2304
Anal. Chem. 1985,57,2304-2308
Luminescence Lifetime Measurements. Elimination of Phototube Time Shifts with the Phase Plane Method E. R. Carraway, B. L. Hauenstein, Jr.,l and J. N. Demas* Department of Chemistry, University of Virginia, Charlottesville, Virginia 22901 B. A. DeGraff* Department of Chemistry, James Madison University, Harrisonburg, Virginia 22807
An adaptatlon of the phase plane (PP) method of luminescence lifetime deconvolution is described, which ellmlnates wavelength-dependent time shifts by using a reference emItter. Derlvative and Integral PP methods are described and tested with both experimental and synthetic data. Data on the accuracy and precision of the methods for different noise levels, reference lifetimes, and fltting regions are presented. Both methods are accurate and precise over a wide range of sample and reference lifetimes, atlhough the Integral approach Is somewhat more accurate and computationaiiy faster. Selection of the optimum reference ilfetlme Is dlscussed.
The measurement of luminescence lifetimes is important in many disciplines. If the excitation pulse is comparable to the sample lifetime, 7,simple semilogarithmic methods of evaluating T are no longer applicable because there is no region free of continued pumping by the flash (I). By measuring the observed excitation pulse, E ( t ) ,and the sample decay, D ( t ) , as functions of time, t , under the same experimental conditions, one can use simple deconvolution methods to determine 7.
For the simpler deconvolution methods, it is usually necessary for E ( t ) and D ( t ) to have a common zero time. In practice, however, acquiring E ( t )and D ( t ) to meet this criterion may be impossible. For example, our lifetime instrument uses 337 nm excitation from a pulsed nitrogen laser, but our RCA C7164R sample photomultiplier tube (PMT) is “blind” to this wavelength. Therefore, direct measurement of E ( t ) is impossible. A more universal problem is caused by wavelength-dependent transit times exhibited by most PMTs. Even if both E ( t ) and D ( t ) are measured with the same detector, a time shift of 100 ps to >1ns can arise for wavelengths differing by 100-300 nm, and the assumption of a common zero time is violated (2). Complex nonlinear least squares methods or time-consuming trial and error shifting can be used to correct for this error, but the most common approach has been to use a reference molecule of known lifetime that emits at the same wavelength as the molecule of interest (2). From the observed reference decay, R(t), and the known reference lifetime 7R,E ( t ) can be calculated by using a derivative approach. Since R ( t ) is measured at the same wavelength as D ( t ) ,the calculated E ( t ) exhibits no time shift and conventional deconvolution methods proceed as usual. However, the normal noise on R ( t ) is magnified in the derivative calculation of E(t),and accuracy of the deconvolutions may suffer. This paper demonstrates the use of a phase plane (PP) method of luminescence lifetime deconvolution coupled with a reference emitter to overcome the problem of phototube time ‘Present address: Bard Critical Care Division, C. R. Bard, Inc., Billerica, MA 01821.
shifts. The PP method has been shown to be useful in evaluation of simple exponential lifetimes ( 3 , 4 )and deconvolution of lifetimes when the sample response is exponential (5) or described by Forster kinetics (6). In addition, the PP method can be used to deconvolute an exponential lifetime in the presence of a scatter component from E ( t ) (7,8) and in evaluation of kinetic exponential decays with a constant but unknown base line (9). The main advantage of the PP method in these applications is that a closed form solution to the deconvolution problem is obtained, and thus the method is easy to program and yields results much more quickly than iterative nonlinear least squares methods. Also, the linear PP plots (vide infra) give a quick, easy, visual indication of the optimum fitting region and of the validity of the assumption of single exponential behavior. With the present implementation of the PP method these advantages are again realized. We will describe two approaches called the derivative and the integral methods. In the first of these, the actual E ( t ) is calculated by a derivative approach. In the second only the integral of E(t)is calculated. The accuracy and relative merits of these approaches are evaluated by deconvolution of single exponential lifetimes from both real and simulated single photon counting decays over a range of conditions. Finally, the criteria for selection of a reference emitter are discussed.
THEORY We begin with a derivation analogous to the basic PP equations (5). The convolution of a single exponential decay impulse response and a generalized excitation profile is given by
D ( t ) = K exp(-t/T) ~ ‘ E ( xexp(x/r) ) dx 0
(la)
E ( t ) = 0 for t I0 Ob) where D ( t ) and E ( t ) are the observed decay and excitation profiles, K is a scaling factor, T is the single exponential lifetime, and x is a dummy variable of integration. This expression assumes the absence of any time shift. Taking the derivative of this expression with respect to t yields d D ( t ) / d t = K E ( t ) - (1/7)D(t) (2) Integration of eq 2 over the interval 0 to t yields
D ( t ) = K s t E ( x )dx - ( l / i ) l b ( a ) dx 0
0
Rearranging we obtain two PP equations Z ( t ) = -7W(t) + K7 W(t)= -(1/7)Z(t) + K
(3) (44 (4b)
Z ( t ) = JrD(x) 0 d x / l t0 E ( x ) dx
(4c)
W ( t ) = D ( t ) / l 0t E ( w )dx
(4d)
0003-2700/85/0357-2304$01.50/00 1985 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 57, NO. 12, OCTOBER 1985
Computationally, one simply evaluates W ( t ) and Z ( t ) a t different times from the observed E ( t ) and D ( t ) using numerical integration (e.g., trapezoidal rule). The linear plots of W ( t )vs. Z(t) or Z ( t ) vs. W ( t )are then fit by linear least squares to yield slopes and intercepts that permit evaluation of K and 7 as indicated by eq 4a and 4b. The fit is normally done by unweighted linear least squares. Equation 4b is preferred over eq 4a (IO). We adopt eq 4b in subsequent discussion. The above expression works well when both E ( t ) and D ( t ) can be obtained at the same wavelength. However, as pointed out earlier, this is frequently inconvenient or impossible. To avoid time shift problems, it is useful to employ a reference emitter. This compound must emit a t the same wavelength as the sample and have a known single exponential lifetime. The observed decay curve for the reference emitter is R ( t ) , which is measured under the same conditions as D(t) including the same emission wavelength. Minor modifications of the basic PP equations permit use of this reference emitter to avoid having to directly measure E(t). In one approach we use a derivative method to determine E ( t ) from R(t). In the other, we evaluate the J E ( x ) dx from R(t). We refer to these as the derivative and integral methods, respectively. The integral of D(t) requires no correction and is evaluated directly from D ( t ) using trapezoidal rule integration. In the derivative approach, R ( t ) is calculated from a rearranged form of eq 2. We assume without loss of generality that K = 1 and substitute R ( t ) for D ( t )
E ( t ) = ( ~ / T R ) R+( ~[dR(t)/dtl )
(5)
where 7 R is the known reference lifetime. Computationally, the derivative at every time tiis evaluated by fitting a parabola to R(t) at the three points R(ti-l),R(ti),and R(ti+J.The slope is then evaluated analytically at ti from the defined curve to yield (2)
E(ti) = ( 1 / 7 ~ ) R ( t+J (l/sAt)[R(ti+l) - R(ti-1)l (6) where At is the time interval between points. This procedure of using three rather than two points to evaluate the numerical derivative decreases the sensitivity of the method to noise. W(t)and Z(t) in the standard PP equations are then evaluated from E ( t ) and D ( t ) using a trapezoidal rule integration and eq 4c and 4d. In the integral approach, we can avoid calculating E(t)since the calculation of W ( t )and Z(t) requires only the integral of E(t)at each point. Thus, JE(x) dx is calculated for all points directly using a rearranged form of eq 3 with trapezoidal rule integrations
The integral approach avoids the extra calculation of E ( t ) in the derivative method and the inherent introduction of derivative noise. To permit visual judging of the results, we calculated D ( t ) from the fit parameters and compared this result with the observed decay. In the derivative approach we obtained D ( t ) from T , K , and the back calculated E ( t ) (eq 6) using the Demas-Crosby formula (11). It is less direct to calculate D(t) in the integral method. D(t) was calculated using eq 3. f E ( x ) dx was obtained from eq 7 and J D ( x ) dx from D ( t ) using trapezoidal rule integrations. EXPERIMENTAL SECTION Materials. All compounds were used as received. Coumarin 440, coumarin 535,POPOP (1,4-bis(5-phenyloxazol-2-yl)benzene), and PPO (2,5-diphenyloxazole) were obtained from Exciton Chemical Co. Anthracene (puriss., 99.9%) was obtained from from Sigma. Aldrich and DPH (1,6-diphenyl-1,3,5-hexatriene)
2305
Mallinckrodt AR cyclohexane and AAPER Alcohol and Chemical Co. absolute ethanol were also used as received. Aerated or deaerated solutions of POPOP, PPO, anthracene, and DPH in cyclohexane were prepared in concentrations ranging from about to lo-' M. Aeration or deoxygenation was achieved by bubbling with solvent-saturated air or nitrogen. Deoxygenated solutions of POPOP and the coumarins in absolute ethanol were prepared at similar concentrations. Instrumentation. Luminescence decays were recorded with a Tektronix 7912 ultrahigh speed transient digitizer interfaced to an Altair 8800B microcomputer. The transient recorder is essentially a digitized oscilloscope with all the distortions of an oscilloscopic system. The system is described in detail elsewhere (12). To reduce the effects of drift, pairs of decays (one sample decay, one reference decay) were acquired temporally as close as possible using the same instrument settings. PMT voltage, recorder sensitivityand time base setting (100,50,or 20 ns full scale), and emission monochromator slit width were held constant for each pair of experiments. Only slit height was adjusted between sample and reference decays. At least two pairs of decays were acquired for all sample-reference combinations. Data reduction was performed on a Hewlett-Packard 85A computer using a BASIC program (available upon request). Simulations. All synthetic data were generated and reduced with BASIC programs on the HP-85. The reduction routines are identical with those used with real data. The data generation routines perform three major tasks: (1) generation of E ( t ) , (2) generation of D ( t )and R(t)from E(t),and (3) the addition of noise to simulate single photon counting (SPC) statistics. E ( t ) was generated from E ( t ) = exp(-t/A) - exp(-t/B); t 2 0 (8) where the constants A and B are equal to 6.6 ns and 5.5 ns, respectively. This procedure yields a flash of approximately 14 ns fwhm and a general shape typical of actual excitation pulses (1). Data were generated every 0.5 ns for 101points over the 0-50 ns range. The synthetic sample and reference decays were then generated using the Demas-Crosby method or the closed form solution obtained by substitution of eq 8 into eq 1. Gaussian noise was added to simulate SPC statistics. First the noise-free decay was scaled to a given peak count (e.g., lo4). By use of the Box, Muller, Marsaglia algorithm, the computer's uniformly distributed random numbers were transformed into the multiplier, N , which exhibits a Gaussian distribution with a mean of zero and a standard deviation of 1. Gaussian noise, obeying single photon counting statistics, was then added to the noise free R(t) or D ( t ) by C,(t) = C ( t ) + N[C(t)]lI2 (9) where C,(t) and C(t) are the noisy or noise free synthetic R ( t ) or D ( t ) . These data were then reduced by the integral and derivative methods. RESULTS AND DISCUSSION Experimental Results. To obtain E ( t ) or the integral of E(t)from the reference decay, it was necessary to assume that the shape of the excitation pulse was constant. T o test this assumption 35 decays of the shortest lived molecule, POPOP in cyclohexane, were acquired in several sets and deconvoluted against each other treating one decay as the actual flash and the other as the sample decay. Slit height was also varied to determine if this had any effect on the decay curve shapes. Neutral density filters were used to reduce emission intensities as the slit height was increased. Assuming the POPOP response is constant, the deconvolution should yield lifetimes near zero. Lifetimes deviating appreciably from zero indicate variations in E @ ) ,instrumental triggering instability, and noise. Decay curve pairs were reduced by the PP method or by a simple moments method, which uses only the zeroth and first moments ( I ) . With the moments method, cutoff corrections could be ignored. Similar results were obtained by both approaches, but the moments method was faster and not subject to any ambiguities because of differing fitting regions.
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ANALYTICAL CHEMISTRY, VOL. 57, NO. 12, OCTOBER 1985
Table I. Summary of Results of Experimental Data samplea DPH/CH anthracene (N,)/CH anthracene (air)/ CH Coumarin 535/EtOH Coumarin 440/EtOH POPOP/CH POPOP/EtOH PPO/CH
b
reference
TIit
POPOP/CH POPOP/EtOH PPO/CH POPOP/CH PPO/CH POPOP/CH PPO/CH POPOP/CH POPOP/EtOH POPOP/CH POPOP/EtOH POPOP/EtOH PPO/CH POPOP/CH PPO/CH POPOP/CH POPOP/EtOH
12.4 (13) 5.23 (14) 4.10 (14) 3.9 (15) 4.8 (15) 1.10 (16) 1.28 (16) 1.42 (14)
ne
Tderiv
12 2 3 12 3 6 2 5 2 2 2 4 5 4 2 5 2
12.87 f 0.37 11.88 f 0.12 12.85 f 0.08 5.29 f 0.15 5.10 f 0.04 3.81 f 0.16 3.66 f 0.02 2.67 f 0.27 2.80 f 0.06 3.52 f 0.06 4.36 f 0.06 1.32 f 0.07 1.03 f 0.06 1.06 f 0.09 1.18 f 0.10 1.47 f 0.07 1.50 f 0.10
(X2red)”2
(X2red)1’2
Tinteg
1.49 1.12 1.36 0.99 0.69 0.95 1.36 1.52 0.99 1.40 1.03 0.86 0.82 0.89 0.92 0.75 0.91
12.88 f 0.38 11.88 f 0.12 12.85 f 0.08 5.31 f 0.14 5.11 f 0.03 3.83 f 0.15 3.67 f 0.02 2.69 f 0.29 2.78 f 0.08 3.53 f 0.07 4.36 f 0.06 1.32 f 0.09 1.05 f 0.05 1.07 f 0.10 1.22 f 0.12 1.50 f 0.06 1.50 f 0.10
1.77 1.28 1.57 1.37 0.90 0.87 2.12 2.50 1.31 2.37 1.26 2.18 5.52 2.62 1.93 4.30 1.51
“CH, cyclohexane; EtOH, absolute ethanol. *All T’S in ns. e n = number of sample, reference pairs. Average deviation from mean reported where n = 2, otherwise n reported. Calculated over full time scale.
The last point was a problem with the PP method because of the many crossover points present in nearly identical decay curves. Approximately 85% of the 200 data pairs reduced by the moments method yielded lifetimes less than 200 ps. Also there was no correlation between lifetime and slit height. This procedure provides a convenient way of judging instrumental stability. From these results, we estimate instrumental stability (pulse shape reproducibility and jitter) at better than k0.2 ns. Table I summarizes all experimental data indicating sample-reference combinations, the number of pairs acquired, the literature lifetimes, the lifetimes obtained by the derivative and integral methods, and average square root of reduced x2 for both methods. In all cases the reference molecule chosen has a relatively short lifetime. The fitting region for nearly all reductions is from the peak of D ( t ) to the end of the acquired data. For the majority of decays this region corresponds to the early part of the linear region of the W ( t )vs. Z(t) plot. However, for several decays the linear region began a t earlier times. For these decays, variation of the fitting region to include earlier points did not significantly alter the calculated lifetime. The square root of the reduced x 2 is a goodness of fit indicator. The reduced x2 is calculated by using standard equations and assuming the variance of the data points, q2,is unity. For physically meaningful units, the square root of this quantity, ( x ~ ~ is~reported. ~ ) ~ / ~ , The data in Table I show that the derivative and integral PP methods work well with lifetimes over the range 1to 12 ns with a flash exhibiting a fwhm of about 10 ns. Even the shortest lifetime of 1.1ns is deconvoluted with about 15-20% accuracy. The only significant disagreement with the literature lifetimes occurs for the coumarin dyes where slightly different solvents were used (absolute vs. 95% ethanol in the earlier work) (15). Figure 1 shows a typical data set and the results used in the reductions of an anthracene vs. POPOP system. R(t),D(t), the back calculated E ( t ) , the derivative PP plot, and the derived D ( t ) are shown. The integral method PP plot and its fit to D ( t ) are indistinguishable visually from the data for the derivative method. While R(t)is relatively noise free, there is appreciable noise introduced on E ( t ) by the derivative method. In spite of the noisy E ( t )the calculated PP plot is remarkably noise free; we attribute this to the smoothing ability of integration. For some of the shorter lived decays, the calculated lifetime is rather sensitive to fitting region. This is particularly noticeable with some of the decays for the POPOP-PPO in
A.
-251 .20
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L
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I
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I
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2 (t)
B.
r
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?
3
60
W
2;
40
c < 1 w K
28
0
TIME
(ne)
Figure 1. Derivative PP plot (A) and experimental data (B) for anthracene in deoxygenated cyclohexane vs. POPOP in deoxygenated cyclohexane. The E ( t ) is back calculated from eq 7. Noise is clipped by the x axis. The +’s are the fit calculated from the PP parameters.
cyclohexane pairs (two out of five data pairs). Close inspection of the decay curves shows that even though there is a finite difference between the two lifetimes (approximately 0.3 ns), the curves experience crossovers where there is still significant intensity. For these short lived pairs, the fits to D ( t ) were not especially good using the derivative method but were extremely poor after the crossover point for the integral method. This inability of the integral method to reconstruct D(t)’s for short-lived systems is due to its sensitivity to instrumental distortions; we do not see this behavior for the integral method with simulated noisy SPC data. However, if one restricts the fitting region such that it begins where the PP plot first linearizes (about at the peak of D ( t ) ) and ends before the crossover, satisfactory fits are obtained up to the crossover
ANALYTICAL CHEMISTRY, VOL. 57, NO. 12, OCTOBER 1985
3. 00
>
t.
I
2307
T
u)
ac
t W
z
c _I
E W
1
-5.00
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0.2
(ne)
Flgure 2. Simulated decays for T = 5 ns, T~ = 10 ns, and lo4 countskhannel for both R ( t ) and D(t). E ( f )is back calculated by the derivative method and noise is clipped as in Figure 1. The +'s are the calculated fit using the PP parameters from fit region b.
points. Further, the calculated lifetimes are in reasonable agreement with the literature values. Given the analog nature of our instrument and the fact that these short lifetimes are an order of magnitude shorter than the width of E ( t ) , we consider this level of performance quite satisfactory. With the data for POPOP/cyclohexane-PP0/ cyclohexane, which exhibited a crossover problem, we used this restricted fitting region to calculate the lifetimes reported in Table I. The poor fits indicated in the table for the integral methods arise because of the two data seta exhibiting crossover coupled with the fact that (X2&/' are calculated over the entire curves. The lifetimes obtained by the derivative and integral methods are essentially identical. A difference between the methods is seen in the average values of (X'r&''. The derivative method results in a lower value in all cases. This is because in the derivative method, D ( t ) is calculated directly from the explicit form of E ( t ) while in the integral method D ( t ) is calculated indirectly. On the other hand, since the integral method bypasses one calculation step, it is somewhat faster than the derivative method. The calculation of the W(t) and Z(t) arrays takes about a third longer with the derivative method (about 2 min vs. 1.5 min for 512 points) on our system. Since the lifetime obtained is the same regardless of the method used, the integral method may be the better choice for many applications. An important point is illustrated by the POPOP/ ethanol-PPO/cyclohexane and POPOP/cyclohexane-POPOP/ethanol systems. The lifetime obtained by deconvolution is essentially equal to the reference lifetime. Thus, the reference and sample lifetimes are too close to be distinguished by the PP method using data from our analog instrument; the differences are 0.14 ns and 0.18 ns, respectively. These results are consistent with our 0.2 ns instrumental stability. This conclusion is supported by the ability of the PP methods to successfully resolve such close lifetimes when used with higher quality synthetic SPC data that mimic the same relative pulse widths and lifetimes. Synthetic Data. Synthetic data were generated for three sample lifetimes, 1,5, and 10 ns, which roughly cover the range of our experimental data. The simulated E ( t ) exhibited a fwhm of 14 ns. Sample lifetimes were evaluated using ( t ) ' s having TB'S ranging from much shorter to much longer than the sample T . The peak counts per channel were usually set to lo4 before adding noise; the noise level is comparable to that usually encountered with SPC instruments. Figure 2 shows representative data and the E ( t ) calculated by the derivative method. For each sample-reference combination, ten pairs of decays were generated and reduced. An average lifetime 7 and
0.5
1.5 Iff
1
I
2.0
5.0
10.0
(n.)
Flgure 3. Relative errors and standard deviations in deconvoluted vs. T ~ The . data are simulated with T = 1 ns and the peak counts in D ( t ) and R ( t ) being lo4: 0, derivative method; *, integral method. Error bars are f l relative standard deviation. Points for the same T~ are slightly offset for clarity and the T R axis is nonlinear. TIS
T
-1.50'
'
1.0
'
4.0
'
4.5
'
'
5.5
'
7.0
10
'
50
I
'
100
-, 200
T
I
500
I R (n.)
Flgure 4. Relative errors and standard deviation for simulation data for T = 5 ns. Otherwise conditions are the same as those given in Figure 3.
T .90
E
.30
0
2
0.00
N
-. 30 -.90 -1.50
1
1
1.0
5.0
8.0
I
9.0 IR
9.5
10.5
11.0
12.0
50
(ne)
Flgure 5. Relative errors and standard deviation for simulation data for T = 10 ns. Otherwise conditions are the same as those given in
Figure 3.
standard deviation a were calculated. Figures 3-5 present percent error (difference between t and assumed sample lifetime) and relative standard deviation (a expressed as a percentage of the mean) for both derivative and integral methods. The figures show that the derivative and integral methods are essentially identical with respect to standard deviation. The integral method shows only random errors and the cal-
2308
ANALYTICAL CHEMISTRY, VOL. 57,
NO. 12, OCTOBER 1985
Table 11. Effect of Reference Lifetime Errors on Deconvolution Accuracy with a 5-11sSample Lifetime"
1 f 0.1 4 f 0.1 6 f 0.1 10 f 0.1 50 f 1 100 f 2 200 f 4 500 f 10
4.986 4.982 4.980 4.980 4.979 4.980 4.980 4.980
f 0.104 f 0.105 f 0.094 f 0.070
f 0.091 f 0.054 f 0.030 f 0.012
4.998 f 0.106 5.000 f 0.106 5.000 f 0.094 5.000 f 0.070 5.000 f 0.090 5.000 f 0.054 5.000 f 0.030 5.000 f 0.012
Fit region 20-100 in a l l cases.
culated value is within one standard deviation of the true lifetime. The derivative method tends to underestimate the lifetime, especially for the shortest 7's where the calculated values are outside one standard deviation of the mean. We have confirmed the existence of these errors in the derivative method by increasing the peak counts to lo6. For the shortest sample lifetime and longer lived reference (10 ns), the systematic errors are about -1.5%. Thus, the integral phase plane method will be somewhat more accurate than the derivative method when used with high-quality SPC data. Both methods become more accurate and precise as the sample lifetime increases. This result is to be expected since shorter lifetimes are harder to deconvolute because of the greater contribution of the flash to the total decay. We turn now to the selection of the reference emitter. Our results (Figures 3-5) show that accuracy and precision are best for reference lifetimes roughly equal to or shorter than the sample lifetime, although the variation with T~ is not large. There appears to be a somewhat higher standard deviation for a reference lifetime of 0.5 ns in the deconvolution of a 1 ns decay. Thus, the optimum T~ may depend on the details of such experimental parameters as excitation pulse width and full scale time. The selection of the reference will, therefore, depend more on the availability of a reference with a suitable emission spectrum rather than on its actual lifetime. Figure 4 shows the effect of choosing a TR with a very long lifetime (50-500 ns). The noise level increases by roughly a factor of 2 on going from the optimum short-lived reference to these very long TR's. Thus, long-lived references do not yield the best precision. As we will show, however, using a long TR significantly enhances accuracy when one considers errors in the assumed TR. Further simulations were performed to investigate the effect of errors in the assumed reference lifetime on the accuracy of deconvolutions. If one assumes a T~ longer than it actually is, E ( t )appears too early in time, and the calculated 7 is too long. Conversely, too short an assumed T~ yields calculated 7's that are too short. To determine the effect of errors in T R , we deconvoluted decay curves assuming both the correct and erroneous Q'S. We have assumed noise-free data in the calculations so that noise does not obscure the effect of propagation of errors, but additional simulations have demonstrated that errors in the assumed T~ have the same effect on the accuracy of noisy data. ~ ~ of ' 1,4,6,10,50,100,200, 9 and 500 ns were used to generate R(t). A sample 7 of 5 ns was used in all simulations. For TR'S less than or equal to 10 ns, we assumed errors of hO.1 ns. For the larger reference lifetimes, we assumed an error of i 2 % . Table I1 summarizes these results. The correct T R and the assumed errors are shown. Symmetric errors in T~ produced symmetric errors in 7. The deconvoluted T'S for the correct TR'S are shown. The spread in 7 for the indicated errors in
is denoted by the i values. The i O . l ns errors resulted in approximately the same absolute error in the deconvoluted 7's. For the longer lived references, we find that as T~ increases, the 2% uncertainty in T~ has less effect on the accuracy of the deconvoluted 7's. Indeed for the 500-ns reference, the 2% error in T~ affects the calculated 7 by only 0.24% for the integral method. This is not surprising given the fact that for long decay times, R ( t ) directly approaches S E ( x ) dx required in the integral PP method (see eq 7). The derivative method shows a similar decreasing sensitivity to the accuracy of T ~ but , there do appear to be small systematic errors similar to those found for shorter lived references (Figures 3-5). Long-lived references have a clear advantage over shortlived ones since T~ can be measured directly. Thus, one does not have to depend on the accuracy of a literature value, which may not be applicable to one's sample and conditions. Suitable long-lived references would be pyrene for the blue to green region and Ru(I1) complexes for the red and nearinfrared regions. A final set of simulations was done to investigate the effect of varying the fitting region. A sample lifetime of 5 ns, reference lifetime of 10 ns, and lo4 peak counts/channel were used. Ten sets of synthetic curves were generated and reduced for each of three fitting regions. These correspond, as shown by Figure 2, to beginning the fit where (a) D ( t ) rose to half the peak value, (b) D ( t ) peaked, and (c) R ( t ) peaked. All fitting regions ended with the last datum point. We found little difference among the three regions except for a slight decrease in standard deviation for the later starting point. We suspect this is due to small errors associated with the use of unweighted least squares fitting since there are errors in both W ( t )and Z ( t ) , especially at short times. Since one would normally try to use as much of the sample decay curve as possible, we recommend carrying out the fits from the peak of either D ( t ) or R(t),whichever comes first. TR
ACKNOWLEDGMENT We thank Clara Colby, who worked on the computer simulations.
LITERATURE CITED Demas, J. N. "Excited State Lifetime Measurements"; Academic Press: New York, 1983. Wahl, P.; Auchet, J. C.; Donzel, B. Rev. Sci. Instrum. 1974, 45, 28. Bernalte, A.; LePage, J. Rev. Sci. Instrum. 1969, 4 0 , 71. Huen, T. Rev. Sci. Instrum. 1969, 4 0 , 1067. Greer, J. M.; Reed, F. W.; Demas, J. N. Anal. Chem. 1981, 5 4 , 710. Love, J. C.; Demas, J. N. Rev. Sci. Instrum. 1983, 5 4 , 1787. Love, J. C.: Demas, J. N. Anal. Chem. 1984, 56, 82. Jezequel, J. Y.; Bouchy, M.; Andre, J. C. Anal. Chem. 1982, 5 4 , 2199. Bacon, J. R.; Demas, J. N. Anal. Chem. 1983, 5 5 , 653. Reed, F. W.; Demas, J. N. In "Time-Resolved Fluorescence Spectroscopy In Biochemistry and Blology"; Cundall, R. B., Dale, R. E., Eds.; Plenum Press: New York, 1983; p 285. Demas, J. N.; Crosby, G. A. Anal. Chem. 1970, 4 2 , 1010. Turley, T. J. M.S. Thesis, University of Virginia, 1979. Berlman, I . B. "Handbook of Fluorescence Spectra of Aromatic Molecules"; Academic Press: New York, 1965. Lampert, R. A,; Chewter, L. A,; Phillips, D.; O'Connor, D. V.;Roberts, A. J.; Meech, S. R. Anal. Chem. 1983, 55, 68. Richardson, J. H.: Stelnmetz, L. L.; Deutscher, S. B.; Bookless, W. A.; Schmelzinger, W. L. 2.Naturforsch. A 1978, 33a, 1592. Rayner, D. M.; McKlnnon, A. E.; Szabo, A. G.; Hackett, P. A. J. Can. Chem. 1976, 5 4 , 3246.
RECEIVED for review March, 18,1985. Accepted May 24,1985. We gratefully acknowledge the support by the National Science Foundation (Grant No. 82-06279)and the donors of the Petroleum Research Fund, administered by the American Chemical Society.
