Fragmentation pathways of nanofractal structures on surfaces

We present a theoretical analysis of the post-growth processes occurring in nanofractals grown on a surface. For this study we have developed a method that accounts for the internal dynamics of particles in a fractal. We demonstrate that the detachment of particles from the fractal and their diffusion within the fractal and over the surface determines the shape of the islands remaining on a surface after the fractal fragmentation. We consider different scenarios of fractal post-growth relaxation and analyze the time evolution of the island's morphology. The results of our calculations are compared with available experimental observations, and experiments in which the post-growth relaxation of deposited nanostructures can be tested are suggested.

Figure 1

Island of deposited particles on hexagonal grid. Processes essential for a fractal formation on a surface are shown by arrows, $\Gamma$ is the free particle diffusion rate, ${\Gamma }_d(m,n)$ is the diffusion rate along the periphery of fractal or island, and ${\Gamma }_e\left(l\right)$ is the particle detachment rate from fractal or island. The diffusion rate along the periphery depends on the number of broken bonds $\left(m\right)$ and the number of maintained neighboring bonds $\left(n\right)$. The particle detachment rate depends on the number of broken bonds $\left(l\right)$. In the depicted example $m=3$, $n=1$, $l=2$.

Figure 2

(a) Fractal structure simulated using the DLA method; (b) structure of silver cluster fractal experimentally grown by clusters deposition technique on graphite surface (Ref. 15).

Figure 3

Evolution of fractal structure on a $650\times 750$-nm${}^2$ substrate with periodic boundary conditions. The initial fractal structure shown in the middle undergoes fragmentation in different final states depending on the interparticle interactions in the system. Numbers above the corresponding images indicate the values of $E_b$ and $\Delta \epsilon$ used in the simulations (in units of $kT$), $\Delta \mu =2$$kT$ in all cases. The simulation time $t$ is given for each path of the fragmentation.

Figure 4

Time evolution of the number of fragments $N_{\mathrm{fr}}$, $\langle R_{\max }\rangle$ introduced in Eq. (19), and of the $\langle S/P\rangle$ ratio introduced in Eq. (27) calculated for the fractal structure shown in Fig. 3. The fractal fragmentation has been analyzed at $\Delta \mu =2$$kT$ for different values of the binding energy $E_b$ and the barrier energy $\Delta \epsilon$. Plots (a)–(c) show the results of calculation obtained for $\Delta \epsilon =0.2E_b$ and the different values of the binding energies between two particles. Lines 1–6 correspond to $E_b=(1,2,3,4,5,6)kT$, respectively. Plots (d)–(f) represent the results obtained at $E_b=4kT$ for different values $\Delta \epsilon =(0,0.4,0.8,1.0,3.2,4)$$kT$. The direction of growth of $\Delta \epsilon$ is shown in these plots.

Figure 5

Dependence of $N_{\mathrm{fr}}$ (squares, left scale) and $\langle R_{\max }\rangle$ (dots, right scale) on the binding energy $E_b$ calculated for the barrier energy $\Delta \epsilon =0.2E_b$ after 4-s simulation, corresponding to the dependencies shown in Figs. 4a and 4b.

Figure 6

Distributions of island sizes formed on the substrate after 4 s of simulation. The distributions were calculated at the fixed values of $E_b=4$$kT$, and $\Delta \mu =2$$kT$ for different values of $\Delta \epsilon$ as indicated.

Figure 7

Size distributions of islands calculated at different stages of the fractal fragmentation (see Fig. 4) for $E_b=2kT$, $\Delta \epsilon =0.4kT$, and $\Delta \mu =2kT$. The corresponding simulation time is given in the insets to the plots.

Figure 8

Similar to Fig. 7, size distributions of islands calculated at different stages of the fractal fragmentation (see Fig. 4) for $E_b=4kT$, $\Delta \epsilon =0.4kT$, and $\Delta \mu =2kT$. The corresponding simulation time is given in the insets to the plots.

Figure 9

Time evolution of the number of fragments/nucleation islands on a surface, $N_{\mathrm{fr}}$, during the fractal fragmentation process (line 1) and during the nucleation process of randomly distributed particles (line 2). The calculations have been performed for a $650\times 750$-nm${}^2$ substrate with periodic boundary conditions. Plots (a) and (b) have been calculated at different values of the model parameters: (a) $E_b=1$$kT$, $\Delta \epsilon =0.2$$kT$, $\Delta \mu =2$$kT$; (b) $E_b=4$$kT$, $\Delta \epsilon =0.4$$kT$, $\Delta \mu =2$$kT$. The insets show the morphology of the system at the end of the simulation.

Figure 10

Time evolution of the island size distributions calculated for the nucleation process of randomly distributed particles (a) and for the fractal fragmentation process (b). The distributions have been calculated for the same values of the model parameters as in Fig. 9b. The initial fractal shape has been chosen the same as in Fig. 2a.

Figure 11

$S/P$ ratio distributions calculated after the fractal fragmentation with different sets of the model parameters and the corresponding distributions of island sizes. Distributions (a) and (b) are calculated with $E_b=3$$kT$, $\Delta \epsilon =0.6E_b$, $\Delta \mu =10$$kT$; (c) and (d) with $E_b=4$$kT$, $\Delta \epsilon =0.4E_b$, $\Delta \mu =2$$kT$. Insets show the results of experimental measurements for silver fractal fragments created via annealing (a) and (b), and by adding oxide impurities to silver clusters (c) and (d) (Ref. 14).