Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

King C.
Lai
* and
James W.
Evans
*

Division of Chemical & Biological Sciences, Ames Laboratory, USDOE and Department of Physics & Astronomy, Iowa State University, Ames IA 50011, USA. E-mail: kclai@iastate.edu; evans@ameslab.gov

Received
10th July 2019
, Accepted 8th September 2019

First published on 10th September 2019

Diffusion and coalescence of supported 3D metal nanoclusters (NCs) leads to Smoluchowski Ripening (SR), a key pathway for catalyst degradation. Variation of the NC diffusion coefficient, D_{N}, with size N (in atoms) controls SR kinetics. Traditionally, a form D_{N} ∼ N^{−β} was assumed consistent with mean-field analysis. However, KMC simulation of a stochastic model for diffusion of {100}-epitaxially supported fcc NCs mediated by surface diffusion reveals instead a complex oscillatory decrease of D_{N} with N. Barriers for surface diffusion of metal atoms across and between facets, along step edges, etc., in this model are selected to accurately capture behavior for fcc metals. (This contrasts standard bond-breaking prescriptions which fail dramatically.) For strong adhesion, equilibrated NCs are truncated pyramids (TP). Local minima of D_{N} sometimes but not always correspond to sizes, N_{TP}, where these have a closed-shell structure. Local maxima generally correspond to N ≈ N_{TP} + 3 for N = O(10^{2}). For weak adhesion, equilibrated NCs are truncated octahedra (TO), and local minima of D_{N} occur for sizes close or equal to those of just a subset of closed-shell structures. Analytic characterization of energetics along the NC diffusion pathway (which involves dissolving and reforming outer layers of facets) provides fundamental insight into the behavior of D_{N}, including the strong variation with N of the effective NC diffusion barrier.

(1) |

Our focus is on diffusion of epitaxially supported 3D NCs where diffusivity is generally lower than for non-epitaxially supported 3D NCs.^{6} Detailed analysis of the epitaxially supported case is limited. One Kinetic Monte Carlo (KMC) simulation study for 3D epitaxial NCs,^{7} and another for the analogous 2D case,^{8} found an oscillatory variation of D_{N} with N, and suggested that minima correspond to sizes with closed-shell structures. However, subsequent analysis revealed a much more complex scenario for the 2D case,^{9,10} and no detailed analysis exists in 3D. Thus, our goal here is to characterize and provide fundamental insight into the “fine structure” in the variation of D_{N} with N for 3D epitaxial NCs.

Specifically, we develop a stochastic lattice-gas model for diffusion of {100}-epitaxially supported metallic fcc NCs mediated by diffusive transport of metal atoms across the surface of the NC. We emphasize that for realistic modeling, not just NC thermodynamics, but also the multiple barriers for surface diffusion across facets, along step edges and around kinks or corners, between layers or facets, etc., must be chosen to realistically capture behavior for fcc metals.^{5,11} KMC simulation of such stochastic modeling allows direct access to the relevant time scale for surface diffusion, and precise characterization of the variation of D_{N} with size N and other key control parameters such as temperature, T. However, for deeper insight into the observed complex behavior, we also identify NC diffusion pathways (which involve dissolving and reforming outer layers of facets). We provide an analytic assessment of the associated variation of atomistic-level NC energetics along such minimum energy pathways, as well as a corresponding coarse-grained continuum analysis.

This article is organized as follows. The stochastic model is described in the next section, as well as procedure for reliable extraction of NC diffusion coefficient, D_{N}. Then, we present detailed KMC results for D_{N}versus N for the case of strong NC adhesion to the substrate, as well as a complementary analytic characterization. Next, we more briefly analyze NC diffusion behavior for the case of weak NC adhesion. Finally, we provide additional Discussion including assessment of related systems, and brief Conclusions.

With regard to NC thermodynamics, interactions within the fcc metal NC cluster are described by an effective NN attraction of strength ϕ > 0. This prescription was shown in recent DFT analysis to effectively capture NC surface energetics for many fcc metals.^{12} In fact, the value of ϕ extracted from this DFT analysis reasonably recovers surface energies, but is much weaker than that extracted from bulk energetics as one sixth of the cohesive energy, E_{c} (e.g., ϕ = 0.225 eV versus E_{c}/6 = 0.49 eV for Ag).^{11} In fact, the bulk cohesive energy is not incorporated as a parameter into our model. This is not unreasonable since NC diffusion is controlled by surface rather than bulk thermodynamics and kinetics. Each atom in the bottom {100} NC layer is regarded as supported by 4 atoms in the top {100} substrate layer. Adhesion is described by a NN attraction of strength ϕ_{s} = fϕ between NC and substrate atoms. Thus, f measures the strength of adhesion, and f = 1 corresponds to homoepitaxy. We focus on the regime of strong adhesion choosing f = 0.75 (where supported NCs resemble truncated pyramids), but for contrast more briefly consider weak adhesion where f ≤ 0.05 (where the supported NCs resemble unsupported Wulff shapes).

The total NC energy, E_{N} < 0, is obtained as the sum of the total interaction energy within the NC, and the total adhesion energy. Ground state NC configurations have the minimum E_{N}. Equilibrated NCs exist in excited states with finite probability as determined by a Boltzmann factor based on E_{N}. In the large-size continuum limit, fluctuations around a well-defined equilibrium shape vanish. For negligible adhesion (f ≈ 0), this equilibrium NC shape corresponds to the Wulff shape of an unsupported NC. For our model, the Wulff shape corresponds to a regular truncated octahedron (TO) where all edges have equal length.^{13} Equilibrium shapes of {100}-epitaxially supported NCs with significant adhesion are determined from the Winterbottom construction which truncates a portion of the unsupported Wulff cluster adjacent to a {100} facet.^{13} Specifically, in the continuum regime, when measured from the center of the unsupported Wulff cluster, the distance to the top {100} facet, h_{100}, and to the substrate, h_{sub}, are related by h_{100}/γ_{100} = h_{sub}/(γ_{100} − β_{ad}). Here, γ_{100} = 2ϕa^{−2} denotes the surface energy of {100} facets, and β_{ad} = 4ϕ_{s}a^{−2} denotes the adhesion energy. It follows that h_{sub}/h_{100} = 1 − 2f. Negative values mean that the location of the substrate for the supported NC is above the center of the unsupported NC. See Fig. 1. In our atomistic model, equilibrium shapes mimic continuum shapes. See below.

Diffusion of supported NCs is sensitively dependent on the prescription of the kinetics of adatom surface diffusion. In stochastic lattice-gas modeling, often a simple IVA bond-breaking prescription is applied to determine the activation barriers for hopping in diverse local surface environments.^{14,15} However, these (and alternative Metropolis type prescriptions) fail dramatically to describe key features of surface diffusion barriers on fcc metal surfaces, e.g., the relative magnitude of terrace diffusion on different facets, of terrace versus step edge diffusion, and of intra- versus inter-layer diffusion.^{11} Consequently, we apply a refined BEP formalism with sufficient flexibility to capture all key diffusion barriers, as described in the next subsection.^{5,11}

For specificity, our model parameters are chosen to correspond to Ag. Supported 3D Ag NCs have been studied on graphite,^{16} and on various oxide surfaces^{17,18} including Al_{2}O_{3},^{19,20} TiO_{2},^{21,22} and MgO.^{23} Theoretical analysis indicates that Ag NC on MgO(001) exhibits cube-on-cube {100} epitaxy at least for N = 40 to 2800, rather than {111} epitaxy or a decahedral structure.^{24} Thus, Ag/MgO(001) falls within the class of systems described by our model.

The C_{α} are selected to recover precise value for terrace, edge, and interlayer diffusion for the metal of interest. See the ESI sec. 1.† For Ag, we select C_{TD100} = 0.425 eV, C_{TD111} = 0.10 eV, C_{ED111B} = 0.30 eV, and C_{ED111A} = C_{ED100} = 0.275 eV for intralayer diffusion.^{11} Here, A and B indicate close-packed {100}- and {111}-microfaceted steps on {111} facets, respectively. The local geometry of the former is the same as for close-packed steps on {100} facets. For interlayer diffusion for Ag, our C_{α} also incorporate an additional Ehrlich-Schwoebel barrier for downward interlayer diffusion of δ_{ES} = 0.1 eV for close-packed (but not kinked) steps on {100} facets, and for A (but not B) steps on {111} facets. We use the same C_{α} for hopping between the lowest and next highest layer of the NC. For atoms at the periphery of the lowest layer hopping around the contact line of the NC: (i) the same C_{α} are used for strong adhesion; and (ii) C_{α} are selected by neglecting substrate atoms for weak adhesion. We also allow atoms to hop to second NN sites in order to round corners of the contact line base the NC.

As indicated above, our model dynamics allows diffusion of metal atoms across the NC surface, but not detachment of atoms from the NC at its base (which would be followed by diffusion across the substrate, and possible reattachment to the NC). The rationale is as follows. The effective barrier for NC surface diffusion (detachment) is determined by the sum of: (i) the energy change upon moving an atom at a kink site on the NC to a facet on the NC (to the substrate); and (ii) the terrace diffusion barrier across the facet (across the substrate). The effective barrier is significantly higher for detachment justifying our neglect of this process. See the ESI sec. 2† for further discussion.

Fig. 2 shows KMC simulation results for a typical diffusion trajectory for the CM of an NC of size N = 50 atoms for the case of strong adhesion with f = 0.75 at 900 K. During diffusion, the NC can be regarded as remaining in equilibrium state. However, as noted above, the NC in this state does not just correspond to a fixed ground state NC configuration, but rather samples excited states. In the ESI sec. 4,† we show snapshots which illustrate the NC evolving through a sequence of configurations during diffusion, and we also provide a movie of NC diffusion. Further discussion of key configurations accessed during diffusion is provided in sec.3 and 4. For the system under consideration here, our analysis will reveal a complex dependence of D_{N} on size N. It will prove instructive to consider an effective activation barrier, E_{eff} > 0, for NC diffusion where D_{N} ∼ exp[−E_{eff}/(k_{B}T)] for fixed N at various T. In contrast to the MF treatment, E_{eff} = E_{eff}(N) will depend strongly on N, and this dependence will in fact induce the complex variation of D_{N} with N.

Fig. 2 Diffusion trajectory from KMC simulation for a closed-shell Ag TP with size N = 50 for the case of strong adhesion with f = 0.75 at 900 K. |

(2) |

(3) |

Ground state TP_{m×n,k} structures have m = n square or m = n + 1 near-square bases, and also k ≤ min(n,m) − 1 so that the top layer is 2 × 2 or larger. Corresponding, ground states sizes in bold font are:

N
_{
m×n,1} = 4, 6, 9, 12 (degen.), (but not 16,…) for m × n = 2 × 2, 3 × 2, 3 × 3, 4 × 3, (but not 4 × 4,…);

N
_{
m×n,2} = 13, 18, 25, 32, 41, (but not 50,…) for m × n = 3 × 3, 4 × 3, 4 × 4, 5 × 4, 5 × 5, (but not 6 × 5,…);

N
_{
m×n,3} = 29, 38, 50, 62, 77, 92, 110, 128, 149 (degen.), (but not 170,…) for m × n = 4 × 4,…, (but not 9 × 8,…);

N
_{
m×n,4} = 54, 68, 86, 104, 126, 148, 174,… for m × n = 5 × 5,…,8 × 8,…

N
_{
m×n,5} = 135, 160,… (but not 90, or 110) for m × n = 7 × 7, 8 × 7,… (but not 6 × 6, or 7 × 6)

etc.

for single- double-, triple-, quadruple, quintuple-layer NCs, etc. The particular stability of these sizes is quantified below. Sizes are also indicated which are not ground states, and N = 12 & 149 are degenerate with higher-layer structures.

Finally, we note that while many previous studies of NC diffusion have focused on simple size scaling, D_{N} ∼ N^{−β}, such a routine analysis is not applicable here given the complex oscillatory behavior. However, one could consider the partial scaling of D_{N} with N, e.g., restricting N to local maxima (max) of D_{N}. Based on the three maxima at N = 65, 80, and 95 in Fig. 3, one obtains β_{max} ≈ 3.1, 2.6, and 2.1 at 700 K, 800 K, and 900 K, respectively, which are higher than but shifting towards β_{MF} = 1.33 for increasing T.

The above process can be quantified analytically by tracking the energy change, ΔE(q), as a function of the number, q, of atoms transferred between different sides of the NC for the minimum energy path (MEP). The MEP has the smallest ΔE(max) = max_{q} ΔE(q). By symmetry, ΔE(q) = ΔE(q_{tot} − q), where q_{tot} = k(2n − k + 1)/2 is the total number of atoms in the facet supplying atoms with base with of n atoms and height of k layers. Examples are provided in Fig. 4 (black curves) for TP_{5×5,3} (N_{TP} = 50), TP_{7×6,4} (N_{TP} = 104), and TP_{8×8,5} (N_{TP} = 190) with ΔE(max) = 4ϕ, 5ϕ, and 6ϕ, respectively. ΔE(max) gives a measure of the difficulty of mass transfer, and thus of NC diffusion (but it does not account for the details of surface diffusion kinetics or entropic effects).

Analysis is more complex for non-closed or open-shell TPs which one generally can regard as a closed-shell TP with an additional incomplete layer on one facet. Consider the following two-stage diffusion pathway. In the first stage, atoms are transferred to complete an incomplete 2D layer on a facet labeled 2 from a complete facet labeled 1 on the opposite side. In the second stage, the incomplete layer now existing on facet 1 is transferred to facet 2. Then, the original NC structure is recovered, but with shifted CM. ΔE(q) versus q has a different form in each stage, but vanishes at the end of each stage. See below. Other cases involve shifting the incomplete layer to an adjacent (rather than the opposite) side facet. Even more complicated scenarios can occur in some cases where the ground state NC structure is k-layer, but the diffusion path with the lowest ΔE(max) goes through a (k − 1)-layer NC configurations. One example is for N = 66 where the ground state is a 4-layer structure, but the optimum diffusion pathway goes through 3-layer structures. See ESI sec. 6.†

Fig. 5 provides comprehensive analytic results for ΔE(max) versus N up to N = 110 for both open- and closed-shell TPs. ΔE(max) for closed-shell TPs correspond to local maxima, and the ΔE(max) variation with N correlates reasonably with that of D_{N}. Fig. 5 also indicates a slow increase in ΔE(max) with N, somewhat obscured by the strong oscillatory behavior. This feature will be elucidated below. ΔE(max) just characterizes the thermodynamics rather than the kinetics of the NC diffusion process.

Fig. 5 Variation of ΔE(max), ΔE(max+), E_{eff}(analytic), and E_{eff}(KMC) with N. Vertical lines indicate sizes for closed-shell ground state TPs. |

To assess kinetics, we note that ΔE(q) just describe energies relative to the ground state after each transfer of an adatom between facets and incorporation into a growing 2D layer. Thus, naturally the system energy is higher mid-atom transfer, and thus the actual energy barrier which the system must surmount for NC diffusion is higher. For large NCs, the increased barrier will reflect energy difference of 3ϕ between an isolated atom in transit at a 3fh site on a {111} facet and that atom incorporated into a kink site at the periphery of an incomplete layer. However, for smaller NCs, this energy difference is often 2ϕ. Fig. 4 also shows the energy profile along the MEP incorporating energies mid-atom transfer as a dashed red line, and Fig. 5 also shows the corresponding boosted ΔE(max+) versus N. Finally, the actual effective barrier for E_{eff} for the above process must add a diffusion barrier which is typically but not always equal to that for terrace diffusion on {111} facets of 0.1 eV for Ag. This analytic estimate of E_{eff}(analytic) versus N is also shown in Fig. 5. Again we caution that this analysis does not account for entropic effects. The analytic treatment is successful in capturing the key features of our KMC estimate, E_{eff}(KMC) (reproduced from Fig. 2).

ΔE(x)|_{CS} = 3σl[(1 − x)^{1/2} + x^{1/2} − 1] with x = q/q_{tot}, | (4) |

ΔE(max)|_{CS} = (√2 − 1)3σl ≈ 1.24σl ∼ N^{1/3}, | (5) |

Next, we more briefly present a continuum treatment for open-shell (OS) TPs. As discussed above, the mechanism for mass transfer leading to NC diffusion has two stages. First, atoms are transferred from a complete facet 1 to grow an initially incomplete layer on facet 2 (which becomes complete). Second, the incomplete layer remaining on facet 1 is transferred to facet 2. Fig. 7 shows the case where the dimensions of the initial incomplete layer are smaller by a factor of r than those of the complete facet. Here, x_{1} (x_{2}) denotes the fraction of atoms transferred in the first (second) stage. Evaluating the energy change as a function of the amount of material transferred (see ESI sec. 8†), one finds that ΔE(max1)|_{OS} = [√2(1 + r^{2})^{1/2} − r − 1]3σl in the first stage, and ΔE(max2)|_{OS} = 3(√2 − 1)rσl in the second. The effective barrier ΔE(max)|_{OS} = max_{j} ΔE(max_{j})|_{OS} for the open-shell TP is below ΔE(max)|_{CS} for a closed-shell TP for all 0 < r < 1, consistent with the lower ΔE(max) for open-shell TPs shown in Fig. 5. The smallest value of ΔE(max)|_{OS} = 3(√2 − 1)σl/(2√2) ≈ 0.439σl occurs for r = 1/(2√2) ≈ 0.354.

D_{N} ∼ (Ω_{TS}/Ω_{0})exp[−E_{eff}(analytic)/(k_{B}T)], | (6) |

As noted above, local maxima in D_{N} tend to occur for N = N_{TP} + 3 for larger sizes with N = O(10^{2}). Consistently, E_{eff} displays a local minima for these sizes corresponding to local minimum in ΔE(max) for N_{TP} + 3 or N_{TP} + 4. For these sizes, the ground state corresponds to a small 2D cluster of 1, 2, 3,… atoms on a facet of a closed-shell TP. The presence of this small 2D cluster naturally facilitates initial transfer of atoms from another complete side of the NC, thereby reducing ΔE(q), ΔE(max), and related quantities.

Finally, we briefly comment of behavior of D_{N} for larger sizes, which is reported in the ESI sec. 5.† The basic features observed in the smaller size range up to N ≈ 190 are preserved, i.e., complex oscillatory decay of D_{N}versus N. However, it should be noted that local minima in D_{N} do not occur for all sizes with closed-shell ground states, but only at or near a subset of these. A similar feature is manifested for the case of weak adhesion described in sec. 4. An effective criterion to assess the subset of closed-shell sizes corresponding to local minima in D_{N} is again provided by (the occurrence of local minima in) the readily calculated quantity δE. See ESI sec. 5.†

N^{TO}(n) = 16n^{3} − 33n^{2} + 24n − 6 = 38, 201, 586,… for n = 2, 3, 4,… | (7) |

N^{TO+}(n) = 16n^{3} − 63n^{2} + 84n − 38 = 79, 314, 807,… for n = 3, 4, 5,… | (8) |

A recent analysis revealed 49 additional sizes of various asymmetric closed-shell TO (including TO+) between each consecutive pair of magic regular TO sizes, which are also relatively stable compared to non-closed-shell structures.^{28} For each of these sizes between N^{TO} = 38 and 201, there is a corresponding size between N^{TO} = 201 and 586, etc., so structures repeat quasi-periodically. For example, the closed-shell structure for N = 244 corresponds to that for N = 52; the N = 314 TO+ corresponds to the N = 79 TO+.

Excitation of a closed-shell structure by moving a corner atom to a {111} ({100}) facet increases the energy by E_{ex} = 3ϕ (2ϕ). Given the restricted number, M, of adsorption sites on facets for the NC sizes considered here, it follows that exp[−E_{ex}/(k_{B}T)]M ≪ 1 so the ground state structure predominates for 700 K.

Fig. 8 Top: δE = [E_{N} − E_{N}(cont)]/N where^{28}E_{N}(cont)/ϕ = −1.59 + 0.061N^{1/3} + 7.554N^{2/3} − 6N recovers E_{N} for regular TO with N = 38, 201, 586, and 1289. Middle: KMC results for D_{N}versus N for an Ag TO at 700 K for f = 0.05 (black) and f = 0 (red). Bottom: Analytic results for −ΔE(max)/ϕ, where the negative sign is included so that peaks and valleys correspond to those of D_{N}. Note: vertical lines correspond to sizes for closed-shell structures, with those for TO, TO+, and other particularly stable structures, indicated as thicker lines. |

The additional challenge here compared to the analysis for TP with f = 0.75 is the variety of possibilities for transferring atoms which must be considered to determine the MEP and ΔE(max). The general principle, as for f = 0.75, is to select atoms to transfer which break the minimum number of bonds and to arrange them on the receiving facet to create the maximum number of bonds. As an example, for N^{TO} = 38, one removes atoms from a {111} then a {100} then a {111} and finally a {100} facet, and builds up layers in the reverse sequence (first on a {100} facet, etc.) to obtain ΔE(max) = 5ϕ. See ESI sec. 10† for further discussion and detailed analysis for N^{TO} = 201.

For the less symmetric closed-shell TO, {100} facets generally have multiple different sizes and shapes. In this case, a separate analysis of ΔE(q) is performed for each possible type of supporting facet. The minimum ΔE(max) from these different possibilities is selected. Results from extensive analysis of ΔE(max) for regular and less symmetric closed-shell ground state TO are reported in Fig. 8 (lower frame).

We do not present a detailed continuum analysis here. However, from the formalism presented for f = 0.75, it is clear that irrespective of the detailed pathway for atom transfer, one has that ΔE(max) ∝ ϕl/a ∼ N^{1/3}. This slow increase in ΔE(max) with N is reflected in Fig. 8 (bottom), again somewhat hidden by oscillatory structure.

The observed oscillatory decay of D_{N} with N appears in marked contrast to the traditional picture of algebraic scaling D_{N} ∼ D°exp[−E_{eff}/(k_{B}T)]N^{−β}, where mean-field analysis gives that E_{eff} is size-independent, and β_{MF} = 4/3. In fact, our analysis shows that the effective barrier for diffusion has a strong oscillatory size dependence, with an overall increase with N like E_{eff} ∼ N^{1/3}. This corresponds to faster than algebraic asymptotic decay of D_{N} for large N. Analogous observations have been made for NC reshaping.^{14,29} Despite this feature, we have shown that for strong adhesion, f = 0.75, an effective β can be reasonably extracted from selected peaks of D_{N} (for a fixed range of sizes). Reported results show this β_{eff} decreasing from 3.1 at 700 K to 2.1 at 900 K. Furthermore, oscillations in D_{N} disappear for sufficiently high T, and classic mean-field scaling β_{eff} → 4/3 of D_{N} is recovered. This behavior can be understood given that the structure of supported NCs become less facetted and more irregular with a randomly rough surface as T increases. In this regime, surface hopping becomes uncorrelated as assumed in the mean-field analysis. This type of recovery of mean-field behavior for increasing T is a general phenomenon applying for any adhesion strength.

As an aside, for diffusion of 2D epitaxial NCs, mean-field scaling is recovered asymptotically as N → ∞ for any T (not just as T → ∞). This fundamental difference arises from the feature that 2D NCs, in contrast to 3D NCs, are not facetted in the large-N continuum regime.

One could consider {111}- rather than {100}-epitaxially supported 3D fcc metal NC. Now, each atom in the lowest layer is regarded as supported by three atoms in the substrate, again with an effective NN substrate-NC atom attraction of ϕ_{s} = fϕ. Then, the distance to the top {111} facet, h_{111}, and to the substrate, h_{sub}, are related by h_{sub}/h_{111} = 1 − 2f. By analogy with f = 0.75 for the {100}-supported case, the Winterbottom shape for f = 2/3 is a truncated hexagonal pyramid where the edge length for the top hexagonal {111} facet matches that of the edges between the six alternating {100} and {111} side facets. For f ≈ 0, the equilibrium shape is a Wulff TO, now supported on a {111} facet. Behavior of diffusivity in these cases is analogous to that for {100}-supported NCs with complex oscillatory decay in D_{N}versus N, and where again detailed insight comes from an atomistic-level analysis of energetics along the MEP or from a continuum analysis.

Finally, some additional comments are appropriate on T-dependence of D_{N}, and in particular on the disappearance of oscillations for high T. Oscillatory structure relies on the distinction between closed-shell and other NC structures. For sufficiently low T, NCs with closed-shell ground states will most likely be found in those configurations as discussed above accounting for excited-state Boltzmann factors and configurational degeneracy. However, for high T, the NC will most likely be in a non-closed-shell excited state. This type of entropic effect diminishes distinction between open- and closed-shells, and thus degrades the oscillatory fine structure of D_{N}versus N.

For diffusion of either supported 2D or 3D epitaxial NCs exhibiting oscillatory decay, it is natural to anticipate that local minima correspond to sizes with closed-shell NC ground state structures with sizes N = N_{CS}.^{7,8} Our analysis does reveal some correlation between local minima in D_{N} and closed-shell structures. However, for strong adhesion, we find local minima for sizes N = N_{CS} − 1 and N_{CS} − 3 and explain this feature as due to higher ground state degeneracy for these sizes relative to closed-shell NCs. For moderate and weak adhesion, local minima only correlate strongly with a subset of closed-shell structures. Thus, a comprehensive and fundamental understanding of the fine structure of D_{N}versus N necessarily requires the type of analytic characterization of energetics along MEP for diffusion provided here.

Finally, we remark that our results for D_{N}versus N can provide input to analysis of SR kinetics of distributions of supported NCs. We have noted the importance of this process for catalyst degradation. However, it might also be noted that under reaction conditions, adsorption of reactants on the NC surface can alter both NC surface thermodynamics and diffusion kinetics, and thus NC diffusivity. Nonetheless, the current study of NC diffusion in a vacuum environment is a valuable precursor to understanding of such behavior in more complex environments.

- E. Ruckenstein and B. Pulvermacher, Kinetics of Crystallite Sintering during Heat Treatment of Supported Metal Catalysts, AIChE J., 1973, 19, 356–364 CrossRef CAS.
- P. Wynblatt and N. A. Gjostein, Supported Metal Crystallites, Prog. Solid State Chem., 1975, 9, 21–58 CrossRef CAS.
- T. W. Hansen, A. T. DeLaRiva, S. R. Challa and A. K. Datye, Sintering of Catalytic Nanoparticles: Particle Migration or Ostwald Ripening?, Acc. Chem. Res., 2013, 46, 1720–1730 CrossRef CAS PubMed.
- Y. Dai, P. Lu, Z. Cao, C. T. Campbell and Y. Xia, The Physical Chemistry and Materials Science behind Sinter-resistant Catalysts, Chem. Soc. Rev., 2018, 47, 4314–4331 RSC.
- K. C. Lai, Y. Han, P. Spurgeon, W. Huang, P. A. Thiel, D.-J. Liu and J. W. Evans, Reshaping, Intermixing, and Coarsening for Metallic Nanocrystals: Non-equilibrium Statistical Mechanical and Coarse-grained Modeling, Chem. Rev., 2019, 119, 6670–6768 CrossRef CAS PubMed.
- P. Jensen, Growth of Nanostructures by Cluster Deposition: Experiments and Simple Models, Rev. Mod. Phys., 1999, 71, 1695–1735 CrossRef CAS.
- L. Li, P. N. Plessow, M. Rieger, S. Sauer, R. S. Sánchez-Carrera, A. Schaefer and F. Abild-Pedersen, Modeling the Migration of Platinum Nanoparticles on Surfaces using a Kinetic Monte Carlo Approach, J. Phys. Chem. C, 2017, 121, 4261–4269 CrossRef CAS.
- J. Heinonen, I. Koponen, J. Merikoski and T. Ala-Nissila, Island Diffusion on Metal fcc (100) Surfaces, Phys. Rev. Lett., 1999, 82, 2733–2736 CrossRef CAS.
- K. C. Lai, J. W. Evans and D.-J. Liu, Communication: Diverse Nanoscale Cluster Dynamics: Diffusion of 2D Epitaxial Clusters, J. Chem. Phys., 2017, 147, 201101 CrossRef PubMed.
- K. C. Lai, D.-J. Liu and J. W. Evans, Diffusion of 2D Epitaxial Clusters on Metal(100) Surfaces: Facile versus Nucleation-mediated Behavior and their Merging for Larger Sizes, Phys. Rev. B, 2017, 96, 235406 CrossRef.
- K. C. Lai and J. W. Evans, Reshaping and Sintering of 3D fcc metal Nanoclusters: Stochastic Atomistic Modeling with a Realistic Surface Diffusion Kinetics, Phys. Rev. Mater., 2019, 3, 026001 CrossRef CAS.
- L. T. Roling, L. Li and F. Abild-Pedersen, Configurational Energies of Nanoparticles Based on Metal−Metal Coordination, J. Phys. Chem. C, 2017, 121, 23002–23010 CrossRef CAS.
- C. R. Henry, Morphology of Supported Nanoparticles, Prog. Surf. Sci., 2005, 80, 92–116 CrossRef CAS.
- N. Combe, P. Jensen and A. Pimpinelli, Changing Shapes in the Nanoworld, Phys. Rev. Lett., 2000, 85, 110–113 CrossRef CAS PubMed.
- T. H. Lim, D. McCarthy, S. C. Hendy, K. J. Stevens, S. A. Brown and R. D. Tilley, Real-Time TEM and Kinetic Monte Carlo Studies of the Coalescence of Decahedral Gold Nanoparticles, ACS Nano, 2009, 3, 3809–3813 CrossRef CAS PubMed.
- I. M. Goldby, L. Kuipers, B. Issendorff and R. E. Palmer, Diffusion and Aggregation of Size-selected Silver Clusters on a Graphite Surface, Appl. Phys. Lett., 1996, 69, 2819–2821 CrossRef CAS.
- M. Chen and D. W. Goodman, Oxide-supported Metal Clusters, in Chemical Physics of Solid Surfaces, Elsevier, Amsterdam, 2007, ch. 5, vol. 12 Search PubMed.
- S. L. Hemmingson and C. T. Campbell, Trends in Adhesion Energies of Metal Nanoparticles on Oxide Surfaces: Understanding Support Effects in Catalysis and Nanotechnology, ACS Nano, 2017, 11, 1196–1203 CrossRef CAS PubMed.
- N. Nilius, N. Ernst and H.-J. Freund, Photon Emission Spectroscopy of Individual Oxide-Supported Silver Clusters in a Scanning Tunneling Microscope, Phys. Rev. Lett., 2000, 84, 3994–3997 CrossRef CAS PubMed.
- W. Benten, N. Nilius, N. Ernst and H. J. Freund, Photon Emission Spectroscopy of Single Oxide-Supported Ag-Au Alloy Clusters, Phys. Rev. B: Condens. Matter Mater. Phys., 2005, 72, 045403 CrossRef.
- D. A. Chen, M. C. Bartelt, S. M. Seutter and K. F. McCarty, Small, Uniform, and Thermally Stable Silver Particles on TiO2(110)-(1×1), Surf. Sci., 2000, 464, L708–L714 CrossRef CAS.
- K. Luo, T. P. St. Clair, X. Lai and D. W. Goodman, Silver Growth on TiO2(110) (1×1) and (1×2), J. Phys. Chem. B, 2000, 104, 143050–143057 Search PubMed.
- J. H. Larsen, J. T. Ranney, D. E. Starr, J. E. Musgrove and C. T. Campbell, Adsorption Energetics of Ag on MgO(100), Phys. Rev. B: Condens. Matter Mater. Phys., 2001, 63, 195410 CrossRef.
- R. Ferrando, G. Rossi, A. C. Levi, Z. Kuntová, F. Nita, A. Jelea, C. Mottet, G. Barcaro, A. Fortunelli and J. Goniakowski, Structures of Metal Nanoparticles Adsorbed on MgO(001). I. Ag and Au, J. Chem. Phys., 2009, 130, 174702 CrossRef PubMed.
- New Trends in Kramer's Reaction Rate Theory, ed. P. Talkner and P. Hanggi, Kluwer, Dordrecht, 1995 Search PubMed.
- D. Wales, Energy Landscapes, Cambridge University Press, Cambridge, 2003 Search PubMed.
- R. L. Whetten, J. T. Khoury, M. M. Alvarez, S. Murthy, I. Vezmar, Z. L. Wang, P. W. Stephens, C. L. Cleveland, W. D. Luedtke and U. Landman, Nanocrystal Gold Molecules, Adv. Mater., 1996, 8, 428–433 CrossRef CAS.
- K. C. Lai, X. Zha, J. W. Evans and A. Travesset, Structure of Polydisperse fcc Nanocrystals: Implications for Crystal Fractionalization, J. Phys. Chem. C, 2019, 123, 9528–9537 CrossRef CAS.
- W. W. Mullins and G. S. Rohrer, Nucleation Barrier for Volume-Conserving Shape Changes of Faceted Crystals, J. Am. Ceram. Soc., 2000, 83, 214–216 CrossRef CAS.

## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9nr05845a |

This journal is © The Royal Society of Chemistry 2019 |