Continuum percolation threshold for interpenetrating squares and cubes
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Abstract:
Monte Carlo simulations are performed to determine the critical percolation threshold for interpenetrating square objects in two dimensions and cubic objects in three dimensions. Simulations are performed for two cases: (i) objects whose edges are aligned parallel to one another and (ii) randomly oriented objects. For squares whose edges are aligned, the critical area fraction at the percolation threshold phi(c)=0.6666+/-0.0004, while for randomly oriented squares phi(c)=0.6254+/-0.0002, 6% smaller. For cubes whose edges are aligned, the critical volume fraction at the percolation threshold phi(c)=0.2773+/-0.0002, while for randomly oriented cubes phi(c)=0.2168+/-0.0002, 22% smaller.Keywords:
Percolation (cognitive psychology)
Volume fraction
Cube (algebra)
Square (algebra)
We evaluate the percolation threshold values for a realistic model of continuum segregated systems, where random spherical inclusions forbid the percolating objects, modeled by hardcore spherical particles surrounded by penetrable shells, to occupy large regions inside the composite. We find that the percolation threshold is generally a nonmonotonous function of segregation, and that an optimal (i.e., minimum) critical concentration exists well before maximum segregation is reached. We interpret this feature as originating from a competition between reduced available volume effects and enhanced concentrations needed to ensure percolation in the highly segregated regime. The relevance with existing segregated materials is discussed.
Percolation (cognitive psychology)
Percolation Theory
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Percolation (cognitive psychology)
Percolation Theory
Local structure
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Large-scale modelling of short-fibre metal-loaded composites is described. The aim was to establish the relationship between fibre aspect ratio (A) and the critical volume fraction (Vc ) at which conductivity via a percolation mechanism occurs. For typical geometries, Vc was found to be proportional to (I/A)1 · 5 for 50 A 500. This gave good agreement with a first-order theoretical model (which predicts a I/A dependence for Vc ) for 150 A 500.
Volume fraction
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Aspect ratio (aeronautics)
Percolation Theory
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Percolation (cognitive psychology)
Volume fraction
Percolation Theory
Matrix (chemical analysis)
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The conductivity of two-dimensional inhomogeneous Au films near the percolation threshold is investigated in the temperature interval 0.5–55 K. The percolation threshold was approached by irradiation of initially continuous films (of thickness ∼ 10 nm) by Ar ions. It is found that near the percolation threshold (on the insulating side of the metal–insulator transition), the film exhibits one-dimensional hopping conduction. The fluctuations of hopping conductivity were also observed. It is shown that the conductivity of a two-dimensional percolation film in the vicinity of the percolation threshold can be determined by a single conducting chain. The presence of narrow insulating bridges in this chain determines the one-dimensional behavior of hopping conduction of the entire system as well as the observed resistance fluctuations. These fluctuations can be treated as a peculiar manifestation of incoherent mesoscopic effects.
Mesoscopic physics
Percolation (cognitive psychology)
Directed percolation
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Monte Carlo simulations are performed to determine the critical percolation threshold for interpenetrating square objects in two dimensions and cubic objects in three dimensions. Simulations are performed for two cases: (i) objects whose edges are aligned parallel to one another and (ii) randomly oriented objects. For squares whose edges are aligned, the critical area fraction at the percolation threshold phi(c)=0.6666+/-0.0004, while for randomly oriented squares phi(c)=0.6254+/-0.0002, 6% smaller. For cubes whose edges are aligned, the critical volume fraction at the percolation threshold phi(c)=0.2773+/-0.0002, while for randomly oriented cubes phi(c)=0.2168+/-0.0002, 22% smaller.
Percolation (cognitive psychology)
Volume fraction
Cube (algebra)
Square (algebra)
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Citations (127)
Percolation (cognitive psychology)
Aspect ratio (aeronautics)
Percolation Theory
Matrix (chemical analysis)
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Percolation (cognitive psychology)
Constant (computer programming)
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Traditionally, isotropically conductive adhesives (ICAs) have been prepared using a high volume fraction of solid silver flakes and particles in a non-conducting adhesive matrix. However, this paper focuses on newly developed ICAs containing a large volume fraction of silver coated polymer spheres (Ag-PS) as the filler in a non-conductive adhesive matrix. This paper mainly investigates the effect of thickness/morphology of the silver coating on the electrical conductivity and percolation threshold of such ICAs. For this study 30μm polymer spheres with four different coating thicknesses have been used. B oth the percolation threshold and the resistivity for a given volume fraction have been found to decrease with increasing thickness of the Ag coating.
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Computer‐generated pictures are presented of the connected component (''infinite cluster'') found at concentrations just above the threshold for 2D site percolation in large (400×400 site) lattices. For each case, we also show the ''backbone'' of the cluster, the smaller set of sites through which a current may flow. The simulations are contrasted with the model of conduction just above threshold due to Skal and Shklovskii and to de Gennes. That model is found to be inconsistent with the observed critical behavior of the conductivity in 2D and 3D models, but may apply to percolation in 4D and above. We show that a proper treatment of inhomogeneity on scales smaller than the coherence length is necessary to account for the observed conductivity and backbone volume just above threshold, and introduce a self‐similar model which accounts reasonably well for these properties.
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Percolation Theory
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