Molecular dynamics simulation of nanofilament breakage in neuromorphic nanoparticle networks

The breaking process of an 8 Å wide, 60 Å long cubic NW between two Au₁₄₁₅ clusters is illustrated in figure 2. The temperatures stated under figure 2 are nominal temperatures that come correspond to the timestep of each frame, assuming linear scaling between the starting and ending temperatures. The typical evolution of the NW in the simulated systems can be summarised in the following stages: At low temperatures, the shape of the NW is changed minimally due to the thermal vibrations of the system. The NW loses its crystallinity and its outer most atoms begin to diffuse towards the clusters in a slow and gradual manner (figures 2(b)–(d)). As the temperature rises, the atomic movement towards the clusters becomes more pronounced, leaving the middle part of the NW thinner and more unstable (figures 2(e), (f)). Eventually, the NW breaks, and all the atoms left in the middle are pulled towards the clusters quickly and are assimilated into them. Figures 2(f) and (g) show the two trajectory frames before and after the breaking event in our simulation. After the breaking, the aggregation continues but in a much slower manner (figure 2(h)) until the whole system melts and becomes amorphous.

Zoom In Zoom Out Reset image size

The investigation of the time effect at constant temperatures on the breaking would be an interesting study which would require a new set of simulations of these nanostructures at a series of temperatures close to the critical points mentioned in the work. This is a major undertaking beyond which is the scope of this work. However, we do believe that, even in a long simulation, this NW would not break at 550 K since it is clear from the figure that, although largely non-crystalline, the NW is still thick enough to hold.

In this study, we consider fragmentation of a nanowire on the atomistic level by means of the MD approach. This imposes limitations on the sizes of the studied systems, which are much smaller than the systems simulated in [47, 48] by means of the kinetic Monte Carlo method. The cited studies demonstrated that micrometer-sized, metal NWs can break thermally into smaller islands which then and coalesce, a phenomenon analogous to the Rayleigh instability for a liquid cylinder. In our simulations, the NWs always break at the middle, and not disperse into multiple islands. This is caused by the strong surface diffusion current towards the clusters, which reduces the thickness of the NW in a gradual way.

A structural analysis can reveal further details of the mechanism of the breaking of the NW. Figure 3 shows the results of this analysis for some critical frames of the breaking process of NWs with different shapes and widths. The atoms of the inner shells of the Au₁₄₁₅ clusters and the connecting NWs are identified being in the fcc structure initially. The atoms of the outer shells, on the other hand, cannot be assigned by the CNA algorithm to any of the crystalline structures due to their reduced coordination number and are therefore classified by the algorithm as 'amorphous', despite them being visibly arranged in a crystalline formation.

Zoom In Zoom Out Reset image size

In the thinner 4 Å-wide cubic NW (see figures 3(a1)–(a3)), the surface atoms constitute a large proportion of the whole NW, therefore only a single row of atoms in the center appears crystalline from the beginning. The algorithm can hardly show the structure change of the NW in this case. However, as more atoms aggregate towards the clusters, atoms marked as fcc or bcc will appear on both ends of the NW.

For the 8 Å-wide cubic NW (figures 3(c1)–(c3)) and 8 Å-wide truncated cubic NW (figures 3(b1)–(b3)), the structural changes during the breaking process are similar. During the perturbation stage, phase transitions from fcc to hcp structure appear diagonally at 2 and 3 places, respectively, dividing the NWs in equally sized parts. As the temperature rises and more atoms aggregate towards the NPs, the structural transitions between fcc and hcp become more frequent and irregular. The middle part of the NW becomes too thin for the algorithm to recognise the structure, but inner atoms at the ends of the NW remain crystalline. These phase transitions are caused by the shearing strength generated by the perturbation. Marszalek et al [62] confirmed experimentally that the sliding of crystal planes within the gold NWs will change the local structure from fcc to hcp. This phenomenon is known for plastic deformation in fcc metals.

In the studied systems, parts of the NW, specifically their two ends, remain crystalline during the whole process, a finding that differs from previous studies of the melting of fcc NWs [44, 45]. This is owed to the presence of the clusters at the ends, which act as a recrystallisation area. The amorphisation of the NW and re-crystallisation at the two ends also visible by the radial distribution function. Figure 4 shows snapshots of 8 Å-wide 40 Å-long cubic NW between two clusters and the corresponding radial distribution function. The nearest neighbour distance ($\sigma \sim 2.8\mathring{\rm{A}}$) can be read from the first peak. Figure 4(a) shows a crystalline state. During the perturbation region of the NW, the $\sqrt{2}\sigma$ peak of the function begins to decrease (figure 4(b)). The $\sqrt{2}\sigma$ peak reaches its minimum (figure 4(c)) and the form of the functions tends resembles a liquid state. In figure 4(d), the NW is broken and the $\sqrt{2}\sigma$ peak gradually appears again. In general, the disappearance of the $\sqrt{2}\sigma$ peaks of happens ∼10 ÷ 30 K before the breaking temperatures.

Zoom In Zoom Out Reset image size

The average breaking temperatures of the simulated systems with the different lengths and widths of the NWs are shown in table 1 and illustrated in figure 5. The breaking temperatures in most cases vary from 400 to 650 K and show a dependence on the width, length and structure of the NW.

Zoom In Zoom Out Reset image size

Wider NWs will generally break at higher temperatures and this becomes more pronounced when the NWs are shorter. The breaking points of 4 Å- and 8 Å-wide 80 Å-long cubic NWs differ by ∼25 K. This difference however is increased to over 70 K for a NW length of 60 Å and to 100 K for 40 Å. The breaking point's dependence on the length varies according to the NW width and structure.

There is an exponential decrease of the breaking temperature with the increasing length for the 8 Å NWs, with this behaviour being more pronounced for the cubic NW. The breaking point of the truncated NW remains the same for lengths ≥40 Å (±10 K). The 4 Å NWs always break at more or less the same temperature (±25 K) regardless of their length. The slight increase in the breaking temperature of the 4 Å-wide cubic NW is within the expected fluctuations in an MD simulation. Finally, cubic NWs are more robust than the truncated ones of the same width for lengths ≤60 Å and the difference is increasing as the length is being reduced. However, they appear to break at the same temperature at a length of 80 Å.

The 8 Å-wide 20 Å-long cubic NW is an interesting case, as it retained its integrity during the whole simulation, until the whole the system ended in one completely melted amorphous structure at about 850 K.

Experimental studies [48] show a relation between the breakup temperature of NWs and the corresponding bulk melting temperature of the material given by:

$$\begin{eqnarray}&&\begin{array}{c}\displaystyle \frac{{T}_{mn}}{{T}_{mb}}={C}_{1}-\displaystyle \frac{{C}_{2}}{{d}_{NW}},\end{array}\end{eqnarray} \tag{ 1 }$$

where ${T}_{mn}$ is the observed breaking temperature of the NW and ${T}_{mb}$ is the melting temperature of the corresponding bulk material. ${C}_{1}$ and ${C}_{2}$ are constants related to the material (equal to 0.34 and 0.10 nm, respectively, in this case) and ${d}_{NW}$ is the initial diameter of the NW. For the pure gold NWs with a 4 Å and 8 Å diameter respectively, the predicted breaking temperatures are 120 K and 290 K respectively. This is a rather large difference with our simulation result, that is owed to the fact that the clusters have a large role in stabilizing the NW in the middle.

The Rayleigh instability model, which describes the breaking up of liquid cylinders, has been used to describe the fragmentation process of NWs [46]. The applicability of this model is questionable in a fragmenting NW [47], however, the behaviour observed in our MD simulations agree qualitatively with the breakage behaviour of a liquid cylinder described by means of the Rayleigh model.

We can consider a short period around the breaking event as the stable state applicable to the Rayleigh model, e.g. according to our description of the breaking process in figures 2(f)–(h), the state 0.2 ns after the breaking of NW. We measure the distance $\lambda$ between the two parts of the broken NW, the diameter of the rest of the NW after the breaking event ${d}_{sp},$ and its initial radius ${R}_{0},$ calculated by the average distance between the center axis and the outer atoms and calculate the $\lambda /{R}_{0}$ and ${d}_{sp}/{R}_{0}$ ratios for the more cylindrical 8 Å-wide truncated cubic NW are 8.2 and 2.5, respectively. Nichols and Mullins predicted $\lambda /{R}_{0}$ and ${d}_{sp}/{R}_{0}$ values equal to 8.89 and, assuming constant volume, 3.76 as the criteria for surface diffusion dominated breakup of a cylindrical wire as result of harmonic surface perturbations [63].

Thus, our results are in good agreement with the Rayleigh model suggesting that surface diffusion dominates the process. Deviations from the theoretical predictions are to be expected since our simulated system is still not of the ideal shape and isotropy which are assumed in the Rayleigh model with atoms of the NW diffusing towards the clusters on the sides.