Effects of Reorientation in Vibrational Sum-Frequency Spectroscopy

8902

J. Phys. Chem. C 2007, 111, 8902-8915

Effects of Reorientation in Vibrational Sum-Frequency Spectroscopy† John T. Fourkas,*,§,‡,⊥ Robert A. Walker,§,⊥ Suleyman Z. Can,§ and Erez Gershgoren§ Department of Chemistry and Biochemistry, UniVersity of Maryland, College Park, Maryland 20742, Chemical Physics Program, UniVersity of Maryland, College Park, Maryland 20742 and Institute for Physical Science and Technology, UniVersity of Maryland, College Park, Maryland 20742 ReceiVed: December 30, 2006; In Final Form: February 7, 2007

Vibrational sum-frequency spectroscopy (VSFS) has become a widely used tool for studying molecules at interfaces. While VSFS data are often interpreted in terms of static structure, Wei and Shen demonstrated that orientational dynamics that occur on the time scale of vibrational dephasing can influence the strength of the VSFS signal (Wei, X.; Shen, Y. R. Phys. ReV. Lett. 2001, 86, 4799). In this report, we consider the orientational averages relevant to VSFS under different polarization conditions in terms of the infrared transition dipole and the Raman tensor when the infrared and Raman transitions are simultaneous and when they are separated by a delay that is greater than the orientational correlation time. We examine specific cases of orientational averaging that apply to commonly studied functional groups. This analysis makes possible the identification of the circumstances under which reorientation should be taken into account when using VSFS to determine molecular orientation at interfaces.

Introduction Vibrational sum-frequency spectroscopy (VSFS) is a powerful technique that makes possible the selective spectroscopic interrogation of vibrational modes of molecules at interfaces.1-4 In this technique, an infrared (IR) pulse at frequency ωIR and a visible pulse at frequency ωvis are incident on an interface, generating a signal at frequency ωSF ) ωIR + ωvis. The sumfrequency signal is enhanced when ωIR is in resonance with an IR-allowed vibration, providing a species-specific spectroscopic probe of interfacial molecules. In this resonant situation, the IR pulse can be thought of as creating a vibrational coherence that is probed via a Raman transition. Since the lifetime of a vibrational coherence can potentially be many picoseconds, it is possible, when ωIR is on vibrational resonance, for the Raman transition to occur significantly later than the IR transition. As revolutionary as VSFS has been for our understanding of structure at interfaces, it is still young enough of a technique that many of its fundamental aspects remain to be explored in full. For instance, VSFS data are often interpreted as giving information on static structure. However, Wei and Shen have demonstrated that molecular reorientation that occurs on the time scale of or faster than vibrational dephasing can lead to significant changes in VSFS signals even when the equilibrium structure of the interface remains unchanged throughout the VSFS experiment.5 They showed that fast reorientation effectively factorizes the orientational averages involved in the VSFS signal. Rather than orientational averaging over three polarizations (those of the two input pulses and the signal) at a single time, orientational randomization between the IR and Raman transitions converts the average into a product of the orientational averages for the two individual transitions.5 †

Part of the special issue “Kenneth B. Eisenthal Festschrift”. * To whom correspondence should be addressed. E-mail: fourkas@ umd.edu. § Department of Chemistry and Biochemistry. ‡ Institute for Physical Science and Technology. ⊥ Chemical Physics Program.

The influence of reorientation on VSFS data remains relatively unexplored. Such dynamic effects are often not taken into consideration in orientational analyses based on VSFS data. Furthermore, reorientation can lead to counterintuitive effects in VSFS. For instance, Wei and Shen predicted for free OH groups at the air/water interface that reorientation should increase the signal under some polarization conditions.5 Here, we explore the orientational averages relevant to VSFS in two limits, when the IR and Raman transitions are time coincident and when they are separated by a time that is much greater than the orientational correlation time. We express these orientational averages in terms of the IR transition dipole and the Raman tensor for the vibrational mode of interest, allowing us to evaluate the importance of reorientation in terms of quantities that can be derived from IR and Raman spectra. We use this analysis to develop guidelines for when reorientation may have a significant influence on VSFS data, and we discuss and interpret some commonly observed features of VSFS data from liquid surfaces in light of these guidelines. Theory The VSFS signal is proportional to the orientational average of the hyperpolarizability βijk, which depends on the infrared transition dipole vector µ and the polarizability tensor r of the vibration. Such averages have been reported many times previously but generally in terms of βijk.6-8 We are interested in what happens when the infrared excitation step is not necessarily simultaneous with the Raman probing step, and so it is more illuminating to determine the elements of µ and r that contribute to the vibrationally resonant signal under different polarization conditions. We will consider azimuthally isotropic interfaces. To perform the orientational average, we begin in the molecular frame of reference, with the moiety of interest (e.g., the 3-fold axis of a methyl group) pointing along the z-axis. The molecule is rotated by angle φ about the z-axis, tilted by angle θ about the y-axis, and then rotated by angle χ about the z-axis (Figure 1).9 [Note that the angle χ is not related to

10.1021/jp0690401 CCC: $37.00 © 2007 American Chemical Society Published on Web 04/21/2007

Effects of Reorientation in VSFS

J. Phys. Chem. C, Vol. 111, No. 25, 2007 8903 (2) transitions). For R (2) xxz ) R yyz, we have

R (2) xxz(0)∝

(

3

)(

3

)

3

∑ ∑ Rγ,1Rδ,1r′γδ β)1 ∑ Rβ,3 µ′β γ)1 δ)1

(3)

The appropriate cubic averages of products of elements of the rotation matrix over angle χ are listed in the Supporting Information. The result of the averaging is

Figure 1. Axis system for orientational average calculations. The molecular axis system is (x, y, z), and the lab axis system is (x′, y′, z′). The molecule is first rotated by angle φ around the z-axis, then by angle θ about the y-axis, and finally by angle x around the z′-axis.

elements of the second-order susceptibility tensor, χ(2).] The z-axis for a given vibrational moiety is chosen as the one that lends itself most readily to averaging over φ. For instance, in a methyl group, the z-axis would be the bond axis between the carbon atom and the remainder of the molecule, and for a methylene group, the z-axis would be perpendicular to the plane defined by the carbon atom and the two hydrogen atoms. For simultaneous IR and Raman transitions, the elements of the second-order response function (the time-domain analogue of the second-order susceptibility) at the surface of an otherwise isotropic medium are given by

(∑ ∑ 3

R (2) ijk (0) ∝

3

γ)1 δ)1

)( ∑

)

3

Rγ,iRδ,jrγδ(q)

Rβ,k µβ(q)

β)1

(1)

where the overbar indicates an average over angle χ, the argument of the response function is the delay time, and q is the vibrational coordinate of interest. The Ra,b are elements of the Euler rotation matrix

[

cos φ cos θ cos χ sin φ sin χ

cos φ sinχ + sin θ cos χ sin φ cos θ cos χ

-sin θ sin χ R ˆ ) -cos φ cos θ sin χ - cos φ cos χ sin φ cos χ sin φ cos θ sin χ -cos φ sin θ

-sin φ sin θ

cos θ

]

(2)

(where a is the column and b is the row), and the numbers 1, 2, and 3 correspond to x, y, and z, respectively. From here on, we will make the usual approximations that µi(q) ) (∂µi/∂q)q0‚ (q - q0) and rij(q) ) (∂rij/∂q)q0(q - q0), and we will write these derivatives as µ′i and r′ij, respectively. The nonzero elements of R(2)(τ) are those that contain x an even number of times, y an even number of times, and z an (2) (2) (2) (2) (2) odd number of times: R (2) xxz ) R yyz, R xzx ) R zxx ) R yzy ) R zyy (2) and R zzz. Under SSP polarization conditions (where the first polarization is the signal, the second is the visible beam, and the third is the IR beam), the R (2) xxz element is probed; under SPS and PSS conditions, the χ(2) element is probed, and under xzx (2) (2) (2) PPP conditions, a combination of the R (2) yyz, R yzy, R zyy, and R zzz elements is probed. Here, we have assumed that both the visible and signal frequencies are far from electronic resonance, such (2) that R (2) ijk ) R jik . We now consider the orientational averages for different tensor elements of R(2)(0) (i.e., for simultaneous IR and Raman

1 3 3 R (2) xxz(0) ∝ µ′1r′11〈cos φ sin θ - cos φ sin θ〉 + 2 µ′1r′12〈(sin φ - sin3 φ)sin3 θ〉 + µ′1r′13〈cos2 φ(cos3 θ - cos θ)〉 + 1 µ′ r′ 〈(cosφ - cos3 φ)sin3 θ - cos φ sin θ〉 + 2 1 22 1 µ′1r′23〈sin φ cos φ(cos3 θ - cos θ)〉 - µ′1r′33〈cos φ sin3θ〉 + 2 1 µ′ r′ 〈(sin φ - sin3 φ)sin3 θ - sin φ sin θ〉 + 2 2 11 µ′2r′12〈(cos φ - cos3 φ)sin3 θ〉 + µ′2r′13〈sin φ cos φ(cos3 θ - cos θ)〉 + 1 µ′ r′ 〈sin3 φ sin3 θ - sin φ sin θ〉 + 2 2 22 1 µ′2r′23〈sin2 φ(cos3 θ - cos θ)〉 - µ′2r′33〈sin φ sin3 θ〉 + 2 1 µ′ r′ 〈cos2 φ cos3 θ + sin2 φ cos θ〉 + 2 3 11 µ′3r′12〈sin φ cos φ(cos3 θ - cos θ)〉 + µ′3r′13〈cos φ(sin θ - sin3 θ)〉 + 1 µ′ r′ 〈sin 2 φ cos3 θ + cos2 φ cosθ〉 + 2 3 22 1 µ′3r′23〈sin φ(sin θ - sin3 θ)〉 + µ′3r′33〈cos θ - cos3 θ〉 (4) 2

Here, 〈 〉 denotes an orientational average, and we have made use of the fact that, for the off-diagonal elements of the r′ tensor, (2) (2) (2) rij ) rji. We next consider R (2) xzx ) R zxx ) R yzy ) R zyy, for which we have

R (2) xzx(0) ∝

(

3

)(

3

)

3

∑ ∑ Rγ,1Rδ,3r′γδ β)1 ∑ Rβ,1 µ′β γ)1 δ)1

(5)

Performing the averaging yields 1 3 3 R (2) xzx(0) ∝ µ′1r′11〈cos φ sin θ - cos φ sin θ〉 + 2 1 µ′1r′12 (sin φ - sin3 φ)sin3 θ - sin φ sin θ + 2 1 µ′1r′13 cos2 φ cos3 θ + (sin2 φ - cos2 φ)cos θ + 2 1 µ′ r′ 〈(cos φ - cos3 φ)sin3 θ〉 + 2 1 22 µ′1r′23〈sin φ cos φ(cos3 θ - cos θ)〉 + 1 µ′ r′ 〈cos φ(sin θ - sin3 θ)〉 + 2 1 33 1 µ′ r′ 〈(sin φ - sin3 φ)sin3 θ〉 + µ′2r′12 (cos φ - cos3 φ)sin3 θ 2 2 11 1 cos φ sin θ + µ′2r′13〈sin φ cos φ(cos3 θ - cos θ)〉 + 2 1 µ′ r′ 〈sin3 φ sin3 θ - sin φ sin θ〉 + µ′2r′23 sin2 φ cos3 θ 2 2 22 1 1 (sin2 φ - cos2 φ)cos θ + µ′2r′33〈sin φ(sin θ - sin3 θ)〉 + 2 2 1 µ′ r′ 〈cos2 φ(cos3 θ - cos θ)〉 + 2 3 11 µ′3r′12〈sin φ cos φ(cos3 θ - cos θ)〉 + 1 1 µ′ r′ 〈cos φ(sin θ - 2sin3 θ)〉 + µ′3r′22〈sin2 φ(cos3 θ - cos θ)〉 + 2 3 13 2 1 1 µ′ r′ 〈sin φ(sin θ - 2sin3 θ)〉 + µ′3r′33〈cos θ - cos3 θ〉 (6) 2 3 23 2

〈

〈

〉

〈

〉

〉

〈

〉

8904 J. Phys. Chem. C, Vol. 111, No. 25, 2007

Fourkas et al.

(2) For R zzz , we must calculate

(∑ ∑ 3

R (2) zzz(0) ∝

3

)(∑ ) 3

Rγ,3Rδ,3r′γδ

γ)1 δ)1

Rβ,3µ′β

(7)

β)1

The resultant average is given by 3 3 R (2) zzz(0) ∝ -µ′1r′11〈cos φ sin θ〉 +

2µ′1r′12〈(sin3 φ - sin φ)sin3 θ〉 + 2µ′1r′13〈cos φ(cos θ - cos3 θ)〉 + µ′1r′22〈(cos3 φ - cos φ)sin3 θ〉 + 2

2µ′1r′23〈sin φ cos φ(cos θ - cos3 θ)〉 + µ′1r′33〈cos φ(sin3 θ - sin θ)〉 + µ′2r′11〈(sin3 φ - sin φ)sin3 θ〉 + 2µ′2r′12〈(cos3 φ - cos φ)sin3 θ〉 + 2µ′2r′13〈sin φ cos φ(cos θ - cos3 θ)〉 - µ′2r′22〈sin3 φ sin3 θ〉 + 2µ′2r′23〈sin2 φ(cos θ - cos3 θ)〉 + µ′2r′33〈sin φ(sin3 θ - sin θ)〉 + µ′3r′11〈cos2 φ(cos θ - cos3 θ)〉 + 2µ′3r′12〈sin φ cos φ(cos θ - cos3 θ)〉 + 2µ′3r′13〈cos φ(sin3 θ - sin θ)〉 + µ′3r′22〈sin2 φ(cos θ - cos3 θ)〉 + 2µ′3r′23〈sin φ(sin3 θ - sin θ)〉 + µ′3r′33〈cos3 θ〉 (8)

To understand, in the most general sense, how reorientation can affect the VSFS signal, we should consider what happens to the orientational averages at the opposite time extreme, that is, when the IR and Raman transitions are separated by a long enough time that all memory of the orientations during the first transition has been lost. Note that we can still assume that the distribution of orientations remains unchanged between the IR and Raman transitions. Under these conditions, we can write

(∑ ∑ 3

R (2) ijk (∞) ∝

) (∑

3

Rγ,iRδ,jr′γδ(τ ) ∞) ×

γ)1 δ)1

)

3

Rβ,kµ′β(τ ) 0) (9)

β)1

where τ is the time after the IR transition. In other words, when τ is much greater than the reorientation time, the orientational averages can be factored into an orientational average of the IR transition dipole at zero time and an orientational average of the Raman tensor at a delay time that is long enough that all orientational memory has been lost, as was shown by Wei and Shen.5 We begin by considering the long-time behavior of the (2) orientational average for R (2) xxz ) R yyz, in which case we have

(∑ ∑ 3

R (2) xxz(∞) ∝

3

γ)1 δ)1

) (∑ )

1 2 2 R (2) xxz(∞) ∝ µ′1r′11〈cos φ sin θ〉〈cos φ sin θ - 1〉 + 2 µ′1r′12〈cos φ sin θ〉〈sin φ cos φ sin2 θ〉 µ′1r′13〈cos φ sin θ〉〈cos φ sin θ cos θ〉 + 1 µ′ r′ 〈cos φ sin θ〉〈sin2 φ sin2 θ - 1〉 2 1 22 1 µ′1r′23〈cos φ sin θ〉〈sin φ sin θ cos θ〉 - µ′1r′33〈cos φ sin θ〉〈sin2 θ〉 + 2 1 µ′ r′ 〈sin φ sin θ〉〈cos2 φ sin2 θ - 1〉 + 2 2 11 µ′2r′12〈sin φ sin θ〉〈sin φ cos φ sin2 θ〉 µ′2r′13〈sin φ sin θ〉〈cos φ sin θ cos θ〉 + 1 µ′ r′ 〈sin φ sin θ〉〈sin2 φ sin2 θ - 1〉 2 2 22 µ′2r′23〈sin φ sin θ〉〈sin φ sin θ cos θ〉 1 1 µ′ r′ 〈sin φ sin θ〉〈sin2 θ〉 - µ′3r′11〈cos θ〉〈cos2 φ sin2 θ - 1〉 2 2 33 2 µ′3r′12〈cos θ〉〈sin φ cos φ sin2 θ〉 + 1 µ′3r′13〈cos θ〉〈cos φ sin θ cos θ〉 - µ′3r′22〈cos θ〉〈sin2 φ sin2 θ - 1〉 + 2 1 µ′3r′23〈cos θ〉〈sin φ sin θ cos θ〉 + µ′3r′33〈cos θ〉〈sin2 θ〉 (11) 2

We next consider the long-time behavior of the orientational (2) (2) (2) average for R (2) xzx ) R zxx ) R yzy ) R zyy. Here, we must calculate

R (2) xzx(∞) ∝

Rβ,3µ′β

3

)( ×

)

3

∑ Rβ,1µ′β β)1

(12)

Each element of the first two rows of the rotation matrix contains terms with a single factor of either sin χ or cos χ; therefore, we conclude that the second average in eq 12 must be zero. The physical interpretation of this result is that it is reorientation about the laboratory z-axis (i.e., angle χ) that can cause the complete decay of R (2) xzx. Finally, we consider the long-time (2) behavior of the orientational average for R zzz , for which we must calculate

(∑ ∑ 3

R (2) zzz(∞) ∝

3

γ)1 δ)1

) (∑ ) 3

Rγ,3Rδ,3r′γδ ×

Rβ,3µ′β

(13)

β)1

The orientational average over the transition dipole is the same as it was for R (2) xxz; therefore, we will again be able to factorize the orientational averages. We find that 2 2 R (2) zzz(∞) ∝ -µ′1r′11〈cos φ sin θ〉〈cos φ sin θ〉 -

2µ′1r′12〈cos φ sin θ〉〈sin φ cos φ sin2 θ〉 + 2µ′1r′13〈cos φ sin θ〉〈cos φ sin θ cos θ〉 µ′1r′22〈cos φ sin θ〉〈sin2 φ sin2 θ〉 + 2µ′1r′23〈cos φ sin θ〉〈sin φ sin θ cos θ〉 - µ′1r′33〈cos φ sin θ〉〈cos2 θ〉 µ′2r′11〈sin φ sin θ〉〈cos2 φ sin2 θ〉 -

3

Rγ,1Rδ,1r′γδ ×

(

3

∑ ∑ Rγ,3Rδ,1r′γδ γ)1 δ)1

2µ′2r′12〈sin φ sin θ〉〈sin φ cos φ sin2 θ〉 + 2µ′2r′13〈sin φ sin θ〉〈cos φ sin θ cos θ〉 -

(10)

β)1

Note that the average over the IR transition dipole contains only elements of the third row of the rotation matrix, none of which depend upon the angle χ. Thus, in the corresponding average at zero time (eq 3), the Ri,3 terms factor out of the integrals for the orientational averages. The orientational average of R (2) xxz(∞) should, therefore, be equivalent to our previous result for R (2) xxz(0) except that the orientational averages will be replaced by a product of two factored orientational averages. Performing this average, we find that

µ′2r′22〈sin φ sin θ〉〈sin2 φ sin2 θ〉 + 2µ′2r′23〈sin φ sin θ〉〈sin φ sin θ cos θ〉 - µ′2r′33〈sin φ sin θ〉〈cos2 θ〉 + µ′3r′11〈cos θ〉〈cos2 φ sin2 θ〉 + 2µ′3r′12〈cos θ〉〈sin φ cos φ sin2 θ〉 2µ′3r′13〈cos θ〉〈cos φ sin θ cos θ〉 + µ′3r′22〈cos θ〉〈sin2 φ sin2 θ〉 2µ′3r′23〈cos θ〉〈sin φ sin θ cos θ〉 + µ′3r′33〈cos θ〉〈cos2 θ〉 (14)

Results One of the most striking facets of the results presented above is the fact that, when the IR transition is x- or y-polarized,


J. Phys. Chem. C, Vol. 111, No. 25, 2007 8905

reorientation can cause the response function to decay to zero, as was also found by Wei and Shen.5 The entire signal is subject to this effect under SPS polarization conditions. Under PPP (2) polarization conditions, there are contributions from R (2) yyz, R yzy, (2) (2) R zyy, and R zzz; the second and third of these response functions are also subject to this effect. In the case of z-polarized IR transitions, the situation is more complex. As we saw above, this IR polarization leads to factorization of the orientational averages. The degree to which the response function is affected by reorientation depends upon how much the factorized orientational averages differ from the corresponding unfactorized averages. For the most general form of the response functions, this is a complex problem that involves many different orientational averages. This issue is further complicated for averages that contain functions of both φ and θ since, in general, the orientational distribution over one angle may depend upon the other angle. To gain insight into this problem, we will focus on simpler situations. The second contribution to the elements of the second-order susceptibility tensor comes from Raman transition moments. Traditionally, VSFS spectra are interpreted in terms of a vibrational mode’s IR activity, but the treatment above shows clearly that the symmetry of the r′ tensor must play an important role when considering how the VSF response changes as the period between the corresponding IR and Raman transitions lengthens. In an isotropic system, the Raman spectrum under any set of polarization conditions can be described in terms of a linear combination of the Fourier transforms of the isotropic and anisotropic time correlation functions (TCFs). The isotropic TCF of r′ is given by9,10

〈r′iso(τ)〉 )

〈91 Tr{r′(0)}Tr{r′(τ)}〉

(15)

where Tr denotes a trace and the angular brackets denote an ensemble average. The anisotropic TCF of r′ is given by9,10 〈r′aniso(τ)〉 )

〈23 PP[r′(0),r′(τ)] - 21 Tr{r′(0)}Tr{r′(τ)}〉 (16)

where the pairwise product PP is given by

PP[a,b] )

aijbij ∑ i,j

(17)

Both the trace of a second-rank tensor and the pairwise product of two tensors are orientationally invariant. Since the trace is orientationally invariant, reorientation does not affect the isotropic TCF. Furthermore, PP[r′0),r′(τ)] is orientationally invariant at any particular time τ. However, reorientation affects r′, thereby causing the pairwise product to decay with time. Indeed, complete orientational randomization will cause the anisotropic TCF to decay to zero. If, for a given vibrational mode, r′xx(τ) ) r′yy(τ) ) r′zz(τ) for all times τ, then the mode is isotropic (completely polarized), and the anisotropic correlation function will be zero at all times. Conversely, if Tr{r′(τ)} ) 0 for all times τ, then the isotropic correlation function will be zero at all times, and the mode is said to be completely depolarized. The degree of depolarization of a mode is measured by the depolarization ratio,9,10

F(ω) )

3〈r′aniso〉ω IVH ) IVV 45〈r′iso〉ω + 4〈r′aniso〉ω

(18)

where IVH refers to the intensity of the depolarized Raman spectrum (in which the excitation polarization is vertical and

detection polarization is horizontal), IVV refers to the intensity of the polarized Raman spectrum, and the subscript ω indicates a Fourier transform of the TCF. A mode that changes the principal axis system of the polarizability tensor symmetrically about its equilibrium position, such as the bend of CO2 or the asymmetric stretch of a methyl group, must be depolarized. A mode that does not change the principal axes of the polarizability tensor, such as the symmetric stretch of CO2 or the symmetric stretch of a methyl group, may still have a depolarized component. By its very nature, the VSFS signal arises from interfacial anisotropy. Furthermore, VSFS depends upon a TCF between an IR transition moment at one time and a Raman tensor at another rather than on a TCF for a Raman tensor at two different times. We therefore should not expect to be able to describe the VSFS signal solely in terms of the isotropic and anisotropic portions of the polarizability tensor. Nevertheless, there is considerable insight to be gained into the VSFS spectrum of a molecule by considering the symmetry properties of its Raman spectrum. For instance, one might imagine that the VSFS signal from a highly polarized mode would tend to be relatively immune to reorientation, whereas that from a weakly polarized mode would be highly susceptible to reorientation. To assess such predictions, we will consider the zero- and infinite-time orientational averages of the VSFS response functions for five specific cases. Case 1: The r′ tensor is diagonal, and all φ are equally likely. These conditions describe the stretching motion of a rotationally invariant functional group, such as the CtN stretch of a nitrile group or the symmetric stretch of a freely rotating methyl group. Case 2: The IR transition moment is along the molecular x and/or y axes, the r′ tensor is not diagonal, and either all φ are equally likely or r′11 ) r′22 and µ′1 ) µ′2. These conditions apply to the asymmetric stretch of a methyl group. Case 3: The IR transition moment is along the molecular z-axis, the r′ tensor is diagonal, but not all φ are equally likely. This situation applies to the symmetric stretch of a rotationally hindered methyl group and related species (such as a trichloromethyl group). Case 4: The IR transition moment is along the molecular x-axis, and the r′ tensor is diagonal. These circumstances apply to the symmetric stretching motion of a methylene group. Case 5: The IR transition moment is along the molecular y-axis, and the r′ tensor is not diagonal. This situation applies to the asymmetric stretching motion of methylene groups. In applying the constraints imposed by Cases 1-5 to the general response functions for simultaneous IR and Raman transitions (eqs 4, 6, and 8) and temporally separated transitions (eqs 11 and 14), we will consider how reorientation can influence spectral intensities observed under the polarization conditions commonly employed in VSFS experiments. Of particular importance will be the impact that reorientation can have on the terms used to determine average molecular orientations of functional groups and, by inference, overall molecular structure. We first consider Case 1. Under these conditions, for simultaneous IR and Raman transitions, eq 4 reduces to

1 R (2) xxz(0) ∝ µ′3(r′11 + r′22 + 2r′33)〈cos θ〉 + 4 1 µ′ (r′ + r′22 - 2r′33)〈cos3 θ〉 (19) 4 3 11 and for transitions separated by a long delay time, eq 11 becomes


Fourkas et al.

and

where the Θ are the angles made by the beams to the surface (2) normal and the L are Fresnel factors. Since R (2) xzx ) R zxx, Θvis = ΘSF and Lii(ωvis) = Lii(ωSF), we can expect that the second and third terms in eq 24 largely cancel one another. The remaining two terms are of opposite sign. Since the terms that are proportional to r′11 + r′22 - 2r′33 are also of opposite sign in (2) R (2) xxz and R zzz, there is no set of angles Θ that can be chosen to remove the reorientation-dependent portion of the signal under PPP polarization conditions. On the other hand, if r′11 + r′22 and r′11 + r′22 + 2r′33 are of the same sign, it may be possible to remove the reorientation-independent portion of the response by suitable choice of the angles Θ. Having determined which portions of the response functions are sensitive to reorientation for different tensor elements of the response function when r′ is diagonal and all angles φ are equally likely, we now address the sensitivity of the R (2) xxz and (2) R zzz response functions to reorientation. The orientational average of any function of θ can be evaluated by integrating the function multiplied by a weighting function over all possible values of θ. Here, we model the weighting function with a modified Gaussian distribution function

1 R (2) zzz(∞) ∝ µ′3(r′11 + r′22)〈cos θ〉 2 1 µ′ (r′ + r′22 - 2r′33)〈cos θ〉〈cos2 θ〉 (23) 2 3 11

P(θ) ) exp[-(θ - θ0)2/2σ2θ]/ 1 π exp[-(θ - θ0)2/2σ2θ]sin θdθ (25) 2 0

Here again, the first term is immune to reorientation, while the second term gets factorized. Several features of eqs 19-23 are noteworthy. First, only the z component of the IR transition dipole survives averaging. For modes that meet the criteria of a diagonal r′ tensor and all angles φ being equally likely, the IR transition dipole is generally along the z-axis, and therefore, the disappearance of the x and y components in the orientational averaging does not pose a significant spectroscopic constraint. Second, for all three unique tensor elements of the second-order response, the term that is affected by reorientation is proportional to r′11 + r′22 2r′33. We can draw a parallel between this quantity and the polarizability anisotropy. In fact, when r′11 ) r′22 (which is often, but not necessarily, the case when all φ are equally likely), the anisotropic portion of the polarizability can be written as r′11 - r′33. Thus, when r′11 ) r′22, r′11 + r′22 - 2r′33 is proportional to the polarizability anisotropy. Third, terms proportional to r′11 + r′22 or r′11 + r′22 + 2r′33 are not affected by reorientation. We can, therefore, draw an analogy between the isotropic portion of the polarizability in Raman spectroscopy and r′11 + r′22 and r′11 + r′22 + 2r′33 in VSFS, at least when the r′ tensor is diagonal and all angles φ are equally likely. Under SSP polarization conditions, in which only R (2) xxz contributes to the VSFS signal, there will be a component of the response that is immune to reorientation and one that is dependent on reorientation. Under SPS polarization conditions, in which only R (2) xzx contributes to the VSFS signal, the response decays completely upon orientational randomization. Under PPP polarization conditions, the situation is more complex. In this case, the response function is given by

where θ0 is the mean orientational angle and σθ is the standard deviation in the distribution. The denominator in eq 25 serves to normalize the probability distribution since θ can only range from 0 to π radians. In Figure 2a, we plot / as a function of σθ and θ0. This ratio describes the extent to which reorientation can change the observed VSF response functions relative to the limit in which the IR and Raman transitions occur simultaneously. For reference, in Figure 2b, we also plot as a function of σθ and θ0. The larger the spread in orientations, the greater the effect that reorientation can have on the VSFS signal. We can assess the importance of reorientation on VSF spectral intensities by observing how the / ratio changes with σθ for a given value of θ0. The earliest efforts to determine molecular orientation from second-order nonlinear optical spectra assumed a delta function for P(θ).1,11-13 In Figure 2a and b, the dependence of the response function on θ for such a distribution would be described by a contour spanning 0-2π in θ for σθ ) 0. For a larger, but still modest, value of σθ in the range of roughly 10° or less, the ratio / remains close to unity except in regions of θ0 for which is close to zero. Of course, if is close to zero, the VSFS signal will be extremely weak. Thus, for modest spreads in the orientational angle, reorientation is expected to have little effect on a strong VSFS signal for SSP or PPP polarization conditions. In agreement with the results of Wei and Shen, for larger values of σθ, reorientation will have an effect on the VSFS signal under these polarization conditions. As σθ grows larger, there is an increased effect of reorientation on the VSFS signal in these polarization combinations, but for all values of for which a VSFS signal is likely to be detected, the effect of reorientation is relatively small. Note that the ratio / is always positive and has a maximum value of unity. Reorientation can, therefore, only act to decrease a term containing . However, whether the decrease in going from to leads to a decrease in the VSFS signal

1 R (2) xxz(∞) ∝ µ′3(r′11 + r′22 + 2r′33)〈cos θ〉 + 4 1 µ′ (r′ + r′22 - 2r′33)〈cos θ〉〈cos2 θ〉 (20) 4 3 11 The first term in each expression is identical and is, therefore, immune to the effects of reorientation, whereas the second term has an orientational average that gets factorized at long times. For R (2) xzx, we find that 1 3 R (2) xzx(0) ∝ µ′3(r′11 + r′22 - 2r′33)〈cos θ - cos θ〉 4

(21)

We know that this response function goes to zero upon (2) reorientation. Finally, for R zzz , we have

1 R (2) zzz(0) ∝ µ′3(r′11 + r′22)〈cos θ〉 2 1 µ′ (r′ + r′22 - 2r′33)〈cos3 θ〉 (22) 2 3 11

(2) R PPP (τ) )

-Lxx(ωSF)Lxx(ωvis)Lzz(ωIR)cos ΘSF cos Θvis sin ΘIR R (2) xxz(τ) Lxx(ωSF)Lzz(ωvis)Lxx(ωIR)cos ΘSF sin Θvis cos ΘIR R (2) xzx(τ) + Lzz(ωSF)Lxx(ωvis)Lxx(ωIR)sin ΘSF cos Θvis cos ΘIR R (2) zxx(τ) + (2) (τ) (24) Lzz(ωSF)Lzz(ωvis)Lzz(ωIR)sin ΘSF sin Θvis sin ΘIR R zzz

∫


J. Phys. Chem. C, Vol. 111, No. 25, 2007 8907 As we have seen, this response function goes to zero upon (2) response function, we orientational randomization. For the R zzz have 3 R (2) zzz(0) ∝ (µ′1r′13 + µ′2r′23)〈cos θ - cos θ〉 + 1 1 µ′ (r′ + r′22)〈cos θ〉 - µ′3(r′11 + r′22 - 2r′33)〈cos3 θ〉 2 3 11 2 (29)

and

1 R (2) zzz(∞) ∝ µ′3(r′11 + r′22)〈cos θ〉 2 1 µ′ (r′ + r′22 - 2r′33)〈cos θ〉〈cos2 θ〉 (30) 2 3 11

Figure 2. Values of (a) / and (b) for a Gaussian distribution of angles with mean value θ0 and standard deviation σθ (see text for details of the distribution function).

depends upon the sign of the term containing , which, in turn, also depends on the signs and magnitudes of r′11, r′22, and r′33. We also note that, since all φ are equally likely, any reorientation that affects the signal involves θ. We next turn to Case 2. For the R (2) xxz response function, we have

1 3 R (2) xxz(0) ∝ (µ′1r′13 + µ′2r′23)〈cos θ - cos θ〉 + 2 1 µ′ (r′ + r′22 + 2r′33)〈cos θ〉 + 4 3 11 1 µ′ (r′ + r′22 - 2r′33)〈cos3 θ〉 (26) 4 3 11

Note that the infinite-time results are identical to those when the r′ tensor is diagonal because the off-diagonal terms go to zero upon orientational randomization. The zero-time results are identical to the long-time limits found above except for the existence of a term proportional to µ′1r′13 + µ′2r′23. This term brings in x and y components of the IR transition dipole but only in conjunction with off-diagonal elements of r′. Thus, if a mode does not have any component of its IR transition dipole along the molecular z-axis, then when all φ are equally likely, only off-diagonal terms of r′ will contribute to the signal, and R(2) will decay to zero with orientational randomization. This situation applies to the doubly degenerate asymmetric stretching modes of methyl groups. Note that it is the factorization of terms such as into 2 that leads to the decay of the signal with reorientation under these conditions. In other words, rotation of a methyl group about the bond its carbon atom makes with the remainder of a molecule is sufficient to cause the VSFS signal for an asymmetric methyl stretch to decay to zero even if the molecules themselves are not reorienting. On the other hand, for other vibrational modes that satisfy Case 2, any component of the IR transition moment that is along z behaves as in Case 1. We next consider Case 3. For R (2) xxz, we find

1 R (2) xxz(0) ∝ µ′3(r′11 + r′22 + 2r′33)〈cos θ〉 4 1 µ′ (r′ - r′22)〈cos 2φ cos θ〉 + 4 3 11 1 µ′ (r′ + r′22 - 2r′33)〈cos3 θ〉 + 4 3 11 1 µ′ (r′ - r′22)〈cos 2φ cos3 θ〉 (31) 4 3 11 and

1 R (2) xxz(∞) ∝ µ′3(r′11 + r′22 + 2r′33)〈cos θ〉 + 4 1 µ′ (r′ + r′22 - 2r′33)〈cos θ〉〈cos2 θ〉 (27) 4 3 11

1 R (2) xxz(∞) ∝ µ′3(r′11 + r′22 + 2r′33)〈cos θ〉 4 1 µ′ (r′ - r′22)〈cos 2φ〉〈cos θ〉 + 4 3 11 1 µ′ (r′ + r′22 - 2r′33)〈cos θ〉〈cos2 θ〉 + 4 3 11 1 µ′ (r′ - r′22)〈cos 2φ cos2 θ〉〈cos θ〉 (32) 4 3 11

For the R (2) xzx response function, we have

For R (2) xzx, we find

1 3 R (2) xzx(0) ∝ (µ′1r′13 + µ′2r′23)〈cos θ〉 + 2 1 (µ′ r′ + µ′3r′22 - 2µ′3r′33)〈cos3 θ - cos θ〉 (28) 4 3 11

1 3 R (2) xzx(0) ∝ µ′3(r′11 - r′22)〈cos 2φ(cos θ - cos θ)〉 + 4 1 µ′ (r′ + r′22 - 2r′33)〈cos3 θ - cos θ〉 (33) 4 3 11

and


Fourkas et al.

Figure 3. Values of (a) /, (b) , (c) /, and (d) as a function of θ0 and σφ. Here, we have chosen σθ ) 30° and φ0 ) 0° (see text for details of the distribution function). (2) For R zzz , we find

1 R (2) zzz(0) ∝ µ′3(r′11 + r′22)〈cos θ〉 + 2 1 µ′ (r′ - r′22)〈cos 2φ cos θ〉 2 3 11 1 µ′ (r′ - r′22)〈cos 2φ cos3 θ〉 2 3 11 1 µ′ (r′ + r′22 - 2r′33)〈cos3 θ〉 (34) 2 3 11

P(θ,φ) ) exp[-(θ - θ0)2/2σθ2]exp[-(φ - φ0)2/2σφ2(θ)] 1 4π

and

R (2) zzz(∞)

In this situation, we have averages that are factorized and contain functions of both φ and θ. In general, the distribution of angles φ should depend upon θ, and this dependence will vary considerably among different chemical species. One common scenario is that, when θ is near 0 or π, the distribution of φ is broad, whereas when θ is near π/2, the distribution of φ is narrower. To explore the effects of reorientation within this scenario, we model the orientational distribution using

1 ∝ µ′3(r′11 + r′22)〈cos θ〉 + 2 1 µ′ (r′ - r′22)〈cos 2φ〉〈cos θ〉 2 3 11 1 µ′ (r′ - r′22)〈cos θ〉〈cos 2φ cos2 θ〉 2 3 11 1 µ′ (r′ + r′22 - 2r′33)〈cos θ〉〈cos2 θ〉 (35) 2 3 11

Note that, if r′11 ) r′22 (i.e., if the r′ tensor is cylindrically symmetric about φ, as is the case, for instance, for a symmetric methyl stretch), then we recover our results from the case in which the r′ tensor is diagonal and all angles φ are equally likely. If r′11 * r′22, then new terms appear that depend upon r′11 - r′22 and orientational averages that include cos 2φ. If φ is independent of θ (which cannot be assumed to be the case in general), then some of the new terms will be unaffected by reorientation. Otherwise, all of the new terms will be affected by reorientation.

∫

2π

0

dφ

∫

π

0

exp[-(θ - θ0)2/2σθ2]exp[-(φ-φ0)2/2σφ2(θ)]sin θ dθ

(36)

where we assume that the standard deviation in φ depends upon θ according to σφ(θ) ) σ(0) φ /sin θ. In Figure 3a, we plot /, and in Figure 3c, we plot / . For reference, the denominators of these two expressions are plotted in Figure 3b and d, respectively. In these representative plots, we have set φ0 ) 0° and σθ ) 30°. In both cases, the unfactorized orientational average has decayed considerably by the point that σφ ) 30°; therefore, we do not include larger values of σφ in the plots. We find that / is near unity for all θ0 when σφ is small and that this ratio grows modestly with increasing σφ. Similar results are observed for larger and smaller σθ. The unfactorized orientational average becomes vanishingly small as φ0 approaches 45° and then grows again until φ0 reaches 90°. The behavior of the ratio remains similar throughout the range of values of σφ. In the case of /, the ratio can be either greater or less than unity, depending on



Figure 4. Values of (a) /, (b) , (c) /, and (d) . Here, we have chosen σθ ) 30° and φ0 ) 0° (see text for details of the distribution function).

θ0. The ratio is greater in regions in which the unfactorized orientational average is greater. For small values of σφ, the ratio is less than one, and it grows with increasing σφ. Similar behavior is observed for other values of θ0 (not shown). Once again, the unfactorized orientational average goes to zero as φ0 approaches 45°; it then grows again until φ0 reaches 90°. The behavior of this ratio also remains similar throughout the range of values of σφ. On the basis of these results and eqs 31-35, we can conclude that, when the r′ tensor is diagonal but not all φ are equally (2) likely, R (2) xxzand R zzz may change modestly with reorientation. We note that, for the symmetric stretch of a methyl group, r′11 ) r′22, and therefore, it is only the factorization of with reorientation that would cause any change in the response function. Thus, the symmetric methyl stretch is sensitive only to reorientation in θ. We next consider Case 4. For R (2) xxz, we find

1 R (2) xxz(0) ∝ - µ′1(r′11 + r′22)〈cos φ sin θ〉 + 2 1 µ′ (r′ + r′22 - 2r′33)〈cos φ sin3 θ〉 + 4 1 11 1 µ′ (r′ - r′22)〈cos φ cos 2φ sin3 θ〉 (37) 4 1 11 and

1 R (2) xxz(∞) ∝ - µ′1(r′11 + r′22)〈cos φ sin θ〉 + 2 1 µ′ (r′ + r′22 - 2r′33)〈cos φ sin θ〉〈sin2 θ〉 + 4 1 11 1 µ′ (r′ - r′22)〈cos φ sin θ〉〈cos 2φ sin2 θ〉 (38) 4 1 11

Here, the first term is immune to reorientation, and the remaining terms are affected by it. For R (2) xzx, we find

1 R (2) xzx(0) ∝ µ′1(r′33 - r′11)〈cos φ sin θ〉 + 2 1 µ′ (r′ + r′22 - 2r′33)〈cos φ sin3 θ〉 4 1 11 1 µ′ (r′ - r′22)〈cos φ cos 2φ sin3 θ〉 (39) 4 1 11 (2) For R zzz , we find

R (2) zzz(0) ∝ -µ′1r′33〈cos φ sin θ〉 1 µ′ (r′ + r′22 - 2r′33)〈cos φ sin3 θ〉 + 2 1 11 1 µ′ (r′ - r′22)〈cos φ cos 2φ sin3 θ〉 (40) 2 1 11 and

R (2) zzz(∞) ∝ -µ′1r′33〈cos φ sin θ〉 1 µ′ (r′ + r′22 - 2r′33)〈cos φ sin θ〉〈sin2 θ〉 + 2 1 11 1 µ′ (r′ - r′22)〈cos φ sin θ〉〈cos 2φ sin2 θ〉 (41) 2 1 11 Here again, the first term is immune to reorientation, and the other terms are not. Note that, for R (2) xxz, the in-plane components of the r′ tensor appear in the term that is immune to (2) reorientation, whereas in R zzz , it is the out-of-plane component that appears in this term.


Fourkas et al. We can conclude from these results and eqs 37-41 that, when the r′ tensor is diagonal and the IR transition dipole is in the x direction, the VSFS signal can be influenced strongly by reorientation. Terms that contain are predominantly sensitive to reorientation in θ, as evidenced by the minimal influence of σφ in the ratio of factorized to unfactorized orientational averages. Terms that contain are sensitive to reorientation about both θ and φ. On the basis of this analysis, we can generally expect symmetric methylene stretches to have considerable sensitivity to reorientation. Finally, we turn to Case 5. For R (2) xxz, we find

1 R (2) xxz(0) ∝ - µ′2(r′11 + r′22)〈sin φ sin θ〉 + 2 1 µ′ (r′ + r′22 - 2r′33)〈sin φ sin3 θ〉 + 4 2 11 1 µ′ (r′ - r′22)〈sin φ cos 2φ sin3 θ〉 + 4 2 11 1 µ′ r′ 〈sin φ sin 2φ sin3 θ〉 (42) 2 2 12 and

1 R (2) xxz(∞) ∝ - µ′2(r′11 + r′22)〈sin φ sin θ〉 + 2 1 µ′ (r′ + r′22 - 2r′33)〈sin φ sin θ〉〈sin2 θ〉 + 4 2 11 1 µ′ (r′ - r′22)〈sin φ sin θ〉〈cos 2φ sin2 θ〉 + 4 2 11 1 µ′ r′ 〈sin φ sin θ〉〈sin 2φ sin2 θ〉 (43) 2 2 12 For R (2) xzx, we find Figure 5. Values of (a) / and (b) . Here, we have chosen σθ ) 30° and φ0 ) 90° (see text for details of the distribution function).

In Figure 4a, we plot /, and in Figure 4c, we plot / . For reference, the corresponding unfactorized orientational averages are shown in Figure 4b and d, respectively. In these plots, we have chosen σθ ) 30° and φ0 ) 0°. For the case of /, we find that the ratio is relatively insensitive to σφ. Although the unfactorized orientational average goes to zero as φ0 approaches 90°, the ratio itself is not so sensitive to the value of φ0 (not shown). The ratio does depend strongly on θ0, declining significantly as θ0 moves away from 90°. For smaller values of σθ, there is a broader range of values of θ0 for which the ratio is near unity; for larger values of σθ, the range is correspondingly narrower. We find that / depends strongly on both σφ and θ0. Both the ratio and the unfactorized orientational average decline rapidly with increasing σφ and as θ0 moves away from 90°. Because the expression contains both cos φ and cos 2φ, the unfactorized average and the ratio depend strongly on φ0; the unfactorized average goes to zero for φ0 ) 45 and 90°, for example. As was the case above, for smaller values of σθ, there is a broader range of values of θ0 for which the ratio is near unity, and for larger values of σθ, the range is correspondingly narrower.

1 3 R (2) xzx(0) ∝ µ′2(r′11 + r′22 - 2r′33)〈sin φ sin θ〉 + 4 1 µ′ (r′ - r′22)〈sin φ cos 2φ sin3 θ〉 4 2 11 1 µ′ (r′ - r′33)〈sin φ sin θ〉 + 2 2 22 1 1 µ′ r′ 〈sin φ sin 2φ sin3 θ〉 - µ′2r′12〈cos φ sin θ〉 (44) 2 2 12 2 (2) For R zzz , we find

R (2) zzz(0) ∝ -µ′2r′33〈sin φ sin θ〉 1 µ′ (r′ + r′22 - 2r′33)〈sin φ sin3 θ〉 2 2 11 1 µ′ (r′ - r′22)〈sin φ cos 2φ sin3 θ〉 2 2 11 µ′2r′12〈sin φ sin 2φ sin3 θ〉 (45) and

R (2) zzz(∞) ∝ -µ′2r′33〈sin φ sin θ〉 1 µ′ (r′ + r′22 - 2r′33)〈sin φ sin θ〉〈sin2 θ〉 2 2 11 1 µ′ (r′ - r′22)〈sin φ sin θ〉〈cos 2φ sin2 θ〉 2 2 11 µ′2r′12〈sin φ sin θ〉〈sin 2φ sin2 θ〉 (46) (2) Once again, for R (2) xxz and R zzz, there is a term that is immune to reorientation. In the former case, this term depends on the in-

Effects of Reorientation in VSFS plane components of the r′ tensor, and in the latter case, it depends on the out-of-plane component. In all cases, the offdiagonal term in the r′ tensor, which is of great importance for asymmetric methylene stretches, is affected by reorientation. The ratio / must act identically to /, except that the unfactorized orientational average is greatest for φ ) 90° rather than for φ ) 0°. By the same token, the ratio / must behave identically to / , except that the unfactorized orientational average is at its largest when φ ) 54.7° rather than at 0°. We plot the final ratio, / in Figure 5a and the corresponding unfactorized orientational average in Figure 5b. Here, we have chosen σθ ) 30° and φ0 ) 90°. The unfactorized orientational average is negative. Both the ratio and the unfactorized orientational average are the greatest at θ0 ) 90° and diminish rapidly with increasing σφ. The dependence range of θ0 over which the ratio is near unity for small σφ is greater for smaller values of σθ (not shown). When the IR transition dipole is along y and the off-diagonal terms of the r′ tensor in the xy plane are nonzero, we conclude that the VSFS signal is affected strongly by reorientation. The terms that contain the average are most sensitive to reorientation in θ, whereas the terms that contain the averages or are sensitive to reorientations in both angles. We note that the off-diagonal term of the r′ tensor is associated with , which is largest when φ ) 54.7°. Asymmetric methylene stretches are thus sensitive to reorientation in both θ and φ and give the strongest signals when φ is at the “magic” angle. Discussion It is important to stress that reorientation includes all processes that serve to randomize the orientation of the functional group under consideration within the constraints of the equilibrium orientational distribution. This definition encompasses functional group reorientation, conformational change, overall molecular reorientation, and Fo¨rster energy transfer between surface molecules. It is also important to note that Fo¨rster transfer between the interface and the bulk can lead to a time dependence in the fraction of vibrationally excited surface molecules that will affect the VSFS signal under all polarization conditions. Reorientation can affect the VSFS signal regardless of the manner in which the spectroscopy is performed. In experiments using nanosecond lasers, both the IR and visible lasers have narrow bandwidths, and the time delay between IR and Raman transitions needs only to fall within the time window of vibrational dephasing. In broad-band VSFS, the IR pulse is short (tens to a few hundred femtoseconds) and has a large bandwidth. The visible pulse in broad-band VSFS is considerably longer (up to several picoseconds) and has a narrow bandwidth. In this case, the possible delay times between the IR and Raman transitions depend both on the vibrational dephasing time and the duration of the visible pulse. In time-resolved VSFS, both the IR and visible pulses are tens of femtoseconds in duration, and the signal is collected as a function of delay time between the two, yielding the response function directly. While the special cases discussed above do not represent an exhaustive list of the possible situations, these cases are relevant to a majority of reported VSFS studies in media in which reorientation can play a role in this spectroscopy. We will summarize our essential findings and then consider the interpretation of common trends in VSFS data based on these results.

J. Phys. Chem. C, Vol. 111, No. 25, 2007 8911 Under SPS polarization conditions, reorientation always causes the VSFS signal to decay to zero. The influence of reorientation on the VSFS signal under SSP or PPP polarization conditions depends upon the nature of the mode being probed, however. For cylindrically symmetric vibrational modes, such as CtN stretches, reorientation has a small effect on the VSFS signal in conditions under which the signal is strong. The same generally holds true for symmetric methyl stretches since r′11 ) r′22 in this case. Symmetric methylene stretches, on the other hand, are expected to be sensitive to reorientation in general. The VSFS signal from asymmetric methyl stretches is sensitive to reorientation of the methyl group (or the entire molecule) around φ, which can cause the response function to decay to zero. Asymmetric methylene stretches are also sensitive to reorientation and, from symmetry considerations, tend to give weak VSFS signals even in the absence of reorientation. In the context of vibrational spectra from molecules on dielectric surfaces, these findings provide a new way of interpreting many common observations in VSFS experiments. In particular, data acquired under SSP conditions generally have an approximately 3- to 10-fold larger response than data acquired under SPS conditions. SSP spectra tend to be dominated by vibrational moieties that fall under Case 1 (a diagonal r′ tensor and enough symmetry that all φ are equally likely), Case 3 (a z-directed IR transition moment, a diagonal r′ tensor, but not all φ equally likely), and Case 4 (an x-directed IR transition dipole and a diagonal polarizability tensor). SPS spectra generally highlight vibrational moieties that fall under Case 2 (x- and/or y-directed IR transition dipole, off-diagonal terms in the r′ tensor, and all φ equally likely) or Case 5 (y-directed IR transition dipole and off-diagonal terms in the r′ tensor but not all φ equally likely). These sorts of cases have been considered previously and have motivated the development of different sets of selection rules than can also include allowances for different experimental geometries.7,8,12,14-19 However, virtually all previous treatments of molecular orientation (and vibrational band intensities) based on experimental VSFS data have considered coincident IR and Raman excitations and have not allowed for the possibility that these two events can be temporally separated. Theoretical efforts that explore the magnitude and origin of molecular nonlinear optical responses do not have this limitation.4,20-25 The most common method of deducing molecular orientation from polarization-dependent VSFS measurements is to consider a quantity known as the orientation parameter, D,26-42 D)

〈cos3 θ〉 〈cos θ〉

(47)

(Some authors define D as the inverse of the quantity in eq 46.) In a common procedure to find D, one measures the secondorder nonlinear optical response of a given vibrational mode under different polarization conditions, fits the data according to versions of eqs 4, 6, and 8, and then takes ratios of different elements of the nonlinear susceptibility. If one then assumes an orientational distribution that is a δ function, then D can be derived from these ratios. Accurate measurement of the ratios needed to determine D for a δ-function distribution presents challenges, however, because it is relatively rare for a given vibrational feature to appear unambiguously in SSP, SPS, and PSS spectra. Consequently, a more common strategy is to compare ratios of the χ(2) tensor resulting from SSP and PPP spectra.6,7,43 Equivalent results are obtained for Cases 1-5 when the zero-time nonlinear


Figure 6. Values of (a) and (b) / for a Gaussian distribution of angles with mean value θ0 and standard deviation σθ (see text for details of the distribution function).

response functions, R (2) ijk (0), are compared. In other words, as long as the IR and Raman transitions responsible for a VSFS signal occur simultaneously, the results presented in this work are equivalent to those used in previous analyses that attempted to determine molecular orientation from the polarization dependence of VSFS responses. More sophisticated methods for determining average orientation that include a distribution of orientations (σθ * 0) have been proposed recently, but again, these approaches still assume coincident IR and Raman transitions.17,19,44-46 Before considering how reorientation can affect the determination of average orientation using D, it is useful to consider how accurate the analysis of an orientational distribution based upon D is likely to be. The average of was plotted in Figure 2b for the orientational distribution of eq 25. In Figure 6a, we plot for this distribution, and in Figure 6b, we plot D. It is clear from Figure 6b that most values of D are compatible with a considerable spread of values of θ0 and σθ. For values of θ for which and are large enough to see a VSFS signal readily, it is generally the case that the larger the value of D, the smaller the potential spread in values of θ0 and σθ. The range of possible average orientations and widths of the orientational distribution for a given value of the ratio D is similarly large for other reasonable choices of the form of the orientational distribution function. A similar conclusion was reached previously by Rowlen and co-workersforsecond-harmonicgeneration(SHG)experiments.47-49

Fourkas et al. Combining results from linear and nonlinear surface experiments measuring adsorbates’ polarization-dependent electronic responses, they sought to explain the plethora of SHG experiments reporting an average orientation of 39° between the electronic transition moments of adsorbates and the surface normal. The authors found that the oft-reported “magic angle” calculated from numerous SHG measurements from a wide variety of systems was not uniquely determined and could result from a variety of combinations of average orientation () and distribution widths (σθ).47 Given that a particular value of D can be consistent with a range of possible average orientational angles, we must assess the applicability of the common assumption of a δ-function distribution of orientations to media in which reorientation may play a role in VSFS. From a potential energy landscape point of view, at a liquid/vapor interface, a significant range of configurations is expected to be accessible with thermal energy. From the complementary dynamic perspective, dynamics in most cases would be expected to be faster at a liquid/vapor interface than in the bulk. If orientational dynamics can occur rapidly at a liquid/vapor interface, then a considerable range of orientations must be available to interfacial molecules. In more constrained environments, such as liquid/solid interfaces or monolayers on liquids, the range of available orientations is likely to be smaller but can still be significant. For instance, most solid interfaces are neither atomically smooth nor chemically homogeneous, and even tightly packed monolayers on liquids are subject to orientational perturbations from capillary waves. We must, therefore, conclude that we cannot, in general, assume a δ-function distribution of orientations in a fluid medium; this conclusion has important ramifications for the importance of orientational dynamics in VSFS. In fluid media, the distribution of orientational angles for any vibrational moiety might be expected to span a range of up to tens of degrees. As is clear from Figure 6, even in the static limit, the potential breadth of the orientational distribution complicates the determination of average orientation. However, as we saw above, broad orientational distributions generally lead to an increased sensitivity to the effects of reorientation. Reorientation, therefore, can present a considerable additional complication in the determination of average orientation in fluid media. For instance, in the limit that reorientation allows the second-order susceptibility to be factorized into separate IR and Raman transition moments, the measured VSFS response can change by almost a factor of 2 for axially symmetric stretching motion (Cases 1 and 3) and by considerably more for vibrational modes having lower symmetries (Cases 2, 4, and 5). A common observation in VSF spectra is that features appearing in an SSP spectrum are largely absent in an SPS spectrum and vice versa. For example, an SSP spectrum of a well ordered monolayer at an air/liquid interface is dominated by the methyl symmetric stretch, r+, near 2870 cm-1 and a higher-frequency Fermi resonance band (near 2940 cm-1) arising from an overtone of CH bending motion.18,50-54 In such instances, the band assigned to the methyl asymmetric stretch (r-) is largely absent in the SSP spectrum but shows up strongly in the complementary SPS spectra. Typically, the r- band in the SPS spectrum has only about 25-30% of the intensity of the r+ band in the SSP spectrum, a result that can be understood largely based on differences in the Fresnel factors. (Quantitative comparisons between SSP and SPS spectra will depend sensitively upon the details of the experimental geometry.46) Occasionally, however, data in SPS spectra are much weaker than



Figure 7. VSFS data for the CDE/vapor interface at room temperature under SSP and SPS polarization conditions.

one would expect based on high the intensities observed in the corresponding SSP spectra.45,54,55 Figure 7 shows the SSP and SPS spectra from the air/solid interface of a molecular glass,56,57 o-cresolphthalein dimethyl ether (CDE). These data were obtained at room temperature, well below the glass-transition temperature of 37 °C. The CDE was prepared as described elsewhere,57 and data were acquired using a broad-band VSFS assembly and procedures described in previous reports.58,59 The SSP spectrum is dominated by an intense band at 2925 cm-1 that we assign to the symmetric stretch of methyl groups bound to aromatic rings. The methoxy r+ feature appears weakly at 2820 cm-1, and two unresolved aryl CH stretches appear between 3020 and 3170 cm-1. Despite rather short acquisition timessa comparable response from wellordered monolayers would require 5 to 10 times longer samplingsthe CDE spectrum shows a remarkably good signalto-noise ratio and easily assigned features. The aromatic r+ band shows clear signs of destructive interference with a neighboring vibrational feature, as evidenced by the steep reduction in intensity on the high-frequency side of the band. The origin of this interference is uncertain. Alkyl groups attached to aromatic rings experience little anharmonic coupling between CH stretching modes and overtone/combination bands.60,61 The vibrational spectra of alkyl groups attached to ether linkages, however, have complex spectra with many more features than can be accounted for simply by considering the number of CH oscillators.62,63 Overtones that are accessible through Fermi resonance couplings share the same symmetry and phase relationships as the fundamental transitions that provide the bulk of the feature’s oscillator strength. Thus, a likely candidate responsible for interference with the r+ band of the aromatic methyl group is a Fermi resonance band of the methoxy methyl groups. The SPS spectrum from the CDE air/solid interface is considerably more congested than the SSP spectrum. Aromatic CH stretching modes appear above 3000 cm-1. Below 3000 cm-1 is a broad, partially resolved collection of at least four vibrational bands. The maximum intensity occurs at 2950 cm-1, but the aromatic methyl r+ band is still readily apparent as a shoulder. If r+ from the methoxy groups contributes to the spectrum, its intensity is buried under a weakly rising shoulder on the low-frequency side of the spectrum. Given the strong intensity of the aromatic r+ band in the SSP spectrum, the SPS features at 2950 and 2980 cm-1 may arise from the two antisymmetric CH stretching combinations that are usually degenerate in freely rotating methyl groups. (2966 and 2995 cm-1)60

Figure 8. VSF band intensities of the r+ and r- bands of linear alcohol monolayers adsorbed to the aqueous/vapor interface as a function of chain length (bottom) and the area/molecule of these monolayers as determined from surface pressure measurements (top). Band intensities were normalized relative to the r+ response acquired under SSP conditions from a neat DMSO liquid/vapor interface. Additional details about these experiments can be found in refs 54 and 70.

What is most remarkable about the SPS spectrum is its approximately 15-fold reduction in intensity relative to that of the r+ band of the aromatic methyl group in the SSP spectrum. Again, the large SSP response would lead one to believe that the solid/vapor interface of the glass is highly ordered with the aromatic methyl groups projecting away from the solid with large projections of their C3 along the surface normal. However, such an arrangement would be expected to lead to a relatively strong r- response in the SPS spectrum, a prediction that is not consistent with our observations. However, this disparity can be reconciled if the methyl groups are undergoing internal rotation on a time scale that is fast relative to the dephasing time of the coherent vibrational excitation. Methyl rotation in toluene is effectively barrierless in the gas phase.64-66 Toluene derivatives substituted in the ortho position experience barriers to large amplitude motion on the order of ∼100 cm-1, which is still smaller than the available thermal energy.67-69 In other words, even though the CDE molecules themselves are not capable of overall molecular motion, given the rigidity of the glass, individual functional groups at the surface can rotate readily and can selectively reduce signals observed in an SPS spectrum relative to those appearing in an SSP spectrum. Reorientation of the methoxy groups about the bond to the aromatic groups may similarly play a role in the reduction in intensity of the methoxy r+ band relative to the aromatic r+ band. A second example in which reorientation can have a significant impact on the intensities of bands observed in VSF spectra is the crystallization of two-dimensional monolayers at the air/water interface. Figure 8 shows variation in both the r+ and r- band intensities for linear alcohol monolayers at their terminal monolayer coverages. Also shown at the top of the plot is the variation in molecular area for monomers in these monolayers. Alcohol chain length varied from eight carbons (1-octanol) to sixteen carbons (1-hexadecanol). Details of the experimental conditions and data acquisition can be found in separate reports.54,70 Representative data from the octanol and hexadecanol monolayers appear in Figure 9. Despite the fact


Figure 9. Representative VSF spectra from monolayers of 1-octanol and 1-hexadecanol acquired under SSP and SPS polarization conditions. Monolayers of 1-hexadecanol were formed from solid 1-C16OH in equilibrium with the monolayer at the equilibrium spreading pressure at 22 °C. Monolayers of 1-octanol formed spontaneously from aqueous solutions saturated with the alcohol. Additional details about these experiments can be found in refs 54 and 70.

that these monolayers form spontaneously from both soluble (C8, C9, C10, C11, and C12) and insoluble (C16) surfactants, all of the monolayers have similar terminal surface coverages of approximately 20 Å2/molecule. For the soluble alcohols, the r+ and r- band intensities remain fairly constant from C8-OH through C11-OH, before jumping sharply in the spectra of C12-OH. Band intensities rise again between C12-OH and C16-OH. Surface pressure isotherms show that the 1-hexadecanol monolayer exists as a twodimensional, crystalline solid at the 22° ( 1 °C conditions under which these experiments were conducted.54 Previous VSF and X-ray scattering experiments report that C12-OH has a surfacefreezing transition temperature of 39 °C,50,71,72 whereas the surface-freezing transition of C11-OH monolayers (28 °C) is close to experimental temperatures.52,71 In VSF measurements, surface freezing of monolayers is accompanied by an abrupt jump in the intensities associated with methyl group vibrational transitions and a corresponding loss in intensity in modes associated with methylene groups.50,52,73 This effect is interpreted in terms of increased conformational order resulting from the elimination of gauche defects and a corresponding (slight) increase in surface coverage. Such arguments cannot be applied to the data reported in Figure 8. For example, the 12% increase in surface coverage (and the corresponding reduction in surface area per molecule) observed when comparing C11-OH monolayers (4.6 × 1014 molecules/cm2) and C12-OH monolayers (5.2 × 1014 molecules/cm2) would lead to a predicted 25% increase in the r+ and r- intensities observed in the SSP and SPS spectra of these monolayers, respectively. Instead, the r+ intensity increases by more than 70%, and the r- intensity increases by a factor of 8. Given that the two different molecules share similar conformations (based on similar surface coverages) and that the optical properties (e.g., Fresnel coefficients) of these films should change little with such a small change in surface coverage, there are few causes one can invoke to rationalize such dramatic changes based upon monolayer “structure”. Reorientation provides plausible mechanism to explain the observed behavior. Alcohol molecules in a two-dimensional

Fourkas et al. liquid maintain rotational freedom about φ, despite adopting well-ordered, upright orientations. Molecules that have begun to aggregate to form a two-dimensional solid, however, will lose this degree of freedom. The resulting slowed reorientation greatly extends the time necessary to reach the R (2) ijk (∞) limit. This change in reorientation will have a greater impact on the data acquired under SPS conditions than those under SSP conditions. While these experimental results do not constitute proof that reorientation plays a role in all VSFS experiments, the data do provide support for a mechanism that can lead to significant disparities in spectral band intensities in data acquired under different polarization conditions. Similar results have been observed in other systems where a large, resonant response observed under one set of polarizations (e.g., Case 1) is not accompanied by an anticipated large response for a complementary vibration (e.g., Case 2) under the appropriate alternative conditions. Examples include the SSP and SPS responses of the neat DMSO and acetone liquid/vapor interfaces,65,74-78 as well as at polymer interfaces77,79-81 and in monolayers formed from neutral, asymmetric surfactants.6,7,54,70 In each instance, the SPS signal is much weaker relative to strong SSP signals than one would expect based simply on differences in Fresnel factors. Conclusions We have considered the effects of reorientation on VSFS spectra for vibrational modes that are commonly employed to determine average molecular orientation at interfaces. Our theoretical results demonstrate that the extent to which reorientation can affect the VSFS signal intensity is highly dependent on the symmetry of the vibrational mode and the polarization conditions under which the VSF spectra are recorded. A general feature for all cases studied is that, the greater the width of the orientational distribution, the greater the degree to which reorientation can decrease the signal intensity. We have also presented experimental VSFS data from methyl stretches at the solid/vapor interface of a molecular glass and alcohol monolayers on water. These data exhibit a number of features that are not consistent with interpretations of VSF spectra based upon static orientational distributions. All of these features can be explained, however, in terms of specific orientational dynamics that occur on a time scale that is comparable to that of vibrational dephasing. The results presented here indicate that orientational dynamics can have a significant influence on the intensities of features in VSF spectra under many circumstances. The influence of reorientation should be considered in estimating the potential range of error in average orientations determined using VSFS. Further work will be required to develop a means of quantifying the effects of reorientation on VSFS data. Acknowledgment. This work was supported by the National Science Foundation, Grant CHE-0628178. Supporting Information Available: Details of the orientational averaging calculations. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Eisenthal, K. B. Chem. ReV. 1996, 96, 1343. (2) Richmond, G. L. Chem. ReV. 2002, 102, 2693. (3) Shen, Y. R.; Ostroverkhov, V. Chem. ReV. 2006, 106, 1140. (4) Perry, A.; Neipert, C.; Space, B.; Moore, P. B. Chem. ReV. 2006, 106, 1234.

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Effects of Reorientation in Vibrational Sum-Frequency Spectroscopy

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