The Dynamics of Fullerene Structure Formation : Order out of Chaos Phenomenon

The buckminsterfullerene C₆₀ molecular structure is basically a three dimensional quasiperiodic Penrose tiling pattern which incorporate the golden mean t =(1+Ö 5)2 @ 1.618 in its geometry. More than a decade after its discovery, we still lack a completely satisfying understanding of the fundamental chemistry, that takes place during fullerene formation. It is now felt that any reasonable reaction mechanism must include a kinetics driven process that goes directly to fullerene, rather than other carbon allotropes (J.W. Mintmire 1996: Science 272, 45). C₆₀ molecules form spontaneously during vaporization of carbon associated with intense heating and turbulence such as in electrical arcs or flames. Self-organization of fluctuations in the highly turbulent (chaotic) atomized carbon vapor appears to result in the formation of the stable structure of C₆₀ and therefore may be visualized as order out of chaos phenomenon. The geometry of C₆₀, namely, the selfsimilar quasiperiodic Penrose tiling pattern implies long-range spatiotemporal correlations. Such nonlocal connections in space and time are ubiquitous to dynamical systems in nature and is recently identified as signatures of self-organized criticality (P.C. Bak et. al. 1988: Phys. Rev. A 38, 364). For example, atmospheric flows exhibit long-range spatiotemporal correlations on all spatial and temporal scales manifested as the fractal geometry to the global cloud cover pattern concomitant with inverse power law form for power spectra of temporal fluctuations documented and discussed in detail by Lovejoy and his group (Y. Tessier et al. 1993: J.Appl.Meteor.32(2) , 223). Mary Selvam (1990: Can.J.Phys. 68, 831) and Selvam and Fadnavis (1999: Chaos, Solitons and Fractals 10(8) 1321): http://xxx.lanl.gov/abs/chao-dyn/9806002 )  have proposed a cell dynamical system model for turbulent fluid flows where it is shown that the observed self-organized criticality is a signature of quantum-like mechanics governing flow dynamics. The model is based on the concept that large scale fluctuations can be visualized as the integrated mean of enclosed small scale fluctuations. Therefore the energy spectrum of fluctuations (eddies) follows the statistical normal distribution. The square of the amplitude of eddy fluctuations represents the probability density. Such a result that the additive amplitude of eddies, when squared represent probability densities is observed in the subatomic dynamics of quantum systems. Therefore macroscale turbulent fluid flows follow quantumlike mechanical laws. The model also predicts an overall logarithmic spiral trajectory for the flow pattern with the quasiperiodic Penrose tiling pattern for the internal structure. It is proposed that the intense turbulent fluctuations of vaporized carbon atoms may be visualized as turbulent fluid flows with spontaneous formation of quasicrystalline structure of the Penrose tiling pattern. The theoretical concept enables to derive fundamental constants of atomic physics such as the fine structure constant and the ratio of electron to proton mass in terms of the universal Feigenbaum’s constants a and d. The C₆₀ structure formation is therefore a dynamical process of self-organization of sub-quantum level turbulent (chaotic) fluctuations leading to ordered structure formation and is directly related to the newly emerging field of nonlinear dynamics and chaos. El Naschie (1997: Chaos, Solitons and Fractals 8(11) , 1873-1886) has shown mathematically the fractal nature of space time fluctuations in quantum systems.

Acknowledgements

The authors are grateful to Dr. A. S. R. Murty for his keen interest and encouragement during the course of the study.