Theoretical background
In this chapter, theoretical background of IR electroabsorption spectroscopy is described in detail. Some of the equations presented here will be used in Chapter IV to analyze experimental data. There are three distinct molecular responses to an externally applied electric field: orientational polarization, electronic polarization, and equilibrium shift. Mathematical expressions are derived for IR absorbance changes arising from these responses.
II-1. IR Absorbance difference (A) spectra
When an electric field is externally applied to the sample, changes in absorption intensity are induced. The absorbance change (A) is calculated from the intensity change intensity spectra of the transmitted IR light through the sample with and without the applied electric field, respectively.
II-2. Three distinct types of molecular responses
An absorbance change originates from different molecular responses to an applied electric field. Here, we consider three mechanisms of molecular responses to an externally applied electric field, namely, orientational polarization, electronic polarization, and equilibrium shift [24]. In what follows, we derive expressions for the A spectrum arising from each molecular response and see how those molecular responses contribute to the overall A spectrum.
II-2-a. Orientational polarization
Consider an ensemble of randomly-oriented polar molecules having a permanent dipole
moment μp. Upon application of an external electric field, the dipole moments align along the direction of the electric field, giving rise to orientational anisotropy, although very small. This induced polarization can contribute to changes in absorption spectrum.
(1) Normally incident nonpolarized light unit vector designating the direction of the electric field of the incident light. In Figs. II-1 and II-2, we set the molecule-fixed coordinate system such that the z-axis coincides with the direction of the applied electric field and the propagation direction of the IR light. The orientations of the permanent dipole moment μP and the transition moment μT are specified by a set of angles (θ, φ, ψ). In Eq. II-2, there are two integrands to be evaluated explicitly; one is the spatial distribution function f
, and the other is the square of the inner product of the transition moment and the unit vector, (e μ T)2.The distribution function f
is proportional to the probability of finding the dipole moment μP in the direction θ with respect to the applied electric field F. Using the coordinate system shown in Figs. II-1 and II-2, we havep p
exp , with E being the dipolar interaction energy
T
Here C is a normalization factor, T is the temperature, and kB is the Boltzmann constant, and F is the electric field strength. Note that the electric field F in Eq. II-6 is not the external field but local field which is actually exerted on individual molecules. The parameter γ reflects the magnitude of the electrostatic interaction. The factor C is determined by the normalization condition
II-8
The scalar product of μT and e can be calculated as follows. The electric field vector e, of the incident light lies in the xy-plane, and a projection of μT onto the xy-plane is related to
2
(μ eT ) . μT is expressed as
T T
cos cos cos sin sin sin sin cos sin cos sin cos cos sin cos sin sin sin sin cos
sin cos sin cos cos
By expanding exponential functions in a Taylor series and retaining terms up to second-order in γ, we have
To confirm that this approximation is valid, suppose that a voltage of 50 V is applied across liquid acetone 5 μm thick. The electric field strength is F 1107Vm1. For simplicity, we do not consider the local field correction. Using the dipole moment of acetone, P = 2.7 D (1 D = 3.33564 × 10−30 C m), we obtain 0.02, for which 1 holds.
The absorbance change caused by the applied electric field is the difference between Aon (Eq. II-15) and Aoff (Eq. II-13). The absorbance change ratio is thus
Making use of Eqs. II-2, II-8, II-9, and II-18, we end up with the absorbance change ratio of the form
absorbance change for angle α can be decomposed into its parallel ( 0o) and perpendicular orientational polarization signal probed with the normal incidence
AAgain
2 6 in the present study, so the first term in the denominator of the right-hand side of Eq. II-21 is safely neglected. Therefore, we are left with
1 3cos
nonpolarized light on the xy-plane is parallel to the sample cell. In other words, χ is equal to 90°, where χ is the angle between the applied electric field F and the electric field vector e of the incoming IR light (see Fig. II-3) When p-polarized light whose electric field vector e has only x-component is incident upon the sample with angle χ, the absorbance change ratio is proportional to the absorption spectrum A, so that it appears as its zeroth derivative (intensity change). An important implication of Eq. II-23 is that the dipole moment μP can beexperimentally determined from the zeroth derivative component of the A spectrum [27, 29].
II-2-b. Electronic polarization
An absorbance change also arises from electronic polarization, which is associated with the changes in molecule’s electronic properties such as the dipole moment and the polarizability. A general theory of the electronic polarization signal was established by Liptay and co-workers [12-13]. In general, the A(~) spectrum can be formulated as [30, 31] higher order terms in Eq. II-24, but they are all proportional to the fourth or higher powers of the electric field, which we do not detect. A( ) comprises the zeroth, first, and second
where the transition hyperpolarizability B is neglected. g is equal to p. A denotes the transition polarizability. μ and α denote the changes in dipole moment and polarizability tensor between the vibrational ground state (g) and an excited state (e), respectively, i.e.,
e g
μ μ μ and α αeαg. mˆ is a unit vector in the direction of the transition dipole moment m (for consistency with the literature, we prefer to use m for the transition moment instead of T). αgm and αm are the components of the ground-state polarizability and the polarization contribution, which we already derived above. The first-derivative component depends on both μ and α and is responsible for the peak shift, as illustrated in Fig. II-4. The second-derivative component, which is characterized solely by μ, shows the change in the band width of the absorption spectrum (see Fig. II-5).
II-2-c. Equilibrium shift
A shift of chemical equilibrium caused by an external electric field can also contribute to the A signal. If the electrostatic interaction differs among molecular species coexisting in equilibrium, the equilibrium would shift towards more stable species [24, 28]. Here let us take 1,2-dicholoroethane as an example for better understanding of such A signal. In 1,2-dicholoroethane, the trans and gauche conformers coexist in equilibrium. The trans conformer is nonpolar, while the gauche is polar. When an electric field is applied to the liquid, the gauche conformer gets stabilized via the electrostatic (dipolar) interaction. On the
other hand, the nonpolar trans conformer is not affected by the electric field. As a result, the equilibrium shifts towards the gauche conformer. Thus IR absorption of the gauche conformer is expected to increase, while that of the trans conformer should decrease.
Since this equilibrium shift A signal is a change in absorption intensity, it has the same shape as the absorption spectrum and hence contributes to the zeroth-derivative component as is the orientational polarization signal. However, the equilibrium shift signal is independent of the angle χ. It is thus possible to distinguish between the orientational polarization and equilibrium shift contributions to the A spectrum through angle χ dependence of the A spectrum.
II-3. Conclusion
In this chapter, the theoretical basis for IR electroabsorption spectroscopy has been introduced. IR electroabsorption signals appear as the changes in intensity, peak position, and band width of the absorption band. Equations II-24–27 are employed for data analysis in Chapter IV.
Figure ΙΙ-1. Coordinate system used in derivation of the orientational polarization signal. α is the angle between μP and μT. This figure corresponds to the α = 0° case, where μP is parallel to μT.
Figure II-2. Coordinate system used in derivation of the orientational polarization signal.
This figure corresponds to the α = 90° case, where μP is perpendicular to μT.
Figure ΙΙ-3. Angle χ between the applied electric field F and the electric field vector e of the incoming IR probe light.
Figure ΙΙ-4. (a) Electric field effect in the vibrational ground and excited states. (b) An absorption peak shifts to the lower frequency (black → red), and the resulting A spectrum (blue line) exhibits a first derivative line shape. > 0 causes such a lower wavenumber shift.
(a)
(b)
Figure ΙΙ-5. (a) Electric field effect on the distribution of the v = 1 ← 0 transition frequency.
(b) An absorption peak is broadened (black → red), and the resulting A spectrum (blue line) shows a second derivative line shape. is responsible for such a broadening.
(a)
(b)
Chapter III
Experiment and Analysis
The apparatus for IR electroabsoption spectroscopy used in the present study was originally developed by Hiramatsu and Hamaguchi [23] and subsequently reconstructed at NCTU by us. The principle of IR electroabsorption spectroscopy and our experimental setup are described in this chapter, followed by the details of our home-built sample cell. It is highlighted that, owing to a combination of a dispersive spectrometer and an AC-coupled amplification technique, the detection limit of absorbance change is as low as A~10−7. Sample preparation for IR electroabsorption and FT-IR measurements are also described. An analytical method for A spectra using singular value decomposition (SVD) is also presented in this chapter with some mathematical accounts.
III-1. Experimental setup—IR electroabsorption spectrometer
The experimental setup for IR electroabsorption spectroscopy is described in this section.
A schematic of the experimental setup is shown in Fig. III-1. The system consists of a light source, a home-built sample cell, an optical chopper (Stanford Research System Inc. SR540), a dispersive IR monochromator, a photoconductive InSb detector (Kolmar Technologies, KISDP-1-J1/DC), an AC-coupled amplifier, and a lock-in amplifier (Stanford Research System Inc., SR844). The probe light source used to illuminate the sample was a ceramic mid-IR emitter (JASCO). As shown in Fig. III-1, the optical chopper and the sample cell were set at the co-focus of two ellipsoidal mirrors and at the other focus of the second ellipsoidal mirror, respectively. A chopper blade with 6 windows generated a modulation of 240 Hz to the probe light. A function generator (IWATSU, FG-330) produced a 25 kHz sinusoidal wave and, after amplified by a power amplifier, the AC voltage was applied across the sample about 6 μm thick.
By combining the dispersive IR monochromator and the AC-coupled technique, the sensitivity to absorbance changes induced by electric field modulation can reach as high as 1107, which is better than that achieved by the latest FT-IR method. The spectral resolution was 8 cm−1 for all A measurements.
In this paragraph, the AC-coupled technique is briefly outlined. It is a powerful technique to detect a small AC component lying on top of a large DC offset. In the present case, the intensity of the transmitted IR probe light corresponds to the DC offset and an intensity change due to electric field modulation is the AC component [Fig. III-2(a)]. The amplitude of the AC component is typically three or even higher orders of magnitude smaller than that of the DC offset. In order to detect such small AC amplitude, we used a low-noise preamplifier to remove the DC offset [Fig. III-2(b)] and amplified only the AC component. Then the output of the preamplifier was amplified once again [Fig. III-2(c)] by the main amplifier [Stanford Research System Inc. SR560, gain 1–5000 (variable)] and fed by the lock-in amplifier. In this way, only the intensity change due to electric field modulation can be detected with a wide dynamic range.
In electroabsorption measurements, three spectra are measured. (i) The intensity spectrum, I0, of the IR probe light without the sample. (ii) The intensity spectrum, I, of the IR probe light with the sample. I0 and I are obtained using a digital sampling oscilloscope (LeCroy, LC334-DSO) and the mechanical chopper operating at about 240 Hz. In these measurements, no electric field is applied across the sample. (iii) The AC-coupled detection technique combined with a lock-in amplifier is employed to detect the intensity difference spectrum, I, recorded with an electric field turned on. The A spectrum is computed using Eq. II-1. In the following chapters, we will present our data in the format of A spectra.
IR absorption spectra were also recorded on a JASCO FT/IR-6100 spectrometer. using a sample cell composed of two CaF2 windows and a lead spacer (thickness = 25 μm). 64 runs were averaged for each IR absorption spectrum, and a wavenumber resolution of 8 cm−1 was used.
III-2. Sample cell
The configuration of our home-built sample cell is schematically shown in Fig. III-3. The sample cell consists of a brass cell holder (A/A’) , two Si windows (B/B’), and a polyethylene terephthalate (PET) thin film (C) as a spacer. The PET film (Diafoil® ) was a gift from Mitsubishi Plastics. The Si windows used (Pier Optics) were p-type boron doped Si plates (resistivity = 0.8–2 Ω cm), so they also serve as electrodes. Because one side of the Si window was coated by a SiO2 layer (thickness = 0.3 μm, resistivity >1010 Ω cm), the electrodes were electrically insulated from the sample. The resulting transmission of the Si wafers is about 60%
in the mid-IR region. The thickness of the PET film must be thin enough to avoid using high voltages, and a 6 μm film was our choice of the spacer. Between A’ and B’, we put chemically durable perfluoroelastmer O-rings (As568A-008) to prevent a liquid sample from leaking out of the flow system during measurement. Flowing the sample was required in order to avoid sample evaporation.
Accurate estimation of the cell gap and the applied voltage is essential for calculating the external electric-field strength. We can estimate the actual cell gap from an interference fringe pattern that appears in the absorption spectrum of a vacant cell. The peak positions of two
1
In order to suppress unwanted work at the cell caused by nonzero resistance between A-A’
(VAA’) and B-B’ (VBB’), we should decrease the contact resistance between them as small as possible. We scratched the surface of the Si plate at two points with a distance of ~2 cm to physically remove the naturally coated SiO2 layer. On those points was pasted indium-gallium alloy (Ga 40%), making electric contacts with the brass cell holder. The resistance between the two points was nominally smaller than ~20 Ω. It depends on the doping properties of the Si plates. A large resistance 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 A signals. This phenomenon has been explained by regarding the sample cell as forming an RC circuit [23]. Figure III-4 shows an RC circuit equivalent to the sample cell. R1
and R2 are the resistances between A and B and between A’ and B’, respectively, and CC is the capacitance between the electrodes. The exact voltage across the sample (VBB’) is related to the applied voltage (VAA’) as [18] we apply. The amplitude and the phase retardation are thus given by
AA'
capacitance CC varies depending on the concentration and the dielectric constant of the sample.
The magnitude of an externally applied electric field, Fext, can be calculated from the applied voltage and the cell gap. However, Fext is not the exact field strength that acts on the molecules in the sample. It is the local field Flocal that is actually exerted on the molecules [23].
Therefore we need to consider the relation between the local and external electric fields. The local electric field Flocal is related to the external field as
local ext
According to the Onsager theory, for instance, the local-field correction is given by
where r is the dielectric constant of the medium. By definition, r takes on values from unity to infinity, resulting in the value of f″ between 1.0 and 1.5. However, this theory is based on a simple model of intermolecular interactions and liquid structures, and there are many limitations to broad applications. Thus, values of molecular properties are often quoted in the units of f″ in previous studies [34-35].
III-3. Sample preparation
Molecular sieves (Type 3Å , 8–12 mesh, J. T. Baker) were put into 1,4-dioxane (>99%, J.
T. Baker) in order to remove residual water, followed by filtration (pore size = 0.2 μm) in
order to remove dusts or particles in 1,4-dioxane. We dissolved deionized water in 1,4-dioxane at concentrations of 0.050, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.50, 0.75, 1.0, 1.25, 1.50, 1.75, and 2.0 M for FT-IR measurements. These molar concentrations correspond to water mole fractions of 0.042, 0.085, 0.013, 0.017, 0.021, 0.025, 0.029, 0.033, 0.041, 0.060, 0.079, 0.096, 0.11, 0.13, and 0.15. We used 1.0 M for IR electroabsorption measurements.
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
fact may give rise to artificial A peaks although there is no intensity change of an IR