Particle aggregation

Particle aggregation is association of initially separate particles into larger clusters. For colloids, both the rate of Brownian motion encounters and the probability that an encounter produces attachment matter. Strong attraction with negligible repulsive barriers can give diffusion-limited cluster aggregation; a barrier that makes attachment unlikely can give reaction-limited cluster aggregation. Reversible association and bond rearrangement require additional kinetic assumptions.

Association of dispersed particles into larger clusters, often after repulsive stabilization is weakened. Salt screening, reduced surface charge, polymer bridging or depletion force can promote it. Clusters may sediment rapidly; preventing aggregation and preventing gravitational separation are distinct stability requirements.

In the standard single-seed model, individual particles execute random walks until first contact with a growing cluster, where they stick irreversibly. Exposed tips intercept more particles than screened inner regions, producing branched clusters. This differs from diffusion-limited cluster aggregation, in which whole existing clusters move and join one another. The two models need not share the same mass fractal dimension.

In reaction-limited cluster aggregation, encounters occur frequently compared with successful attachment: an activation or interaction barrier gives a small sticking probability. Repeated unsuccessful approaches allow clusters to sample more relative configurations before bonding, commonly producing denser aggregates than diffusion-limited cluster aggregation. Typical three-dimensional isotropic irreversible cluster models give mass fractal dimension around $2.1$. This structural contrast does not require previously formed bonds to break and restructure; it can arise from unsuccessful encounters alone. ^[ref]

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.