Cluster formation in two-dimensional random walks: Application to photolysis of silver halides

As a model for the growth of silver aggregates on the surface of silver halide crystallites, we have studied the following problem: Independent ions perform random walks on a closed two-dimensional lattice containing, initially, one trap; when a walker reaches a site adjacent to a trap, that site itself becomes a trap. Several simplifications are necessary and justifiable to enable us to use standard random-walk theory: they include replacing the actual crystallite surface by a toroidal one, averaging over all starting points at an early stage, and replacing the irregularly growing cluster by a simply shaped one. Upper and lower limits on the rate of cluster growth are obtained, and results are fitted to experimental observations.