CHjSCN
-
3489
J . Phys. Chem. 1990, 94, 3489-3494 CHj(%2Az”)
+ NCS(B22+),
6.49 eV (5)
The calculated thresholds are evaluated from the enthalpies of f o r m a t i ~ nand ~ . ~the ~ electronic energies of the A and B states of N C S 2 The onsets observed at 408 and 443 nm correspond to the NCS(A-X) emission. The observed onsets 7.5 f 1.0 eV for CH3NCS and 6.8 f 0.5 eV for CH3SCN are 0.9 and 0.5 eV, respectively, higher than the thresholds calculated for reactions forming NCS(A). On the basis of the Franck-Condon principle, the larger difference between the observed onset and the calculated threshold for CH3NCS is probably ascribed to the direct excitation into a steeper potential surface leading to dissociation. This originates in the fact that the geometry of the NCS(A) state is very different from the bond length of the N-C-S skeleton for CH,NCS.’ The difference between the onset and the threshold is available as the translational and internal energies among the fragments produced. Thus, it is expected that the vibrational structures from CH3NCS is more enhanced than those from CH3SCN. There is, however, no noticeable difference between the emission spectra from CH3NCS and from CH3SCN as discussed previous section. (24) Wagman. D. D.; Evans, W. H.; Parker, V. B.; Schumm, R. H.; Halow, 1.; Bailey, S. M.; Churney, K. L.; Nuttall, R. L. J. Phys. Chem. Ref. Dora 1982, I / , Suppl. 2.
Concluding Remarks We have studied the NCS emission in the 330-500-nm region produced by low-energy electron impact on CH3NCS and CH3SCN. The vibrational progressions well developed in the 390-500-nm region are attributed to the NCS(A-X) transition. The emission below 330 nm suddenly disappears. This is probably caused by the predissociation of vibrational levels 4000 cm-’ above the lowest level of the A state. The decay curves of the NCS emission can be represented by three decaying components. The lifetimes for the fast decaying component from both parents are attributed to the perturbed NCS(A) state, while the lifetimes for the middle decaying component are probably correlated with the NCS(B) state. On the other hand, the predominant slow decaying component (4-10 ps) from both parents is probably described to slow formation steps for NCS via metastable parent states below 15 eV. This is consistent with the result that the emission spectra from both molecules show very similar vibrational structures. In slow formation steps, the available energy can be distributed among all internal states in a statistical way and then the produced N C S no longer maintains the geometrical information of the parent molecule. Acknowledgment. I.T. thanks the Computer Center, Institute for Molecular Science, for the use of the HITAC S810/M680H computer and Library Program GAUSSIAN 82.
Recovery of Fluorescence Lifetime Distributions: Application to Forster Transfer in Rigid and Viscous Mediat Brian D. Wagner and William R. Ware* Photochemistry Unit, Department of Chemistry, University of Western Ontario, London, Ontario, Canada N6A 587 (Received: July 19, 1989; In Final Form: October 31, 1989)
Fluorescence lifetime distributions are recovered from a dipole-dipole energy-transfer system (donor phenanthrene, acceptor acridine) in rigid and viscous media. In the case of a rigid medium, the well-known Forster equation is verified to high precision. An exact equation for the rate constant distribution predicted by Forster theory is obtained and is shown to agree well with the experimentally recovered distribution. The value of the critical transfer distance Ro is calculated from the maximum of the recovered distribution, the result agreeing with that obtained from the fit to the Forster equation. In the case of viscous media, several popular models are compared by establishing an empirical relationship between the maximum of the recovered rate constant distribution and Ro, in a series of viscous solvents.
Introduction For dipole-dipole electronic energy transfer in condensed media there are two limiting cases. In the first case where no diffusion occurs or diffusion is unimportant, Forster kinetics are obtained. At the opposite extreme of low-viscosity solvents, ordinary Stern-Volmer kinetics are observed and the fluorescence decay of the donor follows a single exponential. The intermediate case is complex, and the useful working equations are obtained by approximate methods. In both the high-viscosity case and the intermediate case the donor decay does not follow a single exponential, and in the latter the time dependence is quite involved. In both of these cases, one intuitively expects to find a distribution of lifetimes. I n fact, it is important to be able to recognize the typical shapes of distributions that arise from Forster transfer. There have been a number of studies reported in the literature concerning the validity of Forster kinetics. Millar et a1.I showed the validity of the donor decay law from 1 ps to IO ns after excitation for a donor with an unquenched lifetime of 3 ns, and they found that the critical transfer distance R, (defined as the ‘Contribution No. 426.
0022-3654/90/2094-3489$02.50/0
donor-acceptor distance at which the probability of energy transfer is equal to t h i probability of decay of the isolated donor) was constant over a 1000-fold range of acceptor concentration. However, the donor decay curve was only collected to 3 X IO4 counts in the peak channel (CPC). This level of precision may not reveal small deviations. Struve et aL2 collected donor decay curves to IO5 CPC but only looked at donor-donor interactions. Eisenthal .et a1.j verified the donor decay law using time-resolved ground-state absorption but did not directly collect and fit the donor decay. Thus, in spite of the capability of the time-correlated single-photon technique to generate very accurate donor decay curves, a study using such decay curves does not appear to have been done at very high precision. In the course of this work, donor decay curves on the order of lo6 CPC were collected, and these were used to rigorously test the validity of the Forster donor decay law. ~
~~
~~
( I ) Millar, D. P.; Robbins, R. J.; Zewail, A. H. J . Chem. Phys. 1981, 75, 3649. (2) Hart, D. E.; Anfinrud, P. A.; Struve, W. S . J. Chem. Phys. 1987,86, 2689. (3) Rehm, D.; Eisenthal, K . B. Chem. Phys. Lett. 1971, 9, 387.
0 1990 American Chemical Society
3490
The Journal of Physical Chemistry, Vol. 94, No. 9, 1990
In this paper we will describe the application of distribution analysis to the cases of high and intermediate viscosity. This approach offers an alternative to the two conventional methods for determining Rol spectral overlap measurements and decay curve analysis. This IS especially important to the case of intermediate viscosity, where equations describing the donor decay are complex and not ideal for iterative least-squares reconvolution to obtain Ro.' In our approach, only a convolution of the decay law with the pump profile is required, which is much easier to perform than curve fitting by an iterative reconvolution procedure. The latter requires the use of a fitting algorithm such as the Marquardt, which requires derivatives of the equation with respect to each fitting parameter. This procedure would become extremely complex for some of the equations used. Our method is also more general in that any form of the donor equation can be used, without the need to calculate all the derivatives and write a new deconvolution program. The spectral overlap method is applicable regardless of medium viscosity but requires accurate, quantitative donor emission and acceptor absorption spectra, as well as an accurate value for the donor fluorescence quantum yield. These quantities may not be readily available for some systems. The system selected for study consisted of the donor phenanthrene and acceptor acridine in poly(methy1 methacrylate) (PMMA) plastic at room temperature. In addition, experiments were made in several viscous solvents corresponding to the intermediate case. One reason for selecting this system was the rather extensive data available in the literature for this particular combination of donor and acceptor in a variety of The critical concentration (co)for this system has been reported5 to be 0.0274 M; thus, acceptor concentrations ( c A )on the order of M were used to obtain significant nonexponential behavior. M were used to Donor concentrations (cD) on the order of enable the collection of a very large number of CPC. With these concentrations, there is a possibility of complications arising from donor-donor interactions. It has been shown theoretically6 and experimentally' that Forster kinetics will be observed only if the reduced donor concentration y D = cD/cDo is much less than the reduced acceptor concentration yA = cA/c0, where CDO
=
3000 ~ x ~ / ~ N ( R ~ ~ - ~ ) ~
3000 co = ~ T ~ / ~ N R ~ ~
Theory For systems of high viscosity the well-known Forster decay Iaw9J0 is as foIIows
+ br'/2)]
From the point of view of lifetime distributions, the fluorescence decay law is also given by the Laplace transform of the distribution function F(k) ^rn
I F ( f ) = s ' 0J F ( X ) f L ( 1 - A) dX
(4) Birks, J . B.; Georghiou, S. Chem. Phys. Let?. 1967, I . 355. (5) Birks, J . B.; Georghiou, S. J . Phys. I?, Ser. 2 1968, I , 958. (6) Loring, R. F.: Andersen, H . C.; Fayer, M . D. J . Chem. Phys. 1982, 76, 2015. ( 7 ) Miller. R. J. D.; Pierre, M.; Fayer, M. D. J . Chem. Phys. 1983, 78, 5138. ( 8 ) Berlman, I . Energy Transfer Parameters of Aromaric Compounds; Academic Press: New York, 1973. (9) Forster, Th. Z . Narurforsch. 1949, 49, 321, (IO) Birks, J. B. J . Phys. B, Ser. 2 1968, I . 946.
(5)
(6)
F(k) is thus the inverse Laplace transform of J ( X ) . In the case of J ( X ) given in eq 3 the inverse transform is known, and F(k) =
b
exp{-b2/4(k - a ) ]
2[7r(k -
(7)
The maximum in the k distribution is given by k,,,
=a
and thus Ro is related to k,,,
Ro =
(
+ b2/6
(8)
by
3000(6(k,,,
-
kF))'/'
4(a3k~)'/'NC~
(9)
I n fluid media where diffusion is important the dipole-dipole transfer kinetics have been treated by Yokota and Tanimoto." The time evolution of the distribution of donors and acceptors is determined by diffusion and the decay of the donor. They employ a Pad6 approximation to obtain the following decay law
where 1
B=(
+ 1 0 . 8 7 ~+ 1 5 . 5 ~ ~ 1 + 8.743~
( 1 1)
x =
&-1/3t2/3
(12)
= kFR2
(13)
and
cy
D is the diffusion coefficient. This treatment will be referred to as the YT Pad& The limiting case achieved in rigid media corresponds to 7 > Ro, where f is the ~. and average diffusion length, defined as ( ~ D T ~ ) ' /Yokota Tanimoto thus treated the case where p is approximately equal to R,, Le., where ?/Ro = 1 . Gosele et al.'2t13have pointed out a probable inaccuracy in the Yokota and Tanimoto Pad6 approximation in the case of large diffusion effects, Le., for 7/Ro > 1. They proposed a new Pad6 approximation, which is accurate in the limit of large diffusion effects. Their equation for the donor decay is identical with eq 10, with a new expression for B:
(3)
(4)
F(k) exp(-kt) dk
and since deconvolution is in general required, the actual observed decay data I F ( t ) assume the form
BG
where a = kF, the reciprocal of the unquenched lifetime, and b =2(k~)'/~y~
J,
JF(t)=
(2)
N is Avagadro's number, and is the critical distance for donor-donor transfer. Literature values for RopD and Ro in hydrocarbon solvent were found to be 8.77* and 24.3 A,5 respectively. Using the experimental concentrations cD = 2.83 X lo-) M and cA = 2.37 X we obtained the values yo = 4.22 x 1 0-3 and Y~ = 0.762. Thus, yA is over 3 orders of magnitude greater than yD, and therefore donor-donor interactions will be neglected.
JF(t) = J o exp{-(at
Wagner and Ware
=
(
1
+ 5 . 4 7 ~+ 4 . 0 0 ~ ~ 1 + 3.34x
(14)
In addition, Gosele et al. derived an expression for the donor decay which has the form of eq 3, but with a new value for a: aG = 4aDN '( 0.676( kFR:/
D) 'I4)
( 1 5)
This expression is derived for the long-time limit in systems in ( I I ) Yokota, M.; Tanimoto, 0. J . Phys. SOC.Jpn. 1967, 22, 779. (12) Gosele, U.;Hauser, M.;Klein, U. K. A.; Frey, R. Chem. Phys. Le??. 1975, 34, 519. ( 1 3 ) Klein, U.K. A.; Frey, R . ; Hauser, M.;Gosele, U . Chem. Phys. Le??. 1976, 41, 139.
The Journal of Physical Chemistry, Vol. 94, No. 9, 1990 3491
Recovery of Fluorescence Lifetime Distributions
-
TABLE I: Methods for Calculation of R o for the Ph* Ac System eqs in method of model text form of decay law obtaining Ro Forster 2-4 I ( t ) = Io X fit decay, obtain Ro for b exp(-at - brl/21 Foster 9 I ( t ) = Io x recover distribution, expl-at - b f ' / * ] obtain Ro from k,,, YT Pad6 10-13 I ( f ) = Io X empirical plot of expl-ar - bB(t)t'/*] k,,, vs Ro G H K F Padt IO, 12-14 I ( t ) = Io X empirical plot of expl-ar - bB(f)tl/21 k,,, vs Ro GHKF(tota1) 2-4, 15 I ( r ) = Io X fit decay, obtain expl-at - br1/2] R, from a
which long-range transfer dominates over contact quenching.I4 Equation 10 with B defined by eq 14 will be referred to as the G H K F Padt treatment, and eq 3 with aG defined by eq 15 will be referred to as the GHKF(tota1) treatment, to be consistent with the 1 i t e r a t ~ r e . lIt~ can be seen from these various equations for the donor decay that it is not possible to use some function of R,/D as a fitting parameter. A closed-form inverse Laplace transform of eq 10, with B given by either eq 11 or eq 14, appears out of the question. Nevertheless, there will be a distribution of fluorescent lifetimes and the maximum must be related to R,. Thus, rather than fit eq 10 to the decay data, it is proposed to establish the empirical relationship between k,,, and Ro for a set of D values. Thus, in the YT Pad6 and G H K F Padt treatments, Ro for the experimental data will be obtained from an empirical relationship between k,,, and R, established by using simulated data. In the GHKF(tota1) treatment, the donor decay will be fit to eq 3, and Ro will be obtained from the recovered value of aG, using eq 15. These approaches are summarized in Table I. The recovery of shapes of distributions of fluorescence lifetimes from decay points presents a nontrivial problem in numerical analysis. Inversions of this type are notoriously ill-conditioned. Similar problems arise in light scattering,I5 image recovery in astrophysics,16 etc. While methods involving the minimization of xz are commonly used, the maximum-entropy method (MEM) has recently gained p o p ~ l a r i t y . ' ~ -Inherent ~~ in the method is a lack of bias and the potential for recovering the coefficients of an exponential series with fixed lifetimes which are free of correlation effects and artificial oscillations. Up to 200 terms can be used in the present version of MEM. The MEM maximizes the function Q = S - XC where S is the entropy-like function (Shannon-Jaynes entropy18) N
S = -,Eak In k
(ak/atot)
(16)
In these equations N is the number of exponents in the exponential series
Eaj exp{-kjt) with fixed lifetimes 7F = 1/ k F logarithmically spaced. Initially, all the ai are equal in magnitude, thus giving an unbiased starting point. n , and n2 are the initial and final channels containing the (14) Faulkner, L. R. Chem. Phys. Lett. 1976, 43, 552.
(15) Livesey, A. K.; Licino, P.; Delaye, M. J. Chem. Phys. 1986,84, 5102. (16) Gull, S. K.; Daniell, G.J. Nature 1978, 272, 686. (17) Skilling, J.; Bryan, R. K. Mon. Not. R. Astron. SOC.1984, 211, 1 1 1. (18) Smith, C. R.; Grady, W. T., Jr., Eds. Maximum Entropy and Bayesian Methods in Inverse Problems; Reidel: Boston, 1985. (19) Livesey, A. K.; Skilling, J . Acfa Crysrallogr. 1985, A41, 113. (20) Livesey. A. K.: Brochon, J. C. Biophys. J . 1987, 52, 693.
digital data. Virepresents the number of counts in the ith channel, and D f is the convolution matrix element
D k = l " I L ( t j - A) exp{-h/rk) dX 0
The function Q is maximized subject to the constraint that C zz 1.OO. We employ a computer program which utilizes the method of conjugate gradients.]' This same program can be used to obtain distributions based only on the minimization of C, Le., x2. This modified algorithm, for what is a new version of the exponential series method (ESM), is robust and in general free of oscillations and artificial peaks and furthermore lacks the strong dependence on an arbitrary Ax2 cutoff found in our earlier ESM algorithm. Thus, this new version of the ESM represents a significant improvement over that described previously in work from this laboratory.21s22 In fact, this particular version has been found to produce distribution recovery virtually identical with that from the MEM, and in this study both have been e m p l ~ y e d . ~The ~?~~ new ESM runs with about 1/10 the CPU time of the MEM. Both the MEM and ESM methods used in this work have been tested with simulated data with added Poisson noise and with real data including distributions made up from the addition of large numbers of files such that the actual distribution shapes are known exa~tIy.~~J~ Experimental Section Acridine (Aldrich, 99%), phenanthrene (Aldrich, zone-refined, 99.5+%), benzoyl peroxide (BDH), and isobutyl alcohol (Fisher, spectra grade) were used as received. Methyl methacrylate (BDH) was distilled ( T b = 99 "C) to remove the quinol stabilizer. Cyclohexanol (BDH) was vacuum-distilled (p = 20 Torr, T = 56 "C) to remove fluorescent impurities. The PMMA samples were prepared by dissolving the desired mass of donor and acceptor directly in the monomer, adding a small crystal of benzoyl peroxide to 2 mL of the solution in a small sample vial, purging with dry N2, sealing the vial, and heating in a 60 OC oven for 3 days. A volume decrease was observed after polymerization; comparison of the monomer and polymer densities gave a value of 19% for this decrease, and this value was then used to correct the concentration. The samples in viscous solvents were purged with dry N2 for 15 min and then placed in 1-mm quartz cuvettes. Emission was observed from the front face. Fluorescence lifetime measurements were carried out using a Coherent based laser system consisting of an argon ion laser mode locked and synchronously pumping a Rhodamine 6G dye laser. The output frequency of the dye laser was doubled. Decay curves were determined with the time-correlated single-photon counting technique. This apparatus has an instrument response function of about 230-ps fwhm.25 Cavity dumping assured a pulse repetition rate low enough to prevent repumping of the system. The excitation wavelength was 290 nm; the phenanthrene emission was observed at 358 nm through a monochromator (bandwidth 25 nm) and 320-nm cutoff filter. Magic angle polarization was used in all measurements. The combined deconvolution-distribution analysis was carried out on the University of Western Ontario Vax computer. In addition, a nonlinear least-squares program was used to recover the a and b parameters in eq 3 by iterative least-squares reconvolution. Results and Discussion Before examining the distributions of lifetimes that result from Forster energy-transfer systems, it was considered important to examine critically the degree to which eq 3 describes the behavior of the fluorescence decay in rigid media. While there have been (21) James, D. R.; Ware, W. R. Chem. Phys. Lett. 1986, 126, 7. (22) James, D. R.; Liu, Y.-S.; Petersen, N. 0.; Siemiarczuk, A,; Wagner, B. D.; Ware, W. R. Proc. SPIE I n f . SOC.Opt. Eng. 1987, 743, 117. (23) Siemiarczuk, A,; Wagner, B. D.; Ware, W. R. Proc. SPIE Inr. SOC. Opt. Eng. 1989, 1054, 54. (24) Siemiarczuk, A,; Wagner, B. D.; Ware, W. R. J . Phys. Chem., in press. (25) Ware, W. R.; Pratinidhi, M.;Bauer, R. K. Rev. Sei. Insfrum. 1983, 5 4 , 1148.
r;
Wagner and Ware
3492 The Journal of Physical Chemistry, Vol. 94, No. 9, 1990
a
X'
'!i'.-----. 5
LOG
16
INT
j
12-
8-
I
0
4-
100
200
l/ns
b C
d TABLE 11: Parameters for the Fit of the Decay in Figure 2 to Various Modelsn
model 2-exp 3-exp 4-exp
'
i 1 2 1 2 3 1 2 3 4
ai
T,bS
0.542 0.458 0.296 0.402 0.302 0.178 0.238 0.338 0.245
11.14 34.23 4.92 17.74 37.01 2.78 8.51 20.95 38.17
x2
e
31.26 3.152
Figure 2. Fit of the decay of phenanthrene quenched by acridine in
2.683
PMMA tovarious models: (a) decay curve (---) laser profile, (-) fit to eq 3; (b) residuals for fit to eq 3; (c) residuals for two-exponential fit; (d) residuals for three-exponential fit; (e) residuals for four-exponential fit. [Ph] = 0.0028 M, [Ac] = 0.0237 M, 9 X lo5 CPC, 256 channels, 1100 ps per channel. (See Table I1 for fit parameters.) (e-)
.
Equation 3: a = 0.01 82 ns-I; 6 = 0.21 97 ns-'I2; x 2 = 3.1 50.
b:
I'
several studies reported in the l i t e r a t ~ r e ' - ~that ~ ~ ~have 2 ~ examined this question, none employed the high-precision data used in this study. Typically, we collected 3 X IO5 counts in the peak channel (CPC) for this particular analysis. Iterative reconvolution was used with JF(t) = Jo exp(-(at
+ bt")}
(20)
and a series of numerical analyses made with floating b but fixed n and the value of a fixed at the measured unquenched reciprocal lifetime. The value of n was then varied and the analysis repeated to finally generate a set of fits as a function of n. The system consisted of phenanthrene quenched by acridine in PMMA at room temperature. Figure 1 illustrates the results for the observed x 2 as a function of n. The validity of eq 3 is clear from this data. The fit of the (x*,n)data to a parabolic function gave a minimum at n = 0.503. The fit to eq 3 with both a and b as free floating parameters and n = 0.5 recovered the same value of b, and the value of a obtained compared favorably with the value of kF obtained for the unquenched donor. Using the value of b obtained and eqs 2-4, we calculated Rofor this system to be 25.5 A. This is in excellent agreement with the value of 25.3-25.6 8, reported for alcoholic solvent^.^ For the sake of comparison with the conventional fitting procedures, Figure 2 shows the decay curve for this experimental system deconvoluted with eq 3 (a and b free floating) as well as with two-, three-, and four-exponential models, along with the various statistical tests of goodness of fit. The fit parameters are listed in Table 11. Only the four-exponential model gave a better fit than eq 3; this is probably a reflection of the much larger ( 2 6 ) Pandey, K . K.; Joshi, H. C . ; Pant, T. C . J . Lumin. 1988, 42, 197. (27) Pandey, K. K.; Joshi, H. C.; Pant, T. C. Chem. Phys. Lerr. 1988, 148, 472.
5
1
9
1
5
9
ki/io7 s-' Figure 3. Rate constant distributions recovered by ESM and MEM analysis of simulated and experimental decay of phenanthrene quenched by acridine in PMMA: (a) ESM, simulated data; (b) MEM, simulated data; (c) ESM, experimental data; (d) MEM, experimental data. Simulated decay: eq 3 with a = 0.018 ns-I, 6 = 0.22 ns-i/2. Experimental decay: [Ph] = 0.0028 M, [Ac] = 0.0237 M. The distribution given by eq 7 is shown as 9.99 X los CPC, 512 channels, 1036 ps per channel. -e.
number of fit parameters (eight for the four-exponential model vs two for eq 3). The k distribution recovered by ESM and MEM are shown in Figure 3 both for experimental data and for a simulated decay based on eq 3, using the experimental values of a and b. Figure 3 also shows the theoretical k distribution given by eq 7. The ESM result for the simulated decay agrees extremely well with the theoretical distribution, illustrating that ESM can successfully recover the underlying distribution in these energy-transfer systems. The ESM result for the experimental data also agrees well with the theoretical distribuiton, although not quite as well as in
The Journal of Physical Chemistry, Vol. 94, No. 9, 1990 3493
Recovery of Fluorescence Lifetime Distributions
la
II
22
tb
1
'
D = 5.4
i
l0
1
t I
2
10
,
50
10
I I
50
t au/ns Figure 4. Lifetime distributions recovered by ESM analysis of real and simulated decay data for phenanthrene quenched by acridine in PMMA: (a) simulated three-exponential model; (b) simulated four-exponential model: (c) simulated eq 3; (d) experimental data ([Ph] = 0.0028 M, [Ac] = 0.0237 M). Simulations constructed using parameters in Table 11. Discrete lifetimes in (a) and (c) are shown as 9.99 X IO' CPC, 512 channels, 1036 ps per channel. e-.
the case of the simulated data. Although the problem of background counts was properly treated,23*24other instrument errors such as electronic and radio-frequency noise and drift of the laser profile may have interfered with the distribution analysis. The distributions recovered by MEM for both the simulated and experimental decays are significantly distorted, probably a result of the program overfitting the data. For the results described below the ESM was therefore used for the distribution analyses. It can be seen from Figure 3 that this ESM does recover the correct value of k,,,, the parameter needed to obtain Ro from the distribution. By use of a parabolic approximation to the distribution data in the region of the maximum, a value of 2.66 X lo7 s-I was obtained for k,,,, which gives a value of Ro from eq 9 of 25.7 A. This is in excellent agreement with the value of Ro calculated from the fit of the decay curve (25.5 A). This analysis therefore illustrates the potential usefulness of obtaining values of Ro from distribution results. It is perhaps instructive to examine the utility of distribution analysis for distinguishing Forster-type distributions from threeand four-component models that might under some circumstances be considered as alternative descriptions of a system under study. To this end, the recovered parameters listed in Table I1 derived from three- and four-component fits to the decay data measured in rigid media were used to generate simulated decays and these, after addition of Poisson noise, were subjected to ESM analysis. The results are shown in Figure 4. It is clear that, had the real system been adequately described by a three- or four-component model, the ESM results would have provided evidence of this and would not have indicated a Forster-type distribution. The change to log T space for the examination of this question was motivated by the desire to spread out the three of four components to display more clearly the resolution in the case of the discrete lifetimes. I n the intermediate case of diffusion-influenced transfer, the complexity of the donor decay law (eqs 10-14) makes it difficult to obtain the value of Ro by fitting the decay curve. An alternative approach is to obtain an empirical relationship between k,,, and Ro using simulated decays and then obtain Ro from the experimental value of k,,,. This requires that a curve of k,,, vs Ro be obtained for each medium of interest. The simulations were constructed by convoluting eq 10 with a typical laser profile using eq 6 and then adding Poisson noise. These simulations were then analyzed with ESM, and the maximum was calculated by using a parabolic approximation. Figure 5 shows the resulting plots of k,,, vs Ro for the fluid systems listed in Table 111, with the simulated decays based on both the YT Padt and the GHKF Padt treatments. The eight sets of k,,,, Ro data were fit to a fourthorder polynomial function, with Ro as a function of k,,,. The experimental value of Ro was then calculated from this polynomial
15
20
25
30
35
40
Ro/A Figure 5. Plots of k,,, vs Ro for simulated decay data for phenanthrene quenched by acridine in media of various viscosities, analyzed by ESM. Simulations: 3 X IO' CPC, 256 channels, various picoseconds per channel, [Ac] = 0.0137 M. (-) YT Pad& (---) GHKF Pad& D given in units of
lod
cm2 s-I.
-
TABLE 111: Physical Properties of the Pb* Ac System at 25 OC in Cvclobexanol-Isobutvl Alcohol Mixed Solvents and in the Pure Liauids Da/lO" medium sa/cP cm2 s-I r,/ns ?/A Ro" ?JRa PMMA -0 53.1 =O b =O C6H,10H 64.6 0.31 53.3 18.2 25.3 0.72 4:l C6Hl10H-i-BuOH27 0.74 52.1 28.1 25.3 1.11 1:l C6HIIOH-i-BuOH 10.5 1.9 49.3 45.0 25.6 1.78 3.69 5.4 52.8 25.9 25.6 3.0 i-BuOH Reference 5. Not available. TABLE I V Summary of Recovered Values of k, and Ito for Phenanthrene Quenched by Acridine at 25 OC in Cyclohexanol-Isobutyl Alcohol Mixed Solvents and the Pure Liquids
medium C6HI10H
411 C6HllOHi-BuOH 1:l C6HIIOHi-BuOH i-BuOH
RJA GHKF
GHKF
(Pade)
(total)
3.37 4.18
ref 5 26.8 f 0.7 28.0 f 0.6 29.2 f 2.2 25.3 25.1 f 1.3 27.4 f 1.0 27.0 f 1.8 25.3
6.03
23.9 f 1.5 27.3 f 1.1
krn,Xl1O7 S-I
10.2
YT Pad6
27.1 f 1.7 25.6
23.9f2.6 28.1f2.1 26.5f1.6 25.6
by using the value of k,,, obtained from the experimental distribution. The experimental values of k,,, for each solvent are listed in Table IV, as well as the values of Ro obtained from the empirical curves. Table IV also lists the values of Ro obtained by using the GHKF(tota1) treatment, i.e., fitting the experimental decay to eq 3 and obtaining Ro from the recovered value of aG via eq 15. For comparison, the values of Ro obtained by the spectral overlap method (taken from ref 5) are also included. A comparison of the recovered values of Ro and the spectral overlap values shows several trends. The YT Padi gives better results than the two G H K F treatments for systems with F/Ro = 1. As the ratio r/Roincreases, however, the Y T Padt begins to overestimate the quenching (yielding lower values of Ro than expected), although the increasing relative error makes this conclusion somewhat uncertain. The GHKF Pad&treatment gives a fairly constant but high value of Ro,independent of the solvent used. The GHKF(tota1) treatment gives better results than the two Pad6 approximations for systems with larger values of ?/Ro. This may be more pronounced in systems with L/Ro= 10 or larger, but such systems were not investigated in this study.
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The Journal of Physical Chemistry, Vol. 94, No. 9, 1990
D and the decay curve fit parameter a (obtained from fit statistics), based on eq 15. I t should be noted that this analysis has neglected the effect of molecular rotations on the recovered value of RW This effect is usually treated by including in the equation for donor decay the averaged orientation factor ( K ~ ) ,where K = cos @ D A - 3 cos $D cos (21)
a:
b
2
4
6
8
k i/107s-’ Figure 6. Rate constant distributions recovered by ESM analysis of simulated and experimental decay data for phenanthrene quenched by acridine in cyclohexanol: (a) experimental decay ([Ph) = 0.0028 M, [Ac] = 0.0137 M); (b) YT Pad6 simulation, Ro = 26.8 A; (c) GHKF Padt simulation, Ro = 28.0 A. 3 X IOs CPC; 512 channels, 646.9 ps per channel.
In cases where diffusion is important the k-distribution shape changes such that the maximum shifts to larger k values (shorter T values) and the probability for large k values decreases significantly. One can in fact imagine the continuation of this trend as the viscosity is reduced to that of a nonviscous medium where one should get only a narrow distribution corresponding to a single quenched lifetime (Stern-Volmer limit). For comparison of the distribution shapes from the YT and G H K F PadO models, R, values reported in Table 1V were used to generate simulated decay curves by use of the cyclohexanol data. After the addition of Poisson noise, these decays were analyzed with ESM. As can be seen from Figure 6, the predicted and observed distribution shapes are very close, especially for the YT model. Comparison to Figure 3a,c shows clearly the marked changes in the distributions as diffusion becomes an important factor in determining the time evolution of the donor. The major source of error in this method is the uncertainty in the values of the diffusion coefficient D. Ideally, these should be directly measured for the system in question. The values were calculated from the tracer diffusion measurement of Miller et al. of anthracene in a variety of solvents.2s Two assumptions were made: (a) D is the same for anthracene, acridine, and phenathrene, and (b) D is inversely proportional to viscosity (Le., OD = constant). Miller et al. quote an upper limit on the error in 7D for anthracene of f7%. Given these two assumptions and the small uncertainties in the viscosities, a value of *lo% was taken for the error in the D values for acridine and phenanthrene. Thus, a total error of f 1 4 % was used for the error in the total D value (=DAc DPh). To calculate the sensitivity of the recovered value of Ro to the input value of D, a plot of Ro vs D at the fixed value of the experimentally observed k,,, was made for each solvent, and the slope of the tangent line at the value of D used for that solvent was then determined. This gave a reasonable estimate of the dependency of the recovered value of Roon the input value of D, and using the f 14% error assumed for D,one could determine a range for the calculated value of Ro. These ranges are also included in Table 1V. The relative errors in the GHKF(tota1) results were estimated by a propagation of the relative errors in
+
(28) Miller, T. A.; Prater, B.; Lee, J. K.; Adams, R. N. J . Am. Chem. Soc. 1965, 87. 122.
Wagner and Ware
I$DA is the angle between the donor and acceptor dipole moments; 4 D and 4A are the angles between the dipole moments and the D-A direction. Millar et al.’ include a numerical factor gin the t ‘ / * term in the donor decay law, where g = (3/2(!2))1/2. They give the value of g as ranging from 1 .O in the limit of infinitely rapid reorientation (7 = 0) to 0.845 in the limit of static molecules ( 7 = m). This gives a maximum possible error of 15% in comparing rigid and nonviscous media. In the case of the viscous alcohol media used in this study, the total change in viscosity is relatively small (7 = 3.7-65 cP), and thus the error resulting from neglecting medium dependence of molecular rotation will be smaller than the uncertainty from other sources. These same ideas that we have applied to the F/Ro = 1 case of Forster transfer should be easily extended to systems predicted to exhibit complicated decay Considerable simplification is achieved because the convolution of the assumed decay law with the lamp function is easily performed regardless of the complexity of the decay law and generates simulated decay data which then can be analyzed by ESM. One then approaches the real experimental data both by examining the similarity between the predicted and recovered distributions and by recovering k,,, as a function of one or more of the system parameters as described above. While a similarity in distribution shape between simulated and experimental data should not be considered proof of the validity of a model, it should be pointed out that such proof is not easily achieved with complicated decay laws under any circumstances. On the other hand, large discrepancies between recovered and predicted distributions may be useful in casting doubt on a postulated model. If Ro is known a priori, then the position of the maximum in the k distribution may be related to some other parameter and used empirically to recover values from experimental data. Conclusions There are several conclusions which may be drawn from the results of this work: (a) the Forster equation (eq 3) was found to accurately describe the high-precision donor decay data collected for this dipole-dipole energy-transfer system in a rigid medium; (b) the distribution of rate constants underlying the Forster equation is given by eq 7, and this distribution is accurately recovered by the exponential series method; (c) the recovery of distributions in energy-transfer systems provides an alternative approach to decay curve fitting for the recovery of Ro, which is especially useful in the case of intermediate diffusion effects; (d) the YT Padi: approach, in conjunction with distribution analysis, is successful in recovering Rovalues for systems with F/Ro = 1; (e) the GHKF(tota1) approach, which involves fitting the donor decay curve, is successful at recovering Rovalues for systems with higher values of ?/Ro;and (f) the MEM method for recovering distributions was found to be somewhat inferior to the ESM method.
Acknowledgment. The authors acknowledge the support of the Natural Sciences and Engineering Research Council of Canada. Registry No. Phenanthrene, 85-01-8; acridine, 260-94-6. (29) Twardowski, R.; Kusba, J.; Bojarski, C. Chem. Phys. 1982,64,239. (30) Ferreira, L. F. V. J . Lumin. 1986, 34, 235. (31) Sienicki, K.; Winnik, M. A. Chem. Phys. 1988, 121, 163. (32) Levitz, P.; Drake, J. M.; Klafter, J. Chem. Phys. Letr. 1988, 148, 557. (33) Kaschke, M.; Kittelman, 0.;Vogler, K.; Graness, A. J . Phys. Chem. 1988, 92, 5998.
