Abstract
MD simulations combined with the embedded-atom method have been applied to study the structural and melting properties of gold nanorods (AuNRs) of different sizes. The simulation results for the actual structure of AuNRs obtained after energy minimization processes revealed that the AuNRs with largest cohesive energies tend to be structurally more stable than those with smallest ones. Then, it was found that each actual structure of AuNR is classified as an irregular structure composed of a crystalline gold core covered by an amorphous gold shell. In addition, the results showed that the melting of the AuNR surface is an inhomogeneous, gradually occurring process. Besides, it was established that the premelting ratio is inversely correlated with the AuNR size, indicating that the premelting phenomenon is more pronounced in large NP sizes than in small ones.
Keywords
- gold nanorods
- size effect
- thermodynamic properties
- melting temperature
- premelting temperature
- MD simulations
1. Introduction
Gold-based nanorods have attracted and continue to attract the attention of a vast amount of scientists from all over the world thanks to their fundamental and pragmatic significance. Due to their exciting properties that are found to be absent in corresponding bulk counterpart, the gold nanorods (AuNRs) are useful nanoobjects in many applications [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. Especially, considerable attention has been paid over the past decades to AuNRs, because of their great importance in fabricating the new generation of nanomolecular and molecular electronics as mentioned in Ref. [15]. Besides, AuNRs can be added to base fluids in order to better improve the effective thermal properties of the nanofluids [16]. It is worth noting here that the technological properties of molecular electronic devices and the thermal property enhancement of nanofluids strongly depend on both the state of AuNR surface and temperature at which their structures change. Nonetheless, the temperature-dependent surface structure of single metal nanoparticles was ignored by several authors, where they did not take into account the surface premelting stage during the analysis of the melting process in the nanoparticles [17, 18, 19]. Therefore, it is necessary to understand the AuNR melting properties before fabricating new nanodevices based on them or suspending them in an energetic system (fluid). However, to our knowledge, the investigation on the melting properties of these nanoobjects is still lacking in the theoretical and experimental reports and, thus, needs intensive studies. To this end, the MD simulations combined with the embedded-atom method (EAM) will be applied to study the melting behaviors of AuNRs. For this purpose, a series of results will be presented in the present work that is organized as follows: In Section 2, there is a short description of the computational method and the calculation procedure. Section 3 deals with the simulation results of AuNRs. Finally, the main conclusions arising from this work are summarized in Section 4.
2. Computational methodology
All MD simulations were performed with the LAMMPS package [20], and the atomistic visualizations were carried out by Ovito [21]. First of all, eight different model systems of rod-shaped nanoparticles were constructed using LAMMPS, where the ratio between their diameters to their lengths is more than 1:10. (Example of structure can be seen in Figure 1.) The detailed parameters of all rod-shaped nanoparticles are listed in Table 1. Furthermore, embedded-atom method (EAM) potential was adopted to describe the interatomic interaction that has been developed from the density functional theory (DFT) by Daw and Baskes [22]. Moreover, this potential has been successfully used by our group in a variety of computational disciplines [23, 24, 25, 26]. The EAM potential used here was parameterized by Grochola et al. [27] to describe the gold–gold interactions by adjusting its parameters with experiments and ab initio calculations.

Figure 1.
Structure of rod-shaped Au nanoparticle generated by lamps (D = 6.3 nm, Lz = 61.86 nm, D:lz = 0.1, and 111147 atoms). The atomic structures are visualized by Ovito.
Nanoparticles | NP1 | NP2 | NP3 | NP4 | NP5 | NP6 | NP7 | NP8 |
---|---|---|---|---|---|---|---|---|
2.03 | 3.84 | 4.64 | 5.07 | 6.3 | 7.32 | 8.42 | 9.44 | |
19.43 | 30.43 | 41.25 | 50.55 | 61.86 | 72.43 | 83.34 | 93.04 | |
8907 | 13,907 | 42,375 | 74,164 | 111,147 | 149,133 | 194,139 | 243,880 |
Table 1.
Diameter D, length lz, and total number atoms (N) for the AuNRs.
The total energy given by the EAM is written as a sum of an embedding function
where the factor
In order to investigate the melting properties of AuNRs of different sizes, a series of MD simulations were carried out. First, all models considered in this work are equilibrated over 500,000 time steps in the NVT ensemble at zero temperature using the velocity Verlet method with a fixed time step of 1 fs and neglecting the periodic boundary conditions (PBCs) in three dimensions. Then after energy minimization process, the actual model systems at 0 K were obtained and characterized using two structural techniques, namely, the coordination number (CN) analysis and the common neighbor analysis (CNA) method. The final point of the simulations is that all the simulated equilibrium samples are heated by increasing the temperature from 0 to 2300 K using NVT.
In order to predict the melting temperatures of AuNRs, the variation of thermodynamic properties (including the potential energy per atom (
where
Besides the thermodynamic properties, the melting temperature of NPs is also explored by analyzing the variation in other physical quantities in terms of the temperature [29]. In this context, the Lindemann index is regarded as one of the most important of these quantities. Additionally, this index is a tool to examine the NP in the layer form in order to determine from where the NP melting begins. The Lindemann index of each layer,
where
3. Results and discussions
Although the initial structures of gold nanorods can be generated by using LAMMPS, these original structures are not the nanorods with equilibrium states (i.e., metastables). Therefore, it is necessarily to get the actual model systems (i.e., the real calculation models) before studying the structural analysis.
Using the MD simulation, the temporal evolution of each nanostructure passes through several states until it reaches a more stable one, corresponding to minimum total energy, but remains metastable, resulting from its surface atoms always wanting to aggregate to reduce surface energy as was indicated in Ref. [5]. After this pretreatment stage, the structure stability of AuNRs was investigated by the size-dependent cohesive energy (the absolute value of total energy) of the AuNR, and the results were plotted in Figure 2.

Figure 2.
The cohesive energy per atom as a function of the AuNR size (N) after energy minimization.
As expected, it is found that the calculated cohesive energies of the particles are lower than that of the bulk Au which was obtained previously [23, 31]. It was also established that the cohesive energy of AuNR is very sensitive to the size. The cohesive energy value increases with increasing AuNR size. This behavior seems to indicate that the AuNRs of higher energies tend to be structurally more stable and it can be explained in terms of reducing the influence of surface free energy (i.e., energy of all of the dangling bonds in the surface) resulting from the decrease in the dangling bonds to total bonds ratio when increasing the size.
Within the framework of the local structural characterization of AuNRs, the common neighbor analysis (CNA) technique was adopted. For more details on this method, see Refs. [23, 32, 33]. Once AuNRs in their equilibrium states were obtained (see Figure 3a which displays the section of [100]-oriented equilibrium AuNR at different sizes), the two categories of Au atoms in each NR were calculated by the CNA technique; then their percentage were illustrated in Figure 3b. It was found from Figure 3b that for the smallest AuNR (D = 1.622 and Lz = 15.26 nm), there is a significant percentage of unidentified structures (47.5%), while proportion of

Figure 3.
(a) Sections of [100]-oriented AuNRs after energy minimization. (b) The atomic structures are visualized by Ovito. The percentage of each category of atoms in AuNRs is counted by using the CNA method, yellow for fcc structures and purple for unidentified structures.
In order to describe in more details the local atomic-level structures in different size of AuNRs, the coordination number (CN) about each individual atom was calculated; then the results are presented in Figure 4. It was shown that each actual structure of the AuNR obtained after energy minimization process reveals the core-shell structure composed by a crystalline gold core (full-coordinated atoms with the CN = 12) covered by an amorphous gold shell. This last part is divided into two layers: the first contains atoms located on the subsurface layer (CN = 11 and 10), the second consists of atoms located on the free-surface layer, corresponding to both low-index facets [i.e., {110} and {100} facets with the CN = 7 and CN = 8 and their perimeter (edge and corner with the CN < 7)]. Apparently, the shell part of the NP is inherently different from the core one, where its atoms possess structure intermediate between the bulk liquid structure and the bulk solid one. It is worth mentioning that the classification of AuNRs in core-shell constructions obtained here supports the previous experimental and simulated studies carried out on single metal nanoparticles [23, 24, 25, 26, 34, 35]. Furthermore, it was seen from Figure 4 that for all simulated AuNRs, the fraction of liquid-like atoms located in the shell part is smaller than that of solid-like atoms located in the core part. It was also found that for all actual structures of AuNRs, the liquid-like atoms located in the area of low-index facets (7 and 8 coordinated atoms) exhibit the highest fraction in the shell part. In addition, when the AuNR size increases, the fraction of atoms’ core part increases from 51 to 83%, while the total fraction of shell atoms decreases from 49 to 17%. These findings are consistent with those previously reported by the CNA analysis. The results of the current study indicates that the ratio of low-coordinated atoms to full-coordinated atoms is inversely correlated with the AuNR size.

Figure 4.
The distribution of the coordination numbers of AuNR for different sizes, orange for
Figure 5 shows the typical trend of potential energy (

Figure 5.
Variation of the heat capacity and potential energy per atom of AuNR for different sizes, during the heating process.
Now we will turn to Figure 6, which shows the temperature-dependent Lindemann index value of free-surface atoms (

Figure 6.
Temperature-dependent Lindemann index of first layer for AuNRs.
Once again according to Figure 6, the onset of premelting temperature (
Our attention will now be drawn to identifying how the ratio between

Figure 7.
Premelting ratio as a function of the particle size (N).
4. Conclusions
In summary, the structural and melting properties of AuNRs have been investigated by means of MD simulations combined with the embedded-atom method (EAM) potential. The simulation results for the actual structure of AuNRs revealed that AuNRs of smaller cohesive energies tend to be structurally less stable than those larger ones. Then, the local structure characterization of the actual structure of AuNRs carried out by using the CN and CNA techniques showed that each AuNR is classified as an irregular structure which is composed of a crystalline gold core that is covered by an amorphous gold shell. Further, it was shown that melting temperatures of the actual structure of AuNRs are below that in bulk state. In addition, it was found that the surface melting in the actual structure of AuNRs is an inhomogeneous, gradually occurring process. Besides, the result of the size-dependent premelting ratio proved that the premelting phenomenon is less pronounced in small AuNRs than in large ones.