Fractal Percolation Research Articles

A.C. conductivity measurements in polymeric insulator conductor systems

Measurements of the a.c. conductivity are reported in three types of polymeric insulator-conductor composites for frequencies between d.c. and 9.4 GHz. The a.c. conductivity is shown to follow the general law σ = ω x for frequencies higher than a critical frequency. For every concentration of the conducting additive a frequency region has been found where the predictions for anomalous diffusion in fractal percolating clusters can be considered as fulfilled. At higher frequencies a stronger conductivity dependence has been found which has been interpreted as being due to the polarization of the polymeric matrix between clusters.

Study of recombination reactions of particles adsorbed on fractal and multifractal substrata

Recombination reactions of adsorbed particles on fractal and multifractal media are discussed within the framework of the random walk arguments. Theoretical results, which predict anomalous reaction orderX>2 in the low coverage regime, are checked by means of Monte Carlo simulations on two-dimensional structures and good agreement is found. Thermal desorption experiments on rough surfaces are simulated by studying temperature programmed reactions on fractal percolating clusters. For this case the simulations giveX≅2.5, i. e. the fractal reaction order is greater than the classical one (X=2). The influence of chemisorbed impurities (poison) on the recombination reaction is also studied and the reaction order is found to increase beyondX=2.5 when increasing the concentration of poison. Isothermal (recombination) desorption from energetically heterogeneous surfaces is simulated on two-dimensional substrata with multifractal distributions of activation energy of diffusion. For this caseX (withX>2) depends on the energetic heterogeneity of the substrata (X=2 for an homogeneous substratum). The obtained results point out that the fractal chemical kinetic behaviour is not only restricted to the low concentration regime, but it also covers the medium coverage regime, i.e. it holds for a monolayer surface coverageθ≦0.4 in fractal percolating clusters.

Fractal description of ferromagnetic glasses and random Josephson networks

The well-known analogies between X-Y ferromagnets and superconducting junction arrays are discussed in the light of data on critical exponents of the latter. The role of disorder is analysed. It is shown that, although the Harris criterion seemingly predicts that disorder is irrelevant, its application to fractal percolating clusters predicts exponent values in very good agreement with experiment.

Diffusion and spin correlation in fractal percolation clusters.

We have performed Monte Carlo simulations and theoretical calculations of the time decay behavior of the dipolar and spin exchange autocorrelation functions ${G}_{N}$(\ensuremath{\tau}) over a range of concentrations p (0<p<1) of randomly diluted magnetic species diffusing in a lattice. For p below the percolation threshold ${p}_{c}$, ${G}_{N}$(\ensuremath{\tau}) decays exponentially, at long time, with an effective time constant \ensuremath{\tau}(p) increasing with the concentration. We show that this result is characteristic of a bounded diffusion in a single average cluster explored in the average time 〈\ensuremath{\tau}(p)〉.At the percolation threshold p=${p}_{c}$, ${G}_{N}$(\ensuremath{\tau}) tends asymptotically towards the same power law ${\ensuremath{\tau}}^{\mathrm{\ensuremath{-}}{d}_{\mathrm{eff}}({p}_{c})/2}$ either for a dipolar or a spin exchange interaction. Here ${d}_{\mathrm{eff}}$(${p}_{c}$) is an effective dimensionality, which appears to be lower than the Euclidean dimensionality d. A theoretical calculation of the kinetics of reencounters of two spins diffusing in a distribution of clusters leads to the relation ${d}_{\mathrm{eff}}$(${p}_{c}$)=2-${d}_{s}$(3-${\ensuremath{\tau}}_{a}$) in terms of the spectral density ${d}_{s}$ and the critical exponent ${\ensuremath{\tau}}_{a}$ for the distribution of clusters in the theory of percolation. This relation gives ${d}_{\mathrm{eff}}$(${p}_{c}$)=0.76, 0.95, and (4/3 for d=2, d=3, and d=6, respectively. The asymptotic expression found for ${G}_{N}$(\ensuremath{\tau}) is then coherent with a process of anomalous diffusion in a reduced spatial dimensionality. Finally for p above ${p}_{c}$, one finds for each value of p, a power law G(\ensuremath{\tau})\ensuremath{\propto}${\ensuremath{\tau}}^{\mathrm{\ensuremath{-}}{d}_{\mathrm{eff}}(p)/2}$. There exists a crossover for ${d}_{\mathrm{eff}}$(p) between ${d}_{\mathrm{eff}}$(${p}_{c}$) and d for times sufficiently long that the temperature-dependent diffusion length exceeds the concentration dependent correlation length. This solves the seeming contradiction appearing in recent interpretation of nuclear-magnetic-resonance (NMR) results in mixed paramagnetic compounds.

Dynamic percolation in an insulator-conductor composite: Polyethyleneoxide-polypyrrole, as studied by a.c. conductivity measurements

We report measurements of a.c. conductivity in the Insulator-Conductor polymeric system Polyethyleneoxide-Polypyrrole for frequencies between d.c. and 9.4 GHz. The a.c. conductivity is shown to follow the general law σ(ω) ⋍ ω x for frequencies higher than a critical frequency. For every concentration of polypyrrole a frequency region has been found where the predictions for anomalous diffusion in three-dimensional fractal percolating clusters (x = 0.6) can be considered as fulfilled. The calculated critical frequencies exhibit a dependence on the concentration of the sonducting polypyrrole additive as expected by the theory through the expression for the correlation length of the Random Resistor Network of Miller and Abrahams. This model was previously employed to describe the d.c. behaviour of these composite systems.

The fractal dimension and other percolation exponents in four and five dimensions

For percolation the fractal dimension df, which is identical to the magnetic field scaling power yh, has never been calculated for hypercubic lattices of dimension d=5. Percolation is studied for systems of high dimensionality using the method of large-cell Monte Carlo position-space renormalisation group. The estimate df=yh=3.69+or-0.02 (d=5) and 3.12+or-0.02 (d=4) is obtained. The thermal scaling power yT=1/ nu where nu is the correlation length exponent is calculated. yT-1= nu =0.51+or-0.05 (d=5) and nu =0.64+or-0.02 (d=4). The results are compared with the epsilon expansions of df and nu .

On the spreading dimension of percolation and directed percolation clusters

The authors study the number AN of sites that are accessible after N steps at most on clusters at the percolation threshold. On a Cayley tree AN is of order N2 if the origin belongs to a larger cluster, whereas its average over all clusters is of order N. This suggests that the intrinsic spreading dimension d , defined by AN approximately Nd, is equal to two for fractal percolation clusters in space dimensions d>or=6 and depends on d for d<6. For directed percolation clusters they argue that d is related to usual critical exponents by d=( beta + gamma )/ nu /sub ///. Monte Carlo data that support this relation are presented in two dimensions. Analogous results are derived for lattice animals d=2 on the Cayley tree and d=1/ nu /sub /// for directed animals in any dimensions.

On intrinsic properties of fractal lattices and percolation clusters

Certain properties of fractal lattices are independent of the Euclidean embedding. The implications of this invariance are discussed for various problems: the conductivity of percolation clusters, inequalities between intrinsic dimensions, and the diffusion coefficient in a flow through a porous medium Certaines proprietes des reseaux fractals sont independantes du plongement euclidien. On discute les consequences de cette invariance pour divers problemes: conductivite d'amas de percolation, inegalites entre dimensions intrinseques, coefficient de diffusion dans un milieu poreux en presence d'un ecoulement