Liquid-Liquid Phase Coexistence in Gold Clusters: 2D or Not 2D?

The thermodynamics of gold cluster anions (${\mathrm{Au}}_N^{-}$, $N=11,\dots ,14$) is investigated using quantum molecular dynamics. Our simulations suggest that ${\mathrm{Au}}_N^{-}$ may exhibit a novel, freestanding planar liquid phase which dynamically coexists with a normal three-dimensional liquid. Upon cooling with experimentally realizable cooling rates, the entropy-favored three-dimensional liquid clusters often supercool and solidify into the “wrong” dimensionality. This indicates that experimental validation of theoretically predicted ${\mathrm{Au}}_N^{-}$ ground states might be more complicated than hitherto expected.

Figure 1

Dynamical coexistence of 2D and 3D liquid clusters. Time evolution of planarity parameter $D_{\mathrm{rms}}$, for (a) ${\mathrm{Au}}_{11}^{-}$ and (b) ${\mathrm{Au}}_{14}^{-}$. The insets in (a) are representative snapshots of the ${\mathrm{Au}}_{11}^{-}$ geometry in different phases. (c) Time evolution of the potential energy $E_{\mathrm{pot}}$ for ${\mathrm{Au}}_{14}^{-}$. Inset: the corresponding histogram of $E_{\mathrm{pot}}$ short time averages (same energy scale as main panel). (d) rms bond length fluctuation $\delta$ for ${\mathrm{Au}}_{14}^{-}$. Insets: 300 ps atomic trajectories at high- and low-$E_{\mathrm{pot}}$ regimes. For both ${\mathrm{Au}}_{11}^{-}$ and ${\mathrm{Au}}_{14}^{-}$ data were taken from 3.5 ns microcanonical DFTB simulations corresponding to average temperatures of $T\sim 750\mathrm{K}$, where panels (b)–(d) correspond to the same simulation.

Figure 2

Characteristic barriers of ${\mathrm{Au}}_{14}^{-}$. A portion of the DFTB potential energy landscape at $T=0$, showing a few 2D and 3D local minima (marked by $m$), appearing also at high-temperature DFTB molecular dynamics runs. The very low $2\mathrm{D}/2\mathrm{D}$ barrier ($m^{\prime }/m^{\prime \prime }$) represents a partial rotation of the 2D ground state, where the inner square rotates inside the 10-atom ring.

Figure 3

Cooling simulations. (a) Planarity parameter $D_{\mathrm{rms}}$ versus cooling time (top axis, counted backwards) and total energy (bottom axis). Runs were started at $E=3$, 3.3, 3.5, and 3.8 eV above the ground states of $N=11$, 12, 13, and 14, respectively, corresponding roughly to $T=1000\mathrm{K}$. (b) rms bond length fluctuation $\delta$ for the $N=14$ run. In the dynamical 2D-liquid–3D-liquid coexistence region the fluctuation is well above the conventional criterion $\delta >0.1$ for liquid clusters. The cluster enters the 3D-liquid–3D-solid coexistence region near $E_{\mathrm{tot}}=1.1\mathrm{eV}$ and becomes 3D solid around $E_{\mathrm{tot}}=0.77\mathrm{eV}$. (c) Caloric curve for $N=14$ (averaged over five simulations and separated into 2D and 3D phases using $D_{\mathrm{rms}}$). The planar clusters form the hot, low-potential energy phase. The 3D liquid phase has the heat capacity of $c_v=4.05k_B$ and the 3D solid phase $c_v=3.06k_B$ per atom. A similar decrease of the heat capacity is obtained for the 2D solid-liquid transition. (d) The partial entropies for 2D and 3D phases versus potential energy. The entropy curves (from multiple histogram calculations [23, 24]) crossover around $E_{\mathrm{tot}}=1.5\mathrm{eV}$.