Deconvolutiori of Fluorescence Decay Curves. A ... - ACS Publications

Deconvolution of Fluorescence Decay Curves

measurement is large but no detectable nonlinearities were observed.

Acknowledgment. The support of the National Science Foundation through Grant GP-28436 and an Undergraduate Research Participation Grant EPP 75-04388 is gratefully acknowledged. References and Notes (1) G. R. Stevenson find A. E. Alegrla, J. Phys. Chem., 79, 1042 (1975). (2) G. R. Stevenson, A. E. Alegria, and A. McB. Block, J . Am. Chem. SOC.,97, 4859 (1975). (3) M. T. Jones, J . Chem. Phys., 38, 2892 (1963). (4) R. G. Kooser, W. Y.Volland, and J. H. Freed, J. Chem. Phys., 50, 5243 (1969). (5) M. P. Eastman, G.V. Bruno, and J. H. Freed, J. Chem. Phys., 52, 2511 (1970). (6) S. A. Goldman, G.V. Bruno, and J. H. Freed, J. Chem. Phys., 59, 3071 (1973). (7) C. Jolicoeur and H. L. Freidman, Ber. Bunsenges. Phys. Chem., 75, 248 (1971). (8) M. T. Jones, M. Komarynsky,and R. D. Rataiczak, J. Phys. Chem., 75, 2769 (1971), (9) M. T. Jones and M. Komarynsky, J. Chem. Phys., 56,4404 (1972). (10) A. S. Mason and W. H. Bruning, Chem. Phys. Lett., 15,299 (1972). (11) M. T. Jones, Chem. Phys. Lett., 20, 151 (1973).

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(12) M. T. Jones, J. Mag. Reson., 11, 207 (1973). (13) Note the value reported in ref 14 was not corrected for the error reported by Allendoerfer (ref 15). All other g values reported therein were so corrected. (14) M. T. Jones, T. C. Kuechler, and S. Metz, J . Mag. Reson., 10, 149 (1973). (15) R. D. Allendoerfer, J. Chem. Phys., 55, 3615 (1971). (16) 6.G. Segal, M. Kaplan, and G. K. Fraenkel, J . Chem. Phys., 43, 4191 (1965). (17) M. T. Jonesand W. R. Hertler, J. Am. Chem. Soc., 86, 1881 (1964). (18) R. D. Rataiczak and M. T. Jones, J. Chem. Phys., 56, 3898 (1972). (19) D. J. Cram and R. A. Reeves, J. Am. Chem. Soc., 80, 3099 (1958). (20) W. Moser and R. A. Howle, J. Chem. SOC.A , 3039 (1968). (21) C. J. Pederson, J . Am. Chem. SOC.,89, 7017 (1967). (22) C. J. Pederson and H. K. Frensdorff, Agnew. Chem., 11, 16 (1972). (23) R. A. Roblnson and R. H. Stokes in “Electrolyte Solutions”, Academic Press, New York, 1955, p 389. (24) M. T. Jones and T. C. Kuechler, J. Phys. Chem., 81, 360 (1977). (25) R. A. Howie, L. S. D. Glasser, and W. Moser, J . Chem. SOC.A , 3043 (1968). (26) E. Dalgard and J. Llnderberg, Int. J. Quantum. Chem.,59, 269 (1975). (27) E. Dalgard and J. Linderberg, J . Chem. Phys., 65, 292 (1976). (28) M. T. Jones, S. A. Trugman, V. Rapini, and R. Hameed, J . Phys. Chem., 81, 664 (1977). (29) This phenomena was first discovered by R. Ahmed and Professor S. I. Weissman at Washington University, St. Louis. We are Indebted to Professor Weissman for his interest in and help with thls portion of this study.

Deconvolutiori of Fluorescence Decay Curves. A Critical Comparison of Techniquest D. V. O’Connor, W. R. Ware,* The! PhotochemlstfyUnit, Department of Chemistry, Universlty of Western Ontario, London, Ontario, Canada, N6A 587

anid J. C. Andre Laboratoire de Chimie (.%%&ale, E R A . No. 136 du C.N.R.S., 54042-Nancy, Revised Manuscript Received November 20, 1978)

Cedex, France (Received September 7, 1978;

Several methods of deconvoluting fluorescence decay curves are compared. Real data for which the decay times were known provide tests of each method’s ability (1)to deconvolute two-component decays having decay times of 1 and 5 ns, (2) to cope with distortions in the data resulting from instrumental artifacts, (3) to analyze subnanosecond decay times, and (4)to separate two closely spaced decays. A new method based on Fourier transformationof the observed time profiles is included in the study, and some modifications of existing techniques are tested. It is found that all methods are satisfactory for undistorted one-component data but least-squares iterative reconvolution is most suitable for analysis when distortions are present. Only iterative reconvolution techniques and the method of modulating functions resolved satisfactorily two closely spaced decays.

I. Introduction The observed fluorescence decay curve of a molecule excited by a short pulse of light is a convolution of the pulse shape of the excitation (distorted by the detection system) with the 8-pulse response of the luminescent system, i.e. f(t) = l t g ( t ’ ) d(t - t’) dt’ 0

in which f(t) represents the measured decay curve, g(t) represents the measured intensity-time profile of the exciting pulse, and d(t) represents the undistorted decay law of the system. Difficulties in recovering d(t) from the observed f(t) and g(t) may arise from (i) distortions in the data of which the most frequently encountered are the wavelength depenPublication No. 211 from the Photochemistry Unit, Department

of Chemistry, University of Western Ontario, London, Canada.

dence of the excitation profile, the dependence of the response of the detector on the energy of the incident photons and on the area of the photocathode illuminated, interference of radiofrequency noise and scattered exciting light, walk and jitter in the electronics, and the time dependence of the excitation profile and (ii) the mathematical treatment of eq 1. Although some of the factors listed under (i) may be treated in the deconvolution process, it is generally preferable to arrange the experimental conditions in such a way that distortions are almost negligible and that the validity of eq 1 may be safely assumed. A variety of deconvolution techniques have been devised for the analysis of data collected with the single photon counting technique. In a recent publication’ McKinnon et al. compared the most common of these with each other and recommended the use of the iterative convolution method (vide infra) as the most reliable technique. This publication dealt almost entirely with simulated data; the

0022-3654/79/2083-1333$01 .001Q 0 1979 American Chemical Society

1334

D. V. O’Connor, W. R. Ware, and J. C. Andre

The Journal of Physical Chemistry, Vol. 83, No. IO, 1979

one set of real data used was assumed to follow a double exponential decay law, but, as it resulted from the fluorescence of a solution containing two different types of molecules, it is not certain that this assumption is valid. In this paper, we present the results of a comprehensive study of the deconvolution of real data, including both known single and double exponential decays. We have used the fluorescence decay of n-hexane2 to test the ability of the deconvolution routines to recover extremely short lifetimes. In addition, we have included in the present study a new deconvolution method based on the fast Fourier transform (FFT).3Each single photon counting datum has attached a Poissonian error, which may be represented by

These random fluctuations, arising from the counting process, have hindered for many years the application of Fourier transforms to the solution of eq 1. As we shall show, the new FFT method recovers a decay law reasonably free of spurious oscillations. Although the results of this study indicate that iterative convolution should be the technique of choice when the decay law is known, we have found that the FFT technique and the exponential series method (vide infra) perform, in general, quite satisfactorily and we can recommend their use in the analysis of data for which the decay law is unknown.

than some arbitrarily chosen amount, it is assumed that the minimum has been reached. We use the Marquardt algorithm6p6in order to speed up the search. Leastisquares fitting may be extended to analyze decay curves contaminated with scattered light. In this case, one uses as a fitting function f(t) = &‘g(t? d(t - t ? dt’+ ag(t)

(5)

Alternatively, fitting with eq 3 may be carried out over a region of the decay curve in which such distortions are expected to be small. (ii) Method of Moments. Starting with a method, based on the moments of the decay curve, first formulated by Bay’ and later applied by Brody8 to single exponential decay curves, Isenberg and co-workers have developed and refined a technique, known as the method of moments, for the analysis of multiexponential decay. data.g-14 The principle of the method is as follows: if the kth moments of the exciting pulse and the decay curve are defined by

and if d(t) is assumed to be a sum of exponentials n

d(t) =

(7)

j=l

it can be shown that

11. Methods

Deconvolution methods are usually divided into two groups: (a) those that assume a specific functional form for the decay law and (b) those that make no such assumption. When the functional form of the decay law is indeed known, the former methods are to be preferred. But, often, more complicated decay kinetics lead to decay laws that are not simple exponentials or sums of exponentials. For these decays, so-called deconvolution is required. (i) Least-Squares Iterative Reconvolution. This is perhaps the most widely used analysis technique for single and double exponential decay laws.4 If one has a set of observations, y(t,), one can use the method of linear least squares to obtain the best fit of a chosen function yo(ti,a,), linear in the parameters a,, to y(t,). One does this by calculating the weighted sum of the residuals, x 2

in which n

w,= j=l C(aj7jq

(9)

By taking an appropriate number of moments of the experimental curves, one can use eq 8 and 9 to calculate a, and ~ j Equation . 8 is arrived at by virtue of the limits of integration in eq 5 and 6. Because the observed decays extend only to a finite time T, they must be extrapolated to allow an accurate determination of the moments. Usually, the lamp profile has a negligible intensity at T, and the mk are calculated by means of the equation

& tk g(t) dt T

mk =

For the moments of the decay curve, one may write

1

in which oiis the error in the ith data point, and by varying the a, until a minimum in x2 is attained. For fluorescence decay curves, the fitting function yo(ti) = Jtig(t’) d(t - t’,aj) dt’

Upon substitution for f(t) using eq 1 and 7, one finds

(3)

is nonlinear in the parameters ai. It is linearized by ex-

panding in a Taylor series as a function of the parameters

in which

a,. At each step, the minimization criterion

Sj =

ax2/aa, = 0 leads to values for the parameter increments, 6aj, in the expansion

1g(t?e-t’/7jdt’ T

0

It follows that n

pkm

= XajSjCkj j=l

(4) New values for a,, aj,, = aj,n-l + 6aj, and a new value of x2 are calculated. When successivevalues of x2 differ by less

in which

(14)

The Journal of Physical Chemistty, Vol. 83,No. 70, 1979


kkTvalues are first calculated and lead to estimates of a,

and T ~ .These are substituted in eq 15 to yield, through eq 14 and 11, new values of the & and hence new values The iterations are continued-until of a; and T; and successive values of the moments differ by a predetermined increment. In practice, k is chosen as 0, 1 , 2 , etc. It can be shown12 that by increasing k by 1 or 2, the effect on the analysis of distortions in the data arising from scattered light or from a shift in zero time may be counteracted. Although no weighting of the data points is used in this method, the counting error may be carried through to calculate errors on the final values of the parameters. It should be mentioned that in this error calculation higher moments, which are attended with computation errors, must be employed. (iii) Laplace Transforms. By use of the definition of the Laplace transform of a function x(t)

notify the technique in such a way as to allow us to neglect the rising edge of the decay in the analysis procedure. Suppose one has simple exponential decay kinetics; one can write

where a is a normalization constant. If we wish to use only the portion of the curve beginning at t = tl, we let t’ = t - tl and obtain

F(s) = I m e - S tf(t9 ’ dt’

j-x ( t ) P dt -m

eq 1 may be transformed to yield F(s) = G(s).D(s)

(16)

(17)

In principle, division of eq 17 by G(s) and inverse transformation lead to d(t ) . However, inverse Laplace transformation of an unknown function is difficult to accomplish; therefore, in the application of this technique to single photon counting data the decay law is assumed to be of a specific form and fitting of parameters is made in Laplace space.15-1’ Our use of the technique, similar to that used by Gafni et a1.,l6is designed for the analysis of single- or multiexponential decay laws. The Laplace transform of eq 7 is

By substituting eq 10 into eq 17 and using an appropriate number of s values, one can calculate a, and T;. As in the method of moments, a cutoff correction must be used in the calculation of Laplace transforms of the observed data. One may write

& f(t)e-6td.t + I T m f ( t ) Pdtt T

F(s) =

= FT(s)

+ Fm(s)

(23)

0

G(s) =

Lacst‘ g(t dt ’ ’)

(24)

Since the transform of eq 22 leads to sF(s) - f(t1) (1/T)f(S) = CYG(S)

+

^rn

L[x(t)] = X(s) =

1335

(25)

one can calculate T by using two values of s. But because f(tl) is a noisy data point, it is preferable to eliminate it from the calculations by using three values of s. As will be seen in section IV, this method gives quite satisfactory results when tl is small. However, by neglecting times shorter than tl, one loses information about the lamp shape; consequently, one finds that this method is inferior to least-squares iterative reconvolutian as a means of analyzing selected portions of decay curves. Care must be taken in choosing the correct values of s.15J7For single exponential decays, eq 18 predicts that a plot of 1/D(s) vs. s is linear. In practice, deviations from linearity in such plots are encountered owing to fluctuations in the data and the limit on resolution imposed by the channel width.15 It is necessary to choose s from a linear region of the plot but a preferable procedure, especially in the analysis of two-component data, is to vary s until a range of s values is found over which the calculated parameters are independent of s. (iu)Method of Modulating Functions. For the purposes of the present discussion, a modulating function is defined as any function, $ ( t ) ,that, along with its first derivative $ ( t ) ,is zero-valued at time t = 0 and at some other time t = 2”. Extending the original proposals of Loeb,l9 Valeur and MoirezZ0have developed a technique of deconvoluting multiexponential decays based on the following considerations. One starts with eq 1 and 7

Substitution of eq 1 and 18 into eq 19 yields n

in which C; is the contribution of the exponential component j to the intensity in the decay curve at time T.

It can be shown that (4

f(t) + Alf(t) + A2f(t) + ... + A,f(t) = aIg(t) + An iterative procedure, similar to that described in (ii), is employed until some convergence criterion for F(s) is satisfied. Scattered light distortions and distortions arising from a shift in the zero time may be corrected for with this technique. To correct, for scattered light, for instance, one takes a furthers value and substitutes eq 5 for eq 1 in the derivation of the final expressions for the parameters. Because distortions arising from the wavelength dependence of the photomultiplier response are thought to occur at early times in the decay curve,18we attempted to

CY&(^)

+ ... + CY,

(n-1)

in which A1 =

011

= Ald(0)

ET; ;

+ A2d(0) + ... + A Z O )

g(t) (26)

1336

The Journal of Physical Chemistry, Vol. 83,No. 10, 1979

a2 = Azd(0)

D. V. O'Connor, W. R. Ware, and J. C. Andre

+ A3d(0) + ... + AXO)

~ U N=

Ad(O)

As with Laplace transforms, convolution in real space becomes multiplication in transform space, so that

d(O) = C a j 1

If eq 26 is multiplied by the modulating function, &(t), and integrated between t = 0 and t = T', one obtains

By choosing a suitable number of modulating functions, it is possible to solve eq 27 for the a] and 7,. The extension of the technique to analyze data distorted by scattered light according to eq 5 is straightforward.21 We have followed the suggestions of Valeur and Moire9 as to the choice of modulating functions. We let T' = T and use functions of the form

4(t) = tn(T- t)P

(28)

where n and p are integers. (u) Exponential Series Method. So-called deconvolution techniques require in theory no a priori assumptions about the functional form of d(t). They have as their goal the accurate representation of d(t), which may then be examined from the point of view of various physical models. Reasoning that the decay laws almost always encountered in photophysics may be accurately represented by the sum or difference of exponentials, if enough exponentials are used, Ware et a1.18 proposed that d(t) could be expressed in the form k=l

The inverse transform is

D(u) =

(33)

a

2riu

+1

/ ~

(34)

therefore

D(u) = DR(u) + iDI(u)

(29)

in which the a k may be either positive or negative and no physical or mechanistic significance attaches to the individual ak and Y k . In the usual applications of the technique, the Yk are fixed and, by the method of least squares, the ak are varied until the best fit is obtained between the convolution of g(t) with d(t) and the observed decay f(t). If too many exponentials are used, random fluctuations in the data lead to oscillations or other misbehavior in the recovered d(t), whereas too few exponentials fail to give an accurate representation of d(t). Although decay times associated with G(t) do not need to be in the set of Y k used for deconvolution,18we have observed that the best fit to f(t) is obtained when the Y k are evenly spaced about the actual decay times. In general, we have used sets of ten exponentials in which the ~k are so spaced; examples will be shown in section IV. For the deconvolution of the n-hexane decay ( T = 200 ps), we used set I1 (n = 6) from ref 18. (ui) Fourier Transforms. The Fourier transform of a function x(t) is F[x(t)] = X(u) = ~ ~ x ( t ) e - 2 "dt 'Yt

d(t) = ae-t/r its Fourier transform is

therefore

n

d(t) = Cake-t/rk

F(v) = G(u).D(u) (32) from which D(u) and hence d(t) may be determined. It has long been recognized that Fourier transforms offer, in theory, a simple route to d(t), but their application to single photon counting data has been plagued by the magnification of random fluctuations (Poisson noise) always present in real data. We have recently proposed3 an extension of a method published by Wild et al.,22which employs the fast Fourier transform (FFT) and which greatly reduces the oscillations that have hitherto accompanied inverse transformation. This is achieved by truncating the Fourier series D(u) at frequency v1 and extrapolating it by the Fourier transform of a single exponential. Fourier transformation requires integration to infinity but we have found that the finite time scale of the measurements leads to no serious errors in the deconvoluted d(t), provided that f(t) has decayed at time t = T to about 1%of its maximum value. If the functional form of the decay is known, fitting of the parameters may be performed in Fourier space.22 (a) Exponential Decay. If d(t) is given by

(30)

In FFT, we use discrete values for for T is r=n, - 1 n=2

-(

u,

and the expression

)""

(37)

r ( n - 1 ) DR(n)

in which n is the frequency 1/T X n', n'an integer, and nl is a chosen upper limit to n. For small values of nl, eq 37 leads to acceptable T values. But for nl > 20, the calculation proves unsatisfactory. However, by taking account of the Poissonian noise in the data, one can use a revised expression for T , which leads to more satisfactory results. Wild et a1.22have derived expressions for the ) q ( u ) for DR(u) and DI(v) on standard deviations ~ ( u and the basis of the counting error in F(t) and G(t). One may ) rI(u) to calculate the error u7(u)in the value use ~ ( u and of T . Then

n=2

Much more satisfactory results are obtained with this


expression, as will be seen in section IV. (b) Double Exponential Decay. For two-component decays

Letting alrl = B1 and a2r2= B2, one has D(v) = DR(v) DI(v)

+

Through fitting in Fourier space, in either DR(v) or DI(v), one may determine the parameters. In fact, the use of eq 39 does not lead to alcceptable results. But, if one combines DR(v) and DI(u), viz. (B1 + B2) + 4a2V2[B1T12+ B2~2~1 -2ru- DUV) = -+ B272 + 4'T2V2[&T1722+ Bzri2r21 DR(v) (40) the r values obtainled are somewhat more accurate (see section IV). It must, be admitted, however, that fitting in Fourier space does not appear to be as worthwhile an approach as more conventional methods. 111. Experimental Section (i) Choice of Data. In wishing to analyze real data, and at the same time to be reasonably sure of the decay times with which we were dealing, we faced the difficulty of obtaining two-component decays in which the lifetimes were known. It was decided to construct two-component data from the addition of single exponential curves the lifetimes of which could be determined with some confidence. In order to have a lamp profile that could be used in the analysis of the one-component and the resultant two-component data, we chose fluorescing solutions that absorbed and emitted in the same wavelength regions. Because the measurement of short decay times is sometimes difficult to accomplish, we chose a molecule with a r value in solution of about 1 ns for one of the decays. This was 1,4-bis[2-(5-phenyl)oxazolyl]benzene (POPOP). Its lifetime in benzene solution is 1.26 Anthracene in cyclohexane ( r = ca. 5.2 ns) absorbs and emits in the same wavelength regions as POPOP and was therefore suitable as the second component in the twocomponent curve. Upon analysis of one of the anthracene decay curves with the least-squares iterative convolution technique, a good fit was obtained over the section of the decay curve beginning at six channels past the maximum but a poor fit resulted when the entire curve was analyzed. We included this curve in one of the data sets, hoping it would furnish a means of demonstrating the differences between the least-squares fitting procedures and the more analytical techniques. For the resolution of two closely spaced decays, we had available decay curves of a-cyanonaphthalene (CNN) in two solvents, dimethoxyethane (DME) and ethyl acetate (EtAc), the lifetimes of which are about 9.5 and 11.5 ns, respectively. Therefore, we present the results for three sets of two-component data: (a) anthracene in cyclohexane and POPOP in cyclohexane the decay curves of which are reasonably free of rising edge distortions, (b) the same

The Journal of Physical Chemistty, Vol. 83, No. 10, 1979

1337

solutions with some large distortions, the origin of which is uncertain, on the rising edge of the anthracene decay, (c) CNN in DME and CNN in EtAc with lifetimes of about 9.5 and 11.5 ns and with nonnegligible but fairly minor distortions in the rising edges of both curves. In addition, we include the results of the analysis of a very short-lived single exponential decay, neat n-hexane. This decay curve was measured on a vacuum ultraviolet single photon instruments2 Details about the sample and the measuring instrumentation have been published else~here.~>~~ (ii) Preparation and Purification of Solutions. Anthracene, purchased from Harshaw Chemicals, was recrystallized from toluene. POPOP, puriss grade from Fluka AG, was used without further purification. CNN was vacuum-sublimed three times. Cyclohexane (Cyh) was passed down a column containing 60-120 mesh silica gel, impregnated with silver nitrate, and basic alumina in equal amounts. The middle fraction was retained. DME and EtAc were refluxed for 12 h over calcium hydride in an atmosphere of dry nitrogen and then fractionally distilled. The solution of anthracene in cyclohexane was 4 X M and was deaerated by four freeze-pump-thaw cycles. M in DME and EtAc. Both solutions CNN was 5 X were saturated with dry nitrogen. A saturated solution of POPOP in cyclohexane was filtered and diluted by 100. It was not deaerated. (iii) Instrumental Conditions. Decay curves were collected with a standard single photon counting appar a t ~ s Excitation .~~ was with a hydrogen flash lamp. The detector was a Philips 56 DUVP-03 photomultiplier. A monochromator on the excitation selected light at 340 nm for irradiation of the POPOP and anthracene solutions. Emission from these solutions was viewed through an Oriel 72-3900 filter. The lamp profile, measured at 405 nm, was obtained by scattering off a ground Suprasil plate. CNN solution decay curves were measured as described previou~ly.~~ The time profile of the excitation was again measured at the emission wavelength, 330 nm. An aluminum plate was used as the scatterer. A total of 20000 counts was collected in the maximum of each lamp and decay curve. For four of the curves (3, 4, 5 , and 6; see section IV) lamp curves were collected before and after the measurement of the two fluorescence curves. The two lamp curves were then added. For the other two decays, the lamp was measured between the measurement of the two decays.

IV. Results and Discussion (i) Exponential Decay. In Table I are listed the r values obtained with each of the analysis techniques for six decay curves. Also tabulated are values for the statistic, x:, obtained upon reconvolution of the decay law with the lamp profile g(t). xu2 is defined as n2

C ([f(t) - fc(tJI/gi)'

i=n,

'X

=

(n2- nl - 1) - p

p is the number of fitting parameters (=2 for single exponential decays) and f&) is the reconvoluted curve. For data collected with a Poissonian counting error, : x =1

ideally. LSQ A denotes the least-squares iterative reconvolution method with fitting from six channels past the maximum of the decay curve. For all decays, the xy2 is reasonably close to 1and indicates that this portion of the decay follows a single exponential. Curves 1 and 3 were measured with the same sample as were curves 2 and 4. The agreement between the lifetimes is quite satisfactory.

1338


D. V. O'Connor, W. R. Ware, and J. C. Andre

TABLE I: Comparison of Methods for Data Which Follows a Single Exponential Function curve

method LSQ A xu2

LSQ B

xu2

moments C

xu2

Laplace D

XY2

Laplace E

Xu2

mod fns F

xu2

exp series G

xuz

exp series H xu2 Fourier I xu2

1 2 3 4 5 6 A i n POPOP A i n POPOP CNN CNN Cvh inCvh Cvh inCvh in in EtAc . , 5.28 1.15 5.26 1.13 9.69 11.80 1.00 1.10 0.94 0.86 1.20 1.11 5.28 1.12 5.45 1.12 9.62 11.76 1.14 1.55 18.8 2.11 1.07 1.37 5.29 1.12 5.48 1.10 9.63 11.77 1.14 1.62 18.9 1.29 1.40 2.13 5.29 1.12 5.50 1.10 9.62 11.77 1.14 1.61 19.1 1.31 1.40 .2.13 5.25 1.25 5.19 0.98 9.37 11.58 1.24 6.90 10.8 20.5 2.78 1-80 5.27 1.11 6.81 1.08 9.60 11.75 1.14 1.71 24.7 2.25 1.43 2.17 5.30 1.17 5.26 1.17 9.65 11.81 1.01 1.32 0.94 2.09 1.22 1.11 5.29 1.20 5.24 1.16 9.66 11.77 1.14 3.81 24.9 3.15 1.83 2.13 5.30 1.15 5.44 1.15 9.80 11.85 1.01 1.10 18.8 2.42 2.38 2.54 5.25 1.07 6.00 1.09 9.51 11.71 1.24 2.80 52.9 1.90 2.04 2.48

Fourier J xu2 a For (a) and (b) see text. Cyh = cyclohexane.

/ ..

. ".

*. 40

80 120 160 CHANNEL NUMBER

xo

I

240

Figure 1. Decay of POPOP in cyclohexane (curve 2) (0.25 nslchannel): (-) decay curve, f(t); (a) excitation profile, g(t), full width at half-maximum (fwhm) = 3.5 ns; (b) reconvoluted curve, f,(t) (method A).

LSQ B is the least-squares iterative reconvolution method with fitting over the entire decay curve. For all the curves, there is an expected increase in x,2 as compared to method A, but only in the case of curve 3 is the increase large enough to cause concern about the quality of the data. For curve 1,the x,2 value is still very acceptable and the fact that there is no change in the lifetime determined with methods A and B indicates that distortions in these data are negligible. Method B leads to a slight decrease in the T values for all the other curves except curve 3; this is compatible with a small amount of scattered light in the decay. Figure 1shows plots for curve 2, from which it can be seen that a T value of 1ns may be measured even when

0

002

0.04

a06 S , N SEC.'

0 08

10

I

Figure 2. Plots of [D(s)]" vs. s: (a) real data, curve 1; (b) computer-simulated data, real lamp convoluted with exp(-t/5.286), fo(t ) = g(t)* exp(-t/5.286); (c) computer-simulated data of (b) with added Gaussian noise.

the lamp profile is fairly broad. Curve 3 is anomalous in that the lifetime increases in going from method A to method B and the xu2 becomes unacceptable. This curve will be considered in detail later. Method C is the method of moments used without any index displacement. All the lifetimes recovered with this technique are virtually the same as those that result from LSQ B. If an index displacement of 1 is used, the recovered T values approach those of LSQ A. It may be concluded from these results that this method is quite adequate for deconvoluting single exponential decay curves. This conclusion may also be reached with regard to the Laplace transform technique. The lifetimes obtained with it are all in good agreement with the LSQ B method and give values close to the LSQ A method when analysis for scattered light is performed. Precautions must be taken if this technique is to be successful. Figure 2 contains plots of [D(s)]-l vs. s for real data and for simulated data. According to eq 18 such plots are expected to be linear. As can be seen, the presence of noise leads to gross deviations from linearity. It is essential to choose s from a linear portion of the [G(s)]-l vs. s plot. Tests with artificial data demonstrated that more accurate results were obtained when s was in channel-1 rather than ns-l. However, for the time scales of interest in this study, the choice of channel-l as units entails larger s values and hence greater absolute values for the argument in exp(-st). Special care must be taken to avoid numerical overflow errors in the computer for larger values of s t . This consideration imposed a further limit on the range of s values that could be used in this study; s values in the range 0.005-0.01 channel-' were found to be suitable for all the decays. Method E is the Laplace transform method with integrations begun at time t = tl # 0. As was stated in section 11, eq 25, which requires the inclusion of f(tl) in sF(s) - f(tJ

1 + ;F(s)

= CXG(S)

(25)

the calculation of T when only two s values are used, leads to erroneous values of T for large tl, as the results in Table I1 show, When f(tl) is eliminated from the calculation, by the use of three s values, an increase in the accuracy of the recovered lifetime for large tl is found, as is shown also in Table 11. However, it may be noted that once tl exceeds the time of the maximum in the decay the quality of the analysis decreases. It must be concluded therefore that this method is not a satisfactory means of avoiding early



1339

TABLE 11: Laplace Transformation with x (t)dt, t # 0 .rt"e-st -1 posi(method: eq 25)with tion -- 2 s values 3 s values of -max t,, tl, (chan- chanchancurve nels) nels T . ns nels T. ns 1 69 20 5.28 20 5.26 5.27 40 5.25 4a1 601 5.13 60 5.25 loa1 17.7 80 5.10 100 5.20 120 5.64 140 6.82 2 64 2a1 1.11 20 1-08 401 1.11 40 1.06 601 0.99 60 1.25 80 2.23 80 1.30 100 5.82 100 -8.17 TABLE 111: Exponential Series Representation of d(t) from Curve 5 Yk

1 2 3 4 5

1.0 3.0 5.0 7.0 9.0

ak

0.0&36

0,0041 -0.015 0.0214 0.017

k 6

7 8 9 10

7k

ak

9.5 10.0 11.0 13.0 16.0

-0.11 0.14 -0.043 0.012 -0.0022

Flgure 3. d(t) recovered from curve 5 by the exponential series, method G (0.478 ns/channel): (.-) d ( t ) , (-) straight-line fit.

times in the decay. The lifetimes in Table I were obtained for tl = time of maximum of lamp profile. Method F is the method of modulating functions used without correcting for scattered light. In eq 28 the n,p

4(t)= t y T - t ) P

(28)

pairs used for all the analyses in Table I were 4 and 31 and 4 and 14. The recovered 7's are reasonably close to those found with methods B, C, and D, except for that from curve 3. If the LSQ A method is taken as determining the correct lifetimes, the modulating functions method may appear to be slightly less satisfactory than the methods discussed so far. We did not attempt to optimize the n, p pairs for each decay; consequentlythe values listed could probably be improved upon. In any event, all the values are within 5% of the "true" value and better agreement is attained when correction for scattered light is performed. Method G is the explonential series method with analysis of the part of the decay curve beginning six channels past the maximum, whereas method H uses the whole curve in the analysis. Having determined d(t) as a sum of exponentials, we performed a straight-line fit to calculate the values in Table I. Figure 3 shows the d(t) (method G) and the straight-line fit for curve 5. In Table I11 are listed the Tk that were chosen for the analysis of this decay, together with the coefficients that resulted from the least-squares fitting technique. While there is some misbehavior at long times in the recovered d(t), as Figure 3 illustrates, from the center section of d ( t ) a single 7 value clearly may be obtained. Misbehavior at long times is expected since, in the decay curves, the first 40 or 50 channels are used for collection of backgrouiid noise and consequently there are no constraints on the behavior of the exponential series in the last 40 or 50 channels (i.e., there is a shift in the zero time on going from f(t) to d(t)). Although this method would not normally be used for single exponential decays, the data in Table I show that it gives excellent results for this simple type of analysis. Method I is the Fourier transform method in which d(t) is recovered by inverse transformation and subjected to further analysis. The procedure by which the frequency

Figure 4. Fourier transform (method I) analysis of curve 1 (0.25 nslchannel): (.-) observed decay, f(t); (a) observed lamp profile, g(t); (b) recovered d(t) for v, = 12; (c) reconvoluted f,(t) = g(t)* d(t).

window 0-vl is chosen in Fourier space has been outlined in a former p~blication.~ As the data in Table I show, this method achieves very good results for single exponential decays. Figure 4 illustrates graphically the quality of the analysis for curve 1and the absence of serious oscillations in the recovered d(t). This routine has now been tested with a large number of exponential decays and has yielded, in most instances, satisfactory results. However, it is apparently not suitable for the deconvolution of very short lifetimes (vide infra) and may yield a d(t) with serious oscillations if the observed decay does not conform exactly to eq 1. In method J, fitting is made in Fourier space. As

1340

The Journal of Physical Chemistry, Vol. 83, No. IO, 1979


-0-

20

60

40

80

100

1

120

20

CHANNEL NUMBER

40

60

I

100

80

I

120

CHANNEL NUMBER

Flgure 5. Decay of anthracene in cyclohexane (curve 3) (0.209 nskhannel): (.-) observed decay, f(t); (a) observed lamp profile, g ( t ) (fwhm = 3.4 ns); (b) reconvoluted f,(t), method A.

(-)

TABLE IV: Analysis of Curve 3 by Use of Equation 5

TABLE V: Recovered r Values of Liquid n-Hexane method r , ns xYz method r , ns xV2

method

r , ns

xvz

remarks

A K C

5.26 5.23 5.25

0.94 2.11 31.7

a index displacement

D

5.25

22.4

of 1 S = 0.0, 0.005,

F

5.21

34.7

0.01 (channels)-’ n , p pairs: 4, 31; 4, 8; 4, 5

G

5.26

0.94

a Least-squares iterative reconvolution, with eq 5 as the fitting function and analysis over the whole curve.

was stated previously, the standard deviations of DR(v) and DI(u) must be considered explicitly if acceptable values for the decay times are to be recovered with this technique. When eq 38 was used to calculate T for curves 1 and 2, no change in T was observed when nl was varied between 5 and 30. It should be mentioned that the cutoff in the data at time Twill influence the results of this technique. In method J, we have not used any cutoff corrections; it is likely that the inclusion of such corrections would improve the results. It can be seen in Table I that the analysis of curve 3 by most of the techniques leads to poor results. The determination of the correct value by method H may be fortuitous, although it is possible that the distortion in the early times in the data has been taken care of by the first two or three exponentials, which may be avoided in a straight-line fit. Figure 5 shows this anthracene decay curve and the fit using method A. It is not apparent from these plots that the decay curve is distorted; hence, the reconvolution of the recovered parameters with the lamp profile and the comparison with the observed decay is necessary for all the techniques. Because the same sample was used for the measurements of curves 1and 3, one may safely assume that the sample itself was not contaminated. Two-component analysis over the whole curve with the least-squares iterative reconvolution technique led to a x:

Figure 6. Decay of N,saturated n-hexane, neat liquid (0.12 nskhannel): lamp profile, g ( t ) (fwhm = 3.0 ns); (.-) observed decay curve, f(t).

A B C D E

0.154 0.216 0.209 0.208 0.460

1.11 3.47 5.62 5.69 53

F G H I J

0.206 0.237 0.193

5.70 1.45 4.04

0.189

4.27

value of 19, confirming that the distortion is instrumental rather than arising from impurities. In Table IV, we present the results of the analysis of this curve with the assumption that the decay obeys eq 5. f(t) = &‘g(t) d(t - t ? dt’+ ag(t)

(5)

With this analysis, the results are in excellent agreement with the “true” value obtained from curve 1. However, a,the fraction of scattered light in the decay, is found to be negative. Because of this, we are of the opinion that the most reliable means of analyzing such a decay is with the iterative reconvolution method A. Sometimes, of course, we may know that the decay is contaminated with scattered light and on such occasions the other methods of Table IV may be used with confidence. It is a defect of the Fourier transform technique, from which the other deconvolution method, the exponential series method, does not suffer, that it cannot be easily modified to analyze for scattered light. (ii) Subnanosecond Lifetimes. In Figure 6 is illustrated the decay of a sample of N2-saturated liquid n-hexane.2 Although the distorted lamp profile has a full width at half maximum of about 2.9 ns, subnanosecond lifetimes may be extracted from alkane decay curves. Lifetimes obtained with the techniques of Table I are presented in Table V; the reconvoluted f,(t), from method A, is illustrated in Figure 7. On the basis of repeated experiments by Lyke and Ware: it may be stated that T = 155 f 60 ps. Methods A, B, C, D, and F, therefore, achieve satisfactory results. In methods E and J, the choice of tl and vl, respectively, greatly influences the calculated T . As an illustration of the choice of the set of exponentials for the deconvolution



1341

TABLE VI: Exponential Series Representation of d( t ) for n-Hexane Decava

I

I

coefficient ul in d(t) :=: ale-t/71 + a2e-t/T2 The a values were estimated by generating, with the known g ( t ) , 71, and 7 2 , the functions t

yl(t) =

0

g(t’)e-(t-t‘)/71dt’ I

I

I

I

In Table VIII, we have assembled the results of the two-component analyses. Curve 7 may be assumed to be reasonably free of distortions. It was andyzed satisfactorily by all the methods, although with somewhat reduced accuracy for fitting in Fourier space. A plot of d(t), re-

and using a linear least-squares fit over the section of y(t) beginning a t six channels past the maximum to calculate al and u2. TABLE VIII: Two-Component Decays

curve 8 (3

i - ( 1 t 2) a1

method T , , ns A 1.14 B 1.09 C 1.21

D F G

H I J

1.12 1.09 0.97 1.19 1.16 0.97

T

~

as , (a,

5.29 5.2‘7 5.40 5.3B 5.29 5.211 5.22 5.37 4.95

I

+ a,)

0.70 0.70

0.72 0.72 0.71 0.75 0.63 0.73 0.69

+ 4)

9 (5

a11

xv2

T ~ ,

1.00 1.26 1.75 1.44 1.31 1.96 13.95 3.27 4.57

1.14 1.27 1.26 1.29 1.34 1.14 0.91 1.04 1.02

ns

r , , ns (a, + a,) xv2 5.27 0.70 0.86 5.35 0.68 1.85 1.87 5.33 0.68 5.37 0.69 1.88 5.63 0.71 3.55 5.47 0.77 11.02 5.26 555 0.86 5.22 0.75 57 3.72 0.55 246

rl,ns

T

10.05 6.84 7.04

11.76 11.06 11.06

~

ns,

+ 6) all (a, + a,) 0.58 0.11 0.11

1.29 2.53 2.81

0.57

4.44

xv2

noseparation 9.06

11.64

noseparation no separation no separation no separation

1342



TABLE IX: Exponential Series (Method G ) Representation of d( t ) for Curve 7 1

0.5

-0.081

2 3

1.0 1.1

8

4 5

1.5

-0.043 0.32 -0.12

3.5

0.058

10

6 7

9

4.5 5.0 5.5

6.0 7.0

io5,

-0.079

0.11 -0.088

0.093 -0,017

covered from this decay curve with the Fourier transform technique, is shown in Figure 8, together with the leastsquares fit to a sum of exponentials. Although the number of channels with useful information has been greatly reduced in the deconvolution, the 70 channels remaining could be easily analyzed for two components. For the analysis of two-component decays by the exponential series method, we used sets of ten exponentials. The Tk used for curve 7 and the ak determined with method G are listed in Table IX. It should be emphasized that, although our knowledge of r1 and r2 from the twocomponent least-squares fitting guided our choice of Tk, a suitable choice of Tk may be based on merely a rough estimate of the decay times, such as might be obtained from inspection of a log f(t) vs. t plot. Whereas the data of curve 7 present few difficulties to any of the methods of analysis, those of curve 8, which have as one contribution the distorted anthracene decay curve 3, were analyzed satisfactorily only by the least-squares iterative convolution method A. As with curve 3, when methods C, D, and F were used to correct for scattered light distortions, the accuracy of the recovered parameters increased significantly. However, in the moments and Laplace transform techniques, the correction for scattered light in two-component decays leads to reduced precision in the computations. For example, no result could be obtained with the Laplace transform technique when the cutoff corrections were calculated with eq 20 and 21. But when the contribution of component j to the decay curve a t time T (C, of eq 21) was estimated from the shape of decay at time T - t21s without calculation of the integrals required by eq 21, values of 1.25 and 5.36 were determined for the lifetimes. Therefore, the lack of precision in the integrations leads to meaningless results in the division F(s)/ G(s) while the use of simplified cutoff corrections entails a lack of accurcy in the calculated parameters. It should also be mentioned that when higher moments or a greater number of s values are used in the analysis of two-component decays, the time required to reach convergence increases considerably. In the method of moments, for instance, 86 iterations were needed when no correction for scattered light was made; when the index of the moments was increased by 1, convergence was attained after 240 iterations, the T values determined being 1.16 and 5.28 ns. As was mentioned previously, we had no a priori reason for believing that curve 8 was contaminated with scattered light. It is of interest that the index displacement in the method of moments, proposed by Isenberg12 specifically to deal with contamination from stray exciting light in the fluorescence decay, leads to the same results as method A, in which early times in the decay are ignored in the fitting procedure. Because a negative value was determined for (Y in eq 5 and because at the same time the correction for scattered light with the method of moments yielded good values for 71 and r2,it might be expected that index displacement is a general method for avoiding any distortion at early times in the decay curve. This conclusion receives some justification from the observation that the higher moments, say J:t3 f(t) dt, give more weight

Figure 9. Modulating functions used in the analysis of curve 8 (0.209 nslchannel): 6s.) lamp profile, g ( t ) (fwhm = 3.4 ns); (---) decay curve, f ( t ) . q5 = P ( T - t ) P : 4 , , n = 4, p = 31; q52, n = 4, p = 14; 43, n - 4, p = 8 ; c $ ~ , n = 4, p = 5; I # J ~ , n = 5, p = 4.

-

to later times in the decay than the lower moments, say Jkt f(t) dt. Similar reasoning may be advanced to account for the results obtained with the modulating functions technique. By using five n,p pairs for the analysis of curve 8, one obtains T values of 1.15 and 5.28 ns, in excellent agreement with the “true” values. In Figure 9, the normalized lamp and decay of curve 8 are plotted on a linear scale together with the five modulating functions that were used to correct for scattered light. Functions q51 to q54 were used in the simple two-component analysis. It can be seen that the function 46modulates the decay curve at later times, thus giving more weight to the data that are least distorted. However, accurate analysis is aided by functions, such as &, with steep rising edges;20when $2 to q56 were used in the simple analysis T values of 1.22 and 5.33 ns were obtained, with the predicted loss of accuracy. It should be pointed out that analysis of undistorted data with the method of moments or the method of modulating functions with the correction for scattered light applied yields less accurate results than a simple analysis without correction. Apparently, decreased precision in the calculation of the higher moments reduces the accuracy in the calculations by the method of moments, whereas in the modulating functions technique, an increase in the order of the matrix used in the calculations leads to less accurate values for the parameters. In spite of the recovery of a good value for r1 with method G and reasonable values for r2 with methods H and I, neither the exponential series method nor the Fourier transform method could be regarded as a suitable means of analyzing curve 8; the recovered d(t) from both methods had an unrealistic form and the final values of the parameters depended, to a large extent, on the region of d(t) chosen for the fitting. Curve 9 with its two closely spaced decays presented insurmountable difficulties to most of the techniques. A plot of this decay is illustrated in Figure 10 together with the reconvolution of the decay law in Table VI for the iterative convolution method A. The d(t), recovered with


Flgure 10. Analysis of curve 9 (0.478 nskhannel): (.-) decay curve, f( t); (a) reconvoluted f,(t), method (A); (b) d( t ) recovered with exponential series method (G).

the exponential seriles method, is shown on the same figure. Both least-squares fitting over the whole curve and the method of moments recovered reasonable r2 values from this decay, but the values for T~ and al were in both cases unacceptable. Only least-squares fitting over a part of the curve and the method of modulating functions analyzed the decay with reasonable success. For resolving capability, it may be concluded, therefore, that these two methods, are superior to the other tested techniques.

IV. Cost of Computations Because the amount of computation time will vary for most of the methods from one analysis to another, it is difficult to compare the relative cost of the methods. For simple two-component analysis of data obeying eq 1, as for instance in the analysis of curve 7, it was found that the longest time was taken by the exponential series method with the iterative convolution and Fourier transform I (including fitting of d(t)) methods requiring half and the remaining methods about one-fifth of that time. For the analysis of curve 8, on the other hand, the method of moments with an index displacement of 1 cost as much as the exponential series method. We were particularly gratified that the Fourier transform method compares quite favorably with the other techniques in this regard. V. Conclusion An examination of the results presented in section IV reveals that all the tested methods of deconvolution are capable of satisfactorily analyzing undistorted data, provided that the decay times, in two-component decay curves, are not too close to each other. Fitting in Fourier space does not attain the accuracy of the other methods and is therefore not so suitable for simple exponential or double exponential analysis. While the cost of the exponential series method might prevent its use as an analysis technique for simple decays, the good results obtained for such decays give some confidence that the method would be successful in the extraction of less simple or unknown decay laws from fluorescence decay curves. The position of the lower limit of integration in the Laplace

The Journal of Physical Chemistty, Vol. 83, No.

IO, 1979 1343

transform method in which the transform is integrated from a time t # 0 has a large influence on the recovered T value; this method is, therefore, not recommended, although it might prove useful in some applications. In the Fourier transform method, spurious oscillations in the recovered d(t) are eliminated by a suitable choice of a frequency window [O,vl] in Fourier space and extrapolation of D(v) by the Fourier transform of a single exponential. Since all the information about d(t) resides in the first few Fourier coefficients, the choice of v1 has little effect on the recovered decay parameters. By truncating D(v) one merely eliminates from d(t) the unwanted oscillations that make the further analysis of d(t) difficult to accomplish. As has been demonstrated in section IV, the Fourier transform method yields acceptable results even in the analysis of distorted data. It is also of reasonable cost and should prove a suitable companion method to the exponential series technique in the deconvolution of curves having complex decay laws. Its main disadvantage appears to be the lower limit of about 0.5 ns on the T values that it can recover from distorted lamp functions of the type encountered in this worka3 Least-squares iterative reconvolution, moments, Laplace transforms, and modulating functions are used frequently in deconvolution. In the present study, it was found that all are good methods of analysis. The least-squares and modulating function methods appear to have the greatest ability in resolving closely spaced decay times. Finally, because the least-squares technique can be used with no loss in accuracy, to fit any chosen section of the decay curve, it is recommended as the technique of preference for the analysis of simple decay laws. Acknowledgment. The authors wish to acknowledge the financial assistance of the National Research Council of Canada and ARO-Durham.

References and Notes (1) A. E. McKinnon, A. G. Szabo, and D. R. Miller, J . Phys. Chem., 81, 1564 (1977). (2) R. L. Lyke and W. R. Ware, Rev. Sci. Instrum., 48, 320 (1977). (3) J. C. Andre, L. M. Vincent, D. V. O’Connor, and W. R. Ware, submitted for publication in J . Phys. Chem. (4) (a) A. Grinvald and I.Z. Stelnberg, Anal. Biochem., 59, 583 (1974); (b) J. H. Easter, R. P. De Toma, and L. Brand, Biophys. J., 16, 571 (1976). (5) D. W. Marquardt, J . SOC. Ind. Appl. Math., 11 (2), 431 (1963). (6) P. R. Bevington, “Data Reduction and Error Analysis for the Physical Sciences”, McGraw-Hill, New York, 1969. (7) Z. Bay, Phys. Rev., 77, 419 (1950). (8) S. S. Brody, Rev. Sci. Instrum., 28, 1021 (1957). (9) I.Isenberg and R. D. Dyson, Biophys. J . , 9, 1337 (1969). (10) R. D. Dyson and I.Isenberg, Biochemistry, 10, 3233 (1971). (11) R. Schuyler, I.Isenberg, and R. D. Dyson, Phofochem. Photobiol., 15, 395 (1972). (12) (a) I.Isenberg, J. Chem. Phys., 59, 5696 (1973); (b) E. W. Small and I.Isenberg, J . Chem. Phys., 66, 3347 (1977). (13) 1. Isenberg, J . Chem. Phys., 59, 5708 (1973). (14) I.Isenberg, R. D. Dyson, and R. Hanson, Biophys. J., 13, 1090 (1973). (15) W. P.. Helman, Int. J . Radiat. Phys. Chem., 3, 283 (1971). (16) M. Almgren, Chem. Scr., 3, 145 (1973). (17) A. Gafni, R. L. Modlin, and L. Brand, Biophys. J., 15, 263 (1975). (18) W. R. Ware, L. J. Doerneny, and T. L. Nernzek, J . Phys. Chem., 77, 2038 (1973). (19) J. Loeb, ”Identification experimentale des processus industriels”, Dunod, Paris, 1967. (20) (a) B. Valeur and J. Molrez, J . Chlm. Phys. Phys.-Chim. Biol., 70, 500 (1973); (b) 6.Valeur, Chem. Phys., 30, 85 (1978). (21) D. V. O’Connor, Thesis, University of Western Ontario, 1977. (22) U. Wild, A. Holzwarth, and H. P. Good, Rev. Sci. Instrum., 48, 1621 (1977). (23) I.B. Berlman, “Handbook of Fluorescence Spectra of Aromatlc Molecules”, Academic Press, New York, 1965. (24) R. L. Lyke and W. R. Ware, Chem. Phys. Lett., 24, 195 (1974). (25) W. R. Ware in “Creation and Detectlon of the Excited State”, Vol. l A , A. A. Lamola, Ed., Marcel Dekker, New York, 1971. (26) D. V. O’Connor and W. R. Ware, J . Am. Chem. Soc., to be submitted for publication.

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