Conductivity of the Two-Dimensional Rayleigh Model Near the Percolation Threshold: A Subthreshold Region of Concentrations

Abstract

The conductivity of the two-dimensional Rayleigh model near a critical point, the percolation threshold, is investigated. The effective conductivity of the model with a metal–prefect conductor phase transition is calculated in the binary (pair) approximation. For the alternative model with a metal–insulator phase transition the corresponding effective conductivity is determined from the Keller–Dykhne reciprocity relation.

APPENDIX

Knowing the complete system of eigenfunctions also allows other problems related to the inclusion under consideration—a macroscopic body, for example, about its dipole polarizability, to be solved.

In the case of a body placed in a uniform electric field of strength E₀, the corresponding potential has the following asymptotics (at D = 2):

$$r \to \infty {\text{:}}\,\,\,\varphi ({\mathbf{r}}) \eqsim - ({{{\mathbf{E}}}_{0}} \cdot {\mathbf{r}}) + 2\frac{{({\mathbf{p}} \cdot {\mathbf{r}})}}{{{{r}^{2}}}} + ....$$

(A.1)

Here,

$${\mathbf{p}} = \hat {\Lambda }{{{\mathbf{E}}}_{0}}$$

(A.2)

is the dipole moment and $\hat {\Lambda }$ is the dipole polarizability tensor. According to [5, 6], for the components of this tensor we have

$${{\Lambda }_{{\alpha \beta }}} = - 4\pi (1 - h)\sum\limits_\nu ^{} {\frac{{{{d}_{{\nu \alpha }}}{{d}_{{\nu \beta }}}}}{{h + {{\varepsilon }_{\nu }}}},} $$

(A.3)

where d_ν is an analog of the dipole moment in the asymptotics of the polarization eigenfunction:

$$r \to \infty {\text{:}}\,\,\,{{\psi }_{\nu }}({\mathbf{r}}) \eqsim 2\frac{{({\mathbf{r}} \cdot {{{\mathbf{d}}}_{\nu }})}}{{{{r}^{2}}}} + ...$$

(A.4)

In Eq. (A.3)h = ε₂/ε₁ is the ratio of the permittivities for the body and the surrounding medium. For the pair of circles under consideration the functions $\psi _{{2n}}^{{(e)}}$(r) and $\psi _{{1n}}^{{(e)}}$(r) have a dipole behavior when r → ∞. For the corresponding dipole moments we have

$${{{\mathbf{d}}}_{{2n}}} = n{{( - 1)}^{n}}c{{B}_{n}}{{{\mathbf{i}}}_{x}},\quad {{{\mathbf{d}}}_{{1n}}} = n{{( - 1)}^{n}}c{{A}_{n}}{{{\mathbf{i}}}_{y}}.$$

(A.5)

For the components of the polarizability tensor $\hat {\Lambda }$ we obtain

$${{\Lambda }_{{xx}}} = - 2{{c}^{2}}\sum\limits_{n = 1}^\infty {n\frac{{1 - h}}{{h + \coth n{{\xi }_{0}}}}\frac{{{{e}^{{ - n{{\xi }_{0}}}}}}}{{\sinh n{{\xi }_{0}}}}} ,$$

(A.6)

$${{\Lambda }_{{yy}}} = - 2{{c}^{2}}\sum\limits_{n = 1}^\infty {n\frac{{1 - h}}{{h + \tanh n{{\xi }_{0}}}}\frac{{{{e}^{{ - n{{\xi }_{0}}}}}}}{{\cosh n{{\xi }_{0}}}}} .$$

(A.7)

Note that Eqs. (A.6) and (A.7) satisfy the equalities

$${{\Lambda }_{{xx}}}(h) = - {{\Lambda }_{{yy}}}(1{\text{/}}h),\quad {{\Lambda }_{{yy}}}(h) = - {{\Lambda }_{{xx}}}(1{\text{/}}h),$$

(A.8)

which are a corollary of the reciprocity relation (see [4]).