A carbon-filled polymer composite conducts electricity only if the conductive filler…

PHYSICS · Subjective · Fermi University Physics · Computational Physics · Monte Carlo and lattice simulation of physical systems · Monte Carlo and lattice simulation of physical systems

Computational Physics, Ch Rules on a lattice: diffusion, percolation and growth

Problem

A carbon-filled polymer composite conducts electricity only if the conductive filler particles, each occupying a lattice site independently with probability pp (volume fraction), form a connected chain joining the top face to the bottom face of a moulded sample (four-neighbour site percolation). (a) The 6×66\times6 cross-section below (top row first; 1=1= filler particle, 0=0= insulating matrix) is one moulded sample. Decide, by tracing a chain of adjacent filler sites, whether it conducts from top to bottom.011001010011110110100100101101111001\begin{array}{cccccc} 0&1&1&0&0&1\\ 0&1&0&0&1&1\\ 1&1&0&1&1&0\\ 1&0&0&1&0&0\\ 1&0&1&1&0&1\\ 1&1&1&0&0&1 \end{array} (b) Simulations of many independent samples at four sizes LL give the filler fraction p1/2(L)p_{1/2}(L) at which half the samples conduct: p1/2(8)=0.6078±0.0009,p1/2(16)=0.6003±0.0006,p_{1/2}(8)=0.6078\pm0.0009,\quad p_{1/2}(16)=0.6003\pm0.0006, p1/2(32)=0.5966±0.0004,p1/2(64)=0.5947±0.0002.p_{1/2}(32)=0.5966\pm0.0004,\quad p_{1/2}(64)=0.5947\pm0.0002. Fit a straight line in 1/L1/L through the pair L=8,16L=8,16 and extrapolate to obtain one estimate of the percolation threshold pcp_c; repeat through the pair L=32,64L=32,64 to obtain a second estimate. (c) Propagate each pair's point uncertainties into the uncertainty on its own intercept, state the combined value of pcp_c with its uncertainty, and explain why a manufacturer aiming for a reliably conductive composite should specify a filler fraction well above pcp_c, not merely above it.

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