Two-Dimensional Correlation Analysis in Application to a
Kinetic Model of Parallel Reactions
THOU-LONG CHIN and KING-CHUEN LIN*
Department of Chemistry, National Taiwan University, and Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 106, Taiwan, Republic of China
By applying generalized two-dimensional (2D) correlation analysis as reported by Noda, we have systematically studied a kinetic model of parallel reactions. Given the related rate constants and absorp-tion coef cients, the correlaabsorp-tion between reactant and products are analyzed. The reactant–reactant, reactant–product, and product– product pairs are found to be synchronously correlated, and their intensities increase with increase of the rate constant and the ab-sorption coef cient. On the other hand, only the reactant–product pairs show in the asynchronous spectra. Their intensities also de-pend proportionally on the rate constant and the absorption coef- cient. The in uence of signal-to-noise ratio (S/N) and overlapped spectra are further discussed. The resulting synchronous and asyn-chronous correlation spectra for the kinetic model appear to be weakly in uenced by poor quality of the signal when the referen ce spectrum is set at zero. The ratio of asynchronous to synchronous correlation intensity yields a coherence spectrum. This spectrum remains a constant intensity for all the correlated peaks, being free from the in uence of rate constant and absorption coef cient as well as being weakly disturbed by a small S/N ratio. It also provides a way to evaluate the extent of spectral overlap between two peaks. The coherence spectrum is useful to characterize the type of parallel reactions.
Index Headings: Two-dimensional correlation spectrum; Coherence
spectrum; Kinetic model; Parallel reaction.
INTRODUCTION
The basic concept of two-dim ensional (2D) correla-tion infrared (IR) spectroscopy was rst introduced more than a decade ago.1Given a perturbation-induced time-dependent IR signal, a cross-correlation analysis was used to construct a 2D IR spectrum. The extension of a second dimension provides advantages such as simpli cation of the complicated spectra with over-lapped peaks, enhancement of the spectral resolution, and identi cation of various intra-molecular and inter-molecular interactions through selective correlation of the IR peaks.1 –3 Nevertheless, when the 2D correlation analysis of IR spectroscopy was rst reported, the time-dependent IR signal was limited to perturbation of a sinusoidal sm all amplitude. The method was applied mostly to the vibrational spectral analysis of polymers and liquid crystals as perturbed by m echanical or elec-trical force.2 –7 In 1993, Noda developed a generalized 2D correlation spectroscopy, which can be applied to analysis of any spectral intensity uctuations of an ar-bitrary function of tim e or any other physical variable.8
Received 8 July 2002; accepted 25 September 2002. * Author to whom correspondence should be sent.
It has become a universal spectroscopy applicable to a very wide range of studies.9 –14
The 2D correlation plot is obtained by the correlation intensity as a function of two independent spectral vari-ables.8,13,14 The synchronous correlation intensity charac-terizes the degree of coherence between two signals that are measured simultaneously. It reaches the maximum if the variations of two dynamic spectra are totally in phase with each other and the minimum if they are antiphase with each other. The peaks along the diagonal line are referred to as autopeaks, of which the correlation inten-sity corresponds to an autocorrelation of a dynamic spec-trum at a particular spectral variable. The peaks located at off-diagonal positions are referred to as the crosspeaks of the dynamic spectra. The crosspeaks reveal the infor-mation on the inter- or intra-molecular interactions among the functional groups. On the other hand, the asynchronous correlation intensity indicates the indepen-dent or mutually decoupled nature of the dynamic spectra under a perturbation. It becomes the maximum if the two dynamic signals are orthogonal to each other and vanish-es if they are in phase or antiphase with each other. The asynchronous correlation spectra consist of only the crosspeaks that are antisymmetric with respect to the di-agonal line.13
The 2D correlation analysis provides advantages such as those mentioned above. Nevertheless, the change of band position and band width, the baseline shift, and the quality of signal/noise ratio may complicate the interpre-tation of the correlation spectra.15–20 The in uence of these parameter variations on the 2D correlation spectra has been thoroughly investigated to effectively simplify the resultant contour maps, from which more information may be readily extracted.15–20
In this work, we attempt to systematically investigate the 2D correlation analysis for a kinetic model of parallel reactions. Most studies on the kinetic system emphasize the decay behavior of the reactants but are less concerned with the ways to generate the products. In the parallel reactions studied, only one reactant is considered, with three absorption bands, while three different products are involved, each with one absorption band. The 2D corre-lation spectra are analyzed as functions of reaction rate constant and absorption coef cient. The in uence of sig-nal-to-noise (S/N) ratio and the overlapped spectral bands are discussed. Furthermore, when a ratio of asynchro-nous/synchronous correlation intensity is treated, the re-sultant coherence spectra appear to be independent of the reaction rate constant and the absorption coef cient. The
pattern recognition for the parallel reactions becomes sig-ni cant.
CORRELATION ANALYSIS
The generalized 2D correlation analysis may be ap-plied to any signals uctuating as an arbitrary function of time or any other physical variable.8,13 In this work, the 2D correlation analysis is applied to the following parallel reactions: k12 A1 ® A2 k13 A1 ® A3 e k1n A1 ® An (1)
where kijindicates the corresponding rate constant for the
species Ai to form Aj. The concentration changes can be
monitored by molecular absorption spectroscopy at a par-ticular spectral variable, vi. The integrated form of the
concentration of the reactant A1 and the products Aj ( j 5
2, 3, . . . n) may be readily obtained. The concentration change of each species is characterized by a form of ex-ponential function. For conversion of 2D correlation, the dynamic spectral intensity y(v1, t) at the absorption spec-tral variable v1is rst Fourier transformed to yield Y1(w); w is the Fourier frequency. The conjugate of the Fourier
transform, Y2*(w), of the other dynamic spectral intensity y(v2, t) observed at the absorption spectral variable v2 is likewise obtained. Given the Fourier transforms, Y1(w) and Y2*(w), of the dynamic spectral intensities of y(v1, t) and y(v2, t), the Fourier frequency-dependent synchro-nous and asynchrosynchro-nous correlation spectra, so-called co-spectrum and quad-co-spectrum, respectively, may be ob-tained as:8,13
1 Re Re Im Im
fw(v , v ) 51 2 T[Y (w)Y (w) 1 Y (w)Y (w)]1 2 1 2 (2)
1 Im Re Re Im
cw(v , v ) 51 2 T[Y (w)Y (w) 2 Y (w)Y (w)]1 2 1 2 (3) where YRe and YIm indicate the real and imaginary
com-i i
ponents of the complex variable Yi; T is the observed
period for the dynamic spectral intensities of y(v1, t) and y(v2, t). The synchronous and asynchronous correlation spectra, F(v1, v2) and C(v1, v2), may then be expressed by: ` 1 F(v , v ) 51 2
E
fw(v , v ) dw1 2 (4) p 0 ` 1 C(v , v ) 51 2E
cw(v , v ) dw1 2 (5) p 0Note that F(v1, v2) and C(v1, v2) denote the real and imag-inary components, respectively, of the complex 2D cor-relation intensities of the two dynamic spectral variations. In this work, a kinetic model of parallel reactions is
made up of three elementary steps. The reactant A1 is assumed to have three absorption bands with absorption coef cients «11, «12, and «13 observed at v11, v12, and v13, respectively. For the products A2, A3, and A4, each has one absorption band with a corresponding absorption co-ef cient of «21, «31, and «41 observed at v21, v31, and v41, respectively. The 2D correlation spectra are studied as a result of variation of rate constant, absorption intensity, and the S/N ratio.
For studying the noise effect, a xed deviation pro-duced by a random number generator was added to each signal point, so that the S/N ratio may be held at different levels. For calculating the S/N ratio, the signal is counted from the averaged baseline to the averaged peak of each spectrum. MATLAB 6.1, implem ented on a personal computer, was used to carr y out all the cor-relation analysis. The time evolutions of the reactant and the products were computed num erically by using the Runge–Kutta–Fehlberg method. Then an FFT pro-gram was called to Fourier transform these data to the frequency coordinate.
RESULTS AND DISCUSSION
Two-Dimensional Correlation Analysis for a Par-allel Reaction System. Given the parPar-allel reactions as in
Eq. 1 with three products, the related rate constants k12, k13, and k14 are all assumed to be equal to 1 s21; the absorption spectral variables for the reactant A1 are at v11 5 15, v12 5 45, and v13 555, and those for the products A2, A3, and A4 are at v21 5 35, v31 5 65, and v41 5 85, and the corresponding absorption coef cients are all set at 1. The initial concentration of A1 is assumed to be 10 000, while the product concentrations are zero. The resulting 2D correlation analysis for synchronous and asynchronous spectra are shown in Fig. 1. In the syn-chronous correlation contour map, two aspects may be discerned. First, the magnitudes of the synchronous tra are all positive. Here we consider the reference spec-trum to be equal to zero, such that the resulting synchro-nous peaks become positive. If an arbitrary reference spectrum is taken, then the synchronous peaks may change sign.15 Second, all the reactant and products ex-hibit autopeaks in the diagonal line and the reactant–re-actant, reactant–product, and product–product pairs show the crosspeaks. When the reactant decreases, the products may increase. The reactant is closely associated with the product. The products are also related to each other. The branching ratio of the product concentrations relies on the ratio of their corresponding rate constants. The inten-sities of reactant–reactant correlation appear to be stron-ger than those of reactant–product and product–product correlations.
As shown in Fig. 1, the asynchronous correlation spectrum is composed of reactant–product correlations, which include the 15–35, 15–65, 15–85, 45–35, 45–65, 45–85, 55–35, 55– 65, and 55–85 pairs. Their intensi-ties appear to be the same as a result of equal amounts of the related rate constants and absorption coef cients. Since the rising time for all the products are the sam e, their asynchronous correlations becom e zero. The ab-sorption spectral variables at 15, 45, and 55 stem from
FIG. 1. Contour maps of (a) synchronous and (b) asynchronous
cor-relation spectra for a kinetic model of parallel reactions. (c) Coherence spectra, obtained by the ratio of the asynchronous to synchronous cor-relation spectra. The absorption spectral variables for the reactant A1
are at v115 15, v125 45, and v13 5 55; those for the products A2, A3,
and A4 are at v21 5 35, v31 5 65, and v41 5 85. The rate constants k12
5 k13 5 k14 5 1 s2 1; the corresponding absorption coef cients are all
set at 1.
FIG. 2. Contour maps of (a) synchronous and (b) asynchronous
cor-relation spectra for a kinetic model of parallel reactions. (c) Coherence spectra. The conditions are the same as in Fig. 1, except for the rate constants k125 1, k135 2, and k145 3 s2 1.
the same reactant A1, and thus the reactant–reactant asynchronous correlations also disappear. If the ratio of asynchronous to synchronous correlation intensity is treated, the resultant coherence spectrum is made up of cylinder-like peaks with equal intensities for both pos-itive and negative components. The result is also shown in Fig. 1. Note that a blinding lter is used to remove the noise from the asynchronous correlation spectra in the normalization process. The treatment is similar to that used for the global 2D phase map by Ozaki and co-w orkers.19 If the asynchronous correlation intensity is smaller than a threshold value, then the weak signal
or noise will be ignored. Thus, the 2D pattern of the coherence spectra looks like the asynchronous spectra. The arctangent of the coherence actually corresponds to the global phase angle.19
The intensities of synchronous and asynchronous cor-relations should be subject to the variation of rate stants. We now consider the following case. The rate con-stants are changed to k12 5 1, k13 5 2, and k14 5 3 s21, while the absorption coef cients remain the same. As shown in Fig. 2, the product A4, with a larger rate con-stant, results in a larger intensity in the synchronous cor-relation for either autopeaks or crosspeaks, while A2, with a smaller rate constant, results in a smaller intensity. On the other hand, a larger rate constant such as k14may lead to a larger asynchronous correlation intensity. Therefore, the pairs of 85–15, 85– 45, and 85–55 show larger peaks
FIG. 3. Contour maps of (a) synchronous and (b) asynchronous
cor-relation spectra for a kinetic model of parallel reactions. (c) Coherence spectra. The rate constants are k12 5 1, k13 5 2, and k14 5 3 s21. The
absorption coef cients for the reactant are changed to «125 3 at v125
15, «135 2 at v13 5 45, «145 1 at v145 55, and those for the products
are changed to «215 1 at v215 35, «315 2 at v315 65, and «415 3 at v415 85.
FIG. 4. Noise effect on (a) amplitude and (b) phase of Fourier
trans-form of y1 5 exp(23t) for three replicates under a signal-to-noise (S/
N) ratio equal to 2.
than the pairs of 35–15, 35–45, and 35–55. Again, the coherence spectra give rise to cylinder-like peaks with equal intensity, independent of the variation of rate con-stants.
To understand the in uence of the absorption coef -cient on the 2D correlation intensities, the rate constants are all xed at 1 s21, and then the absorption coef cients for the reactant are changed to «12 5 3 at v12 5 15, «13 5 2 at v13 5 45, and «14 5 1 at v14 5 55, and those for the products are changed to «21 5 1 at v21 5 35, «31 5 2 at v31 5 65, and «41 5 3 at v41 5 85. As shown in Fig. 3, for the synchronous spectrum, the autopeak at 15–15 has the largest intensity. In spite of the same absorption
coef cient, the autopeak for the product–product pair at 85–85 (or 65–65 and 35–35) is smaller than that for the reactant–reactant pair at 15–15 (or 45– 45 and 55–55). Its interpretation arises from the fact that the depleted amount of reactant is distributed into three channels of products and each channel gains only a fraction. For the crosspeaks, a larger absorption coef cient may lead to stronger correlation intensity. Thus, the reactant–reactant pair at 15– 45 yields the strongest intensity, while the re-actant–product pair at 55–35 is the weakest. For the asyn-chronous spectrum, the only existing reactant–product correlation shows that the 15–85 peak is the strongest, while the 35–55 peak is the weakest. Similar to the de-pendence of the rate constants, the intensities for the 2D correlation spectra are thus subject to the absorption co-ef cient associated with each correlated pair. The coher-ence spectrum is also characterized by cylinder-like peaks with equal intensity, which is similar to those in Figs. 1 and 2.
By solving the related equations, we attempt to analyze why the product–product asynchronous spectra disappear and the coherence spectra retain the same peak intensities. For the parallel reactions, the solutions to Eq. 1 give:
FIG. 5. Comparison of co-spectrum and quad-spectrum correlating the
kinetic equations of y1 5 exp(23t) and y2 5 exp(27t) for three
repli-cates under the conditions of (a) S/N 5 5 and (b) S/N 5 2.
FIG. 6. Time evolution of parallel reactions as described in the text
under the conditions of (a) S/N 5 5 and (b) S/N 5 1. The related parameters are k125 1, k135 1, k145 1 s2 1, «12 5 1 at v12 5 15, «135 1 at v13 5 45, «145 1 at v14 5 55, «215 1 at v21 5 35, «315 1 at v31 5 65, and «415 1 at v415 85. 2k tt A 5 A k e1 10 t A k10 1n 2k t t A 5n k (1 2 e ) (6) t n k 5t
O
k1i (7) i51where A10 indicates the initial concentration of the reac-tant A1; k1 i is the rate constant for A1®Ai (i 5 2, 3, . . .
n), and kt, a summation of all the individual rate constants
of the products, is indicative of the rising rate for the products. Note that all the products have identical kt
val-ues. It is the same values of kt, but not the same values
of individual k1 i of the products, that lead to the zero intensity of product–product asynchronous peaks. This fact may be realized from Figs. 2 and 3, in which dif-ferent k1 i of products are considered.
Following the Fourier transform of reactant and prod-ucts, their synchronous and asynchronous correlation may be simpli ed as:
F(v , v ) 5 k A (v )A (v )B(k )1 n 1n 1 n t (8) C(v , v ) 5 k A (v )A (v )C (k )1 n 1n 1 n t (9)
where A(v1) and A(vn) are the absorption intensities of the
reactant and the product at spectral variable v1 and vn,
respectively. B(kt) and C(kt) are the kt-dependent
for-mulae obtained from the correlation analysis of Fourier transform of A1 and An in Eqs. 6 and 7. The values of
k1 n, A(v1), and A(vn) are obtained by the multiplication of
the coef cient terms in Eqs. 6 and 7, remaining the same in the synchronous and asynchronous correlation spectra. Accordingly, the ratio of C(v1, vn) to F(v1, vn) should give
rise to a constant value R, i.e.:
C (k )t
R 5 (10)
B(k )t
Noise Effect on Two-Dimensional Correlation Analysis for Parallel Reactions. The poor quality of
the obtained spectra may severely affect the reliability of the 2D correlation analysis and complicate the in-terpretation.1 5– 19 B efore exam ining the noise effect on the parallel reactions, we demonstrate its in uence on the correlation analysis between two kinetic equations,
y(v1, t) 5 exp(23t) and y(v2, t) 5 exp(27t). W hen the signal-to-noise ratio (S/N) is assumed to be 2, the am-plitude and phase angle of the Fourier transform of
FIG. 7. Contour maps of (a) synchronous and (b) asynchronous
cor-relation spectra and (c) coherence spectra for the parallel reactions given in Fig. 6 under the condition of S/N 5 1.
FIG. 8. Time dependence of reactant and product intensity for a
kinetic reaction under the condition of S/N 5 5 in (a) a spectral coordinate and (b) a temporal coordinate. The related param eters are
k12 5 k135 1 s2 1, «12 5 3 at v125 25, «13 5 1 at v135 45, and «21
5 0.3 at v215 30.
three replicates are given in Fig. 4. The noise appar-ently causes severe uctuation of the phase angle in the region of high-frequency components, at which the am-plitudes are very sm all. Thus, the resulting 2D corre-lation spectrum becomes slightly affected even with a poor signal quality. Figure 5 shows a com parison for fw(v1, v2) and cw(v1, v2) between S/N 5 2 and 5 for three replicates. They reveal sim ilar qualities for both correlation spectra.
The parallel reactions, made up of the same kinetic equations as in the previous section, are regarded with different S/N ratios for inspection of the noise effect. As the related parameters remain the same as in Fig. 1, the time evolution of reactant and products for S/N 5 5 and S/N 5 1 are compared in Fig. 6. The decrease of the S/ N ratio makes it hard to gain any quantitative information or even discern the type of reaction. However, the
cor-responding synchronous and asynchronous correlation spectra are surprisingly weakly disturbed by the poor sig-nal quality. Figure 7 shows the case for S/N 5 1. The noise-free nature of these correlation spectra has been explained earlier. Furthermore, the coherence spectra (Fig. 7c) give rise to constant intensities for all the cor-related peaks, which are also weakly disturbed by the poor S/N ratio. One should note that the reference spec-trum is taken as zero in this work. Under this condition, Czarnecki also came to a consistent conclusion that the 2D correlation spectra may be free from heavy distortion of noise and/or baseline uctuations. While taking an ap-propriate reference spectrum into account, the number of correlation peaks may be possibly reduced to simply the 2D spectra.15
In uence of O verlapped Spectra in Parallel Re-actions. The 2D correlation analysis is advantageous
to enhance the separation for the overlapped spectral bands. This advantage is also found in the analysis of a kinetic system. Given a simple parallel reaction, the absorption coef cients for the reactant are assumed to be «12 5 3 at v12 5 25 and «13 5 1 at v1 3 5 45, while the absorption coef cient for the product is «21 5 0.3 at v2 1 5 30. The rate constants are all assumed to be 1
FIG. 9. Contour maps of (a) synchronous and (b) asynchronous
cor-relation spectra, and (c) coherence spectra for the kinetic reaction given in Fig. 8.
s21. The full width at half-maxim um (FW HM) of all the peaks remains 5, with a signal-to-noise ratio at 5. Due to a weak absorption of the product and the severe spectral overlap between v 5 25 and 30, it is hard to distinguish the product from the reactant, as shown in Fig. 8. After the analysis of 2D correlation, two peaks for the reactant and the product can be clearly recog-nized at the spectral variable 25 and 30 from the achronous correlation spectrum, but not from the syn-chronous correlation spectrum yielding all the positive peaks (Fig. 9). It may not be easy to evaluate the over-lapped region of two peaks from either synchronous or asynchronous correlation spectra. However, when the coherence spectrum is treated, one may easily identify the extent of the spectral overlap from the shaded area of the contour map shown in Fig. 9. The coherence spectrum plays a role of pattern recognition for the type
of parallel reactions. The spectrum remains at constant intensity for all the correlated peaks, being free from the in uence of the reaction rate constant and absorp-tion coef cient, as well as being weakly disturbed by a poor quality of signal. It also provides a way to eval-uate the extent of spectral overlap between two peaks.
CONCLUSION
Two-dimensional correlation analysis has been sys-tem atically applied to a kinetic model of parallel re-actions. Given the related rate constants and the ab-sorption coef cients, the temporal evolution of reactant and products are determined and their correlations are then analyzed. The reactant–reactant, reactant–product, and product–product pairs are found to be synchro-nously correlated and their intensities increase with an increase of the rate constant and the absorption coef- cient. On the other hand, only the reactant–product pairs show in the asynchronous spectra. The intensities also depend proportionally on the rate constant and the absorption coef cient.
The co-spectra and quad-spectra are analyzed, re-vealing that the major contribution to the amplitude lies in the low Fourier frequency region. The noise effect results in a marked uctuation of the phase angle in the high Fourier frequency region. The amplitude con-tribution in this region is relatively sm all, such that the resulting 2D synchronous and asynchronous correlation spectra may be weakly in uenced by a poor quality of signal. The coherence spectrum retains constant peak intensity, irrespective of a small signal-to-noise ratio and variation of the rate constants and absorption co-ef cients. This spectrum may well characterize the type of parallel reaction, even when the temporal or spec-tral-dependent spectra of the species in the reaction are hardly discerned due to serious noise effect or spectral overlap.
ACKNOWLEDGMENT
This work was supported by the National Science Council, Republic of China, under contract no. NSC 90-2113-M-002-036.
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