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.
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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.
Please contact:
Mária Vicsek - SZTAKI
Tel: +36 1 1665 783
E-mail: m.vicsek@sztaki.hu
