Irreversible diffusion-limited cluster aggregation: The behavior of the scattered intensity

In this paper, we discuss the evolution of the scattered intensity I(q) during irreversible diffusion-limited cluster-cluster aggregation. We propose a mean field model to describe the correlation among different clusters that develops during the irreversible aggregation process. The model is based on two coupled differential equations, controlling the growth of the average cluster mass and the time dependence of the probability of finding pairs of clusters as a function of their distance. The model predicts a moving and growing peak in the scattered intensity at a wave vector ${\mathit{q}}_{\mathit{m}}$. For growing compact clusters, as in the case of late-stage decomposition under deep-quench conditions, we recover the expected results both for the scaling law of the scattered intensity, i.e., ${\mathit{q}}_{\mathit{m}}^{\mathit{d}}$I(q/${\mathit{q}}_{\mathit{m}}$)=f(q/${\mathit{q}}_{\mathit{m}}$), and for the growth of the average cluster mass. For growing clusters with fractal dimension ${\mathit{D}}_{\mathit{f}}$, the model predicts no scaling for I(q), particularly in the initial stage of the aggregation. Only in the late stages, an approximate scaling of the scattered intensity in ${\mathit{q}}_{\mathit{m}\mathit{f}}^{\mathit{D}}$I(q/${\mathit{q}}_{\mathit{m}}$) holds. We compare the prediction of the model with the recent experimental results of Carpineti and Giglio [Phys. Rev. Lett. 68, 3327 (1992)] on colloidal aggregation and with data from Brownian dynamics simulations. The agreement between analytical and experimental results is excellent.