Beyond Magic Numbers: Atomic Scale Equilibrium Nanoparticle

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Beyond Magic Numbers: Atomic Scale Equilibrium Nanoparticle Shapes for Any Size J. Magnus Rahm, and Paul Erhart Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b02761 • Publication Date (Web): 09 Aug 2017 Downloaded from http://pubs.acs.org on August 12, 2017

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Beyond Magic Numbers: Atomic Scale Equilibrium Nanoparticle Shapes for Any Size J. Magnus Rahm and Paul Erhart∗ Chalmers University of Technology, Department of Physics, S-412 96 Gothenburg, Sweden E-mail: [email protected] Abstract In the pursuit of complete control over morphology in nanoparticle synthesis, knowledge of the thermodynamic equilibrium shapes is a key ingredient. While approaches exist to determine the equilibrium shape in the large size limit (≳ 10−20 nm) as well as for very small particles (≲ 2 nm), the experimentally increasingly important intermediate size regime has largely remained elusive. Here, we present an algorithm, based on atomistic simulations in a constrained thermodynamic ensemble, that efficiently predicts equilibrium shapes for any number of atoms in the range from a few tens to many thousands of atoms. We apply the algorithm to Cu, Ag, Au and Pd particles with diameters between approximately 1 and 7 nm and reveal an energy landscape that is more intricate than previously suggested. The thus obtained particle type distributions demonstrate that the transition from icosahedral particles to decahedral and further into truncated octahedral particles occurs only very gradually, which has implications for the interpretation of experimental data. The approach presented here is extensible to alloys and can in principle also be adapted to represent different chemical environments.

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Keywords Nanoparticle shape, Wulff construction, Monte Carlo simulation, icosahedron, decahedron, truncated octahedron

The size and shape selective synthesis of nanoparticles is of paramount importance in nanotechnology. 1–4 While the kinetics of the particle growth process is crucial in this context, 5 the underlying thermodynamics plays an equally important role as it determines driving forces as well as the thermodynamic stability of different nanoparticle shapes. 6–9 It is therefore crucial that methods are available that can efficiently predict the shape of nanoparticles in a wide size range. The Wulff construction 10 and its generalizations 11,12 have for a long time been valuable tools in this strive. 13–15 Given a set of calculated or measured surface energies, these methods yield well-defined shapes in the continuum limit. Yet, while the Wulff construction may be enhanced by, e.g., including edge effects 16 or allowing for lattice parameter dependent surface energies, 17 it remains fundamentally a continuum description. As a result, it is inevitably limited when it comes to capturing important atomic scale contributions. Many studies have therefore used atomic scale simulations to find energy minima, often by density functional theory, 18,19 the computational effort of which, however, puts a limit to the size and number of particles that may be studied. The size restriction may be overcome with empirical potentials, and larger particles have been investigated considering so-called magic numbers only, 20,21 or by minimization methods such as simulated annealing 22,23 or genetic algorithms, 24 the latter sometimes coupled to TEM images to reproduce experimentally observed structures. 25,26 While particles with hundreds or thousands of atoms have been assessed in this fashion, the particle sizes thus sampled are sparsely spaced and very little information is available for the intermediate regions. The small cluster regime has been extensively studied with global energy minimum techniques, without restriction to crystal lattices. 27–29 The potential energy surface has, however, 2 ACS Paragon Plus Environment

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extraordinarily many dimensions already for very small clusters, and the minimization problem is to date virtually unsolvable for anything but clusters with up to a few hundred atoms. The overall picture of shape as a function of size has thus remained incomplete. Here, we remedy this situation by introducing a computational procedure based on Monte Carlo simulations in the variance constrained semi-grand-canonical (VCSGC) ensemble. 30 This approach allows us to efficiently predict the shape of particles for any size in the range from tens of atoms to thousands and we apply it here to Cu, Ag, Au, and Pd particles with diameters of up to 7 nm. Using atomistic simulations, we can resolve the edge and corner effects that the regular Wulff construction fails to capture, and we comply to the constraint that any shape cannot be constructed with any number of atoms. The search is constrained by assuming that the stable structures are close to certain pre-defined structural motifs, which significantly expands the size regimes that may be explored. (a) FCC: Oh

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Figure 1: Symmetrical nanoparticle shapes considered in this work, with {111} and {100} facets in light blue and green, respectively, and edges/corner atoms in dark blue. All particles in the upper row are FCC-based octahedra with different levels of truncation: (a) octahedron, not truncated, (b) regular truncated octahedron for which the {111} surfaces are equilateral hexagons and (c) cuboctahedron for which the {111} surfaces are equilateral triangles. All particles in the lower row are strained and twinned: (d) icosahedron with {111} surfaces only, (e) decahedron with {111} surfaces only, (f) a truncated decahedron with truncated edges and re-entrant surfaces, commonly referred to as a Marks decahedron. 31


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Figure 2: Volumetric strain per atom as calculated and visualized with OVITO 32 for a regular truncated octahedron (a,d), an icosahedron (b,e), and a Marks decahedron (c,f), all based on Ag. Particles (a-c) contains about 6,000 atoms, roughly corresponding to a diameter of 6 nm, while (d-f) comprise about 400,000 atoms, corresponding to 25 nm. Dark blue colors indicate the absence of strain relative to the bulk lattice constant; red colors represent compressive strain. For materials that adopt face-centered cubic (FCC) crystal lattices in the bulk, three different structural motifs have been identified as particularly important: truncated octahedra (TO), icosahedra (Ih) and decahedra (Dh) 31,33–36 (Fig. 1). The close-packed {111} surface is the lowest-energy surface for the late transition metals considered here. Truncating a defect-free FCC crystal using only {111} surfaces yields an octahedron, which is, however, energetically unfavorable due to the undercoordination of the atoms at its corners (Fig. 1a). The energy of the particle (while fixing the total particle volume/number of atoms) can therefore be lowered significantly by truncating the corners (Fig. 1b-c; TO), even though this introduces higher energy {100} surfaces. Particles with only {111} surfaces can also be obtained by introducing interior interfaces, specifically twin planes, and accepting some level of strain (Fig. 2). Icosahedral particles (Fig. 1d; Ih) consist of twenty tetrahedra sharing a common vertex while decahedral particles (Fig. 1e; Dh) comprise five tetrahedra sharing a common edge. 33,34 For the decahedral motif, truncating the edges along with re-entrant surfaces at the boundaries between the tetrahedra (Fig. 1f; Marks decahedron) commonly yields even lower energies. 31 Since neither decahedra


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nor icosahedra can be created by stacking regular tetrahedra, particles based on these motifs are by geometrical necessity strained. The effect is most pronounced for icosahedra, as shown in Fig. 2 where the volumetric strain is visualized for relaxed silver particles. For small particles (Fig. 2a-c), surface tension leads to compression in all motifs considered here. For large particles (Fig. 2d-f) in contrast, strain persists for Ih and Dh, while TO particles approach the lattice parameter of the bulk material. This reflects the general principle that due to their symmetry Dh and Ih cannot be expanded to the infinite bulk limit. The TO structural motif is therefore ultimately always the most favorable for large enough particles. While analytic expressions that take into account surface, twin boundary and strain effects have been developed, 20,37 the cohesive energy can empirically be expressed as a thirddegree polynomial in N 1/3 (N being the total number of atoms), 38

Ecoh = aN + bN 2/3 + cN 1/3 + d.

(1)

Here, a through d are material and shape-dependent coefficients; strain enters along with the bulk-like cohesive energy in a, surface and twin boundary energies in b, edge and corner contributions in c and d, respectively. By dividing both sides of Eq. (1) with N , it is apparent that the size dependence of the cohesive energy per atom should be given by a third degree polynomial in N −1/3 . Using this expression, one can fit the cohesive energy per atom of, e.g., regular truncated octahedra practically exactly. Below, this provides us with a convenient reference energy to which any other structure can be referred. As a result of the competition between the contributions noted above, different motifs are the most stable in different size regimes. Experimentally, all three motifs can be observed but their proportions vary depending on experimental conditions, including synthesis route and chemical environment (the latter altering the surface energies). For example, gold particles produced by sputtering in one study contained very few icosahedra and almost one third TO particles, 39 while particles of similar size synthesized via a wet chemical route in


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another contained no TO particles but almost two thirds Ih. 40 Most computational studies agree that icosahedra are stable for small particle sizes, decahedra for intermediate sizes and truncated octahedral particles for all sizes above a certain limit, but the estimates of where the crossovers occur for specific materials differ wildly. For Au, which arguably is the most well-studied material in this context, the predictions of the the Ih to Dh crossover range from 80 atoms to around 40,000 and the Dh to TO crossover from 500 atoms to almost 100,000. 22,41,42 In this context, it is important to note that the symmetrical shapes described here can only be created with certain fixed numbers of atoms, commonly referred to as magic numbers. The smallest five icosahedra, for example, have 13, 55, 147, 309 and 561 atoms each, and the smallest regular truncated octahedra (RTO), i.e. truncated octahedra having equilateral hexagonal {111} faces (Fig. 1b), have 38, 201, 586, 1289 and 2406 atoms. Since the magic numbers for icosahedra and RTOs differ, direct comparisons are elusive. By varying truncation, however, one may argue that the magic numbers are much more densely spaced. In particular, it is possible to truncate octahedral particles such that they have exactly the same number of atoms as the magic number icosahedra. This results, however, in cuboctahedra (Fig. 1c), which have large {100} faces and are thus quite far from the ideal Wulff shape for most materials. It is therefore not unlikely that a TO like particle with less truncation but lower symmetry is a better low-energy competitor to the magic number icosahedron. The inherent discreteness of particle shape versus number of atoms hence makes comparisons between different structural motifs complicated for small and medium-sized particles. In this work we overcome this difficulty by allowing the system to adapt its preferred shape for a given size, with the only restriction being the adherence to an underlying lattice (while still allowing for local atomic relaxations). We thus create particles of different motifs with any number of atoms, making comparisons between different motifs viable and fair for any size. Approaches that aim at predicting global minimum structures need to effectively consider a continuous 3N -dimensional space, where N is the particle size. Techniques that operate


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Define a lattice, occupy with atoms and relax

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Set 𝑇 = 2000 K

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Figure 3: Flow chart of the algorithm developed in this work.


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in the fully continuous space are therefore computationally demanding and typically limited to particle sizes of a few hundred atoms. 28,29 While for small particles below approximately a couple of hundred atoms such approaches are necessary, for larger particles the more constrained structural motifs described above emerge. A full sampling of the continuous configuration is therefore no longer indicated. These considerations motivate the present approach, in which we construct nanoparticles by optimizing the distribution of atoms and vacancies on a predefined lattice, while still allowing for local relaxations of the atoms. The system is initialized by setting up a large spherical FCC crystal, icosahedron or decahedron, which is first filled with atoms and relaxed until all forces are less than 10−6 eV/Å, and then occupied by vacancies save for a small seed of atoms in the center. Different particle sizes are obtained by varying the number of sites occupied by atoms (and conversely vacancies). While the present setup thus models nanoparticles in contact with vacuum, we note that the method can in principle be generalized to describe different chemical environments. The binary vacancy–atom system is sampled using Monte Carlo (MC) simulations in the variance-constrained semi-grand-canonical (VCSGC) ensemble 30 as implemented 43 in the molecular dynamics code lammps. 44 The VCSGC ensemble represents a key component in the present approach, as it enables us to access any particle size (unlike the semi-grandcanonical ensemble) while maintaining a tolerable acceptance rate (unlike the canonical ensemble). When sampling the VCSGC ensemble, trial steps in the MC algorithm consist of changing one atom into a vacancy or vice versa, and the probability for accepting a swap is calculated using the following Metropolis criterion,

P = min {1, exp (−∆UVCSGC /kB T )} ,

(2)

where kB is the Boltzmann constant, T is the temperature, and ∆UVCSGC is the change in


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the thermodynamic potential as defined by ( ) ¯ 2. UVCSGC = U + NS κ ¯ N/NS + ϕ/2

(3)

Here, U is the potential energy, NS is the total number of sites and N is the number of atoms. The parameters ϕ¯ and κ ¯ constrain the average and variance of the number of atoms, respectively. Unlike the semi-grand-canonical ensemble, in which no compositions inside a miscibility gap can be stabilized, we may sample any particle size using a sufficiently large value for κ ¯ (see Ref. 30 for a detailed discussion). In this work we used κ ¯ = 104 and by sweeping ϕ¯ from 0.001 to −2.0, we sampled particle sizes ranging from a few atoms up to more than 10,000 atoms corresponding to the number of sites in the initial spherical region. A considerable increase in the acceptance probability is achieved by considering only trial swaps with vacancies with a non-zero number of non-vacancy neighbors. By excluding vacancies far from the atoms, the computational efficiency does not suffer from a large amount of rejected trial swaps that correspond to inserting atoms in the vacuum. We note that by increasing the size of the initial sphere, this approach can also be straightforwardly extended to much larger particles with, e.g., several ten thousand atoms. The potential energy U was calculated using embedded atom method potentials developed by Mishin et al. 45 for Cu, Williams et al. 46 for Ag, and Marchal et al. 47 for Au and Pd. The energetics of the particle shapes are most sensitive to the surface and twin boundary energies. 20,37 Hence the potentials considered here were selected to provide a good description of these quantities, comparable to the level of density functional theory calculations based on the generalized gradient approximation. To further speed up the sampling of low energy structures, the temperature was varied in a quenching procedure, in which T was successively lowered from 2000 to 100 K in steps of 50 K. To reach the minimum energy structures, atomic relaxations were carried out in regular intervals using conjugate-gradient minimization in order to allow for local displacements away


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from ideal sites, e.g., near corners, edges, and the top surface layers. The atomic relaxations were deemed converged when the forces on all atoms were less than 10−6 eV/Å, after which energy and atomic positions of the final relaxed structure were recorded. Since, especially for smaller particles, relaxation can lead to surface reconstructions, the system was restored to its state prior to relaxation before returning to the MC stage. The algorithm is summarized in Fig. 3. Approximate particle diameter (nm) 2.5

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Figure 4: Energy landscape for Ag in the small particle regime as calculated with our method. The energy is measured relative to a third degree polynomial fit of low-energy symmetric particles from the respective motifs. The particles shown as insets were color-coded for simpler identification of surface facets, edges and corners. (a) TO energy landscape along with some representative snapshots. All RTOs, marked with orange circles, were found by the algorithm. (b) Dh energy landscape along with snapshots viewed from the side and from above. The energy is measured relative to a fit of symmetric Marks decahedra that are outside the range of this figure. (c) Ih energy landscape along with magic number icosahedra (which were all found in the simulations). The region marked by the rectangle is magnified in (d), where local minima are marked with orange numbers, signifying how many {111} faces are capped. Snapshots corresponding to the first eight minima are shown as insets, with the capping atoms colored in yellow and red. 10 ACS Paragon Plus Environment

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Figure 5: (a) Relative stability of the three different motifs for Ag, Au, Pd and Cu. Orange, white and blue stripes indicate that for the given size Ih, Dh or TO motifs are, respectively, energetically the most stable. The bars in the upper part of each subpanel indicates the regions of stability that would be predicted from fitting magic number particles to Eq. (1). Note that some transitions fall outside of the region shown here. (b) Fraction of the different shapes assuming Boltzmann distributions at 300 K and particle sizes with a dispersity given by a normal distribution in the number of atoms with a standard deviation of σ = 0.05N0 , where N0 is the average number of atoms. For both (a) and (b), the intervals are split into 40–1,000 atoms to the left and 1,000–10,000 to the right for clarity of visualization. Approximate diameters in nm are indicated by labels at the top of each subpanel. We start by considering results for Ag particles in the size regime up to about 1300 atoms (Fig. 4). Firstly we note that for all motifs our algorithm succeeds in finding the known energy minima in this size regime, i.e., RTOs for TO based structures as well as magic number clusters for Dh and Ih. For TO derived structures, while the RTOs are among the lowest energy structures, the energy landscape quickly flattens out with increasing size and the energy difference between intermediate particles and RTOs is usually well below 1 meV/atom, effectively extinguishing the magic of magic numbers. The same applies for the Dh landscape (Fig. 4b), which is almost as flat as in the TO case, at least for particles with ≳ 700 atoms. Here, one can observe the formation of particles with convex and inverted surfaces (so-called Marks decahedra), and it is apparent how the inverted edges are repeatedly filled and depleted as the particle grows (see insets in Fig. 4b). In contrast, in the case of Ih particles (Fig. 4c), there is a very clear difference between magic-number particles and intermediate ones. This is apparent by the complete icosahedra


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being separated from each other by large “bumps” in the energy landscape (akin to the behavior of the mixing energy in an immiscible binary alloy). These results suggest that the growth of Ih particles is associated with high energy intermediate structures, whereas the landscape for TO or Dh particles is much smoother. A closer look at the transition from one complete icosahedron to another reveals the origin of the coarse energy landscape (Fig. 4d). In adding the new layer of atoms that is needed to complete the larger particle, the twenty {111} faces of the icosahedron are essentially capped one by one (visualized by the insets in Fig. 4d, where yellow and red atoms belong to the new layer). The partial layers introduce steps on the surface, increasing the energy significantly. Each completely capped face constitutes a local energy minimum, in particular when the capped faces can share many edges, such as when five faces are capped. TO and Dh, in contrast, offer more possibilities for building larger particles without the need for heavily stepped surfaces. Indeed, by capping, say, just one {100} face of a TO particle, the new shape will have the same number of edges and corners as before, thus keeping the energy low. It will, however, loose the octahedral or five-fold symmetry of magic-number TO or Dh particles. The intermediate low energy particles of our simulations accordingly lack the symmetry of the parent lattice, being slightly elongated or skewed. The observation that the most stable structures between magic numbers are often asymmetrically terminated provides a clue as to how symmetry breaking can arise in the growth of anisotropic nanostructures such as nanorods. A comprehensive understanding of this phenomenon does, however, require taking into account various additional factors, most importantly kinetics. 48 The analysis detailed above for small Ag particles can be readily extended to both larger sizes and other materials (Fig. S1 of the Supporting Information). The general features of Fig. 4 are the same for all metals studied here as the results differ only quantitatively. The results can be compiled into a stability map that shows which structural motif is the most stable for each size between a few hundred up to tens of thousands of atoms (Fig. 5a). If, as commonly done, the stability map is constructed using the interpolated energetics of


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particles with optimal atom counts (indicated by orange circles in Fig. 4), it appears that Ih, Dh and TO motifs dominate for small, intermediate and large sizes, respectively, and are separated by sharp transitions (as visualized by the upper bars in the panels of Fig. 5a). By contrast, the comprehensive sampling of the size axis achieved here, reveals a more complex picture and a much more gradual transition between the structural motifs. Most notably the bumpy icosahedral energy landscape limits the stability of Ih clusters to small size intervals close to the magic numbers (orange regions in Fig. 5a). In fact, Ih particles have in some studies been surprisingly rare 39,49 and while kinetic effects may be the most important reason for that, one should be aware that the experimental size distribution typically spans a wider size range than the range of stable icosahedra. Such particle ensembles thus contain many sizes where Ih is not the most stable motif, even if the average size corresponds to the magic number icosahedron. The same considerations apply also when comparing Dh and TO, which are competitive in the whole size regime studied here. Under such circumstances, one needs to refrain from using the energy of the average of the size distribution as a measure of the stability for the full particle ensemble. Here, we have instead assumed a particle ensemble with a normal distribution of sizes and a standard deviation relative to the number of atoms, for which the motifs are distributed according to Boltzmann statistics at 300 K (see the Supporting Information for more details). The thus obtained particle type distributions (Fig. 5b) clearly demonstrate that one has to expect more than just one structural motif present at practically any size. This has been indicated also by experimental observations. For example, Wang and Palmer 50 observed icosahedral Au particles with 923 ± 23 atoms to transform into roughly 30% TO and 70% Dh under electron irradiation, which is in qualitative as well as quantitative agreement with our predictions. A simple fit to magic numbers, on the other hand, would predict Dh particles to be the only stable ones, as the crossover to TO particles occurs for particles with several thousand atoms. Our results reveal some differences between the materials considered here that are difficult or impossible to capture with simple fits to magic numbers. To begin with, Cu is the material


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that most readily prefers the Ih motif. This is true not only at the magic numbers, but the Ih stability range is also the widest in Cu. Further, if simple fits are considered neither Ag nor Pd are predicted to yield stable TO particles with diameters below 7 nm. Yet our simulations identify many sizes for which TO particles are the most stable. Finally, while one can observe that around 3 nm Au, Pd and Cu have similar fractions of Dh, these fractions decrease with different rates for the different materials, and particularly slowly for Pd, with a high degree of sensitivity to the small energy differences between the motifs. The sensitivity also explains why the different metals, while having similar qualitative behaviors, exhibit quite different shape distributions; small differences in surface energy, twin boundary energy, strain energy or other contributions lead to large differences in the resulting distribution. These observations demonstrate that predictions of a distinct crossover between particle types have to be treated with great caution. Small shifts in energy may shift the crossover size several nanometers, which may explain the great diversity in previous such predictions. Here, the present approach provides a powerful and generalizable avenue to obtain much more comprehensive thermodynamic information. It is worth mentioning that cuboctahedra (Fig. 1c) were never encountered in our simulations. In fact, due to the large {100} faces, the energy of perfectly symmetric cuboctahedra is several meV/atom higher than the slightly less symmetric TO particles found in our simulations (Fig. S2 of the Supporting Information). Comparing cuboctahedral to icosahedral particles is therefore highly unfair, even if they have the same number of atoms. Finally, it should be noted that our results are strictly valid at zero temperature. The different motifs have different vibrational properties, and temperature may as a consequence stabilize some motif relative to the others. 42,51,52 We also note that our simulations correspond to particles in vacuum. The chemical environment is bound to affect particle energetics and thereby size distributions, and should be considered in future work. We furthermore note that our method may readily study the coupling between shape and composition in nanoalloys 53,54 by generalizing the algorithm from a binary to a ternary system, consisting


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of two elemental species and vacancies. In conclusion, we have explored the energy landscape of TO, Ih and Dh nanoparticles up to 10,000 atoms (6 or 7 nm) by efficient sampling in the variance constrained semi-grandcanonical ensemble. We have demonstrated that low energy TO and Dh particles close to the convex hull can be found not only for magic numbers but for any number of atoms if the particles contain at least a couple of hundred atoms. We have also shown that a symmetrical truncation may lead to substantially higher energies than an asymmetrical one if the truncation is too deep, as in the case of cuboctahedra. For Ih particles, successive filling of shells leads to a coarser energy landscape than for TO and Dh, and as a result magic-number particles are more pronounced. Furthermore, one can expect the transition path from Ih to TO or Dh to be rugged and steep, providing a rationale for the remarkable kinetic stability of this particle shape. 9 The present results have implications for both theory/modeling and experiment. Firstly, when comparing the stability of different nanoparticle shapes, great care must be taken in choosing shapes. In particular, asymmetric particles should be considered, and one should be careful when interpolating energies between sparsely distributed sizes since the energy landscape in between can exhibit vastly different characteristics for different motifs. With regard to experiments, one need not always invoke kinetics to rationalize the observation of a poly-disperse ensemble of particle shapes. Since the different motifs typically have very similar energies with multiple energy crossovers, they can all be present in thermodynamic equilibrium in sizable proportions. Contrary to what is sometimes encountered in the literature, one would in such cases need kinetics as an explanation of a mono-disperse ensemble rather than a poly-disperse one. Thus we believe that our results help interpreting the outcome of current and future particle synthesis methods and we hope that our work provides guidance when further pushing the boundaries of nanoparticle shape control.


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Supporting Information Available Methodological details and further visualization and description of results (PDF).

Acknowledgement This work was funded by the Knut and Alice Wallenberg Foundation (KAW), the Swedish Foundation for Strategic Research (SSF) and the Swedish Research Council (VR). Computer time allocations by the Swedish National Infrastructure for Computing at NSC (Linköping) and PDC (Stockholm) are gratefully acknowledged.

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Beyond Magic Numbers: Atomic Scale Equilibrium Nanoparticle

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