Fractal Growth Models
by Mária Vicsek and Tamás Vicsek
The rich variety of complicated patterns in nature can be successfully modelled by simple fractal growth models which capture the essential physics behind the associated phenomena. Computer simulations of such aggregation models have been playing an important role in our understanding of far-from-equilibrium growth processes which are in close relation to many processes of practical importance (eg solidification of alloys, secondary oil recovery etc). Growth models with simulation programs have been developed in cooperation between SZTAKI and Eötvös Loránd University (Department of Atomic Physics).
During the last 20 years it has widely been recognized by natural scientists working in diverse areas that many of the structures common in their experiments possess a rather special kind of geometrical complexity. The particular geometrical properties of these structures have been shown to be related to and described by fractals - objects with non-integer (fractal) dimensions.
The physics of far-from-equilibrium growth phenomena represents one of the main fields in which fractal geometry is widely applied. Computer models based on growing clusters made of identical subunits (particles) provide a particularly useful tool in the investigation of fractal growth and in determination of the most relevant factors affecting the geometrical properties of a growing object.
The formation of large clusters by aggregation of identical subunits (particles) is the characteristic feature of many important processes in physics, chemistry, biology and engineering. A wide variety of materials, like colloids, polymers, aerosols ceramics, glasses and thin films are formed by aggregation. Aggregation takes place when identical particles are joined into clusters according to some rule. Generally, the simulations are carried out on regular lattices and the diameter of the particles is assumed to be the same as the lattice spacing, but many variations of this basic idea can be simulated. Two main geometries are mostly considered: the aggregation may take place along an interface (in a strip) or start from a single seed particle.
Aggregation almost always leads to ramified structures with fractal geometry. It should be pointed out that in the simulations the relevant details of the models are dictated by the physics of the aggregation process being simulated. In particular, the trajectory of the particles along which they are brought together plays a decisive role. If the particles move along straight lines the process is called ballistic aggregation. In the other limit the particles undergo a diffusive motion (random walk) and the resulting structures are quite different from the ballistic case.
Our programs include a variety of aggregation models. For example the variations of ballistic aggregation processes can be used to simulate thin film growth by vapor deposition. The models lead to complex clusters in spite of the simplicity of the related algorithms.
Figure: Ballistic aggregation on a seed.
The first aggregation model which was shown to lead to fractal structures is based on joining randomly walking particles to a growing cluster and is called diffusion - limited aggregation (DLA). The process starts with a single seed particle at the origin. A particle is released from a distant point and is allowed to undergo a diffusive motion until it arrives at a site adjacent to the seed, where it sticks permanently to the seed. Further particles are released one by one and are attached to the growing aggregate in the same way.
It is easy to show that the probability of finding a randomly walking particle at a given point in space and time satisfies the Laplace equation and that the attachment of the particle to the cluster corresponds to a moving boundary condition.
On the other hand, we know that many growth phenomena in nature are described by the diffusion equation which under some approximations becomes equivalent to the Laplace equation for the probability, pressure, temperature or electric potential, depending on the physical process to be described. In this way the examples related to DLA include aggregation, fingering in two phase viscous flows, solidification (snowflakes) and dielectric breakdown (lightening) and many other phenomena.
In many cases the study of interfacial growth phenomena in which the motion of the unstable interfaces is dominated by surface tension leads to non-linear partial differential equations. The instability and the extreme complexity of the solutions usually makes impossible to find them analytically (except special cases). Since numerical solution of the above mentioned equations provides an alternative and promising approach to the description of fractal growth phenomena we intend to continue our work in this direction.
Mária Vicsek - SZTAKI
Tel: +36 1 1665 783