MONTE CARLO RENORMALIZATION GROUP FOR PERCOLATION OF LINEAR CHAINS



Duygu Balcan and Aye Erzan

Department of Physics, Faculty of Sciences and Letters, Istanbul Technical University, Maslak 80626, Istanbul, Turkey.



ABSTRACT


       Steady state fluorescence experiments performed on a bulk linear polymer reveal a sharp phase transition characterized by percolation exponents, as a function of the chain length [1]. Although percolation exponents are expected in a gelation phase transition [2] and indeed have been experimentally observed by the same experimental method to a very high accuracy [3], there is no a priori reason to expect such a transition in a system of linear chains [4]. We therefore set out to compute the critical exponents of a system of growing linear chains, via Monte Carlo Renormalization Group. We model the polymerization process by growing non-self intersecting chains from randomly chosen lattice sites. The Monte Carlo RG [5,6] is implemented by determining the average fraction of occupied sites at a given time step, in both the original and coarse grained configurations. The fixed point is determined from requiring the two densities to be the same, and the correlation function exponent is determined from the slope of this transformation. Our results on two dimensional lattices are not distinguishable from the percolation value. Work is in progress on three dimensional lattices.

 

REFERENCES

[1] Y. Yilmaz, O. Pekcan and A. Erzan, "Percolation exponents at the entanglement transition," cond-mat/0111339.
[2] D. Stauffer and A. Aharony, Introduction to Percolation Theory, (Taylor and Francis, London, 1991).
[3] Y. Yilmaz, A. Erzan, O. Pekcan, Phys. Rev. E, 58, 7487 (1998).
[4] A. Coniglio and M. Daoud, J. Phys. A, 12, L259 (1979).
[5] R. H. Swendsen, Phys. Rev. Lett. 47, 1159 (1981); and in T. W. Burkhard and J. M. J. van Leeuwen eds., Real Space Renormalization (Springer Verlag, Heidelberg, 1982).
[6] P. J. Reynold, H. E. Stanley, W. Klein, Phys.Rev. B, 21, 1223 (1980).