COMPLEX PATTERN FORMATION

In the earlier chapter we have discussed the formation of colour bands in moving wave fronts, stationary structures which are governed by coupling of reaction and diffusion. In this chapter we will be concerned with pattern formation governed by the process of mass flow and diffusion related to precipitation as crystals, electro-deposits, bacterial colonies and diffusion. Just as in the former case, in the present case also we come across very complex patterns, depending on the experimental conditions. However in the present case it is possible to rationalize the complex structure with use of new mathematical concepts of fractal geometry. 

A fractal is a geometrical structure that at first seems to be complicated, irregular and random. A fractal pattern is one that repeats itself at smaller and smaller scales. When viewed carefully, one begins to realize the presence of tractable properties that are inherent in it and helps us to systematically study them. Following are the principal objectives of fractal growth studies (i) characterization and quantification of hidden order in complex pattern and (ii) analysis of correlation in the development of order in seemingly disordered state. Ferns are one example. They are made up of branches that also look like individual ferns and in turn each of these is made up of even smaller branches that also look the same and so the patterns goes on. 

A typical property of fractals is that they are locally (asymptotically) self-similar of small-length scales. Fractals are shapes that look more or less the same on all or many 

Self-similar objects can be divided into N parts. Consider a two-dimensional  

Let us consider the following simple structures, where D is non-integer, called fractals. Let us first consider contour set. We can have a set of lines as shown in Fig. 13.4. A line is divided into three parts. One middle part is removed. Then again, we divide the remaining two parts into three subparts and the middle part in the two cases is also removed. The process is repeated again and again and finally the set of lines called contour set is obtained. 

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Yoo PJ, Lee HH (2008) Complex pattern formation by adhesion-controlled anisotropic wrinkling. Langmuir 24 6897-6902... 

Thus, chemotaxis gene products are involved in two very different modes of multicellular behavior in E. coli and Salmonella swarming motility and the complex pattern formation discussed above. 

Gutnick, D. and Ben-Jacob, E. (1999). Complex pattern formation and cooperative organization of bacterial colonies, in Microbial Ecology and Infectious Disease (E. Rosenberg, ed.). Am. Soc. Microbiol., Washington, D.C. 

Page, K., Maini, P.K., Monk, N.A.M. Complex pattern formation in reaction-diffusion systems with spatially varying parameters. Physica D 202(1-2), 95-115 (2005). http //dx.doi.org/10.1016/j.physd.2005.01.022... 

Gutmann, J.S., Muller-Buschbaum, P., Stamm, M. Complex pattern formation by phase separation of polymer blends in thin films. Faraday Discuss. 112, 285-297 (1999)... 

Obhque drop impact was studied with phospholipid mono-layers on both drop and target liquid surfaces. These experiments visualize the rheological properties of mono-layers giving rise to complex pattern formation. During... 

Over the past 40 years, numerous modeling studies have focused on the cellular slime mold Dictyostelium discoideum. This is because D. discoideum provides an experimentally accessible and relatively simple system for studying key developmental processes like chemotaxis, cell sorting, and complex pattern formation. Early studies adopted continuous models [77-80]. More recent studies used elegant hybrid approaches that combined CA models and partial differential equations to model 2D and 3D problems involving aggregation and self-organization ofD. discoideum [81-84]. 

Zhao, Q., Qian, )., Gui, Z., An, Q., and Zhu, M. (2010) Interfacial self-assembly of cellulose-based polyelectrolyte complexes pattern formation of fractal trees . Soft Matter, 6, 1129-1137. 

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