J. Phys. Chem. 1985, 89, 5450-5458
5450
conditions. A comparison of the ~ ~ ~ values ~ ( ~ for 'these 0 )ions shows that the composite rotational motion involving the fluctuations in the EFG at oxygen is much slower in the carbonate than in the nitrate anion (see Figure 3). A comparison of the plots of log 7eff('70)against l / T in Figure 3 for the two anions shows a larger activation energy for the carbonate ion. This is in contrast to the Raman study where the activation energies for end-over-end rotations were found to be identical within experimental error. Consequently, it is interesting to consider the cause for the difference in our E , values in Figure 3. The random error in the I7Orelaxation times, given by the computer fits to the recovery curves, is ca. 1%. Because the variation in the quadrupolar coupling constant and asymmetry parameter with temperature is expected to be small,I3 we are confident that the relative error in the E , values is less than 5%. The I7O N M R results indicate that the E , values for in-plane and/or end-over-end rotations are greater in the carbonate anion relative to the nitrate anion. If the Raman data are reliable, the implication of our work is that the observed difference in the activation energies is due to effects on the in-plane rotation. A larger E, (Figure 3) for the C 0 3 2 -ion may be caused by complexation with Na', because the anion with the higher negative charge will be expected to form a stronger complex with a cation. An equilibrium constant, K, where log K = 0.55, for the formation of a complex between Naf and C03*-in aqueous solution has been r e p ~ r t e d , ~which ' ~ ? ~suggests ~ that over 50% of all C03*- should be complexed with Na+. However, the thermodynamic data cannot provide any information of the dynamic nature of the complex. Thus, a given (NaC03)- anion may not survive a collision and rotate as a rigid unit. Nevertheless, the larger activation energy for the carbonate anion, obtained in this work, supports the claim of Perrot and co-workers that the formation of'; dynamic complex, NaC0,-, is responsible for the slower reorientation of C03*- relative to NO3-. (33) Nakayama, F. S. J . Znorg. Nucl. Chem. 1971, 33, 1287.
We have also measured NMR Tl(23Na)values of 49.8 and 24.2 ms, at 23.5 "C, in aqueous solutions of 1 M NaNO, and 1 M Na2C03,respectively. For an infinitely dilute solution, T ,(23Na) = 57.0 ms. The much more efficient rate of relaxation for the carbonate solution relative to the nitrate solution (a factor of 2.1) is a result of a larger effective quadrupolar coupling constant, q ~ c ( ~ ~ and/or N a ) Teff value.34 In any case, it appears that the sodium cation is more strongly associated with the carbonate than with the nitrate. Conclusions We have shown that nuclear relaxation rates measured by N M R can be used to study the rotational dynamics of the nitrate and carbonate anions in dilute aqueous solution. Rotational correlation times for the end-over-end rotation of the C3axis of the nitrate anion are only in fair agreement with the Raman data. Using I7O N M R T I data we have been able to probe motions of the nitrate and carbonate anions parallel to the C3 symmetry axis. The in-plane rotations are found to be slower than the end-over-end rotations of the C3 axis in both cases. The slower rotations and larger associated activation energy of the carbonate anion relative to that of the nitrate anion is evidence for the existence of a complex between Naf and C0,2in aqueous solution.
Acknowledgment. All N M R spectra were obtained at the Atlantic Regional N M R center, Dalhousie University. We are grateful to NSERC for financial support in helping establish and maintain this center and for an operating grant (R.E.W). We thank Professor R. J. Boyd for assistance with the MO calculations and the referees for several helpful comments. Registry No. NO3-, 14797-55-8; C032', 3812-32-6. (34) (a) Eisenstadt, M.; Friedman, H. L. J . Chem. Phys. 1967, 46, 2182. (b) Eisenstadt, M.; Friedman, H. L. J. Chem. Phys. 1966, 44, 1407. (c) Delville, A,; Detellier, C.; Gerstmans, A,; Laszlo, P. J . Magn. Reson. 1981, 42, 14 and references therein.
Multiexponential Fluorescence Decay of Indole-3-alkanoic Acidst Douglas R. James and William R. Ware* Photochemistry Unit, Department of Chemistry, The University of Western Ontario, London, Ontario, Canada N6A 587 (Received: March 22, 1985; In Final Form: July 2, 1985)
The fluorescence decay of a series of indole-3-alkanoic acids has been studied as a function of pH. Each successive member of the series varies only by the addition of a methylene group to the unbranched alkane side chain. In the pH region near the acid pK,, extra fluorescence decay lifetimes appear which cannot be assigned to only the ionized and neutral acid species. The fluorescence decay behavior is explained by a dynamic interaction of the side chain with the solvated indole group during the excited-state lifetime. Implications of these observations are discussed in connection with models for the fluorescence decay properties of tryptophan-like molecules.
Introduction The fluorescence properties of indole derivatives have been extensively studied.'J This is in part a consequence of the occurrence of the indole moiety in tryptophan, a commonly occurring amino acid in proteins. Tryptophan recently has been shown to exhibit complex fluorescence decay behavior.3-" The Occurrence of multiple fluorescence lifetimes for a single fluorophore is not common, and several explanations have been put forward to account for this ~ b s e r v a t i o n . ~The - ~ ~currently popular explanation proposes that the indole moiety interacts with the amino and carbonyl groups of the alanyl side chain, thus generating two different types of charge-transfer (CT) interactions. The two 'Publication No. 341.
0022-3654/85/2089-5450$01 S O / O
interacting states yield two lifetimes. This explanation is called the conformer model4 or, in a more recent formulation, the (1) R. Lumry and M. Hershberger, Photochem. Photobio/.,27, 819-840 (1978). (2) D. Creed, Photochem. Photobiol., 39, 537-562 (1984). (3) D. M. Rayner and A. G.Szabo, Can. J . Chem., 56,743-745 (1978). (4) A. G. Szabo and D. M. Rayner, J . Am. Chem. Sot., 102, 554-563 (1980). (5) G . R. Fleming, J. M. Morris, R. J. Robbins, G. J . Woolfe, P. J . Thistlewaite, and G . W. Robinson, Proc. Natl. Acad. Sci. U.S.A., 75, 4652-4656 (1978). (6) G. S. Beddard, G.R. Fleming, Sir G. Porter, and R. J. Robbins, Phi/os. Trans. R. SOC.Lond A, 298, 321-334 (1980). (7) M. C. Chang, J. W. Petrich, D. B. McDonald, and G.R. Fleming, J. Am. Chem. SOC.,105, 3819-3824 (1983).
0 1985 American Chemical Society
Fluorescence Decay of Indole-3-alkanoic Acids
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 5451
modified conformer model (MCM).7,8 There are some shortcomings to the MCM, however. Recent investigations by Templeton et al.loJ1have shown that models which depend only on the ground-state conformations of the alanyl side chain are insufficient to account for all the details of the decay process. They thus extend the conformer model to include the solvation and solvent relaxation processes which occur during the excited-state lifetime. It is apparent from the aforementioned investigations that the photophysics underlying the multiexponential fluorescence decay of tryptophan is very complex. As part of a program to critically examine the MCM and other models of multiexponential decay, we have investigated a “simplified” tryptophan. We have studied the steady-state spectra and lifetimes of a series of indole-3-alkanoic acids. This class of compounds has been previously investigated by steadystate and exhibited a decrease in emitted intensity with variation in pH at pH’s too high for proton quenching. No fluorescence lifetime measurements appear to exist which relate in detail to the influence of pH on fluorescence intensity for these compounds.“J4 As will be shown, even these singly substituted indoles can exhibit photophysical behavior consistent with at least a triple-exponential fluorescence decay behavior. Materials and Methods Indole-3-acetic acid (IA), indole-3-propanoic acid (IP), indole-3-butanoic acid (IB), and 3-methylindole (3MI) were obtained from Sigma Chemical. These materials were recrystallized from ethanol/water. Stock solutions of all materials were made to ca. M in deionized quartz distilled water. The stock solutions were prepared every 3 days and were stored refrigerated in the dark; no deterioration of the solutions was noted. Aliquots of the stock solutions were further diluted to a final concentration of about 104-10-5 M with water. The desired pH was obtained by adding HCl and NaOH. The pH meter was calibrated to better than 0.02 unit against standard buffers. Sometimes in the case of p H s between 4 and 9 an addition was made of M acetate, phosphate, or borate buffer. At these concentrations, the pH was stabilized but no significant fluorescence quenching by the buffer was observed.13 Fluorescence lifetimes were determined by the time-correlated single-photon counting method using either a PRA 3000 fluorescence lifetime instrument or a sirnilar instrument modified to use laser excitation from a Coherent mode-locked, synchronously pumped, frequency-doubled dye laser/argon ion laser combination, the details of which appear e1~ewhere.I~ The full width at half-maximum of the instrument response function for the laser system was about 250 ps. The upper limit of lifetimes determined by the laser system was 2 ns due to the problem of the incomplete decay of longer lived species between excitation pulses at the mode-locked laser frequency employed. Excitation was at 283 nm, and emission typically was observed at 340 nm. Fluorescence decay curves were analyzed as a sum of exponentials by utilizing either the PRA iterative reconvolution program or a program based on the Marquardt algorithm which also used iterative reconvolution16but where parameters could be fixed if desired. The fluorescence decay intensity Z ( t ) was assumed to have the form Z(t)
=
Cui exp(-t/ri) i
(1)
(8) J. W. Petrich, M. C. Chang, D. B. McDonald, and G. R. Fleming, J . Am. Chem. Soc., 105, 3824-3832 (1983). (9) E. Gudgin, R. Lopez-Delgado, and W. R. Ware, Can. J . Chem., 59, 1037-1044 (1981). (10) E. Gudgin, R. Lopez-Delgado, and W. R. Ware, J. Phys. Chem., 87, 1559-1565 (1983). (11) E. F. Gudgin-Templeton and W. R. Ware, J . Phys. Chem., 88, 4626-4631 (1984). (12) J. Feitelson, Isr. J . Chem., 8, 241-252 (1970). (13) R. W. Ricci and J. M. Nesta, J . Phys. Chem., 80, 874-980 (1976). (14) I. Weinryb and R. F. Steiner, Biochemistry, 7, 2488-2495 (1968). ( 1 5) W. R. Ware, M. Pratinidhi, and R. K. Bauer, Rev. Sci. Inrtrum., 54, 1148 (1983). (16) D. R. James, D. R. M. Demmer, R. E. Verrall, and R. P. Steer, Rev. Sci. Instrum., 54, 1121-1130 (1983).
m
f
Y
C
0
2
4
6
8
1
0
PH
Figure 1. Dependence on pH of indole-3-alkanoic acid fluorescence lifetimes recovered from an unconstrained two-exponential analysis. (A) IA, (B) IP, (C) IB. The lines are calculated from the best-fit values of Table I by means of eq 7 followed by a two-component fit to simulated three-component decay curves. The lines are shown only when the preexponentials exceed 0.01.
where ai are the preexponentials corresponding to the observed fluorescence lifetimes Ti. Steady-state spectra were collected on a Perkin-Elmer 640-50 fluorometer, typically with 1-nm spectral bandwidths. Absorption spectra were recorded on a Cary 15 spectrophotometer. The fluorescence quantum yield of IB as a function of pH was determined from the integrated inten~ities.’~J~ All experiments were done at 20 f 2 “C.
Results Data analyzed with all parameters floating yielded results shown in Figure 1 where the lifetime dependencies of IA, IP, and IB on pH are given. In all cases the decays were fit adequately by a single exponential at pH 25 and pH 1 2 , but two lifetimes were required for 2 IpH I5. The pattern of the two lifetimes seen in Figure 1 is complex. The lifetimes appear to reflect excited-state quenching by H*. The lines drawn through the data will be discussed later. The dependence of the data upon pH is not readily explained as the longer lifetime begins to shorten at a pH too high for significant quenching by a proton. Examination of Figure 1 suggested a potential problem. It is w e l l - k n ~ w n ’that ~ ~ ~two ~ lifetimes differing by less than a factor of 2 may be artifactually correlated in the analysis procedure. In Figure 1 the short lifetime r2,appears to closely approach the long lifetime, rl. Hence, it seemed possible that the two curves might actually cross but that a correlation artifact had disguised this event. A preliminary check over the range 2 IpH I5 revealed that if one lifetime was held constant at a value somewhere between those recovered from the unconstrained fit, the other three parameters could change by significant amounts. Typically, near the region of closest approach the unconstrained lifetime could vary by up to 3 ns and the preexponentials could vary by up to (17) C. A. Parker and W. T. Rees, Analyst (London) 85,587-600 (1960). (18) J. N . DemasandG. A. Crosby,J. Phys. Chem., 75,991-1024 (1971). (19) M. G. Badea and L. Brand, Methods Enzymol., 61, 371 (1979). (20) J. R. Knudtson, J. M. Beechem, and L. Brand, Chem. Phys. Lett., 102, 510-507 (1983).
5452
James and Ware
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 SCHEME I
IO
k . IU+.
8
f-
6
II h
A L
2
a-
0
h. k:
.
IHtl
k;
I1
' AH
IO 8
I
6
T/ns L
2 0 10
8 6
4 2
0 )
2
L
6
8
1
0
PH Figure 2. Dependence on pH of indole-3-alkanoic acid acid fluorescence lifetimes recovered from a constrained two-exponential analysis. The constraint conditions are discussed in the text. (A) I A , (B) IP, (C) IB. The lines through the short lifetimes are calculated assuming SternVolmer quenching by H+with quenching constants of about 1 X 1Olo M-' s-I. The lines through the long lifetimes are calculated from an empirical model (see text for details). 0.3 unit while the reduced x2 increased by less than 0.1 unit. This implied that the recovered parameters were highly correlated near the region of closest approach. Further tests for correlation artifacts seemed warranted. Because the two lifetimes are separated by about a factor of 2 between the range 4 IpH I10 and because the longer lifetime quenched with decreasing pH, we used an empirical model to provide a systematic constraint for one of the lifetimes. In this empirical model we fit the long lifetime to the equation 1/71 = A B[H+], where A is the reciprocal of T~ a t pH 10. B was adjusted to fit the data between 4 IpH I 10. In the subsequent reanalysis of the raw data between pH 2 and 4, 71 was constrained by this empirical model and the other three parameters were unconstrained. Good fits were recovered with 1.0 Ix2 I 1.2. Only the crossover region, 2 5 p H I4, was constrained by the empirical model. The results of this analysis are presented in Figure 2. Note that because the observed lifetimes for IA are so widely separated, the unconstrained data were considered satisfactory. Note also that over the range 2 I pH I 4 the empirical model for 71 was essentially equivalent to a straight-line extrapolation of the unconstrained data for 71 through the region 3.5 IpH I 4.5. Hence, the analytical form chosen for the constraint of 71 was convenient but not essential. Thus, the result of these calculations was that the single-exponential data below pH 2 and the T~ data above pH 3 (for IP and IB) could be connected with a smooth curve. Proceeding on the hypothesis that the curves did cross, the low-pH data for 7 2 were fit to a proton quenching model yielding apparent quenching constants of about 1 X 1Olo M-' s-l for all three compounds. As a check of this value, the proton quenching constant for 3MI was determined and also yielded a value of 1 X 1O'O M-' s - I . Thus, the testing by constrained deconvolution suggested that another data set where the two lifetimes crossed might be a valid representation of the behavior of the system, especially since the proton
+
quenching constant for 72 seemed reasonable when compared to the proton quenching constant for 3MI. Starting from the simplest viewpoint, both the constrained and unconstrained fluorescence lifetimes for the indole-3-alkanoic acids were subjected to a preliminary analysis in terms of a coupled two excited-state, two ground-state model (Scheme I). The species A- and A H are the ground-state anion and protonated carboxylic acid, respectively, and similarly A-* and AH* denote the excited state. The constraints associated with this kinetic scheme are klg, the diffusion-controlled association rate constant for a proton to a carboxylic acid in water; k2g, the corresponding dissociation rate constant; k2 and k,, the similarly defined dissociation and association rate constants for the excited-state species; kRA,kNRA,and kQA,the radiative, nonradiative, and proton quenching rate constants for the excited-state anions; and kRAH, kNRAH, and kQAH,the similarly defined constants for the excited-state protonated acid. The rate equations for Scheme I may be solved with boundary conditions such that the relative initial excited-state populations are given by the ground-state populations. Hence, [AH] = [A-][H+]/KB where KB = k2g/klg. The solution is 2x1,~= (kA Ul
+ kAH)
[(kA - kAd2 + 4k2k1[H+1)11'2 (2)
=
I - k,] kRAIXl - kAH- k2[H+]/KB] ~ R ~ ~ [ H ' ] [ (-XkA)/Kg [XI - X2][kRA+ ~ R ~ ~ [ H ' ] / K ~ ] (3) a2 = 1 - al (4) where X1,2 are the reciprocals of the observed fluorescence decay rate Constants 71 2, kA = kRA kNRA (k1 ~ Q ~ ) [ H +and ] , kAH = kRAH+ kNRAIlt-t k2 + kQAH[H+].Implicit in Scheme I is the postulate that A- generates only the long-lived species at high pH and A H generates only the short-lived species at low pH. As is usual for coupled excited-state kinetic schemes, neither XI nor X, can be assigned to a given species, at least in the intermediate pH region. Note that by setting k l = k2 = 0 and/or klg = k2g = 0 the same kinetic scheme can be tested for ground state and/or no coupling at all. Equations 2-4 were used in a computer program to generate simulated 7;s and u,'s as a function of pH. First, the data from Figure 2 were analyzed for ground-state coupling only. It was assumed that kRA= kRAH,and therefore the difference in observed lifetimes results from kNRAz kNRAH.2' The parameters used were kA = 1.1 1 X lo8 sC1 + k A[H+],kAH = 2.86 X lo8 s-l kQAH[H+],klg = 2 X 10" M-' s,' k2g = 3.56 X lo5 SKI,and k, and k2 = 0. Using these values as parameters in eq 2, we varied kQAand kQAHto fit the data. It was found that although kQAHwas approximately correct for diffusion-limited proton quenching, only a value of kQA= 2.6 X 10" M-' s-' would yield a fit to the data. This value for kQAis the same as that found for the empirical constant B used to generate the constrained fit. It is well-known that the proton association rate constant for carboxylic acids is (1-5) X 1Olo M-I s-1.22 Two aspects are in contradiction. The observed apparent quenching constant for 71 is approximately 10 times too large. Also, one must postulate the unlikely differential quenching by a proton of the two ionic forms
+
(21) R. Ricci, Phorochem. Phorobiob, 12, 67-75 (1970). (22) M. Eigen and L. De Maeyer, "Techniques of Organic Chemistry", Vol. 7, Part 2, A. Weissberger, Ed., Interscience; New York, 1963, Chapter 18, pp 895-1054.
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 5453
Fluorescence Decay of Indole-3-alkanoic Acids
SCHEME I1 k),
a
A kNk, kg lHtl
a
k t , k N i , kgiHtl
7 f
't
*:*
2
t
I.:
6t
.... --.
*:.
I/-
*
#-
SCHEME I11
4
2
[
1
[
1
0 ,/*: 0
kA kA
R'
d
2
4
6
PH
8
100
2
6
L
8
k0
10
PH
Figure 3. Fit of Scheme I to the data recovered from the unconstrained two-exponential analysis allowing both ground- and excited-state coupling. The lines are calculated by means of eq 2 with fixed parameters as defined in the text. The varied parameters are (a) kQA= AH = 0.0 M-l S-I, (b)kQA = kqAH = 1.0 X 10'' M-' S-I, (C) kqA = kQA kg = 2.6 X 10" M-I d,and (d) kQA = 2.6 X 10" M-'s-' and kQAH = 1.0 X 10" M-I s-I. The lines are shown only when the preexponentials exceed 0.01.
of the indole-3-alkanoic acid. These facts eliminate the coupled ground-state model. Similar results are obtained for a model coupled in the ground and excited states. Thus, Scheme I fails in its entirety to fit the data of Figure 2. A similar treatment of the unconstrained data of Figure 1 by eq 2 failed to fit the data for the above reasons. As well, there was a generally poor fit of the calculated curves to the data. For example, Figure 3 shows a representative fit of the unconstrained data by Scheme I allowing both ground- and excited-state coupling. The parameters used in eq 2 were the same as the ground-state model (vide supra) except that k l = 2 X 1O'O M-' s-' and k2 = 3.56 X lo5 s-I. The calculated lines clearly fail to fit the data. These arguments led to the conclusion that Scheme I was inconsistent with the data. The elimination of Scheme I leads to another problem: if two-state models cannot account for either the constrained or the unconstrained data sets, then the validity of both sets must be further examined. There is no question that two exponentials are seen in the range 2 Ip H I5. Several observations were noted. First, the recovered fits to the data of pH >3.5 yield lifetimes well enough separated that correlation artifacts are not a problem. Thus, the prematurely descending portion of the T~ curve at pH >3.5 was not particularly subject to correlation artifacts even through the two curves might be correlated near the region of closest approach. This is particularly true for IA where the correlation artifact is expected to be negligible over almost all the range of unusual T~ behavior. Second, as noted above, the recovered value for kQA= B is physically unreasonable, yet only values of B about equal to 2.6 X 10" M-' s-' (for IP and IB) generated a satisfactory constrained data set. This observation eliminates any constrained model where the two curves cross. Thus, one is left with the problem of accounting for the unusual pH behavior of the unconstrained data set. The most unexpected feature of the unconstrained data set is the pH dependence of T~ at p H >3.5. Since neither proton quenching nor any two-state model can account for this, one must look for some other physically reasonable mechanism to be the cause of the rl behavior. The most likely circumstance is that fluorescence emission from three or more species is mixing in a fashion which averages, yieMing a two-exponential decay with an unusual pH dependence.
k":,
k;?
NR'
1
k a IH'l
Q
AHO k F o , kNR
kp
IH'I
k r c [HI1 hu
The elimination of Scheme I also requires one to reconsider the molecular interactions implicit in the application of Scheme I. It was assumed that the species A H was entirely interacting without considering the nature of the interaction. Examination of molecular models suggested that at least two conformational forms might need to be considered: an open, noninteracting form and a closed, interacting form. This is coupled to the acid-base equilibria to yield Scheme 11, a model with four potentially fluorescing species. In Scheme I1 the open and closed forms are denoted respectively by AH,* and AH,* for the protonated acid and similarly for the anion. We have introduced an opening rate constant, k,, and a closing rate constant, k,, to account for the time dependence of this intramolecular process. Thus, Scheme I1 allows for AH,* to not have the same k N Ras AH,*. It is assumed that 4-* will not have a lifetime significantly different from &-*since the anion of carboxylic acids is a very weak quencher.13 All species are also assumed subject to proton quenching on the indole moiety. Generally, Scheme I1 will result in a four-exponential fluorescence decay from the four coupled states. This is a complex analytical problem, and simplifications were sought in order to make the problem more tractable. First, no consideration was made of electronic interactions between the acid moiety and the indole ring. This is experimentally justified for IB as there was only ca. a 1-nm shift in the fluorescence spectra as a function of pH, and none was observed for the absorption spectra. This justification is less valid for IA and IP since both the absorption and emission spectra show changes as a function of pH. For IA the fluorescence emission for the A- form is 10 nm red-shifted with respect to the A H form, whereas IP is shifted about 3 nm. Second, k R is assumed to be constant for all species. This is justified by experiment.21 Thus, kNR carries all the important photophysical variations. Third, since the anion of carboxylic acids is a very weak quencher, A;* and &-* can be expected to have the same lifetime." Using this information, one can approximate Scheme I1 by Scheme 111. In this model [&-*I + [&-*I = [A-*I.
5454
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985
-
-
kl'[H+] corresponds to the two consecutive processes &-* ,4-* AH,* of Scheme 11, and k i corresponds to the reverse process. The opening and closing rate constants, k, and k,, between AH,* and AH,* have been retained from Scheme 11. Scheme I11 is thus a generalized three-state model, one step more complex than Scheme I. Scheme 111 may be solved for the concentration dependence of each excited-state species as a function of time and pH by imposing the boundary conditions that the initial excited-state populations correspond to the unperturbed ground-state populations. Hence, [A-] = [AH,]k2(l + k,/k,)/(k,[H+]) and [AH,] = [AH,] k,/k,. The general solutions are not simple algebraic forms but rather depend on the numerical solution to the matrix equation formed from the differential equations for each species. The general solutions are given in the Appendix. The form of the equations is given by
where x(t) corresponds to the concentration of an excited-state species at time t , ui corresponds to specific coefficients determined from the rate constants and initial excited-state populations, and the Xi are the roots of the cubic equation generated from the matrix equation (vide infra). Equation 5 may be written as
Multiplication by the radiative lifetime for each species X(t), summation of species, and collection of terms yield the predicted fluorescence intensity Z(t) =
kRAN(A,X 1 )
kRAHoN( AH,,
1)
+ kRAHcN(AH,,Xl) e-XI' +
D(X1) ~R~N(A,+ X ~ )R ~ ~ " N ( A H , , X ~kRAHcN(AH,,X2) ) e-hlf + D(X2) ~ R ~ N ( A , X ~kRAH"N(AH,,h3) ) + ~ R ~ ~ ' N ( A H , , X ~ (7) ) DO31
+
Thus 3
z(t) =
Cai exp(-Xit) i= 1
(8)
which is the expected three-exponential fluorescence decay function. Lastly, eq 8 can be integrated to give quantum yields. Division by the steady-state intensity at high pH yields the relative fluorescence quantum yield Qplwhere
@pi=
i m Z ( p H , t )d r / l m0 Z ( p H 10,t) dt
(9)
Equation 9, with the value of Ricci and Nesta13 of +f N 0.38 for the absolute quantum yield of IB at pH 10, gives a calculated absolute quantum yield &, where 3
3
& = 0 . 3 8 i[=Cl a i ~ ,,/ECl a i ( p H10) T,(PH lo)]
(IO)
Equations 5-10 were used to numerically investigate Scheme 111. This was accomplished with a prbgram which evaluated the a, and X i as a function of pH by means of eq 7 using trial estimates of the various rate constants. These trial a, and Xi were then used to generate by means of eq 8 a set of synthetic decay curves covering the pH range 0.5-10. The trial impulse decay curves were convoluted with a laser instrumental response function (cf. ref 16) to generate stimulated experimental decay curves. Poisson noise was added, and these trial decay curves were analyzed as double-exponential decays by using an unconstrained fitting procedure. The goodness of the fit of the recovered ai's and 7;s
James and Ware TABLE I: Best-Fit Rate Constants for Scheme 111 for the Data of Figure 1
rate constant (XlO-*) kRA, kRAHo,kRAHc/S-' kNRA, kNRAH"/S-I kNRAHC/S-I k,, k,'/M-' s-l k,, k,'/s-l ka/M-' s-l k,/s-' kc/s-l
' K A G = 1.78
IA" 0.438 0.812 3.56 200 0.00356 100-200 6b
IP" 0.422 0.688 2.44 200 0.00356 200 0.1-0.3 0.8-1.1
IBa 0.422 0.688 2.44 200 0.00356 200 0.08-0.20 0.5-1.1
M. bk, 2 30k,.
X
to the data was evaluated over the entire pH range to determine the appropriateness of the trial rate constants. The entire cycle was repeated as necessary in order to get a satisfactory match to the data over the entire pH range. The investigation was guided by several factors. (1) The fit was not strongly sensitive to the values of the proton quenching constant other than to shift the curves up- or downrange on the pH scale. It turned out that kQA = kQAHc= kQAHa= k l = k, yielded the best fit when these values were set at the diffusion limit for the proton in water; Le., k, = 2 X 1O'O M-I SKI.(2) The value of I/(kRA kNRA)= 8 or 9 ns was determined from the high pH value of 7 , . (3) The value of l/(kRAH+ kNRAH) 3.5 ns for protonated I P and IB was determined from the observed value for IP ethyl ester.4 The value of ca. 2.5 ns was determined for protonated IA from the lifetime of IA ethyl ester. (4) It was generally assumed that kRA = kRAHa= kRAHcand that kNRA = kNRAHa # kNRAH'; this assumption was tested (vide infra). (5) kRA/(kRA+ kNRA) = & N 0.38.13 For the case where all the radiative lifetimes are constant, the exact value of 4f is inconsequential (cf. eq 7; the radiative rate constants factor out). This is no longer true if the radiative rate constants vary. (6) In recognition of the fact that Scheme I11 is an approximation to Scheme 11, k( was set such that kl'[H+] Ik,. This approximation accounts for the fact that the rate-limiting step for the process AH,* AH,* changes from protonation to closing at A,-* low pH. The model was tested to see whether this approximation skewed the results. It was found that k,' has almost no effect on the fitting of the model except at low pH. This is outside the region of multiexponentiality, and thus the potentially deleterious effects of k,' can be safely neglected. (7) k2/ was set equal to k2 for the reverse process because the dissociation of the carboxylic acid was always slower than the estimated value for k,. (8) The initial excited-state populations were determined by the ground-state population since the excitation wavelength was at a pH-insensitive portion of the absorption spectra. (9) It was deemed appropriate to set the ground-state equilibrium k,g/k,g equal to the excitedstate equilibrium as there is no evidence to propose any other value. (10) it was assumed that pK equals pK,*. This is justified for IB because the observed absorption and emission spectral changes are small. This justification may be less valid for IP and perhaps especially for IA. Thus, although Scheme 111 has a great many parameters, it is highly limited by knowledge of the rate constants determined from other experiments. The only significant parameters not predetermined are k,, k,, and to some degree kRAb. The latter must be considered a less well-known parameter since the nature of the emitting state is not well understood. However, it is not expected that kRAHcwill differ greatly from kRAas the results of Ricciz' indicate that most of the changes will occur in kNR. The results of this investigation for IB are given in Figures 1 and 4-7, and the overall results for the series are given in Table I. Figure 4 shows a typical simulated three-exponential decay fit by two exponentials. Figure 5 shows as a function of pH the predicted values of a, and T~calculated from eq 7 for the constants of Table I. The solid curves of Figure 6 show the results of the three-exponential analysis of eq 7 applied to the two-exponential unconstrained fits of IB when analyzed by the three-exponential convolution-two-exponential deconvolution procedure. Figure 6 indicates that the details of the data are reasonably fit by the
+
- -
-
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 5455
Fluorescence Decay of Indole-3-alkanoic Acids
r
I
I
1
6=I
02LOG I
0-,
I
k
0
8.0
I
12.0
I
18.0
I
24.0 NS
I
30.0
I
I
38.0
42.0
,
,
,
,
,
-
00-,
02
,
,
,
,,
1
48.0
+ 2.7
RESID
-
2.7
ACORR
te--------.
Figure 4. Analysis of a simulated three-exponentialdecay as two exponentials. Input values: a, = 0.67, T~ = 3.34 ns, az = 0.22, iZ = 5.65 ns, a3 = 0.1 1, r 3 = 8.89 ns. Recovered values: a , = 0.77, T~ = 3.53 ns, az = 0.23, T~ = 7.75 ns, xz = 1.01. The laser profile is shown by (----), and the simulated data with added Poisson noise are shown by (-). The plots of weighted residuals and autocorrelationfunction of the weighted residuals2sare labeled RESID and ACORR, respectively. The ordinate of the ACORR plot spans the range -1 to +1, and the abscissa is displayed expanded by a factor of 2. r
1
PH
PH
Figure 5. Calculated fluorescence lifetimes and preexponentials as a function of pH for IB by use of eq 7 and the values from Table I.
three-exponential model. Particularly indicative is the behavior of the preexponentials which closely resemble the experimental data. Although smooth curves have been drawn to represent the fitted a,)s, approximately as much scatter occurred for the simulations as for the real data. Similar results were obtained for all three compounds. The lines plotted through the data of Figure 1 are calculated by means of eq 7 from the best-fit values of Table I by using the three-exponential convolution-two-exponential deconvolution procedure. Figure 7 shows the application of eq 10 using the values of Table I to determine df for IB. Figure 7 shows that the features of df are well accounted for by the three-exponential model. In recognition of the complexities of Scheme 111, it was deemed necessary to test the effect of a changing kRAH.This is the most important assumed parameter both because of its potential effect upon the kinetic scheme and, since the nature of the A&* emitting
'
' O'*: 0.1
f' I/
0 ,
0
2
4
6
8
1
0
PH Figure 7. Fit of the fluorescence quantum yield as a function of pH for IB to the curve calculated from eq 10 and the values from Table I. The error bar is representative of the entire data set.
species is not well characterized, because the parameter might be expected to change somewhat. Simulations were carried out varying kRAHcwhile keeping kRAHc kNRAHcconstant and setting k, = k,. This procedure tests for the effects of k R A H c without perturbations from the rate of intramolecular motion. As expected, the 7,'s remained unchanged but the preexponentials and +f varied. It was found that, in order to get reasonable values for the T,'s, k, = k, N lo8 s-l. However, over the range of kRAHctested ((3.9-10.0) X lo7 sd) no satisfactory fit to either the short T or the preexponentials was found. Reasonable fits to q4f did occur, likely indicating that c $is~ not particularly sensitive to relatively small variations of kRAHc.Overall, it appears for IB that kRAHc needs to be constant and equal to kRAHo for the kinetic scheme to fit the data. This is in accord with the insensitivity of the spectra to variations in pH. The larger spectral variation with pH for IA and IP indicates changed electronic distributions in both the ground and excited states, and it is likely that k R A H c does change somewhat. Unfortunately, it is not possible to unequivocably determine the relative changes to kRAHcand kNRAHc from +f since there are multiple exponentials contributing to the fluorescence decay.
+
Discussion In view of the complexity of Schemes I1 and 111, it is reasonable to question the validity of the quantitative arguments presented above. It is therefore pertinent to reiterate several points. First, in order to derive a minimal model capable of accommodating the data, it was necessary to use at least a three-exponential decay. Two-state models were first examined and shown to be inadequate. Second, in spite of the apparent complexities of the system, most rate constants are known from other systems. The number of free parameters is thus reduced to only k, and k,. Numerical searches on two parameters are reasonable procedures and not likely to be subject to many local solutions. No unusual rate constants for proton quenching or proton equilibria were needed. In IB the acid group is far removed from the indole ring and the acid pK,
5456
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985
is assumed equal to that of butanoic acid. Finally, although the kinetic model has been derived from a conformational viewpoint, the “opening“ rate constant could more generally be called a “loss of interaction” rate constant, and similarly so for the “closing” process. In other words, the three-exponential model rationalizes the observed data in terms of states and interconversions between states. The important issue is that more than two states are necessary. The specific values of the rate constants are not fundamental to this argument. The fact that the data can be fit with three exponentials implies that Scheme I11 is a reasonable approximation to the minimal four-state model proposed in Scheme 11. The data quality does not warrant accounting for other additional sources of nonexponentiality such as diffusion-controlled rate constant^.^^^^^^^ The overall experimental result is that three or more excited states interconverting on the 107-108-s-’ time scale are necessary to account for the data. This is true for all three molecules which implies that the length of the alkyl carboxylate side chain is not the determining factor in the kinetic scheme. However, the numerical differences observed within the series indicate that k, and k , vary somewhat with chain length. If the carboxylic acid chain did not interact with the indole chromophore, one would of course expect to observe one lifetime independent of pH until an acid concentration was reached where quenching by H+ was kinetically possible (for k, = 10” M-l s-’, T = 8 ns, a pH of -3 is required). If both the COO- and COOH groups “quenched” the chromophore, then because of the multiplicity of conformations possible, one might expect to see a number of species emitting with characteristic lifetimes, perhaps modified by interconversion kinetics. In fact, a decay curve requiring a number of exponentials would not be surprising in systems of this type. It is inferred from dynamic quenching results that COO- is a poor quencher whereas COOH is expected to quench with quite high efficiency.13 Thus, for the compounds in this study, at pH >> pK,, one might expect a single lifetime characteristic of the unperturbed chromophore, Le., 8-9 ns. As one lowers the pH, more complex photophysics is expected as the COOH forms in both the ground and excited states and interacts with the excited indole. How complex the behavior is should then be determined by the number of distinct quenching conformations, their interconversion rates, and quenching efficiency. The data at high pH could be fit to one exponential with the lifetime of unperturbed indole, and the expectation of negligible quenching interaction between the COO- and the excited indole chromophore was thus realized. However, coincident with the appearance of the COOH and at a pH too high for diffusioncontrolled protonation or quenching to reduce the lifetime, a significant drop in the long lifetime was observed as well as a shorter component. As the pH was further lowered, two-component fits were found to give satisfactory x 2 values, autccorrelation plots,25and random residuals even when the lifetimes closely approached one another. However, as detailed above, simple mechanisms which are normally invoked to explain two-component decays failed completely to provide an explanation for the observed lifetimes and preexponentials unless totally unreasonable rate constants were used. It was therefore necessary to postulate that the sharp drop in lifetime at pH -4.5 was due to a more complex mechanism and that three or more lifetimes were being averaged in the data analysis to give an apparent two-component decay. As described above, this more complex scheme provided a good fit to both the steady-state and lifetime data but required three emitting sources A-*, AH,*, and AH,* where the subscripts “on and are meant to imply only absence or presence of some interaction giving rise to a shorter lifetime. It is of course realized that these systems may be considerably more complex than assumed in Scheme 111. However, this scheme gives a good deOC”
(23) w. R. Ware and J. C. Andre, “Time-Resolved Fluorescence Spectroscopy in Biochemistry and Biology”, R. B. Cundall and R. E. Dale, Eds.. Plenum Press, New York, 1983, pp 363-392. (24) R. W. Wijnaendts Van Resandt. Chem. Phys. Letr., 95, 205-208 (1983). (25) A. Grinvald and I. Z. Steinberg, Anal. Biochem. 59, 583-598 (1974).
James and Ware scription of the variation of lifetimes, preexponentials, and quantum yield with pH. The notion of two forms of the free acid A H will now be discussed in the context of the broader question of the multiexponential fluorescence decay behavior of many indole derivatives. From the results of this study, one can conclude that these acids most probably are to be classed with the many indole derivatives which exhibit a multiplicity of decay times. In this case we have what appears to be two characteristic decay times for the free acid, modified by interconversions, plus an additional decay time characteristic of both the anion and a noninteracting form of the acid-presumably an extended form where the COOH is well removed from the indole ring. In some respects these results are consistent with the conformer model of S ~ a b o modified ,~ by Fleming and c o - w o r k e r ~(the ~~~ so-called modified conformer model, Le., MCM). In the MCM the multiplicity of decay times for various indole derivatives is attributed to various noninterconverting conformations that enter into donor-acceptor interactions in the excited state and lead to quenching. If one considers the class of molecules given by the structure
then if n = 1 and if R, and/or R2 are good electron acceptors, conformations with shortened liftimes are predicted on the basis of this model. If R, and R2 are identical and quench, one lifetime is predicted by the MCM model, whereas if R, and R, differ in their quenching ability the model predicts two lifetimes. However, this model ignores the interconversion of conformations, the profound influence of the solvent on the disposition of excited states in indole derivatives, the effects of alkyl chain length, and as well, the question of extended forms of molecules where R, = H and R2 is a quenching group. In the case of the IP, the MCM model would lead one to predict only two lifetimes for the free acid which would be attributed to the two over-the-ring conformations, one with -COOH in close proximity and one with -H in close proximity. The MCM model would predict that one lifetime would be observed independent of conformation for the anion due to the weak or nonexistent quenching property of this group. Thus, there are elements of agreement between MCM model and the observed behavior of these compounds. However, Scheme I11 requires the interconversion between conformations to be kinetically significant, a phenomenon not included in the MCM. This is true for all values of n-the chain length is not a determining factor for the qualitative observations. Hence, rotation about the C,-C, bond7%* is of secondary importance in explaining the observed results. Also, the conformation where the -H rather than the -COOH is in close proximity to the ring is assumed in our model to have the lifetime of the extended or noninteracting form, modified by the rate of interconversion. These observations may be contrasted to those for certain indole derivatives exhibiting one lifetime. For example, the ethyl ester of IP is reported to have one lifetime,4 but one can imagine conformations where the ester group is well removed from the indole ring. Such conformations should have a long lifetime (ca. 8-9 ns) easily observable in the presence of a 2-3-11s component which results from the ester-indole interaction. It is unlikely that the ester-substituted side chain moves more rapidly than the free acid so the appearance of a single lifetime is puzzling. It is of interest that the same qualitative behavior is seen for all three acids. With indole-3-acetic acid some through-bond interaction is expected, and in fact the parameters required to fit this case are somewhat different than required for the other two acids. The observed photophysical behavior becomes essentially identical, once two or three methylene groups separate the quenching group from the chromophore, suggesting that folded conformations, which now become possible, are sufficiently similar in the propanoic acid and butanoic acid cases to give identical photophysics. One also sees significant spectral evidence of in-
Fluorescence Decay of Indole-3-alkanoic Acids teraction only in the acetic acid case suggesting that the through-bond interaction is more effective in modifying the position of the energy levels than the intramolecular through-space or through-the-solvation-shell interactions from the folded conformations of the longer chain acids. There remain a number of interesting questions associated with the photophysics of these quite fascinating systems: (a) What is the nature of the nonradiative process when the presence of interacting groups shortens the lifetimes? Furthermore, do the radiative lifetimes change? There is considerable evidence to suggest strong solvent interaction with the indole electronic states resulting in long radiative lifetimes in polar solvents. Meech and Phillips attribute this to the stabilization of the L&T state in polar solvents.26 It therefore seems unlikely that in polar solvents the acid side chain will further influence the radiative lifetime, especially if it operates through the solvent shell surrounding the indole ring. In systems exhibiting complex decay kinetics it is not easy to study this question, but we are currently examining cases where one observes side chain quenching but single-exponential decay. Thus, the shortening of the lifetime is mast likely to be the result of enhanced nonradiative processes. Photoionization is known to occur in molecules of this class when excited into the first absorption band, and recently it was shown that all or at least part of the photoionization was from the fluorescent state rather than from a more energetic unrelaxed prec~rsor.2~It has already been s u g g e ~ t e d 'that ~ ~ the quenching mechanism involves the electron acceptor properties of the R, and R2 groups of the structure given above, and the report of competition between quenching and photoionization strengthens this argument. Thus, it is possible that the shortened lifetime is due to electron transfer to the side chain or to the solvent via the side chain. This is to be compared with the case where the emitter is the intramolecular C T complex, and the shortened lifetime would then be an intrinsic property of the complex. (b) What is the relative importance of the dynamic (side chain movement) vs. static (side chain in place) quenching? This study on indole alkanoic acids suggests that one must include in the kinetic description a dynamic process involving the formation and breakup of an interacting conformation. This interaction is treated as a first-order process with a time-independent rate constant. There is some justification in this approach from Monte Carlo calculationsZswhich suggest that end-to-end quenching processes follow approximately a single-exponential kinetic law. The observed values of k, and k, E 107-108 s-l are very intriguing. These values are much too slow to be solvent relaxation about the excited-state dipole. These constants are slow even for many intramolecular motions. For example, these values are about 10 times slower than the rise times reported for the intramolecular excimer formation of (N,N-dimethy1anilino)aIkylanthracenederivatives in hexane and various alcohol^.^^^^^ Moreover, the values for k , and k, are about the same for I P and IB but k, is significantly faster for IA. Does this imply that the intrinsic rate of motion of the longer alkyl side chains in water is about 107-10s s-' or does it imply the occurrence of a weakly complexed side chain to the indole ring perhaps with solvent involvement? However, the indole acids are not ideal cases for a study of this question because of the multiplicity of processes, Le., dynamic proton quenching, dynamic acid-base equilibria, and side chain conformational changes. A large number of systems thought to offer a more definitive opportunity to study excited-state conformational changes are currently being investigated in our labora tory. (26) S. R. Meech, D. Phillips, and A. G . Lee, Chem. Phys., 80,317-328 (1983). (27) C . M. Previtali, Photochem. Phorobiol, 40, 689-692 (1984), and references therein. (28) J. A. Nairn and C. L. Braun, J . Chem. Phys., 74,2441-2447 (1981). (29) M. Migita, M. Kawai, N. Mataga, Y. Sakata, and S. Misumi, Chem. Phys. Lett., 53.67-70 (1978). (30) M. Migita, T. Okada, N . Mataga, N. Nakashima, K. Yoshihara, Y. Sakata, and S. Misumi, Chem. Phys. Lett., 72, 229-232 (1980).
The Journal of Physical Chemistry, Vol. 89. No. 25, 1985 5457 (c) A persistent question in this area is associated with through-bond vs. through-space or through-solvent interactions. In view of the evidence available for studies in rigid systems,!'*32 through-bond interaction may be important in electron transfer or intramolecular C T systems; the question is now being studied in our laboratory in connection with indole derivatives.
Conclusion The appearance,of multiexponential decay for a series of indole-3-alkanoic acids shows clearly that changes in the interactions of the side chain with the indole ring occur on the time scale of the excited-state lifetime. The qualitative equivalence of the results for all compounds implies that the side chain length is not the major factor determining the photophysical behavior. This does not agree with the MCM which assumes no motion of the side chain on the time scale of excited-state lifetime. It is suggested that a complete model of side chain interactions with a chromophore requires consideration of dynamic as well as static processes. It is, of course, entirely possible that the actual kinetic behavior of these three intramolecular quenching systems is more complex than assumed in the model we use to explain the data. The lifetimes we measure may represent averages over distributions of lifetimes which result from a multiplicity of interactions associated with a multiplicity of conformation^.^^ The model required to explain our observations is already so complex that it takes the data to its limit, and it appears quite impossible to obtain information regarding more sophisticated models from this experimental approach. Acknowledgment. D.R.J. thanks W.R.W. for the provision of a post-doctoral fellowship. We thank the Natural Sciences and Engineering Research Council of Canada for continuing financial support.
Appendix Transient Fluorescence Decay of Three Coupled Emitters. From consideration of Scheme 111, the equations describing the rate of excitation loss from the three coupled emitters is given by d[A-*] -- - [ A - * ] [ ~ R+~ kNRA + [H+](kqA k1 kl')] dt [AHo*]k2 [AH,*]ki ( A l )
+ + +
+
+ + k, + kaAHo[H+]]+ [AH,*]k, (A2) d[AH,*l -- [A-*]k,'[H+] + [AH,*]k, dt [ A H , * ] [ ~ R ~+~~ , N R +~ k,~ +' k2' + kaAHc[H+]] (A3) kNRAHo k2
With collection of terms, eq Al-A3 may be written as d[A-*] -- - -[A-*]~A+ [AHo*]k2 [AH,*]ki (A4) dt
+
d[AHO*l dt
--
- [A-*]k,[H+] - [AH,*]~AH,+ [AH,*]k,
(A.5)
To facilitate the simultaneous solution of eq A4-A6, we take the Laplace transforms of these equations and cast the resulting rearranged equations into a matrix equation:
(31) P. Pasman, F. Rob, and J. W. Verhoeven, J . Am. Chem. Soc., 104, 5 127-51 33 (1 982). (32) R. S. Davidson, R. Bonneau, J. Joussot-Dubien, and K. Toynae, Chem. Phys. Lett., 63, 269-272 (1979). (33) D. R. James and W. R. Ware, Chem. Phys. Letr., in press.
James and Ware
The Journal of Physical Chemistry, Vol. 89, No. 25, I985
5458
The solution of eq A7 is obtained by application of Cramer’s rule. Define A as the determinant of the 3 X 3 matrix in eq A7. A[A], A[AH,], and A[AHc] are defined as the determinants of the similar matrix obtained by substitution of the first, second, and third columns, respectively, of the matrix in eq A7 by the right-hand side of eq A7. After expansion and collection of terms one obtains
Thus
and similarly so for =*(s) and G C * ( s ) . The inverse Laplace transform of equ A13 has a solution of the general form
+
A = S3 S z ( k A + kAH, + kAK) + S[kAkAH, + kAkAH, + [A-*(t)] = kAH$AH, - k ~ -k(kikz ~ + k~’k2)[H+ll+ [ k ~ k ~ ~- , k ~ ~ , k ~ k c k -, (kikckz’+ kzkoki’ + kAH,kl’kZ‘ ~ A H ~ ~ I ~ Z ) [ H + I I
a0
2 ~ + a z X 1 2 e-hir + (YO - ~ r l X 2+ ( ~ 2 xe-h2r +
- alX1 -
- X2)(X3-
-
which, after collecting terms in obvious notation, gives A =
$3
+ s2e2 + se, + eo
(A81
Similarly AA
where the Xi are the roots of eq A8. Similar expressions can be written for AH,*(t) and AH,*(t). For convenience relabel the numerators and denominators of eq A14 to give
=
f2[A-*(0)1 + s[[A-*(o)l(kAH, + kAH,) + [AHO*(0)lkZ + [AH,*(O)Ik,’I [[A-*(0)I(kAH,kAH, - kcko) [AHO*(O)](~AH,~Z + kckz’) + [AHC*(~)I(~AH,,~,’ + k2k0)1 (A9) The initial excited-state populations are determined from the ground-state populations. If the extinction coefficient is equal for all conformers, then [A-I = [A]:
+ [A,-]
= [AHoIKAG(l
+ &L)/
[AH,] = [AH,]kc/k, = KcLIAHo], where
KAG
which, after substitution of [A-*(t)], [AH,*(t)], and [AH,*(t)] and rearrangement, yields
= k2/kl
Z(t) =
AA = ~ ’ K A C+( KcL)/[H+I ~ + s[(~AH, + ~AH,)KAG +( ~ KcL)/[H+I + kz + kz’Kc~1+ [ ( ~ A H , ~ A-HkckoW~c(1 , + K c d / [ H + l + ( ~ A H , ~+z U 2 ’ ) + ( k ~ ~ , k+2 /k2kO)fkrl
+
= S’LY~
+
S(YI
(YO
I(t) = k ~ ~ [ A - * ( t ) ] i k ~ ~ ~ ~ [ A H , * ( tk) ~] ~ ~ ~ [ A H , * ( t ) l (A1 6)
[Hfl
Substitution into eq A9 yields
AA
Assuming no spectral discrimination of the emission of the three emitters, the decay of fluorescence intensity is given by
kRAN(A,x I )
= s2 + s [ ( k A
[(k~ki’+ ~ I ~ A H , ) K A G KCL) ( I + (kAkAH, - k2’ki’[H+I) (klk2’[H+1 + kAko)KCL1 (A1 1) + Po AAH, = S’KCL+ s [ ( ~ A+ ~AH,)KCL + kc + k i ’ K ~ ~ (+1 KcL)] + [(kikc + ~ A H , ~ I ’ ) K A+GKCL) ( ~ + ( k ~ +k AAH,
= s2Pz + sP1
k2ki’[H+I) + (k,kAHo - kikz[H+I)Kc~I AAH,
= s2y2 +
+ YO
(‘412)
e-hll
+
k~~N(A,k+ 2 ) ~ R ~ ~ ” N ( A H ,+, X~ ~R)~ ~ ‘ N ( A H , , X Z ) e-A21
+
D(U ~ R ~ N ( A ,+ X ~ )R ~ ~ ” N ( A H ,+, Xk ~ )~ ~ ” ( A H , , h e-hil 3) DO,) (A171
(A101
+ AH,) + k i K A G ( 1 + KCL) + MCLI +
1)
W I )
The same considerations yield AAH,
+ k R A H oAH,, ~ ( X + kRAHcN( AH,, X
or Z(t)
+
+
= a(Xl)e-A1r a(Xz)e-A2r a(X3)e-”“ =
3
Eaie& i= I
(A18)
Finally, the integrated intensity is given by 3
I = Cai/Xi i= 1
from which the quantum yield of the photostationary process can be calculated in the manner of eq 10. Registry No. IA, 87-51-4; IP, 830-96-6; IB, 133-32-4; 3M1, 83-34-1.
