a2
Anal. Chem. 1984, 56,82-85
expected since we were exciting into a radial acoustic resonance mode of the cell. The long cell had the advantage that we could directly compare the PAS spectrum with an FTIR spectrum of the same sample. The PAS spectrum had about two to three times better SIN. An advantage of the PAS spectrum is that there is no interference from atmospheric absorptions such as water. The sensitivity advantage of PAS was more dramatic with shorter path length samples. Over the range 0.3-4.4 cm, the photoacoustic signal was practically independent of the sample length, so that for a 3-mm sample the detection limit ( S I N = 1)was an absorbance of 1 X a factor of about 50 better than a typical FTIR spectrum. The shortest path length feasible with our current cell and microphone design was 3 mm. Presumably the sample could be made considerably thinner. The observed noise level a t the output of our lock-in amplifier was very close to the expected Johnson noise of our 22 Ma preamplifier input resistor over the band-pass of our lock-in amplifier. This noise determines the detection limit. Higher sensitivities can be achieved by increasing the laser power or cooling the input resistor to the preamplifier. An advantage of operating at these high frequencies is that our detector is able to operate in noisy surroundings and even with flowing samples with little interference. Furthermore using radial acoustic modes we can achieve substantial rejection of interference from window absorption.
-
CONCLUSIONS IR absorption spectra of weak absorbers and trace contaminants in liquids can be observed by using a photoacoustic detection and a modest power F-center laser. The moderate Q natural acoustic resonances of typically sized liquid cells are used to enhance the signal. Even with a modest power laser (3.6 mW) absorbances in the range of 1 x; corre-
sponding to a sample concentration for a typical absorber in the OH region of a few parts per million and a sample path of 3 mm are detectable. The current apparatus is about 50 times more sensitive than a typical FTIR. The limiting noise is thermal (Johnson) noise in the input resistor, which is of course independent of signal. The signal scales linearly with the laser power, so the sensitivity should scale as the laser power. Thus higher sensitivities will be possible as more powerful IR lasers become available. The high frequency used makes the technique useful in high ambient noise applications and for flowing samples. It also provides discrimination against contributions to a background signal from window absorption.
ACKNOWLEDGMENT We thank James Mitchell and Timothy Harris for their interest and aid in obtaining a duplicate of the Tam and Patel microphone. The initial experiments which led to this research were done in collaboration with T. Harris.
LITERATURE CITED (1) McClelland, J. F. Anal. Chem. 1983, 55, 89A. (2) Rosencwaig, A. "Photoacoustics and Photoacoustic Spectroscopy"; Why-Interscience: New York, 1980. (3) Voigtman, E.; Jurgensen, A.; Winefordner, J. Anal. Chem. 1983, 53, 1442. (4) Patel, C. K. N.; Tam, A. C. Rev. Mod. Phys. 1981, 53,517. (5) Patel, C. K. N.; Tam, A. C. Appl. Phys. Lett. 1979, 34, 760. (6) Nelson, E. T.; Patel, C. K. N. Appl. Phys. Lett. 1981, 3 9 , 537. (7) McDonald, P. A.; Shirk, J. S. J. Chem. Phys. 1982, 7 7 , 2355. (8) Morse, P. "Vibration and Sound"; McGraw-Hill: New York, 1948; p 398f. (9) Fisher, M. R.; Fasano, D. M.; Nogar, N. S.Appl. Spectrosc. 1982, 3 6 , 125.
RECEIVED for review August 11, 1983. Accepted October 7 , 1983. This work was initially supported by the NSF under Grant No. CHE 79-09380.
Phase Plane Method for Deconvolution of Luminescence Decay Data with a Scattered-Light Component J. C. Love and J. N. Demas* Department of Chemistry, University of Virginia, Charlottesville, Virginia 22901
A three-parameter version of the phase-plane method for deconvolutlon of lumlnescence decay data Is presented that corrects for contrlbutlons of scattered exclted llght In the observed decay. Through computer slmulatlons, we tested the method with numerous comblnatlons of Ilfetlmes, noise levels, and scatter coefflclents. The modlfled equatlon was found to be computatlonally rapld and yielded excellent preclslon and accuracy In the decay parameters.
Excited state lifetime measurements are pervasive and provide crucial information in analytical chemistry, photochemistry, photophysics, and photobiology (1,2). The most common approaches to lifetime measurements involve exciting the sample with a short optical pulse and monitoring the sample decay. If the excitation source is short enough and the detection system responds quickly enough, the observed
sample decay is the desired sample impulse response. Frequently, however, the sample decays on a time scale comparable to the excitation pulse width and the response time of the detection system. The observed decay is then a complex function given by the convolution of the system response and the sample impulse (2). The process of extracting decay parameters from the observed excitation and decay profiles is called deconvolution. The most important deconvolution problem involves Samples with impulse responses that are single exponential decays. The observed sample decay, D ( t ) , is then given by
D ( t ) = K exp(-t/r)JfE(x) 0
exp(x/r) dx
E ( t ) = 0; t I O
+ aE(t)
(1) where D ( t ) is the observed decay signal vs. time, t , K is the proportionality constant, E ( t ) is the observed excitation profile, and 7 is the sample lifetime (1-4). The first term on
0003-2700/84/0356-0082$01.50/00 1983 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 56, NO. 1, JANUARY 1984
the right-hand side accounts for the contribution of the sample decay. The aE(t) term accounts for any leakage of the scattered excitation pulse to the detector. Especially for bright emitters with a large wavelength difference between excitation and emission monitoring, a can frequently be negligibly small. In eq 1 any distortions of E ( t ) and D ( t ) are assumed to be linear and the same for both measurements. Furthermore, there is no time shift between E ( t ) and D(t). A variety of methods are used to extract information from data of the form of eq 1especially where a = 0 (i.e., negligible scatter) (1-5). Many of these methods are complex and computationally time-consuming. For the a = 0 case the phase-plane (PP)method, which was originally developed for simple exponential evaluations (6, 3,has proved a successful deconvolution method (2-4). It linearizes the data for viewing, is computationally exceptionally fast, and gives good precision and accuracy in decay parameters over a wide range of data (2-4). The PP method has recently been extended to decays with a scatter contribution (i.e., a > 0 in eq l),a sum of two exponentials (8),or decays involving nonradiative intermolecular energy transfers (Forster kinetics) (9). Independently, we had arrived at the deconvolution of eq 1with a scatter component. Jezquel et al. (8)reported only limited data on the accuracy, precision, and limitations of the PP method with a scatter correction. We report here a more extensive analysis of the limitations of the PP method with a scatter correction. I n particular, we assess the accuracy and precision of the method for a range of 7,a,and K as well as for differing noise levels.
THEORY We derive the phase-plane equations following the approach used for the normal PP equation without scatter. Taking the derivative of both sides of eq 1 with respect to t yields dD(t)/dt = -(1/7)D e x p ( - t / 7 ) x t E ( t ) exp(t/T) dt
+
KE(t) + a d E ( t ) / d t (2) The first term on the right is simplified by replacing it with a rearranged form of eq 1, yielding -(l/~)[D(t )aE(t)]+ a d E ( t ) / d t (3) dD(t)/dt Reintegrating eq 3 over the interval of 0 to t and rearranging yields
Y ( t )= A0 + AlX,(t) + A2X2(t) A, = K + a / 7
(4b)
AI = 1 / 7 A2 = CY
C(Y(t,)- [A, + A J 1 , + A2X21112
83
(5)
where X1, and X,, are X1 and X, evaluated at t,. The summation is over all data points used in the data fitting. Taking the partial derivative with respect to Ao,AI, and A, and setting them equal to zero yields the “normal” equations necessary for a minimum. In a matrix form the normal equations are
(
z;y:X2) ~ X I I X , , ZX2LZ
2x11 E1 CX1IZ 2XlI ZXZI
(ii)
= fc::;Xl)
(6)
X X ( tAX2 I
The summations are over the points used in the fitting. Solution of this system of normal equations yields the best Ao, Al, and A, which, in turn, gives K , 7, and a from eq 4. Equation 6 is analogous to that used by Jezequel et al. (8).
EXPERIMENTAL SECTION To analyze the effects of different noise levels, lifetimes, and scatter coefficients on the accuracy and precision of the new equation, we have applied the PP method to the reduction of digitally simulated decay curves. The synthetic data used had noise levels similar to that found in both single photon counting (SPC) and analog instrumentation. Synthetic decay curves were generated by using eq 1 with K = 1and a range of a’s. E(t) was simulated as described earlier (2-4) with A = 6.0 and B = 6.6 ns which yielded an E(t)that peaks in -5 ns, exhibits a full width at half maxima (fwhm) of 14 ns, and decays exponentially at long times with a 6.64-1slifetime. The calculated decays were then scaled to give peak values in E(t)and in D(t) of lo2, lo3,and lo4. a’s used were 0.5, 2.5,5.0, and 10.0. A total of 201 data points were generated at a 0.25-11s spacing, and T’S of 2.5 ns and 15 ns were used. These T’S were selected as representative. The short 7 is much shorter than the flash width, and the long 7 is appreciably longer, but D ( t ) is still significantly affected by E ( t ) . Synthetic data were generated for all combinationsof a,7,and peak channel counts in E(t)and D(t). Gaussian noise with a standard deviation equal to the square root of D ( t ) was added to the suitably scaled E(t)’s and D(t)’s as described earlier (4). For values of D ( t ) 1 25 this distribution is essentially Poisson and mimics SPC statistics. For peak values of lo4,the data correspond to typical SPC decays. Peaks of IOs to lo4 represent reasonable analog data, while peak values of lo2 would correspond to poor analog data. Typical noise levels are shown in ref 4. The integrals necessary for the evaluation of Y ( t ) XI@), , and X&) were evaluated by the trapezoidal rule (2-4).
(44 J t ’ o ( x ) dx = (At/2)R,
(74
(44
R, = (D(t,)+ D(t,-J) + RL-l,i 1 1, Ro = 0
(7b)
(44
J ” E ( x ) dx = (At/2)Q,
(74
Q, = (E(t,)+ E(4-1)) + Q,-1, i 1 1, Qo At = t,+l - t ,
Y ( t )= W ( t )= D ( t ) / l 0t E ( x )dx
=3
(7d) (74
W ( t ) ,Y ( t ) ,and Z(t) are thus given as
X,(t) = E ( t ) / l0t E ( x )dx
(40
Y(t),X,(t), and X,(t) are easily evaluated functions of the observed D ( t ) and E@). W ( t )and Z ( t ) are the original definitions given by Demas and Adamson (3). The use of Y(t)’s a n d X(t)’s here is a more consistent nomenclature for generalization. The formula is analogous to the one used by Reed and Demas (10) that minimized systematic errors that occur for noisy data fit by normal unweighted least squares (11). Equation 4 is a linear function of two independent variables. Standard linear least-squares methods can be used to fit the observed data to obtain the desired parameters. In an unweighted least-squares sense, one strives to minimize
Ut,) = 2D(t,)/(AtQ,)
@a)
xz(t,)= Wt,)/(AtQ,)
(8b)
XI(t,) = R,/Q, (8c) The three linear equations of eq 6 were generated and solved by matrix algebra to yield the K , a,and T values of the simulated curves. All fits were carried out from t corresponding to the peak of the noise free D(t)to the 201st point. This range was selected to give as much information as possible about the sample decay while minimizing the contribution from the scatter. Further, since there is no weighting, inclusion of data from earlier times degrades the results. See the plots in ref 3 and 4. For data with a large scatter component, the observed peak of D ( t ) will be at shorter times than the peak for the scatter free D ( t ) . With a little experience or by generating a few simulated decays, however, the
84
ANALYTICAL CHEMISTRY, VOL. 56, NO. 1, JANUARY 1984
0
10
20
30
40
50
t (ne)
Scatter free D ( t )(X’s)and a E ( t ) terms for T = 2.5 ns. The a for each E ( ? ) term is indicated by the curve.
Flgure 3.
0
10
20
30
40
50
3. 0
t (ns)
2.
Scatter free D ( t )and a € ( ? terms ) for T = 15 ns. The a for each E ( t ) term is indicated by the curves. Flgure 1.
3 2. a
2. 5r
5
8.5 0.0
0
2
4
6
8 1 0 1 2 1 4
X I (t)
Flgure 4. Normal linearized PP plots for the data of Figure 3. a is indicated by each curve. Data are plotted from the time of the peak of the noise and scatter free D ( t ) . x 1 (t)
Table I. Relative Errors and Standard Deviations in 7
Figure 2. PP plots for the data of Figure 1. a is indicated by each curve. Data are plotted from the time of the peak of the noise and scatter free D ( t ) .
operator can easily determine where the fit should actually begin. For every set of parameters, 20 noisy decay curves were generated and reduced. The average T , a, and K and their respective standard deviations were calculated. For each set of decay parameters, we determined a relative error, RE, and a relative standard deviation, RSD, given by RSD % = 1OOu/V
(9b)
where V is the decay parameter used to simulate the decays, V is the average calculated parameter, and u is the standard deviation. All simulations were performed on an HP-85A desktop computer.
RESULTS AND DISCUSSION Figure 1 shows the scatter free sample decay (T = 15 ns) and the scaled scatter contributions for different a’s. Naise levels are for curves of the indicated amplitude. Note, however, that the data reduced had the noise added only after generating and scaling the composite D(t). a = 0.5 the distortions are relatively small but the contributions to the observed decays for a Z 2.5 are quite serious. Figure 2 shows the corresponding normal phase plane plots of Y(t)vs. X,(t) for the same data. In the absence of scatter these plots should be linear with slopes equal to (-l/~).In the original formulation (3) these plots would be W(t)vs. Z(t)plots. As expected for a’s 2 -0.5, the slopes of these normal PP plots would yield inaccurate estimates of T. Figures 3 and 4 show corresponding data for T = 2.5 ns. Even for LY = 0.5, significant distortions of the observed decays arise from the scatter. For a = 2.5, the peak amplitudes of D ( t ) and the scatter contributions are comparable. Clearly use of the normal PP method for estimating T’S is worthless. Indeed, for cy = 10 the scatter peak appears to completely overwhelm the sample decay, but, as we will show, useful information still can be extracted from such data. Figure 4 shows the Y(t)vs. X l ( t ) plots. As expected, use of the normal PP equation without a scatter correction would not yield
and’D(t) 100
1000
10 000
(Y
0.5 2.5 5.0 10.0 0.5 2.5 5.0 10.0 0.5 2.5 5.0 10.0
RE
RS D
RE
RSD
-0.53 -25.1 -17.0 -24.7 -1.4 -6.1 -1.7 -16.6 0.44 -0.07 0.93 4.0
19.3 32.8 33.7 46.4 8.8 16.5 23.3 38.5 3.3 3.5 9.5 16.3
-0.54 -2.8 -4.8 -12.0 -0.75 -1.0 -0.98 -1.1 0.02 0.11 -0.26 0.06
5.5 5.4 7.4 14.4
2.3 1.9 2.1 3.1 0.83 0.80 0.59 0.80
reliable estimates of T for a 2 -0.5. Table I summarized our relative error and standard deviation in T for the range of parameters. For brevity we have included only the data where the peaks in E ( t ) and D ( t ) were equal. Where the peak values were unequal, the results generally fell between those entries in the table for the two corresponding equal peak cases. Because the lifetime is usually the important parameter estimated, we have not reproduced our data for a and K. The full set of calculations is available on request. All the errors, however, were typically within 1-2 standard deviations of the generating parameters for the lo3 and lo4peak data. Precisions in K and T declined somewhat as the amount of scatter detected by the system rose, but acceptable fitting will occur for any reasonable degree of scatter. Single photon counting quality data typically yielded a K and T accuracy of 5 ~ 2 % or better; even noisy analog data yielded results within 1 9 % of the correct values, except for the 100 peak data and the shortest lif&ime. Precision in a improved as the precisions of K and T declined. This is to be expected, since the exponentiality of the decay curve is influenced by the degree of scatter, yielding a somewhat more complex decay scheme (eq 4), but a t the same time providing more information on the a component. Even the apparently impossible problem of deconvoluting a 2.5-11s lifetime with a = 10 yielded noisy but useful results. The RSD in this case was only 16% for lo4peak counts which
Anal. Chem. 19a.4, 56,85-88
is the standard signal level in many SPC experiments. We have also verified that the parameters calculated from the PP method yield statistically valid fits. Using the least squares K , r , and a, we regenerated D ( t ) using eq 1 and compared the resultant curve with the original data using a x2 test (12). The reduced x2’s typically ranged from 0.9 to 1.1 which is acceptable for 200 data points (12). All data fits are unweighted, and in view of the excellent results, we recommend the use of unweighted fits for the range of parameters and fits indicated. Unweighted fits are easier to do, and the much more complex weighted data treatment appears unwarranted. This conclusion is the same as that reported by Greer et al. ( 4 ) for deconvolution of single exponential decays using the phase-plane method and also is consistent with the results of the PP method derived for base line correction of exponential decays (13). There is a potential problem with the treatment presented here when used with SPC data. These instruments actually measure the integral of the number of photons collected over the preceding time window rather than the instantaneous intensity. If too low a density of points is used, then the PP method, which is an integral method, gives systematic errors in the phase-plane plots and the evaluated parameters. However, for virtually all SPC data, the density of the data points is high enough so that the error should be negligible. Jezequel et al. (8)pointed out this error source and described a simple way to circumvent it should the errors ever become significant. When the PP method is used with analog data, this error source is not present. We conclude that for a reasonable range of decay parameters and fitting regions, the modified PP equation is accurate and precise. The method can be easily programmed and is
a5
computationallyvery fast. These features give the PP!method distinct advantages over much slower nonlinear leasbsquares methods, especially in view of the increasing use of microcomputers in data acquisition systems.
ACKNOWLEDGMENT We thank J. C. Andre for providing preprints of his manuscript and J. Y. Jezequel for helpful comments.
LITERATURE CITED (1) Blrks, J. B. “Photophysics of Aromatic Molecules”; Why-Interscience, New York, 1970. (2) Demas, J. N. “Excited State Lifetime Measurements”; Academic Press: New York, 1983. (3) Demas, J. N.; Adamson, A. W. J. Phys. Chem. 1971, 75, 2483. (4) Greer, J. M.; Reed, F. W.; Demas, J. N. Anal. Chem. 1961, 53, 710. (5) Isenberg, I.; Dyson, R. D. Blophys. J. 1969, 9 , 1337. (6) Bernalte, A.; Lepage, J. Rev. Scl. Instrum. 1969, 4 0 , 71. (7) Huen, T. Rev. Sci. Instrum. 1969, 4 0 , 106. (8) Jezequel, J. Y.; Bouchy, M.; Andre, J. C. Anal. Chem. 1962, 5 4 , 2199. (9) Love, J. C.; Demas, J. N. Rev. Scl. Instrum. in press. (10) Reed, F. W.; Demas, J. N. “Time Resolved Fluorescence Spectroscopy in Biochemistry and Biology”. Cundall, R. B., Dale, R. E., Eds.; Plenum Press: New York, 1983; p 285. (11) Knuth, P. E. “Semlnumerical Algorithms. The Art of Computer Programming”; Addison-Wesley: Readlng, MA, 1969; Voi. 1. (12) Bevington, P. R. “Data Reduction and Error Analysis for the Physical Sciences”; McGraw-Hill: New York, 1969. (13) Bacon, J. R.; Demas, J. N. Anal. Chem. 1983, 55, 653.
RECEIVED for review July 29,1983. Accepted October 3,1983. We gratefully acknowledge the donors of the Petroleum Research Fund, administered by the American Chemical Society, the Air Force Office of Scientific Research (Chemistry) (Grant AFOSR 78-3590), the Department of Energy for SERI Grant DE-FG02-CS84063, and the National Science Foundation (CHE 82-06279).
Optimization of Anion Separation by Nonsuppressed Ion Chromatography Dennis R. Jenke and Gordon K. Pagenkopf* Department of Chemistry, Montana State University, Bozeman, Montana 5971 7
The effect of eluent pH and eluent species concentration on the nonsuppressed ion chromatographic separatlon of anions has been studied. The retentlon times of CI-, Br-, NO,-, SO:-, and SO , :were determined over a large range of eluent composltlons and the data were utilized to construct window diagrams. These window dlagrams were used to optimize eluent composition for the separation of two or more anaiytes. I n virtually every case resolytlon is ilmlted by the separation of the NO,-/Br- pair.
Since its introduction in 1975, ion chromatography has rapidly evolved into a widely accepted method for the quantitative determination of anions in aqueous samples (I). Single column or nonsuppressed ion chromatographic techniques have been successful in a variety of applications (2). As is the case with all chromatographic techniques, the most effective utilization of the ion chromatographic process requires accurate characterization of the analyte retention times and identification of analytical variables that affect the relative 0003-2700/84/0358-0085$0 1.50/0
retention characteristics of the analytes. Of particular interest is the development of a predictive capability and optimization of ion chromatographic resolution, while minimizing analysis time. This can be complicated by the potential existence of multiple optima and the large range of eluent related variables that control chromatographic separation. The concept of “window diagrams” has been utilized to locate optimum conditions in gas chromatography ( 3 , 4 ) . This technique also has been applied to high-performance liquid chromatographic separations (5-8). This study utilizes the technique to optimize the separation of inorganic ions and to predict the behavior of selected analytes under various experimental conditions.
EXPERIMENTAL SECTION The chromatographic system employed consisted of a Perkin-Elmer Series 3B liquid chromatograph, a Vydac Model 3021 C4.6 anion separator column, a Vydac Model 6000CD conductivity detector, and a Sargent Welch XKR strip chart recorder. Injector sample loop volume was 0.10 mL and samples of 0.5 mL were injected with a Hamilton Co. Model 750 microliter syringe. Laboratory temperature was maintained at 22.5 f 2.0 OC and the 0 1983 American Chemlcal Society
