Multicluster growth via irreversible cooperative filling on lattices

Abstract

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. Disciplines

Random and cooperative sequential adsorption

“…Consider ID semideterministic lattice models. Simple extension of the arguments used above to calculate P, =P, shows that P"(t) =exp[ (n -1)k-ot]P" for n I, which is consistent with shielding of single empty sites (Evans et al , 1986). For the 1D continuum problem, one can readily, but unconventionally, calculate the probability, P (x), of finding an empty stretch of length x.…”

“…It is instructive to consider a class of lattice models which bridge the gap between the continuum grain growth and lattice CSA models (Evans et al , 1986). In the simplest such models, nucleation or birth of islands occurs randomly at constant rate k0 at lattice sites.…”

Two-parameter sequential adsorption model applied to microfiber clustering

“…A useful consequence of the competing influence of 3NN and 4NN cooperativity on domain size is that the parameters may be varied in tandem to increase the coverage at saturation without simultaneously increasing the average domain size. In previously studied models of cooperative sequential adsorption that include only one level of cooperativity (typically NN enhancement or NN exclusion and 2NN enhancement 19,20 ), increasing the rate of cooperative adsorption increases both the saturation coverage and the size of domains. In contrast, if the system exhibits cooperativity that induces adsorption of particles on a different sublattice as in the current model, then increasing this rate (β in the current model) increases the saturation coverage θ * while decreasing the measures of domain size, s av and m av .…”

“…Our model is similar to previously studied CSA models on a square lattice with close-neighbor cooperative effects. 16,[18][19][20][21] Previous studies, however, have focused on a single level of cooperativity with rates k i = α i k 0 , α > 1 defined at each site when i is the number of occupied closest neighbor sites. (This rate choice is called a multiplicative or Arrhenius rate, in contrast to the Eden rate choice, k i = αk 0 , α > 1 for i ≥ 1.)…”

Nonequilibrium structure in sequential assembly

“…, directly from System (5) as s i → 1 ∀ i. This is a known solution for clustering by irreversible filling [40,41] at equal filling rates. Another limiting case occurs when each site can accept only one particle, (that is p 0 = 1 and p i = 0 for i > 0).…”