Article pubs.acs.org/JPCC
Periodic Trends in Adsorption and Activation Energies for Heterometallic Diffusion on (100) Transition Metal Surfaces Handan Yildirim, Subramanian K.R.S. Sankaranarayanan,* and Jeffrey P. Greeley* Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, United States S Supporting Information *
ABSTRACT: A first-principles analysis of trends in metal-on-metal hopping diffusion for 64 admetal/substrate systems is presented. Focusing on the (100) facets of various transition metal substrates, we demonstrate that the calculated hopping diffusion barriers may be interpreted in terms of the cohesive energies of the admetals and substrates, as well as the lattice constants of the substrates. We further show that general linear relationships exist between the diffusion barriers and the corresponding adsorption energies on each transition metal substrate. The slopes in these Brønsted−Evans− Polanyi relationships are related to the degree of resemblance between the initial states and the transition states for hopping diffusion, and the slopes are found to depend sensitively on the nature of the transition metal substrate. Substrates with higher cohesive energies and smaller lattice constants generally exhibit smaller slopes and, therefore, a closer correspondence between the transition states and the initial states. These relationships, in addition to providing fundamental insights into trends in diffusion across different transition metal surfaces, give a powerful and convenient means of predicting diffusional kinetics from purely thermodynamic quantities. The results may ultimately provide a useful input to kinetic Monte Carlo (kMC)-type simulations, enabling efficient and accurate studies of heteroepitaxial metal-on-metal growth.
I. INTRODUCTION Metal-on-metal surface diffusion is central to both the basic physics of crystal and thin film growth and to a variety of technologically important fields, including catalysis, microelectronics, and corrosion. In spite of these diverse and significant applications, however, fundamental knowledge of the kinetics and dynamics of diffusing metal adspecies is far from complete, and nearly all atomistic studies of these phenomena have focused on the diffusion of specific metals across specific substrates.1−6 While such studies, involving both experimental7,8 and computational techniques, have identified many important principles of surface diffusive processes, a more general understanding of atomic-scale trends in surface diffusion across different admetals and substrates is lacking. The development of such a generalized understanding, in turn, could be of significant benefit in controlling the structure of alloys during growth or dealloying processes and in designing bimetallic materials for desired applications. Theoretical surface science studies have emerged in recent years as a powerful tool to elucidate the kinetics, dynamics, and atomistic details of the mechanisms governing surface processes. Such studies, based primarily on periodic Density Functional Theory (DFT) calculations, have found extensive uses in a variety of applications.9−15 These calculations have been shown, for example, to be useful for obtaining linear energy scaling relationships for a variety of molecular adsorbates on transition metal surfaces.16 Additional linear relationships are of the Brønsted−Evans−Polanyi (BEP) type,17−19 which describes the correlations between the kinetics © XXXX American Chemical Society
of elementary surface processes and the corresponding thermodynamics.20,21 For complex catalytic reactions, these relationships have permitted the description of fundamental reactivity trends across diverse catalyst surfaces using just a few independent parameters, or descriptors.22−27 To a significantly lesser extent, simplified forms of these general classes of linear correlations have been used to describe metal adatom diffusion on transition metal substrates.28−31 In particular, adatom diffusion on corrugated surfaces such as (100) has often been studied,28−31 and some correlations between the adsorbate binding strengths and the bulk bond energies have been suggested for self-diffusion processes32 wherein the admetal and the substrate have the same elemental identity (these systems are also referred to with the term “homodiffusion” in the remainder of this article). A linear relation between the hopping barriers over step edges and the (111) terrace adsorption energies has also been reported for a few heterodiffusive processes, wherein the elemental identities of the admetal and substrate are different.33 Similarly, a relatively recent firstprinciples study, using related ideas, has reported that BEP-type correlations exist for the diffusion of several atomic and molecular adsorbates (C, N, NO) on close-packed transition metal surfaces.34 In spite of the advances described above, there are relatively few general principles that have been identified for describing atomistic details of heterodiffusion of metal adatoms on metal Received: September 7, 2012
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the bottom two layers are held fixed). The (100) surface contains three adsorption sites: top, 4-fold, and bridge. To test the energetically most stable adsorption sites for the adatoms, we calculated the adsorption energies at each site for selected adsorbate−substrate pairs (Au, Ag, Pd, and Pt adatoms on Cu(100) and Au and Pt adatoms on Ag(100), Au(100), Pd(100), Pt(100), and Ir(100)). Our results suggest that the 4fold site (FF) is the energetically most stable adsorption site, while the bridge is the next most stable and the top site the least stable, and we therefore focus on FF calculations in what follows. The activation barriers for the adatom diffusion via hopping mechanism on (100) are calculated using either the drag method (1D scan of the potential energy surface) or the nudged elastic band method56 (see also discussion in the text). For the drag method, the potential energy surface is scanned with about 20 intermediate points between the two FF sites to obtain the highest energy transition state configuration. The NEB calculations are performed using five images, which is found to be sufficient for such a symmetric diffusion pathway.
substrates. This lack of generality, in turn, is largely due to the absence of a comprehensive, trends-based, first-principles database of energetic information for diffusion of different classes of transition metal adspecies on metal substrates. Such results would be highly useful in identifying periodic trends in adsorption and activation energies for hetero metal-on-metal diffusion and in developing correlations between the thermodynamics and kinetics of diffusion for use in longer time scale kinetic Monte Carlo (kMC)-based simulations.35−43 In this paper, we report on first-principles, periodic DFT calculations of the adsorption energies and activation barriers for hopping diffusion of the metal adatoms Cu, Ag, Au, Pd, Pt, Rh, Ru, and Ir on the terraces of the unreconstructed (100) surfaces of Cu, Ag, Au, Pd, Pt, Rh, Ni, and Ir, for 64 systems in total (we do not explicitly consider diffusion via exchange mechanisms although exchange processes are known, in some cases, to lead to surface alloying on (100) surfaces44). We confirm that the energetics of self-diffusion for homodiffusion systems are closely related to the bulk cohesive energies of the studied metals, and we demonstrate that more general classes of correlations, in the form of BEP relationships, exist between the kinetics and thermodynamics of metal diffusion in many hetero systems. We discuss how the slopes of these relationships are related to the degree of correspondence between the transition states and the stable initial states on various metals, and we further analyze how knowledge of substrate lattice constants and the cohesive energies of the adsorbates and substrates can be used to obtain a qualitative understanding of the presented trends. Finally, we briefly mention how these results could be extended to provide the necessary input for predicting the nature of growth modes associated with each system.45,46
III. RESULTS AND DISCUSSIONS III.1. Diffusion Barriers and Binding Energies. We begin by demonstrating that hopping diffusion barriers calculated using the computationally costly, yet accurate, NEB and the more efficient drag method (see the description in the theoretical section) yield very similar results. In Figure 1a,b,
II. THEORETICAL METHODS First-principles calculations have been carried out within the periodic Density Functional Theory framework, as embodied in the Vienna ab initio Simulation Package (VASP).47−50 The Perdew−Wang 91 (PW91) gradient-corrected potential is used to represent the electron exchange-correlation functional in the generalized gradient approximation (GGA).51,52 We choose the GGA approximation over the local density approximation (LDA) as it gives activation barriers that better match experimental results.53 For all the calculations, a kinetic-energy cutoff of 400 eV is used for the wave functions to obtain converged results. Brillouin zones are sampled using 14 × 14 × 14 Monkhorst−Pack k-point meshes for determining the bulk lattice constants and 5 × 5 × 1 k-point meshes for the surface calculations.54 The optimization of the total energies is achieved via the conjugated-gradient (CG) algorithm55 with the force criterion on each atom set for the convergence to be around 0.02 eV/Å. Spin-polarized calculations are performed for calculations involving Ni and are also employed to obtain the energies of the isolated metal atoms in the gas phase. The bulk lattice constants obtained with these sets of parameters are 3.52, 3.64, 3.84, 3.88, 3.96, 3.99, 4.16, and 4.18 Å for Ni, Cu, Rh, Ir, Pd, Pt, Ag, and Au, respectively. For all the calculations, the (100) surfaces are constructed using a super cell with a vacuum region of 15 Å between the surfaces. For all the slabs considered, five layers of a 5 × 5 surface unit cell with an adatom adsorbed on only one side of the slab are used to model the adsorbate−substrate systems. The structure (surface with adsorbed atom) is optimized using the conjugate-gradient method, and during the optimization we fixed the bottom layer of atoms (the results are unchanged if
Figure 1. Diffusion barriers as a function of the reaction coordinate for selected admetals on Cu(100) obtained via (a) the drag method and (b) the NEB method.
we plot activation barriers on Cu(100) calculated using the drag (Figure 1a) and the NEB (Figure 1b) methods as a function of the diffusion coordinate for five adsorbates; the agreement between the methods is clearly satisfactory. In light of these results, we use the computationally efficient drag calculations for the remainder of this study. Next, we analyze the process of self-diffusion via hopping in several homodiffusion systems. In Figure 2a, we plot the selfdiffusion barriers as a function of the bulk bond energies (1/6 of the cohesive energy) associated with each metal. We find that these barriers follow the order, from the lowest to highest, of Ag/Ag(100) < Cu/Cu(100) < Au/Au(100) < Pd/Pd(100) < Ni/Ni(100) < Rh/Rh(100) < Pt/Pt(100) < Ir/Ir(100). These results are completely consistent with available first-principles reports32 and, when plotted against a measure of the bulk bond B
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diffusion barriers and Ir the highest. Although a broad relationship between the barrier and the cohesive energy can be seen when the adsorbate/substrate pairs are considered in aggregate, the clearest trends are observed for the diffusion on a given substrate, when the geometric and electronic/energetic properties of the substrate do not vary. For all of the adsorbates, the largest diffusion barriers are in general found on Au(100) substrates, while the smallest barriers are on Ni(100) substrates. Au has the largest lattice constant of all the systems studied, and it is one of the least cohesive metals. Ni has the smallest lattice constant, and it is quite cohesive in comparison to Au. Although the trends are not completely regular for metals between these extremes (see also discussion below), these observations suggest that, for heteroepitaxial systems, the cohesive energy and lattice constants are useful parameters for describing qualitative trends in diffusional properties. When considering the inverse set of trends to that described above, corresponding to diffusion of a single admetal element across different substrates, trends in diffusional properties become less well-defined. For example, a plot of diffusion barrier against the substrate element (plot not shown) for given admetals, with the substrate elements listed in order of increasing cohesive energy, does not show a clearly defined monotonic trend; this result is due to the fact that, when moving from one substrate element to the next, there are substantial changes in both the substrate’s electronic and energetic properties, as well as in the substrate lattice constant. Nevertheless, a limited number of trends governing the diffusion of given adsorbate species across multiple substrate metals can be deduced from the results in Figure 2b. Most significantly, adsorbates with higher cohesive energies, such as Pt, Rh, Ru, and Ir, show a broader range of diffusion barriers across different substrates (the vertical spread in Figure 2b increases accordinglynote that the point for Ir/Au(100) is omitted since a certain amount of surface alloying was observed in this case). This increased sensitivity of diffusion barrier to the substrate element is likely due to the fact that more cohesive metals will naturally interact more strongly with the surrounding metal atoms, leading to a greater sensitivity to the properties of the substrate. In addition to kinetic quantities such as diffusion barriers, trend-based analyses may also be performed for thermodynamic quantities such as admetal adsorption energies. In Figure 2c, we plot the adsorption energies at the FF site as a function of the elemental identities of the admetal adsorbates, in order of increasing cohesive energy of the adsorbate metals (a larger adsorption energy, in this figure, denotes a stronger admetal− substrate bond). As with the diffusion barriers, there is an approximately monotonic increase in the strength of adsorption as the cohesive energy of the adsorbate atoms increases while keeping the identity of the substrate unchanged; a small number of admetals, howevermost notably Ptdeviate from this trend. Additionally, we note that there is a general trend for less cohesive substrates, such as Ag(100), to exhibit weaker adsorption of all adsorbates as compared to substrates with higher cohesive energies, such as Ir(100). This trend is more clearly consistent in the case of the FF adsorption energies than in the case of the diffusion barriers, as will be further discussed below. Finally, as is the case for the diffusion barriers, there seems to be a variable range of adsorption energies for a given admetal on different metal substrates, with the largest range exhibited by the most cohesive admetals.
Figure 2. (a) Self-diffusion barriers (homodiffusion systems) as a function of one-sixth of the bulk cohesive energies and of the surface adsorption energies at the 4-fold (FF) and diffusional transition state adsorption sites, (b) diffusion barriers for all of the admetal/substrate pairs as a function of the adsorbate identities, listed in order of increasing adsorbate cohesive energy, and (c) adsorption energies at the FF site for all the pairs as a function of adsorbate identities. A more positive adsorption energy indicates a more strongly bound adatom. (b) and (c) exclude the value for Ir/Au(100), as Ir penetrates into the substrate and hence cannot be considered to have the same diffusion mechanism. The cohesive energies of the metals in (a), (b), and (c) are 2.94, 3.48, 3.84, 3.90, 4.44, 5.76, 5.82, 6.74, and 6.96 eV for Ag, Cu, Au, Pd, Ni, Rh, Pt, Ru, and Ir, respectively (from reference 57).
energy obtained from experimental cohesive energies,57 clearly confirm that homodiffusion is related to the metals’ bulk cohesive energies, with the least cohesive metals having the lowest diffusion barriers. To further analyze the relationship between the kinetics and thermodynamics of surface diffusion in these systems, we extend this analysis by plotting the diffusion barriers against one-sixth of the adsorption energies at the FF (EFFbind), and diffusional transition state, TS (ETSbind, found at bridge positions), sites (the factor of 1/6, in these cases, is for a consistent comparison with the cohesive energy analysis). A linear relation is also observed, demonstrating that surface thermodynamic properties can also be used to understand trends in admetal diffusion barriers. To our knowledge, generalized analyses of surface diffusional properties, discussed above for homodiffusion systems, have not previously been investigated for a wide variety of hetero systems from first principles. The elucidation of such patterns, in turn, would provide useful atomistic insights into simulations of dealloying, electrodeposition of heterometallic films, and the surface stability of metallic alloys, among other processes.58−61 As a first step in the determination of these trends, we plot, in Figure 2b, the calculated diffusion barriers as a function of the adsorbate (admetal) identities in order of increasing cohesive energy of the metal adsorbate. While we discuss these trends primarily in terms of lattice constants and cohesive energies, we note that analysis of other electronic and energetic properties could also provide insights into diffusional properties (see Supporting Information for an example of an alternative description). For a given substrate element, the barriers exhibit a generally monotonic increase with increasing cohesive energy of the admetal, with Ag adspecies generally having the lowest C
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Figure 3. Periodic plots constructed for the (a) diffusion barriers and (b) admetal adsorption energies at FF sites. Corresponding numerical values of the diffusion and adsorption energies are given in (c) and (d). The figures exclude the value for Ir/Au(100), as Ir significantly perturbs the structure of the Au(100) surface.
Figure 3 shows an alternative presentation of the kinetic and thermodynamic diffusion data. In Figure 3a, the hopping diffusion barriers are summarized in a periodic grid, with the elements listed in order of increasing cohesive energy from left to right and from top to bottom. The shading of each box reflects the magnitude of the diffusion barrier of each adsorbate−substrate pair, with the darkest-colored boxes corresponding to the highest diffusion barriers. Figure 3b shows the FF adsorption energy data in a similar periodic arrangement. Consistent with our discussion of Figures 2b and 2c above, a general increase in diffusion barriers and adsorption strength is clearly seen when moving across the rows, corresponding to changing the admetal with a given substrate. For the diffusion barriers, moving down the columns, corresponding to changing the substrate element for a fixed admetal, does not yield such well-defined trends; as discussed above, this effect is likely due to the simultaneous changing of electronic (cohesive energy) and geometrical (lattice constant) effects. The fact that the diffusion barriers do not show a clear trend when changing the substrate, while the qualitative trend is more clearly established in the case of the FF site adsorption energies, is likely due to the fact that the scale of the diffusion barriers, which are effectively differences between the energies of the diffusional transition states and the FF adsorption energies, is much smaller than the corresponding scale of the adsorption energies (see also discussion below). III.2. Correlation between Diffusion Barriers and Binding Energies. The similarities between the trends, described above, for heteroatom diffusion barriers and adsorption energies suggest that monotonic relationships may exist between these two quantities across different admetal/ substrate pairs, especially for diffusion of atoms across a given substrate metal. Demonstrating the existence of such BEP relationships, in turn, would not only extend a valuable fundamental concept from catalytic surface science to surface
diffusive processes but also provide an important practical tool for rapidly and accurately estimating diffusion barriers from thermodynamic data alone. In Figure 4a, we plot the diffusion
Figure 4. (a) Linear correlations obtained for the admetal diffusion barriers as a function of the adsorbate adsorption energies at 4-fold sites, (b) correlations between the adsorption energies at the transition states (ETSbind) and those at the most stable adsorption sites (EFFbind), and (c) linear fits for the BEP-type relationships for admetal diffusion on each metal substrate.
barriers as a function of the adsorption energies at the FF sites for each considered adsorbate on each metal substrate (total of 64 systems). The overall correlation, considering all 64 admetal/substrate pairs together, is only qualitative in nature; while a general trend is apparent, the scatter is considerable. However, when admetal diffusion and adsorption are D
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Table 1. Calculated Lattice Constants and Experimental Cohesive Energies (from reference 57) for Main Group Transition Metals property
Ni
Cu
Rh
Ir
Pd
Pt
Ag
Au
lattice constant (Å) cohesive energy (eV)
3.52 4.44
3.64 3.48
3.84 5.76
3.88 6.96
3.96 3.90
3.99 5.82
4.16 2.94
4.18 3.84
considered only on a given substrate, the correlations become much more quantitative, with standard errors of 0.04, 0.05, 0.03, 0.02, 0.01, 0.02, 0.02, and 0.02 eV for diffusion on Ag, Au, Cu, Ir, Ni, Pd, Pt, and Rh(100), respectively; these errors are well within the standard uncertainties of DFT calculations. The fact that better correlations exist for given metal substrates is not surprising in light of the discussions above; for given substrates, both the geometrical and the energetic/electronic properties of the substrate are fixed, and all remaining variations are due only to changes in the nature of the admetal. The existence of these correlations, in turn, provides a powerful tool for combining the accuracy of DFT calculations with accelerated strategies for simulating metal-on-metal growth or dealloying processes; with only the computationally inexpensive determinations of adsorption energies, diffusional properties can be determined with great efficiency. The slopes of the correlations on Ag(100) and Au(100) are the largest (dimensionless values of 0.33 and 0.36, respectively), followed by Pt(100) and Pd(100), with slopes of 0.23 and 0.26, by Cu(100) and Ni(100), with values of 0.24 and 0.20, and finally by Ir(100) and Rh(100), with slopes of 0.15 and 0.16. These slopes, in turn, provide insights into the degree of correspondence between the transition and initial states of diffusion on different substrate metals. Since the hopping diffusion barrier is the difference in adsorption energies between the diffusing adatom at the transition state and the most stable FF adsorption sites (Figure 1), a smaller slope in these relationships implies a closer relationship between the energy at the transition state and that at the initial (FF) state. A slope of zero would suggest that there is a constant diffusion barrier, which in turn indicates that there is merely a constant energy offset between the FF and transition states for all adatoms on a given metal substrate; such a constant, or nearconstant, offset has generally been interpreted, in the BEP literature, as evidence that the transition states resemble, in either a geometric or an energetic sense, the corresponding thermodynamically stable states.62,63 As with our analysis in Figures 2 and 3 above, we discuss the linear correlations in Figure 4a in terms of two properties, the substrate lattice constant (a geometric property) and the substrate cohesive energy (an energetic/electronic property). These properties provide an important qualitative understanding of the BEP behavior, which is in line with other largely qualitative interpretations that have been given for the existence of BEP relationships in the heterogeneous catalysis literature62,63 (we again note, however, that related properties, such as the metal substrate electronic d-band centers and densities of states, could have similar explanatory power). The correlations with the largest slopes are generally associated with the metal substrates that have the largest lattice constants and smaller cohesive energies (Au(100) and Ag(100), Table 1). A larger slope also implies, on average, that there is a weaker magnitude of binding of the admetal species at FF sites on the substrates. In contrast, smaller slopes are often associated with those substrate metals with smaller lattice constants (Ir and Rh have the third smallest lattice constants among all the systems)
and higher cohesive energies; these smaller slopes also indicate a higher average strength of binding of the admetals to the substrates. Pt and Pd have both correlation slopes and lattice constants/cohesive energies in between these extremes. These associations suggest that substrates with smaller lattice constants or larger cohesive energies exhibit more energetic similarities between the diffusional transition states and the corresponding FF adsorption sites. Both factors could be intuitively thought to promote closer interactions with the substrates, in a geometric and an energetic sense, respectively, thus leading to the closer correspondence. While consideration of the substrate lattice constants and cohesive energies provides useful understandings of numerous characteristics of the diffusion/adsorption correlations on a number of transition metal substrates, neither of these parameters provides a universal description. Diffusion on the Cu(100) substrate illustrates this point. If we consider only the Cu lattice constant, we would expect the slope of the correlation to be lower than the slopes of the corresponding correlations for Ir and Rh substrates, according to the arguments presented above. An analysis based solely on cohesive energies, on the other hand, would suggest a slope close to that of the Au or Ag substrates. Figure 4a shows, however, that neither prediction corresponds perfectly to the calculated results; indeed, the slope is somewhat higher than what we would expect based on lattice constant trends and somewhat lower than the expectation based on cohesive energies. This different behavior likely results from a coupling of the lattice constant and cohesive effects; the result may, in turn, be related to the unusual combination of Cu’s small lattice constant and its low cohesive energy (Table 1). Ni is another interesting case, with the smallest lattice constant of any of the substrates but an intermediate value of the cohesive energy; again, the calculated slope of the BEP relationship does not correspond perfectly to the qualitative arguments made based on either lattice constant or cohesive energy. We note that this result is not surprising, given that a full explanation of linearity in BEP relationships remains elusive even for systems that have been extensively studied, such as the reactions of organic species on metal surfaces.16,22,23,34 An alternative formulation of BEP relationships involves an explicit correlation of adsorption energies of transition states with adsorption energies of the stable initial states; this formulation more directly illuminates the fundamental links between transition state and initial state characteristics. In Figure 4b, such a relationship is plotted for the hopping diffusion calculations. The results confirm that a linear relationship between the adsorption energies at the FF and the TS sites exists, even considering the broad range of adsorption energies (from 2.0 to 7 eV) corresponding to the 64 systems in this analysis. The reason that a reasonable linear relationship can be obtained in this case for all heterodiffusional systems, whereas a similar relationship does not exist when plotting diffusion barriers against adsorption energies (Figure 4a), is simply related to the different choice of dependent variable in Figures 4a and 4b. The larger range of the transition E
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Energy Sciences, under Contract No. DE-AC02-06CH11357. The authors also acknowledge the use of the computational facilities provided by CNM-ANL (Carbon Cluster) and Fusion Clusters.
state adsorption energies, compared to the corresponding range of diffusion barriers, enhances the linearity of the relationships. We note that this enhancement is also likely related to the more obvious relationship between adsorption energy and cohesive energy, described in Figure 2c, compared to the corresponding relationship for diffusion barriers and cohesive energy (Figure 2b). In spite of the overall linearity of the generalized BEP relationship in Figure 4b, it is possible to obtain even better relationships by generating separate correlations for each metal substrate (Figure 4c). The ordering of the slopes of these correlations is reversed compared to the ordering in Figure 4b, but the interpretation of the results in terms of the correspondence between the transition states and the initial states, described earlier, remains unchanged.
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IV. CONCLUSIONS We have used periodic density functional theory calculations to screen the diffusion and adsorption energetics for hetero metalon-metal diffusion via a hopping mechanism on the (100) surfaces of 64 admetal/metal substrate systems. The results demonstrate that relationships between hopping diffusion barriers and metal cohesive energies, previously established for homodiffusion systems, can be generalized to heterometallic systems, and the resulting relationships can be described by considering the cohesive energies of the admetals and substrates, as well as the lattice constants of the substrates. In addition, there is a linear relationship between the adsorption energies and the diffusion barriers when the adsorption is considered on a given substrate. The slopes of these linear relationships indicate the degree of similarity between the diffusional transition states and initial states, and the slopes are at least partly determined by the cohesive energies and lattice constants of the metal substrates. The relationships provide fundamental insights into the links between the diffusional transition state and the corresponding initial state properties, and they yield a powerful and convenient means of predicting diffusional kinetics from purely thermodynamic quantities. Similar studies could be carried out to describe hopping and exchange diffusional processes on other surface geometries, and taken together, these results may ultimately provide a very useful input to KMC-type simulations, enabling studies on dynamics and time evolution of the growth processes for these adsorbate−substrate heterometallic pairs.
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ASSOCIATED CONTENT
S Supporting Information *
A simple computational decomposition procedure that separates the contribution of lattice constant changes from changes in the electronic/alloying properties of the systems is shown. This material is available free of charge via the Internet at http://pubs.acs.org.
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REFERENCES
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected];
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS A DOE Early Career Award for J.G., together with use of the Center for Nanoscale Materials, was supported by the U.S. Department of Energy, Office of Science, and Office of Basic F
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