Molecules form fractals
(appeared in April 2015)

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Patterns that are made up of patterns, which are themselves made up of patterns, and so on, are of special interest, says S.Ananthanarayanan.

This kind of mathematical entity, called a Fractal, is one where any small part of a pattern, if expanded, is seen to consist of the same pattern as the original whole. And then, any small part this part of the pattern also shows the same pattern, when expanded, and so on. These patterns occur in nature, in traditional decoration motifs and are important in aesthetics, mathematics, science and engineering.

Jian Shang, Yongfeng Wang, Min Chen, Jingxin Dai, Xiong Zhou, Julian Kuttner, Gerhard Hilt, Xiang Shao, J. Michael Gottfried and Kai Wu, of the Peking University, from Marburg,Germany, Heifei in China and Singapur, report in their paper in the journal, Nature Chemistry, that they have got organic chemical molecules to self-assemble into the form of a well studied fractal, the Sierpinsky triangle, a fractal so far created only in theory, by drawing lines on paper.

Instances of fractals that we encounter in everyday life are the branches of a tree or the capillaries of blood vessels, which keep dividing, showing the same pattern as we go to finer and finer scales. Another handy example is the profile of a coastline. At the usual scale of tens of kilometers, the outline shows indentation, but detail at the scale of tens of meters is not visible. If a portion is expanded, to the scale of a kilometer, the detail in tens of meters comes into view, displaying the same kind of indentation as first outline showed. Refine the map to the scale of meters, and there is similar indentation at the level of centimeters! It is apparent that when the pattern that is seen at one scale is nearly the same as what is seen at other scales, the measurement of length at the finer scale would be much greater than what a longer yardstick could reveal.

A more formal example is the so called Sierpinsky triangle, named after Waclaw Sierpinsky, a Polish mathematician. This shape is constructed by dividing a triangle with equal sides into four similar triangles by joining the midpoints of the sides, and then knocking the middle triangle out. The same operation is then carried with the three remaining, newly added triangles, to yield nine new triangles. And the process is repeated with these new triangles, and so on. In fact, this figure could be created by starting with any triangle or even other shapes, a square, for example, and the result would be the Sierpinsky triangle. The triangle appears from any starting shape because the process of itself amounts to iterations that retain points that belong to the fractal shape.

The process could equally well be carried out in three dimensions, to yield the Sierpinsky pyramd or the Menger sponge, which is a cube that ends up with infinite surface area but zero volume!

Another example of a fractal shape is the Koch snowflake. This shape again starts from an equal sides triangle, but adds a triangle, a third of the size of each side, on to each of its sides. Each side thus now has four straight segments, where it first had one. Now, each of these segments add a triangle equal to a third of the new segments, and so on. This pattern keeps adding on third to the length of the outline and three new line segments to each existing one, at each iteration. The length of the outline hence increases rapidly and approaches infinity.

The ‘self-similarity’ quality of fractals make these shapes fundamentally different from ordinary, orderly, arrangements of entities. The difference is that the change of scale reveals a new level of complexity that is not there at a different scale. We know, for instance, that doubling the dimensions of any plane figure will increase its area by a factor of four, which is the factor by which dimensions are increased (ie, 2) to the power of the dimension of the space in which the figure lies (which is 2).In the case of a solid figure, the containing space is 3-D and the volume increases by the cube, eight times, if we double the dimensions. But if a fractal’s one dimensional lengths are doubled, the spatial content of the figure increases by a power that is somewhere between integers. It is this different ‘space filling’ quality of fractals that is their characteristic, and mechanical methods of measuring areas, like filling the closed figure with sand, would show different results as we change the size of the grains of sand. It is this fractional nature of the dimension that gives rise to the name, fractal.

Doing it in the laboratory

The authors of the paper in Nature Chemistry, observe attempts to use synthetic chemical compounds which combine in fractal shapes has not met with any significant results. But for a first time, they had succeeded in getting the organic chemicals, dibromo-terphenyl and dibromo-quaterphenyl to deposit as Sierpinsky triangles on a silver substrate, which provides leads as what kind of substances, on what bases and under what conditions can form fractals.

The authors note that the chemical structure of the two bromo compounds has the correct angular orientation and is formed with bonds of bromine and hydrogen, which allows self assembly into a network of triangular voids. The relative strength of the bonds allows mobility of the molecules to self correct and fall into the triangle pattern and also to remain stable in that form. The researchers also note that the structure of the substrate should have three-fold symmetry, which matches the structure of the cyclic bromine-hydrogen bonds in the building blocks.

The method used was to create a silver single crystal, with the appropriate orientation of silver atoms at the surface, and the bromine compounds were deposited by condensation of the vapour. As soon as the deposit formed, the assembly was cooled, with liquid helium, so that the carbon-bromine bonds in the film did not degrade. The patterns that that the deposit took were measured using the Scanning Tunneling Microscope and the images acquired were processed with special software

The result was a whole series of molecularly assembled and defect-free Sierpinsky triangles, the largest being with 48 participating molecules. Tests to assess the dimension of the figures, by counting pixels, show a fractional value, which confirms that the figures are fractals. Fractals, apart from being theoretical marvels, represent the optimization, for economy and stability that is found in nature. Sea-shells, broccoli, the pine-cone, leaf and petal formation, lightning bolts, ice crystals, biological tissue formation, all follow fractal design. Techniques to harness fractal design by creating man-made fractals at the molecular level may lead to advances in material science, efficiency of chemical processes, nanotechnology and even new basic designs. “A full understanding of such amazing molecular fractals awaits sophisticated theoretical treatments at multiple scales and levels,” the authors say.



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