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Diffusion Limited Aggregation with Relaxation for Complex Morphology and Visualization Studies

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 :

KPZ equation

\tilde{h}  Is the RMS roughness of the top surface, x the vector position, and t the particle diffusion cycles.

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.

Site under construction

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 = "" }

2D   DLAr  image gallery

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 $F \propto 1/{r^2}$. 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.  

3D   DLAr  image gallery 

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