J. Phys. Chem. 1996, 100, 14103-14108
14103
Dynamic Quenching of Luminescence in Reversed Micelles under High Pressure Masami Okamoto Faculty of Engineering and Design, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606, Japan ReceiVed: March 6, 1996; In Final Form: April 25, 1996X
Dynamic quenching of luminescence of tris(2,2′-bipyridyl)ruthenium(II) chloride by potassium hexacyanoferrate(III) in the sodium bis(2-ethylhexyl)sulfosuccinate (AOT)/water/heptane reversed micellar system was studied as a function of pressure from 0.1 to 300 MPa at 25 °C. From the analysis of the decay curves in solutions with W0 ()[H2O]/[AOT]) of 10.3, 14.0, and 22.0, it was found that the quenching rate constants, kq, in micelles were decreased significantly with increasing pressure, while the rate constants, kex, for the quencher exchange by collision between micelles were found to increase strongly with increasing pressure. The activation volumes for kq were almost independent of W0 (16 cm3/mol), and those for kex were decreased monotonically from -24 (W0 ) 10.3) to -54 cm3/mol (W0 ) 22.0) depending on W0. From the results of the pressure dependence of kq, the microenvironment inside the micelles is discussed. The largely negative activation volumes for kex are interpreted on the basis of an encounter or a collision complex formation between micelles prior to the quencher exchange, whose dissociative backward process exhibits larger pressure dependence than the forward one.
Introduction Water droplets in reversed micelles, which can solubilize many types of molecules in the water core and/or near the interface, have been well characterized by scattering,1 spectroscopy,2 and photochemical probe3-11 methods. It has been shown that the size of the micelles increases with increasing molar ratio of water to surfactant, W0, and that the microenvironment in micelles such as viscosity and dielectric constant depends remarkably on W0. Solubilizate exchange, which occurs on collision between micelles and/or Via the homogeneous phase in the micellar system, has been observed by monitoring the quenching of luminescence4-17 and a fast thermal reaction.18 In particular, photochemical dynamics including solubilizate exchange has been studied extensively, and the parameters associated with the micellar system have been determined from the analysis of the decay curves with quencher.5-17 In reversed micellar systems, Robinson et al.7,18 revealed that the interaction between micelles, which causes infinite clusters of micelles to form at the phase separation limit, is an important factor for the solubilizate exchange. Recently, Almgren et al.10 studied the dynamic behavior of reversed micelles by time-resolved phosphorescence-quenching measurements and proposed a kinetic model including clusters of micelles in which the quencher exchange occurs. The clustering in their model was concluded to occur due to the intermicellar interaction. More recently, the rate constants for probe migration and quencher exchange between micelles were determined by time-resolved fluorescence quenching of sodium 1-pyrenesulfonate as a function of W0 for a reversed micellar system in the presence of a cosurfactant in toluene.11 The rate constants were found to become higher than the diffusion-controlled limit in a collision process of micelles at W0 larger than 16, and this result was ascribed to clustering of the micelles. Thus, the solubilizate exchange is probably related to clustering of reversed micelles, but the details of the mechanism still remain unresolved. The present work focuses on the pressure dependence of the microenvironment in reversed micelles and of the rate of the X
Abstract published in AdVance ACS Abstracts, July 15, 1996.
S0022-3654(96)00683-1 CCC: $12.00
quencher exchange between reversed micelles, from the analysis of time-resolved luminescence-quenching measurements. The quenching reaction in micelles is expected to be sensitive to pressure, since the microviscosity inside the micelles and possibly their size may be affected strongly by compression. In fact, previous research on aqueous micelles has involved the pressure dependence of the microviscosity from time-resolved fluorescence decay measurements,19 although such a study in reversed micelles has not been reported. The quencher exchange by collision between micelles is also expected to be sensitive to pressure since the solvent viscosity is increased strongly by the application of high pressure. In this work, the dynamic luminescence quenching of tris(2,2′-bipyridyl)ruthenium(II) chloride by potassium hexacyanoferrate(III) in reversed micelles of sodium bis(2-ethylhexyl)sulfosuccinate (AOT)/water/heptane was investigated as a function of pressure from 0.1 to 300 MPa at 25 °C. The probe and quencher in this system are completely solubilized in micelles and do not migrate into the oil phase.4,7,18 Therefore, the quencher exchange is initiated by collision between micelles and might occur via the formation of an encounter complex or a collision complex between micelles. This implies that the rate of the quencher exchange is related not only to the collision frequency between micelles in heptane but also to microviscosity, the micelle size, the location of the probe and the quencher in the micelle, and any specific interactions of the surfactant molecules with the probe and quencher. The aim of the present work was to investigate the pressure dependence of the rate constant for the quenching reaction in micelles, as well as that for the exchange in order to provide some insight into the detailed mechanism of these processes. Experimental Section Sodium bis(2-ethylhexyl)sulfosuccinate (AOT) was the specially prepared reagent for testing water (Nakarai Chemicals, Ltd.) and tris(2,2′-bipyridyl)ruthenium(II) chloride, Ru(bpy)3Cl2, was obtained from Aldrich. Water and heptane of a nonfluorescent grade were purchased from Dojin Chemicals. These materials were used without further purification. Potassium © 1996 American Chemical Society
14104 J. Phys. Chem., Vol. 100, No. 33, 1996
Okamoto
TABLE 1: Rate Constants, Aggregation Number, N, and Radius of the Water Core, r, Associated with the AOT/Water/ Heptane Reversed Micellar System at 25 °C ([AOT] ) 0.22 M at 0.1 MPa) pressure, MPa
k0, 106 s-1
kq, 106 s-1
kq′, 108 M-1 s-1
kex, 108 M-1 s-1
N
r, nm
3.0 ( 0.3 5.0 ( 0.4 6.5 ( 0.8 8.1 ( 0.9 9.4 ( 0.9 9.0 ( 1.1 8.8 ( 1.2
105 ( 10 111 ( 11 115 ( 17 113 ( 16 115 ( 16 100 ( 16 94 ( 14
2.0 ( 0.1 2.0 ( 0.1 2.0 ( 0.1 2.0 ( 0.1 2.0 ( 0.1 1.9 ( 0.1 1.9 ( 0.1
0.1 50 100 150 200 250 300
1.79 1.79 1.81 1.82 1.83 1.90 1.90
21.9 ( 0.3 15.6 ( 0.4 12.5 ( 0.4 11.5 ( 0.4 11.4 ( 0.4 12.1 ( 0.6 12.1 ( 0.7
W0 ) 10.3 4.3 ( 0.5 3.1 ( 0.4 2.6 ( 0.5 2.3 ( 0.5 2.2 ( 0.4 2.1 ( 0.5 1.9 ( 0.4
0.1 50 100 150 200 250 300
1.78 1.81 1.81 1.82 1.85 1.82 1.88
14.7 ( 0.9 10.8 ( 0.5 9.0 ( 0.3 7.9 ( 0.2 6.6 ( 0.5 7.0 ( 0.2 7.9 ( 0.5
W0 ) 14.0 6.2 ( 1.4 4.8 ( 1.0 4.4 ( 0.6 3.9 ( 0.5 3.3 ( 0.8 3.0 ( 0.4 3.2 ( 0.9
2.4 ( 0.4 4.2 ( 0.7 8.4 ( 1.2 11.8 ( 1.7 15.5 ( 2.5 18.4 ( 1.1 17.9 ( 2.0
169 ( 26 179 ( 28 200 ( 17 206 ( 18 211 ( 35 184 ( 14 175 ( 35
2.6 ( 0.2 2.6 ( 0.2 2.7 ( 0.1 2.7 ( 0.1 2.7 ( 0.2 2.6 ( 0.1 2.6 ( 0.2
0.1 50 100 150
1.74 1.76 1.79 1.80
5.9 ( 0.7 4.6 ( 0.5 3.8 ( 0.5 3.4 ( 0.4
W0 ) 22.0 6.5 ( 2.7 5.2 ( 2.0 5.1 ( 2.1 5.3 ( 2.0
3.1 ( 1.4 8.5 ( 2.7 16.0 ( 9.5 25.9 ( 9.9
278 ( 81 298 ( 80 354 ( 98 420 ( 99
3.5 ( 0.4 3.6 ( 0.4 3.7 ( 0.4 3.9 ( 0.4
Figure 1. Plot of ln kq(H2O) against pressure in aqueous solution (0), and plots of ln kq′ against pressure in reverse micelles (0.22 M AOT in heptane) with W0 of 10.3 (b), 14.0 (O), and 22.0 (9) at 25 °C. The solid lines were drawn by assuming that ln ki ) A + BP + CP2 for W0 of 10.3 and 14.0 and ln ki ) A + BP for W0 of 22.0 and aqueous solution. The activation volumes, ∆Vq′q and ∆Vqq(H2O), were determined by eq 9 from the least-squares slopes of the solid lines at 0.1 MPa.
hexacyanoferrate(III), K3Fe(CN)6, of a guaranteed grade (Wako Chemicals, Ltd.) was recrystallized from acetone-water mixtures. Reversed micellar solutions with W0 ()[H2O]/[AOT]) of 10.3, 14.0, and 22.0, containing the probe and the quencher, were prepared as follows: the calculated amounts of the aqueous probe and quencher solutions were injected into a concentrated heptane solution of AOT (ca. 0.5 M) with use of a microsyringe. The mixture was shaken for a few minutes until a visually clear solution was obtained. To this solution the additional amount of water was added to adjust W0, and then the solution was diluted with heptane to the concentration of AOT examined. The concentration of the probe was adjusted to be 5 × 10-5 M. The sample solution was deoxygenated by bubbling nitrogen gas under nitrogen atmosphere. A high-pressure time-resolved luminescence measuring system and the associated experimental techniques have been described elsewhere.20 A nitrogen laser (337.1 nm/8 ns pulse width) was used as the excitation light source. The luminescence intensities were measured at right angles to the direction
Figure 2. Luminescence decay curves (dots) of Ru(bpy)32+ in reversed micelles (W0 ) 10.3, 0.22 M AOT in heptane) with quencher, Q, at 25 °C and five pressures; [Q] ) 1.22 × 10-3 M and [Ru(bpy)32+] ) 5 × 10-5 M at 0.1 MPa: 0.1 (a), 50 (b), 100 (c), 150 (d), and 300 MPa (e). The solid lines are the best-fit decay curves calculated according to the function in eq 1 by a nonlinear least-squares method.
of the exciting light by use of a Hamamatsu R 928 photomultiplier through a Ritsu MC-25NP monochromator. The output signal was digitized with an Iwatsu TS 8123 storagescope. All data were analyzed by a NEC PC9801 microcomputer which was interfaced to the digitizer. Pressures were measured by a calibrated Manganin wire. Temperatures were controlled at 25 ( 0.1 °C. Results The luminescence decay curves in aqueous solution were measured with and without quencher. It was found that the decay curves were single exponential under the experimental conditions examined up to 300 MPa. The decay constants obtained without quencher increased from 1.78 × 106 s-1 (0.1 MPa) to 1.99 × 106 s-1 (300 MPa). The quenching constant in aqueous solution, kq(H2O), which was determined from the dependence on the quencher concentration up to 8.4 × 10-4 M, decreased monotonically from (3.25 ( 0.04) × 1010 M-1 s-1 at 0.1 MPa to (2.61 ( 0.04) × 1010 M-1 s-1 at 300 MPa (Figure 1), suggesting that the quenching is diffusion-controlled. Figure 2 shows typical examples of the decay curves in reversed micelles at five pressures. As seen in Figure 2, the
Quenching of Luminescence in Reversed Micelles
J. Phys. Chem., Vol. 100, No. 33, 1996 14105
initial rapid nonexponential decay is followed by a singleexponential one at each pressure, whereas the decay curves were exponential in the absence of quencher. The decay constants, k0, with no quencher are listed in Table 1. The luminescence intensities, I(t), at time t after a δ-pulse excitation in the micellar solution with quencher are expressed by4,12,15
I(t) ) A1 exp[-A2t - A3{1 - exp(-A4t)}]
(1)
The decay curves obtained (Figure 2) were fitted to the function in eq 1 by a conventional nonlinear least-squares method. The solid lines shown in Figure 2 are the best-fit curves at five pressures. The variations in the recovered values of A2 and A3 with the concentrations of quencher, [Q], at three pressures are shown in Figure 3. The plots of A2 against [Q] (Figure 3a) are linear, and it was found that the extrapolated value at zero concentration of quencher is approximately equal to k0 at each pressure. The plots of A3 against [Q] (Figure 3b) are also linear, almost passing through the origin at each pressure. On the other hand, the values of A4 at a fixed W0 were almost independent of [Q] over the concentration range examined at each pressure. However, the mean values of A4 varied with pressure from (22.6 ( 0.3) × 106 s-1 at 0.1 MPa to (14.3 ( 0.7) × 106 s-1 at 300 MPa for the solution with a W0 of 10.3. The values of A2 are dependent on pressure and increase linearly with increasing [Q] at each pressure as seen in Figure 3a, and the probe and quencher exist only in micelles. The evidence suggests that the quenchers exchange on collisions between micelles at a rate depending on pressure. In this case, A2, A3, and A4 are given by eqs 2, 3, and 4,4,15b respectively. In eqs 2-4, kex, kq, and [M] are the bimolecular rate constant for quencher exchange by collision between micelles, the quenching constant in micelles, and the concentration of micelles, respectively.
A2 ) k0 +
A3 )
kqkex
[Q]
(2)
[Q]
(3)
kq + kex[M] kq2
(kq + kex[M]) [M] 2
A4 ) kq + kex[M] kq )
A3A42 A3A4 + A2 - k0
(4) (5)
The value of kq was determined from eq 5 using the values of A2, A3, and A4 together with the value of k0 at each pressure (Table 1). Using the values of kq and A4, kex was evaluated from the least-squares slope in the plots of A2 against [Q] (Figure 3a), and [M] from the least-squares slope in the plots of A3 against [Q] (Figure 3b) at each pressure. The aggregation number, N, is given by eq 6. In the determination of N by eq 6, [AOT] was corrected by taking into account the compressibility of heptane,21 and the cmc was neglected since [AOT] . cmc in the present study (the cmc in heptane ≈ 1 mM1e). The parameter values obtained are summarized in Table 1.
N)
[AOT] - cmc [M]
(6)
Equations 2-4 are derived assuming that the micelle size is monodisperse.4,15b In order to confirm the monodispersity, the values of A2, A3, and A4 were measured in 0.11 and 0.32 M
Figure 3. Plots of A2 (a) and A3 (b) against [Q] in reversed micelles (W0 ) 10.3, 0.22 M AOT in heptane) at 25 °C and three pressures: 0.1 (b), 150 (O), and 300 MPa (9).
Figure 4. W0 dependence on the radius, r, of the water core in reversed micelles (AOT in heptane) (this work, [) together with the results measured by various techniques at 0.1 MPa: fluorescence depolarization (b),2b fluorescence quenching (O),6,9 SANS (9),1c and ultracentrifugal (0)23 techniques.
AOT solutions with a fixed W0 of 10.3 up to 300 MPa. Plots of A2 and A3 against [Q] were linear, and it was found that the values of N so determined were almost independent of [Q] and/ or [AOT] at each pressure. The results suggest that micelles are monodisperse in size.10a,22 The bimolecular quenching constant, kq′ in micelles and the radius of the water core, r, of the micelle were calculated using the data listed in Table 1 by eqs 7 and 8, respectively. In eq 8, V0 is the volume of a water molecule. The values of kq′ and r are also listed in Table 1, and the values of r at 0.1 MPa are compared in Figure 4 with the results by other workers. It can be seen in Figure 4 that the values of r in this work agree well with the results measured by various methods.
kq′ ) 18W0Nkq/103 (M-1 s-1)
(7)
r ) (3W0NV0/4π)1/3
(8)
The activation volume, ∆Viq, for the rate process with ki was determined from eq 9, where i ) 0, q, q′, or ex. The plot of
(∂ ln ki/∂P)T ) -∆Viq/RT
(9)
ln kq′ against pressure is shown in Figure 1 together with the pressure dependence of kq(H2O) in aqueous solution, and the plot of ln kex against pressure is also shown in Figure 5. Values of the activation volumes were determined from the least-squares
14106 J. Phys. Chem., Vol. 100, No. 33, 1996
Okamoto
Figure 5. Plot of ln kdiff against pressure in heptane (0), and plots of ln kex against pressure in reversed micelles (0.22 M AOT in heptane) with W0 of 10.3 (b), 14.0 (O), and 22.0 (9) at 25 °C. The solid lines were drawn by assuming that ln ki ) A + BP + CP2. The activation volumes, ∆Vexq and ∆Vdiffq ()∆Vηq), were determined by eq 9 from the least-squares slopes of the solid lines at 0.1 MPa.
TABLE 2: Volumes of Activation, ∆Viq (cm3/mol), for the Rate Parameters Associated with the AOT/Water/Heptane Reversed Micellar System at 25 °C and 0.1 MPaa W0
∆V0q
∆Vqq
∆Vq′q
∆Vexq
∆V-1q - ∆V2q
10.3 14.0 22.0
-0.6 ( 0.1 -0.4 ( 0.1 -0.5 ( 0.1
16 ( 2 17 ( 2 15 ( 2
13 ( 2 11 ( 2 3(2
-24 ( 2 -37 ( 2 -54 ( 4
46 ( 3 57 ( 2 77 ( 7
a ∆V q(H O) ) 2.6 ( 0.1 cm3 /mol. ∆Vdiffq ()∆Vηq) was estimated q 2 to be 21 ( 2 cm3/mol at 0.1 MPa.
slopes of the plots shown in Figures 1 and 5. Similarly, the activation volumes for k0 and kq were determined. The results are listed in Table 2. The following points from the present results are worthy of note: (1) the agreement between the experimental data and theoretical curve predicted from eq 1 is very good, (2) A2 and A3 are linearly related to [Q] at each pressure, (3) A4 is almost independent of the quencher concentration at each pressure, (4) the micelle concentrations determined by the decay analysis are linearly related to [AOT] at each pressure, and (5) the micelle size is monodisperse. The evidence suggests that the luminescence-quenching kinetics in the present system, which has been confirmed at 0.1 MPa, is still maintained at high pressure. At a first glance of the data summarized in Table 1, the values of k0 are slightly dependent on W0 and pressure, and it is also found that the values of N are dependent on W0, but independent of pressure at a fixed W0 within the experimental errors. The discussion is, therefore, focused on the rate processes for kq and kex, both of which show significant pressure dependence. Discussion 1. Quenching Constants, kq and kq′. Since the quenching may be diffusion-controlled in micelles as expected from the results in aqueous solution, kq should depend on the micelle size. In fact, as can be seen in Table 1, the values of kq decrease with increasing N at each pressure. One can also see in Table 1 that kq decreases with increasing pressure at a fixed W0. This may be due to the increase in local viscosity induced by pressure in the interior of micelles where the quenching occurs. However, the activation volume, ∆Vqq, for kq (Table 2) is almost independent of W0, which may be attributed mainly to the pressure effects on the microviscosity and also the aggregation number at a given W0. Hence, the microenvironment inside the micelles should be discussed from the pressure dependence of kq′.
The bimolecular quenching constant, kq′, which was taken into account the aggregation number (eq 7), decreased monotonically with increasing pressure (Table 1). As seen in Table 2, the activation volume, ∆Vq′q, for kq′ decreases with increasing W0 and seems to approach the value of ∆Vqq(H2O). This result implies that the quenching occurs in an environment that becomes closer to water in micelles with increasing W0. Three types of environment in a micelle have been distinguished:2f the interfacial region, the bound water layer, and the free water layer in the core of the micelle; the latter is increased with increasing W0. The average number of bound waters per AOT molecule increases with increasing W0 up to 18, above which it remains constant, while the number of free water molecules remains constant between W0 ) 10 and 18 followed by the sharp increase with further increase in W0. These observations are parallel to the W0 dependence of the activation volume for kq′ (Table 2). Therefore, it may be concluded that the quenching occurs in the environment of the bound water layer at W0 of 10.3 and 14.0 and that it occurs mainly in the free water layer at W0 of 22.0. In general, the rate constant, kdiff, for the diffusion-controlled reaction can be calculated approximately by eq 10 in solvent with viscosity, η. The values of microviscosity in micelles
kdiff ) 8RT/3000η (M-1 s-1)
(10)
estimated from eq 10 by equating kdiff ) kq′ were 15, 11, and 10 cP for solutions with W0 of 10.3, 14.0, and 22.0 at 0.1 MPa, respectively. These results are compared with those estimated from the fluorescence depolarization measurements using 1-aminonaphthalene-4-sulfonic acid (24 cP at W0 ) 9.2)2b and tetrasodium 3,4,9,10-perylene-tetracarboxylate (3.9 cP at W0 ) 9.40)2c as probes in reversed micelles of AOT/water/isooctane. The main discrepancies in the estimation of the microviscosity may be due to the location of the probe molecules in micelles as well as the size of probes used. Similarly, it can be seen from the pressure dependence of kq′ in Table 1 that the microviscosity increases monotonically with increasing pressure at a fixed W0. 2. Quencher Exchange Rate Constant, kex. As seen in Table 1, the values of kex are almost independent of W0 at 0.1 MPa, whereas they increase significantly with increasing pressure. The values of ∆Vexq (Table 2) are largely negative and strongly dependent on W0. This increase in kex on the application of high pressure might look surprising at first glance since the quencher exchange is related to the diffusion of the micelles in heptane, but it can be interpreted as described below. The bimolecular rate constant, kex, is defined as the rate constant for the quencher transfer from a micelle containing a quencher molecule to another micelle.4,15b Since the quenchers in the present system are solubilized completely in micelles and do not migrate into the oil phase,4,7,18 an encounter or a collision complex between micelles should be formed prior to the quencher exchange. From these facts, together with the result in this work that the micelle size is monodisperse, the mechanism illustrated in Scheme 1 is rationalized as the quencher exchange processes. In Scheme 1, Pn* is a micelle containing an excited probe and n quencher molecules, Mj is a micelle with j quencher molecules, and Pn*Mj is the complex between the corresponding species. A similar mechanism was suggested by Robinson et al.,7,18 whose model involves the coalescence of micelles followed by fission into new micelles (fusion-fission process) in the step of k2. In this case, the micelles are expected to be also monodisperse if the fused state is short-lived. Almgren et al.10 observed clusters of the micelles by using probes with longer
Quenching of Luminescence in Reversed Micelles
J. Phys. Chem., Vol. 100, No. 33, 1996 14107
SCHEME 1
Figure 6. Plots of ln(k-1/k2) against ln η in reversed micelles (0.22 M AOT in heptane) with W0 of 10.3 (b), 14.0 (O), and 22.0 (9) at 25 °C, where the solvent viscosity, η, was changed by the application of high pressure.21
lifetimes of microseconds and found that the clusters are polydisperse whereas the micelles are not. They suggested that the fusion-fission process participates in the step of k2. On the basis of Scheme 1, the observed kex is expressed approximately by eq 11 (see Appendix). The steps of k1 and k-1 which may involve the diffusion of micelles in heptane are expected to depend significantly on the pressure-induced solvent viscosity. In the low viscosity limit (k-1 . k2), kex ) k1k2/k-1. In the high-viscosity limit (k-1 , k2), however, eq 11 is not applicable since the reverse process of k2 is probably significant in this limit.
kex )
k1k2 k-1 + k2
(11)
Figure 5 shows the pressure dependence of kdiff calculated by the Debye equation (eq 10) using the known values of the solvent viscosity, η,21 together with that of kex. As seen in Figure 5, kex seems to approach kdiff in solutions with W0 of 14.0 and 22.0 at higher pressures. We assume here that k1 ) kdiff. From eq 11 using the values of kex and kdiff, the efficiency for the quencher exchange, F ()k2/(k-1 + k2)), and also F-1 1 ()k-1/k2) were evaluated at each pressure (Figure 6). The activation volumes, ∆V-1q - ∆V2q, obtained from the leastsquares slopes in the plots of ln(k-1/k2) against pressure, are listed in Table 2. It can be seen in Table 2 that ∆V-1q - ∆V2q is largely positive and significantly dependent on W0. The step of k2 is related to the intermicellar mobility of quencher, which may depend mainly on the formation of transient holes or the fusion followed by the diffusion of the quencher molecules. The hole formation or the fusion process is probably related to the interfacial tension of the micelles. Although there is no direct information, the pressure dependence of the interfacial tension for the water/benzene and the water/
decane systems is not significant.24 Hence, the step of k2 is assumed to be mainly due to the diffusion of quencher between the micelles in the pairs or in the fused state, which may be affected by microviscosities in the interfacial region as well as inside the micelles. Unfortunately, there is no available information about the pressure effect on such microviscosities, but the activation volume, ∆V2q, for k2 is probably positive. If we assume that ∆V2q is equal to ∆Vq′q, reflecting the microviscosity in micelles where the quenching occurs, ∆V-1q was calculated to be 59 ( 4, 68 ( 4, and 80 ( 9 cm3/mol at W0 values of 10.3, 14.0, and 22.0, respectively. The value of ∆V-1q estimated thus is much larger than ∆Vdiffq in heptane (21 cm3/ mol; see Figure 5) even if ∆V2q ) 0, meaning that the reverse process is retarded by increase in pressure much more than the fully diffusion-controlled complex formation. Similar observation has been found for the pressure dependence on the reverse reaction of pyrene excimer (36-41 cm3/mol)25 formation in organic solvents, and the largely positive activation volumes have been explained by assuming that the reverse reaction involves processes associated with the solvent viscosity as well as the bond-breaking processes when the excimer diffuses apart into the reactants. For reversed micellar solutions, there is evidence for the interactions between micelles which promote clustering.26,27 These facts may suggest that ∆V-1q for k-1 involves contributions not only from the solvent viscosity but also from the breaking of the interaction between the micelles on separation, and the W0 dependence of ∆V-1q also suggests that the interaction increases with increasing W0. The latter conclusion is supported by the fact that the clustering is enhanced by the increase in W0.1b,26 Figure 6 shows the plots of ln(k-1/k2) against ln η. It can be seen in Figure 6 that ln(k-1/k2) decreases linearly with increasing ln η, and the values of R in eq 12 were found to be 1.82 ( 0.07, 2.77 ( 0.05, and 3.48 ( 0.09 at W0 values of 10.3, 14.0, and 22.0, respectively.
k-1/k2 ) Aη-R
(12)
The linear correlation shown in Figure 6 is evidence that k-1/ k2 is related directly to the pressure-induced solvent viscosity. Since the quencher exchange occurs when two micelles are in contact (see Scheme 1), it seems likely that k2 does not depend significantly on the solvent viscosity. This is supported by the fact that the rate of the fusion process is almost independent of the solvents used.10c Consequently, the speculation may suggest that the observed R-value in eq 12 reflects the power dependence of k-1 on η. In general, the solvent viscosity dependence on the rate constant, km, for diffusion-controlled or nearly diffusion controlled reactions is given phenomenologically by28
km ) Bmη-βm
(13)
where Bm is a constant and the exponent, βm, of η ranges from zero to unity. If the rate constants, k-1 and k2, are viscosity
14108 J. Phys. Chem., Vol. 100, No. 33, 1996 dependent and expressed by eq 13, that is, k-1 ) B-1η-β-1 and k2 ) B2η-β2, R in eq 12 is given by the equation R ) β-1 - β2. Hence, β-1 ) R + β2, meaning that the value of β-1 is equal to or larger than that of R observed since β2 g 0. Therefore, it may be concluded that the power dependence of k-1 on η, β-1, is significantly larger than unity and increases with increasing W0. This suggests that extra contribution of the solvent viscosity due to the interaction between micelles as well as the solvent viscosity dependence expressed by eq 13 participates in the step of k-1. Finally, a factor controlling the quencher exchange was demonstrated in an approach from high-pressure study. The clustering of micelles due to the interactions between them is known to cause eventually a phase separation.1b,18,26 From the present work, it seems likely that the clustering which leads to the phase separation occurs as a result of larger retardation of k-1 compared to k1 with increasing pressure. The solutions with W0 of 10.3 and 14.0 were transparent over the experimental conditions examined, but the solution with W0 of 22.0 turned turbid on going from 200 to 250 MPa. This phase change induced by pressure gives further evidence of the existence of attractive interactions between micelles. Acknowledgment. This work was partly supported by Japanese Government Grant-in-Aid No. 06640651. Appendix One can obtain eqs 11a and 11b by the stationary state approximation, where k ) k1k2/(k-1 + k2), y ) k2/k-1, and P(j) is the Poisson distribution.
{
}
(1 + y)j d[P0*]/dt ) - k0 + k[M]∑ P(j) [P0*] + j)0 (1 + jy) 1+y P(j)[P1*] (11a) k[M]∑ j)0 {1 + (1 + j)y} (1 + y)j d[Pn*]/dt ) k[M]∑ P(j)[Pn-1*] j)0 {1 + (n - 1 + j)y} (1 + y)(j + n) k0 + nkq + k[M]∑ P(j) [Pn*] + j)0 {1 + (n + j)y} (1 + y)(n + 1) P(j)[Pn+1*] (11b) k[M]∑ j)0 {1 + (n + 1 + j)y}
{
}
When y ) 0, eqs 11a and 11b have the same forms as those derived by Tachiya.15b Equations 11a and 11b were solved for the several sets of k, k0, and kq numerically as a function of the number average of quenchers in the micelle, 〈n〉, and y by using the Runge-Kutta-Gill algorithm in fourth order, and the decay curves calculated thus were fitted to the function in eq 1 in order to determine the mean values of kex by using eqs 2-5. It was found that kex/k decreased monotonically from 0.99 (〈n〉 ) 0.1) to 0.96 (〈n〉 ) 0.5) for y ) 0.1 and from 0.98 (〈n〉 ) 0.1)
Okamoto to 0.89 (〈n〉 ) 1.0) for y ) 0.5. Therefore, it was concluded that the observed kex is expressed approximately for the low value of 〈n〉 by eq 11. References and Notes (1) (a) Zulauf, M.; Eicke, H.-F. J. Phys. Chem. 1979, 83, 480. (b) Kotlarchyk, M.; Chen, S.-H.; Huang, J. S. J. Phys. Chem. 1982, 86, 3273. (c) Robinson, B. H.; Toprakcioglu, C.; Dore, J. C.; Chieux, P. J. Chem. Soc., Faraday Trans. 1 1984, 80, 13. (d) Toprakcioglu, C.; Dore, J. C.; Robinson, B. H.; Chieux, P. J. Chem. Soc., Faraday Trans. 1 1984, 80, 413. (e) Kotlarchyk, M.; Huang, J. S.; Chen, S.-H. J. Phys. Chem. 1985, 89, 4382. (f) Howe, A. M.; Toprakcioglu, C.; Dore, J. C.; Robinson, B. H. J. Chem. Soc., Faraday Trans. 1 1986, 82, 2411. (2) (a) Wong, M.; Thomas, J. K.; Nowak, T. J. Am. Chem. Soc. 1977, 99, 4730. (b) Zinsli, P. E. J. Phys. Chem. 1979, 83, 3223. (c) Keh, E.; Valeur, B. J. Colloid Interface Sci. 1981, 79, 465. (d) Martin, C. A.; Magid, L. J. J. Phys. Chem. 1981, 85, 3938. (e) D’Aprano, A.; Lizzio, A.; Turco Liveri, V.; Aliotta, F.; Vasi, C.; Migliardo, P. J. Phys. Chem. 1988, 92, 4436. (f) Jain, T. K.; Varshney, M.; Maitra, A. J. Phys. Chem. 1989, 93, 7409. (3) (a) Wong, M.; Thomas, J. K.; Gra¨ntzel, M. J. Am. Chem. Soc. 1976, 98, 2391. (b) Rodgers, M. A. J.; Becker, J. C. J. Phys. Chem. 1980, 84, 2762. (4) (a) Atik, S. S.; Thomas, J. K. Chem. Phys. Lett. 1981, 79, 351. (b) Atik, S. S.; Thomas, J. K. J. Am. Chem. Soc. 1981, 103, 3543. (5) Gelade´, E.; De Schryver, F. C. J. Photochem. 1982, 18, 223. (6) Bridge, N. J.; Fletcher, P. D. I. J. Chem. Soc. Faraday Trans. 1 1983, 79, 2161. (7) Howe, A. M.; McDonald, J. A.; Robinson, B. H. J. Chem. Soc., Faraday Trans. 1 1987, 83, 1007. (8) Verbeeck, A.; De Schryver, F. C. Langmuir 1987, 3, 494. (9) Lang, J.; Jada, A.; Malliaris, A. J. Phys. Chem. 1988, 92, 1946. (10) (a) Jo´hannsson, R.; Almgren, M.; Alsins, J. J. Phys. Chem. 1991, 95, 3819. (b) Almgren, M.; Jo´hannsson, R. J. Phys. Chem. 1992, 96, 9512. (c) Jo´hannsson, R.; Almgren, M. Langmuir 1993, 9, 2879. (d) Almgren, M.; Jo´hannsson, R.; Eriksson, J. C. J. Phys. Chem. 1993, 97, 8590. (11) Gehlen, M. H.; De Schryver, F. C.; Bhaskar Dutt, G.; van Stam, J.; Boens, N.; Van der Auweraer, M. J. Phys. Chem. 1995, 99, 14407. (12) Infelta, P. P.; Gra¨tzel, M.; Thomas, J. K. J. Phys. Chem. 1974, 78, 190. (13) Atik, S. S.; Nam, M.; Singer, L. Chem. Phys. Lett. 1979, 67, 75. (14) Gehlen, M. H.; Boens, N.; De Schryver, F. C.; Van der Auweraer, M.; Reekmans, S. J. Phys. Chem. 1992, 96, 5592. (15) (a) Tachiya, M. Chem. Phys. Lett. 1975, 33, 289. (b) Tachiya, M. J. Chem. Phys. 1982, 76, 340. (16) Almgren, M.; Lo¨froth, J.-E.; van Stam, J. J. Phys. Chem. 1986, 90, 4431. (17) Gehlen, M. H.; Van der Auweraer, M.; Reekmans, S.; Neumann, M. G.; De Schryver, F. C. J. Phys. Chem. 1991, 95, 5684. (18) (a) Fletcher, P. D. I.; Robinson, B. H. Ber. Bunsen-Ges. Phys. Chem. 1981, 85, 863. (b) Fletcher, P. D. I.; Howe, A. M.; Robinson, B. H. J. Chem. Soc., Faraday Trans. 1 1987, 83, 985. (19) Turro, N. J.; Okamoto, M.; Kuo, P.-L. J. Phys. Chem. 1987, 91, 1819. (20) Okamoto, M.; Teranishi, H. J. Phys. Chem. 1984, 88, 5644. (21) Brazier, D. W.; Freeman, G. R. Can. J. Chem. 1969, 47, 893. (22) Almgren, M.; Lo¨froth, J.-E. J. Chem. Phys. 1982, 76, 2734. (23) Robinson, B. H.; Steytler, D. C.; Tack, R. D. J. Chem. Soc., Faraday Trans. 1 1979, 75, 481. (24) (a) Michaels, A. S.; Hauser, E. A. J. Phys. Chem. 1951, 55, 408. (b) Harvey, R. R. J. Phys. Chem. 1958, 62, 322. (c) Jennings, H. Y., Jr. Colloid Interface Sci. 1967, 24, 323. (25) Okamoto, M.; Sasaki, M. J. Phys. Chem. 1991, 95, 6548. (26) Maritra, A.; Mathew, C.; Varshney, M. J. Phys. Chem. 1990, 94, 5290. (27) Halle, B. Prog. Colloid Polym. Sci. 1990, 82, 211. (28) Hirayama, S.; Yasuda, H.; Scully, A. D.; Okamoto, M. J. Phys. Chem. 1994, 98, 4609 and references cited therein.
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