Percolation in Systems Containing Ordered Elongated Objects
Romiszowski Piotr *, Sikorski Andrzej
Department of Chemistry, University of Warsaw
Pasteura 1, 02-093 Warsaw, Poland
*E-mail: prom@chem.uw.edu.pl
Received:
Received: 21 February 2013; revised: 13 April 2013; accepted: 16 April 2013; published online: 22 May 2013
DOI: 10.12921/cmst.2013.19.02.115-121
OAI: oai:lib.psnc.pl:463
Abstract:
We studied the percolation and jamming of elongated objects using the Random Sequential Adsorption (RSA) technique. The objects were represented by linear sequences of beads forming needles. The positions of the beads were restricted to vertices of two-dimensional square lattice. The external field that imposed ordering of the objects was introduced into the model. The percolation and the jamming thresholds were determined for all systems under consideration. The influence of the chain length and the ordering on both thresholds was calculated and discussed. It was shown that for a strongly ordered system containing needles the ratio of percolation and jamming thresholds cp/cj is almost independent on the needle length d.
Key words:
jamming, Monte Carlo method, percolation, random sequential adsorption
References:
[1] D. Stauffer, A. Aharony, Introduction to Percolation Theory (Taylor
and Francis, London 1994).
[2] L.N. Lisetski, S.S. Minenko, A.P. Fedoryako and N.I. Lebovka, Dis-
persions of multiwalled carbon nanotubes in different nematic meso-
gens: The study of optical transmittance and electrical conductivity,
Physica E 41, 431 (2009).
[3] J.G. Meier, C. Crespo, J.L. Pelegay, P. Castell, R. Sainz, W.K. Maser,
A.M. Benito, Processing dependency of percolation threshold of
MWCNTs in a thermoplastic elastomeric block copolymer, Polymer
52, 1788 (2011).
[4] Matoz-Fernandes , D.H. Linares, A.J. Ramirez-Pastor, Europhys.Lett.,
Determination of the critical exponents for the isotropic-nematic
phase transition in a system of long rods on two-dimensional lattices:
Universality of the transition, Europhys.Lett. 82, 50007 (2008).
[5] L.G. Lopez, D.H. Linares, A.J. Ramirez-Pastor, S.A. Cannas, Phase
diagram of self-assembled rigid rods on two-dimensional lattices:
Theory and Monte Carlo simulations, J. Chem. Phys. 133, 134706 (2010).
[6] V. Cornette, A.J. Ramirez-Pastor, F. Nieto, Dependence of the per-
colation threshold on the size of the percolating species, Physica A
327, 71 (2003).
[7] V. Cornette, A.J. Ramirez-Pastor, F. Nieto, Percolation of polyatomic
species on a square lattice, Eur. Phys. J. B 36, 391 (2003).
[8] J.W. Evans, Random and cooperative sequential adsorption, Rev.
Mod. Phys. 65, 1281 (1993).
[9] J. Talbot, G. Tarjus, P.R. Van Tassel, P. Viot, From car parking to
protein adsorption: an overview of sequential adsorption processes,
Colloid. Surface A 165, 287 (2000).
[10] P. Adamczyk, P. Polanowski, A. Sikorski, Percolation in polymer-
solvent systems: A Monte Carlo study, J. Chem. Phys. 131, 234901 (2009).
[11] M. Pawłowska, S. Żerko, A. Sikorski, Note: Percolation in two-
dimensional flexible chains systems, J. Chem. Phys. 136, 046101 (2012).
[12] R.D. Vigil, R.M. Ziff, Random sequential adsorption of unoriented
rectangles onto a plane, J. Chem. Phys. 91, 2599 (1989).
[13] R.M. Ziff, R.D. Vigil, Kinetics and fractal properties of the random
sequential adsorption of line segments, J. Phys. A: Math. Gen. 23,
5103 (1990).
[14] N. Vandewalle, S. Galam, M. Kramer, A new universality for random
sequential deposition of needles, Eur. Phys. J. B 14, 407 (2000).
[15] G. Kondrat, A. Pekalski, Percolation and jamming in random se-
quential adsorption of linear segments on a square lattice,Phys. Rev.
E 63, 051108 (2001).
[16] G. Kondrat, A. Pekalski, Percolation and jamming in random bond
deposition, Phys. Rev. E 64, 056118 (2001).
[17] G. Kondrat, Influence of temperature on percolation in a simple
model of flexible chains adsorption, J. Chem. Phys. 117, 6662 (2002).
[18] P. Adamczyk, P. Romiszowski, A. Sikorski, A simple model of stiff
and flexible polymer chain adsorption: The influence of the internal
chain architecture, J. Chem. Phys. 128, 154911 (2008).
[19] E.J. Garboczi, K.A. Snyder, J.F. Douglas, M.F. Thorpe, Geometrical
percolation threshold of overlapping ellipsoids, Phys. Rev. E 52, 819
(1995).
[20] J.-S. Wang, Series expansion and computer simulation studies of
random sequential adsorption, Colloid. Surface A 165, 325 (2000).
[21] X. Wang, A.P. Chatterjee, Connectedness percolation in athermal
mixtures of flexible and rigid macromolecules: Analytic theory, J.
Chem. Phys. 118, 10787 (2003).
[22] Y.B. Yi, A.M. Sastry, Analytical approximation of the percolation
threshold for overlapping ellipsoids of revolution, Proc. Royal Soc.
Lond. A 460, 2353 (2004).
[23] A. P. Chatterjee, Percolation thresholds for rod-like particles: poly-
dispersity effects, J. Phys.: Condens. Matter 20, 255250 (2008).
[24] V.A. Cherkasova, Y.Y. Tarasevich, N.I. Lebovka, N.V. Vygornitskij,
Percolation of aligned dimers on a square lattice, Eur. Phys. J. B 74,
205 (2010).
[25] N.I. Lebovka, N.N. Karmazina, Y.Yu. Tarasevich, V.V. Laptev, Ran-
dom sequential adsorption of partially oriented linear k-mers on
a square lattice, Phys. Rev. E, 85, 029902 (2012).
[26] A. Ghosh, D. Dhar, On the orientational ordering of long rods on
a lattice, Eur. Phys. Lett. 78, 20003 (2007).
[27] P.Kählitz, H.Stark, Phase ordering of hard needles on a quasicrys-
talline substrate, J.Chem.Phys. 136, 174705 (2012).
[28] Y.Yu. Tarasevich, N.I.Lebovka, V.V. Laptev, Percolation of linear
k-mers on a square lattice: From isotropic through partially ordered
to completely aligned states, Phys. Rev. E 86, 061116 (2012).
[29] J. Hoshen, R. Kopelman, Percolation and cluster distribution. I. Clus-
ter multiple labeling technique and critical concentration algorithm,
Phys. Rev. B 14, 3438 (1976).
[30] M. Dolz, F. Nieto, A.J. Ramirez-Pastor, Dimer site-bond percolation
on a square lattice, Eur. Phys. J. B 43, 363 (2005).
We studied the percolation and jamming of elongated objects using the Random Sequential Adsorption (RSA) technique. The objects were represented by linear sequences of beads forming needles. The positions of the beads were restricted to vertices of two-dimensional square lattice. The external field that imposed ordering of the objects was introduced into the model. The percolation and the jamming thresholds were determined for all systems under consideration. The influence of the chain length and the ordering on both thresholds was calculated and discussed. It was shown that for a strongly ordered system containing needles the ratio of percolation and jamming thresholds cp/cj is almost independent on the needle length d.
Key words:
jamming, Monte Carlo method, percolation, random sequential adsorption
References:
[1] D. Stauffer, A. Aharony, Introduction to Percolation Theory (Taylor
and Francis, London 1994).
[2] L.N. Lisetski, S.S. Minenko, A.P. Fedoryako and N.I. Lebovka, Dis-
persions of multiwalled carbon nanotubes in different nematic meso-
gens: The study of optical transmittance and electrical conductivity,
Physica E 41, 431 (2009).
[3] J.G. Meier, C. Crespo, J.L. Pelegay, P. Castell, R. Sainz, W.K. Maser,
A.M. Benito, Processing dependency of percolation threshold of
MWCNTs in a thermoplastic elastomeric block copolymer, Polymer
52, 1788 (2011).
[4] Matoz-Fernandes , D.H. Linares, A.J. Ramirez-Pastor, Europhys.Lett.,
Determination of the critical exponents for the isotropic-nematic
phase transition in a system of long rods on two-dimensional lattices:
Universality of the transition, Europhys.Lett. 82, 50007 (2008).
[5] L.G. Lopez, D.H. Linares, A.J. Ramirez-Pastor, S.A. Cannas, Phase
diagram of self-assembled rigid rods on two-dimensional lattices:
Theory and Monte Carlo simulations, J. Chem. Phys. 133, 134706 (2010).
[6] V. Cornette, A.J. Ramirez-Pastor, F. Nieto, Dependence of the per-
colation threshold on the size of the percolating species, Physica A
327, 71 (2003).
[7] V. Cornette, A.J. Ramirez-Pastor, F. Nieto, Percolation of polyatomic
species on a square lattice, Eur. Phys. J. B 36, 391 (2003).
[8] J.W. Evans, Random and cooperative sequential adsorption, Rev.
Mod. Phys. 65, 1281 (1993).
[9] J. Talbot, G. Tarjus, P.R. Van Tassel, P. Viot, From car parking to
protein adsorption: an overview of sequential adsorption processes,
Colloid. Surface A 165, 287 (2000).
[10] P. Adamczyk, P. Polanowski, A. Sikorski, Percolation in polymer-
solvent systems: A Monte Carlo study, J. Chem. Phys. 131, 234901 (2009).
[11] M. Pawłowska, S. Żerko, A. Sikorski, Note: Percolation in two-
dimensional flexible chains systems, J. Chem. Phys. 136, 046101 (2012).
[12] R.D. Vigil, R.M. Ziff, Random sequential adsorption of unoriented
rectangles onto a plane, J. Chem. Phys. 91, 2599 (1989).
[13] R.M. Ziff, R.D. Vigil, Kinetics and fractal properties of the random
sequential adsorption of line segments, J. Phys. A: Math. Gen. 23,
5103 (1990).
[14] N. Vandewalle, S. Galam, M. Kramer, A new universality for random
sequential deposition of needles, Eur. Phys. J. B 14, 407 (2000).
[15] G. Kondrat, A. Pekalski, Percolation and jamming in random se-
quential adsorption of linear segments on a square lattice,Phys. Rev.
E 63, 051108 (2001).
[16] G. Kondrat, A. Pekalski, Percolation and jamming in random bond
deposition, Phys. Rev. E 64, 056118 (2001).
[17] G. Kondrat, Influence of temperature on percolation in a simple
model of flexible chains adsorption, J. Chem. Phys. 117, 6662 (2002).
[18] P. Adamczyk, P. Romiszowski, A. Sikorski, A simple model of stiff
and flexible polymer chain adsorption: The influence of the internal
chain architecture, J. Chem. Phys. 128, 154911 (2008).
[19] E.J. Garboczi, K.A. Snyder, J.F. Douglas, M.F. Thorpe, Geometrical
percolation threshold of overlapping ellipsoids, Phys. Rev. E 52, 819
(1995).
[20] J.-S. Wang, Series expansion and computer simulation studies of
random sequential adsorption, Colloid. Surface A 165, 325 (2000).
[21] X. Wang, A.P. Chatterjee, Connectedness percolation in athermal
mixtures of flexible and rigid macromolecules: Analytic theory, J.
Chem. Phys. 118, 10787 (2003).
[22] Y.B. Yi, A.M. Sastry, Analytical approximation of the percolation
threshold for overlapping ellipsoids of revolution, Proc. Royal Soc.
Lond. A 460, 2353 (2004).
[23] A. P. Chatterjee, Percolation thresholds for rod-like particles: poly-
dispersity effects, J. Phys.: Condens. Matter 20, 255250 (2008).
[24] V.A. Cherkasova, Y.Y. Tarasevich, N.I. Lebovka, N.V. Vygornitskij,
Percolation of aligned dimers on a square lattice, Eur. Phys. J. B 74,
205 (2010).
[25] N.I. Lebovka, N.N. Karmazina, Y.Yu. Tarasevich, V.V. Laptev, Ran-
dom sequential adsorption of partially oriented linear k-mers on
a square lattice, Phys. Rev. E, 85, 029902 (2012).
[26] A. Ghosh, D. Dhar, On the orientational ordering of long rods on
a lattice, Eur. Phys. Lett. 78, 20003 (2007).
[27] P.Kählitz, H.Stark, Phase ordering of hard needles on a quasicrys-
talline substrate, J.Chem.Phys. 136, 174705 (2012).
[28] Y.Yu. Tarasevich, N.I.Lebovka, V.V. Laptev, Percolation of linear
k-mers on a square lattice: From isotropic through partially ordered
to completely aligned states, Phys. Rev. E 86, 061116 (2012).
[29] J. Hoshen, R. Kopelman, Percolation and cluster distribution. I. Clus-
ter multiple labeling technique and critical concentration algorithm,
Phys. Rev. B 14, 3438 (1976).
[30] M. Dolz, F. Nieto, A.J. Ramirez-Pastor, Dimer site-bond percolation
on a square lattice, Eur. Phys. J. B 43, 363 (2005).
