Mixing effects in theA+B→0 reaction-diffusion scheme

Abstract

We study analytically the irreversible reaction A +B-• 0 under laminar stirring, for equal overall particle concentrations. In contrast to the usually found algebraic decays, here the reactant concentrations display exponential patterns, which stem from the mixing kinetics. The crossover from this short-time behavior to the classical kinetic law in a fully mixed system is also investigated. Furthermore, disorder effects are fundamental, since for strictly regular patterns the short-time decay shows an exponentially exponential time dependence.PACS numbers: 82.20.-w, 05.40.+J In recent years diffusion-controlled reactions have received much attention, triggered by the realization that often the decay laws deviate quantitatively from the chemical-kinetics forms. Thus for the A+B-+0 reaction with equal numbers of A and B particles the kinetic scheme predicts at long times a / ~l behavior, while more consistent theoretical approaches and numerical simulations find (in the absence of stirring) more general patterns, such asa/ -<//4 law (where d is the spatial dimension), a fact due to the random fluctuations in the particle distribution. 1 " 6 The usual assumption is that under vigorous stirring the standard classical kinetics should again apply. In this Letter we show that the situation is not so simple in that stirring does not necessarily lead to algebraic decay.Here we mimic stirring by laminar mixing models. 7 " 11 Basically, mechanical mixing of two initially separated liquids often produces a quasi-one-dimensional lamellar structure. 7 " 9 Thus the irreversible A+B->0 reaction was investigated in Refs. 10-13 by first creating the striation pattern and then letting the reaction develop. Using Ranz's methods 14 we solve here the problem of simultaneous reaction and mixing by using so-called warped-time transformations. Note that one may envisage mixing by picking particles randomly and placing them in new positions; 1516 although this procedure leads to the classical-kinetics results, it does not mirror real mixing closely.In standard situations, laminar mixing of two liquids gives rise to a structure of alternating striations, which get thinner in time because of folding and squeezing. In fact, thinning of lamellar arrangements of striations occurs generally under mixing conditions and may be found even in turbulent flow. The striation thickness distribution (STD) is random, its mean decreasing, in general, exponentially with time. 9 We use here the fact that the so-called warped-time transformation 9,14 allows one to map the diffusion problem in a steadily thinning array of striations onto a diffusion problem in a static lamellar system. 10 " 12 This allows us to implement our recent analytical results for the static problem 13 to investigate the A+B-•() reaction under mixing, both in the case of infinite local reaction rates and also for finite ones. Specifically, we display the decay behavior under mixing and we determine the dependence of the overall decay rate on the reacting species (concentrations, diffus...

Chaotic mixing of a competitive–consecutive reaction

“…The utility of the lamellar model was demonstrated by Clifford et al [13], who found reasonable agreement with results from two-dimensional simulations of the full advection-reactiondiffusion problem [6], although, as we have already remarked, at large Péclet number the latter simulations are known to be inaccurate [27]. Of particular note are the works of Sokolov and Blumen [41,42], who analysed a single reaction taking place in a lamellar structure, with special emphasis on 'fast' reactions, for which a great deal of analytical progress is possible, and of Muzzio and Ottino [19,20], who examined the evolution of a lamellar structure in which only the primary reaction takes place (i.e., the case k 2 = 0), at infinite rate. The latter focused on the evolution of the distribution of striation thicknesses, which is not our direct concern here, and commented that it would be interesting to include fluid mechanical mixing effects into their lamellar model.…”

“…The chemical species diffuse (in the x-direction) and react; they are also advected by a straining flow (u, v) = (−µx, µy), which allows us to retain the effect of mixing in stretching the interface exponentially in time, but without the corresponding folding [2,12,13,[16][17][18][40][41][42]. The appropriate dimensionless governing equations are then…”

“…Most analytical and computational advantage is gained by considering the lamellar array to be one-dimensional [3,4,[11][12][13][14][15][16][17][18][19][20][21][35][36][37][38][39][40][41][42][47][48][49][50], although clearly in reality the striations possess significant curvature where they are folded over by the flow, and this curvature can impact on the yield. As with the single-interface model above, the effects of fluid mixing in compressing and stretching the lamellae in (approximately) normal and parallel directions may be considered or not.…”

Study of a bimolecular annihilation process for coarsening reactants

“…[1][2][3][4] This regime is due to the spontaneous separation of reactants and makes the AϩB→0 scheme a very interesting model, e.g., for the investigations of effects of mixing procedures applied to diffusion-controlled reactions which are otherwise slow and ineffective. 5,6 The reaction schemes considered in such a context are normally oversimplified because no particles interactions ͑except those leading to the reaction͒ are considered. However, in many situations such interactions can be of great importance.…”

Fluctuation-Dominated Kinetics under Stirring

“…The classical form is obeyed when the concentrations are homogeneous at all times, which can be achieved through efficient procedures such as mixing by dilatational flow [11], tossing [12], or unbounded shear flow [13][14][15]. In general realistic mixing flows, especially in two dimensions, are less effective [1].…”