J. Phys. Chem. 1991,95, 56845689
5684
They are shifted downfield in the presence of micellar SDS. It is the hightemperature,nonpolar conformers of the PEO blocks that are s t a b i l i i by the micelle surface. The nonpolar segments are envisioned to replace water in contact with the hydrophobic core between the charged headgroup, with loop of more polar segments interacting with water and the charges. The obscwcd shielding of excited pyrene from quenching by oxygen is also explained in this way. 5. For the shifts of the alkyl chain carbons of SDS,the most notable effect from interactions with the EPEs is a periodicity of three, reflecting the period of the PPO chain.
EPE molecule. For comparison, the micelles first formed in PEO are small with an aggregation number around 20 and increase in size to an aggregation number around 50 at saturation, i.e., at the concentration where free micelles with an aggregation number of about 60 begin to appear.'* 3. The I3C chemical shift measurements report on the states of the PPO and PEO blocks and the surfactant tails. The methyl carbons in PPO show similar shifts in F68 aqueous solution as in neat PPO and also in nonmicellar L64 at 25 O C . They are shifted strongly downfield in the large micelles of L64 at 40 OC and about half as much downfield in the mixed micelles. We take this as an conformational effect, ranging from compact curls in the neat state and in the unimer micelles in water to a highly extended state in the large L64 micelle. 4. The methylene carbons of the PEO blocks show similar shifts in aqueous solutions of PEO and EPEs and also in micellar L64.
Acknowledgment. This work was supported by the Swedish Natural Science Research Council. Reg&@ No. (EO)(PPO) (block copolymer), 106392125; sodium dodecyl sulfate, 151213.
Stochastk Model for Fluorescence Quenching in Monodisperse Mkelies with Probe Migration M.H.Cehlei~,**~ M.Van der AuweraerJ S .
M.G. Neumann,: and F. C. De Schryver**t Chemistry De artment, Fatholieke Universiteit Leuuen, Celestijnenlaan ZOOF, 3001 Leuven, Belgium, and Instituto de isica e Quimica de Siio Carlos, Uniuersidade de Siio Paulo, Brazil (Received: August 27, 1990; In Final Form: February 15, 1991)
Fe
Fluorescence quenching in monodisperse micelles with probe migration and Poissondistributedquencher molecules is analyzed theoretically and by numerical simulation. An exact solution for the fluorescence decay is derived and expressed as a series of generalized selfconvolution products of an InfeltaTachiya type equation. Based on Almgren's approach and data analysis of synthetic sample decays, an approximated solution to the fluorescence decay is also derived.
I. Introduction
which leads to a Poisson distribution of the quencher molecules over the micelles.
Fluorescence quenching in micellar solution has been one of the most useful methods in the determination of structural and kinetic parameters such as the mean aggregation number and the rate constantsof a micelltprobequencha system. The expression for the observed fluorescence decay (eq 1) after a delta pulse
+ A3(exp(A4t)  1))
At) = A, exp(A2t
A"
(1)
excitation in the case of an "immobile" probe (probe with a small probability to undergo migration between micelles during its excitedstate lifetime) and a mobile quencher, was derived by Infelta' and Tachiya.2 In eq 1 the A parameters are expressed explicitly as AI A i = k, A3
=A01
(2)
+ kqkA(kq+ kJ'
(3)
= Rk,'( kq + k J 2
(4)
kq + k
A4
(5)
with k, the deactivation rate constant of the fluorescent probe in the absence of quenchers, kp the firstorder rate constant of the intramictllar quenching, k, the exit rate constant for a quencher from a micelle, and R the average number of qucnchers per micelle. Equation 1 describes the quenching dynamics in the micelles if the quencher molecules exchange via the aqueous phase, as described by Mn + Q
k+ (,
+
' &I
n
0, 1 2,
.a.
(1) Infeltr, P.P.;Gratzcl, M.;Thomsr,J. K.J. Phys. Chem. 1974,78,190. (2) Tachiya, M. Chem. Phys. Lett. 1975, 33, 289. (3) Maatri, M.; Infdta, P.P.;Gratta, M.J. Ch" Phys. 197&69,1522. (4) Van der Auwerau, M.; Dederen, J. C.; Gclade, E.; De Schryver, F. C . J. Chem. Phys. 1981,74, 1140. (5) Almgren, M.;Lofroth, J. E. J . Chem. Phys. 1982.76, 2734. (6) Warr, G. G.; Griscsr, F. J . Chem. Soc., Faraday Trans. I 1986,82, 1813. (7) Kahlweit, M.J. Colloid I n r e f l i e Sci. 1982, 990, 92. , J. Phys. Chem. 1989, 93, 10. (8) Jada, A.; Lmg, J.; Z ~ M R. (9) HglSlein, A,; F b h , Th.&r. BunsenGes. Phys. Chem. 1978,82,471. (10) Eicke, H. F.; Shepherd, J. C. W.; Steinemam, A. J. c d l d d I n t e t f i Scl. 1976. 56. 168. (1 1) Fletcher, P. D. I.; Robinran, B. H.Ber. BunsenGes. Phys. Chem. 1981,85,863. Infelta, P. P.; Graglia, R.; Minero, C.; Peliuetti, E. Co//dds SurJ 1987, 28, 289.
(6)
Kptholicke Univeniteit Leuven. tUnivmidade de Slo Paulo.
OO223654/91/20955684$02.50/0
...
P,, =  exp(R) n = 0, 1, 2, (7) n! When the exchange rate constant of the quencher (k) is much smaller than the fluorescence decay constant (ko),the quencher can be considered as an immobile species and in this case, eq 1 reduces to a more simple form.) At) =AO) exp(k,t + n[exp(kqt)  11) (8) Some extensions of the above treatment have been worked out to take into account static quenching4and polydispersity,5I6still retaining the initial condition that the probe can be considered as an immobile species. However, compartmentalized reactions at high micelle concentration or in the presence of additives,' in reversed micelles, microemulsions? and in certain polyelectrolyte solutions, may have to take into account probe migration or exchange of both specia during the lifetime of the excited probe. Mechanisms such as hopping9of the probe or the socalled fusionfission processlall
(b
1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No. 14, 1991 5605
Fluorescence Quenching in Monodisperse Micelles
By use of the mass balance to express the total micelle concentration m
OD
= CM! =
@i
10
m
c M l + PEMl* 10 O
(13)
and the definition of the density of micelles containing one excited probe and n quenchers at time r OD
P,* = M,*/CM]*(O)
(14)
10
the following stochastic equation in terms of P,* is obtained dP,* = (k, k)P,*  nkqP,* + kP,EP,* (15) dt 10 where k = kpMo and P, is the Poisson distribution given by eq 7. Considering that the efficiency of the excitation of a probe is independent of the quencher occupgtion number for a delta pulse excitation, the initial number of quenchers in a micelle that contains an excited probe also may be described by a Poisson distribution with the same average. The initial condition associated with eq 12 will then be given by

Figure 1. Examples of probe migration by a hopping mechanism: (a) in micellar solution; (b) between two microdomains in a polyelectrolyte.
have been mentioned as possible effective exchange processes. Examples of the former process are shown in Figure 1. The hopping rate constant of a charged species in micellar systems have been calculated in the mean first passage time approximation and it has been shown that this pnxxss can be very important when the probe is a singly or doubly charged species with an opposite charge to that of the micelle surface.I2 In this work, theoretical considerations are presented concerning the problem of probe migration assuming that the quencher molecules remain in the same micelle during the excitedstate lifetime of the probe. The same problem has been analyzed by T a ~ h i y a , using ' ~ the Laplace transform method. The approach introduced by this paper is based on an integral equation formalism.
+
m
P,*(O) = M,*(0)/CMl*(O) = (iin/n!) exp(ii)
Stochastic equations are most readily solved by using a generating function." For the process considered, the generating function can be expressed as m
G(s,t) =
by k?
n , j = 0 , 1 , 2 ,...
(9)
where M,* stands for a micelle containing one excited probe and n quenchers, whereas MIstands for a micelle with j quenchers but without an excited probe. k, is the sccondorder rate constant for the probe migration process. The possibility of probe migration substantially modifies the topology of the system, since now all molecules are connected via eq 9. The stochasticity of the probe migration process is due to the statisticaldistribution of quenchers, assumed to be a Poisson distribution, which leads to a time fluctuation in the number of quenchers that can interact with the probe during its excitedstate lifetime. Two additional assumptions are necessary. First, no more than one excited probe should occupy a micelle simultaneously, and this condition is fulfilled experimentally by using a low probe concentration. Second, the concentration of the probe in the continuous phase should be at all times negligible. This condition is fulfilled by adjusting the hydrophobicity of the probe or by charging it with a charge opposite to that of the micelle surface. To complete the kinetic scheme, the unimolecular decay of the probe and the fluorescencequenching process, linearly dependent on the number of quenchers n, should be included.
M,*
M,
If M,* and M, denote, respectively, the concentrations of micella with n quenchers and with or without an excited probe, the following set of rate equations may be derived: dM,* (ko + nk,)M,*  k , , M n * b l k,,M,,h41* (12) dr J0 10


+
~~
~
s
s 10