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A Virtual Curve-Fitting Instrument for Interactive Analysis of Excimer Dynamics Giles Henderson Department of Chemistry, Eastern Illinois University, Charleston, IL 61920;
[email protected] Birks and Christophorou reported the unusual concentration dependence of the luminescence of electronically excited pyrene in 1963 (1). They found that 10{6 M pyrene solutions irradiated with UV light exhibit simple fluorescence emission. However, at higher concentrations the spectra display additional longer wavelength features assigned to pyrene excimers. An excimer is a dimer composed of an electronically excited molecule complexed with a ground-state molecule. Excimer formation is possible only at concentrations sufficiently high that an encounter of these species can occur within the excited molecule’s radiative lifetime. Birks subsequently characterized the dynamics of excimer formation and decay from time-resolved emission studies of pyrene solutions (2). More recently, Muenter and Deutsch (3) have developed an undergraduate laboratory experiment in which students use nanosecond pulses from an inexpensive nitrogen laser to generate excited-state pyrene molecules at various concentrations in cyclohexane. Fast kinetics data for this system are obtained at various wavelengths with a simple monochromator or bandpass filters, using a photomultiplier (PMT) detector and a digital oscilloscope to record time resolved luminescence curves. Rate constants for the various reaction channels of the pyrene system can in principle be extracted from these data. However, the analysis is somewhat complicated because the differential equations describing the time evolution of excited pyrene monomers and excimers are coupled (2). This paper describes the use of National Instrument’s LabVIEW software (4) to create a virtual instrument (VI) panel that allows the user to directly compare and optimize the fit of calculated monomer and excimer concentration profiles with experimental data. Each theoretical parameter is assigned to a panel control that can be independently adjusted while its influence on the calculated waveforms is observed. Other examples of LabVIEW applications in the curriculum have been described (5–8). Theory
d[P*] = k DM[PD*] – k M[P*] – k MD[P][P*] dt
(4)
d[PD*] = k MD[P][P*] – k D[PD*] – k DM[PD*] dt
(5)
Muenter and Deutsch have suggested using a numerical method to solve these coupled differential equations (3). They suggest replacing the differential change in concentrations with finite differences evaluated at corresponding small time increments, δt : [P*]n+1 = [P*]n + (kDM [PD*]n – k M [P*]n – k MD[P][P*] n)δt (6) [PD*]n+1 = [PD*]n + (kMD[P][P*] n – k D[PD*]n – k DM[PD*]n)δ t (7)
In the perfect world, the measured time-resolved emission profiles of P* and PD* are proportional to the their respective concentration curves, as described by eqs 6 and 7. In practice, these emission signals are subject to significant distortions by the detector system (9). Simple computer codes can be written to evaluate eqs 6 and 7 and convolute the corresponding theoretical intensity profiles with instrument response characteristics: t
Muenter and Deutsch’s experiment uses a laser pulse, which is short compared to pyrene’s excited state lifetime, to prepare an initial concentration of excited pyrene molecules, P* (3). The radiative decay of P* is then measured with a monochromator or filter to select a fluorescence wavelength, a PMT detector, and a digital oscilloscope. Additional measurements at an emission wavelength of the excimer, PD*, reveal the profile of a transient reactive intermediate in which there is an onset followed by a decay in PD* concentration and emission intensity. The dynamics of this system are characterized by the following reactions: P* k→ P + hν1 M
P* + P
kMD kDM
PD*
PD* → PD + h ν2 kD
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Equation 1 describes simple monomer fluorescence emission at ν1 characterized by a first-order rate constant, kM. Equation 2 describes the reversible bimolecular excimer formation in which kMD and kDM are the respective formation and nonradiative dissociation rate constants. The decay of the excimer is described by eq 3 and is characterized by the firstorder rate constant kD. Although kM and kD contain both radiative and nonradiative contributions (3), these components are not separated or distinguished in this experiment. Rate equations for the excited state monomer and excimer can be written in accord with the reactions above:
(1) (2) (3)
Dt = Σ Sx Rt –x x =0
(8)
where D(t) is the instrumentally distorted signal, S(t) is the true signal and R(t) is the instrument’s measured response function. Nonlinear least squares (10) or simplex (11) algorithms can be used to determine the four rate constants that best fit the convoluted theoretical luminescence profiles to the observed data. However, it is perhaps pedagogically more beneficial to fit these curves with an interactive computer graphics tool. Excimer Dynamics Analysis Control Panel A custom LabVIEW program has been developed, which uses eqs 6 and 7 to calculate the monomer and excimer intensity profiles for assumed values of the unknown rate constants. The VI control panel (Fig. 1) has a convolution switch,
Journal of Chemical Education • Vol. 76 No. 6 June 1999 • JChemEd.chem.wisc.edu
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Figure 1. A virtual instrument panel designed for interactive curve fitting. All the panel controls can be operated with a computer mouse. The convolution switch enables the convolution of theoretical calculated curves with instrument response functions and also enables the variance of fit meter. The curves and legend are color coded when displayed on a VGA computer monitor. In this example the curves are identified as (a) experimental excited pyrene monomer concentration; (b) experimental excimer concentration; (c) calculated excited pyrene monomer concentration; (d) calculated excimer concentration.
Figure 2. When the convolution function is enabled, the calculated signals are subjected to the same instrumental distortions as the measured experimental signals. A computer mouse can be used to interactively adjust the theoretical parameter values while observing their effects on the calculated curves displayed on the waveform chart. The quality of the fit is measured by a variance of convoluted fit meter.
which enables eq 8 and subjects the calculated intensity profiles to the same instrumental distortions as those experienced by the experimental profiles. The VI simultaneously displays the calculated and experimental curves and allows the operator to interactively adjust the values of the unknown rate constants until the calculated curves best fit the observed curves as measured by a variance of fit meter. A δ t control allows the user to specify the time increment between data points (in nanoseconds). Our students use the method described by Muenter and Deutsch (3) to collect 501 data points at 0.5-nanosecond intervals for each data set. The dt control is then set at 0.5 ns so that the experimental and calculated waveforms are digitized at the same interval. The Phase 1 and Phase 2 controls allow the user to adjust the origin of the time axis of the calculated monomer and excimer profiles to properly align them with the respective experimental profiles. The waveform chart in Figure 1 compares experimental monomer and excimer profiles with corresponding calculated signals, S(t) in which no convolution of instrumental distortion has been included. The same parameters are used in Figure 2, but the convolution switch has been enabled so that the calculated true signals, S(t), have been convoluted with the instrument response function R(t) in accord with eq 8 to give the calculated instrumentally distorted signal, D(t). The fit is optimized by adjusting the parameters to obtain a minimum variance.
virtual control panel describe above is depicted in Figure 3. This diagram serves as the source code and is a pictorial algorithm for the solution to eqs 6–8. This top-level program or VI contains subprograms called subVIs. Data and parameters are transferred to and from subVIs and other operation modules (icons) through connecting wires. The top-level VI (Fig. 3) contains a subVI read data, which reads three experimental time-resolved emission data files from a user-specified path: (i) the excited pyrene monomer; (ii) the pyrene excimer; and (iii) the scattered laser light from an empty cell used to characterize the detector response function, R(t). The amplitudes of the monomer and excimer waveforms are then normalized before being displayed on the waveform chart.
LabVIEW Program (G-code) LabVIEW software uses a graphical programming language, G, to create programs in block diagram form. The block diagram for the
Figure 3. LabVIEW G code block diagram of the top-level virtual instrument panel shown in Figures 1 and 2.
JChemEd.chem.wisc.edu • Vol. 76 No. 6 June 1999 • Journal of Chemical Education
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Values for the four rate constants are defined by their respective controls and flow through connecting wires to a subVI, DIF EQNS. The subVI’s block diagram, Figure 4, solves eqs 6 and 7 inside an i = 0 to N = 500 FOR LOOP. Convolution of these profiles with the detector response function is determined by the status of the convolution switch. When the boolean variable is TRUE, the calculated profiles are convoluted with the instrument response function in accord with eq 8 by the convolution subVI, Figure 5. The del^2/n subVI calculates the variance between a calculated and experimental curve. The sums of the monomer and excimer variance values are then added and transferred to the variance of fit indicator. Results After closely examining the role of the rate constants defined in eqs 1–3, students are able to make qualitative predictions of how a change in their values will affect [P*] and [PD*]. In some instances, the results are in conflict with their intuition as a consequence of the coupled concentration dependence. The analog nature of this virtual instrument provides an opportunity to interactively explore the behavior of the coupled kinetics system. The use of this tool provides the student with an insight that is lacking in traditional curvefitting algorithms. The scale limits of each control can be changed by placing the operating tool on the scale and typing a new minimum or maximum value into the display. However, the default scale limits have been selected to assure reasonable initial values for each rate constant. Students are then able to obtain reasonable results without guidance. Typical student optimized values of the rate constants for a 0.01 molar solution of pyrene in cyclohexane can be seen in FigTable 1. Typical Values for ure 2 and are compared Optimized Rate Constants with literature values in Value Rate Table 1. Constant Student Literature (3) At the suggestion of kM 4.9 × 106 2.1 × 106 a reviewer, the software is kD 2.1 × 107 2.3 × 107 capable of operating in a 6 mode where only the calk DM 3.5 × 10 2.5 × 106 culated curves are disk MD 7.9 × 109 5.0 × 109 played. This option allows students to preview the behavior of the system before performing the experiment. In addition, this mode could be effectively used by students even when the experiment is not performed and provides a good example of the effects of coupled second-order rate equations. Distribution The virtual curve fitting instrument software requires an 80386 or 80486-based PC with coprocessor, Windows 3.1 or higher operating system, and a minimum of 8 MB of memory. Upon request, I will provide an executable runtime program and samples data sets at no cost to readers who provide a self-addressed envelope capable of holding a 31/2-in. disk. Acknowledgments I wish to express my gratitude to the National Science Foundation, Instrumentation and Laboratory Improvement
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Figure 4. LabVIEW G code block diagram of the subVI DIF EQNS. This algorithm caries out a numerical solution of the coupled differential rate eqs 6 and 7.
Figure 5. LabVIEW G code block diagram of the subVI CONVOLUTE. This algorithm caries out a numerical convolution of the a theoretical time-resolved luminescence profile with an instrument response function as described by eq 8.
Grant # DUE-9551644, for supporting the development of laser spectroscopy in our undergraduate curriculum and to Mark McGuire, Douglas Klarup, and John Muenter for their comments and suggestions. Literature Cited 1. Birks, J. B.; Christophorou, L. G. Spectrochim. Acta 1963, 19, 401–410. 2. Birks, J. B. Photophysics of Aromatic Molecules; Wiley Interscience: London, 1970. 3. Muenter, J. S.; Deutsch, J. L. J. Chem. Educ. 1996, 73, 580–585. 4. LabVIEW for Windows; National Instruments Corp., 6504 Bridge Point Parkway, Austin, TX 78730-5039. 5. Gostowski, R. J. Chem. Educ. 1996, 73, 1103–1107. 6. Drew, S. M. J. Chem. Educ. 1996, 73, 1107–1111. 7. Muyskens, M. A.; Glass, S. V.; Wietsma, T. W.; Gray, T. M. J. Chem. Educ. 1996, 73, 1112–1114. 8. Ogren, P. J.; Jones, T. P. J. Chem. Educ. 1996, 73, 1115–1116. 9. Demas, J. N. Excited State Lifetime Measurements; Academic: New York, 1983. 10. Kim, H. J. Chem. Educ. 1970, 47, 120–122. 11. Hecht, H. G. Mathematics in Chemistry: An Introduction to Modern Methods; Prentice Hall: Englewood Cliffs, NJ, 1990; pp 259–263.
Journal of Chemical Education • Vol. 76 No. 6 June 1999 • JChemEd.chem.wisc.edu
