Shape control of size-selected naked platinum nanocrystals

Controlled growth of far-from-equilibrium-shaped nanoparticles with size selection is essential for the exploration of their unique physical and chemical properties. Shape control by wet-chemistry preparation methods produces surfactant-covered surfaces with limited understanding due to the complexity of the processes involved. Here, we report the controlled production and transformation of octahedra to tetrahedra of size-selected platinum nanocrystals with clean surfaces in an inert gas environment. Molecular dynamics simulations of the growth reveal the key symmetry-breaking atomic mechanism for this autocatalytic shape transformation, confirming the experimental conditions required. In-situ heating experiments demonstrate the relative stability of both octahedral and tetrahedral Pt nanocrystals at least up to 700 °C and that the extended surface diffusion at higher temperature transforms the nanocrystals into equilibrium shape.

Snapshots from a growth simulation at T=700 K and deposition rate of 1 atom/ns Supplementary Figure 11 Snapshots from growth simulations at T=900 K and deposition rate of 1 atom/ns.

SD-1 Excess Energy of Different Structural Motifs
In Supplementary Figure 1, we report the excess energy for regular truncated octahedra, octahedra, truncated tetrahedra and tetrahedra of sizes up to 5000 atoms. Different degrees of symmetric truncation of the tetrahedron are considered. The results in Supplementary Figure 1 clearly show that regular truncated octahedra are the most energetically stable structures, followed by octahedra, truncated tetrahedra and tetrahedra. Energies are calculated by means of the Gupta potential as specified in the Methods section.
To further confirm the out-of-equilibrium character of the tetrahedral crystalline nanoparticles we compare some perfect complete and truncated tetrahedra to truncated octahedra of the same size (see Supplementary Table 1). The sizes considered are magic sizes for the tetrahedra or truncated tetrahedra but not for the truncated octahedra, which therefore present some defects (rectangular basis, asymmetric truncations, excess atoms accommodated on small islands on the surface). However truncated octahedra are by far more energetically favourable in all cases, as shown by the negative values of the energy difference ΔE and the corresponding different excess energies.

Supplementary Figure 1 | Excess energy of nanoparticles with different structural
motifs and schematic representation of the growth sequence. Excess energy Eexc is shown as a function of number of atoms (N) of nanoparticles, for regular truncated octahedra (squares), octahedra (diamonds), truncated tetrahedra and tetrahedra (triangles) respectively. Lesser and lesser truncations give tetrahedral structures of higher energy. truncated octahedra of the same atomic size (N). We show the energy difference ΔE between truncated octahedra and tetrahedra/truncated tetrahedra and the excess energy Eexc of the two structures under comparison, for a range of atomic sizes. We consider three complete perfect tetrahedra (first three rows of the table) and two pairs of truncated tetrahedra with different degrees of truncation (taken from the green and cyan curves of the graph in Supplementary  Figure 1, respectively). ΔE and Eexc are in eV. The same data in (b) replotted as a function of the inverse of the He flow rate in the log scale, demonstrating a single relationship. The error bar in the y-axis is from the mass resolution of the mass filter. The blue and orange arrows point to the contrasting growth conditions under which cluster beams showing significant shape transformation from octahedra to tetrahedra. The results demonstrate not only that the shape can be tuned but also the transformation occurs for the nanoparticle beams of similar current but increasing atomic size, suggesting an adatom condensation mechanism.

SD-3 Procedures of Capturing Images for Shape Identification and Their Uncertainty
For shape identification, one small region on the grid is selected randomly and imaged by STEM at low magnification of 600k, for example, as shown in Supplementary Figure 3 (a1) and (b1) for Pt5000 and Pt20000. The area is then imaged in sequence with higher magnifications of 2M to include every Pt particle within the area for visual identification, as shown in Supplementary Figure 3(a2), 3(a3) and 3(b2). For those particles cannot be recognised easily by eyes, images with atomic-scale details are taken at even higher magnifications, e.g.
10M in Supplementary Figure 3(b3) and analysed as described in the main text. About 100 Pt nanoparticles are imaged at 10M for Pt5000 and Pt20000, respectively. All imaged nanoparticles are classified as three groups: tetrahedra, octahedra, and other structures, and their populations are shown in Fig. 2e.
For assessing the uncertainty of the structure grouping, three regions (R1-R3) in different areas are chosen to repeat the above procedures. Supplementary Table 2 displays the number of Pt nanoparticles with different shapes identified in each area. The variation in the percentage counts of the same shape in the different regions is used to estimate the counting uncertainty.

SD-4 Tetrahedron and Octahedron Image Libraries
The deposited Pt nanoparticles with the same size and shape can display different projected images. To help with identification of the experimental HAADF-STEM images, we generate structural libraries for tetrahedron and octahedron, showing simulated STEM images of the same models but rotated systematically along the xand y-axis, as shown in Supplementary Figure

Supplementary Figure 6 | Representative HAADF-STEM images of experimental Pt
nanoparticles with 'other structures' as described Fig.2. The images at the upper row are Pt5000 nanoparticles and at the bottom row are Pt20000 nanoparticles. (a1) a regular five-fold decahedron; (a2) a particle with a twin structure; (a3), (b2) and (b3) particles with flat hexagonal shape and triangular shape; (a4) and (b4) elongated particles; (b1) a rounded shape particle with fcc structure. These "other structures or shapes" occupy low populations, compared with octahedra and tetrahedra. Few structures with elongated or hexagonal shape are also grown in the molecular-dynamics simulations.

SD-6 Molecular Dynamics Simulations for Different Sizes and Temperatures
The final structures observed in the growth MD simulations are reported in Supplementary Table 3 In Supplementary Table 3 we consider seven different growth temperatures (300, 400, 500, 600, 700, 800 and 900 K) and, for each temperature, two deposition rates (1 atom/ns and 0.1 atom/ns) are considered. For all pairs (temperature, deposition rate), five independent simulations are made, for a total of 70 independent simulations.
From the results in Supplementary Table 3, it turns out that, depending on simulation temperature and deposition rate, different final outcomes of the growth process are observed. At the lowest temperature, irregular structures with rough surface are observed at the end of the simulations. In the temperature range 400-600 K, tetrahedra become the most frequently observed motif, with more and more regular surfaces as temperature increases. From 700 K onwards, growth takes place mostly within the octahedral motif, with truncations becoming more and more present at the highest temperatures. A few twin structures are also grown, in agreement with the experimental observations. At 600 K, a hexagonal structure, of the type shown in the experimental images of Supplementary Figure 6(a3) and 6(b2), is grown. The only octahedral structure grown at 500 K is indeed quite elongated, with a shape resembling that of Supplementary Figure 6(a4) and 6(b4).
The effect of slowing down the deposition rate is to produce structures that are somewhat closer to the equilibrium. This effect is rather clear at low temperatures, while at high temperatures the results seem independent of the growth rate (at least in this interval of rates).
We also made simulations for larger sizes, beginning with a truncated octahedron of 586 atoms and depositing on top of it at a rate of 1 atom/ns to reach size 3000. We made 5 independent simulations at 600 K and at 700 K. At 600 K, we obtained (as final results at 3000 atoms) 4 tetrahedra and 1 hexagonal structure, whereas at 700 K we obtained 4 tetrahedra and 1 octahedron. Snapshots from a sequence at 700 K are given in Supplementary Figure 10.
In Supplementary Figure 11 we show snapshots of simulations of the growth at 900 K. In Supplementary Figure 11a we show the sequence truncated octahedron → octahedron → tetrahedron starting from about 1300 atoms and ending to about 10000 atoms. In Supplementary Figure 11b we show a sequence starting with a perfect octahedron of about 4600 atoms, then evolving to a deeply truncated tetrahedron for sizes around 10000 atoms and finally growing to a sharper tetrahedron close to 15000 atoms. We note that these sequences perfectly match the experimental sizes.
In summary, these results show that, as expected, temperature increase and slowing down of deposition rate decrease the importance of kinetic trapping phenomena at a given size, thus allowing the production of growth shapes that are closer and closer to equilibrium. On the other hand, temperature increases and slower deposition rates shift the out-of-equilibrium phenomena leading to the growth of tetrahedra to larger sizes.
Supplementary Table 3: Final structures of molecular dynamics simulations at different temperatures from 300 to 900 K and for two deposition rates (1 and 0.1 atom/ns). Simulations start from a truncated octahedral structure of 201 atoms and are stopped when the nanoparticle size reaches 1001 atoms. As regards twins and hexagons, the only structure with hexagonal shape (with a shape very similar to that of the experimental structure of Supplementary Figure 6(a3) and 6(b2) is the one grown at 600 K, whereas the structures grown at 800 and 900 K are triangular twins. For each size, the same structure is shown from two different perspectives. Atoms colored in white correspond to the starting truncated octahedron of 201 atoms, while blue atoms are those completing the octahedral tips on top of it, to form an octahedron of 231 atoms. Atoms in yellow, green, orange and cyan correspond to the tetrahedral tips which would grow on that octahedron of 231 atoms, as in the case of the sequence at 400 K of Fig. 3 of the main text. Light grey atoms correspond to further layers in fcc stacking, whereas dark grey atoms are those of islands in hcp stacking. At variance of the sequence at 400 K, growth is still within the octahedral motif up to about a size of (a) 470 atoms, where an island in hcp stacking is formed (atoms in dark grey). (b-d) This island triggers the fast growth of the three tetrahedral tips over four (those in yellow, green and cyan) that have borders in common with it, while the orange tip, which has no faulted islands at its border, does not grow. (e-f) A second faulted island is formed at the border of the orange tip, at its right, and the orange tip starts growing one layer. (g-h) This second faulted island reverts back to fcc stacking, and the growth of the orange tip stops again, no further layers nucleating beyond the second. The final structure is of tetrahedral type, but with three sharp tips and one truncated tip. Note that another small faulted island on the left side of the lower image in (f-g), reverts back to fcc stacking (h) producing a tetrahedron with one more fcc plane. . Blue horizontal lines on the graph point out the number of atoms at which the different layers of the tips are completed, whereas red vertical lines mark the steps of the growth at which islands in hcp stacking are formed. Before the formation of the first faulted island, the green and yellow tips have completed two layers whereas the cyan and orange tips have completed only one layer. After the formation of the island, the green, yellow and cyan tips grow quickly until they complete the last layer. The growth of the orange tip, which is the only one not connected to the first faulted island, is much slower: the tip is blocked at the first layer until a second faulted island is formed at its border.

SD-9. Growth
Supplementary Figure 10 | Snapshots from a growth simulation at T=700 K and deposition rate of 1 atom/ns. The simulation is started from a truncated octahedron of 586 atoms, which grows to an octahedron and then to a tetrahedron of 2900 atoms.