Figure 1

Ball-and-stick representation of C30 on spherical (Lennard-Jones) catalysts. Yellow outer spheres represent the catalyst radius $R$, and orange inner spheres correspond to $R-2^{1/6}\sigma$ where there is infinite repulsion.Reuse & Permissions

Figure 2

Variation in cohesive ($E_T$) and adhesive ($E_{\mathrm{LJ}}$) energies of C30 as $R$ increases from 3.2 Å to 7.2 Å in increments of 0.01 Å. Note that in (a) $E_{\mathrm{LJ}}$ is plotted versus $R$ for ${\epsilon }_s=1$ at three different temperatures; and in (b) $E_T$ is plotted versus $R$ at 1000 K for three different values of ${\epsilon }_s$. All the $E_T(R)$ and $E_{\mathrm{LJ}}(R)$ values were equilibrated and averaged over ${10}^5$ time steps (0.5 fs each). We find the trends to be reversible—subsequent gradual shrinking of $R$ yields the same behavior. Adjusting the temperature has some effect on the general shape of the trends, with the lift-off transition being sharper at lower temperatures, but the location of $r_e$ and $R_{\mathrm{crit}}$ remains largely unchanged. $E_T(R)$ and $E_{\mathrm{LJ}}(R)$ remain roughly constant for $R>R_{+}$, where $R_{+}$ is typically between $R_{\mathrm{crit}}$ and $R_{\mathrm{crit}}+1\AA$. In the inset we plot the maximal strain energy, i.e. $\Delta E_T=E_T(R_{\mathrm{crit}})-E_T(r_e)$, versus the work of adhesion that is overcome during lift-off, i.e., $\Delta E_{\mathrm{LJ}}=E_{\mathrm{LJ}}(R_{+})-E_{\mathrm{LJ}}(R_{\mathrm{crit}})$. We find $\Delta E_{\mathrm{LJ}}\approx \Delta E_T$ for a wide range of temperatures, catalyst binding strengths, and different cap structures (i.e., $\mathrm{C}30+$, C39 and C110) defined later in the text.Reuse & Permissions

Figure 3

$E_T$ versus $R$ for C110 and C39 at 1000 K. Arrows indicate transitions to complete or partial lift-off.Reuse & Permissions

Figure 4

$E_T$ versus $R$ for a defective C30 cap. Simulations at 1000 K do not exhibit a consistent trend, whereas those at 500 K do and it is possible to identify the lift-off transition.Reuse & Permissions

Figure 5

Schematic of our continuum model for cap lift-off. The catalyst is a rigid sphere of radius $R$, and the cap is a thin, spherical membrane of area $A$ and adjustable radius $r$.Reuse & Permissions

Figure 6

The ratio $R_{\mathrm{crit}}/r_e$ is shown to increase with the specified work of adhesion ($\Delta E_{\mathrm{LJ}}$), and the trends are fitted with Eq. (2). From our simulations of a complete C60 fullerene at 1000 K, we estimate the occupied area per atom $\Omega \sim 3{\AA }^2/\mathrm{atom}$, which yields $\kappa \sim 2.5\mathrm{eV}$ for C30. Adding an extra ring of 10 carbon atoms (see $\mathrm{C}30+$) shifts the trend and raises the rigidity to $\kappa \sim 19\mathrm{eV}$. C39, which has a zigzag edge, is even more rigid with $\kappa \sim 37\mathrm{eV}$.Reuse & Permissions