J . Phys. Chem. 1986, 90, 1160-1 168
1160
ponents: a rotational-translational one (MMI), a vibrational excitation one (EXC), and zero-point energy one (ZPE). The first is independent of temperature, whereas the vibrational terms are very sensitive to T. The relative EXC and ZPE contributions to H R R are dependent on the temperature, but for many cases Z P E contribution dominates. The H R R term does not have the same effect as the ZPE one when, for a bimolecular reaction, the new modes and reactant contributions contribute oppositely. This occurs when the reactant substituted has a global relaxation of force constant. In this case the ZPE contribution associated with reactants tends toward a normal K I E and the new modes contribution tends always toward an inverse KIE. On the other hand the primary KIE’s are very dependent on the temperature because the important value of the ZPE term. In a similar form to the above conclusions, the comparable interand intramolecular additions can have different KIE’s because in the intermolecular reaction there are no new vibrations generated in the transition state. So, when the substitution in the
free radical additions leads to parallel contributions for the new modes and the reactant modes, that is, when the substituted center suffers a global increasing of the force constants in the path from the reactant to the transition state, an equivalent result will be obtained for the similar intramolecular addition. An example of this is in the substitutions in the radical center. Nevertheless, if the substitutions are in the double bond a noticeable contribution associated to the reactant modes in the free methyl radical addition to olefin does not exist, and a very small inverse KIE is obtained even at low temperatures, whereas for the pent-4-enyl radical cyclization a considerable normal effect associated to reactant is found.
Acknowledgment. W e thank to Dr. S. Nagase for the communication of unpublished results. Computer time was provided by the C.P.D. of the Ministerio de Educacidn y Ciencia and made available through the terminal of the Centre de C5lcul of the Universitat PolitEcnica de Catalunya.
Time-Resolved Emission Spectra, Decay-Associated Spectra, and Species-Associated Spectra+ Jan-Erik Lofroth Department of Physical Chemistry, Chalmers University of Technology and University of Goteborg, S-412 96 Goteborg, Sweden (Received: March 12, 1985; In Final Form: August 9, 1985)
An approach to derive the steady-state fluorescence spectra of species in excited-state reactions is described. Such spectra we call species-associated spectra, SAS. Two algorithms are given to derive SAS: a simple approach, which might be used
in many studies, and a general approach to be used when the overlap of the spectra of the individual excited species is annoying. Experimentally, single-photon-counting data were globally deconvoluted to generate time-resolved emission spectra and decay associated spectra, from which the species associated spectra were obtained. SAS were calculated for 1 ,?;-his( 10-acetoxy9-anthry1)ethane (DADAE) in toluene at 21.0 O C . It is shown that the photophysics of DADAE can be described by a three-state excited-state reaction involving intramolecular excimer formation and also that the ground state shows heterogeneity. Also, studies of a mixture of POPOP, anthracene, and diphenylanthracene in ethanol (ground-state heterogeneity) have been included, as well as a study of @-naphtholin water at pH 3.0. Computer-simulated two-state and three-state excited-state reactions have been included for comparison and for critical tests of the proposed algorithms.
Introduction Fluorescence steady-state spectroscopy to describe excited-state reactions is often limited in its description due to more or less pronounced overlap between the spectra of the individual species in the studied system. One alternative to a steady-state spectrum is a time-resolved emission spectrum, TRES. In such spectra the intensities at different wavelengths are presented as functions of time, but they may serve only as a qualitative description of the total fluorescence and its development in time. (Different approaches to record TRES has been compared by Meech et a].’) The spectra describing each individual species have, however, seldomly been presented. When this has been done, ground-state heterogeneity or the case with slow excited-state reactions generally has been assumed. In an important paper concerning these questions, Knutson et al. introduced the concept of decay-associated spectra, DAS.* Briefly, the DAS for a decay constant ki may be defined by
where f ( X , t ) is the deconvolved total fluorescence intensity at wavelength X a t time t after &pulse excitation andf;(A,t) is the + A preliminary account of this work was presented at the British Biophysical Discussion Meeting, December 20, 1984, London.
0022-3654/86/2090-1160$01.50/0
intensity of species “i”. It is seen that the different excited-state spectra may be obtained as
provided that the studied system is of the ground-state heterogeneity type or if the excited-state reactions are slow. Knutson et al. showed one procedure with which the DAS for the different rate constants could be calculated from TRES obtained with the single-photon-counting technique. Also, the application to a fast two-state excited-state reaction was discussed. However, examples of the applications of these ideas to fast excited-state reactions have so far not yet been shown, and the steady-state spectra of the participating species that can be obtained have not been derived. It must be emphasized that for a fast multiple-excited-state reaction, the observed decay constants cannot directly be attributed to the fluorescence lifetimes of the ( 1 ) S . R. Meech, D. V. O’Connor, A. J. Roberts, and D. Philips, Phorochem. Photobiol., 33, 159 (1981). (2) J. R. Knutson, D. G. Walbridge, and L. Brand, Biochemistry, 21,4671 ( I 982).
@ 1986 American Chemical Society
Species-Associated Spectra individual species, as is the case for ground-state heterogeneity. Equation 2 is then not applicable. In current work on substituted dianthrylethanes we have found that the excited-state reactions at room temperature of the diacetoxy compound in the solvents so far utilized must be described with at least three excited states. Since the emission spectra of these states strongly overlap, it was necessary to develop data analysis algorithms, by which the individual spectra could be calculated. The complexity of these systems also required that the algorithms were tested on well-characterized reactions as well as on computer-simulated reactions. Thus, in this work excited-state reactions involving two and three species have been studied. For comparison a study of a known mixture of three chromophores (ground-state heterogeneity) has been included. Single-photon-counting data were deconvoluted with the global-reference method (global deconvolutions of data with quenched references) which recently was pre~ented.~From observed decays, sampled typically every fifth to tenth nanometer, DAS were generated. From the DAS the individual spectra of the excited species were then calculated. Such calculated spectra we call species-associated spectra, SAS. Since the fluorescence intensity at any wavelength of a SAS decays with the same characteristics, the different S A S can be illustrated either as steady-state spectra of the individual species, or as fluorescence intensity decays at any wavelength of each SAS. The method to derive SAS has been described in detail in this paper with the well-characterized @-naphthol/H20system as an e ~ a m p l e .Extensions ~ to three-state excited-state reactions have then been exemplified with computer simulations and with one study of 1,2-bis(IO-acetoxy-9-anthryl)ethane,DADAE, in toluene. However, the calculated SAS for these systems were possible to derive with a simplified, not general, approach. It was based on the fact that it was experimentally possible to record fluorescence intensity decays at wavelengths where the intensities originated from only one of the species. This is not always possible and has, for example, been verified for DADAE in acetonitrile. Therefore, a general algorithm has been derived and has been illustrated with a computer-simulated two-state excited-state reaction. The usefulness of deriving species-associated spectra was clearly revealed by this work.
Experimental Section Materials. PPO (2,5-diphenyloxazole, Fluka puriss.), POPOP (1 ,4-bis(5-phenyl-2-oxazolyl)benzene,Fluka puriss.), DIMPOPOP (1,4-bis(4-methyl-5-phenyl-2-0xazolyl)benzene,Fluka puriss.), N a I (Merck p.a.), anthracene (Merck p.a.), DPA (9,lO-diphenylanthracene, ICN), and @-naphthol(Fischer Scientific) were used as supplied. DADAE (1,2-bis( 10-acetoxy-9-anthry1)ethane) was synthesized as described e l ~ e w h e r e . ~Appropriate known volumes of stock solutions of POPOP, anthracene, and DPA in ethanol (99.5%) were mixed to give solutions with absorbances around A,,, = 0.1 at 356 nm. NaI-quenched ethanol solutions of PPO and DIMPOPOP (lifetimes around 0.25 ns) were prepared as described b e f ~ r e .These ~ two solutions were used as references in the global deconvolutions. @-naphtholin water (millipore) was adjusted to pH 3.0with HCl(aq) to give an absorbance of A],, = 0.4 at 313 nm. DADAE was dissolved in toluene (BDH, Aristar) which first had been destilled over Na(s) to remove water. The absorbance was A,,, = 0.2 at 356 nm. All measurements were carried out at 21.0 OC. Only the DADAE solution was deoxygenated (N2(g) bubbling for 5 min with an N,(g) atmosphere around the cuvette during the measurements). Time-Resolved and Steady-State Fluorescence Studies. Time-resolved single-photon-counting data were recorded with (3) (a) J.-E. Lofroth, paper presented at NATO AS1 on Excited State Probes in Biochemistry and Biology Acireale, Italy, Sept, 1984. (b) J.-E. Lofroth, Eur. Eiophys. J . , 13, 45 (1985). (4) (a) A. Weller, Z . Elektrochem., 56, 662 (1952); (b) M. Almgren, Chem. Scr., 6, 193 (1974). (c) W. R. Laws and L. Brand, J . Phys. Chem., 83, 795 (1979); (d) J. van Stam and J.-E. Lofroth, J . Chem. Educ., in press. (5) H.-D. Becker, T. Elebring, and K. Sandros, J . Org. Chem., 47, 1064 ( 1982).
The Journal of Physical Chemistry, Vol. 90, No. 6, 1986 1161 a time division of 0.345 ns channel-' with an instrument described in detail elsewhere.6 The lamp was an Edinburgh Instruments Ltd. flashlamp, Model 199F, filled with N2(g) at 0.5 atm, a gap of 1 mm, and run at 30 kHz at 7.0 kV. The excitation and emission wavelengths were selected by monochromators (Jobin Yvon HlOUV and HlOVIS) with 16- and 8-nm fwhm, respectively. Polarizers were set at 0' and 55' to the vertical before and after the thermostated sample holder. The emissions from the samples and the references were concurrently recorded at 500 Hz, and sampled to lo3 or lo4 counts at the peaks (samples) and 5 X lo3 or 2 X lo4counts (references). (In the following discussion these decays will be referred to as lo3 decays, etc.) The photomultiplier was a Hamamatsu R928 with a modified base. The electronics of the rest of the instrument were standard Ortec modules, while the multichannel analyzer was a ND8 12, interfaced to a Texas Silent terminal. The terminal was connected to GDC (Goteborgs Computer Centre) where the deconvolutions were run on an IBM 3081D. The steady-state spectra were recorded on an Aminco S P F 500 corrected spectra version, with the emission bandwidth set at 8-nm fwhm, the same as in the time-resolved measurements. Computer-Simulated Data. A steady-state spectrum for an excited species was generated as the sum of appropriate Gaussian functions. The sum of such spectra gave the total, observable fluorescence spectrum of the system. In the case when the general algorithm was tested, the individual spectra were first quenched according to the equations given in the Results section. Decays of the concentration of a species X*, [X*](t)/[X*](O), were simulated by assuming realistic values for the rate constants in the actual reaction scheme. The choice of the rate constants was dictated by reported values for similar systems or by results obtained in this laboratory in other studies. From the individual concentration decays and the individual steady-state spectra, the fluorescence intensity decays were obtained by convolution with an excitation lamp profile obtained with a scattering solution. The intensity decays were calculated at time intervals of 0.043 ns channel-' at every nanometer over the spectra. The total fluorescence intensity decays, f(X,t), at different wavelengths were then obtained as sums of individual decays and normalized to lo3 or lo4 counts at their peaks. Poissonian noise was finally simulated by adding G(X,t)lf(X,t)1/2, where G(X,t) were chosen at random with a number generator from a Gaussian distribution with a mean of zero and a standard deviation of one. Reference decays were simulated by convolution of exp(-t/0.25 ns) with the scattered data, normalizing to 2 X lo4 or 5 X lo3 counts at the peaks and adding Poissonian noise. Data Analysis. The total fluorescence intensity decays at different wavelengths were deconvoluted with the global-reference m e t h ~ d .Thus, ~ deconvolved decays did not suffer from problems like time-dependent color shifts etc. The deconvolution program was based on a modified Levenberg-Marquardt algorithm which did not need any explicitly calculated derivatives.' Typically four to five decays at different wavelengths with lo4 counts at their peaks were first deconvoluted globally with 2 X lo4 reference decays by assuming one, two, three, or four exponentials. During the analysis the reference lifetime was also estimated and the value compared to what was expected from the known concentration of N a I and from separate experiments of unquenched reference solutions. The chosen references have been shown to have predictable quenched lifetimes that are independent of ~ a v e l e n g t h . ~ Only every eighth channel of the decays was used when the computer-simulated data were analyzed, giving a time division of 0.344 ns channel-', comparable to the real experiments. In the final deconvolutions the lo3 decays sampled typically every fifth to tenth nanometer were deconvoluted with 5 X lo3 references together with the lo4 decays and the 2 X lo4 references. During the first part of the analysis, the decay constants and the reference lifetime were held constant at the values found in the (6) J.-E. Lofroth, Ph.D. Thesis, University of Goteborg, 1982. (7) International Mathematical and Statistical Libraries, Inc., Houston TX, routine ZXSSQ.
Lofroth
1162 The Journal of Physical Chemistry, Vol. 90, No. 6, 1986
'O0l
\
1A
ANTHRACENE
z
0
w
W
a
!
l
l
l
400
!
l
!
l
l
l
l
l
I
!
I
l
500
WAVfLENGTH/nm
--
"."
LOO
l
500
VIAVCiENCTHinm
l
l
l
400
l
l
l
l
l
l
l
WAVELENGTHim
~
1
1
1
500
Figure 1. Results from a study of a mixture of three chromophores. Global deconvolution of 22 wavelengths. lo4 decays sampled at 400, 425, 450, and 500 nm. ( A ) Relative contribution to the total fluorescence intensity. (B-D)Species-associated spectra (filled circles) compared to individually recorded steady-state spectra (full drawn curves).
first part of the analysis. This gave reliable starting values for the preexponential terms for the rest of the analysis, when nothing was held constant. The best fit to the data was assumed when CR(K)*had been minimized. R ( K ) was defined as
The deconvolved decays at each wavelength were presented as YM(X,t) = Cni(X)exp(-wit), where xlni(X)I = 1. The DAS(X,wi) were then calculated as (4)
(3)
+
with I = 1, 2, ..., N, J = 1, 2, ..., L; K = ( J - 1)N I . In the present studies, L equaled the number of deconvoluted decays, and N was the number of channels fitted at the wavelengths. The fitting ranges were chosen to start and stop in channels containing 50 to 100 counts. Thus in a typical global deconvolution N was of the order of 200, while L varied between 20 and 40. This mapping of the residuals R ( K ) into one vector was a somewhat modified version of the global analysis technique as presented by Knutson et a].* and was most convenient to use. The algorithm has been shown to converge to the same minimum for starting values far away from the true values of the estimate^.^ Typical CPU time (central processor unit time) in the IBM 3081D for N = 200, L = 36, three exponentials, and one reference lifetime was 3-4 min and required approximately 5.5-Mbyte memory. Once the minimum in CR(f12was found, the reduced x 2 values (xu2),z values from test runs, weighted residual plots, and estimated reference lifetimes were used to judge the fit. The tests were always done on the residuals for each individual decay. Standard deviations of the estimates were calculated from the curvature of xu2in the region of the best fit.
(8) J. R. Knutson, J . Beechem, and L. Brand, Chem. Phys. Lett., 102, 501 ( 1983).
where P ( X ) was the observed total steady-state fluorescence intensity. With this normalization of$bsd(X,t)the deconvolved as required. decays fullfil Jf(A,t) dt = CDAS(X,w,)/w, = FS(X),
Results Ground-State Heterogeneity. It has generally been argued that a very high number of counts are necessary in the recorded decays to accurately deconvolute single-photon-counting data. According to our earlier opinion, lo4 counts a t the peaks should be the minimum to deconvolute decays described by single exponentials, although this number of counts sometimes had been sufficient for double or even triple exponential recovery.' With the globalreference technique it was verified that IO4 counts were enough for triple exponentials, if, e.g., experiments at different wavelengths were a n a l y ~ e d .However, ~ it has been suggested'O that, with the global analysis technique, even a smaller number of counts could be enough also for three-exponential deconvolutions. This would be an advantage when generating TRES with a flashlamp. Thus, as a first study, a mixture of POPOP, anthracene, and DPA was investigated. The contribution to the total fluorescence, Figure (9) (a) D. Biddle and L. L. Chapoy, Mucromo[ecules, 17, 1751 (1984); (b) J. Baudier, J. Tyrzyk, J.-E. Lofroth, and P. Lianos, Biochem. Eiophys. Res. Commun., 123, 959 (1984). (10) L. Brand, paper presented at NATO AS1 on Excited State Probes in Biochemistry and Biology, Acireale, Italy, Sept, 1984.
The Journal of Physical Chemistry, Vol. 90, No. 6, 1986 1163
Species-Associated Spectra SCHEME I A*
-
ha
[A*](O)k3
aB*(x)
w1 - w2
B"
WI
- w2
(10)
1** A -
(o)(kA - w2) -
DAS(X,w,) = 43
DAS( X,WJ =
aA*(X)[A*l(o)(wlWI
-
kA)
+
aB*(X)[A*l(o)k3 w1 -
w2
w2
0
Finally, the species-associated spectra, defined by
lA, from each species was calculated as ujJCuf;whereA is the fluorescence steady-state spectrum and ui is the volume of stock solution "i" used to prepare the mixture. Recorded lo4 and lo3 decays were globally deconvoluted with data from a quenched DIMPOPOP (expected lifetime = 0.28 ns) to give ni(X) and w,, i = 1 , 2, 3. The DAS for each component was then determined as described by eq 4. For the ground-state heterogeneity case, eq 2 was applicable and the calculated SAS are shown in Figure 1, B, C, and D, together with the observed spectra. For clearness, the individual recorded spectra have been normalized to unity heights and displayed in separate parts of the figure. The SAS have been normalized with the same normalization constants as their steady-state spectra. The lifetimes recovered in the global analysis agreed within their standard deviations with the individually determined lifetimes (POPOP 1.25 ns, anthracene 4.27 ns, and DPA 6.21 ns). It is seen that even with the low contribution of anthracene fluorescence above 430 nm, the calculated SAS indicates the presence of the component. Clearly, it is the total number of counts of all analyzed decays that determines how accurate and reliable the deconvolutions of lo3 decays will be. However, for the intention of the rest of this work, the accuracy obtained in this ground-state heterogeneity study was accepted as a measure of what can be obtained with lo3 decays. Two-State Excited-State Reactions. The general two-state excited-state reaction to be discussed in this paper is described in Scheme I (the case when B is also directly excited is commented on in the last section and has recently been discussed in detail together with applications elsewherell): k l and k2 are rate constants that include deactivations by fluorescence and nonradiative transitions, while k3 and k4 may include, e.g., a p H dependence (for &naphthol we have k4 = k,'[H+]). Formally, we may write the fluorescence intensity of a species X* as
fxt0\4 = a x 0 ) [X*I(t)
(5)
It is seen that Jax*(A) dX = kRkI,wherefm is the rate constant for fluorescence and k, is an instrument constant. Thus,fx.(X,t) should be regarded as the observed intensity of species X*, if it could have been recorded exclusively. The integrated forms of the rate equations of Scheme I, assuming 6-pulse excitation and [B*](O) = 0, are then given by
+ k3, kB = k2 + k,, and w1 and w2 are given by ((kA - k ~ ) '+ 4k3k4)'" k A + kB = ___ + (8) 2 2
where kA = k, WI
kA w2=--
+ kB 2
((kA - kg)2 + 4k3k4)'I2 2
(9)
For the decay-associated spectra of wI and w2 we have in this case
are given by
as it should be if the total intensity decays are described by eq 1. Also, eq 6 and 7 can be written in terms of SAS:
W.W.
These equations have later been used to simulate the experiments testing the general algorithm. SAS,.(h) can be calculated if kB is known. Since w I + w2 = kA kB, SASB*(X) can also be calculated, either directly with eq 15 or by difference as P ( h ) - SASA*(X).For later discussions it is convenient to define an apparent kB:
+
The definition is obtained from eq 14 by replacing SASA*(X)with F S ( h ) .It is seen that kBaPP(X)= kB when P ( X ) = SASA*(X). Thus from a plot of kBaPP(X)vs. A it might be possible to estimate kB. (Replacing SASB*(X)with P(X)in eq 15 will for this model only give kAapp(X)= w I + w2, independent of the wavelength.) Figure 2 shows the results obtained for /3-naphthol at pH 3.0, the deconvolutions carried out with quenched PPO (estimated lifetime = 0.24 ns). At this pH, the excited-state reaction was reversible (pK* = 2.8), while only A was directly excited (pK = 9.6).4 The reciprocal of the rate constants w1 and w2 were estimated to 4.26 f 0.08 and 8.36 f 0.06 ns, compared to 4.05 and 8.8 ns reported earlier in a study at different pH.4C The DAS(X,wi), Figure 2A, clearly reveals that above 390 nm, the preexponential term of w I infB*(h,t) dominated over the wl term in fA.(h,t), leading to a negative DAS(X,w,), eq 10. If the studied reaction not was disturbed during the experiments, e.g., by quenching of oxygen, this means that the fluorescence intensity from B* was higher than that from A* at these wavelengths. The plot of kBaPP(X),Figure 2B, also shows that the two emissions were well separated and the extrapolated value of kB was determined to 0.150 ns-I, giving kA = 0.204 ns-' (= w1 w2 - kB). Values of kA and kB were reported by Laws and Brand4cas 0.208 and 0.106 47[Hf] ns-I. At pH 3.0, kB should then have been 0.153
+
+
(1 I ) J.-E. Lofroth, Anal. Instrum. ( N . Y . ) ,in press.
ns-'.
The Journal of Physical Chemistry, Vol. 90, No. 6, 1986
1164
Lofroth SCHEME I1 A*
-
k4
-
B*
-
'6
C*
I C
A-B
1
Y
It is noteworthy that the 0.150-ns-' extrapolated value of k B in the present report also was the highest possible value. A higher value had given SASA*(X)> P(X), eq 14 and 19. Also, the lowest mathematically possible value of kB (kB > w2 = 1/8.36 ns-I) was clearly unphysical, giving a highly improbable and unrealistic structured SASB*(X) for this model. The situation is illustrated in Figure 2C, where the SAS for the two extremes have been displayed. Since SAS, as well as DAS, are quantities that are calculated with a proposed model, one should indeed expect that a wrong model should be reflected in an unphysical behavior of the SAS. The same should apply if the rate constants were estimated to false values. Thus, ifa value of kB = 0.120 ns-I (=w2) had been obtained in the extrapolation of kBaPP(X),the SAS,*(X) (open squares in Figure 2C) had indicated that wrong model most probably had been assumed in the deconvolutions. In the present study, the behavior of SASB*(X) for k , = 0.150 ns-I supported the estimated value. The agreement between S A S obtained in this study and steady-state spectra extrapolated from spectra obtained at different pH values was excellent.M The small hump in SAS,*(X) around 355 nm was also found in'this other study and was attributed to Raman scatter. The possibility of using decay-associated spectra to remove or isolate the Raman scatter was discussed by Knutson et Three-State Excited-State Reactions. During the past few years, the photophysics and photochemistry of bis(anthry1)alkanes have been the subject of several studies.I2 We have recently complemented steady-state fluorescence intensity studies with time-resolved measurements on substituted bis(anthry1)ethanes. A full account of these studies will be presented elsewhere. In the present report results are presented only from one such study of the DADAE in toluene system at room temperature. For reasons which will be discussed later, reaction Scheme I1 was assigned. Moreover, as a first approximation it was assumed that only A was directly excited and that k5
t z W k
z W 0
z
W 0 (I)
w
a
0 3 U A
0
5
TIM" ,E
s
10
15
Figure 4. Results from a study of DADAE in toluene at 21.0 O C . Global deconvolutionof 34 wavelengths. lo4 decays sampled at 400, 420, 450, and 500 nm. (A) Calculated decay-associated spectra (symbols). The curves were drawn to smoothly join the calculated points. (B) Plot of kCaPP(X)(open triangles) calculated according to eq 23. The full drawn line (B) corresponds to w2 + wj. (C) Species-associated spectra calculated with kc = 0.256 ns-I. The full drawn curve is the observed total steady-state spectrum. (D) Decays of the peaks of the SAS of Figure 4C.
seen that DAS(X,w,) was negative even at very long wavelengths, and that DAS(X,w2) was negative only at the very red end of the spectrum. With an excitation wavelength of 420 nm (suggested excitation of B), DAS(X,w,) was negative already at quite short wavelengths (results not shown). Also, estimated rate constants agreed with this choice of excitation with those obtained with excitation at 356 nm. The support for Scheme I1 with the suggested approximations thus seemed to be strong. However, some difficulties arose when kc was estimated from kCaPP(X), Figure 4B. Only at the very blue end of the emission spectrum a reliable kc = 0.256 ns-' seemed to be obtained. These problems were even more accentuated for DADAE in acetonitrile, for which a plot of the apparent kc was essentially constant over the whole emission spectrum. If a kc = 0.256 ns-' for DADAE in toluene was accepted, the SASB*(X) and SASc.(X) could be constructed, Figure 4C. Finally, the decays of the peaks of the different species-associated spectra are shown in Figure 4D. The decays were calculated from equations derived in the same way as eq 17 and 18 for the two-state excited-state reaction. However, as discussed in the Introduction, any wavelength of the spectra
could have been chosen for these plots. The only difference would have been the multiplication of the intensities with three different constants, independent of time. The typical anthracene character of the locally excited state A* was indicated in SASA+). When the temperature was lower, the contribution from A* was more pronounced, which explained the structured emission indeed observed in the total steady-state spectrum at low temperatures. For example, at -93 OC, SASA*()\) dominated at all wavelengths, with e.g., sAs,.(426)/sAsB*(426) = 18.2, SAS,.(490)/SASB.(490) = 3.8, and SAS,.(X) = 0 at all wavelengths. The dominating species in the present study at 21.0 "C was B*, giving rise to only a slightly structured total emission. However, the signs of structure in the emission of C* was not in accord with C* being an excimer. It might be argued that the precision in these measurements was not better and that the results had to be accepted. On the other hand, the precision which is obtainable with IO3 decays was high enough to reveal the anthracene character of A* in the DADAE study as well as the anthracene in the ground-state heterogeneity study. The structure of SASc.(X) could not be removed by assuming higher
The Journal of Physical Chemistry, Vol. 90, No. 6, 1986 1167
Species-Associated Spectra
TABLE 11: Values of Rate Constants of the Simulated Two-State Excited-State Reaction kqA = 3 M-I nsd, kqB = 5 M-' ns-I 0.010 0.000 0.020 0.040 [QI/M 0.260 0.200 0.230 kA/ns-l, inputo 0.320 0.200 0.250 0.350 0.150 kB/ns-I, inputa
/ ns-' , outputb X = 425 nm X = 450 nm X = 482 nm
0.080
0.060 0.380 0.450
0.440 0.550
0.100 0.550 0.650
0.750 0.819 1.474 2.02 2.02 f 0.02 2.98 2.98 f 0.01 0.480 0.482 0.197 0.150 1.505 0.206 0.143
0.865 0.940 1.622 1.71 1.65 f 0.02 2.47 2.50 f 0.01 0.640 0.656 0.169 0.106 1.490 0.200 0.149
0.977 1.050 1.747 1.47 1.41 i 0.02 2.12 2.16 i 0.01 0.800 0.826 0.147 0.078 1.490 0.200 0.149
kBapp( X)
( l / w , ) / n s , input ( l / w l ) / n s , output' ( l/w2)/ns, input ( l / w 2 ) / n s , output' (wI w 2 - wlo - w:)/ns-l, (wI+ w 2 - wlo - w:)/ns-I,
+
0.446 0.513 1.162 4.00 4.02 f 0.09
10.00 9.97 f 0.06
input output"
FA4A)'
FB*(X)' (kAok,, kB0kqA)/M-',outputf kAO/ns-l, output8 kB0/ns-l1 outpu tg
+
0.000 0.000 0.437 1.ooo
0.494 0.562 1.205 3.48 3.53 f 0.06 7.01 6.97 f 0.04 0.080 0.078 0.355 0.613 1.501 0.205 0.144
0.544 0.61 1 1.255 3.07 3.10 f 0.04 5.43 5.41 f 0.03 0.160 0.159 0.303 0.416 1.496 0.202 0.146
0.639 0.706 1.337 2.46 2.52 f 0.03 3.8 1 3.76 f 0.02 0.320 0.314 0.238 0.234 1.518 0.21 1 0.137
"Calculated according to eq 24 and 25. *Obtained from deconvolutions of lo4 decays at indicated wavelengths. Calculated according to eq 19. Plotted according to eq 28: best straight line through the origin has slope = 5.349 M-I ns-I (= kqB). CObtainedfrom deconvolutions of IO4 decays. Reference lifetime estimated to be 0.24 ns. "Plotted according to eq 26: best straight line through the origin has slope = 8.17 M-I ns-I (= K). 'Relative heights of the simulated steady-statespectra (eq 12 and 13). fcalculated according to eq 27 and k 4 =~ 5.349 M-' ns-l and k,, = 2.822 M-' ns-I from b and d. 8Estimated fromfand kAo + kBo= wIo + w: = 0.349 ns-I (estimated in c).
or lower values of kc within the mathematical limits (w2 > kc > w3).One explanation to the structure might be that also B was directly excited. Another, although less probable, explanation is that a more complicated reaction scheme must be assumed. Also, an induced charge-transfer character in C* could result if there was some interaction between the solvent and the diacetoxy groups of the molecule. These and similar questions are presently under investigation. A General Algorithm f o r Two-State Excited-State Reactions. The problems met to estimate kc, specially for the DADAE/ acetonitrile system, forced us to develop a general algorithm to obtain SAS. For simplicity only a two-state excited-state reaction similar to that of (3-naphthol will be discussed. The idea was simple: if a quencher is added to a system, the participating species would generally be quenched with different efficiencies. Thus, rate constants for quenching, k,, and kqB,can be introduced in Scheme I:
where kAo and kBo are given by kAo = kl + k3 and kBo = k2 + k4, i.e. the same as for the unquenched reaction. [Q] is the total concentration of added quencher. w1 and w 2 are given by eq 8 and 9, while the unquenched decay constants wlo and w? are given by eq 8 and 9 with kAoand kBosubstituted for k A and k,. Since w I w 2 = k A kB and similarly for the unquenched constants, we have
+
+
WI
+ ~2
- wI0 - wIo = K[Q]
+
where K = kqA kqB. Also, since kAokBo= wlowzo+ k,k,, we find after some algebra that
kAok,B
+ kBok,A = (WlW2
-
WloW2)K2
K(w1
- kqAkqB(Wl
+ WZ
+ W2 - W l o - Wz0)'
- w ~ O- ~ 2 ' )
(27)
For the fluorescence intensities of the individual species, for the DAS and the SAS, eq 6-18 still apply. It is also possible to define kBapP(X)according to eq 19. In these equations there is now also a dependence of [Q]. However, since DAS(X,w,) + DAS(X,w2) = a,*(X)[A*](O) (eq 10 and l l ) , there is no quenching dependence in the sum of DAS. Therefore, again after some algebra, we find kBapp(X)- kBaPP.'(X)= k ¶B [Q]
(28)
where kBaPP3'(X) = P 3 0 ( X ) ~ I o ~ ~ /and ~ DPAo (SX ) is the total,
unquenched, steady-state spectrum. Simulated, unquenched species associated spectra, SAS,.(X,O) and SASB*(X,O),were generated as before from Gaussian functions. Values of the rate constants are given in Table 11. Quenched SAS and quenched total, observable fluorescence spectra were then generated for six quencher concentrations according to eq 12 and 13, where a,.(X)[A*](O) was given by SASA*(X,O)W1°W~/kBo and similarly for aB*(X)[A*](()). The quenching constants kqA and k,,, Table 11, were of the order expected for diffusion-controlled quenching with, e g , I- in nonviscous solvents. Some five to ten times higher values could have been chosen to mimic quenching with oxygen. The individual fluorescence intensity decays were then simulated with eq 17 and 18. Generations and deconvolutions of the total intensities were carried out as already described. The following strategy was then applied to determine the rate constants and the SAS: (a) k,B was determined at different wavelengths according to eq 28 and 19. In principle, these plots of kBaPp(X) - kBaPP.O(X) should have the same slope for all wavelengths, but in practice some variation was expected. (b) K was determined according to eq 26. The plots according to (a) and (b) should be forced through the origin. (c) Once kqBand K were determined, k,, was calculated as k,, = K - kqB. (d) kAoand kBowere determined at different quencher concentrations (different w 1 and w2)according to eq 27 and kA0 kBo= wIo + w:. Again, some variation in the values was expected. (e) SASA*(X) and SASB*(X)were calculated with eq 14 and 15 where k , and k B are given by eq 24 and 25. The results of the calculations are shown in Table 11. The estimated values of w I and w 2agreed excellently with the inputs, as well as K obtained from the plot of w I w 2 - wIo y ovs. [Q]. It is noteworthy that even at the highest concentration of quencher, the global analysis found the decay constants giving a reliable straight line to estimate K = 8.17 M-l. The rate constants, needed to calculate the apparent k B for the plots of kBaPP(X)- kBaPP-O(X), were estimated by global deconvolutions at each [Q] of the lo4 decays at 425, 450, and 482 nm. A more sophisticated analysis would have been a global analysis, linked not only by w1 and w 2 but also by kqBat all [Q]. With present analysis technique this latter linking was done by fitting a straight line according to eq 28. The slope, 5.349 M-' ns-I, was in acceptable agreement with the known input, kqB= 5.000 M-l ns-I. In Figure 5A, DAS(X,w,,[Q]) are illustrated for three different quencher concentrations. It is seen that each individual spectrum showed a complex quencher dependence, but with X D A S being
+
+
1168 The Journal of Physical Chemistry, Vol. 90, No. 6, 1986
Lofroth unquenched spectrum. This was also revealed by kBapp.O(X) which never was within the mathematical limits w , > k B > w2 at any wavelength. The simulated and calculated unquenched species associated spectra are finally shown in Figure 5B.
AI
I
4w
WAVELENGTHInm
550
Figure 5. Results from a study of a computer-simulated two-state excited-state reaction. Global deconvolution of 16 wavelengths. lo4 decays were sampled at 425,450, and 482 nm for seven quencher concentrations. (A) Decay-associated spectra for three quencher concentrations. For clarity only the curves that smoothly join the calculated points are shown. (B) Unquenched species-associated spectra (symbols). The full drawn curves are the expected individual and total spectra.
independent of quencher concentration, as required. A slight shift to the blue was observed in the spectra as [Q] increased, in accordance with a more efficient quenching of B*. Moreover, DAS(X,w,,O) was never positive at the analyzed wavelengths, which indicated that B* fluorescence was dominating the total,
Summary Complex decay behavior of fluorescence intensity from a system is often due to heterogeneity in the ground state or in the excited state. Spectroscopic techniques exist with which highly accurate and reliable data can be obtained. The concept of decay-associated spectra was introduced some years ago and used to calculate the spectra of individual species of ground-state mixtures. In this paper an extension of the ideas has been exemplified with the concept of species-associated spectra for excited-state reactions. Applications of the algorithms were shown both for computer-simulated and real systems: @naphthol in water was chosen as an example due to its well-characterized photophysics, while three-state excited-state reactions were exemplified with results from a current study on substituted bis(anthry1)alkanes. The agreement between expected and derived SAS in both the P-naphthol and in the simulated experiments also presented was excellent. However, some questions arose concerning the reliability of the SAS for the DADAE system. We are presently calculating SAS by assuming excitation in more than one of the ground-state species. Briefly, ultimately this will be done by global analyses of apparent rate constants as functions of both emission and excitation wavelengths. For a two-state excited-state reaction this means that, if both A and B are directly excited, the apparent constants at short and long wavelengths have values that depend on k4 and kj. One application has recently been presented." Also. the general approach to overcome problems with overlapping spectra, will be applied. Quenching with oxygen will then be one experimental approach. The prerequisites for a correct interpretation of complex fjuorescence intensity data have, in our opinion, increased by the SL4Sconcept. Since the SAS are derived quantities, relying on a model for the observed intensities, they can also be used as tests of the models. Also, with excitation sources with high repetition rates, like the laser or the synchrotron, data can be sampled to give TRES of very high quality, from which derived DAS and SAS might be used, e.g., for fluorescence quantum yield determinations and other quantitative work.
Acknowledgment. Dr. Kjell Sandros is gratefully acknowledged for stimulating discussions and valuable advice, Mr. Alpo Karppinen is thanked for aid with the illustrations and for technical assistance. This work was financially supported by grants from the Swedish Natural Science Research Council. Registry No. DADAE, 58382-04-0; POPOP, 1806-34-4; anthracene. 120-12-7; diphenylanthracene, 28351-02-2; 0-naphthol. 135-19-3.
