Diffusion-controlled death ofA-particle andB-particle islands at propagation of the sharp annihilation frontA+B→0

Abstract

We consider the problem of diffusion-controlled evolution of the A -particle-island- B -particle-island system at propagation of the sharp annihilation front A+B-->0 . We show that this general problem, which includes as particular cases the sea-sea and island-sea problems, demonstrates rich dynamical behavior from self-accelerating collapse of one of the islands to synchronous exponential relaxation of both islands. We find a universal asymptotic regime of the sharp-front propagation and reveal the limits of its applicability for the cases of mean-field and fluctuation fronts.

Self-similar evolution of theA-particle island–semi-infiniteB-particle sea reaction-diffusion system

“…(18) and considering that x m − x f ∼ r, we find that at the exponential evolution stage, the relative front width increases by the law η ∼ (e τ τ 3/2 /kr 2 ) 1/3 . Thus, defining the time boundary of the sharp front regime by the condition η ∼ 0.1 [10], we obtain τ | η=0.1 ∼ ln(η 3 kr 2 /τ 3/2 η ). Following [10,12], we shall estimate the applicability limit of the sharp front approximation for a perfect 3D diffusion-controlled reaction with dimensionless (in units of D/L 2 b 0 ) constant k ∼ r a L 2 b 0 , where r a is the reaction radius.…”

“…Thus, defining the time boundary of the sharp front regime by the condition η ∼ 0.1 [10], we obtain τ | η=0.1 ∼ ln(η 3 kr 2 /τ 3/2 η ). Following [10,12], we shall estimate the applicability limit of the sharp front approximation for a perfect 3D diffusion-controlled reaction with dimensionless (in units of D/L 2 b 0 ) constant k ∼ r a L 2 b 0 , where r a is the reaction radius. Substituting here r a ∼ 10 −8 cm,L ∼ 0.1 cm, and b 0 ∼ 10 20 cm −3 , we obtain k ∼ 10 10 and find τ | η=0.1 ∼ ln(10 7 r 2 /τ 3/2 η ), from which we conclude that at a sufficiently large r the exponential relaxation stage is reached in the sharp-front regime.…”

“…Recently, a new line in the study of the A + B → 0 front dynamics has attracted significant attention under the assumption that the particle number of one or both species is finite (island-sea and island-island systems) and, therefore, in the final state one or both islands disappear completely [9][10][11][12][13][14]. It has been established that in the sharp-front regime these systems exhibit rich scaling behavior, and though in these systems the QSA is always asymptotically violated, at large initial particle numbers and a high-reaction constant the vast majority of particles die in the sharp-front regime over a wide parameter range.…”

“…To complete the picture outlined, we have to reveal the applicability limits for the key condition of the sharp annihilation front [10,12],…”

Complex dynamics of interacting fronts in a simpleA+B→Creaction-diffusion system

“…In real experiments, the solutions of reactants are, however, confined into a limited region of space leading to finite-size effects when the front reaches one of the system boundaries. Finite-size effects also naturally arise in the context of multiple A + B → C fronts when, for instance, the solution of one of the reactants is initially confined between solutions of the other reactant [24][25][26][27]. The formation of two or more localized reaction zones, randomly separated in space, is indeed expected to be much more likely than the formation of a single isolated one in natural environments.…”