CHAOTIC FRACTAL MODELS GENERATED BY RECTANGULAR CELLS*
Novikov V. V. 1, Wojciechowski Krzysztof W. 2, Privalko V. P. 3
1 Odessa National Polytechnical University, 1 Shevchenko Avenue, 65044 0dessa, Ukraine
e-mail: novikov@te.net.ua
2Institute of Molecular Physics, Polish Academy of Sciences
M. Smoluchowskiego 17, 60-179 Poznań, Poland; e-mail: kww@man.poznan.pl
3 Institute of Macromolecular Chemistry, National Academy of Sciences of Ukraine
263160 Kiev, Ukraine
Received:
Rec. 18 December 2003
DOI: 10.12921/cmst.2004.10.01.91-100
OAI: oai:lib.psnc.pl:563
Abstract:
Properties of some chaotic fractal models constructed on hierarchies of rectangular cells (the latter being rectangular subsets of the square lattice) are investigated. Fractal dimensionalities and average neighbour numbers of structures generated by small rectangular cells Lx x Ly (2 ≤ Lx ≤ 4, 1 ≤ Ly≤ 4) are derived. Generating probability functions and critical indices for the correlation length as well as for the percolation cluster density are calculated for the models considered. The calculations show that structures generated by anisotropic (rectangular) initial cells show much broader range of critical indices and other characteristic parameters than structures generated by ‘isotropic’ (square) initial cells.
References:
[1] B. B. Mandelbrot, Fractals: Form, Chance, andDimension (Freeman, San Francisco, 1977).
[2] P. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979).
[3] D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd edition (Talor & Francis,
London, 1992).
[4] M. Sahimi, Applications of Percolation Theory (Taylor & Francis, London, 1994).
[5] V. P. Privalko and V. V. Novikov, The Science of Heterogeneous Polymers Structure and Thermophysical Properties (-Wiley, Chichester, 1995).
[6] P. Meakin, Fractals, Scaling and Growth far from Equilibrium, Cambridge Nonlinear Science
Series, Vol 5 (Cambridge University Press, Cambridge, 1998)
[7] I. V. Zolotukhin, L. I. Yanchenko, and E. K. Belogonov, -ETP Letters 67, 720 (1998).
[8] S. S. Narine and A. G. Marangoni, Phys. Rev. E 59, 1908 (1999).
[9] T. H. Solomon, R. R. Hartley, and A. T. Lee, Phys. Rev. E 60, 3063 (1999).
[10] L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, Phys. Rev. Lett. 84, 2275 (2000).
[11] M. Lattuada, H. Wu, and M. Morbidelli, Phys. Rev. E 64, 61404 (2001).
[12] G. Reiter, G. Castelein, and –U. Sommer, Phys. Rev. Lett. 86, 5918 (2001).
[13] F. Yang and F. Pan, Phys. Rev. E 64, 51402 (2001).
[14] Z. W. Chen, X. P. Wang, S. Tan, S. Y. Zhang, -. G. Hou, and Z. Q. Wu, Phys. Rev. E 63, 65413
(2001).
[15] P. Cicuta and I. Hopkinson, Phys. Rev. E 65, 041404 (2002).
[16] F. Barra, B. Davidovitch, and I. Procaccia, Phys Rev E 65, 46144 (2002); see also references
therein.
[17] P. -. Reynolds, H. E. Stanley, andW. Klein, Phys. Rev. B 21, 1223 (1980).
[18] V. V. Novikov and V. P. Belov, -. Exp. Theor. Physics 106, 780 (1994).
[19] V. V. Novikov, K. W. Wojciechowski, D. V. Belov, and V. P. Privalko, Phys. Rev. E 63, 36120
(2001).
[20] K. W. Wojciechowski and V. V. Novikov, TASK Quarterly 5, 5 (2001).
[21] V. V. Novikov and K. W. Wojciechowski, -. Exp. Theor. Physics 122, 462 (2002).
[22] V. V. Novikov and K. W. Wojciechowski, Physics ofthe Solid State 44, 2055 (2002).
[23] K. E. Evans, I.-. Hutchinson, and S.C. Rogers, Nature 353, 124 (1991).
[24] K. W. Wojciechowski, “Monte Carlo Simulations of model particles forming phases of negative Poisson ratio”, in Properties and Applications of Nanocrystalline Alloys from Amorphous Precursors, edited by B. Idzikowski, P. Svec, and M. Miglierini (Kluwer Academic Publishers, Dordrecht, 2004) pp. 237-248; see also references therein.
Properties of some chaotic fractal models constructed on hierarchies of rectangular cells (the latter being rectangular subsets of the square lattice) are investigated. Fractal dimensionalities and average neighbour numbers of structures generated by small rectangular cells Lx x Ly (2 ≤ Lx ≤ 4, 1 ≤ Ly≤ 4) are derived. Generating probability functions and critical indices for the correlation length as well as for the percolation cluster density are calculated for the models considered. The calculations show that structures generated by anisotropic (rectangular) initial cells show much broader range of critical indices and other characteristic parameters than structures generated by ‘isotropic’ (square) initial cells.
[1] B. B. Mandelbrot, Fractals: Form, Chance, andDimension (Freeman, San Francisco, 1977).
[2] P. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, 1979).
[3] D. Stauffer and A. Aharony, Introduction to Percolation Theory, 2nd edition (Talor & Francis,
London, 1992).
[4] M. Sahimi, Applications of Percolation Theory (Taylor & Francis, London, 1994).
[5] V. P. Privalko and V. V. Novikov, The Science of Heterogeneous Polymers Structure and Thermophysical Properties (-Wiley, Chichester, 1995).
[6] P. Meakin, Fractals, Scaling and Growth far from Equilibrium, Cambridge Nonlinear Science
Series, Vol 5 (Cambridge University Press, Cambridge, 1998)
[7] I. V. Zolotukhin, L. I. Yanchenko, and E. K. Belogonov, -ETP Letters 67, 720 (1998).
[8] S. S. Narine and A. G. Marangoni, Phys. Rev. E 59, 1908 (1999).
[9] T. H. Solomon, R. R. Hartley, and A. T. Lee, Phys. Rev. E 60, 3063 (1999).
[10] L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz, Phys. Rev. Lett. 84, 2275 (2000).
[11] M. Lattuada, H. Wu, and M. Morbidelli, Phys. Rev. E 64, 61404 (2001).
[12] G. Reiter, G. Castelein, and –U. Sommer, Phys. Rev. Lett. 86, 5918 (2001).
[13] F. Yang and F. Pan, Phys. Rev. E 64, 51402 (2001).
[14] Z. W. Chen, X. P. Wang, S. Tan, S. Y. Zhang, -. G. Hou, and Z. Q. Wu, Phys. Rev. E 63, 65413
(2001).
[15] P. Cicuta and I. Hopkinson, Phys. Rev. E 65, 041404 (2002).
[16] F. Barra, B. Davidovitch, and I. Procaccia, Phys Rev E 65, 46144 (2002); see also references
therein.
[17] P. -. Reynolds, H. E. Stanley, andW. Klein, Phys. Rev. B 21, 1223 (1980).
[18] V. V. Novikov and V. P. Belov, -. Exp. Theor. Physics 106, 780 (1994).
[19] V. V. Novikov, K. W. Wojciechowski, D. V. Belov, and V. P. Privalko, Phys. Rev. E 63, 36120
(2001).
[20] K. W. Wojciechowski and V. V. Novikov, TASK Quarterly 5, 5 (2001).
[21] V. V. Novikov and K. W. Wojciechowski, -. Exp. Theor. Physics 122, 462 (2002).
[22] V. V. Novikov and K. W. Wojciechowski, Physics ofthe Solid State 44, 2055 (2002).
[23] K. E. Evans, I.-. Hutchinson, and S.C. Rogers, Nature 353, 124 (1991).
[24] K. W. Wojciechowski, “Monte Carlo Simulations of model particles forming phases of negative Poisson ratio”, in Properties and Applications of Nanocrystalline Alloys from Amorphous Precursors, edited by B. Idzikowski, P. Svec, and M. Miglierini (Kluwer Academic Publishers, Dordrecht, 2004) pp. 237-248; see also references therein.
