Monte Carlo simulation has been performed to study the size, shape, composition and temperature dependent surface segregation behaviors and atomic-scale structure of Au–Ag nanowires (NWs). Segregation of Ag to the surface is observed in all the NWs considered. The surface segregation of prism-like nanowires with tetragonal cross-section (T-NWs) is enhanced by the increasing of the particle sizes (or Ag compositions) and the decreasing of temperatures. For Au3Ag T-NWs, variation range of surface Ag fraction is smaller than that of truncated octahedron nanoparticles (TO-NPs) with the same diameter, however, AuAg and AuAg3 T-NWs show the opposite character. The surface Ag fraction of T-NWs is higher than that of cylindrical nanowires (C-NWs) and hexagonal cross-section nanowires (H-NWs) due to T-NWs’ owning more low-coordination sites while surface Ag fraction of C-NWs and H-NWs are close to each other under every composition, size and temperature. It is found that the most stable mixing configuration of NWs is Ag-rich-surface, Au-rich-subsurface and alloyed-core structures. Accordingly, the calculated alloying extents for NWs are consistent with those of Au–Ag NPs. The possible effects of size and shape distribution of surface Au atoms on tuning the catalytic activity and selectivity of bimetallic NWs are also discussed.Highlights• The segregating behavior of Au–Ag nanowires is different from that of nanoparticles. • Ag-rich surface, Au-rich subsurface and alloyed-core are the most thermodynamically favorable structure for Au–Ag nanowires. • The surface Ag fraction depends on the cross-section of Au–Ag nanowires. • Composition and temperature of NWs play a key role in surface Au atoms distributions.

Monte Carlo simulation of the order–disorder transition revealed that the transition temperature of Co–Pt nanowires increases with wire diameter, approaching the bulk value if the size is large enough. The transition temperature is affected by the shape of cross-section, though the shape effect is less significant than the size effect. It is showed that the rise of transition temperature in nanowires is largely due to the decrease of surface area compared with nanoparticles. The phase separation and tetragonalization are discussed by introducing mixing parameter and asphericity parameter. It is also found that the order–disorder transition starts from the surface and then to the core, indicating that the order–disorder transition of nanowires is a surface-dominant phenomenon, governed by the atomic under coordination.Highlights► We simulate the order–disorder transition of Co–Pt nanowires by Monte Carlo method. ► The depression of transition temperature depends on the size and shape of cross-section. ► The transition can be predicted by Bond-Energy model. ► The simulation confirms the order–disorder transition to be a surface-dominant process.

The size and temperature dependent Gibbs free energies of titanium nanosolids (nanoparticles, nanowires, and nanofilms) are calculated through the optimization of our previous Gibbs free energy model, and then the size–temperature phase diagrams of titanium nanosolids have been obtained. It is found that Gibbs free energy of titanium nanosolids reveals a strong size effect in small size ranges, and it increases with the decrease of size and temperature. The size dependent Gibbs free energy and structure transition temperature of titanium nanosolids can be expressed by the universal relationship accurately, i.e., Pn = Pb(1 – K/D), where Pb denotes the corresponding bulk properties, and K is the material constant. The calculated results indicating the variation ratio among nanoparticles, nanowires, and nanofilms satisfies 3:2:1. Significantly, besides the HCP to FCC and HCP to BCC transitions, we predict an unobserved structure transition between FCC and BCC structures in the size and temperature ranges between 8.1 nm and 1715 K to 27.3 nm and 1156 K for nanoparticles, 5.5 nm and 1705 K to 18.6 nm and 1152 K for nanowires, and 2.7 nm and 1701 K to 9.2 nm and 1150 K for nanofilms. The present calculations agree well with experimental results.

The present study provides a simple model to predict the favored phase of Cu2S nanosolids upon variation of their size, shape, and temperature. Insight into the phase stability at the nanoscale will aid in the design of new nanomaterials. A study of the solid–solid phase transition demonstrated that the low-chalcocite to high-chalcocite transition temperature of Cu2S nanowires increases with increasing wire diameter consistent with recent experimental data.(31) At ambient temperature, the high-chalcocite phase is the most stable phase for nanofilms with a thickness below 1.5 nm, nanowires with a diameter below 3 nm and nanoparticles smaller than 4.5 nm, all in agreement with experimental observations.

On the basis of our previous work, the Debye model has been generalized for the size- and shape-dependent order–disorder transition and surface order–disorder transition of FePt nanoparticle (NPs) by considering the configuration entropy. The ordering temperature of FePt NPs decreases with a decrease of the particle size for a specific shape and decreases with a decrease of the shape factor at fixed size. The particle size is the dominant factor while the shape is the secondary one in affecting the transition. The ordering process starts from the surface and then propagates to the core, indicating the order–disorder transition is a surface-dominant process. To induce ordering, the annealing temperature should be lower than the surface ordering temperature, which suggests our calculation can be used to determine the highest annealing temperature. The calculation results also suggest that, at a small size and low temperature, the ordered NPs are stable but, at a large size and high temperature, the disordered NPs are stable. The present predictions agree well with the available literature data.

Based on the rigorous consideration of the bond broken rule and surface relaxation, a model for the size-dependent surface free energy of face-centered-cubic nanoparticles and nanocavities is presented, where the surface relaxation is calculated by the BOLS relationship. It is found that the surface free energy of nanoparticles and nanocavities represents a reverse size effect—the surface free energy of nanoparticles decreases with the decrease of particle size while it rises with the shrinkage of cavities. The size effect on the surface free energy of nanoparticles and nanocavities is not evident in large size ranges, while it becomes more and more distinct with decreasing size, especially for sizes smaller than 10 nm. The present predictions are in good agreement with the available literature data.

The previous model on surface free energy has been extended to calculate size dependent thermodynamic properties (i.e., melting temperature, melting enthalpy, melting entropy, evaporation temperature, Curie temperature, Debye temperature and specific heat capacity) of nanoparticles. According to the quantitative calculation of size effects on the calculated thermodynamic properties, it is found that most thermodynamic properties of nanoparticles vary linearly with 1/D as a first approximation. In other words, the size dependent thermodynamic properties Pn have the form of Pn = Pb(1 − K/D), in which Pb is the corresponding bulk value and K is the material constant. This may be regarded as a scaling law for most of the size dependent thermodynamic properties for different materials. The present predictions are consistent literature values.

The Gibbs free energy of HCP, FCC, and BCC structures is calculated, and the size–temperature phase diagram is obtained for hafnium nanoparticles. It is found that FCC, HCP, and BCC are small-size, low-temperature, and high-temperature stable phases, respectively. The temperature-induced structure transition is caused by the relative magnitude of lattice vibration (characterized by the Debye temperature) for different structures, while the size-induced structure transition originates from the different molar volumes. The observed HCP to BCC and HCP to FCC structure transitions are consistent with our model predictions. More importantly, we predict that there exists a new structure transition from FCC to BCC in the size and temperature ranges 3.6–14.6 nm and 1766–1801 K for spherical nanoparticles (α = 1) and 1586–1742 K and 4–14.8 nm for tetrahedral ones (α = 1.49), which has not been reported in the literature.

A model has been developed to account for size, shape, surface segregation, composition and dimension dependent cohesive energy of bimetallic nanosolids, and further been extended to predict the size dependent thermodynamic properties, such as melting temperature, Curie temperatures, ordering temperature and phase diagram. The cohesive energy, melting temperature, Curie temperatures and ordering temperature of bimetallic nanosolids decrease with decreasing the particle size. The depression is dramatic in the lower range of size, while it becomes smoothly in large size. For nano phase diagram, the solidus and liquidus curves drop and the two-phase zones become small, as the size of the nanosolids decreases. The two-phase zones of the nano phase are always lower than the regions indicated in the bulk Ag–Pd alloy phase diagram, and they may deteriorate into a curve at a critical size. It is also found that the thermodynamic properties of nanosolids not only depend on the compositions, the atomic diameter and the cohesive energy of each component, but also depend on the size and the shape. The model predictions are consistent with the corresponding simulation, semi-empirical model and experimental data.

The cohesive energy of metallic nanoparticles has been studied by Lennard–Jones potential. It is found that the Lennard–Jones potential can be used to calculate the cohesive energy of metallic nanoparticles by considering the size-dependent potential parameters. It is predicted that the cohesive energy of small particles decreases with decreasing the particle size, which is consistent with the experimental values of Mo and W nanoparticles.

A new model accounting for the particle size and shape dependent melting temperature of metallic nanoparticles is proposed in this paper, where the particle shape is considered by introducing a shape factor. It is shown that the particle shape can affect the melting temperature of nanoparticles, and the particle shape effect on the melting temperature become larger with decreasing of the particle size. The present calculation results on the melting temperature of Sn, Pb, In and Bi nanoparticles are well consistent with the corresponding experimental values and better than these given by liquid drop model.

The previous model on surface free energy has been extended to calculate size dependent thermodynamic properties (i.e., melting temperature, melting enthalpy, melting entropy, evaporation temperature, Curie temperature, Debye temperature and specific heat capacity) of nanoparticles. According to the quantitative calculation of size effects on the calculated thermodynamic properties, it is found that most thermodynamic properties of nanoparticles vary linearly with 1/D as a first approximation. In other words, the size dependent thermodynamic properties Pn have the form of Pn = Pb(1 − K/D), in which Pb is the corresponding bulk value and K is the material constant. This may be regarded as a scaling law for most of the size dependent thermodynamic properties for different materials. The present predictions are consistent literature values.

Based on the rigorous consideration of the bond broken rule and surface relaxation, a model for the size-dependent surface free energy of face-centered-cubic nanoparticles and nanocavities is presented, where the surface relaxation is calculated by the BOLS relationship. It is found that the surface free energy of nanoparticles and nanocavities represents a reverse size effect—the surface free energy of nanoparticles decreases with the decrease of particle size while it rises with the shrinkage of cavities. The size effect on the surface free energy of nanoparticles and nanocavities is not evident in large size ranges, while it becomes more and more distinct with decreasing size, especially for sizes smaller than 10 nm. The present predictions are in good agreement with the available literature data.