Cluster survival and polydispersity in aggregation

– We study the probability, PS(t), of a cluster to remain intact in one-dimensional cluster-cluster aggregation when the cluster diffusion coefficient scales with size as D(s) ∼ s . PS(t) exhibits a stretched exponential decay for γ < 0 and the power laws t −3/2 for γ = 0, and t−2/(2−γ) for 0 < γ < 2. A random walk picture explains the discontinuous and non-monotonic behavior of the exponent. The decay of PS(t) determines the polydispersity exponent, τ , which describes the size distribution for small clusters. Surprisingly, τ(γ) is a constant τ = 0 for 0 < γ < 2. Many models of aggregation phenomena lead to scale-invariance: the average cluster size increases as a power law, S(t) ∼ t, which defines a dynamical exponent z. This kind of behavior is met in various contexts ranging from chemical engineering to materials science to atmosphere research to, ultimately, even astrophysics [1]. It is of interest to explore the statistics of aggregation as a dynamical process, beyond the lengthand timescales defined through z. In this letter we introduce a new quantity in aggregation systems, the cluster survival, defined as the probability PS(t) that a cluster present at t = 0 remains unaggregated until time t. This is a first-passage problem [2] in a many-body system and analogous to persistence which is often studied by measuring the fraction of a system that preserves its initial condition for all times [0, t] [3]. The cluster survival turns out to decay in a nontrivial and counterintuitive manner. The behavior can be understood by a mean-field–like random walk analysis. Even on the mean-field level the question reduces to a novel, unsolved random walk problem, which we analyse in the long-time limit. More importantly, by solving the decay of the cluster survival we are able to determine the polydispersity exponent characterizing the cluster size distribution. We concentrate on a common and important example: diffusion-limited cluster-cluster aggregation (DLCA) [4]. In the lattice version of DLCA any set of nearest-neighbor occupied lattice sites is identified as a cluster. Each of these performs a random walk with a sizedependent diffusion constant, D(s) ∼ s , where γ is the diffusion exponent. Colliding clusters are irreversibly merged together and the aggregate diffuses either faster (γ > 0) or slower (γ < 0) than a monomer. In the following, cluster survival is investigated in one dimension for numerical and analytical simplicity. One can discern three separate cases: i) 0 < γ < 2, which results in a power law decay for the survival, PS(t) ∼ t−θS , ii) γ = 0, which is exactly solvable [5], and iii) γ < 0, when PS(t) ∼ exp[−CtβS ]. Here θS(γ) is the survival exponent, C > 0 a constant, and βS(γ) the stretching exponent. For γ > 2 the system has a gelation transition and is not of interest here.