Ligand-receptor interactions in chains of colloids: When reactions are limited by rotational diffusion

We discuss the theory of ligand receptor reactions between two freely rotating colloids in close proximity to one other. Such reactions, limited by rotational diffusion, arise in magnetic bead suspensions where the beads are driven into close contact by an applied magnetic field as they align in chainlike structures. By a combination of reaction-diffusion theory, numerical simulations, and heuristic arguments, we compute the time required for a reaction to occur in a number of experimentally relevant situations. We find in all cases that the time required for a reaction to occur is larger than the characteristic rotation time of the diffusion motion tau(rot). When the colloids carry one ligand only and a number n of receptors, we find that the reaction time is, in units of tau(rot), a function simply of n and of the relative surface alpha occupied by one reaction patch alpha = pi rc(2)/(4 pi r(2)), where r(c) is the ligand receptor capture radius and r is the radius of the colloid.