Selforganization and stability of fractals

An important goal of nanotechnology is the development of controllable, reproducible and industrially transposable, nanostructured materials . In this context, controlling of the final architecture of such materials by tuneable parameters is a fundamental problem. The investigation of the dendritic structures (fractals) has attracted considerable attention of many scientists . The formation of such systems provides a natural framework for studying disordered structures on surface, because fractals are generally observed in far from equilibrium growth regime. During the last years fractal shape have been recordered for a variety of systems. Fractals consisting of Ag clusters and C60 molecules have been fabricated on graphite surface with the use of the cluster deposition technique [Lando, Kebaili, Cahuzac, Masson, Brechignac, Phys. Rev. Lett. 97, 133402 (2006);Böttcher, Weis, Jester, Löffler, Bihlmeier, Klopper, Kappes, Phys. Chem. Chem. Phys. 7, 2816 (2005)].

The post-growth transformation of silver cluster fractals to compact droplets on graphitic surface was experimentally studied [see e.g. Lando, Kébaïli, Cahuzac, Colliex, Couillard, Masson, Schmidt, Bréchignac, Eur. Phys. J. D 43, 151, (2007)] . It was demonstrated that depending on the experimental conditions the shape and the size of the stable silver droplets changes significantly.

Theoretical study of nanofractal stability on the surface

Recently [Dick, Solov'yov, Solov'yov, Phys. Rev. B 84, 115408 (2011)] we performed a detailed theoretical analysis of the post-growth processes occurring in a nanofractal on surface. For this study we developed a method which describes the internal dynamics of particles in a fractal and accounts for their diffusion and detachment. We demonstrate that these kinetic processes control the final shape of the islands on surface after the post-growth relaxation.

The example below illustrates the analysis of silver cluster fractal post-growth relaxation performed using the stochastic Monte-Carlo-based dynamics module of MBN Explorer (see our MBN Explorer webpage). The fractal structure shown in Figure 1 has been chosen for the further investigation of the post-growth relaxation processes. The diameter of the fractal is 1.2 µm, which corresponds to the diameter of the experimentally grown structures, and consist of more than 38,000 Ag500 particles.

fractals_experiment_theory.jpg
Figure 1. Fractal structure used in the study of microscale pattern evolution on a surface (left); structure of silver cluster fractal grown by clusters deposition technique on graphite surface (right);

Video 1. Evolution of a loosely bound fractal structure on a 4.3×5.0 µm2 substrate with periodic boundary conditions.

 

The stability of silver cluster fractals on graphite surface has been studied on the timescale of seconds. Note that the simulation stepsize Δt used in stochastic Monte-Carlo-based dynamics in MBN Explorer is significantly larger than the stepsize of conventional molecular dynamics (where it is typically taken as 1-2 fs), and, therefore, allows to perform simulations on significantly larger time scales. Stability of silver cluster fractals on graphite surface is of great relevance to experiment, as the post-growth relaxation of silver fractals has been extensively studied over the last decade [see e.g. Lando, Kébaïli, Cahuzac, Colliex, Couillard, Masson, Schmidt, Bréchignac, Eur. Phys. J. D 43, 151, (2007)] . To make simulations consistent with experimental observations the model parameters were taken close to experimental values: the diameter of a particle has been taken equal to 2.5 nm, and the diffusion coefficient of a silver cluster on graphite at room temperature was reported to be equal to 2•10-7cm2/s [3], leading to Δt = 78 ns.

The rate of fractal decay depends on the interparticle interaction, which in turn defines the morphology of the fragments that are formed during the process. An illustrative video demonstrating the fragmentation process of a fractal is shown in Video 1. This video shows how the fragments morphology depends on the interparticle interaction energies: the particle interaction energy Eb and peripheral diffusion barrier Δε. For convenience, the model parameters Eb and Δε are defined in units of kBT (1 kBT = 0.026 eV) at room temperature (293 K).

For Eb=2 kBT, Δε =0.2 kBT the entire fractal structure melts and forms a large compact droplet. The binding energy Eb between the particles is small, allowing an easy detachment of particles, but at the same time it is large enough to make the characteristic particle detachment time comparable with the characteristic particle nucleation time, thereby preventing the system from entire fragmentation. Thus, the fragmentation path at Eb=2kBT goes via the rearrangement of the entire system, and the formation of large stable droplets.

Video 2. Evolution of a strongly bound fractal structure on a 4.3×5.0 µm2 substrate with periodic boundary conditions.

 

An increase of the interparticle interaction energy leads to the change of the fractal fragmentation pattern. As seen in Video 2 at Eb=4 kBT the fractal fragments into several compact droplets, while the overal profile of the initial fractal structure is preserved. The analysis of morphology of the created patterns leads us to the following main conclusions: (i) the growth of Eb leads to the decrease of the average fractal-core branch width during the fragmentation process. This happens because the detachment of particles from the fractal becomes energetically unfavorable process, and the fractal fragments slower. (ii) The increase of the peripheral diffusion barrier energy Δε suppresses the diffusion of particles, resulting in a slower evolution and fragmentation of the fractal shape.

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Figure 2. The important processes which govern pattern formation in the course of particles' random walk dynamics are indicated by arrows: Γ is the diffusion rate of a free deposited particle, Γd is the diffusion rate of a particle along the periphery of an island on surface, and Γe is the detachment rate of a particle from the island.

This example illustrates the analysis of the post-growth processes occurring in a nanofractal on a surface using the method implemented in MBN Explorer, which models the internal dynamics of particles in a fractal and accounts for their diffusion and detachment, as illustrated in Figure 2. These kinetic processes are responsible for the shape of the islands created on a surface after the post-growth relaxation. The present example studies particle dynamics in 2D, while MBN Explorer allows performing stochastic Monte-Carlo-based dynamics in 3D. This is especially interesting to do, because there are many examples of three dimensional fractal systems in biology, where the dendritic shapes are rather common.

Recent Publications

Efficient 3D Kinetic Monte Carlo Method for Modeling of Molecular Structure and Dynamics, Mikhail Panshenskov, Ilia A. Solov'yov, Andrey V. Solov'yov, Journal of Computational Chemistry, 35, 1317-1329, (2014)
Thermally induced morphological transition of silver fractals, Ilia A. Solov'yov, Andrey V. Solov'yov, Nouari Kébaili, Albert Masson, Catherine Bréchignac, Physica Status Solidi B, 251, 609-622, (2014)
Simulation of nanofractal dynamics with MBN Explorer, Ilia A. Solov'yov, Andrey V. Solov'yov, Journal of Physics: Conference Series, 438, 012006-(1-12), (2013)
Fragmentation pathways of nanofractal structures on surfaces, Veronika V. Dick, Ilia A. Solov'yov, Andrey V. Solov'yov, Physical Review B, 84, 115408-(1-14), (2011)
Nanoparticles dynamics on a surface: fractal pattern formation and fragmentation, Veronika V. Dick, Ilia A. Solov'yov, Andrey V. Solov'yov, Journal of Physics: Conference Series, 248, 012025-(1-11), (2010)