PATTERN-FORMATION BY GROWING DROPLETS - THE TOUCH-AND-STOP MODEL OF GROWTH

We investigate a novel model of pattern formation phenomena. In this m odel spherical droplets are nucleated on a substrate and grow at const ant velocity; when two droplets touch each other they stop their growt h. We examine the heterogeneous process in which the droplet formation is initiated on randomly distributed centers of nucleation and the ho mogeneous process in which droplets are nucleated spontaneously at con stant rate. For the former process, we find that in arbitrary dimensio n d the system reaches a jamming state where further growth becomes im possible. For the latter process, we observe the appearance of fractal structures. We develop mean-field theories that predict that the frac tion of uncovered material PHI(t) approaches to the jamming limit as P HI(t) - PHI(infinity) is similar to exp(C(t)d) for the heterogeneous p rocess and as a power law for the homogeneous process. Exact solutions in one dimension are obtained and numerical simulations for d = 1-3 a re performed and compared with mean-field predictions.