Physics and Astronomy, Mathematics, Chemistry, Ames Laboratory
Journal or Book Title
Physical Review A
Consider irreversible cooperative filling of sites on an infinite lattice where the filling rates ki depend on the number, i, of occupied sites adjacent to the site(s) being filled. If clustering is significantly enhanced relative to nucleation (k1/k0≡ρ≫1), then the process is thought of as a competition between nucleation, growth, and (possible) coalescence of clusters. These could be Eden clusters with or without permanent voids, Eden trees, or have modified but compact structure (depending on the ki, i≥1).
Detailed analysis of the master equations in hierarchial form (exploiting an empty-site shielding property) produces results which are exact (approximate) in one (two or more) dimensions. For linear, square, and (hyper)cubic lattices, we consider the behavior of the average length of linear strings of filled sites, lav=J∞s=1 sls/J∞s=1 ls, where ls is the probability of a string of length s [lav=(1−CTHETA)−1 for random filling, at coverage CTHETA].
In one dimension, ls=ns gives the cluster size distribution, and we write lav=nav. We consider the scaling lav∼A(CTHETA)ρω as ρ→∞ (with CTHETA fixed), which is elucidated by the introduction of simpler models neglecting fluctuations in cluster growth or cluster interference. For an initially seeded lattice, there exists an upper bounding curve lav+ for lav (as a function of CTHETA), which is naturally obtained by switching off nucleation (setting k0=0). We consider scaling of lav+ as the initial seed coverage ε vanishes. The divergence, lav∼C(1−CTHETA)−1 as CTHETA→1, is also considered, focusing on the cooperativity dependence of C. Other results concerning single-cluster densities and ls behavior are discussed.
American Physical Society
Evans, James W.; Bartz, J. A.; and Sanders, D. E., "Multicluster growth via irreversible cooperative filling on lattices" (1986). Physics and Astronomy Publications. 448.