Diffusion-Limited Aggregation and Growth

Leclanche cells (see Fig. 2.7), whilst others strengthen the rubber in car tyres, increasing the mileage by a factor of a hundred over pure rubber tyres. Some silicas have the same properties (see Fig. 2.16). 

The field governing the growth of such a cluster is either the probability field for the presence of a random walker, or equivalently a concentration field set up by diffusion from a constant and inexhaustible source located far from the seed (concentration C = Cq far from the cluster) towards an infinite sink which is the surface of the growing object (concentration C = 0 at the cluster boundary). The law governing this diffusion is just Pick s law = dC/dt. 

As the diffusion can be considered to be quasi-static. Pick s equation reduces to Laplace s equation 

Moreover, as the probability of growth at a given site on the cluster depends on the flux of particles arriving at that location (or the flux of probability of a walker being present there), it is Pick s first law which determines the growth rate 

This just says that growth will be greatest where the gradient is greatest, i.e., where the equiprobability curves are the closest together (see Fig. 2.18). 

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