Anal. Chem. 2001, 73, 4486-4490
Optimized Gating Scheme for Rapid Lifetime Determinations of Single-Exponential Luminescence Lifetimes Sing Po Chan,† Z. J. Fuller,† J. N. Demas,*,† and B. A. DeGraff*,‡
Department of Chemistry, University of Virginia, Charlottesville, Virginia 22904-4319 and Department of Chemistry, James Madison University, Harrisonburg, Virginia 22807
The rapid lifetime method (RLD) for determining excitedstate lifetimes uses the ratio of the areas under two regions of the decay. To get good precision with the standard method, prior knowledge of the lifetime is essential to selecting the integration regions. As will be shown, the usual method of selecting integration regions is far from optimal. An optimal gating scheme that is more precise and much more forgiving in the selection of integration region than any of the prior methods will be shown. Monte Carlo simulations were performed to determine the optimal gating. Experimental data was used to confirm the capabilities of the optimized RLD. The speed of the optimal RLD is similar to the standard RLD but without the necessity of matching the integration interval to the lifetime for precise results. Time-resolved luminescence lifetime measurements have been used to study various photophysical phenomena and for chemical analysis. Because luminescence lifetimes can be very sensitive to the immediate environment of the luminophore, they are widely used to study the environment of molecular probes.1 These probes can provide information that includes the rotational correlation times, molecular separations, diffusion, and kinetics of excitedstate reactions (e.g., proton transfer, energy transfer). Lifetime measurements in which the lifetime of the sensing molecule depends on analyte concentration also have analytical applications. While intensity measurements can sometimes serve to measure these quantities, lifetime measurements are preferred because they are less sensitive to instrumental changes and sample properties. For example, the lifetime is largely independent of excitation intensity or a small amount of sample decomposition while luminescence intensity is not. A common method of luminescence lifetime decay evaluation is to acquire a multipoint decay curve and fit it by least-squares methods. This approach requires considerable computational time and is, therefore, unsuitable for real-time systems such as HPLC and for imaging (e.g., microscopy, flames) where obtaining multiple time points on the image is difficult.2 The rapid lifetime determination method (RLD) is a family of evaluation schemes used to quickly analyze the decay data and allow real-time data *Corresponding authors. J.N.D.: (phone) 434-924-3343; (fax) 434-924-3710; (e-mail)
[email protected]. B.A.D.: (phone) 540-568-6246; (fax) 540-568-7938; (e-mail)
[email protected]. † University of Virginia. ‡ James Madison University. (1) Lakowicz J. R. Principles of Fluorescence Spectroscopy; Plenum Press: New York, 1983. (2) Ni, T. Q.; Melton, L. A. Appl. Spectrosc. 1993, 47, 773-781.
4486 Analytical Chemistry, Vol. 73, No. 18, September 15, 2001
analysis of the lifetime.,3-5 For a single-exponential decay, the lifetime is evaluated from the ratio of integrated areas under two different regions of the decay curve. The two integrated areas can be determined by the accumulated photon intensity or, in the case of a gated CCD camera, directly from the pixel intensity. If the two time intervals (gates) are equal, there is a simple closed form solution for a single-exponential luminescence decay. An extensive amount of analysis has been done to determine and improve the precision of the RLD.6-9 While the RLD has been traditionally used for single-exponential decays, it was recently extended to the double-exponential case.10 The standard RLD belongs to a broad class of similar data reduction methods, and we will give the general solution. We will present an optimized RLD that is far more forgiving in terms of the experimental conditions that must be used for good accuracy and precision. Further, this optimized RLD has better precision under a wide variety of conditions than the standard RLD has under the best experimental conditions. Both experimental data and theoretical Monte Carlo simulations will be used to show the capabilities of the optimized RLD. THEORY The gating scheme and nomenclature for the standard RLD with equal contiguous gating is shown in Figure 1a where the D’s are areas under the indicated portion of the decay curve. For a single-exponential decay, the lifetime τ is calculated from
τ)
-∆t ln(D1/D0)
(1)
In general, however, there is no reason the integration periods must be equal. Also, as has been shown, the intervals do not have to be contiguous.6,7 We show the general case in Figure 1b. The first integral runs from 0 to ∆t while the second starts at Y∆t and (3) Ashworth, H. A.; Sternberg D.; Debiran, A., Seton Hall University, South Orange, NJ, unpublished work, 1983. (4) Wang, X. F.; Uchida, T.; Coleman, D. M.; Minami., S. Appl. Spectrosc. 1991, 45, 360-366. (5) Woods, R. J.; Scypinski, S.; Cline Love, L. J.; Ashworth, H. A. Anal. Chem. 1984, 56, 1395-1400. (6) Waters, P. D.; Burns, D. H. Appl. Spectrosc. 1992, 47, 111-115. (7) Tellinghuisen, J.; Wilkerson, C. W. Anal. Chem. 1993, 65, 1240-1246. (8) Soper, S. A.; Legendre B. L., Jr. Appl. Spectrosc. 1993, 48, 400-405. (9) Ballew, R. M.; Demas., J. N. Anal. Chem. 1989, 61, 30-33. (10) Sharman, K. K.; Periasamy, A.; Ashworth, H. A.; Demas, J. N.; Snow N. H. Anal. Chem. 1999, 71, 947-952. 10.1021/ac0102361 CCC: $20.00
© 2001 American Chemical Society Published on Web 08/15/2001
Figure 1. Graphs showing the integrated areas used for single-exponential decay with standard RLD using equal contiguous intervals (a) and the generalized RLD (b).
runs to Y∆t + P∆t. Y is the fractional delay in units of ∆t and can assume values from 0 to >1. P is the width of the second integration period in units of ∆t and can assume values greater than 0. For Y ) P ) 1, we have the standard contiguous RLD while for Y * 1 and P ) 1, we have the family of noncontiguous delayed standard RLDs including those with overlapping gates. The general expression is derived as follows:
D0 ) D0 )
∫
∫
∆t
0
A exp(-t/τ) dt
Y∆t+P∆t
Y∆t
A exp(-t/τ) dt
(2) (3)
where the decay is assumed to be given by A exp(-t/τ), A is the preexponential factor, t is time, τ is the lifetime, and Y and P are the adjustable coefficients for the integration intervals. The ratio of D1 to D0 is then given by
D1 exp(-R(P + Y)) - exp(-YR) ) D0 exp(-R) - 1
(4)
R ) ∆t/τ
(5)
simulations. The integrated number of counts under the entire decay curve was 106. For our treatment, we assumed that both integrals were taken from the same decay. This required special treatment for the overlapping case (Y < 1). It was necessary to use three integrals.
S0 )
∫
S1 )
∫
S2 )
Y∆t
0
∫
∆t
Y∆t
A exp(-t/τ) dt
(6)
A exp(-t/τ) dt
(7)
Y∆t+P∆t
∆t
A exp(-t/τ) dt
(8)
For each simulation, Poisson noise was added to each integral. For the overlapping case, D0 ) S0 + S1 and D1 ) S1 + S2. For nonoverlapping intervals (Y g 1), we used eqs 2 and 3. Reagents and Chemicals. The luminophore was aqueous 50 mM Na3[Tb(dpa)3)] (dpa ) 2,6-pyridinedicarboxylic acid) at pH 7.11 The structure is shown below.
For contiguous and equal time intervals (P ) 1 and Y ) 1), eq 4 reduces to the standard contiguous RLD in eq 1. If, however, the time intervals are not equal, P * 1, R cannot be isolated and τ cannot be obtained in closed form. Nevertheless, R and then τ can be calculated rapidly and efficiently by solving eq 4 using an iterative Newton-Rhapson method. EXPERIMENTAL SECTION Monte Carlo Simulations. Monte Carlo simulations were first performed to investigate different gating schemes for precision and accuracy. These simulations were performed using programs in Mathcad 8 Professional (Mathsoft, Inc., Cambridge, MA 02142). In these simulations, artificial exponential decays were generated, Poisson noise was added to the simulated experimental data, and the lifetime was computed from the selected algorithm. The process of adding noise and reducing the data was repeated 1000 times and the mean, standard deviation (σ), and relative standard deviation (σ/τ) were computed. Simulations were repeated for ∆t/τ values from 0.25 to 10 in increments of 0.25. The number of simulations make it possible to produce very smooth, wellcharacterized curves for computed parameters for each set of
This complex was available from earlier work, has a long easily measured luminescence lifetime (∼2.0 ms), is photochemically (11) Metcalf, D. H.; Snyder, S. W.; Demas, J. N.; Richardson, F. S. J. Am. Chem. Soc. 1990, 112, 5681-5695.
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Figure 2. Percent standard deviation of lifetime at various ∆t/τ using Monte Carlo simulations. Six simulations are shown with varying integration periods involving the optimized new methods and previously used methods. The first integral runs from 0 to ∆t while the second starts at Y∆t and runs to Y∆t + P∆t.
and thermally very stable, and has a low temperature coefficient of the decay time near room temperature (
