Anal. Chem. 1995,67,1377-1380
Modeling of Luminescence Quenching=Based Sensors: Comparison of Multisite and Nonlinear Gas Solubility Models J. N. Demas,**t B. A. Deora*,*#*and Wenying Xut Departments of Chemistty, University of Virginia, Chadoftesville, Virginia 2290 1, and James Madison Universiv, Hatrisonburg, Virginia 22807
Quenching-based luminescence sensors generally are supported in organic or inorganic polymers and exhibit nonlinear Stern-Volmer quenching behavior. ' h o common explanations of the nonlinearity are either multisite binding or the nonlinear solubility properties of the analyte in the sensor. Both models have three fitling parameters. These two models are compared, and the merits of each are discussed. It is shown that while the underlying physical bases of the two models are radically different and chemically incompatible, the two models are mathematically equivalent for data fitting. The correspondence of parameters in the two models is given. The nonlinear solubility model is probably computationally slightly simpler to use, but the two-site model seems to better approximate the underlying chemistry of systems studied to date. Increasingly,luminescence-based sensor materials are finding their way into a variety of areas. For example, they lend themselves to remote sensing of oxygen, pH, pCOz, temperature, and For oxygen sensors, a particularly attractive method is based on the luminescence quenching of a luminophore. The lifetime and emission intensity decreases as the oxygen concentration increases. Either of these changes can be quantitated and provide the basis for a relatively simple, inexpensive analysis method. Typically the sensor molecule is supported in a gaspermeable, solvent-impermeablematrix to protect the sensor from interacting with other species in the environment that could adversely disturb the oxygen response. Polymer-supportedsystems frequently have nonlinear response curves. There are two fundamentally different models for quantitating this nonlinear quenching data. We will explore the basis for these two equations, their similarity and differences, and ways to differentiate between the two models and, finally, show that from a purely mathematical standpoint both models give exactly equivalent equations for fitting experimental data even though the underlying physical models are incompatible. University of Virginia. $James Madison University. (1)Woltbeis, 0.Molecular Luminescence Spectroscopy Methods andApp1icafion.s: Part 2; Schulman, S. G., Ed.: John Wiley and Sons: New York, 1988; pp 129-281. (2) Hough, H.J.; Demas, J. N.; Wadley, H. N. G. In Proceedings of Review of Progress in Quantitative Nondestructive Evaluation;Thomas, D. O., Chimenti, D. E., Eds.: Plenum Press: New York, 1993; Vol. 12, pp 1137-45. (3) Demas, J. N.: DeGraff, B. A. In Advances in Fluorescence Spectroscopy; Lakowicz, J., Ed.; Plenum Press: New York, 1994; pp 71-107.
Quenching-based luminescence sensors utilize Stern-Volmer bimolecular quenching kinetic^.^ The kinetic scheme is given by
D+hv-D* D*
(14
- D + hv or A
k,
0)
+ Q*
k2
(24
D* + Q - D
-D-+Q+
(2d)
where D is the luminophore, Q is a quencher, kl is the rate constant for unperturbed decay of the excited state, including both radiative and nonradiative processes, and kz is the bimolecular rate constant for the sum of all bimolecular quenching processes that can deactivate the excited state. Equation l a shows excitation, and l b shows the decay paths unperturbed by added quencher. Equations 2a-d show the possible bimolecular processes involving the quencher Q. These include energy transfer to give excited Q (2a), catalytic deactivation of D* without excitation of Q (2b), and excited state redox processes that involve electron transfer reactions (2c,d). Equation 2c denotes oxidative quenching, and eq 2d is reductive quenching. This kinetic scheme gives the normal Stem-Volmer intensity quenching equations. In the subsequent analysis, we assume that the quencher is oxygen since this is currently the most widely used system. However, our results clearly apply to any quenching-based scheme that rests on bimolecular quenching. The normal Stem-Volmer equation is
+
0003-2700/95/0367-1377$9.00/0 0 1995 American Chemical Society
where Z is the emission intensity, KSV is the Stem-Volmer quenching constant, and z is the luminescence lifetime. The subscript 0 denotes the value in the absence of quencher. A linear calibration curve results for a plot of ZO/Z versus [021. (4) Lakowicz, J.
R Principles of Fluorescence Spectroscopy; Plenum Press: New
York, 1983.
Analytical Chemistry, Vol. 67, No. 8,April 15, 1995 1377
Equation 3 ignores the possibility of static or associational q u e n ~ h i n g .This ~ ~ ~occurs where the quencher and luminescent molecules associate in the ground state and form a nonluminescent complex. This has not been observed for oxygen quenchers in sensing media, and we ignore this complexity. 6-7 The luminscence decay curves in the ideal Stem-Volmer case are all single exponential with a measured quencher-dependent lifetime given by z = qI/u
+ K,,[O,I)
(44
zo/z = 1+ K,,[O,I (4b) where the second form is the normal lifetime form of the SternVolmer equation. A plot of rg/r versus [021 would be linear with a slope equal to KSV. Comparison of eqs 3a and 4b shows that in the absence of static quenching the lifetime and intensity StemVolmer plots are coincident. In solution, most systems satisfy the ideal Stem-Volmer equation. However, most sensors require that the sensor molecule be supported, typically on a polymer. This is necessary since virtually all luminescent sensors will respond to species other than oxygen (e.g., proteins, surfactants, solvents, metal ions, oxidants, reductants, etc.). Thus, the sensor must be isolated from these interferents while still providing full access to oxygen. Further, even if the media monitored is gaseous, the difficulty of working with a solvent can present problems in instrumental design. To isolate the sensor molecule from the environment, one typically supports the sensor molecule in a gas-permeable, solventimpermeable polymer membrane. In this way, nongaseous solvent-bome interferences are eliminated. The disadvantage of supported systems is that, unlike fluid solutions, virtually all polymer-supported sensors exhibit at least some degree of heterogeneity due to the differences in sites occupied by the sensor molecules and the failure of these sites in the rigid media to be averaged over the excited state lifetime of the sensor molecule. Heterogeneity manifests itself in nonlinear downward-curved Stem-Volmer intensity quenching plots and, where data are available, by nonexponential luminescence decays. Two conflicting models for fitting quenching in these microheterogeneous supports have been proposed. As we shall show, while their underlying physical models are completely different, these quite disparate models yield functionally identical equations.
foj
i 1
+ Kpwfl0,
where thefo’s are the fractional contributionsto the unquenched steady-state emission at the monitoring wavelength, and the KSV’S are “effective” quenching constants based on extemal gas concentrations (superscript C) or pressure (superscript p). The subscript j s refer to individual quenching constants for the different sites; I021 is the external gas concentration. Since 1021 is proportional to pressure, then the pressure form (eq 5b) is equivalent to eq 5a except for the units. In the literature it is common to use the pressure form and to omit the superscript. The summation is over all sites contributingto the luminescence. Equation 5 is based on the assumption that gas solubility in the polymer obeys Henry‘s law. This use of apparent KSV’Sis necessitated by the general absence of information on actual gas solubility in the polymer coupled with the fact that it is really only the extemal concentration or partial pressure that is of interest for analysis. Assuming that Henry’s law holds for the different binding regions in the polymer, then eq 5 can be related to fundamental parameters by substituting [ 0 2 1 = k&02 for each region.
&/I =
Kpwj = G
;k,
where K$ is the true Stem-Volmer constant, k~ is the Henry‘s law constant, and j corresponds to an index for the different binding sites or regions. We have made the assumption that different binding regions or domains might exhibit different local solubilities (i.e., Henry‘s law constants). For a two-site model, the simplest form that gives a nonlinear Stern-Volmer equation, we have
RESULTS AND DISCUSSION
We first describe the two different models and their experimental ramiikations. We then discuss the correspondence between the two models and conclude with a view of where the two models fit into analysis and sensor design. Multisite (Two-Site) Model. In this model, the sensor molecule can exist in two or more sites each with its own characteristic quenching c o n ~ t a n t .The ~ . ~ intensity Stem-Volmer plot then becomes
(5) Parker, C. A. Photoluminescence in Solutions; Elsevier Publishing Co.: New York, 1968. (6) Xu,Wenying; McDonough, R C., III;Lmgsdorf, B.; Demas. J. N.; DeGraff, B. A. Anal. Chem. 1994,66,4133-41. (7)Carraway, E. R; Demas, J. N.; DeGraff, B. A; Bacon, J. R Anal. Chem. 1991,63,337-42. (8) Sacksteder, L.; Demas, J. N.; DeGraff, B. A. Anal. Chem. 1993,65,348083. 1378 Analytical Chemistry, Vol. 67, No. 8, April 15, 1995
Even if both regions have the same oxygen solubilities ( k ~ = 1 k ~ z ) we , still have no reduction in the number of parameters necessary to fit data since we still have the same number of unknown Ksv’s. A limiting case of eq 7 has been used with one of the StemVolmer quenching constants equal to zero. ’Ihis is probably most appropriate when there is an uncompensated constant sample background. In this case, however, the model has no fundamental importance but is used to compensate for inadequate instrumentation or samples. In the multisite model, the decay curves will typically be a sum of exponentials. If the emission spectra from the different sites are the same, and the radiative rate constants for all sites are the same, then the sample impulse response is given by
In the simplest version of the nonlinear solubility model, the luminescence decays will be a single exponential with the lifetime given by zj = zw/(l+ g v j Poz)
(ab)
A very real problem in trying to study these systems is the poorly poised nature of the data-fittingproblem. For lifetime data, two or three lifetimes can fit a large number of fundamentally different data sets including distributions, multiple distributions, and large numbers of discrete sites. For intensity data, the same is true.639-11Thus, just because a model fits experimental data does not mean it is true. Indeed, in complex polymer systems, simple models will, at best, be approximations of a much more complex underlying reality. Nonlinear Solubility Model. The second model assumes that all nonlinearity in the Stem-Volmer plot arises from the nonlinear solubility of oxgyen in the polymer.'2-l5 The nonlinear quenching is then assumed to rest on the presence of a single quenchable form of the sensor, which detects the average dissolved oxygen concentration.16J7This nonlinear solubility is attributed to a Langmuir adsorption of gas in microvoids in addition to the normal Henry's law solubility. The nonlinear solubility of gases in polymers is given by
= k&
+ C,'bp/(l + bp)
(9)
where CHand CLare the Henry's law and Langmuir contributions to the solubility, respectively. kH, CL', and b denote Henry's law parameter, the Langmuir adsorption capacity, and the af6nity constant of the gas for the Langmuir sites, respectively. Based on these assumptions, one arrives at a nonlinear Stem-Volmer intensity equation16
where A and B are composite parameters including gas solubility parameters and rate constants. b is the parameter in the nonlinear gas solubility equation. This equation is that of E and Wong,16 but we have omitted the possibility of dissociation of the encounter complex without energy transfer. However, omission of this complication has no affect on the functionalform of their equation. Depending on the relative Henry's law and Langmuir contributions to quenching, the Stem-Volmer plots will range from linear to concave downward and rising to a plateau. (9) Simiarczuk, A; Ware, W. RJ. Phys. Chem. 1989,93, 7609-18. (10) Carraway, E. R; Demas, J. N.; DeGraff, B. A Anal. Chem. 1991,63,3326. (11) Demas, J. N.; DeGraff, B. A. Sens. Actuators 1992,11, 35-41. (12) Vieth, W. R; Amini, M. A Polymer Science and Technology, Vol. 6:
Permeability of Plastic Films and Coatings: Hopfenberg, H. B., Ed.; Plenum Press: New York, 1974, pp 49-61. (13)Fenelon, P.J. Polymer Science and Technology,Vol. 6:Permeability ofPlastic Films and Coatings;Hopfenberg, H. B., Ed.; Plenum Press: New York, 1974; pp 285-299. (14) Kamiya, Y.; Muoguchi. IC;Naito, Y.; Hirose, T.J. Polym. Sci, Part B: Polym. Physi. 1986,24, 535-47. (15)Sangani, A. S. J. Polym. Sci., Part B: Polym. Phys. 1986,24, 563-75. (16) Li, X.-Y.; Wong, K-Y. Anal. Chim. Acta 1992,27-32. (17) Li,X.-M.; Ruan, F.-C.; Wong, K-Y. Analyst 1993,118, 289-92.
z=
TO
1 + APo, + (BPo,/(1
+ bPoz))
(11)
Correspondence of Nonlinear and Two-Site Quenching Models. Comparison of eqs 7 and 10a suggests that both models yield completely different intensity quenching equations. However, if one fits the same intensity data sets with both models, one discovers that no matter what the data set, the best fits caculated by both models are identical. That is, the functions are clearly mathematically equivalent. Further, the quality of the fits using either model is excellent, as can be seen from the data presented in the literatu~-e.~J~ The correspondence between the parameters in the two models is given by
The equivalence between eqs 7 and 10a with this correspondence of parameters can be verified by substituting them into eq loa, subtracting the right-hand side of eq 10a from the right-hand side of eq 7, and simphfying to yield zero. The inverse transformation from A, 6,and B tofol, Ksvl, and KSVZ is much less tractable and is not shown here. Initially, we note that both models are at best only approximations to reality. Polymer-supported systems exhibiting heterogeneity are clearly extremely complex. At best one can hope for a fit with a set of parameters that are lumped quantities averaged over the multiple sites and distributions of the polymer. To the extent that the model resembles the true chemical nature of the system, these lumped parameters can provide insight into the chemistry and binding characteristicsof the sensor in the polymer. Having said that, we have yet to see a quenching system that was not fit quantitatively in the intensity data by these models. Thus, from the standpoint of data analysiswhere the goal is simply fitting the data with no regard to chemical significance, there is no reason for preferring one model over the other. Indeed, the nonlinear solubility model is possibly a little simpler to fit and to use for calculations. However, we prefer the two-state model for the following reasons, especially where any attempt is being made to physically interpret the data. First, there are fundamental reasons for generally rejecting the nonlinear solubility model in favor of the two-site model. The nonlinear solubility equations generally only apply to glassy polymers where the microvoids are trapped by polymer rigidity. For polymers above the glass temperature or rubbery polymers, Henry's solubility law seems to be obeyed12-15and the nonlinear solubility model is inappropriate. The nonlinear solubility model does, of course, depend on parameters that can be verified independently. In particular, b can be obtained from oxygen solubility measurements as a function of a pressure. Unfortunately, at this time no workers in sensor design seem to be making these measurements. It would be extremely interesting both for testing the nonlinear solubility model and for providing the necessary solubilities to allow calculation of fundamental rate constants for these measurements to be made. Analytical Chemistry, Vol. 67, No. 8, April 15, 1995
1379
Table 1. Correspondencebtween Oxygen Quenching Parameters of ReL(CO)aNCR+in R N - I 18 Slllcone for Two-Slte and Nonlinear Solubllity Models.
complex L/Rbb bpy/t-Bu phenlt-Bu
CMephen/t-Bu SMephen/t-Bu SClphen/t-Bu Mezphen/t-Bu Mezphen/n = 3 Mezphen/n = 11 Medphen/t-Bu Me4phen/n = 3 Merphen/n = 7 Medphen/n = 11
Phzphen/t-Bu
hl 0.21 0.50 0.78 0.82 0.83 0.90 0.90 0.92 0.95 0.95 0.95 0.95 0.85
two-site parameters (Torr-') Ksvz Von-9
A V0rr-l)
&VI
0.0083 0.0711 0.075 0.0879 0.0064 0.1204 0.0909 0.0708 0.1568 0.1519 0.2043 0.1421 0.1823
0.0010 0.0030 0.0035 0.0039 0.0025 0.0034 0.0031 0.0018 0.0032 0.0033 0.0060 0.0037 0.0108
nonlinear solubility parameters b Vorr-')c B Vorr-1)
0.00123 0.00576 0.01365 0.01802 0.00506 0.02711 0.02372 0.01741 0.04612 0.04672 0.07702 0.04951 0.05390
0.007f 0.001 0.037f 0.006 0.019 f 0.004 0.019f 00.005 0.0032 f 0.0004 0.015 f 0.006 0.012 f 0.005 0.007 f 0.004 0.011f 0.008 0.011 f 0.007 0.016f 0.010 0.011f 0.007 0.037f 0.009
0.00131 0.0313 0.0456 0.0548 0.00068
0.0816 0.0584 0.0479 0.103 0.0978 0.1174 0.0857 0.1027
a Two-site model data taken from ref 8. R = CHs(CHz), or f-Bu = tert-butyl and L is an a-diiiine given by one of the following abbreviations: bpy = 2,2'-bipyridine,phen = l,l@phenanthroline,4Me hen - 4methy!-l,l@phenanthroline,5Mephen = 5methyl-l,l@phenanthroline,4Clphen = khloro-1,lO-phenanthroline, Mezphen = 4,7-dimethyfl,l@~henanthro!ine,Me4phen = 3,4,7,&tetramethyl-l,lO-phenanthroline, and Phzphen = 4,7-dipheny!-l,lO-phenanthroline. Estimated standard deviation based on assumed absolute error of 0.05 infol, and 15% fractional error in Ksvl
and KSVZ.
Second, in its pristine form, the nonlinear solubility model predicts a single-exponential decay at all quenching conditions. At least in our experience, systems exhibiting nonlinear SternVolmer intensity curves invariably show nonexponential decays (analyzable as a sum of two or three exponentials). On the other hand, the two-site model predicts a double-exponentialdecay with both decays being quenched independently. True, close inspection of virtually all quenching lifetime data indicates that one must fit the data with more than two sites. However, this is a further refinement that can be accommodated by the general form of eq 5a with more than two sites. It is possible for the nonlinear solubility model to accommodate a double-exponential decay. One merely assumes a different quenching constant for the Henry's law region and for the Langmuir region. This would add another parameter to the model, however, and in spite of the extra parameter would do no better than the two-site model in explaining multiple-exponential decays. Also, if more than two exponentials must be explained, one can assume inequivalence of either the Henry's law region or the Langmuir sites. However, now one has a model that is a hybrid between the nonlinear solubility model and the multisite model. Finally, the nonlinear model predicts that different sensor molecules in the same support should all yield the same b parameter. We have published data on a series of R e 0 complexes (ReL(CO)sNCR+ with R = an aliphatic group and L = a-diiiine) in RTV-118 silicone rubber? which allows us to test the suitability of the nonlinear solubility model. Table 1 shows the original two-site fitting parameters and the nonlinear solubility parameters calculated from eq 11. For b we have estimated the uncertainty based on conservative estimates of absolute uncertainties of 10.05 in fol and relative uncertainties of 15%in and Ksv~.It is clear that b fluctuates widely (greater than a factor of 10) over the range of complexes. For example, compare Re-
1380 Analytical Chemistry, Vol. 67, No. 8, April 15, 1995
(phen) (CO)3NC(t-Bu)+and Re(Ph2phen) (CO)~NC(~-BU)+ with Re(SC1-phen)(C0)3NC(t-Bu)+. It is clear that these data cannot be reconciled with a constant b. Thus, in this case, the nonlinear solubility model is a less satisfactory model than the two-site model. As a further benefit of the two-site model, the parameter variations can be explained on the basis of a physically plausible structural modeL6 Earlier Li et al.17 did show that, for a single metal complex in a rubbery silicone polymer, b was fairly constant with changes in complex concentration while A and B varied more strongly. However, because of the use of only one sensitizer, their ability to vary the quenching properties was nowhere near as large as in our study, and it would be easier to mask systematic changes in b. Also, of the three parameters, inspection of the equations suggests that b might be less sensitive to variation in the fundamental parameters than A and B. In conclusion, we suggest that, for general fitting of intensity quenching data, the two-site model is preferable to the nonlinear solubility model. This is especially true if one is looking for fundamental information concerning the nature of the sites responsible for the quenching. However, if one is only interested in quantitatively fitting the data, the nonlinear solubility model is identical in fitting quality and may have a small computational advantage. ACKNOWLEDQMENT We gratefully acknowledge support by the National Science Foundation (CHE 91-18034 and 9419074). Received for review February 7,1995.
November 8,
1994.
Accepted
AC941004V @
Abstract published in Advance ACS Abstracts, March 15, 1995.
