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Percolation of aligned rigid rods on two-dimensional triangular lattices.

P Longone1, P M Centres1, A J Ramirez-Pastor1

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Percolation threshold for k-mers on triangular lattices increases with k-mer size, unlike square lattices. The system belongs to the random percolation universality class.

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Area of Science:

  • Statistical Physics
  • Materials Science
  • Computational Physics

Background:

  • Percolation theory studies the formation of connected clusters in random systems.
  • Understanding how particle size and lattice geometry influence percolation is crucial for materials design.

Purpose of the Study:

  • To investigate the percolation behavior of aligned rigid rods (k-mers) on two-dimensional triangular lattices.
  • To determine the relationship between k-mer size and percolation threshold.
  • To analyze the effect of anisotropy and universality class.

Main Methods:

  • Numerical simulations were employed to study the percolation behavior.
  • Finite-size scaling analysis was used to determine critical exponents.
  • Connectivity analysis was performed by calculating the percolation probability R_{L,k}(p).

Main Results:

  • The percolation threshold p_{c}(k) increases with k-mer size on triangular lattices, following p_{c}(k)=A+B/(C+sqrt[k]).
  • This contrasts with square lattices where the threshold decreases with k-mer size.
  • Anisotropy favors percolation along the alignment axis in finite systems, but the threshold is direction-independent in the thermodynamic limit.

Conclusions:

  • The phase transition belongs to the standard random percolation universality class for all k-mer sizes.
  • The findings provide insights into the influence of particle shape and lattice structure on percolation phenomena.