Abstract
Motivated by recent studies of gas permeation through polymer networks, we consider a collection of ordinary random walks of fixed length ℓ, placed randomly on the bonds of a square lattice. These walks model polymers, each with ℓ segments. Using computer simulations, we find the critical concentration of occupied bonds (i.e., the critical occupation probability) for such a network to percolate the system. Though this threshold decreases monotonically with ℓ, the critical “mass” density, defined as the total number of segments divided by total number of bonds in the system, displays a more complex behavior. In particular, for fixed mass densities, the percolation characteristics of the network can change several times, as shorter polymers are linked to form longer ones.
Similar content being viewed by others
References
P.J. Flory, J. Am. Chem. Soc. 63, 3083 (1941); S.R. Broadbent, J.M. Hammersley, Proc. Camb. Phil. Soc. 53, 629 (1957); J.M. Hammersley, Proc. Camb. Phil. Soc. 53, 642 (1957). For more recent reviews, see, e.g., J.W. Essam, Rep. Prog. Phys. 43, 833 (1980) and R. Zallen, The Physics of Amorphous Solids (Wiley, NY, 1983)
Laot C.M., Marand E., Schmittmann B., Zia R.K.P.: Macromolecules 36, 8673 (2003)
Schmittmann B., Gopalakrishnan M., Zia R.K.P.: J. Phys. C: Cond. Matt. 17, S1817 (2005)
C.M. Laot, Gas transport properties in polycarbonate – Influence of the cooling rate, physical aging, and orientation (PhD thesis, Virginia Polytechnic Institute and State University, 2001) http://scholar.lib.vt.edu/theses/available/etd-12012001-133140/
A very similar problem was studied by V. Cornette, A.J. Ramirez-Pastor, F. Nietoa, Eur. Phys. J. B. 36 (2003) 391399. Their polymers are more like self-avoiding walks rather than simple random walks. Thus, there is no distinction between ρ and p
Gopalakrishnan M., Schmittmann B., Zia R.K.P.: J. Phys. A: Math. Gen. 37, L337 (2004)
Antal T., Hilhorst H.J., Zia R.K.P.: J. Phys. A: Math. Gen. 35, 8145 (2002)
Hoshen J., Kopelman R.: Phys. Rev. B. 14, 3438 (1976)
Wu Y., Schmittmann B., Zia R.K.P.: J. Phys. A: Math. Gen. 41, 025004 (2008)
Hughes B.D.: Random Walks and Random Environments, vol 1. Random Walks. Clarendon, Oxford (1996)
Sciutto S.J.: J. Phys. A.: Math. Gen. 28, 3667 (1995)
Xia W., Thorpe M.F.: Phys. Rev. A. 38, 2650 (1988)
Yi Y.-B., Sastry A.M.: Phys. Rev. E 66, 066130 (2002)
Y. Wu, B. Schmittmann, R.K.P. Zia, J. Stat. Mech. P04002 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zia, R.K.P., Wu, Y. & Schmittmann, B. Percolation of a collection of finite random walks: a model for gas permeation through thin polymeric membranes. J Math Chem 45, 58–64 (2009). https://doi.org/10.1007/s10910-008-9367-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-008-9367-6