Volume 149, number 1 PHYSICS LETI’ERS A 10 September 1990 A two-dimensional model of trapping reactions with Gaussian coils G.S. Oshanin, A.V. Mogutov and S.F. Burlatsky Institute ofChemical Physics, USSRAcademy of Sciences, Kosygin Street 4, 117334 Moscow V-334, USSR Received 19 June 1990; accepted for publication 29 June 1990 Communicated by V.M. Agranovich We study the kinetics of trapping reactions A+B-.B in the special case when traps are attached to segments of immobile Gaussian coils randomly distributed on a surface. We obtain novel expressions for the trap-free cavity size distribution and the dependences describing the temporal behavior of the survival probability ofthe diffusing particles. We show that these analytical dependences are in good agreement with numerical simulation results. Within the last years there seems to have been a considerable progress in the theory of diffusion-con- P(R, t) = 4 ~p 2(j) exp [ — p2 (j)Dt/R2], (3) trolled chemical reactions. While earlier works [1— 4] were focussed on refinements of the Smolu- where D is the diffusion coefficient of particle A, the chowsky concentration gradient method, more re- eigenvalues pCi) are easily determined dimension- cent developments have tended to emphasize the less constants. For instance, for d= 2 the constants consequences of spatial fluctuations and fluctuation- ~U)are the zeroes of the Bessel function 4(x). The induced kinetics [5—13]. In particular, it has been averaged survival probability of particles A is dom- discovered that the kinetics of the trapping reaction mated by A + B—’ B is drastically dependent on the spatial dis- tribution of traps B. It was shown that for random ,, ~. f d ~ i R ~ R ,1 placement of traps the Smoluchowsky approach is not valid at large times and the trapping kinetics ex- ° hibits unusual behavior [5,8—12], Maximizing the value of the exponents in eq. (4) with respect to R with I fixed we get that at large times ln CA(t) ~ _td/((~~+2)C~f(~2). (1) CA(t) is defined by the decay of eq. (1). Interest- ingly, this heuristic scheme (2)—(4) entails the ex- One may evaluate the decay of eq. (1) by means act asymptotic solution to a many-body reaction— of the following considerations [8—11]. For a Pois- diffusion problem, as was proved by the penetrating sonian distribution of traps the existence probability analysis of Donsker and Varadhan [9] and the sim- of a spherical trap-free cavity of radius R is ilar but much more succinct “method of bounds” of Kayser and Hubbard [10]. P~.. (A) = exp ( — VdCBR”), (2) One can consider that the remarkable result of eq. (1) is a mere abstraction and is irrelevant to real where Vd is the volume of the d-dimensional unit physical systems since it can be observed at astro- sphere and CB is the mean concentration of traps. nomically large times when the survival probability The probability P(R, z) that a given diffusing par- of particle A drops below 10—13 [141. However, some tide A survives for a time tin a trap-free cavity of recent studies of trapping reactions proceeding in re- radius A enclosed by a trapping boundary is stricted geometries — in dense percolation-like sys- 0375-9601/90/S 03.50 © 1990 — Elsevier Science Publishers B.V. (North-Holland) 55