Sum Frequency Generation Spectroscopy and Electrochemical

Downloaded by CORNELL UNIV on June 29, 2012 | http://pubs.acs.org Publication Date (Web): December 18, 2009 | doi: 10.1021/bk-2009-1030.ch020

Chapter 20

Sum Frequency Generation Spectroscopy and Electrochemical Analysis of the 1-Butyl-3-methylimidazolium Bis{(trifluoromethyl)sulfonyl}amide Double Layer Structure on the Platinum Electrode Selimar Rivera-Rubero and Steven Baldelli* Department of Chemistry, University of Houston, Houston, Texas 77204-5003

Sum frequency generation vibrational spectroscopy (SFG), the vibrational Stark effect of CO on Pt, and electrochemical impedance spectroscopy (EIS) were used to determine the ionic arrangement of 1-butyl-3-methylimidazolium bis{(trifluoromethyl) sulfonyl}amide ionic liquid at a platinum electrode. A Stark shift of 24 cm-1 V-1 was found for CO adsorbed on Pt. From EIS, a capacitance of 0.15 F m-2 was found at the potential of zero charge (~ -500 mV vs Ag(QRE)). Results indicate a double layer thickness of ~ 4 Å, suggesting that the ions are organised in a Helmholtz-type layer at the electrode surface. Analysis of the cation orientation as a function of the applied potential indicates a reorientation of the ions as the electrode surface charge changes.

Introduction For the past few years, there has been an increase in studies of ionic liquids in electrochemical systems (1-8). Ionic liquids are salts with a melting point close to or below room temperature. Usually formed by an organic cation and a © 2009 American Chemical Society In Ionic Liquids: From Knowledge to Application; Plechkova, N., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2010.

291


292 weakly coordinating anion, these salts offer a range of tuneable physical properties depending on the selected ionic combination. This tunability also applies to their electrochemical properties, such as conductivity and electrochemical window. Understanding the influence of the different ions on the electrochemical properties will facilitate the appropriate selection of ions for a specific application. There is vast information on the molecular-level description of the liquidelectrode interface for aqueous electrolytes and molten salts (9,10). Surprisingly, molten salts were found to possess a potential of zero charge (PZC) similar to aqueous electrolytes (9,10). However, molten salts are usually formed by inorganic, spherically symmetric ions (e.g. NaCl) and these results cannot be related to non-spherical polarisable ionic liquids. In this report, a molecular-level description of the ionic arrangement at the metal/ionic liquid interface is provided for an imidazolium-based ionic liquid. The first step in the description of the interfacial structure is the definition of the double-layer structure of the system and its PZC. The double layer is defined by the interface region at which the charge transfer occurs or at that range of potential decay at which the ions are influenced by the electrode potential. However, it is essential to know if the potential at the electrode is positive or negative with respect to the PZC. This will indicate if the electrode is positively or negatively charged, and provide a correlation to ions being repelled or attracted to the metal surface. Even though the models for the double-layer structure are based on dilute aqueous systems, we recently reported a double layer structure for 1-butyl-3methylimidazolium tetrafluoroborate ionic liquid as one-ion-layer thick, which is consistent with a Helmholtz model (8). This analysis was performed combining sum frequency generation (SFG) vibrational spectroscopy of the Stark effect of carbon monoxide on platinum in the ionic liquid, and electrochemical impedance spectroscopy (EIS). In this report, the double layer structure of 1-butyl-3-methylimidazolium bis{(trifluoromethyl)sulfonyl}amide ([C4mim][NTf2]) is analysed. SFG as a function of potential, within the double layer region, will provide information on the cation orientation as the electrode surface goes from being negatively to positively charged. In addition, SFG analysis of the vibrational Stark effect of CO adsorbed to the Pt electrode in [C4mim][NTf2] allows a correlation on the strength of the local fields to the double layer thickness. Using EIS, the capacitance (C) obtained as a function of potential provides the PZC for the system at which the capacitance is at its minimum. In addition, the capacitor thickness is equal to the double layer thickness allowing a direct correlation of the ionic liquid capacitance with the double layer thickness: C∝

1 d

SFG is a second order nonlinear vibrational spectroscopic technique, sensitive to molecules in a non-centrosymmetric environment such as the liquid/electrode interface. For a detailed description of SFG, the reader is referred to several excellent reviews (11-17). In general, by overlapping a fixed

In Ionic Liquids: From Knowledge to Application; Plechkova, N., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2010.

293 frequency visible beam and a frequency tuneable infrared beam at the surface, a third beam is generated at the sum frequency of the incident beams. The intensity of the generated light is proportional to the square of the induced polarisation:

ISF ∝ P (2) = χ (2) : Evis EIR

⎡

2

⎤ ⎥ ⎢⎣ωIR − ωq + iΓq ⎥⎦


χ (2) = χ nr + ∑ ⎢

N β ( 2)

( 1)

(2)

for which E refers to the electric field of the visible and infrared input beams and χ(2) is the second order susceptibility tensor with χnr arising from the nonresonant component of the surface. The hyperpolarisability, β(2), contains the Raman polarisability and the infrared dipole transition averaged over the molecular orientation, indicated by the brackets 〈 〉. N is the number of modes contributing the SFG signal. The ωIR and ωq correspond to the frequency of the incoming IR radiation and the normal mode of vibration, respectively, with Γq as the damping constant for the qth vibrational mode. With a polarisation analysis of the interface, the molecular orientation with respect to the surface normal is deduced (18).

Experimental The synthesis of [C4mim][NTf2] have been described previously (19). To determine the electrochemical window to be analysed, a PINE Instrument Potentiostat Model AFCBP1 was used. The same potentiostat was also used to control the electrochemical potential at which the SFG was acquired. EIS measurements were performed using an EG&G PAR Model 263A potentiostat/ galvanostat in combination with a PAR M5210 lock-in amplifier controlled by PowerSINE software. The frequency range used was from 100 kHz to 1 Hz with an applied sinusoidal signal of 5 mV. Impedance data were fitted to an equivalent circuit, R(QR), by ZSimpWin software. The SFG spectroscopy system consisted of an Ekspla picosecond Nd:YAG laser which pumped (1064 nm) the optical parametric generation/amplification system, OPG/OPA (LaserVision). The OPG/OPA system generated the second harmonic 532 nm and the tuneable infrared beam (2000-4000 cm-1). The visible beam (532 nm) was at 56± from the surface normal and the infrared beam was at 63.7° at the ionic liquid/Pt interface. The SFG spectrum was collected while scanning over the desired infrared energy using a LabVIEW program. Each datum point was an average of 20 shots at 1 cm-1 s-1. The laser beams passed through a quartz prism and a thin layer of ionic liquid (~10 μm thick), followed by the Pt electrode. The SFG signal was reflected out of the cell to the detection system, and the IR beam was reflected and directed towards the reference channel, which was collected concurrently


294 with each SFG spectrum. The reference channel consisted of powdered KTP in glass at which the reflected IR signal was mixed with 532 nm light generating a sum frequency signal sensitive to the infrared fluctuations. An important feature of these experiments is the custom-made vacuum-tight electrochemical cells used, which can hold vacuum to > 2μ10-5 Torr. For the cyclic voltammetry and impedance measurements, a glass cell with a flat Teflon stopcock was used, Figure 1. The working electrode was a polycrystalline Pt bead, with a contact area of 0.713 cm2, along with a Pt counter electrode and a Ag wire reference electrode with a potential of ~ -200 mV vs. NHE (Normal Hydrogen Electrode).


Pt working electrode Pt counter electrode

Ag reference electrode

ionic liquid translational Teflon stopcock

Figure 1: Sketch of the electrochemical cell used for the cyclic voltammetry and impedance measurements All three electrodes were attached to the cell by a vacuum-tight thread seal. A translational Teflon stopcock with Kalrez O-rings controls the height of the ionic liquid inside the cell for the liquid-electrode contact, Figure 1. To prepare the sample, the ionic liquid was placed over the Teflon stopcock and pulled away from the level of the electrodes. The Ag electrode was cleaned with ammonium hydroxide and rinsed thoroughly before each experiment. The Pt working and counter electrodes were flame annealed in a hydrogen-air flame for one minute, and then cooled in the electrochemical cell under an argon flow. After the preparation of the electrodes, the cell was evacuated to > 2μ10-5 Torr. Once the ionic liquid was dried, the level of the liquid was raised to allow contact with the electrodes. Cyclic voltammetry was acquired at a scan rate of 100 mV s-1. EIS was collected for a potential range of -1000 to +1000 mV. For the potential controlled SFG spectroscopy, a glass cell with a Kel-F shaft and a quartz prism was used, Figure 2. An important feature of this cell is the way the Pt working electrode was attached to the Kel-F shaft. An O-ring around the electrode allowed a vacuum-seal when compressed between the threaded glass and the Kel-F cap, Figure 2B. The working electrode was a polycrystalline Pt rod with a diameter of 0.25 cm. The counter and reference electrodes were a Pt and Ag wire, respectively.


295 Prior each experiment, the Pt working electrode was flame annealed in a hydrogen-air flame and cooled in a glass tube with an argon flow for ~10 min

Pt working electrode teflon swage fitting

IR-quartz prism

Pt electrode with Kalrez O-ring Threaded glass

Pt counter electrode vacuum tight screw cap

reference electrode

Threaded K-lef cap Kalrez O-ring for vacuum seal Threaded K-lef shaft

Kalrez O-ring


translation

Kel-F shaft electrical contact

A B Figure 2: (A) Sketch of the SFG-electrochemical cell. (B) Detailed description of the inside shaft design (B). (20). Immediately after removing the electrode from the glass tube, a drop of ionic liquid was placed on the surface of the Pt electrode and an O-ring was positioned over the electrode. A Kel-F cap, with a central hole for the Pt electrode, was used to attach the electrode to the translational shaft while making a vacuum tight seal by compressing the O-ring, Figure 2B. The shaft was then introduced into the SFG electrochemical cell. Subsequently, the SFG cell was evacuated with a vessel containing the dried [C4mim][NTf2] attached to it. Once the system was evacuated for approximately 4 h (at ~60 ±C), the [C4mim][NTf2] flowed into the SFG cell without exposing the liquid to air. The Pt electrode was pressed against the quartz prism and the spectrum was acquired while controlling the potential applied to the system. SFG was acquired for a potential range of -1000 to +1500 mV at every 500 mV while scanning the infrared energy from 2750 to 3300 cm-1. Each SFG spectrum corresponds to an average of 5 scans. For the Stark shift analysis, the latter procedure was also followed except that the vessel containing the dried [C4mim][NTf2] was backfilled with CO instead of Ar. Furthermore, the potential of the system was hold at -200 mV for 1 h before the Pt electrode was pressed against the prism to allow the CO adsorption to the electrode surface. SFG was acquired for a potential range of -800 to +1600 mV at every 400 mV while scanning the infrared energy from 1950 to 2150 cm-1. Each CO spectrum is an average of 3 scans.


296

Results and discussion From cyclic voltammetry of the [C4mim][NTf2], Figure 3, an electrochemical window of approximately 4 V is observed, from which a window of 2.5 V was selected for analysis (-1 to +1.5 V).

Current density /mA/cm


2

0.4

[C4mim][NTf [BMIM][imide] 2]

0.3

-5

10 Torr

0.2 0.1 0.0 -0.1

Potential window of SFG study

-0.2 -0.3

-2

-1

0

1

2

Applied Potential / V vs. Ag/Ag-imide

Figure 3: Electrochemical window of [C4mim][NTf2] from cyclic voltammetry analysis scan rate =100 mV s-1 The SFG spectra obtained for [C4mim][NTf2], Figures 4 and 5, are similar to those previously reported by our group for [C4mim][PF6] and [C4mim][BF4] (7). For the ppp spectra, there are three peaks at 2870, 2930 and 2970 cm-1 assigned, respectively, to the symmetric, Fermi resonance and antisymmetric modes of CH3 from the butyl chain. There are four peaks between 3000 and 3200 cm-1 from the imidazolium ring. At 3027 cm-1, a peak due to the interaction of the anion with the ring C-H bonds is observed, and at 3076 cm-1 there is the ring C(2)-H peak. The peaks at 3160 and 3200 cm-1 are the ring H-C(4)C(5)-H antisymmetric and symmetric peaks respectively.


297

[C4mim][NTf2] ppp

6.0

5.0 4.5 4.0 3.5 3.0

4.0 3.5 3.0 2.5

-1

Wavenumber /cm

[C4mim][NTf2] ppp

5.0

6.0

[BMIM][imide] ppp +500 mV

5.5 SFG Intensity /a.u.

4.5

-1

B

A

SFG Intensity /a.u.


4.5

2.0 2700 2800 2900 3000 3100 3200 3300

2.5 2700 2800 2900 3000 3100 3200 3300 Wavenumber /cm

-500 mV

5.0 SFG intensity /a.u.

SFG Intensity /a.u.

5.5

[C 4mim][NTfppp 2] ppp [BMIM][imide]

5.5

[BMIM][imide] ppp -1000mV

4.0 3.5 3.0 2.5

[C4mim][NTf2] ppp [BMIM][imide] ppp +1500 mV

5.0 4.5 4.0 3.5 3.0

2.0 2700

2800

2900

3000

3100

Wavenumber /cm

C

3200

-1

3300

2.5 2700 2800 2900 3000 3100 3200 3300 -1

Wavenumber /cm

D

Figure 4: SFG ppp polarised spectra of [C4mim][NTf2] at the Pt electrode for different applied potentials: A -1000 mV; B -500 mV; C +500 mV; D +1500 mV For the ssp spectra, five resonances are observed, Figure 5. At 2879 and 2940 cm-1 are the butyl CH3 symmetric and Fermi resonance in addition to the CH2 antisymmetric mode observed at 2914 cm-1. The imidazolium ring peaks are observed at 3170 and 3210 cm-1 for the H-C(4)C(5)-H antisymmetric and symmetric modes.


298

[C4mim][NTf2] ssp [BMIM][imide] ssp -1000mV

0.5

0.3 0.2 0.1

[C4mim][NTf2] ssp [BMIM][imide] ssp -500mV

0.6

SFG intensity /a.u.

SFG Intensity /a.u.

0.4

0.7

0.5 0.4 0.3 0.2 0.1 0.0 2700

2700 2800 2900 3000 3100 3200 3300 -1 Wavenumber /cm

2800

2900

3000

3100

Wavenumber /cm

A

3200

3300

-1

B [C4mim][NTf2] ssp

0.5 0.4

[C4mim][NTf2] ssp

0.5

[BMIM][imide] ssp +500mV

0.4

0.3

SFG Intensity /a.u.

SFG Intensity / a.u.


0.0

0.2 0.1 0.0

[BMIM][imide] ssp +1500mV

0.3 0.2 0.1 0.0

2700

2800

2900

3000

3100

Wavenumber /cm

3200 -1

3300

2700 2800 2900 3000 3100 3200 3300 Wavenumber /cm

-1

D C Figure 5: SFG ssp polarised spectra of [C4mim][NTf2] at the Pt electrode for different applied potentials: A -1000 mV; B -500 mV; C+500 mV; D +1500 mV. The convolution given by the intense non-resonant background and the peaks interaction with the non-resonance, and within themselves, complicates the analysis of the spectra. When compared with previous results (21), different peaks are observed in the spectra, as well as the positions of the peaks being slightly different, possibly due to interaction of the molecule with the platinum electrode. Furthermore, the flexibility of the butyl chain does not allow correlating the butyl CH3 orientation with the overall cation. Therefore, analysis of the imidazolium ring peaks as a function of the potential applied provides a more accurate relation to the overall cation orientation. Analysis of the spectra will be focussed over the changes observed for the H-C(4)C(5)-H peaks. As the potential changes from -1000 to +1500 mV, changes in the H-C(4)C(5)-H peak intensities are observed for the ppp as well as the ssp spectra, Figures 4 and 5. At -1000 mV in the ppp spectra, the H-C(4)C(5)-H symmetric peak at 3200 cm-1 is more intense than the antisymmetric peak at 3160 cm-1, while the opposite is observed in the ssp, Figure 4A and 5A. As the potential becomes more positive, a difference on the intensity ratio of the two



299 peaks is observed. For the potentials of -500 and +500 mV, this change is more pronounced in the ssp than the ppp polarised spectra. At -500 mV, Figure 5B, the intensity of the 3170 cm-1 peak, in the ssp polarised spectra, is about twice the intensity of the peak at 3210 cm-1, but when the potential varies to +500 mV, Figure 5C, the 3210 cm-1 peak becomes more intense with an intensity similar to that observed for the 3170 cm-1 peak. Furthermore, a change in potential to +1500 mV, Figure 5D, increases the intensity of the 3210 cm-1 peak even more, the peak being now double the intensity of the 3170 cm-1 peak, Figure 5. In the ppp polarised spectra, there is an increase in the peak’s intensity, although a significant change is not observed. Only a clear definition of the H-C(4)C(5)-H antisymmetric and symmetric peaks is observed for the +1500 mV ppp spectra, which can be associated with the increase in intensity, Figure 4D. For a determination of the cation orientation at the ionic liquid/Pt interface, the same coordinate definition used by Hirose was applied to relate the molecular coordinate (a,b,c) to the surface coordinate (x,y,z) (15,16). The orientation of the molecules is expressed by a tilt (θ) angle of the c axis with respect to the z axis and a twist (φ) along c, Figure 6. For the orientation analysis, a simulation of the intensity ratio as a function of the tilt and twist angles is compared with the experimental values using the ssp and ppp polarised spectra. Changing the electrode surface charge from negative to positive does not have an effect in the tilt of the cation ring as previously observed for [BF4](7). Instead, the twist seems to be affected by the potential applied mimicking the behaviour previously observed for [PF6]- (21). A wider range of twist angles are found when the electrode is negatively charged. For positive surface charge, the twist of the imidazolium ring is constrained, keeping the imidazolium charge away from the surface plane. For a negative surface charge, the cation is expected to dominate the surface, correlating to the wide range of possible tilt angles found. On the other side, positive surface charge repels the imidazolium ring from the surface giving surface access to the anion and limiting the possible orientations of the ring. The same analysis for [C4mim][PF6] indicates a similar potential dependent orientation (7). For both ionic liquids, the tilt orientation is relatively constant from negative to positive potentials. The cation of [C4mim][PF6] was oriented at about 12-26° at -500 mV and 23-34° at +1000 mV (21). For [C4mim][NTf2], the orientation of the cation changes from 15-32± at 1000 mV to 35-60° at +1500 mV. However, the twist angle changes dramatically for both ionic liquids, from ~0-35° at negative potentials to about 65-90° at positive potentials, Figure 7. Further, the effect is different when compared to [BF4]-, most likely due to the size and/or charge density (7). The idea of the cation reorienting as the electrode surface potential becomes positive to allow the anion access to the surface was recently confirmed in an analysis of [C4mim][N(CN)2], performed by Aliaga and Baldelli (22).


300

tilt

Surface normal

c

twist z y

Figure 6: Description of the imidazolium ring molecular coordinate axes (a,b,c) and surface coordinate axes (x,y,z). Twist 5 25 45 65 85

30

SFG Intensity Ratio ppp/ssp(sym) H-C(4)C(5)-H


x

25 20

[C4mim][PF6] [C4mim][NTf2]

[BMIM][PF6] [BMIM][imide]

-1000 mV -500 mV

15 10

+1000 mV 5

+1500 mV

0 0

20

40

60

80

100

Tilt angle(θ)

Figure 7: Simulation curve for the ppp/ssp intensity ratio of the H-C(4)(C(5)-H of [C4mim][PF6] and [C4mim][NTf2]. The structure and thickness of the double layer is an important aspect in the understanding of the electrolyte influence in the rate of the redox process at the electrode. Therefore, to establish a molecular model of [C4mim][NTf2] at the electrode surface, the thickness of the double layer structure must be determined. To estimate the double layer thickness of the system, the same approach previously used for [C4mim][BF4] with the Stark effect of CO on Pt is followed (8). From the SFG of CO on Pt as a function of the applied potential, the vibrational Stark effect is related to the double layer thickness of the system. The SFG spectra of CO on Pt in the [C4mim][NTf2], Figure 8, are found to have a change of 24 cm-1 V-1. To determine the double layer thickness, the local electric field Stark effect {∂νCO/∂Eloc ~ 1 x 10-6 cm-1 (V cm-1)-1} is related to the applied voltage:


301 ⎛ ∂ν CO ∂ν CO ⎞ = d = (5 ± 1) x 10-10 m ⎜⎜ ⎟⎟ / ⎝ ∂Eloc ∂φ ⎠

(3)

- 800 V - 400 V 0V + 400 V + 800 V + 1200 V + 1600 V

2110

SFG intensity /a.u.


8

2100 -1

Frequency (cm )

10

2090 2080 2070 2060 2050

6

2040 -1.0 -0.5 0.0 0.5 1.0 1.5 Potential (V) vs. Ag (QRE)

2.0

4 2 0 1950

2000

2050

2100 -1

2150

Wavenumber /cm

Figure 8: SFG spectra of CO on a Pt electrode in [C4mim][NTf2] electrolyte. Insert: Plot of peak position as a function of electrode potential. A double layer thickness of 4.2 Å is found for [C4mim][NTf2], which is comparable to the 3.3 Å previously reported for [C4mim][BF4] and 3.8 Å also found for [C4mim][PF6] (8). In addition, EIS measurements are acquired as a second technique used in the analysis of the double layer thickness, Figure 9A. An equivalent circuit R(QR) is used to fit the impedance data and obtained the capacitance as a function of potential, Figure 9B. From the capacitance curve a minimum is observed at ~ -500 mV vs. Ag(QRE) which is defined as the potential of zero charge for the system. The EIS capacitance, C, is related to the thickness of the double layer by:

d=

εε 0 C

(4)

where ε and ε0 are the relative and vacuum permittivity, respectively. Using the EIS capacitance of 0.15 F m-2, a double layer thickness of 4 Å was determined in accord with the 4.2 Å found with the Stark effect analysis. Using the dimension of the imidazolium cation of (10 x 4 x 1) Å3, a thickness of ~ 4 Å is an indication of a one-ion-thick double layer (8). These results support previous findings for [C4mim][BF4], for which a Helmholtz-type layer was also found (8).


302 Previous findings for [C4mim][N(CN)2] present a double layer thickness of 25 Å (22). The difference in the results can be attributed to the strong interaction and adsorption of the dicyanamide anion to the platinum electrode. 75000

0.20

60000

[BMIM][imide]/Pt [C 4mim][NTf2]/Pt

Zimg

45000

-1000 mV -750 mV -500 mV -250 mV 0 mV +250 mV +500 mV

30000 15000 0

0

Capacitance / F/m

2

0.19

4000 8000 12000 16000 20000


Zre

0.18 0.17 0.16 0.15 0.14

-1000

-500

0

500

1000

Potential /mV vs. Ag (QRE)

A

B

Figure 9: Capacitance as a function of applied potential for a Pt electrode in [C4mim][NTf2] It is interesting to compare these results with recent theoretical analysis of the interfacial structure of ionic liquids near a charged electrochemical interface. Oldham used the Gouy, Chapman, and Sterns approach, where the dilute electrolyte is replaced by the pure ionic fluid (23). Another approach was based on statistical mechanics and mean field theory simulations (24-26). Both results reach conclusions that appear contradictory to the experimental observations, but their essential approach must be true. The theoretical work makes, as its primary link to experiment, the capacitance-voltage curve. The C-V plots usually give a “U” shape where the minimum corresponds to the PZC of the electrode interface (10,27,28). This is the point where the double layer is the thinnest, and the ion concentration at the interface is the same as the bulk electrolyte, i.e. no surface excess. This is the situation for the dilute electrolyte solution, and is not expected to be a valid model for the ionic liquids. However, the recent experimental results do not indicate a maximum in the C-V curves for these ionic liquids (8,22,29-37) with one exception (36). The results presented here using in situ spectroscopy (SFG and vibrational Stark effect) combined with electrical measurements (EIS) provide the view that the ‘double layer’ structure is very thin - approximately one ion layer thick. Since ionic liquids as a class of chemicals have widely variable physical and chemical properties depending on the ion type in the salt, many more experiments will be needed before a general conclusion can be reached.

Conclusion A double layer one-ion-thick, Helmholtz-type layer, was found for [C4mim][NTf2] at a Pt electrode. The structure at the electrode-liquid interface is dominated by the cation; at a negative surface charge (potential < -500 mV) the imidazolium cation is oriented over a wide range of possible twist angles.


303 This orientation is influenced by the interaction of the imidazolium ring positive charge with the negatively charged electrode. As the electrode becomes positive (potential > -500 mV), the charged ring twists away from the electrode surface, indicating the repulsion of the positive charges and allowing the anion to compensate for the electrode charge.

Acknowledgements


This project was possible thanks to the support provided by the Welch Foundation (E-1531) and the Petroleum Research Fund.

References 1. 2. 3. 4.

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