Fractal

How It all Started Plato sought to explain nature with five regular solid forms.

Newton and Kepler bent plato’s circle into an ellipse.

Modern Science analysed plato’s shapes into particles and waves and generalised the curves of Newton and Kepler to relative probabilities, still without a single “rough edges”.

Now more than 2000 years of Plato and 300 years of Newton and Kepler, Benoit Mandelbrot that ranks these rough edges with the laws of regular motion.

The Fractal was coined in 1975 by the polish/american Mathematician Benoit Mandelbrot to describe the shapes which are detailed at all scales. He took the word from Latin root fractus, suggesting Fragmentation, broken and Discontinuous.

Why are we basically studying irregularities?

Nature has irregularities in itself.

It deals in with non uniform shapes and rough edges.

We can take the human form, there is certain symmetry about it but it is still indescribable in terms of Euclidean Geometry.

To study about the patterns in Nature we think are irregular, Does it have an Regularities or Order?

What happening here?

We see some order in Nature

This order is very intuitive.

This order in a state of disorder is called Chaos.

The study which deals with Chaos and Fractals is called Non Linear Dynamics

Non Linear Dynamics To study about NLD, we need to primarily know about Dynamical systems

What are Dynamical System??

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

When the system is not obeying to the basic laws of Linearity.

It is a Non Linear System, which means that the final result is dependent on the initial conditions.

When we talk about NLD, we need mention the theory which governs every aspect of Non Linear Dynamics. It is called Chaos Theory.

What is Chaos Theory??Chaos theory concerns deterministic systems whose behaviour can in

principle be predicted. Chaotic systems are predictable for a while and then appear to become random. The amount of time for which the behaviour of a chaotic system can be effectively predicted depends on three things:—

How much uncertainty we are willing to tolerate in the forecast.

How accurately we are able to measure its current state.

A time scale depending on the dynamics of the system, called the Lyapunov time.

Chaos theory is a field of study in mathematics, with applications in several disciplines including meteorology, sociology, physics, engineering, economics, biology, and philosophy.

Chaos theory studies the behaviour of dynamical systems that are highly sensitive to initial conditions — a response popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general.

Butterfly Effect In chaos theory, the butterfly effect is the sensitive dependency on initial conditions in which a small change at one place in a deterministic nonlinear system can result in large differences in a later state.

The name of the effect, coined by Edward Lorenz, is derived from the theoretical example of the details of a hurricane (exact time of formation, exact path taken) being influenced by minor perturbations equating to the flapping of the wings of a distant butterfly several weeks earlier

Lorenz discovered the effect when he observed that runs of his weather model with initial condition data that was rounded in a seemingly inconsequential manner would fail to reproduce the results of runs with the unrounded initial condition data.

FractalsA fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. If the replication is exactly the same at every scale,it is called a self-similar pattern

Fractals can also be nearly the same at different levels.

Fractals also includes the idea of a detailed pattern that repeats itself.

Fractals are different from other geometric figures because of the way in which they scale. Doubling the edge lengths of a square scales its area by four, which is two to the power of two, because a square is two-dimensional. Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two to the power of three, because a sphere is three-dimensional. If a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power of two that is not necessarily an integer. This ratio is called the “fractal dimension” of the fractal, and it usually exceeds the fractals's topological dimension.

Bifucations

Some more examples of Fractals

Cantor Set

Koch Curve

Koch curve• The Koch curve is one of the Swedish mathematician

Helge von Koch in 1904 imagined example of an everywhere continuous , but nowhere differentiable curve .

They also concern her one of the first formally described fractal objects.

The Koch curve is one of the most cited examples of a fractal and has been in the discovery as a monster curve.

The Koch curve is also known in the form of cooking rule snowflake that is produced by a suitable combination of three Koch curves.

We can construct this kind of a curve by the means of Iterative process.

And eventually find the fractal dimension of the Koch curve.

How is it relevant to Materials Science

For selection of materials for specific applications and for the assessment of their performance in service. It is imperative that materials scientists, a number of procedures for the evaluation of Microstructures( i.e Low level structures) and Fractographs( i.e Fracture surfaces)are available. Most procedures involve examination of these materials at a high magnification using optical and electronic microscopes and general knowledge to rate the quality of the materials.

Fractal mathematics can provide a quantitative tool for evaluation of microstructures and fractographs.

Materials have hierarchically organised complex structures at different lengths.

Materials are evaluated at various length scales to know their properties at global scale.

Quantification of SEM images at Nano and scale length can help in developing correlations with materials properties at micro and global length scales.

For quantification of low level images, Fractal dimensions are often useful since they are known to indicate the extent of complexity in images

Fractal Dimensions(D) is a non-integer power law exponent which correlates the measures length L, with its the scale of Measurement k and the correlation is given by

L= N0.(k)1-D

Where N0 is a constant obtained for k = 1

Dimensions

Euclidean Dimensions

Topological Dimensions

Fractal Dimensions.

Fractal DimensionsThere really was a reason to fear pathological entities like the Koch coastline and Peano's monster curve. Here were creations so twisted and distorted that they did not fit into the box of contemporary mathematics. Luckily, mathematics was fortified by the study of the monsters and not destroyed by them. Whatever doesn't kill you only makes you stronger.

It is also called Hausdorff Dimensions is often difficuilt to calculate in practice. A simpler way to measure the fractal dimensions od a curve is called the “Box counting Method”. Cover the curve with a grid of little squares and count how many squares it passes through.

Repeat this process with smaller and smaller squares. In this limit for the fractal curve, the rate at which the proportion of filled squares decreases gives the Fractal Dimensions.

Box Counting Method

D = lim h-0 Log N(h) Log

(1/h)

Uses of Fractal Geometry in Materials Science

Metallic Materials :- To analyze the dislocation patterns, Grain boundary morphology, slip lines and Void structures

Porous materials: surface fractal dimension, an important parameter reflecting the roughness of pore surfaces is the aim of characterization of different porous systems . For instance Al2O3-SiO2 membranes with application in filtration ,mesoporous carbons with application as electrode materials in electric double-layer capacitors , SiO2 ceramics , palladium-alumina ceramic membrane, and soil structure.

Thin-film materials: surface roughness characterization by means of fractal geometry has been reported for gold films deposited on quartz crystals, copper tungsten thin film deposited on silicon wafers for application in electronic devices, thin films of BaTiO3 for electro-ceramic applications, and silver oxide film used as a photocathode.

Crack paths: analysis of the crack geometry using fractal dimensions is reported for alumina−zirconia composites.

Fine particles: irregularity of the structure of metal powder grains, pigments, wear particles, colloidal particles in solid-liquid suspension, and aerosol particles like carbon black and diesel soot has been quantified by fractal geometry.

Image Processing

In materials science images play very important role in the area of the quantitative description of the microstructure. In order to perform measurements on images of microstructures image processing is the first step to be taken. Image processing, by definition, include those methods that start with an image (an array of pixels each with a gray-scale value) and end with an image.

Various kinds of methods intend to produce a modified image that emphasizes some aspects of the original image, for instance filtering technique selects certain kinds of image data to be analyzed.

The operations performed on images of microstructures in order to enhance them or to make them more accessible for quantitative analysis, are described as image processing.

The process of image processing can be viewed as consisting of three steps:

1. Preprocessing or basic image enhancement

2. Segmentation or discrimination of features in microstructure

3. Post processing

Material Description

The samples which will be used in this work will be galvanostatic anodic titanium oxide ones prepared in oxalyc acid solution. In this electrochemical preparation method, a titanium plate is the anode in a two electrode electrochemical cell. A platinum plate will be used as cathode.

Then, the anode is polarized under constant current condition and an oxide film starts to form over the anode following the

equation: Ti + 2H2O TiO2 + 4H+ + 4e-. It is important to stress that TiO2 is formed by the direct reaction between the metal and water over the metal.

The surface morphology of the oxide is sensible to the experimental conditions used. In the present case, it will be used for different values of applied current, oxalyc acid concentration and solution temperature.

Using a 23 factorial design, it were performed 8 titanium oxide anodizations, generating, therefore, 8 classes of samples. From each class, it were acquired images from 8 different regions on their surface. Therefore, we have a total of 64 samples to be used in the model building. Each sample is a rectangular piece of the plate, which is measured through SEM-FEG technique generating a matrix(image) with a resolution of 3072 x 2060 pixels. shows one image per class, illustrating the general aspect of the dataset.

The images are analyzed using suitable Matlab programs. The purpose of this whole presentation is to find the Fractal dimensions of various processed images and it can be computationally explained. After image processing these images are read using suitable tools in Matlab.

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