2908
J . Phys. Chem. 1990, 94. 2908-29 I 4
Diffusion-Controlled Reactions: Transient Effects in the Fluorescence Quenching of Indole and N-Acetyltryptophanamide in Water C. C. Joshi,+ R. Bhatnagar, S. Doraiswamy, and N. Periasamy* Chemical Physics Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Bombay 400 005, India (Receioed: June 19, 1989: I n Final Form: September 22, 1989)
Time-domain fluorescence quenching studies of indole and N-acetyltryptophanamide (NATA) in water were carried out at different temperatures using KI and acrylamide as quenchers. The fluorescence decay was nonexponential in all the cases as expected for a diffusion-controlled bimolecular reaction, and the data were analyzed according to different decay equations. The data for indole/Kl were consistent with the Smoluchowski-Collins-Kimball (SCK) decay equation and the limiting values for the fundamental parameters-diffusion coefficient, D, reaction distance, R, and the absolute rate coefficient, k,-are reasonably close to the expected values. The SCK decay equation was tested by an "assumption-free" method and by using the "long-time" approximation. The results obtained by the two methods are consistent. Analysis of the quenched fluorescence data of indole/KI/H20 to fit the long-time SCK decay equation (mathematically similar to the Smoluchowski equation) gave reasonable values for D and the distance parameter, R: It is generally observed that the values of D and R'are correlated. The experimentally determined values for D and R' for four fluorophore-quencher systems at different temperatures are reported in this paper and discussed. The results of D and R'obtained at different temperatures are consistent with the identification that indole/KI system follows the SCK model and indole/acrylamide system is closer to the Smoluchowski model. The results of D and R' for NATA/KI and NATA/acrylamide systems were not favorable for identifying these systems with either model.
Introduction Transient effects in the diffusion-limited bimolecular reactions are well-known.'-3 Theoretical models predict various equations for the time-dependent rate ccefficient."I0 Rigorous experimental tests to verify the theoretically predicted equations can be done for fluorescence quenching reactions."-'* So far, the tests using fluorescence quenching data have not revealed any reaction that satisfies the simple Smoluchowski model. The modified model, known as Smoluchowski-Collins-Kimball (SCK) model, has been found to adequately explain the experimental data for several fluorescence quenching reaction^.'^-'^ A successful test of a bimolecular reaction to the model should give reasonable values for the fundamental parameters of the model, namely, reaction distance, R, and diffusion coefficient, D. For the SCK model a reasonable value for the absolute rate coefficient k, is also mandatory. An experimentally determined value of k , for intermolecular reactions is an extremely significant quantity, for the reason that k, is decoupled from the diffusion limitation, and hence k , is related to other experimentally determined rates such as that of the two reactants at a fixed distance as obtained in covalently linked molecule^'^ or in frozen media.20 Prior to 1987, the SCK model was tested by using the "longtime" approximation for the rate expression and not by using the complete or "all-time'' equation [see next section for the equations]. Recently, Lakowicz et al have used the all-time equation of the SCK model for the analysis of the frequency-domain quenched fluorescence data of several systems,l6-lBand the SCK model was found to be adequate especially at low concentrations of the quencher. Independently, a satisfactory deconvolution method was developed for the analysis of time-domain fluorescence data for fitting the all-time equation of the model,2' and the method was successfully applied to an experimental system of ionic reactants and reasonable values for R, D,and k, were obtained.I5 In view of the importance of such new experimental approaches, both to verify a theoretical equation and to extract a value for the fundamentally significant rate coefficient, our investigation is extended to neutral-ion and neutral-neutral reactants. The fluorescence quenching of indole and N-acetyltryptophanamide (NATA) in water by KI and acrylamide at different temperatures was investigated by the time-domain technique. These systems were chosen so that the results obtained at room temperature may *Author for correspondence. 'Permanent address: Department of Physics, Garhwal University, Srinagar. Garhwal (U.P.) 246 174, India.
be compared with those reported in the literature recently by Lakowicz et a1.I6 and by others.I3 Such a comparison is also necessary to dispel the doubts expressedI6 concerning the capability of the time-domain experiment by the technique of time-correlated single photon counting (TCSPC) in unraveling the intricacies involved in testing the SCK model. The method of analysis of quenched fluorescence data in the frequency domain or the time domain to fit the decay equation predicted by the SCK model has not been a straightforward one like, for example, fitting the data to a one-exponential decay equation. A free fit of the all-time SCK equation has not been possible for the time-domain dataI5s2' as well as, perhaps, for the frequency-domain data." Because of this difficulty Lakowicz et al." have chosen to fix the value of one fundamental parameter (reaction velocity, K (cm S K I ) ,which is related to k , through R) of the SCK model so that physically realistic values can be obtained for other parameters. The time-domain data were analyzedI5 by holding one parameter (the time shift 6) as fixed. 6
( 1 ) Rice, S. A. I n Comprehensive Chemical Kinetics: Dijfusion Limited Reactions; Bamford, C. H., et al., Eds.; Elsevier: London, 1985; Vol. 25. (2) Keizer, J. Chem. Reu. 1987, 87, 167. (3) Noyes, R. M . Prog. React. Kinet. 1961, I , 129. (4) Smoluchowski, M . V . Z . Phys. Chem. 1917, 92, 129. ( 5 ) Collins, F. C.; Kimball, G. E. J . Colloid Sci. 1949, 4, 425. (6) Hong, K. M . ; Noolandi, J. J . Chem. Phys. 1978, 68, 5163, 5172. (7) Flannery, M . R. Phys. Reu. A 1982, 25, 3403. (8) Cukier, R. I . J . Chem. Phys. 1985, 82, 5457. (9) Green, N . J . B. Chem. Phys. Lett. 1984, 107, 485. (IO) Lee, S.; Karplus, M. J . Chem. Phys. 1987, 86, 1883. ( 1 I ) Nemzek. T. L.; Ware, W. J . Chem. Phys. 1975, 62, 477. ( 1 2 ) Ware, W.; Andre, J. C. In Time-Resolued Fluorescence Spectroscopy in Biochemistry and Biology; Cundall, R. B., Dale, R. E., Eds.; NATO AS1 Ser. A,; Plenum: New York, 1980; Vol. 69. (13) Wijnandts van Resandt, R. W. Chem. Phys. Lett. 1983, 95, 205. (14) Periasamy, N.; Doraiswamy, S.; Maiya, G. B.; Venkataraman, B. J. Chem. Phys. 1988, 88, 1638. (15) Periasamy, N.; Doraiswamy, S.; Venkataraman, B.; Fleming, G. R. J . Chem. Phys. 1988, 89, 4799. (16) Lakowicz, J. R.; Johnson, M . L.; Gryczynski, 1.; Joshi, N.:Laczko, G. J . Phys. Chem. 1987, 91, 3277. (17) Joshi, N.; Johnson, M. L.; Gryczynski, 1.; Lakowicz, J. R. Chem. Phys. Lett. 1987, 135, 200. (18) Lakowicz, J. R.; Joshi, N . B., Johnson, M. L.; Szmacinski, H.; Gryczynski, I . J. Biol. Chem. 1987, 262, 10907. (19) Closs, G. L . ; Miller. J . R. Science 1988, 240, 440. (20) Miller, J . R.; Beitz. J . V.; Huddleston. R. K. J . Am. Chem. Sot. 1984, 106. 5057. (21) Das, R . ; Periasamy, N . Chem. Phys. 1989, 136. 361.
C 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990 2909
Diffusion-Controlled Reaction was varied through a reasonable range and all parameters were optimized by using a set of empirical criteria. No assumption was made about the values of the fundamental parameters of the SCK model, but it was found necessary to use simulated data to prescribe the criteria. The time-shift parameter was found to be a significant parameter for the analysis of time-domain data even when an MCP-PMT was used, whereas this parameter was ignored in the analysis of frequency-domain data. For these reasons, it is necessary that a suitable method of analysis of quenched fluorescence data be developed which is totally free of any assumption on the values of parameters. An "assumption-free" method of data analysis is described and used in the present study. The results reported in this paper demonstrate that (i) timedomain studies by the TCSPC technique is suitable for the investigation of diffusion-limited reactions, (ii) results of time-domain studies reported earlier13 are reproducible, (iii) time-domain fluorescence data can be analyzed to fit the SCK decay equation by an assumption-free method to yield limiting values for the fundamental parameters of the SCK model, (iv) the indole/ Kl/water system is an example following the SCK model and indole/acrylamide/water is a likely example for the Smoluchowski model, and (v) temperature variation of D follows reasonably well the T/T relation predicted by the Stokes-Einstein equation.
Smoluchowski model A = Io p = 4aRDCq
v =~(TD)I/~R~C~ A = Io exp(bC,/c2)
(12)
p = ac,
(13)
It is to be noted that eq 12 for the preexponent term A differs from the one given in the literature" by including a concentration-dependent exponent term. The correctness of eq 12 is explained in ref 22. The intensity decay equation for the SCK model (eq 2) is
exp(c2t) erfc ( ~ t ' / -~ )1
[
k(t) = 4nRD 1
+(:;:;2]
where a =
ka[ 1
a(a
D=-
+ b exp(c2t) erfc (ct1l2)
"?-I
+ 4nRD
(2)
-I
(3)
+-
Experimentally it is possible to verify the applicability of eq 8 or eq 15 for a particular fluorescence quenching reaction and obtain values for p and u in the case of eq 8, or a, b, and c in the case of eq 15. Using these values one calculates the values for the fundamental parameters of the models using eqs 16-1 8. SCK model R = [a2(l + b/a)3]1/3[4nbcz]-1/3 (16)
and in the case of the SCK model k(t) = a
( 1 1)
long-time SCK model
Theory The time-dependent rate coefficient equations for a diffusionlimited reaction of initially randomly distributed reactants have been obtained as exact solutions of the partial differential equations for two simple models: Smoluchowski and Smoluchowski-Collins-Kimball (SCK) models. The basic features of the models are described in detail in ref 1. In the case of the Smoluchowski model
(9) (10)
+ b)
4aRb
ka=a+b (18) For the long-time SCK model, it is convenient to define a distance parameter R'by eq I9 so that eqs 13 and 14 are written in Smoluchowski form as eqs 20 and 21: R ' = R[1
+ 4aRD/ka]-'
(19)
p = 4xR'DCq (4)
(20)
u = 8(nD)112R'zCq
(21)
Experimental values of p and u can be used to calculate D and R':
In the above equations, D (cm2 s-l) is the sum of the diffusion coefficients of the reactants and R (cm) is the "encounter" distance at which reaction proceeds with an absolute rate of k, (cm3 s-' molecule-'); k, = m for the Smoluchowski model. It is usual to approximate eq 2 for ct1/2>> 1 to give the so-called long-time, Smoluchowski-type equation for k(t):
For a diffusion-limited fluorescence quenching reaction the intensity decay equation is given by eq 7
R'=
P 4rDCq
-
It can be easily seen that, for k , > 102(4xRD), R' R and the long-time SCK equation has to be identified as the Smoluchowski equation. Conversely, if the experimentally determined value of R'is equal to the expected value of R then it may be inferred that Smoluchowski model is applicable. Experimental Section
I ( t ) = Io exp[-trO-l - x ' k ( t ) C , dt]
(7)
where I, is the intensity at t = 0, r0 is the lifetime of the excited state in the absence of quenching and Cq~(molecules/cm3)is the bulk concentration of the quencher. The intensity decay equation for the Smtluchowski or long-time SCK model is I(1) = A exp[-t(ro-l
where
+p)-
(8)
The time-correlated single photon counting (TCSPC) experimental setup has been described previ~usly.'~The PMT XP202OQ has been replaced with Hamamatsu MCP-PMT 156411-01 or 2809U-01 used with C F D s manufactured by Tennelec Inc., USA. The instrumental response function has a width of 80-90 ps. Indole (Aldrich) and potassium iodide were recrystallized from (22) Periasamy, N.; Joshi, G. C.; Das, R. Chem. Phys. Lerr. 1989. 160, 451.
2910
The Journal of Physical Chemistry, Vol. 94, No. 7 , I990
Joshi et al. TABLE I: Results of Analysis of Fluorescence Decay Data (10.4 pdchannel) of Indole in Water at 25 O C 0 T , ns 6, ps D W P resc X2 -fitting range A I
In
5 6 7 8
I
n
c
P w.0
ab
6 1-450 7 1-450 8 1-450 9 1-450 104-450 104-450 154-450 154-450'
2 3 4
4
11.0 11.0 11.0 11.0 11.0 11.0 11.0
4.36 4.36 4.33 4.33 4.33 4.33 4.31 4.33
-11.1 -10.2 -1.5 -5.0 14.8
1.10 1.11 1.12 1.14
O.Od O.Od O.Od
D
1.14 1.25 1.71 1.72 1.73 I .72 1.76 1.72
E F G
H I
'The channel numbers corresponding to the foot of the rising edge of excitation data, the peak of the excitation data, and the peak of the emission data are 61, 77, and 104, respectively. bDurbin-Watson parameter. 'Residual distribution curve in Figure l . dThis value for 6 was held fixed in the analysis of the data. 'Analysis by using a 6function as the excitation function.
10 4 pr/ch A
> 150
5 E 10.0 z
TABLE 11: Fluorescence Quenching of Indole by KI in Water at 25 " C Results of One- and Two-Exponential Fits to the Quenched Fluorescence Data
50
one expo 00
1.68 1.52 1.10 1.09
I
1.04
2 08
3.12
4.16
5w
mM
TIME INANO SECONDS)
Figure 1. Fluorescence emission of indole at 360 n m in water at 25 OC (curve B, with noise) and the results of analysis for various fitting ranges.
The emission data were fitted to the one-exponential decay equation for different fitting ranges (see Table I) and the distribution of weighted residuals are shown by curves D-I. The percentage of weighted residuals between +2 and -2 was greater than 91% for curves D and E and greater than 94% for others. Curve C is the calculated emission curve for the parameters: A = I 1 .O, T~ = 4.33 ns. and 6 = -5.0 ps obtained for the range 91-450 channels. Curve A is the instrumental response function measured a t 295 nm. double-distilled water which is also the solvent used in this study. Acrylamide was recrystallized from spectroscopy grade benzene and NATA (Sigma Chemicals) was used as received. No photon counts above the noise level were detected at 360 nm when 200 mM K I or acrylamide solution was excited by 295-nm laser pulse, indicating the absence of fluorescent impurities and the absence of interference due to scattering in the quencher solution. The and concentration of indole or NATA was between 0.5 X 1 X M for all the experiments. The fluorescence decay data obtained in the TCSPC experiment are deconvoluted by the iterative nonlinear least-squares method in order to optimize the values of the parameters in I ( t ) , and also the time-shift parameter b appearing in eq 24. F ( r ) and R(r)
F(r) = l'!?(s+6) 0 I(r-s) ds
(24)
in eq 24 are the experimentally measured fluorescence emission data at 360 nm and instrument response data at 295 nm, respectively. The details of the deconvolution analysis are given e l ~ e w h e r e . ' ~The - ~ ' uncertainties in the optimized values of the parameters are estimated from the diagonal elements of the error matrix calculated after completion of the ~ p t i m i z a t i o n . ~Large ~ uncertainties indicate correlation of parameters. Results
Fluorescence Decay of Indole in Warer. The fluorescence decay of indole in water at 25 "C measured at 10.4 ps/channel is shown in Figure 1. The fluorescence decay was fitted to a one-exponential decay equation (eq 25) for various fitting ranges and the I ( t ) = A exp(-t/r,)
c,
(25)
results are given in Table I. The residual distributions for a few fitting ranges are also shown in Figure 1. The lifetime remains constant (4.33 & 0.02 ns) for all the fitting ranges and agrees with that obtained for the fitting range 154-450 channels, in which (23) Bevington, P. R. Data Reduction and Analysis in Physical Science; McGraw-Hill: New York. 1969.
50 100 150 200 500
T,
6,
ns
ps
1.57 -13.1 0.973 -4.1 0.737 0.3 0.522 11.2 0.215 2.0
'/(I)=
A
two expb
x2 1.45 1.81 2.21 3.16 6.82
6, A , , A2
ps
0.7, 0.3 -4.5 0.43,0.57 2.1 0.834, 0.166 5.8 0.15, 0.85 -9.7 0.005, 0.995 2.9
exp(-r/r). * l ( r ) = A , exp(-r/r,)
T,,
x2
r 2 , ns
1.71, 1.22 1.32 1.14,0.823 1.22 0.772, 0.421 0.99 0.720, 0.471 0.97 0.912, 0.207 1.65
+ A2 exp(-t/r2).
region there is no overlap of the excitation function with the emission function. It is noticed that the time-shift parameter 6 is not a significant parameter (see lines 5 and 6 in Table I) when the fitting range begins from the peak of the emission. When the fitting range includes part or all of the rising edge of the emission, it is necessary to optimize also the time-shift parameter for a good fit of the decay data. It is also noticed that the fitting of eq 25 to the early part of the rising edge is not satisfactory. Adjusting the angle of the polarizer between the sample and the monochromator at various angles, including 0 and 90°, resulted in decay curves which were nearly identical with that observed at the magic angle of 54.7'. This indicates an extremely rapid mechanism of depolarization of the observed emission, confirmed recently24 to have a time constant of 2.6 ps. The photochemistry of indole in water and in other solvents has been studied very extensively in the past [reviewed in refs 25 and 261 and the consensus seems to be that several "states" of the excited molecule have to be invoked in order to explain all the known experimental observations. The fluorescence emission itself is believed to be due to the exciplex state of indole and water, in which case the decay equation ought to be a two-exponential equation with a grow-in component which may be extremely rapid. Another well-known aspect of the indole emission is the sensitivity of the quantum yield and the lifetime to temperature in aqueous s o I ~ t i o n s . ~ ~In. ~view * of the existing information on the photochemistry of indole it was decided to select a practically meaningful range of time for which the emission decay is unambiguously fitted to a one-exponential function. As seen in Figure 1 and Table I, a fitting range that includes a part of the rising edge (100-150 ps before the peak of the emission) seems to be adequate. Based on this result a fitting range beginning at 100 ps before the peak of the emission and ending either at -4 ns or at a time when the intensity has dropped to -20 counts, whichever is less, was chosen for the analysis of the quenched fluorescence data. Indole/Kllwater: Analysis of Data f o r Different Decay Equations. Fluorescence decay data were obtained at 360 nm
-
~
~
~~~~~
~
(24) Ruggeiro, A,; Todd, D.; Fleming, G . R. In Ultrafasr Phenomena VI; Yajima, T., et al., Eds.; Springer-Verlag: Berlin, 1989; pp 477-479. (25) Lumry, R.; Hershberger, M. Photochem. Phorobidl. 1978, 27. 819. (26) Creed, D. Phorochem. Photobiol. 1984, 39, 537. (27) Kirby, E. P.; Steiner. R. F. J . Phys. Chem. 1970, 74, 4480. ( 2 8 ) Eftink. M . R.; Ghiron, C. A . Anal. Biochem. 1981. 114, 199.
The Journal of Physical Chemistry, Vol. 94, No. 7, I990
Diffusion-Controlled Reaction
2911
TABLE 111: Results of Analysis of the Quenched Fluorescence Data of Indole/KI/Water (25 'C) Using E4 8' I O ~ D , R: IO"R'D, P3 c, mM ns-l cm2 s-I A cm3 S-I X2
;
e
50 50
W
n
IO0 IO0
W
I-
?!
I50 150 200 200
Y
0.066 0. I35 0.101
0.175 0.124 0.360 0.538
4.65 3.48 3.30 4.09 3.32 4.33 2.73 1.91
2.23 2.87 2.95 2.42 2.74 2.16 3.57 4.78
10.4 10.1
9.8 10.0
9.2 9.4 9.8 9.2
1.22 1.27 1.18 1.36 I .02 I .53 2.18 5.15
"Data fitting range begins from 104 ps (IO channels) before the peak of the emission and extends to -4 ns.
10.4 pr/ch
I
TIME
0.39 I 0.378 0.739 0.745 1.03 1.06 1.48 1.38
(NANOSECONDS)
Figure 2. Quenched fluorescence emission of indole/KI (100 mM) in water at 25 OC (curve B, with noise) and the results of analysis for different decay equations. Curves D and E show the distributions of weighted residuals for the one-exponential decay equation. Curve D is obtained for the free fit of all parameters, A , r, and 6, and curve E is obtained when 6 is held fixed. The residual distributions are nonrandom in both the cases. Curves F, G, and H show the residual distributions obtained for the fit of two exponentials, eq 8 and eq 15, respectively. The distributions of residuals seen in curves F, G, and H are acceptably random. Curve A is the instrumental response function and curve C (smooth) is calculated emission curve for the parameters obtained in the analysis using eq 15 for c = 5.0 ns-1/2.
with a time resolution of 10.4 ps/channel for five concentrations of potassium iodide: 50, 100, 150, 200, and 500 mM. The fluorescence decay data were analyzed to fit one-exponential and two-exponential equations, eq 8, and eq 15, and the results are described below. The mathematical procedures used for the fitting of the nonexponential equations, eqs 8 and 15, are discussed in detail in refs 14 and 21. Table I1 gives a summary of the results for the fitting of oneand two-exponential equations. One-exponential equation is inadequate to fit the data because the distribution of weighted residuals is nonrandom (curves D and E in Figure 2 and high value of the reduced x2). The two-exponential function fits well (curve F in Figure 2) but the results of the fitting, namely, lifetimes r l , r2, and amplitude ratio ( A I / A 2 ) ,do not have any physical significance because there is no theoretical model which predicts Z(t) to be a two-exponential equation. A good fit of the two-exponential equation for the quenched fluorescence data was also observed previously in time-domainI4 and frequency-domainI6 experiments. These results are included here with the expectation that in future it may become possible to approximate eq 15 (or any other equation for I ( ? ) predicted by a more realistic model) to two exponentials. The fit of eq 8 to the data (Table 111 and curve G in Figure 2) is adequate for C, up to 150 mM. Using the optimized values of p and u for each concentration we calculated the values of D and the distance parameter, R', using eqs 22 and 23. It is observed that D and R'are correlated such that the product DR'is relatively constant. Rejecting the result for C, = 200 mM (high x 2 ) , we obtain average values for D = 3.86 f 0.57 cm2 s-I and R'= 2.56 7 0.34 A. (The signs f and F indicate correlation between D and R'.) We note that the value of R'is far less than the expected value for R (-6 A) and hence the Smoluchowski model is not valid. The good fit of eq 8 to the data must therefore be considered as an indication that SCK equation may be valid. Since eq 8 is a valid approximationZ2to eq 15 only at long time after the initiation of the reaction, eq 8 must be fitted to the quenched fluorescence data after a cutoff time. The cutoff time is, however, unknown. Therefore, the decay data for all con-
TABLE IV: Average and Standard Deviation of the Values of D and R' Obtained in the Analysis To Fit Eq 8 of Quenched Fluorescence Decay Data of Indole/KI/Water for Four (25 "C) and Three (5 "C) Concentrations (Two Experiments for Each Concentration) for Different Fitting Ranges Beginning at NB and Ending at -4 ns" IO~D, IOI~R'D, NB cm2 s-I R', A cm3 s-l T = 25 O C peak - 104 ps 3.86 f 0.57b 2.56 T 0.34b 9.80 f 0.45b peak 3.1 1 f 0.33 3.12 7 0.33 9.66 f 0.45 3.42 7 0.51 9.71 f 0.53 peak + 104 ps 2.84 f 0.46 peak + 208 ps 3.01 f 1.25 3.4 =F 0.96 9.50 f 0.54
peak - 104 ps peak peak + 104 ps peak + 208 ps peak + 312 ps peak + 416 ps
T = 5 T 3.18 f 1.02 2.26 7 0.55 2.31 & 0.33 2.76 =F 0.31 1.98 f 0.34 3.17 7 0.36 1.78 f 0.36 3.52 7 0.52 1.54 f 0.37 4.02 7 0.73 1.34 f 0.35 4.58 7 0.98
'The peak of the emission was -2 concentrations: see Table 111.
X
6.47 f 0.28 6.33 f 0.31 6.22 i 0.36 6.13 f 0.40 5.98 f 0.44 5.84 f 0.50
lo4 counts. bAverage of three
centrations were analyzed for different fitting ranges varying in the initial time. The average values of D and R'obtained for each fitting range are given in Table 1V. It is observed that the expected values for D (2.7 X IOT5 cm2 s-I at 25 OC and 1.5 X cm2 SKIat 5 "C) are obtained only if the data in the region of the peak and before are necessarily excluded from the fitting range. The method used to fit eq 15 to the data is similar to that described in refs 15 and 21. Briefly, a free fit of eq 15 to the decay data was unsuccessful in yielding a set of unique values for the five parameters: A (=lo), p (=aC,), q (=bC,), c, and 6 which appear in eqs 15 and 24. It is necessary to keep one parameter constant for optimizing the remaining four. In this paper the data are analyzed by keeping c as a fixed parameter, and a globally (for all concentrations) consistent set of values for a , b, and c is sought. In contrast to the method used p r e v i o ~ s l y this ' ~ ~ method ~~ is assumption-free. As a fixed parameter, c is varied over a range, 0.2-100 ns-li2. Using the optimized values of p and q for each concentration, we calculated a and b. It is expected that for one value of c, a and b will be constant for all concentrations. In other words, the plots of a vs c and b vs c for all concentrations may converge to a single value for a and b at some value for c, thus establishing globally consistent values for a, b, and c. In Figure 3 we have plotted b vs c in the top panel, a vs c in the middle panel, and x2 vs c in the bottom panel, for all concentrations. The variation in the values of A is marginal. 6 was found to vary by 1-2 ps for 0.5 < c < 100 ns-Ii2. The variations of A and u are not shown in the figure for simplicity. The variation in the value of DWP leads to a conclusion similar to that arrived at by the x2 vs c plot, namely, that the statistical test on the weighted residuals is not helpful to choose a value for c. It is observed that the values of a are more spread out for c < 2, and the spread is less for c > 2. This is helpful to conclude that c < 2 ns-'I2 is not acceptable. Examining the b vs c plot we find that the scatter in the values of b is apparently less in the domain 2 < c < 8, and the scatter is large outside this domain. (The values of b obtained
2912
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990
Joshi et al.
I
24
20 O
.
4
-
x
n
1
15 0
5
0
0
i.
PL 0 X
14 3
4
35
4 8
3
-8
=
Q
3 1
i
2
3
a
N
2
0
x
B I
I
I
-
2
I
X
0
0
I
1 1 0 e
0
0.5
n
0
I .4
I
0
O
0
0 0 0 0 0 0
0
0
I.2 N
X
1.0 (
0
.
e
e...).
I I
1
1
10
100
Figure 3. Plot of x 2 , a (cm3 s-I molecule-'), and 6 (cm3 s-' molecule-') vs c (ns-'12) obtained in the analysis of the experimental data for indole/Kl/water ( 2 5 "C) for four concentrations of KI: 50 mM ( X ) , I00 m M (O), I50 m M (O), and 200 m M (0). The convergence parameter q5 is plotted in the top panel.
G I
IO
I
c(
IOC
ns"'2)
Figure 4. Plot of x2, a (cm3s-I molecule-'), and b (cm3 s-' molecule-') vs c (ns-I12) obtained in the analysis of the simulated data for indole/ Kl/water at 25 OC for four concentrations of KI: 50 mM (X), 100 mM (0). 150 mM (e),and 200 mM (0). The convergence parameter $J is plotted in the top panel. 6 is minimum for c 3 3 ns-'12; the expected value is c = 3.85 n d 2 .
for C, = 200 mM do seem to deviate from the results for other concentrations, indicating perhaps an upper limit to the concentration for the validity of the SCK model or for the constancy of the values of the fundamental parameters.) The absence of a unique convergence in the plots of a vs c and b vs c is disappointing. However, it is gratifying to note that the value of b does seem to converge in the region 2 < c < 8. The convergence of the values of b for all concentrations was examined quantitatively by calculating the value of a convergence parameter, (b, using the squares of the relative deviations in the values of b as in eq 26.
i>j
I n eq 26 n is the total number of concentrations or experiments ( n = 4 in our case). 4 is minimum for the best convergence of b. A plot of CI# vs c is shown in Figure 3 (top panel). It is observed that (b attains a minimum for c 5 ns-'/* and remains unchanged for higher values of c. It has not been possible to get a unique value for c but a lower limit for c is clearly indicated.
-
In order to derive confidence in the above conclusion for the experimental data, analysis was also carried out for the quenched fluorescence data simulated for the quencher concentrations C, = 50, 100, 150, and 200 mM. The results of this analysis are plotted in Figure 4. The variations of a, b, and x 2 with c are similar to those shown in Figure 3. The values of b are more scattered for c < 2 and for c > 8 ns-'12, and a convergence in the region of c centered at c = 4 ns-'12 is clearly indicated. The arrows in the figure show the values of a = 9.16 X cm3 molecule-' s-I, b = 6.24 X cm3 molecule-' s-I, and c = 3.85 ns-'12 which were used for the simulation of data. It is satisfying to note that
c
(ns-"2)
Figure 5. Variation of R (A), D (cm2 s-l), and k, (cm3s-I molecule-') with c (ns-'/*) for the indole/Kl/water system at 25 "C. The value of c = 5 ns-'/* is considered optimum (see 4 vs c plot in Figure 3).
the plot of the convergence parameter, (b vs c indicates that c 3 3 is the acceptable result. The analysis of experimental or simulated data by the assumption-free method leads to a unique lower limit for c. This lower limit is perhaps the best optimized value as suggested by the result of analysis of simulated data. The range of values for D, R , and k , that is allowed for c in the range of 3-6 ns-'12 for the indole/Kl/water system was calculated. For each value of c, a and b are calculated as the average of four values for C, = 50, 100, 150, and 200 mM. The values of R, D, and k , are calculated by using eqs 16-18. Figure 5 shows the variations in D, R, and k , with c. It is noted that R varies substantially, but the variations of D and k, are relatively marginal. For c = 5 ns-'12, we get a = b = 1.2 X IO-" cm3 s-I molecule-', D = 2.84 X
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990 2913
Diffusion-Controlled Reaction
TABLE V Experimental Values of k,, ~ T R ' DD, , and R' for Various FluorophoreQuencherSystems at 5, 25, and 45 "C, and the Expected Values for D (=DF DQ) and R ( h p + ' Q ) at 25 "c" expected valuesb 10-9k,, 4aR'D X 1 05D, 10sD, rF + rQ, system T , "C M-' s-I M-1 s-I cm2 s-l R', A cm2 s-I A indole/ K1/ water 5 5.4 4.6 1.8 k 0.4 3.5 7 0.5 1.5c 7.5 1.3 2.8 0.5 25 3.4 r 0.5 2.7 6.0 1 .8d Id 9.4 10.6 45 0.55 f 0.06 6.9 F 0.6 2.9 3.1 indole/acrylamide/water 5 5.6 6.9 1.8 5.8 1.2 f 0.1 6.2 7 0.6 25 1.5' 6.5' l-lSd 6-7' 8.6 6.0 F 0.6 11.0 1.9 f 0.3 45 1.2 0.2 3.8 3.5 2.5 6.6 3.8 r 0.4 NATA/Kl/water 25 2.5 6.5 7 0.4 3.3 0.5 f 0.05 NATA/acrylamide/water 5 4.1 5.9 0.73 0.06 7.4 'f 0.6 1.6 6.4 25 6.1 6.8 7 0.5 9.2 45 1.3 f 0.2
+
*
*
*
" k , was obtained by using the "lifetimes" of one-exponential fits and the Stern-Volmer equation, k, = (7-l - ~,yl)/C,. Lifetimes (T,,) for indole: 5.85 ns (5 "C); 4.33 ns (25 "C); 1.83 ns (45 "C). Lifetimes (T,,) for NATA: 3.85 ns (5°C); 2.85 ns (25OC); 1.93 ns (45 " C ) . 4rR'D (=a), D,and R'were obtained by using eqs 13, 22, and 23, respectively. *Values of DF,DQ, rF, and rQ are from Table VI. 'Calculated from D at 25 "C;D l / D 2 = T 1 q 2 / T 2 q 1'20 . OC, ref 16. 'Reference 36.
cm2 s-l, R = 6.74 A, and k, = 2.4 X IO-" cm3 s-l molecule-' (1.44 X 1Olo M-l s-I). The Long- Time Domain f o r Practical Applications. The analysis of the data to test the SCK equation (eq 15) by the method using the long-time approximation is the simplest and least expensive. This method gives reasonable estimates of D and R' (DR'is more reliable) even when the applicable long-time region is not known a priori. It will be instructive to examine the validity of the long-time SCK equation and the applicable time domain. The long-time equation is valid only after a lapse of time given by the condition c t 1 I 2> 7, so that the t 1 / 2term in eq 8 is a good approximation22 ( - 1% error in the transient term in eq 15). Thus, for c = 5 ns-1/2, applicable for the indole/KI/water (25 "C) system, the long-time equation is valid only for t > 1.96 ns, whereas time-domain experimental data for f < 1.96 ns have been fitted adequately by using eq 8 (Tables 111 and IV) and reasonable values for D and R' have been obtained by using a cutoff time of only a few hundred picoseconds. This apparent contradiction between theory and experiment is clarified when one considers the precision of the time-domain experimental data and the associated error. The error in the intensity data in a TCSPC experiment is N'12(one standard deviation), where N is the number of counts.29 In order to differentiate at short times between eqs 8 and 15 the TCSPC error has to be much less than the error incurred in using eq 8 as an approximation of eq 15. The notional TCSPC errors for the decay data [defined as the ratio of the standard deviation, N'/2,to the intensity, N = ZI5(t)]are calculated by using eq 15 and the parameters of the indole/Kl/water (25 cm2 s-I, R = 6.74 A, k , = 2.4 X "C) system: D = 2.84 X cm3 s-' molecule, C, = 0.1 M , and T~ = 4.35 ns. In Figure 6, curves A and B show the variation of TCSPC error for lo = 2 X IO4 and 1, = 1 X 1O4 counts, respectively. The error involved , l,,(t) is the by using eq 8 (error = l s ( t ) - Z l s ( t ) l / l l s ( t )where intensity calculated by using eq n ) is shown by curve C. The TCSPC errors (curves A and B) exceed curve C for t 3 140 and t 2 65 ps, respectively, which explains the adequacy of eq 8 for the experimental data of t < 1.96 ns. Curves E and D are the TCSPC error (lo= 2 x IO4 counts) and approximation error for 190 ps. Thus, the indole/Kl system at 5 OC, and they cross at t in practice eq 8 will give an adequate fit to the data of practical importance (lo 2 X IO4) at a cutoff time far less than the theoretical value. Results for Other Systems. We have also investigated the quenching in NATA/KI, indole/acrylamide, and NATA/ acrylamide in water at different temperatures. The time-domain 1 X IO4 counts at 20.8 or 41.6 ps/channel) for these data (peak
-
-
-
(29) Demas, J. N. Excifed Stare Lijefime Measuremenfs;Academic: New York, 1983.
0
0.2
0.4 0.6 0.8 TIME
IO
(NS)
Figure 6. Time dependence of the relative errors incurred in the longtime approximation of eq 15 to eq 8 (curves C and D), and the notional experimental TCSPC errors (curves A, B, and E). The parameters used for the calculations are relevant to the indole/Kl/water system at 5 and 25 " C . In practice, eq 8 will be a "good" approximation to eq 15 after the cutoff times indicated by the crossings of curves A and C (for 25 "C) and curves D and E (for 5 "C) for a peak count of 2 X IO4 at t = 0. For a peak count of 1 X IO4 the cutoff time is less as shown by the crossing of curves B and C.
systems were obtained for four concentrations (50, 100, 150, and 200 mM) of the quencher. The data were analyzed by using eq 8 with a fitting range beginning from the peak and 100 or 200 ps after the peak of the emission. In all the cases satisfactory fits were obtained for eq 8. The optimized values of p and u were used to calculate D and R'and the average of all values for D and R' for each case is given in Table V. In the case of indole/Kl/water at 45 OC widely divergent values for D and R'were obtained, averaging was unacceptable and therefore omitted. However, the value of R'D was more reliable and hence the values of 47rR'D in the unit of M-' s-l are given in Table V. The signs of the standard deviations, f for D and F for R', indicate that they are correlated. The expected values for the sum of radii and sum of diffusion coefficients, D = DF + Dp, are given in Table V, along with values considered by others. The values of the quenching rate constants ( k , ) calculated by using 'lifetimes" of one-exponential fits and Stern-Volmer equation are given for comparison with the experimental values of 4rR'D. In all cases k , > 4nR'D, which is the expected result for diffusion-limited reactions.
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Discussion The relevance of the SCK model or the Smoluchowski model for the various fluorophore-quencher systems has to be examined by comparing the experimentally determined D and R'with the values expected for these systems. Table VI gives the values of
2914
The Journal of Physical Chemistry, Vol. 94, No. 7, 1990
TABLE VI: Values of Radii and Diffusion Coefficients at 25 OC in Water for the Fluorophores, Ouenchers, and Related Molecules radius, diffusion molecule/ion A coeff. cm2 s-l indole 3.2" 7.8 X IOda
3.8"
NATA
6.4
tryptophan
X
IOda
6.6 X IOdb
1I-
1.0 x
2.8c
10-5"
1.91 x 10-5d 1.87 x 10-5'
KI KI acrylamide propionamide
1.91 X I .o x I 0-5 a 1 . 1 x 10-56
2.6"
"Edward's theory.'O Reference 32. CReference33. dReference 31. CReference34. /Reference 35. radii and diffusion coefficients in water at 25 OC. The experimental values for tryptophan and propionamide are reasonably close to those calculated for indole and acrylamide, respectively, using Edward's empirical theory.30 The self-diffusion coefficient3' of I- (or K') is nearly the same as that of KI. We use D = 1.9 x cm2 s-l for I-. The analysis of the fluorescence decay data of indole/Kl/water at 25 "C to fit the SCK equation (eq 15) gave values for D 2.8 X cm2 s-l, R 6.7 A, and k, 2.4 X IO-" cm3 molecule-' s-l. These values are in reasonable agreement with the cm2 s-I and R = 6 expected values (Table V): D = 2.7 X A. The use of eq 8 has given D = (2.8 f 0.5) X cm2 s-' and R ' = 3.4 i0.5 A, and these values are consistent with the above values. The SCK model is therefore valid for this system. The results for the indole/acrylamide or NATA/acrylamide system (25 "C) indicate that the value for the distance parameter R ' is reasonably close to the expected value for R. However, the value of D is considerably less in both cases (33% for indole and cm2 s-' 56% for NATA). I f the estimate of D = (1-1.5) X for indole/acrylamide made by Lakowicz et aLi6is correct, then it is reasonable to conclude that indole/acrylamide is perhaps the first example of a diffusion-limited reaction for which the Smoluchowski model is valid. For NATA/KI the value for the distance parameter is substantially lower than R and therefore it has to be interpreted as R'of the long-time SCK model, but the value of D is also substantially low for an unambiguous identification of this system with the SCK model. The substantially lower value of D for NATA/KI and NATA/acrylamide (-50% in both the cases) is intriguing. The value for R'is in agreement with that for the corresponding indole system. This low value for D is related to the result that the quenching constant, k,, or the value of 47rR'D for the NATA systems is substantially less than the corresponding values for the indole systems. It appears that the indole moiety in NATA is quenched less efficiently when compared with free indole. This reduced quenching efficiency for NATA is less likely to be due to a decrease in the diffusion coefficient of NATA, since the extra pendants of indole in NATA do not drastically change the radius or the diffusion coefficient (Table VI). It is possible that the reactivity of NATA and the quencher is anisotropic, in which case the Smoluchowski or SCK model is not applicable. The correlation between the experimentally determined values of D and R' makes it difficult to examine rigorously the tem-
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(30) Edward, J. T.J . Chem. Educ. 1970, 47, 261. (31) Marcinkowsky, A. E.; Nelson, F.; Kraus, K. A. J . Phys. Chem. 1965, 69, 303. (32) Landoldt-Bornstein Zahlenwerte und Funktionen; Springer-Verlag: Berlin, 1969; Band 11, Teil 5 , pp 61 1-653. (33) Marcus, Y . Ion Solvation; Wiley: London, 1985. (34) CRC Handbook of Chemistry and Physics, 55th ed.; Chemical Rubber Co.: Cleveland, OH 1975. ( 3 5 ) Kamakura. K . Bull. Chem. Soc. Jpn. 1982, 55, 3353. (36) Eftink, M. R.; Ghiron, C. A . J . Phys. Chem. 1976, 80, 486.
Joshi et al. perature dependence of D. It is expected that D a T / q according to the Stokes-Einstein equation. For the case of indole/KI/water the values of D q / T obtained at 5 and 25 OC are (9.8 f 2.2) X and (8.4 f 1.5) X cm2 s-' K Pi,respectively. For the case of indole/acrylamide the values of Dq/T are (3.0 f 0.3) x 1O-Io, (3.6 f 0.4) X and (3.6 f 0.4) X cm2 s-l K P I at 5, 25, and 45 O C , respectively. These results are consistent with the D a T / q relation. In this paper we have shown that it is possible to use eq 8 for the indole/Kl/water system and get a good fit to the time-domain data and obtain reasonable values for the fundamental parameters of the model. However, Lakowicz et aLi6 have found that the use of eq 8 for indole/KI/water at 20 OC gave D = (0.4-0.9) X IO-5 cm2 s-I, which is far less than the expected value. The reason for this discrepancy between the results of time-domain data and frequency-domain data is not clear. On the other hand, the values of D, R, and k, obtained by Lakowicz et a1.I6 using eq 15, especially the results of simultaneous analysis of data for C , = 25-150 mM [ D = 2.84 X cm2 s-I, R = I A, and k, = 47rR2~ = 1.54 X IO-" cm3 s-I molecule-'], are in reasonable agreement with our values, noting that k, was held fixed for the analysis of frequency domain data. We assume that the differences in temperature (20 vs 25 "C) and medium (buffered vs unbuffered) in the two experiments are not drastically significant. Wijnandts van ResandtI3 has carried out a time-domain investigation of the fluorescence quenching of NATA/KI/water at 22 OC and analyzed the data to fit the Smoluchowski model. cm2 s-' and R = 3.40 It is reported that D = (1 .I 2 f 0.05) X f 0.12 A for this system. Our values of D and R'for this system at 25 OC (Table V) are satisfactorily close to the above values. This reproducibility of the results in the analysis of time-domain quenched fluorescence data is gratifying. The results presented in this and previous papers' '-I5 for the various fluorophorequencher systems demonstrate that the time-domain method is capable of verifying transient effects in diffusion-controlled reactions.
Conclusions The time-domain quenched fluorescence decay data of indole/KI/water system at different temperatures are consistent with the decay equation predicted by the SCK model and the experimental values for the fundamental parameters are reasonably close to the expected values. The Smoluchowski model may be applicable to the fluorescence quenching of indole/acrylamide/ water system. The fluorescence quenching of NATA by KI or acrylamide cannot be identified with either of the models. The time-domain fluorescence data by the TCSPC technique was shown to be eminently suitable for the investigation of diffusion-controlled bimolecular reactions. The results of the time-domain fluorescence study of NATA/KI/water reported previouslyi3 are reproducible. The doubts raisedi6 concerning the capability of the TCSPC technique to study the transient effects in diffusion-limited reactions are unwarranted. The analysis of quenched fluorescence data to fit the SCK decay equation (eq 15) is difficult because of correlation of parameters. It is usual to assume a fixed value for one or two parameters" or rely on the results of analysis of simulated A method free of such preconditions or assumptions is described and used in this paper. This method gives limiting values for the fundamental parameters of the SCK model.
Acknowledgment. We thank the Department of Science and TecHnology, Government of India, for financial assistance to set up the Unit for Chemical Dynamics and Picosecond Spectroscopy. Thanks are due to Prof. B. Venkataraman for the critical reading of the manuscript and for the numerous helpful suggestions. Registry No. N A T A , 10346-41-5; K1, 7681-1 1-0; indole, 120-72-9; acrylamide, 79-06-1.
