Effective Conductivity and Critical Properties of a Hexagonal Array of Superconducting Cylinders

TL;DR

This paper analyzes the effective conductivity of a 2D hexagonal array of superconducting cylinders, providing new series expansions, critical properties, and a formula valid for any volume fraction, with results aligning well with theoretical expectations.

Contribution

It introduces a series expansion for effective conductivity, calculates the critical amplitude and next-order term, and derives a general formula for arbitrary volume fractions.

Findings

Critical amplitude estimated between 5.14 and 5.24

Next order constant term calculated as -6.229

Effective conductivity formula valid for all volume fractions

Abstract

Effective conductivity of a 2D composite corresponding to the regular hexagonal arrangement of superconducting disks is expressed in the form of a long series in the volume fraction of ideally conducting disks. According to our calculations based on various re-summation techniques, both the threshold and critical index are obtained in good agreement with expected values. The critical amplitude is in the interval $(5.14,5.24)$ that is close to the theoretical estimation $5.18$. The next order (constant) term in the high concentration regime is calculated for the first time, and the best estimate is equal to $(-6.229)$. Final formula is derived for the effective conductivity for arbitrary volume fraction.