82
Anal. Chem. 1981, 53,92-96
where the relative error of a quantity, a, err (a) = Il& a ~ ~ /the ~ ~ bara indicating ~ ~ , the perturbed values. W and h2 are the terms appearing in the Householder algorithm (see eq 43,44). Since multiplication by an orthogonal matrix leaves the norm of a vector invariant, the norm of AN is easily calculated in terms of W.
As W is a small triangular matrix, the computation of its condition and its norm by determining its singular values is easier. Very popular (but not always realistic) estimates of llWll and cond (W)are given by the following formulas:
max lwijl llWll 2 . .max
,...,r
1J = 1
Iwij(
cond (W) 1
J,' (47) min lwiil 1
Thus with the Householder algorithm, the terms appearing on the right-hand side of eq 45 are obtained as an easily
calculable byproduct allowing the determination of an upper bound for the error entering the columns kl,..., k, of the K matrix.
LITERATURE CITED Kaisec, H. Rxe Appl. Chem. 1973, 34, 35-61. Saxberg, Bo E. H.; Kowalskl, 8 . R. Anel. Chem. 1979, 57, 1031- 1038. Ratzlaff, K. L. Anel. Chem. 1070, 57, 232. Stewart, G. W. "Introductkn to MaMx Computations"; Acedemk Ress: New York, 1973. Stow, J. "Ehfihnng in die Numerlsche Mathemat& I", springer: Bertln-Heldelberg-New York. 1976; 2. Autlage. Guest, P. G. "Nunerlcel Methods of Cuve FMng"; Unlverslty Press: C a m , 1961. Banodale. 1, Appl. %f. 1966, 77, 51-57. Lawson, L.: Hetanson, R. "SOMng Least Squares Pr&lems"; P r m Hell: Englewocd Crma, KI, 1974.
RECEIVED for review March
13,1980. Accepted September 29, 1980. This work was supported in part by the Office of Naval Research. The work was also supported in part by the
Deutscher Akademischer Austauschdienst and Deutsche Forschungsgemeinschaft.
Simultaneous Multicomponent Rank Annihilation and Applications to Multicomponent Fluorescent Data Acquired by the Video Fluorometer C.-N. Ho, G. D. Christian, and E. R. Davidson' Department of Chemlsby, Unlversky of Washington, Seattle, Washington 98195
A procedure based upon the Fletcher-Powell algortthm for mHmlzatkn of functions of m e than one variable has been Incorporated Into the method of rank annlhlatlon to allow slmuttaneous cakulatlon of the concentratlons of several known species from approprlate mdlkofnponent data. l h k new method, called sknutlaneous muttkomponent rank annlhllatlon, provkles the flexlblltty of analyslng for m e constltuents wtthout requlring complete knowledge of the qualltatlve composltlon of the sample. The method was appHed to a set of six-component polynuclear aromatk hydrocarbon solutions by us8 of data acquked by the video fluorometer In the form of an excttatlon-emlsskn matrlx.
Simultaneous multicomponent analysis has been a subject of intense investigation in atomic spectroscopy (1-4). In molecular fluorescence, several groups (5-10) have presented a variety of approaches to perform such analysis. The method of rank annihilation (11, 12) has been shown to be useful for analysis of multicomponent fluorescence data acquired by the video fluorometer (13) in the form of an excitation-emission matrix (EEM). With this method, concentrations of known components are computed independently from the EEM of a sample whose complete qualitative composition is unknown. By incorporating the Fletcher-Powell (14) multivariate minimization algorithm, we have generalized the rank annihilation procedure to allow simultaneous computation of the concentrations of all known components. This method, called simultaneous multicomponent rank annihilation (SMRA), is 0003-2700/81/0353-0092$01 .OO/O
efficient and yields satisfactory results.
THEORY The theory of the method of rank annihilation for the determination of one known species present in a mixture of more than one component was presented in previous work (11, 12). To extend the method for simultaneous multicomponent analysis using the Fletcher-Powell algorithm (14) requires only minor modification of the previous algorithm. For a single component when absorbances at all wavelengths are kept low the EEM is ideally of the form Mij = C Y X J ~ (1) where xi is proportional to the number of fluorescencephotons emitted a t wavelength A,, y, is proportional to the number of photons absorbed a t wavelength Xi, and a is a concentration-dependent factor. For a mixture of r components, we may write k=l
(2)
where k denotes the kth component, xk and Yk are normalized column vectors containing all the xu's and y,kk respectively, and T denotes matrix transposition. For a real experimental matrix of rank r, M can be a p proximated as
M
=
where @ 1980 American Chemlcal Society
kEkUkVkT
k=l
(3)
ANALYTICAL CHEMISTRY, VOL. 53, NO. 1, JANUARY 1981
93
(5) Ideally, the number of nonzero eigenvalues i k equals the number of components present and (u),(x) and (VI, (ul, respectively, span the same vector spaces. Suppose the kth standard EEM is written as
N'k' = a k % k y k T
aslap, = C t , T ( a e / a B j ) t i
(15)
t
which allows us to obtain the gradient analytically. From eq 9
a e p q / aB j = - ajYapjbqj€q
n
xBkN'k' k-1
then first order perturbation theory gives
(6)
and standards are known for n components of an r-component mixture. If f l k is the relative concentration of component k, the residual matrix when all knowns are subtracted from the EEM may be defined as
E=M-
Define an eigenvector t k such that
(7)
+ a q j b p j t p ) + a j % Bkk a k ' S k j ( a p k a q j
a p p q d
(16)
The resulting correlation matrix is
Substit.uting eq 16 into eq 15 and defining
The matrix elements of EE"' in the basis of vectors (u)which spans the nonnull space of M W and EE"' are given by
we obtain
e , = upTEETuq =g
a p ,
-
(10)
All eigenvalues of the matrix e are nonnegative and ideally n of them should become zero for the correct choice of B's. For real data these eigenvalues will not actually be zero, but they should have distinct minima near the correct choice of the fl's. In the case of rank annihilation for a single known component, one eigenvalue of the residual matrix is minimized. For simultaneous multicomponent rank annihilation, the s u m of n eigenvalues can be used instead. Because of errors in the experimental data, it is sometimes desirable to w R vectors u where R is greater than r. In this case e will ideally have R - r additional zero eigenvalues for all choices of the B's. For this situation we followed a procedure analogous to that used in the previous rank annihilation work (12) and minimized the sum of the smallest R - r + n eigenvalues of e. Note that this criterion is equivalent to minimizing the trace of EETfor the special case when all components present in the solution are included in the simultaneous multicomponent rank annihilation and the set (u)spans the nonnull space. This, in turn, is identical with the least-squares criterion for choosing the 6's. Hence this method must reproduce the least-squares results when a complete qualitative analysis of the solution is available. In this limiting situation, however, the least-squares algorithm is much more efficient than the nonlinear procedure described here. The Fletcher-Powell algorithm requires that the gradient of the function to be minimized be computed. Let
(12) where pi is the ith eigenvalue of the matrix e. The gradient is then
The details of the experimental procedure, data acquisition, and reduction have h e n previoualy described (12). The solutea used were perylene, fluoranthene, tetracene, 9,10-dimethylanthracene, chrysene, and anthracene. As an indicator of the nature of the mixture, the EEM of a mixture of these six compounds as well as individual excitation and emission spectra are presented in Figures 7-9, respectively, of ref 12. Three seta of data for mixed solutions of these compounds including the set of six-component mixtures used in the rank annihilation study (12) were employed to teat the present procedure and to obtain direct comparison between the two methods.
RESULTS AND DISCUSSION Verification of the Procedure. T o test the validity of the procedure, we used a set of seven binary solutions of perylene and tetracene. In Table I, the resulta of simultaneous multicomponent rank annihilation and those of least squares are tabulated. Although not included here, the results obtained assuming only one component is present in the binary mixtures using this algorithm are identical with those of single-component rank annihilation. The results in Table I show that even when a set of only two u vectors is used, the calculated concentrations of simultaneous multicomponent rank annihilation are comparable with thoee obtained by using the method of least squares with the whole 60 x 60 EEM. Analysis of Tertiary Mixtures. For a more stringent teat of the algorithm, a set of tertiary mixtures of anthracene, tetracene, and perylene was carefully analyzed. Figure 1 is a contour plot of an EEM of a mixture in which fluoreacence intensity from each component is roughly equal. Table I1 tabulates the overlaps (strictly defined in footnote a) between the standards. From this table, we can see that there is a large overlap between the excitation spectra of perylene and tetracene and a small overlap between anthracene and tetracene. These different overlaps cause different correlations between the resulta of least squares and those of simultaneous multicomponent rank annihilation. Table III presents the results for the method of least squares assuming only two out of the three components in the eamplea are known. Let us first consider the calculated results for anthracene. At the higher anthracene signal, the calculated concentration is close to the correct answer, whether the other
94
ANALYTICAL CHEMISTRY, VOL. 53, NO. 1, JANUARY 1981
Table I. Relative Concentrations for Binary Mixtures of Perylene and Tetracene Using the Method of Least Squares and Simultaneous Multicomponent Rank Annihilation (SMRA)" perylene least squares 0.974 0.913
sample
a
expected 1 1.000 2 1.000 3 1.000 4 1.000 5 1.000 6 0.120 7 0.040 T w o u vectors are used.
1.010 1.029 1.020 0.114 0.038
SMRA
expected
tetracene least squares
0.955 0.971 1.007 1.027 1.019 0.114 0.038
1.000 0.330 0.120 0.040 0.013 0.330 0.040
0.998 0.339 0.118 0.044 0.015 0.308 0.039
Table 11. The Overlaps in Excitation (Unprimed)and Emission (Primed)for Perylene, Anthracene, and TetraceneO perylene anthracene tetracene
P A T
1.0000 0.2265 0.8177 1.0000 0.3464 0.5860
P' A'
T
anthracene"
1
1.0000
1.0000 0.0840
1.0000
1.010 0.354 0.141 0.056 0.022 0.309 0.040
Table 111. Results of Least Squares for Tertiary Mixtures of Anthracene, Tetracene, and Perylene Assuming Only Two Components am Present sample
1.0000 0.1773
SMRA
2
a The unprimed rows are yhTyl (see eq 11) and the primed rows are xhTxI.
3
4
tetracene"
5.0 x lo-' M 1.0 x M 0.421 2.781 0.383 *.. ... 1.199 (0.400) (1.20) 0.080 2.956 ... 0.040 ... 1.304 (0.040) (1.20) 0.058 2.975 0.018 .. . ... 1.316 (0.020) (1.20) 0.041 2.944 0.001 ... ... 1.291 (0.002) (1.20)
perylenea
1.2 x lo-' M
. . .b 1.145 1.023 (1.00)
... ...
1.020 (1.00)
.. .
1.213 1.022 (1.00)
...
1.205 1.016 (1.00)
a The reported concentrations are all relative to the molar concentrations of the standards given in the first row. The expected concentrations are given in parentheses. Component omitted in calculation.
I
10
20
30
44 WNNELS
50
80
MISSION 1. Contw pbt of a sample of a tertiary rnixtue of anthracene, tetracene, and perylene.
component analyzed simultaneously is perylene or tetracene. However, for the lower concentration samples, the results for anthracene are better if the other component computed simultaneously is perylene. This is because anthracene overlaps more with perylene. Therefore if they are analyzed together, the result is more nearly correct. When they are determined together, the signal contributed by perylene in the region of overlap will not be regarded as due to anthracene as is the case when only anthracene and tetracene are computed simultaneously. This trend is especially obvious for tetracene. The results using the method of least squares for tetracene are too high when it is assumed that tetracene and anthracene are the only two components present in the tertiary mixtures. However, when tetracene and perylene are assumed to be present simultaneously in the mixtures, the results for tetracene improve greatly even with the method of least squares. Perylene does
not seem to be affected as much by the mistaken assumption that there are only two components present in a tertiary mixture. Table IV tabulates results for the same set of data by using the method of simultaneous multicomponent rank annihilation. The results for the two constituents simultaneously determined are tabulated a c r m each row and recorded under the column of the particular compound. For instance, in the first row of each sample, the results tabulated are those when anthracene and tetracene are computed simultaneously. The first column under each compound lists the results using a basis of 3 u vectors while in the second column 10 vectors are employed. The observations made relating overlapping spectra to the calculated resulta by use of the method and least squares are not applicable here. The method of simultaneous multicomponent rank annihilation is a great deal more stable than least squares and noticeably less affected by assumptions as to which components are known in the mixture. Table V is another set of data for the same tertiary system but now the concentration of perylene is varied in the first three samples and the concentrations of anthracene and tetracene are lowered in the last two samples to keep the total intensity within the range of the instrument. The results of both Tables IV and V illustrate the superiority of the method of simultaneoua multicomponent rank annihilation over least squares when only a partial qualitative analysis is available. These results also show that the method has great flexibility and can provide a self-check because the analyst can use different combinations to calculate the results of one com-
ANALYTICAL CHEMISTRY, VOL. 53, NO. 1, JANUARY 1981
95
Table IV. Results of Simultaneous Multicomponent Rank Annihilation for Tertiary Mixtures of Anthracene, Tetracene, and Perylene Assuming Only Two Components Are Present
anthracene
tetracene
sample 1
0.386 0.386
...
2
[ 0.3861e
0.041 0.042
0.041 0.042
(0.04) 0.018 0.019
...
4
...
( 0.40)d
...
3
0.382 0.386
(0.02) 0.004 0.001
...
i n nnai
...
[ 0.0431
0.019 0.020
...
[0.0191 0.002 0.003
...
rn nni FII
perylene
(B)
(A) 1.338
1.321
...
... 1.233 1.3291 1.411
1.241 (1.20) 1.424
...
...
1.323 (1.20) 1.444
1.319 1.4371 1.430
... f
,
..
1.243 1.4481 1.411-
(1.20) 1.425
...
...
1.456
1.240
(A)
. . .c
(B)
...
0.989 0.974 (1.00)
0.987 0.975 [0.999]
1.029 1.016 (1.00)
1.027 1.016 [ 1.0351
1.037 1.041 (1.00)
1.034 1.034 l1.0451
1.035 1.039
1.032 1.029
...
...
...
...
...
...
r 1.0261
Expected relative cona Three u vectors are used. Ten u vectors are used. Component omitted in the calculation. centration in parentheses. e Single-component rank annihilation results from best of 4-10 vectors are in brackets. f No meaningful results obtained. Table V. Results of Simultaneous Multicomponent Rank Annihilation for Tertiary Mixtures of Anthracene, Tetracene, and Perylene Assuming Only Two Components Are Resent anthracene tetracene perylene sample 5
(A)" 0.408 0.400
... 6
(0.40)d 0.402 0.403
...
7
8
9
(0.40) 0.401 0.400 (0.40) 0.081 0.081 (0.08) 0.056 0.082
(B) 0.408 0.409
...
[0.408Ie 0.402 0.403
...
[0.401] 0.401 0.400
...
[0.401] 0.082 0.081
...
[0.0811 0.082 0.082
(A) 1.254
...
1.221 (1.20) 1.215
...
1.195 (1.20) 1.226
...
1.180 (1.20) 0.047
...
0.047 (0.04) 0.050
...
(B) 1.251
...
1.222 [ 1.2521 1.213
...
1.197 [1.2101 1.227
...
1.196 [1.235]
f. . . 0.034 [ 0.0451
f
...
(A)
. . .c
(B)
...
0.210 0.199 0.20)
0.208 0.199 [0.214
0.085 0.082 0.080)
0.081 0.080 [0.085
0.029 0.026 0.020)
0.020 0.020 [0.026
...
... ...
0.034 0.036 (0.04)
...
... ...
...
0.050 0.044 [0.036]
...
0.006 0.002 ... ... 0.044 0.044 0.005 0.0047 (0.08) [0.0821 (0.04) [0.039] (0.004) [ 0.00101 a Three u vectors are used. Ten u vectors are used. Component omitted in the calculation. Expected relative concentration in parentheses. e Single-component rank annihilation results from best of 4-10 vectors are in brackets. f No meaningful results obtained.
Table VI. The Compositions of the 10 Six-Component Mixturesa sample
perylene 1.97 x lo-'
fluoranthene 9.65 x
tetracene 2.50 x
dimethylanthracene 2.80 x lo-'
chrysene 1.93 x lo-'
anthracene 9.80 x
1 0.80 0.80 1.00 0.40 0.40 1.00 2 0.80 0.80 1.00 0.40 0.40 0.25 3 0.80 0.80 1.00 0.40 0.40 0.125 4 0.16 0.16 0.20 0.080 0.080 0.20 5 0.016 0.016 0.020 0.008 0.008 0.020 6 0.48 0.48 0.60 0.24 0.24 0.125 7 0.048 0.048 0.060 0.024 0.024 0.0125 8 0.16 0.16 0.20 0.080 0.080 0.125 9 0.080 0.080 0.10 0.040 0.040 0.025 10 0.080 0.080 0.10 0.040 0.040 0.025 a The concentrations of the components are relative to the corresponding standards, whose actual molar concentrations are given in the first row.
ANALYTICAL CHEMISTRY, VOL. 53, NO. 1, JANUARY 1981
06
Table VII. Results of Simultaneous Multicomponent Rank Annihilation for Sample 5 Assuming All Six Components Are Present and Using the Indicated Number of u Vectors no. of dimethylvectors perylene fluoranthene tetracene anthracene chrysene anthracene 5 6 8 10 15 20
0.0149 0.0149 0.0149 0.0149 0.0149 0.0149 0.016
expected
0.0214 0.0215 0.0215 0.0215 0.0215 0.0215 0.020
0.0184 0.0184 0.0183 0.0183 0.0183 0.0183 0.016
Table VIII. Relative Concentrations by Simultaneous Multicomponent Rank Annihilation for Sample 4 Using Six u Vectors Assuming Only T w o Components are Known F
Pa
T
D
C
0.00699 0.00700 0.00701 0.00699 0.00699 0.0 0 6 9 9 0.008
0.196 0.090 0.380 0.231
C A
A
1 2 3 4 5 6 7 8 9 10
0.159 0.160 0.161 0.161 0.180 0.175 0.165 0.161 0.197 0.195 (0.200) 0.196 0.197 0.0783 0.0892 (0.080) 0.0820 0.0830 0.0225 0.124 0.226 (0.080) 0.111 0.186 0.251 0.21 (0.200) 0.203
r
a Components computed simultaneousl with components listed on the left most column. Concentrations for component given in the left column are reported. The expected concentration for the component is in parentheses.
Table IX. Results of Simultaneous Multicomponent Rank Annihilation for Anthracene Assuming It and One Other Component Are Known in the Five Samples of Six-Component Mixturesa sample
Pb
F
T
D
C
expected
1
0.957 0.263 0.140 0.103 0.130
0.937 0.249 0.123 0.118 0.134
0.986 0.272 0.148 0.136 0.132
0.906 0.203 0.134 0.100 0.123
1.005 0.278 0.154 0.154 0.136
1.000 0.250 0.125 0.125 0.125
2 3 6 8 a
Using six u vectors.
Other component.
Table X. Results of Simultaneous Multicomponent Rank Annihilation for Perylene in Sample 4 Assuming Perylene and Other Components Are Known in the Six-Component Mixture no. of components known 2
3
4
5
Table XI. Results of Simultaneous Multicomponent Rank Annihilation for Dimethylanthracene in Sample 4 Assuming Dimethylanthracene and Other Components Are Known and Using Eight u Vectors no. of components known 3
4
5
0.656 0.395 0.430 0.0813 0.0135 0.263 0.0205 0.0687 0.0138 0.00710
0.817 0.799 0.817 0.165 0.0188 0.516 0.0524 0.1 61 0.0837 0.0823
0.939 0.248 0.137 0.198 0.0219' 0.137 0.0139 0.124 0.0254' 0.0159
See Table VI for expected relative concentrations. Using Using 6 u vectors unless otherwise indicated. 10 u vectors. a
Table XIII. Results of Simultaneous Multicomponent Rank Annihilation for All 10 Six-Component Mixtureso for Components Fluoranthene, Dimethylanthracene, Chrysene, and Anthracene fluoran- dimethylsample thene anthracene chrysene anthracene 1 2 3 4 5 6 7 8 9 10
0.778 0.777 0.797 0.160 0.0209 0.504 0.0513 0.156 0.0847 0.0822
0.395 0.376 0.384 0.0720 0.0309 0.231 0.0207 0.0728 0.0372 0.0317
0.692 0.367 0.393 0.0775 0.0051 0.260 0.0256 0.042 0.0150 0.0148
0.890 0.226 0.113 0.197 0.0245 0.125 0.0132 0.122 0.0262 0.0245
See Table VI for expected relative concentrations. Using 6 u vectors.
a
6
avconcnb 0.159 0.158 0.155 0.152 0.147 std dev 0.0025 0.0041 0.0044 0.0052 The average of results for all a Using six u vectors.' possible combinations with perylene as one of the components. Expected concentration of perylene is 0.160.
2
0.0215 0.0215 0.0215 0.0215 0.0215 0.0215 0.020
Table XII. Results of Simultaneous Multicomponent Rank Annihilation for All 10 Six-Component Mixturesa for Components Fluoranthene, Chtysene, and Anthraceneb sample fluoranthene chrysene anthracene
P b (0.160)c 0.155 F 0.183 (0.160)
T D
0.0130 0.0130 0.0131 0.0128 0.0127 0.0127 0.008
6
av concna 0.0859 0.0803 0.0786 0.0700 0.0714 std dev 0.00415 0.00401 0.00599 0.00126 The average of results for all possible combinations with dimethylanthracene as one of the components. Expected concentration of dimethylanthracene is 0.080.
Table XIV. Results of Simultaneous Multicomponent Rank Annihilation for All 10 Samplesa for Components Perylene, Fluoranthene, Chrysene, Dinethylanthracene, and Anthracene dimethylfluoran- anthrachrysample perylene thene cene sene anthracene 1 2 3 4 5 6 7 8 9 10
0.787 0.771 0.776 0.149 0.0146 0.475 0.0446 0.146 0.0786 0.0769
0.761 0.772 0.797 0.164 0.0187 0.505 0.0522 0.160 0.0863 0.0859
0.388 0.371 0.381 0.0714 0.00714 0.229 0.0202 0.0714 0.0375 0.0315
0.721 0.771 0.397 0.0820 0.0115 0.269 0.0275 0.0901 0.0143 0.0155
0.893 0.228 0.115 0.196 0.0212 0.125 0.0132 0.122 0.0260 0.0241
See Table VI for expected relative concentrations. Usine 6 u vectors,
a
ANALYTICAL CHEMISTRY, VOL. 53, NO. 1, JANUARY 1981
97
Table XV. Results of Rank Annihilation Using Six Basis Vectors for the 1 0 Samples Listed in Table VI dimethylsample perylene fluoranthene tetracene anthracene chrysene anthracene 1 2 3 4 5 6 7 8 9 10
0.816 0.774 0.779 0.161 0.0161 0.481 0.0473 0.157 0.0836 0.0844
0.769 0.806 0.832 0.176 0.0205 0.533 0.0529 0.169 0.0939 0.0928
1.034 1.058 1.380 0.197 0.0219 0.601 0.0597 0.198 0.130 0.102
ponent of interest. The algorithm seems to have excellent dynamic range and sensitivity. Analysis of Six-Component Mixtures. The same set of six-component polynuclear aromatic hydrocarbon mixtures discussed in ref 12 is utilized to test the methodology further. This set of data provides varying degrees of overlap and fluorescence intensities among the components to allow o b servations on how well the algorithm fares with regard to overlaps, dynamic range, and sensitivity. The compositions of the 10 samples are tabulated in Table VI. All sample number references made in the discussion refer to this table. In this study, the number of components assumed to be known to the analyst is varied from 2 to 6. For each case, simultaneous multicomponent rank annihilation is applied to each permutation of the species; that is, 15 permutations for 2, 20 for 3, 15 for 4, and 1 for 6. For each permutation again, the number of u vectors used to represent the mixture EEM’s is varied from 5 to 20. Table VI1 lists the concentration values for sample 4 computed by use of 5-20 vectors. The degree of overlap (see eq 21 of ref 12 and the discussion in ref 11)converges with five basis vectors, except for chrysene. Increasing the basis set does not change the calculated concentrations by any significant amount. Even for chrysene, the calculated concentration improves only negligibly, if any, by this fourfold increase in the basis set. This again seems to emphasize the fact that the signal content of an EEM is adequately represented by the first few significant eigenvalues. The results for chrysene also illustrate an important aspect of the method of rank annihilation in general; that is, the degree of overlap should be quite close to unity before one can hope to achieve any reasonable quantitation. If one cannot obtain a large value for the degree of overlap by increasing the basis set up to the limit (the dimension of the matrix), the component fluorescence is present only weakly or not at all. The quantitative results are only as good as the experimental data. However, for chrysene, the results of simultaneous rank annihilation are better than those obtained by the use of the method of least squares and singlecomponent rank annihilation of the same basis set. Table VI11 tabulates, for sample 4, the results of simultaneous multicomponent rank annihilation, assuming only two components are known a t a time. The results for perylene and tetracene not only are accurate but also are highly precise regardless of assumptions made as to which components are present together with them. This is true not only with this sample but with all samples and with all combinations. This may be due to the fact that both perylene and tetracene are more separated from the rest of the components in the EEM’s. The other components all overlap with each other quite substantially. The presence of fluoranthene, which has a broad band emission spectrum, causes substantial overlap. The calculated results of these four compounds are thus somewhat scattered depending on their relative fluorescence in the sample and their overlap. Chrysene, which in addition to large
0.454 0.395 0.404 0.093 0.0116 0.245 0.0268 0.0873 0.0368 0.4 12
0.198 0.282 0.300 0.136 0.216 0.127 -0.19 0.212 0.0178 0.062
1.217 0.302 0.191 0.203 0.0216 0.124 0.0158 0.138 0.0284 0.0270
overlaps is also only weakly fluorescing, exhibits the greatest variation. Table IX tabulates the results for simultaneous multicomponent rank annihilation of anthracene in the indicated samples. The concentration of anthracene is decreased steadily in samples 1to 3. Then it is kept constant in samples 3,6, and 8 while the concentrations of the other components are decreased in that sample. In general, the concentration values obtained for anthracene improve as its concentration increases relative to the other components. Although not displayed here, the same holds true for chrysene whose calculated concentrations become better and less scattered as its concentration increases relative to the others. Tables X and XI tabulate the results of simultaneous multicomponent rank annihilation analysis of sample 4 for perylene and dimethylanthracene, respectively. The concentrations tabulated are the averages of the results from all possible permutations for the different assumptions made regarding the number of components present in the sample, with one of them being perylene and one dimethylanthracene in Tables X and XI, respectively. These computed values show that the algorithm yields consistent results and in general is not adversely affected by the assumptions regarding the number and type of components present. These results also show that as more components are known simultaneously, the calculated concentrations tend to be lowered. Tables XII-XIV tabulate the results for all 10 samples by use of the indicated combinations. Table XV tabulates the results of single component rank annihilation by using six vectors for these six compounds for comparison. We can see from Table XI1 that when fluoranthene, anthracene, and chrysene are analyzed together the results for chrysene are quite acceptable even a t lower concentrations.
CONCLUSION The method of simultaneous multicomponent rank annihilation has been demonstrated to be a viable approach to simultaneous multicomponent fluorescence analysis. I t is rapid and efficient in terms of computation and offers the analyst the flexibility of using different numbers of components up to the total number in a sample. The method commands an obvious advantage over the conventional leastsquares approach in that it does not seem to be affected by overlapping spectra and it does not suffer from the requirement that exact composition of the sample be known. Furthermore, this method is self-consistent in terms of results obtained with different assumptions regarding the species present. This approach should be readily applicable to analytical systems such as GC/MS where the data can be put in a similar matrix representation. ACKNOWLEDGMENT The authors wish to express their gratitude to James B. Callis, Leon W. Hershberger, and Mary Lu Cianelli for their assistance and stimulating discussions.
98
Anal. Chem. 1981, 53,98-102
LITERATURE CITED (1) F e d , ti. 1.; Pardue, H. L. Anel. Chem. 1078, 50, 802-800. Jackson, K. W. AM/. Chem. 1075, 47, (2) A b , K. M.;Mchell. D. 0.; 1034- 1037. Monlson, G. H. Anel. Chem. 1074. 46, (3) Busch, K. W.; Howell, N. 123 1-1 238. (4) Knapp, D. 0.;Omenetto. N.; Hart, L. P.; Ptenkey. F. w.; whefadner, J. D. Anal. CMn.A& 1074, 69, 455-480. (5) O'kver, T. C.; Parks, W. M. Anal. Chem. 1074. 46. 1888-1804. (8) VoMnh, T. Anal. Chem. 1978. 50, 306-401. (7) Wehry, E. L.; Mementov, 0.Anel. Chem. 1979, 57. 843A. (8) Myeno, M. J.; K h , K.-Y. ANI/. Chem. 1077, 49, 555-550. (0) T a M , Y.; mer,D. c.; ~adamac,J. R.; s a n ~w. , h i . m.1978, 50, 036A.
a;
(10) W a r m . I. M.; D a w , E. R.; CMstten, G. D. Anel. Chem. 1077, 49, 2155-2150. (11) Ho, GN.; W t t e n , G. D.; Davldson, E. R. Anal. Chem. 1078, 50, 1108- 1113. (12) Ho, GN.; CMstten, G. D.; Davfbson, E. R. Anel. chem.1080, 52, 107 1-1070. (13) Jdnson, D. W.; ca.S,J. B.; CMsUan, G. D. AMI. Chem. 1977, 49, 747A. (14) Fletchec, R.; Powell, M. J. D. Cornput. J . 1083, 6, 183-167.
RECEIVED for review August 15,1980. Accepted October 14, 1980. This work is supported in part by NIH Grant No. GM-22311.
Oxygen Probe Based on Tetrakis(alky1amino)ethylene Chemiluminescence Thompson M. Freeman and W. Rudolf
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DepeNnenf of W m k b y . UnhmHy of New Hempshke, Dwhem, New Hanpshke 03824
An oxygen probe based on the chemllumlnescence (CL) of 1 , 1 ' , 3 , 3 ' - t e t r a e t h y A z ~ ~ ~ (EIA) ~~nb e )described. The devlce codst8 of a r e m chamber contalnhg 10% ETA In hexane separated from the analyte by a Tefkn membrane. O2 dmuses through the membrane and reacts wlth EIA yleldhg CL wtikh b measured wlth a photomuHlpUer tube. Steadydate CL lntenSny b propoctknal to the pertlal pressure of oxygen. Reqxmse to pwe O2 decays gradualty over a 124 perktd. Thb decay ts assodated wlth EIA consunptkn. The detectlam llmn of O2 In the gas phase Is esthated to be 1 ppm (v/v). The devlce atso responds to oxygen In water, but the skpe of the analytkal curve b decreased due to slow mass transfer of Os to the membrane surface.
Tetraaminoethylenes without aromatic functions react with molecular oxygen to produce bright chemiluminescence (CL). The first example of this reaction was the oxidation of tetrakis(dimethy1amino)ethylene (TMAE), reported by Pruett and co-workers (I) in 1950, in work with polyfluoroolefins. During the late l W s , there was interest in these compounds, primarily TMAE, as possible CL light sources. During this time, the reaction mechanism and the effects of reaction conditions (2-1 7) were investigated in detail. The reaction between tetraaminoethylenes and O2is shown. y 3
y 3
CHZ
CHZ
CH3
CH3
In general, the only isolatable product from this reaction is the corresponding imidazolidone (16), the exception being the oxidation of TMAE. The substituted urea is the dominant product of a large mixture (4). The CL reaction of tetraaminoethylenes is unusual in that bright CL is possible at very high reagent concentrations or 0003-2700/81/0353-0008$01.00/0
even using the neat liquid (18). This is an advantage for lighting applications since it increases the light capacity of a given volume of solution. However, there are problems which have prevented the development of a practical light source. The products of the reaction are known to quench the emission (18). Much of the research on this reaction has been directed a t methods for overcoming this problem, either by selective solvation (7) or by precipitation of the products (14). The reaction is also sensitive to the presence of protolytic substances, which accelerate the reaction and quench the emission (2,3,8). Finally, the CL efficiency of the reaction is not very high, estimated on the order of einstein/mol TMAE reacted (3). The CL intensity accompanying the oxidation of these compounds is directly proportional to the concentration (partial pressure) of O2and to the square of the tetraaminoethylene concentrations up b high concentrations (18). The linear relationship between O2 and CL intensity and the fact that low level CL is readily measured make this reaction an interesting candidate for the quantitative analpie of OP We have therefore undertaken a study to evaluate the possibility of O2 analysis based upon a tetraaminoethylene CL reaction. Other CL methods for oxygen have been developed on the basis of the luminol reaction and bacterial bioluminescence, but they have not been practical for general use (19, 20). We have ~ ~ n ~ t r ua~device t e d in which the oxygen diffuses through a Teflon membrane before reacting with the CL reagent. This should exclude most potential interferences aa well as preventing reagent evaporation. Unfortunately, TMAE is synthesized by a bomb technique which we could not exploit easily. Other compounds of this class can be synthesized under mild conditions (21). We chose to work with 1,1',3,3'-tetraethyl-Aw-bi(imidazolidine)(EM). We have since found that TMAE is commerically available from Armageddon Chemical Co., Durham, NC.
EXPECTED CHARACTERISTICS The O2 measuring device consists of a reagent chamber separated from the analyte by a Teflon membrane. When the Teflon membrane contacta an 02-containing sample, CL will be obse~edas oxygen diffueea through the membrane to react with the tetraaminoethylene. Steady state will be achieved 0 1080 Amerlcan chemlcel sodety
