J. Phys. Chem. 1995,99, 4181-4186
4181
Triplet-Triplet Annihilation in Micelles Including Triplet Intermicellar Migration Marcel0 H. Gehlen* Instituto de Quimica de Siio Carlos, Universidade de Siio Paulo, 13560-250, Siio Carlos-SP, Brazil Received: August 3, 1994; In Final Form: December 1, 1994@
The problem of triplet-triplet annihilation in micelles is analyzed under conditions where triplet molecules are mobile species migrating between micellar confinements. Using a stochastic treatment, it is demonstrated that the triplet decay is characterized by a faster component due to the intramicellar annihilation followed by a slow component ascribed to the intermicellar migration and self-decay of the triplets. The slow process can be properly fitted by a decay function resulting from a competition of second-order and first-order processes of the classical chemical kinetics. The method is extended to experimental data obtained from laser flash photolysis of safranine in sodium dodecyl sulfate (SDS)solution. Analysis of triplet decays of the monoprotonated safranine at several different dye concentrations in 10 mM SDS, pH = 10 and ionic strength 0.1, yields a triplet mean residence time of about 10.4 f 2.7 ps. The triplet decay time of the dye in these experimental conditions is 23.4 f 2.2 ps.
1. Introduction
The study of photoreactions involving triplet excited states in microheterogeneous systems like micelles, reverse micelles, and microemulsions is a basic research topic of areas of the applied photochemistry like emulsion photopolymerization, solar energy conversion, and photocatalysis.l s 2 Triplet molecules have also been employed as probes in micellar systems to obtain information on polarity, molecular partition coefficients, and rate constants of the entrylexit dynamics of probe and quenche r ~ .The ~ triplet-triplet absorption in laser flash photolysis, delayed fluorescence from the triplet-triplet annihilation ('ITA) process, and phosphorescencemeasurements with single-photon timing are the time-resolved techniques used to monitor the triplet concentration. The micellar kinetics of triplets of neutral compounds like ketone^^,^ and a r e n e ~ ~as- ~well as of ionic dyes1°-13 has been the topic of systematic investigations. The formulation of models of triplet dynamics in micellar systems has been based mainly on classical chemical kinetics, and only a few models have been described in terms of a stochastic treatment.14-" However, the statistical formulation of the problem of chemical reactions in micelles is essential, and this point was demonstrated in several studies of luminescence quenching. *, In the present contribution, triplet-triplet annihilation ('ITA) in micellar systems is analyzed under conditions where triplet molecules are mobile species migrating between micelles. Using a stochastic treatment, it is demonstrated that the decay profiles are characterized by a fast decay due to the intramicellar annihilation (usually a diffusion-controlled process), followed by a slow decay ascribed to the intermicellar migration and natural triplet decay. We shall see that the slow process can be properly fitted by a decay function resulting from a competition of second-order and first-order processes of the classical chemical kinetics. Experimental results obtained from laser flash photolysis of safranine in sodium dodecyl sulfate solution are analyzed with the proposed model, and the rate constant of triplet intermicellar migration is determined.
2. Experimental Section Safranine T chloride (Merck) and acridine orange chloride (Sigma) were recrystallized from ethanol before use. Sodium ~
~~~
~
@Abstractpublished in Advance ACS Abstracts, February 15, 1995.
0022-365419512099-4181$09.0010
dodecyl sufate (Sigma) was used as received. For the fluorescence measurements, the solutions were air equilibrated. For triplet studies they were deoxygenated by nitrogen bubbling. A Nd-YAG laser (Spectron SL 401) operating at 532 nm by frequency doubling was used for excitation. The optical path of the analyzing beam in the irradiated volume of the sample was 3 mm. Triplet absorption transients were determined with an Applied Photophysics laser kinetic spectrometer. Transients were detected by a Hamamatsu R928 photomultiplier and were recorded and averaged on a HP 54504A digitizing oscilloscope. The triplet decay of monoprotonated safranine T was observed at 820 nm. The extinction coefficient at this wavelength was 18500 M-' cm-'. Steady-state fluorescence experiments were performed on the Edinburgh CD 900 photon counting spectrofluorimeter. The changes of the critical micellar concentration of SDS upon addition of NaCl were determined from fluorimetric tritation methods using acridine orange as a fluorescent probe. At 0.1 M NaC1, the cmc value found was 1.4 x M. This value is in agreement with the previous value of 1.48 x M determined by Emerson and H o l t ~ e r .The ~ ~ micellar aggregation number of SDS in a 10 mM solution with 0.1 M NaCl was assumed as 82.26 For a 10 mM SDS solution the corresponding concentration of micelles is 0.1 mM. In the data treatment of the triplet transients, the time-dependent average number of triplets per micelle was calculated by dividing the monitoring triplet concentration by the micelle concentration of 0.1 mM. 3. Model
-
The stochastic treatment of an irreversible bimolecular reaction 2A C in small volumes was first studied by McQuarrie.20 Triplet-triplet annihilation in micelles was considered by Rothenberger et a1.I6 as a typical example of this kind of reaction. They solved the respective Master equation of the problem subject to the boundary condition given by a Poisson distribution. The model was then used to study the triplet decay of l-bromonaphthalene dissolved in an aqueous micellar solution of cetyltrimethylammonium bromide. Exchange of the solubilizate 1-bromonaphthalene between micelles was neglected due to the high hydrophobic character of this probe. The model of the triplet kinetics in micelles considered 0 1995 American Chemical Society
Gehlen
4182 J. Phys. Chem., Vol. 99, No. 12, 1995
here is an extension of Rothenberger's model with addition of triplet intermicellar migration and triplet self-decay processes. Let the random variable Y(t) be the number of triplet A* molecules in a given micelle at time t. The probability of finding a micelle containing yA* at time t, Py*(t)= Prob{Y(t) = y}, is a function of the annihilation, unimolecular decay, and intermicellar migration dynamics. The stochastic equation for this problem may be written as
where 0,and 0,are the reaction and migration contributions, respectively. The expression of 0,include the transition probabilities of a pure death process 2o and the transition probabilities of the annihilation process,20being written as
0,= ko[o,+ l)Py+,* - yPy*] +
this situation, the equilibrium condition for interphase exchange including the bulk phase contributionz3
k+[A*l(k-
e,*
dt = k,P,* dP* k -- - "[o, 2
+ PPI* - kPo*(y)
+ 2)o, + l)Py+z* - y o , - l)Py*l -
[k(Y) + PYIPY* + k(Y)Py-,* +Po,
k"re + 2)o, + 1)Py+2* - Y o , - l>py*l ( 2 ) 2
-
(8)
should be considered. In eq 8, [A*] is the total concentration of triplet molecules in solution. When the probe molecule has a high affinity with the micellar phase, triplets will be generated almost exclusively in the micellar pseudophase and its concentration in the aqueous phase will be very small. Considering eqs 1, 2, and 6, the stochastic equation of this problem can be written as
dt
where /Q and ka are the unimolecular decay rate constant, and the annihilation rate constant, respectively. It is assumed in eq 2 that the intramicellar 'ITA process is 2A* 2A. In order to define the transition probabilities of Om,a mechanism for A* migration should be assumed. Migration of a solubilizate via the aqueous phase can be represented by19
+ k+[MI)-' = (y)
+ l)Py+l* y = 1 , 2 , 3 , ... (9)
+
where /3 = k ko. The initial condition of TTA kinetics in micelles was discussed in detail by Rothenberger et a1.16 If the probe molecules in the ground state are distributed according to a Poisson law with average and if the efficiency for a given probe to be converted into its triplet state upon light excitation is the same for all probes in the irradiated volume, then the initial condition isI6
(3) where k- is the fist-order rate constant for the exit of an A molecule from a micelle and k+ is the second-order rate constant for the entry of an A molecule into a micelle. My and A, denote a micelle containing y solubilizates and the solubilizate in the aqueous phase, respectively. On the other hand, migration of probe molecules during micelle interaction may be described by21 YkP MY
+
iz+MY-l+
M,+1
(4)
where K is a parameter that depends on the molecular properties of the probe (extinction coefficient at the excitation wavelength and triplet quantum yield) and on the experimental conditions such as laser pulse intensity. A general time-dependent solution of eq 9 is unknown. However, an analytical solution of the limiting case where only ka is non-zero was obtained by using the generating function method.16 The time-dependent average number of triplets per micelle is given by
where kp is the second-order rate constant for migration. These ideal migration dynamics were discussed in detail by Tachiya.Ig In both cases, the equilibrium distribution of solubilizates among micelles obeys Pdsson statistics
n=l where n = 1, 2, 3,
-
2n - 1 6 d r(G-n+11/21 B, = -e - Y K Z j = n U - n)! r(u n 1)/2) 2"
(5) where is the average number of A molecules per micelle. The same migration dynamics may be assumed for A*. Considering eq 4,the expression for 0,is z2
0,= k [ o , + l)Py+,* - yPy* + (Py-l* - Py*)(Y)l,
... and
+ +
+
+
with j = n, n 2, n 4,.... 00, eq 11 converges to
-
r is the gamma function.
(12) For
t
y = 1 , 2 , 3 , ... (6)
where (y) is the time-dependent average number of A*'s per micelle,
(Y) =
cy,*
(7)
Y
k is a fist-order migration rate constant defined as k = k,[M]. [MI is the total concentration of micelles. When eq 3 is considered as the migration mechanism, 0,still has the same mathematical form as in eq 6 but with k replaced by k-. In
To study the problem of TTA in the presence of triplet intermicellar migration, eq 9 was solved numerically by using the Runge-Kutta method with adaptive step-size controLZ4 3.1. Decay in the Particular Case Where ko = 0. Figure l a shows 8-pulse excitation decays at several different values of the triplet migration constant, k, but at constant annihilation rate constant, ka. The decay curve shows two distinct parts when ka >> k. An initial fast decay, solely controlled by k,, and a slow component which is a function of k are observed. More
J. Phys. Chem., Vol. 99, No. 12, I995 4183
Triplet-Triplet Annihilation in Micelles
o'oll
-I
,
0.0
-0.5 -1.0
-
~;
-1.5 -
-2.0 -2.5 -
1.1
0
-2.5
0.10
5
10
20
15
5
0
25
10
15
20
25
20
25
time (ps)
time (ps) 7,
01' 0
"
5
"
10
"
15
"
20
'
time ( ps) Figure 1. (a, top) Triplet decay in the presence of the triplet intermicellar migration process: k, = 2.0 ,us-'; , k = 0.0, 0.02, 0.05, and 0.10 ,us-'. Plot of the inverse of the triplet average micellar occupancy as a function of time.
interesting is that a plot of the inverse of the triplet concentration as a function of time develops a well-defined linear part at long times (see Figure lb). Figure 2 shows the effect of triplet occupancy at time zero in the decay. A faster decay is observed at a higher initial concentration of triplets. On the other hand, plots of the inverse of the triplet concentration, as shown in Figure 2b, are still linear at long times with a slope independent of the initial triplet concentration. The slope and intercept at t = 0 of the linear part of the inverse plots were analyzed considering several different combinations of the kinetic parameters of the model. A typical set of results is shown in Figure 3. When k, is much larger than k, the slope of the linear part converges to 2k, while the intercept approaches the value of (y(m))-l given by eq 13. Such results indicate that the decay at long times when ka >> k may be given by a second-order law written as
(y>-' = (y(-))-'
+ 2kt
5
"0 25
(14)
Figure 4 shows the inverse of the triplet concentration as a function of time at several different values of the ratio kJk. For values of k as large as 4ka, (y)-l is a linear function of time in the whole time domain. Since the slope converges to ka, the mean field approximation of the decay function reads
When ka and k are of the same order, the long time behavior of
10
15
time ( ps-' Figure 2. (a, top) Effect of the triplet initial concentration on the 'ITA kinetics: k, = 2.5 ps-l; k = 0.1 ,us-', and y(0) = 0.5, 1.0, and 2.0. (b, bottom) Second-order plot of the decay profiles of part a.
H
11-
H
H 10-
A A
0.9-
A
A
Z
A
I
slopel2k
A
A 0.8
A 1
2
3
4
,
5
6
In(k, I k)
Figure 3. Slope and intercept of the second-order plots as a function of kJk. See text for details.
the decay function will be defined by both processes. As seem from Figure 4, the plots of the inverse of the mean triplet occupancy versus time are linear at long times even in the situation of ka = k. If a stationary condition between annihilation and migration processes is assumed, then the decay function would be
where
Gehlen
4184 J. Phys. Chem., Vol. 99, No. 12, 1995 ,"
k,lk=
25 -
20
20 -
0-
a
-l 6
15-
4
10-
2
t .
0 0
5-
10
5
15
a=-
0
20
time ( ps ) Figure 4. Plot of the inverse of the triplet average micellar occupancy as a function of time for several different ratios of kJk.
2kka 2k k,
+
and 4 has its value between the limits (JK)-~and (y(m))-l. If eq 17 is valid, then a plot of a-l as a function of ka-l at constant k should yield a straight line with slope 1 and intercept 1/2k. This assumption was tested and c o n f i i e d . A typical set of results obtained from analysis of synthetic decays is shown in Figure 5 . A slope of 0.97 and an intercept of 2.32 were found. The theoretical value of 1/2k in the simulated data is 2.5. 3.2. Decay with ko f 0. Figure 6 shows the decay profiles considering only annihilation and natural decay of the triplet (in the condition where ka >> h).At long times the decay becomes exponential with a rate constant h,as would be expected. When the triplet migration process is included, the decay on a log scale is no more linear at long times (see Figure 7). Figure 8a shows the decay as a function of the initial triplet concentration at constant values of k,, h,and k. The plots of the inverse of the triplet concentration are included in Figure 8b. The lack of a clear linear part is due to the presence of the self-decay of the triplet state. Analysis of the tail of the synthetic decays generated in the situation where ka is much larger than ko and k was performed by using a fitting function resulting from the competition of the reactions A* products and 2A* products in a homogeneous medium,
-
-1
-
0
5
10
15
20
1
ki' Figure 5. Inverse of the angular parameter of second-order plots of synthetic decays as a function of ka-'. Linear regression yielded a slope = 0.97and a linear coeficient = 2.32.
Ot-1 -1
-\
-2
. -
-s . h
h
h V
c.
.
h
-=
k, = 0.0
I
-3
\ \
-
0.10
-5 0
5
10
15
20
25
time ( ps )
Figure 6. Triplet decay in the absence of triplet intermicellar migration: k, = 2.0 ps-'; k~ = 0.0, 0.02, 0.05, 0.10, and 0.15 ps-l.
s
O -t l
-1
h
. . E x
-2-
h
-C
'
-3
-
-4-
where bl, b2, and bf are adjustable parameters. The fits of eq 18 to the tail of the synthetic decays were good in all cases analyzed, as indicated by the x2 values obtained. The results indicate that in the limit of ka >> k~ and k, the parameters bl, b2, and b3 converge to h,(y(m))-l, and 2Wh, respectively. When annihilation is much faster than the other processes, the average number of triplets per micelle hits rapidly the value (y(-)) given by eq 13. At this point, micelles are singly occupied or empty. Migration of a triplet to a nonempty micelle results in fast annihilation with a rate equal to 2k. On the other hand, when k >> k,, b3 converges to kalko, and when k, = k, b3 is given approximately by d k o . These results are, respectively, the mean field approximation and the stationary condition, as discussed in part 3.1.
4 0 30 10
20
40
time ( ps ) Figure 7. Triplet decay considering intermicellar migration and selfdecay processes: k, = 2.0 ,us-'; kO = 0.04 ,us-'; k = 0.02, 0.04, 0.08, and 0.16 ps-'.
4. Experimental Results and Discussion The triplet decays of safranine T in SDS micelles were analyzed according to eq 18. Fitting parameters bl, b2, and b3 were obtained from a nonlinear least squares program based on the Marquardt algorithm. The time window considered in the analysis was 3 to approximately 90 ,us. Triplet decay curves of experiments at different concentrations of safranine were
J. Phys. Chem., Vol. 99, No. 12, 1995 4185
Triplet-Triplet Annihilation in Micelles
TABLE 1: Values of ko and k Obtained from the Fitting of Safranine Triplet Decay in SDS Micelles Using Eq 18 and Assuming bl = ko and b3 = 2kk0 [safranine], 10-4M ko, s-' k, s-l 0.3 4.02 x 104 1.17 x 105 0.6 1.o 1.5
3.91 x 104 4.46 x 104 4.81 x 104
1.30 x 105 7.50 x 104 8.60 104
than migration, the observed first-order rate constant in a heterogeneous medium should converge to k0 k, (y), where k, is the self-quenching rate constant. For the experimental system studied, the average value of the triplet migration rate constant is (1.02 f 0.26) x lo5 s-l, and the triplet mean residence time (average of llk) is 10.4 & 2.7 ps. The average decay time ( l / h ) is 23.4 f 2.2 ps. It is well-known that TTA rate constants in homogeneous solution approach diffusion control when spins-statistics are taken into a c c ~ u n t .The ~ same behavior should be observed in the case of the TTA reaction in micelles. The preferential location of the safranine triplet in SDS micelles is the Stem layer of the micelle due to the positive charge of this probe. If the annihilation process involves tangential diffusion of the triplets, then the rate constant at long times in the diffusion controlled limit, b,can be calculated by25
+
10
0
20
30
40
time ( ps )
cy>-'
k,(w) = S1(Sl
time ( ps ) Figure 8. (a, top) Log plot of the normalized average triplet occupancy as a function of time (considering migration and self-decay processes) for several different initial occupancies: k, = 2.5 ,us-'; ko = 0.04; k = 0.10; y(0) = 0.5, 0.7, 1.0, and 1.5. (b, bottom) Inverse plots of the decay profiles of part a.
[Triplet] / [MI 0.08
0 04
1 41
0.00
000
where D and R are the diffusion coefficient and the distance from the micelle center to the zone of preferential location of the triplet probe, respectively. The factor sl(sl+l) can be evaluated by the following empirical correlationz5 ln[s,(s,
+ l)] = 0.613 ln[R,R-']
002
004
006
008
010
time (ms)
Figure 9. Triplet decay of monoprotonated safranine in a 10 mM SDS solution, pH = 10, ionic strength 0.1. The solid line is the fitting curve calculated from eq 18. [MI is the micelle concentration, which in the present experimental condition is 0.1 mM. Dye concentration is 0.1 mM.
recorded and analyzed. Figure 9 shows a typical experimental decay of the safranine triplet. The solid line in Figure 9 is the fitting curve calculated from eq 18. In order to calculate k and k~,,it was assumed that ka >> k, leading to bl = k0 and b3 = 2k/k~.This assumption will be discussed later. Table 1 shows the values of k , ~and k obtained from the analysis of the safranine triplet decay experiments at different dye concentrations in SDS micellar solution. The observed increases of k0 with dye concentration would be related to a self-quenching process of the triplet by its ground state. If the self-quenching is slower
- 0.374
(20)
where R, is the tri let encounter distance. Assuming realistic values of R = 20 R, = 6 A, and D within the range (0.2-1) x cm2 s-l, the value of the inverse of the intramicellar annihilation rate constant falls in the interval 122-610 ns. This result means that for triplet probes with lifetime of several microseconds, the decay profiles will be characterized by an initial fast component followed by a slow one. This prediction is supported by the results of FlamigniI3 in a nanosecond laser flash photolysis study of the intramicellar TTA process of erythrosin B in a cetyltrimethylammonium bromide (CTAB) aqueous system. A fast reaction, occurring within 1 p s after excitation, and a slow reaction, occurring within hundreds of microseconds, were observed following erythrosin B triplet decay in the cationic micellar system of CTAB. Rate constants ranging from 5.5 x lo6 to 3.6 x lo6 s-l were obtained for the fast intramicellar triplet decay process.13 The inverse of these limits corresponds to 182 and 278 ns. These values are within the range of decay times of a diffusion controlled intramicellar annihilation process. However, the slow process was ascribed to the first-order deactivation process of the triplet occurring in the single occupied micelles, and a possible effect of intermicellar migration was neglected. Comparing the experimental value of the mean residence time of safranine triplets with the inverse of the intramicellar annihilation rate constant in the diffusion controlled limit, one concludes that the condition k, >> k holds in the present system, and the recovered parameter b3 is then 2Wh. The constant for association of safranine T to SDS micelles in 0.1 M NaCl was determined from fluorescence quenching data of pyrene by the dye. Application of the Encinas-Lissi26
1,
0.16,
D + 1)-R2
4186 J. Phys. Chem., Vol. 99, No. 12, I995
method yielded an association constant of the ground state, K = (9.4 f 1.3) x lo3 M-l. To correlate K with the first-order rate constant for the exit of the safranine triplet from a micelle, it is necessary to assume that ground and excited state association constants are not very different. Considering that K = k+/k-, and taking the values of K and k- determined, a rate constant for entry of the dye into the micelle of 9.6 x lo8 M-' s-l is calculated. In a study of mobility and exchange of triplet probes in reverse micelles with monitoring based on a P-type delayed fluorescence signal upon laser excitation, Bohne et a L 3 s 9 considered a model where the time-dependent emission intensity is a quadratic function of the triplet concentration. Data were fitted to the following equation
where [TO]is the initial triplet concentration, kl is the firstorder decay rate constant, 2km.4 is the second-order TTA rate constant, and a is an experimental parameter that incorporates emission quantum yields, the quantum yield of singlet formation in the 'ITA process, and the detection sensitivity. The model was used to describe the kinetics of triplets of pyrene derivatives in 0.2 M solutions of AOT in isooctane. In experiments with sodium 1-pyrenesulfonate and 1-pyrenedodecanoic acid, values of 2kpA were (2.4 f 0.5) x lo9 M-' s-l and (3.0 f 0.4) x lo9 M-' s-l, re~pectively.~,~ Note that the term inside brackets in eq 21 is similar to eq 18. Dividing [To] by the micelle concentration, [MI, yields the average occupancy of triplets per micelle at time zero. Such scaling of the function results in the pseudo-fist-order rate constant 2km~[M]in eq 21, which is equivalent in meaning to 2k if annihilation is faster than migration. But if the annihilation rate constant is of the same order as the migration rate constant, then the experimentally observed annihilation rate constant should approach 2kka/(2k ka), where k, is the first-order rate constant of the intramicellar 'ITA process.
+
5. Conclusions In this paper, a stochastic model of triplet-triplet annihilation in micelles including triplet self-decay and triplet intermicellar migration is presented. Analysis of simulated data based on the numerical solution of the corresponding Master equation of this problem demonstrated that the triplet decay at long times can be properly described by a decay function resulting from a competition of second-order and first-order processes of the classical chemical kinetics. The use of this simple decay function allows the determination of the triplet intermicellar
Gehlen migration rate when migration is much slower than TTA. Application of this method to experimental data obtained from laser flash photolysis of safranine in sodium dodecyl sulfate solution (10 mM, pH = 10, and ionic strength 0.1) indicated that the triplet dye molecule migrates between micelles with an exit rate constant of about (1.02 I'C 0.26) x lo5 s-l, while the rate constant for the entry of the triplet dye molecule into the micelle is 9.6 x lo8 M-' 5-l. Acknowledgment. Financial support by CNPq and FAPESP (Brazil) is gratefully acknowledged. References and Notes (1) Kalyanasundaram, K. Photochemistry in Microheterogeneous Systems; Academic Press: New York, 1987. (2) Gratzel, M. Heterogeneous Photochemical Electron Transfer; CRC Press: Boca Raton, FL, 1989. (3) Bohne, C.; Barra, M.; Boch, R.; Abuin, E. B.; Scaiano, J. C. J. Photochem. Photobiol. A: Chem. 1992, 65, 249. (4) Scaiano, J. C.; Selwyn, J. C. Can. J . Chem. 1981, 59, 2368. (5) Leigh, W. J.; Scaiano, J. C. J. Am. Chem. SOC. 1983, 105, 5652. (6) Gosele, K.; Klein, U. K. A.; Hauser, M. J. Mol. Struct. 1982, 84, 353. (7) Bolt, J. D.; T w o , N. J. J. Phys. Chem. 1981, 85, 4029. (8) Almgren, M.; Grieser, F.; Thomas, J. K. J. Am. Chem. SOC. 1979, 101, 279. (9) Bohne, C.; Abuin, E. B.; Scaiano, J. C. J. Am. Chem. SOC. 1990, 112,4226. (10) Scaiano, J. C.; Kim-Thuan, N.; Leigh, W. 3. J. Phorochem. 1984, 24, 79. (1 1) Mackay, R.; Gratzel, M. Ber. Bunsen-Ges. Phys. Chem. 1985,89, 526. (12) Faughinel, E.; Ortman, W.; Behrmann, K.; Willscher, S.; Turro, N. J.; Gould, I. R. J. Phys. Chem. 1987, 91, 3700. (13) Flamigni, L. J. Phys. Chem. 1992, 96, 3331. (14) Rothenberger, G.; Infelta, P. P.; Gratzel, M. J. Phys. Chem. 1979, 83, 1871. (15) Hatlee, M. D.; Kozak, J. J. J. Chem. Phys. 1980, 72, 4358. (16) Rothenberger, G.; Infelta, P. P.; Gratzel, M. J. Phys. Chem. 1981, 85, 1850. (17) Rothenberger, G.; Moser, J.; Gratzel, M.; Serpone, N.; Sharma, D. K. J. Am. Chem. SOC.1985, 107, 8054. (18) Gehlen, M. H.; De Schryver, F. C. Chem. Rev. 1993, 93, 199. (19) Tachiya, M. In Kinetics of Nonhomogeneous Processes; Freeman, G. R., Ed.; Wiley: New York, 1987; p 575. (20) McQuarrie, D. A. J. Appl. Prob. 1967, 4, 413. (21) Tachiya, M. J. Chem. Phys. 1982, 76, 340. (22) Gehlen, M. H. Chem. Phys. 1994, 186, 317. (23) Barzykin, A. V.; Lednev, I. K. J. Phys. Chem. 1993, 97, 2774. (24) Press, H. W.; Teukolsky, S. A. Comput. Phys. 1992, 6, 188. (25) Van der Auweraer, M.; Dederen, J. C.; Gelad6, E.; De Schryver, F. C. J. Chem. Phys. 1981, 74, 1140. (26) Encinas, V.; Lissi, E. A. Chem. Phys. Lett. 1982, 91, 5 5 . (27) Emerson, M. F.; Holtzer, A. J. Phys. Chem. 1967, 71, 1898. (28) Literature values of SDS aggregation number in the presence of 0.1 M NaCl are 75,2983,30and 88.31 (29) Turro, N. J.; Yekta, A. J. Am. Chem. SOC. 1978, 100, 5951. (30) Coll, H. J. Phys. Chem. 1970, 74, 520. (31) Mysels, K. J.; Princen, L. H. J. Phys. Chem. 1959, 63, 1696. JP942019S
