J. Phys. Chem. 1981, 85, 1850-1856
1850
selves, are coprecipitated with PbHP04, thus increasing the total yield. It may be assumed that the heterogeneous nucleation occurs also in precipitation from dilute H3P04solutions (Table X, series K and L), though no apatite was actually precipitated at pH N 2, as the trend of total yield was similar to that at pH 5, Le., with a definite minimum at 10 ppm U concentration (Figure 9). The competition between the intrinsic nucleation and the preferred heterogeneous nucleation a t the first stage of the precipitation of alkaline earth and lead carbonates and phosphates in the presence of uranyl could account
for all of the complex framework of the experimental facts presented. This concept of active heterogeneous nucleation forming definite and complete (or incomplete) nuclei, the latter being dependent on the cation/impurity ratio, and the competition between the heterogeneous nucleation and the intrinsic (homogeneous) nucleation may have practical applications in the field of crystallization.
Acknowledgment. S. Tandy is grateful to the Eshkol Foundation for the Ph.D. studies grant allotted to this study of crystallization mechanisms.
Dynamics and Statistics of Triplet-Triplet Annihilation in Micellar Assemblies Guldo Rothenberger, Plerre P. Infelta, and Mlchael Gratzel’ Institui de Chimie Physique, €cole Polytechnique F M r a i e de Lausanne. Lausanne, Switzerland (Received: December 17, 1980)
A statistical model describing the distribution of excited solubilizate molecules among micelles resulting from flash photolysis in micellar surfactant solutions is presented. This model provides the basis for the kinetic analysis of intramicellar triplet-triplet annihilation processes. Laser flash photolysis experiments carried out with 1-bromonaphthalene dissolved in aqueous cetyltrimethylammonium bromide solutions led to the estimation of the rate constant for the intramicellar annihilation of a pair of triplets. The value k / 2 = 1.4 X lo7 s-’ was obtained. An extinction coefficient e* = 11500 M-’ cm-’ at X = 425 nm for the triplet state of 1-bromonaphthalene was also estimated from these experiments.
1. Introduction
Recently increasing attention has been payed to the study of fast photochemical processes in colloidal assemblies such as micelles,’ vesicles, and finely dispersed redox catalysts.2 These investigations are of prime importance in the understanding of light energy harvesting and conversion in biological and artificial systems. Micelles, for example, are relatively simple and well-defined model systems to mimic a more complex biological en~ironment.~ A point of particular interest is how in such assemblies the restriction of spatial extent and the reduction of dimensionality of the reaction space influence the kinetics of photophysical and photochemical processes4such as triplet energy transfer5 or electron transfer6 processes. The interaction of two triplet molecules in solution gives rise to triplet-triplet annihilation (TTA).7 This process
(1) (a) Turro, N. J.; Gritzel, M.; Braun, A. M. Angew. Chem., Int. Ed. Engl. 1980,19, 6755. (b) Kalyanasundaram, K. Chem. SOC.Rev. 1978, 7, 453. (2) Kiwi, J.; Gratzel, M. Nature (London) 1979, 281, 657. (3) (a) Tanford, C. “The Hydrophobic Effect”; Wiley: New York, 1973. (b) Fendler, J. H.; Fendler, E. J. “Catalysis in Micellar and Macromolecular Systems”; Academic Press: New York, 1975. (4) Hatlee, M. D.; Kozak, J. J.; Rothenberger, G.; Infelta, P. P.; Gratzel, M . J . Phys. Chem. 1980, 84, 1508. (5) Rothenberger, G.;Infelta, P. P.; Gratzel, M., J. Phys. Chem. 1979, 83, 1871.
(6) Maestri, M.; Infelta, P. P.; Gratzel, M. J. Chem. Phys. 1978, 69, 1522.
may lead to the formation of an excited singlet molecule ‘M* and ultimately to P-type delayed monomer fluorescence or may lead to the formation of a singlet excimer ‘D* responsible for the delayed excimer fluorescence. The formation of a triplet and the radiationless deactivation of the triplet pair can also be envisaged. The different pathways for TTA are represented in Scheme I.7r Ottolenghi and co-workers8 have pointed out that the con-
Scheme I 3M*
-
k + 3M* & {3M*3M*]
--
k-1
1M + 1M* or ID* ‘M lM
+ ‘M + 3M*
(7) (a) Birks, J. B. “Photophysics of Aromatic Molecules”; Wiley: New York, 1970. (b) Birks, J. B., Ed. “Organic Molecular Photophysics”; Wiley: New York, 1976; Vols. 1, 2. (c) Bonneau, R.; Faure, J.; JoussotDubien, J. Chem. Phys. Lett. 1968,2, 65. (d) Stevens, B. Ibid. 1969, 3, 233. (e) Razi Naqvi, K. Ibid. 1968, I , 561; 1970,5, 288; 1978,54, 49. (0 Razi Naqvi, K.; Behr, J. P.; Chapman, D. Ibid. 1974, 26, 440. (9) Razi Naqvi, K.; van Willigen, H.; Capitano, P. A. Ibid. 1978,57, 197. (h) van Willigen, H. Ibid. 1975,33,540. (i) Capitano, D. A.; van Willigen, H. Ibid. 1976,40,160. 6)Wyrsch, D.; Labhart, H. Ibid. 1971,8,217; 12,373,378. (k) Spichtig, J.; Bulska, H.; Labhart, H. Chem. Phys. 1976, 15, 279. (1) Lendi, K.; Gerber, P.; Labhart, H. Ibid. 1976,18,449; 1977,20,145. (m) Yekta, A.; Turro, N. J. Mol. Photochem. 1972,3,307. (n) Prasad, P. N. Chem. Phys. Lett. 1973, 20, 507. (0)Nickel, B. Ibid. 1974, 27, 84. (p) Tfibel, F.; Lindqvist, L. Chem. Phys. 1975, I O , 471. (4) Subudhi, P. C.; Lim, E. C. Chem. Phys. Lett. 1978, 58, 62. (r) Butler, R. P.; Pilling, M. J. J. Chem. SOC., Faraday Trans. 2 1977, 73,886. (9) Fang, T. S.; Fukuda, R.; Brown, R. E.; Singer, L. A. J. Phys. Chem. 1978,82, 247. (8) Lachish, U.; Ottolenghi, M.; Rabani, J. J. Am. Chem. SOC.1977, 99, 8062.
0022-3654/81/2085-1850$01.25/00 1981 American Chemical
Society
Triplet-Triplet Annihilation Micellar Assemblies
The Journal of Physical Chemistry, Vol. 85, No. 13, 1981 1851
finement of excited triplet states of molecules by micellar assemblies considerably enhances the yield of triplettriplet interactions. The present work is devoted to a quantitative study of this effect. A model which takes into account both the statistical and kinetic aspects of T”A in micellar systems is first elaborated and subsequently tested with laser flash photolysis experiments using l-bromonaphthalene as aromatic probe molecule and cetyltrimethylammonium bromide as micelle-forming surfactant.
2. Experimental Section Materials. Cetyltrimethylammonium bromide (CTAB, Fluka purum) was recrystallized 5 times from a methanol/ether mixture and once from water. The critical micelle concentration (crnc) was obtained by surface-tension measurements as 1.1 X MS9 An aggregation number of u = 92 as determined previously5 was used for the calculations. 1-Bromonaphthalene (Fluka purum) was distilled twice under reduced pressure. Deionized water was distilled from alkaline permanganate and twice from a quartz still. Sample Preparation. Aliquot of a stock solution of bromonaphthalene in hexane were transferred into dry flasks, and the solvent was evaporated. The detergent solution was added, and the solution first stirred for 3 h at 30 “C and then for at least 12 h at room temperature. The probe concentration was checked spectrophotometrically. The samples were degassed by using the freezepump-thaw technique (3-4 cycles). Pure N20 (CARBA, 510 ppm 0,) was added when the presence of an electron scavenger was necessary. Apparatus. A JK-2000 frequency quadrupled (265 nm) neodymium laser with a pulse half-width of 15 ns was used for excitation. The energy fluence of the laser pulse (total energy per unit area) was varied up to 250 mJ/cm2 by adjusting the electrical energy dissipated in the flash lamps of the laser amplifier. This fluence was measured with a Laser Instrumentation Ltd. Model 20 radiometer through a 3-mm diameter hole adjusted at the center of the laser beam. The concentration of the excited species was measured by T-T absorption, the analyzing light beam being perpendicular to the laser beam. The length of the analyzing light path in the irradiated volume of the sample was adjusted to 1 = 3 mm by means of a slit placed in front of the laser beam. The band-pass of the monochromator was set to 1.6 nm for measuring the spectrum displayed in Figure 1 and to 2.5 nm for all other measurements. A RCA 1P28B or Hamamatsu R 928 photomultiplier operated with five dynodedo was used for the detection. The time course of the absorption was recorded with a Tektronix WP 2221 wave form acquisition system. 3. Statistics of the Probe Distribution among the Micelles Probe Distribution prior to the Photolysis. A number of assumptions are made in the derivation of the present model for TTA in micellar systems. We suppose that the micelles are identical, and the variation of the size and the properties of the aggregates around a mean value is neglected. The surfactant and the experimental conditions have to be chosen such that the aggregation number of the micelles does not depend on the surfactant concentration. Aromatic molecules, hereafter called “probes”, are solubilized in the micelles. We suppose that these species are (9) This is somewhat higher than the values 8 X 104-9.8 X lo-’ M given for the cmc of CTAB in: Mukerjee, P.; Mysels, J. J. Natl. Stand. Ref. Data Ser. (US., Natl. Bur. Stand.) 1971, 36. (10)Beck, G. Reu. Sci. Instrum. 1976, 47, 537.
1002
40.01
..
300
400
bl
500
Figure 1. T-T absorption spectrum of 1-brornonaphthalene (1.29 X M) in CTAB solution (2 X lo-* M). The scale for the triplet extinction coefficient was obtained from t * = 11500 M-’ cm-’ at 425 nrn (see text).
hydrophobic enough to be associated almost exclusively with the micellar aggregates. Nevertheless, an exchange of the probe molecules from the micelles to the aqueous bulk and vice versa may take place, leading to a dynamical equilibrium of the probes with the micelles. This exchange is supposed to be slow enough such that, during the short time scale of observation (2 IS),only a negligible fraction of the micelles is affected by it. Furthermore, it is assumed that the rate describing the entry of the probe molecules in micelles is independent of the micellar occupancy, whereas the rate of exit of probes from micelles depends linearly on the number of probes that a micelle contains. This supposes a negligible interaction between solubilized probes. As stated in ref I l d , this kinetic scheme implies that the distribution law for the probes among the micelles is a Poisson law: F(rt,i) = M i / M = (rti/i!)e+ (1) rl =
c o / M = cou/([surfactant]
- cmc)
(2)
where F is the fraction of the micelles which contain i probe molecules, Mi is the concentration of these micelles and M is the total concentration of the micelles in the solution, rt is the average number of probe molecules per micelle, v is the average aggregation number of the micellar assemblies, cmc is the critical micelle concentration, and co is the probe concentration in the solution. The Poisson distribution law has been shown to allow a satisfactory description of the experimental results of several t h o r ~ . ~ J l J ~ Distribution of the Triplets among the Micelles at the End of the Laser Pulse. Upon photolysis, probe molecules are excited and a fraction of them is converted into the triplet state via intersystem crossing. In order to interpret the kinetic data, it is desirable to select the aromatic probe such that the intersystem crossing is fast compared with the laser pulse duration. The absorbance of the solution at the laser wavelength has to be low enough so that the excitation can be considered as spatially homogeneous (11) (a) Khuanga, U.;Selinger, B. K.; McDonald, R. Aust. J. Chem. 1976,29, 1. (b) Rodgers, M. A. J.; Wheeler, M. D. F. Chem. Phys. Lett. 1978,53, 165. (c) Yekta, A.; Aikawa, M.; Turro, N. J. Ibid. 1979, 63, 543. (d) Infelta, P. P.; Gratzel, M. J. Chem. Phys. 1979, 70, 179. (e) Infelta, P. P. Chem. Phys. Lett. 1979, 61, 88. (12) (a) McQuarrie, D. A. J. Chem. Phys. 1963,38,433. (b) McQuarrie, D. A.; Jachimowski, C. J.; Russel, M. E. Ibid. 1964, 40, 2914. ( c ) McQuarrie, D. A. Ado. Chem. Phys. 1969,15,149. (d) Ishida, K. J.Chem. Phys. 1964,41,2472. (e) Darvey, I. G.; Ninham, B. W.; Staff, P. J. Ibid. 1966,45,2145. (0 Darvey, I. G.; Ninham, B. W. Ibid. 1967,46, 1626. (9) Staff, P. J. Ibid. 1967,46,2209. (h) Thakur, A. K.; Rescigno, A.; DeLisi, C . J.Phys. Chem. 1978, 82, 552. (i) Vass, Sz. Chem. Phys. Lett. 1980, 70, 135. 6)Hatlee, M. D.; Kozak, J. J. J. Chem. Phys. 1980, 72, 4358. (13) Almgren, M.; Grieser, F.; Thomas, J. K. J. Am. Chem. SOC.1979, 101, 279.
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Rothenberger et al.
throughout the irradiated volume of the solution. Let us define a as the probability for a given probe molecule to be converted into a triplet upon laser excitation. If co is the total probe concentration in the irradiated volume and if co* is the concentration of the triplets in the same volume, co* being measured just after the laser pulse and before any decay mechanism has affected the triplet concentration, then a = co*/co (3)
In case 1, since the triplets will react in a pairwise fashion, the micelles which initially contained an odd number of triplet molecules will contain one triplet molecule upon completion of all TTA. All triplets will react in micelles which contained an even number of triplet molecules. Thus
The value of a depends on the properties of the probe molecule (extinction coefficient a t the excitation wavelength, quantum yield for triplet formation, etc.) and on the experimental conditions, i.e., the number of photons per laser pulse. We make the assumption that the value a is the same for all probe molecules in the irradiated volume. This means that the probability for the conversion of a probe into a triplet is independent of its chemical environment. (This assumption would not be fulfilled, for example, if the formation of an excimer via lM* 'M lD* competes efficiently with intersystem crossing in micelles with multiple probe occupancy.) If the above-mentioned conditions are fulfilled, the probability for a micelle which had i probes prior to excitation to have x ( x I i ) probes in the triplet state after excitation is given by the binomial distribution P(x,i,a) = [ i ! / x ! ( i- x)!]ax(I (4)
where x = 0, 1,2,3, ... and the fraction of triplet molecules which do not undergo a T-T annihilation is
+
-
and the fraction of the micelles which contain x triplets immediately after photolysis, irrespective of the number of ground-state molecules which they contain, is given by m
K(x,fi,a)= C F(fi,i)P(x,i,a) = [ ( r i ~ ) ~ / x ! ] (e5-) ~ i=x
Equation 5 describes the initial distribution of the triplets among the micelles; this means that the distribution which results after excitation and which holds before any reaction involving triplets has taken place. It is interesting to note that
c xK(x,ri,a)= ria
x=o
m
c,* = M C [(fi~)~""/(2x+ l)!]e-m = Me-m sinh (iia) x=o
(9)
In case 2, the micelles which initially contained at least one triplet molecule will contain one triplet molecule on completion of all TTA processes. Thus m
c,* = M C [ ( r i ~ ) ~ / x ! ] = e -M ~ "( l - e+) x=l
(11)
and the fraction of the surviving triplets is c,* 1 - e-na f2
= - =CO*
(12)
ria
A * @ ) , the transient absorbance of the solution, was measured. If the experimental conditions are such that only the triplet state contributes to this absorbance, then A * ( t ) = c*c*(t)l (13) where E* is the (decadic) triplet extinction coefficient at the observation wavelength, c * ( t ) is the triplet concentration, and 1 is the length of the analyzing light path in the irradiated volume (in our case 1 = 3 mm). Applying eq 13 to eq 10 or 12 leads to the calculation of A,*, the absorbance of the solution after completion of the intramicellar TTA reaction. In case 1, one gets
(6)
Thus, the average number of excited triplet states per micelle is ria. Distribution of the Triplet after Completion of Intramicellar Triplet-Triplet Annihilation. In order to calculate how the distribution function of the triplets is affected by the TTA process, we suppose in this section that all other decay mechanisms are slow in comparison with the TTA process and neglect them on the fast time scale of observation. We admit also that on such a fast time scale the TTA process occurs only if the two triplets involved in the annihilation event are located in the same micelle. It is convenient to distinguish two limiting situtations concerning the products of the annihilation reaction. In case 1, one assumes that no triplet is formed in any step following a TTA event: 3M* + 3M* product (7)
-
in case 2, the TTA always leads to the formation of a triplet. The overall reaction is described by (8) 3M* + 3M* 3M* + product
-
Let c,* be the triplet concentration in the irradiated solution after completion of the TTA process (Le., at time t = a). c,* can be calculated for both limiting cases as follows.
where
m = l/([surfactant] - cmc)
(15)
In case 2, a similar relation is obtained:
The determination of the absorbance A,* as a function of the surfactant concentration of the solution at constant probe concentration co and at invariant irradiation conditions (a = constant) will allow us to compare the predictions of eq 14 or 16 with experimental data. 4. Kinetics of Intramicellar TTA Classical chemical kinetics used to describe the time course of reactions in homogeneous solutions do not apply to the case of intramicellar reactions. The partition of the reacting species over small volumes and the s m d number of these species present in each volume require the application of a stochastic model of chemical reactions. Several authors have dealt with such problems.12 In particular, McQuarrie et al.12b,c have solved the stochastic model for the second-order reaction 2A C, the reactants being compartmentalized in a single volume. A similar model is adequate to describe the present situation. Let us define p , ( t ) as the probability that a micelle contains x triplet molecules a t time t . We have
-
The Journal of Physical Chemistry, Vol. 85, No. 13, 1981 1853
Triplet-Triplet Annihilation Micellar Assemblies
where M x ( t )is the concentration at time t of the micelles with x triplet hosts. M is the total micelle concentration. We suppose that the triplets react in a pairwise fashion according to reaction 7, the (first-order) rate constant for the annihilation reaction in micelles containing two triplets being k. In micelles containing x triplets, the rate for the transition x x - 2 (i.e., the triplet disappearance rate) is assumed to be given by 1 / 2 k x ( x- 1). Among the x triplets, one can react with the x - 1 others, and l/zx(x 1)is the number of ways that a pair of reactant molecules can be chosen from a total of x reactant molecules. These hypotheses lead to the following set of differential-difference equations: dpx(t)/dt = f/,k(x + 2)(x + l)p,+z(t) - '/zkx(x - l)Px(t) (18) where x = 0, 1, 2, ... and k is the rate constant. These equations are subject to the boundary condition given by eq 5: p,(O) = [(~ta)"/x!]e-" a t time t = 0 (19)
0'04
1
-
From this, it is possible to calculate an expression for the time evolution of the triplet concentration in the irradiated solution (see the Appendix):
where n = 1, 2, 3,
... and (j-n+l\
-
+
+
with j = n, n 2, n 4, .... is the gamma function. For t m , eq 20 reduces to eq 10. 5. Results and Discussion
Some Photolysis Aspects of 1-Bromonaphthalene Solubilized in CTAB. 1-Bromonaphthalene dissolved in CTAB solutions was used as probe molecule. According to ref 13, the distribution constant K between the micellar and aqueous phases for the 1-bromonaphthalene/CTAB system is -1.7 X lo6 M-l. From this, it is estimated that, under our experimental conditions, at most 1.5% of the probes are present in the aqueous phase. This small fraction was neglected. Bromonaphthalene when solubilized in the CTAB micelles displays a weak fluorescence. Single photon counting experiments showed that the singlet lifetime is shorter than 1ns. This finding is pertinent to the mechanistic considerations made above. Since, during a time period of only 1 ns, there occurs practically no diffusional displacement of the probe, the formation of excimers from an excited singlet via 'M* + 'M lD* in micelles with multiple bromonaphthalene association cannot take place. Hence apart from radiative and ratiationless deactivation or photoionization, the only reaction pathway available for lM* is triplet formation. For low-energy fluences, a transient absorption spectrum with a maximum at X = 425 nm and characteristic of T-T transitions for 1-bromonaphthaleneis observed (see Figure 1). At higher exciting light intensities, an additional much broader absorption band centered around 715 nm appears. This band is attributed to hydrated electrons resulting from the photoionization of bromonaphthalene. It disappears within -3 11s. Addition of N 2 0 (the solutions are
-
Flgure 2. Transient absorbance at h = 425 nm and A = 715 nm (measured just after the laser pulse) as a function of the laser energy M; [CTAB] = 2 X lo-' fluence. [l-bromonaphthalene] = 1.29 X M (A = 0.06). The absorbance at h = 425 nm was measured with a N,O-saturated solution.
saturated at 1.3 atm) causes the elimination of any absorption in the 700-nm region within the laser pulse duration. No transient absorption due to the radical cation of bromonaphthalene was observed (the radical cation of naphthalene is known to have an absorption maximum around 700 nm14). This is attributed to a fast subsequent reaction of the cation radical with the medium. N2O was added to the samples whenever the contribution of the hydrated electrons to the transient absorption at wavelengths where triplets were observed had to be avoided (ce-(aq) = 2500 M-' cm-' at X = 425 nm15). The effect of the energy fluence on the absorbances a t X = 425 and 715 nm (measured just after the laser pulse) is displayed in Figure 2. One notices that, upon increasing the laser dose, the contribution of the 715-nm signal increases at the expense of the triplet absorption at 425 nm. This is consistent with a biphotonic mechanism for the photoionization process. The temporal behavior of the absorbance at X = 425 nm is dependent on both the energy fluence of the laser pulse and the average occupancy of the micelles. Similar features are obtained for low average occupancies or low-energy fluences. A transient absorbance curve illustrating the latter case is depicted in Figure 3a. One observes a sharp rise of the transient absorbance, which decays afterwards according to a pseudo-first-order law (kd = 2 X lo4 9-l). This decay is attributed to quenching of the triplets, notably by impurities, or to phosphorescence. The slow rate of triplet diappearance observed under these conditions is indicative of a micellar inhibition of TTA. Thus TTA does not play any significant role on the 2-11 time scale of observation. A t the low-energy fluence employed, the probability for having two excited states in the same micelle is small. Furthermore, the fact that no fast decay is observed in Figure 3a shows that the quenching of triplets by ground-state probes is a slow process. This agrees with results of a study by Takemura and co-workers,16who determined the rate constant for the triplet-ground state reaction for 1chloronaphthalene in isooctane. The value obtained is -400 times slower than the diffusion-controlled rate constant. (14) (a) Shida, T.; Iwata, S. J. Am. Chem. SOC.1973, 95, 3473. (b) Lesclaux, R.; Joussot-Dubien, J. In ref 7b, Vol. 1, Chapter 9. (15) Hart, E. J.; Anbar, M. "The Hydrated Electron"; Wiley: New York. 1970. (16) Takemura, T.; Aikawa, M.; Baba, H.; Shindo, Y. J. Am. Chem. SOC.1976, 98, 2205.
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The Journal of Physical Chemistty, Vol. 85, No. 73, 7987
a
O
f
time [ p s e q
laser pulse
0'01
i
b
-0.5
I
I
0
150
1
in Ch4-y
300
Flgure 4. Long-time behavior of the absorbance observed at 425 nm ( A m o )as a function of m . [l-bromonaphthalene] = 5.46 X lo-' M; fluence = 130 mJ/cm2; N20 = 1.3 atm. Ao' and the curves are calculated (eq 14 or 16; see text).
O +
time [psec]
laser pulse
Flgure 3. Transient absorbance observed at X = 425 nm of a solutiin containing 5.46 X lo-' M 1-bromonaphthalene and 4.55 X M CTAB (ri = 1.45). The solution was saturated with N,O at 1.3 atm. Fluence of the exciting laser pulse: (a) 10 and (b) 130 mJ/cm2.
Evaluation of the Statistics of Intramicellar TTA. The feature of the temporal behavior of the triplet absorption at 425 nm change drastically when solutions with high average micellar occupancy are excited with high-fluence laser pulses. An example is displayed in Figure 3b. Here a fast and partial decay is followed by a plateau region which corresponds to a second, much slower decrease. The fast component of the decay is attributed to T-T annihilation which occurs in micelles containing at least two triplet molecules. The slower decrease follows a pseudofirst-order rate law on the 2-ps time scale of observation with kd = 5 X lo4 s-'. This higher value for k d (as compared with k d = 2 X lo4 s-' obtained for low laser fluence) is attributed to an enhancement of intermicellar TTA as the triplet concentration in the solution is increased. In order to compare eq 14 or 16 with experimental data, eight solutions with a 1-bromonaphthalene concentration of 5.46 X M and surfactant concentrations between 4.55 X M (ii = 0.42) were M (ii = 1.45) and 1.30 X photolyzed at a laser energy fluence of 130 mJ/cm2. The conditions of invariant laser fluence and constant probe concentration were chosen in order to meet the assumption that the probability for triplet formation a remains a constant. The absorbance A,* was determined by extrapolating back to time t = 0 (the beginning of the laser pulse) the plateau region of the transient absorbance curves measured at X = 425 nm. The eight experimental values for Am*were fitted to either eq 14 or eq 16 by using a least-squares method. The following parameters were used: v = 92, cmc = 1.1X M, 1 = 0.3 cm (see the Experimental Section); t* and a are adjustable parameters. In case 1 (eq 14) optimal agreement was found with t* = 11500 M-' cm-' and a = 0.60. In case 2 (eq 16) a best fit was obtained with t* = 5750 M-' cm-' and a = 1.20. This set of data is shown in Figure 4 where log (A,*/Ao*)is reported as a function
of m (eq 15). The solid line and Ao* are calculated according to eq 14 or 16 with the respective optimal parameters. The broken lines are calculated with a *lo% deviation from the optimal value for a. A decision as to the mechanism of TTA operative in our system may be made on the basis of limits for the probability of triplet formation a. Evidently a cannot exceed unity. In addition, one has to take into account that a fraction of the probe molecules undergoes photoionization. This fraction was determined from Figure 2. Knowing the extinction coefficient of the hydrated electron = 18500 M-' cm-' a t 715 nm15), one calculates that -12% of the bromonaphthalene molecules are photoionized a t an exciting light fluence of 130 mJ/cm2. Therefore, the upper limit for a is reduced to 0.88. It is evident that the a value of 1.20 derived by applying the mechanism according to reaction 8 (case 2) to the experimental data exceeds by far this limit for a. The good agreement of the data and the calculated curve in Figure 4 excludes an error as high as 30% in the determination of a. One concludes that the probability for reaction 8 to occur in the present system is low. The alternative mechanism according to reaction 7 (case 1)leads to a much better description of the system. The values a = 0.60 and e* = 11500 M-' cm-' are therefore an acceptable set of parameters. It should however be noted that the latter value is only an upper limit for the extinction coefficient as a small contribution of reaction 8 to the overall TTA cannot be excluded on the basis of the present analysis. The high probability for the occurrence of reaction 7 may be interpreted in terms of an efficient radiationless deactivation of either the triplet-triplet pair or an eventual singlet excimer (see Scheme I). Such an enhancement of radiationless deactivation could be induced by the local environment in the micelle/water interface, in particular the high Br- concentration (3-6 M) present in the Stern layer of the micelle. It is instructive to compare the results obtained from the foregoing analysis of intramicellar 'M'A with the laser photolysis data displayed in Figure 2. We recall that the latter illustrates the effect of laser intensity on the end of pulse absorbance a t 425 and 715 nm under conditions of very low micellar occupancy (a = 0.06). In this case, the contribution of intramicellar TTA to the triplet decay processes is negligible. From the maximum absorbance value at 425 nm (Ao* = 2.97 x obtained a t a laser fluence of 130 mJ/cm2, one derives the fraction of
The Journal of Physical Chemlstry, Vol. 85, No. 13, 1981 1855
Triplet-Triplet Annihilation Micellar Assemblies
It is noted that the triple-triplet annihilation process observed by Turro et al.Is in the case of 1-chloronaphthalene in micellar solutions is much slower than the TTA process described in the present paper. We believe that the reported decay process occurring at a rate of 8 X lo7 M-l in the system 1-chloronaphthalene/CTAB arises from intermicellar TTA. This is substantiated by the fact that second-order kinetics are observed. -*nu-**-******=
-05
1
0
I
0 25
time
[ps]
05
Figure 5. Kinetics of the T-T annihilation process. [I-bromonaphthalene] = 5.46 X M. [CTAq ( * ) 4.55 X lo-? (0)4.88 x io-? (*) 5.43 x io3; (u)5.95 x i o : (0)6.75 x IO-? (A)8.23 x io-? (0) 1.02 x io-*; (e) 1.30 x io-2.
ground-state molecules converted into triplets. A ratio co*/co= 0.67 is obtained if we use the extinction coefficient t * = 11500 M-' cm-' determined in the analysis of intramicellar T T A according to reaction 7 (case 1). The same analysis yielded the value a = 0.60 for the probability of triplet formation at 130 mJ/cm2. The two values agree within 11% , which illustrates the coherence of the data. The value of the triplet extinction coefficient for 1bromonaphthalene derived above may be compared with ern= values obtained previously for related molecules. Bensasson, Land, and Amand found extinction coefficients which were about twice as large for naphthalene (E* = 24500 M-I cm-' at 415 nm17a)and 1-chloronaphthalene(e* = 20700 M-' cm-l at 420 in cyclohexane, using pulse radiolysis and triplet energy transfer methods. The difference may be explained in part by the fact that for 1bromonaphthalene the absorption band at 425 nm is broader (the width a t half-height is -23 nm) than the corresponding band at 415 nm for naphthalene (the width at half-height is -8 nm, see ref 17a). Evaluation of the Kinetics of Intramicellar TTA. The model for the kinetics pertaining to case 1 has been developed in section 4. This model will now be compared with the measured decay curves. Figure 5 shows a plot of log (A*(t)/Ao*)as a function of time. The experimental data have been corrected numerically for the slow pseudo-first-order decay process observed on a longer time scale. This correction affected the absorbance at most by 2.5% (kd = 5 X lo4 s-l). The solid lines drawn through the experimental points in Figure 5 were calculated from eq 20, with a = 0.60. The rate constant k giving an optimal fit with the experimental data was found to be (2.8 f 0.3) X lo7 s-l. The value of k / 2 (two 3M*'s annihilate per encounter) is of the same order of magnitude as k,, the rate constant for irreversible intramicellar triplet energy transfer in CTAB ( k , = 1.5 X lo7 s-' for the system MPTH/trans-stilbene5 and k , = 1.0 X lo7 s-l for the system chrysene/trans-stilbene4). A plausible explanation for this similarity would be that, for both processes, an average diffusion time of k;' = 60-100 ns has to elapse before an encounter complex of the two reactant molecules is formed, the annihilation or transfer reaction taking place afterwards being extremely efficient.
6. Conclusion The present paper explores the kinetic and statistical aspects governing triplet-triplet annihilation in micellar assemblies. In comparison with homogeneous solutions, one notices that, for low light intensities and for low probe occupancies, the micellar aggregates tend to stabilize the triplets. In addition to a shielding effect against aqueous quenchers, the micelles slow down the TTA process. The picture changes drastically as soon as the light excitation produces simultaneously several excited states per micelle. Such a situation arises if the light intensity is high and multiple occupancy of the surfactant aggregates by the probe molecule prevails. Here, the yield of TTA is greatly enhanced with respect to homogeneous solution, and it follows also a different rate law. With respect to the earlier work on intramicellar triplet energy t r a n ~ f e r ,we ~ note that the intramicellar TTA process requires a different kinetic approch. In the former case, one triplet donor was reacting with an ensemble of ground-state acceptors, whereas, in the latter case, the interactions in an ensemble of molecules in the triplet state have to be considered. From the present analysis of TTA in micellar systems, three results emerge. The first relates to the statistics of the excited probe distribution among the micelles. It is found that the distribution follows a Poisson law if the ground-state probe distribution prior to the excitation obeyed also Poisson statistics. Secondly, from this analysis one derives an approximate value for the extinction coefficient of the probe triplet state. An upper limit E* = 11500 M-' cm-I a t X = 425 nm was found for 1-bromonaphthalene. A third and important parameter obtained from this analysis is the rate constant for the annihilation of a pair of triplets in a micelle. The value k/2 = 1.4 X lo7 s-I derived indicates that this process is controlled by the diffusion of triplets in the micelle. Hence, this type of analysis provides a further method to obtain information on the diffusional displacement of molecules in micellar aggregates.
Acknowledgment. Acknowledgment is made to the Swiss National Science Foundation for support of this work under grant No. 4.339-0.79.04. Appendix By making use of the probability generating function m
F(s,t) =
(AI)
x=o
where s is an arbitrary variable, one may transform eq 18 into the partial differential equation aF/at = y2k(1 - ~ 2 a2F/as2 ) (A2) A general solution of eq A2 can be expressed as12b m
F(s,t) =
AnCn-'12(s)exp[-'/,kn(n
-
Ut]
(A3)
n=O
where C;1/2(s) (17)(a) Bensasson, R.; Land, E. J. Trans, Faraday Soc. 1971,67,1904. (b) Amand, B.; Bensasson, R. Chem. Phys. Lett. 1975, 34, 44.
c p,(t)s"
is a Gegenbauer polynomial of degree n.
(18) Turro, N. J.; Aikawa, M.
J. Am. Chem. SOC.1980, 102, 4866.
J. Phys. Chem. 1981, 85, 1856-1864
1856
The coefficients A , can be calculated from the boundary condition eq 19 m
F(s,o) = C sXp,(o)= em(s-l) x=o
(A41
Inserting eq A4 into eq A3 and taking the derivative with respect to s yields m
aF(s,O)/as = C A,, dC,,-lI2/ds = iiuenn(s-l) (A5) n=O
A, = 1 - 2n l:emsP,..l(s) ds 2 Expanding ems in a power series leads to
Using the parity properties of si and Pn-l,and formulas 8.14.15 and 6.1.18 of ref 20, one can obtain the following expression for An:
Sincelg
(P,(s) is the Legendre polynomial of degree n), one gets m
C A,P,-,(s)
n=l
= -fiuem(s-l)
(A61
+
Of
Legendre PolPomi-
ais19
+
m
Multiplying both sides of eq A6 by P,,-l(s)and integrating between -1 and +1 yields
Using the orthogonality property
+
where j = n, n 2, n 4, n 6, ..., and r is the gamma function (n = 1, 2, 3, ...). ( x ( t ) ) , the mean number of triplets per micelle a t time t , can be written as ( x ( t ) ) = [(dF/as)ls=i =
C xp,(t) x=l
(All)
and from eq 6, ( ~ ( 0 ) )= AU, one obtains finally c*(t)/co* = ( x ( t ) ) / ( x ( O ) ) =
l I ' , ( s ) P,,(s) ds = 0
m
for m # n = 2/(2n 1) for m = n
-(l/Au) C A,, exp(-'/,kn(n - 1)t) (A13)
+
n=l
one can write eq A7 as
Equation A13 is identical with eq 20 and eq A10 is identical with eq 21 when setting A , = -B,.
(19) Gradshteyn, I. S.; Ryzhik, I. M. "Table of Integrals, Series and Products", 4th ed.; Academic Press: New York, 1980.
(20) Abramowitz, M.; Stegun, I. A. "Handbook of Mathematical Functions"; Dover: New York, 1972.
147-nm Photolysis of Phosphine and Phosphine-d,' J. Blarejowskl and F. W. Lampe' Department of chemistry, The Pennsylvania State University, University Park, Pennsylvania 16802 (Received: December 26, 7980; In Final Form: March 6, 1981)
The 147-nm photodecompositions of PH3 and PD3 result in the formation of Hz(Dz)and PzH4(P2D4)in the gas phase and a solid deposit which is probably a mixture of phosphorous and polymeric phosphorous hydrides (PH), and (PD),. The quantum yields for the gaseous products and for depletion of reactants were measured and were found to increase with increasing pressure of phosphine and to decrease with increasing light intensity. All quantum yields for PD3 are lower than those for PH3. Isotope distribution analysis of the diphosphines formed in the photolysis of PH3-PD3 mixtures indicates that diphosphine is formed principally via combination of PH, (PD,) radicals. A mechanism involving these primary photodissociations is proposed which is in accord with the experimental facts. The thermal decomposition of PzH4at ambient temperature (-300 K) is shown to be second order with a specific reaction rate of (4.25 f 0.18) X cm3/(molecule s).
Introduction The photochemistry of phosphine has been the subject of many studies since the early years of this century: a d much information concerning both the Hg-photosensitized (1) US.Department of Energy Document No. DE-AS02-76ER0341617. (2) A. Smits and A. H. W. Aten, 2. Elektrochem., 16, 264 (1910).
0022-3654/81/2085-1856$01.25/0
and the direct photolysis of PH3 and PD3 has been published by Melville and co-workers3* during the period 1932-37. At that time some problems regarding the (3) H. W. Melville, Nature (London), 129, 546 (1932). (4) H. W. Melville, Proc. R. SOC.London, Ser. A, 138, 374 (1932). (5) H. W. Melville, Proc. R. Soc. London, Ser. A, 139, 541 (1933). (6) H. W. Melville, J. L. Bolland, and H. L. Roxburgh, Proc. R. Soc. London, Ser. A, 160, 406 (1937).
0 1981 American Chemical Society
