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Molecular orientation from sum frequency generation spectroscopy: case study of water and methyl vibrations

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Molecular orientation from sum frequency generation spectroscopy: case study of water and methyl vibrations

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Content Molecular Orientation from Sum Frequency Generation Spectroscopy: Case Study of Water and Methyl Vibrations by Muhammet Mammetkuliyev A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Chemistry) May 2021 Copyright 2021 Muhammet Mammetkuliyev Acknowledgements I would like to thank my advisor Alexander Benderskii to for guiding me throughout my graduate school time and always showing me parts of my academic work that I did not realize. I would also like to thank my committee members Jahan Dawlaty, Sumumu Takahashi and Vitali Kresin for sharing their knowledge and insight in relation to my academic work. During my time in graduate school, I was surrounded by caring and supportive community that included our chair Stephen Bradforth, graduate student advisor Magnolia and Michele and general chemistry class coordinator Catherine. My friends at USC that made my graduate school time more interesting include Chayan, Angelo, Purnim, Ariel, Gaurav, Joel, Deepak and many others. I thank them all. Finally, I would like to thank my family, my wife Oguljan and my daugther Aynur for giving me time to work on my school related things even when I should be spending time with them. ii Table of Contents Acknowledgements ii List Of Tables v List Of Figures vi Abstract viii Chapter 1: Introduction 1 1.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter 2: Molecular orientation at the air/water interface: convergence of theory and experi- ment 4 2.0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.0.2 Spectral Features of Interfacial Water Bend Vibration . . . . . . . . . . . . . . . . 6 2.0.3 Orientational Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.0.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1 Chapter 2. Supplementary information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 Experimental details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.2 Spectral fitting of the SFG data. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.3 Water bend hyperpolarizability values . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.4 Fit Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1.5 Free OH Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.1.6 SFG Intensity curves for different Gaussian FWHM values . . . . . . . . . . . . . 29 2.1.7 MB-MD Simulations and SFG Details . . . . . . . . . . . . . . . . . . . . . . . . 29 2.1.8 Normal Mode Analysis and Orientational Analysis . . . . . . . . . . . . . . . . . 30 2.1.8.1 Bending Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.8.2 Free OH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.9 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Chapter 3: Optimal experimental geometry for co-propagating reflection SFG experiment 36 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 SFG Intensity and Incoming Laser Angles . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.3 Experimental Geometry to Detect Change in Orientation . . . . . . . . . . . . . . . . . . 45 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 iii Chapter 4: Increased Accuracy in Molecular Orientation with near Total Internal Reflection (TIR) Geometry 50 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3.1 SSP spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.3.2 PPP spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3.3 SPS spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4.1 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Appendix A MATLAB Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 iv List Of Tables 2.1 Parameters used for MB-pol calculated spectra. . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Fit results for water bend spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Experimental and calculated amplitudes for stretch and bend vibrations. . . . . . . . . . . 18 2.4 Sensitivity of Type I and II water species orientation onb values . . . . . . . . . . . . . . 25 2.5 Ab initio calculated hyperpolarizability and experimental depolarization ratio values for water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6 Fitting parameters used with MB-pol calculatedc (2) . . . . . . . . . . . . . . . . . . . . . 28 3.1 Parameters used for calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Optimal IR and visible laser angles for SSP, PPP, SPS and PSS polarizations . . . . . . . . 45 4.1 Hyperpolarizability values for CH vibrations . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Orientation angles from SFG and XRD experiments . . . . . . . . . . . . . . . . . . . . . 59 4.3 Fit results for air/stearic acid and prism/stearic acid spectra . . . . . . . . . . . . . . . . . 59 v List Of Figures 2.1 Orientation of free OH and C2v axis of water . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Histogram of up and down oriented water . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Water bend SFG spectra and fitted function. SFG spectra calculated with MB-MD simula- tion with MB-pol function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4 Molecular coordiante axes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Orientation curves for water bend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.6 Free OH orientation possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.7 Orientation angle and FWHM possibilities for water bend . . . . . . . . . . . . . . . . . . 17 2.8 Orientation of water bend at the interface . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.9 Pictorial depiction of water orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.10 c (2) components of PPP spectra plotted separately . . . . . . . . . . . . . . . . . . . . . . 23 2.11 Molecular axis of water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.12 Possible hyperpolarizability values from depolarization ratio. . . . . . . . . . . . . . . . . 24 2.13 Imaginary and real parts ofc ijk from fitted spectra . . . . . . . . . . . . . . . . . . . . . . 26 2.14 Imaginary part of two resonant Lorentzians from water bend spectra . . . . . . . . . . . . 27 2.15 Calculated vibrational SFG spectra of the water stretch mode at the air/water interface using MB-Pol potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.16 SFG intensity curves for Gaussian distribution withDq=30 andDq=50 . . . . . . . . . . 29 3.1 Definition of IR and visible laser angles used in the calculation . . . . . . . . . . . . . . . 37 3.2 SFG intensity contour lines for OH stretch . . . . . . . . . . . . . . . . . . . . . . . . . . 41 vi 3.3 SFG intensity contour lines for water bend . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4 SFG intensity contour lines for water bend . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5 dg dq contour plot for free OH stretch and water bend vibration . . . . . . . . . . . . . . . . 46 3.6 dg dq contour plot for CH3 symmetric stretch vibration . . . . . . . . . . . . . . . . . . . . . 47 4.1 Stearic acid orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 SFG setup with prism and phase matching liquid . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Phase matching liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4 SFG spectra at air/stearic acid and prism/stearic acid geometries. . . . . . . . . . . . . . . 54 4.5 SFG amplitude vs orientation. R values used in calculation are given in the Table 4.1. Refractive index for air is 1 while for CaF 2 , it is 1.43 for visible and SFG frequencies and 1.41 for IR. Interfacial refractive index was calculated using the equation given in [6]. . . . 58 4.6 Orientation angle plots for air/stearic acid and prism/stearic acid . . . . . . . . . . . . . . 58 vii Abstract Interfaces can be formed from the intersection of two immiscible media such as gas/liquid, liquid/liquid, gas/solid, liquid/solid or solid/solid. Even within a given interface, i.e. air/water, numerous variations can be studied: air/salt water, air/water under electric field or air/acidic water. Sum frequency generation (SFG) is a coherent, second order technique where two beams are overlapped both temporally and spatially at the interface. The signal generated travels in the phase matching direction. SFG is generated in non- centrosymmetric medium where inversion symmetry is broken. At interfaces, non-centrosymmetry is satisfied because molecules have a preferred orientation. Euler angles relate molecular orientation angles to the laboratory coordinates. Using the transformation of laboratory to molecular coordinates, averaged molecular orientation angles can be extracted from experimental spectra. Extracting molecular orientation angles from the experiments is the focus of this thesis. Understanding the air/water interface has been the focus of both experimental and theoretical groups. Experimentally, accurate phase sensitive spectra of the air/water interface has been challenging. Theoreti- cally, molecular dynamics treatment of the air/water interface is being refined to get good agreement with experiments. Our work directly benefits the molecular dynamics simulation efforts. We use homodyne SFG to record the spectra of air water interface and extract average orientation angles of interfacial water molecules. Molecular dynamics simulation community then can use our calculated orientation angles to evaluate the orientation angle predictions of their numerical models. In addition to orientation studies of air water/interface, we studied the dependence of the SFG intensity to the incoming laser angles. We determined that PPP SFG intensity is sensitive to the incoming laser angles. Finally, we propose a new low cost approach to experimentally maximize the SFG intensity. viii The approach uses prism to reach experimental conditions that is close to the total internal reflection geometry. This new approach is evaluated experimentally and considerable increase in SFG signal intensity is observed. SFG spectra are then analyzed to extract orientation angles. ix Chapter 1 Introduction Surface specific spectroscopy techniques started getting attention at around 1980s with second harmonic spectra of air/water interface reported by Heinz group in 1988 [1] and subsequently sum frequency gener- ation spectra reported by Shen group in 1993 [2]. Since then, many groups used second harmonic and sum frequency generation techniques to study surfaces. Since the technique is probing second order polarizabil- ity and requires the medium to be non-centrosymmetric, the signal is weaker than the linear spectroscopy such as UV-Vis and FTIR. However, in the seemingly stringent requirement of non-centrosymmetry is hidden the strength of this surface technique: all interfaces inherrently is non-centrosymmetric and thus second harmonic and sum frequency active. Moreover, this non-centrosymmetric medium requirement can be extended to interfaces of micron scale biological vesicles suspended in a solution. This extension of interface makes surface techniques very attractive but with a caveat: small size and shape of the surface of biological vesicles will present a challenge. Phase matching at the surface will still occur and nonlin- ear signal generated but it will go into different directions since the surface is not flat. Such application of surface technique is quite appropriately referred to as a sum frequency scattering spectroscopy. For flat surfaces on the other hand, combination of two incoming light and phase matching condition tells us exactly where the nonlinear signal will go. With the approximation that the surface nonlinear signal is only due to the second order polarization, it is possible to model the surface sum frequency and second harmonic signal. Specifically, it is possible 1 to extract molecular orientation from surface sum frequency and second harmonic signals. In this thesis, all the experiments are done using sum frequency generation spectroscopy. As such, all the molecular orientation calculations are limited to the sum frequency generation. The second chapter of the thesis is about sum frequency generation spectroscopy of the air/water interface. In this work, water stretch and bend spectra are analyzed. Orientation angles for the transition dipoles of OH stretch and water bend vibrations are calculated. Since both OH stretch and water bend vibrations are due to the same water molecules at the interface, a unified molecular orientation picture of the water molecules at the interface is presented. The third chapter of the thesis is about the numerical study of SFG signal. Specifically, the following questions were addressed: What should the incoming laser angles for IR and visible beams be so that the SFG signal from the interface is maximized? What are the laser angles for which change of molecular ori- entation at the interface is easiest to detect? These questions are answered for the air/water and air/ethanol interfaces and conclusions are extended to the general case of any air/material interface. The fourth chapter of the thesis is about increasing accuracy in determining the molecular orientation angles. This is done by using a prism to reach near total internal reflection geometry (TIR) in sum fre- quency generation. Traditionally, to reach TIR geometry, sample is deposited on a CaF 2 prism and then probed. Here, we propose and demonstrate a cheaper and easier alternative. Instead of CaF2 prism, the sample (e.g. stearic acid) is deposited on a CaF2 window and a phase matching liquid is used to close air gap between the prism and the window. Our spectra show clear advantage of working at near TIR condi- tions. Moreover, calculated molecular orientation for vibrational transition dipoles of CH 3 symmetric and asymmetric vibrations for prism/stearic acid and stearic acid/CaF 2 window geometries were compared. Finally, in the Appendix, MATLAB code is presented. This code is used for calculation in chapter 3. The code can be adapted to calculate orientation angles. 2 1.1 References 1. M. C. Goh, J. M. Hicks, K. Kemnitz, G. R. Pinto, K. Bhattacharyya, K. B. Eisenthal and T.F. Heinz. Absolute orientation of water molecules at the neat water surface J. Phys. Chem. 1988, 92, 18, 5074–5075 2. Q. Du, R. Superfine, E. Freysz, and Y . R. Shen. Vibrational Spectroscopy of Water at the Vapor/Wa- ter Interface. Phys.Rev.Lett. 70, 2313 (1993) 3 Chapter 2 Molecular orientation at the air/water interface: convergence of theory and experiment 2.0.1 Introduction Like a ship sailing on the ocean’s waves with sails up and keel down, water molecules at the air/water interface have preferential orientation. This alignment is the manifestation of the fundamental asymmetry that defines the boundary between two media. Besides its practical importance as the most common inter- face on Earth, the air/water interface is of fundamental interest: its asymmetry is created not by electric fields or molecular interactions with the other medium, but simply by geometric termination of the ex- tended 3-dimensional hydrogen bond network of bulk water. Practical manifestations of this termination are the many unusual physical and chemical properties of the water surface including high surface tension, viscosity, surface wetting and hydrophobic effect, specific ion partitioning, and surface acidity/basicity. 1 It is therefore a crucial testing ground of our theoretical understanding of the intermolecular interactions and statistical mechanics of water, to be compared with experiment before considering more complex aqueous interfaces such as mineral surfaces, electrochemical systems, heterogeneous catalysts, and biomembranes. Assuming azimuthally isotropic surface, two angles are necessary to describe 3-dimensional orienta- tion of a water molecule relative to the surface plane like the ship’s motion described by pitch and roll. 4 Conveniently, water has two vibrational modes, OH-stretch and HOH-bend, which in principle can pro- vide the two complementary pieces of information to completely describe its orientation. However, the OH-stretch spectroscopy in H 2 O is complicated by intra- and intermolecular coupling between the OH oscillators. The intra-molecular coupling in the gas-phase water molecules results in the symmetric and antisymmetric stretch modes with orthogonal directions of the transition dipoles. In liquid water, the sym- metric and antisymmetric modes are strongly mixed, implying that the direction of the transition dipole in the molecular frame becomes uncertain. Furthermore, intermolecular coupling leads to extensive delocal- ization of thevibrational excitation, over as many as 10 water molecules. 2 Thus, the OH-stretch mode in bulk water is not a local probe of H-bonding or molecular orientation. While bulk water molecules have on-average 3.4-3.7 H-bonds per molecule (2 donor and 2 acceptor H-bonds, one of which may be temporarily broken), 3 a unique hydrogen bonding motif is found at the air/water interface, the so-called free OH or dangling OH species, with only one of the hydrogens par- ticipating in a donor H-bonding, and the other ‘free’, generally pointing towards the air phase. 4;5 These water species constitute 25% of the top monolayer of water. 4;6;7 Serendipitously, they are largely free of the complications due to the intra- and intermolecular vibrational coupling, due to their spectral separation from other OH oscillators, and orientation that tends to be perpendicular to most nearest neighbors. 6 Thus, the orientation of the free OH provide the first piece of the puzzle (angles, Figure 2.1). The water bend is less affected by intermolecular couplings due to its smaller transition dipole; there is no intramolecular coupling because there is only one HOH bend mode per molecule. 8 The spectral con- tributions of the water bend lineshape at the air/water interface have been recently analyzed, and provide a way to quantify orientation of the different hydrogen bonding species. 911 It was concluded that bending vibration of water molecules with one OH dangling in air contribute mostly to peak at the redder side whereas molecules where both OH are H-bonded mostly represented by the peak at the bluer side. Thus, bending vibration of water molecules with one OH dangling in air will provide a second piece of the puzzle (Figure 2.1, angleq). 5 However, without the rules governing the puzzle, pieces alone are not sufficient to get accurate orienta- tion picture of the interfacial water. vSFG spectra of free OH stretch and bend are recorded independently. Orientationss andq are extracted from the experiments. While these are two independent measurements, s andq are not completely independent of one another: maximum difference between them is 1/2 of the water molecule’s H-O-H angle which is 52 . This is illustrated in Figure 2.1, where the physically allowed region on (s,q) plane is represented by the rectangle along the anti-diagonal. In other words, Figure 2.1 constitute rules by which orientationss andq are bound. In this chapter, we present a comprehensive picture of the interfacial orientation of water molecules by combining the results of the OH-stretch (angles, Figure 2.1) and HOH bend (angleq, Figure 2.1) spec- troscopy with theoretical modeling using classical many-body molecular dynamics (MB-MD) simulations 1214 of the vibrational sum-frequency generation (vSFG) spectra with the MB-pol potential energy function for water. 1517 2.0.2 Spectral Features of Interfacial Water Bend Vibration Surface-selective vSFG spectra of the water bend at the air/water interface were first reported by Vinaykin and Benderskii. 8 Nagata and coworkers 10 have calculated the vSFG spectra of the water bend mode using MD simulations and reported the experimental and theoreticaljc (2) j 2 spectra at the air/water interface. According to their calculations, the water bending region consists of a negative peak around 1645 cm 1 and a positive peak around 1730 cm 1 . Ni and Skinner 11 developed a mixed quantum classical approach to calculate the spectral signature of different types of H-bonding species at the air/water interface. They also predicted negative amplitudes of the free or weakly H-bonded water molecules (1 N , 2 S and 3 S hydrogen bond classes as defined by Tainter et al. 18 , here we refer to these as Type I species) in the less blue shifted (with respect to the gas phase 19 frequency of 1595 cm 1 ) region and a positive amplitude for strongly hydrogen bonded species (4 D , 3 D hydrogen bond classes as defined in Ref.18. Here we call these Type II) in the SSP polarization. Medders and Paesani’s 13 MD simulation of air/water interface with MB-pol potential reveals that about 29% of the water molecules at the air/water interface are free OH or Type I 6 Figure 2.1: Orientation of the water molecule at an interface defined by two angles,s (OH-stretch mode) andq (HOH bend mode). Due to the 104.5 bond angle of water, the (s,q) space is limited by the light- shaded rectangle. 7 species, which agrees with previous studies. 4;7 Moreover, histogram of Type I and II water species with negative and positive amplitudes indicate a frequency distribution around Figure 2.2. Histogram of up (red) and down (blue) oriented water molecules, with the SSP SFG spectrum (black trace) overlaid. The up oriented waters have a maximum around 1660 cm 1 and the down oriented water molecules have a maximum near 1680 cm 1 as shown in Figure 2.2. Figure 2.2: Histogram of up (red) and down (blue) oriented water molecules, with the SSP SFG spectrum (black trace) overlaid. The up oriented waters have a maximum around 1660 cm 1 and the down oriented water molecules have a maximum near 1680 cm 1 We validated the theoretically calculated spectra and provided a rationale for the apparent frequency shift between SSP and PPP polarization spectra, as shown in Figure 2.3, in a recent letter. 9 If two Lorentzians with opposite amplitudes interfere, the frequency of peaks in the spectra will change as the magnitude of two amplitudes change. Bend amplitudes of Type I and II species in SSP configuration is different from PPP configuration. As a result, higher frequency peak in SSP is slightly blue shifted than PPP spectra as shown in Figure 2.3. Moreover, we have shown that when the air/water interface is covered with sur- face active molecules (e.g., surfactants), intensity of Type I species (1625 cm 1 ) is suppressed as Type I species are converted to Type II species by H-bonding to surfactant molecules. 31 vSFG spectra of the 8 water bending mode at the air/water interface have been measured experimentally in SSP and PPP po- larizations, however SPS spectra have not been reported so far. In SPS spectra the dipole components parallel to the interface can be probed. However, due to random orientation of the water molecules at the interface, the in-plane component of the dipole moment is canceled, on average, as qualitatively explained by Raymond et. al. 20 In addition to dipole moment cancellation, orientational dynamics of the interfacial water molecules causes SPS lineshapes to broaden as was shown in several studies 26;5;35;36 . Influence of orientational dynamics on SPS spectra will be discussed later in the text. We present the vSFG spectrum of water bend mode at the air/water interface measured at different polarization combinations (SSP, PPP, and SPS for SFG, visible, and infrared, respectively) and provide a detailed quantitative analysis of the water orientation at the air/water interface based on our experimental data. Spectral lineshapes for water bending are mostly dispersive with the resonant features sitting on top of a broad non-resonant background. The overall spectra have a negative feature around 1625 cm 1 followed by a positive feature around 1665 cm 1 due to Type I and Type II species, respectively. As water bending is affected from both the OH groups on a single water molecule, orientational analysis using water bending vibration on the Type I or Type II species may provide more localized information, complementary to that of OH stretch spectroscopy. vSFG spectra measured for all three polarization combinations (SSP, PPP, and SPS) of the water bend vibration at the air/water interface are shown in Figure 2.3 (left panels). Data acquisition time were same for the three spectra, which are normalized by the maximum intensity of the SSP spectra. For both SSP and PPP polarizations, the 1625 cm 1 peak representing the free OH species (Type I) is negative while the 1665 cm 1 peak for the H-bonded species (Type II) is positive. In contrast, the SPS spectrum has opposite signs for these two peaks. The two resonant Lorentzian fit might be an oversimplification of the fact that several H-bonding classes with different spectral shape and average orientation contribute to the overall spectra as predicted by theory. 10;11 Due to smaller transition dipole and weaker coupling, the water bend lineshapes will be least effected by the neighboring H-bond networks. Non-Condon effects on the water bend vibration are also expected to be small. The middle panels in Figure 2.1 show the corresponding vSFG spectra obtained by fitting frequency 9 Figure 2.3: Left Panels: vSFG spectra of the water bend mode at the air/water interface for SSP (blue), PPP (red), and SPS (green) polarization combinations of SFG, visible, and infrared. Black solid lines show fit described in the text. Middle Panels: vSFG spectra calculated at the same polarization combinations using classical MB-MD simulations with MB-Pol including a nonresonant background as a fitting parament. Right Panels: Same as left panel but nonresonant background was not included as shown in Table 2.0.2. 10 dependentc (2) calculated from classical MB-MD simulations with MB-pol potential as shown in Equation (2.1) I SSP =BjF yyz T yyz (A yyz NR e if yyz +c (2) yyz;MBpol )j 2 I SPS =BjF yzy T yzy (A yzy NR e if yzy +c (2) yzy;MBpol )j 2 I PPP =BjF yyz T yyz (A yyz NR e if yyz +c (2) yyz;MBpol )+F zzz T zzz (A zzz NR e if zzz +c (2) zzz;MBpol )j 2 (2.1) Non-resonant background was included as a fitting parameter. Laboratory axis is defined such that z axis is parallel to the surface normal while all light beams propagate in xz plane. F ijk and T ijk in Equation 2.1 are abbreviations for prefactors. For SSP polarization, F xxz =L xx (w)L xx (w 1 )L zz (w 2 ) T xxz = cosa SF cosa vis sina IR (2.2) In right panels, non-resonant background was not included in the fit as indicated in Table 2.0.2. Thus, the extent of MB-pol potential’s ability to capture the non-resonant background is evaluated. From right panels in Figure 2.3, it is seen that without the non-resonant background, MB-MD simulation with MB-pol potential does not reproduce the dispersive shape of experiment. The theoretical lineshapes, where non- resonant background is included in a fitting (middle column, Figure 2.3), are in quantitative agreement with the experimentally measured spectra, except for the SPS combination where the calculated relative signal intensity is slightly higher. Note that spectra calculated with MB-pol potential using classical MB-MD simulation have been redshifted by 60 cm 1 to account for the neglect of nuclear quantum effects. 13 Water stretch spectra from MB-MD simulation with MB-pol potential is shown in Supporting Infor- mation. Briefly, calculation captures the qualitative features of the experimental spectra of Ref 5. Quanti- tatively, free OH stretch amplitude ratio A SSP :A PPP :A SPS of Type I species from Ref 5 is 1 : 0.280.04 : 11 MB-pol calculated Re(c (2) ) and Im(c (2) ) fitted to experimental bend spectra With NR Background Without NR Background A NR f NR B A NR f NR B YYZ 1.4 180 7.5 0 0 70 YZY -0.6 280 7.5 0 0 70 ZZZ 0.8 260 7.5 0 0 70 Table 2.1: Fitting parameters used for MB-pol calculatedc (2) shown in Equation 1. 0.05 0.02. These ratios are used in the orientational analysis in next section. The MB-MD simulations yield the ratio of 1 : 0.23 : 0.11. Note that experimental free OH amplitudes in Ref 5. are reported as A SSP : A PPP :A SPS instead of as shown in Equations 3 and 4. Since experimental amplitudes included prefactors F ijk and T ijk , they were also included in calculated amplitudes. 2.0.3 Orientational Analysis vSFG signal intensity is highly polarization dependent allowing a quantitative analysis of molecular ori- entation at the interface. To this end, the SSP and SPS lineshapes were fitted using I SSP=SPS =jF ijk T ijk j 2 A ijk NR e if ijk NR + å n A ijk n ((ww ijk n )+iG ijk n ) 2 (2.3) while PPP was fitted using I PPP = F xxz T xxz A yyz NR e if yyz NR + å n A yyz n ((ww yyz n )+iG yyz n ) +F zzz T zzz A zzz NR e if zzz NR + å n A zzz n ((ww zzz n )+iG zzz n ) 2 (2.4) where F ijk is the product of Fresnel factors due to three wavelengths, A NR is the non-resonant back- ground, f is the relative phase between the resonant and the non-resonant background, and b n , w n and G n are the amplitude, frequency and width of the n th resonant vibrational mode. Frequency dependence of the refractive index of water was used as was given by Hale and Querry 29 . The experimental spectra 12 were initially fit with two Lorentzians and a complex-valued background. Next, fit parameters for two resonant Lorentzians only (not the background amplitude and phase) were frozen and spectra were refit by including two more Lorentzians to account for broad spectral features of the background, presumably existing from overtone and combination bands of librations. In PPP spectra fit, results from SSP fit was used sincec (2) yyz =c (2) xxz . Moreover, we ignored A 1 A 2 A 3 A 4 w 1 w 2 w 3 w 4 G 1 G 2 G 3 G 4 A NR f NR YYZ -32.32.0 58.41.5 308 23.4 1625 1665 1231 1734 40 45 174 50 3.19 0.46 YZY -37.51.0 51.10.7 417 32 1625 1665 1324 1744 40 40 57 63 4.67 2.83 ZZZ 8.80.1 -16.70.4 177 162 1625 1665 1589 1739 34 41 66 129 -1.75 2.81 Table 2.2: Four Lorentzian were fitted to the experimental spectra as shown in Equation 3 and 4. Result of SSP fit was included in PPP fit. Therefore, only parameters related toc (2) zzz were extracted from PPP fit. contribution from F zxx .T zxx .c (2) zxx and F xzx .T xzx .c (2) xzx since their difference is negligible as shown in Sup- porting Information. The fitting results for each polarization combination are given in Table 2.0.3. It should be noted that fit result is sensitive to the amplitude and phase of the non-resonant background. To extract orientation of water from bend spectra, resonant vibrational amplitude A ijk was calculated using the expression for C 2v symmetric stretch given by Wang et. al. 22 A xxz =A yyz = 1 4 N s ((b aac +b bbc + 2b ccc )< cos(q)>+(b aac +b bbc 2b ccc )< cos 3 (q)>) A yzy = 1 4 N s ((b aac +b bbc 2b ccc )< cos(q)>< cos 3 (q)>) A zzz = 1 2 N s ((b aac +b bbc )< cos(q)>(b aac +b bbc 2b ccc )< cos 3 (q)>) (2.5) A ijk is the vSFG amplitude obtained from the fit,b ijk are hyperpolarizability tensor elements in molec- ular coordinates (Figure 4.), q is the tilt angle between surface normal and C 2v axis of water and <> brackets are orientation angle averaging defined as < cos(q)>= R p 0 cos(q)f(q)sin(q)dq R p 0 f(q)sin(q)dq (2.6) 13 Figure 2.4: Molecular coordinate axes. Orientation distribution is represented by f(q) function. Hence, the variation of A ijk as a function of orientation angle can be determined by Equation 5. To calculate A ijk for different polarizations, it is necessary to know the values ofb aac , b bbc andb ccc for the bending (n 2 ) vibrational mode of water. We used experimental depolarization ratior=0.55 which is within the range of values obtained by Walrafen and Blatz 23 and close to the value (r = 1/1.92) reported by Pavlovic et.al. 27 Using this depolarization ratio and the formulae r= 3g 2 45a 2 + 4g 2 a= b aac +b bbc + 1 3 g 2 = [(b aac b bbc ) 2 +(b aac 1) 2 +(b bbc 1) 2 ] 2 (2.7) we extracted values asb aac = -1.82,b bbc = -0.5 andb ccc = 1. Detailed analysis of the effect ofb values on the orientation of water bend is given in the Supporting Information. This orientational analysis was limited to electric dipole approximation since quadrupolar polarizabil- ity for water bend was not available to us. It is worth noting that heterodyne water bend spectra reported by Kundu et.al. 37 showed entirely positive spectra in Im(c (2) ssp ) contradicting our fit results. They ascribed this observation to considerable quadrupolar contribution to water bend spectra. Figure 4.6 shows calculated amplitudes of the vSFG signal for three A ijk components as a function of the average tilt angleq. It was assumed that f(q) is Gaussian distribution withDq (FWHM) indicated in Figure 4.6. Experimental bend amplitude ratios (A yyz :A zzz :A yzy See Table 2.0.3) are -1 :-1.160.08 : 0.270.02 for Type I species. For Type II species, these amplitude ratios are 1 : 0.880.03 : -0.290.01. 14 Figure 2.5: Calculated SFG amplitude plots for threec components. Experimental amplitudes are color coded X signs. Vertical dashed green lines indicate an angle where experimental and calculated amplitudes match for c yyz and c zzz . c yzy amplitude was not used in orientation calculation since it’s amplitude is affected by orientation dynamics. Dashed line on water molecules indicate H-bonding. Experimental amplitudes are shown in Figure 4.6(A,B) as color-coded X signs. Note that calculation captures the sign change of amplitude when going from SSP to SPS polarization. The reason for the change can be ascribed to hyperpolarizability tensor elements and the relation between molecular and lab frames for different polarizations. Comparing the experimentally determined amplitude ratios to that of calculated SFG amplitudes, we obtain an average tilt angle for the bending dipole of 70 2 for Type I and 117 2 for Type II species. Orientation angle changes slightly when differentDq value is used as shown in Figure 4.6B. In this analysis, Gaussian distribution was assumed for tilt angles of Type I and II water species. Distribution widths (Dq) for the orientation of the C 2v axis of the water molecules at the air/water interface are 45 for both the Type I and II water species. Caution must be exercised when deciding on a distribution type since two different f(q) type can give the same SFG amplitude for all three polarizations as was the case with free OH orientation 28;24 . Gan et.al. 24 recorded vSFG spectra of air/water interface in stretching region. From their orientational analysis, they concluded that free OH stretch is oriented at aroundq 0 = 30 with Gaussian distributionDq = 30 as shown in Eq. 2.0.3 15 f(q)=N s e 4ln(2)(qq 0 ) 2 Dq 2 (2.8) In their analysis, Gan et.al. 24 used polarization intensity ratio (PIR) method with PPP and SSP inten- sities to extract the free OH orientation. Intensity ratio was calculated as a function of two variablesq and Dq. Therefore, it is expected that many (q,Dq) pairs will yield the same PPP/SSP intensity ratio as shown in Figure 2.6A. The gray shaded area is due to the experimental error in amplitude as reported in Ref.5. Sun et.al. 28 , on the other hand, recorded air/D 2 O spectra in the stretching region. From their orientational analysis, they concluded that free OD orientation can be described by an exponential distribution as shown in Eq. 2.0.3 f(q)=N s e q q 0 (2.9) Free OH orientation calculation with exponential distribution for experimental amplitudes from Ref.5 yields an average angle of<q>= 49 6 (q 0 = 31 5 ). The experimental error in amplitude is shown as a horizontal gray area in Figure 2.6B. Considering the competing interpretation for free OH orientation, our Gaussian distribution assumption for water bend orientation will be evaluated in the next paragraph. Figure 2.6: Free OH orientation possibilities for Gaussian (A) and exponential (B) distributions. 16 In this analysis, PIR ratio method was used to extract orientation for Type I and II species. Amplitudes from PPP and SSP spectra were used. When Gaussian distribution is assumed, multiple (q, Dq) values reproduce the experimental intensity ratio as shown in Figure 2.7. It was ensured that PPP and SSP intensities corresponding to points in Figure 2.7 have appreciable values. This is a necessary check since the ratio of two numbers obscures the magnitude of the numbers. PPP and SSP curves for various Dq values are shown in Supporting Information. For Type I species, orientation angle can range between 60 -77 when Dq is varied between 25 - 50 . For Type II species, orientation angle can vary between 110 -122 for the sameDq range as for Type I species. Exponential distribution, on the other hand, does not reproduce the experimental amplitudes for Type I or II species. For Type I species, sign of A yzy is opposite while for type II species, sign A zzz is opposite to the experimental amplitudes as shown in Table 2.0.3. In fact, exponential distribution with anyq 0 value in the range 0 -180 does not capture the sign of experimental amplitudes. As a result, agreement between experimental and calculated bend amplitudes A yyz and A zzz can only be obtained if Gaussian distribution is used. When evaluating calculated amplitudes against experiment, only A yyz and A zzz were considered since orientational dynamics affects A yzy . This point will further be considered in Discussion section. Figure 2.7: Orientation angle and FWHM possibilities for Type I and II species. 17 A SSP A PPP A SPS Reorientation factor TYPE I Water (Free OH) Stretch Experiment (Ref. 5) 1 0.280.04 0.05 0.02 Calculation Gaus. Dist. Dq= 30 ,q= 30 1 0.27 0.21 5.02 Exp. Dist. <q>= 49 ,q= 33 1 0.27 0.21 5.02 TYPE I Water Bend A yyz A zzz A yzy Experiment, this study -1 -1.160.08 0.27 0.02 Calculation Gaus. Dist. Dq= 45 ,q= 70 -1 -1.23 1.84 6.80.3 Exp. Dist. <q>= 66 ,q= 60 1 -0.30 -0.50 TYPE II Water Bend A yyz A zzz A yzy Experiment, this study 1 0.880.03 -0.29 0.01 Calculation Gaus. Dist. Dq= 45 ,q= 117 1 0.88 -1.53 5.30.2 Exp. Dist. <q>= 66 ,q= 60 1 -0.30 -0.50 Table 2.3: Experimental and calculated amplitudes for stretch and bend vibrations. Experimental YZY am- plitude is smaller than calculated. Scaling factor for YZY amplitude is defined to be the ratio of calculated to experimental amplitudes. 2.0.4 Discussion Based on combined OH-stretch and bend orientational analysis, a comprehensive orientation picture emerges of water molecules at the interface. Type I species straddle the interface with C 2v axis having Gaussian distribution around an angle 70 2 with Dq of 45 . Free OH of Type I species is pointing to air with an average angle ofq=30 if Gaussian distribution withDq=30 is assumed or with an average angle of <q > = 49 6 if exponential distribution is used. These conclusions were illustrated in Figure 2.8 where bend-free OH distributions are Gaussian-exponential in the main figure and Gaussian-Gaussian in the inset. Considerable density of Type I species in either case is aggregated close to the dashed line. This line represents water molecules whose molecular plane makes 90 with the interfacial plane. In other words, Type I species are straddling the interface with an average angle of 90 between planes spanned by water molecule and the interface. Type II species, on the other hand, are pointing towards the bulk with an aver- age angle of 117 between C 2v symmetry axis and surface normal. Orientations of Type I and II species are shown pictorially in Figure 2.9. Since A yzy of free OH stretch is affected by orientational dynamics, it has not been used to calculate orientation of free OH of Type I 18 species 5;25;28 . Similarly, bend amplitude A yzy for both Type I and II species were small that it was sus- pected that orientational dynamics is broadening resonant peaks in SPS geometry. Therefore, orientation calculation for water bend was done by only considering resonant amplitudes A yyz and A zzz . Figure 2.8: Orientation of the water molecule at an interface defined by two angles,s (OH-stretch mode) andq (HOH bend mode). Main figure: free OH distribution is exponential withq = 33 and bend distribu- tion is Gaussian with = 70° and = 45°. Inset: free OH distribution is Gaussian withq = 30 andDq = 30 and bend distribution is Gaussian withq = 70 andDq = 45 . Vinaykin and Benderskii 26 studied the effect of orientational dynamics on SSP, PPP and SPS SFG line- shapes and found that the orientational dynamics has a negligible effect on SSP and PPP while considerably weakens the SPS peak. Effect of orientational dynamics on SPS peak can be accounted by calculating a scaling factor if the diffusion time constant is known as shown by Vinaykin and Benderskii 26 . Diffusion time constant for C 2v symmetry axis of interfacial water (pertinent to Type I and II species) is not available in literature. In bulk water, pure dephasing and total dephasing for water bend vibration was reported to 19 Figure 2.9: Orientation of Type I and II species from water stretch and bend vSFG spectra. be 1.4 ps 33 and 170 fs 32 , respectively, which indicate that Lorentzian amplitude of dynamic species will 8.2 times weaker than the static case. On the other hand, Hsieh et.al. 34 reported free OH diffusion time constant for Type I water species at the interface. Using reported time constants (D q ( rad 2 ps ) = 0, D phi ( rad 2 ps ) = 1.25) for free OH stretch and weak confinement model of Vinaykin and Benderskii 26 , scaling factor of 6.2 is obtained. Scaled free OH stretch amplitude ratios A SSP :A PPP :A SPS of Ref 5 is 1 : 0.280.04 : 0.310.02 which is in good agreement with the static orientation calculation given in Table 2.0.3. Experimental amplitudes A yzy for Type I (stretch and bend) and II (bend) species are tabulated in Table 2.0.3 along with the amplitudes calculated with static orientation formulas as given in Equation 4. Fur- thermore, reorientation factor is defined for A yzy as the ratio of calculated to experimental amplitudes. It is interesting to note that reorientation factor for bend amplitudes of Type I and II species are quite similar and are with the range of reorientation factor for free OH stretch of Type I species as shown in Table 2.0.3. This may indicate that similar reorientation time constant describe the dephasing of all vibrations: free OH stretch and bend of Type I and II species. In conclusion, we have employed a quantitative orientational analysis at the air/water interface to deter- mine the hydrogen bonding structure and orientation of water molecules using water bending vSFG spec- tra for different polarization combinations. Our analysis along with results obtained from water stretch spectroscopy 4;5;24 provide a complete, molecular-level picture of the hydrogen bonding structure at the 20 air/water interface. As water bend spectra provide a more localized and complementary picture of wa- ter structure at the interface to that available from water stretch spectra, it can be applied to study water orientation at charged interfaces as well as other complex chemical and biological interfaces. Author Contribution: C.D. performed the vSFG experiments, M.M., C.D., and A.V .B. performed the orientation analysis. D.R.M., S.C.S, and F.P. carried out the MB-Pol simulations. All authors contributed to the final version of the manuscript. Acknowledgements: This research is supported by AFOSR Grant FA9550-15-1- 0184 and NSF Grant CHE-1153059. 2.1 Chapter 2. Supplementary information 2.1.1 Experimental details. Our setup for the current experiment is same as depicted before. 34;27 We are using a broad-band infrared pulse centered at 1650 cm 1 as generated by an optical parametric amplifier (OPA) followed by a dif- ference frequency generator (NDFG). The sum-frequency signal is upconverted by a narrow-band visible pulse as generated by a home built 4-f stretcher with frequency resolution of 15 cm 1 . 2.1.2 Spectral fitting of the SFG data. The frequency domain SFG data is recorded with a monochromator and an LN-cooled CCD detector. The expression for the broad-band SFG intensity is I(w)=j[c(w) ˙ E(w)]:E(w)j (2.10) 21 Where the second order frequency domain response function (c (2) ) is a sum of resonant and non- resonant contributions. The expression for PPP intensity spectra is I PPP =L xx (w)L xx (w 1 )L zz (w 2 )cosa SF cosa vis sina IR c xxz L xx (w)L zz (w 1 )L xx (w 2 )cosa SF sina vis cosa IR c xzx +L zz (w)L xx (w 1 )L xx (w 2 )sina SF cosa vis cosa IR c zxx +L zz (w)L zz (w 1 )L zz (w 2 )sina SF sina vis sina IR c zzz (2.11) Here the lab coordinate system has been chosen such that the z-axis is the surface normal and the lasers beams are in the xz plane. In the main text and here, Fresnel factors will be succinctly written as F ijk =L ii (w SF )L jj (w vis )L kk (w IR ) (2.12) and functions as T xzx = cosa SF sina vis cosa IR (2.13) while preserving the order of frequencies. The bolded expressions corresponding toc xzx andc zxx has opposite signs thus the addition will be shown to be negligible. To this end, frequency dependentc (2) yzy andc (2) yyz were extracted from SPS and SSP spectra, respectively. Using the fact thatc (2) xzx =c (2) yzy =c (2) zyy =c (2) xxz andc (2) xxz =c (2) yyz we will show that jF xxz T xxz j:c (2) xxz >>(jF xzx T xzx jjF zxx T zxx j):c (2) xzx (2.14) As seen in Figure 2.1, difference of c (2) zxx and c (2) xzx is sufficiently flat that it will not affect amplitude extracted form fit to PPP spectra. 22 Figure 2.10: Solid line is the contribution from XZX and ZXX while dashed line is from XXZ to the PPP spectra. When PPP spectra is fitted, the contribution from XZX and ZXX was assumed to be negligible as can be seen above. The solid lines have features that are negligible when compared to the dashed lines. 2.1.3 Water bend hyperpolarizability values Accuracy of molecular orientation depends on the hyperpolarizability values. For water bend orientational analysis,b aac ;b bbc andb ccc values are needed. Figure 2.11: Molecular axis of water used in text. Here, c is parallel to C 2 axis of water bend, a extends from one H to another H while b is perpendicular to the plane of water. Hyperpolarizability values can be approximated using ab-initio calculations for gas 23 phasemolecules as was done by Li et. al. 5 or can be extracted from Raman depolarization (r) measurement of water bend. Raman depolarization ratio is r= 3g 2 =(45a 2 + 4g 2 ) (2.15) where a= a 0 aa +a 0 bb +a 0 cc 3 = b aac +b bbc b ccc 3μ 0 c g 2 = (a 0 aa a 0 bb ) 2 +(a 0 aa a 0 cc ) 2 +(a 0 bb a 0 cc ) 2 2 = (b aac +b bbc ) 2 +(b aac +b ccc ) 2 +(b bbc +b ccc ) 2 2μ 02 c For a value ofr= 0:55, a range of values forb aac andb bbc can be found as shown in Figure 2.12 by solving forb aac for a given value ofb bbc assumingb ccc = 1. This approach amounts to Figure 2.12: Possible hyperpolarizability values for depolarization ratio of 0.55. Blue line has a slope of 1. For a givenb bbc value, twob aac values are found. 24 factoring outb 2 ccc from the expression above and cancelling. Blue line has a slope of 1. We choser = 0.55 value since it is between two experimental values given by Walrafen 23 and is close to the value reported be Pavlovic et.al. 2 Other hyperpolarizability values reported in literature are given in Table 2.1.3. Out of all possible combinations, we narrowed the range forb aac (-1.1, -11.2) andb bbc to (-1, 4) using the following considerations: • Polarizability derivativea 0 c c must have opposite sign toa 0 aa •ja 0 bb j<ja 0 aa j • Hyperpolarizability values must yield same sign for c (2) zzz and c (2) yyz while opposite sign for c (2) xzx in agreement with the experimental PPP, SSP and SPS spectra A yyz A zzz A yzy Type I bend, Experiment 1 1.160.08 -0.27 0.02 Type II bend, Experiment -1 -0.88 0.03 0.29 0.01 b aac b bbc b ccc Type I bend Type II bend Calculated A yzy (q;Dq) (q;Dq) (Type I,II) -2.84 0 1 70,45 118,45 (1.44, -1.21) -1.82 -0.5 1 70,45 117,45 (1.80, -1.53) -1.1 -1 1 68,45 117,45 (2.01, -1.74) -5.51 1 1 71,45 118,45 (1.03, -0.87) -8.30 2 1 73,45 119,45 (0.88, -0.72) -11.2 3 1 74,45 120,45 (0.81, -0.63) Table 2.4: Sensitivity of Type I and II water species orientation onb values extracted fromr = 0.55 value. In the Table 2.1.3, orientation angle and gaussian FWHM was recalculated for other values ofb ijk . It can be concluded that within the range ofb ijk we are considering, orientation angle and gaussian FWHM does not change. Otherb ijk values for water bend found in the literature is given in Table 2.1.3. 25 b ccc (C 2 axis) b bbc b aac (H-H) r Calculated Ab initio 1 -2.47 -6.62 0.26 Skinner 11 Ab initio 1 -2.47 3.94 0.60 Skinner 11 Ab initio 1 -0.43 -1.26 0.65 Avila 39 r Experimental Gas 1 -0.231 -0.985 0.74 Murphy 38 Gas 1 -0.075 -0.735 0.74 Murphy 38 Bulk 0.59 Walrafen 23 Bulk 0.53 Walrafen 23 Bulk 1/1.92 (= 1 r Pavlovic 2 Table 2.5: Ab initio calculated hyperpolarizability and experimental depolarization ratio values for water reported in literature. 2.1.4 Fit Results Figure 2.13: Imaginary and real parts ofc ijk from fitted spectra. 26 Figure 2.14: Imaginary part of two resonant Lorentzians and background defined as c NR = L3 + L4 + A NR exp(if) are shown for A)c (2) YYZ , B)c (2) YZY and C)c (2) ZZZ . 27 2.1.5 Free OH Spectra Figure 2.15: Calculated vibrational SFG spectra of the water stretch mode at the air/water interface us- ing MB-Pol potential. First Row: Classical MB-MD simulations with MB-Pol including a nonresonant background as a fitting parament. Second Row: Same as first row but nonresonant background was not included as shown in Table 2.1.5. MB-pol calculated Re(c (2) ) and Im(c (2) ) fitted to experimental stretch spectra of Shen 3 With NR Backround Without NR Background A NR f NR B A NR f NR B SSP -2.5 -20 1 0 0 1 SPS -1.3 -90 0.9 0 0 1 PPP -5 -30 1 0 0 1 Table 2.6: Fitting parameters used with MB-pol calculatedc (2) shown in Equation 1. 28 2.1.6 SFG Intensity curves for different Gaussian FWHM values Figure 2.16: SFG intensity curves for Gaussian distribution withDq=30 andDq=50. AsDq increases, PPP polarization intensity profile flattens. 2.1.7 MB-MD Simulations and SFG Details Classical MB-MD simulations were performed using the MB-pol potential energy surface on a box with dimensions of 26x26x100 A in 3D periodic boundary conditions, containing 512 water molecules. This setup provides a slab of water molecules parallel to the xy plane. This system had previously been equili- brated at 298.15 K in the NVT ensemble using Nos´ e-Hoover chains of four thermostats with the equations of motion propagated using the velocity-Verlet algorithm with a timestep of 0.02 fs. 42 These simulations provided 32 initial conditions for NVE simulations. Each of the 32 initial conditions were run for 250 ps resulting in8 ns of NVE trajectories which were used to calculate the real and imaginary parts ofc (2) in the PPP, SSP, and SPS polarizations. 8 For the calculation of sum frequency spectra from the corre- lation function, the truncated-cross correlation function (TCF) method was implemented as described in Ref.41. In order to analyze only water molecules near the surface of the slab, and to account for the double interface setup of the slab system, the screening function was used, 29 g sc (z)=sign(z)x 8 > > > > > > > > < > > > > > > > > : 0 ifjzjz c1 cos 2 p(jzjz c2 ) 2(z c1 z c2 ) ifz c1 <jzjz c2 1 ifjzj>z c2 with z c1 = 7 A and z c2 = 8 A. The one-body and N-body induced contributions (“1B+NB”) approx- imation of the dipole moment surface and “1B” approximation to the polarizability tensor were used in calculating the TCF. Only contributions from pairs of water molecules whose oxygen atoms were no more than 4 A were counted. For further details on the implementation of the TCF method with classical MB- MD using MB-pol, see Ref.13 2.1.8 Normal Mode Analysis and Orientational Analysis Individual frames were extracted from the 8 ns of NVE trajectories, then their configurations optimized. From these optimized configurations, the normal modes were calculated by diagonalization of the Hessian matrix of the simulation box in periodic boundary conditions. 4 0 The resulting frequencies and displace- ments for each normal mode from 8 independent frames were analyzed in the subsequent analysis to determine orientations of water molecules in the bending region and fractions of free-OH species. 2.1.8.1 Bending Region The same screening function and values for z c1 and z c2 given in Equation 12 were used to select only water molecules near the surface normal. For each remaining water molecule near the surface, the angle along each water molecule’s C 2v axis creates with the surface normal was calculated. To approximately identify which molecules contribute to the vibrational mode, the total displacement D i of the oxygen molecule was calculated to provide an approximate measure of the water molecule’s activity, D i = å j=x;y;z jd (i) j j 2 (2.16) 30 Here,d (i) j is the displacement of the oxygen atom of the ith water molecule along the jth Cartesian axis in the kth normal mode. The D i values in each frame were normalized with respect to the largest value over all modes. As a computational cost saving method, all water molecules with a D i value less than 0.01 were not included in the following analysis. Note that the displacement is an approximate method of estimating each molecule’s contribution to the SFG signal. The average angle of the remaining water molecules was calculated using their corresponding D i values as weights. The average frequency of the sets of up and down oriented water molecules was also calculated. 2.1.8.2 Free OH The following criteria were used to determine whether a water molecule contains a free-OH bond: 1. Normal modes with frequencies larger than 2800 cm 1 . 2. For this analysis a hard cutoff of 8 A above the center of mass was used instead of the screening function g sc (z). 3. The two angles created between the surface normal and the two OH bond vectors in each water molecule were calculated. The smallest must be less than 90 . 4. The distance between the hydrogen of the smallest angle and the next nearest oxygen molecule must be greater than 2.5 A. If a molecule satisfies all four conditions, it is included and cross checked with the list of molecules from the bending region for each frame. From this cross checked list, the fraction of water molecules in the bending region with free-OH bonds was calculated to be 0.29300.0220. 2.1.9 Acknowledgements C.D. performed the vSFG experiments, M.M., C.D., and A.V .B. performed the orientation analysis. D.R.M., S.C.S, and F.P. carried out the MB-Pol simulations. All authors contributed to the final version of the 31 manuscript. Acknowledgements: This research is supported by AFOSR Grant FA9550-15-1- 0184 and NSF Grant CHE-1153059. Chayan Dutta 1; performed the vSFG experiments, Muhammet Mammetkuliyev 1; , Chayan Dutta 1; and Alexander V . Benderskii 1; performed the orientational analysis, Daniel R. Moberg 2; , Shelby C. Straight 2 and Francesco Paesani 2;3;4; carried out the MB-pol simulations. 1 Department of Chemistry, University of Southern California, Los Angeles, CA 90089, U.S.A. 2 Department of Chemistry and Biochemistry, 3 Materials Science and Engineering, 4 San Diego Supercomputer Center, University of California San Diego, La Jolla, CA 92093, U.S.A. 2.1.10 References 1 Saykally, R. J. Air/water interface: Two sides of the acid-base story. Nat. Chem. 5, 82-84 (2013). 2 Bakker, H. and Skinner, J. Vibrational spectroscopy as a probe of structure and dynamics in liquid water. Chem. Rev. 110, 1498-1517 (2009). 3 Eaves, J., Loparo J.J., Fecko C. J., Roberts S. T., Tokmakoff A. and Geissler P.L. Hydrogen bonds in liquid water are broken only fleetingly. Proc. Natl. Acad. Sci. 102, 13019- 13022 (2005). 4 Du, Q., Superfine, R., Freysz, E. and Shen, Y . Vibrational spectroscopy of water at the vapor/water interface. Phys. Rev. Lett. 70, 2313 (1993). 5 Wei, X. and Shen, Y . 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L. and Tuckerman, M. Nos´ e–Hoover chains: the canonical ensemble via continuous dynamics. J. Chem. Phys. 97, 2635-2643 (1992). 35 Chapter 3 Optimal experimental geometry for co-propagating reflection SFG experiment 3.1 Introduction Sum-frequency generation (SFG) spectroscopy involves mixing of three electric fields (visible, IR and SFG) in the medium where inversion symmetry is broken. A polarization layer is formed, which emits the electric field with the frequency equal to the sum of two incoming beam frequencies. Sum-frequency beam contains the spectroscopic information of the medium where it was generated. Among many possible experimental geometries, most frequently chosen is where two fundamental beams are co-propagating, and the reflected portion of sum-frequency beam is recorded. Such reflected geometry SFG avoids the complication due to the SFG generated in the bulk 1;2;3 . Since SFG is a three-wave mixing process, product of three Fresnel factors, constants from projection of laser polarization vector onto the lab coordinates and surface specificc (2) determine the intensity of the SFG signal. Whilec (2) originates from medium probed and therefore independent of the experimental geometry, Fresnel factors and constants arising from projection of polarization vector onto the lab coordinates are not. At certain incoming IR (b 1 ) and visible (b 2 ) laser angles shown in Figure 3.1, the product of these three Fresnel factors and constants from projection reaches a maximum and so does the SFG signal. Further away from these optimal angles, the vSFG intensity decreases. Many groups have previously noticed 36 Figure 3.1: Definition of IR and visible laser angles used in the calculation. the vSFG intensity dependence on the visible and IR angles. Hong-fei Wang et.al. 4 observed that the PPP spectra of symmetric stretch of CH 3 group of methanol went from featureless flat spectrum to the spectrum with two peaks when visible angle is moved from 62 (IR at 53 ) to 37 (IR at 51 ). vSFG intensity dependence on incoming laser angles was also observed by Tanaka et.al. 6 for octadecanothiol adsorbed on gold and Baldelli et.al. 7 for CO adsorption on platinum surface. However, a systematic study of vSFG intensity dependence on the incoming laser angles is not available in literature. Another interesting question to study is which experimental geometry is most sensitive to the change in molecular orientation. In vSFG experiment, molecular orientation can be extracted from the ratio of resonant intensities in PPP and SSP polarized spectra or by seeking polarization angle of visible beam where SFG signal goes to zero. The former technique is commonly referred to as polarization intensity ratio (PIR) and the latter as polarization null angle (PNA). Gan et.al. studied PNA method and its sensitivity to orientation change at different incoming laser angles 5 . PNA method has not been as widely used in literature as PIR method since computationally, it is not as straightforward. Moreover, PNA method can only be used if the resonant peak does not have spectral interference as was noted by Gan et.al. 5 . Here, we present a computational study that will seek experimental geometry that yields optimal SFG signal 37 intensity and is sensitive to the molecular orientation change. Here, visible light is 800 nm and medium 1 is air. Therefore, refractive indices for all laser beams in this medium is set to 1. Molecular systems considered are free OH stretch, water bend and CH 3 symmetric stretch which cor- respond to C infv , C2v and C 3v molecular symmetries. Other molecular systems or interface types can be studied by adapting the attached Matlab code. 3.2 SFG Intensity and Incoming Laser Angles SFG intensity is expressed as I(w)= 8p 3 w 2 I(w 1 )I(w 2 ) c 3 n 1 (w)n 1 (w 1 )n 1 (w 2 ) jc (2) ef f j 2 sec 2 (b)=Kjc (2) ef f j 2 sec 2 (b) (3.1) where n 1 (w 1 ) is refractive index of air, c is the speed of light, I(w i ) is the intensity of incoming light, w andb are frequency and angle of SFG signal from surface normal. All the constants that do not depend on experimental geometry is represented by K. Effective susceptibility, c (2) ef f , for different polarization combination of three electric fields is expressed as I PPP =L xx (w)L xx (w 1 )L zz (w 2 )cosbcosb 1 sinb 2 c xxz L xx (w)L zz (w 1 )L xx (w 2 )cosbsinb 1 cosb 2 c xzx +L zz (w)L xx (w 1 )L xx (w 2 )sinbcosb 1 cosb 2 c zxx +L zz (w)L zz (w 1 )L zz (w 2 )sinbsinb 1 sinb 2 c zzz I SSP =L yy (w)L yy (w 1 )L zz (w 2 )sinb 2 c yyz I SPS =L yy (w)L zz (w 1 )L yy (w 2 )sinb 1 c yzy I PSS =L zz (w)L yy (w 1 )L yy (w 2 )sinbc zyy (3.2) where 38 L xx (w i )= 2n 1 (w i )cosg i n 1 (w i )cosg i +n 2 (w i )cosb i L yy (w i )= 2n 1 (w i )cosb i n 1 (w i )cosb i +n 2 (w i )cosg i L zz (w i )= 2n 2 (w i )cosb i n 1 (w i )cosg i +n 2 (w i )cosb i n 1 (w i ) n 0 (w i ) 2 (3.3) where n 0 (w i )= s n 1 (w i ) 2 +n 2 (w i ) 2 + 4 2(n 1 (w i ) 2 +n 2 (w i ) 2 + 1) (3.4) b and g are incident and refracted angles, respectively, as light passes from medium 1 to medium 2. Expression for the interfacial refractive index n 0 is from Kundu et.al. 8 . c (2) i 0 j 0 k 0 is second order nonlinear susceptibility in laboratory coordinates. Euler transformation relatesc (2) i 0 j 0 k 0 to molecular hyperpolarizability b ijk c (2) i 0 j 0 k 0 =N Så ijk <R i 0 i R j 0 j R k 0 k >b ijk (3.5) Where N S is number of molecules, R i 0 i , R j 0 j and R k 0 k are matrix elements of Euler transformation matrix and is given in Supplemental Information. For example, expression for SSP polarization of free OH stretch vibration with C infv symmetry is 39 I ssp =Kjc (2) ef f j 2 sec 2 (b) =K sec 2 (b)jL yy (w)L yy (w 1 )L zz (w 2 )sin(b 2 )c (2) yyz j 2 = 1 4 KN 2 s sec 2 (b)jL yy (w)L yy (w 1 )L zz (w 2 )sin(b 2 )((b ccc +b aac )< cos(q)>(b ccc b bbc )< cos 3 (q)>)j 2 = 1 4 KN 2 s jsec 2 (b)jL yy (w)L yy (w 1 )L zz (w 2 )sin(b 2 )j 2 j((b ccc +b aac )< cos(q)>(b ccc b bbc )< cos 3 (q)>)j 2 = 1 4 KN 2 s jsec 2 (b)jL yy (w)L yy (w 1 )L zz (w 2 )sin(b 2 )j 2 jMolecule specific parametersj 2 (3.6) whereb aac ,b bbc andb ccc are hyperpolarizability tensor elements,q is the orientation angle from surface normal and<> brackets are orientation angle averaging defined as < cos(q)= R p 0 cos(q)f(q)sin(q)dq R p 0 f(q)sin(q)dq > (3.7) where f(q) is the orientation distribution function. Similar expressions for other polarizations can be found in Supplemental Information. Note that molecule specific parameters include orientation angle q, distribution function f(q) and hyperpolarizability tensor elementsb ijk as shown in Eq.5. Distribution function f(q) was assumed to be Gaussian function f(q)=exp( 4ln(2)(qq 0 ) 2 Dq 2 ) (3.8) n 1 (w i ) n 2 (w 1 ) n 2 (w 2 ) n 2 (w) R 1 = b aac b ccc R 2 = b bbc b ccc q Dq C infv , free OH (3700 cm 1 ) 1 1.329 1.1863 + 0.0212i 1.332 0.32 0.32 30 30 C 2v , H2O bend (1600 cm 1 ) 1 1.329 1.2977 + 0.1215i 1.3307 -1.82 -0.5 70 45 C 3v , CH3 stretch (2875 cm 1 ) 1 1.3565 1.3887 + 0.0453i 1.3593 1.7 1.7 30 20 Table 3.1: Parameters used for calculation. Refractive indices of air for all three frequencies are 1. Re- fractive indices for water are from Hale et.al.[9] and for ethanol are from Sani et.al. 11 . Hyperpolarizability values for free OH is from [10], water bend from [12] and CH 3 symmetric stretch from [13]. 40 Molecular parameters and refractive indices used in calculation are given in Table 3.2. Results SFG intensity at different IR and visible laser angles for free OH stretch are plotted in Figure 3.2 where x-axis is the angle for IR beam and y-axis is the angle for visible beam from surface normal. Single contour line shown for each polarization delineate an area where SFG intensity is within 85% of maximum SFG (optimal IR-visible angles) which is indicated as a dot. SFG intensities at optimal angle are normalized by the lowest intensity of four polarizations. In Figure 3.2B,C,D dependence of optimal angles on molecular parameters q, Dq and b ijk for PPP polarization is shown. For the other three polarizations (PSS, SPS, SSP), SFG signal is due to a single surface susceptibility componentc (2) i 0 j 0 k 0 . Therefore, changing molecular specific parameters q, Dq and b ijk changes c (2) i 0 j 0 k 0 and SFG intensity but does not change optimal laser angles. This separation of laser angles from molecular specific parameters is shown in Eq. 3.6 for SSP polarization. Figure 3.2: SFG intensity contour lines for free OH stretch. B, C and D plots show dependence of PPP intensity on q, Dq or b. Contour lines delineate an area where SFG intensity is within 85% of SFG maximum. 41 SFG intensities for SSP and PPP polarizations at optimal angles are an order of magnitude greater than intensities for SPS and PSS polarizations. Consequently, weak free OH peak in SPS and PSS polarizations becomes an intrinsic property so that many experimental spectra show very weak peak for free OH peak at these polarizations 23;24 . When molecular orientation angle q approaches the plane of interface, ideal experimental geometry requires visible laser angle to be parallel to the surface normal as shown in Figure 3.2B. No dependence of optimal angles onDq or R (= b aac b ccc = b bbc b ccc ) is observed for free OH as shown in Figure 3.2C,D. This observation has the following practical consequence: if a system has multiple free OH types, experimental geometry can be chosen to discriminate based on orientation angles alone (noDq or R dependence). If laser angles are near upper right contour lines in Figure 3.2B, then free OH oriented at or above 60 from surface normal will minimally contribute to the SFG signal in PPP polarization. A general observation is that for PPP polarization, optimal IR laser angle does not change when molecular parameters are varied but visible laser angle does. Experimental SFG intensity spectra for water stretch at different geometries was reported by Feng et.al. 19 . It was shown that SSP spectra of water taken at different geometries resulted in same SFG intensity for the entire spectral region when geometry specific Fresnel factors and sec(b) were removed. This indicates that experimental geometry factors into the SFG intensity for SSP, PSS and SPS polarizations through Fresnel factors and sec(b) term. Consequently, for the three polarizations considered, optimal laser angles where SFG intensity is maximum depends only on geometry specific factors and independent of molecular system studied. Optimal laser angles for water bend are shown in Figure 3.3. For molecular parameters given in Table 3.2, SFG intensities for SPS and PSS polarizations are greater than for SSP and PPP polarizations at optimal laser angles as shown in Figure 3.3A. Contour line for PPP polarization is well separated from other three polarizations and it shows dependence onDq andq but not on R. Ratios of hyperpolarizability values R 1 and R 2 are not equal for water bend. Therefore, depolarization ratior=0.55 was chosen based on available experimental values 16;17 and was used to extract R 1 and R 2 using Eq. 3.9 r= 3g 2 45a 2 + 4g 2 a= R 1 +R 2 + 1 3 g 2 = [() 2 +(R 1 1) 2 +(R 2 1) 2 ] 2 (3.9) 42 Figure 3.3: SFG intensity contour lines for water bend. B, C and D plots show dependence of PPP intensity onq,Dq orb. Contour lines delineate an area where SFG intensity is within 85% of SFG maximum. Extracted values and respective contour lines are shown in Figure 3.3D. For q and Dq dependence, optimal laser angles aggregate at two different locations. Whenq< 80 orDq 45 , visible angle closer to surface normal is preferred as shown in Figure 3.3B,C. Similar to free OH vibration, water bend optimal angle for IR laser in PPP polarization aggregates around one value in all cases but the angle for visible laser changes when molecule specific parameters are changed. Experimental water bend spectra in SSP and PPP polarizations were first reported by Vinaykin and Benderskii 18 at laser angles 67 and 62 for visible and IR, respectively. Spectra had a dispersive shape indicative of strong interference with either non-resonant background or another resonant signal or both which makes it difficult to compare simulation result with experiment. To study the optimal laser angles for CH 3 symmetric stretch, reference values for pure ethanol 11 was used. Normalized SFG intensities for SSP and PPP polarizations are 2 orders of magnitude greater than intensities for SPS and PSS polarizations as shown in Fig. 3.4. Optimal angles for PPP polarization are not 43 Figure 3.4: SFG intensity contour lines for water bend. B, C and D plots show dependence of PPP intensity onq,Dq orb. Contour lines delineate an area where SFG intensity is within 85% of SFG maximum. dependent onDq (5 -50 ),q (15 -60 ) or R (1-3.4) within the values indicated. CH 3 symmetric stretch has been studied extensively in literature 20;21;22 . General observation is that CH 3 symmetric stretch peak at around 2875 cm 1 is stronger in SSP spectra than in PPP spectra which is in agreement with our calculation shown in Figure 3.4. Specifically, Wolfrum and Laubereau 20 report SFG intensity spectra of hexadecanol monolayer on water recorded with IR and visible laser angles at 60 and 50 , respectively. Laser angles used by Wolfrum and Laubereau 20 are very close to the blue dot in Figure 3.4. Therefore, it is expected that CH 3 symmetric stretch peak in SSP polarization is much stronger than it is in PPP polarization. Experimental ratio of intensities I ssp I ppp is 15 which is in very good agreement with the qualitative prediction from Figure 3.4. For a quantitative comparison, intensity ratio I ssp I ppp was calculated by adapting system and geometry used by Wolfrum and Laubereau 20 which yields 16.8. Small disagreement can be attributed to the uncertainty in the molecular parametersq,Dq andb ijk . Optimal laser angles for all polarizations are given in Table 3.2. Visible and IR laser angles for SSP polarization for all three systems is 55 2 and 60 2 , respectively. Small deviation from mean indicates that difference of molecular parameters from system to system has little effect. Small deviation from mean angle also holds for SPS and PSS polarizations. 44 SSP PPP PSS SPS Vis IR Vis IR Vis IR Vis IR C infv , free OH 55 62 73 67 64 54 65 46 C 2v , H2O bend 57 59 1 58 65 41 66 31 C 3v , CH3 stretch 54 60 1 58 64 45 65 33 Table 3.2: Optimal IR and visible laser angles for SSP, PPP, SPS and PSS polarizations. This is because optimal angles for SPS, SSP and PSS polarizations do not depend on molecular pa- rameters. Instead, they only depend on refractive indices of three beams in medium 2 when medium 1 is air. These three refractive indices for the three systems are very similar as shown in Table 3.2 that it should not come as a surprise that optimal angles for these systems are also very close to one another. 3.3 Experimental Geometry to Detect Change in Orientation Similar to SFG intensity, change in molecular orientation can be easily detected at certain laser angles. Intensity ratio of resonant peaks from SSP and PPP spectra is used to extract molecular orientation in vSFG spectroscopy and is defined as g(q)= I ppp (q) I ssp (q) (3.10) Explicit expression for I ssp (q) is given in Eq.5 while I ppp can be found in Ref.4. The rate of change of this ratio as a function of laser angles is needed to find experimental geometry where orientation change is easily detected. To this end, dg dq at reference values given in Table 3.2 was calculated as a function of laser angles. Result for free OH stretch is shown in Figure 3.5 where magnitude of dg dq at contour lines is also given. Red line shows regions where dg dq is zero. Where it is not zero, dg dq indicate that a given geometry is sensitive to change in q. However, dg dq contours must be complemented with SFG intensity contours in PPP and SSP polarizations to decide on experimental geometry because I ppp I ssp can measured only if a peak appears in PPP and SSP spectra. Therefore, SFG intensity contours for SSP and PPP polarizations were also included as shown in Figure 3.5. For free OH stretch, laser geometry at the intersection of SSP 45 and PPP contours produce SFG intensity that is within 85% of SFG max while dg dq is close to the optimal possible value. Experimental geometry that will not be sensitive to the orientation change is where the visible laser angle is 8 or 40 and is irrespective of IR laser angle as indicated by red contour line in Figure 3.5. In fact, dg dq is more sensitive to visible laser angle than to IR. Figure 3.5: dg dq contour plot for free OH stretch (left) and water bend vibration (right). Change in molecular orientation cannot be detected at certain laser angles indicated by red contour lines. SFG intensity for various IR and visible laser angles are also included (black and blue contours) Settling on an experimental geometry for water bend is more challenging than free OH stretch when Figure 3.5 is analyzed. Change in molecular orientation, dg dq , is not sensitive to IR angle while SFG intensity is optimal when IR angle is 60 for both SSP and PPP polarizations. Visible angle around 30 delivers a compromise for SFG intensity in two polarizations but value is relatively small as shown in Figure 3.5. Change in molecular orientation dg dq and SFG intensity contour for CH3 symmetric stretch of ethanol is shown in Figure 3.6. Contour lines for dg dq is more sensitive to visible laser angle than to IR as was the case for free OH stretch and water bend. Therefore, IR angle at 60 yields good SFG intensity and intersects lines where dg dq is non-zero. Visible angle around 35 together with IR angle at 60 is ideal combination. Red contour lines point to visible angle of 10 and 65 where change in molecular orientation cannot be detected experimentally. 46 Figure 3.6: dg dq contour plot for CH3 symmetric stretch vibration. Change in molecular orientation cannot be detected at certain laser angles indicated by red contour lines. SFG intensity for various IR and visible laser angles are also included (black and blue contours) 3.4 Discussion Optimal laser angles for SFG intensity can be determined even without knowing molecule specific param- eters (q, distribution type f(q,b ijk ) for SSP, SPS and PSS polarization since expressions containing laser angles can be factored out as shown in Eq. 3.6 for SSP polarization. In other words, optimal angles for SSP, SPS and PSS polarizations depend only on the refractive indices of three beams in medium 1 and 2 through Fresnel factor expressions given in Eq. 3.3. Therefore, SFG intensity contours for SSP, SPS and PSS polarizations overlap considerably since refractive indices given in Table 3.2 are similar. It is not the case for PPP polarization because four susceptibility components interfere in PPP spectra. As such, SFG intensity in PPP polarization will depend on molecular parameters. Effective susceptibility component in SSP polarization isc (2) yyz while in PPP polarization,it can be nar- rowed toc (2) xxz andc (2) zzz sincec (2) zxx =c (2) xzx and have opposite signs. Consequently, IR laser angle factors into both SSP and PPP polarizations throughjL zz (w 2 )sin(b 2 )j 2 factor which peaks at IR angle of 60 . There- fore, optimal angle for IR laser in all systems considered is near 60 for SSP and PPP polarizations. Optimal angle for visible light, on the other hand, is either 1 or 75 for PPP polarization. Dual possibility for optimal angle of visible light can be attributed to the competition betweenc (2) xxz andc (2) zzz . Visible angle 47 factors into PPP polarization throughjL xx (w 1 )cos(b 1 )j 2 andjL zz (w 1 )sin(b 1 )j 2 which peak at visible an- gles 1 and 75 , respectively. SFG beam is very similar in frequency (ww 1 ) and angle (bb 1 ) to visible that factors for SFG and visible beam are equal. Therefore, effective factor becomejL xx (w 1 )cos(b 1 )j 2 andjL zz (w 1 )sin(b 1 )j 2 which still peak at visible angles 1 and 75 , respectively. Note that this analysis applies to system where medium 1 is air. Other systems can be studied by adapting the attached Matlab code (Appendix A). 3.4.1 References 1. R. Shen, Appl.Phys.B, 68, 295, (1999) 2. X. Wei, S. Hong, A.I. Lvovsky, H. Held and Y .R. Shen, JPC B, 104, 3349, (2000) 3. H. Held, A.I. Lvovsky, X. Wei and Y .R. Shen, Phys.Rev. 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Chem. Lett.,4, 801, 2017 16. Walrafen, G.E. and Blatz, L.A. Weak raman bands from water. J. Chem. Phys. 59, 2646-2650 (1973). 48 17. Pavlovic, M., Baranovic, G. and Lovrekovic, D. Raman study of the bending band of water. Spect. Acta, 47A, 897-906 (1991). 18. Vinaykin, M. and Benderskii, A. V . Vibrational sum-frequency spectrum of the water bend at the air/water interface. J. Phys. Chem. Lett. 3, 3348-3352 (2012). 19. Feng R., Guo Y ., Lu R., Velarde L. and Wang Hf. Consistency in the Sum Frequency Generation Intensity and Phase Vibrational Spectra of the Air/Water Interface. J. Phys. Chem. A. 115, 6015 (2011) 20. Wolfrum K. and Laubereau A. Vibrational sum-frequency spectroscopy of an adsorbed monolayer of hexadecanol on water. Destructive Interference of adjacent lines. Chem. Phys. Lett. 228, 83 (1994) 21. Guyot-Sionnest P., Hunt J.H. and Shen Y .R. Sum-Frequency Vibrational Spectroscopy of a Lang- muir Film: Study of Molecular Orientation of a Two-Dimensional System Phys. Rev. Lett. 59, 1597 (1987) 22. Ye S., Noda H., Nishida T., Morita S. and Osawa M. Cd2+-Induced Interfacial structural changes of Langmuir Blodgett films of stearic acid on solid substrates: a sum frequency generation study Langmuir 20, 357 (2004) 23. Gan, W., Wu, D., Zhang, Z., Feng, R.-r. and Wang, H.-f. Polarization and experimental configu- ration analyses of sum frequency generation vibrational spectra, structure, and orientational motion of the air/water interface. J. Chem. Phys. 124, 114705 (2006). 24. Wei, X. and Shen, Y . Motional effect in surface sum-frequency vibrational spectroscopy. Phys. Rev. Lett. 86, 4799 (2001). 49 Chapter 4 Increased Accuracy in Molecular Orientation with near Total Internal Reflection (TIR) Geometry 4.1 Introduction Vibrational sum-frequency generation (vSFG) spectroscopy involves mixing of three electric fields (vis- ible, IR and SFG) in the medium where inversion symmetry is broken. A polarization layer is formed, which emits the electric field with the frequency equal to the sum of two incoming beam frequencies. Sum- frequency beam contains the spectroscopic information of the medium where it was generated. Specifi- cally, vibrational signatures of molecules at the interface make up vSFG spectrum. A prerequisite for SFG active medium is that it is non-centrosymmetric. This prerequisite is met when molecules in the medium have a preferential orientation. In fact, vSFG signal intensity is dependent on molecular orientation. For example, if a molecule lies in the plane of the interface, SFG signal goes to zero because factors due to Eu- ler transformation from molecular coordinates to laboratory coordinates vanishes. In vSFG experiments, molecular orientation is found by equating the ratio of experimental vibrational amplitudes from different polarizations to the calculated amplitude. A ssp exp A ppp exp = A ssp calc (q) A ppp calc (q) 50 The angle q at which equality was satisfied becomes the orientation angle of the transition dipole moment with respect to the surface normal. For CH 3 symmetric and asymmetric stretches for example, vi- brational transition dipole moment is parallel and perpendicular to the C 3v axis of CH 3 group, respectively as shown in Figure 4.1. Figure 4.1: Stearic acid backbone is perpendicular to the surface. In this orientation, CH 3 symmetric stretch transition dipole is oriented at 35 while CH 3 asymmetric transition dipole is oriented at 130 from the surface normal. At low surface pressures (e.g. < 20 mN/m), gauche defects are observed on the backbone of stearic acid which is associated with CH 2 symmetric/asymmetric stretches. To extract the molecular orientation, resonant vibrational amplitudes must be isolated in the vSFG spectra since the spectra has contributions from non-resonant background and resonant vibrations. Disen- tangling resonant signal from non-resonant background is done by fitting combination of constant back- ground and Lorentzian functions I= A ijk NR exp(if ijk NR )+ å n A ijk n ((ww ijk n )+iG ijk n ) (4.1) where A ijk NR andf ijk NR are amplitude and phase of non-resonant background while A ijk n is the vibrational amplitude,G ijk n is the FWHM of the resonant peak,w ijk n is the resonant frequency. If the resonant vibra- tion is weak, fitting becomes very sensitive to non-resonant background. Consequently, uncertainty inA ijk n increases. To address this problem, prisms have been used to achieve Total Internal Reflection (TIR) condi- tion at the interface. It has been demonstrated TIR condition amplifies signal intensity considerably.(cite) 51 However, TIR condition using prism was seldom employed in vSFG experiments probably because of the cost and inconvenience since sample to be studied must be deposited onto the prism. Here, we propose and demonstrate a more affordable and easy to use modification to the vSFG experiments with prism. Specifically, we deposit sample to be studied on an inexpensive CaF 2 window and use an index matching fluid (IMF) between CaF 2 prism and window to reach near Total Internal Reflection condition as shown in Figure 4.2. Figure 4.2: SFG setup with prism and phase matching liquid 4.2 Experimental Monolayer of stearic acid was deposited on clean CaF 2 windows using Langmiur-Blodgett trough which was equipped with microbalance, two movable barriers and a motorized anchor. CaF 2 windows were initially cleaned with copious amounts of acetone and distilled water. Next, they were exposed to UV light under ozone for 15 min. Windows were deemed clean if no considerable SFG peaks appeared in the methyl stretching region. Clean windows were immersed into LB trough pocket so that 75% was submerged in a distilled water. Predetermined volume of stearic acid in hexane solution was slowly and incrementally introduced to the surface of water to yield 20 ˚ A 2 per stearic acid molecule for trough area of 8400 mm 2 and 1mg/mL stearic acid solution. Stearic acid monolayer dispersed on water was kept for 20 min for complete evaporation of hexane. Monolayer was compressed with barriers while recording the surface pressure with Wilhelmy plate hung on the microbalance. When the desired surface pressure 52 was reached, barriers were stopped for 15 min to allow for monolayer equilibration. Next, CaF 2 window was slowly raised from water while monitoring the surface pressure. Pressure drop resulting from stearic acid monolayer deposition onto CaF 2 was compensated by moving the barriers. Monolayer deposited CaF 2 windows were kept under 200 mmHg N2 atmosphere to allow for drying before their spectra were recorded. Three samples at surface pressures of 14, 25.4 and 30 mN/m were prepared. These pressures were chosen because orientation angles for arachidic acid (CH 3 (CH 2 )18COOH) at similar pressures at air/water interface was reported using X-ray diffraction experiments 2 . Air/stearic acid spectra were taken by probing the stearic acid deposited surface of CaF 2 window directly. Subsequently, to record prism/stearic acid spectra, the window was flipped and phase matching liquid was smeared onto the CaF 2 prism and placed on the flipped CaF 2 window. Dimethylformamide (Figure 4.3) was used as a phase matching liquid since it’s refractive index is very close to CaF 2 and has low vapor pressure. Figure 4.3: Phase matching liquid with refractive index values matching that of CaF 2 . SFG laser setup was described elsewhere 1 . Briefly, half of 60 fs 800 nm visible with 2 mJ/pulse was used to pump optical parametric amplifier (OPA) to produce fs IR pulses with FWHM of 300 cm 1 while the other half was passed through etalon to yield a ps visible pulse with FWHM 5 cm 1 . Incoming angles for visible and IR beams are approximately 67 and 63 , respectively. Reflected portion of SFG beam was 53 directed to 500 mm monochromator (Acton ARC-SP-2558) and an LN-cooled CCD detector (SPEC-10, 2048 x 512 pixels, Princeton Instruments). Experimental spectra are shown in Figure 4.4. Within each window, two spectra which are air/stearic acid and prism-DMF-window/stearic acid are shown. All spectra are normalized for IR laser profile and solid lines are fitted functions. Figure 4.4: SFG spectra at air/stearic acid and prism-DMF-CaF 2 window/stearic acid geometries. Dimethylformamide (DMF) is used as a phase matching liquid. Monolayer surface pressure at which stearic acid was deposited on the CaF 2 window are 14, 25.4 and 30 mN/m. Solid lines are fit to the spectra. All spectra have been normalized by the IR spectral profile. 54 4.3 Results To demonstrate that TIR condition amplifies the resonant vibrations, air/stearic acid and prism/stearic acid interfaces were studied experimentally. Since the CH 3 stretching region of the stearic acid was to be probed, there was a possibility that thin layer of dimethylformamide absorb part of the incoming IR light. It is even possible that SFG is generated at the CaF 2 /dimethylformamide interface. However, fitting analysis of prism/stearic acid spectra and subsequent orientation calculation suggest that experimental spectra are minimally affected by the phase matching liquid. CH stretching region consists of the following resonant vibrations listed in order of increasing fre- quency: CH 2 symmetric stretch (d+, 2850 cm 1 ), CH 3 symmetric stretch (r+, 2880 cm 1 ), CH 2 asym- metric stretch (d-,2910 cm 1 ), Fermi resonance between CH 3 symmetric stretch and bend overtone (rFR, 2940 cm 1 ) and CH 3 antisymmetric stretch (r-, 2970 cm 1 ). These vibrations are amplified with prism/stearic acid geometry is used. Spectra for different SFG polarizations (SSP, SPS and PPP) show different contributions from the resonant vibrations listed above. The analysis of the experimental spectra is more informative if it is done with the calculated SFG signal at different orientation angles. These calculations are shown in Figure 4.5 where x-axis is the orientation of the respective transition dipole and y-axis is the calculated SFG amplitude. One plot for each resonant vibration is shown. For experimental peak due to Fermi resonance, calculation was not done since it is the interference of symmetric stretch and bend overtone. Polarizations are color-coded and y-axis intensities can be compared within calculated plots. These calculations will be referred below when experimental spectra are analyzed. 4.3.1 SSP spectra SSP spectra for different stearic acid surface pressures are shown in Figure 4.4. Using prism-DMF- CaF 2 /stearic acid geometry, signal intensity increases by an order of magnitude. From the calculated 55 intensities (Figure 4.5), it is seen that CH 2 -as (2850 cm 1 ) stretch will least likely be observed in the spec- tra. Analysis of the experimental spectra does show small peak at 2850 cm 1 for surface pressure of 14 mN/m. The appearance of this peak has the following obscured information: at low surface pressures such as 14 mN/m, neighboring stearic acid molecules are separated enough that gauche defects on the backbone of the stearic acid forms as shown in Figure 4.1. The intensity of 2850 cm 1 peak signals the number density of these gauche defects. As the surface pressure increases, the distance between the neighboring stearic acid molecules decrease. Consequently, stearic acid molecules have straight backbone without any gauche defect. In this geometry, CH 2 symmetric/asymmetric stretch vibrations are cancelled because of centrosym- metry. At 25.4 mN/m, SSP spectra does not have 2850 cm 1 peak even when prism is used. However, it does appear for 30 mN/m when spectra is take with prism. This could indicate the quality of the stearic acid monolayer deposited on CaF 2 window. CH 3 symmetric stretch appears at 2880 cm 1 . Calculated amplitude predicts strong intensity for this peak. Moreover, it increases as the orientation angle approaches the surface normal or as the surface pressure increases. Experimental spectra does show strong peak at 2880 cm 1 . The direct relationship between stearic acid surface pressure and the orientation angle of CH 3 symmetric stretch is observed as will be discussed below. Calculation shows that CH 2 and CH 3 asymmetric peaks (2910 cm 1 and 2970 cm 1 ) are opposite in amplitude to symmetric peaks. Small shoulder at around 2970 cm 1 may come from CH 3 asymmetric vibration. Peak at 2940 cm 1 is due to fermi resonance between CH 3 symmetric stretch and bend overtone. 4.3.2 PPP spectra Prism-DMF-CaF 2 /stearic acid and air/stearic acid spectra are drastically different. In air/stearic acid spec- tra, the only prominent peak is due to CH 3 asymmetric stretch. In prism-DMF-CaF 2 /stearic acid spectra, 56 resonant peaks due to CH 3 symmetric stretch (2870 cm 1 ) and Fermi resonance (2940 cm 1 ) also ap- pear. One significant benefit using the prism is that two strong CH 3 symmetric peaks in SSP and PPP polarizations should allow precise determination of CH 3 transition dipole orientation. 4.3.3 SPS spectra In SPS spectra, the only prominent peak is due to CH 3 asymmetric stretch. This observation is supported by the calculation where all other resonant vibrations (CH 2 -ss, CH 3 -ss and CH 2 -as) have very small am- plitudes. 2850 (CH 2 -ss,d + ) 2880 (CH 3 -ss,r + ) (CH 2 -as,d ) 2940 (CH 3 -fr, r + FR ) 2970 (CH 3 -as,r ) Hyperpolarizability values b aac 0.16 5 2.5 3 ,1.66 4 - - b bbc 0.82 5 2.5 3 ,1.66 4 - - b ccc 0.49 5 1 3 ,1 4 - - b aca - - 2 1 6 1 Table 4.1: Observed resonant peaks in experimental spectra and corresponding hyperpolarizability values for each vibration. 1 These values do not affect the shape of the orientation curves given in Figure 4.5 since they are constant factors in the expression. Fit results are given in Table 4.3. Orientation angle of the stearic acid at both interfaces was calculated using the experimental ratio of CH 3 symmetric stretch from SSP and PPP spectra and CH 3 asymmetric stretch from SPS and PPP spectra. Two orientation angles should be self-consistent since the angle between transition dipole moments of CH 3 symmetric and asymmetric stretch is about 95 . So, if the orientation angle for CH 3 symmetric stretch is 40 , then the angle for CH 3 asymmetric stretch should be around 135 . Calculated PPP/SSP and SPS/PPP curves for symmetric and asymmetric stretches are shown in Figure 4.6. On the same plot, experimental ratios for 14mN/m stearic acid sample are indicated as X and corre- sponding orientation angles and angular distribution width is specified. Similar analysis was done for all spectra and the resulting orientation angleq and angular distribution widthDq are tabulated in Table 4.2 Close analysis of the results in Table 4.6 will reveal that orientation angles for CH 3 symmetric and asymmetric stretches are consistent for prism/stearic acid geometry and not consistent for air/stearic acid 57 Figure 4.5: SFG amplitude vs orientation. R values used in calculation are given in the Table 4.1. Refrac- tive index for air is 1 while for CaF 2 , it is 1.43 for visible and SFG frequencies and 1.41 for IR. Interfacial refractive index was calculated using the equation given in [6]. Figure 4.6: Orientation angle plots for air/stearic acid and prism/stearic acid for CH 3 symmetric and asymmetric stretches. Y-axis are the ratio of SFG intensities PPP/SSP for symmetric stretch and SPS/PPP for asymmetric stretch. Red X is the experimental ratio. Corresponding orientation angle and gaussian distribution width of orientation angle is specified on the top. 58 Stearic acid (SFG) Air/stearic acid Prism/stearic acid CH 3 -ss CH 3 -as CH 3 -ss CH 3 -as Surface pressure (mN/m) q Dq q Dq q Dq q Dq 14 45 55 170 55 125 60 50 155 50 95 25.4 25 55 160 30 135 35 40 140 40 105 30 N/A N/A 140 30 NA 30 20 135 20 105 Arachidic Acid (XRD) Surface pressure (mN/m) 15.9 21.6 25 Air/water interface 24 15 0 Table 4.2: Tilt angles from SFG spectra of stearic acid deposited on CaF 2 window and from X-ray diffrac- tion studies of arachidic acid at air/water interface 2 . XRD results are for the orientation of the arachidic acid backbone. 14 mN/m 25.4 mN/m air/stearic acid prism/stearic acid air/stearic acid prism/stearic acid PPP SSP SPS PPP SSP SPS PPP SSP SPS PPP SSP SPS A1 0.34 0.77 - 2.95 2.7 - 4.17 0 - 8.17 2.26 - A2 0.42 5.1 -4528.35 8.92 10.4 -1500.22 0.37 10.4 -409.71 8.63 10.4 -1965.69 A3 423.63 -0.08 49.06 8.07 -47.52 18.89 1.56 -2.41 530.51 7.08 -63.98 8.28 A4 -4.81 2.8 -9.1 7.03 8.41 -3.4 3.96 16.1 -114.1 8.72 10.79 -9.4 A5 8.06 0 11.6 3.77 -0.08 9.6 4 -4.46 13.1 5.55 -0.04 11.5 F1 2858 2847 - 2849 2847 - 2849 2847 - 2849 2847 - F2 2880 2879 2860 2876 2879 2860 2876 2879 2860 2876 2879 2860 F3 2906 2920 2880 2906 2756.1 2885 2906 2965.1 2885 2906 2564.3 2885 F4 2945 2943 2944 2938 2941 2941 2937 2941 2941 2937 2941 2941 F5 2966.5 2950 2959 2963 2955 2960 2965 2955 2971 2963 2955 2971 W1 2.9 4 - 8.8 6.5 - 61.6 16.5 - 19.1 21.5 - W2 2.8 5 1025.7 5.1 6 664.1 26.8 6 108.6 5.8 6 866.3 W3 1120.7 7814.3 104.2 83117.8 194.6 35.5 18.7 3.9 115.5 19.6 115.3 64.7 W4 39.7 7.4 7.5 6.8 7 4.3 39.1 10.2 52.1 11.5 9.7 10.1 W5 4.6 37.4 6.1 5.5 588.6 6.1 6.9 8.9 7.1 5.2 1993.8 8 X1 -0.43 -0.52 4.15 -0.31 0.39 2.17 -0.14 -0.31 0.35 -0.28 0.16 2.43 Y1 0.44 0.88 -1.42 0.68 3.68 -1.45 1.82 0.51 -0.88 0.19 10.45 -1.41 30 mN/m air/stearic acid prism/stearic acid PPP SSP SPS PPP SSP SPS A1 20.02 0.78 - 4.03 2.3 - A2 -15.5 5.56 -115.81 7.08 9.6 -0.17 A3 1.61 -0.23 6.72 8.55 -33.7 5.45 A4 0.7 11.13 -4.6 6.2 10 -13.1 A5 10.5 -14.25 5.5 4.23 -2.39 9.3 F1 2873.8 2847 - 2853 2855 - F2 2873 2879 2860 2876 2877 2860 F3 2920 2920 2885 2906 2920 2885 F4 2937 2943 2941 2937 2941 2941 F5 2965 2950 2958 2965 2962 2960 W1 18.8 15.6 - 15.7 7.2 - W2 15 5.8 226.5 5.7 6.2 12822.6 W3 11.8 16.3 61.4 89321.5 126.7 29.5 W4 6.7 9.7 7 7.6 10 25.1 W5 6 29.2 6.9 4.5 4 8 X1 0.31 -0.21 0.6 -0.25 -0.35 0.17 Y1 2.46 -0.03 -1.04 0.86 0.79 -0.79 Table 4.3: Fit results for air/stearic acid and prism/stearic acid spectra. A1=CH 2 symmetric stretch, A2=CH 3 symmetric stretch, A3=CH 2 asymmetric stretch, A4=Fermi resonance and A5=CH 3 asymmetric stretch. 59 geometry. In other words, difference between orientation angles between asymmetric and symmetric stretches should be around 95 as is the case for prism/stearic acid geometry. This indicates that signal amplification for prism/stearic acid interface increases the accuracy of orientation calculation. This signal amplification and corresponding gain in accuracy of molecular orientation will be even more important for surfaces with only one resonant vibration. In such case, self-consistent check between two orientation angles is not possible. 4.4 Conclusion Recording vSFG spectra with prism amplifies resonant vibrational peaks. It is especially true for PPP polarization. Using prism geometry, extra peaks appeared in PPP spectra (shown as x in Table 4.1) that can refine molecular orientation determination. Indeed, for prism/stearic acid geometry, the accuracy of molecular orientation was excellent. Moreover, prism geometry reduces spectral acquisition time. In our case, it was observed that data acquisition time can be reduced at least by an order of magnitude without affecting the S/N ratio. 4.4.1 Acknowledgements Ariel Nessl 1 and Kyowon Koo 1 helped acquire vSFG spectra and prepare stearic acid samples on CaF 2 window. 1 Department of Chemistry, University of Southern California, Los Angeles, CA 90089, U.S.A. 4.4.2 References 1. Bonderyuk A.N. and Benderskii A.V . Spectrally- and temporally-resolved vibrational surface spec- troscopy: Ultrafast hydrogen-bonding dynamics at D 2 O/CaF 2 interface J.Chem. Phys. 122, 134713 (2005) 2. Kjaer K., Als-Nielsen J., Helm C.A., Tippman-Krayer P. and Mohwald H. Synchrotron X-ray Diffrac- tion and Reflection Studies of Arachidic acid monolayers at the air-water interface. J. Phys. Chem. 93, 60 3200 (1989) 3. X. Zhuang, P. B. Miranda, D. Kim, and Y . R. Shen. Mapping molecular orientation and conformation at interfaces by surface nonlinear optics. Phys. Rev. B. 59, 12632 (1999) 4. K.Wolfrum and A.Laubereau. Vibrational sum-frequency spectroscopy of an adsorbed monolayer of hexadecanol on water. Destructive interference of adjacent lines. Chem. Phys. Lett. 228, 83 (1994) 5. X. Wei, S. Hong, X. Zhuang, T. Goto, and Y . R. Shen, Nonlinear optical studies of liquid crystal align- ment on a rubbed polyvinyl alcohol surface. Phys. Rev. E. 62 5160 (2000) 6. X. Zhuang, P. B. Miranda, D. Kim, and Y . R. Shen. Mapping molecular orientation and conformation at interfaces by surface nonlinear optics. Phys. Rev. B 59, 12632 (1998) 61 Appendix A MATLAB Code 1 % This code follows xyz axis definition as in paper 2 % International Reviews in Physical Chemistry, Vol. 24, No. 2, A p r i l June 2005, 191 256 3 4 % Run without changing anything and it will 5 % generate PPP contour plot of water bending 6 % same as in paper Fig 2. top left. 7 clear all 8 % close all 9 10 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 11 % REGION 12 % 2 = free OH % 13 % 1 = water bending, % 14 % 3 = C3v CH3 symmetric stretch 15 % 4 = custom 16 region = 4; % 17 dist_type = 2; % 1 = step, 2 = gaussian, 3 = sigma % 18 vis_nm = 800; % 19 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 20 21 %constants 22 c = 3 * 10ˆ8; 23 cm1toGHz = 100 * c / 10ˆ9; 24 25 if region == 1 % BENDING 26 IR_cm1 = 1650; 27 % converted to coordinate system consisted with the equations here 28 beta_ccc = 1; 29 beta_aac = -1.8; 30 beta_bbc = -0.5; 31 32 n1_omega1 = 1; n1_omega2 = 1; n1_omega = 1; 33 n2_omega1 = 1.3290; n2_omega2 = 1.2977 + 0.1215*1i; n2_omega = 1.3307; 34 ni_omega1 = 1.1482; ni_omega2 = 1.1362 + 0.0510*1i; ni_omega = 1.1489; 35 36 theta = 70*pi/180; dist = 45; 37 38 elseif region == 2 % FREE OH 39 IR_cm1 = 3700; 40 beta_aac = 0.32; beta_bbc = 0.32; beta_ccc = 1; 41 42 n1_omega1 = 1; n1_omega2 = 1; n1_omega = 1; 43 n2_omega1 = 1.329; n2_omega2 = 1.1863+0.0212*1i; n2_omega = 1.3320; 44 ni_omega1 = 1.1482; ni_omega2 = 1.0872 + 0.0093*1i; ni_omega = 1.1494; 45 46 theta = 30*pi/180; dist = 30; 62 47 48 elseif region == 3 % C3v CH3 symmetric stretch 49 IR_cm1 = 2875; 50 beta_aac = 1.7; beta_bbc = 1.7; beta_ccc = 1; 51 52 n1_omega1 = 1; n1_omega2 = 1; n1_omega = 1; 53 n2_omega1 = 1.3565; n2_omega2 = 1.3887 + 0.0453*1i; n2_omega = 1.3593; 54 ni_omega1 = 1.1596; ni_omega2 = 1.1729 + 0.0185*1i; ni_omega = 1.1608; 55 56 theta = 30*pi/180; dist = 20; 57 58 elseif region == 4 % custom for C3v/C_inf symmetry 59 60 IR_cm1 = 3700; 61 beta_aac = 0.16; beta_bbc = 0.82; beta_ccc = 0.49; 62 63 n1_omega1 = 1; n1_omega2 = 1; n1_omega = 1; 64 n2_omega1 = 1.329; n2_omega2 = 1.1863+0.0212*1i; n2_omega = 1.3320; 65 ni_omega1 = 1.1482; ni_omega2 = 1.0872 + 0.0093*1i; ni_omega = 1.1494; 66 % 67 % n = n2_omega1; 68 % ni_omega1 = ((nˆ4 + 5*nˆ2)/(4*nˆ2 + 2))ˆ0.5; 69 % n = n2_omega2; 70 % ni_omega2 = ((nˆ4 + 5*nˆ2)/(4*nˆ2 + 2))ˆ0.5; 71 % n = n2_omega; 72 % ni_omega = ((nˆ4 + 5*nˆ2)/(4*nˆ2 + 2))ˆ0.5; 73 74 theta = (20)*pi/180; dist = 10; 75 76 end 77 78 79 % wavelength in GHz 80 IR_um = 10ˆ4 / IR_cm1; 81 vis_um = vis_nm/1000; 82 omega1 = c / (vis_um * 1000) ; %GHz, visible 83 omega2 = IR_cm1 * cm1toGHz; %GHz, infrared 84 omega = omega1 + omega2; %sfg 85 sfg_um = c/(omega * 1000); 86 87 %IR and visible incoming angles 88 %%%%%%%%%%%%%%%%% 89 ir_initial = 1; 90 ir_final = 90; 91 vis_initial = 1; 92 vis_final = 90; 93 %%%%%%%%%%%%%%%%% 94 increment = 2; 95 range2 = floor((ir_final - ir_initial)/increment) + 1; 96 range1 = floor((vis_final - vis_initial)/increment) + 1; 97 98 ssp = zeros(range1, range2); % ssp_d = zeros(range1, range2); 99 sps = zeros(range1, range2); % sps_d = zeros(range1, range2); 100 pss = zeros(range1, range2); % pss_d = zeros(range1, range2); 101 ppp = zeros(range1, range2); % ppp_d = zeros(range1, range2); 102 103 104 l=1; 105 for vis_angle = vis_initial:increment:vis_final 106 107 k = 1; % laser_angle(k) = vis_angle; 108 for ir_angle=ir_initial:increment:ir_final 109 % 110 63 111 %convert to radians 112 beta1 = vis_angle*pi/180; % visible 113 beta2 = ir_angle*pi/180; % IR 114 beta = asin((omega1*sin(beta1) + omega2*sin(beta2))/omega); 115 116 Lxx_omega = fresnel_factor(’xx’, n1_omega, n2_omega, ni_omega, beta ); 117 Lxx_omega1 = fresnel_factor(’xx’, n1_omega1, n2_omega1, ni_omega1, beta1); 118 Lxx_omega2 = fresnel_factor(’xx’, n1_omega2, n2_omega2, ni_omega2, beta2); 119 Lyy_omega = fresnel_factor(’yy’, n1_omega, n2_omega, ni_omega, beta ); 120 Lyy_omega1 = fresnel_factor(’yy’, n1_omega1, n2_omega1, ni_omega1, beta1); 121 Lyy_omega2 = fresnel_factor(’yy’, n1_omega2, n2_omega2, ni_omega2, beta2); 122 Lzz_omega = fresnel_factor(’zz’, n1_omega, n2_omega, ni_omega, beta ); 123 Lzz_omega1 = fresnel_factor(’zz’, n1_omega1, n2_omega1, ni_omega1, beta1); 124 Lzz_omega2 = fresnel_factor(’zz’, n1_omega2, n2_omega2, ni_omega2, beta2); 125 126 % CALCULATE C AND D VALUES 127 if region == 1 || region == 4 %water bending 128 Const = 1; 129 %SSP 130 sspC1 = Lyy_omega * Lyy_omega1 * Lzz_omega2 * sin(beta2) * 0.25; 131 sspC2 = (beta_aac + beta_bbc + 2*beta_ccc); 132 sspC3 = (beta_aac + beta_bbc - 2*beta_ccc); 133 sspC = -sspC3/sspC2; 134 sspD = sspC1*sspC2; 135 %SPS 136 spsC1 = - Lyy_omega * Lzz_omega1 * Lyy_omega2 * sin(beta1) * 0.25; 137 spsC2 = (beta_aac + beta_bbc - 2*beta_ccc); 138 spsC3 = -(beta_aac + beta_bbc - 2*beta_ccc); 139 spsC = -spsC3/spsC2; 140 spsD = spsC1*spsC2; 141 %PSS 142 pssC1 = - Lzz_omega * Lyy_omega1 * Lyy_omega2 * sin(beta) * 0.25; 143 pssC2 = (beta_aac + beta_bbc - 2*beta_ccc); 144 pssC3 = -(beta_aac + beta_bbc - 2*beta_ccc); 145 pssC = -pssC3/pssC2; 146 pssD = pssC1*pssC2; 147 %PPP 148 pppAf = Lxx_omega * Lxx_omega1 * Lzz_omega2 * cos(beta) * cos(beta1) * sin( beta2); 149 pppBf = Lxx_omega * Lzz_omega1 * Lxx_omega2 * cos(beta) * sin(beta1) * cos( beta2); 150 pppCf = Lzz_omega * Lxx_omega1 * Lxx_omega2 * sin(beta) * cos(beta1) * cos( beta2); 151 pppDf = Lzz_omega * Lzz_omega1 * Lzz_omega2 * sin(beta) * sin(beta1) * sin( beta2); 152 153 pppK = beta_aac + beta_bbc + 2*beta_ccc; 154 pppL = beta_aac + beta_bbc - 2*beta_ccc; 155 pppM = beta_aac + beta_bbc; 156 157 pppC1 = -pppAf*pppK/4 + pppBf*pppL/4 - pppCf*pppL/4 + pppDf*pppM/2; 158 pppC2 = -pppAf*pppL/4 - pppBf*pppL/4 + pppCf*pppL/4 - pppDf*pppL/2; 159 pppC = -pppC2/pppC1; 160 pppD = pppC1; 161 162 163 elseif region == 2 || region == 3 %free OH, CH3 symmetric stretch 164 Const = 1; 165 R = beta_aac/beta_ccc; 166 % SSP 167 sspC = (1-R)/(1+R); 168 sspD = Lyy_omega * Lyy_omega1 * Lzz_omega2 * sin(beta2) * beta_ccc * (1+R) * 0.5; 169 %SPS 64 170 spsC = 1; 171 spsD = Lyy_omega * Lzz_omega1 * Lyy_omega2 * sin(beta1) * beta_ccc * (1-R) * 0.5; 172 %PSS 173 pssC = 1; 174 pssD = Lzz_omega * Lyy_omega1 * Lyy_omega2 * sin(beta) * beta_ccc * (1-R) * 0.5; 175 %PPP 176 pppA = Lxx_omega * Lxx_omega1 * Lzz_omega2 * cos(beta) * cos(beta1) * sin( beta2); 177 pppB = Lxx_omega * Lzz_omega1 * Lxx_omega2 * cos(beta) * sin(beta1) * cos( beta2); 178 pppC = Lzz_omega * Lxx_omega1 * Lxx_omega2 * sin(beta) * cos(beta1) * cos( beta2); 179 pppD = Lzz_omega * Lzz_omega1 * Lzz_omega2 * sin(beta) * sin(beta1) * sin( beta2); 180 181 pppC1 = 0.5 * beta_ccc * (-pppA * (1+R) - pppB * (1-R) + pppC * (1-R) + 2 * pppD * R); 182 pppC2 = 0.5 * beta_ccc * (1-R) * (pppA+pppB-pppC+2*pppD); 183 pppC = -pppC2/pppC1; 184 pppD = pppC1; 185 186 end 187 188 ssp(l,k) = abs(sec(beta)*sspD*(expect(dist_type, dist, ’cos’, theta) - sspC* expect(dist_type, dist, ’cos3’, theta)))ˆ2; 189 sps(l,k) = abs(sec(beta)*spsD*(Const * expect(dist_type, dist, ’cos’, theta) - spsC*expect(dist_type, dist, ’cos3’, theta)))ˆ2; 190 pss(l,k) = abs(sec(beta)*pssD*(Const * expect(dist_type,dist, ’cos’, theta) - pssC*expect(dist_type, dist, ’cos3’, theta)))ˆ2; 191 ppp(l,k) = abs(sec(beta)*pppD*(expect(dist_type, dist, ’cos’, theta) - pppC* expect(dist_type, dist, ’cos3’, theta)))ˆ2; 192 193 % ppp(l,k) = abs(Lzz_omega * Lzz_omega1 * sin(beta) * sin(beta1))ˆ2; 194 % ppp(l,k) = abs(Lxx_omega * Lxx_omega1 * cos(beta) * cos(beta1))ˆ2; 195 196 k=k+1; 197 end 198 199 200 l=l+1 201 end 202 203 if 1==0 204 description = ’write_your_description_here’; 205 filename = [’ssp_’,description,’.mat’]; save(filename, ’ssp’); 206 filename = [’ppp_’,description,’.mat’]; save(filename, ’ppp’); 207 filename = [’sps_’,description,’.mat’]; save(filename, ’sps’); 208 filename = [’pss_’,description,’.mat’]; save(filename, ’pss’); 209 end 210 211 % Find optimal IR and Visible angle for PPP 212 % and plot 15% contour line 213 [a,b]= max(ssp); 214 sfg1=max(a) 215 [a,b]= max(ppp); 216 sfg2=max(a) 217 [a,b]= max(sps); 218 sfg3=max(a) 219 [a,b]= max(pss); 220 sfg4=max(a) 221 for i=1:length(ssp(:,1)) 222 for j=1:length(ssp(:,1)) 65 223 if ssp(i,j)== sfg1 224 sprintf(’ssp’) 225 ir_index = j 226 vis_index = i 227 end 228 if ppp(i,j)== sfg2 229 sprintf(’ppp’) 230 ir_index = j 231 vis_index = i 232 end 233 if sps(i,j)== sfg3 234 sprintf(’sps’) 235 ir_index = j 236 vis_index = i 237 end 238 if pss(i,j)== sfg4 239 sprintf(’pss’) 240 ir_index = j 241 vis_index = i 242 end 243 244 end 245 end 246 hold on; 247 figure(1); 248 contour(ppp, [sfg2*0.85 sfg2*0.85],’k’,’LineWidth’,3); 249 % [c,h]=contour(ppp, [sfg2*0.85 sfg2*0.85],’k’,’LineWidth’,3); 250 % 251 % xlabel(’IR laser angle’); ylabel(’Visible laser angle’); 252 % v=[0.0]; clabel(c,h,v);ax = gca; 253 % ax.XAxisLocation = ’bottom’; ax.YAxisLocation = ’origin’; 254 % ax.XGrid = ’off’; ax.YGrid = ’off’; 255 % ax.FontSize = 16; 256 % xticks([10 20 30 40 50 60 70 80 90]) 257 % xticklabels({’10’, ’20’, ’30’, ’40’, ’50’, ’60’, ’70’, ’80’, ’90’}) 258 % yticks([10 20 30 40 50 60 70 80 90]) 259 % yticklabels({’10’, ’20’, ’30’, ’40’, ’50’, ’60’, ’70’, ’80’, ’90’}) 260 % 261 % 262 contour(ssp, [sfg1*0.85 sfg1*0.85],’b’,’LineWidth’,3); 263 contour(sps, [sfg3*0.85 sfg3*0.85],’r’,’LineWidth’,3); 264 contour(pss, [sfg4*0.85 sfg4*0.85],’g’,’LineWidth’,3); 265 266 267 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 268 269 % This code follows xyz axis definition as in paper 270 % International Reviews in Physical Chemistry, Vol. 24, No. 2, A p r i l June 2005, 191 256 271 272 % Run without changing anything and it will 273 % generate PPP contour plot of water bending 274 % same as in paper Fig 2. top left. 275 clear all 276 277 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 278 % REGION 279 % 2 = free OH % 280 % 1 = water bending, % 281 % 3 = C3v CH3 symmetric stretch 282 % 4 = custom region with C3v/C_infv symmetry 283 region = 4; % 284 dist_type = 2; % 1 = step, 2 = gaussian, 3 = sigma % 285 vis_nm = 800; % 66 286 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 287 288 %constants 289 c = 3 * 10ˆ8; 290 cm1toGHz = 100 * c / 10ˆ9; 291 292 if region == 1 % BENDING 293 IR_cm1 = 1650; 294 295 beta_ccc = 1; 296 beta_aac = -1.82; 297 beta_bbc = -0.5; 298 299 n1_omega1 = 1; n1_omega2 = 1; n1_omega = 1; 300 n2_omega1 = 1.3290; n2_omega2 = 1.2977 + 0.1215*1i; n2_omega = 1.3307; 301 ni_omega1 = 1.1482; ni_omega2 = 1.1362 + 0.0510*1i; ni_omega = 1.1489; 302 303 theta_start = 65; 304 theta_end = 75; 305 theta_step = 1; 306 dist = 45; 307 elseif region == 2 % FREE OH 308 IR_cm1 = 3700; 309 beta_aac = 0.32; beta_bbc = 0.32; beta_ccc = 1; 310 311 n1_omega1 = 1; n1_omega2 = 1; n1_omega = 1; 312 n2_omega1 = 1.329; n2_omega2 = 1.1863+0.0212*1i; n2_omega = 1.3320; 313 ni_omega1 = 1.1482; ni_omega2 = 1.0872 + 0.0093*1i; ni_omega = 1.1494; 314 315 theta_start = 25; 316 theta_end = 35; 317 theta_step = 1; 318 dist = 30; 319 320 elseif region == 3 % C3v CH3 symmetric stretch 321 IR_cm1 = 2875; 322 beta_aac = 1.7; beta_bbc = 1.7; beta_ccc = 1; 323 324 n1_omega1 = 1; n1_omega2 = 1; n1_omega = 1; 325 n2_omega1 = 1.3565; n2_omega2 = 1.3887 + 0.0453*1i; n2_omega = 1.3593; 326 ni_omega1 = 1.1596; ni_omega2 = 1.1729 + 0.0185*1i; ni_omega = 1.1608; 327 328 theta_start = 25; 329 theta_end = 35; 330 theta_step = 1; 331 dist = 20; 332 333 elseif region == 4 % custom for C3v/C_inf symmetry 334 335 IR_cm1 = 2875; 336 beta_aac = 1.7; beta_bbc = 1.7; beta_ccc = 1; 337 338 n1_omega1 = 1; n1_omega2 = 1; n1_omega = 1; 339 n2_omega1 = 1.3565; n2_omega2 = 1.3887 + 0.0453*1i; n2_omega = 1.3593; 340 ni_omega1 = 1.1596; ni_omega2 = 1.1729 + 0.0185*1i; ni_omega = 1.1608; 341 342 % n = n2_omega1; 343 % ni_omega1 = ((nˆ4 + 5*nˆ2)/(4*nˆ2 + 2))ˆ0.5; 344 % n = n2_omega2; 345 % ni_omega2 = ((nˆ4 + 5*nˆ2)/(4*nˆ2 + 2))ˆ0.5; 346 % n = n2_omega; 347 % ni_omega = ((nˆ4 + 5*nˆ2)/(4*nˆ2 + 2))ˆ0.5; 348 % 349 theta_start = 20; 67 350 theta_end = 40; 351 theta_step = 1; 352 dist = 20; 353 354 end 355 356 357 % wavelength in GHz 358 IR_um = 10ˆ4 / IR_cm1; 359 vis_um = vis_nm/1000; 360 omega1 = c / (vis_um * 1000) ; %GHz, visible 361 omega2 = IR_cm1 * cm1toGHz; %GHz, infrared 362 omega = omega1 + omega2; %sfg 363 sfg_um = c/(omega * 1000); 364 365 %IR and visible incoming angles 366 %%%%%%%%%%%%%%%%% 367 ir_initial = 60; 368 ir_final = 60; 369 vis_initial = 62; 370 vis_final = 62; 371 %%%%%%%%%%%%%%%%% 372 increment = 1; 373 range2 = floor((ir_final - ir_initial)/increment) + 1; 374 range1 = floor((vis_final - vis_initial)/increment) + 1; 375 376 ratio = zeros(range1, range2); 377 angle = zeros(1, floor((theta_end-theta_start)/theta_step) +1); 378 ssp_r = zeros(1, floor((theta_end-theta_start)/theta_step) +1); 379 pss_r = zeros(1, floor((theta_end-theta_start)/theta_step) +1); 380 sps_r = zeros(1, floor((theta_end-theta_start)/theta_step) +1); 381 ppp_r = zeros(1, floor((theta_end-theta_start)/theta_step) +1); 382 383 l=1; 384 for vis_angle = vis_initial:increment:vis_final 385 386 k = 1; % laser_angle(k) = vis_angle; 387 for ir_angle=ir_initial:increment:ir_final 388 % 389 390 %convert to radians 391 beta1 = vis_angle*pi/180; % visible 392 beta2 = ir_angle*pi/180; % IR 393 beta = asin((omega1*sin(beta1) + omega2*sin(beta2))/omega); 394 395 Lxx_omega = fresnel_factor(’xx’, n1_omega, n2_omega, ni_omega, beta ); 396 Lxx_omega1 = fresnel_factor(’xx’, n1_omega1, n2_omega1, ni_omega1, beta1); 397 Lxx_omega2 = fresnel_factor(’xx’, n1_omega2, n2_omega2, ni_omega2, beta2); 398 Lyy_omega = fresnel_factor(’yy’, n1_omega, n2_omega, ni_omega, beta ); 399 Lyy_omega1 = fresnel_factor(’yy’, n1_omega1, n2_omega1, ni_omega1, beta1); 400 Lyy_omega2 = fresnel_factor(’yy’, n1_omega2, n2_omega2, ni_omega2, beta2); 401 Lzz_omega = fresnel_factor(’zz’, n1_omega, n2_omega, ni_omega, beta ); 402 Lzz_omega1 = fresnel_factor(’zz’, n1_omega1, n2_omega1, ni_omega1, beta1); 403 Lzz_omega2 = fresnel_factor(’zz’, n1_omega2, n2_omega2, ni_omega2, beta2); 404 405 % CALCULATE C AND D VALUES 406 if region == 1 %water bending 407 Const = 1; 408 %SSP 409 sspC1 = Lyy_omega * Lyy_omega1 * Lzz_omega2 * sin(beta2) * 0.25; 410 sspC2 = (beta_aac + beta_bbc + 2*beta_ccc); 411 sspC3 = (beta_aac + beta_bbc - 2*beta_ccc); 412 sspC = -sspC3/sspC2; 413 sspD = sspC1*sspC2; 68 414 %SPS 415 spsC1 = - Lyy_omega * Lzz_omega1 * Lyy_omega2 * sin(beta1) * 0.25; 416 spsC2 = (beta_aac + beta_bbc - 2*beta_ccc); 417 spsC3 = -(beta_aac + beta_bbc - 2*beta_ccc); 418 spsC = -spsC3/spsC2; 419 spsD = spsC1*spsC2; 420 %PSS 421 pssC1 = - Lzz_omega * Lyy_omega1 * Lyy_omega2 * sin(beta) * 0.25; 422 pssC2 = (beta_aac + beta_bbc - 2*beta_ccc); 423 pssC3 = -(beta_aac + beta_bbc - 2*beta_ccc); 424 pssC = -pssC3/pssC2; 425 pssD = pssC1*pssC2; 426 %PPP 427 pppAf = Lxx_omega * Lxx_omega1 * Lzz_omega2 * cos(beta) * cos(beta1) * sin( beta2); 428 pppBf = Lxx_omega * Lzz_omega1 * Lxx_omega2 * cos(beta) * sin(beta1) * cos( beta2); 429 pppCf = Lzz_omega * Lxx_omega1 * Lxx_omega2 * sin(beta) * cos(beta1) * cos( beta2); 430 pppDf = Lzz_omega * Lzz_omega1 * Lzz_omega2 * sin(beta) * sin(beta1) * sin( beta2); 431 432 pppK = beta_aac + beta_bbc + 2*beta_ccc; 433 pppL = beta_aac + beta_bbc - 2*beta_ccc; 434 pppM = beta_aac + beta_bbc; 435 436 pppC1 = -pppAf*pppK/4 + pppBf*pppL/4 - pppCf*pppL/4 + pppDf*pppM/2; 437 pppC2 = -pppAf*pppL/4 - pppBf*pppL/4 + pppCf*pppL/4 - pppDf*pppL/2; 438 pppC = -pppC2/pppC1; 439 pppD = pppC1; 440 441 442 elseif region == 2 || region == 3 || region==4 %free OH, CH3 symmetric stretch 443 Const = 1; 444 R = beta_aac/beta_ccc; 445 % SSP 446 sspC = (1-R)/(1+R); 447 sspD = Lyy_omega * Lyy_omega1 * Lzz_omega2 * sin(beta2) * beta_ccc * (1+R) * 0.5; 448 %SPS 449 spsC = 1; 450 spsD = Lyy_omega * Lzz_omega1 * Lyy_omega2 * sin(beta1) * beta_ccc * (1-R) * 0.5; 451 %PSS 452 pssC = 1; 453 pssD = Lzz_omega * Lyy_omega1 * Lyy_omega2 * sin(beta) * beta_ccc * (1-R) * 0.5; 454 %PPP 455 pppA = Lxx_omega * Lxx_omega1 * Lzz_omega2 * cos(beta) * cos(beta1) * sin( beta2); 456 pppB = Lxx_omega * Lzz_omega1 * Lxx_omega2 * cos(beta) * sin(beta1) * cos( beta2); 457 pppC = Lzz_omega * Lxx_omega1 * Lxx_omega2 * sin(beta) * cos(beta1) * cos( beta2); 458 pppD = Lzz_omega * Lzz_omega1 * Lzz_omega2 * sin(beta) * sin(beta1) * sin( beta2); 459 460 pppC1 = 0.5 * beta_ccc * (-pppA * (1+R) - pppB * (1-R) + pppC * (1-R) + 2 * pppD * R); 461 pppC2 = 0.5 * beta_ccc * (1-R) * (pppA+pppB-pppC+2*pppD); 462 pppC = -pppC2/pppC1; 463 pppD = pppC1; 464 465 end 69 466 i=1; 467 for theta=(theta_start:theta_step:theta_end)*pi/180 468 angle(i) = theta*180/pi; 469 ssp_r(i) = abs(sec(beta)*sspD*(expect(dist_type, dist, ’cos’, theta) - sspC* expect(dist_type, dist, ’cos3’, theta)))ˆ2; 470 sps_r(i) = abs(sec(beta)*spsD*(Const * expect(dist_type, dist, ’cos’, theta) - spsC*expect(dist_type, dist, ’cos3’, theta)))ˆ2; 471 pss_r(i) = abs(sec(beta)*pssD*(Const * expect(dist_type,dist, ’cos’, theta) - pssC*expect(dist_type, dist, ’cos3’, theta)))ˆ2; 472 ppp_r(i) = abs(sec(beta)*pppD*(expect(dist_type, dist, ’cos’, theta) - pppC* expect(dist_type, dist, ’cos3’, theta)))ˆ2; 473 i=i+1; 474 end 475 476 ratio_grad = gradient(ppp_r./ssp_r); 477 ratio(l,k) = ratio_grad(5); 478 479 k=k+1; 480 end 481 l=l+1 482 end 483 484 if 1==0 485 description = ’write_your_description_here’; 486 filename = [’ppp_over_ssp_’,description,’.mat’]; save(filename, ’ratio’); 487 end 488 hold on; 489 [c,h]=contour((ratio(:,1:90)), [0.0 0.0], ’LineWidth’,2); clabel(c,h, ’FontSize’, 14); 490 xlabel(’IR laser angle’); ylabel(’Visible laser angle’); 491 492 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 493 494 function expr = expect(dist_type, dist, type, angle) 495 496 % this function calculates the expectation values 497 if dist_type == 2 %GAUSSIAN DIST 498 499 delta_angle = dist*pi/180; 500 sigma = delta_angle/2.35; 501 theta0 = angle; 502 503 % sin(x) * Gaussian 504 distribution = @(x) sin(x).* (1/(2*pi*sigmaˆ2)ˆ0.5) .* exp(-(x-theta0).ˆ2 / (2*sigma ˆ2)); 505 if strcmp(type,’cos’) 506 fun = @(x) cos(x).* sin(x).*(1/(2*pi*sigmaˆ2)ˆ0.5) .* exp(-(x-theta0).ˆ2 / (2* sigmaˆ2)); 507 expr = integral(fun, 0,pi)/integral(distribution,0,pi); 508 else 509 fun = @(x) (cos(x)).ˆ3 .* sin(x).*(1/(2*pi*sigmaˆ2)ˆ0.5) .* exp(-(x-theta0).ˆ2 / (2*sigmaˆ2)); 510 expr = integral(fun,0,pi)/integral(distribution,0,pi); 511 end 512 end 513 514 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 515 516 function y = fresnel_factor(type,n1, n2, ni, theta) 517 % this function computes fresnel factors 518 % type is xx, yy or zz, 519 % n1, n2, ni are refractive indices 520 521 gamma_here = asin(n1*sin(theta)/n2); 522 if strcmp(type, ’xx’) 70 523 y = 2*n1*cos(gamma_here) / (n1*cos(gamma_here) + n2*cos(theta)); 524 elseif strcmp(type, ’yy’) 525 y = 2*n1*cos(theta) / (n1*cos(theta)+n2*cos(gamma_here)) ; 526 else 527 y = 2*n2*cos(theta) * (n1/ni)ˆ2 / (n1*cos(gamma_here)+n2*cos(theta)); 528 end 529 530 end 71

Abstract (if available)

Abstract Interfaces can be formed from the intersection of two immiscible media such as gas/liquid, liquid/liquid, gas/solid, liquid/solid or solid/solid. Even within a given interface, i.e. air/water, numerous variations can be studied: air/salt water, air/water under electric field or air/acidic water. Sum frequency generation (SFG) is a coherent, second order technique where two beams are overlapped both temporally and spatially at the interface. The signal generated travels in the phase matching direction. SFG is generated in non-centrosymmetric medium where inversion symmetry is broken. At interfaces, non-centrosymmetry is satisfied because molecules have a preferred orientation. Euler angles relate molecular orientation angles to the laboratory coordinates. Using the transformation of laboratory to molecular coordinates, averaged molecular orientation angles can be extracted from experimental spectra. Extracting molecular orientation angles from the experiments is the focus of this thesis. ❧ Understanding the air/water interface has been the focus of both experimental and theoretical groups. Experimentally, accurate phase sensitive spectra of the air/water interface has been challenging. Theoretically, molecular dynamics treatment of the air/water interface is being refined to get good agreement with experiments. Our work directly benefits the molecular dynamics simulation efforts. We use homodyne SFG to record the spectra of air water interface and extract average orientation angles of interfacial water molecules. Molecular dynamics simulation community then can use our calculated orientation angles to evaluate the orientation angle predictions of their numerical models. In addition to orientation studies of air water/interface, we studied the dependence of the SFG intensity to the incoming laser angles. We determined that PPP SFG intensity is sensitive to the incoming laser angles. Finally, we propose a new low cost approach to experimentally maximize the SFG intensity. ❧ The approach uses prism to reach experimental conditions that is close to the total internal reflection geometry. This new approach is evaluated experimentally and considerable increase in SFG signal intensity is observed. SFG spectra are then analyzed to extract orientation angles.

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Asset Metadata

Creator Mammetkuliyev, Muhammet (author)

Core Title Molecular orientation from sum frequency generation spectroscopy: case study of water and methyl vibrations

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Chemistry

Publication Date 02/11/2021

Defense Date 01/04/2021

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag air/water interface,molecular orientation,OAI-PMH Harvest,optimal experimental geometry,sum frequency generation spectroscopy,total internal reflection SFG

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Takahashi, Susumu (committee chair), Benderskii, Alexander (committee member), Dawlaty, Jahan (committee member), Kresin, Vitaly (committee member)

Creator Email mammetku@usc.edu,muhammed237@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-422027

Unique identifier UC11667422

Identifier etd-Mammetkuli-9267.pdf (filename),usctheses-c89-422027 (legacy record id)

Legacy Identifier etd-Mammetkuli-9267.pdf

Dmrecord 422027

Document Type Dissertation

Rights Mammetkuliyev, Muhammet

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA