Mind the gap: testing the Rayleigh hypothesis in T-matrix calculations with adjacent spheroids.

The T-matrix framework offers accurate and efficient modelling of electromagnetic scattering by nonspherical particles in a wide variety of applications ranging from nano-optics to atmospheric science. Its analytical setting, in contrast to purely numerical methods, also provides a fertile ground for further theoretical developments. Perhaps the main purported limitation of the method, when extended to systems of multiple particles, is the often-stated requirement that the smallest circumscribed spheres of neighbouring scatterers not overlap. We consider here such a scenario with two adjacent spheroids whose aspect ratio we vary to control the overlap of the smallest circumscribed spheres, and compute far-field cross-sections and near-field intensities using the superposition T-matrix method. The results correctly converge far beyond the no-overlap condition, and although numerical instabilities appear for the most extreme cases of overlap, requiring high multipole orders, convergence can still be obtained by switching to quadruple precision. Local fields converge wherever the Rayleigh hypothesis is valid for each single scatterer and, remarkably, even in parts of the overlap region. Our results are validated against finite-element calculations, and the agreement demonstrates that the superposition T-matrix method is more robust and broadly applicable than generally assumed.

Structure, thermodynamics, and rearrangement mechanisms in gold clusters-insights from the energy landscapes framework.

We consider finite-size and temperature effects on the structure of model AuN clusters (30 ≤ N ≤ 147) bound by the Gupta potential. Equilibrium behaviour is examined in the harmonic superposition approximation, and the size-dependent melting temperature is also bracketed using molecular dynamics simulations. We identify structural transitions between distinctly different morphologies, characterised by various defect features. Reentrant behaviour and trends with respect to cluster size and temperature are discussed in detail. For N = 55, 85, and 147 we visualise the topography of the underlying potential energy landscape using disconnectivity graphs, colour-coded by the cluster morphology; and we use discrete path sampling to characterise the rearrangement mechanisms between competing structures separated by high energy barriers (up to 1 eV). The fastest transition pathways generally involve metastable states with multiple fivefold disclinations and/or a high degree of amorphisation, indicative of melting. For N = 55 we find that reoptimising low-lying minima using density functional theory (DFT) alters their energetic ordering and produces a new putative global minimum at the DFT level; however, the equilibrium structure predicted by the Gupta potential at room temperature is consistent with previous experiments.

Structure and Thermodynamics of Metal Clusters on Atomically Smooth Substrates.

We analyze the structure of model NiN and CuN clusters (N = 55, 147) supported on a variety of atomically smooth van der Waals surfaces. The global minima are mapped in the space of two parameters: (i) the laterally averaged surface stickiness, γ, which controls the macroscopic wetting angle, and (ii) the surface microstructure, which produces more subtle but important templating via epitaxial stresses. We find that adjusting the substrate lattice (even at constant γ) can favor different crystal plane orientations in the cluster, stabilize hexagonal close-packed order, or induce various defects, such as stacking faults, twin boundaries, and five-fold disclinations. Thermodynamic analysis reveals substrate-dependent solid-solid transitions in cluster morphology, with tunable transition temperature and sometimes exhibiting re-entrant behavior. These results shed new light on the extent to which a supporting surface can be used to influence the equilibrium behavior of nanoparticles.

Impurity effects on solid-solid transitions in atomic clusters.

We use the harmonic superposition approach to examine how a single atom substitution affects low-temperature anomalies in the vibrational heat capacity (CV) of model nanoclusters. Each anomaly is linked to competing solidlike "phases", where crossover of the corresponding free energies defines a solid-solid transition temperature (Ts). For selected Lennard-Jones clusters we show that Ts and the corresponding CV peak can be tuned over a wide range by varying the relative atomic size and binding strength of the impurity, but excessive atom-size mismatch can destroy a transition and may produce another. In some tunable cases we find up to two additional CV peaks emerging below Ts, signalling one- or two-step delocalisation of the impurity within the ground-state geometry. Results for Ni74X and Au54X clusters (X = Au, Ag, Al, Cu, Ni, Pd, Pt, Pb), modelled by the many-body Gupta potential, further corroborate the possibility of tuning, engineering, and suppressing finite-system analogues of a solid-solid transition in nanoalloys.

Grand and Semigrand Canonical Basin-Hopping.

We introduce grand and semigrand canonical global optimization approaches using basin-hopping with an acceptance criterion based on the local contribution of each potential energy minimum to the (semi)grand potential. The method is tested using local harmonic vibrational densities of states for atomic clusters as a function of temperature and chemical potential. The predicted global minima switch from dissociated states to clusters for larger values of the chemical potential and lower temperatures, in agreement with the predictions of a model fitted to heat capacity data for selected clusters. Semigrand canonical optimization allows us to identify particularly stable compositions in multicomponent nanoalloys as a function of increasing temperature, whereas the grand canonical potential can produce a useful survey of favorable structures as a byproduct of the global optimization search.

Quasi-combinatorial energy landscapes for nanoalloy structure optimisation.

We formulate nanoalloy structure prediction as a mixed-variable optimisation problem, where the homotops can be associated with an effective, quasi-combinatorial energy landscape in permutation space. We survey this effective landscape for a representative set of binary systems modelled by the Gupta potential. In segregating systems with small lattice mismatch, we find that homotops have a relatively straightforward landscape with few local optima - a scenario well-suited for local (combinatorial) optimisation techniques that scale quadratically with system size. Combining these techniques with multiple local-neighbourhood structures yields a search for multiminima, and we demonstrate that generalised basin-hopping with a metropolis acceptance criterion in the space of multiminima can then be effective for global optimisation of binary and ternary nanoalloys.

Structure prediction for multicomponent materials using biminima.

The potential energy surface of a heteroparticle system will contain points that are local minima in both coordinate space and permutation space for the different species. We introduce the term biminima to describe these special points, and we formulate a deterministic scheme for finding them. Our search algorithm generates a converging sequence of particle-identity swaps, each accompanied by a number of local geometry relaxations. For selected binary atomic clusters of size N = N(A) + N(B) ≤ 98, convergence to a biminimum on average takes 3 N(A)N(B) relaxations, and the number of biminima grows with the preference for mixing. The new framework unifies continuous and combinatorial optimization, providing a powerful tool for structure prediction and rational design of multicomponent materials.

Degenerate Ising model for atomistic simulation of crystal-melt interfaces.

One of the simplest microscopic models for a thermally driven first-order phase transition is an Ising-type lattice system with nearest-neighbour interactions, an external field, and a degeneracy parameter. The underlying lattice and the interaction coupling constant control the anisotropic energy of the phase boundary, the field strength represents the bulk latent heat, and the degeneracy quantifies the difference in communal entropy between the two phases. We simulate the (stochastic) evolution of this minimal model by applying rejection-free canonical and microcanonical Monte Carlo algorithms, and we obtain caloric curves and heat capacity plots for square (2D) and face-centred cubic (3D) lattices with periodic boundary conditions. Since the model admits precise adjustment of bulk latent heat and communal entropy, neither of which affect the interface properties, we are able to tune the crystal nucleation barriers at a fixed degree of undercooling and verify a dimension-dependent scaling expected from classical nucleation theory. We also analyse the equilibrium crystal-melt coexistence in the microcanonical ensemble, where we detect negative heat capacities and find that this phenomenon is more pronounced when the interface is the dominant contributor to the total entropy. The negative branch of the heat capacity appears smooth only when the equilibrium interface-area-to-volume ratio is not constant but varies smoothly with the excitation energy. Finally, we simulate microcanonical crystal nucleation and subsequent relaxation to an equilibrium Wulff shape, demonstrating the model's utility in tracking crystal-melt interfaces at the atomistic level.

Communication: a new paradigm for structure prediction in multicomponent systems.

We analyse the combinatorial aspect of global optimisation for multicomponent systems, which involves searching for the optimal chemical ordering by permuting particles corresponding to different species. The overall composition is presumed fixed, and the geometry is relaxed after each permutation in order to relieve local strain. From ideas used to solve graph partitioning problems we devise a deterministic search scheme that outperforms (by orders of magnitude) conventional and self-guided basin-hopping global optimisation. The search is guided by the energy gain from either swapping particles i and j (ΔEij) or changing the identity of particles i (ΔEi). These quantities are derived from the underlying (arbitrary) energy function, hence not constituting external bias, and for site-separable force fields each ΔEi can be approximated simply and efficiently. In our self-guided variant of basin-hopping, particles are weighted by an approximate ΔEi when randomly selected for an exchange, yielding a significant improvement for segregated multicomponent systems with modest particle size mismatch.

Filling a nanoporous substrate by dewetting of thin films.

Following a simple thermodynamic model, which predicts that an array of non-wettable pores can be filled by dewetting of sufficiently thin films, we use molecular dynamics to simulate the rupture of nanometre-thick liquid Au films on nanoporous substrates. Our simulations clearly exhibit spinodal dewetting and hole nucleation, and some of the metal is indeed absorbed by non-wettable pores solely as a virtue of the Laplace pressure acting on dewetted droplets and rivulet-like structures. Finally, we show that the fraction of absorbed Au can be increased through patterning of the initial film.

Electronic effects on the melting of small gallium clusters.

Motivated by experimental reports of higher-than-bulk melting temperatures in small gallium clusters, we perform first-principles molecular dynamics simulations of Ga(20) and Ga(20)(+) using parallel tempering in the microcanonical ensemble. The respective specific heat (C(V)) curves, obtained using the multiple histogram method, exhibit a broad peak centered at approximately 740 and 610 K--well above the melting temperature of bulk gallium (303 K) and in reasonable agreement with experimental data for Ga(20)(+). Assessment of atomic mobility confirms the transition from solid-like to liquid-like states near the C(V) peak temperature. Parallel tempering molecular dynamics simulations yield low-energy isomers that are ~0.1 eV lower in energy than previously reported ground state structures, indicative of an energy landscape with multiple, competing low-energy morphologies. Electronic structure analysis shows no evidence of covalent bonding, yet both the neutral and charged clusters exhibit greater-than-bulk melting temperatures.

Electronic shell structure in Ga12 icosahedra and the relation to the bulk forms of gallium.

The electronic structure of known cluster compounds with a cage-like icosahedral Ga(12) centre is studied by first-principles theoretical methods, based on density functional theory. We consider these hollow metalloid nanostructures in the context of the polymorphism of the bulk, and identify a close relation to the α phase of gallium. This previously unrecognised connection is established using the electron localisation function, which reveals the ubiquitous presence of radially-pointing covalent bonds around the Ga(12) centre--analogous to the covalent bonds between buckled deltahedral planes in α-Ga. Furthermore, we find prominent superatom shell structure in these clusters, despite their hollow icosahedral motif and the presence of covalent bonds. The exact nature of the electronic shell structure is contrasted with simple electron shell models based on jellium, and we demonstrate how the interplay between gallium dimerisation, ligand- and crystal-field effects can alter the splitting of the partially filled 1F shell. Finally, in the unique compound where the Ga(12) centre is bridged by six phosphorus ligands, the electronic structure most closely resembles that of δ-Ga and there are no well-defined superatom orbitals. The results of this comprehensive study bring new insights into the nature of chemical bonding in metalloid gallium compounds and the relation to bulk gallium metal, and they may also guide the development of more general models for ligand-protected clusters.

Throwing jellium at gallium--a systematic superatom analysis of metalloid gallium clusters.

Motivated by recent developments in the field of so-called "superatom complexes", as well as by the challenge posed to theory in understanding the many polymorphs of gallium, we analyse the electronic structure of several previously synthesised ligand-protected gallium clusters and their model derivatives using density functional theory. The calculated electron charge densities within the respective gallium cores are shown to be consistent with the jellium superatom model, exhibiting well-defined global spherical shells and wide HOMO-LUMO gaps--indicating enhanced chemical stability. It is demonstrated that the HOMO-LUMO gaps are widened due to the presence of covalent gallium-ligand bonds and a closed electron shell (i.e. electron "magic" number). The tendency of retaining a filled electron shell is shown to be particularly apparent in two closely-related clusters, with one derived from the other simply via substituting a doubly negative charge by a single protective moiety containing a lone electron pair. This analysis verifies that spherical electron shells can influence the chemical stability of ligand-protected gallium clusters, and also demonstrates the significant stabilising effects of metal-ligand interactions-something that is poorly accounted for in the current superatom model.

Uptake and withdrawal of droplets from carbon nanotubes.

We give an account of recent studies of droplet uptake and withdrawal from carbon nanotubes using simple theoretical arguments and molecular dynamics simulations. Firstly, the thermodynamics of droplet uptake and release is considered and tested via simulation. We show that the Laplace pressure acting on a droplet assists capillary uptake, allowing sufficiently small non-wetting droplets to be absorbed. We then demonstrate how the uptake and release of droplets of non-wetting fluids can be exploited for the use of carbon nanotubes as nanopipettes. Finally, we extend the Lucas-Washburn model to deal with the dynamics of droplet capillary uptake, and again test this by comparison with molecular dynamics simulations.

Molecular dynamics study of the melting of a supported 887-atom Pd decahedron.

We employ classical molecular dynamics simulations to investigate the melting behaviour of a decahedral Pd(887) cluster on a single layer of graphite (graphene). The interaction between Pd atoms is modelled with an embedded-atom potential, while the adhesion of Pd atoms to the substrate is approximated with a Lennard-Jones potential. We find that the decahedral structure persists at temperatures close to the melting point, but that just below the melting transition, the cluster accommodates to the substrate by means of complete melting and then recrystallization into an fcc structure. These structural changes are in qualitative agreement with recently proposed models, and they verify the existence of an energy barrier preventing softly deposited clusters from 'wetting' the substrate at temperatures below the melting point.

Dynamics of capillary absorption of droplets by carbon nanotubes.

We consider the capillary absorption of liquid metal droplets by carbon nanotubes using molecular dynamics simulations and the steady-state flow model due to Marmur [A. Marmur, J. Colloid Interface Sci. 122, 209 (1988)]. We find an exact solution to Marmur's evolution equation for the height of the absorbed liquid column as a function of time, and show that this reproduces the dynamics observed in the simulations well. The simulations show that the flow of the metal exhibits a large degree of slippage at the tube walls, with slip lengths of up to 10 nm depending on the wettability of the nanotube. The results support the use of the Lucas-Washburn approach for modeling capillary absorption at the nanoscale.

Capillary absorption of metal nanodroplets by single-wall carbon nanotubes.

We present a simple model that demonstrates the possibility of capillary absorption of nonwetting liquid nanoparticles by carbon nanotubes (CNTs) assisted by the action of the Laplace pressure due to the droplet surface tension. We test this model with molecular dynamics simulation and find excellent agreement with the theory, which shows that for a given nanotube radius there is a critical size below which a metal droplet will be absorbed. The model also explains recent observations of capillary absorption of nonwetting Cu nanodroplets by carbon nanotubes. This finding has implications for our understanding of the growth of CNTs from metal catalyst particles and suggests new methods for fabricating composite metal-CNT materials.

Superheating and solid-liquid phase coexistence in nanoparticles with nonmelting surfaces.

We present a phenomenological model of melting in nanoparticles with facets that are only partially wet by their liquid phase. We show that in this model, as the solid nanoparticle seeks to avoid coexistence with the liquid, the microcanonical melting temperature can exceed the bulk melting point and that the onset of coexistence is a first-order transition. We show that these results are consistent with molecular dynamics simulations of aluminum nanoparticles which remain solid above the bulk melting temperature.

Transition from icosahedral to decahedral structure in a coexisting solid-liquid nickel cluster.

We have used molecular dynamics simulations to construct a microcanonical caloric curve for a 1415 atom Ni icosahedron. Prior to melting, the Ni cluster exhibits static solid-liquid phase coexistence. Initially, a partial icosahedral structure coexists with a partially wetting melt. However, at energies very close to the melting point the icosahedral structure is replaced by a truncated decahedral structure that is almost fully wet by the melt. This structure remains until the cluster fully melts. The transition appears to be driven by a preference for the melt to wet the decahedral structure.

Static, transient, and dynamic phase coexistence in metal nanoclusters.

Molecular dynamics simulations are used to examine static and dynamic coexistence between solid and liquid phases in nanoscale silver, copper, and nickel clusters. We find static coexistence in the 561-atom copper icosahedron, the 561-atom silver icosahedron, and the 923-atom nickel icosahedron, and in cluster sizes above these thresholds, but not in smaller clusters. Nonetheless, in smaller clusters we typically observe either dynamic coexistence between fully solid and liquid states or transient coexistence which is essentially dynamic coexistence between a fully solid state and a solid-liquid state.