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Entropic rigidity of randomly diluted two- and three-dimensional networks
M Plischke1, D C Vernon, B Joós
1Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6.
Summary
Diluted networks with central forces remain rigid above the percolation threshold at finite temperatures. This contrasts with zero-temperature networks and supports universality class conjectures.
Area of Science:
- Physics
- Materials Science
- Network Theory
Background:
- Previous work showed site-diluted triangular networks are rigid above percolation at finite temperatures.
- Zero-temperature networks exhibit vanishing shear modulus at a higher concentration.
Purpose of the Study:
- Investigate rigidity of other diluted networks (bond-diluted triangular, site-diluted square, site-diluted simple-cubic).
- Determine the critical behavior of the shear modulus near the percolation threshold.
- Test de Gennes' conjecture on universality class.
Main Methods:
- Finite-temperature simulations of diluted central-force networks.
- Analysis of shear modulus (mu) near the percolation concentration (p(c)).
- Approximate renormalization group calculations.
Main Results:
- All simulated diluted networks (bond- and site-diluted) are rigid for p > p(c).
- Shear modulus scales as mu ~ (p-p(c))^f near p(c).
- Exponent f is approximately 1.3 for 2D and 2 for 3D simple-cubic lattices.
Conclusions:
- Diluted central-force networks are rigid above the percolation threshold at finite temperatures.
- Results support de Gennes' conjecture: universality class matches random resistor networks.
- Renormalization group calculations confirm this conclusion.