Dendrite (fractale)

Dimension fractale

A rigorous mathematical definition of a dendrite would be a figure Fractale of dimension fractale 1,77.

Data-processing modelings

With the advent of data processing, many scientists sought to reproduce dendritic growths by means of computer. Among them, Thomas Witten and Leonard Sander , two American mathematicians of the years 1980, established a model which from now on bears their name.

Model of Witten and Sander

This model is based primarily on dendrites of Cuivre which one observes after a electrolysis of a solution containing of the Ion S cuivre.
In this experiment, the ions move by chance in the solution, and when one of the ions passes near the electrode, this one changes, via a Réaction of oxydoreduction, out of solid copper, and is stuck to the electrode, making it grow bigger. This process leads to the creation of metal dendrites centered on the electrode.

Witten and Sander took as a starting point this reality to create their model, of which here main features:

a central point fixes
mobile particles moving
by chance the shocks between the variable components are ignored
If particles mobile passes near a fixed point, the particle sets at it
One generally adopts a cylindrical space, or spherical, so that the mobile particles leaving with dimensions of the workspace rerentrent with dimensions opposite
there simulation finishes when all the mobile particles were immobilized.

It will be understood easily that this simulation can give good performances only with coherent parameters: The ions copper do not have a infinite Solubilité, and, consequently, the number of ions per unit of volume is raised. This is why a simulation carried out in a small space, with a great number of mobile particles, cannot correspond to reality, and cannot thus give acceptable results.

The main advantage of this algorithm is the simplicity of its Mise in work, and its linear complexity (what means, coarsely, that it is carried out in a very reasonable time).

Influence grid

The grid determines the way in which one authorizes the particles to be driven. The standard grid is the square grid. With each movement of the particle, one considers that it can move either in top, or on the right, or in bottom, or on the left. That thus made four movements possibles.
One of work of Witten and Sander was to show, through their simulations, that the grid did not influence the results of simulations: That one uses a square grid with four possible movements, a hexagonal grid with six possible movements, etc, the final figure will always have a dimension fractale of 1,77.

Disadvantages of the model of Witten and Sander

Actually, there is often simultaneous formation several distinct clusters, which, possibly, at the end of a certain time, meet the ones the others. Indeed, our first model is unaware of the interactions between the free particles. However, the particles interact between them, and, according to the type of particles, are pushed back, attracted each other, and, sometimes, combine between them. In general, more the cluster is large, more it is attractile. Thus, of the secondary clusters can be formed, and nothing is opposed so that they exceed in the face the principal cluster. For this reason, a model based on the electrolysis of water is not universal, because it comprises a single attractile point, whereas in nature, there exist many situations or several aggrégats is formed at the same time.

Colloidal model

A second possible model rests on the taking into account of the collisions between particles. Characteristics:

Of the mobile particles moving by chance.
When two particles return in collision, they amalgamate to form a cluster of two particles.
When two clusters return in collison, they amalgamate to form a cluster including/understanding all the particles of the two clusters of origins. The same applies if a particle returns in collision with a cluster.
One generally adopts a space cylindrical, or spherical, so that the particles or the outgoing clusters of with dimensions of the workspace rerentrent there with dimensions opposite one.
By choice, simulation finishes when there remains only one single cluster, gathering all the initial particles.

This model, if it is more universal, with the disadvantage of being more complex. Its implementation is thus more difficult, and its slower execution.

Experiments

Such Fractale S can be observed, for example:

By injecting, using a syringe, water in solidified plaster.
By carrying out a electrolysis of a solution of Copper: Around the electrode of copper, will be formed metal copper dendrites.
While observing, with the electronic Microscope, a sheet of Gold. The formations observed are called Colloïdes of gold.

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