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J . Phys. Chem. 1993,97, 2114-2711

Fluorescence Quenching Kinetics in Monodisperse Micellar Solution with Exchange of Probes and Quenchers A. V. Barzykin' and I. K. Lednev Department of Photochemistry, Institute of Chemical Physics, Russian Academy of Sciences, Chernogolovka, Moscow Region 142432, Russia Received: September 17, 1992; In Final Form: November 17, 1992

A stochastic model for fluorescence quenching in monodisperse micellar solution with exchange of probes and quenchers via a one-particle mechanism during the excitation lifetime is addressed and extended to account for the probe solubility in the bulk phase. An exact solution to the fluorescence decay kinetics is derived. The general solution reduces to that obtained previously for a completely solubilized probe. In the case of a n immobile probe, the overall decay consists of the sum of two independent decays corresponding to monoexponential homogeneous quenching in the bulk phase and Infelta-Tachiya quenching within the micelles, weighted with the interphase partition coefficients. The general expression for the relative fluorescence quantum yield is also derived.

Introduction

Fluorescence quenching methods have been extensively used for probing the structure and dynamics of micellar systems and similar self-aggregated molecular assemblies.',* The basic model to describe the fluorescence quenching kinetics in micellar solutions has been originally proposed by Infelta et al.3 and later rigorously defined and treated by T a ~ h i y a .The ~ model assumes small monodisperse micelles, within which quenching follows pseudo-first-order kinetics, and the distribution of solubilizates among micelles obeys Poisson statistics. The probe is considered as immobile, remaining in the same micelle during the excitedstate lifetime, and the quenchers are allowed to migrate between micelles via the bulk phase. The model has been widely discussed and extended by several authors to include static quenching and migration of quenchers through collisions between micelle^,^ as well as to account for limited solubilization capability cf the micelle,6 polydi~persity,~.* and effects of micelle ~ h a p e . ~Almgren ,'~ et aLI1have developed a general model, where both probe and quencher may change their host confinement during the excited-state lifetime. Within the framework of the approximation suggested, the fluorescence decay has been shown to follow the Infelta-Tachiya type of kinetics with a generalized interpretation of the parameters. The model has been further extended to the case where there is a limit to the number of solubilizates in any one micelle.I2 Recently TachiyaI3 and Gehlen et al.I4 have succeeded in deriving an exact solution to the fluorescence decay kinetics with migrating probes and immobile quenchers. Their approach has been elaborated upon by Gehlen et al.I5 and Barzykin,I6 who have obtained an exact expression for the decay with exchange of both probes and quenchers using the generating function technique. Almgren's approximation has been further discussed in detail," and generalization to the case of micelle limited solubilization capability has been carried out.18 The foregoing models all concern only the case of a completely solubilized probe. In general, the probes are partitioned between the micelles and the bulk, and the solubility in the latter phase may be substantial. Such probe-micelle systems are not unknown. The contribution of excited probes in the bulk phase to the total fluorescence then has to be taken into account.ld The decay should be very sensitive to the migration capability of a probe, whether it can change the bulk environment with low quenching probability to a micellar microdomain, where quenching is usually much more effective, during the excited-state lifetime.

In this paper, we address the problem of migration-assisted fluorescence quenching in micellar solution and explicitly consider the partitioning of both probes and quenchers between the bulk phase and the micelles. The following assumptions are accepted in full accord with the previous 1iterature:'b.c (I) Micelles are small and monodisperse, and within them quenching obeys pseudo-first-order kinetics and the rate constant for quenching of a probe is proportional to the number of cohabitant quenchers in the micelle. (11) The equilibrium distribution of reactants among micelles is Poissonian, implying dimensionless and noninteracting molecules. I (111) The exchange of probes and quenchers between micelles and between phases occurs via a one-particle mechanism with the rate constant for exit of a molecule from a micelle being proportional to the number of these molecules and the entrance rate being independent of the number of reactants in the micelle. (IV) The excitation efficiency of a probe is sufficiently low, so that one micelle cannot contain more than one excited probe, and is independent of the number of quenchers in the micelle with the probe. (V) The absorption cross section and the radiative rate constant of the probe are not influenced by the environment (generalization to the case where both absorption and radiative constants are different in different phases is straightforward). (VI) Quenching and exchange acts are treated independently via the formal kinetic scheme given below. Exact expressions for both time-resolved and steady-state fluorescence observables are derived within the framework of the model. The general solution is shown to reduce to that obtained previously15.l6for a completely solubilized probe. Theory

Consider an ensemble of luminescent probes and quenchers dissolved in a monodisperse micellar solution and assume that the equilibrium distribution of reactants among micelles obeys Pois -7 statistics. The concentration of micelles containing n que,.-.ters and j probes, [M,,,], is given by

W,,I = [MIP(n,fi,)p v q where [MI is the total micelle concentration and

(1)

P(n,fi,) = [ ( ~ , ) " / n !exp(-fi,), ] cy = p, q (2) A, is the mean number of a molecules per micelle; a = p, q corresponds to probe and quencher, respectively.

QQ22-3654/93/2Q91-2714~Q4.QQ/Q 0 1993 American Chemical Society

The Journal of Physical Chemistry, Vol. 97,No. 11, 1993 2115

Fluorescence Quenching in Micelles A, is related to the associated equilibriumconstant for interphase exchange, kz/ky, as follows: fi, = k:[B,]/k*

(3)

where k: is the rate constant for entry of an a molecule into a micelle from the bulk phase, k_"the rate constant for exit, and [B,] the concentration of a molecules in the bulk phase:

[B,] = [T,lkY(kY

+ k",MI)-'

(4)

[Tu] is the total concentration of a molecules in the solution. Equations 3 and 4 imply one-particle exchange mechanism assuming that the exit rate constant from the micelle with n molecules is nk:, and the entrance rate is independent of the number of probes and quenchers in the micelle. The distribution of probes and quenchers among micelles with excited probe at time zero (immediately after &pulse excitation) follows the equilibrium distribution

where k: and k! are the self-decay rate constants in the bulk phase and in micelles, respectively, k: denotes the second-order rate constant for quenching in the bulk phase, and k y is the pseudo-first-order rate constant for intramicellar quenching. During the excited-state lifetime, the environment of the excited probe may be changed due to the interphase migration processes: kP

M*, e M , kt

Bfi, = [ M * ] ( O ) / [ M ]