Abstract:We report an extensive finite-size study of polymer networks near the percolation threshold, using numerical techniques. The polymers are modeled by random walks occupying the bonds of a two-dimensional square lattice. We measure the percolation threshold and critical exponents of the networks for various polymer lengths. We find that the critical occupation probability is a decreasing function of the polymer length, and the percolation of polymers with a fixed polymer length belongs to the same universality c… Show more
“…All P ( φ ) curves are typical, i.e., S-shaped and the increase of the chain length shifts the curve toward lower polymer concentrations. Qualitatively the same behavior was found for other two-dimensional polymer chains: for short chains on the square lattice [17], long linear chains with explicit solvent molecules on the triangular lattice [26] and the square lattice [21] and for off-lattice hard ellipsoids [11]. One can also observe that the slope of curves decreases with the chains length which is caused by the finite size of the system.Fig.…”
Section: Resultssupporting
confidence: 73%
“…The second method of the determination of the percolation threshold for an infinite system is based on the finding that percolation probability curves P ( φ ) for systems of different size intersect in one point [11, 21, 36]. Figure 4 shows the percolation probabilities P as functions of the polymer concentrations φ for the chain N = 10 calculated for some sizes of the Monte Carlo box L .…”
The structure of a two-dimensional film formed by adsorbed polymer chains was studied by means of Monte Carlo simulations. The polymer chains were represented by linear sequences of lattice beads and positions of these beads were restricted to vertices of a two-dimensional square lattice. Two different Monte Carlo methods were employed to determine the properties of the model system. The first was the random sequential adsorption (RSA) and the second one was based on Monte Carlo simulations with a Verdier-Stockmayer sampling algorithm. The methodology concerning the determination of the percolation thresholds for an infinite chain system was discussed. The influence of the chain length on both thresholds was presented and discussed. It was shown that the RSA method gave considerably lower thresholds for longer chains. This behavior can be explained by a different pool of chain conformations used in the calculations in both methods under consideration.FigureThe percolation cluster (in red) in the system consisting of long flexible chains
“…All P ( φ ) curves are typical, i.e., S-shaped and the increase of the chain length shifts the curve toward lower polymer concentrations. Qualitatively the same behavior was found for other two-dimensional polymer chains: for short chains on the square lattice [17], long linear chains with explicit solvent molecules on the triangular lattice [26] and the square lattice [21] and for off-lattice hard ellipsoids [11]. One can also observe that the slope of curves decreases with the chains length which is caused by the finite size of the system.Fig.…”
Section: Resultssupporting
confidence: 73%
“…The second method of the determination of the percolation threshold for an infinite system is based on the finding that percolation probability curves P ( φ ) for systems of different size intersect in one point [11, 21, 36]. Figure 4 shows the percolation probabilities P as functions of the polymer concentrations φ for the chain N = 10 calculated for some sizes of the Monte Carlo box L .…”
The structure of a two-dimensional film formed by adsorbed polymer chains was studied by means of Monte Carlo simulations. The polymer chains were represented by linear sequences of lattice beads and positions of these beads were restricted to vertices of a two-dimensional square lattice. Two different Monte Carlo methods were employed to determine the properties of the model system. The first was the random sequential adsorption (RSA) and the second one was based on Monte Carlo simulations with a Verdier-Stockmayer sampling algorithm. The methodology concerning the determination of the percolation thresholds for an infinite chain system was discussed. The influence of the chain length on both thresholds was presented and discussed. It was shown that the RSA method gave considerably lower thresholds for longer chains. This behavior can be explained by a different pool of chain conformations used in the calculations in both methods under consideration.FigureThe percolation cluster (in red) in the system consisting of long flexible chains
“…10,27 In our investigation the percolation threshold contrary to both mentioned works decreases rapidly with the increase in the chain length in the regarded range. Similar results were obtained by means of the Monte Carlo simulations and the RSA technique.…”
Section: Discussioncontrasting
confidence: 70%
“…18 He showed that the percolation threshold in such a system exhibits a minimum for certain temperatures. [26][27][28] The system containing stiff molecules ͑rods͒ and spheres was also recently simulated and it was shown that the introduction of spheres diminished the percolation thresholds of rods. Another study, which sug-a͒ Tel.…”
In this study we investigated the percolation in the system containing long flexible polymer chains. The system also contained explicit solvent molecules. The polymer chains were represented by linear sequences of lattice points restricted to a two-dimensional triangular lattice. The Monte Carlo simulations were performed applying the cooperative motion algorithm. The percolation thresholds and the critical exponents of chains and solvent molecules were determined. The influence of the chain length on the percolation was discussed. It was shown that the percolation threshold decreased strongly with the chain length, which is closely connected to changes in chains' structure with the decreasing polymer concentration. The critical exponent beta for all chains under consideration and for solvent molecules was found almost constant and close to the theoretical value 5/36.
“…Various different correlated percolation models have been studied before, see for example [9][10][11][12] and also a case of "bootstrap" percolation in [13].…”
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