Orientational Time Correlation Functions for Vibrational Sum

Feb 23, 2015 - We find that after an initial rapid, inertial decay, there is little coupling among the orientational degrees of freedom of the methyl ...
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Orientational Time Correlation Functions for Vibrational SumFrequency Generation. 3. Methanol Shule Liu†,⊥ and John T. Fourkas*,†,‡,§,∥ †

Department of Chemistry & Biochemistry, ‡Institute for Physical Science and Technology, §Maryland NanoCenter, and ∥Center for Nanophysics and Advanced Materials, University of Maryland, College Park, Maryland 20742, United States S Supporting Information *

ABSTRACT: Molecular dynamics simulations have been used to study the orientational dynamics of methanol at the liquid/vapor and liquid/silica interfaces. Orientational timecorrelation functions for the symmetric and asymmetric methyl stretches of methanol have been calculated to assess the role of reorientation in the vibrational sum-frequency generation (VSFG) spectroscopy of these modes at the two interfaces. We find that internal methyl rotation plays a significant role in suppressing the intensity of the asymmetric methyl stretches at the liquid/vapor interface. The broad orientational distribution of the methyl groups and the properties of the Raman spectra of the asymmetric stretches are also major contributors to the low intensity of these modes in VSFG spectra at this interface. We find that after an initial rapid, inertial decay, there is little coupling among the orientational degrees of freedom of the methyl group, which suggests that time-dependent VSFG spectroscopy may be used to probe interfacial orientational dynamics in this and other hydrogen-bonded liquids. chain alcohols. Sung and Kim17,18 proposed that rapid internal rotational dynamics of the methyl group of methanol are responsible for this disparity in signal intensities. We have previously used molecular dynamics (MD) simulations to study methanol at the LS interface.25 Here, we use these simulations to compute the orientational TCFs relevant to the VSFG spectroscopy of the methanol methyl stretches at both the LV and LS interfaces. We find that reorientation does not have an appreciable influence on the spectroscopy of the symmetric methyl stretch under most polarization conditions, but does lead to the decay of the signal from the asymmetric stretch over the course of a few picoseconds at both interfaces. This effect alone is not sufficient to account for the absence of the asymmetric methyl stretches from experimental VSFG spectra of methanol at the LV interface. We show that other factors, including a broad orientational distribution, contribute to the lack of a measurable signal for the asymmetric methyl stretches in the VSFG spectrum at the LV interface.

I. INTRODUCTION The organization and dynamics of a liquid in the vicinity of an interface can differ remarkably from these same properties in the bulk liquid.1,2 Such differences in behavior have broad implications for a wide range of chemical and physical processes in liquids. Interface-selective, nonlinear optical techniques3 give a unique window into the changes in behavior that occur when a liquid is in close proximity to an interface. Vibrational sumfrequency generation (VSFG) spectroscopy, in which a vibrational coherence created in interfacial molecules using infrared light is probed with a subsequent Raman transition, has become a mainstay for studying interfacial liquids. The vibrational selectivity of VSFG allows this technique to be used to probe the orientational distribution of specific functional groups of target molecules.4−14 Although VSFG is used predominantly to study properties of orientational distributions at interfaces, it is also sensitive to orientational dynamics.14−23 A number of groups have derived expressions for the relative strength of the VSFG signal under different polarization conditions in the limit of rapid reorientation.15,18,19 More recently, we have derived orientational time correlation functions (TCFs) relevant to the VSFG spectroscopy of commonly studied vibrational modes, including the symmetric and asymmetric stretches of methyl groups and methylene groups.22,23 We used these orientational TCFs previously to evaluate the importance of orientational dynamics in the VSFG spectra of acetonitrile22 and propionitrile23 at the liquid/vapor (LV) and liquid/silica (LS) interfaces. VSFG has been used by several groups to study the orientation of the terminal methyl groups of straight-chain alcohols at the LV interface.17,18,24 The signal from the asymmetric methyl stretch of methanol at this interface is anomalously weak as compared to the same mode in longer© 2015 American Chemical Society

II. THEORY The methyl group of methanol is not a free rotor. The preferred conformation of the molecule is staggered, which gives it Cs symmetry (Figure 1). This symmetry breaks the degeneracy of the methyl asymmetric stretches.26−28 The inplane asymmetric stretch and the symmetric stretch both transform as A′, which causes these modes to mix to some extent. As an indication of this mixing, a “pure,” in-plane, asymmetric stretch would have a completely depolarized Received: January 10, 2015 Revised: February 19, 2015 Published: February 23, 2015 5542

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the angle of rotation of the methyl group about the Z axis, and χ as the rotation angle of the Z axis about the z axis. Because VSFG is a second-order nonlinear process, only elements of the second-order response function, R(2), that contain the laboratory axis z an odd number of times can be non-zero.3 For an azimuthally isotropic interface, each laboratory axis x and y must appear an even number of times (i.e., twice or not at all). We therefore need consider only three unique tensor elements: xxz (which is equivalent to yyz), xzx (which is equivalent to zxx, yzy, and zyy), and zzz. Here, the indices are listed in the order of the signal beam, the probe beam, and the infrared beam. Experimentally, the three most commonly used polarization combinations are SSP, SPS, and PPP, where P indicates polarization in the plane of incidence and reflection and S indicates polarization perpendicular to this plane. The SSP signal is dependent on the xxz tensor element of R(2), the SPS signal is dependent on the xzx tensor element of R(2), and the PPP signal is dependent on all three of the independent tensor elements (with the exact mixture of these tensor elements being a function of the experimental geometry).30 We describe the tensor elements of the second-order response function in terms of the derivatives of the dipole moment and polarizability tensor (μ′ and α′, respectively) with respect to the vibrational coordinate of interest. Based on our previous derivations,23 for the xxz tensor element of the symmetric methyl stretch we have

Figure 1. Coordinate system used here for the methyl group of methanol. The molecule is depicted in its most stable Cs geometry, with the mirror plane shown. The in-plane asymmetric methyl stretch has a transition dipole that is within the mirror plane, whereas the outof-plane asymmetric methyl stretch has a transition dipole that is perpendicular to this plane.

Raman spectrum (i.e., a Raman depolarization ratio29 ρ of exactly 0.75). However, the experimental value of ρ for this mode is 0.594 in the liquid phase and 0.150 in the vapor phase.27 These measurements indicate that there is modest mixing between the symmetric and in-plane asymmetric stretches, in agreement with quantum calculations that suggest that each mode takes on about 10% of the character of the other pure mode.26 Based on the relatively small degree of mixing, we will determine orientational TCFs for the pure modes, but it is worth noting that reorientation will affect the symmetric methyl stretch slightly more, and the in-plane asymmetric stretch slightly less, than our treatment indicates. We have derived the orientational TCFs for hindered methyl groups previously,23 so here we employ our prior results that are relevant to the analysis of the VSFG spectroscopy of the methyl stretches of methanol. The functional forms of the orientational TCFs used here are given Table 1, using the same

⎛1 ⎞ 1 (2) R xxz (τ ) ∝ μZ′ αI′⟨cos θ ⟩ + μZ′ αa′⎜ ⟨cos θ ⟩ + C1(τ )⎟ ⎝ 12 ⎠ 4 1 ′ − αYY ′ )C5(τ ) − μZ′ (αXX (1) 8

Here, αI′ is the isotropic portion of the polarizability derivative tensor, which is given by one-third of the trace of the tensor, and αa′ is the modified polarizability anisotropy, which is given by

Table 1. Shorthand Notation Used for the Orientational TCFs Used notation

time correlation function

C1(τ) C2(τ) C3(τ) C4(τ) C5(τ) C6(τ) C7(τ) C8(τ) C9(τ)

⟨cos θ cos 2Θ⟩ ⟨sin θ sin 2Θ cos χ cos X⟩ ⟨sin θ sin 2Θ cos φ cos Φ⟩ ⟨(cos Θ + cos θ cos 2Θ) cos φ cos Φ cos χ cos X⟩ ⟨cos θ cos 2Φ − cos θ cos 2Θ cos 2Φ⟩ ⟨sin θ sin 2Θ cos2 Φ cos χ cos X⟩ ⟨sin θ sin 2Θ sin2 Φ cos χ cos X⟩ ⟨sin θ sin 2Θ sin φ sin Φ⟩ ⟨cos Θ cos φ cos Φ sin χ sin X + cos θ cos 2Φ sin φ sin Φ cos χ cos X⟩ ⟨cos Θ sin φ sin Φ sin χ sin X + cos θ cos 2Φ cos φ cos Φ cos χ cos X⟩

C10(τ)

αa′ =

′ + αYY ′ − 2αZZ ′ αXX 2

(2)

The modified polarizability anisotropy cannot be determined directly from the experimental depolarization ratio. However, we have shown previously that for a symmetric methyl stretch the modified polarizability anisotropy must be less than or equal to the actual polarizability anisotropy and that the two quantities should be close in magnitude.23 We can therefore use the true polarizability anisotropy in place of the modified polarizability anisotropy in our calculations to determine the maximum possible influence of reorientation on the VSFG signal for this mode. Furthermore, for this mode we expect α′XX − α′YY to be small. In addition, we find that C5(τ) is 2 orders of magnitude smaller than C1(τ) for this liquid at the LV and LS interfaces (Supporting Information, Figure S1). We therefore will use

nomenclature as previously.22,23 The coordinate system used is shown in Figure 1. Lowercase Cartesian axes are in the lab frame, and uppercase Cartesian axes are in the molecular frame. Lowercase Greek letters refer to the angle at the time of the infrared interaction (which we define as time 0), and uppercase Greek letters refer to the angle at the time of the Raman interaction (which we define as time τ).We define θ as the angle between the methyl Z axis and the laboratory z axis, φ as

⎛1 ⎞ 1 (2) R xxz (τ ) ∝ μZ′ αI′⟨cos θ ⟩ + μZ′ αA′ ⎜ ⟨cos θ ⟩ + C1(τ )⎟ ⎝ 12 ⎠ 4 (3)

to determine this response function. In the case of the xzx tensor element, we have 5543

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The Journal of Physical Chemistry C 1 1 (2) ′ C6(τ ) − μZ′ αYY ′ C7(τ ) R xzx (τ ) ∝ − μZ′ αXX 2 2 1 ′ C2(τ ) + μZ′ αZZ 2

1000 methanol molecules, with simulation box dimensions of Lx = 45.60 Å, Ly = 43.88 Å, and Lz = 150.00 Å. Slab-corrected Ewald 3D sums were used for treatment of electrostatic interactions in our system.32 The OPLS-AA force field33 was used to describe the intramolecular and intermolecular interactions of methanol molecules. MD simulations in the NVT ensemble were performed using the DL_POLY 2.18 package34 with a time step of 1 fs. After the system had been equilibrated for 300 ps at T = 298 K, configurations of the system were recorded every 15 fs. The total sampling lasted for 450 ps and gave a trajectory file with a total of 30000 configurations for analysis. Orientational TCFs were computed in the same manner as in our previous work,22,23 i.e., using the shifting origin algorithm35 with a time window of 15 ps (1000 configurations) over the entire trajectory, such that each data point in the trajectory can be used as the initial point for averaging the TCFs. For the values of our TCFs at the maximum time 15 ps, 29000 independent time origins were used in the calculations. We computed orientational TCFs for the LS, LV, and bulk regions. Our previous MD simulation study showed that methanol molecules form a bilayer-like structure at the methanol/silica interface.25 There are different molecular orientations in the two sublayers, as indicated by the methanol singlet density profile characterized by the z coordinate of the O−H bond center. Therefore, the boundary locations of the LS and LV interfaces, of the bulk liquid, and of the LS sublayers, can be defined on the basis of the z coordinate of the methanol O−H bond center, as shown in Table 2. The boundary

(4)

Our simulations show that for both the LV and LS interfaces, C6(τ) = C7(τ) = C2(τ)/2 for this liquid (Supporting Information, Figure S2), so we can rewrite this response function as 1 (2) R xzx (τ ) ∝ − μZ′ αa′C2(τ ) 2

(5)

For computing this response function, we will again use the true polarizability anisotropy in place of the modified polarizability anisotropy. Finally, for the zzz tensor element we have ⎛1 ⎞ 1 (2) (τ ) ∝ μZ′ αI′⟨cos θ ⟩ − μZ′ αA′ ⎜ ⟨cos θ ⟩ + C2(τ )⎟ R zzz ⎝6 ⎠ 2 (6)

where we have once more replaced the modified polarizability anisotropy with the true polarizability anisotropy. Note that eqs 3, 5, and 6 are directly analogous to the results for a freely rotating methyl group.22 To calculate the relative magnitudes of the above response functions, we need to know the relative values of α′I and α′A. Using a method we have discussed previously, 22 the experimental value of 0.007 for the Raman depolarization ratio of this mode in liquid methanol implies approximate values of 1.3 and 0.42 for α′I /α′ZZ and α′A/α′ZZ, respectively. For the in-plane asymmetric methyl stretch, we have 1 (2) ′ C3(τ ) R xxz (τ ) ∝ − μX′ αXZ 2

(7)

(2) ′ C10(τ ) R xzx (τ ) ∝ μX′ αXZ

(8)

Table 2. Boundary Locations along the z Direction in the Methanol/Silica Simulation, Defined by the z Coordinate of the Methanol O−H Bond Center25

and (2) ′ C3(τ ) R zzz (τ ) ∝ μX′ αXZ

(9)

For the out-of-plane asymmetric methyl stretch we have 1 (2) ′ C8(τ ) R xxz (τ ) ∝ − μY′ αYZ 2

(10)

(2) ′ C9(τ ) R xzx (τ ) ∝ μY′ αYZ

(11)

location 0−3.1 Å 0−1.5 Å 1.5−3.1 Å 20−30 Å 32−40 Å

locations of LS sublayers are based on the first two minima in the methanol singlet density profile. The LV interface is the region in which the value of singlet density drops from 90% to 10% of the bulk density, as in our previous studies of acetonitrile36 and propionitrile.37

and (2) ′ C8(τ ) R zzz (τ ) ∝ μY′ αYZ

region LS interface first sublayer second sublayer bulk LV interface

(12)

IV. RESULTS The orientational TCFs relevant to the symmetric methyl stretch of methanol are shown in Figure 2, along with error bars. We first consider C1(τ), which is involved in the xxz and zzz tensor elements of the second-order response. As shown in Figure 2a, this orientational TCF is zero in the bulk liquid, as expected. At the LV interface, C1(τ) is initially positive but becomes negative in less than 1 ps before slowly approaching its long-time value of −0.047. This behavior is similar to what we have observed previously in simulations of both acetonitrile22 and propionitrile,23 although the asymptotic value of C1(τ) is substantially greater for methanol than for the other two liquids. These results indicate that the methyl group has a greater tendency to point into the vapor phase in methanol than it does in acetonitrile or propionitrile, which

Our simulations further show that C3(τ) = C8(τ) and C9(τ) = C10(τ) for this liquid at the LV and LS interfaces (Supporting Information, Figure S3), so that both of these modes have the same orientational TCFs. The corresponding response functions need not have the same magnitudes for the two asymmetric stretches, as these response functions depend on different elements of the dipole and polarizability derivatives.

III. SIMULATION DETAILS MD simulations of liquid methanol at a planar hydrophilic silica surface were performed using the same system as in our previous work.25 The hydrophilic silica surface, which was constructed on the basis of an idealized ß-cristobalite (C9) crystal structure,31 was placed at z = 0. The system consisted of 5544

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between C1(τ) and C2(τ) arises from the fact that the former TCF depends only on θ, which has a nonzero average value, whereas the latter also depends on χ, for which the average value is zero due to the azimuthally isotropic nature of the system. The time dependence of C2(τ) is also similar to that of C1(τ) at the LS interface, with an initial rapid decay and a small recurrence. C2(τ) has faster relaxation at long time than does C1(τ). This phenomenon again arises from the dependence of the former TCF on χ. Due to this dependence on χ, at the LS interface the second sublayer also plays a much greater role in the decay of C2(τ) than it does for C1(τ) (Supporting Information, Figure S4b). The time dependences of the different tensor elements of the second-order response for the symmetric methyl stretch are plotted in Figure 3. At the LV interface (Figure 3a),

Figure 2. Orientational TCFs (a) C1(τ) and (b) C2(τ) for the symmetric methyl stretch of methanol in the bulk liquid and at the LV and LS interfaces. Error bars represent one standard deviation of the TCF at that value of τ.

may be a reflection of the hydrogen bonding in the bulk methanol liquid. At the LS interface, a significant component of C1(τ) decays on a time scale of a few hundred femtoseconds, after which there is a small recurrence at approximately 600 fs. After this recurrence, the orientational TCF decays to its equilibrium value of approximately 0.069 on a time scale of tens of picoseconds. The behavior of this TCF is dominated by the contribution of the first sublayer (Supporting Information, Figure S4a), with the TCF for the second sublayer showing little time dependence. For both acetonitrile and propionitrile, C1(τ) exhibits a small, subpicosend relaxation component at the LS interface. However, the magnitude of the subpicosend decay is considerably larger for methanol, and neither acetonitrile nor propionitrile exhibits any sort of recurrence in this TCF. The subpicosend dynamics in methanol are likely indicative of an inertial component to the orientational dynamics. As we have shown previously,25 the methyl groups of the first sublayer tend to reside in a region of the liquid in which the density is otherwise depleted. This situation may favor a larger inertial component of the methyl group dynamics at the LS interface in methanol than in either of the nitriles we have studied.22,23 The existence of a recurrence further suggests that there is some librational character to the orientational dynamics at the LS interface that is not as important at the LV interface, at which the molecules are subject to fewer orientational constraints. The orientational TCF C2(τ) is plotted for the bulk liquid and the LV and LS interfaces in Figure 2b. This TCF is also zero in the bulk liquid, as expected. At the LV interface the decay time of C2(τ) is similar to that of C1(τ), but the long-time value of C2(τ) is zero. The difference in long-time values

Figure 3. Tensor elements of the second-order response function for the symmetric methyl stretch of methanol at (a) the LV interface and (b) the LS interface.

reorientation causes the xxz and zzz response functions to change in amplitude by approximately 10% over a time scale of a few ps. The xzx response function has a much smaller initial amplitude and a much larger fractional decay than do the other two response functions. All three response functions decay to their limiting values on roughly the same time scale. At the LS interface (Figure 3b), all of the tensor elements of the response have a minimal dependence on time. The effects of inertial reorientation are evident in the first few hundreds of fs, after which the response functions all decay much more slowly than the typical time for other mechanisms of vibrational dephasing in liquids.38 The time-invariant portions of the response functions are much larger than the time-dependent portions, and so the recurrences observed in the TCFs are so small as not to be readily visible in the response functions. The xxz and zzz response functions are again of much greater 5545

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The Journal of Physical Chemistry C magnitude than the xzx response function, whereas the latter response function experiences a considerably greater fractional decay over the 4 ps time window examined here. We next turn to the orientational TCFs relevant to the asymmetric methyl stretches (Figure 4). C3(τ) is shown for the

spondence again suggests that there is some degree of coupling between θ and φ. The orientational TCF C10(τ) for the bulk liquid and at the LV and LS interfaces is plotted in Figure 4b. As with the other TCFs studied here, C10(τ) is zero in the bulk liquid. At both interfaces there is a rapid initial decay over the first 200 fs or so, followed by a weak recurrence and then a slower decay on a time scale of several ps. This TCF has a notably larger amplitude and longer decay time at the LS interface as compared to the LV interface. Interestingly, at the LS interface the second sublayer makes almost no contribution to this TCF (Supporting Information, Figure S5b). The tensor elements of the second-order response for the asymmetric methyl stretches are shown in Figure 5a for the LV

Figure 4. Orientational TCFs (a) C3(τ) and (b) C10(τ) for the inplane asymmetric methyl stretch of methanol in the bulk liquid and at the LV and LS interfaces. Error bars represent one standard deviation of the TCF at that value of τ.

bulk liquid and the LV and LS interfaces in Figure 4a. As expected, this TCF is zero in the bulk. At the LV interface, C3(τ) has an initial rapid decay, after which it relaxes to zero over the course of a few ps. This TCF also exhibits a small recurrence at approximately 350 fs. This recurrence may arise in part from the fact that ⟨sin θ sin 2Θ⟩ grows in amplitude up until about this time before decaying (Supporting Information, Figure S7a), which may again be indicative of inertial dynamics in θ. There is also a small shoulder in ⟨cos φ cos Φ⟩ at this time (Supporting Information, Figure S7a), which may be due to slight coupling between the θ and φ degrees of freedom. It is also possible that the shoulder in ⟨cos φ cos Φ⟩ arises in part from periodic, hindered internal rotation, as the torsional barrier in our model is on the order of 0.75 kBT.33 In the case of the LS interface, C3(τ) again has an initial rapid decay, after which it relaxes to zero smoothly over a somewhat longer time scale than for the LV interface. In this case, both sublayers at this interface exhibit similar dynamics but contribute to the orientational TCF with opposite sign (Supporting Information, Figure S5a). C3(τ) has a considerably greater initial magnitude at the LS interface than at the LV interface. This orientational TCF also has a very slight shoulder at approximately 300 fs, which corresponds to the time at which a small-amplitude, rapid inertial decay of ⟨sin θ sin 2Θ⟩ is complete (Supporting Information, Figure S7b). This corre-

Figure 5. Tensor elements of the second-order response function for the asymmetric methyl stretches of methanol at (a) the LV interface and (b) the LS interface.

interface and Figure 5b for the LS interface. At the LS interface, a recurrence at approximately 300 fs is evident in each tensor element of the response. Each response function decays by a factor of roughly one-half over the first few hundred femtoseconds and then relaxes to zero more slowly, over the course of several picoseconds. Similar behavior is observed at the LS interface, but the fast component accounts for a smaller fraction of the total decay and the slow component relaxes more slowly than at the LV interface. We next consider the factorization of the orientational TCFs. In Figure 6 we compare the full version of C2(τ) (⟨sin θ sin 2Θ cos χ cos X⟩) with its factorized version (⟨sin θ sin 2Θ⟩ ⟨cos χ cos X⟩). At the LV interface (Figure 6a), the behavior of the factorized TCF is similar to that seen in propionitrile.23 Both versions of the orientational TCF match for a few hundred femtoseconds before diverging, with the factorized TCF having a smaller fast component and then decaying on 5546

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Figure 7. Unfactorized (black) and factorized (red) versions of C3(τ) and C10(τ) for the symmetric methyl stretch of methanol at (a) the LV interface and (b) the LS interface.

Figure 6. Unfactorized (black) and factorized (red) versions of C2(τ) for the symmetric methyl stretch of methanol at (a) the LV interface and (b) the LS interface.

of C3(τ) (Supporting Information, Figure S7a). These results suggest that this recurrence arises from internal methyl group rotation, which may mediate modest coupling between θ and φ but does not couple χ and φ to any appreciable extent. The relaxation of both factorized TCFs is dominated by dynamics in φ, although dynamics in χ make an appreciable contribution as well (Supporting Information, Figures S7a, S8a, and S9a). The agreement between the factorized and unfactorized orientational TCFs is also good at the LS interface (Figure 7b), suggesting that there is only modest dynamic coupling among the degrees of freedom. The recurrence at approximately 600 fs observed in C2(τ) is evident in the χ-dependent portions of the factorized TCFs (Supporting Information, Figures S7b, S8b, and S9b) but is not readily apparent in the full factorized TCFs. The recurrence at roughly 300 fs appears in both TCFs but less prominently than for the LV interface. This recurrence appears to have its origin in the φ-dependent portion of the factorized TCFs (Supporting Information, Figures S7b, S8b, and S9b), which again suggests that it arises from hindered internal rotation of the methyl group. After a rapid inertial relaxation over a few hundred femtoseconds, the decays of the factorized TCFs are dominated by dynamics in φ.

roughly the same time scale as the unfactorized TCF. These results are indicative of some degree of coupling between the dynamics in θ and χ on a short time scale. The decays are dominated by dynamics in χ, with dynamics in θ making a minimal contribution (Supporting Information, Figure 6a). The relatively broad distribution of angles θ (Supporting Information, Figure S10) is an important constraint for models of interfacial reorientation in this system.23 The correspondence between the unfactorized and factorized versions of C2(τ) at the LS interface is also reminiscent of the behavior observed for propionitrile.23 The two versions of the TCF are similar to one another but have different overall amplitudes. The fact that the initial values of the factorized and unfactorized TCFs differ indicates that there is coupling between the θ and χ degrees of freedom. Once again, the majority of the relaxation arises from dynamics in χ rather than dynamics in θ (Supporting Information, Figure S6b). The recurrence at approximately 600 fs appears in both ⟨sin θ sin 2Θ⟩ and ⟨cos χ cos X⟩. These results again suggest that there is coupling between the dynamics in θ and χ and that both degrees of freedom are involved in any librational motion at this interface. The factorized and unfactorized versions of C3(τ) and C10(τ) are compared in Figure 7. At the LV interface (Figure 7a) there is close agreement between the two versions of the orientational TCFs. The recurrence at approximately 300 fs is more prominent in the unfactorized TCFs than in the factorized TCFs, which suggests that there is some dynamic coupling between θ and φ. The recurrence appears in the φ-dependent portions of the factorized TCFs but not in the χ-dependent portion (Supporting Information, Figures S7a, S8a, and S9a) and corresponds with a maximum in the θ-dependent portion

V. DISCUSSION Based on the data presented above, we can evaluate the proposal of Sung and Kim17,18 that the near absence of methyl asymmetric stretches in the SSP and PPP VSFG spectra of methanol at the LV interface arises from reorientational effects. Overall, there are four possible reasons for the relatively small contribution of these modes: (1) an orientational distribution that favors the symmetric stretch signal; (2) reorientation selectively suppressing the signal from the asymmetric stretches; (3) a smaller IR transition dipole and/or polar5547

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contribute to the difficulty in observing them in the VSFG spectrum for the LV interface. All four of the above phenomena are likely to contribute to the absence of clear asymmetric stretching peaks in the VSFG spectrum of the methyl group of methanol at the LV interface. Reorientation unquestionably plays a significant role, but the combination of the other four effects is likely to be even more important. It has also been found previously that the C−H stretching VSFG spectrum of methanol at the LS interface is quite weak compared to those of other linear alcohols.39 It is clear from the response functions in Figures 3b and 5b that reorientation is unlikely to be the primary reason for this weak signal. The orientational distributions from our simulations are also not consistent with a weak signal, which suggests that either our model of silica does not match the experimental substrate or additional experimental conditions, such as the presence of small amounts of water, can have a strong effect on interfacial organization, and therefore on the strength of the VSFG spectrum. Recent simulations do indicate that even at quite small mole fractions, water in methanol partitions to the LS interface and thus could have a substantial influence on the orientational distribution of methanol.40 The recurrences that we have observed in the orientational TCFs did not appear in the corresponding TCFs of either acetonitrile22 or propionitrile.23 For the symmetric methyl stretch, a recurrence is evident at the LS interface but not at the LV interface. This observation and the apparent involvement of both the θ and χ degrees of freedom suggest a librational character in the recurrence. Some of the molecules may rotate about hydrogen bonds with the silica surface, but rapidly collide with neighboring molecules and reverse direction. This effect could be less important at the LV interface because most molecules would be expected to make two hydrogen bonds and because the local environment of the methyl groups is less tightly packed. In the case of the asymmetric stretches, a recurrence is observed most prominently at the LV interface. Hindered rotation of the methyl groups presumably competes with other collisional processes at the LS interface. The less crowded environment of the methyl groups at the LV interface provides less external hindrance to internal rotation. We can also use our data to assess whether tractable analytical models of interfacial orientational diffusion, such as the wobbling-in-a-cone model employed by Vinaykin and Benderskii20 and Gengeliczki, Rosenfeld and Fayer,41 can be used to extract information on orientational dynamics directly from the VSFG spectrum of methanol. The existence of recurrences in our orientational TCFs indicates that a diffusive model cannot capture the orientational dynamics completely. The rapid, inertial relaxation of the orientational TCFs at early time similarly cannot be described by diffusion. Furthermore, the broad orientational distributions at both interfaces (Supporting Information, Figure S10) indicate that a wobbling-in-a-cone model is not likely to provide a good description of the dynamics. On the other hand, it is also worth noting that after the inertial dynamics are complete, the unfactorized and factorized versions of each orientational TCF studied here resemble each other strongly, following appropriate scaling. Furthermore, the long-time decay in each case is nearly exponential. The correspondence between the two versions of each TCF is considerably stronger than for the previous two liquids that we

izability change for the asymmetric stretches than for the symmetric stretch; and (4) broader line widths for the asymmetric stretches than for the symmetric stretch. We now consider each of these possibilities. Orientational Distribution. Based on Figures 3a and 5a, the initial magnitudes of the xxz and zzz response functions (scaled by the respective elements of the polarizability and IR transition dipole) for the symmetric methyl stretch are a factor of 4−5 larger than those of the asymmetric methyl stretches, whereas the initial magnitudes of the xzx response functions for these modes are comparable to one another. Given that the homodyne VSFG signal is proportional to the square of the response function, this difference in magnitudes alone could account for a factor of approximately 20 between the symmetric and asymmetric stretch signal amplitudes for SSP and PPP polarization conditions, depending on the experimental geometry. It is therefore clear that the orientational distribution plays an important role in the differences between the intensities of the symmetric and asymmetric methyl stretches in the VSFG spectrum. Part of the reason that this phenomenon was not believed to be important by Sung and Kim is that they considered only a narrow distribution of methyl group orientations at the LV interface,18 when in fact our simulations (Supporting Information, Figure S10a) and those of Ishiyama, Sokolov, and Morita28 both indicate that this distribution is quite broad. Reorientation. It is evident from Figure 3a that reorientation has virtually no influence on the xxz and zzz tensor elements of the VSFG response function for the symmetric methyl stretch at the LV interface. On the other hand, Figure 5a shows that reorientation can cause all of the tensor elements of the asymmetric methyl stretches to decay to zero. The exponential time scale for this decay is on the order of 1.1 ps (Supporting Information, Figure S11a). As suggested by Sung and Kim,18 the time scale on which reorientation causes the response functions to decay is comparable to what they estimate to be a 700 fs time scale for dephasing due to other processes, and so can also act to diminish the intensities of the asymmetric stretching peaks. However, the time scale of these dynamics is sufficiently long that reorientation alone should not suppress these peaks completely. Transition Dipole/Polarizability. All of the C−H stretching modes of methanol have appreciable infrared activity.27 Yu et al. used a highly sensitive, polarization-selective Raman technique to study the methyl stretches of methanol in the gas phase and in the liquid phase.27 For the liquid they found that the peak intensity of the in-plane, asymmetric stretch in the depolarized Raman spectrum is a factor of more than 30 times smaller than the peak intensity of the symmetric stretch in the polarized spectrum. The peak intensity of the outof-plane asymmetric stretch was only slightly greater than that of the in-plane asymmetric stretch in the depolarized spectrum. Thus, the low Raman activity of the asymmetric stretches must play a major role in their weak intensities in the VSFG spectrum of methanol at the LV interface. Line Widths. The Raman spectra of Yu et al. show that the in-plane, asymmetric methyl stretch peak is somewhat broader than the symmetric stretch peak.27 The out-of-plane asymmetric stretch peak is very broad, with a width on the order of 200 cm−1. The width of the out-of-plane peak suggests that it undergoes extremely rapid dephasing. The widths of two peaks for these asymmetric stretching modes also are likely to 5548

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The Journal of Physical Chemistry C studied.22,23 In methanol, the noninertial dynamics in the θ coordinate are quite slow at both interfaces, which leads to the decoupling of dynamics in this degree of freedom from those in the other degrees of freedom. These slow dynamics in θ probably are the result of hydrogen bonding of the methanol. The dynamics in φ are also well decoupled from the dynamics in χ. Together, these results suggest that it may be possible to use VSFG spectra to gain insight into the time scales of interfacial orientational dynamics in hydrogen-bonded liquids.

Present Address

VI. CONCLUSIONS We have used MD simulations to calculate the orientational TCFs and response functions relevant to the VSFG spectroscopy of the methyl C−H stretches of methanol at the LV and LS interfaces. At both interfaces, reorientation has a greater effect on the asymmetric methyl stretches than on the symmetric methyl stretch under SSP and PPP polarization conditions and affects both types of mode under SPS polarization conditions. At the LV interface, internal rotation of the methyl group plays a role in diminishing the signal from the asymmetric stretches. We find that the broad orientational distribution of molecules and the spectral properties of the asymmetric stretching transitions also play major roles in determining the signal strengths of these modes. We examined the orientational dynamics of methanol at the LV and LS interfaces using orientational TCFs for VSFG that we derived previously for interfacial acetonitrile22 and propionitrile.23 There are a number of recurring themes in these studies. The simulations are generally in good agreement with experiment,17,18,24,42−44 except in the case of the methanol LS interface.39 In each case, reorientation plays only a minor role in the response functions for symmetric stretches. The largest influence of reorientation on the signal from symmetric stretches occurs when the relevant orientational TCF depends on the azimuthal angle χ, which plays a role in the xzx tensor element of the second-order response. Reorientation generally plays a more significant role for the VSFG spectroscopy of asymmetric stretches, particularly when internal rotation is rapid. Our simulations also indicate that the orientational degrees of freedom of methanol at these interfaces experience less coupling than do those of acetonitrile and propionitrile, particularly after the rapid inertial orientational relaxation is complete. This diminished coupling is likely due to hydrogen bonding among molecules and raises the prospect of being able to use the time dependence of the VSFG signal to probe orientational dynamics in hydrogen-bonded liquids at interfaces.

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⊥

Department of Chemistry, University of Chicago, 5735 S Ellis Ave, Chicago, IL 60637.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation Collaborative Research in Chemistry program (Grant No. CHE-0628178). We thank John Weeks for helpful discussions.

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ASSOCIATED CONTENT

S Supporting Information *

Orientational time-correlation function C5(τ), comparison of related orientational TCFs, contributions of the two sublayers at the liquid/silica interface to orientational TCFs, time dependence of the individual factorized orientational TCFs, orientational distributions at the LV and LS interfaces, and exponential fits to the decays of the response functions for the asymmetric methyl stretches. This material is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 5549

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