Effects of epitaxial strain on the melting of supported nickel nanoparticles

We use molecular dynamics to investigate the effects of substrate-induced epitaxial strain on the melting temperature and equilibrium structure of supported metal nanoparticles. Our model system comprises Ni clusters supported on strained graphene. The clusters are modeled using an embedded atom potential, and the nickel-carbon interactions are described by a Lennard-Jones field with one parameter varied to control the substrate binding strength. We find that, after adjusting for curvature effects due to the clusters’ free surface, the melting temperature of supported Ni clusters can shift by hundreds of degrees depending on the cluster-substrate epitaxial relationship. The order of magnitude of this effect is shown to be consistent with prior predictions based on thermodynamic modelling. We also find that sufficiently strong substrate binding leads to a solid-solid transition from icosahedral to lamellar-twinned fcc particles, which occurs via a melt-freeze process. These results illustrate how substrate-induced epitaxial strain can be used to control the phase of metal nanoparticles.

Figure 1

When a particle of fixed volume is placed on a substrate, the resultant contact angle and curvature (of the free surface) will, in general, depend on the phase of the particle.

Figure 2

An illustration of how $\varepsilon$ affects the geometry of molten Ni clusters on graphene. We extracted (time-averaged) $w$ and $h$ from our simulations and then computed the contact angle (${\theta }_l$) and the curvature radius ($R_l$) using the geometry of spherical caps: $R_l=\tilde{w}\equiv w/2$ if $\tilde{w}\ge h$ or $R_l=h(1+{\tilde{w}}^2{h}^{-2})/2$ if $\tilde{w}<h$; $\cos {\theta }_l=1-h{R}_l^{-1}$. The plot shows that $\cos {\theta }_l\propto \varepsilon$ for all Ni${}_N$ clusters considered here, and the trend is independent of cluster size $N$.

Figure 3

The internal energy $E$ and the time-averaged number of hcp ($N$${}_{\mathrm{hcp}}$), fcc ($N$${}_{\mathrm{fcc}}$), and ico ($N$${}_{\mathrm{ico}}$) atoms are plotted versus temperature for Ni${}_{1415}$ on graphene with different $\varepsilon$ values. Solid-solid transitions lead to a drop in $\langle$$N$${}_{\text{ico}}\rangle$ correlating with an increase in $\langle$$N$${}_{\mathrm{fcc}}\rangle$ or $\langle$$N$${}_{\mathrm{hcp}}\rangle$, and they occur near the melting temperature $T_m$ of unsupported Ni${}_{1415}$.

Figure 4

Snapshots of graphene-supported Ni${}_{1415}$ ($\varepsilon =0.03$ eV) illustrating the icosahedral fivefold symmetry axes (ico atoms are large red spheres) vanishing between 1360 and 1370 K. The substrate is depicted in gray with the constituent atoms not shown.

Figure 5

Time evolution of the potential energy $E$${}_{\mathrm{pot}}$ of Ni${}_{1415}$ on graphene with $\varepsilon =0.03$ eV at different temperatures. The spike in $E$${}_{\mathrm{pot}}$ at 1370 K corresponds to the icosahedral structure melting almost entirely and then recrystallizing into an fcc and hcp arrangement. The sequence of cluster snapshots illustrate this process, with larger golden spheres representing the solid Ni atoms and smaller blue dots representing the liquid Ni atoms. Substrate atoms not shown.

Figure 6

A sample from the simulated data that was used to characterize the melting transition in supported Ni${}_{923}$ for different substrate binding strengths ($\varepsilon$). A steep rise in the equilibrium liquid fraction $\langle {\phi }_L\rangle$ correlates with the absorption of latent heat in the corresponding caloric curves. Each melting temperature ($T_m$) is defined by the point when $\langle {\phi }_L\rangle =0.9$.

Figure 7

The melting temperature of graphene-supported Ni clusters versus the curvature ($R_l^{-1}$) of the free surface. Unsupported closed-shell icosahedra and decahedra—the lowest-energy structures for this size range in the absence of a substrate—are also included for comparison. All four panels show exactly the same data for supported clusters, but sorted differently. In the top panels the data are divided into subsets with the same number of constituent atoms ($N$), and the trends illustrate how $T_m$ changes when $R_l$ is altered through variation in $\varepsilon$. In the bottom panels the data sets are sorted by the specified $\varepsilon$, demonstrating changes in $T_m$ when $R_l$ is altered through variation in $N$ ($\varepsilon$ constant). Panels on the right show that in both cases the subsets roughly follow $T_m\propto R_l^{-1}$. Dashed curves show Eq. (1), with both $T_m$ and $\mathcal{C}$ fitted, and serve merely as a guide to the eye.

Figure 8

(a) Melting temperatures ($T_m$) of supported Ni${}_{923}$ clusters as a function of curvature radius of the liquid phase ($R_l$). In the approximately strain-free data (red circles), the graphene lattice spacing was fixed while $\varepsilon$ was varied. For the specified $\varepsilon$ values, namely, $\varepsilon =0.03$ and 0.04 eV, the data points were obtained by straining the substrate (i.e., changing $\delta$ from about $-0.3$ to $0.2$). $T_m$ values for unsupported icosahedral (ico) and decahedral (dec) clusters of different sizes are included for comparison. Dashed curves correspond to Eq. (1) with both $T_c$ and $\mathcal{C}$ fitted. (b) Strain-induced shifts in melting temperature ($\Delta T_m$) of Ni${}_{923}$ as a function of specified substrate strain $\delta$. Each $\Delta T_m$ was calculated by taking $T_m$ of Ni${}_{923}$ on substrates with specified $\delta$ and then subtracting the value of the strain-free $T_m\left(R_l\right)$ trend at the corresponding $R_l$. Dashed curves are quadratic fits and serve as a guide to the eye. Note that filled symbols correspond to clusters that remained icosahedral until the melting transition, while unfilled symbols correspond to clusters that were in the lamellar-twinned fcc phase.

Figure 9

The radius of curvature ($R$) and contact angle ($\theta$) are shown to increase ($\Delta \theta \sim {15}^{\circ }$) as the liquid fraction grows during the melting of Ni${}_{923}$ on graphene (with minimal epitaxial strain). Evolution of the instantaneous liquid fraction (${\phi }_L$) indicates a broad temperature range over which large liquid fractions remain stable (for 0.6 ns). Snapshots illustrate the atomic structure of Ni${}_{923}$ with $\varepsilon =0.04$ eV at different temperatures, with each atom classified as either “solid” (gold spheres) or “liquid” (blue dots) using the Steinhardt ${\overline{q}}_6$ parameter (see Sec. 2b).