Pattern formation by growing droplets: The touch-and-stop model of growth

Yu A. Andrienko, N. V. Brilliantov, P. L. Krapivsky

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22 Citations (Scopus)


We investigate a novel model of pattern formation phenomena. In this model spherical droplets are nucleated on a substrate and grow at constant velocity; when two droplets touch each other they stop their growth. We examine the heterogeneous process in which the droplet formation is initiated on randomly distributed centers of nucleation and the homogeneous process in which droplets are nucleated spontaneously at constant rate. For the former process, we find that in arbitrary dimension d the system reaches a jamming state where further growth becomes impossible. For the latter process, we observe the appearance of fractal structures. We develop mean-field theories that predict that the fraction of uncovered material Φ(t) approaches to the jamming limit as Φ(t)-Φ(∞)∼exp(Ctd) for the heterogeneous process and as a power law for the homogeneous process. Exact solutions in one dimension are obtained and numerical simulations for d=1-3 are performed and compared with mean-field predictions.

Original languageEnglish
Pages (from-to)507-523
Number of pages17
JournalJournal of Statistical Physics
Issue number3-4
Publication statusPublished - May 1994
Externally publishedYes


  • fractals
  • nucleation and growth
  • Pattern formation


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