VOLUME 71, NUMBER 21 PH YSICAL REVIEW LETTERS 22 NOVEMBER 1993 Scaling Anomalies in Reaction Front Dynamics of Confined Systems Mariela Araujo, * Hernan Larralde, t Shlomo Havlin, i and H. Eugene Stanley 'Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02115 Physics Department, Bar-Ilan University, Ramat Can, Israel (Received 1 February 1993; revised manuscript received 13 August 1993) We study the kinetics of the reaction front for diffusion-reaction systems of the form A+ B — + C which are confined to one dimension, and in which the reactants are initially separated, For the case in which both A and B diffuse, the spatial moments of the reaction front are characterized by a hierarchy of exponents, bounded by the exponents o = 1/4 and b = 3/8 characterizing the asymptotic time dependence of the distance Ized(t) between nearest neighbor 2 and B particles and the fluctuations of the midpoint m(t) between them, respectively. We argue that this behavior arises from confinement effects and will appear in other confined systems. PACS numbers: 82.40. — g, 82.20.Db, 82.30. — b Diffusion-reaction systems of the form A + B C(inert) in which the reactants are initially separated in space have been the subject of many experimental and theoretical studies [1 — 13]. The behavior of the "reac- tion front" is well understood within a "mean field" the- ory when the dimension of the system is greater than or equal to 2 [5,7]. When the system is confined to one di- mension, however, the correlations in the concentration fIuctuations that arise from the confinement of the reac- tants invalidate the "mean field" assumptions. Thus, the question of how to describe such systems is still open. The reaction front R(x, t) is the localized region where the reaction takes place; it is defined as the average number of C particles produced at position 2: at time t. Most of the analytical results for the reaction front kinetics to date are based on the mean field assump- tion that the probability for a reaction to occur at a given time and place is proportional to the product of the average concentrations of the reactants [1]. This as- sumption leads to a scaling form for the reaction front R(x, t) t ~f(~ ~x/t ) with o. = 1/6 and P = 2/3. Deviations from the mean field behavior of the reac- tion front appear in true 1D systems, due to correlations among the fIuctuations associated with the diffusive mo- tion of the reactant species. The correlations, which give rise to an unexpectedly complex behavior of the reaction front, arise from the confinement of the reactants upon a line, and may be present in other confined systems (i.e. , quasi-one-dimensional systems and fractals). In this paper we undertake the study of the kinetics of the reaction front for the limiting case of true 1D systems. We consider systems in which initially the A species is located to the left of the origin and the B species to the right. We assume that both species have the same diffusion constant, and we compare our results with the case in which one of them is static. Though the correlations that arise from confinement effects will be present for any nonzero reaction proba- bility, they are maximized when the reaction probability is 1. In this limiting case the presence of a particle of species A at position x precludes the possibility of find- ing particles of species B at any position to the left of x. We therefore focus on the situation in which the reaction probability is 1. Thus, in the systems we are consider- ing, reactions can only take place between the rightmost A particle (RMA) and the leftmost B particle (LMB). As the system becomes less confined, it becomes easier for particles to walk around each other without meet- ing. As the "thickness" of the systems goes to infinity, the correlations become negligible and the mean field as- sumption of uncorrelated overlapping concentrations be- comes valid. This is what makes the mean field form of R(x, t) valid only for systems whose dimension is d ) 2, and explains why an expression for R(x, t) in terms of the average reactant concentrations cannot be written for 1D systems. To properly describe the systems we are considering, we use the fact that the only possibility of reaction is between the RMA and the LMB. We define the coordi- nate m(t) to be the midpoint between the RMA and the LMB; m(t) is closely related to the reaction front since every time a reaction occurs, it perforce must occur at a position given by m(t). Thus, the fluctuations in the value of m(t) will be an important contribution to the width of R(x, t) [14]. The midpoint m(t) may be defined for any system, but as the confinement increases, the connection between m(t) and the reaction front becomes more significant. To study the statistics of m(t), we perform Monte Carlo simulations. The number of particles on each site is chosen from a Poisson distribution with the same con- centration, co — — 1, for both species. We consider the case in which both reactants have the same diffusion constant D~ = D~ = 1/2, so that each particle can move to one of its neighbors with equal probability. At each unit step, we move the particles sequentially keeping track of the position of the RMA and LMB, m(t) and I~~, then we check for the occurrence of reaction in the re- 3592