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Bond-percolation processes are studied for random lattices on the surface of a sphere, and for their duals. The estimated threshold is 0.3326 ± 0.0005 for spherical random lattices and 0.6680 ± 0.0005 for the duals of spherical random lattices, and the exact threshold is conjectured as 1/3 for two-dimensional random lattices and 2/3 for their duals. A suitably defined spanning probability at the threshold, E_p(p_c), for both spherical random lattices and their duals is 0.980±0.005, which may be universal for a 2-d lattice with this spanning definition. The shift-to-width ratio of the distribution function of the threshold concentration and the universal values of the critical value of the effective coordination number can be extended from regular lattices to spherical random lattices and their duals. The results of critical exponents are consistent with the assertion from the universality hypothesis. Finite-size scaling is also examined.

We consider site percolation on a correlated bi-colored simple cubic lattice. The correlated medium is constructed from a strongly alternating bi-colored simple cubic lattice due to anti-site disordering. The percolation threshold is estimated. The cluster size distribution is obtained. A possible application to the double 1:1 perovskites is discussed.

In this paper, we present a Monte Carlo study of four explosive bond percolation models on the square lattice: (i) product rule which suppresses intrabonds (PR-SI), (ii) sum rule which suppresses intrabonds (SR-SI), (iii) product rule which enhances intrabonds (PR-EI) and (iv) sum rule which enhances intrabonds (SR-EI). By performing extensive simulations and finite-size scaling analysis of the wrapping probability $R^{(x)}$ and a ratio $Q$ for PR-SI, SR-SI, PR-EI, and the composite quantities $Z_R$ and $Z_Q$ for SR-EI (defined by $R^{(x)}$ and $Q$ corresponding to two different $p$-values, respectively), we determine the thresholds $p_c$ of all models with best precision. We also estimate the critical exponents $\beta ∕\nu$ and $\gamma ∕\nu$ for PR-SI, SR-SI and PR-EI by studying the critical behaviors of the size of the largest cluster $C_1$ and the second moment $M_2$ of sizes of all clusters. For SR-EI, from $C_1$ and $M_2$, we only obtain pseudo-critical exponents, which are nonphysical. Precisely at $p_c$, we study the critical cluster-size distribution $n(s,L)$ (number density of the clusters of size $s$) for all models and find that it can be described by $n(s,L)\sim s^{-{\tau }^{\prime }}ñ(s∕L^{d^{\prime }})$, where ${\tau }^{\prime }=1+d∕d^{\prime }$ with $d^{\prime }=d_{\mathrm{\text{F}}}$ (fractal dimension) for PR-SI, SR-SI, PR-EI, $d^{\prime }=d$ (spatial dimension) for SR-EI, and $ñ(x)$ with $x=s∕L^{d^{\prime }}$ is an universal scaling function. Based on critical cluster-size distribution, we conjecture the values of $\beta ∕\nu$ (and $\gamma ∕\nu$ with the help of a scaling relation) for SR-EI. It is found that the exponents for PR-SI and SR-SI are consistent with each other, but the ones for PR-EI and SR-EI are different. Our results disclose two facts: (1) all models investigated here undergo continuous phase transitions, since their behaviors can be described by typical scaling formulas for continuous phase transitions; (2) PR-SI and SR-SI belong to a same universality class, however, PR-EI and SR-EI belong to different universality classes, and all of them differ from which random percolation belongs to. This work provides a testing ground for future theoretical studies.

Site and bond percolation thresholds are calculated for the face centered cubic, body centered cubic and diamond lattices in four, five and six dimensions. The results are used to study the behavior of percolation thresholds as a functions of dimension. It is shown that the predictions from a recently proposed invariant for percolation thresholds are not satisfactory for these lattices.

The universal law for percolation thresholds proposed by Galam and Mauger (GM) is found to apply also to dynamical situations. This law depends solely on two variables, the space dimension d and a coordinance number q. For regular lattices, q reduces to the usual coordination number while for anisotropic lattices it is an effective coordination number. For dynamical percolation we conjecture that the law is still valid if we use the number q₂ of second nearest neighbors instead of q. This conjecture is checked for the dynamic epidemic model which considers the percolation phenomenon in a mobile disordered system. The agreement is good.

Polymer–metal composites with different fillers, such as nanocrystalline nickel (n-Ni), core shell n-Ni and nickel oxide (NiO)[n-Ni@NiO] were prepared under the same processing conditions with polyvinyledene fluoride matrix. The larger value of critical exponents (s and s') and percolation threshold (f_c ~ 0.30) for n-Ni@NiO composites as compared to n-Ni composites (f_c ~ 0.07) and a comparable effective dielectric constant (ε_eff ~ 300) with low loss tangent (tan δ ~ 0.1) at 100 Hz in case of percolative n-Ni@NiO composite was observed. The core shell structure [n-Ni@NiO] also shows a very high value of ε_eff ~ 6000 with tan δ ~ 8 at 40 Hz. The results have been explained by using boundary layer capacitor effect and the percolation theory. The difference in f_c and critical exponents is attributed to NiO insulating layer that gives rise to different extent of continuumness at f_c and have been explained with the help of Swiss cheese model.

Carbon nanotubes (CNT) is turning out to be a replacement to all the existing traditional sensors due to their high gauge factor, multidirectional sensing capability, high flexibility, low mass density, high dynamic range and high sensitivity to strains at nano and macro- scales. The strain sensitivity of CNT-based strain sensors depends on number of parameters; quality and quantity of CNT used, type of polymer used, deposition and dispersion technique adopted and also on environmental conditions. Due to all these parameters, the piezoresistive behavior of CNT is diversified and it needs to be explored. This paper theoretically analyses the strain sensitivity of CNT-based strain sensors. The strain sensitivity response of CNT strain sensor is a result of change in total resistance of CNT network with respect to applied strain. The total resistance of CNT network consists of intrinsic resistance and inter-tube resistance. It has been found that the change in intrinsic resistance under strain is due to the variation of bandgap of individual, which depends on the chirality of the tube and it varies exponentially with strain. The inter-tube resistance of CNT network changes nonlinearly due to change in distance between neighboring CNTs with respect to applied strain. As the distance $d$ between CNTs increases due to applied strain, tunneling resistance $R_{\mathrm{\text{tunnel}}}$ increases nonlinearly in exponential manner. When the concentration of CNTs in composite is close to percolation threshold, then the change of inter-tube resistances is more dominant than intrinsic resistance. At percolation threshold, the total resistance of CNT networks changes nonlinearly and this effect of nonlinearity is due to tunneling effect. The strain sensitivity of CNT-based strain sensors also varies nonlinearly with the change in temperature. For the change of temperature from $-20^{\circ }$C to 50${}^{\circ }$C, there is more than 100% change in strain sensitivity of CNT/polymer composite strain sensor. This change is mainly due to the infiltration of polymer into CNTs.

Dielectric properties of composite materials made of epoxy resin filled with different concentrations of goethite (Fe₂O${}_3\cdot$ H₂O) nanorods and manganese ferrite (MnFe₂O₄) are presented in a wide frequency range. The electrical percolation threshold in these composites is close to 30vol.%. Electrical conductivity in the composites occurs both below and above the percolation threshold due to the finite conductivity of the epoxy resin under high temperatures.

Based on the percolation threshold theory of conductive-dielectric composites, the dielectric constant can be improved more by adding a certain mass fraction of conductive particle into polymer matrix. However, the dielectric loss increases with the increasing mass fraction of conductive particle. In this paper, conductive Polyaniline (PANI) with different particle sizes is utilized to illustrate that reducing particle size can improve the dielectric properties. The dielectric constant is increased from 319 to 540, and dielectric loss is decreased from 2.34 to 0.85 when PANI with smaller particle size is used. Moreover, PANI coated with an insulating surfactant layer can further improve the dielectric properties, the experimental results show that the dielectric constant of the composite could be more than 1000, while the dielectric loss is 0.35 at 1KHz.

This paper presents the results of an experimental study of the percolation behavior of the complex dielectric constant of ceramic matrix composites (CMC). Samples of piezoactive CMC were obtained by joint sintering of synthesized PZT piezoceramic powder (matrix) and crushed particles of sintered PZT piezoceramics (filler) of different compositions. The experimental dependencies of real and imaginary parts of the complex dielectric constant of CMC on the porosity of piezoceramic matrix and volume content of ceramic filler particles were measured and analyzed. It was shown that the additional porosity of the ceramic matrix resulting from sintering of the CMC masks the dielectric percolation transition, that actually occurs at the volume concentration of ceramic filler close to percolation threshold (V ∼ 1/3).