable under various conditions. In addition to these factors, surface chemistry is also known to be influential and is a topic for future work. Acknowledgment. This work has been supported by the Glasstone Benefaction at the University of Oxford. Computational resources for this project have been supplied by the MSCF in EMSL (a national scientific user facility sponsored by the U.S. DOE, OBER and located at PNNL) and the University of Oxford Campus Grid, OxGrid. References and Notes

Figure 4. Approximate free energy of formation for isovolumetric gold nanorods as illustrated in Figures 1 and 2, as a function of R at ambient temperatures.

the computational results of Diao et al.03,46 and Wang et al.47 and arises because of the fact that R ∝ q. A high R will give a high q, which increases to total free energy via the surface contribution. We can also see that although the Marks decahedral nanorod is not the lowest energy twinning configuration at low R, an energetic preference of this morphology over the alternatives is introduced and increases with increasing R. An interesting feature of these results is that at R J 10 the results indicate that it is just as preferable for some twinned gold nanorods to be longer as it is for them to be shorter, even under ambient conditions. As q is increasing and raising the total free energy, so too is the contribution from ∑φθφηxφ, which has a stabilizing influence.31 Although axial growth is likely to be controlled by kinetics, these calculations suggest that, at high aspect ratios, axial growth of Marks decahedral nanorods and the longitudinally (and possibly radially) twinned 〈011〉 nanorods will be thermodynamically encouraged. Without the twin defect, the kinetic (axial) growth will be thermodynamically unfavorable (see results for pristine 〈011〉 nanorods in Figure 4), and shorter nanorods could be expected. These results are reasonable, considering thermodynamics have been used experimentally to control aspect by Park et al.,23 where the authors systematically changed R by adjusting the reaction temperature in the seed-mediated synthesis of the gold nanorods, in the range 315-276 K. Furthermore, it is thought that these findings may help to explain the persistence of large high-aspect pentagonal nanorods, at sizes well above those that would be intuitively expected for these defective nanostructures.2 4. Conclusions Presented here are results of an analytical shape-dependent thermodynamic model, with parameters calculated using relativistic density functional theory, examining the relative stability of different twinning configurations in unsupported gold nanorods as a function of size, temperature, and aspect ratios. The results point to a close competition between nonconvex pentagonal nanorods and 〈011〉 oriented fcc nanorods containing multiple longitudinal twin planes, in excellent agreement with experimental studies reported in the literature. Although kinetics will undoubtedly play a part,48 the results of these calculations indicate that temperature, aspect ratio, and twinning configuration are all likely to be influential in determining whether a particular type of gold nanorod may be

(1) Barnard, A. S. Nat. Mater. 2006, 5, 245. (2) Elechiguerra, J. L.; Reyes-Gasga, J.; Jose´-Yacama´n, M. J. Mater. Chem. 2006, 16, 3906. (3) Foss, C.; Hornyak, G.; Stockert, J.; Martin, C. J. Phys. Chem. 1994, 98, 2963. (4) Yu, Y.-Y.; Chang, S.-S.; Lee, C.-L.; Wang, C. R. C. J. Phys. Chem. 1997, 101, 6661. (5) Link, S.; Mohamed, M.; El-Sayed, M. J. Phys. Chem. B 1999, 103, 3073. (6) Jana, N. R.; Gearheart, L.; Murphy, C. J. J. Phys. Chem. B 2001, 105, 4065. (7) Link, S.; El-Sayed, M. A. J. Phys. Chem. B 2005, 108, 11876. (8) Pe´rez-Juste, J.; Pastoria-Santos, I.; Liz-Marza´n, L. M.; Mulvaney, P. Coord. Chem. ReV. 2005, 249, 1870. (9) Gulati, A.; Liao, H.; Hafner, J. H. J. Phys. Chem. B 2006, 110, 22323. (10) El-Sayed, M. A.; Volkiv, V.; Link, S. Chem. Phys. Lett. 2000, 317, 517. (11) Link, S.; El-Sayed, M. A. Int. ReV. Phys. Chem. 2000, 19, 409. (12) Chang, S.-S.; Shih, C.-W.; Chen, C.-D.; Lai, W.-C.; Wang, C. R. C. Langmiur 1999, 15, 701. (13) El-Sayed, M. A.; Ismail, K. Z.; Link, S. J. Phys. Chem. B 1998, 102, 9370. (14) Link, S.; Burda, C.; Mohamed, M. B.; Nikoobakht, B.; El-Sayed, M. A. J. Phys. Chem. A 1999, 103, 1165. (15) Wang, Z. L.; Mohamed, M. B.; Link, S.; El-Sayed, M. A. Surf. Sci. 1999, 440, L809. (16) Link, S.; Burda, C.; Nikoobakht, B.; El-Sayed, M. A. J. Phys. Chem. B 2000, 104, 6152. (17) Gardea-Torresday, J. L.; Tiemann, K. J.; Go´mez, G.; Dokken, K.; Tehuacanero, S.; Jose´-Yacama´n, M. J. Nanopart. Res. 1999, 23, 1. (18) Canizal, G.; Ascencio, J. A.; Gardea-Torresday, J.; Jose´ Yacama´n, M. J. Nanopart. Res. 2001, 3, 475. (19) Johnson, C. J.; Dujardin, E.; Davis, S. A.; Murphy, C.; Mann, S. J. Mater. Chem. 2002, 12, 1765. (20) Murphy, C. J.; Sau, T. K.; Gole, A.; Orendorff, C. J. MRS Bull. 2005, 30, 349. (21) Jana, N. R.; Gearheart, L. A.; Obare, S. O.; Johnson, C. J.; Edler, K. J.; Mann, S.; Murphy, C. J. Mater. Chem. 2002, 12, 2909. (22) Gou, L.; Murphy, C. J. Chem. Mater. 2005, 17, 3668. (23) Park, H. J.; Ah, C. S.; Kim, W.-J.; Choi, I. S.; Lee, K.-P.; Yan, W. S. J. Vac. Sci. Technol., A 2006, 24, 1323. (24) Wu, H.-Y.; Chu, H.-C.; Kuo, T.-J.; Kuo, C.-L.; Huang, M. H. Chem. Mater. 2005, 17, 6447. (25) Busbee, B. D.; Obare, S. O.; Murphy, C. J. AdV. Mater. 2003, 15, 414. (26) Gao, J.; Bender, C. M.; Murphy, C. Langmuir 2003, 19, 9065. (27) Chen, H. M.; Peng, H.-C.; Liu, R.-S.; Asakura, K.; Lee, C.-L.; Lee, J.-F.; Hu, S.-F. J. Phys. Chem. B 2005, 109, 19533. (28) Hu, M.; Hillyard, P.; Hartland, G. V.; Kosel, T.; Pe´rez-Juste, J.; Mulvaney, P. Nano Lett. 2004, 4, 2493. (29) Gai, P. L.; Harmer, M. A. Nano Lett. 2002, 2, 771. (30) Barnard, A. S.; Curtiss, L. A. J. Mater. Chem. 2007, 17, 3315. (31) Barnard, A. S. J. Phys. Chem. B 2006, 110, 24498. (32) Unless otherwise stated, all of the results in the proceeding section have been calculated under ambient conditions, with T ) 298.15 K and Pex ) 101.33 kPa. (33) Page, K.; Proffen, T.; Terrones, H.; Terrones, M.; Lee, L.; Yang, Y.; Stemmer, S.; Seshadri, R.; Cheetham, A. K. Chem. Phys. Lett. 2004, 393, 385. (34) Barnard, A. S.; Lin, X. M.; Curtiss, L. A. J. Phys. Chem. B 2005, 109, 24465. (35) Rosengaard, N. M.; Skriver, H. L. Phys. ReV. B 1993, 47, 12865. (36) Crampin, S.; Hampel, K.; Vvedensky, D. D.; Maclaren, J. M. J. Mater. Res. 1990, 5, 2107. (37) Murr, L. E. Scr. Metall. 1972, 6, 203. (38) Barnard, A. S.; Curtiss, L. A. Chem. Phys. Chem. 2006, 7, 1544.

1390 J. Phys. Chem. C, Vol. 112, No. 5, 2008 (39) Guggenheim, E. A. J. Chem Phys. 1945, 13, 253. (40) Guggenheim, E. A. Thermodynamics, 4th ed.; North Holland: New York, 1993. (41) Grosse, A. V. J. Inorg. Nucl. Chem. 1962, 24, 147. (42) Somorjai, G. A. Introduction to surface chemsitry catalysis; John Wiley & Sons, Inc.: New York, 1994. (43) Buffat, Ph.; Borel, J.-P. Phys. ReV. A 1976, 13, 2287.

Barnard (44) Qi, W. H.; Wang, M. P. Mater. Chem. Phys. 2004, 88, 280. (45) Diao, J.; Gall, J.; Dunn, M. L. Nature Mater. 2003, 2, 656. (46) Diao, J.; Gall, J.; Dunn, M. L. Phys. ReV. B 2004, 70, 075413. (47) Wang, Y.; Teitel, S.; Dellago, C. Nano Lett. 2005, 5, 2174; Wang, Y.; Teitel, S.; Dellago, C. J. Comput. Theor. Nanosci. 2007, 4, 282. (48) Chui, Y. H.; Grochola, G.; Snook, I. K.; Russo, S. P. J. Comput. Theor. Nanosci. 2007, 4, 291.