J. Phys. Chem. 1988, 92, 4479-4483
4479
Fluorescence Quenching Dynamics In Rodllke Micelles Mats Almgren,* Jan Alsins, Emad Mukhtar, and Jan van Stam The Institute of Physical Chemistry, University of Uppsala, P.O. Box 532, S-751 21 Uppsala, Sweden (Received: December I , 1987; In Final Form: February 12, 1988)
The fluorescence decay curves obtained from hydrophobic probes and quenchers solubilized in long, rodlike micelles are d d . An expression is derived for the decay when quenching in infinite cylinders occurs. The influence of migration in this case is explored. It is shown that the decay curves obtained for quenching by nonmigrating quenchers in both very long rods and in shorter but highly polydisperse rods can be. very similar to decay curves for migrating quenchers in small monodisperse micelles. As an illustrative example, some results are presented for measurements on pyrene fluorescence quenched by benzophenone in micellar solutions of cetyltrimethylammonium chloride in a region where a sphere-to-rod transition is accomplished by the addition of sodium chlorate.
Introduction The confinement of fluorescent probes and quenchers in small discrete aggregates as in a micellar solution have two important effects on the overall fluorescence decay kinetics: a proximity effect that makes the quenching rapid in micelles containing both an excited species and quenchers, and a shielding effect that hinders the quenching of excited molecules created in micelles without quenchers. The latter effect depends on the statistical distribution of quenchers over the micelles and is important only when the average number of quenchers per micelle is small, on the order of unity or less. The possibility to distinguish between micelles with and without quenchers is the basis for most methods to determine micellar aggregation number by fluorescence quenching techniques.'-' The basic model for the analysis of fluorescence quenching data from micellar systems was formulated by Infelta et a1.* and has been discussed and elaborated upon by several a u t h o r ~ . ~ The J~ simple model has several limiting restrictions. It applies to quenching in monodisperse micelles with quenchers and probes randomly distributed over the micelles (a Poissonian distribution). The quenchers are allowed to migrate between the micelles; if they are stationary, the tail of the decay is exponential with the time constant l/roof the unquenched fluorescence decay. With migrating quenchers an exponential final decay is still obtained but with a larger decay constant. It is further assumed, in the Infelta model, that the fluorescence decay from a set of micelles with x stationary quenchers will be exponential with decay constant 1/ T o xk,. The decay that results from a polydisperse set of micelles will not follow the Infelta equation." If the quenching is sufficiently rapid and migration does not occur, it is still possible to analyze the decay so that a "quenching average" ( N ) Qof the micelle size is obtained. The variation of this average with quencher concentration allows the determination of the weight-average aggregation number ( N ) , and width and skewness parameters of the micelle size di~tribution."-'~
+
Turro, N. J.; Yekta, A. J . A m . Chem. SOC.1978, 100, 5951. Atik, S. S.;Singer, L. A. Chem. Phys. Lett. 1978, 59, 519. Infelta, P. P. Chem. Phys. Lett. 1979, 61, 88. GTieser, F. J. Phys. Chem. 1981, 85, 928. Llanos, P.; Zana, R. J . Phys. Chem. 1980, 84, 3339. Almgren, M.; Lafroth, J.-E. J. Colloid Interface Sci. 1981, 81, 486. De Schryver, F. C.; Croonen, Y.; Gelade, E.; Van der Auweraer, M.; Dederen, J. C.; Roelants, E.; Bcens, N. In Solution Behavior of Surfactants; Mittal, K. L., Lindman, B., Eds.; Plenum: New York, 1984; Vol. I, p 662. (8) (a) Infelta, P. P.; GrBtzel, M.;Thomas, J. K . J. Phys. Chem. 1974, 78, 190. (b) Infelta, P. P.; GrBtzel, M. J. Chem. Phys. 1983, 78, 5280. (9) Tachiya, M. Chem. Phys. Lett. 1975, 33, 289. (10) Van der Auweraer, M.; Dederen, J. C.; Gelade, E.; De Schryver, F. C. J. Chem. Phys. 1981, 74, 1140. (11) Almgren, M.; Mfroth, J.-E. J. Chem. Phys. 1982, 76, 2734. (12) LXroth, J.-E.; Almgren, M. ref 7, Vol. I, p 627. (1) (2) (3) (4) (5) (6) (7)
0022-36S4/88/2092-4479$01.50/0
More general migration possibilities of both probe and quencher, e.g., as due to exchange on micelle collisions, were recently treated for monodisperse mi~e1les.I~To a good approximation the decay can be described by an Infelta type of equation:
F(t) = F(0) exp(-A2t
+ A3 (exp(-A4t) - 1))
(1)
with
A3 = n(l - ( x ) , / n ) 2
(3)
(4) where n is the mean number of quenchers per micelle and ( x ) , the mean number of quenchers per micelle with excited probe during the stationary stage of the decay. In Infelta's case
( x ) , = k-n/(k,
+ k-)
(Infelta)
(5)
where k- is the quencher exit rate constant. Equation 1 describes a decay where a rapid initial phase, characterized by parameters A3 and A4, is followed by a final exponential tail (at A4t >> 1) with amplitude 17(O)e-~3and decay constant A2. Unfortunately, decays from many different experimental situations can be described quite well by such an equation. The mere fact that an experimental decay curve fits well to (1) does not prove that a micellar system with exchanging probes and/or quenchers is at hand. In the following the quenching in rodlike confinements will be discussed. For sufficiently long rods the end effects are negligible, and quenching in infinite rods can be considered. When the excited probe and a quencher are close in such a system, i.e., separated by about a cylinder radius, the quenching probability per unit time kq is comparable to that in a spherical micelle with the same radius. The transport in the cylindrical micelle to this reaction zone can be described as a one-dimensional diffusion process. Although the resulting expression for the fluorescence decay is very different from ( l ) , it turns out that some experimental decay curves may be fitted about equally well to both. A distinctive feature of micelles which grow into long rods is thought to be a very pronounced size polydispersity; simple arguments predict an exponential size di~tribution.'~-~O The im(13) Warr, G. G.; Grieser, F. J. Chem. SOC.,Faraday Trans. 1 1986, 82, 1813. Warr, G. G.; Grieser, F.; Evans, D. F. J. Chem. SOC.,Faraday Trans. I 1986, 82, 1829. (14) Almgren, M.; Lafroth, J.-E.; Van Stam, J. J. Phys. Chem. 1986, 90,
Ad? . 1
(15) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. SOC., Faraday Trans. 1 1976, 72, 1525. (16) (a) Mukerjee, P. J. Phys. Chem. 1972, 76, 565. (b) Mukerjee, P. In Micellization, Solubilization and Microemulsions; Mittal, K. L., Ed.; Plenum: New York, 1977; p 171.
0 1988 American Chemical Society
4480 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
portance of this polydispersity on the fluorescence decay kinetics is demonstrated by simulated decay curves for a system with an assumed broad size distribution. Such decay curves calculated assuming completely stationary probes and quenchers fit well to the Infelta equation with migration. The quenching behavior in systems with very long rods is exemplified by fluorescence decay measurements on pyrene quenched by benzophenone in micelles formed by CTAC, 40 mM, in the presence of varying concentration of NaC103. Fluorescence decay measurements by others2' on this system were interpreted with the Infelta equation. The results indicated that the micelles were rather small and the pyrene migration was rapid. Two of us discussed these results in a previous publication22and showed that rapid migration of hydrophobic solubilizates is excluded by evidence found on measurements of the deactivation of the long-lived triplet state of 9-methylanthracene by guaiazulene (1,4-di-
Almgren et al. 5.0
4.0
--
3.0
42
U.
m
2.0
1.0
0.0
0
600
300
Fluorescence Quenching in Infinite Cylinders An excited molecule and a quencher confined in a sphere of radius R will react with first-order rate constant k,, which also gives the encounter frequency of the pair if the quenching is diffusion-controlled. Simulationsz3 and theoretical considerat i o n ~ show ~ ~ -that ~ ~the decay is roughly exponential with decay constant l/To + k,. The quenching constant k, = y D / R 2 where y is a numerical factor of the order unity. In a long cylinder, the encounter frequency for a pair with initial separation I would be of the order D/12. Typical experimental values for spheres are l / T o = 2 X lo6 s-', k, = 1 X lo7 s-l, and R = 20 A. In cylinders with length I = 1000 A the slowest encounter frequencies would be 4 X IO3 s-l. Such a cylinder would appear unlimited on the time scale of the fluorescence experiment. Consider an infinite cylinder of radius R. An excited molecule is created at t = 0 at x = 0. Assume that only one excited molecule is present in the cylinder (which is in accord with the experimental conditions of the time-correlated photon-counting method). The fluorescence decay is built up by the emission from an ensemble of such systems. The quenchers are assumed to act independently so that the excited molecule can be regarded as fixed at x = 0 and the quenchers move with diffusion constant D, = DQ Dp (6) When the quencher is close to the excited molecule, all three dimensions of the reaction space are important. This problem is circumvented by the following artifice. When a quencher is in a cylinder segment of length ( 2 / 3 ) Ron both sides of the excited molecule, it is assumed to react with a first-order rate constant k, (which would be similar in magnitude to that in a sphere with radius R). With c(0,t) quenchers per unit length in this reaction zone, the fluorescence deactivation is given by d[P*]/dt = - ( l / ~ o + (4/3)Rc(O,t)kq)[P*] (7)
+
or, formally In ( [ P * ] / [ P * l 0 )= -
900
1200
time Ins1
methyl-7-isopropylazulene).
t / ~- ~( 4 / 3 ) R k ,
L'c(0,t') dt'
(8)
Figure 1. Simulated decay curve according to (15) with T,, = 400 ns, co = 2 X lo* m-I, h = 5 X lo8 m-I, and D, = 5.7 X lo-" mz s-I. The full-drawn curve is the best fit according to (l), x 2 = 2.75.
the number density c(x,t) of quenchers around the surviving excited molecules evolves according to the diffusion equation
ac(x,t) -=Dat
(9)
with the initial condition
c(x,O) = co
(10)
and the boundary conditions c ( m , t ) = CO
It is enough to consider x > 0 for symmetry reasons. The artificial introduction of a reaction zone with a finite reaction rate constant k, is an approximate way to account for the decay at short times. The solution of (9) with conditions (10-12) is found in standard texts on diffusion, e.g., Crank's2' treatment of diffusion in a semiinfinite medium with surface evaporation conditions. The resulting expression for (8) is In ([P*I/[P*Io)= To
h
2 ehZDmr erfc (h(D,t)'lZ) - 1 + -h(D,t)'12 *1/2
where
h = 2kqR/3D,
(14)
The time evolution of the fluorescence signal, which is proportional to [ P * ] , from the initial value F(0) is thus described by only two parameters B, = 2 / h and B2 = h2D in addition to the known value of 1 / 7 0 ;
Initially, the number density of quenchers in the cylinder is uniform, co. After the creation of an excited molecule at x = 0 (17) Tanford, C. The Hydrophobic Effect; Wiley: New York, 1980. (18) Wennerstrom, H.; Lindman, B. Phys. Rep. 1979, 52, 1. (19) Porte, G., Appell, J. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum: New York, 1984;Vol. 11, p 805. (20) Mazer, N. A,; Benedek, G. B.; Larey, M. C. J . Phys. Chem. 1976, 80, 1075. Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J . Phys. Chem. 1980,84, 1044. (21) Malliaris, A.; Lang, J.; Zana, R. J . Phys. Chem. 1986, 90, 655. (22) Almgren, M.; Alsins, J. Prog. Colloid Polym. Sci. 1987, 74, 55. (23) Gosele, U.;Klein, U. K. A.; Hauser, M. Chem. Phys. Lett. 1979,68, 291. (24) Hatlee, M. D.; Kozak, J. J.; Rothenberger, G.; Infelta, P. P.; Gratzel, M. J . Phys. Chem. 1980,84, 1508. (25) Van der Auweraer, M.; De Schryver, F. C. Chem. Phys. 1987, 107,
a2C(X,t) ax2
exp(B2t) erfc ( ( B z t ) ' l 2 )-
For B2t >> 1 one obtains
A truly exponential tail will obviously never evolve. It is nevertheless possible to obtain a good fit to the Infelta model by a simulated decay curve according to ( 1 5 ) with added statistical noise, as shown in Figure 1 . The limited time range which is
1 n5
(26) (a) Sano, H.; Tachiya, M. J . Chem. Phys. 1981, 75, 2870. (b) Tachiya, M. In Kinetics of Nonhomogeneous Processes; Freeman, G. R., Ed.; Wiley: New York, 1987;p 575.
(27) Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon: Oxford, 1975.
Fluorescence Quenching Dynamics in Rodlike Micelles
The Journal of Physical Chemistry, Vol. 92, No. 15, 1988 4481
TABLE I: Parameter Values Obtained in the Fit of (1) to the Simulated Decpy Curves from Figure 2 [Ql/[Sl X 10.' (x),/n n kq x 10*/s-' X2 1.25 2.5 5.0 7.5 10.0
0.171 0.177 0.120 0.065 0.017
0.517 0.972 1.943 2.891 3.754
1.63 1.77 1.77 1.77 1.83
0.99 1.06 1.11 1.28 1.39
accessible experimentally makes it impossible to distinguish between a truly exponential tail and the behavior according to (15). In the application of these equations to long but finite micelles it must be realized that the parameter co is the mean concentration of quenchers per length unit of the entire system and is not affected by the statistical variation of the number of quenchers per micelle. This is so because the equations apply only for time periods that are limited so that only those quenchers are effective which initially are found within a distance from the excited state that is much shorter than the length of the micelles. The statistical distribution of quenchers over this length is already taken care of by the diffusion equation.
Rodlike Micelles of Finite Length. Polydispersity Effects When pairs of excited states and quenchers are confined together in finite volumes, the time evolution of the system will always lead to an exponential decay as the final, although not necessarily experimentally accessible stage. This conjecture has not been proven, but it is in accord with a number of numerical simulations and theoretical studies of the quenching in finite volumes of various shapes and dimension^.'^^^^-^^ Some simple simulation in one-dimensional systems-random walks of quenchers and the excited molecule until collision on a one-dimensional lattice-resulted in this behavior of the fluorescence decay. The quenching constant of the final exponential phase was furthermore proportional to the number of independent quenchers on the lattice and inversely proportional to the square of the length (the number of lattice points). For the experimental study of quenching in rodlike micelles, however, the pronounced p o l y d i ~ p e r s i t y ' ~is- ~most ~ important. It may be assumed that the average number of quenchers in micelles of a given aggregation number, N , is proportional to the aggregation number, and that the distribution of quenchers over micelles of a certain size is Poissonian. For a given size distribution, and with an assumption on how the quenching constant depends on N , the fluorescence decay for nonmigrating probes and quenchers may be calculated by adding Infelta-type contributions from all micelle sizes. Simulated decay curves of this type are shown in Figure 2, based on the micelle size distribution shown in Figure 3. It was assumed that the quenching constant varied according to
where M was chosen as 450. This formula gives an interpolation between kq(N) 0: 1 / N for small N a n d kq(N) 0: l / p for large
N. Simulated decay curves calculated for several quencher concentrations gave quite good fits to the Infelta equation. The resulting parameter values are collected in Table I. The results suggest, quite erroneously, that a rapid migration of probes and quenchers occurred between micelles of moderate size. It would not be possible to conclude from fluorescence quenching measurements alone that the system in reality was composed of long polydisperse micelles. The last entry in Table I gives a lower bound to the quenching-averaged aggregation number, obtained by extrapolation to t = 0 with a straight line of slope - l / ~ ~ from , the final portion of the decays. The values are much too small, in particular at the lowest quencher concentrations. The true ( N)Qvalues should extrapolate to ( N ) , at [A]/[S] = 0, and the slope would give an estimate of the width u of the distribution from the expansionI3
1.0
0.0
N,'
W)Q
413 389 389 386 375
448 436 412 387 355
1 '
1
0
500
1000
1500
time Ins)
Figure 2. Simulated decay curves for polydisperse micelles with a micelle size distributionf(N) as shown in Figure 3. The decay curves were ( N / ( N ) ) f(N) explN([Q]/ calculated from F ( t ) = F(O)dfl7o [S])(exp(-k,(N)t - l)), where kq(N) is given by (17). The parameter values were T~ = 400 ns, k,(450) = 2 X lo6 s-', and quencher concentrations as shown in Table I. 2.0
I
1.5
-
1.0
--
I
I
0
g
r. --z
0.5 -
0.0
0
8
1
300
600
900
1200
N
Figure 3. Micelle size distribution used for decay curves in Figure 2 was calculated fromf(N = [1/(N, - N2)] [exp(-[(N-No)/N,] - exp(-[(N - No)/Nz])], with N, = 400, N2 = 50, and No = 50.
( N ) Q= (N)w - ( 1 / 2 ) ~ [ Q l / [ S l+ (1/6)~dQ1/[S1)~.Thevalues of these parameters for the distribution used are (N), = 825 and u = 403, whereas the values in Table I suggest (N), = 480 and u = 150. Thus, the procedure gives too small values but does show that the micelles are large and polydisperse. From the experimentalist's point of view, these results underline the importance of using probes and quenchers that are known to be stationary and extend the measurements to a time where the l/s0decay is reached. Migration between Rodlike Micelles Let us return again to a system of infinite rods and consider a case when there is an equilibrium concentration [Qlr of quenchers in the intermicellar solution. The equilibrium concentration of quenchers in the rods are co per unit length. The exchange of quenchers between the rods and the surrounding solution is assumed to be diffusion-controlled. At equilibrium the inward and outward flows j + and j - are equal and determined by the solution of the steady-state diffusion equation with boundary
4482 The Journal of Physical Chemistry, Vol. 92, No. 15, 1988
Almgren et al.
TABLE 11: Estimates of k, and Aggregation Number from Fitting the Infelta Model to Curves 1-3 in Figure 5, with A 3 = 1 / =~ 3.48 ~ from Curve 0 ICIOl-l /mM k, x 104/s-l n N 2 0 lZlb 0
19.0
34 42
10.0
0.89 f 0.02 1.58 f 0.02 2.40 f 0.03
5.7
80
1.17
142
2.07
216
2.06
X
IO6 s-l
0.92 6.39 2.02
"Reduced x 2 test. bRuns test. .conditions [Q] = 0 at r = R, i.e., at the cylinder surface, and [Q] = [QIfat r = R,, where R, defines a cylindrical cell surrounding a rod and is determined from the volume fraction 4 of the hydrophobic phase.
R/R, =
fd
12
(18)
,
One obtains
CI 0;
cI- ,
4O,mM,CTJ,
,
L
2
where Dois the diffusion coefficient of the quencher in the intermicellar solution. The net inflow at x, per unit length of the rod, at a concentration c ( x ) in the rod is
j = j + ( l - +)/eo) (20) The quenching process in the rod is still treated as a one-dimensional diffusion. For simplicity an infinite reaction rate is assumed at x = 0, so that the boundary conditions are c(0) = 0
at t
>0
(21)
= co (22) As above only x > 0 is considered. The diffusion equation has now to be complemented by a term from the influx from the surroundings. c(a,t)
- - - - - - --
OO
- - 4 - - - -A - - - - - 100
50 Cs,lt/mM
Figure 4. Relative viscosity, s,I = ?/?(H,O), for solutions with 40 mM CTAC and various added concentrations of sodium chlorate (0) or so-
dium chloride (A). This truly stationary stage of the decay will be observable only if T~ is sufficiently long and if the constant term c0D,,,7, in (29) is not much larger than unity. At times much shorter than 7, (27) can be approximated by
(?)"*(+ &)
In ( F ( t ) / F ( O ) )= - t / T O - 4c0
1
(30)
which shows the first perturbation of the migration on the intramicellar decay (still with the assumption that the quenching reaction is infinitely fast for probe and quencher at the same x ) .
The substitution Y = (1
- c/co)
and 7e-1
+ = j= co
+
2rDo[Q1f co In ( W R )
transform (23) into
with conditions
y=O
at x > O , t = O
y = l
at x = l , t > O
y=O
at x = m
The solution of this problem was given by Danckwerts28and reproduced by Crank.27 The resulting fluorescence decay will be given by In (F(t)/F(O)) = - t / T O ~ C ~ ( D , , , T , ) ~ / ~ (y2) ( ~ erf / T ,( t / 7 , ) 1 / 2 ( t / ( r T e ) ) l / 2 e - f / r e ) (27) An interesting feature of this equation is that at t >> 7 , a steady state will evolve with
+
+
c ( x ) / c o = 1 - exp(-x(7$,)-1/z)
(28)
The limiting fluorescence decay is exponential: In ( F ( t ) / F ( O ) ) = 4 / 7 0 - c o ( ( D , ~ , ) ' /+ ~ 2t(D,,,/~,)'/~) t
Experimental Results and Discussion For an experimental illustration of the quenching behavior, the system CTAC NaClO, is particularly well suited. In this system, Figure 4, a transition from globular micelles to very long rodlike micelles occur already at rather low additions of ClO,-. The fluorescence decays could thus be recorded for systems with equal concentrations of CTAC, probe (pyrene), and quencher (benzophenonez9), and where only the size of the micelles was varied with the CIO,- concentration. Such fluorescence decay curves are shown in Figure 5 . At low C103-concentration the globular micelles give decay curves with very well developed exponential tails: The final decay constants are very close to the values without quenchers, showing that no appreciable migration of benzophenone (BP) occurs. At higher C103- concentrations the exponential tail becomes less well developed, and at concentrations of 49 mM C103- and above the logarithmic plot shows curvature over the whole time window. For concentrations of 64 mM C10, and more the curves are indistinguishable, which means that the rods are long enough to be effectively infinite already at 64 m M C103-. The decays up to 42 mM C103- were analyzed with the Infelta ~ . results are shown in Table 11. The equation and Az = 1 / ~ The Infelta equation with Az 2 1 / fits ~ well ~ to the decays at 42-64 mM C103-. The curves for 64-100 mM NaC10,- are fitted about as well by the Infelta model with migration as by (1 5 ) . An interesting feature can be noted in the initial portions of the decays shown in Figure 5 , which are almost identical. This part of the decay reflects the quenching of excited probes created close to quenchers. During the first few moments the diffusion of the probe and quencher will be independent of all space lim-
>> 7 , (29)
(29) Russel, J. C.; Wild, U. p.; Whitten,
(28) Danckwerts, P. V. Trans. Faraday SOC.1950, 46, 300.
1319
D. G . J . Phys. Chem. 1986, 90,
4483
J. Phys. Chem. 1988, 92, 4483-4490 4.00
k, and D, estimates were obtained on the basis of an assumed radius of 21 A for the cylindrical micelles:
I
D, = DBp+ Dp = 9 kq = 2.4
1.00
'
0
I
300
I
600
,
9 00
_.
.. I
X
X
lo-" m2 s-l
lo7 s-l
These results are very reasonable. The decay data gave no indication on a migration of benzophenone during time window studied. A result quoted by Russell et alez9allows an estimation of the distribution of benzophenone between water and micelle and from that the importance of migration. The distribution constant for benzophenone between octanol and water is given as KD = 1514. Equation 25 can be written
.I
1200
t i i i e Ins1
Figure 5. Fluorescence decay curves of pyrene, 2 X M, in micelles of CTAC, 40 mM. Curve 0 represents the decay without quencher, and 1-5 represent decays with 4.45 X lo4 M benzophenone as quencher. Sodium chlorate was added in decays 2-5: 34 mM in 2, 42 mM in 3, 49 mM in 4, and 64 mM in 5. Additions of 72 and 100 mM NaC103 gave curves very similar to curve 5 .
Insertion of Do = 5 X mz s-I, 4 = 0.01 (40 mM CTAC), R , = 20 A, and KD = 1500 gives T , = 1.4 X s. Since the time window utilized is about 1.4 ~ s t ,
