Segregation at a Single Trap in the Presence of Fields
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SEGREGATION AT A SINGLE TRAP IN THE PRESENCE OF FIELDS HAIM TAITELBAUM AND GEORGE H. WEISS Physical Sciences Laboratory, Division of Computer Research and Technology, National Institutes of Health, Bethesda, MD 20892 Abstract There have been a number of recent investigations of segregation properties of diffusing particles in the presence of a single static trap in low dimensions. We study these properties when the diffusing particles are subject to different forms of external fields: global constant bias, random bias fields (Sinai model) and random transition rates. We discuss two measures of segregation, the distances from the trap either to the point at which the concentration profile reaches a specified fraction of its bulk value, or to the nearest unreacted particle. For the cases of global bias (both away from, and towards the trap) and random fields, we found that both measures of segregation have the same asymptotic temporal behavior, while for random transition rates they differ. We explain this difference by relating the nearest-neighbor distance measure to properties of hardcore diffusion in these systems. We also found anomalous spatial shapes for the profile in the vicinity of the trap in the random systems, as well as anomalous reaction rates. 1. Introduction When chemical reactions take place in confined geometries, the reactants tend to segregate, namely to form domains with an enhanced concentration of one or the other of the reactants [1-11]. The first quantitative model of this phenomena was that of Smoluchowski [12]. He analyzed the reaction A + B --* B, in which a single spherical static trap (B) is surrounded by a swarm of diffusing point particles (A), initially uniformly distributed throughout the space. The occurrence of A - B reactions creates a depletion zone around the trap, which segregates the bulk of the diffusing particles (A) from the trap (B). Physical examples of such systems are provided by exciton trapping, quenching or fusion, electron-hole and soliton-antisoliton recombination, phonon upconversion and free-radical scavenging. Examples of reactions in confined geometries include quasi one-dimensional crystals grown inside pores and microcapillaries, polymer chains in dilute blends, catalytic surface reactions, and heterofusion in ultrathin molecular wires, filaments and pores [13-15]. How can one characterize quantitatively the kinetic properties of the segregation phenomena? There are at least two possible ways of measuring the local repulsion in the vicinity of the trap. The first is the distance from the trap to a point at which the concentration profile of the diffusing particles, P(x, t), reaches an arbitrary fraction 0, (0 < 0 < 1), of its bulk value. This distance, xe(t), which will be later referred to as the 9-distance, is defined through the equation
P(XG(t), t) = Oc, Mat. Res. Soc. Symp. Proc. Vol. 290. @1993 Materials Research Society
(1)
352
where c is the initial constant concentration of the A particles, which for simplicity will henceforth be set equal to u
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