J. Phys. Chem. 1983, 87,1566-1571
1566
Studies on the Nature of Solute Distributions in Molecularly Doped Polystyrene. Triplet-Triplet Absorption of 1,2-Benzanthracene Excited by Nanosecond Laser Pulses R. D. Burkhart Depattment of Chemistty, Unlversity of Nevada, Reno, Nevada 89557 (Received: July 6, 1982; In Flnai Form: December 9, 1982)
The kinetics of the decay of 1,2-benzanthracene (BNZ) triplets produced by 337-nm pulses from a nitrogen laser have been studied at ambient temperature. The test molecule was studied in a polystyrene matrix and the decay was monitored by triplet-triplet absorption at 485 nm. In sufficiently dilute samples the results could be understood on the basis of concurrent first- and second-order reactions of triplets. The latter process is presumably responsible for the observed delayed fluorescencefrom these samples by triplet-triplet annihilation and the associated specific rate constants range from 5 X lo3 to 2 X lo4 M-' s-' depending upon solute concentration. At sufficiently short times following the excitation pulse and in sufficiently concentrated samples, the apparent rate constants for triplet-triplet annihilation decreased monotonically with time. It is proposed that this time dependence arises from a nonuniform distribution of solute molecules such that rates of energy migration are relatively large in concentrated regions giving rise to proportionately large rates of triplet-triplet annihilation. Thus, triplets residing in concentrated regions would have shorter lifetimes than those in more dilute regions so the average intermolecular distance between solute molecules in those regions occupied by triplets should increase with time. Model calculations based on the electron-exchange mechanism for triplet migration indicate that apparent rate constants for triplet-triplet annihilation are very sensitive to the intermolecular separation distance.
Introduction During the last several years, work in these laboratories on the triplet exciton migration in amorphous polymer films has been especially concerned with the interesting rigid solutions consisting of 1,2-benzanthracene (BNZ) in polystyrene.lS These rigid matrices which, to some extent, mimic the light-harvesting structures of the photosynthetic apparatus, exhibit triplet decay kinetics which have proved quite challenging to interpret in terms of a simple mechanism. Spectroscopic studies of the triplet luminescence from BNZ show that delayed fluorescence (410 nm), delayed excimer fluorescence (520 nm), and phosphorescence (605 nm) are all easily ~ b s e r v e d . It ~ has also been shown that both delayed fluorescence and delayed excimer fluorescence occur by triplet-triplet annihilation.* The best conditions for observing luminescence decays in these matrices involve the use of dilute samples and observation times which commence at relatively long times following the excitation pulse. For example, a 0.034 M sample yielded an excellent exponential phosphorescence decay over a t least three lifetimes. At ambient temperatures a phosphorescence lifetime of 225 f 11ms was found in such experiments. On the other hand, if the observation of the phosphorescence decay is begun only 2-3 ms after the excitation pulse, then a distinct nonexponentiality is observed which becomes more prominent as the solute concentration is raised. The comments just made about the phosphorescence decays also apply to delayed fluorescence decays. If anything, the onset of nonexponential behavior for delayed fluorescence is even more sensitive to solute concentration and delay time than is phosphorescence; however, it was found that a delayed fluorescence lifetime of 95 f 5 ms was consistently observed by using optimum conditions. It was decided to undertake a detailed investigation of these rigid matrices in an attempt to understand the (1) Burkhart, R. D. Chem. Phys. 1980, 46, 11. (2) Burkhart, R. D.; Abia, A. A. J. Phys. Chem. 1982, 86, 468. (3) Nickel, B. Chem. Phys. Lett. 1974, 27, 84.
0022-3654f 83f 2087- 1566$01.50f 0
reasons for the departure from ideal behavior of these luminescence decay signals when the solute and/or triplet concentrations are sufficiently large. As a starting point, one must recognize that triplets generated in the excitation pulse must, a t a minimum, disappear by a combination of first-order and second-order processes. Wyrsch and Labhart4 demonstrated that, in fluid solutions of 1,2-benzanthracene, the existence of these two decay routes could be used to evaluate the bimolecular rate constant for triplet-triplet annihilation. In the rigid solutions with which we are concerned, the specific rate constant for triplet-triplet annihilation may not be uniquely defined as it is for the fluid solutions, as was pointed out by Naqvi Thus, the probability that an annihimany years lation event will occur depends not only upon the proximity of the two triplets involved but also upon the rates of exciton migration which bring the two triplets within the required annihilation radius. A great deal of additional light was shed on the problem by the results of recent experiments on the time-dependent optical anisotropy of phosphorescence and delayed fluorescence from these matrices.2 When pulsed excitation from a polarized laser source was used, it was found that the polarizations of both of these luminescences increase in absolute value with time following the excitation pulse. The clear indication was that dipole randomization is occurring most rapidly in those regions of the sample where the triplet lifetimes are the smallest. Presumably these would be the most concentrated regions where the triplet exciton migration would be most rapid. These experiments also demonstrated that no translational migration of the solute species can be occurring on a time scale of a few seconds. It was concluded from these experiments that some of the complexity associated with the kinetics of the luminescence decays must be due to nonuniform solute distributions. In the present study, the method of triplet-triplet absorption has been applied to this problem by using 6-11s (4) Wyrsch, D.; Labhart, H. Chem. Phys. Lett. 1971, 8, 217. (5) Naqvi, K. R. Chem. Phys. Lett. 1968, 1, 497.
0 1983 American Chemical Society
Solute Distributions in Molecularly Doped Polystyrene
The Journal of Physical Chemistry, Vol. 87,No. 9, 1983
W I
citation was provided by a Lambda Physik excimer laser operated with nitrogen gas. This laser provides a 6-11s pulse at 337 nm which is sufficiently energetic to produce So S2transitions in 1,2-benzanthracene. The analyzing beam was first passed through a 1-mm slit at a wedge interference filter (Jena Glaswerk Schott and Gen., Mainz) set at 485 nm, before traversing the sample. The transmitted beam was then passed through the 1.25-mm slit of a Spex Model 1670 monochromator (5-nm band-pass) also set at 485 nm. The use of a double-monochromator system for the analyzing beam reduces the stray light intensity to a negligible value. Care also had to be taken to ensure that the shape and the size of the analyzing beam were such that it was completely contained within the area of the sample illuminated by the excitation beam. The radiant power of the analyzing beam was measured by an EM1 9789B photomultiplier in association with a Brandenburg Model 472R constant-voltage power supply. The photomultiplier output was monitored by a Nicolet Model 1072 instrument computer. The signal was collected in 256 channels of computer memory using dwell times per channel varying between 50 ps and 5 ms. A small portion of the pulsed laser beam was tapped off by a fiber optic light guide and sent to an auxiliary photomultiplier. The signal rise in this secondary photomultiplier was used to initiate a delay circuit which triggered the Nicolet after a preset delay time. The delay time was set equal to the dwell time in every case. Data Handling and Analysis. Since the analyzing beam was on continuously and since the start time for a measurement event was initiated (after a predetermined delay) by the laser pulse, it was easy to compare the intensity of the analyzing beam in the presence and in the absence of triplets simply by selectively blocking off the laser beam between the fiber optic tap and the sample. In most cases,32triplet decay events were collected in one quarter of the Nicolet memory, and then 32 additional events were recorded in a second quarter of memory but with the excitation beam blocked. A signal dc level adjustment was made prior to signal collection with both analyzing and excitation beams blocked. This assured that the dark condition gave a zero signal. If we let Io represent the analyzing beam intensity in the absence of triplets and I(t) represent this intensity as a function of time following an excitation event, then A(t) = log l o / I ( t )is the time-dependent absorbance due to triplet-triplet transitions and T(t) = A(t)/tl is the time-dependent triplet concentration. The quantities t and 1 are, of course, the molar absorptivity of triplets at 485 nm and the thickness of the absorbing layer, respectively. For each experiment, the value of Io was recorded and the 256 values of I(t) were transmitted to the University’s computer (a CDC Cyber 172-8) using punched tape. A locally developed computer program called ABSORB was used to convert the raw data into triplet concentrations. Determination of Sample Thickness. The thickness of these samples was determined by optical absorption as follows. First, a solution of 1,2-benzanthracene in toluene was prepared at a concentration of 0.0439 M. Using 0.1mm path length absorption cells manufactured by Aminco, we carefully measured the absorbance at a peak in the spectrum occurring at 3860 A. A Cary 14 UV-visible spectrophotometer was employed. It was decided that a rather concentrated solution should be used, close to that of the test samples, because of the small deviations in Beer’s law which may occur at the large concentrations employed. Having determined t for 1,2-benzanthracene at 3860 A (the actual value is 831.4 M-l cm-l), we deter-
-
‘w-[-~ signa I avera g e r
Figure 1. Overview of experimental apparatus used for triplet-triplet absorption. W is a tungsten lamp, M is a mirror, S is the sample, and F is a wedge interference filter.
excitation pulses from a nitrogen laser to produce the triplets. The molar absorptivity for 1,2-benzanthracene triplets has been determined at 485 nm by several different workers and is thought to be reliably known6 When this value is coupled with the optical path length of our samples and the measured absorbances, absolute values of triplet concentrations may thus be determined. Specific rate constants for both first- and second-order disappearance of triplets may then be determined. The investigations have been carried out with different solute concentrations and different time regimes following the excitation. Once again, we find that the essential results may be interpreted in terms of nonuniform solute distribution.
Experimental Section Chemicals. The 1,2-benzanthracenewas purchased from Aldrich Chemical Co. I t was recrystallized 3 times from ethanol and was then subjected to two resublimations under vacuum. Toluene was washed with concentrated H2S04and then with dilute base and water before being dried over anhydrous sodium sulfate and distilled. The polystyrene used was purchased from Scientific Polymer Products. I t was purified by repeated reprecipitations using benzene as solvent and methanol as nonsolvent. It was finally treated 3 times with boiling methanol. Sample Preparation. The samples used in this study consisted of thin polystyrene films (usually about 0.003 cm thick) sandwiched between optically flat quartz plates. They were made by dissolving 200 mg of polystyrene and from 1 to 20 mg of 1,2-benzanthracene in toluene. The solution was then purged with oxygen-free nitrogen in a glovebox for 1 h after which the solvent was evaporated off. Still operating in a nitrogen atmosphere, we transferred the molten polystyrene matrix to a preheated quartz plate where it was allowed to flow out into a smooth film. Then a cover plate was placed over the film to produce a sandwich and the sample was allowed to cool before being exposed to the atmosphere. The sample disks were placed in a copper sample holder and mounted in the spectrometer by a threaded connection at the end of a cryotip assembly. The cryotip was simply a convenient mounting device, all of the present experiments being carried out at room temperature. Spectroscopic Equipment. An overview of the spectroscopic apparatus is given in Figure 1. It will be noted that the sample is positioned at a 45O angle to both the laser excitation beam and to the analyzing beam. The two beams intersect at 90° at the center of the sample. Ex~~~~
(6) Labhart, H.; Heinzelmann, W. “Organic Molecular Photophysics”; Birks, J. B., Ed.; Wiley: New York, 1973; Vol. 1, p 297.
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Burkhart
The Journal of Physical Chemistry, Vol. 87, No. 9, 1983
1568
-1
-1
.g
20
r0
F H R Nr;”o” ~L
120
190
160
Figure 3. Graph of the natural logarithm of the concentration of 1,2-benzanthracenetriplets vs. time for a sample 0.345 M in solute. One channel equals 1 ms. TABLE I: Values of k Z a Calculated from Eq 4 for Three Samples Having Different Solute Concentrations and at Selected Times Following the Excitation Pulse mr( 0
b
-=.
c”-1
0.
\+\ :=, *’+
U 0 --I-1
\ ,
0.8 1.6 2.4 3.2
\\
*-+
3
\ \
P*, \
E.
+ *
**&*+. -2
\ , +**+++ * **
0’
\
++++*y;\ +.+a
-2
53
delay time, ms
20
E0
-
*+\ +*?
mean values a
0.127 M 2.4 1.1 1.6
1.5 1.7 ?: 0.4
10-4k, 0.260 M 3.2 3.8 3.7 4.2 3.7
0.345 M 5.2 6.6
i:
0.3
5.9 5.5 5.8
i:
0.5
Units of M-I s-’.
+
100
tion, then at least two decay processes must be considered for removal of triplets
T
mined the absorbances of the samples a t this same wavelength and applied Beer’s law to find the optical path length for each sample. An assumption in this method is that the t value is the same in toluene and in polystyrene. Toluene was chosen for this experiment because of its chemical similarity, as a monomeric compound, to polystyrene. The sample thicknesses determined in this way varied between 2.75 X and 4.26 X cm.
Experimental Results Figure 2a is a graph of the natural logarithm of the triplet concentration vs. channel number for the triplet decay occurring with a 0.127 M sample at 5 ms per channel. The solid line is the weighted least-squares best fit. It is clear that for nearly three lifetimes an excellent fit is obtained. Averaged over four such experiments the triplet lifetime was found to be 213 f 5 ms, which is in very good agreement with the 225-ms lifetime found earlier for the phosphorescence of 1,2-benzanthracene at ambient temperature in a polystyrene matrix. Figure 2b also is a graph of the natural logarithm of the triplet concentration vs. channel number but in this case a 0.345 M sample was used. Here a solid line is superimposed correspondingto the triplet lifetime found for dilute samples. The nonlinear nature of the experimental curve is quite apparent. Even more striking is a similar curve shown in Figure 3 for which a dwell time of 1 ms per channel was used. Clearly a single exponential relationship does not account for all of the data especially for the more concentrated samples. Since it has already been shown in earlier work1v4that delayed fluorescence arises from triplet-triplet annihila-
+ T 2 ‘S* + lS0
(2) In these equations T is intended to symbolize a molecule in the triplet state, IS*is the first excited singlet, and ‘So is the ground-state molecule. From the corresponding differential equation
-d[T]/dt = k,[T]
+ k2[TI2
(3)
it is clear that a single exponential decay will be found only if kl >> k,[T]. The solution of eq 3 may be written in the form kz = ~IK[TIo/[TI)exp(-k,t) - 1l/[Tl0ll - exp(-ht)l (4)
where [TI, is the triplet concentration at an arbitrarily selected time zero. Our first attempt to reconcile the nonexponential decays, therefore, makes use of eq 4. In Table I a summary is given of k2 values calculated from eq 4 using kl = 4.4 s-l and at selected times following the excitation pulse. These particular experiments were carried out with a dwell time of 50 ps per channel. Over the time span of 2.4 ms which is given here, there is no consistent change in the calculated values of k2 with time. For this reason, we have simply calculated an average k 2 for each sample. It is clear that, within the accuracy of the k2 values and in this concentration range, these rate constants are proportional to the solute concentration. The lack of any time dependence in the k2 values of Table I is, evidently, just a result of viewing a small time interval. If one examines a larger span of time, a definite time dependence of k2 values appears as is seen in Figure 4. Here the triplet-triplet absorption experiments were made with a dwell time of 0.2 ms and a total time period
Solute Distributions in Molecularly Doped Polystyrene
The Journal of Physical Chemistry, Vol. 87, No. 9, 1983
1569
Triplet exciton mobility has usually been discussed in terms of an electron-exchange mechanism wherein some finite overlap of neighboring molecular orbitals between donor and acceptor molecules is required. The theory was put into a form suitable for experimental applications by Inokuti and Hirayama,' who expressed the number, n,of exciton jumps per second in the form n = ~ 0 - exp[y(l l - 1/10)]
0
0
8
I
0
16
24 TIME ( M t e c )
32
40
46
Flgure 4. Graph of k p , in M-' s-' X vs. time following the excitation pulse. The symbol X Is used for the 0.345 M sample, open circles are for the 0.260 M samples, and closed circles for the 0.127 M sample.
of 48 ms was probed. The k, values are essentially time invariant from 16 to 48 ms for the 0.127 M sample and the average k2 value is 5 X lo3 M-' s-l. For the two more concentrated samples the apparent Itz values show a time dependence which becomes more obvious as the solute concentration increases and, similar to the results of Table I, the long-time values of k, increase with increasing solute concentration. It is interesting to compare the values of the quotient k,[T]/kl for the most dilute and most concentrated samples at 48 ms. The triplet concentration is actually smaller for the most concentrated sample at this time. Its value is 4.1 X lo4 M compared with 4.7 X M for the dilute sample. On the other hand, the order of the k2 values is just the reverse with the concentrated sample having a value of 1.9 X lo4 M-' s-l as opposed to 4 X lo3 M-' s-l for the most dilute. Thus, the quotient turns out to be 1.8 for the concentrated sample and 0.43 for the dilute one. Of course, at longer times when the triplet concentration has dropped even further, the second-order component, at least for the dilute sample, will have decreased in magnitude even further giving rise to the pseudo-first-order behavior observed in Figure 2a. When an attempt was made to calculate k2 values from eq 4 using an even larger time span, the results were very inconsistent and yielded wide fluctuations in apparent k2 values. At least a part of this problem is due to the fact that the quantity ([T],/[T]) exp(-k,t) is negligibly different from unity at long times so that the numerator of eq 4 fluctuates among positive, negative, and zero values according to the experimental uncertainties in the triplet concentrations. Thus, the calculations associated with eq 4 put rather stringent demands on the precision required for the triplet concentrations.
Discussion An important aspect of this investigation which sets it apart from solution or gas-phase studies is the fact that translational mobility of the reactant species is most certainly frozen out, at least in the relevant time scale of 1 s or less. The observed persistence of delayed luminescence polarization at long times2 is the best evidence of this restricted solute mobility. Thus, the bimolecular process of eq 2 has to be understood in terms of excitonic migration of the triplets.
(5)
where T~ is the triplet lifetime in the absence of quenchers, 1 is the intermolecular separation distance, and lo is an intermolecular separation distance such that the probability of triplet decay to the ground state equals the probability that the exciton will make a migratory jump. The quantity y is equal to 2Z0/Zb where lb is a dimension called the effective Bohr radius of the molecule and is related to the spatial extension of donor and acceptor molecular orbitals. Inokuti and Hirayama obtained values of 1.3 and 13 A, respectively, for lb and lo for the molecule 1-bromonaphthalene. Using an estimate based on molecular models, we would suggest 2.0 A as the lb value for 1,2-benzanthraceneand 16 A as an estimate for lo. For the most concentrated solution used in this study the solute concentration of 0.345 M leads to an average intermolecular separation distance of 9.4 A compared with 13.1 A for the most dilute solution.8 When these estimates are combined with the triplet lifetime of 1,2-benzanthracene, one obtains jump frequencies of 165 and 4.1 s-l for concentrated and dilute samples, respectively, using eq 5. For a random flight diffusion of excitons, the diffusion coefficient is related to the jump frequency by
D = 6n12
(6)
and yields values of 8.7 X and 4.2 X cm2s-l for concentrated and dilute samples, respectively. These diffusion coefficients may then be inserted into the Smoluchowski equation along with an estimated encounter radius to determine the specific rate constant for a triplet-triplet annihilation occurring on every encounter within this radius. BirksQhas discussed these processes in some detail and suggests 15 A as a reasonable encounter radius. Thus, using k = ~TR$NA/~C)OO
(7)
where R, is the encounter radius and NA is Avogadro's number, one finds values of 9.9 X lo3 and 4.8 X lo2 M-' s-l for the rate constants of the most concentrated and most dilute samples, respectively. Thus, the use of reasonable estimates for the relevant quantities yields a rate constant for the most concentrated sample which is within a factor of 2 of the experimentally observed rate constant obtained a t a long time (48 ms) after the excitation pulse. For the dilute sample, the same values of lb and lo yield a rate constant which is too small by 1 order of magnitude. In view of the way in which lb and lo were estimated, even a mismatch by a factor of 10 would not be too surprising when taken alone; however, such a result is difficult to reconcile with the rather good agreement between theory and experiment for the concentrated sample. That is, the internal consistency of these (7) Inokuti, M.; Hirayama, F. J. Chem. Phys. 1965,43, 1978. (8) This calculation is based on a model of randomly distributed molecules as presented by: Chandrasekhar, S. Rev. Mod. Phys. 1943,15, 1. We are grateful to a referee for calling thia reference to our attention. (9) Birks, J. B. 'Photophysics of Aromatic Molecules";Wiley: New York, 1970; p 390.
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The Journal of Physical Chemistry, Vol. 87, No. 9, 1983
results is unsatisfactory. Furthermore, the internal consistency is not significantly improved by adjusting lo and Ib within reasonable limits. It seems likely that neither the electron-exchangetheory nor the theory of encounter-controlledprocesses is basically inapplicable to the excitonic migrations being monitored in this work. Much more likely areas to explore in order to understand the results involve assumptions concerning the nature of these polymer matrices and, possibly, the degree to which the exciton migrations may be treated as random flights. Let us consider the latter of these two possibilities first and characterize in some detail the basic features expected of the exciton migration. Perhaps the most important characteristic is suggested by eq 5. That is, the probability of exciton migration increases as the intermolecular separation distance decreases. Thus, given an environment consisting of neighboring molecules at varying distances from the site of the exciton, the migratory step will occur to the closest molecule with greatest probability. This rather obvious statement has interesting consequences if one follows the course of an exciton’s path. Consider, for example, the case in which the transfer occurs from site A to a different site B to a different site C etc. and that in each instance the jump is to the closest molecule. The implication is that at each site there is a molecule closer to the current site of the exciton than to the site from which the exciton just migrated. But suppose this is not the case. That is, if the recipient of the exciton is closer to the donor molecule than to any other molecule in its immediate vicinity, then back-transfer will be the preferred migratory direction. Thus, neighboring molecular pairs for which no molecule is closer to either member of the pair than is its partner represent unique configurations in the molecular distribution which will tend to trap excitons and inhibit their migration. The term “secluded pair” is used to describe this special configurational arrangement. The efficiency of the trap will be determined by how secluded the pair is, that is, by the distance to the next nearest neighbor compared with the distance separating the two members of the pair. Experimental evidence supporting this model has already been presented.l It will be noted, however, and the existence of such traps would give rise to experimental diffusion rates which are less than would be predicted by the random flight model. Thus, if this trapping process were the major cause of the mismatch between theory and experiment, one would expect experimental rate constants to be less than theoretical ones when, in fact, just the reverse is true. The lack of internal consistency which was noted above suggests that the intermolecular separation distances calculated for a random distribution of solute molecules may be at fault. If the solute molecules formed clusters, for example, which are essentially isolated from each other, then the average intermolecular separation distances would be smaller than predicted by a random distribution. It was found to be instructive to insert into eq 7 the experimentally observed specific rate constants for triplet-triplet annihilation and, by combining with eq 5 and 6, obtain estimates for the intermolecular separation distance as a function of time following an excitation pulse. In carrying out these calculations we have set the encounter radius R, equal to the intermolecular separation distance. This seems to be a reasonable approach since, according to earlier work? anytime two excitons reside at adjacent sites, the annihilation process will occur if the separation distance is equal to or less than 15 8, and the average distances calculated here are well below this value for all times
TABLE 11: Calculations of Intermolecular Separation Distances Needed To Fit Observed Rate Constants for Triplet-Triplet Annihilation Assuming an Encounter-Controlled Process and Random Flight Triplet Migration time, ms
4 16 28 48
10-4k(exptl),a M-I s - l
4.8 3.0 2.0 1.9
10-3n(calcd), S“
5.0 2.0
1.o 0.9
I(calcd),
x
6.0 6.9 ’i.6 r -
1.1
a These rate constants are taken from the results of the 0.345 M sample shown in Figure 4.
involved in the calculation. Table I1 gives the results for selected times using the parameters 1, = 16 A and Ib = 2.0
A.
The major point which should be made concerning the results of Table I1 is that the long-time value of the calculated intermolecular distance is significantly smaller than the value of 9.4 A found by using random distributions. A similar calculation yields a value of 9.7 A for the intermolecular separation distance in the most dilute sample compared with 13.1 A for the value based on random distributions. Of course, the value of I ultimately found depends upon the choice of lo and lb For example, a larger value of 1, improves the agreement between experimental values of 1 and those obtained by assuming random distributions. However, 16 A is already a very large transfer distance for a triplet exciton. It happens that 17.1 and 18.4 A would be required to fit the data for concentrated and dilute samples, respectively, keeping Ib constant. These are distances on the order of eight or nine molecular radii and represent rather sizable dimensions for an electron exchange. Another question raised by the results of Table I1 concerns the time dependence of k2 and the resulting time dependence of the separation distances. The excitation pulses used in these studies were on the order of 7 mJ each and the optical path lengths were on the order of 0.003 cm. Thus, the number of photons traversing the sample at each pulse is the same order of magnitude as the number of molecules in the light path. Under these conditions the gradient of the triplet concentration along the light path will be small even for a large Beer’s law absorbance. It is for this reason that the major contribution to the time dependence of the k2 values is thought to arise from inhomogeneities in the solute distribution rather than to a gradient in triplet concentrations. The sort of inhomogeneity which may well be involved here has already been discussed in relation to earlier results of time-resolved delayed luminescence polarization. That is to say, the solute molecules seem to be present as clusters in which the solute concentration is much higher than the average. In this model, those triplets produced in regions of high solute concentration would tend to have lifetimes less than the average because of higher than average triplet-triplet annihilation rates. The triplet-triplet annihilation rates in these concentrated regions would be large not only because of the enhanced triplet concentrations but also because of enhanced rates of exciton migration resulting from the relatively small intermolecular separations there. Thus, the triplets remaining after long times following the excitation pulse would tend to be present in regions of ever increasing average intermolecular separation. This model not only provides an understanding for the present kinetic results but also explains earlier experiments which showed an increase in delayed
J. Phys. Chem. 1983, 87, 1571-1579
luminescence polarization with time following an excitation pulse.2 The observation of time-dependent rate constants for bimolecular processes does not, of course, mean that a nonhomogeneous distribution of solute molecules is necessarily present. Starting with an equilibrium distribution of molecules it has been shown that k2 would have a t-’/2 dependence in the very early stages of the reaction depending on the magnitude of the diffusion coefficient.1° Such transient terms in bimolecular rate constants are well-known. It should also be pointed out that an earlier detailed study of the Inokuti-Hirayama equation also uncovered some difficulties in applying the equation over a large time regime.” These results also indicated that n(R) was larger a t small times than would be predicted from values of 2lO/lbfound at long time. That is, the sense of the incompatibility with the Inokuti-Hirayama equation was the same as that observed in the present study. It was proposed that the rate of short-range triplet migration is (10)Noyes, R. M. B o g . React. Kinet. 1961,1, 129. (11)Yamamoto, K.; Takemura, T.; Baba, H. J. Lumin. 1977,15,445.
1571
underestimated by the Inokuti-Hirayama equation and a correction theory was presented which gave a better fit. Thus, the effects which have been observed in this work and ascribed to nonhomogeneous solute distribution might instead be due to a breakdown of the migration theory for very short triplet transfer distances. However, since our model of solute clusters is also consistent with the earlier results of time-dependent delayed luminescence polarization and the luminescence decay kinetics, it seems to us to be a preferred explanation for the results a t this time. In fact, computer modeling of various types of solute distributions within such clusters is currently underway in these laboratories in an attempt to simultaneously satisfy the time dependence of k2 values and the time dependence of the delayed luminescence polarizations.
Acknowledgment. The laser used in this work was obtained on loan from the San Francisco Laser Center supported by NSF Grant No. CHE79-16250 to the University of California, Berkeley, and to Stanford University. Registry No. 1,2-Benzanthracene,56-55-3;polystyrene, 9003-53-6.
Can Intrinsic Barriers Be Obtained from Curvature Measurements? Joseph R. Murdoch Deparlment of Chemistry and Blochemistty, University of California, Los Angeles, California 90024 (Received: Ju& 14, 1982)
In the past several years, the intrinsic barrier concept has assumed a central spot in efforts to determine how the barrier of a reaction is dependent on AGO for the reaction and to understand the factors contributing to differences in the barrier height for reactions with the same AGO. The fact that the intrinsic barrier associated with a series of related reactions can be empirically related to the curvature seen in a plot of AG* vs. AGO has prompted Saunders to call attention to an important problem: there are different choices for the relationships relating the intrinsic barrier to observed curvature (Marcus equation, Marcus BEBO equation, Rehm-Weller equation, etc.), and Saunders has reported that the derived intrinsic barriers are highly dependent on the choice of model. In fact, using models based on intersecting Morse functions, Saunders has shown that the curvature resulting from these models leads, through the Marcus equation,to intrinsic barriers which are too low by factors as high as 10. Consequently, Saunders has suggested that the Marcus equation may underestimate intrinsic barriers derived from curvature measurements and that this “flaw”in the Marcus equation may be responsible for at least some of the anomalously low intrinsic barriers reported in the past. The curvatures obtained by Saunders could be termed “local” curvatures since they are closely related to the value of the second derivative, (a2AG*/dAG02)AC*o, at a specific value of AGO. Saunders’ conclusions regarding the model dependence of the intrinsic barriers is confirmed for other empirical models when “local”curvaturesare used to obtain the intrinsic barriers. However, it is also possible to define “global”measures of curvature which incorporate information over large spans of AGO. It is shown in the present paper that “global” measures of curvature lead to intrinsic barriers which are not highly dependent on the model, and it is found that the intrinsic barriers derived from “global” curvature measurements are in agreement to 15% or better for two models based on intersectingMorse functions, the Rehm-Weller equation,the Marcus BEBO equation,an arctan equation, and the Marcus equation.
I. Introduction A . The Intrinsic Barrier. The intrinsic barrier of a reaction is rapidly becoming a focal point in efforts to determine how changes in the overall free energy of a reaction propagate into the transition state and affect the barrier to the reaction.1-6 The idea of an intrinsic barrier (1) (a) R. A. Marcus, J. Chem. Phys., 24, 966 (1956); (b) J. Phys. Chem., 72,891(1968);(c) A. 0.Cohen and R. A. Marcus, ibid., 72,4249 (1968). 0022-365418312087-1571$01.50/0
originated from Marcus’ equationla A G * = AGO*
+ ‘/,AGO + (AGo)2/16AGo*
(1)
where AGO*is the intrinsic barrier and AGO is the free (2)(a) M. M. Kreevoy and D. E. Konasewich, Adu. Chem. Phys., 21, 241 (1971); (b) M. M. Kreevoy and Sea-Wha Oh, J. Am. Chem. SOC.,95, 4805 (1973);(c) A. I. Hassid, M. M. Kreevoy, and T. M. Laing, Faraday Symp. Chem. SOC., 10,69(1975);(d) W.J. Albery and M. M. Kreevoy, Adu. Phys. Org. Chem., 16, 87 (1978); (e) M. M. Kreevoy, Faraday Discussion No. 74 on Proton and Electron Transfer Reactions, Southampton, England, Sept 1982.
0 1983 American Chemical Society
