(31) K. A. Zachariasse, Ph.D. Thesis, Vrije Universiteit te Amsterdam, 1972. (32) A . S. Cherkasov and I. E. Obyknovennaya. Dokl. Akad. Nauk SSSR,
research by the u.S. Research Office-Durham, is gratefully acknowledged. This material was presented in Paper No. 16 a t the 165th National Meeting of the American Chemical Society (Analytical Division), Dallas, Texas, April 1973; and in Paper No. 14 a t the 27th Southwestern Regional Meeting of the American Chemical Society (Analytical Division), San Antonio, Texas, December 1971.
173, 867 (1967).
RECEIVEDfor review June 13, 1974. Accepted September 13, 1974. This is paper number XX in the series “Electrogenerated Chemiluminescence.” Financial support of this
Computerized Kinetic Luminescence Spectrometry: TimeResolved and Component-Resolved Phosphorescence Spectrometry R. Marshall Wilson and Theodore L. Miller Deparlment of Chemistry, University of Cincinnati, Cincinnati, Ohio 4522 1
A computer-controlled laser phosphorimeter is described. The form of data acquisition utilized by this system is unique in that the phosphorescence spectra are recorded on magnetic tape as signal-averaged, families of decay curves. The utility of this data format is illustrated in the analysis of phosphorescence spectra of samples exhibiting nonexponential decay. An extremely flexible and simple procedure for the generation of time-resolved spectra is described. The technique of co’mponent-resolved spectrometry is discussed in detail and illustrated by the resolution of the phosphorescence spectra of the constituents from the emission of a binary mixture. The system described is extremely versatile in that the decay of phosphorescence spectra can be analyzed in terms of any one of a wide variety of kinetic models. The results of this kinetic analysis may be displayed in the form of calculated time-resolved and component-resolved spectra for comparison with observed timeresolved spectra and spectra of pure components.
Phosphorescence spectra obtained by conventional means are frequently rendered difficult to interpret because of the superposition of emission from two or more excited species. Most efforts have been devoted to the simplification of these complex emission spectra through either tedious purification schemes ( I ) or application of elaborate mechanical ( 2 ) and electronic devices (3-7). I t has only been recently that serious attempts have been made to study this phenomenon. It now appears that there are several situations that can give rise to the formation of more than one excited species and, thus, to complex emission spectra. Perhaps the most common situation is that in which the components of a mixture of substances are emitting independently in the same spectral region (7-10). In certain instances, a single pure substance can afford complex emission spectra. The following explanations for this type of behavior have been offered, but in many cases have not been confirmed experimentally. Simultaneous emission from both n,a* and K,T* states has been demonstrated in some cases ( 1 1 ) and suggested for others ( 1 2 ) .(For a summary of the complex emission behavior of aryl carbonyls, see Reference 12. ) Emission from different states of aggregation has been demonstrated ( 1 3 ) .Emission from a single substance in different conformations (14, 1 5 ) or different 256
A N A L Y T i C A L CHEMISTRY, VOL. 47, NO. 2, FEBRUARY
solvent or matrix environments (12) might account for complex emission. Finally, a t very low temperatures emission has been observed from the three sub-levels of the triplet state (16, 17). Workers in this area have relied heavily upon kinetic information which was acquired by recording the relative emission intensity as a function of time a t one or, a t most, a few different wavelengths. While the analysis of decay data of this kind can provide one with an estimate of the number of emitting species and the manner in which they undergo change, kinetic analysis alone provides little information concerning the nature of the species involved. To make judgements obtaining to chemical structure, one also requires spectrometric information in which the relative emission intensity is recorded as a function of wavelength. Conventional time-resolved luminescence spectrometry combines this type of kinetic and spectrometric data in a very qualitative fashion; rigorous kinetic analysis of conventional time-resolved spectrometric data is not possible. Consequently, quantitative kinetic and spectrometric information are acquired in two independent sets of measurements. We have assembled a computerized kinetic luminescence spectrometry system which unifies kinetic and spectrometric analysis in a novel and rigorous fashion. In principle, the kinetic spectrometry system described here might be used to study any sample exhibiting complex emission behavior. However, it was felt that the only way to evaluate the capabilities of the approach used in this system was to apply the technique to a known mixture of emitting substances. For this purpose we have selected a mixture of anthrone (ca. 8.8 X 10-4M) and benzophenone (ca. 3.8 X 10-4M) to illustrate in detail the technique of kinetic spectrometry.
THE DATA ACQUISITION SYSTEM The unification of kinetic and spectrometric information requires the recording of relative luminescence intensity as a function of both time and wavelength simultaneously. This can be accomplished by taking advantage of the high speed data acquisition capability of an on-line computer to permanently record a complete family of emission decay curves taken a t small wavelength intervals over the entire spectrum range. It then becomes possible by again employing a computer to reassemble these decay data into a com1975
TEST PFITTERN
Figure 1. Luminescence life history assembled from decay data
plete life history of the luminescence as illustrated in Figure 1. The emission decay curves are represented by the heavy lines in Figure 1. Only a few of these are shown to illustrate how the decaying spectrum can be reconstructed from a family of such decay curves. The computerized laser phosphorimeter diagrammed in Figure 2 has been assembled and can readily acquire sufficient decay data for the generation of luminescent life histories of compounds emitting with lifetimes in the 1 msec to 10 sec range. The exciting source (1) is a pulsed nitrogen laser which delivers ultraviolet light a t 337.1 nm with a pulse width of 20 nsec, peak power of about 100 kW, and repetition rates of up to 100 pulses per second. The short pulse width eliminates the necessity for mechanical shutters in the excitation beam and allows the time delay between excitation and data acquisition to be as small as a few microseconds. The light emitted by the sample (2) is then analyzed by a monochromator (3) and detected with a photomultiplier (4). The photomultiplier signal is amplified (5), and can be observed with an oscilloscope (6). The computer ( 7 ) acquires the data and controls the data acquisition. T o obtain relatively noise-free data which are essential for the subsequent analysis, about 100 decay curves are collected for each wavelength and automatically averaged by the computer. Data acquisition for a typical sample proceeds as follows. The monochromator is set a t an initial value. The computer records the background line voltage on magnetic tape and then fires the laser. The emission decay is recorded by the computer and, after sufficient time has elapsed for the complete decay of the emission, the computer triggers the laser again. The process is repeated about 100 times. The 100 decay curves are signal averaged, and the averaged curve is stored on magnetic tape. The computer then signals a monochromator drive system (8) which moves the monochromator to the next wavelength. About 150 to 200 wavelengths are examined a t 1-nm intervals. This entire
Figure 2. Diagram of phosphorimeter
process requires about 1 hour for a sample with a lifetime of 30 msec. During that time, about 15,000 decay curves or about 20 million data points have been incorporated into the data recorded on magnetic tape.
EXPERIMENTAL Signal Amplifier. The combination electrometer-amplifier is shown in Figure 3. This component is connected to the photomultiplier (RCA 1P28) uia a short RG-58/U cable (35 cm) to minimize induced noise. The system consists of a variable gain current-tovoltage converter (Keithley Model 3021, a variable gain voltage amplifier, an inverter with a voltage offset and a driver amplifier. The second stage also provides a filter network t o optimize the signal-to-noise ratio for different decay rates. The amplified signal is transmitted over a Belden 24AWG cable to the computer. Computer. A Raytheon 704 computer is used for data acquisition and instrument control. This system consists of a 16K memory, two magnetic tape units, a cartridge disk unit, a high speed paper tape reader, a teletype, memory oscilloscope and x-y recorder display units, an eight-channel mutiplexed analog-to-digital converter (ADC) and a four-channel digital-to-analog converter (DAC). The ADC accepts single-ended f l 0 - V bipolar inputs and
A N A L Y T I C A L CHEMISTRY, V O L . 4 7 , NO. 2 , F E B R U A R Y 1975
257
(11 05uF r-1 R22 5M ; * - - , -w
RII 2211
Y
RI IOK
,
1
L-
n
%
36K
l
0
'l5V
Figure 3. Schematic diagram of signal amplifier
converts these inputs to a 12-bit binary code a t a 100-kHz throughput rate. Emission decay data are collected in real-time and automatically accumulated so that the signal-averaged decay curve is transferred to magnetic tape a t each wavelength. The realtime operation system for the 704 is a disk-based operation system which provides for rapid response t o real-time events concurrent with automatic batch processing on a time-available basis. The time base for the acquisition routine is provided by programmable clock interrupts. The time between data points is variable over a 17- to 32,767-rsec range and is linear and reproducible to a few parts in lo5. Synchronization pulses ( + l o V) from the DAC are used to trigger the laser and drive the monochromator. Control Circuitry. A single pnp transistor is used to amplify the laser trigger pulse. The monochromator (American Instrument Company) emission motor circuitry has been modified so that it can be operated remotely by the computer. A thyristor (2N4442) switching device is used to activate a relay and in turn operate the motor when a positive voltage is applied. Therefore, the interval between wavelength observations is established by the width of the DAC pulse (adjustable from 10 wsec to 0.4 sec) and the emission motor scan rate. In practice, the greatest precision is obtained when the scan rate is adjusted with the maximum DAC pulse width. Adjusting the scan rate to obtain a I-nm wavelength interval is a rather tedious task but, once calibrated, the scan is reproducible to a few nm's over the total spectrum range so that the wavelength interval is 1.00 f 0.04 nm. While emission monochromators with stepping motor drive systems ( 4 , 18 ) would be ideally suited for computerized control, the relay system described here offers a convenient and inexpensive alternative for modifying a conventional monochromator. Test Circuitry. One final, yet most important, aspect in the development of this system was the recognition and minimization of all sources of signal distortion. In emission work, one is dealing with a signal that should in theory decay exponentially. In actual practice, deviations from exponentiality are quite common among materials decaying on a millisecond time scale. Therefore, the use of chemical decay standards for the purpose of testing the data acquisition circuitry was avoided. A most convenient alternative has been found to be the electronically generated exponential signals provided by the discharge of a high quality polystyrene capacitor. Table I summarizes the results of such a test using a 0.1 LLFf 1% capacitor which has been charged with a 9-V battery and allowed to discharge through a 147-kR f K% resister. The capacitor decays exponentially with an RC time constant of 14.7 msec f 1%.When this decay voltage is sampled directly by the ADC and subsequent-
ly fitted using the Marquardt minimization routine (vide infra ), the average time constant (run 1-3, Table I) was determined t o be 14.810 msec. This value differs from the RC product by 0.75% and is well within the 1%tolerance value. The same RC network was used to test the signal amplifier and the transmission cable. Since the discharge voltage across the 147-kR resistor was coupled to the amplifier via a 22-MQ resistor which is effectively in parallel with the RC network, the impedance becomes 146.02 kR and the new RC time constant is 14.602 msec It 1%. The observed time constants for the amplifier (run 4, Table I) and the amplifier plus cable (run 5, Table I) were 14.669 and 14.638 msec, respectively, well within the 1%tolerance limit. Similar results were observed using RC networks with time constants of 1.47 and 6.01 msec. Through the application of this simple device, it has been possible to minimize any distortion introduced by system components after the signal leaves the photomultiplier. Sample Preparation. Anthrone (Aldrich Chemical Company), benzophenone (actinometry standard), and 3-naphthil (19) were all recrystallized in Craig tubes a minimum of four times in at least two different solvents. Anthrone and 8-naphthil were purified initially by silica gel (Brinkmann less than 0.06 mm) column chromatography eluting with benzene. Anthrone was sublimed before the final recrystallization. The 2-methyltetrahydrofuran was refluxed for several hours over Na-K alloy and distilled under an argon atmosphere collecting the center fraction. After this process had been repeated four times, the solvent was sealed under vacuum and stored in the dark at 0 OC. MCIP was prepared from five parts methylcyclohexane (Matheson, Coleman and Bell, used without purification) and one part isopentane (Matheson, Coleman and Bell, used without purification). The samples were dried for several hours under vacuum. The Pyrex sample tubes (6 mm) were sealed after the fifth freeze-thaw degassing cycle at less than 10+ Torr. Procedure. The matrix was maintained by liquid nitrogen (77 OK) in a quartz dewar that had been thoroughly cleaned to prevent nitrogen bubbling. The photomultiplier was driven at -800 V provided by a Kepco regulated dc power supply. The gain on the electrometer-signal amplifier combination was adjusted to amplify the photomultiplier output to 20 V (-10 to +10 V) at the wavelength of maximum emission intensity. During the data acquisition, 1400 points along the decay curve were sampled. The time between points was adjusted at the emission maximum so that these 1400 points were acquired within a voltage range of -10 to +9.95 17. Therefore, the data are recorded over six halflives in the decay of the luminescent signal. To achieve the -10- to +lo-V signal range, the base line was offset to +10 V when the photomultiplier was dark. This "background voltage" was sampled and recorded just prior to the data acquisition at each wavelength in the spectrum. These background values were averaged and subtracted from each data point at each wavelength. The absolute value of' this dif'ference was the luminescent intensity for that point. The averaged background voltage never varied more than +5 mV during the time that it took to record the entire spectrum.
TIME-RESOLVED EMISSION SPECTROMETRY With 1400 points of signal,-averaged decay data per wavelength cataloged by wavelength and permanently stored on magnetic tape, it is a simple task t o regenerate at one's leisure the conventional phosphorescence spectrum on a Calcomp plotter. T h e alpha diketone P-naphthil provides a n interesting example of this procedure for a relatively long-lived-carbonyl emission (Figure 4a ). This spec-
Table I. Results of Capacitor Discharge Tests 93% Coniidence limits
Time constant,
$0
Difference
Run
msec
Rate, rnsec-l
Lower
Upper
X 2
from RC
1"
14.814 14.801 14.815 14.669 14.638
67.502 67.561 67.500 68.172 68.310
67.4 67.5 67.4 67.9 68.0
67.6 67.7 67.6 68.4 68.6
0.0082 0.0086 0.0069 0.0552 0.0794
0.78 0.69 0.78 0.46 0.25
2" 3" 4* 5c
a One discharge directly into the ADC. b One discharge through the signal amplifier into the ADC. One discharge through the signal amplifier across the transmission line and into the ADC. (
~~
258
~~~~~
A N A L Y T I C A L C H E M I S T R Y , VOL. 47, NO 2, FEBRUARY 1975
trum consists of a plot of the time integrated intensity or the area under the decay curves a t each wavelength in the spectrum us. wavelength. Thus, in this method of time-resolved spectrometry, the computer serves as the shutter. The “on” and “off” parameters recorded a t the top of Figure 4a are the time limits of the integration which generated the spectrum, and are quite analogous to the opening and closing of a very fast shutter. In contrast to the conventional forms of time-resolved spectrometry which require that a new set of data be acquired for each time window that one might wish to use, the experimental data set acquired by the aforementioned technique can be analyzed using any one of no less than 979,300 different time windows. Thus, by simply changing the “on” and “off” parameters, one can look a t the early stages of the emission (Figure 4 b ) where the spectrum of the short-lived species predominates or the later stages of the emission (Figure 4 c ) where the spectrum of the longlived species predominates. I t should be noted that since the time delay between excitation and observation can be greatly reduced and the window made of much shorter duration than would be possible using conventional techniques, time-resolution of fast emissions is possible as illustrated for @-naphthil in Figure 4a-c. (For a comparison with spectra obtained using conventional chopper techniques see Reference 14.) A further illustration of the time-resolving power of this technique may be illustrated using a mixture of anthrone and benzophenone. These two substances emit with lifetimes differing by only a factor of about 3.1, and yet the spectra associated with these two substances can be resolved reasonably well as shown in Figure 5,a-c. Figure 5a is the spectrum of the early emission, and is derived mostly from the faster and shorter wavelength (higher energy) emission of anthrone. Figure 5b is the spectrum of the emission a t an intermediate time where both anthrone and benzophenone contribute about equally to the spectrum. Figure 5c is the spectrum of the later emission, and is derived mostly from the slower and longer wavelength (lower energy) emission of benzophenone. These spectra are not generated by the time integration approach described previously, but are constructed by simply plotting a single data point from each of the decay curves in the life history set of experimental decay curves. The time index of these data points is noted a t the top of each spectrum. Therefore, these spectra represent the shortest time window, a “time slice”, attainable by this technique. The data acquisition loop requires about 17 Fsec to cycle and therefore establishes the lower limit of the time-resolving capabilities of this system. However, the time required for the ADC to sample the voltage is much shorter. So, for all practical purposes, a time slice represents a single point in time.
COMPONENT-RESOLVED EMISSION SPECTROMETRY With the same data set tape used to generate the timeresolved spectra illustrated above, one can extend the investigation of the emission spectrum into the unique domain of component-resolved emission spectrometry. To do this, one must first be able to resolve each of the decay curves that make up the emission life history into the contributions from the emitting components. In the ideal case, this amounts to the successful analysis of the sums of exponentials which is a problem that has long intrigued and frustrated many workers (20). The emission from the anthrone-benzophenone mixture can be successfully analyzed using a two-component decay model in which the predicted
m 3.0 MSEC T I M E OFF U20.0 HSEC TIME
0 0
a
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i
NRPHTHIL 7E
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b
T I M E ON
366.6 HSEC
T I M E OFF 3 7 2 . 6 HSEC
C
NAVELENGTH I N H l NRPHTHIL 7E
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Figure 4. ( a ) Total emission spectrum of P-naphthil in MCIP. Timeresolved spectra of @-napthil, ( b ) Spectra of short-lived emission, ( c )Spectra of long-lived emission. Data derived from 30 signal-averaged decay curves per wavelength
ANALYTICAL CHEMISTRY, VOL. 47, NO. 2, FEBRUARY 1975
259
1.0
TIME
MSEC
value of the intensity a t the ith data point would be derived from the following expression:
I i = Ioexp(-kti)
+
Io’exp(-k’ti)
(1)
The problem is to compute estimates of the parameters IO, IO’, k , and k ’ which will accurately describe the entire
a
RNTHRONE-BENZOPHENONE
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TIRE
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I
\
emission life history. Traditionally, attempts to resolve multicomponent emissions have resorted to the graphical method (21). In this approach, data acquired from an oscilloscope trace are used to construct a semilogarithmic plot of relative intensity versus time. The linear portion a t the end of the decay curve is extrapolated to some arbitrary time zero and the values of I O and k are determined for the more slowly decaying component. The rapidly decaying component is evaluated in a similar fashion from the residue remaining after subtracting the slowly decaying signal from the total decay curve. This graphical method has been adapted t o the computer (22). However, we have found that even when applied to relatively noise-free, signal-averaged data using iterative computer methods, this procedure proved to be unsatisfactory for all but simple systems with large differences in decay rate constants. Minimization methods (23, 2 4 ) for fitting equations to nonlinear data have been used for the complete and accurate separation of the components in the system described here. These techniques have also been applied to multicomponent radioactive decay (25-28 ), and more recently the maximum likelihood method (29) and the method of moments (30) have been described. Previously, we have used the grid search method for the analysis of emission decay data (31), but this method converges more slowly than the method used in this work (uide infra). These procedures are more efficient and flexible than the computerized graphical method, and use data from the entire decay curve, rather than just a small segment of the curve, to compute estimates of the decay parameters. Basically, minimization techniques employ a leastsquares fitting procedure which compares the ith measurement, y L ,to the value y ( x J . y ( x J is estimated using trial parameters in a given fitting function. The sum of the differences between the measured values and the calculated values squared is defined as xz,Equation 2.
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The problem of determining the best estimates for the parameters IO, Io’, k , and lz’ (Equation l) reduces to one of calculating estimates for these parameters which will minimize x 2 with respect to each of the parameter simultaneously, Equation 3.
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Figure 5. Time-resolved spectra of a mixture of anthrone and benzophenone in 2-MTHF; time slices at (a) 1.0 msec, (b) 3.8 msec, (c) 7.3 msec. Data derived from 100 signal-averaged decay curves per wavelength
260
where yi is the i t h data point and y ( x ) is the fitting function with parameters ai. T o solve Equation 3, x 2 must be considered a continuous function of the parameters a; and describes a surface in ndimensional space. T o establish those parameters which give the best fit to the experimental data, one must search the x2 surface to find the minimum. Most algorithms used for this type of nonlinear least-square estimation employ some variation of either the Taylor series method or the steepest descent (gradient search) method. Both methods involve inherent difficulties. The steepest descent approach converges rapidly for the first few iterations until the region of the minimum is entered, but it converges very slowly in the region of the minimum. On the other hand,
ANALYTICAL CHEMISTRY, VOL. 47, NO. 2, FEBRUARY 1975
the Taylor series method cannot be relied upon to approach the minimum when the x2 surface is not approximately parabolic as it is in the vicinity of the minimum. The maximum neighborhood method of Marquardt (32 ) combines the best features of the two previous methods. In Marquardt’s algorithm, an interpolation between the steepest descent and Taylor methods is optimized by increasing the diagonal terms of the curvature matrix by a factor A. When operating far from the minimum, the rapidly converging steepest descent method controls the fit but, as the minimum is approached, the more accurate Taylor series method assumes control. Marquardt’s method, as adapted in the Meeter-Wood program (24) (IBM/S 36013.6.007) and modified by us t o automatically fit each decay curve in the life history data set, has been used in this work. To evaluate the feasibility of component-resolved emission spectrometry, this x 2 technique has been applied in the resolution of the same anthrone-benzophenone mixture data set t h a t was used to generate Figures 5,a-c. The evaluation and reconstruction of an emission life history requires a x2 analysis of the decay curve a t each wavelength in the spectrum, or about 150 x2 analyses. Since it is not practical to conduct such extensive computations with a small laboratory computer, the original data set tape was transferred to an IBM 370 computer for analysis. Less than five minutes of CPU time are required for the analysis of a complete spectrum. The x2 analysis provides the fitted parameters ( l o , Io’, h, and K ’ ) and statistical information which are stored on disk for later use by the plotting programs. Finally, it must be emphasized that the following component-resolved plots are generated from the fitted parameters and thus constitute calculated spectra rather than the observed spectra which are represented in Figures 5,a-c.
The first step in the analysis of any emission spectrum is to decide which of many possible kinetic expressions best describes the decay characteristics without making any fundamental assumptions. A preliminary and qualitative idea of the nature of the decay may be obtained by simple inspection of the time-resolved slices, Figures 5a, 56, and 5c. This set of spectra would seem to indicate that the emission from the mixture was composed of a relatively intense, rapidly decaying component (anthrone) and a less intense, slowly decaying component (benzophenone). A second novel and most useful tool in estimating the number of components present is the lifetime dispersion of the mixture emission as determined using the single component kinetic model (Figure 6). Ideally, this dispersion should be a single horizontal trace for single component emission. The fact that it osciliates with a large amplitude indicates that more than one component is present or that the emission is not a simple sum of exponential terms. In this particular case, a comparison of this dispersion (Figure 6) with the time slices (Figures 5a and 5 c ) is most instructive. The minima in the lifetime dispersion (Figure 6) occur a t 393, 426, 457, and 496 nm and are coincident with the bands in the rapidly emitting species (Figure 5 a ) which occur at 400,425,455, and 492 nm. Furthermore, the maxima in the lifetime dispersion (Figure 6) occur a t 413, 443, 478, and 520 nm and are coincident with the bands of the slowly emitting species (Figure 5 c ) which occur a t 413, 443, 475, and 520 nm. Therefore, an oscillating lifetime dispersion of the form illustrated in Figure 6 indicates that there are twCJ emitting species: a rapidly emitting species with emission maxima that correspond to the minima in the dispersion and a slowly emitting species with emission maxima that correspond to the maxima in the dispersion. While
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these two types of information, the time slice plots and one component lifetime dispersion plot, are very useful in deciding how many decay elements are present, they are only qualitative in nature. Therefore, the next step in the component resolution is to obtain a quantitative measure of the likely decay schemes a t a few selected wavelengths in the spectrum. T o do this, one conducts a x 2 analysis a t the selected wavelengths using the appropriate kinetic models. At present, the kinetic models incorporated into the x 2 program are exponential decay, the sum of two and three exponentials, and competitive first- and second-order decay such as might be expected in triplet-triplet annihilation (33).I t is a trivial matter to incorporate any kinetic expression that might be applicable to the systems under examination. For example, one might analyze the mixture decay at wavelength maximum (428 nm) in terms of a single exponential [I = Io exp ( - h t ) ] or in terms of a sum of two exponentials (Equation 1).Analysis in terms of a single exponential affords a x2 minimum of 1.508 and the residual distribution plot shown in Figure 7a. Analysis in terms of the sum of two exponentials affords a x2 minimum of 0.017 and the residual distribution plot shown in Figure 7b. In general, if the incorporation of an addition exponential term into the kinetic model does not bring about a reduction in x2,based on 100 data points, by an order of magnitude or more, one cannot make a decision between the two kinetic expressions with certainty. Since, in this instance, the reduction in x2 upon going to the two-component model is nearly two orders of magnitude, the decay clearly is better described by a dual emission model. This decision is supported by the residual plots (Figures 7a and b) in which the abcissa is the fitted emission intensity ( I ) and the ordinate is the residual (difference between the observed and fitted intensity). Figure 7a exhibits a systematic distribution of the residuals which signals an improper fit of the decay data using the one-component kinetic model. On the other hand, Figure 76 exhibits a much more random distribution of smaller residuals indicating that the “goodness” of the fit is approaching the noise level of the data. On the basis of this type of quantitative analysis of a few decay curves, one can select the most appropriate kinetic expression and proceed with the analysis of the entire spectrum. In this case, Equation l was chosen.
ANALYTICAL CHEMISTRY, VOL. 4 7 , NO. 2, FEBRUARY 1975
261
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Figure 7. Residual plots for the anthrone-benzophenone mixture generated by ( a )a one-component kinetic model and ( b ) a two-component kinetic model 262
A N A L Y T I C A L CHEMISTRY, VOL. 47, NO. 2, FEBRUARY 1975
A
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(
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Figure 8. Luminescence life history of the anthrone-benzophenone mixture
Figure 8 is a reconstruction of the life history of the emission from the anthrone-benzophenone mixture. One can readily observe the more rapid decay of those bands associated with the anthrone emission. The dispersion of lifetimes ( 7 = l/k) for anthrone and benzopheone as determined from the mixture are illustrated in Figures 9a and 9b, respectively. The mean lifetimes as well as their standard deviations over the entire spectrum range are recorded a t the top of each of these dispersion plots. A x 2 analysis of pure anthrone and pure benzophenone with a single component kinetic model afforded lifetimes of 1.60 f 0.09 and 4.98 f 0.10 msec, respectively. These values which were obtained in 2-methyltetrahydrofuran compare well with Winefordner's values for anthrone, 1.7 msec, and benzophenone, 4.8 msec, recorded in EPA ( 1 0 ) . I t should be noted that the ratio of these lifetimes is about three. The resolving power of this technique is not clear a t this time. If the resolution program is tested with synthetic data, it is possible to resolve components with lifetime ratios approaching 1.0. Clearly, when real data are examined, low noise levels and exact kinetic models are the critical factors essential for the resolution of components with small lifetime ratios. Finally, one should note that while the lifetime dispersions in Figures 9a and b still exhibit some fluctuation above 400 nm, the fluctuations are approaching the noise level of the data. The dip in the lifetimes below 400 nm is indicative of a typical problem encountered in this type of analysis. The same problem is exhibited in a more obvious fashion in the zero time slice of the reconstructed spectrum (Figure 10). In Figure 10, the most intense trace is the total emission spectrum of the mixture and the two traces of lower intensity represent the contributions to the total emission due to anthrone (middle trace) and benzophenone (lower trace). Below 400 nm, the component spectra are very erratic; whereas, above 400 nm, these spectra are quite good representations of the emission from the two pure substances. The reason for the dispersion dip and the erratic spectra is simply that benzophenone does not emit below
about 400 nm. Thus, below 400 nm the emission consists of one component and above this wavelength the emission consists of two components; however, the computer has been instructed to analyze the entire spectrum using only a two-component kinetic model. When analyzing a true single exponential as two components, the x 2 routine will divide the total intensity between two equal rate constants. In this case, the ratio of the two rate constants drops from about 3.4 over the enitire spectrum to about 1.7 below 400 nm. Therefore, the two rate constants do tend to converge below 400 nm, but apparently the decay of anthrone deviates slightly from that of a single exponential, so the rate constants never completely converge. This same type of behavior can be observed throughout the entire spectrum of pure anthrone. The source of this difficulty is most clearly indicated by the behavior of x2.Table I1 shows the values for x 2 a t a few wavelengths in the erratic region. Below 400 nm, a single emitting species describes the emission characteristics about as well as does two emitting species. This is indicated Table 11. x 2 Values for t h e Anthrone-Benzophenone Mixture Emission Fitted Using One- a n d Two-Component Kinetic Models Wavelength,
nm
Model l ( x t )
385 389 393 397 400 405 409 413 417 42 1 425
0.0032 0.0064 0.0434 0.0923 0.1627 0.5351 0.9617 1.2297 1.6996 1.9273 1.9262
Model 2(X
z2)
0.0013 0.0035 0.0119 0.0099 0.0132 0.0082 0.005 1 0.0079 0.0091
0.0080 0.0161
x:/xz2
2.5 1.8 3.6 9.2 12.3 65.3 188.6 155.7 186.8 240.9 119.6
A N A L Y T I C A L C H E M I S T R Y , VOL. 47, NO. 2 , F E B R U A R Y 1975
263
1.50
L I F E T I M E IMSECI
10.10
a
,'
WRVELENCTH
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ubo.00
uio.00
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L I F E TIME
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b
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WRVELENCTH INMI
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RNTHRdNE-BENZJPHENONE 2RTHF rlE I21
Figure 9. Lifetime dispersions for the anthrone-benzophenone mixture generated by a two-component kinetic model; ( a ) dispersion of anthrone, ( b )dispersion of benzophenone
by the fact that the addition of a second emitting species does not lead to a significant (order of magnitude) improvement in x2. Above 400 nm, this is no longer true. In this region, the addition of a second emitting species leads to an improvement in x 2 that levels off a t well over two orders of magnitude. Since the difficulty is clearly due to a hybrid single and dual component spectrum in which the great majority of the spectrum is two components, the problem is easily overcome. To do this, one simply performs a second x' analysis using the original data set and lifetimes that have been fixed a t the average values determined in the first analysis. (One might consider analyzing the region below 400 nm using a one-component model and the region above 400 nm using a two-component model. However, even when examining a known mixture such as described here, it is not clear a t which wavelength the second component becomes significant. The net result is that it is extremely tedious t o mate the results of the two analyses without introducing a discontinuity a t the interface. Such an operation would be even more difficult and of questionable validity when dealing with an unknown mixture.) 264
INMl
I21
Figure 10. Time slice of the anthrone-benzophenone mixture at 0 msec generated with variable rate constants
0
aso.oo
R N T H R O N E - B E N Z O P ~ E N O N E fMTHF
Figure 11,a-c are time slices reconstructed from the parameters calculated in the second x* analysis. In this series of time slices, the erratic behavior below 400 nm has been completely eliminated and the benzophenone intensity drops to zero a t about 400 nm as expected. The time slices generated from the original data (Figures 5 4 - c ) were recorded a t the same times as these reconstructed slices (Figure l l p c ) , and therefore, may be compared with each other. (The time recorded a t the top of Figure 5,a-c is the real time; whereas, the time recorded a t the top of Figure 11,a-c is time relative to the beginning of the x 2 analysis of the data.) Simple comparison of the spectra in Figure 5,a-c with the corresponding reconstrdcted spectra in Figure 11,a-c indicates that the fitted spectra are an accurate representation of the time-resolved spectra. A quantitative comparison of these two sets of spectra is provided in Table 111. These data indicate that the fitted time slices are most accurate a t the intermediate decay times (3.8 msec), and are least accurate a t the later decay times (7.3 msec). In our somewhat limited experience, we have observed that the most sensitive test of the validity of the kinetic model used in the x2 analysis is such a comparison of the time slices taken late in the decay. Alternatively, one can evaluate the goodness of fit over the entire decay time by comparing the integrated spectrum obtained from the original data set (Figure 12a ) with the integrated spectrum generated from the two component kinetic model incorporating the calculated parameters (Figure 12b ). Table 111. Comparison of Intensities of the Time-Resolved Slices with Fitted Slices Time, I,nrn
A N A L Y T I C A L CHEMISTRY, VOL. 47, NO. 2, FEBRUARY 1975
401 401 401 427 427 427 441 441 441 454 454 454
msec
Cbsd I
Fitted I
1.0
6.030 1.030 0.166 9.107 1.797 0.4 10 5.020 1.734 0.704 6.303 1.542 0.492
5.974 1.031 0.160 8.964 1.785 0.398 5.005 1.728 0.703 6.24 1 1.555 0.482
3.8 7.3 1.0 3.8
7.3 1.0 3.8 7.3 1.0 3.8 7.3
Residual
0.056 - 0.001
0.006 0.143 0.012 0.012 0.015 0.006 0.001 0.062 - 0.013 0,010
11,
Error1
0.928 0.097 3.61 1.57 0.667 2.92 0.299 0.346 1.42 0.984 0.843 2.03
MSEC
T I M E OFF 1 8 . 0
a
MSEC
a
J
8' 420.00
RNTHRONE-BENZOPHENONE 2MTHF
1.0
TIME O N
T I M E (MSECIO
i2FKl
360.00
WQVELENGTH UU0.00 ubO.00I N H IS 2 0 . 0 0
