Non a Priori Analysis of Fluorescence Decay Surfaces of Excited-State

2047

J . Phys. Chem. 1991, 95, 2047-2058 The interaction of 2-anilinonaphthalene in cyclohexane with ethanol has been investigated by DeToma and Brand.') Nanosecond fluorescence decay curves were collected a t various concentrations of ethanol. The process was written as being concentration dependent in the forward step. From the published values of the rate parametersi4 the relaxation times at two different, concentrations can be calculated. The rate constants are then recalculated by using eqs 26 and 27. The following two possible solutions for the system matrix are obtained

values. Consequently, AI and A2 are related-by a similarity transformation, i.e.

A2 = H-IAIH

where H = UIU2-1with U, and U2 the matrices of which the columns are respectively the eigenvectors of A, and A2 If Ai,Bi,Ci and A2,B2,C2are realizations of the same # ( t ) then the following relations hold

B2 = H-IB,; C2 = CIH

A i = (-(0.22'+ 2.lh21.t) 0.13

-(0.16

+ 0.13)

= (-(0.29 + 2.lh21.k) 0.06 2. h21,k -(0.16

+ 0.06)

2*1h21,k A2

where the rate constants are expressed in ns-' and the ethanol concentration is denoted by h2i.k. The characteristic polynomials of Ai and A2 are the same irrespective of the concentration value h2,,k. This means that A, and A2 have always the same eigen(13) DcToma, R. P.; Brand, L. Chem. Phys. Lor?. 1977, 47, 231. (1 4) Badea, M. G.; Brand, L. In Merhods in Enzymology; Him,C. H. W.,

(C. 1)

(C.2)

For the considered experimental system, the rate koi can be determined directly by measuring the fluorescence decay in the absence of ethanol. This yields kol = 0.22 ns-l so that only AI can be retained as the exact solution. A second example deals with a similar recalculation performed with the rate parameters obtained in the study of the excited-state reaction of 2-naphthoLi5 In the discussion below, the compartment correspondin to 2-naphthol is labeled as compartment 2. The parameters k2, and kO2can readily be obtained. The quadratic equation in kol gives two positive solutions: 0.2085 and 0.1056 ns-l. The first solution yields a negative value for ki2(i.e. -0.0325 ns-l). Obviously, only the second solution for koi can be retained with the corresponding value of 0.0704 ns-' for k i 2 .

1

Timasheff, S.N., Eds.; Academic Press: Orlando, FL, 1979; Vol. 61, part

H.

(IS) Laws, W. R.;Brand, L. J. Phys. Chem. 1979,83, 795.

Non a Priori Analysis of Fluorescence Decay Surfaces of Excited-State Processes. 2. Intermolecular Exclmer Formation of Pyrend ROM Andriessen,* Nod Boens,**t Marcel Ameloot,i and Frans C. De Schryvert Department of Chemistry, Katholieke Universiteit Leuven, 8-3001 Heverlee- Leuven, Belgium, and Limburgs Universitair Centrum B-3590 Diepenbeek, Belgium (Received: July 2, 1990)

The so-called global compartmental approach to analyze the kinetics of excited-state processes (Ameloot et al., preceding paper) is tested on simulated and real fluorescence data of the excimer formation of pyrene in a nonviscous solvent. The identifiability problem of a bicompartmental system is verified. The results of the bicompartmental analysis are compared to the results of the classical analysis. The following rate constant values for the excited-state processes of pyrene in isooctane at room temperature are obtained by the bicompartmental analysis: koi = 2.17 X IO6 ri,kli = 7.2 X lo9 s-I M-I, kO2= 14.8 X IO6 s-l, and ki2 = 5.7 X lo6 s-l. The evidence for the nonexistence of ground-state dimers of pyrene in isooctane for concentrations lower than 1.5 X IO-) M is demonstrated. The species associated emission spectra (SAEMS) of the monomer and the excimer emission are calculated. It is shown that the unsubstituted intermolecular pyrene excimer is emitting at 375 nm.

I. Introduction Timeresolved fluorescence spectroscopy is an excellent physical technique to study molecular systems exhibiting dynamic behavior on a subpicosecond to microsecond time scale. Many processes, such as excited-state processes of various kinds,-' and solvent relaxation$ can be studied with this technique. The three most important time-resolved fluorescence techniques used at present are the single photon timing technique,' streak camera detection$ and phase shiftfmodulation fluorometry.' Although single photon timing has already been used for a long time, important improvements have been realized in recent years. The use of 'Dedicated to Prof. A. Weller on the occasion of his birthday. 8 Katholieke Univenitcit Lcuvcn. I Limbursi Univenitair Centrum.

0022-3654/91/2095-2047502.50/0

mode-locked, synchronously pumped, cavity-dumped picosecond dye lasers as excitation sources* combined with fast detectors like microchannel plates9 increases the time resolution and decreases ( I ) Mauer, R.; Vogel, J.; Schncider, S.Phorochem. Phorobiol. 1987, 46,

247.

(2) (a) Zachariasse, K.A.; Duveneck, GoJ. J . Am. Chem. Soc. 1987,109, 3790. (b) Andriessen, R.; De Schryver, F. C.; Boens, N.; Ikeda, N.; Masuhara, H. Macromolecules 1989, 22, 2166. (c) Weller, A. In The Exclplex: Gordon, M.; Ware, W. R., Eds.; Academic Press: New York, 1975; p 23. (3) (a) Holzwarth, A. R.; Wendler, J.; Sutter, 0. W. Blophys. Chem. 1987, 51, I . (b) Sparrow, R.; Evans, E. H.; Brown, R. G.;Shaw, D. J . Phorochrm. Photoblol., B 8101. 1989, 3, 65. (4) (a) DeclCmy, A,; RulliCre, C. Chrm. Phys. Lett. 1988, 146, 1. (b) Barbara, P. F.; Jazeba, W. In Advances In Phorochemlsrry; Wiley: New York, 1990; Vol. IS. (c) Van Damme, M.; Hofkens, J.; De Schryver, F. C. Tetrahedron 1990, IS, 4693.

Q 1991 American Chemical Society

2048 The Journal of Physical Chemistry, Vol. 95, No. 5, 1991

considerably the time needed to record a single fluorescence decay curve. This allows for a fast registration of fluorescence decay surfaces measured along different experimental axes, such as emission/excitation wavelength, concentration, pH, ... Similar advances have been made in the frequency-domain technique.I0 Improvements have also been achieved in the methodology associated with the analysis of the decay curves. In the simultaneous analysis of related fluorescence decay curves, Le., the global analysis of a fluorescence decay surface," it is possible to take into account the relations between various parameters which appear in the different decay fitting functions. This, so-called linking procedure results in a more accurate parameter recovery and a better discrimination between possible models. An additional improvement is achieved by using the reference convolution methodI2 to correct for the wavelength dependence of the response function of the instrument. This paper extends the recently introduced so-called global compartmental analysis of fluorescence decay surfaces.13 T$is method of analysis allows fitting directly to the rate constantsof excited-state processes. In addition, the emission spectra associated with each species involved in the excited-state process can be recovered. This compartmental analysis has been applied using the single photon timing technique for the study of the excited-state acid-base equilibrium of 0-naphthol." The identifiability problem of two-state excited-state processes has been discussed by Ameloot et aI.l4 In both studies, it was assumed that the experimental decay curves were properly normalized and that the fraction of the excited light absorbed by each species in the ground state was known. The method of compartmental analysis has been extended in the preceding paper.ls This new approach does not require properly normalized decay traces. Furthermore, the relative absorbances of the ground-state species do not have to be known a priori, but are fitting parameters instead. These two new features expand the applicability of compartmental analysis. In this paper the new implementation of compartmental analysis is illustrated and compared to alternative approaches of analysis. The excimer formation of pyrene in a nonviscous solvent has be& used as the experimental system because it is well understood. 11. Theory 1 . Introduction. To assure full accessibility to the method,

a short theoretical discussion of a two-state excited-state system will be given as a specific application of the theory for a n-state excited-state system discussed in the preceding paper. ~~

(5) O'cbnnor, D. V.; Phillips, D. Time-correluted Single Photon Counting, Academic Press: London, 1984. (6) (a) Tsuchiya, Y. IEEE J . Quantum Elecrron. 1984, 20, 1528. (b) Yamashita, M.; Tomono, T.; Kobayashi, S.;Torizuka, K.; Aizawa, K.; Sato, T. Phorochem. Photobiol. 1988, 47, 189. (7) Lakowicz, J. R.; Laczko, G.; Gryczynski, 1.; Szmacinski, H.; Wiczk, W. Ber. Bunsen-Ges. Phys. Chem. 1989, 93, 316. (8) (a) Spears, K. G.; Cramer, L. E.; Hoffland, L. D. Reo. Sci. Instrum. 1w%,49,255. (b) Phillips, D. A.; Drake, R. C.;OConnor, D. V.; Christensen, R. L.Anal. Instrum. 1985,14,267. (c) van Hock, A.; Visser, A. J. W. Anal. Insfrum. 1985, 14, 359. (d) McKinnon, A. E.; Szabo, A. G.;Miller, D. R . J . Phys. Chem. 1977,81, 1564. (9) (a) Yamazaki. I.; Tamai, N.; Kume,K.; Tsuchiya, H.; Oba,K. R m . Sei. Inslrum. 1985,56, 1187. (b) Bebelaar. D. Rev. Sci. Instrum. 1986,57, 116. (c) Anfinrud, P. A.; Struve, W. S.Rev. Sei. Instrum. 1989, 60, 800. (IO) (a) Fedderen, B.; Beechem, J. M.; Fishkin, J.; Gratton, E. Biophys. J . 1986, 53, 404. (b) Lakowicz, J. R.; Laczko, G.In Time-resolved Loser Specrroscopy in Biochemistry II; Lakowicz, J. R., Ed.; Proc. S.P.I.E. 1204: SPIE: 1990; p 13. (1 I ) (a) Eienfeld. J.; Ford, C. C. Biophys. J. 1979,26,73. (b) Beechem, J. M.; Knutson. J. R.; Brand, L. Photochem. Photobiol. 1983, 37, 520. (c) Knutson, J. R.; k h a m , J. M.; Brand, L. Chem. Phys. f e r r . 1983,102,501. (d) Ameloot, M.; k c h e m , J. R.; Brand, L. Bophys. Chem. 1986,23, 155. (12) (a) Ameloot, M.; Hendrickx, H. J . Chem. Phys. 1982,76,4419. (b) Lafroth, J. E. Eur. Biophys. J . 1985, 13, 45. (13) k h e m , J. M.; Ameloot, M.; Brand, L. Chem. Phys. fert. 1985,120. 466. (14) Ameloot, M.; Beechem, J. M.; Brand, L. Chem. Phys. f e r r . 1986,129, 21 I .

(IS) Ameloot, M.;

Ameloot et al. SCHEME I

'Consider a system (seeScheme I) with an equilibrium between two absorbing species 1 and 2 which form upon excitation the excited species 1 * and 2*, respectively. These excited states can depopulate via fluorescence (kF), internal conversion (k,& and intersystem crossing ( k , x ) . The deactivation rate constants are represented by kol (=kFI + klcl k,scl)and kO2(=kF2 + klc2 + klsc2) for 1* and 2*, respectively. The rate constants of exchange between the excited states are denoted by kll for the process I * 2* and by k I 2for the process 2* I*. The latter two rate constants can be concentration dependent. In Scheme I, kll is assumed to be dependent on the concentration of a molecule M. 2. General Description of T w d t a t e Excited-State Processes. Suppose the system shown in Scheme I is excited with a &pulse. The concentration dependence as a function of time for the two excited species is given by

+

-

-

The concentrations of 1 * and 2* as a function of time are given by

[I*l(t) = P I I~ X P ( - Y+I PI* ~ ) exp(y2f)

(3)

[2*1(t) = 821 ~ X P ( W )+ P22 exp(y2t)

(4)

The exponential factors y I and y2are related to the decay times bY y, = -l/rl (5) and are given by Y1,2

= - Y 2 I V 1 + Xz) 7 [((XI - X2)2 + 4~21~12[M1)11'21 ( 6 )

The expressions for the preexponential factors are

PI1 = I[~*lf.O(XI + 72) - [2*lf-ok12I/(Y2 - 71)

- 71) P21 = l[2*lf-o(~2+ 72) - [~+1f~0~21[M11/(Y2 - 71) 6 2 2 = -I[~*I~-o(X~ + 71) - [ ~ * 1 ~ - & 2 1 [ M ] ~-/ (71) ~~ PI2

= -I[~'lf-o(Xl + 71) - P*lf.Okl2l/(Y2

(7) (8) (9) (10)

with XI = kol + k21[Ml

(11)

x2 = k02 + k12

(12)

The spectral emission density at wavelength hem,p(Xm), is defined as

.m FAAem)

p(xcm)

=

(13)

with F, the steady-state fluorescence spectrum. The fractionf, of the detected emission as a function of time

The Journal of Physical Chemistry, Vol. 95, No. 5, 1991 2049

Fluorescence Decay Surfaces over ihe emission band AXm for each of the two emitting species is given by

fi(Am,O = ~ F I [ ~ * I ( ~ ) L ~ dXCm ~ P I =( c1 ~ (Acm) ' ~ ) [ 1*I(t) ji(Xm,t)

/

(14)

= ~ 2 * 1 ( r ) S , , _ p ~ ( ~ cdm ~) = m c2(Xcm)P*l(t) (15)

02

which become after substitution of eqs 3 and 4 fi(Xcm,Acx,t) = cl(~cm)[811(~cx) expht)

+ Bl2(XcX) exp(y2t)l (16)

f2(XCm,XQ9r) = c ~ ( A ~ ) [ & I ( X ~ exp(yl0 ~) + 822(Acx) exp(y2t)l (17)

cI and c2 depend only on the emission wavelength XCm, and Bij depends on the excitation wavelength XeX. y I and y2 are independent of both emission and excitation wavelengths. The total detected fluorescence as a function of time at a given excitation and emission wavelength is then given by

+

= fi(Xem,hcx,t) f2(Xcm,Xcx,t)

f(Xcm,Xu,t)

= [c1811 + c4211 exp(y1t) + [CIS12 + c28221 exP(Y2r) which can be formally written as f(Xcm,Xcx,r) = al exp(ylt)

+ a2 exp(y2t)

(18) (19)

3. CfassicafAnalysis. In this analysis the decay traces are analyzed in terms of preexponential factors aIand a2(eq 19) and decay times -yI-' and -y2-I (eq 6). The parameters of interest to be retrieved, namely the rate constants, the spectral emission density, and the ratio [I *],=o/[2*],,o, can be calculated using eqs 6-19. The rate constants can be calculated from the eigenvalues y I and y2. A detailed description of the methods for calculating the rate constants will be given below. Global analysis in terms of exponential decay times (eq 19) allows linking of the decay times over different emission wavelengths and time increments. LofrothI6 has shown how to calculate cI and c2 once the rate constants k, are known. Global analysis according to eq 19 will not necessarily improve the accuracy of cI and c2 because the information is retrieved from the preexponential factors al and a2,which are functions of the concentration of M and therefore cannot be linked. The ratio [ 1*]rlo/[2*],M can only be calculated if the emission of 1 * and 2* can be measured separately. If c2 = 0, only emission from species 1 * is detected, and when cI = 0, only emission from spectra 2* is detected (eqs 7-10). Therefore, assumptions have to be made about the spectral densities of the fluorescence spectrum. Since the ratio [ 1 *],=o/[2*],,o has to be obtained from the preexponential factors a I and at, its accuracy will not necessarily improve by global analysis according to eq 19. 4. Application of the Classical Approach to the Intermolecular Excimer Formation of Pyrene. The system studied in this paper is the intermolecular excimer formation of pyrene in a nonviscous solvent. [MI is the concentration of pyrene, 1* denotes the locally excited state of pyrene, and 2* is the excimer state. Doller and F6rsterI7 have shown that [2*],,0 = 0. This means that, within the studied concentration range, no preformed excimer is present in the ground state (see Scheme 11). The equations for the preexponentials pij are then given by XI + Y I

-ff2= - =81-2ffl

XI + Y2

PI1

if only excimer emission (cI = 0) is detected. There are several ways to determine the four rate constants kol, k21,kO2,k I 2by use of the decay parameters obtained from biexponential fits. In a first method,l* denoted as method I, y1and y2are determined as a function of pyrene concentration [MI. The theoretical curves according to eq 6 are visually adjusted to the experimental data with suitably chosen values for the rate constants. A second method of evaluating the rate constants1*(method 11) involves fluorescence decay measurements at only two concentrations. At very low [MI, the decay is monoexponential with -yI = kol. At higher [MI where the decay is biexponential, yl, y2, and the ratio q / a 2are used to obtain the rate constants by using eqs 6 and 20. The ratio aI/a2equals /311//312(eq 20) only if the monomer emission can be monitored separately at some emission wavelength. A third method of calculating the values of the rate constantsI9 (method 111) consists of plotting -(yl y2)and yIy2as a function of [MI. The slopes and the intercepts of these plots yield four , equations with four unknown parameters, which makes it possible to calculate all rate constants. This method needs at least two different excimer forming concentrations. These methods will be compared to the global compartmental analysis in which the date are analyzed directly in terms of the parameters of interest. . 5. Compartmental Analysis. As has been described above, the fitting parameters of a global biexponential analysis (eq 19) are the preexponential factors al and a2and the eigenvalues y I and y2. The parameters of interest are hidden within these fitting factors. Therefore, it would be preferable if the analysis could determine those parameters directly. This is achieved by the so-called global compartmental a n a l y ~ i s . ~ ~ ? ~ ~ Define the following matrices

+

*

[

k2l[Ml)

I ; ; ;2;

kl2 -(k02

+ kl2)

A is called the transfer matrix of the system. Equations 1 and 2 can be rewritten as dX*/dt = AX*(t) (24) The fraction b, of the excitation light absorbed by species i is given by b, = W l O (25) where Io is the incident light flux and la,idenotes the light flux absorbed by species i. Equation 25 can be written aslS

if only monomer emission (c2 = 0) is detected and -ff2= - = -822 I a1

821

where e, denotes the molar extinction coefficient of the ith species, d is the path length, and [ I ] and [2] are the concentrations of ~

(16) Lbfroth, J, E.J. Phys. Chem. 1966, 90,1160. (17) Dlller, E.; Fanter, Th. Z . Phys. Chcm. Munfch 1961, 34, 132.

(IS) Birks, J . B. Rep. Prog. Phys. 1975, 38, 903. (19) Cheung, T.C.; Ware, W. R. 1. Phys. Chem. 198%87.466.


species 1 and 2 in the ground state. The boundary conditions at time t = 0 are given by

Ameloot et al. The steady-state fluorescence spectrum F,can be decomposed into the underlying spectra of the two emitting species, the so-called species associated emission spectra (SAEMS).I5 The SAEMS for species i is given by

The matrix B is defined as

B=

[ :I

The inverse of the transfer matrix A is given by

Assuming delta function excitation at time t = 0, the boundary conditions for eq 24 are given by

X*(O) = B The solution of eq 24 is given by

(29)

X*(t) = UerfD

(30) The columns of U are the eigenvectors of A. U is defined by

AU = ur and I' is the eigenvalue matrix of A given by

(31)

(33)

The column matrix 29 and 30):

D is determined by the initial conditions (eqs

D = U-IB Equation 30 can be written as

(34)

X*(t) = UerfU-'B (35) U and $'are functions of the rate constants kol, k21,kO2,and kI2 and [MI, while B depends on Xcx and [MI. Defining the row matrix C as

c = [CI

czl (36) the fluorescence &-responseat emission wavelength XCmis given by f(Xcm,Xcx,t) = C(Xcm)UerfU-lB(~cx)

(37)

which is equivalent to eq 18. C(Xem)depends only on the emission wavelength and B(XCX) depends on the excitation wavelength. UerrU-' depends on the rate constants of the system and the concentration of M. The estimated C(Xm) can be used to calculate the species associated emission spectrum (SAEMS) of each of the two emitting specie^.'^ The estimated B(Xex) allows the calculation of the ratio [ I *lr=0/~2*lr=0. Experimentally the fluorescence b-responsef(Xcm,hcx,t)(eq 37) can be determined within a proportionality constant. Therefore, we use with elements f, defined as

e

pi

=

C,/(CI

+ cz)

(38)

Note that the E elements are different from the previously definedI3 SAS (species associated spectra) elements c. Similarly, the matrix B is defined with elements 6,

6, = b,/(bl + 62)

(39) The use of and b allows one to link T I at a given emission wavelength and 6 , at a given excitation wavelength and a concentration [MI, irrespective of the collection time of the considered decay traces. Equation 37 can now be written as em, Xex,t) = KQ hem) U e ' W 8 ( ex) (40)

with

K

a proportionality constant.

111. Experimental Methods

I . Program Implementation. The global compartmental analysis of the fluorescence decay surface of species undergoing excited-state processes was implemented in the existing general global analysis program. The generalized global mapping table approach described previously20allows simultaneous analysis of experiments done at different excitation/emission wavelengths and at multiple concentrations. Convolution with measured excitation pulses or monoexponential reference decay traces (see below) is possible. Any or all decay parameters can be kept fixed during the fitting, or may be freely adjustable to seek optimum values. The program allows the analysis of two-state and three-state compartmental systems. Consider the two-state excited-state process depicted by Scheme I. The global fitting parameters are kol, k02, k12,k2,, E l , 61, and possibly the reference lifetime 7, (see below). The only local fitting parameters are the scaling factors. Once a kinetic model (two-state or three-state excited-state process) is chosen to describe the decay data surface, the corresponding transfer matrix A can be constructed. Note that this transfer matrix is unique for the model under consideration. The eigenvalues y and the associated eigenvectors are determined by using routines from EISPACK, Matrix Eigensystem Routines.21 The scaling matrix D (eq 34) for th_e eigenvectors is computed according to the actual values of B. This involves the solution of a linear system of two or three equations in two or three unknowns respectively. The fluorescence &response of the sample, f,(Xcx,Xcm,f), is then biexponential or triexponential n

fs(XcxJcmlt) =

&I

is I

exp(yit)

(43)

with a, defined as a, = d i ( t U i )

(44) d, is the ith element of the column matrix D (eq 34) and U i is the ith column of U (eq 31). In an ideal single photon timing experiment, the time-resolved fluorescence profile of the sample ds(Xcx,Xcm,t), obtained by excitation at wavelength Xcx and observed at wavelength XCm, can be written as

d,(r) = ~ r0 ~ ( X e x , A e mfs(Xex,Xcm,t-~) ,~) ds

(45)

where u(Xcx,Xem,t)denotes the instrument response function. We used the reference convolution method22 to correct for the wavelength dependence of the instrument response function. In ~~

(20) Boens, N.: Janssens, L. D.; De Schryver, F. C. Biophys. Chem. 1989, 33, 77. (21) Smith, B. T.; Boyle, J. M.; Garbow, 8. S.;Ikeke, Y.; Klema, V. C.; Moler, C.B. In Lecture Notes in Computer Science; Coos, G., Hartmanis,

Eds.; Springer Verlag: Heidelberg, FRG, 1974; Vol. 6. (22) (a) Gauduchon, P.; Wahl, P. Biophys. Chem. 1978'8, 87. (b) Wijnaendt van Resandt, R. W.; Vogel, R. H.; Provencher, S. W. Reu. Sci. Instrum. 1982, 53, 1392. (c) Lbfroth, J . E. Eur. Blophys. J . 1985, /3, 45. (d) Zucker, M.; Szabo, A. G.;Bramall, L.; Krajcarski, D. T.; Selinger, 8 . Reo. Sci. Instrum. 1985, 56, 14. (e) Van den Zegel, M.; Boens, N.; Daems, D.: De Schryver, F. C.Chem. Phys. 1986, 101, 31 I . (f)Boens, N.; Janssens, L. D.;De Schryver, F. C. Springer Ser. Chem. Phys. 1988, 48,486.


Fluorescence Decay Surfaces

TABLE I: Rate Constants for Pyrene Photophysics in Cyclohexane Accordinn to Birks" ko2 = 15.5 X 106 hi kol = 2.25 X IO6 s-I k12= 6.5 X lob s-I kZ1= 6.7 x 109 M-1 5-1

this method the parameters of the decay of the samplef;(XCX,Xm,t) are obtained from the measured fluorescence decays of sample d,(Acx,Xcm,f) and reference d,(Xw,Xm,t)observed under identical instrumental conditions:

d,(t)

~ '0d , ( X c X , A mx(XeX,XCm,t-s) ,s) ds

(46)

TABLE 11: Calculated Fluorescence h y Times, Based on the Rate Constants Shown in Table I. at Different Pyrene Concentrations [MI,mol L-I yI-', ns - y c ' , ns 443.56 45.45 1 x 10-6 161.43 40.40 I x 10-3 93.08 29.76 3 x 10-3 70.91 12.96 I x 10-2

If the reference compound decays with single-exponential kinetics

= a, exP(-t/Tr) (47) (where a, is a scaling factor) t) that satisfies eq 47 is given by

x(

where 6 ( t ) is the Dirac delta function andfl(t) denotes the time derivative. Specgying initial guesses for the scaling factor, kol, kO2,k I 2 , k21,f l , bl, and 7, (Scheme I), or for_the-scaling factor, kol,kO2. k039 k12, kI33 k21, k23, k31, k32r f l , Z2, bl, b2, and T,, allows one to calculate the predicted time-resolved fluorescence response as a function of Xcx, Xem, concentration, and timing calibration. Incremental changes of the parameters are obtained by using Marq~ardt's'~algorithm. This requires the computation of the partial derivatives of the predicted response with respect to all parameters. A two-point formula was used to numerically evaluate all derivative^.^^ Using this approach, experiments done at different excitation/emission wavelengths and at multiple concentrations are linked by all rate constants defining the system. Of course, this linkage is valid only in the concentration range where the proposed kinetic model is valid. The E elements are the same in those experiments collected at the same emission wavelength Aem, irrespctive of concentration and excitation wavelength Acx. The 6 elements are dependent on Acx and on concentration, irrespective of ACm. The fitting parameters were determined by minimizing the global reduced chi-square x: (49) where the index I sums over q experiments and the index i sums over the appropriate channel limits for each individual experiment. YO,, and PI,denote respectively the observed (experimentally measured or synthetic) and calculated values corresponding to the ith channel of the Ith experiment, and w, is the corresponding. statistical weight. v represents the number of degrees of freedom for the entire multidimensional fluorescence decay surface'. The statistical criteria to judge the quality of the fit included both graphical and numerical tests. The graphical methods comprise plots of surfaces ("carpets") of the autocorrelation function values vs experiment number, and of the weighted residuals vs channel number vs experiment number. A good fit should produce carpets free of pronounced "creases". The numerical statistical tests include the calculation of the global reduced chi-square statistic x; and its corresponding Zx2, Since Z d is standard normally distributed, theoretical probabilities of ZQzvalues occurring within a given range can be easily obtained from the cumulative standard normal distribution. Using Zx, the goodness of fit of analyses with different Y can be readily compared. 2. Strategy toward Compartmental Analysis. The sequence in the analysis of fluorescence decay curves was as follows. First, the decay curves were analyzed individually as either mono- or biexponential functions. Second, the decays' arising from solutions with identical pyrene concentration were analyzed globally in terms of preexponential factors and decay times (eqs 5 and 19) with the latter being linked. To obtain an optimum fit, as judged by the statistical goodness of fit parameters, one has t o adjust the (23) Msrquardt, D. W. J . Soc. Indust. Appl. Math. 1963, 1 1 , 431. (24) Dorn, W. S.; McCracken, D. D. Numerical methods with Fortran IV Case Studies; Wilcy: New York, 1972; pp 221-236.

'

backgrounds of sample and reference of the individual decays. Possibly a zero-time shift of the reference is needed. Third, initial guesses for the rate constants were calculated from the estimated decay times according to method 111. Finally, all the fluorescence decays were combined in a compartmental analysis. 3. Synthetic Data Generation. Synthetic reference and sample decays were generated by convolution off,(t) andfs(t), respectively, with the same nonsmoothed measured instrument response function u(t). Many different experimental u(t) were used in the simulations. Each matching ds,d,pair was generated with different Poisson noise. The preexponential factors were adjusted to obtain the desired number of counts. All computer-simulated decays had l/'K or 1/2K data points with 104 counts in the peak channel. The time increment per channel was 1,2, or 3 ns and the reference lifetime was 15 ns. Full details of the decay data simulations are given elsewhere.25 The synthetic data generations and all individual and global analyses were done on an IBM 6150-125 computer with single precision. 4. Instrumentation. Fluorescence decay curves were obtained by using the 325-nm excitation of a Spectra-Physics synchronously pumped, mode-locked, cavity-dumped, frequency-doubled DCM dye laser with single photon timing detection. All fluorescence decay curves were collected in 1/2Kdata points of the multichannel analyzer and contained between 5 X IO3 and IO' peak counts. Details of the fluorescence lifetime apparatus and the associated optical and electronic components are described elsewhere.20 Fully corrected steady-state fluorescence spectra were recorded in the front-phase configuration on a SPEX Fluorolog 212. 5. Chemicals. Pyrene (zone refined) and isooctane (Merck, ' Uvasol) were used as received. The solutions were degassed using five freeze-pumpthaw cycles. The correct pyrene concentrations were recalculated after degassing by measuring the absorbances at 304 nm (e = 12000 M-' cm-I).

IV. Results and Discussion I , Computer-Generated Data. To investigate the performance of the new implementation of the compartmental analysis, fluorescence decays of pyrene at several concentrations in hexane were simulated, using the rate constants at room temperature published by Birks et (Table I). The photophysics of pyrene at concentrations higher than M in a nonviscous solvent obeys Scheme I1 after excitation at 343 nm. By use of the rate constants reported in Table I, the decay times - l / y l and -I/y2 were calculated for different pyrene concentrations from eq 6. These calculated decay times are listed in Table 11. In Figure 1 the theoretical ratio of the preexponentials, R (eqs 18, 19), given by

is plotted against E2 (= 1 - t l )for the three highest pyrene concentrations of Table I I . R, calculated from the estimated preexponentials obtained by single-curve analyses, is also given in Figure 1 , Equation 51 reduces to eq 20 if only monomer (25) Van den Zegel, M.;Boens, N.; Daems, D.; De Schryver, F. C. Chem. Phys. 1986, 101, 311. (26) Birks, J. B. Phoiophysics of Aromatic Molecules; Wiley: London, 1970.

Ameloot et al.


: c

SCHEME III: Llaking Scheme for a Global BkxponenH.1 Analysis of the Decays at Each Concentration [MY

i x n

I

a Boxed decay parameters are linked, while X denotes unlinked parameters K C X , and K C X where ~ K denotes a scaling factor (eq 19). See text for more details.

SCHEME I V Linking Scbew for a Global Two-State Compartmental A ~ l y of~ the b Decays at i Different Concentrations and I ) Different Emission Wavelengths AN Collected at a Single Excitation Wavelength X" a

18

~

I 16

-7

-11_ _ _ _U--__a 1 . 1

I A

14-

7c. w

-

YI

'

2

12-

x

8

I

6 T

0 02

0

0 06

0 04

0 08

0 1

1 -F1

6 4 -

6.2

-

':j

~,

,

,

,

,

,

,

,

5 9

0

0 02

0 04

I I I

*Boxed_parameters are linked, while X denotes the unlinked scaling factors. b, depends on the concentration [MI,. See text for more details.

0

6.3 -

3 L3 k02

A

A

r

0 06

0 08

0 1

1-7,

Figure 2. Calculation of the rate constants as a function of Fz (=1 - F,) when method I1 is used (see text for more details) for a 0.01 and IO4 kO;, M simulated solution of pyrene: (--) theoretical values (A) (0) (m) k021, (A)kl~',and (A)k12S; (B) (0)k2,', and (e) k2,g. s = single curve exponential analysis and g = global exponential analysis.

emission is detected, and to eq 21 when only excimer emission is detected. Figure 1 shows that, a t a pyrene concentration of 0.01 M, the ratio R is strongly dependent on the spectral overlap between excimer and monomer. For instance, at 1 - E, = 0.01, there is already a decrease of 8.5% in the ratio R of the preexponentials, in comparison to the value at 0% spectral overlap (E, = I ) . This means that, if the kinetic parameters are calculated according to method 11, there should be 90 spectral overlap. Otherwise an incorrect value for R will be used, resulting in wrong values for the rate constants. This is shown in Figure 2 where the rate constants are plotted as a function of Z2. Compared to the true values for the rate constants calculated with R at 2, = I , kO2and k I 2show a negative deviation, whereas kZl shows a

positive deviation if the R factor is used for E, < 1, Le., when spectral overlap exists. If the fluorescence decays of the 0.01 M pyrene solution are analyzed globally as a biexponential function by linking the two decay times 1 / r l and 1 /y2 at the different E , values, the same deviation of the rate constants is observed when the R-factor is used for f l < 1 (Figure 2). Furthermore, as the pyrene concentration is increased, the excimer band progressively dominates the emission spectrum and the probability of excimer overlap at the monomer region increases. As the pyrene concentration is lowered, the R factor becomes gradually smaller at high f, values (Figure I), making an accurate recovery of the R factor at E l = 1 more difficult. In method 111, only the decay times are needed to calculate the rate constants. Two fluorescence decays at two different concentrations at any emission wavelength suffice to calculate the rate constants of the system. Hence, spectral overlap will not effect the result of the calculation. Schemes 111 and 1V compare the linking schemes of global biexponential and global two-state compartmental analyses of experiments at a single excitation wavelength. As can be seen from these schemes, global compartmental analysis allows a more extensive linking of the fitting parameters as compared to global biexponential analysis. Global compartmental analysis creates the possibility to link experiments at

I



SCHEME V Possible Minimum Linking Schemes To h i v e a General Two-State Excited-State Reaction (Scheme 1)o maaurementa: 2 concentrations, 1

measuremanta: 2 concentrations, 3 A"

biexponential

global tin-state compartamntal

global biexponential

global two-state compartmental

8 parameters

9 parameters

16 parameters

15 parameters

measurements: 3 concentrations, 1 biexponential

hem


yl[

measurements I 3 concentrations, global biexponential 12 parameters

11 parameters


Xem

Y

global two-state compartmental X

X

hem

Y

X

measurements: 2 concentrations, 2

2

global two-state cosnp.rtmental

X

Y

X

,

X

18 parameterr

12 parameters

15 parameters

12 parameters

OX denotes the preexponentials ~a~and K U ~ .Ydenotes the unlinked scaling factors. y I ,y2, and 6, are concentration dependent. El depends on the emission wavelength XCm only.

different concentrations, resulting in an additional gain in accuracy and model discrimination. Because the rate constants are independent of both the concentration and the emission wavelength, they can be linked over the emission wavelengths and the concentrations. As has been shown before, f l depends only on the emission wavelength and therefore can be linked over the concentrations at a given Am. b, depends on the excitation wavelength and the concentration and therefore can be linked over the emission wavelengths at a given concentration. Global biexponential analysis (eq 19) of the decay surface comprised of experiments at different concentrations does not allow one to link the decay times because they are concentration dependent (eq 6). For optimum recovery of the decay times, experiments at different time increments and at different emission wavelengths can be simultaneously analyzed at each concentration.20 As has been shown by Daller and Farsterl' for the pyrene system, the light is only absorbed by the monomer, which means that 6 , equals 1, The use of eq 20 for the calculation of the rate constants implies the following assumptions: ( I ) there is no preformed excimer present in the ground state and (2) monomer emission can be measured without spectral, overlap. These assumptions are not required in the approach ddscribed in this paper. Therefore, one can check if there is preformed excimer in the ground state in the concentration range studied and if there is spectral overlap at the observation wavelength. In the accompanying paper15 the identifiability problem of a two-state excited-state process was discussed. To exemplify this theory, the identifiability problem of the pyrene system will be discussed. In the following sections, we investigate the experimental conditions to determine hl,E l , and the rate constants of the pyrene

system. Because unnormalized fluorescence decay curves are used, the scaling factor, for each decay curve has also to be determined. A schematic overview is given in Scheme V, where the number of estimated parameters from a global biexponential analysis are compared to the number of estimated parameters from a global two-state compartmental analysis. Scheme V is the general linking scheme for a two-state compartmental system as depicted in Scheme 1. For the intermolecular excimer formation of pyrene, the assumption of the nonexistence of a preformed excimer in the ground state can be-made within the studied concentration range. This mean? that b, equals 1 for all pyrene conc-entrations. Therefore, b, can be treated as a variable or fixed (b, = 1) parameter linkcd over the different pyrene concentrations. This specific case is shown in Scheme VI. Starting from the biexponential fitting parameters, the preexponentials ai and the eigenvalues Y/, equations can be constructed which lead to the solution of the system. The sum and product of the two decay times at two different pyrene concentrations yield four equations from which the rate constant values can be derived (method 111). For every other pyrene concentration added, no new independent equations are obtained from the decay times only. As has been shown in the preceding paper, for each decay curve two independent equations can be constructed (eqs 52 and 531,

mo = Y(a, + az)

(52)

ml = Y ( W l + a2721

(53)

where Y denotes the experimentally estimated scaling factor. Those two equations can k used to calculate hl, E l , and the scaling


Ameloot et al.

TABLE III: Recovered Panmeter Values (est) Accordtmg to Schemes V and VI from Simulated F'ylvllc Data' [MI, mol L-l

fl

true

k019

6,est

est

XI@

s-l

k21, X1O4 M-I

k02,

s-I

X l o d 5-1

k121

Xlod

s-'

A

0.7 f 1.7

8.5 f 1.7

13 f 1.4

3.9f 2

0.75 f 0.3

If6 If5 0.9 f 3 (I)

2*2

6.7 f 3

17.3 f 4

4.9 f 1

0.75

0.75 f 0.2

0.9 f 1 (I)

2

* 1.6

6.7 f 1.4

15.6 f 2

6.6 f 0.4

0.75

0.745 f 0.006

1 (c)

2.0 f 0.3

6.8 f 0.4

15.4 f 0.2

5.8 f 0.9

0.001 0.003 0.0I 0.001 0.003 0.01 0.001 0.003 0.0I 0.001 0.003 0.01 0.001 0.003 0.01 0.001 0.003 0.01

0.95

0.95 f 4

1.0 f 8 0.96 f 13 0.99 f 34

2.2 f 0.1

6.7 f 0.1

15.5 f 0.1

6.3 f 0.3

0.95

0.943 f 0.006

1.08 f 0.05 (I)

2.37 f 0.05

6.6 f 0.1

15.6 f 0.2

6.6 f 0.3

0.95

0.951 f 0.003

1 (c)

2.35 f 0.05

6.7 f 0.1

15.5 f 0.2

6.6 f 0.3

0.05

0.1 f 0.8

1.1 f 1.1 1.1 f 1.2 1.1 f 1.3

2.27 f 0.06

6.77f 0.09

15.54f 0.01

6.8 f 0.2

0.05

0.03 f 0.03

0.98 f 0.03 (I)

2.29 f 0.06

6.7 f 0.09

15.55 f 0.08

6.8 f 0.2

0.05

0.044f 0.002

1 (c)

2.31 f 0.05

6.72f 0.07

15.57 f 0.08

6.8 f 0.16

0.001

0.95 0.9

0.84 f 0.05 0.81 f 0.05

1.5 f 0.8

0.5 f 1.3

8.7 f 1.4

13f 1

3.9 f 1.5

0.95 0.5

1 .O f 0.08 0.5 f 0.03

0.9 f 0.6 1.1 f 0.4

2.8 f 0.2

5.6 f 0.5

17f 1

7.1 f 0.4

0.95 0.05

0.88 f 0.04 0.02f 0.08

1.1 f 0.3 1.3 f 0.2

2.2 f 0.1

6.8 f 0.2

15.5 f 0.3

6.6 f 0.2

1.3 f 0.2 1.14f 0.06

2.25 f 0.08

6.7 f 0.1

15.54f 0.08

6.6 f 0.2

1.17 f 0.07 1.03 f 0.03 (I)

2.36 f 0.07

6.6 f 0.1

15.56 f 0.08

6.7 f 0.2

6.68 f 0.08

15.52 f 0.07

6.6 f 0.2

0.001 0.003 0.001 0.003 0.001 0.01 0.001 0.01

1

0.9 f 0.9

0.75

B

C 0.003 0.001 0.003 0.001 0.003 0.001

0.01 0.001

0.95 0.05

0.931 f 0.008 0.I5 f 0.03

0.95 0.05

0.949 f 0.005 0.07f 0.03

0.95 0.05

0.952f 0.004 0.044 f 0.002

I (c)

2.32 f 0.06

0.9 0.75 0.5

0.85 f 0.04 0.73 f 0.02 0.54 f 0.05

1.1 f 0.2

2.4 f 0.4

6.4 f 0.8

15.7 f 1.2

6.4 0.7

1.3 f 0.6 1.01 0.06

2.3 0.3

*

6.6 0.2

*

15.5 f 0.2

6.5 f 0.9

1.1 f 0.3 1.03 f 0.02

2.29 f 0.08

6.7 f 0.1

15.54 f 0.06

6.7 h 0.2

1.05 f 0.02 1.03 f 0.02 (I)

2.37 f 0.07

6.62 0.08

* 0.06

6.8 f 0.2

0.01

0.001

0.01

D 0.001 0.003 0.001

0.01 0.001

0.01 0.001 0.01

0.75 0.5

0.88 f 0.03 0.14 0.03 0.50 f 0.01

0.95 0.5 0.05

0.946 f 0.004 0.499 f 0.004 0.077 f 0.017

0.95 0.5 0.05

0.952 f 0.004 0.498 f 0.004 0.067 f 0.017

0.9

*

*

*

15.58

*



TABLE 111 (Continued)

[MI?

fl

6,est

mol L-I

true

est

0.001

0.95 0.5 0.05

0.952 f 0.004 0.495 f 0.004 0.044 f 0.002

1 (4

2.32 f 0.06

0.95 0.9

0.89 f 0.03 0.85 f 0.03

1.2 f 0.12

6.67 f 0.07

15.54 f 0.06

6.64 f 0.02

2.2 f 0.7

6.7 f 0.1

15.4 f 0.14

6.3 f 0.3

1.08 f 0.04

2.32 f 0.04

6.67 f 0.06

15.59 f 0.06

6.76 f 0.13

1.08 f 0.04 1.10 f 0.05 1.03 f 0.02 (I)

2.33 f 0.04

6.67 f 0.06

15.56 f 0.06

6.69 f 0.13

1 (4

2.31 f 0.03

6.71 f 0.05

15.53 f 0.05

6.65 f 0.12

0.01

E 0.001 0.003 0.0 I 0.001

1.2 f 0.2 1.5 f 0.4 0.95 0.05

0.941 f 0.005 0.11 f 0.03

0.003 0.0 I 0.001

0.95 0.05

0.949 f 0.004 0.07 f 0.02

0.95 0.05

0.952 f 0.003 0.044 f 0.002

0.003 0.01 0.001 0.003 0.01

true is I for all experiments. The linked 6, are marked with (I). A 6, followed by (c) means that this parameter is kept constant during the calculation. All errors are 2 X SD. SCHEME VI: Possible Minimum Linking Schemes To Solve the Pyrene System According to Scheme 11"

Xem

m a r u r e m n t i I 2 concentrationi, I hem

m a i u r m e n t s : 2 concentrationi, 3

biexponential

global two-atate compartmental


global two-atate compartmental

8 prmuterr

8 parameter8

16 parameter8

14 parameter8

.

m a a u r e m n t s I 3 concentrationa, 1 Xem biexponential


maiurement8i 3 concentrationa, 2 global biexponential 1 2 parmuterm

hem

global two-itate compartmental

9 parmetern X

m a s u r e w n t a r 2 concentrations, , 2hem g l o b 1 biexponential

12 parmuters


3

X

X

18 p a r m t e r s

13 parmutera

11 p r m u t e r a

" X denotes the preexponeatials KaI and K ( T ~ . Y denote the scaling factors. y I and y 2 are concentration dependent while 6,can be linked over the concentration range. 2, depends on the emission wavelength Xem only.

2056 The Journal of Physical Chemistry, Vol. 95, No. 5, 1991 factor Y (Scheme V). %and mI will be used for different pyrene concentrations and emission wavelengths. In the first set of experimental conditions (Scheme VA), fluorescence decays at two different pyrene concentrations measured at the same emission wavelength are considered. The four rate constants can be determined from the foutdecay times. The remaining unknown parameters are F,, two b, values, and two scaling factors Y , a total of five. These have to be determined from four equations (eqs 52 and 53). Hence, the data set depicted in Scheme VA is not sufficient to solve the system. This is shown in Table IIIA. Linking without fixing bl (Scheme VIA) makes one unknown parameter disappears and the system should be solvable. However, no accurate parameter recovery is obkained (Table IIIA) by using the pyrene concentrations 0.001 and 0.003 M. A better set of concentrations, 0.001 and 0.01 M pyrene, results in a more_accurate but not yet acceptable parameter recovery. Fixing bl at unity (Scheme VIA, b, = 1) yields a good accuracy of all recovered parameters (Table IIIA). In this latter case, four equations (eqs 52 and 53) are available and only three parameters have to be estimated (F1 and two scaling factors Y). In the second experimental setup, Scheme VB, the fluorescence decays of three different pyrene concentrations at one single emission wavelength are used in the global bicompartmental analysis. At first sight, this set of experiments should suffice to calculate all parameters. Indeed, 12 parameters are available from the biexponential fitting (six cyi and six r,),whereas 1 1 parameters (four k,, E l , three bl values, and three scaling factors Y) have to be estimated from the bicompartmental analysis. However, as has been pointed out before, the decay times of a third concentration do not lead to any new independent equations. The four rate constants can be calculated by using the two decay times at two different concentrations. Equations 52 and 53 allow us to construct six more equations. Hence, the bicompartmental linking Scheme VB does not allow the estipation of all the parameters of interest (Table IIIB). Linking bl (Scheme VIB) diminishes the number of unknown parameters to five, which can be delermined from the six available equations (Table IIIB). Fixing b, at unity slightly improves the accuracy of the recovered parameters (Table IIIB), because there is one parameter less to be determined (six equations for four unknown parameters). In the third experimental setup, the fluorescence decays of two different pyrene concentrations each at two different emission wavelengths are combined in one global bicompartmental analysis according to Scheme VC. The rate parameters can be determined from the two sets of decay times. Equations 5 2 and 53_allow the construction of 8 equations for the calculation of two bl values, two F, values, and four scaling factors Y,a total of eight unknown parameters. As was pointed out in the previous paper, this set of experimental conditions can give rise to two different solutions. However, th_e restrictions of positive values for the rate constants, and P I and bl being limited to the range beyeen zero and unity, can lead to the unique solution for the pyrene system. Changing the initial guesses for the bicompartmental analysis had no influence on the estimated values. As is shown in Table IIIC, the accuracy of the parameter recovery improves by combining experiments with extreme 2, values. For real experiments, this implies that including fluorescence decays collected at the edges of the emission spectrum will contribute the most to the improved parameter recovery.20 Increasing the concentration difference enhances th; accuracy of the parameter recovery (Table IIIC). Linking of b , (Scheme VIC) reduceslhe number of parameters to be estimated by one. Fixing of bl reduces the number of parameters to be estimated by two. Both have a beneficial effect on the accuracy of the parameter recovery (Table IIIC). The fourth experimental setup uses the fluorescence decays of two different concentrations at three different emission wavelengths (Scheme VD). The four rate constants can be calculated from the two sets of decay times. Twelve equations can be constructed from eqs 52 and 53 for the calculation of 1 I parameters: two b, values, three f, values, and six scaling factors Y. As shown in the preceding paper, this set of experimental conditions has a unique solution. This was indeed found for the simulated pyrene data as is shown in Table HID. Linking b, (Scheme VID), fixing

Ameloot et al. using extreme f,, and increasing the concentration difference improve the accuracy of the parameter recovery (Table IIID). The fifth set of experimental conditions (Scheme VE), three concentrations and two emission wavelengths, should have a unique solution as was shown in the accompanying paper.I5 The rate constants can be calculated from the decay times. Equations 52 and 53 can be used to cocstruct 12 equations for the calculation of 1 1 parameters: three bl values, two El values, and six scaling factors Y. This is shown in Table HIE. Similar conclusions can be drawn as in the third and the fourth set of experimental conditions (Scheme VIE and Table IIIE). The results of Table 111 can be summarized as follows: (1) The number of independent equations should match or exceed the number of parameters to be determined. (2) Increasing the concentration difference, increasing !he difference between the ?, values, and linking and/or fixing 61 improve the accuracy of the parameter recovery. Furthermore, the validity of the condition for identifiability’5given by

(54)

nconcnem 2 nconc -I-nem

was demonstrated. n, denotes the number of fluorescence decays at different concentrations included in the bicompartmental analysis. ncmis the number of fluorescence decays at different emission wavelengths. For n,,, = 2 and ncm= 1 , condition 54 is not valid and the set of experimental conditions is not sufficient to solve the system. This was indeed found (Scheme VA). The same was true for nWnc= 3 and nem= 1 (Scheme VB). For the set of experimental conditions according to Scheme VC (n,,, = 2, nem= 2), VD (nmnc= 2, ncm= 3), and VE (nconc= 3 , ncm= 2), eq 54 is valid and the system is identifiable, as was indeed found. The simulated fluorescence decays of pyrene at concentrations 0.001,0.003, and 0.01 M, each at nine different P I values together with a simulated fluorescence decay of pyrene at 10” M were analyzed globally (a total of 28 simulated fluorescence decays) as a bicompartmental system according to Scheme IV. The recovered rate constants are listed in Table IV and are in perfect agreement with the input values. Figure 3 shows the recovered P I values when they are linked over the different concentrations. The standard deviations on the recovered F, are approximately 0.003. To investigate the advantage of using normalized matrices, the same 28 simulated decay curves were globally analyzed as a bicompartmental system with unlinked and unnormalized c elements. In this case the scaling factor is included in the c elements. Afterwards, the estimated cI values were normalized to unity and compared to the linked f, estimates (Figure 3). The standard derivatives of these normalized cI estimates are approximately-0.5. This demonstrates the advantage of using normalized C matrices. 2. Real Data. Three solutions of pyrene in isooctane were prepared with concentrations of 1.53 X 1.45 X 10-4, and l P M. Upon excitation at 325 nm, the solutions with the two highest concentrations exhibited excimer emission at room temperature. The ratio of the fluorescence quantum yields of excimer and monomer, @FE/@FM = @p2/GFI, was equal to 9 for the 1.53 X IO-’ M solution and 0.82 for the 1.45 X M solution. The fluorescence decays of the 1.53 X and 1.45 X IO4 M solutions wcre collected every IO nm starting from an emission wavelength of 375 nm. The decays were analyzed globally by using the linking scheme shown in Scheme 111. The rate constants can be calculated from the estimated decay parameters (see above). Assuming no spectral overlap at 375 nm, the concentrations 1.53 X IO-’ M/lOd M and 1.45 X IO4 M/lOd M can be used in method 11. The concentrations 1.53 X 1 O-’ M/ 1.45 IO4 M are needed to calculate the rate constants according to method 111. The recovered decay times are listed in Table V. The ratios of the preexponentials, R (eq 5 I ) , are plotted as a function of emission wavelength in Figure 4. This figure can be compared to Figure 1 for the simulated data. For the 1.45 X M solution, the decays collected between 375 and 405 nm could all be described by monoexponentials as is shown in Figure 4. I n this case, method I1 cannot be used to

e

The Journal of Physical Chemistry, Val. 95, No. 5, 1991 2051

Fluorescence Decay Surfaces TABLE I V Estimated Rate Colrrtrnt Values for the Simulated Pyrene System by the Two-State Compartmental Analysis Method using 28 Decay Curves"

k kol, XlO"

s-I

X104 M-' s-I k02. x 106 s-I k I 2 , X l P s-I k2l.

'See text

input

estimated

2.25 6.7 15.5 6.5

2.22 f 0.01 6.76 f 0.02 15.52 0.02 6.55 i 0.06

*

for more details. All errors are 2

X

SD. I

I

TABLE V Globally Estimated Decay Time8 and CalcuLtd Rate Constants for Pvnne in Isooctane at Room TemeMtunO [MI,mol L-I -yI-I. ns Ti1, ns 10-6 b 461 f 1.5 48.4 0.4 345.4 f 0.4 1.45 X IO-' 39.5 4 1.5 122.4 f 0.2 1 . 5 3 x 10-3 method I1 method 111 IOd M IO" M 1.45 X lO-'M rate const 1.45 X IO-' M 1.53 X M 1.53 X M 2.17 f 0.1 2.18 f 0.1 k o l , XlO" s-I 2.17 f 0.1 5.67 f 0.3 7.2 f 0.2 5.0 f 0.2 kZ1.X104 M-l s-I 20.7 1.3 18.1 f 1.4 14.8 f 1.1 kO2. xi04 s-I b 6.7 f 0.2 5.5 f 0.7 k12, XIO" s-I

*

*

! VI

"All errors are 2

X

SD. blmpossibleto calculate.

TABLE VI: Rate Constants for Pyrew in Isooctane at Room Temperature Estimated with Compartmental Analysis" ko1 = (2.17 f 0.004) X IO6 s-I ko2 (14.8 0.02) X 106 PI kZl = (7.2 f 0.1) X IO9 M-I s-l k I 2= (5.7 4 0.07) X I06 s-l

0 O 64 I

02

02

06

0 4

OAll errors are 2 X SD.

08

1-5,

1

Figure 3. El estimated by global bicompartmental analysis as a function of the theoretical E2 ('1 - E I ) at three simulated pyrene concentrations: (0)0.001 M,(A)0.003 M,and (0)0.01 M are obtained from unlinked cI elements; (m) experiments with linked El elements; (--) theoretical values.

OB

0.6 2"-

04-

02-

0-

i , , , . , , , , . , , , , , , ,

-0 2

370

390

410

430

450

470

490

510

530

emasston wavelength (nm)

j

-0.6

I

A

-0.8 -1

370

, , , 390

,

410

,

,

A A, 450 A, : l, 470 l,

430

,

,a,1510, a ,530,

Figure 5. P I as a function of the emission wavelength for pyrene in isooctane: (0) 1.53 X IO-' M, (A)1.45 X IO-' M unlinked cI,and (U) linked E l for the two pyrene concentrations. The error bars are for a 95%

confidence limit.

490

emtsston wavelength (nm)

Fipn 4. The ratio R of the globally estimated preexponentials (q51) as a function of the emission wavelength: ( A ) 1.53 X M and (m) 1.45 X lo-' M pyrene in isooctane.

calculate the rate constants because the R value (eq 51) is equal to zero. For the 1.53 X IO-' M pyrene solution, the R value (eq 51) at 375 nm does not yet reach a constant value (Figure 4). The excimer emission band of this solution is quite intense as is shown in the fluorescence spectrum (Figure 7). This means that at 375 nm spectral overlap with the monomer emission can be expected. Thus, method 11 will most probably yield incorrect rate constant values. The most accurate values for the rate constants, starting from the exponential decay estimates, are probably found by using method I l l . As is shown in Table V, the rate constant kll calculated with method I1 has a lower value compared to the value obtained with method 111. The rate constants kO2and k 1 2calculated according to method I1 are higher in comparison to the values recovered with method 111. The deviations are similar to those found for the simulated pyrene data in the case of spectral overlap (see Figure 2 ) . These results indicate that there is excimer emission at 375 nm. The direct fitting to nonnormalized cI and cl. (and the rate constants) has been done already for the acid-base equilibrium in the excited state of 8-naphth01.l~There was no possibility of linking c, over the different proton concentrations. One of the important advantages of the global compartmental analysis as described in this paper is the possibility of linking f, over the different concentrations. Furthermore, the analysis approach allows us to fit directly to the relative absorbances (6,) of the

species. This represents an additional novelty in the domain of fluorescence decay analysis. The pyrene system was chosen to exemplify the linking of F,. Since only the monomer is absorbing light, the pyrene system is not a veryjnteresting example to demonstrate the fitting to and linking of b,. Nevertheless, it is an interesting system to evaluate the validity of the kinetic scheme (Scheme 11) proposed by Diiller and F6rster.I' Because no assumptions are made about the ground-state equilibri_um, one can test if indeed a value of one will be obtained for b, as would be expected for Scheme 11. Forty-two fluorescence decays were combined in one compartmental analysis experiment yielding a xS2 value of 1.063, corresponding to Zx2 = 4.467. The rate constants (Table VI) are in very good agreement with those calculated with method 111. The 95% confidence limits are smaller than those for the rate constants calculated according to method 111 (compare to Table V). The relative absorbance 6, was linked at each concentration and the recovered values were 0.996f 0.004(2 SD)and 0.998 f 0.003(2SD)for 1.45X IO-'and 1.53 X lo-' M,respectively. This is in perfect agreement with the fact that, in the studied concentration range, there are no ground-state dimers for homogeneous solutions of pyrene in a nonviscous solvent and that the excitation light is absorbed by the monomeric pyrene molecules only. Figure 5 shows P I as a function of the emission wavelength. This figure can be compared to Figure 3 for the simulated data. If the c , elements are not linked over the two concentrations, the error bars (2 SD) on cI normalized to unity are large (Figure 5), as was found for the simulated data (Figure 3). The error bars (2 SD) for the linked P I are smaller than the symbols used (Figure 5). Figure 5 also shows that the linked f l at 375 nm deviates from unity. This is probably due to the fact that the excimer contributes

Ameloot et al.


The isoemissive point of the monomer and excimer emission can be derived from Figure 5. As was pointed out in the p d i n g paper,l5 the ratio of ko, over kO2a t the isoemissive point is given by c2

From Table VI, eq 55 yields a value of 0.15 for kol/ko2.The condition f l / f 2 = 0.15 places the isoemissive point of the pyrene excimer forming system at 430 f 2 nm (Figure 5). This is in close agreement with the value of 425 f 2 nm visually derived from the spectra published by Parker and H a t ~ h a r d . ~ '

emaspon wauelength ( n m )

Figure 6. Decomposition ofthe steady-state fluorescence spectrum of 1.45 X IO4 M of pyrene in isooctane into the SAEMS: ( 0 )monomer emission, (A)excimer emission, and (-)

the steady-state spectrum.

'0°

5

2L

I

V. Summary The global compartmental analysis method makes it possible to estimate directly the rate constants of the excimer formation process of pyrene. The resulting accuracy is better than that of previous methods because experiments at different concentrations and emission wavelengths can be linked. The described compartmental analysis method does not require normalization of the considered decay curves. The method allows fitting directly to and linking the E values. These E values can be used, in combination with the steady-state spectrum, to calculate the corresponding species associated emission spectra (SAEMS). The current appoach also creates the possibility to fit directly to andJink the b values, the relative normalized absorbances. These b values can be used to calculate the equilibrium constant in the ground state when the total absorbance and concentration are known. This means that ground-state complexation can be investigated by using fluorescence decay surfaces.

o.:':..L

80 I

?

KO2

60

40

20

. A

Ab

....

emassion wauelength ( n m )

Figure 7. Decomposition of the steady-state fluorescence spectrum of 1.53 X IO-' M of pyrene in isooctane into the SAEMS: ( 0 )monomer emission, (A) excimer emission, and (-) the steady-state spectrum.

to the emission at this wavelength. This is in agreement with the results of the classical analyses. Figures 6 and 7 show the emission spectra associated with the monomer and the excimer (SAEMS, eq 41). It is evident that the excimer band of pyrene overlaps the monomer band at 375 nm.

Acknowledgment. R.A. is an Aspirant Onderzoeker of the Belgian Nationaal Fonds voor het Wetenschappelijk Onderzoek (NFWO). N.B. is a Bevoegdverklaard Navorser of the Belgian Fonds vmr Geneeskundig Wetenschappelijk Onderzoek (FGWO). The continuing support of the Ministry of Scientific Programming to the K. U. Leuven research group is gratefully acknowledged. (27) Parker, C. A.; Hatchard, Trans. Faraday S&. 1962, 59, 284.

Non a Priori Analysis of Fluorescence Decay Surfaces of Excited-State

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