  Correlations between electrical and thermal conduction in polymer composites
are blurred due to the complex contribution of charge and heat carriers at the
nanoscale junctions of filler particles. Conflicting reports on the lack or
existence of thermal percolation in polymer composites have made it the subject
of great controversy for decades. Here, we develop a generalized percolation
framework that describes both electrical and thermal conductivity within a
remarkably wide range of filler-to-matrix conductivity ratios (Y_f/Y_m),
covering 20 orders of magnitude. Our unified theory provides a genuine
classification of electrical conductivity with typical Y_f/Y_m >= 10^10 as
insulator-conductor percolation with the standard power-law behavior, and of
thermal conductivity with 10^2<= Y_f/Y_m <= 10^4 as poor-good conductor
percolation characterized by two universal critical exponents. Experimental
verification of the universal and unified features of our theoretical framework
is conducted by constructing a 3D segregated and well-extended network of
multi-walled carbon nanotubes in polypropylene as a model polymer matrix under
a carefully designed fabrication method. We study the evolution of the
electrical and thermal conductivity in our fabricated composites at different
loading levels up to 5 vol%. Significantly, we find an ultralow electrical
percolation threshold at 0.02 vol% and a record-low thermal percolation
threshold at 1.5 vol%. We also apply our theoretical model to a number of 23
independent experimental and numerical datasets reported in the literature,
including more than 350 data points, for systems with different microscopic
details, and show that all collapse onto our proposed universal scaling
function, which depends only on dimensionality.
