Analytic scaling functions applicable to dispersion measurements in percolative metal-insulator systems

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Scaling functions, ${F}_{+}(\ensuremath{\omega}/{\ensuremath{\omega}}_{c}^{+})$ and ${F}_{\ensuremath{-}}(\ensuremath{\omega}/{\ensuremath{\omega}}_{c}^{\ensuremath{-}})$ for $\ensuremath{\varphi}g{\ensuremath{\varphi}}_{c}$ and $\ensuremath{\varphi}l{\ensuremath{\varphi}}_{c},$ respectively, are derived from an equation for the complex conductivity of binary conductor-insulator composites. It is shown that the real and imaginary parts of ${F}_{\ifmmode\pm\else\textpm\fi{}}$ display most properties required for the percolation scaling functions. One difference is that, for $\ensuremath{\omega}/{\ensuremath{\omega}}_{c}l1,\mathrm{Re}{F}_{\ensuremath{-}}(\ensuremath{\omega}/{\ensuremath{\omega}}_{c})$ has an \ensuremath{\omega} dependence of $(1+t)/t$ and not ${\ensuremath{\omega}}^{2}$ as previously predicted, but never conclusively observed. Experimental results on a graphite--Boron nitride system are given, which are in reasonable agreement with the ${\ensuremath{\omega}}^{(1+t)/t}$ behavior for $\mathrm{Re}{F}_{\ensuremath{-}}.$ Anomalies in the real dielectric constant just above ${\ensuremath{\varphi}}_{c}$ are also discussed.