Determination of micelle aggregation numbers and micelle fluidities from time-resolved fluorescence quenching studies

Determination of Micelle Aggregation Numbers and Micelle Fluidities from Time-Resolved Fluorescence Quenching Studies M. A L M G R E N AND J . - E . L O F R O T H

Department of Physical Chemistry, Chalmers University of Technology and University of GOteborg, S-412 96 G6teborg, Sweden Received July 8, 1980; accepted October 28, 1980 A critical study of the quenching of the fluorescence of tris(2,2'-bipyridyl)ruthenium(II) by 9-methylantracene in sodium dodecyl sulfate micelles at various ionic strength, in microemulsions and in pentanol-swollen micelles is presented. The results are discussed in terms of aggregation numbers and quenching constants obtained from single-photon counting data and steady-state fluorescence measurements. In connection with the analysis of the time-resolved data, an approach to choose correct time zero is outlined. The choice of the time where the excitation profile has its maximum is suggested. Varying the ionic strength in the micelle solutions resulted in an increase of the mean aggregation number from 63 at zero concentration of sodium chloride to 373 at 0.75 M of the electrolyte. The first-order rate constant for quenching decreased in these solutions from 2 × 107 to 0.2 × l0 Tsec -~ in an inverse proportion to the estimated aggregation numbers. The correct values of the aggregation numbers are probably higher than the estimated values, which is shown in computer simulations assuming a distribution of micelle sizes. The necessity of timeresolved experiments to calculate an aggregation number is demonstrated in measurements on a microemulsion (detergent-pentanol-dodecane). The product of the quenching constant, 0.2 × l0 T sec-~, and the lifetime, 309 nsec, of the fluorescent molecule is too small to justify the use of steady-state data only. The estimated aggregation number for this solution is 306, a factor of three higher than earlier reported. For pentanol-swollen micelles, data are presented where there is a need for deconvolution, the quenching constant, 6 x 107 sec-~, being too large to ignore the finite width of the excitation pulse. For this system it is better to calculate the aggregation numbers from steady-state data than from single-photon counting data once the magnitude of the quenching constant and the lifetime have been deduced. The fluidity of the micellar aggregates are discussed by defining local second-order rate constants. These were calculated to 0.4 × 109 M 1 sec-~ in micelles, to 1.0 x 109 M -~ sec -~ in pentanol-swollen micelles, and to 1.5 x 109 M -1 sec -x in microemulsion particles, compared to a measured value of 4.2 x 109 M -~ sec -~ in a homogenous ethanol solution. INTRODUCTION In 1978 T u r r o a n d Y e k t a (1) p r o p o s e d a m e t h o d b a s e d o n t h e q u e n c h i n g o f fluoresc e n c e to d e t e r m i n e t h e a g g r e g a t i o n n u m b e r , a, of micelles. The system investigated was s o d i u m d o d e c y l sulfate, S D S , a n d t h e y foll o w e d t h e d e p e n d e n c e o f h on i o n i c s t r e n g t h . The same method was utilized for determ i n a t i o n o f ~ o f S D S m i c r o e m u l s i o n s (2) and SDS micelles swollen with pentanol and o t h e r a l c o h o l s (3). The m e t h o d of Turro and Yekta relies on a n u m b e r of basic assumptions:

(i) B o t h t h e f l u o r e s c e n t m o l e c u l e a n d t h e q u e n c h e r m o l e c u l e i n t r o d u c e d to t h e m i c e l l a r s y s t e m ar e s t r o n g l y a s s o c i a t e d w i t h t h e m i c e l l e s so that t h e y stay in o r on a m i c e l l e for a period of time m u c h longer than the u n q u e n c h e d lifetime of the fluorescent probe. (ii) T h e q u e n c h i n g in o r on a m i c e l l e c o n taining b o t h an e x c i t e d p r o b e a n d a q u e n c h e r m o l e c u l e is m u c h f a s t e r t h a n t h e f l u o r e s c e n t d e c a y , so that in e f f e c t f l u o r e s c e n c e is observed only from micelles without quenchers. (iii) T h e d i s t r i b u t i o n o f p r o b e a n d 486

0021-9797/81/060486-14502.00/0 Copyright @ 1981 by Academic Press, Inc, All rights of reproduction in any form reserved.

Journal of Colloid and Interface Science, Vol. 81, No. 2, June 1981

FLUORESCENCE STUDIES OF MICELLES

quencher on the micelles is known; in practice a Poisson distribution is assumed. If these assumptions hold, fi can be determined by measuring the total intensity of fluorescence with and without quencher, F and F0, respectively. The relation between F and quencher concentration, [Q], is given by F = F0 exp{-[Q]/[M]},

,[1]

where [M] is the mean concentration of micellar aggregates. [M] is related to 6 by [M] = {[Det] - [free monomer]}/~, where [Det] is the concentration of detergent and where [free monomer], with good approximation, can be taken as CMC. According to Eq. [1], In Fo/F is proportional to [Q], and 5 can easily be calculated from the slope of this straight line. However, the validity of assumptions (i) and (ii) must be ascertained as Infelta (4) clearly pointed out. In practice, precise time-resolved measurements--which means single-photon counting fluorometry, SPC, - - are required. From such measurements, information about the rate of diffusion, or at least of quenching, on/in the micelle is obtained in addition to information about the micelle size. The kinetic quenching model will be discussed in detail below, but at the moment it should be pointed out that a fourth assumption must be added to the list of conditions: (iv) The rate of quenching in a micelle containing n quencher molecules is n times that in a micelle with just one quencher. With these conditions enforced, a comparatively simple expression is obtained for the fluorescence decay after excitation with a 8 pulse (4), F(t) = F(0) exp{-t/~-0 + h(e -kqt - 1)}, [2]

where ~'0 is the unquenched lifetime, h = [Q]/[M], and kq is the first-order rate constant for fluorescence quenching in a micelle with one quencher. In this paper we present precise (SPC)

487

measurements using the fluorescent probe and quencher suggested by Turro and Yekta. Thus we have undertaken measurements of the fluorescence decay of tris(2,2'bipyridyl)ruthenium(II), Ru(bipy)~ +, in the presence of 9-methylanthracen, 9-MeA, in micelles of SDS at different ionic strength, in pentanol-swollen micelles of SDS, and in microemulsions (SDS, pentanol, dodecane). The results are used to critically test the quenching model, Eq. [2], and to verify or reevaluate some earlier applications of Turro and Yekta's static method. The calculation of "microviscosity" values for the miceUe from measured diffusion-controlled rate constants is critically discussed. EXPERIMENTAL

I. Materials

Sodium dodecyl sulfate (BDH, specially pure) was used as supplied. In preliminary measurements, SDS which had been twice recrystallized from methanol was used but no significant differences in results could be detected. Ru(bipy)~ + was also used without further purification. The 9-MeA was a gift from Dr. Kjell Sandros. Sodium chloride (Merck, p.a. quality), 1-pentanol (Baker Chemical Co.), and dodecane (Matheson Co.) were used as supplied. All solutions were made with distilled water, run through a Millipore system. H. Sample Preparation

In a preliminary study we searched for a simple, reliable, and reproducible method to introduce a hydrophobic molecule into a micellar solution. The method suggested by Shinitzky and Weber (5) gave turbid solutions, probably due to small fragments from the glass beads utilized. The easiest and best method was to use a stock solution of high concentration of the hydrophobic molecule, in this case 10-2 M 9-MeA, in benzene. Appropriate volumes (0-500/xl) of the 9-MeA solution were injected into a 5-ml Journal of Colloid and Interface Science, Vol. 81, No. 2, June 1981

488

ALMGREN AND LOFROTH

SDS solution in a 10-ml volumetric flask. The solution was rigorously stirred for 2 hr at approximately 40°C while dry air was blown over it. The solution was then diluted to the 10-ml mark, the volumetric flask sealed and the solution again stirred for ½ hr. With this method we achieved both precision and accuracy Within 1% with respect to the required concentration of 9-MeA in SDS micelles, as determined by measuring the absorbance at 388.5 nm (where the last peak in the absorption spectrum for 9-MeA was found). No difference in results was observed between a solution used directly and one used the day after. When samples for the quenching experiments were prepared, a stock solution of Ru(bipy)~+ in SDS was used. The concentration for Ru(bipy)~ + was calculated from absorption measurements by using the value E453 = 1.4 × 104 M -1 cm -a (6). When pentanol was needed a concentrated stock solution of 9-MeA in pentanol was used. To vary the ionic strength, appropriate volumes of a 3.000 M solution of NaC1 in water were added before the final dilution. The compositions of the solutions are given in Table I.

III. Steady-State Measurements Absorption spectra were recorded with a Beckman Acta III UV-visible spectrophotometer and emission spectra with an Aminco SPF-500 corrected spectra spectrofluorometer. As the concentration of Ru(bipy)~+ within each series was constant, the relative fluorescence intensities were easily found by measuring the relative heights of the peaks in the emission spectra for a quenched and unquenched solution, the latter being the same during one series of experiments. To eliminate effects due to small concentration variations, the excitation wavelength was chosen from the absorption spectrum of the solution without quencher so as to give an absorbance of 0.70 cm -a which corresponds to the maxiJournal of Colloid and Interface Science, Vol. 81, No. 2, June 1981

TABLE I Composition of the Samples Sample

[9-MeA] × IO~/M

[NaCI]/M

A1 a

A2 A3 A4 A5 A6

1.44 2.78 4.22 5.71 6.97

B1 b B2 B3a B3b B3c B4 B5

3.55 3.55 3.55 5.59 7.32 3.55 3.55

C1 c C2 C3 C4 C5 C6

2.12 3.93 5.90 7.86 9.98

0 0.30 0.45 0.45 0.45 0.60 0.75

D1 d

D2 D3 D4 D5 D6

1.13 2.19 5.62 7.16 8.94

a Series A: [SDS] = 0.046 M, [Ru(bipy)] +] = 7.19 × 10-5 M, CMC = 8.2 x 10-3 M (1). b Series B: [SDS] = 0.070M, [Ru(bipy)~ +] = 9.50 × 10-s M, CMC = 8.2 × 10- 3 M . c Series C: In wt%: 5.17 SDS, 10.32 1-pentanol, 5.17 dodecane, 79.34 water, [SDS] = 0.181 M, [Ru(bipy)] +] = 9.45 × 10-~ M, CMC = 2 × 10-a M (2). a Series D: [SDS] = 0.050 M, [Ru(bipy)] +] = 9.33 × I0 -~ M, [1-pentanol] = 0.40 M, CMC = 0.7 × 10-3 M (3).

mum in the fluorescence intensity vs absorbance curve under the prevailing excitation geometry (7). At the excitation wavelength chosen, 415.5 nm, there was no direct excitation of 9-MeA. The emission intensity was measured around 638 nm.

IV. Time-Resolved Measurements The fluorescence decay of Rb(bipy)~ + was followed with the single-photon counting technique. The lamp used was a gated

489

FLUORESCENCE STUDIES OF MICELLES ns 0

100

200

all channels up to a channel where the statistics for the observed points became too poor. This range is marked by A and B in Fig. 1. The fitting p r o c e d u r e was a nonlinear least-squares method using a modified L e v e n b e r g - M a r g u a r d t algorithm (8). As criteria for a good fit, both the X~ value and a plot of the weighted residuals, WR, were used. X~ and WR are defined as

300

3.5

3

A~ (t)

1

X~---

x

~ (WR(i)) 2,

N-Pi=I

WR(i) o

I

I

I

o

A

100 channel

[

200

I

B

number

FIG. 1. Decay profiles for series A and excitation pulse (1.566 nsec channel-i).

flashlamp from PRA (Model 510B) with a F W H M value of 2 - 3 nsec. The instrument is described in detail elsewhere (8). Figure 1 shows a logarithmic plot of experiments on solutions A 1 - A 6 and also the instrument response, E(t), recorded with the fluorescent solution replaced by a scattering solution and without an emission filter. On this time scale it is obvious that the excitation pulse, selected by a 450-nm interference filter, can be considered a 8 pulse. This means that there is no need to solve the convolution integral,

F°(t) =

E(t - x ) . F ( x ) ' d x ,

[3]

F o - F~

(F~) 112

where N = number of channels (=A - B + 1), v = number of estimated parameters, F~ = calculated intensity value from Eq. [2], and F ° the observed number of counts in channel i. ' Since r0 in Eq. [2] can be estimated in separate experiments without quencher, there are only three parameters to estimate: F(0), h, and k,. F(0) has no physical meaning in these experiments since the estimated value depends on the counting time, counting frequency, etc. F r o m h it is possible to determine [M] and thus h with knowledge of [Q]. H o w e v e r , when estimating h and kq it is necessary to have the "zero-time channel," ZTC, determined. The same problem is met in all situations where the theoretical function is not a single exponential. The correlation between the choice of ZTC and h in Eq. [2] is obvious and was clearly demonstrated experimentally, as will be discussed under Results and Discussion. To our knowledge, no strict TABLE II

with respect to F(t), but one can assign the observed F°(t) values directly to F(t). In Table II the relative heights of the three curves E(t), A1, and A6 are given for some channel numbers. Since the excitation intensity has decreased to 0.5% in channel 37, E(t) does not significantly disturb the fluoresence after this channel. The analysis was therefore started in channel 37 and included

Relative Heights of Excitation and Fluorescence Decays Channel number

E(t)

FAI(t)

FA6(t)

24 30 35 37

1.00 0.10 0.01 0.005

0.52 1.00 0.98 0.97

0.52 1.00 0.86 0.79

Journal of Colloid and Interface Science, Vol. 81, No. 2, June 1981

490

ALMGREN AND LOFROTH

mathematical solution to this problem has yet been published. This problem will not be so serious if a deconvolution is performed but as it is impossible to find a recursion formula for this kind of kinetics in analogy with the one presented by Grinvald and Steinberg (9) the computer time for doing the convolution, Eq. [3], will be too long for practical and economical work. (The zero-time problem discussed here should not be confused with the difficulties to establish a common zero time for E(t) and F°(t) in the deconvolution situation (8).) RESULTS AND DISCUSSION

Results from four series of measurements are presented. In series A the quenching of Ru(bipy)~ + by methylanthracen in SDS micelles is studied. The results are used to find a solution to the ZTC problem discussed above. In series B the ionic strength is varied by addition of NaC1. The results from these time-resolved measurements are compared with the steady-state results of Turro and Yekta (1) on the variation of t~ with ionic strength. Series C, using a SDSpentanol-dodecane microemulsion, shows a case when the quenching is so slow that Turro and Yekta's method may not be applied. Finally, in series D, SDS micelles with pentanol were investigated. In this case the intramicellar quenching was very fast and a deconvolution is necessary to estimate t~ and kq correctly. All measurements were performed at 25°C.

Series A When Ru(bipy)] + in SDS micelles is excited with and without quencher the results from the steady-state measurements plotted according to Eq. [1] give straight lines as shown in Fig. 2. The slope is 1568 M -I from which the mean aggregation number was calculated as 59 +--1 with CMC = 8.2 × 10-~ M. The value of fi agrees very well with values obtained by other techniques (10, 11). The steady-state measurements Journal of Colloid and Interface Science, Vol. 81, No. 2, June 1981

1.G

_c

0

I

r 2

I

I 4

I

I 6

I

(9.MeA) • 1 04//M

FIG. 2. Luminescence quenching of Ru(bipy)~+ by 9-McA in sefies-A solutions plotted according to Eq. [1].

are thus well accounted for by Eq. [1] in this case. The results from the time-resolved measurements (shown as lg F(t) vs t in Fig. 1) show a fast initial decay, the amplitude and rate of which increase with quencher concentration, followed by a much slower linear decay with the same slope = -1/~'o at all concentrations. The behavior is as predicted by Eq. [2], and can be simply understood as follows. (Equation [2] has been derived by several authors independently (4, 12-14) and also in more generalized forms (15a-c). We will present a simple derivation which emphasizes the physical background.) Consider a situation where the concentration of fluorescent molecules and the intensity of the excitation light are so low that rarely more than one excited molecule is found in any micelle. The excited species and the quenchers have long residence times in the micelles compared to the fluorescence lifetime of the probe. The decay rate constant of a fluorescent molecule in a micelle with i quenchers is assumed to be ki = 1/~-0 + ikq. The resulting fluorescence signal is then given by the sum of contributions from micelles with different numbers of quenchers,

F(t) = ~ eiF(0)e -k,t, i=0

[4]

FLUORESCENCE STUDIES OF MICELLES

491

TABLE III Estimated Values of T, h, and kq for Series A Sample

T/channela

kq x 10-Vsec -la

hn

kq x 10 7/sec-ab

X~

hC

A2 A3 A4 A5 A6

16.0 12.7 11.9 12.8 13.2

1.82 2.05 2.19 2.25 2.19

0.216 0.466 0.730 0.959 1.151

1.96 2.12 2.24 2.29 2.20

0.97 0.99 0.95 1.20 1.02

1.18, 1.10

a With r0 fixed at 480 nsec and a = 62. b With ~'0fixed at 480 nsec and T = 13. c With ~'ofixed at 480 nsec and T = 14 and 12, respectively. where Pi is the probability of finding a probe molecule in a miceUe with i quenchers. Assuming a Poisson distribution (16) for the quenchers on the micelles, hie -~

Pi -

it

and that this distribution is unaffected by the presence of probes, we have

F(t) = ~ i=0

hie-~

F(0) exp{-t(1/7o + ik,)}

i! (he-k,9 i

= F(0) exp(-t/~-0 - h) i=0

i!

= F(0) e x p { - t / % + h ( e - k d -

1)}. [2]

If t ~ ~ then F ( t ) = ae -"~°, where

a =- F(O)e -~. If t ~ 0 then

e -kd

=

1 - kqt,

and

F ( t ) = F(O) e x p { - t ( 1 / Z o + h k O } . Thus with this model the logarithmic decay curves for different h, that is for different [Q], at long times should be a family with equal slopes. At short times h o w e v e r the slopes depend on [Q]. For the computer Eq. [2] is rewritten in terms of channels:

V(j) = a'exp{-(j q- h ' ( e - ( j - l + T ) ' k q - -

- 1 + T)/"ro 1)} j = 1, 2, 3, . . . ,

where F(1) is the intensity in that channel

in the multichannel height analyzer where the fit shall begin (37 in our case) and T is the difference between this channel and the ZTC. We first estimate T. Taking the aggregation number of SDS micelles as known, 5 = 62 (11), h may be calculated for different [Q]. Both z0 and h are then regarded as k n o w n constants and a, T, and kq estimated. The results are shown in Table III. The mean value for T is 13 + 2, which means that ZTC is 37 - 13 -- 24. From Table II it is seen that this was the channel where the excitation pulse had its maximum. Once this channel had been selected as t = 0 the analysis could be made by holding r0 and T fixed and estimating a, h, and kq. The results from this analysis and an analysis where the ZTC was shifted one channel before and after channel 24 are also shown in Table III. The significance level, as judged by the X~ value, is as high as about 35%, thus indicating excellent fits. This is also indicated by the W R plots, two of which are shown in Fig. 3. Although it is probably not entirely correct, the choice of the maximum of E ( t ) as the ZTC has the merit of being simple and well defined. The differences in estimated h when shifting the ZTC one channel are insignificant. H o w e v e r , choosing the maximum of F ( t ) as ZTC is not advisable.

Series B U p o n the addition of NaCI or other simple electrolytes to ionic micellar solutions a growth to large aggregates is favored. The Journal of Colloid and Interface Science, Vol. 81, No. 2, June 1981

492

ALMGREN AND LOFROTH

aggregation numbers in SDS solutions (0.07 M) at high NaC1 concentrations (up to 0.6 M) were determined by Mazer et al. (17) with the quasi-elastic light-scattering technique. They found a steep growth to an aggregation number of the order 1000 at the highest NaC1 concentration. Later investigations (18a) have shown the presence of rod-formed micelles in these solutions. On the other hand, Turro and Yekta (1) found only a slight growth of micellar size with their steadystate fluorescence quenching method. The time-resolved measurements provided a test of the validity of Eq. [1]. Our results from both steady-state and time-resolved measurements are presented in Figs. 4 and 5 and in Tables IV and V. It is seen in Fig. 4 that the aggregation numbers at high NaC1 concentrations from the time-resolved measurements are substantially larger than the apparent values from the steady-state measurements. However, they are still much smaller than the values obtained by Mazer e t al. (17). The difference between the steady-state results and the time-resolved results is due to the fact that the quenching in the large aggregates is too slow to make Eq. [1] a good approximation. This limitation to the method of Turro and Yekta is even more

1000

/

9 O0 [

'

l

m 400

I

E == .~

/ /

300

/

o

/

~ 200

/

_~ 100

J

0

I

0

I

0.2

i

t

0.4

i

i

0.6

0.8

(NaCI) / M

FIG. 4. Dependence of aggregation number on ionic strength for 0.070M SDS. Continuous curve is the result from Turro and Yekta (1), dashed curve from Mazer et al. (17), and circles represent the result from this work.

apparent in the microemulsion results discussed below. The main reason to the difference from the results by Mazer et al. is the fact that the quasi-elastic light scattering yields a weightaveraged aggregation number, a w, whereas the fluorescence quenching methods yield the number average, as will be discussed below. The aggregation numbers quoted in (17), which are also indicated in Fig. 4, were shown to be too big due to an impure SDS preparation (18a). The following values from (18b) refer to pure SDS from the same ns source as used in the present investigation: 0 100 200 300 (aw, [NaC1]) 200, 0.5 M; 520, 0.6 M; 1600, J i i i 0.8 M, all at 25°C and 0.0694 M SDS. As WR discussed by Missel et al. (18b) and earlier by Mukerjee (19) a ratio a w / a , = 2 should be expected for large rod-like micelles. The ratio between the aggregation numbers from quasi-elastic light scattering and fluorescence WR quenching is indeed 2 at 0.6 M NaC1. At 0.5 M NaC1, where the aggregation numbers are still quite small, the ratio is less than 2. At 0.8 M the ratio is much larger. The broad size distribution has important I I I I I[ implications for the fluorescence quenching. 0 24 A 100 200 B256 channel number The Poisson distribution will not apply for FIG. 3. WR plot from an analysis of decays for the fluorescent molecules and the quenchers, samples A1 (upper) and A6 (lower) (1.566 nsec even if the microconcentrations in the micelles are independent of aggregate size. channel 1). Journal of Colloid and Interface Science, Vol. 81, No. 2, June 1981

FLUORESCENCE 1.5

1.0

3 0.5

0

2 4 ( 9 MeA ) 104/M

6

493

STUDIES OF MICELLES

8

Fro. 5. L u m i n e s c e n c e q u e n c h i n g o f Ru(bipy)~ ÷ b y 9 - M e A in 0.070 M S D S a n d 0.45 M N a C I p l o t t e d a c c o r d i n g to E q . [1].

The magnitude of these effects were explored by some computer simulations reported in the Appendix. It was assumed that micelles with aggregation numbers a0 and 3 a0 were present simultaneously, with three times as many of the small size as of the large size. Half of the solubilized molecules were assumed bound to each type of micelle, and kq was assumed three times larger in the small micelles than in the big ones, in accordance with the variation of kq with micelle size found experimentally (see' below). Fluorescence decay curves were synthesized by assuming that Eq. [2] applied for each subsystem of micelles. The contributions from the two subsystems were added as well as realistic noise. Equation [2] was fitted to the simulated data thus obtained. It is seen from the results given in the Appendix that the estimated values of h and kq varied with the quencher concentration (note that hok~ = hkq). The estimated aggregation number was larger than the number average (1.5a0) at low quencher concentrations, and somewhat smaller than 1.5a0 at high concentrations. The deviations were larger at a high value ofkq than at a low value. The fits to the data were very good. Only the change of the estimated values with the quencher concentration reveals the presence of different ag-

gregates. The conclusion from the results in the Appendix is that the estimates of a0 from the measurements at the two highest salt concentrations could deviate somewhat, but probably by less than 20%, from the number average aggregation numbers, if the solubilized molecules had no special preference for either small or large aggregates. However, it was shown earlier (20) that pyrene prefers large, rod-like CTAB micelles to small globular ones. A similar preference of 9-MeA for large aggregates would give too low estimates of 4, and seems to be a probable explanation for the difference in results between the number average aggregation number that can be calculated from the results of Mazer et al. and the present findings. Experiments at varying quencher concentrations were performed at 0.45 M NaC1 (besides series A, at zero salt concentration). No trend was observed in the 4, kq data in these measurements. Since the aggregation number at 0.45 M NaC1 was still as low as 135, the micelles were probably close to globular, with a narrow size distribution. Series C

Integration of Eq. [4] yields the relative steady-state fluorescence intensities: TABLE IV E s t i m a t e d V a l u e s of a a n d kq for S e r i e s B

Sample

kq × 10-r/sec-1

fi

B1

63

2.04

B2 B3 B4 B5

104 134 260 373

1.23 0.868 0.376 0.240

3 a X~ = 1.07 w i t h ~

c~e-tl~i,

X~

1.03 1.02 1.02 1.24 ~ 1.33 b

where

a I

=

0.27,

~'1

1

= 57 n s e c ; az = 0.62, ~'3 = 480 n s e c . a X~ = 1.09 w i t h ~

~'2 = 172

a~e-"~i,

nsec;

where

c~3 = 0.11,

cq = 0.30,

zl

1

= 53 n s e c ; ct2 = 0.51, ~-~ = 480 n s e c .

~'~ = 159

nsec;

~3 = 0.19,

Journal of Colloid and Interface Science, Vol. 81, No. 2, June 1981

494

ALMGREN AND L()FROTH

F

Fo

i

Pi

i

2.0

F B

~ 1 + ikq'ro hi = e -~ ~

[5]

i!(1 + ik,r0)

C E

As pointed out by Infelta the value of the product kqr0 is decisive for the steadystate interpretation. This is illustrated in Fig. 6. Computer plots of In (Fo/F) vs h/6 were calculated using Eq. [5] with some values of kq~-0 found from the SPC experiments. Also included are some curves for other k,r0 values. The aggregation numbers in these curves were chosen to give approximately equal initial slopes of the quenching curves. Since h/g~ is proportional to [Q] independently of 5, this type of plot will result from pure steady-state experiments. In order to detect the curvature experimentally, very precise fluorescence intensity measurements are required, that are extended to high degrees of quenching. Due to the limited solubility of 9-MeA, a curvature would probably not be detected at all in those quenching ranges where it is possible to work. The highest concentrations of 9-MeA used in this work give for series A, h/iz. 100 = 1.84, for series B, 1.18, for series C, 0.55, and for series D, 1.81. The errors in h when using Eq. [1] uncritically would be of the order of 10% already at kq~-0 = 10, grow to more than 30% at kq~-0 = 3, and to a factor of 3 - 4 at kq~0 = 1. These facts are also illustrated in Fig. 7 which is a computer plot of In (Fo/F) vs h for kq~-0 = ~ and kqr0 = 0.62. The latter value was obtained in this work for the microemulsion system, as series C. The ~0 TABLE V Estimated Values of 5 and kq for Samples B3a, B3b, and B3c

Sample

5

kq × 10-r/sec-~

X~

B3a B3b B3c

134 135 137

0.868 0.931 0.897

1.02 1.09 1.09

Journal of Colloid and Interface Science, Vol. 81, N o . 2, June 1981

A

o

0

;

'

io

'

;o

T.loo FIG. 6. C o m p u t e r plots of In (Fo/F) vs h/gt for different values ofkq~'0 "F/Fo is given by Eq. [5]. (A) kqr0 = 10.56, 5 = 62; (B) kqr0 = 4.306, ~i = 137; (C) k j 0 = 0.62, h = 306; (D) kq% = 21.06, 5 = 34; (E) kq~-0 = 0.10, ~ = 1400; (F) kqT0 = c~, 6 = 100.

found was 309 nsec and the kq equals 0.2 ___ 0.02 x 107 sec -1. The ratio between the initial slopes is 2.9 which means that the aggregation number for the microemulsion with the same composition studied by Almgren et al. (2) is wrong by a factor of 3. Their data treated with Eq. [1] gave -- 100. The application of Eq. [2] to our SPC data gave fi = 306 using a CMC of 2 × 10-3 M (2) in good agreement with the foregoing discussion. We also recalculated the core radius, given as 37 A by Almgren et al., to be 54 A. The number of pentanol molecules present in one microemulsion droplet was calculated to be 2000. The question of the localization of the pentanol molecules in the droplet was discussed by Almgren et al. With the small value of the radius of the assumed spherical droplet it seemed possible that all alcohol molecules were present at the surface, with their OH group in contact with water. However, the present data would indicate a mean surface area per surface-active molecule (pentanol + SDS) of only about 16 A s. Furthermore, with this localization the pentanol molecules could contribute to the volume of the droplet only in an outer shell of at most 7 A depth, constituting about

495

FLUORESCENCE STUDIES OF MICELLES i

1

5

2

e

,

o

~

~'

~

FIG. 7. C o m p u t e r plots ofln (Fo/F) vs h. F/Fo is given by Eq. [5]. (1) kq~0 = ~; (2) kq~0 = 0.62.

35% of the volume of the droplet, whereas the pentanol molecules comprise more than half of the volume. Thus a substantial fraction of the pentanol must be present in the interior of the microemulsion droplets if the particles really are spherical. If the aqueous phase is assumed saturated with pentanol the estimated particle radius becomes 52/~. The conclusion regarding the localization of some pentanol in the interior of spherical droplets remains. Resulting values of kq and h in series C are presented in Table VI. There is an obvious trend in the values: kq decreases and increases with the quencher concentration, whereas the product kq5 remains essentially constant. This is exactly the type of results that would be expected from the foregoing discussion for a broad size distribution of the particles. It is then probable that the particles are nonspherical, which would allow the majority of the pentanol molecules to be present at the surface of the particles.

Series D Measurements on pentanol-swollen SDS micelles (3) gave 5 = 34 for a solution of composition as in Table I, series D. This was found by applying Eq. [1] to steady-state

data. The SPC measurements justify the use of Eq. [1] in this case as seen in Table VII, the kqT"0 value being 20. It is obvious, however, that the large value of kq, 1/kq = 18 nsec, requires a deconvolution of the SPC data. Such an analysis was performed on one of the decays, using the model function F(t) = ~=~ o~e-tl'i. The result is also given in Table VII and shows that in this case the decay can excellently be divided in two exponential parts. The more long-lived component gives again ~- = 353 nsec, the value without quencher. The value of 1/r~ for the fast component must be some weighted mean of ~]i ikq. We may thus conclude that kq < l/r1 = 6.4 x 107 sec -1. QUENCHING RATES AND THE PROPERTIES OF THE MICELLE

The first-order rate constant kq for the quenching is a measure of the encounter frequency of the pair of molecules confined in the micelle, provided that the quenching process is diffusion controlled. This seems to be the case for 9-MeA quenching of Ru(bipy)~ + fluorescence in homogenous solutions. The second-order rate constant was obtained as 4.2 x 109M -1 sec -1 in ethanol at 25°C. Before discussing the results it is appropriate to comment upon the kind of information that may be obtained about the micelle. Quite often "microviscosities" of micelles or membranes have been calculated from measurements of the rates of diffusion-controlled reactions (in particular the excimer formation of pyrene) or rotaT A B L E VI Estimated Values of h and kq for Series C

Sample

h

fi

kq × lO-r/sec 1

C1 C2 C3 C4 C5 C6

-0.433 0.689 1.085 1.273 1.483

-365 314 329 290 266

-0.170 0.181 0.184 0.201 0.219

×~

~o = 3 0 9 n s e c

1.02 0.86 1.10 1.08 0.95 1.05

Journal of Colloid and Interface Science, Vol. 81, No. 2, June 1981

ALMGREN AND LOFROTH

496

TABLE VII Estimated Values of~ and k, for Series D Sample

h

t~

kq × 10-r/sec-I

D1 D2 D3 D4 D6 D5

-0.067 0.092 0.270 0.372 --

-13.2 21.4 24.2 20.9 --

-7.58 4.42 6.62 3.97 --

X~

~'0 = 373 nsec

al = 0.43 a~ = 0.57

1.01 1.04 1.17 1.11 1.05 1.09a

~-~= 15.5 nsec % = 353 nsec

a o~ and ~-~deconvoluted from the convolution integral. tional relaxation times of various p r o b e s in such structures. These calculations are based on the S t o k e s - E i n s t e i n relation between viscosity of a fluid and the diffusion coefficient of the probe: D = kT/67rgr.

H o w e v e r , the d e n o m i n a t o r in this expression is the friction factor from Stokes' law for a sphere moving in a h o m o g e n o u s fluid of infinite extension, and does not apply to m o v e m e n t in a small, limited volume (21). We are not a w a r e of any calculation of the friction factor for a sphere moving inside, e.g., a spherical volume. It can be anticipated that the corrections to Stokes' law in such a case would depend not only on the properties and dimensions of the confined fluid, but also on the size (and form) of the moving molecule. F u r t h e r m o r e , the corrections would be different for translational and rotational motion. Thus, the " m i c r o v i s c o s i t y " is not a mean viscosity of the micelle or of the m e m b r a n e , and it is not e v e n a p r o p e r t y of the structure itself. It seems m o r e appropriate, therefore, to report this type o f information about the fluidity of the structures in t e r m s o f diffusion coefficients, or rotational relaxation times and encounter frequencies, so that the d e p e n d e n c e on the nature of the p r o b e is clearly revealed. Values of kq h a v e been obtained for 9 - M e A - R u ( b i p y ) ] + in SDS micelles of varying size, Fig. 8, and in S D S - p e n t a n o l Journal of Colloid and Interface Science, Vol. 81, No, 2, June 1981

micelles, the volume of which are not k n o w n but p r o b a b l y are about the same as the normal SDS micelle (3), and in big S D S p e n t a n o l - d o d e c a n e microemulsions particles. The data in Fig. 8 show that kq is a p p r o x i m a t e l y proportional to I/h, with a slight curvature so that kq is s o m e w h a t larger in the small micelles. Since h is proportional to the micelle volume the concentration corresponding to one molecule per micelle is proportional to 1/~. Proportionality of k, to 1/a therefore implies that the process is characterized by the same secondorder rate constant, independently of the micelle size. This is quite u n e x p e c t e d for a diffusion-controlled reaction, and is probably the result of cancellation b e t w e e n opposing effects. One of these is the change on confinement of the friction factor discussed above.

2.0

~

(

1.0

o

0

I 40

r

1/~.

i 80

i

i 120

i 160

lo 4

FIG. 8. Dependence ofkq on mean aggregation number for SDS micelles in solutions of varying ionic strength.

FLUORESCENCE

STUDIES

The friction factor should increase with a decrease in size of the micelle. Another factor, the effect of the confinement upon the diffusion process, was recently investigated by G6sele et al. in an instructive paper (22). They made computer simulations and approximative analytical solutions of the diffusion problem for two spherical molecules confined in a spherical isotropic region. The results are presented as quenching curves. One of the molecules was assumed excited and the other a quencher which quenches the fluorescence on the first encounter. The logarithm of the remaining fraction of excited systems is presented as a function of time for three cases: the excited molecule fixed at the center of the sphere (a), at the surface (b), or diffusing freely within the sphere (c). The quencher is diffusing freely in all cases. Results are given for two values of the pertinent size parameter, R/rAB = 4 and 10. R/rAB is the ratio of the micelle and encounter radii. Of particular interest is the fact that the computer simulations show surprisingly good agreement with Eq. [6] for the encounter-rate constant, k = 4~'(D a

+ DB)rAB ,

[6]

in homogeneous solution at conditions where a steady-state diffusion gradient has time and space to evolve. The deviation in case (c) is only about 20%, but substantially greater in (a) and (b). On decreasing size of the confinement an increase in the secondorder rate constant would be expected both because of the "transient effect" which appears also in homogenous solutions at short times, and because of the spatial requirements of the steady-state diffusion gradient. Such an increase is apparent in the computer simulations. The second-order rate constant that may be calculated from the curves in case (c) increases with some 15% from the large to the small micelle. Both effects are also clearly displayed by the analytical solution in case (a): the transient term enters in the usual way as a second term, depending on t -112, in our Eq. [6] and

OF MICELLES

497

the spatial restriction results in a correction factor (1 + 1.8 x R / ( R - rAB)) to the term given in Eq. [6]. We have thus found that the changes expected in the friction factor and in the rate of diffusive encounters at a given diffusion coefficient on a change of micelle size will counteract each other. Changes in the properties of the micelle, and in the distribution of the reactants, may also occur with a change of size in real systems. The latter may be of particular importance in comparing results for systems with widely different surface to volume ratios like micelles and microemulsion particles. The Ru(bipy)a2+ is an ion with some hydrophobic propert i es-the fluorescence spectrum reveals a strong interaction with the micelle--and may be expected to be present in a surface layer with s o m e extension into the micelle. The quencher, 9-MeA, is probably more evenly distributed over the micelle volume, presumably with some preference for the surface which would be important in small micelles (16, 23). The equilibrium distributions of the solubilized molecules may formally be described by means of their local microconcentrations p ( r ) , normalized so as to represent one molecule per micelle, o~P(r)47rr2dr = 1.

[7]

For a reaction-controlled bimolecular reaction a local second-order rate constant k ( r ) may be defined. We will at once assume k ( r ) = constant = k and obtain for kq kq = k

fo

pA(r)pB(r)4~rZdr = k12.

[8]

The "overlap integral," 12, has the dimension of reciprocal volume. If both A and B are confined within the micelle, and at least one of them is evenly distributed over the entire volume (p = l/l?), then 12 = l/i?. In the present case we assume that methyl= anthracene is evenly distributed within a radius R of the micelle. Ru(bipy)~ + is asJournal of Colloid and Interface Science, Vol. 81, No. 2, June 1981

498

ALMGREN AND LOFROTH

sumed to be present at the surface in such a way that a certain fraction of its distribution is within the radius R. This fraction will be arbitrarily taken as V2; the main point is that there is no reason to assume a change in this fraction when going from a normal micelle to a microemulsion particle. The overlap integral would then be f~ = ½ 17 in all cases. For a normal SDS micelle the volume is close to 10 dm 3 mole-~; the same value will be used for the pentanol-containing micelles. The volume of the microemulsion particles were estimated as 370 dm a mole -~. Using Eq. [8] the following values of k are obtained:

tanol at 25°C are quite close, 1.35, 1.20, and 1.36 cP, respectively) (2). In the small-sized particles the e n c o u n t e r rate constants are certainly lower than in the homogenous solution, and the S D S - p e n tanol micelle is less viscous than the SDS micelle. The result may be compared to "microviscosities" calculated from fluorescence depolarization data (2): SDS micelles, 52 cP, S D S - p e n t a n o l micelles, 0 - 2 cP and microemulsions, 4 - 5 cP. These results show much larger difference between the systems than the second-order rate constants, and also a qualitative difference in that the S D S - p e n t a n o l micelle has the lowest " m i c r o v i s c o s i t y . " The comparison amply demonstrates that the so-called " m i c r o v i s c o s i t y " is not a micelle property.

SDS micelle

0.4 × 109 M -I sec -~

SDS-pentanol micelle

1.0 × 109 M - ' sec -~

Microemulsion particles

APPENDIX: SIMULATED FLUORESCENCE D E C A Y IN A M I X T U R E O F T W O M I C E L L E SIZES

1.5 × 10a M -1 sec -1

Fluorescence decay data for quenching in a micellar system containing micelles of two sizes were synthesized by adding contribution from each subsystem of monodisperse micelles, calculated from Eq. [2]. It was assumed that half of the fluorescent molecules and half of the quenchers were present in each subsystem, that is

Homogenous ethanol 4.2 × 109 M - ' sec -a solution In this calculation o f the second-order rate constant it was thus assumed that the inhomogenous distributions o f the reactants had the same effect as for a reaction-controlled reaction. The results of the computer simulations by G6sele et al. (22) in case (b), when the excited molecule was fixed at the surface, show a more p r o n o u n c e d effect on the encounter rate constant and show in particular that the encounter rate constant decreases with increasing size of the system. H o w e v e r , this case is not quite comparable with the present s i t u a t i o n - - a simulation of a case with the quencher moving freely in the total volume, and the excited molecule within a surface layer of a certain depth would be very interesting. If the unequal distributions of the reactants have a substantially larger effect than assumed the second-order rate constant for the quenching in the microemulsions would approach that found in homogenous solutions (the viscosities of dodecane, ethanol and penJournal of Colloid and Interface Science, Vol. 81, No. 2, June 1981

F ( t) = a e -t/7o. {O.5e n°~e-k~'-l) + 0 . 5 e 3 % (e-k~"3-1)} ;

which otherwise were characterized as follOWS"

Aggregation number

ao

3ao

Micelle concentration

m

m/3

Quencher/micelle

ho

3h0

Quenching-rate parameter

k°

k ~/3

The number average aggregation number is thus h = 1.5a0 and the number average quenching constant kq = k~/1.5. To simulate noise, numbers were chosen at random from a Gaussian distribution with a mean of zero and a standard deviation of one by a number generator for each channel, multi-

499

FLUORESCENCE STUDIES OF MICELLES plied by the square root of the calculated c o u n t s o f t h e f l u o r e s c e n c e d e c a y in t h a t c h a n n e l , a n d finally a d d e d to t h e f l u o r e s -

cence decay. The resulting decay curves w e r e t h e n a n a l y z e d u s i n g E q . [2] in t h e s a m e w a y as for r e a l d a t a . T h e r e s u l t s a r e a s follows.

Input Output z0 = 480 n s e c a = 104

kq

h0

× 10-Z/see -1

h/ho

k~/kq

X~

k~ - 0.2 × 107 s e c -1 0.24 0.30 1.0 3.0 10.0 30.0

0.406 0.506 1.582 4.373 14.731 44.248

0.114 0.115 0.125 0.138 0.136 0.135

1.69 1.69 1.58 1.46 1.47 1.47

1.75 1.74 1.60 1.45 1.47 1.48

0.96 1.00 0.98 1.10 0.96 1.06

k~ = 2.0 x 107 s e c -1 0.3 1.0 3.0

0,534 1.535 3.942

0.930 1.26

1.78 1.54

2.15 1.30

1.02 1.06

1.601

1.31

1.53

1.20

ACKNOWLEDGMENT This work has been financially supported by grants from the Swedish Natural Science Research Council. REFERENCES 1. Turro, N. J., and Yekta, A.,J. Amer. Chem. Soc. 100, 5951 (1978). 2. Almgren, M., Grieser, F., and Thomas, J. K., J. Amer. Chem. Soc. 102, 3188 (1980). 3. Grieser, F., J. Phys. Chem., in press. 4. Infelta, P. P., Chem. Phys. Lett. 61, 88 (1979). 5. Shinitzky, M., Dianoux, A.-C., Gitler, C., and Weber, G., Biochemistry 10, 2106 (1971). 6. Meisel, D., Matheson, M. S., and Rabani, J., J. Amer. Chem. Soc. 100, 117 (1978). 7. Sandros, K., private communication. 8. Biddle, D., and L6froth, J.-E., in preparation. 9. Grinvald, A., and Steinberg, I. Z.,Anal. Biochem. 59, 583 (1974). 10. Coil, H., J. Phys. Chem. 74, 520 (1970). 11. Granath, K., Acta Chem. Scand. 7, 297 (1953). 12. Yekta, A., Aikawa, M., and Turro, N. J., Chem. Phys. Lett. 63, 543 (1979). 13. Atik, S. S., and Singer, L. A., Chem. Phys. Lett. 59, 519 (1978). 14. (a) Infelta, P. P., Gr~itzel, M., and Thomas, J. K., J. Chem. Phys. 78, 190 (1974); (b) Maestri, M.,

15.

16. 17. 18.

19. 20. 21. 22. 23.

Infelta, P. P., and Gr~ttzel, M., J. Chem. Phys. 69, 1522 (1978). (a) Tachiya, M., Chem. Phys. Lett. 33, 289 (1975); (b) Hatlee, M. D., and Kozak, J. J., J. Chem. Phys. 72, 4358 (1980); (c) Dederen, J. C., van der Auweraer, M., and de Schryver, F. C., Chem. Phys. Lett. 68, 451 (1979). Almgren, M., Grieser, F., and Thomas, J. K., J. Amer. Chem. Soc. 101, 279 (1979). Mazer, N. A., Benedek, G. B., and Carey, M. C., J. Phys. Chem. 80, 1075 (1976). (a) Young, C. Y., Missel, P. J., Mazer, N. A., Benedek, G. B., and Carey, M. C., J. Phys. Chem. 82, 1375 (1978); (b) Missel, P. J., Mazer, N. A., Benedek, G. B., Young, C. Y., and Carey, M. C., J. Phys. Chem. 84, 1044 (1980). Mukerjee, P., J. Phys. Chem. 76, 565 (1972). Almgren, M., L6froth, J.-E., and Rydholm, R., Chem. Phys. Lett. 63, 265 (1979). Stigter, D., private communication. G6sele, U., Klein, U. K. A., and Hansel M., Chem. Phys. Lett. 68, 291 (1979). Mukerjee, P., Cardinal, J. P., and Desai, N. R., in "Micellization, Solubilization, and Microemulsion" (K. L. Mittel, Ed.), Vol. 1, p. 241. Plenum, New York, 1977.

Journal of Colloid and Interface Science, Vol. 81, No. 2, June 1981