Understanding the Stability and Electronic and Adsorption Properties

Jul 21, 2014 - Understanding the Stability and Electronic and Adsorption Properties of Subnanometer Group XI Monometallic and Bimetallic Catalysts. Na...
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Understanding the Stability and Electronic and Adsorption Properties of Subnanometer Group XI Monometallic and Bimetallic Catalysts Natalie Austin and Giannis Mpourmpakis* Department of Chemical Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15621, United States ABSTRACT: Density functional theory (DFT) was used to calculate the physicochemical and (CO and O2) adsorption properties of six-atom Ag, Au, and Cu clusters and their bimetallic combinations. The physicochemical properties include the cohesive energy (binding energy per atom), electron affinity (EA), ionization potential (IP), d-band center (dc), charge transfer, and frontier orbital localization on the clusters. Our results demonstrate that the cohesive energy of Au−Cu bimetallic catalysts decreases with increasing Au composition, whereas for Au−Ag bimetallic clusters, there was no variation with composition. Regarding the adsorption properties, we found that CO, acting as an electron donor molecule, preferentially interacts with low coordinated sites of the monometallic clusters that localize the LUMO orbitals. The CO adsorption on the bimetallic clusters is affected by the charge that is transferred between the different metals. O2, acting as an electron acceptor molecule, preferentially interacts with clusters that exhibit dc values closer to the LUMO level of O2. We revealed a linear trend between the average O2 binding energy on the different sites of the clusters and the dc of the clusters. This trend explains the high oxophilicity of Cu compared to the other clusters. Additionally, we identified a cooperative adsorption of CO and O2, with preadsorbed CO enhancing the adsorption of O2. This effect becomes significant in the case of the Cu6 cluster, where O2 adsorption in the presence of CO results to the formation of a partially oxidized cluster. The oxidation of the cluster is supported by charge analysis showing electron density loss from the Cu6 cluster to O2. Overall, this work highlights that unravelling the adsorption behavior on bimetallic catalysts becomes very challenging at the subnanometer scale.

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INTRODUCTION The control of metal−adsorbate interactions is key to the design of catalysts with optimal performance.1−4 Nanosized catalysts designed for the CO oxidation reaction are beneficial for controlling emissions in the environment as well as for purifying hydrogen in the steam re-forming of hydrocarbons. In addition, the fundamental understanding of simple oxidation reactions, such as the CO to CO2 reaction, can serve as a probe to understand mechanisms in more complex oxidation processes on nanomaterials. Gold (Au) catalysts have been proven to be chemically inert in bulk; however, they become exceptionally active at the nanoscale,5 thus highlighting the importance of nanoparticle size on catalytic activity. Haruta was the first to show that Au nanoparticles 2−5 nm in diameter were exceptionally reactive,6 especially in low-temperature oxidation.7,8 The CO oxidation activity of small Au nanoparticles is primarily attributed to their low-coordinated sites (which are mainly located at the edge and corner sites of the nanoparticles).9−11 Theoretical studies have shown that the binding energy (BE) of the adsorbates increases with decreasing surface coordination number (CN) of Au.12,13 As a result, the emerging consensus is that the CO oxidation activity on Au increases with decreasing nanoparticle size.13 However, understanding the catalytic behavior of Au at the subnanometer scale becomes even more challenging, since © 2014 American Chemical Society

support, electronic, and stability effects on the clusters can contribute to a “magic number” activity, with specific cluster sizes being active and others inert.14,15 As an example of this complexity, it has been proposed that the reactivity of small Au clusters is controlled by the shape of the HOMO (highest occupied molecular orbital)−LUMO (lowest unoccupied molecular orbital) orbitals16 as well as by their directionality.12 Bimetallic catalysts often exhibit better catalytic activity than their monometallic counterparts.3,17−21 Particularly, there is an increasing interest in Au-based nanoparticles such as AuAg and AuCu which have shown high activity for CO oxidation compared to monometallic gold.22−24 Currently, a fundamental understanding of the catalytic behavior of these bimetallics is lacking.18 Understanding the bimetallic properties at the subnanoscale increases in complexity compared to the monometallic because, in addition to the determination of properties as a function of nanoparticle size and shape, composition will also be an important factor. The growing interest in the study of bimetallics can potentially reveal unique physicochemical characteristics that can be fine-tuned with metal composition to produce catalysts with optimal activity.18 Received: April 24, 2014 Revised: July 19, 2014 Published: July 21, 2014 18521

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Figure 1. Cohesive energy of M6 clusters (M = Cu, Ag, Au) as a function of Au composition (AuCu: blue; AuAg: red; dashed lines serve as a guide to the eye).

clusters3 as well as the CO oxidation reaction on small Au clusters.15,35 All the clusters and the adsorbates were fully relaxed without any symmetry constraints, and the obtained structures were further verified as minima with frequency calculations. Different spin states have been considered in our calculations. Equation 1 was used to calculate the cohesive energy (binding energy (BE) per atom) of the clusters:

Understanding the chemisorption behavior of CO and O2 on Ag, Au, and Cu clusters is of high importance to control their catalytic properties. Theoretical studies have shown that CO, acting as an electron donor molecule, would preferentially bind the site of the clusters where the LUMO orbitals are localized.16 Previous research has shown that CO preferentially binds to the corners than the edges of Au6 clusters due to the lower coordination of the corners (CN = 2) compared to the edges (CN = 4) and the localization of the LUMO orbital on the corners of the cluster.11 In addition, CO binds much stronger than O2 on neutral and positively charged Au6 clusters, which as a result leads to poisoning by CO and deactivation of the catalyst. However, O2 strongly adsorbs on negatively charged Au clusters, and transient activity is observed.15 Experimental research has shown that on small Au clusters CO and O2 adsorb cooperatively.25,26 Even though a lot of research effort has been focused on the CO and O2 adsorption on Au, the adsorption behavior on Au-based bimetallic clusters is still not well understood. In this work we used density functional theory (DFT) calculations to investigate the stability of M6 (M = Ag, Au, Cu) clusters and their bimetallic combinations. In addition, we investigated the CO and O2 adsorption behavior as a function of the catalyst’s structural11 and electronic16 characteristics. At this subnanometer scale, research has revealed that the catalytic activity is cluster size specific (e.g., Au8 is more active cluster than Au6).15 As a result, understanding the physicochemical properties of small size clusters can provide guidelines for their potential use as effective catalysts.11,27

BE/6 = (E(A xBy) − xE(A) − yE(B))/6

(1)

where x + y = 6 and E is the total electronic energy of the clusters with composition A, B (this can be any combination of Au, Ag, and Cu). The cohesive energy is used as a descriptor of the average metal bond strength on the clusters. This property is used to understand the relative stability between the clusters of different compositions. We accounted for singlet, triplet, and quintet spin states of the neutral clusters, and we found that the singlet was always the energetically most preferred state. Equations 2 and 3 were used to calculate the electron affinity (EA) and the ionization potential (IP) of the clusters, respectively: EA = E(negative cluster) − E(neutral cluster)

(2)

IP = E(positive cluster) − E(neutral cluster)

(3)

The neutral clusters are of singlet spin state, whereas the negative and positive clusters of doublet. Equation 4 was used to calculate the binding energies of the adsorbates on the clusters, where the adsorbates are the CO and O2 molecules:

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BE = Ecluster−adsorbate − Ecluster − Eadsorbate

COMPUTATIONAL METHODS We used the B3LYP hybrid functional combined with the LANL2DZ basis set as implemented in the Gaussian 09 program package28 to investigate the structural, electronic, and adsorption properties of six-atom metal clusters. These M6 clusters, with M = Ag, Au, Cu ( Au6 > Ag6) do not follow the experimental cohesive energies (CE) of the bulk metals (CEAu = 87.9, CEAg = 68.0, CECu = 80.5 kcal/mol).37 At this subnanoscale cluster size, the cohesive energy of Cu is significantly enhanced compared to that of Ag and Au clusters. Figure 2 shows the EA of both the monometallic and bimetallic clusters as a function of Au composition. It was

Figure 3. IP as a function of Au composition of the clusters (color code as in Figure 1).

periodic table, and it does not tend to lose electrons easily. This is also depicted in Figure 2 where Au has lower (more negative) electron affinity values than Cu and Ag. The IPs shown in Figure 3 did not appear to exhibit any trend associated with Au composition. It should be noted that we observed a transition from planar to three-dimensional structures in the positively charged clusters. Since Figure 1 shows that mixing Au with Cu increases the stability of the cluster (resulting to a more negative cohesive energy), we used this trend to analyze CO adsorption on the monometallic clusters and on AuCu bimetallics with 50% Au composition. At this specific composition, eight different structures exist. We selected the clusters with the highest and lowest cohesive energies: AuCu bimetallic clusters with BE/n values of −65.90 and −59.86 kcal/mol associated with the clusters Au3Cu3 and Cu3Au3, respectively (total energy difference between the clusters is ∼36 kcal/mol). The Au3Cu3 cluster has the Au atoms on the corners and the Cu atoms on the edges, whereas the Cu3Au3 cluster has the Cu atoms on the corners and Au on the edges. Table 1 shows our calculated BEs for CO adsorption on the corner and edges of the clusters. These CO adsorption energies

Figure 2. EA of clusters as a function of Au composition (color code as in Figure 1).

determined that Cu3Au3 (Cu on the corners and Au on the edges as shown in Figure 1) had the highest (most negative) EA (−79.91 kcal/mol). The reason we observed this high electron affinity was that the negatively charged Cu3Au3 cluster restructured to the more energetically stable Au3Cu3 structure (Au atoms on the corners). In the latter structure, the negative charge is located at the Au (corner) atoms (Au more electronegative than Cu) reducing the overall electrostatic repulsions on the cluster. This is the only case we observed restructuring on the negatively charged clusters. According to Figure 2, unlike with the cluster BE/n, EA is not a linear combination of the EA of the monometallic clusters. The EA of AuCu is larger (more negative) than that of the AuAg bimetallic clusters in compositions of Au ≥ 50%, and it becomes larger for AuAg bimetallic clusters of Au compositions smaller than 50%. EA is an important property of the cluster since it defines its ability to accept electrons from the metal oxide supports.15,17 The charging of the clusters can in turn affect the adsorption energies of the reactants and the catalytic behavior of the cluster.15,35 We have also calculated the IP of the clusters, as presented in Figure 3, as a function of Au composition. Similar to the EA, the IP is an essential cluster property because it determines the cluster’s ability to donate electrons from interactions with materials (e.g., oxides) that are used as supports in heterogeneous catalysis. The Ag6 (154.79 kcal/mol) and Cu6 (157.61 kcal/mol) clusters have a lower IP than the Au6 (194.99 kcal/mol) cluster. This trend occurs because Au is the most electronegative metal in group XI of the

Table 1. BE of CO on the Corners and Edges of the Clusters (Values in kcal/mol) clusters

BE CO corner

BE CO edge

Ag6 Au6 Cu6 Au3Cu3 Cu3Au3

−6.67 −17.27 −15.94 −10.63 −20.54

−4.51 −9.66 −14.78 −15.56 restructured to Au3Cu3 CO corner

are weaker than the ones calculated for CO adsorption on Pt6 (BE = −62.95 kcal/mol).38 Additionally, the binding energy values of Table 1 are significantly stronger than that for CO adsorption on the top site of a 10-atom cluster which modeled the (111) surface (Cu: −2.66; Ag: −2.28; Au: 1.35 kcal/mol).3 The CO adsorption behavior was analyzed as a function of the coordination number of the adsorption site and the LUMO localization on the clusters. As shown in Table 1, the BE was stronger on the corners than the edges for the monometallic clusters. This trend was expected since we have shown that the lower coordinated sites exhibit stronger CO binding energies.11,12 However, this trend was not observed in the bimetallic clusters. CO binds stronger to the edges of Au3Cu3 than to its corners. When CO was adsorbed on the edge of the Cu3Au3 cluster, it restructured to Au3Cu3 with CO adsorbed on the corner. In order to further understand the CO adsorption 18523

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Figure 6 illustrates NBO charge distribution analysis on the clusters. The positive charge is located at the corners of the monometallic clusters and of the Cu3Au3 cluster. However, in the case of Au3Cu3 the positive charge is located at the edges. The negatively charged corners of Au3Cu3 (electron localization) is the result of the higher electronegativity of Au compared to Cu. CO, which acts as an electron donor, preferentially binds the sites of the cluster that are positively charged or “electron poor”. As a result, this charge distribution analysis can explain the stronger binding of CO on the edges of Au3Cu3. One aspect of Table 1 that we have not discussed is the restructuring of Cu3Au3 with CO adsorbed on the edge to Au3Cu3 with CO adsorbed on the corner. The reason we observed this restructuring is because Au3Cu3 is more stable than Cu3Au3 as shown in Figure 1. This is not surprising, since we have very recently shown that adsorption of CO oxidation species (CO and O2) on the Au6− cluster can change the cluster’s structural characteristics (cluster breathing during reactions).15 In addition, we investigated the BE of CO on the bimetallic clusters as a function of CO coverage. Zhai et al. have shown that CO binding to the apex sites of the Au6 (the three corners) does not significantly disturb the structure of the bare cluster.39 The lack of disturbance to bare cluster is a trend also observed in the Au3Cu3 cluster with placement of CO on the corner sites. As it can be observed from Figure 7 the BE of CO on the

behavior, we analyzed the LUMO orbital localization on the clusters as shown in Figure 4. Figure 4 clearly shows that the

Figure 4. HOMO and LUMO Orbitals of Ag6, Au6, Cu6, Au3Cu3, and Cu3Au3 clusters.

LUMO orbitals are localized at the corners of the clusters. This is important because CO as an electron donor would preferentially bind to the sites of the clusters where the LUMO is localized.16 This observation further supports the preferential binding of CO to the corners of the monometallic clusters. Figure 5 presents the HOMO−LUMO gaps of the clusters with respect to the Cu6 LUMO orbital. Research has shown

Figure 5. HOMO and LUMO gaps of Ag6, Au6, Cu6, Au3Cu3, and Cu3Au3 clusters with respect to the energy level of the Cu6 LUMO orbital. Figure 7. CO coverage effects on Au3Cu3 and Cu3Au3 clusters. The numbers in red represent the binding energy per CO molecule.

that large HOMO−LUMO gaps represent structures of higher stability.11,30 In Figure 5, Au6 and Au3Cu3 exhibit the largest gaps and their HOMO and LUMO energies are positioned at comparable levels. This is because according to Figure 4, these two clusters localize their HOMO and LUMO orbitals on the same (Au) atoms and the orbitals exhibit the similar symmetry. Cu6 and Cu3Au3 can be grouped together since they show the closely related orbital localization/symmetry and HOMO− LUMO gap behavior. However, just as with the coordination number, the LUMO orbital analysis cannot explain the higher BE of CO at the edges of Au3Cu3.

corners of Au3Cu3 is weakly affected by CO coverage, with an average BE per CO molecule approximately at −10 kcal/mol (BE per CO = (Ecluster−COn − E − nECO)/n). In addition to the restructuring that occurred when CO was adsorbed on the edge of Cu3Au3, we also observed restructuring when three CO atoms were adsorbed on the corners of Cu3Au3. This, again, is a result of the higher stability (more negative cohesive energy) of the Au3Cu3 cluster compared to Cu3Au3. As a final note on the CO coverage effects, we observe a symmetry transition from

Figure 6. NBO charge distribution on the clusters. The numbers in red show the total charge transferred within the cluster (Δq). 18524

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D3h to C2v on both Au3Cu3 and Cu3Au3 clusters when two CO molecules bind the edge atoms. This is because the presence of two CO molecules disturbs the symmetry of the cluster in a way to facilitate transition to C2v symmetry. Table 2 describes the BE behavior of O2 on the corners and edges of the metal clusters as well as the average BE on the Table 2. BE of O2 on the Corners and Edges and the Average BE of the Clusters (Values in kcal/mol) clusters

BE O2 corner

BE O2 edge

av BE

Ag6 Au6 Cu6 Au3Cu3 Cu3Au3

−4.95 −3.58 −11.75 −3.16 −12.09

−3.36 −2.41 −7.77 −4.86 −2.48

−4.15 −3.00 −9.76 −4.01 −7.28

Figure 9. O2 average BE on the different clusters vs dc of the clusters (black dashed line serves as a guide to the eye).

cluster. The calculated O2 adsorption energies on Ag6 fall within the range of reported literature values of O2 BEs on Agn clusters (n = 1−7; BE: −3.16 to −25.18 kcal/mol).40 Additionally, O2 binding on Cu6 is stronger than BEs previously reported on Cu2 (−4.7 kcal/mol) but weaker than on Cun (n = 3−5; BE: −14.9 to −39.1 kcal/mol).41 It has been shown by Nørskov’s group that the adsorbate binding energy can be correlated with the d-band center (dc) of the metal.42,43 As a result, we calculated the density of states (DOS) of the d orbitals and the dc of the clusters as presented in reference.44 Figure 8 illustrates the DOS, dc, and HOMO−LUMO levels for the clusters with respect to the O2 LUMO orbital. The reason we chose the LUMO of O2 as energy reference in Figure 8 is because O2 acts as an electron acceptor molecule, and the energy level of its LUMO plays an important role in bonding. According to Figure 8, the overall dc trend is Cu6 > Cu3Au3> Au3Cu3 > Au6 > Ag6. In Figure 9 we plotted the average BE of O2 on the cluster as a function of the cluster dc. The average BE of O2 is linearly related to the dc of the clusters showing that the higher the dc (closer to the O2 LUMO), the stronger the O2

binding. This trend, although developed on subnanometer metal clusters, follows Nørskov’s d-band model for adsorption on periodic metals surfaces.42 It is worth noticing that an attempt to relate the average and site specific CO binding energy on the clusters (Table 1) with the dc did not give clear linear trend as in the case of O2 adsorption. As shown with CO adsorption on the monometallic clusters (Table 1), the BE of O2 on the corners is stronger than on the edges (Table 2). The weaker adsorption of O2 compared to CO has been also observed on neutral and positively charged Au6 clusters.15 Adsorption behavior on the various sites of the cluster was examined using HOMO orbitals and dc of the clusters. O2 acting as an electron acceptor molecule would prefer to bind to sites of the clusters that the HOMO is localized.16 Figure 4 shows that the HOMO is localized primarily on the corner sites of the clusters. Orbital analysis supports higher binding on the corners than the edges of the

Figure 8. d-orbital density of states (DOS), d-band center (dc), and HOMO−LUMO orbitals of the clusters with respect to the energy level of O2 LUMO orbital. Within each color code (representing the different clusters), the lines below the DOS lines represent from left to right the dc, HOMO, and LUMO orbitals of the clusters. 18525

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that CO and O2 adsorb cooperatively instead of competitively.25,26 A profound example in our observations is the case of Cu6, where the BE of O2 on its edge atom is −7.77 kcal/mol and the adsorption is strengthened to −21.27 kcal/mol when CO is preadsorbed on the corner. The reason for this significant energy change is that the Cu6 cluster is oxidized as structurally shown in the inset of Figure 8 and further verified by NBO charge analysis (Cu6 loses one electron). This result highlights the significance of preadsorbed CO on the cluster especially since this oxidation behavior was not observed when O2 was singularly adsorbed to the edge of the Cu6 cluster.

monometallics and the Cu3Au3 cluster. However, this is not sufficient to explain the O2 adsorption behavior of Au3Cu3. To understand the slightly stronger binding of O2 to the edge of Au3Cu3 rather than the corner, we turn back to the O2 adsorption behavior on the monometallic clusters Au6 and Cu6. As shown in Table 2, we found that Cu6 had a higher affinity (−9.76 kcal/mol) to bind O2 than Au6 (−3.00 kcal/ mol) due to the oxophilic nature of Cu compared to Au.22 This is supported by our dc cluster analysis (Figure 8) where we found that the dC of Cu6 (purple line) lies closer to the LUMO level of O2 than the dC of Au (navy blue line). Consequently, the presence of Cu on the edges of the bimetallic Au3Cu3 cluster increases the O2 adsorption energy on this specific site (compared to Au6 cluster). On the other hand, adsorption on the Au atoms of the Au3Cu3 (corner) and Cu3Au3 (edge) bimetallic clusters is significantly weaker compared to the case of Cu6. If we compare the O2 adsorption on the same sites of the clusters that exhibit the same metal composition, we observe that O2 adsorption does not change significantly from monometallic to bimetallic clusters (compare O2 on corners of Au6 vs Au3Cu3, on corner of Cu6 vs corner of Cu3Au3 and on edge of Au6 vs edge of Cu3Au3). Exception from this behavior is the adsorption of O2 on the edge of Cu6, which is −7.77 kcal/ mol and changes to −4.86 kcal/mol on the edge of Au3Cu3 (approximately 40% adsorption energy drop). The reason for this adsorption change is that both the HOMO and dc of Au3Cu3 (orange line) are energetically located further away from the LUMO of O2 than the corresponding levels of Cu6 as shown in Figure 8. As a final note, we observed that O2 adsorption on any site of the clusters did not result to any restructuring, and the clusters retained the D3h symmetry. This is due to the weaker adsorption of O2 on the clusters (BE range = −2.4 to −12.0 kcal/mol) compared to CO (BE range = −4.5 to −20.5 kcal/mol). Finally, to analyze the BE behavior of coadsorbed species, we performed calculations with CO adsorbed on the corner and O2 adsorbed on the edge of the metal clusters. According to Figure 10, the addition of O2 to the clusters with preadsorbed CO

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CONCLUSIONS In this study we investigated the cohesive energy (BE/n), electronic and adsorption properties of six-atom Ag, Au, and Cu, and bimetallic combinations of AuAg and AuCu metal clusters. Our results demonstrated that the cohesive energy of the bimetallic clusters is a linear combination of the BE/n of the monometallic clusters. As a result, the BE/n of small Au clusters can be significantly enhanced when mixed with Cu (formation of bimetallic clusters). We also calculated the electron affinities (EA) and ionization potential (IP) of the clusters. We found that the AuCu bimetallic clusters have higher tendency to receive electrons than the AuAg clusters in compositions of Au ≥ 50%. This trend is reversed in Au compositions smaller than 50%. We did not observe any trend with IP calculations. In addition, we determined that the preferential binding of CO to the corners compared to the edges of monometallic clusters could be explained by the CN and the LUMO localization on the clusters. However, these descriptors are not sufficient to explain the CO adsorption on the bimetallics, and one has to account for the charge distribution among the atoms of the cluster. NBO charge distribution analysis revealed that the edges of Au3Cu3 (Au atoms occupy the corners) are positively charged, consequently resulting in the stronger binding of CO on the edges (CN = 4) than on the corners (CN = 2) of the cluster. The restructuring of Cu3Au3 to Au3Cu3 with CO adsorption highlighted the importance of the cohesive energy of the cluster (Au3Cu3 is more stable than Cu3Au3). When more than one CO molecules adsorbed on the corner atoms of the Au3Cu3 cluster, the BE of CO on the cluster did not vary significantly, thus supporting structural integrity against CO coverage. In addition to CO adsorption, we investigated O2 adsorption and coadsorption (with CO) on the clusters. We found that the average O2 BE on the cluster can be linearly related to the dc of the cluster. Additionally, we determined that the preferential binding of O2 to the corners instead of the edges on the monometallic clusters and bimetallic Cu3Au3 can be explained by the CN and HOMO localization on the clusters. The higher oxophilicity of Cu6 compared to Au6 was explained in terms of dc of the clusters. Our study also revealed that the adsorption of O2 on the corners of Au3Cu3 is significantly weaker than that at the corners of Cu6 (O2 binds in both cases Cu atoms) due to the fact that the HOMO orbital of Au3Cu3 has been energetically shifted away from the LUMO orbital of molecular O2. The coadsorption of CO and O2 showed that the BE of O2 increases with preadsorbed CO, which consequently supports a cooperative interaction between the molecules. Additionally, we discovered that a significant enhancement in the BE of O2 on Cu6 due to preadsorbed CO leads to the formation of a partially oxidized cluster. Charge analysis supported the oxidation of this cluster by illustrating its loss of electron density to O2. Overall, this study demonstrates

Figure 10. Coadsorption of CO and O2 on Ag6, Au6, Cu6, Au3Cu3, and Cu3Au3 clusters. CO is adsorbed on the corner and O2 on the edge of the clusters.

results to an increase in the BE of O2. The values in parentheses in Figure 10 represent the adsorption of O2 when CO is preadsorbed on the clusters. All these adsorption energies are stronger than the corresponding ones presented in Table 2 (edge adsorption). Our theoretical observation is supported by mass spectrometry experiments on small clusters which showed 18526

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a high degree of complexity in the adsorption behavior of subnanoscale bimetallic catalysts.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; phone 412-624-7034 (G.M.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Center for Simulation and Modeling (SAM) at the University of Pittsburgh for computational support. This research has been supported by start-up funds from the University of Pittsburgh.

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REFERENCES

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