Size-Dependent Role of Surfaces in the Deformation of Platinum Nanoparticles

The mechanical behavior of nanostructures is known to transition from a Hall-Petch-like “smaller-is-stronger” trend, explained by dislocation starvation, to an inverse Hall-Petch “smaller-is-weaker” trend, typically attributed to the effect of surface diffusion. Yet recent work on platinum nanowires demonstrated the persistence of the smaller-is-stronger behavior down to few-nanometer diameters. Here, we used in situ nanomechanical testing inside of a transmission electron microscope (TEM) to study the strength and deformation mechanisms of platinum nanoparticles, revealing the prominent and size-dependent role of surfaces. For larger particles with diameters from 41 nm down to approximately 9 nm, deformation was predominantly displacive yet still showed the smaller-is-weaker trend, suggesting a key role of surface curvature on dislocation nucleation. For particles below 9 nm, the weakening saturated to a constant value and particles deformed homogeneously, with shape recovery after load removal. Our high-resolution TEM videos revealed the role of surface atom migration in shape change during and after loading. During compression, the deformation was accommodated by atomic motion from lower-energy facets to higher-energy facets, which may indicate that it was governed by a confined-geometry equilibration; when the compression was removed, atom migration was reversed, and the original stress-free equilibrium shape was recovered.

Section S1: Supporting videos S1 -S5 Still frames of the supporting videos showing the size dependent deformation mechanisms and the role of surface in representative nanoparticles.    Section S3: Homogeneous deformation and shape recovery of a representative nanoparticle below 9 nm One of the nanoparticles (identified as 7.5 nm in the main text, Fig. 1) was compressed many times in order to understand the repeatability of shape recovery. While only one test was quantitatively analyzed for stress and strain, this figure shows a progressive array of still images representing multiple compressions. Seven separate compression tests were performed in the same video, and for each test there is a frame before testing, at maximum compression, and after testing. This set of figures corresponds to supporting Video S3.

Section S4: Flow stress
Some particles exhibited a clear yield plateau (main text, Fig. 1) and therefore allowed the calculation of the average stress during yielding/ While other tests had to be stopped prior to yielding due to limitations of the experimental setup. Consequently, while the stress at 10% strain (main text, Fig. 3) can be computed for all tests, a flow stress cannot be. For those tests of particles greater than 9 nm that DID show a yield plateau, Fig. S8 shows their flow stress.
The flow stress was computed as the average stress in the plateau and it is shown as a function of particle diameter. The trends are consistent with those of the stress at 10% strain Figure S8: Average flow stress for particles with a clear yield plateau. The strength of particles decreases monotonically down to a critical size of 9 nm, where the deformation changes from inhomogeneous to homogeneous.

Section S5: Melting temperature vs size in nanoparticles and nanowires
The semi-empirical thermodynamic model by Kim and Lee 1 was used to estimate the predicted variation in melting temperature ∆T m with size in Pt nanoparticles and Pt nanowires, shown in equations 1 and 2 respectively.
where γ is the surface energy, V s is the room temperature solid molar volume, ∆H m is the latent heat of melting, T m is the melting point of the bulk material, r e is the first nearest neighbor distance for crystalline materials, and r is the particle radius or the cylinder radius in nanowires.
The parameters for Pt are given in Table S1.

Section S6: Calculation of diffusion rate in a small Pt nanoparticle
According to the classical curvature-driven shape equilibration theory 2 , atoms on high-curvature surfaces possess higher chemical potentials and tend to move toward the lower-curvature surfaces with lower chemical potentials. The diffusion rate or diffusion coefficient (D s ) of atoms in the surface is calculated according to equation 3 2,3 .
Where R is the radius of curvature, h is recession height, τ is the relaxation time, δ s is the surface layer thickness, γ is the surface energy, and ω is the atomic volume.
To understand the driving force for atom migration, we first compute the diffusion coeffi-   Fig. 2f (1.2 nm) to Fig.2k (2.1 nm)). The observed relaxation time (7.75 s) for such shape evolution to avoid overheating. The annealing time is just a minor variable in nanoparticle size control as nanoparticles rapidly evolve to a relatively stable state at high temperature, unless heating time is too long, e.g., > 3 hr and nanoparticles are over-coarsened 5 . Different synthesis recipes provide nanoparticles at a wide range of sizes spanning from 5 nm to > 50 nm (low annealing temperature for small nanoparticles, and high annealing temperature for large nanoparticles), which covers the size range studied in this research.

Section S8: Calculation of stress and strain
The applied load F is calculated using the Hooke's law by measuring the AFM tip displacement ∆x multiplied by the spring constant k of the tip (Fig. S11a-b).
To study the strength of the particles, we calculated the true stress σ by dividing the applied load F by the contact area of the particle and the tip. A circular contact region was presumed, and its radius r was measured at each frame (Fig. S11d).
The true strain ϵ was calculated by measuring the initial height h 0 and instant height h of the particle as shown in panels Fig. S11c-d. ϵ = ln h h 0 (6) Figure S11: The reference point is the pre-contact position of the AFM probe, which is where it sits when it is under zero load. All deflections from that point represent a change in position that indicates a load is applied.