Chapter III Experimental and Analysis
III- 4. Analytical method: Singular value decomposition
Singular value decomposition (SVD) is an important factorization method of a complex matrix. This technique can be employed in principal component analysis (PCA); thus it has orthogonal matrix (n × n). The diagonal elements of the matrix W are called singular values.
Equation III-10 can be explicitly written in terms of matrix elements:
11 1 11 1 1 11 1
Let the matrix X represent a set of time-resolved spectra with a row corresponding to the spectrum observed at a given time. In this case, uk represents a time dependence, while vk corresponds to the intrinsic spectrum for the kth component. Contributions of the kth component to the overall matrix X are determined by the singular value Wk: the larger Wk is, the more significantly the kth component contributes. The usefulness of SVD stems from the fact that components with small singular values can be neglected.
How does SVD analysis work? Consider the following two cases typically encountered in spectral analysis using SVD: (1) SVD yields only one or two major singular values [see Fig.
III-5(a)]. (2) SVD yields several non-negligible singular values [Fig. III-5(b)].
In case 1 [Fig. III-5(a)], the presence of the two major singular values indicates that, in principle, two and only two molecular species are responsible for the original data set (matrix X). Such a case can be found, for example, when performing the SVD of a series of absorption spectra in which absorption band 1 increases and concomitantly another band 2 decreases as a function of concentration. We are able to disregard all the other singular values and to focus on the behavior of the two components 1 and 2. Mathematically, this simplification implies
1 2
1 1 mathematical operation and hence there are no physical meanings attached to vectors u and v as they stand. Those vectors need to be reconstructed by taking linear combinations of u and v. At this stage, physics (or chemistry) behind the observed phenomenon comes into play. In order to obtain physically meaningful vectors u′ and v′, we need to assume model functions that u and v are expected to obey. By way of example, in the SVD analysis of χ-dependent IRelectroabsorption spectra [27], the model functions for the -dependent and -independent components have been assumed to be 13cos2
and constant with respect to χ, respectively.Mathematically, this reconstruction corresponds to inserting the product of a transformation matrix K and its inverse K−1 between matrices U and W, namely, inserting a unit matrix E = KK−1. The matrix elements of K are determined by least-squares fitting to the model functions.
1 2
1 1 1
1 2
1In case 2, there are many (typically more than three) singular values with non-negligible amplitudes, as shown in Fig. III-5(b). Such a plot is characteristic of the data in which one component undergoes a continuous shift as a function of a variable such as concentration or time. If an absorption band continuously shifts from red to blue or vice versa with concentration, a plot of the singular values so obtained resembles Fig. III-5(b). In comparison with case 1, SVD analysis is silent in this case. However, it does provide important information on whether the observed data arise from a few components in equilibrium or a single, continuously changing component.
III-5. Electroabsorption (A) and intensity difference (I) spectra
In this section, we would like to demonstrate how the presence of an offset in the I spectrum that is possibly caused by the electronics results in artifacts in the A spectrum. As already described in Chapter II, the A spectrum is calculated from the I spectrum according to Eq. II-1. Equation II-1 shows that the A signal depends not only on I but also on I. This
fact may give rise to artificial A peaks although there is no intensity change of an IR transition induced by an electric field. Suppose that there is a vibrational band at 1150 cm−1. It appears as a dip in the intensity (I) spectrum [Fig. III-6(a)]. Here, we consider four different types of the I spectrum [Fig. III-6(b)]: (i) Neither I signal nor an offset, namely, I = 0 (red spectrum). (ii) There is an absorption intensity change of −4.4 × 10−7 at 1150 cm−1 without any offset (purple spectrum). This is the ideal case. (iii) The I spectrum contains both constant offset of 5 × 10−7 and absorption intensity change of −4.4 × 10−7 at 1150 cm−1 (blue spectrum). (iv) The I spectrum contains a constant offset of 8 × 10−7 but no absorption intensity change at 1150 cm−1 (green spectrum). Unfortunately, we sometimes encounter cases iii and iv, producing an unwanted distortion of the spectrum or emergence of an artifact.
In each case, the A spectrum is simulated by using Eq. II-1. The resulting A spectra for the four cases are displayed in Fig. III-6(c). Needless to say, in case i, there is no A signal everywhere. In case ii, a positive peak is observed at 1150 cm−1, which is what we want to obtain in actual experiments. In case iii, a positive peak is still seen on top of an offset of ~2 × 10−6. It should be noticed, however, that the peak height at 1150 cm−1 is somewhat diminished compared to that in case ii. Furthermore, the peak at 1150 cm−1 accompanies tiny dips at both edges. These features showing up in the A spectrum should have no molecular origin because the corresponding I spectrum shows no such features.
What is worse, the A spectrum of case iv displays a negative peak of similar amplitude at 1150 cm−1. Clearly, this negative peak is an artifact; there is no absorption intensity change in the I spectrum [see Fig. III-6(b), green line].
The simple simulation mentioned above gives an important caveat: if one does not monitor the I spectrum and focus exclusively on the A spectrum, one may be fooled by an artifact originating from an offset in the I spectrum and hence reach a false conclusion regarding the electric-field effects on the molecule. Therefore, it is crucial to confirm that the
I spectrum shows intensity changes at vibrational transition frequencies before interpreting the A spectrum.
Figure ΙIΙ-1. Experimental setup of IR electroabsorption spectroscopy.
Figure ΙIΙ-2. Schematic of an AC-coupled amplification technique. (a) Output of the detector consisting of a large DC offset and a small AC component. (b) The output is AC-coupled and only the AC component remains. (c) The AC component is then amplified.
(c) (b) (a)
Figure ΙIΙ-3. Configuration of our sample cell. A/A’ are brass cell holders, B/B’ are Si plates, and C is a PET film used as a spacer.
Figure ΙIΙ-4. RC circuit equivalent to the sample cell (Figure ΙΙΙ-3.). R1 is the resistance between A and B, R2 is that between A’ and B’, and CC is the capacitance of the capacitor C.
Figure ΙIΙ-5. Plots of singular values typically obtained in singular value decomposition analysis.
(a)
(b)
Figure ΙIΙ-6. (a) Intensity spectrum. (b) I spectra with neither signal nor offset (red line);
with signal only (purple line); with both signal and offset (blue line); and with offset only (green line). (c) Corresponding A spectra simulated using Eq. II-1.
(b) (a)
(c)
Chapter IV
Infrared electroabsorbtion spectroscopic study of
water in 1,4-dioxane
IV-1. Introduction
In this chapter, an IR electroabsorption spectroscopic study of water dissolved in 1,4-dioxane is presented. As already described in Chapter I, the O–H stretch [ν(OH)]
transition of water occurs as an intense, broad band. Because of the featureless band profile, few clues can be obtained from the steady-state ν(OH) absorption spectrum alone. More insights should be gained by investigating water’s responses to an external perturbation, for which we adopt here the application of an external electric field. Since water is polar (P = 2–3 D [36]), it is expected to respond to an externally applied electric field via electrostatic interactions.
Although the ultimate goal is the understanding of pure water, there arise several technical difficulties when conducting IR electroabsorption measurements on pure water.
Since water has a considerably large dielectric constant (r ≈ 80), it causes a significant RC circuit problem described in Chapter III. In addition, the molar extinction coefficient of the ν(OH) band of water is so large that the transmittance in the ν(OH) region can easily fall below the detection limit of the apparatus. A straightforward remedy for this problem would be to make the sample extremely thin, typically thinner than 1 m [37]. However, it would be inconvenient to employ such a thin sample cell. In this work, we dilute water with solvent.
The solvent we used, 1,4-dioxane, is an aprotic solvent with nearly zero dipole moment [36], so the electric-field effects on 1,4-dioxane is expected to be minor.
Water dissolved in 1,4-dioxane, although at a low concentration, is essentially different from pure water. A highly structured network of hydrogen bonding that is present in pure water is lost completely or broken to a great extent in 1,4-dioxane solution. Nevertheless, we believe that it is important as a bottom-up approach to study 1,4-dioxane solution of water, in which water may be thought of as more “free” solute molecules. Such a study can fill a huge gap lying between research for the gas phase and pure liquid. Another important issue relevant
to water in solution is solvation. Solvation is one of the most fundamental phenomena that take place in the solution phase [38-39]. It is an interaction of a solute molecule (water in the present case) with surrounding solvent molecules (1,4-dioxane), which leads to stabilization of the solute molecule in the solution. In a variety of solution-phase chemical reactions, solvation plays an important role in determining the direction and rate of the reactions and in changing the energies of excited as well as the ground state.
First, the concentration dependence of FT-IR absorption spectra of water dissolved in 1,4-dioxane is studied in order to clarify how many water species exist in the solution. The observed concentration-dependent spectra are well-explained by assuming two water species:
one is an isolated water molecule interacting solely with 1,4-dioxane, and the other is the water molecule in a small water cluster. Next, the results of IR electroabsorption measurements on the 1,4-dioxane solution of water are shown. As described in Chapter III, the electric resistance of the sample cell often causes artificial phase retardation with respect to the applied sinusoidal wave. Therefore, the phase dependence of the ΔA spectra is examined in order to find the optimum phase for lock-in detection of the signal that can best compensate the phase retardation. The dependence of the ΔA spectra on the field strength F is also studied.
We performed an SVD analysis of the A spectra measured at different angles and
The experimental apparatus and the sample cell for IR electroabsorption spectroscopy used in this study have already been described in Chapter III. In measuring the angle χ dependence of ΔA spectra, we used a wire-grid mid-IR polarizer to obtain p-polarized light.
Four runs were averaged for each ΔA spectrum, which required about 2 h. The molar concentration of water was 1.0 M for IR electroabsorption measurements.
IV-3. Results and discussion
IV-3-1. Concentration dependence of FT-IR spectra
Figure IV-1(a) shows FT-IR spectra in the 3000–3800 cm−1 region of water dissolved in 1,4-dioxane at molar concentrations of 0.25, 0.50, 0.75, 1.0, 1.25, 1.5, 1.75, and 2.0 M. A major broad feature centered at ~3550 cm−1, which apparently consists of two peaks, is observed.
This band is unambiguously assigned to the O–H stretch of water. There is a small hump at around 3260 cm−1 (see, for example, the 2.0 M spectrum). This band is assigned to the first overtone of the O–H bend, 2OH). The band shape of the (OH) band is found to change with water concentration. To better see the change, we normalized the absorption spectra to the intensity at 3514 cm−1. Figure IV-1(b) shows the resulting normalized spectra. As the concentration increases, the higher-wavenumber peak of the (OH) band decreases in intensity and concomitantly, the lower-wavenumber side of the band appears to be broadened.
We also measured the concentration dependence of FT-IR spectra of water in 1,4-dioxane at lower concentrations of 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, and 0.40 M.
Figure IV-2(a), (b) shows the raw data and the spectra normalized to the intensity at 3514 cm−1, respectively. As can be seen from Fig. IV-2(a), the band shape of the OH band and the intensity ratio of its two peaks look similar in this concentration range. However, the normalized spectra shown in Fig. IV-2(b) do exhibit the same tendency as in Fig. IV-1(b), although somewhat less prominent. These results seem to indicate that the water species responsible for the observed (OH) band may exist even at low concentrations (<0.40 M).
In order to know how many water species are involved in the 1,4-dioxane solution in the concentration range studied, we analyzed the spectra using multivariate curve resolution
(MCR). In MCR analysis, a data matrix X consisting of a series of spectra at different concentrations is decomposed into physically meaningful matrices H and W. The matrix H contains normalized concentration profiles of individual molecular components and the matrix W contains the corresponding intrinsic spectra. The sum squared residual X HW T is minimized by solving alternating least-squares problems iteratively under the constraint that concentration and spectral intensity (i.e., the matrix elements of H and W) are non-negative. In MCR analysis, we need to assume the number of components a priori. Here, we refer to the result of SVD of the dataset shown in Fig. IV-1(a). Figure IV-3 plots the singular values obtained from the SVD. It is clear that there are two principal singular values.
We therefore set the number of components required in MCR as two. Hereafter the major component will be denoted 1 and the minor 2. A randomly generated basis was used as an initial guess for W. 1000 iterations were enough to obtain a good convergence regardless of the initial guess.
What molecular models can be applied to account for the water components 1 and 2?
Since the major component 1 exists even at low concentrations, it would be natural to attribute it to an isolated water molecule surrounded by 1,4-dioxane molecules as depicted in Fig. IV-4(a). On the other hand, we hypothesize that the minor component 2 is assigned to water species inside a small cluster (ensemble) of water molecules that forms hydrogen bonds with neighboring water molecules or 1,4-dioxane molecules outside the cluster [Fig. IV-4(b)].
The concentration profiles (h1 and h2) and intrinsic spectra (w1 and w2) of components 1 and 2 derived from the MCR analysis are displayed in Fig. IV-5. As expected, the w1
spectrum explains the major feature of the O–H stretch band of water, in which two peaks at around 3600 and 3500 cm−1 are dominant. The two peaks are reminiscent of the antisymmetric and symmetric O–H stretches, as(OH) and s(OH), of water. It is not surprising that the w1 spectrum shows two distinct (OH) peaks in contrast with the broad,
featureless (OH) band of pure liquid water, because component 1 has been considered in our model as an isolated water molecule in the 1,4-dioxane medium. In this sense, the w1 spectrum is consistent with our model of water. We fit the w1 spectrum to a sum of three Lorentzian functions plus a baseline represented by a linear function. The two Lorentzian functions at higher wavenumbers account for the as(OH) and s(OH) bands, and the third one with very small intensity for the first overtone of (OH). The best fit is shown in Fig. IV-6(a), and the peak positions and band widths of the three bands are given in Table 1.
The w2 spectrum shows a more complicated band profile, which has a maximum at a lower wavenumber relative to the w1 spectrum and includes at least three peaks. The profile of the w2 spectrum resembles the inhomogeneously broadened (OH) band of pure water.
This similarity in spectral profile is consistent with our model for component 2, in which the component is assigned to a water molecule in a water cluster. In fact, the w2 spectrum can be well reproduced by a sum of two Gaussian functions and one Lorentzian function (plus a baseline) rather than a sum of three Lorentzian functions, implying inhomogeneous characteristics of the w2 spectrum. The best fit is shown in Fig. IV-6(b), and the peak positions and band widths of the three bands are given in Table 2.
IV-3-2. Phase dependence of IR electroabsorption spectra
As described in Chapter III, a large resistance of the sample cell may give rise to a decrease in amplitude of the applied voltage and phase retardation with respect to the applied sinusoidal wave. The latter may result in nonzero out-of-phase ΔA signals. In this section, we show the dependence of ΔA spectra on lock-in detection phase. The ΔA spectrum was measured at four different phases: −90°, −100°, −110°, and −120°. In our setting, if the sample cell is an ideal capacitor with zero resistance, there will be no out-of-phase ΔA signal at −90°.
A 25 kHz sinusoidal wave of an electric field whose amplitude was ~1.2 × 107 V m−1 was applied across the cell gap of about 6 μm. The in-phase and out-of-phase ΔA spectra are shown
in Fig. IV-7(a), (b), respectively. The in-phase ΔA spectrum is less affected by changing the detection phase; only a small variation in signal amplitude can be seen. In the out-of-phase ΔA spectrum, however, the feature at ~3550 cm−1 swings from a negative dip at −90° to a positive signal at −120°. We believe that the observed out-of-phase ΔA signal is not of molecular origin but artifacts produced electrically. At −110°, the out-of-phase signal at ~3550 cm−1 is effectively removed, so we chose to measure all the ΔA spectra presented below at this detection phase.
Regardless of the detection phase, the ΔA spectra shown in Fig. IV-7(a) all have a negative offset of about 5 × 10−7. To eliminate this offset, we subtracted the ΔA spectrum recorded under the same conditions but without electric field [gray line in Fig. IV-8(a)].
Nonzero signals in the F = 0 spectrum may be due to some electrical artifacts. Subtraction of the F = 0 spectrum also removes a negative signal at ~3750 cm−1, which is caused presumably by the artifact illustrated in Fig. III-6. The resulting ΔA spectrum still has a slowly varying baseline [Fig. IV-8(b)], which was fit to a sine function and then subtracted off from the ΔA spectrum. After these corrections, the ΔA spectrum shows a flat baseline fluctuating around ΔA = 0. The ΔA spectra presented hereafter have been subjected, if necessary, to the above-mentioned baseline corrections.
IV-3-3. Electric field strength dependence of IR electroabsorption spectra
Figure IV-9(a) shows the A spectra in the (OH) region of water dissolved in 1,4-dioxane obtained with applied voltages (V0-p) of 50, 70, and 80 V. These voltages are equal to the strength of the external electric field of 8.3, 11.7, and 13.3 MV m−1, respectively.
As the field strength F increases, the A signal of water gets stronger with the spectral shape unchanged. As discussed in Chapter II (see Eq. II-24), A signals observed in our experiment are expected to be proportional to the square of the field strength F. Figure IV-9(b) plots the A signals integrated for the 3516–3572 and 3580–3684 cm−1 intervals as a function the square of
the field strength, F2. It is found that that both positive (3516–3572 cm−1) and negative (3580–3684 cm−1) A signals are proportional to F2, ensuring that what we detect here is the second-order Stark effect.
IV-3-4. Angle dependence of IR electroabsorption spectra
Figure IV-10(a) shows the A spectra in the OH) region of water in 1,4-dioxane (1.0 M, Fext = 1.17 × 107 V m−1) at five different angles = 55°, 63°, 72°, 81°, and 90°. A signal in the O–H stretch region as small as 1 × 10−6 has been successfully detected for the first time.
At = 90° (normal incidence of the IR light to the sample cell), the (OH) A spectrum is predominated by a positive peak centered at ~3540 cm−1 and a broad dent between 3600 and 3700 cm−1. As varies from 90° to 55° (magic angle), the positive peak gradually diminishes, but the negative peak remain almost unchanged. To examine the number of independent components involved in the observed dependence, we performed SVD of the data. The singular values obtained from the SVD are plotted in Fig. IV-10(b), in which at least two major components are found. With singular values alone, it is difficult to determine whether or not the third and later components should be taken into account in subsequent analysis. Referring to the intrinsic spectraof the largest four components [Fig. IV-10(c)], we find that the spectra of the third and fourth components contain only noises with no appreciable vibrational features in the (OH) region. Therefore, we will focus on the largest two components and disregard the remaining components as noises (see Eq. III-13). Like in previous studies [16, 18], we assume that the surviving two singular values are associated with a -independent (constant with respect to ) component and a -dependent component which behaves as 1 3cos 2 . By taking a proper linear combination of the vectors (recall Eqs. III-14 and III-15) based on the model for dependence, we can obtain physically meaningful vectors.
At = 90° (normal incidence of the IR light to the sample cell), the (OH) A spectrum is predominated by a positive peak centered at ~3540 cm−1 and a broad dent between 3600 and 3700 cm−1. As varies from 90° to 55° (magic angle), the positive peak gradually diminishes, but the negative peak remain almost unchanged. To examine the number of independent components involved in the observed dependence, we performed SVD of the data. The singular values obtained from the SVD are plotted in Fig. IV-10(b), in which at least two major components are found. With singular values alone, it is difficult to determine whether or not the third and later components should be taken into account in subsequent analysis. Referring to the intrinsic spectraof the largest four components [Fig. IV-10(c)], we find that the spectra of the third and fourth components contain only noises with no appreciable vibrational features in the (OH) region. Therefore, we will focus on the largest two components and disregard the remaining components as noises (see Eq. III-13). Like in previous studies [16, 18], we assume that the surviving two singular values are associated with a -independent (constant with respect to ) component and a -dependent component which behaves as 1 3cos 2 . By taking a proper linear combination of the vectors (recall Eqs. III-14 and III-15) based on the model for dependence, we can obtain physically meaningful vectors.