Anal. Chem. 1997, 69, 2268-2274
An Adaptation of Kubista’s Method for Spectral Curve Deconvolution Mark F. Vitha, Jeff D. Weckwerth,† Kristopher Odland, Valdemia Dema, and Peter W. Carr*
Department of Chemistry, University of Minnesota, Kolthoff and Smith Halls, 207 Pleasant Street S. E. Minneapolis, Minnesota 55455
A chemometric approach to spectral curve deconvolution is described, evaluated, and applied to micellar systems. The technique is based on the method of principal component analysis of a spectral matrix followed by transformation of the abstract vectors into real spectra and concentrations. The approach reported here is similar to that of Kubista et al. (Anal. Chem. 1993, 65, 994-998). In the present study, however, more spectral information is known about the system of interest. This information is included in the deconvolution, which should, in general, increase the reliability of the method. From this method we obtain very reliable (noise-insensitive) λmax values of indicator molecules in the micellar pseudophase free from contributions of the indicator in the aqueous phase. The water-to-micelle partition coefficients are also determined. The effects of noise and the extent of indicator partitioning on the reliability of the method are evaluated using model data. The application of the method to the study of eight indicators in a prototypical micellar system (sodium dodecyl sulfate) is presented. Extension of the method to other types of chemical studies such as the determination of kinetic rate constants and product spectra is briefly discussed. This paper describes an approach to the deconvolution of UVvisible spectra of certain types of mixed systems. Although we believe that the basic chemometric approach has considerably wider applicability, it was specifically developed to obtain the spectra of molecules in micelles in aqueous micellar solutions. Prior to describing the methodology we will outline the motivation for its development. In the mid 1970s, Kamlet, Taft, and their co-workers used the phenomenon of solvatochromism to develop three independent scales describing the dipolarity/polarizability, hydrogen bonddonating ability, and hydrogen bond-accepting ability of bulk solvents.1-3 These scales are called the π*, R, and β scales, respectively, and are based on the solvent-induced changes in the λmax values of indicator molecules. We are interested in using the π*, R, and β scales to characterize micelles formed by surfactant molecules, acting as if they were a distinct bulk phase, or “pseudophase”. In the study of micellar phases, the presence of water causes complications when the solvatochromic comparison method of Kamlet and Taft is applied. The complication arises from the fact † Eastman Chemical Co., Kingsport, TN. (1) Kamlet, M. J.; Abboud, J. L. M.; Taft, R. W. J. Am. Chem. Soc. 1977, 99, 6027-6038. (2) Taft, R. W.; Kamlet, M. J. J. Am. Chem. Soc. 1976, 98, 2886-2894. (3) Kamlet, M. J.; Taft, R. W. J. Am. Chem. Soc. 1976, 98, 377-383.
2268 Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
that the indicator molecules in the micellar solution are simultaneously present in both the water and micellar pseudophase. Thus, UV-visible spectra of indicator molecules in micellar solutions are actually concentration-weighted sums of the UV-visible spectra of the indicators in water and in the micellar pseudophase. To use the Kamlet-Taft solvatochromic scales to characterize only the micelle phase, however, it is necessary to obtain the spectra of the indicators in the micelle phase free from contributions of the indicators in water. If accurate knowledge of the partition coefficients of the indicators were available, the problem could be solved algebraically by subtraction of the appropriate proportion of the spectrum in pure water. Partition coefficients for the indicators are not generally available, however, nor are they easily measured or estimated. Thus, we needed to find a method of curve deconvolution appropriate for micellar systems. In that regard, we adapted the chemometric approach to curve deconvolution described by Kubista et al.4 Using this approach, one can obtain equilibrium constants and the spectral response and concentrations of the species under study without knowing the spectrum of any component. It is, however, necessary to have an appropriate mathematical model for the phenomenon being studied. One must also know the total concentration of the indicator, the concentration of the specific species that controls changes in the spectrum of the indicator, and a sufficient number of linearly independent spectra so as to calculate the desired number of unknown system parameters. This approach is general for all problems in which these requirements are satisfied. In the present method, we take advantage of the fact that we know the spectrum of each indicator in pure water. In other words, we input more information about our system than is required. Use of the known spectrum in the deconvolution method allows the rapid and independent determination of two of the four elements of a transformation matrix used to convert abstract vectors into real spectra and should, in general, increase the reliability of the method since one of the “results” of the curve deconvolution is input as a known parameter prior to the analysis. The other two elements of the transformation matrix are obtained by linear regressions involving abstract vectors and known system parameters such as the total micelle and indicator concentrations. DECONVOLUTION METHOD The approach taken toward the deconvolution of micellar spectra includes the following steps: (1) defining the system in terms of parameters that are known and those which we ultimately wish to calculate, (2) performing singular value decomposition (4) Kubista, K.; Sjoback, R.; Albinsson, B. Anal. Chem. 1993, 65, 994-998. S0003-2700(96)00942-0 CCC: $14.00
© 1997 American Chemical Society
(SVD) on a data matrix comprised of spectra recorded at different but known surfactant concentrations to obtain two matrices, one that corresponds to abstract spectra and one that corresponds to abstract concentrations, (3) introducing a transformation matrix to transform the abstract matrices into real matrices corresponding to molar absorptivities and concentrations, (4) finding the elements of the transformation matrix by linear regressions of abstract vectors against known system parameters, and (5) using the transformation matrix to calculate the real indicator spectrum and concentration in the micellar phase free from contributions in the aqueous phase. Step 1. The micellar solution is viewed as two distinct phases having volumes Vm (micellar phase volume) and Vw (aqueous phase volume) sharing a total volume VT. Each phase also has a corresponding concentration of indicator molecules given by Ci ) Ni/Vi, where Ni is the number of moles of indicator in phase i and Vi is the volume of the phase. The total concentration of dissolved indicator is given by CT ) NT/VT. Finally, the path length of the micellar phase is bm and the path length of the aqueous phase is bw. Thus, the total path length of the system, bT, is given by bm + bw. The distribution of the indicator molecules between the two phases is defined by the chemical equilibrium
indicatorwater h indicatormicelle
(1)
The corresponding partition coefficient, K, is
K)
[indicator]micelle [indicator]water
)
Cm Cw
(2)
Furthermore, we define b m as the indicator molar absorptivity as a function of wavelength in the micellar phase, b w as the indicator molar absorptivity as a function of wavelength in the aqueous phase, and b aT as the total absorption spectrum recorded when both phases are present. Using these definitions, algebraic substitutions, and eq 3, we can write eq 4 for a single spectrum
(3)
b a T ) cmbmb m + cwbwb w b aT )
KCTbTVm
b m +
(KVm + VT - Vm)
CTbT(VT - Vm)
b w (4)
experimental data matrix of spectra as follows:
[ ]
bw b m] AT ) [
(5)
where AT is an m × n matrix, m being the number of data points per spectrum and n being the number of spectra collected at the various surfactant concentrations employed. b w and b m are m × 1 vectors as defined above, b cw and b cm are 1 × n vectors denoting the indicator concentration in the aqueous and micellar phases, respectively, and N is an m × n matrix comprised solely of noise and is therefore not used in further analysis. Step 2. SVD is applied to the data set AT and the results truncated after the first two factors, yielding
[ ]
b c1 AT ) [e b1 b e 2] b c 2 ) EC
(6)
where E is an m × 2 matrix comprised of the abstract vectors b e1 and b e2, and C is a 2 × n matrix comprised of the abstract vectors b c1 and b c2. We have limited the analysis to the first two factors since our model is based on the assumption that the indicator exists in only two environments, namely, the aqueous and micellar phases. To verify that there are two and only two important factors contributing to our spectra in micellar systems, we plotted the three most important abstract factors. The first two factors showed discernible trends containing systematic information, while the third factor consisted of only random noise. Note that b e1 and b e2 are linearly related, by an appropriate rotation, to the desired b w and b m. Similarly b c1 and b c2 are linearly related to b cw and b cm. The key to converting the abstract spectral vectors (e b1, b e2) and abstract concentrations (c b1, b c2) to real spectra ( bw, b m) and concentrations (c bw, b cm) lies in two facts. First, we know the real spectrum in water ( bw), and second, we know the total indicator concentration (CT) in each solution. This knowledge allows us to determine the elements of a transformation matrix, T, defined in the next step. Step 3. A 2 × 2 transformation matrix, T, and its inverse, T-1, are introduced to transform the abstract matrices E and C into the real (desired) matrices [ bw b m] and
[ ]
(KVm + VT - Vm)
of an indicator in a micellar solution. Equations 3 and 4 will be used in combination with SVD and linear regressions to deconvolve the micellar solution spectra. We emphasize that we are assuming that the micelles act as a bulk phase and therefore the indicators within the micelles experience only one overall chemical environment. In other words, the equations, as we have written them, do not account for the possibility that the indicators may be located in two chemically distinct regions within the micelles (e.g., the headgroup region and the nonpolar, alkane-like core). Support for making this assumption is discussed below. Ultimately we wish to obtain the indicators’ spectra in the micellar phase free from contributions of the indicator in the water phase. In other words, we wish to obtain b m. To do so, we applied SVD to a series of spectra collected in micellar solutions of varying surfactant concentration, keeping the total concentration of indicator molecules constant in each solution. We can represent the
b c wbw +N b c mbm
b c wbw b c mbm
respectively.
T≡
[ ] t1 t3 t2 t4
and T-1 ≡
[
t4 -t3 1 t1t4 - t2t3 -t2 t1
]
(7)
Based on eqs 5-7 we can write
[ ]
bw b m] AT ) (ET)(T-1C) ) [
b c wbw b c mbm
(8)
At this point, T is unknown while E and C are known from the SVD analysis. Thus, if we could determine T, then the spectra of the indicator in the micellar phase, b m, could be obtained. Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
2269
Step 4. We seek now to establish t1, t2, t3, and t4. We do so by imposing the following constraints:
[ ]
ET ) [e b1 b e 2] T-1C )
[
(
t1 t3 ≡ [ bw b m] t2 t4
t4 -t3 1 t1t4 - t2t3 -t2 t1
(9)
])[ ] [ ] b c1 b c b ≡ w w b c2 b c mbm
(10)
The forms of eqs 9 and 10 are suggested by eqs 6-8. In essence, we are determining the elements of T by demanding that they simultaneously satisfy eqs 9 and 10. From eq 9 we can write
To calculate t1 and t2 using eq 11, b w, which is experimentally measurable and thus available, is regressed against b e1 and b e 2, the two m × 1 abstract factor vectors comprising E (obtained by applying SVD). The coefficients of the regression are t1 and t2. We now need to find t3 and t4. From eq 10 we can write eq 12. Since we do not know the vector b cwbw we cannot find t4 and
) (
)
t4 t3 b c1 b c )b c wbw t1t4 - t2t3 t1t4 - t2t3 2
(12)
t3 as simply as we found t1 and t2. But given eqs 3 and 4, we see that b cwbw is equal to the coefficient of b w in eq 4. Thus, we can rewrite eq 12 as eq 13, where b cT is the vector (in contrast to the
(
) (
) (
)
t4 t3 (VT - Vm)bT b c1 b c2 ) b c t1t4 - t2t3 t1t4 - t2t3 (KVm + VT - Vm) T (13)
scalar CT) comprised of the set of the total indicator concentrations in each of the micellar solutions being analyzed since more than a single spectrum is being considered. The vector b cT is known, as are b c1 and b c2, and thus, eq 13 aids in solving for t3 and t4. In a similar manner, by considering the coefficient of b m in eq 4, eq 14 is obtained.
(
) (
) (
)
-t2 t1 KVmbT b c1 + b c2 ) b c t1t4 - t2t3 t1t4 - t2t3 (KVm + VT - Vm) T (14)
Adding eq 13 to eq 14 yields eq 15
p1 b c 1 + p2b c2 ) b cT
(15)
where
p1 )
t4 - t2 (t1t4 - t2t3)bT
(16)
and
p2 ) 2270
t1 - t3 (t1t4 - t2t3)bT
Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
c 1 + q2b c2 ) q1b
(17)
t1b c 2 - t2b c1 b c micV h mic
(18)
where b cmic is a vector containing the concentration of micelles present in each micellar solution being analyzed, V h mic is the molar volume of a micelle, and
(11)
b e 1t1 + b e 2t2 ) b w
(
cT The coefficients p1 and p2 are obtained by linear regression of b against b c1 and b c2 according to eq 15. Dividing eq 13 by eq 14 followed by algebraic rearrangement and substitution yields
q1 ) Kt4 - t2
(19)
q2 ) t1 - Kt3
(20)
The coefficients q1 and q2 are obtained by linear regression of c1 and c2 against the quantity on the right-hand side of eq 13 for which all of the constants necessary for calculating the concentration and molar volume of micelles (the total surfactant concentration, the critical micelle concentration, the molar volume of a micellized surfactant molecule, and the average aggregation number) are known for the specific surfactant being used. Knowing p1, p2, q1, q2, t1, and t2, it is possible to solve for t3 and t4 using
t3 )
t4 )
(p1t1 + p2t2)(t1 - q1) p2(q1 + t2) + p1(t1 - q1) (p1t1 + p2t2)(q1 + t2) p2(q1 + t2) + p1(t1 - q1)
(21)
(22)
Step 5. Now, all of the elements of the transformation matrix T are known and are used to obtain the spectrum of the indicator in the micellar phase free from contributions of the indicator in the aqueous phase through the use of the relationship in eq 9. Additionally, from eq 19 or 20 the partition coefficient, K, can be calculated for each indicator used. In the following section we present tests of the methodology using model data as well as experimental data. The results from experimental data relating to partition coefficients of indicators in sodium dodecyl sulfate (SDS) micellar systems are also presented while the results relating to the micellar spectra of the same indicators are discussed in detail elsewhere.5 EXPERIMENTAL SECTION Chemicals. The following chemicals were purchased from Aldrich and used without further purification: o-nitroanisole (99+%), p-ethylnitrobenzene (99+%), p-nitroaniline (99+%), and SDS (98+%). The p-nitroanisole (97%) used in these experiments was also from Aldrich and was purified using chromatography on silica gel with methylene chloride as the solvent, followed by recrystallization from 2-propanol. N,N-Diethyl-4-nitroaniline was from Frinton Laboratories and was used as received. The two R indicators, 2,6-dichloro-4-(2,4,6,-triphenyl-N-pyridino)phenolate [ET(33)], and bis[R-(2-pyridyl)benzylidine-3,4-dimethylaniline]bis(cyano)iron(II) [Fe(LL)2(CN)2], were prepared and purified using (5) Vitha, M. F.; Weckwerth, J. D.; Odland, K.; Dema, V.; Carr, P. W. J. Phys. Chem. 1996, 100, 18823-18828.
procedures given in the literature.6,7 All water was deionized and passed through Barnstead ion exchange and Organic Free cartridges followed by a 0.45 µm filter. Spectroscopy. A stock solution of each indicator in water was prepared such that the maximum absorbance of the peak of interest was approximately 0.5 (unless solubility limited, in which case an excess of the indicator was stirred in water for approximately two days followed by gravity filtration to remove the undissolved material). Indicator solutions having varying SDS concentrations (0-500 mM) were made by preparing either a 500 or 100 mM SDS solution using the stock indicator solution as the diluent, followed by serial dilution using the stock aqueous indicator solution. Corresponding aqueous SDS solutions were prepared using pure water and used as reference solutions. UV-visible absorbance spectra were collected using a Varian DMS 200 scanning spectrophotometer, 1 cm quartz cells (Starna Cells, Inc.), a scan rate of 20 nm/min, and a slit width of 0.2 nm. A holmium oxide filter was used to calibrate the spectrophotometer. Solutions were allowed to remain in the thermostated sample compartment for a minimum of 10 min before the spectra were collected. The temperature was maintained at 25.0 ( 0.2 °C using a Fisher Scientific Isotemp constant temperature circulator (Model 800) and a Utile Products Neslab Instruments Inc. (Model U-Cool) cooling unit. Data Analysis. Digitized spectra were collected using DMSSCAN software from Varian and analyzed using a program based on the method described above written in Matlab for Windows. We have assumed that the volume of the micelle is 15.5 L/mol, obtained by multiplying the molar volume of micellized SDS monomers (0.25 L/mol)8,9 by the average aggregation number (62).10,11 Following curve deconvolution, the spectra of the indicators in SDS generated by the algorithm were imported into TableCurve and gently smoothed with a 10% FFT method, and the “9/10” methods of Kamlet and Taft applied to find λmax.1 The 10% FFT smooth was found to cause very little perturbation to the overall peak shape but eliminated enough noise to provide a smooth continuous curve near the peak maxima. This greatly aids in using the 9/10 method since it relies on finding the maximum absorbance value.
RESULTS AND DISCUSSION Evaluation of the Deconvolution Method with Two Sets of Model Data. To demonstrate the ability of the above algorithm to deconvolve two-component spectra, two Gaussian curves were created and summed together in known proportions chosen to mimic the theoretical behavior of real indicator molecules in micellar solutions. The two curves model the pure water and pure micellar spectra. Spectrum 1, which simulates the water spectrum, was centered at 450 nm and had a standard deviation of 33 nm and a maximum molar absorptivity of 5000. Spectrum 2, which models the micellar spectrum, was centered at 435 nm and had a (6) Kessler, M. A.; Wolfbeis, O. S. Chem. Phys. Lipids 1989, 50, 51-56. (7) Burgess, J. Spectrochim. Acta 1970, 26A, 1957-1962. (8) Sepulveda, L.; Lissi, E.; Quina, F. Adv. Colloid Interface Sci. 1986, 25, 1-57. (9) De Lisi, R.; Liveri, V. T. Gazz. Chim. Ital. 1983, 113, 371-379. (10) Cline Love, L. J.; Habarta, J. G.; Dorsey, J. G. Anal. Chem. 1984, 56, 1132A1148A. (11) Wennerstrom, H.; Lindman, B. Phys. Rep. 1979, 52, 2-86.
Table 1. Percent Differencea between Calculated and Input Partition Coefficients Based on Model Data as a Function of the Input Partition Coefficient and Surfactant Concentration Range input partition coefficients
a
range (mM)
50
300
1000
0-60 0-50 0-45 0-40 0-35 0-30 0-25 0-20
-0.4 -0.6 -0.6 -0.8 -0.8 -0.6 -0.8 +0.6
-0.20 -0.20 -0.23 -0.23 -0.27 -0.20 -0.23 0.00
-0.16 -0.16 -0.16 -0.17 -0.17 -0.16 -0.16 -0.10
Percent difference, (Kcalcd - Kinput/Kinput) × 100.
standard deviation of 36 nm and a maximum molar absorptivity of 4500. The differences in the standard deviations and maximum molar absorptivities were introduced in anticipation of a change in peak height and width of the indicators in going from water to micelles. No noise was added to the curves in the initial test of the deconvolution method. A solute distribution model based on the equilibrium in eq 1 was used to calculate the fraction of indicator in each phase and thus weight the contribution of each pure spectrum in the total spectrum generated using a multicomponent Beer’s law expression. The total concentration of indicator was kept constant in the creation of the synthesized spectral data since our experimental data are collected under this condition. This is not necessary for the application of the deconvolution method, but a knowledge of the total indicator concentration present in each solution is required. Eleven spectra were created mimicking micellar spectra. We based the model data set on SDS as the surfactant being studied. We have done so simply to make the data set related to a real system. We note that the method is applicable to any surfactant system for which the cmc and molar volume of the micelles are known. The following constants were used in creating the data; critical micelle concentration, 8.1 mM;12 average aggregation number, 62;10,11 molar volume of micellized surfactant molecules, 0.25 L/mol.8,9 The spectra we generated corresponded to SDS concentrations ranging from 0 to 60 mM. The curve deconvolution method requires a minimum of four input spectra to allow the equations to be solved. We have tested the program using the artificial data set as a function of the surfactant concentration range included in the analysis and the magnitude of the partition coefficients of the indicator. The results are summarized in Table 1. The calculated partition coefficients are within 1% of the actual partition coefficients used to create the data independent of the magnitude of the partition coefficient and the surfactant concentration range. More important for our purposes is the agreement between the input pure micellar spectrum and the micellar spectrum generated by the curve deconvolution. In all cases, the pure water and micellar phase spectra obtained from the curve deconvolution overlap the corresponding input spectra (results not shown). This result holds independent of the magnitude of the partition (12) Mukerjee, P.; Mysels, K. J. NSRDS-NBS-36; U.S. Government Printing Office: Washington, DC, 1971.
Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
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Table 2. Percent Differencea between Calculated and Input Partition Coefficients Based on a Second Set of Model Data as a Function of the Input Partition Coefficient and Surfactant Concentration Range
Table 3. Percent Differencea between Estimated and Input λmax Values Based on a Second Set of Model Data as a Function of the Input Partition Coefficient and Surfactant Concentration Range
input partition coefficients
a
input partition coefficient
range (mM)
50
300
1000
range (mM)
50
300
1000
0-60 0-50 0-45 0-40 0-35 0-30 0-25 0-20 0, 30-60 0, 30-50 0, 30-45 0, 20-60 0, 20-50 0, 20-45
-4.1 -5.0 -3.5 -2.1 -7.9 -18.2 +27.4 -16.4 -0.10 -5.9 -4.5 +3.7 +3.2 +5.8
-1.6 -1.9 -1.9 -2.1 -2.5 -3.4 -3.5 -6.3 -0.33 -2.7 -1.0 +0.77 +0.30 +1.2
-0.99 -1.2 -1.1 -1.2 -1.3 -1.6 -1.8 -2.7 -0.98 -1.5 -1.3 +1.1 +0.50 +1.9
0-60 0-50 0-45 0-40 0-35 0-30 0-25 0-20 0, 30-60 0, 30-50 0, 30-45 0, 20-60 0, 20-50 0, 20-45
-0.009 -0.004 +0.009 +0.036 +0.038 -0.022 +0.19 -0.24 +0.004 -0.018 -0.004 +0.031 +0.038 +0.065
+0.002 +0.004 +0.002 +0.007 +0.018 +0.011 -0.011 -0.038 +0.004 +0.007 +0.011 +0.009 +0.011 +0.013
+0.007 +0.004 +0.009 +0.004 +0.011 +0.011 -0.002 -0.007 +0.004 +0.004 +0.054 +0.007 +0.004 +0.009
Percent difference, (Kcalcd - Kinput/Kinput) × 100.
coefficient and the range of the surfactant concentration included in the deconvolution. Overall, for this set of data created to mimic expected micellar behavior, the partition coefficients and spectra generated by the deconvolution agree exceptionally well with the values used to synthesize the data set. Effect of Noise and Spectral Shift. Random noise was added to another set of artificial data generated to more rigorously test the deconvolution method. The curve modeling the indicator’s spectrum in water was centered at 450 nm and had a standard deviation of 33 nm and a maximum molar absorptivity of 5000. The curve modeling the spectrum of the indicator in SDS was centered at 447 nm and had a standard deviation of 36 nm and a maximum molar absorptivity of 5000. Additionally, after the spectra were summed together using a Beer’s law multicomponent expression and the indicator distribution model, random noise of (0.003 absorbance unit was added to each spectrum. Thus, the difference in λmax values has been decreased to 3 nm and noise at the level we observe experimentally has been added. This set of artificial data, therefore, is quite similar to the data obtained in the actual SDS solutions containing solvatochromic indicators and provides a more rigorous test of the method than did the first set of artificial data. The partition coefficients generated by the curve deconvolution method were again examined as a function of the input partition coefficients and surfactant concentration range included in the deconvolution (Table 2). Larger differences between input and output partition coefficients are observed for this set of data than in the first test of the method. Presumably this is due to the added noise and smaller overall spectral shift between the two pure spectra. In general, however, the results are still quite acceptable with most calculated partition coefficients being well within 10% of the true values. The most important observation that should be made regarding the results in Table 2 is that, in general, the error in the partition coefficients, not surprisingly, decreases as data at higher surfactant concentrations are included in the deconvolution. Additionally, omitting data at lower surfactant concentrations also leads to smaller percent differences between the calculated and actual 2272 Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
a
Percent difference, (λmax, calcd - λmax, input/λmax, input) × 100.
partition coefficients. Furthermore, the relative error decreases as the partition coefficient of the indicator increases. These results are expected since the shifts observed with higher surfactant concentrations and larger partition coefficients are larger than those at lower surfactant concentrations and smaller partition coefficients. This makes the mixture spectra more distinct from the water spectrum and thus easier to deconvolve. Since our ultimate interest is in obtaining the spectra of indicators in the micellar phase, and more specifically the λmax of the indicator in the micellar phase, we determined the λmax of the micellar spectra generated by the deconvolution method for this set of created data. The results are presented in Table 3. As can be seen, variations in λmax values obtained as a function of input surfactant concentration range and input partition coefficient are very small, generally having less than 0.1% (0.04 nm) error. The input and output spectra again overlap when plotted together as was observed for the first set of test data. Thus, the deconvolution method is working quite well with regard to determining partition coefficients and more importantly generating the spectrum of an indicator in the micellar phase. The method is relatively insensitive to noise because the noise is forced to appear in abstract factors higher than the second factor. Application of the Method to Experimental Data. Given the favorable results obtained with the artificial test data, we believe that the approach to deconvolving the spectra is sufficiently accurate for analyzing our real spectral data taken in SDS solutions and thus obtaining the spectra of indicators in SDS micelles free from contributions of the indicator in the water phase. In other words, the errors in the λmax values obtained with the artificial data are small enough to have very little influence on the π*, R, and β values determined from them. We therefore applied the deconvolution to eight different indicators in SDS solutions. The λmax values of the indicators in the micellar phase are discussed elsewhere in association with an analysis of the Kamlet-Taft π*, R, and β values calculated using the λmax values.5 The partition coefficients obtained for each indicator as a function of the SDS concentration range included in the curve deconvolution are shown in Table 4. We observe that when the four spectra recorded at the lowest SDS concentrations are the
Table 4. Partition Coefficients Obtained with Each Indicator o-nitroanisole [SDS]a 0-20 0-30 0-40 0-50 0-60 0-70
Kb 59.69 111.93 133.07 149.85 153.65 155.6
p-nitroanisole [SDS] 0-16.6 0-25 0-33.3 0-50 0-100
K 44 77 119 151 161
p-ethylnitrobenzene [SDS] 0-20 0-25 0-30 0-40 0-50 0-75
K 578 597 603 626 624 594
N,N-diethylp-nitrobenzene [SDS] 0-16 0-20 0-25 0-30 0-40 0-50
K 2509 2553 2562 2568 2551 2525
ET(33)
Fe(LL)2(CN)2
[SDS]
K
0-20 0-30 0-40 0-50 0-60 0-70
4.09 × 3.85 × 104 3.82 × 104 3.95 × 104 4.01 × 104 4.00 × 104 104
[SDS]
K
0-40 0-50 0-60 0-70 0-80
9.23 × 5.26 × 104 3.56 × 104 3.74 × 104 3.84 × 104
p-nitroaniline [SDS]
K
0-20 0-30 0-40 0-50 0-60 0-70
481 448 414 396 366 350
104
a SDS concentration range spanned by the spectra input into the curve deconvolution algorithm. The first three spectra are the probe in water, 10 mM SDS, and an SDS concentration between 10 mM and the fourth highest concentration (the first value listed under each probe). b Molar partition coefficient ([probe]micelle/[probe]water) obtained from the curve deconvolution algorithm.
only spectra included in the analysis, the partition coefficients are significantly different than when more data are included. This likely results from the fact that the spectra taken at the lowest SDS concentrations also exhibit the smallest spectral shifts, making curve deconvolution relatively difficult. As spectra with higher SDS concentrations, and therefore larger spectral shifts, are added the method is better able to discern a second component distinct from the dominant water component. In fact, for most indicators we studied, the measured partition coefficients and micellar phase spectra did not vary appreciably after at least six spectra were included in the analysis. Thus, including more data at higher SDS concentrations leads to more reliable determinations of the partition coefficients, as was observed with the synthetic data. Several qualitative comments can be made regarding the partition coefficients. From published linear solvation energy relationships (LSERs),13-15 it is known that, all else being equal, large (hydrophobic) solutes will partition to the greatest extent. This is borne out by the large partition coefficients we obtained for the two largest indicators, ET(33) and Fe(LL)2(CN)2. Additionally, one would predict that p- and o-nitroanisole will partition to a smaller extent than p-ethylnitrobenzene since the methoxy moiety is better able to interact with water than is the ethyl moiety. Thus, the partition coefficients obtained for these compounds make chemical sense relative to one another. We note, also, that for p- and o-nitroanisole, p-ethylnitrobenzene, and N,N-diethyl-pnitroaniline, the observed partition coefficients generally fall within the range predicted by the published linear solvation energy relationships (LSERs) (Table 5). The observed partition coefficient for p-nitroaniline, however, is much larger than predicted by the LSERs. The spectroscopic behavior of this indicator is also inconsistent with SDS LSERs and is discussed elsewhere.5 Overall, given the results of the tests we have performed with model data, we are confident that the partition coefficients generated by the curve deconvolution method are chemically reasonable and reliable. Their internal qualitative consistency and agreement with predicted partition coefficients also lend confidence to the reliability of the deconvolution method. Additionally, (13) Vitha, M. F.; Dallas, A. J.; Carr, P. W. J. Colloid Interface Sci.1997, 187, 179-183. (14) Quina, F. H.; Alonso, E. O. Farah, J. P. S. J. Phys. Chem. 1995, 99, 1170811714. (15) Abraham, M. H.; Chadha, H. S.; Dixon, J. P.; Rafols, C.; Treiner, C. J. Chem. Soc. Perkin Trans. 2 1995, 887-893. (16) Abraham, M. H.; Chadha, H. S.; Whiting, G. S.; Mitchell, R. C. J. Pharm. Sci. 1994, 83, 1085-1100.
Table 5. Partition Coefficients Estimated from SDS LSERsa solute
Carr
Abraham
Quina
p-ethylnitrobenzene o-nitroanisole p-nitroanisole N,N-diethyl-p-nitroaniline p-nitroaniline
475-655 110-150 110-150 1750-2420 22-30
560-1220 170-370 170-370 1880-4110 80-170
620-1130 140-250 140-250 2570-4680 45-80
a SDS LSERs are from refs 13-15. Solute parameters used to estimate the partition coefficients are taken from ref 16 or estimated from values therein.
the micellar spectra obtained and the solvatochromic parameters derived therefrom are also chemically reasonable and consistent with previous literature reports. We note that the method, although generally reliable, may be inappropriate for systems displaying very small overall shifts in λmax arising from small partition coefficients coupled with low surfactant concentrations. Applicability of the Method. As Kubista has stated, these methods are applicable to all equilibrium systems with linear spectral responses for which the equilibrium expressions are known.4 Furthermore, the methodology can be extended to other problems. For example, it can be used to determine kinetic rate constants and spectra of reactants and products in systems for which it is not possible to obtain them directly because the spectra of the reactants and/or products severely overlap or cannot be measured free from other spectral components. To solve these problems, one needs to know the order of the reaction and the corresponding explicit mathematical expression relating the concentration of the species of interest, the amount of time allowed for reaction, and the rate constant. It is also necessary to know the total concentration of one of the species involved as well as the elapsed time of reaction. If these things are known, expressions very similar to those of eqs 13 and 14 in Kubista’s paper4 can be written and used to solve for the desired spectra and rate constant. More generally, the method can be used to solve any curve deconvolution problem provided that an explicit functional relationship is known between the independent variables (such as pH, time, temperature, initial concentrations, etc.) and the soughtafter parameters (such as partition coefficients, rate constants, equilibrium constants, or enthalpies of reaction). Thus, the mathematics presented here and by Kubista et al. represent a general scheme for solving a large family of problems. Analytical Chemistry, Vol. 69, No. 13, July 1, 1997
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CONCLUSIONS We have developed a method of curve deconvolution applicable to micellar systems which is based on the general approach described by Kubista et al.4 From the deconvolution we obtain the partition coefficients of indicator molecules transferring from water to the micellar pseudophase. We also obtain the indicators’ micellar spectra. The deconvolution method was tested with two sets of model spectra and reliably reproduced the input partition coefficients and model spectra. The method was also applied to experimental data taken in SDS micellar systems. The partition coefficients and spectra generated are generally chemically reasonable and consistent with predictions based on previous studies.
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ACKNOWLEDGMENT The authors thank Dr. David Whitman and Professor Sarah Rutan for their helpful discussions. This work was supported by grants from the National Science Foundation and the University of Minnesota.
Received for review September 17, 1996. February 20, 1997.X AC960942F
X
Abstract published in Advance ACS Abstracts, April 15, 1997.
Accepted
