Analysis of Heterogeneous Fluorescence Decays. Distribution of

Anal. Chem. 2001, 73, 5564-5575

Analysis of Heterogeneous Fluorescence Decays. Distribution of Pyrene Derivatives in an Octadecylsilane Layer in Capillary Electrochromatography Yan He and Lei Geng*

Department of Chemistry, University of Iowa, Iowa City, Iowa 52242

The distribution of solute molecules in the stationary phase in capillary electrochromatography (CEC) has been investigated with time-resolved fluorescence in the frequency domain. The analysis of fluorescence decay poses a challenging problem for the complex decay kinetics of heterogeneous systems such as the C18 stationary phase. The nonlinear least-squares (NLLS) method selects the decay model by minimizing the χ2 value. The χ2 criterion, in conjunction with the requirement that the residues should be randomly distributed around zero, frequently leads to a feasible set of multiple decay models that can all fit the data satisfactorily. The maximum entropy method (MEM) further chooses a unique model from the group of feasible ones by maximizing the Shannon-Jaynes entropy. The unique model, however, is not necessarily the most probable one. In this paper, the best model for the fluorescence decays of solute molecules is selected with NLLS using the χ2 statistics, the stability of the fit, and the consistency within replicate experiments. In addition, the recovered lifetime parameters of the true model should display the same trend as the fluorescence decay profiles when an experimental condition is varied. Using these criteria, a Gaussian distribution of fluorescence lifetimes satisfactorily fits the data under all experimental conditions. An additional minor component with a discrete lifetime is attributed to the systematic errors in the measurements. The distribution is a manifestation of an ensemble of heterogeneous microenvironments in the stationary phase of CEC. MEM is not suitable for the modeling of CEC data because of its inaccuracy in recovering broad fluorescence lifetime distributions and its lack of consistency in the replicate measurements in the studies of high-voltage effects. Capillary electrochromatography (CEC) is a hybrid separation technique that combines the advantages of both HPLC and CE.1-4 It is gaining popularity rapidly due to its potential to achieve chemical separation with high speed, high efficiency, and excellent * Corresponding author: (phone) (319)335-3167; (fax) (319)335-1270; (e-mail) [email protected]. (1) Colo´n, L. A.; Burgos, G.; Maloney, T. D.; Cintro´n, J. M.; Rodrı´guez, R. L. Electrophoresis 2000, 21, 3965-3993. (2) Beale, S. C. Anal. Chem. 1998, 70, 279R-300R. (3) Altria, K. D. J. Chromatogr., A 1999, 856, 443-463. (4) Dermaux, A.; Sandra, P. Electrophoresis 1999, 20, 3027-3065.

5564 Analytical Chemistry, Vol. 73, No. 22, November 15, 2001

selectivity, coupled with low sample loading, low solvent consumption, and increased mass sensitivity. To optimize the separation conditions, a detailed understanding of the CEC system, especially the mechanism of solute-stationary phase interaction under high electric field, is required. In situ investigations of stationary phases under pressure-driven conditions in HPLC have been reported using steady-state fluorescence spectroscopy5 and Raman spectroscopy.6 Fluorescence spectroscopy of both covalently attached and physiosorbed probes has revealed the organization, polarity, and microviscosity of the interface in HPLC, as well as the kinetics of solute-stationary phase interactions.7-22 In this work, we use time-resolved fluorescence spectroscopy in the frequency domain to probe the C18 phase microenvironments under separation conditions in CEC. On-the-fly fluorescence lifetime detection in both time and frequency domain has been actively pursued in CE and HPLC for its ability of species identification.23-27 In these applications, the time-resolved fluorescence module is used only as an analyte-specific detector. The rich dynamic information (5) McGuffin, V. L.; Chen, S.-H. Anal. Chem. 1997, 69, 930-943. (6) Doyle, C. A.; Vickers, T. J.; Mann, C. K.; Dorsey, J. G. J. Chromatogr., A 1997, 779, 91-112. (7) Rutan, S. C.; Harris, J. M. J. Chromatogr., A 1993, 656, 197-215. (8) Lochmuller, C. H.; Colborn, A. S.; Hunnicutt, M. L.; Harris, J. M. Anal. Chem. 1983, 55, 1344-1348. (9) Lochmuller, C. H.; Colborn, A. S.; Hunnicutt, M. L.; Harris, J. M. J. Am. Chem. Soc. 1984, 106, 4077-4082. (10) Hansen, R. L.; Harris, J. M. Anal. Chem. 1998, 70, 4247-4256. (11) Wirth, M. J.; Ludes, M. D.; Swinton, D. J. Anal. Chem. 1999, 71, 39113917. (12) Wang, H.; Harris, J. M. J. Am. Chem. Soc. 1994, 116, 5754-5761. (13) Wang, H.; Harris, J. M. J. Phys. Chem. 1995, 99, 16999-17009. (14) Wirth, M. J.; Swinton, D. J. Anal. Chem. 1998, 70, 5264-5271. (15) Swinton, D. J.; Wirth, M. J. Anal. Chem. 2000, 72, 3725-3730. (16) Burns, J. W.; Bialkowski, S. E.; Marshall, D. B. Anal. Chem. 1997, 69, 38613870. (17) Carr, J. W.; Harris, J. M. Anal. Chem. 1987, 59, 2546-2550. (18) Carr, J. W.; Harris, J. M. Anal. Chem. 1986, 58, 626-631. (19) Wong, A. L.; Harris, J. M. Anal. Chem. 1991, 63, 1076-1081. (20) Bogar, R. G.; Thomas, J. C.; Callis, J. B. Anal. Chem. 1984, 56, 10801084. (21) Ståhlberg, J.; Almgren, M.; Alsins, J. Anal. Chem. 1988, 60, 2487-2493. (22) Ståhlberg, J.; Almgren, M. Anal. Chem. 1985, 57, 817-821. (23) Desilets, D. J.; Kissinger, P. T.; Lytle, F. E. Anal. Chem. 1987, 59, 18301834. (24) Li, L.-C.; He, H.; Nunnally, B. K.; McGown, L. B. J. Chromatogr., B 1997, 695, 85-92. (25) Soper, S. A.; Legendre, B. L., Jr.; Willams, D. C. Anal. Chem. 1995, 67, 4358-4365. (26) He, Y.; Geng, L. Anal. Chem. 2001, 73, 943-950. (27) Wang, G.; Geng, L. Anal. Chem. 2000, 72, 4531-4542. 10.1021/ac010293u CCC: $20.00

© 2001 American Chemical Society Published on Web 10/09/2001

contained in the time-resolved fluorescence data has yet been utilized. A major difficulty associated with solute-stationary phase interaction study using time-resolved fluorescence is the analysis of the fluorescence decay or the recovery of the correct model that describes the fluorescence relaxation.28-40 Because of the heterogeneity of both the silica substrate and the structure of the bonded ligands, as well as the effect of experimental factors, fluorescence decays observed were seldom monoexponential for these systems. Standard decay analysis utilizes the nonlinear leastsquares (NLLS) method to fit the data, where a priori selection of a model composed of multiple discrete or distributional components is required. The goodness of fit is justified by the χ2 criteria, but ambiguity often occurs because different models may result in similar χ2 values. One other approach is the maximum entropy method (MEM), which does not require a preselected model and recover a unique lifetime distribution.33,37-39,41 The entropy maximization selects a single solution from the group of equivalent models allowed by the χ2 criterion. However, the MEM solution is not necessarily the most probable one. To justify solutions from different methods of decay analysis, other criteria such as the physical significance and data consistency need to be considered in addition to data statistics. Fluorescence lifetime recovery is even more complicated for kinetic measurements in separation. It was found that exponential analysis is strongly affected by the signal-to-noise ratio (SNR) of the input decay, which is mainly determined by the sensitivity of the equipment.28 For measurements in static systems, SNR can be improved by a large number of averaging. In chemical separations, however, SNR is limited by low sample loading mass, small detection volume, and short solute transient time as well as electronic and optical noises of the instrument. Furthermore, the number and range of data points in the decay also affect the stability of lifetime analysis. One of the most widely used fluorescence probe for chromatographic surfaces is pyrene, whose high fluorescence quantum yield, relatively long fluorescence lifetime, the sensitivity of its monomer emission to its local microenvironment, and the excimer formation make it well suited for such studies.7-9,12,13,16-22,33-36 The fluorescence lifetimes of pyrene adsorbed on or covalently attached to the silica surface and other solid surfaces have been studied extensively. Liu and Ware showed that a bimodal distribution is the best description of the lifetime of pyrene adsorbed on a silica gel surface.33,34 The two distributions represented the association of pyrene molecules with hydrogen-bonded and (28) Istratov, A. A.; Vyvenko, O. F. Rev. Sci. Instrum. 1999, 70, 1233-1257. (29) Draxler, S.; Lippitsch, M. E. Anal. Chem. 1996, 68, 753-757. (30) Neal. S. J. Phys. Chem. 1997, 101, 6883-6889. (31) Neal. S. J. Phys. Chem. 1997, 101, 6890-6896. (32) Krasnansky, R.; Koike, K.; Thomas, J. K. J. Phys. Chem. 1990, 94, 45214528. (33) Liu, Y. S.; Ware, W. R. J. Phys. Chem. 1993, 97, 5980-5986. (34) Liu, Y. S.; Ware, W. R. J. Phys. Chem. 1993, 97, 5987-5994. (35) Liu, Y. S.; Ware, W. R. J. Phys. Chem. 1993, 97, 5995-6001. (36) Bonzagni, N. J.; Baker, G. A.; Pandey, S.; Niemeyer, E. D.; Bright, F. V. J. Sol-Gel Sci. Technol. 2000, 17, 83-90. (37) Brochon, J.-C.; Livesey, A. K.; Pouget, J.; Valeur, B. Chem. Phys. Lett. 1990, 174, 517-522. (38) Shaver, J. M.; McGown, L. B. Anal. Chem. 1996, 68, 9-17. (39) Shaver, J. M.; McGown, L. B. Anal. Chem. 1996, 68, 611-620. (40) Lakowicz, J. R. Principles of Fluorescence Spectroscopy; Kluwer/Plenum: New York, 1999. (41) Geng, L. Ph.D. Dissertation, Duke University, 1994.

isolated surface silanol groups, respectively. Wang and Harris studied pyrene covalently attached to a silica gel surface and found that its fluorescence decay follows a four-exponential model.13 The two longer components reveal the two distinguishable silanol sites with different activities on the silica surface, and the two shorter components are attributed to background fluorescence. Bonzagni et al. showed that the fluorescence decay of pyrene sequestered within xerogels is best described with a four-exponential model.36 Two of the components represent two different pyrene microenvironments within the xerogel, and the other two reflect the background fluorescence. We describe an approach to studying solute-stationary phase interactions in CEC based on kinetic fluorescence lifetime measurement and analysis in frequency domain. The fluorescence probes are pyrene and its derivatives. In this paper, attention is focused on kinetic lifetime analysis under CEC conditions only. Using both real and simulated data, method comparison between NLLS and MEM and discrimination between discrete and distributional models are discussed in detail. The experimental setup, sensitivity improvement of the instrument, and effect of separation conditions in CEC will be presented in the future.42 METHODOLOGY Fluorescence Lifetime Measurement. In frequency domain fluorescence lifetime measurement, when a fluorescent sample is illuminated with sinusoidally modulated light at an angular modulation frequency ω, the florescence emission is also sinusoidally modulated at the same frequency but is phase-shifted and demodulated relative to the excitation beam. For a complex system, which contains multiple discrete noninteracting lifetime components, the observed fluorescence is a sum of exponential decays. The phase, φω, and the demodulation value, mω, are functions of the angular modulation frequency, ω, the individual lifetimes, τi, and their fractional intensity contributions, fi:

φω ) tan-1(Nω/Dω)

(1)

mω ) (Nω2 + Dω2)1/2

(2)

where

Nω )

∑

fiωτi

∑

fi

i

Dω )

i

1 + ω2τi2

1 + ω2τi2

(3)

(4)

Measurements at many different frequencies are necessary to construct a phase and modulation response curve in order to resolve the fluorescence lifetime profile. If background correction is necessary, the corrected values of Nω and Dω are given by40

Nω )

mω,obs sin φω,obs - fBmω,B sin φω,B 1 - fB

(5)

(42) He, Y.; Geng, L., manuscript in preparation.

Analytical Chemistry, Vol. 73, No. 22, November 15, 2001

5565

Dω )

mω,obs cos φω,obs - fBmω,B cos φω,B 1 - fB

(6)

where subscripts obs and B denote the total observed values and the background values, respectively, and fB is the faction of the total signal due to the background. The corrected φω and mω can then be calculated by substituting eqs 5 and 6 into eqs 1 and 2. In the presence of a continuous distribution of fluorescence lifetimes, the summations in eqs 3 and 4 are replaced with integrations.

Nω )

∫

Dω )

∫

∞

f(τ)ωτ 1 + ω2τ2

0

f(τ)

∞

1 + ω2τ2

0

dτ

(7)

dτ

(8)

the model chosen must be physically reasonable; (2) among the same type of models with similar χ2 and residuals, the simplest one with the fewest number of adjusting parameters should be chosen; (3) the solutions should be reproducible under the same experimental conditions; and (4) the solutions should be consistent within a series of experiments, for example, when an experimental parameter is varied. An alternate approach is the maximum entropy method, which is a self-modeling technique without using a preselected model. In MEM, a broad lifetime window that covers the lifetime range of interest is evenly divided into closely spaced discrete exponential terms in linear or log τ domain. The fitting process starts with a flat distribution and then varies the fractional intensity of each term to minimize the χ2. Once a feasible set of models is obtained after χ2 reaches the minimum, a unique solution is chosen by maximizing the Shannon-Jaynes entropy,

S)The fluorescence lifetime distribution can take any preselected functional forms. The most generally used ones are Gaussian

fG(τ) )

1

e-(τ - τc) /2σ 2

2

x2πσ2

(9)

where τc is the center of the fluorescence lifetime distribution and σ is the standard deviation of the Gaussian and the Lorentzian distribution

fL(τ) )

Γ 1 2π (τ - τ )2 + (Γ/2)2 c

(10)

in which Γ is the full width at the half-maximum (fwhm) of the distribution. The width (fwhm) of a Gaussian distribution is related to its standard deviation by w ) 2(2 ln 2)1/2σ. Fluorescence Lifetime Analysis. Given an a priori discrete or distributional model of the system, NLLS analysis based on minimization of reduced χ2 can be used to fit the multifrequency data: 2

χ )

∑ ω

(

(φω - φcω)2 σφ,ω2

+

)

(mω - mcω)2 σm,ω2

/(2N - M + 1) (11)

where φcω and mcω are the calculated phase shift and demodulation value; σφ,ω and σm,ω are the standard derivations of measured phase and modulation responses, respectively; N is the number of frequencies; and M is the number of unknown parameters in the decay model, of which M - 1 are independent. If the model chosen is consistent with the fluorescence decay, and the σφ,ω and σm,ω values truly reflect the experimental variations, χ2 should be close to unity and the residuals in both phase and demodulation should oscillate randomly around zero.43 Unfortunately, for many systems, there may be a number of models that produce χ2 values within the acceptable range and it is difficult to reject any of them statistically. In those situations, criteria other than χ2 and residuals need to be considered in model discrimination. Some general considerations are as follows: (1) 5566

Analytical Chemistry, Vol. 73, No. 22, November 15, 2001

∑p log p i

i

(12)

i

where pi is the factional intensity of term i. Shannon-Jaynes entropy, in contrast to other definitions of entropy, does not introduce any correlation into the analysis that is not inherent in the data. The lifetime distribution recovered by MEM is unbiased by any a priori model that may be oversimplified. However, the unique solution is not necessarily the most probable one. It has been reported recently that the frequency range, noise, and other factors can affect MEM analysis.37-39,41 Therefore, other criteria are still required to justify the validity of the MEM solution. EXPERIMENTAL SECTION Materials. Pyrene and 1-pyrenebutanol were purchased from Aldrich (Milwaukee, WI) and used as received. A 5-µm Spherisorb ODS2 bonded phase was obtained from Phenomenex (Torrance, CA). This monomeric stationary phase was end-capped and had a carbon load of 12%, an average pore size of 80 Å, a surface area of 200 m2/g, and a bonded phase coverage of 2.72 µmol/m2. A 75% acetonitrile and 25% 4 mM Na2B4O7 buffer at pH 8.5 was used throughout the experiments. The buffer solution was prepared in 18 MΩ‚cm distilled and deionized water purified with a Milli-Q system (Millipore, Bedford, MA). Before injection into the CEC column, all solutions were degassed by sonicating for 10 min in an ultrasonicator. An additional objective of sonication was to break up any possible microcrystals of pyrene that could complicate the photophysics of the system and the mechanistic interpretation of CEC separation. Chromatographic Measurements. The capacity factor of pyrene was measured with a chromatographic column packed with ODS2 stationary phase. t0 was determined with potassium nitrate, which was not retained on this column. The HPLC system consisted of a Waters 600E multisolvent delivery system (Millipore, Milford, MA), a UV-visible absorption detector (Waters 484 tunable absorbance detector, Millipore) operated at 355 nm, and a chart recorder (HP 3396A integrator). The capacity factor of pyrene was calculated from three replicate injections to be 6.26. The relative standard deviation in the capacity factor was 0.3%. (43) Bevington, P. R.; Robinson, D. K. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1999.

Fluorescence Measurements in CEC. Instrumentation and experimental details are discussed elsewhere.42 Briefly, a column was prepared by packing a short quartz cell (400-µm i.d.) with 5-µm ODS2 microbeads and then sealing the cell with two extension capillaries containing premade frits. Then the column was inserted into a CEC interface constructed in-house that was coupled to the sample chamber of an SLM 48000 multiharmonic Fourier transform (MHF) multifrequency phase-modulation spectrofluorometer (Jobin Yvon, Edison, NJ). The fluorophore solution prepared in the mobile phase was driven through the cell packed with ODS-2 beads continuously under high voltage and was excited by a 325-nm laser beam focused onto the packing, for the in situ measurements of fluorescence kinetics. The solute concentration was 10 µM in all fluorescence experiments. At this fluorophore concentration, it was ensured that probe-probe interaction did not occur; the solute molecules stayed in the monomer form, without forming ground-state or excited-state dimers. Fluorescence signal from the solutes was collected through two 345-nm long-pass filters. Scattering signal from the beads was selected with a combination of 325-nm band-pass filter and neutral density filters to serve as the lifetime reference. The use of the scattering from the sample as the lifetime reference reduces the effects of photon migration in silica beads, a highly scattering medium.44 The fluorescence instrument was operated in kinetic MHF mode. The base frequency was 4.1 MHz, and the correlation frequency was 7 Hz. At each high voltage, the reference data were an average of 14.3 s and the fluorescence data were collected for 300 s. In fluorescence-quenching studies, solutions containing both the quencher potassium iodide and the fluorescent probe were driven through the column under high voltage. The fluorescence emission spectra were taken at a bandwidth of 2 nm. When a new solution at a different quencher concentration was introduced into the column, 200 column volumes were pumped through the cell with a syringe pump first. A high voltage was then applied across the column, and the fluorescence emission was monitored to ensure the system had reached equilibrium before the fluorescence spectra were measured. The average of five scans at each quencher concentration was used in the analysis. Data Analysis. The intensity profile, the time domain decay, and the frequency domain phase and modulation profiles were extracted from the data using several in-house programs written in C++. The decay profiles were analyzed with either the nonlinear least squares or the maximum entropy method. The NLLS program was obtained from Globals Unlimited (Urbana, IL). The MEM program was purchased from Maximum Entropy Data Consultant Ltd. (Cambridge, U.K.). In simulation studies, programs were written in MatLab (Mathworks Inc., Natick, MA) to generate phase and modulation data for models containing both discrete and continuous distribution of fluorescence lifetimes. Noises were created with a random number generator in MatLab and added to the calculated phase and modulation values. In NLLS analysis, various initial guesses of parameters were used to run the χ2 minimization repeatedly. This was done to ensure that the global minimum of the χ2 surface was found for the best fitting parameters. This procedure was essential for complex models, (44) Hutchinson, C. L.; Troy, T. L.; Sevick-Muraca, E. M. Appl. Opt. 1996, 35, 2325-2332.

Table 1. NLLS Fits of PY300 with Different Models modela

τ1 (ns)b,c

D DD DDD DDDD DDDDF G GD GDF GDD GDDF L LD

24.2 43.3 55.0 64.0 77.4 32.9 43.7 39.6 48.5 42.6 25.3 40.5

τ2 (ns) 3.0 9.9 20.4 31.6Fe 3.1 31.6F 5.9 31.6F 2.6

τ3 τ4 σ (ns) (ns) (ns)c

R1d

1.7 4.9 6.7

0.91 0.79 0.65 0.45

1.8 2.7

1.2 1.5 44.3 35.7 27.5 30.2 26.7 51.9 33.4

0.97 1.41 0.94 1.20 0.98

R2

R3

χ2

5759.21 124.75 0.17 4.32 0.25 0.08 0.99 0.41 0.11 1.32 33.99 0.03 1.25 -0.41 15.59 0.04 0.89 -0.22 0.02 0.95 347.88 28.59

a Models used in NLLS fitting. D, DD, DDD, and DDDD: decays containing 1, 2, 3, and 4 discrete exponential terms, respectively. G, GD, and GDD: a unimodal Gaussian distribution and a Gaussian plus 1 and 2 discrete exponential terms, respectively. L and LD: a unimodal Lorentzian and a Lorentzian plus a discrete exponential term. Subscript F denotes a fit where one of the discrete lifetimes was fixed at the mobile phase value of 31.6 ns. b τ’s, fluorescence lifetimes. c When a continuous distribution is included in the model, the lifetime of the first component (τ1) is designated as the center of the distribution and σ is standard deviation of the Gaussian distribution or the fwhm of the Lorentzian distribution. d R’s, fractional intensity contributions. The fractional intensity contributions sum to unity: ∑iRi ) 1. e F denotes the lifetime that was fixed during NLLS fitting.

especially the four-exponential model whose solutions generally displayed large fluctuations. RESULTS AND DISCUSSION Model Discrimination: Discrete or Distributional. Table 1 summarizes the results of NLLS fits of a typical data file (PY300) to several decay models. The data were collected by flowing 10 µM pyrene in mobile phase through an ODS2 packing under 10 kV. All 300-s data were averaged to obtain the best signal-to-noise ratio and 0.15% linear error was used in the analysis. Upon inspection of Table 1, the single-, double-, and triple-exponential decay models (D, DD, and DDD, respectively) can be eliminated as good models for this set of data based on their large χ2 values. All three models resulted in χ2 values over 4. Similarly, several distributional models, including the unimodal Lorentzian (L), unimodal Gaussian (G), and Lorentzian plus one discrete component (LD) models can be rejected. The Gaussian plus one discrete component (GD), the Gaussian plus two discrete components (GDD), and the four-exponential decay (DDDD) models exhibit the smallest χ2 values. They all fit the data well and have similar residual plots (Figure 1). The GDD and GD are the same type of models. The χ2 is 0.89 for the former and 1.25 for the latter. It seems that the GDD model is a better fit than the GD model based on the χ2 values alone. According to the F statistics40 for comparison between the same type of mathematical models that have 60 degrees of freedom, however, a ratio of χ2 values of 1.84 is necessary to reject the model with higher χ2 with 99% confidence. The ratio of χ2 for the GD model to that for the GDD model is 1.40, which is not large enough for the selection of the GDD model over GD statistically. On the other hand, the GD model has only four independent parameters and the GDD model has six free variables, favoring the selection of GD. In addition, the GDD model cannot generate reproducible results. Table 2 lists eight NLLS fitting runs of PY300 Analytical Chemistry, Vol. 73, No. 22, November 15, 2001

5567

Figure 1. NLLS fits (A) and residual plots (B) for PY300 using G, GD, GDD, and DDDD models. In (A), the filled circles and triangles are the experimentally measured phase and demodulation values, respectively; the dashed line is the fit with G model; the solid line is the overlapping fits of the GD, GDD, and DDDD models. In (B), the open circles, filled squares, filled circles, and filled triangles are the residuals from the G, GD, GDD, and DDDD models, respectively. The data were 300-s average of 10 µM pyrene driven through the ODS2 packing at 10 kV. The decay models G, GD, GDD, and DDDD are a unimodal Gaussian distribution, a Gaussian distribution plus a discrete exponential term, a Gaussian distribution plus two discrete exponentials, and four discrete lifetimes, respectively.

with only slightly different initial guesses. It can be seen that neither the χ2 nor the recovered parameters from the GDD model are stable. Especially of note are fitting runs 5 and 6, where very similar initial guesses produced quite different results with χ2

values of 0.89 and 1.23. In contrast, all fittings with the GD model converged to the same parameters consistently. The lower χ2 from the GDD model is probably a consequence of fitting noise. Therefore, the GDD model is more complex, with two more fitting parameters than the GD model, but is less stable and its χ2 value is not significantly better. Thus, the GDD model is rejected in favor of the GD model. The DDDD model has a smaller χ2 value than the GD model, but has seven independent parameters, three more than the GD model. Since they are two different types of models, neither of them can be simply rejected. In fact, the DDDD model and the GD model are statistically equivalent, based on the F statistics, in the frequency range available using MHF measurements under our current experimental conditions. To examine the two models, a simulated decay was generated using the recovered parameters in Table 1 from the DDDD model and was analyzed in two different frequency ranges with GD and DDDD models, giving GD4D and DDDD4D. In the symbol GD4D, GD is the model used in NLLS analysis and the subscript 4D is the model used to simulate the data: four exponential decays. Similarly, a simulated decay from the GD model was obtained and analyzed, giving GDGD and DDDDGD. The results are listed in Table 3. When the analysis is performed in the frequency range between 1 and 1000 MHz, the χ2 values are 1.02 and 1.03 for and DDDD4D and GDGD, respectively, but they are 1.89 and 12.97 for DDDDGD and GD4D, respectively. If the simulation and fitting use the same model, both results have good χ2 values close to 1.0; when the simulation and fitting use different models, both results have χ2 values much higher than unity. If the DDDD model is an overfit compared to the GD model, DDDDGD should have a similar χ2 as GDGD. Thus, these two models are of different types mathematically and the GD model is not necessary simpler than the DDDD model. When the analysis is performed in the frequency range between 5.012 and 125.9 MHz, which is similar to the range used in the experiment, the χ2 values are 0.86 and 1.38 for DDDD4D and GD4D, and 1.17 and 1.10 for GDGD and DDDDGD, respectively. For all

Table 2. Eight Consecutive NLLS Fits for PY300 Using GDD and GD Models initial valuesa

final valuesb

trial no.

τ1 (ns)c

τ2 (ns)

τ3 (ns)

σ (ns)

R1

1 2 3 4 5 6 7 8

100.0 44.9 45.9 47.0 45.4 45.9 41.1 45.4

10.0 5.6 5.6 5.6 5.6 5.6 5.6 5.6

1.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0

10.0 34.4 28.8 28.9 28.2 34.0 31.8 26.1

1 2 3 4 5 6 7 8

100.0 44.9 45.9 47.0 45.4 45.9 41.1 45.4

1.0 3.1 3.1 3.1 3.1 3.1 3.1 3.1

10.0 34.4 28.8 28.9 28.2 34.0 31.8 26.1

τ1 (ns)

τ2 (ns)

τ3 (ns)

σ (ns)

R1

R2

χ2

0.95 0.96 0.96 0.89 0.88 0.89 0.94 0.97

GDD Model 0.03 46.1 0.03 46.5 0.03 46.0 0.03 48.5 0.03 48.5 0.03 44.1 0.03 47.4 0.03 44.1

4.9 5.0 4.7 5.9 5.9 4.7 5.7 4.5

2.0 2.0 2.0 1.8 1.8 2.9 2.0 2.8

33.1 32.6 33.2 30.2 30.2 35.4 31.6 35.4

0.96 0.96 0.96 0.94 0.94 0.97 0.95 0.97

0.03 0.03 0.03 0.04 0.04 0.01 0.03 0.01

1.01 0.98 1.04 0.89 0.89 1.23 0.91 1.23

0.95 0.96 0.96 0.89 0.88 0.89 0.94 0.97

GD Model 43.7 43.7 43.7 43.7 43.7 43.7 43.7 43.7

3.1 3.1 3.1 3.1 3.1 3.1 3.1 3.1

35.7 35.7 35.7 35.7 35.7 35.7 35.7 35.7

0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97

R2

1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25

a Initial values of the fitting parameters in NLLS analysis. b Final values of the decay parameters after NLLS minimization. c The denotation of all fitting parameters is as in Table 1.

5568 Analytical Chemistry, Vol. 73, No. 22, November 15, 2001

Table 3. NLLS Fits of Simulated Fluorescence Decays frq range (MHz) 1-1000

5-125

modeloriginala

τ1 (ns)b

τ2 (ns)

τ3 (ns)

τ4 (ns)

DDDD4D GDGD DDDDGD GD4D DDDD4D GDGD DDDDGD GD4D

63.6 43.2 71.4 47.5 61.4 43.5 63.3 43.7

18.9 0.0 22.8 0.0 16.4 0.0 17.6 0.0

4.4

1.1

σ (ns) 36.3

4.0

0.7 29.1

3.6

0.8 36.0

3.4

0.4 35.9

R1

R2

R3

χ2

0.66 0.97 0.57 0.95 0.70 0.97 0.67 0.97

0.25

0.07

0.34

0.08

0.23

0.07

0.25

0.07

1.02 1.03 1.89 12.97 0.86 1.17 1.10 1.38

a model original, the subscript original designates the decay model used to generate simulated data; model designates the decay model used to fit the data in NLLS analysis. b The parameters are defined as in Table 1.

four analyses, whether the simulation and fitting use the same or different models, they give similar χ2 values close to unity. So the DDDD and GD models are statistically equivalent in this frequency range. Therefore, the ambiguity between these two models in Table 1 is because of the limited frequency range, which cannot be circumvented in our current MHF measurement. Due to the very long lifetimes of the probe, fluorescence is significantly demodulated at high frequencies; the phase and modulation values collected at these frequencies have low signal-to-noise ratios. Especially of notice is that the DDDD model apparently fits the GD data better (χ2 ) 1.10) than the GD model does (χ2 ) 1.17). The selection of decay models should not be determined by the χ2 values alone. The DDDD and GD models cannot be distinguished easily from an apparent physical background either. If the DDDD model is chosen, it is possible to attribute the two shortest lifetime components to the silica background and assign the two longest lifetime components (64.0 and 20.4 ns) to pyrene molecules distributing into two different microenvironments. Neither microenvironment is in the mobile phase, since the lifetime of pyrene in the buffer is 31.6 ns.42 This is consistent with fluorescencequenching studies, which indicated that less than 1% of the pyrene molecules are accessible to solvent molecules.16,17 The longer lived lifetime component is associated with a less polar environment where the pyrene molecules are completely surrounded by C18 chains. The shorter lived lifetime component is perhaps associated with a more polar environment where the pyrene molecules are close to surface silanol groups. If the GD model is chosen, one can explain that the discrete component is from the background and that the Gaussian distribution indicates a continuous distribution of heterogeneous microenvironments in the C18 bonded phase. Although the χ2 values in Table 1 alone cannot statistically differentiate between the GD and DDDD models, their stability during the fitting and sensitivity against random noise are different. Figure 2 compares the sensitivity of the two models to initial values of fitting parameters. One hundred sets of initial guesses for PY300 were generated randomly for each model and subjected to NLLS minimization. For the DDDD model, 17 out of the 100 runs failed to converge to a χ2 value below 2.17, which is within the 95% confidence interval of the minimum χ2 value shown in Table 1. For the other 83 minimizations that finished below 2.17, the χ2 values scattered around the space. Only 52 runs resulted in a χ2 value that is within one standard deviation of the

Figure 2. Distribution of χ2 values of 100 fits of PY300 with GD (dark column) and DDDD (shaded columns) models, starting from randomly selected initial guesses. χ2 values above 2.25 are not included in the plot.

minimum χ2. For the GD model, only two runs failed to converge within the 95% confidence interval. The remaining 98 runs all finished with the minimum χ2 ) 1.25. The instability of the DDDD model is further confirmed by its χ2 surfaces (Figure 3A). These χ2 surfaces were obtained by systematically varying one lifetime value over a particular range and then rerunning the NLLS fitting to again minimize the χ2, keeping this lifetime value constant while allowing all other parameters to float.36,40 An offset lifetime value is said to be consistent with the data if the χ2 is below a certain value. The dashed horizontal line designates the one standard deviation confidence interval based on the Fχ statistic. The points where the dashed line intersects the χ2 surface mark the range within which the lifetime values are acceptable with 68.3% confidence. The wider the range, the more uncertain the recovered lifetime becomes. Shown in Figure 3A are the χ2 surfaces for the two longest lifetime components (τ1 and τ2) from the DDDD model and for the center of the Gaussian distribution (τc) from the GD model. At least 10 different sets of initial guesses that cover all ranges of the floating parameters were used for NLLS minimization at each point, and the smallest χ2 of all minimizations were selected to construct the χ2 surfaces. It is evident that the GD model has a much steeper χ2 surface. The one standard deviation interval from the χ2 minimum is 59-74 and 1-29 ns for the two longest lifetime components in the DDDD model, respectively, and 42.8-44.8 ns for the center of Gaussian distribution in the Analytical Chemistry, Vol. 73, No. 22, November 15, 2001

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Figure 3. χ2 surfaces for the GD and DDDD models. χ2 surfaces centered around 45 ns: the center of the Gaussian distribution in the GD model. χ2 surfaces centered around 20 and 65 ns: the two longest lifetimes recovered in the DDDD model. (A) Fit of the pyrene data. Horizontal dashed lines: one standard deviation from the minima of the χ2 surfaces. (B) Fits of the pyrene data (solid line), simulated GD data (dashed line), and simulated DDDD data (dashdotted line). Horizontal dashed line: 90% confidence level. The χ2 values are plotted in units of the minimum χ2 values for each surface.

GD model. The lifetime values are determined with higher certainty using the GD model than using the DDDD model. The broad χ2 surfaces of the DDDD model also indicate that its NLLS fit is sensitive to the noise level of the data. The results listed in Table 1 are the analysis of one data file containing 300-s averages, which has a very high signal-to-noise ratio. In real CEC separations, however, it is impossible to use such a long time of averages and each data file is much noisier. To mimic the effect in real separations, 100 phase and demodulation response curves, each with 3-s average and 20 frequencies, were generated from the same data set and analyzed using the GD and DDDD models. Figure 4 shows the fluctuations of τ1 and τ2 from the DDDD model and that of τc and σ from the GD model. When the initial guesses are close to the “true” values (Figure 4A) listed in Table 1, the average and standard derivation for τ1 and τ2 are 63.4 ( 1.8 and 21.4 ( 3.3 ns, respectively, and those for τc and σ are 42.5 ( 2.7 and 37.2 ( 3.2 ns, respectively. Similar χ2 values were obtained in both models for all 100 data files. Again, these replicate fits prove that the two models cannot be distinguished on χ2 values alone. When the initial guesses are far from the “true” values (Figure 4B), the average and standard derivation for τc and σ are still 42.5 ( 2.7 and 37.2 ( 3.2 ns, but those for τ1 and τ2 become 74.1 ( 5.6 and 30.5 ( 5.7 ns, respectively. Thus, both the precision and the accuracy of the recovered parameters for the DDDD model depend on the starting values of the fitting. Remarkably, although the recovered lifetimes for the DDDD fit in Figure 4A (fit A) are wildly different from the DDDD fit in Figure 4B (fit B), the resulting χ2 values of both fits are very similar for every 5570 Analytical Chemistry, Vol. 73, No. 22, November 15, 2001

3-s data set. As shown in Figure 4C, a plot of the χ2 values for fit A versus those of fit B overlays very well with the unity line of the plot. In contrast, the recovered parameters for the GD model are insensitive to initial guesses. The fact that the results from the 3-s averages are close to that of the 300-s average also indicates that the GD model is less affected by the noise level. Therefore, the GD model is more reliable because of its higher certainty and stability in parameter determination. Although these two models are equivalent judging from χ2 values in the frequency range studied, we feel confident in rejecting the model of fourexponential decays. To further examine the characteristics of the GD and DDDD models, we simulated frequency domain decay files using the NLLS parameters obtained from pyrene data (Table 1). In the simulation, the phases and modulation factors were calculated for the GD and DDDD models, respectively, at frequencies identical to those for the pyrene data, with an error level of 0.15%. The simulated GD and DDDD data files, which will be referred to as the GD data and DDDD data, were then fitted with both GD and DDDD models in NLLS analysis. The χ2 surfaces for these fits are examined (Figure 3B). The widths of these χ2 surfaces at the 90% confidence level are listed in Table S1 (Supporting Information). The χ2 surfaces for the fits with GD model are significantly narrower than the DDDD fit, suggesting that the GD model is intrinsically more robust and stable than the model of fourexponential decays. The widths of the χ2 wells at 90% confidence are similar for the GD fit of both GD data (3.4 ns) and DDDD data (3.8 ns). Furthermore, these widths are close to that of the GD fit for the experimental data of pyrene distributed in the C18 stationary phase (3.6 ns). The DDDD fit of the simulated files, however, showed more pronounced differences between the GD data and DDDD data. The DDDD model fits DDDD data significantly better than GD data, as evidenced by the narrower widths of the χ2 surfaces, especially the surfaces of the longest fluorescence lifetime. The DDDD fit for the DDDD data resulted in a width of 22.2 ns, less than half of that for the fit of GD data (46.5 ns). More importantly, the χ2 surface for the fit to GD data are very close to those obtained in fitting the pyrene fluorescence decay with DDDD model, further supporting the conclusion that the fluorescence decay characteristics of pyrene in the C18 phase are better described by a continuous Gaussian distribution. On the basis of data consistency, sensitivity to noise, and the χ2 surfaces, we have selected the Gaussian model for the heterogeneous fluorescence decay of pyrene in C18 stationary phase. We will now examine the physical significance of the model. The Gaussian distribution is attributed to pyrene molecules distributed into the C18 stationary phase. The structural heterogeneity of the bonded phase results in pyrene molecules experiencing a continuous distribution of local environments with different polarities. Fluorescence studies have shown that the grafted chains of the stationary phase have a tendency to cluster,8,9 forming a liquidlike phase that the solute molecules partition into,8,9,20 leading to chromatographic retention. The C18 phase has been found to have low polarity7-9,16-18,22 and a microviscosity of 19 cP.20 There has been evidence that the microenvironment that the solute experiences in the bonded phase is heterogeneous.9,11,13,14,16-18 The origin of microheterogeneity can be (1) distinguishable energetics of the active silanol sites, (2) exposure

Figure 4. Fitting results of 100 three-s data with the GD and DDDD models when the initial guesses are close to (A) and far from (B) the final results. Filled squares and triangles: the longest two lifetimes recovered by the DDDD model. Open circles and crosses: the center and standard deviation recovered in the GD model, respectively. Initial guesses for fits in (A) (fit A): GD model, τ1 ) 43.0 ns, σ ) 35.0 ns, τ2 ) 3.0 ns, and R1 ) 0.95; DDDD model, τ1 ) 62.0 ns, τ2 ) 19.0 ns, τ3 ) 4.3 ns, τ4 ) 1.2 ns, R1 ) 0.67, R2 ) 0.23, and R3 ) 0.07. Initial guesses for fits in (B) (fit B): GD model, τ1 ) 100.0 ns, σ ) 10.0 ns, τ2 ) 1.0 ns, and R1 ) 0.50; DDDD model, τ1 ) 100.0 ns, τ2 ) 30.0 ns, τ3 ) 6.0 ns, τ4 ) 1.0 ns, and R1 ) R2 ) R3 ) 0.25. (C) Comparison of the χ2 values for the DDDD fits in (A) (fit A) and (B) (fit B).

of the solute molecules to the solution phase, (3) different organizational modes of the grafted chains, or (4) partition of organic solvent molecules into the bonded phase. The silanol sites are probably not the primary origin of the heterogeneity due to the deactivation by water or solvent molecules hydrogen bound to these sites18 and the end-capping of the stationary phase. At the 75% acetonitrile/25% buffer mobile phase composition used in this work, acetonitrile partitions into the bonded phase, causing the C18 chains to assume an extended structure rather than a collapsed,18 effectively shielding pyrene from the mobile phase. At the bonded phase coverage of 2.72 µmol/m2, the C18 chain density in ODS-2 is almost optimum for chromatographic retention.45,46 At a higher density, strong steric constraints among chains will lead to an unfavorable entropy when the solute (45) Dill, K. A.; Naghizadeh, J.; Marqusee, J. A. Annu. Rev. Phys. Chem. 1988, 39, 425-461. (46) Sentell, K. B.; Dorsey, J. G. Anal. Chem. 1989, 61, 930-934.

partitions into the stationary phase.45,46 The high density of alkyl chains at 2.72 µmol/m2 coverage indicates a relatively small organizational heterogeneity. Thus, the microheterogeneity revealed by the pyrene probe is mainly caused by a gradient in solvent partitioning into the stationary phase. The grafted chains are known to be more ordered close to the silica surface and flexible at the interface, creating a gradient of probability normal to the silica surface for the partition of solutes or organic modifiers. Statistical mechanical theories predict a “breathing” surface: the thickness of the stationary phase increases upon the partition of organic solvent molecules.47 As the fraction of organic solvent increases in the mobile phase, inclusion of this organic modifier in the stationary phase becomes more extensive.17 The inclusion also perturbs the structure of the bonded phase by inducing chain ordering.48 At the mobile phase composition used in this study, it (47) Martire, D. E.; Boehm, R. E. J. Phys. Chem. 1983, 87, 1045-1062. (48) Dorsey, J. G.; Dill, K. A. Chem. Rev. 1989, 89, 331-346.

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is conceivable that all these structural effects resulted from the extensive uptake of the organic solvent and generate an ensemble of microenvironments with varying micropolarity. This continuous distribution of microenvironments manifests itself in the distribution of fluorescence lifetimes of the probe. To further examine this hypothesis, fluorescence quenching was used to determine the accessibility of the solute molecules to the attack of quenchers in the mobile phase. 10, 30, 60, and 100 mM KI was added to the solution that was driven through the cell packed with ODS-2. Fluorescence emission spectra measured at all quencher concentrations are shown in Figure S1 (Supporting Information). The absence of a broad peak in the longwavelength range indicates that the solute molecules are in the monomer form, without the formation of ground-state dimers or excimers. In the existence of ground-state or excited-state dimers, a broad, unstructured peak would be observed around 470 nm in the emission spectra.20 The emission spectra of pyrene at all quencher concentrations overlap each other very well, suggesting that the solute is not accessible to the quencher, consistent with the results by Carr and Harris,17 and by Burns et al.16 Carr and Harris observed that KI does not quench pyrene fluorescence when the bonded phase is in equilibrium with a mobile phase containing over 50% methanol.17 Burns et al. determined a small quenching efficiency of 0.1% in 60% methanol and of 2 × 10-6 in 100% methanol, indicating that pyrene is not accessible to quencher potassium iodide.16 KI is a quencher that is not retained by the C18 layer under the experimental conditions used in this study and thus quenches the solute molecules in the mobile phase exclusively. The fluorescence emission spectrum of pyrene is known to vary with the solvent and has generally been used as a probe of the polarity of its microenvironments. Especially, the III/I ratio, or the intensity ratio of the third vibronic band to the vibronic origin (the 0-0 transition), increases with decreasing solvent polarity.16,17,49 The invariance of the observed pyrene spectra (Figure S1) and the III/I ratio (Table S2) with quencher concentration suggests that pyrene molecules in the stationary phase, which is inaccessible to KI, contribute all fluorescence intensity. The Gaussian distribution of fluorescence lifetimes thus describes the continuous nature of the microenvironments that pyrene experiences in the C18 phase in our CEC experiments. The possible contribution of the mobile phase solutes is further examined in time-resolved measurements. The fluorescence lifetime of pyrene in the mobile phase in an open capillary was measured to be 31.6 ns and does not change with the electric potential applied in capillary electrophoresis or the solute concentration.42 Mobile phase contribution in the measured fluorescence decays should appear at a lifetime of 31.6 ns. When all the fitting parameters were freely floating in the NLLS analysis, none of the discrete lifetimes appeared close to the value of 31.6 ns, in either the GD or the DDDD model (Table 1). We then tried to force this mobile phase lifetime into the models by including a fluorescence lifetime fixed at 31.6 ns in the NLLS fitting of pyrene fluorescence decays. This value was used in both the GD and DDDD fits of experimental data as a fixed lifetime. The recovered parameters and corresponding χ2 values are also listed in Table 1. It is known that fixing one or more predetermined parameters (49) Kalyanasundaram, K.; Thomas, J. K. J. Am. Chem. Soc. 1977, 99, 20392044.

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Table 4. Effects of Background Subtraction on the NLLS Fit background correction before aftera

model

τ1 (ns)

G GD G GD

32.9 43.7 41.3 47.8

τ2 (ns) 3.1 3.8

ω (ns) 44.3 35.7 34.2 29.5

R1 0.97 0.97

χ2 33.99 1.25 47.53 1.58

a Decay parameters from NLLS fit after the background is removed from the data file using eqs 5 and 6.

in NLLS fits generally reduces the χ2 values. However, fixing a lifetime at 31.6 ns increased the χ2 values in decay analysis with both GD and DDDD models. Especially, a negative fractional intensity was recovered for the discrete lifetime component in the GD model, which is physically meaningless. We further examined several pyrene data sets collected under various CEC conditions, again fixing one lifetime to the mobile phase value of 31.6 ns. The fits yielded widely varying intensity contributions from this fluorescence lifetime that do not display a consistent trend with the experimental parameter being changed.42 In many cases, negative fractional intensity contributions were needed to reach the global minimum of χ2. These results support the conclusion of the quenching studies: an overwhelming majority of fluorescence signal is from pyrene molecules in the stationary phase. Numerous heterogeneous microenvironments exist in the C18 phase, forming a continuous distribution. The source of the discrete component in the GD model, however, is difficult to identify. This component has a lifetime value of 3.1ns and only a small fractional intensity contribution of 3%. Apparently it could be associated with the background fluorescence, because the intensity contribution from the stationary phase background is also 3%. However, the phase and demodulation curves from the pure C18 bonded phase cannot be fit with a single-exponential decay. NLLS analysis indicates that the best model for the background is a four discrete exponential decay, with lifetime components 87.0, 12.5, 4.5, and 1.2 ns and fractional contribution of 0.10, 0.29, 0.38, and 0.23, respectively. Thus, the discrete component cannot account for the background fluorescence. To further verify this, the background contribution is deducted from the data file, according to eqs 5 and 6. The corrected file is subjected to NLLS analysis, and the results are listed in Table 4. It is evident that, after the correction, the GD model is still much better than the G model, though the center and width of the Gaussian distribution recovered from the G model are reasonably close to the true values. Compared to the results of the fitting using the GD model before the correction, the center of the distribution is ∼4 ns longer and σ is ∼6 ns narrower. The lifetime value and fractional contribution of the discrete component, however, are close to those before correction. The χ2 value is larger, indicative of the expected decrease in signal-to-noise ratio. Thus, background correction cannot eliminate the discrete component. The more likely source of this lifetime component is the systematic errors in the frequency domain measurements. It has been shown that reference lifetime errors, timing errors, mismatched intensities, and random noise could all result in an

anomalous component in NLLS analysis.39,50 All these factors exist in our situations and none of them can be eliminated due to experimental limitations. In our measurements, the lifetime of the scattering reference is assumed to be 0.0 ns, but it has been found that a reference lifetime value in the picosecond range for scattering solution is needed for accurate recovery of the fluorescence lifetimes. Our sample and reference signals were collected through different filters, which could introduce wavelengthrelated timing delay caused by the color effect of the PMT. We chose to use scattered light as the lifetime reference instead of fluorophores that emit in a wavelength range similar to that of pyrene. This choice was made to minimize the effects of photon migration, which could substantially influence lifetime measurements in highly scattering media.44 The intensity of sample and reference cannot be perfectly matched because the combination of neutral density filters can only provide discrete attenuation of signals. It has been shown that these experimental factorssthe color effect of the PMT, the imperfect intensity matching, and the deviation in reference lifetimeswill lead to an apparent fluorescence lifetime in the picosecond and lower nanosecond range, with an intensity contribution of up to a few percent.39,50 Therefore, we attribute the discrete component in the GD model to systematic errors in the measurements. Model Discrimination: NLLS or MEM. Although the GD model is the best selection in NLLS analysis, these symmetric functions may be an oversimplification of the true distributions. For such a complex system, it is preferable to use an approach such as MEM that does not require any prior knowledge of the system. It has been reported that, for systems containing one to three discrete components, MEM is more robust against some systematic errors than NLLS.39 Since the discrete component in the GD model largely originates from systematic errors, it was expected that MEM can produce more consistent, noise-free fits. Figure 5A shows the result of lifetime recovery for PY300 using MEM. The fitting used 240 exponential terms between 0.1 and 1000 ns evenly distributed in log τ domain. Three peaks can be observed from the lifetime profile. One large and broad peak is asymmetric and has a long tail extending above 1000 ns. It has a maximum at 46.4 ns and an fwhm of ∼64 ns. Two smaller peaks are observed with maximums at ∼7.0 and ∼1.5 ns, respectively. The simulated GD and DDDD data were also fitted with the MEM for comparison (Figure 5A). Three peaks appeared in the MEM fit of GD data, a broad peak with high intensity at 51.9 ns, and two smaller peaks at 12.2 and 2.6 ns. MEM fit of DDDD data resulted in four peaks with finite widths, centered at 50.0, 15.9, 4.5, and 1.4 ns. The peak widths of the MEM fits for GD data are similar to those for DDDD data. Thus, the self-modeling maximum entropy method splits the broad Gaussian distribution into narrower peaks that have similar widths to fits of discrete fluorescence decays. To examine the consistency and reliability of fitting results, the capability of MEM to recover lifetime distributions was tested. We simulated phase and demodulation data files that obey a unimodal Gaussian distribution decay law. Four models were simulated with centers at 50.0 ns and widths of 5.0, 10.0, 15.0, and 20.0 ns, respectively. Each data set contains 24 frequencies from 1 to 1000 MHz in logarithmic scale. The statistical error was (50) Litwiler, K. S.; Huang, J.; Bright, F. V. Anal. Chem. 1990, 62, 471-476.

Figure 5. Fluorescence lifetime distributions recovered with MEM. (A) Fits for 300-s average of pyrene data (solid line), simulated GD (dashed line), and DDDD data (dash-dotted line). (B) Fits for six pyrene data sets of 50-s averages. A total of 240 exponential terms evenly distributed in the log τ space between 0.1 and 1000 ns were used in the fitting. The fitting was initiated at a flat intensity distribution for all exponential terms.

set to 0.15%, or 0.15° in phase and 0.0015 in modulation. The MEM fitting used 250 exponential terms between 0.1 and 1000 ns evenly distributed in logτ domain. Figure 6 illustrates the lifetime profiles recovered using MEM and using unimodal Gaussian distribution with NLLS. The center and width of each peak from MEM were calculated. To estimate these parameters precisely, interpolation was performed to add points between lifetime cells in the original distribution using Origin (Microcal Software, Northampton, MA). It can be seen that the NLLS fittings match the simulations, as expected, but none of the MEM results is acceptable. When the true width is 5.0 ns or the width/center ratio is 10% (Figure 6A), MEM does recover a Gaussian distribution. The center is 49.5 ns, in agreement with the real value, but its width is 2.8 ns, 55% of the true width. When the width of the simulation is 10.0 ns or the width/center ratio is 20% (Figure 6B), the MEM result starts to deviate from a simple Gaussian distribution. The recovered center is 51.3 ns, quite close to 50.0 ns; the recovered width is 3.4 ns, 66% narrower than 10.0 ns. Besides the main peak, a spurious peak at 26.2 ns is observed. As the width of the simulation increases to 15 ns (Figure 6C), three peaks can be clearly observed. The two large broad asymmetric peaks are centered at 62.4 and 41.6 ns and overlap each other extensively. The small narrow peak has a center at 9.5 ns with a fractional contribution of 1.2%. When the real width is 20 ns (Figure 6D), the lifetime profile recovered by MEM is more complex. In addition to the spurious peak, whose center shifts to 11.7 ns, the broad peak clearly splits into two, one centered at 36.0 ns and the other centered at 70.0 ns. A fourth sharp peak appeared at 1.0 ns. These results indicate that MEM cannot recover broad lifetime distributions. Although the recovered centers are acceptable when the width/center ratio is small, the recovered widths deviate substantially from the true values. When the width/center ratio is large, Analytical Chemistry, Vol. 73, No. 22, November 15, 2001

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Figure 6. Fitting simulated Gaussian distrubtions with the NLLS and the MEM methods: solid lines, true distributions; dashed lines, NLLS fits; dash-dotted lines, MEM fits. The center of the Gaussian is 50.0 ns for all distributions in the simulation; the widths of Gaussian are 5.0, 10.0, 15.0, and 20.0 ns for (A)-(D), respectively.

the appearance of spurious peaks in addition to the main peak is very likely to be taken as real components and can lead to misinterpretation of the data. We suspect that the entropy constraint is too stringent to allow termination of the maximization process when the appropriate distribution is reached and further iterations split the broad distribution into two regions of lifetimes. These observations are consistent with earlier trials of MEM.41 A similar phenomenon has also been observed when MEM was used to recover lifetime distributions of pyrene adsorbed on silica gel surfaces.33 The simulation provides insights into the recovered lifetime profile in Figure 6A. The width of the large peak is narrower than the real value and the ∼7.0-ns peak is very likely a “ghost” peak. With this consideration, the prominent peak in the MEM result agrees with the GD model from NLLS. The nature of the spurious peak is further confirmed by fitting of repetitive measurements. Figure 5B shows the MEM recoveries of six 50-s average data files generated by splitting the original 300-s data file. The position and shape of the large peaks almost overlay each other, but the center of the spurious peak shifts randomly between 1 and 10 ns. Although the position of the “ghost” peak is irreproducible and it has no physical meaning, it does affect the consistency of lifetime recovery. Figure 7 shows the lifetime analysis of 1-pyrenebutanol on C18 stationary phase under various high voltages. A 10 µM 1-pyrenebutanol solution in buffer was pumped through the ODS2 packing under 5, 10, 15, and 20 kV, respectively. Each data file is an average of 300 s. Figure 7A shows the time domain decay profiles obtained from the measurement channel on the MHF instrument under different high voltages. The fluorescence decay 5574 Analytical Chemistry, Vol. 73, No. 22, November 15, 2001

Figure 7. Fluorescence decay curves (A), NLLS (B), and MEM fits (C) of 1-pyrenebutanol. The solute was driven through the ODS2 stationary phase at 5 (solid lines), 10 (dashed lines), 15 (dotted lines), and 20 kV (dash-dotted lines). The concentration of 1-pyrenebutanol was 10 µM. The lifetime window of the MEM fitting was 1-1000 ns, though only lifetimes below 70 ns are shown.

becomes slower from 5 to 20 kV, indicating that the average lifetime becomes longer. Figure 7B shows the lifetime distribution recovered with the GD model using NLLS. The recovered distributions are consistent with the time domain fluorescence decay profiles. Figure 7C shows the lifetime fitting using MEM. The time domain decay of 10 kV is clearly slower than that of 5 kV, but MEM recovery of 10 kV almost overlays the 5 kV in the peak region. The time domain decay curves of 15 and 20 kV almost overlay each other, indicating they probably have similar lifetime distributions, but their MEM recoveries are clearly different. Therefore, the MEM results are inconsistent with the fluorescence decay data. In brief, the lifetime distributions recovered by MEM do not provide more information than the results from the GD model using NLLS. Furthermore, its inaccuracy and inconsistency make the interpretation more difficult. Thus, MEM is not suitable for lifetime analysis of pyrene derivatives on C18 stationary phase. It should be noted that MEM has been successfully applied to

fluorescence lifetime recoveries for some applications. The applicability of NLLS, MEM, or other methods in a particular system depends on the nature of the problem and experimental conditions. There is not one universal fluorescence lifetime analysis method that can be applied to solve all problems.

SUPPORTING INFORMATION AVAILABLE

ACKNOWLEDGMENT We thank the University of Iowa for supporting this research. The authors are grateful to Ms. Man Zhang for her assistance in the measurements of capacity factors of pyrene.

Received for review March 13, 2001. Accepted August 19, 2001.

Additional data as noted in the text. This material is available free of charge via the Internet at http://pubs.acs.org.

AC010293U

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