A membrane engineer models a thin filtration membrane as a square lattice of pores

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 membrane engineer models a thin filtration membrane as a square lattice of pores, each open (able to pass liquid) independently with probability

p

. Liquid crosses the membrane only if a connected chain of open pores joins the top face to the bottom face (four-neighbour site percolation). (a) The

5\times5

grid below (top row first;

1=

open pore,

0=

blocked pore) is one sampled membrane. Decide, by tracing a chain of adjacent open pores, whether liquid spans from top to bottom.

\begin{array}{ccccc} 1&0&1&0&1\\ 1&1&1&0&1\\ 0&1&0&0&1\\ 0&1&1&1&1\\ 0&0&0&1&1 \end{array}

(b) Simulations of many independent membranes at four sizes

L

(pores across) give the porosity

p_{1/2}(L)

at which half the membranes span:

p_{1/2}(20)=0.6010\pm0.0005,\quad p_{1/2}(40)=0.5965\pm0.0004,

p_{1/2}(80)=0.5945\pm0.0003,\quad p_{1/2}(160)=0.5935\pm0.0002.

Using the pair

L=20,40

, fit a straight line in

1/L

and extrapolate to

1/L\to0

to obtain one estimate of the infinite-lattice threshold

p_c

. Repeat with the pair

L=80,160

to obtain a second estimate. (c) Propagate each pair's own point uncertainties into the uncertainty on that pair's extrapolated intercept. Then take the unweighted mean of the two intercepts as the best value of

p_c

, and combine the standard uncertainty of that mean in quadrature with half the spread between them, to quote

p_c

with an overall uncertainty.

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