Chemical Physics 170 (1993) 235-241 North-Holland Behavior of the rate constant for reactions in restricted spaces: case of luminescence quenching in water-in-oil microemulsions P. Lianos University of Patras, School of Engineering, Physics Section, 26500 Patras, Greece zyxwvutsrqponmlkjihgfedcbaZYXWVU J.C. Brochon and P. Taut LURE, CNRS-MEN-CEA, Baf. 2090, Centre Universitaire Paris&d, 91405 Orsay, France Received 3 June 1992; in final form 19 October 1992 The luminescence (fluorescence) decay profiles of ruthenous tris( 2,2’-bipyridine) and pyrenetetrasulfonate in the presence of ferricyanide solubilized in cyclohexane-pentanol-sodium dodecylsulfate quaternary water-in-oil microemulsions have been ana- lyzed with a percolation model applicable to reactions in restricted geometries. The reaction rate is time-dependent and it is given by K(t) =fC,t’-’ -2fCztzJ-’ where Cr and Cz are constants. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH f< 1 and it is related with the dimensionality of the reaction. The importance of the term containing Cz is discussed. The evolution of K with time, nature of the probe, quencher concentration, droplet concentration and temperature is discussed. 1. Introduction The fluorescence quenching reaction between an excited fluorophore and a quencher in restricted en- vironments can be described by models obtained from the theory of random walk in fractal domains. Fluo- rescence (luminescence) quenching is part of the re- action A+B-+products ( [A] << [B] ). For infinitely short pulse excitation, he decay of the fluorescence a intensity is satisfied by t e following equation [ l-8 ] : Z(t)=Zoexp(-bt)exp(-C1tf+C2t2f), (1) where k,, is the decay rate constant in the absence of quenching, C, and C2 are constants and f a non-inte- ger exponent so that 0-cf-c 1. f is related to the di- mensionality of the reaction. When quenching oc- curs by direct energy transfer between an excited immobile donor and an immobile acceptor, then f is proportional to the fractal dimension df of the reac- tion domain [ 9- 111. However, for mobile reactants the dimensionality depends not only on the geometry but also on the path followed, which might involve repeated visits to the same available sites. Then the dimensionality is expressed by the so-called spectral dimension d, which is given by d, = 2dfld,+. = 21; where d, is the fractal dimension of the path [ 12,13 1. The- oretically, the argument of the second exponential in eq. ( 1) is an infinite series [ 1,2,14,15] but one or two terms are sufftcient to adequately describe the decay profiles. Blumen et al. [ 31 have shown that as dimensionality decreases, higher-order approxima- tion is necessary. Klafter and Blumen [ 21 have also shown that the second-order term is proportional to ln2( 1 -p), where p is the occupation probability of the available sites by quenchers, i.e. in our case, it is proportional to the global quencher concentration [ Q ] and inversely proportional to the number of the permitted sites. Thus when p decreases, either by de- creasing [Q] or by increasing the number of sites, 1 -p tends to unity and ln2( 1 -p) tends rapidly to zero. These questions are discussed here on the basis of the present experimental results. The first-order reaction rate can be derived from eq. ( 1) by differentiation, and it is given by K(f)=fC,tf-'-2fC2t2f-'+ . . . . (2) In principle, it also has an infinite number of terms. However, C2 is in most cases much smaller than C, while higher-order terms are considered negligible a priori. Fitting eq. ( 1) to the experimental decay pro- 0301-0104/93/f 06.00 0 1993 Elsevier Science Publishers B.V. All rights reserved.