Nanoparticles of metals with a face-centered cubic lattice have been studied in numerous works of both a theoretical and experimental nature, which are too numerous to list. During the conducted research, a variety of possible geometric forms were found, among which the most common are the truncated octahedron, icosahedron, and dodecahedron. Let's focus on analyzing the data we have obtained for a single metal, specifically silver.
In our earlier works on this topic [18, 19], we conducted a study on the thermal stability of a series of small silver nanoclusters with diameters up to 2.0 nm under the condition of an initial FCC phase. The analysis of the obtained results showed that for silver nanoparticles larger than 200 atoms, there is no spontaneous restructuring of the cluster structure, meaning that the initial FCC structure of such clusters remains thermally stable up to the melting temperature. However, for smaller silver nanoparticles, the situation is much more complex, and numerous cases of thermally induced changes in the cluster structure were observed, often occurring through various scenarios, presumably influenced by various geometric and electronic “magic” numbers [18, 19].
Based on all the experiments conducted using computer modeling methods, it was concluded that the size of the FCC metal cluster in the range of 200–220 atoms (D = 2.0 nm) is likely to be the natural limit above which the initial FCC phase is thermally stable, without the appearance of five-particle modifications of the structure (Ih and Dh) [18–22].
Analyzing the experimental data [12, 13], the following conclusions can be drawn. Firstly, many silver nanoparticles with diameters ranging from 2.0 to 3.5 nm exhibited nearly perfect FCC structures and distinct facetted shapes. This result is not surprising and aligns completely with our molecular dynamics experiments (Fig. 1a). Clusters of larger diameters slightly lost this perfection [12, 13]. Similar results were observed in our molecular dynamics simulations (Fig. 1b‒d).
Figure 1. FCC silver nanoparticles obtained by MD modeling of the crystallization process: a) D ≈ 3.0 nm, b) D ≈ 4.8 nm, c) D ≈ 6.0 nm, d) D ≈ 7.0 nm. Different colors indicate different coordination numbers for the modeled atoms.
Second: in the range of fixed sizes (D = 2.0–10.0 nm) there was a large percentage of nanoparticles with five-foldsymmetry, i.e. icosahedral and decahedral structure [12, 13]. Thus, in the initial state, approximately half of the nanoparticles were observed in this structural state. As a result of annealing at T ≈ 600 K, the number of Ih and Dh particles increased to 80% [13].
Thus, we find a clear contradiction between classical crystallography theory, which forbids five-fold symmetry in volumetric bodies, molecular dynamics modeling that does not support the presence of such structures in nanoparticles with diameters D > 2.0 nm, and real experiments [12, 13].
Let's try to resolve the contradiction that has arisen. We conducted a series of comparative MD experiments for small Ag nanoclusters with an initial FCC and amorphous structure [23–25] and the results obtained convincingly indicate a different evolutionary trajectory of the internal structure of Ag nanoclusters. Thus, under the condition of the initial amorphous structure of an Ag nanocluster, upon annealing it is already capable of forming a five-fold structure of atoms even with a diameter D > 2.0 nm [25].
These results present us with the task of analyzing the processes of formation of Ag nanoparticles synthesized in [12, 13]. As mentioned earlier, the nanoparticles of silver were formed using vacuum-thermal evaporation. In fact, this synthesis method is a variant of condensation from the gas phase [26–29], characterized by the fact that the bulk precursors, which are in a high-temperature state, are converted into a phase of metastable supersaturated vapor, which leads to the formation of nanoparticles.
One of the primary methods of gas-phase synthesis is the evaporation of precursors in an atmosphere of cold inert gas, usually He or Ar, with low density (pressure on the order of 1 mbar). The convective flow of inert gas passes through the precursor evaporation zone and carries the evaporated substance towards collection filters or substrates [26–29]. The walls of the reactor zone are cooled by liquid nitrogen, water, or another substance depending on the required heat dissipation rate [26–29].
The use of this method allows for the rapid removal of excess energy from the system through the elastic collision of inert gas particles and metal atoms. Therefore, in this classical synthesis method, nanoclusters are intensively cooled during their flight to the collection surface, and they usually have a temperature equal to the temperature of the walls of the cooled reactor zone at the moment of impact [26–29].
The method used in [12, 13] had different features in the synthesis process, specifically the absence of active cooling of the reaction zone. In [26], an analysis of the structure of metallic nanoparticles [30] was conducted to determine the possible temperature of the nanoparticles at the moment of their collision with the substrate. The analysis showed that during vacuum-thermal deposition, the substrate was bombarded with liquid droplets and it is reasonable to assume that silver nanoparticles during vacuum-thermal evaporation were also likely to be in a molten state, i.e., they had a primary amorphous structure.
Once on the substrate, they spread over it and took on a rounded shape, which is confirmed by the HRTEM image of a flat 2D nanoparticle [31], as a result of which, after crystallization, it was the FCC structure that was formed there. The formation of the fcc structure at such a small cluster size in all synthesized nanoparticles can only be explained by the fact that at the moment of crystallization the clusters did not have a spherical 3D shape corresponding to the minimum surface energy [26].
It is clear that the greatest volume minimization is determined by the formation of an icosahedron (Ih) Mackay, whose surface consists of 20 equilateral triangles. Larger Mackay icosahedra are formed by adding further Ih shells, leading to the minimization of the cluster's surface area and a corresponding gain in surface energy. However, in the case of a flat 2D nanoparticle, forming a Mackay icosahedron is simply impossible; it must be three-dimensional. Since the difference in binding energy between FCC and five-fold symmetry structures in the case of nanoclusters is usually not significant [20], the reduced dimensionality of Ag nanoparticles is sufficient for the formation of an FCC structure.
Let's now move on to the results of [13]. The dimension of the synthesized Ag nanoparticles is not directly mentioned, but the necessary information can be obtained from another work of these authors [31]. As can be seen from the corresponding histograms of the distribution of Ag particles by size, when evaporating a dose of approximately 2 mg (corresponding theoretical hypothetical thickness of the film 1.3 nm), the predominant cluster size is approximately 9 nm, while when evaporating a larger dose of approximately 11 mg (hypothetical thickness of the film 7 nm), the predominant size is 35 nm [31]. It is clear from this that the Ag nanoparticles synthesized by vacuum-thermal evaporation have a reduced height h relative to the diameter D of the contact area with the substrate. However, as the diameter of the nanoparticles increases, their height also increases. Thus, for Ag clusters with a size of approximately 9 nm, the hypothetical height was only 14.4% of their diameter, while for a size of approximately 35 nm, it was already 20%. However, this is the hypothetical height [31], in reality h may have a higher percentage value.
To verify this assumption, let's consider the crystallization processes with different rates of thermal energy dissipation of silver nanoparticles of various diameters (D = 2.0–10.0 nm) assuming their spherical shape. Using the Andersen thermostat, a gradual cooling of silver clusters from the liquid phase to room temperature T = 300 K was simulated with certain fixed cooling rates corresponding to cooling times of τ = 0.5; 1.0; 2.0 ns, without holding at intermediate temperatures.
The analysis of the results obtained during the cooling of silver nanoclusters of various diameters from the molten state showed a clear correlation between the process of forming a crystalline or amorphous structure and the rate of heat dissipation. It is visible that at a low rate of heat dissipation, corresponding to a cooling time of τ = 2.0 ns, all clusters have a crystalline structure. With a decrease in the cooling time to 0.5 ns, there is a decrease in the percentage of the occurrence of crystalline structures.
FCC and HCP structures were combined into one group during the analysis, as it was difficult to distinguish each of them in a pure form. From the data we obtained, it can be observed that at any cooling rate we used, for D = 2.0 nm, the probability of realizing an FCC/HCP structure in Ag nanoparticles is only about 20%, increasing fairly evenly with further growth in cluster size. This value turned out to be approximately half the probability of forming an FCC/HCP structure in small Ag nanoparticles (about 41%) found in [13]. Such a high value achieved in our computer experiment is only possible at a diameter of Ag nanoparticles D = 4.0 nm. Therefore, for Ag nanoparticles with a diameter D < 3.5 nm, the target value of 41% can only be achieved with significant deformation of the nanoparticles placed on the substrate, hindering the formation of five-fold symmetry (2D form). As demonstrated in our study [32], slight deformation of a metallic cluster resulting from its placement on a carbon substrate cannot hinder the formation of five-fold symmetry.
Thus, at a low rate of heat dissipation, corresponding to a cooling time of 2.0 ns, all clusters had a crystalline structure, with only the FCC modification realized at D ≥ 6.0 nm. With an increase in cooling rates to τ = 1.5 ns, the probability of realizing a crystalline phase slightly decreases for particles with a size of 2.0 ‒ 4.0 nm, and there amorphous structures begin to appear. At τ = 0.5 ns, this process intensifies, and at a diameter of nanoclusters of 10.0 nm, all the investigated particles had only an amorphous structure.
Icosahedral structures are found in Ag nanoclusters with sizes ranging from 2.0 to 4.0 nm, and the probability of their occurrence decreases with the particle diameter growth. This result is expected because the formation of a structure with five-fold symmetry, due to the relatively high contribution of surface energy, is advantageous only for clusters of small size.
The same trends apply to decahedral structures as for Ih morphology, but at D = 4.0 nm, there is a slight increase in the probability of realization. In our view, this behavior can be explained by the intermediate character between Ih and FCC of the Dh configuration.
Therefore, under the synthesis technological parameters of Ag nanoparticles used in [12, 13], and possible rates of further cooling of Ag nanoparticles on a carbon substrate, conditions may well arise where some small clusters (D < 3.5 nm) were deposited on the substrate as liquid 2D droplets, while larger Ag nanoparticles largely retained their 3D structure. As a result of subsequent solidification, the small 2D nanoparticles formed an FCC structure (about 41%), while the remaining small clusters with a 3D form and nanoparticles up to D ≤ 5.0 nm predominantly formed Ih or Dh structures.
However, the computer experiments we conducted were unable to explain such a large percentage (~ 50%) of such structures. In addition, Ih or Dh nanoparticles were observed at larger diameters up to D = 8.0 ‒ 9.0 nm, while our MD modeling did not confirm this. The solution to this contradiction may lie in the fact that at high rates of heat dissipation, many Ag nanoparticles, especially those of large size, were observed by us in an amorphous state. However, such a state is not thermodynamically stable and can evolve into a more favorable atomic structure even at room temperature during annealing.
In this regard, it is also necessary to note that all MD results on the crystallization of Ag nanoparticles from the melt were obtained under the assumption of a microcanonical NVE ensemble, which implies stabilization of the energy of the nanoparticles during the process. However, in real experiments, especially when synthesizing nanoparticles from the gas phase, nanoparticles are often observed in an explicit metastable state [33, 34], which differs significantly in its internal structure from optimized structures with minimal energy. A characteristic feature of such a state is its thermal stability, as a result of which metastable nanoparticles can be observed for a long time in a wide temperature range.
Based on the results mentioned above, let's try to find the probability of the process using MD modeling, in which silver nanoclusters, initially having an amorphous morphology, spontaneously change their structure to an FCC structure, characteristic of bulk material, or Ih/Dh polytypic modification as a result of thermal evolution.
Figure 2. Normalized histogram of the probability of structural transition for Ag nanoparticles with diameters ranging from 3.0 to 10.0 nm after annealing at T = 600 K. The initial structure of the nanoclusters is amorphous. The data were obtained under conditions of thermodynamic equilibrium.
The creation of model nanoclusters with an initial amorphous morphology proceeded through several stages. Primary particles were obtained by cutting them out of an ideal FCC lattice. They were then subjected to a stepwise heating process until the complete destruction of long-range order in them, followed by rapid cooling to a temperature of T = 20 K to “freeze” the disordered phase. Additionally, a selection process was carried out to eliminate any residual crystalline phase nuclei in the particles. After the formation process of an array of silver nanoclusters with primary amorphous structure, they underwent a repeated stepwise heating procedure up to a temperature of 600 K.
Analyzing the results of the conducted modeling, two main trends can be observed (Fig. 2). The first trend is associated with the expected increase in the proportion of FCC/HCP structures as the size of the nanoclusters grows. The second trend involves a simultaneous decrease in the probability of Ih or Dh configurations occurring, with a peak Ih morphology for particles around 3.8 nm in size. Let's delve into the detailed analysis of the obtained data.
Let's start with the Ag nanocluster containing 791 atoms (D ≈ 3.0 nm). In this case, we observe a clear competition between Ih and Dh structures, which overall confirms the trend previously observed for particles ranging from 100 to 200 atoms [20]. However, here, a mixed FCC/HCP structure emerges for the first time. When discussing this structure, it is important to note that for the studied ensemble of particles, it is challenging to obtain any structure in a perfect “pure” form. The investigated nanoclusters typically represent various combinations in the form of twinning or layered structures.
Figure 3. An example of a five-fold structure of Ag nanoclusters obtained through molecular dynamics modeling of annealing an initial amorphous structure at T ≈ 600 K: a) D ≈ 3.0 nm, b) D ≈ 4.8 nm, c) D ≈ 6.0 nm, d) D ≈ 7.0 nm. The different colors represent different coordination numbers for the modeled atoms.
When transitioning to the nanocluster containing 1553 atoms (D ≈ 3.8 nm), there is a sharp increase in the proportion of Ih configurations observed (Fig. 2). For the subsequent ensemble of particles consisting of 3055 atoms (D ≈ 4.8 nm), the proportion of FCC/HCP structures reaches 50%, competing with the Dh configuration. In this case, Dh or Ih particles of relatively large size (D ≈ 6.0 nm) exhibit a fairly “correct” structure (Fig. 3), while FCC/HCP structures may contain inclusions of an amorphous phase or twin boundaries (Fig. 4).
When the nanoclusters reach a size of about 7.0 nm (N = 10005 atoms), a FCC/HCP structure is formed in most cases. The first nuclei of the FCC phase form in the surface layer at T = 100 K, gradually increasing and penetrating deeper into the nanocluster until reaching a critical value, at which a transition from “amorphous” to “crystalline” occurs in the temperature range from 200 to 300 K. During this process, the nanoclusters can have varying degrees of defectiveness, which gradually decreases during further heating to temperatures of approximately 600 K. An interesting example is the Ag nanoparticle shown in Fig. 4d. This nanoparticle represents a transitional form from the Ih structure to the FCC structure, as two parts of this nanocluster simultaneously possess both Ih and FCC structures.
Thus, in the case of an initial amorphous morphology, two groups of silver nanoclusters can be distinguished based on their size. The first group is characterized by competition between Ih and Dh structures (D ≈ 2.0–4.0 nm), while the second group shows a predominance of the mixed FCC/HCP phase, with a nearly complete transition to it for particle diameters exceeding 8.0 nm. Further molecular dynamics (MD) simulations for Ag nanoclusters with a diameter of 10.0 nm did not observe the emergence of five-fold symmetry in them, even with an initial amorphous phase.
In conclusion, we present the results of study [35], which used the method of cluster beam deposition on a quartz substrate based on magnetron sputtering to create an ensemble of Ag nanoparticles with well-controlled sizes. The TEM data from [35] showed that nanoparticles (D = 12.5 ± 1.1 and 24.0 ± 2.0 nm) had a monocrystalline structure with an FCC lattice similar to bulk silver. Silver clusters of these sizes were deposited in a so-called soft landing mode, which ensures that the kinetic energy of the cluster is much lower than the cohesive energy of atoms, resulting in no or very small distortion of the cluster shape, which should be close to spherical [35]. Therefore, this work can be considered clear evidence of our result that the size of an Ag nanoparticle of approximately 10 nm can be considered a barrier above which it is no longer possible to obtain a five-fold inner structure in equilibrium conditions of physical synthesis methods.
The trends we have discovered can be used in the preparation of Ag plasmonic nanoclusters on carbon substrates. Specifically, by slowly cooling an arrays of Ag nanoclusters with a size of D = 6‒10 nm under thermodynamic equilibrium conditions, the nanoclusters in the main mass will be in a relatively uniform crystallographic state (FCC). However, under clearly non-equilibrium crystallization conditions, the formation of nanoclusters with five-fold symmetry (Ih and Dh) will predominate. Technologically, the conditions of crystallization can be influenced by heating the substrate during the synthesis of Ag nanoclusters.