Irreversible bimolecular reactions with inertia: from the trapping to the target setting at finite densities

We investigate numerically pseudo-first-order irreversible bimolecular reactions of the type A + B → B between hard spheres undergoing event-driven Brownian dynamics. We study the encounter rate and the survival probability of A particles as functions of the packing fraction ϕ in the trapping (a single particle diffusing among static non-overlapping traps) and target (many traps diffusing in the presence of a single static target particle) settings, as well as in the case of diffusing traps and particles (full mobility). We show that, since inertial effects are accounted for in our simulation protocol, the standard Smoluchowski theory of coagulation of non-interacting colloids is recovered only at times greater than a characteristic time Δt, marking the transition from the under-damped to the over-damped regime. We show that the survival probability S(t) decays exponentially during this first stage, with a rate 1/τ0 is proportional to φ. Furthermore, we work out a simple analytical expression that is able to capture to an excellent extent the numerical results for t < Δt at low and intermediate densities. Moreover, we demonstrate that the time constant of the asymptotic exponential decay of S(t) for diffusing traps and particles is k(S)(-1), where kS = 4π(DA + DB)Rρ is the Smoluchowski rate. Detailed analyses of the effective decay exponent β = d [log(-logS(t))]/d (logt) and of the steady-state encounter rate reveal that the full mobility and trapping problem are characterized by very similar kinetics, rather different from the target problem. Our results do not allow one to ascertain whether the prediction S(t) is proportional to exp(-at(3/2)) (a = const.) as t → ∞ for the trapping problem in 3D is indeed recovered. In fact, at high density, S(t) is dominated by short encounter times, which makes it exceedingly hard to record the events corresponding to the exploration of large, trap-free regions. As a consequence, at high densities the steady-state rate simply tends to 1/τ0. Finally, we work out an analytical formula for the rate that shows a remarkable agreement with the numerics up φ = 0.4.