Letter Cite This: J. Phys. Chem. Lett. 2017, 8, 5402-5407
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Structure and Thermodynamics of Metal Clusters on Atomically Smooth Substrates M. Eckhoff,* D. Schebarchov,* and D. J. Wales* University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, United Kindom S Supporting Information *
ABSTRACT: We analyze the structure of model NiN and CuN clusters (N = 55, 147) supported on a variety of atomically smooth van der Waals surfaces. The global minima are mapped in the space of two parameters: (i) the laterally averaged surface stickiness, γ, which controls the macroscopic wetting angle, and (ii) the surface microstructure, which produces more subtle but important templating via epitaxial stresses. We find that adjusting the substrate lattice (even at constant γ) can favor different crystal plane orientations in the cluster, stabilize hexagonal close-packed order, or induce various defects, such as stacking faults, twin boundaries, and five-fold disclinations. Thermodynamic analysis reveals substrate-dependent solid−solid transitions in cluster morphology, with tunable transition temperature and sometimes exhibiting re-entrant behavior. These results shed new light on the extent to which a supporting surface can be used to influence the equilibrium behavior of nanoparticles.
N
We model the cohesive energy of XN clusters (X = Ni, Cu and N = 55, 147) using the many-body Gupta34 potential with a standard set of parameter values35 for each metal. Ni was chosen for comparison with the study by Ferrando et al.,25 while Cu provides a broader comparison between different metals with similar bulk lattice spacing. The two cluster sizes correspond to closed-shell Mackay36 icosahedra, which are particularly stable for the selected metals. Adhesion to a substrate is modeled by the Lennard-Jones potential ULJ = 4ϵ∑i r(Ni) 0 , where is the nearest-neighbor separation in bulk Ni. Figure 3b r(Ni) 0 shows analogous energy curves for the Ni(sqr) 147 global minima in Figure 2b−e, with crossovers verifying the aforementioned sequence ICO → TET → DEC → F+H as a decreases. The four curves also flatten in the limit of a → 0, where the F+H structure with hexagonal close packing in the contact layer has the lowest energy. To isolate and quantitatively compare the substrate microstructure effect for a broader set of clusters, we propose the following analysis. For a given local minimum m with energy Um, re-minimizing the cluster coordinates without the substrate will yield a different energy U†m, with ΔUm ≡ Um − Um† quantifying the substrate-induced change in the binding energy. Now consider the ansatz ΔUm ≈ f (a)F(γ ) 5404
(1) DOI: 10.1021/acs.jpclett.7b02543 J. Phys. Chem. Lett. 2017, 8, 5402−5407
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The Journal of Physical Chemistry Letters
cluster contact layer, but for increasing a, the structure can change significantly, especially on substrates with higher γ. The effect of increasing γ is demonstrated in Figure 3c for the Ni(hon) Mackay icosahedron with a0 = 2.6 Å. We also find that 55 decreasing the σ parameter in our substrate model increases the maximal magnitude of f(a). We now consider thermal effects in the harmonic super(hon) position approximation, focusing on Ni55 with a fixed Lennard-Jones parameter ϵ = 35.6 meV (as opposed to constant γ) and several values of a in the range of 2.20−2.52 Å to model (isotropically strained) graphene. Heat capacity (CV) plots based on about 3 × 104 local minima (for each a) are shown in Figure 4a, all exhibiting a large peak at T between 600
Figure 3. Potential energy U and modulating function f(a) of selected minima X(l) N (γ,a0) versus the lattice parameter a, which is varied away from a0 at constant γ and scaled by metal-specific r0. (a) Ni(l) 147 Mackay icosahedron with different contact epitaxies for γ = 10 meV/Å2 and a0 = 2.6 Å. (b) ICO, TET, DEC, and F+H correspond to the minima in Figure 2b−e, respectively. (c) Cyan curves include all minima in Figure 2 as a subset; dark blue highlights the GM of Cu(sqr) 55 for γ = 25 meV/Å2 and a0 = 2.2, 2.4, and 2.6 Å. The dashed and dotted data sets Mackay icosahedra with σ = 2.8 and 2.0 Å, correspond to Ni(hon) 55 respectively; a0 = 2.6 Å. The five curves correspond to γ = 10, 15, 20, 25, and 30 meV/Å2, showing an upward shift in f(a) as γ increases.
Figure 4. (a) Heat capacity (CV) plotted versus temperature for Ni(hon) 55 with ϵ = 35.6 meV and a0 = 2.46 Å, unless indicated otherwise. The two red curves in (a) respectively correspond to expansion (dotted) and contraction (dashed) of the honeycomb lattice by 5% in the zigzag direction, with equal contraction/expansion in the orthogonal (armchair) direction so that ρs remains constant. The occupation probability (Pocc) shares of different motifs are plotted in (b) and (c), corresponding to the thick black and the dashed red CV curves in (a), respectively.
where F(γ) is an energy function of the laterally averaged surface energy density γ, which may satisfy F(γ) = γA(γ), with A(γ) representing an effective interface area, and f(a) is a dimensionless function that describes the effect of epitaxial stresses due to substrate microstructure, parametrized by a. The relation in eq 1 essentially presumes that microstructure effects can be treated as a modulating perturbation of a laterally averaged (i.e., microstructureless) description F(γ), and we expect that lima→0 f(a) = 1. Indeed, the flatlining in Figure 3a,b suggests that f(a) ≈ 1 for a/r0 < 0.5; therefore, we infer f(a) by computing ΔUm(a)/ΔUm(a*), where a* = 1 Å proved small enough. The resulting f(a) is plotted for a variety of minima in Figure 3c, showing significant dispersion for a/r0 > 0.75. Interestingly, f(a) ≥ 1 over the plotted range, implying that atomic roughness generally enhances substrate binding strength. The jumps in some curves correspond to the local minimization step suddenly finding a different minimum, which in some cases is due to a fold catastrophe52 in the underlying potential energy landscape, with the initial local minimum vanishing as a varies away from a0. The changes are less pronounced when a diminishes, which leads to more subtle rearrangements in the
and 1000 K. We attribute this peak to melting because it coincides with a dramatic increase in the occupancy of AMB (AMB) , as shown in Figure 4b where the minima, i.e., Pocc occupation probability shares of the different motifs are plotted for a = 2.46 Åthe in-plane lattice constant of graphite. Note that melting was also diagnosed by canonical molecular dynamics simulations at various temperatures, using the Lindemann53,54 index. Inspection of motif occupancies also helps us to interpret the premelting CV features in Figure 4a. For a between 2.40 and 2.50 Å, there is a single low-temperature feature that changes systematically with a, marking a transition from predominantly F+H to predominantly ICO morphology, with the former represented mainly by the GM and the latter by the closed-shell Mackay icosahedron, which has a significantly higher vibrational 5405
DOI: 10.1021/acs.jpclett.7b02543 J. Phys. Chem. Lett. 2017, 8, 5402−5407
The Journal of Physical Chemistry Letters
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ACKNOWLEDGMENTS This work was financially supported by EPSRC Grant EP/ J010847/1 and the ERC.
entropy than the GM. Note that premelting features are absent for a ≥ 2.50 when the Mackay icosahedron is the GM (as in the unsupported Ni55), and the ICO motif retains the highest occupancy until the melting transition. For a ≤ 2.30 Å, on the other hand, ICO minima become so unfavorable that their combined occupation probability remains insignificant for all temperatures. Hence, the premelting features for a = 2.20 and 2.30 Å in Figure 4a mark a transition between other motifs: F +H and HCP. In Figure 4b, P(ICO) decreases while P(F+H) turns over and occ occ actually increases for T in the range of 400−700 K. This S-bend is indicative of re-entrant behavior, with P(F+H) increasing due occ to higher landscape (as opposed to vibrational) entropy because the ratio of F+H to ICO minima in our largest set of local minima is 8961:339. Despite the S-bend, the most populated motif switches only twice, in the sequence F+H → ICO → AMB, as T increases. Stretching the substrate in the zigzag or armchair direction by 5% while contracting by the same amount in the orthogonal in-plane direction makes the Sbend more pronounced. In this case, the F+H motif has the highest occupation probability over two separate temperature ranges (see Figure 4c), with the most populated motif now switching thrice, F+H → ICO → F+H → AMB, as T increases. Furthermore, the corresponding CV curve exhibits multiple premelting features: a low-temperature peak to mark the F+H → ICO transition as well as a premelting shoulder to mark the re-entrance of the F+H motif. Hence, the re-entrant behavior can to some degree be tuned by controlled distortion of the substrate, showing that atomically smooth and relatively inert substrates, interacting primarily via dispersion-type forces, can influence the equilibrium behavior of supported metal clusters in surprisingly intricate ways, suggesting that more detailed examination of substrate effects is warranted in future work. To summarize, we have shown that the GM morphology and finite-temperature behavior of model metal clusters can be sensitive to the microstructure of an atomically smooth substrate, even when the substrate is relatively inert. Hence, a given metal cluster of particular size can exhibit a rich variety of equilibrium morphologies, depending on the supporting surface, which could be exploited to tailor nanoparticle properties. For instance, controlled straining of the substrate could be used to shift the temperature of thermal instability in a supported cluster or eliminate the instability entirely, which may prove useful in the design of novel and potentially switchable catalysts or plasmonic nanoparticles.
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ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b02543. Criteria defining the morphology classes (motifs) (PDF)
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Letter
AUTHOR INFORMATION
Corresponding Authors
*E-mail: Marco.Eckhoff@web.de (M.E.). *E-mail:
[email protected] (D.S.). *E-mail:
[email protected] (D.J.W.). ORCID
D. Schebarchov: 0000-0002-8385-7186 Notes
The authors declare no competing financial interest. 5406
DOI: 10.1021/acs.jpclett.7b02543 J. Phys. Chem. Lett. 2017, 8, 5402−5407
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