Diffusion-limited cluster aggregation (DLCA)

In diffusion-limited cluster aggregation, mobile clusters undergoing Brownian motion attach with probability close to one when they encounter another cluster. Irreversible first-contact sticking tends to trap particles in exposed outer branches, producing open aggregates. In the common three-dimensional isotropic cluster model, the mass fractal dimension is roughly $1.8$. This is different from depositing individual random walkers onto one fixed cluster. Rotational motion, anisotropic attraction and bond restructuring can change the morphology.

For two dilute spherical species with absorbing contact and negligible interaction barriers, relative diffusion has coefficient $D=D_1+D_2$ and contact radius $R=a_1+a_2$. The steady concentration outside one target is $c(r)=c_{\infty }(1-R/r)$, solving the radial Laplace equation with zero contact concentration. Its inward diffusive flux is $4\pi DR{c}_{\infty }$, proving the displayed bimolecular rate coefficient. For identical particles, converting this encounter coefficient into a rate of loss of cluster number additionally needs the factor that prevents counting the same pair twice.