Cubic lattice percolation

Abstract. The square and cubic lattice percolation problem and the selfavoiding random walk model were simulated by Monte Carlo method in order to obtain new understanding of the fractal properties of branched and hnear polymer molecules. The central point of this work refers to the comparison between the cluster properties as they emerge from the percolation problem on one hand and the random walk properties on the other hand. It is shown that in both models there is a drastic difference between two and three dimensional systems. In three dimensions it is possible to find a regime where the properties converge towards simple non-avoided random walk, while in two dimensions the topological reasons prevent a smooth transition of the properties pertaining to avoided and non-avoided random walks. 

The side chain separation varies in a range of 1 nm or slightly above. The network of aqueous domains exhibits a percolation threshold at a volume fraction of 10%, which is in line with the value determined from conductivity studies. This value is similar to the theoretical percolation threshold for bond percolation on a face-centered cubic lattice. It indicates a highly interconnected network of water nanochannels. Notably, this percolation threshold is markedly smaller, and thus more realistic, than those found in atomistic simulations, which were not able to reproduce experimental values. 

The numerical solution of the system (4.86), by a procedure of the Newton-Raphson type with two variables, requires the calculation of the derivatives dtjds and dt /5Wc . The results we obtained for a square lattice are similar to those by Yonezawa and Odagaki180 for a cubic lattice. The most striking feature is the existence, at low concentration, of a gap in the density of states,179 which isolates the zero energy on which a 3 peak builds up. Thus the HCPA produces a forbidden region of energy for the transport the gap and the 3 peak disappear at a critical concentration, analogous to the percolation threshold of the mean-field of resistances. 

For anisotropic particles, the percolation limit is a function of the aspect ratio. For ellipsoids of revolution, the percolation limit for a simple cubic lattice was studied by Boissonade et al. [85]. They found as the aspect ratio increases from 1 (a sphere) to 15 (a fiber), the percolation limit decreased from a volume fraction of 0.31 to 0.06 and the correlation length (i.e., aggregate size) did not change (i.e., it was the same as that of the sphere). 

The numerical results reviewed above were obtained for infinite lattices. How do the various quantities of interest behave near the percolation threshold in a large but finite lattice This problem has been studied by renormalization methods, which are essentially equivalent to finite-size scaling. For finite lattices the percolation transition is smeared out over a range of p, and one must expect a similar trend in other functions, including the conductivity. Computer simulations by the Monte Carlo method have been carried out for bond percolation on a three-dimensional simple cubic lattice by Kirkpatrick (1979). Five such experimental curves are shown in Fig. 40, each of which corresponds to a cube of size b, containing bonds. In Fig. 40 the vertical axis gives the fraction p of such samples that percolate (i.e., have opposite faces con-... 

Fig. 40. Scaling for finite-sized lattices. Computer calculations of scaling properties for bond percolation on the three-dimensional simple cubic lattice. When p is the fraction of connected bonds, p = p (p,b) is the fraction of cubic samples of edge length b that contain a continuous path of connected bonds (a spanning cluster) which links opposite faces of the sample. From Kirkpatrick (1979).

A quantitative analysis of the failure process was made by Duxbury et al (1987) by modelling the system by a lattice and we shall present their results. The simplest lattices were taken a square lattice in two dimensions and a simple cubic lattice in three dimensions, in which the bonds are all equal resistors to begin with. Each resistor can stand a current up to io. If i > io, the resistor is fused and becomes a perfect insulator. It is believed that the results are not dependent on the type of the lattice as it was proven in the case of percolation. The size of the lattice is L in two dimensions it... 

Fig. 8. Parameter Zoq,qb at the percolation threshold as a function of for the simple cubic lattice. The drawing is constructed by employing the Monte Carlo results (33).

Employing Eq. (24), we may assume that the percolation probability for the sublattice of voids with r > rp is the same as the universal percolation probability for the bond problem (Section II). The latter probability was originally calculated only for regular lattices. The sublattice of voids with r > rp is not regular. In addition, some of the voids with r > rp are not connected with the other voids having r > rp. However, the numerical results obtained by Yanuka (33) for a randomized cubic lattice (see also the discussion in Section II) support the hypothesis on the universality of the percolation probability for both regular and irregular lattices. 

Experimental results of water vapor adsorption. Helium relative permeability, Pr, and water vapor permeability, Pe, for the two alumina pellets are presented in figures 6a and 6b, for water relative pressures up to unity. As the amount of water adsorbed starts to rapidly increase with P/Po, due to capillary condensation, a significant increase of its permeability may also be observed due to the resulting capillary enhancement of flow. At a certain value of P/Po where Vs is close to unity, all pores of the membrane are in the capillary condensation regime and thus follow the capillary enhanced type of flux. At this point water vapor permeability reaches its maximum value while, helium relative permeability decreases rapidly and falls to zero well below the point of saturation. This may be attributed, according to percolation theory, to the fact that in a simple cubic lattice, if -75% of the pores are blocked by capillary condensate, the system has reached its percolation threshold and helium... 

If the grains were spheres on a simple cubic lattice, they would interact at ar = sr/6 the intersection would correspond to a percolation threshold. Because we see no percolation until x <=> 3, the expression we use (corresponding to a lattice of cubes) is the simplest choice. 

A Bethe-tree is a particular case of more general networks considered in percolation theory, which is used to a growing extent to describe transport inside catalysts, as evidenced by a recent review by Sahimi et al [ref 26] Sahimi and Tsotsis [ref. 27] applied percolation theory and Honte Carlo simulation to deactivation in zeolites, represented by a simple cubic lattice. 

In this work, we describe the results of theoretical studies of Poisson s ratio in disordered structures composed of two phases of disparate elastic properties applying a renormalization group approach to a model of percolation on a hierarchical cubic lattice. Although this approach has been described in detail elsewhere [160], we present it briefly here, for completeness. At the percolation... 

The results of calculations of the effective Poisson s ratio vp dependence on the bulk concentration of a rigid phase p at various values of a = log i/C/Au) are shown in Fig. 53. The calculations were made for Poisson s ratios of the phases ranging from 0.1 to 0.4. It can be seen that at percolation threshold Poisson s ratio of the isotropic fractal composite is vp = 0.2, when K jK > 0 it is also independent of the Poisson s ratios of the individual components of the composite. The Poisson s ratio obtained by us near the percolation threshold is in agreement with computer simulation results and the conjecture of Arbabi and Sahimi [161]. It has been shown that an approximate theoretical treatment of percolation on a cubic lattice exactly reproduces the Poisson s ratio obtained in computer simulation at the percolation threshold. This result may encourage one to use this approximation to describe various elastic properties of composites. It is worth noting that some critical indices have been calculated recently with a high degree of accuracy in the context of the present model. 

The MC simulation of Kremer [16] has been referred to as the only numerical estimate for Vp for a number of years, until Lee et al. [17] performed simulations for SAWs on square and cubic lattices at dilution very close to the percolation threshold. Their results for the SAWs critical exponent i/jc rather surprisingly indicate that the exponent i/jc even at pc is close to the pure result. Furthermore, numerical uncertainties in the [16] were indicated namely, it was noted, that the chains used in that simulation are probably not long enough to estimate the critical exponents. 

Fig. 3.14. The fraction of the extended states in the minority subband (%) as a function of the minority element concentration for 8 = >. Curve (a) has been calculated from percolation theory for a simple cubic lattice. Curve (b) is calculated from the approximate localization function F(E) computed within the CPA for simple cubic lattice.

The main topics in lattice theories, which are relevant for the polymer subject are avoided random walk, lattice percolation [3] and lattice spin models. In this work we shall put the emphasis on the numerical investigation of the systems in the framework of lattice percolation methodologies and avoided random walks on square and cubic lattices. 

Linke et al. (1983) and Franz (1984, 1986) developed a different version of the lattice-gas concept. These models emphasize the experimental observation (Secs. 3.4 and 4.4) that thermal expansion of liquids is mainly achieved by a reduction of the average near-neighbor coordination number. A given structure such as the artificial, but mathematically convenient Cayley Tree or Bethe lattices can, when partially populated, be viewed as a crystalline alloy of atoms and vacancies. Tight-binding methods then permit calculation of the electronic structure, in particular the density of electronic states. Franz made use of quantum percolation theory to model the DC conductivity. A more recent model (Tara-zona et al., 1996) employs a body-centered-cubic lattice which, when fully occupied, provides a reasonable approximation to the local structure of liquid metals near the melting point. 

Numerical simulations of percolation structures in a centered cubic lattice... 

Scalar percolation theory deals with the connectivity of a component randomly dispersed in another [7-8]. Examples of percolation are gelation during a polymerization of monomers with multifunctional linkages and the onset of conductivity in blends of conducting and non-conducting materials [9-10]. The percolation threshold p for a finite-sized object is defined as the minimum concentration (of the percolating medium) at which connectivity is established between the top and bottom surface, is different for lattices of different geometry [7]. For Id site percolation, p = 59.20%, while = 31.17% for a cubic lattice. 

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