Anal. Chem. 1987, 59, 2609-2615
ACKNOWLEDGMENT The authors express their appreciation to Jerry Milner of the Baylor Electronics Shop for help in assembling the circuits used in this study. The authors also express their appreciation to Milton Luedke of the Baylor Chemistry/Physics Machine Shop for fabricating various components of the apparatus.
LITERATURE CITED (1) Kirchhoff, G.; Bunsen, R. Philos. Mag. 1860. 20, 89-109. (2) Grant, D. W. I n Gas Chromatography 1958; Desty, D. H., Ed.; Pro-
(3)
ceedings of the 2nd Symposium on Gas Chromatography, Amsterdam, May 1958; p 153. McWllllam, I . G.; Dewar, R. A. I n Gas Chromatography 1958; Desty, D. H.. Ed.: Proceedlnas of the 2nd Svmooslum on Gas Chromatoara-- phy, Amsterdam, May 1958: p 142. Gaydon, A. G. The .Spectroscopy of Flames; Chapman and Hall: London, 1974; pp 221-243. Gaydon, A. 0.;Wdmard, H. G. Flames, Thelr Stfuctwe, Radlatlon and Temperature, 4th ed.; Chapman and Hall: London, 1979; pp 238-259. Plyler, E. K. J . Res. Natl. Bur. Sfand. ( U S . ) 1948, 4 0 , 113. I
(4) (5) (6)
2009
Bailey, C. R.; Lih, K. H. Trans. Faraday SOC. 1929, 25, 29. Nakamoto, K. Infrared Spectra of Inorganic and Coordination Compounds: Wiley: New York, 1963 p 77. Curclo, J. A.; Buttrey, D. V. E. Appl. Opt. 1966, 5 , 231. McGuffin, V. L.; Novotny. M. J . Chromatogr. 1981, 218, 179. McGuffin, V. L.; Novotny,‘M. Anal. Chem. 1981, 53, 948. Julin, B. G.; Vanderborn, H. W.: Klrkland, J. J. J. Chromatogr. 1975, 112, 443. Freed, D. J. Anal. Chem. 1975, 4 7 , 186. Putley, E. H. I n Optlcal and Infrared Detectors; Keyes, R. J., Ed.; Springer-Verlag: Berlin, 1980; Chapter 3. Busch, K. W.; Howell, N. 0.; Morrison, G. H. Anal. Chem. 1974, 4 6 , 1231. Boyd, R. W. Radbmetty and the DetecHon of Optical Radiation : Wiley: New York, 1983; Chapter 10.
r
RECEIVED for review January 27, 1987. Resubmitted July 9, 1987. Accepted July 9,1987. This work was supported by Baylor University Research Grant 021-S85-URC, and was presented at the 42nd Southwest Regional Meeting of the American Chemical Society, Houston, TX, Nov 1986.
Data Processing Techniques to Extract Pure-Component Spectra from Mixture Spectra and Their Application to Polymeric Systems Juwhan Liu and Jack L. Koenig* Department of Macromolecular Science, Case Western Reserve University, Cleueland, Ohio 44106
A nonlinear optimization technique was developed to extract purecomponenl spectra from the mixture spectra. The absorbances of the pure spectra and the concentrations are variables to be optimized In the algortthm proposed. Any prlor information on the concentrations and the reference spectra of the pure components is not required. As constraints, nonnegativlty of absorbances and concentrations are applied where the concentrations are used In terms of fractions. Through tests, three “pure” rotational isomer spectra of poly(ethyiene terephthalate) were obtained. A least-squares procedure was utilized to expand the scope of the proposed method. This technique Is applicable to ail systems described by the Beer-Lambert law.
Often a spectroscopist is confronted with the problem of identifying the unknown componentsof a composite spectrum. The idea of “unmixing” (1)or “spectroscopicpurification” (2) is to isolate the component spectra from the spectra of mixtures. Especially when a spectroscopist needs to determine the spectra of “pure” conformational isomers in a polymer system, an extraction process of the polymer spectra is required. In general, preparations of such “pure”conformational isomers are not possible and they are not modeled by suitable standard compounds. Moreover, additional difficulties arise in the estimation of a component spectrum if a proper measure of the concentrations of the components in the mixtures cannot be made or no other component spectra from a reference library are available. The ratio method, proposed by Hirschfeld (3) and later modified by Koenig and co-workers (4,5), is one of the methods to treat such situations and generally gives good 0003-2700/87/0359-2609$0 1.50/0
results for two-component systems. However, extension of this method to n (13)component cases requires a very restrictive set of conditions because it needs n points, one for each component,at which the absorption coefficient is nonzero for a component while the coefficients for all other components are zero. Another method called factor analysis (6) also deals with the conditions of no reference component spectra and no information on the concentrations. While the rotations of eigenspectra approaches (7-1 3) are based on the physical nonnegativity criteria of the absorbance intensities and/or the amounts of components, another series of approaches terms ”self modeling curve resolution” (14-1 7) are based on the determination of two bounding lines, between which lies a pure-componentspectrum, by using the given condition that the component spectra are normalized by area and by utilizing the fact that the first eigenvector correspondingto the largest eigenvalue has nonzero elements as well as the nonnegative character of absorbances and concentrations. Originally the self-modeling curve resolution method was developed for analysis of two-component mixtures (14),but later the method was expanded for three or more components (15). Also the limiting two boundary bands were reduced to a single band with the help of the concept of entropy minimization (16,17). The accomplishment of the former approaches of factor analysis depends largely on the degree of nonoverlap between component spectra. Similarly, in the latter approaches, it depends on the narrowness of the solution bands, which generally requires a wide range of concentrations for each component. In this paper, a new algorithm for extraction of pure-component spectra is proposed. No a priori knowledge of the spectra of the componentsand their concentrationsis required in this approach. The algorithm is based on a nonlinear 0 1987 American Chemical Society
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ANALYTICAL CHEMISTRY, VOL. 59, NO. 21, NOVEMBER 1, 1987
optimization approach where the absorbances of the pure spectra and the concentration terms are variables to be optimized.
EXPERIMENTAL SECTION Infrared spectra were collected on a Digilab FTS 20 E Fourier transform infrared (FTIR) spectrometer equipped with an MCT detector and transferred to a DEC VAX 11/780 computer operating under VMS for data processing. A series of spectra of poly(ethy1ene terephthalate) (PET)annealed at 230 and 200 O C under various annealing times described elsewhere (23) were analyzed as received to distinguish conformationalisomer spectra. All calculations needed to extract pure-componentspectra were performed by the program PURE written in FORTRAN 77. A Tektronix-type graphics terminal is utilized by the program to monitor the spectrum shapes at each iteration. The statistics on the results as well as the outputs of the optimization process are stored in files for future reference.
THEORY For a multicomponent system which is described by the Beer-Lambert law, the absorbances can be expressed as
i = 1 , 2 , ..., m,k = 1, 2, ..., n) a = (ah,; k = 1, 2, ..., n, j = 1, 2 , ...,p ) C=
(Cik;
(7)
a constrained optimization problem, and a transform technique (19) is used to convert it to an unconstrained one. The variables E and are transformed into the new variables a = (pi& i = 1, 2, ..., m, k = 1, 2, ..., n)and = (wkl;k = 1, 2, ..., n,j = 1, 2, ...,p ] given from the relations
n
where M,,is the absorbance of the ith mixture measured a t a given frequency j , cik the concentrationof the kth component in the ith mixture, ak, the absorbance or extinction coefficient of the kth component measured at frequency j , and n the number of spectroscopicallyindependent components in the mixtures. This equation is applied to spectra measured at unit path length or the values of ak,’s are scaled properly. In matrix notation, eq 1 is expressed more compactly where [MI is an ( m X p ) mixture spectra matrix with ith mixture absorbance spectrum in ith row, in which m is the number of mixtures and p is the number of frequency points in a spectrum, [C] is an ( m x n) concentration matrix, and [A] is an (n X p ) pure-component spectra matrix with a component spectrum in each row. In fact, eq 1 and 2 can be taken as a system of nonlinear equations in which q k ’ s and ak,’s serve as the unknowns. Actually one has ( m x p ) such equations and ( ( m + p ) x n) unknown variables. However the major problem in the analysis of systems described by eq 2 can be expressed by the equation (26) rank [MI = min (rank [C], rank [A]] (3) In other words the system is “underdetermined” even if one arranges so that ( m x p ) > ((m p ) X n). To overcome this problem one needs to acquire additional equations that are generally imposed as constraints. Of course, the conditions that the absorbance values and concentrationsmust be greater than or at least equal to zero constitute such constraints. But, they are not sufficient to isolate the component spectra uniquely, as demonstrated by Ohta (11). So more constraints are needed and such additions could be made if the concentrations are expressed in terms of percentages or fractions, where, for all i
In this way the constraints (eq 6) are all removed. The problem becomes r n P
MINIMIZE F(q,Cij) =
I3 CfiT(Q,U) i=l j=1
(9)
where
Among the various nonlinear least-squares techniques, the Levenverg-Marquardt algorithm (20) shows several advantages (21)such as the algorithm uses a damping parameter, and the Cholesky decomposition can be applied. Generally, most drawbacks in the methods dealing with large scale least-squares problems lie in the fact that it often costs too much to solve the Newton equations with large dimensions. For the system under consideration, however, the Jacobian matrix, [J], and its product [JIt[J] matrices have highly structured forms (Figure l),and their elements take little time to be constructed. Note that the Jacobian matrix has two parts because there are two kinds of variables, p and w. The product matrix [JIt[J] can be partitioned as
+
and its Cholesky decomposition is given by (22)
with
Here [Lll] is obtained from the Cholesky decomposition done on the submatrix [Kll], that is
(4) Fortunately, the constraints given by eq 4 and the nonnegativity of the concentrations, Cik, and of the absorbance intensities, akj, are generally found to be adequate for the extraction of pure-component spectra as will be shown below. Because there are more equations than the number of unknowns, the least-squares approach was adopted. Thus the problem is reduced to solving an optimization situation
[Kill =
[L1lIILlllt
(14)
[L12I-’[K121
(15)
and
[Wl,l =
Finally [L2J is calculated from the Cholesky decomposition of the modified matrix [K221
=
w221
-
[w121t[w121= [L22I[L22It
(16)
ANALYTICAL CHEMISTRY, VOL. 59, NO. 21, NOVEMBER 1, 1987 0.43
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Figure 1.
Matrix structures of [J] and [J]’[J]. Only the shadowed [J]‘[J] represent
parts have nonzero elements. The small squares in (n X n ) blocks.
In other words, the problem with large dimensions can be reduced to several subproblems, which involve Cholesky decompositions and matrix multiplications with lower dimensions. Also note that [K,,] and [K,,] are diagonal block matrices and very sparse (Figure 1). The selection of initial values is important in the sense of rapid convergence. With a set of initial values that are far from the solution, it is sometimes possible to end up with meaningless results. To avoid that, a scheme to select the initial values was devised and described below for two- or three-component cases, but the scheme can be extended to four or more component systems. First, for each mixture spectrum, the number of frequency points where the absorbances of the spectrum under considerationare greatest is the m mixture spectra is counted. Then three mixture spectra with the three greatest such numbers are indexed as spectrum I, II, and III. Each of these selected spectra is likely to contain the greatest amount of one of the components. They can be used as initial values for the absorbances directly (I and I1 for two-component cases, all three for three-component cases), but to obtain better initial values, the following procedure was used. The absorbances of the m mixture spectra are compared at each frequency and two values are calculated in such a way that the first value, say am=, is the maximum absorbance added by the difference between the maximum and median absorbances, and the second value, say amin,is the minimum absorbance subtracted by the difference between the median and the minimum absorbances. Now amaxis assigned to one of the three indexed spectra whose absorbance is equal or nearest to the maximum absorbance and, on the other hand, aminis assigned to the other two indexed spectra. This procedure is repeated throughout all frequency points, and then the modified indexed spectra I and I1 are taken as the initial values for the absorbances in the case of two-component mixtures, while all three spectra are used for three components. When desirable, smoothing could be done to these initial absorbance values to remove possible fluctuations according to the equation fork = 1, 2, ..., n a n d j = 1, 2, ..., p . Now that the initial values for the absorbances are obtained, those values for the concentrations can be simply calculated from the least-squares equation
Flgure 2. (a) First set of mixture spectra (0.07 unit offset). (b) Lorentzian bands extracted from the mixture spectra (dotted lines) and input Lorentzian bands (solid lines). Peak positions are 80, 50, and 20 cm-’ from bottom to top (scale expanded).
Although the least-squares procedure is applied here, the initial values obtained thus far do not satisfy the convergence criteria given below because the initial absorbance spectra are not those of pure components. Iterations, the refinements of the values, become necessary. The idea used in the initialization step is to exaggerate those mixture spectra that are bounding the other spectra. For example, in the self-modeling curve resolution approach, it can be shown, for two-component systems, two extreme mixture spectra are used as “inner boundaries” to limit the solution bands (14). Several criteria were used to control the iteration. As the system converges the value of the sum of squares (SSQ), its relative change, or the changes in the paraqeter values fall within some preset tolerances. These convergence criteria are seldom met when the number of data points become large (usually over 100). In such cases one needs to limit the maximum number of iterations. Because the sum of squares or the root-mean-squares deviations tend to reduce rapidly for the first few iterations and slowly afterward, 20 iterations were usually sufficient. Due to the high dimensions of matrices involved in the algorithm, the practical number of data points in a spectrum is limited up to about 200. A least-squares procedure can be utilized to overcome this difficulty (see eq 19).
RESULTS AND DISCUSSION As a demonstration of this procedure, three sets of test spectra were constructed from three Lorentzian curves with varying degree of overlap. The relative peak distances of the neighboring two peaks in each set of spectra, that is, the ratio of the difference in the peak positions and the s u m of the full width at half heights (fwhh) were set to be 0.375 (Figure 2a), 0.25 (Figure 3a), and 0.125 (Figure 4a). The same combinations were used for the three sets as follows: 0.6,0.2,0.2; 0.2,0.5,0.3; 0.2,0.3,0.5; 0.3,0.4,0.3; 0.1,0.3,0.6. The extracted
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ANALYTICAL CHEMISTRY, VOL. 59, NO. 21, NOVEMBER 1, 1987 0.m
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Flgure 3. (a) Second set of mixture spectra (0.07 unit offset). (b) Lorentzian bands extracted from the mixture spectra (dotted lines) and input Lorentzlan bands (solid lines). Peak posltlons are 70, 50, and 30 cm-' from bottom to top (scale expanded).
Figure 4. (a) Third set of mixture spectra (0.07 unit offset). (b) Lorentzlan bands extracted from the mixture spectra (dotted lines) and input Lorentzian bands (solid lines). Peak positions are 60, 50, and 40 cm-' from bottom to top (scale expanded).
pure-component spectra along with the input Lorentzian bands used in combinations are shown in Figures 2b, 3b, and 4b for each mixture set. The peak positions and the shapes
Flgure 5. (a) FTIR spectra of ethylbenzene-toluene-o-xylene mixtures (0.20 unit offset). The volume fractions are as follows, from bottom to top: 0.4, 0.4, 0.2; 0.1, 0.7, 0.2; 0.4, 0.3, 0.3; 0.2, 0.1, 0.7; 0.6, 0.3, 0.1. (b) Pure spectra extracted from the mixture spectra (dotted lines) and neat spectra (solM lines): bottom, ethylbenzene; middle, toluene; top, o-xylene (scale expanded).
are reproduced for all three cases, though the center peaks get slightly narrower as the degree of overlap increases. These examples show that a good estimation of pure spectra can be obtained from a limited number of mixture spectra where the components show a fair degree of overlap between their spectra and there is no isolated peak. Five spectra of three-component mixtures composed of ethylbenzene, toluene, and o-xylene were used to test the algorithm. The range selected was from 1200to 880 cm-'. The spectra are shown in Figure 5a. The factor analysis showed that there are three components in the five mixture spectra used, as expected. The extracted pure spectra and the pure spectra are shown in Figure 5b. Again the overall shapes and peak positions of the extracted spectra match fairly well with those of the original spectra. Finally this procedure was used for a series of PET polymers annealed a t 230 O C for various annealing times. Spectra of PET reflect the changes in trans and gauche rotational isomers. Former studies based on factor analysis in the fingerprint region (2,23)and C-H stretch region (24) revealed only two spectral components. For later comparison the 1540-1430-cm-' region was used to extract the spectra of the two spectroscopically independent components in that region, which are the gauche and the trans isomers. Clearly the factor analysis showed that there are only two spectroscopically distinguishable components in the region under consideration in agreement with previous results (2,23).The extracted pure spectra are shown in Figure 6. Also using the concurrent concentration data, a least-squares procedure was utilized to construct pure spectra having a wider range of frequencies
1-41 = ([Clt[Cl)-l[Clt[W
(19)
The spectra of the 1700-750-cm-' region (Figure 7a,b) as well as of the C-H stretch region (Figure 7c) obtained in this way
ANALYTICAL CHEMISTRY, VOL. 59, NO. 21, NOVEMBER 1, 1987 2.703
1 8
1-%5
2613
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1430
CM-1
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Figure 6. Conformational isomer spectra of PET extracted from six spectra of PET annealed at 230 OC of the 1540-1430 cm-’ region by assuming two components (scale expanded).
Table I. Results of Factor Analysis of FTIR Spectra of PET Annealed at 230 O C as a Function of Time
annealing time, min
component
1% (eigenvalue)
IE
X 10
IND
X 10
(a) 1430-1360 cm-’ Region 2.5 5.3 40 100 1000 6120
1 2 3 4 5 6
6.654 4.331 2.933 1.896 1.405 1.363
31.76 10.53 5.359 4.670 5.094
3.112 1.140 0.842 1.430 5.580
1
IC
(b) 1060-950 cm-’ Region 2.5 5.3 40 100 1000 6120
1
6.571 4.961 3.473 2.560 2.353 1.783
2 3 4 5 6
52.49 16.20 9.704 9.110 6.634
I 5.143 1.754 1.525 2.789 7.267
(c) Combined Region of a and b 2.5 5.3 40
100 1000 6120
1 2 3 4 5 6
6.915 5.053 3.602 2.766 2.514 2.303
45.62 15.02 9.898 9.644 9.412
4.470 1.625 1.555 2.593 10.31
(d) 920-820 cm-’ Region 2.5 5.3 40 100 1000 6120
1 2 3 4 5 6
6.121 4.491 2.878 2.111 1.408 0.697
31.83 8.518 5.037 3.116 1.988
3.119 0.922 0.791 0.954 2.178
agree well with those obtained by the ratio method (4,23,24). However, Lin and Koenig (23)succeeded in obtaining the pure spectra of the gauche isomer, the trans isomer in crystalline region (crystalline trans), and the trans in amorphous region (amorphous trans) and noted the spectral differences for the 1017-, 1024-, 1408-, and 1412-cm-’ modes. Their results suggest that these three isomers may be distinguished if one uses narrower ranges including the above modes. The results of factor analysis are shown in Table Ia-c, where two spectral regions from 1430 to 1360 cm-’ and from 1060 to 950 cm-’ were analyzed. Also the two regions were combined and analyzed altogether. In both regions as well as in the combined case, the eigenvalues and the imbedded errors (IE) level out after three components indicating three components, and the Malinowski indicator functions (IND) confirm this showing minima at the third component.
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CY-I
Flgure 7. Conformational isomer spectra of PET reconstructed by using the concentration values obtalned along with the spectra in Figure 6. (a) The amorphous gauche isomer (1700-750 cm-’). (b) The crystallineancCamorphoustrans Isomer (1700-750 cm-’). (c) Isomers in the C-H stretch region (3160-2820 cm-’).
Figures 8 and 9 show the PET spectra as function of annealing time in both regions. The spectra of the three isomers were successfully extracted, and though there are the same trends in the concentration values, their scales were found to be different in both regions. These discrepancies were apparently caused by the differences in the degree of overlap between the three “pure” isomer spectra in each region, and the 1430-1360 cm-’ region showed much more overlap than the 1060-950 cm-’ region. Thus it was desired to run the two sections of spectral data simultaneously in order to obtain more reliable concentrationinformation. The extracted “pure” isomer spectra are shown in Figures 8b and 9b for each region, and the concentrationvalues are given in Table IIa. As before, with these concentration values pure spectra spanning a wide range of frequencies were obtained by eq 19. It was found, however, that several spurious peaks appeared in the spectrum of the amorphous trans isomer, notably at 1454 and 898 cm-’. On comparison of the spectra in parts a and b of Figure 7, they obviously came from the contribution of the gauche isomer. On the other hand, there was evidence of “mixing”
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ANALYTICAL CHEMISTRY, VOL. 59, NO. 21, NOVEMBER 1, 1987
Table 11. Values of Concentrations of Isomers in a Series of FTIR Spectra of PET Annealed at 230 OC as a Function of Time
mixture
annealing time log (min)
%
gauche
%
%
crystalline trans
amorphous trans
(a) Values Obtained from the Program 1 2
3 4 5 6
0.398 0.724 1.602 2.000 3.000 3.787
73.3 39.5 31.0 28.4 17.4 14.9
8.5 22.5 24.9 35.5 48.2 65.7
18.2 38.0 44.1 36.1 34.4 29.4
(b) Normalized Values After Modified to “Unmix”the Contribution of the Gauche Isomer in the Spectra of the Amorphous trans Isomer 1 2 3 4 5 6
I
I Y
143.ccO
IsM.ccO
1410.01)
1370.ccO
m-1 Figure 8. (a) FTIR spectra of PET annealed at 230 O C for the times indicated from 1430 to 1360 cm-’ (0.50 unk offset). (b) Spectra of conformational Isomers in PET from 1430 to 1360 cm-‘ extracted from the spectra in part a (scale expanded). The peak at 1372 cm-’ in the amorphous trans spectrum indicates “mixing” by the gauche isomer.
\ IC60
lcoj
.
5%
m-‘ Figure 9. (a) FTIR spectra of PET annealed at 230 O C for the times indicated from 1060 to 950 cm-‘ (0.50 unit offset). (b) Spectra of conformational isomers in PET from 1060 to 950 cm-‘ extracted from the spectra in part a (scale expanded). The rump around 1045 cm-’ in the amorphous trans spectrum indicates “mixing” by the gauche isomer as in Figure 8b.
0.398 0.724 1.602 2.000 3.000 3.787
74.3 43.9 36.9 33.5 22.9 18.3
8.2 20.9 22.8 33.0 45.0 63.1
17.5 35.2 40.4 33.5 32.1 18.6
of the trans isomers in the spectrum of the gauche isomer. Based on these analysis, it was concluded that, in order to obtain the pure conformational spectra, one could safely remove the contribution of the gauche isomer in the spectrum of the amorphous trans isomer and add that portion to the spectrum of the gauche isomer without affecting the concentrations of the crystalline trans isomer. These contributions were determined from the scaling factor needed to subtract the peak a t 1454 cm-’ in the spectrum of the amorphous trans isomer obtained above. The final spectra of the three conformational isomers in PET thus obtained are given in Figure loa-c for the 1700-750 cm-l region and in Figure 10d for the C-H stretch region. The associated concentration values are given in Table IIb. Note that these values were normalized to give percentages. During inspection of the “pure” conformational spectra, it was found that the band at 849 cm-l also shows spectral differences between the amorphous and the crystalline trans isomers. Factor analysis supported this fact, giving three as the number of components in the region of 920-820 cm-l (Table Id). In this way, an unexpected spectral change could be detected. Lin and Koenig (23) extracted the pure spectra of isomers based on the ratio method combined with subtraction. The spectra of the gauche and trans isomers and thus the resulting three-isomer spectra obtained by the ratio method could possibly be mixed in the two regions under consideration. If the three-component assumption is correct, it is interesting to note the similarities, especially in the band modes around 1410 and 1020 cm-’. While the amount of the gauche isomer decreases and that of the crystalline trans isomer continually increases, the amorphous trans isomer shows a maximum with the log (annealing time). To check it further, another set of spectra of P E T annealed at 200 “C were fitted by using the combined region. The least-squares curve-fitting results are shown in Figure 11, where the fraction of each isomer is given as a function of log (annealing time). These results suggest that the gauche component is transformed into both the crystalline and the amorphous trans isomers at the same time in the first stage of the annealing process, and as the crystallization further develops, the transformation from the amorphous trans isomer to the crystalline trans isomer corresponds with that from the gauche isomer to the crystalline trans isomer. When the mixture data contain a lot of noise, the iteration usually reaches the maximum number of iterations (20).
ANALYTICAL CHEMISTRY, VOL. 59, NO. 21, NOVEMBER 1, 1987
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,: ,
80.
1j 43.
B
[ I
3’;
M.-
*
0.
010
015
l!O 115 210 215 W I N ; TIM ( L E M I N K S )
310
315
4.0
Flgure 11. The fractions of the conformational Isomers in PET annealed at 200 O C as function of the annealing time: X, gauche; 0, crystalline trans; A, amorphous trans:
were required to produce the results shown in Figures 2b, 3b, and 4b. When the equal path length condition is not met, normalization of the mixture spectra is feasible.
CONCLUSION A new algorithm utilizing a nonlinear optimization technique was developed to extract pure-component spectra from their mixture spectra. It is not necessary to have information on the concentrationsor reference spectra of pure components. A scheme to select proper initial absorbance values for refinement was introduced. Through tests, the proposed algorithm was applied to a polymer system of PET and the extraction of the “pure” spectra of the gauche, the crystalline trans, and the amorphous trans isomers of PET was achieved. Registry No. PET, 25038-59-9; H3CCH,CsH5, 100-41-4; CGH~CH~, 108-88-3;2-H3CCGH,CH3, 95-47-6.
LITERATURE CITED
Flgure 10. “Pure” spectra of the conformational isomers in FET. (a) The amorphous gauche isomer (1700-750 cm-’). (b) The crystalline trans isomer (1700-750 cm-’). (c) The amorphous trans isomer (1700-750 cm-’). (d) Isomers in the C-H stretch region (3160-2820 cm-’).
Hirschfeld, T. Appl. Opt. 1978, 17, 1400. Antoon, M. K.; D’Esposito, L.; Koenig, J. L. Appl. Spectrosc. 1979, 3 3 , 351. Hirschfeid, T. H. Anal. Chem. 1978, 4 8 , 721. Koenig, J. L.; D’Esposito, L.; Antoon, M. K. Appl. Spectrosc. 1977, 3 1 , 292. Koenig, J. L.; Kormos, D. Appl. Spectrosc. 1979, 3 3 , 349. Malinowski, E. R.; Howery, D. G. Factor Analysls in Chemistry; Wiiey: New York, 1980. Rozett, R. W.; Petersen, E. M. Anal. Chem. 1975, 4 7 , 1301. Swain, C. G.; Bryndza, H. E.; Swain, M. S. J. Chem. I n f . Comput. Sci. lQ7Q. -, 19. 19. Giiette, P. C.; Lando, J. B.; Koenig, J. L. Anal. Chem. 1983, 55, 630. Macnaughtan, D., Jr.; Rogers, L. B.; Wernimont, G. Anal. Chem. 1972. 4 4 , 1421. Ohta, N. Anal. Chem. 1973, 45, 553. Martens, H. Anal. Chim. Acta 1979, 112, 423. Warner, I. M.; Christian, G. D.; Davidson, E R.; Callis, J. B. Anal. Chem. 1977, 49. 564. Lawton, W. H.; Sylvestre, E. A. Technometrlcs 1971, 13, 617. Sasaki, K.; Kawata, S.; Minami, S. Appl. Opt. 1983, 2 2 , 3599. Sasaki. K.; Kawata, S.; Minami. S. Appl. Opt. 1984, 2 3 , 1955. Kawata, S.; Komeda, H.; Sasaki, K.; Minami, S. Appl. Spectrosc. 1985, 3 9 , 610. Kankare, J. J. Anal. Chem. 1970, 42, 1322. Gill, P. E.; Murray, W.; Wright, M. H. Practical Optimlzatlon; Academic: New York, 1961. Marquardt, D. W. J. Soc. Ind. Appl. Math. 1963, 1 1 , 431. Dennis, J. E., Jr. “Some Computational Techniques for the Nonlinear Least Squares Problem”; I n Numerlcal Solutlon of Systems of Nonllnear Algebralc Equations; Byrne, G. D., Hail, C. A,, Eds.; Academic: New York, 1973. George, A. SIAM J. Numer. Anal. 1974, 11, 585. Lin, S.-B.; Koenig, J. L. J. folym. Sci., folym. f h y s . Ed. 1982, 20, 2277. Koenig, J. L. Acc. Chem. Res. 1981, 14, 171.
However, the root-mean-square deviations were found to be less than 2 in most cases, and the correlation coefficients between a mixture spectrum and that reconstructed from the extracted pure spectra were close enough to unity, which means excellent fits. The initialization scheme usually gave useful initial values. For example, only four or five iterations
RECEIVED for review September 12,1986. Resubmitted March 12,1987. Accepted July 2,1987. The authors acknowledge the Polymer Section of the National Science Foundation for the financial support of this work under Grant No. DMR8316884.
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