*http://www.tsi.enst.fr/~marquez/DLA/dla.html *

* *

*Jorge Márquez, Marc Sigelle and
Francis Schmitt *

IMAGE Department,
Ecole Nationale Supérieure de Télécommunications,

46, rue Barrault, 75634, Paris Cedex 13, France

Fractal exponent characterization has proved to be useful in the search
of invariant features, as scale symmetry, in the case of autosimilar phenomena.
Together with many other morphometrical parameters, from simple volume
ratios to complex inter-structural features (for example *mixture quality*
in multi-species amorphous deposition) quantitive features of data volumes
have became more difficult to estimate, as computer models produce more
complex and realistic simulations of physico-chemical or biological phenomena. Design
of new semiductor devices is only an example of such simulations, and a
collaboration in this field stimulated [1]
the present work.

New analitical methods are being required in models of complex morphology
in many Science and Industry applications. From percolation to dendritic
growth, from branching processes to vegetal development, a number of *morphological
paradigms* may be simulated by computer models of non-linear diffusion
process, where random motion of ions, particles or clusters diffuse, aggregate
and interact to shape a wide set of possible morphologies and structures.
Physical or chemical properties of these objetcs depend on
morphology features, such as interface roughness, bulk porosity and isotropy.

Validation of such models by physical realizations has became more meaningful as experimental methods develop, and sophisticated probing tools allow to obtain one-, two- and even three-dimensional samples that may help to test computer models and their theoretical background. Often these samples are related to known, controllable parameters, and may contain diversified information.

An increased need of specific analysis has promted in many fields the
*per-se* study of complex structure generation, as is the case
of *Diffusion Limited Aggregation *(DLA) process.* *
Other pardigms are *Martkov Random Field *realizations for texture
synthesis, *auto-regressive models*,
and fractal, recursive generators with randomization (fractional brownian
motion, and Fourier synthesis are popular examples).

Our final aim is the development and test of morpho-analytic tools for
three dimensional data. In order to analyse different complex configurations
and 3D-textures with features as branching, multiple connectedness, porosity,
mixtures and convoluted surfaces, we developed a DLAr process simulator
in 2D and 3D, incorporating many features to allow for semi-controlled
structure variation. The simulator's features include initial geometry
configuration, topology constraining, displacement through potential fields
and barriers, neighborhood interaction (which allows to emulate relaxation
and surface dynamics after cluster coalescence), anisotropic flux of the
clustering particles, sticking probablities, temperature-like and *reaction*
beahaviour, configuration-specific penalization or favouring, and other.

Interest in difussion process relies in the possibility of mathematical
modeling of important features that may be verified by experiments. In
surface growth, DLA process take place on a substrate layer (for example,
a plane). The expected roughness of the top surface, defined as the
*RMS* deviation from a mean height *h* can be predicted by the
difussion *(Kardar-Parisi-Zheng) *KPZ-equation [2],
which is a non-linear version of the Langevin equation, for brownian motion
:

\tilde{*h*} Is the RMS roughness of the top surface, ** x
**the vector position, and

F is a "drift" offset.

\gamma Is the coefficient associated to surface tension (or viscosity),
and the Laplacian term is known as the *relaxation*, or *diffusion*
*term. *

\lambda Is the coefficient for the *lateral growth contribution*,
which accounts for neighborhood* reaction* interactions. It
is also known as *bias term*.

\eta This is the "input signal", which usually is constituted by gaussian noise.

**BibTeX entry:**

@techreport{MarquezDLA97, author = {Marquez J. and Sigelle M. and Schmitt F.}, institution = {ENST Paris 97}, type = {Rapport interne}, title = {Diffusion Limited Aggregation with Relaxation for Complex Morphology and Visualization Studies}, month = {december}, year = {1997}, keywords = {diffusion limited aggregation, DLA, 3D, fractals, relaxation, texture, microstructure}, annotate = { In preparation }, url = "http://www.tsi.enst.fr/~marquez/DLA/dla.html" }

**Figure 1:** *Computer simulation of two dimensional diffusion
limited aggregation (DLA) in a radial (central) field and a square
grid. A discrete fractal cluster is generated by aggregation
of random-walking particles (a) interacting in a central force field where
.
In a non-fractal solid, mass m increases following a squared law with radius
(b). Mass in a dendritic cluster follows a fractional power law of radius
(c), with "1.27" the corresponding fractal (Hausddorff) dimension. (See
also Fig 8,9).*

*Some examples of 2D realizations of DLA
with relaxation process follow.*

**Figure 2. ** *The classic bush-like growth, with the simplest
DLA model. In this case, \lambda = \gamma = 0 (random deposition).*

**Figure 3**. Introducing changes in particle diffusion,
and neighborhood interactions.

**Figure 4. ** Short-range interactions of deposited particles
generate deep meanders and column-like structures. Gray-levels code layers
of 10K particles.

**Figure 5.** Emulation of other kind of relaxation,
with porous structure formation. Note diagonal anisotropies, intrinsic
to DLA clustering.

**Figure 6. ** A higher relaxation produces more dense aggregation.
Note the diagonal anisotropies and the sparse pores that result from rare
events in which very few particles penetrate into a narrow gulf.

**Figure 7. **Coupled relationships between neighborhood
interactions and relaxation coeffitients give rise to this algae-like organization
with narrow, deep meanders.

**Figure 8. ** Radial DLA. Diffusion
towards central atractor is different of a simple change to polar coordinates. Notice
the relaxation effect in comparison with Figure 1.

**Figure 9**. Radial DLA with a lateral bias.

**Figure 10. ** Staircase distribution of initial
positions for the aggregated particles.

(posted for december 98)

**Figure 11. ** (projection in 3D from above of
the Xmas' distribution of Fig,10) .

**Figure 12**. Trabecular structures arise when lateral
configurations are promoted and "temperature" of random particles
is increased.

* *