Vibrational spectra of H 2 O

In bulk water, the vibrational infrared band of the OH stretching mode is centred at 3,400 cm−1 (Fig. 1) with a linewidth of several hundreds of wavenumbers. The vibrational response of interfacial water can be readily accessed with SFG spectroscopy. The SFG intensity is determined by the second-order susceptibility χ(2), and the imaginary part of the second-order susceptibility, Im[χ(2)], can be determined using phase-resolved methods30,31. Im[χ(2)] constitutes the surface equivalent of the bulk infrared absorption spectrum. Moreover, a positive (negative) band in the Im[χ(2)] spectrum indicates the net up (down)-orientation of the OH stretch transition dipole moment at the air/water interface. The spectral shape of the Im[χ(2)] spectrum of water at the air/water interface is similarly broad as the bulk infrared absorption spectrum of H 2 O (Fig. 1)30. As detailed below, we monitor the vibrational dynamics after excitation with a narrowband infrared pump pulse in bulk using infrared pump-probe spectroscopy and at the air/H 2 O interface using interfacial infrared pump-SFG probe experiments by probing the vibrational response.

Figure 1: OH stretching band of H 2 O. The infrared absorption spectrum of bulk water and the Im[χ(2)] spectrum of the air/water interface in the hydrogen-bonded OH stretch region. Full size image

Bulk water infrared pump-probe results

To study VER in bulk H 2 O, we use a narrow-band infrared excitation pulse to excite a subset of H 2 O molecules of a given frequency. We probe the temporal evolution of the excitation, which results in a modulation of the vibrational spectrum, with a second weak infrared probe pulse that measures the difference spectrum between the excited and non-excited sample, Δα iso . The presence of an excited state population causes a reduction of the absorbance at the fundamental frequency (0→1 transition) of the oscillator (a negative feature in the transient spectra, Δα iso <0) and an induced absorbance at somewhat red-shifted frequencies because of the excited state absorption (Δα iso >0, 1→2 transition). The red-shift of the excited state absorption with respect to the fundamental frequency is a direct consequence of the anharmonicity of the OH oscillator. We note that after VER, persistent non-zero signals contribute to the transient spectra at the fundamental frequency, which originate from a local temperature rise because of the dissipation of the vibrational energy. This increase in temperature gives rise to a blue-shift of the OH stretching band because of a weakening of all H-bonds in water. To circumvent complications due to these thermal effects in determining the lifetime, we probe the excited state absorption at ω probe =2,900 cm−1, where the contribution of these thermal effects is small and thus can be easily corrected for (see Methods and Supplementary Figs 1–3 for details).

To study the frequency-dependent VER, we tune the excitation pulses to seven different frequencies, ω pump . The pump-probe traces are fit with a kinetic model in order to correct for the thermal contribution (that is, the time-dependent contribution of the ‘hot ground state’ to the transient spectra resulting from sample heating is subtracted, see Methods for details). Typical normalized infrared pump/probe traces for bulk H 2 O, which are corrected for the thermalization of the sample, are shown in Fig. 2a. It is apparent from these data that the vibrational relaxation is significantly slowed-down with increasing ω pump . This slow-down is directly observable in the raw data as the thermal contribution to the transient spectra is small (see Supplementary Fig. 1 for details). We extract the decay time of the transient signals, τ 1 , by fitting a single exponential decay to the transient signals shown in Fig. 2a. The variation of τ 1 with ω pump is summarized in Fig. 3. From these measurements, we find τ 1 to increase from ∼250 fs for ω pump <3,300 cm−1 to τ 1 ∼550 fs for ω pump =3,700 cm−1. Remarkably, the value of τ 1 of (560±70) fs for ω pump =3,700 cm−1 is close to the value observed for the OH vibration of HOD molecules in D 2 O (740 fs)32,33. We note that the vibrational relaxation for ω pump =3,500 cm−1 is somewhat slower compared with what has been reported previously23. However, this early study23 used broader excitation pulses, and as a consequence the reported relaxation time represented a weighted average over a wider frequency range (∼200 cm−1). Thus, this report is fully consistent with our present findings. It is also important to point out that the measured transient signals with tunable ω pump and fixed ω probe =2,900 cm−1 contain both contributions from VER and from spectral diffusion as the absorbance of the excited state absorption at this frequency depends on the frequency of the excited oscillator.

Figure 2: Vibrational dynamics at OH stretching frequencies. (a) Normalized infrared pump/probe data for H-bonded OH groups in bulk H 2 O. The delay traces are taken at ω probe =2,900 cm−1 with the pump frequencies centred at ω pump =3,200, 3,450, 3,500 and 3,600 cm−1. The data are corrected for the contribution of the thermalized transient spectrum (see Methods section for details). At 2,900 cm−1, the magnitude of this correction is >0.2 (see Supplementary Figs 1–3 for details). (b) Dynamics of the interfacial water molecules obtained using an infrared pump/HD-SFG probe scheme. The delay traces show data with the pump pulses centred at ω pump =3,500, 3,300 and 3,100 cm−1. The SFG probe frequency is set to spectral ranges where the contribution of thermalization to the signal is negligible. Full size image

Figure 3: Vibrational relaxation time constants of bulk and interfacial H 2 O. Experimentally observed relaxation times τ 1 of the OH stretch vibration of bulk (blue symbols) and interfacial (red symbols) H 2 O as a function of the OH stretch excitation frequency. The open-red symbol corresponds to the vibrational relaxation time of the free (non hydrogen-bonded) OH groups. Error bars for the infrared pump-probe decay times correspond to a 100% increase in the sum of the squared deviations of the fit of the kinetic model to the data shown in Supplementary Figs 2 and 3. Error bars for the infrared pump-SFG-probe decay times correspond to the standard error obtained using a Levenberg–Marquardt fit of a single exponential decay to the experimental data in Fig. 2b. The solid blue curve represents the τ 1 time calculated using the model with the overtone of the bending vibration centred at 3,250 cm−1. The dashed (dotted) red line shows the τ 1 time calculated using the same model with the bending overtone of interfacial water centred at 3,190 cm−1 and a twofold (threefold) reduced spectral diffusion rate (for details, see text and Methods section). Full size image

The slow-down of the decay of the transient signals measured at 2,900 cm−1 with increasing pump frequency shows that full spectral equilibration is not achieved within the timescale of the experiment (∼1 ps). Our results show that vibrational relaxation of neat liquid water varies with frequency, with substantially slower population relaxation for OH stretch oscillators centred at the blue-side of the OH absorption band. Although our experiments are conceptually similar to previous two-dimensional (2D) infrared experiments, the frequency dispersion of τ 1 has not been recognized before12,13,14. This apparent contradiction can be explained by noting that the dispersion of τ 1 is most pronounced at blue-shifted pump-frequency where the transition dipoles of OH oscillators are small34. As a consequence, the slowly decaying signals at high frequencies are easily overwhelmed by the more intense transient signals at the centre of the OH stretching band and can be easily overlooked in commonly used 2D infrared plots. As demonstrated in the analysis presented below, the present results are fully consistent with previous 2D infrared experiments12,13,14.

Interfacial infrared pump-SFG probe experiments

In Fig. 2b, we show the corresponding experiments for the air/H 2 O interface. In these infrared pump/heterodyne-detected SFG (HD-SFG) probe experiments, we excite a subset of OH oscillators with a narrowband infrared pump pulse centred at ω pump =3,100, 3,300, 3,400 and 3,500 cm−1. Subsequently, we detect the modulation of the vibrational response of the interfacial water molecules using a HD-SFG detection scheme29,35,36. Here, we detect the modulation of the imaginary part of the second-order response, ΔIm[χ(2)], which is the surface analogue of the transient absorption spectrum of bulk water (see Methods for details). For the data shown in Fig. 2b, we select the SFG detection frequency (such as ω pump /ω probe =3,100 cm−1/3,100 cm−1, 3,300 cm−1/3,100 cm−1, 3,500 cm−1/3,200 cm−1 and 3,500 cm−1/3,500 cm−1), so that the contribution of the thermalization is zero (that is, the contribution of the heated ground state to the transient spectra at long delay times is negligible)29. Thus, the ΔIm[χ(2)] data in Fig. 2b directly correspond to the dynamics of the excited state population. The signals recorded at ω probe =3,100 cm−1 (Fig. 2b) directly reveal, also at the interface, a pronounced frequency dependence of the vibrational relaxation dynamics: the transient signals decay with a time constant of τ 1 =(350±20) fs and (160±30) fs for excitation at 3,300 and 3,100 cm−1, respectively. For the excitation at 3,500 cm−1, the transient signal decays substantially slower. Note that for these data, our experimental frequency range allows probing the population relaxation of the same type of O–H oscillator both at its fundamental 0→1 transition frequency of 3,500 cm−1 and at its excited absorption 1→2 transition frequency of 3,200 cm−1. At both frequencies, the contribution of the hot ground state is negligible. The relaxation times (τ 1 =(700±50) fs for ω probe =3,500 cm−1 versus τ 1 =(750±70) fs for ω probe =3,200 cm−1) at these two probe frequencies are identical within their respective errors, demonstrating the robustness of the experimental approach. At 3,700 cm−1, which is the resonance frequency of the (non H-bonded) free OH groups, the decay has a time constant of (840±50) fs (refs 37, 38). The population decay times of interfacial water molecules at different excitation frequencies are summarized in Fig. 3. From this figure it is clear that the frequency dependence of the relaxation time is even more pronounced at the air/H 2 O interface compared with bulk H 2 O.

From the measured relaxation times τ 1 for both bulk and surface (Fig. 3), it is clear that there is a remarkably strong dispersion in the relaxation times despite the strong intra- and intermolecular coupling of the OH vibrations of liquid H 2 O. The observation of strongly dispersive relaxation times directly shows the structural heterogeneity of water, both in the bulk and at the interface: On a timescale of ∼1 ps, a variety of differently H-bonded water molecules persist, each with its own distinct relaxation time. We note that this conclusion follows directly from the data, independent of the model presented below.

Modelling dispersive vibrational relaxation

The observed time constants τ 1 are spectrally averaged values of the intrinsic vibrational population relaxation time constants T 1 , and the dispersion in T 1 will therefore be even stronger than that of τ 1 . The value of τ 1 at a particular observation frequency is determined by both the frequency dispersion of the intrinsic relaxation time constants T 1 and the rate of spectral equilibration. To quantitatively describe the experimentally observed dispersion in the relaxation times τ 1 , we use a kinetic model that accounts for spectral diffusion and vibrational relaxation. In this model, we decompose the experimental infrared absorption spectrum of bulk H 2 O into individual Lorentzian oscillators with a linewidth of Γ=150 cm−1 (evenly spaced by 2 cm−1 over frequencies ranging from 3,000 to 3,800 cm−1). The choice for this value of the linewidth will be discussed below. The frequency-dependent transition dipole moment for each Lorentzian was taken from the literature34.

Spectral diffusion in liquid water can originate from distinctively different molecular processes: anharmonic couplings of the OH stretching vibration to lower frequency modes and structural fluctuations of the hydrogen-bonded water structure strongly affect the resonance frequency of an OH oscillator5,14,39. In addition, dipole–dipole coupling results in a transfer of the excitation population to neighbouring OH oscillators19,21,40, which can have slightly different resonance frequencies. Despite the different possible molecular origins of the spectral diffusion process, they all have in common that the modulation rate must slow down with increasing amplitude of the frequency fluctuations: large frequency excursions take more time. For convenience, we thus model spectral diffusion in a phenomenological manner by taking the transfer rate from one Lorentzian oscillator to another with different frequency to be proportional to the spectral overlap integral of both oscillators and proportional to the relative number density of the accepting mode in water multiplied by an intrinsic transfer rate k inter . We adjusted the rate constant k inter of the transfer between OH oscillators to reproduce the spectral dynamics of liquid H 2 O (ref. 12) (see Supplementary Discussion 1). The parameters k inter and the natural linewidth of the Lorentzian oscillator Γ are interdependent; we here set Γ=150 cm−1, but note that a range of combinations of k inter and Γ can reproduce the experimental observations.

As detailed above, our data provide direct evidence for a dispersion of the intrinsic vibrational relaxation time T 1 . This dispersion is likely the result of the energy mismatch between the relaxing OH stretch vibration and the combination of modes that accept the vibrational energy. There are several potential combinations of intra- and intermolecular modes that can accept the energy of the excited OH stretch vibration. However, there is substantial evidence that the overtone of the bending vibration forms the major pathway for VER21,26,27,41. Hence, we assume the vibrational relaxation of the OH stretching vibration to occur via energy transfer to the bending overtone. The overtone of the H 2 O bending vibration is described as a Gaussian band centred at 3,250 cm−1 with a linewidth of 125 cm−1 (ref. 41). We take the intrinsic population relaxation time T 1 (that is, the loss of vibrational excitation) to be proportional to the spectral overlap of each Lorentzian OH oscillator (with a linewidth Γ=150 cm−1) with the Gaussian spectrum of the overtone of the H 2 O bending vibration multiplied with an intrinsic relaxation rate k bend (ref. 18). This approach is quite general: the spectral profiles represent the effects of fluctuations (independent of the origin of the fluctuations) on the energy levels that are needed to compensate the energy mismatch. With increasing central frequency of the Lorentzian OH oscillator, the spectral overlap with the overtone of the bending mode decreases and T 1 becomes longer.

Starting with an initial Gaussian excitation profile (centred at ω pump with a linewidth of σ=100 cm−1), we simulate the temporal evolution of the excitation population numerically with a time step of 0.2 fs. The total population of all excited oscillators as a function of time is then convolved with a 70 fs (full-width half-maximum (FWHM)) instrument response function and fit to a single exponential decay. The resulting time constants for the exponential decay are compared with our experimental results (see Supplementary Fig. 4 for details). Details of the modelling can be found in the Methods section below.

As can be seen in Fig. 3, by only adjusting the intrinsic rate for energy relaxation to the overtone of the bending overtone, k bend , and the intrinsic spectral diffusion rate, k inter , we can accurately reproduce the observed variation of the relaxation time constant τ 1 with frequency (blue solid line in Fig. 3) in bulk H 2 O. The relaxation to the bend overtone directly yields the frequency dependence of the intrinsic T 1 time constant. Within our model, the intrinsic T 1 time constant varies from ∼100 fs at 3,100 cm−1 to ∼1.5 ps at 3,600 cm−1 in a strongly nonlinear manner. This frequency dependence is determined, in the model, by the spectral characteristics of the OH stretch and bend overtone, and the associated values for k inter and k bend . A similar nonlinear frequency dependence of the intrinsic T 1 could have been obtained with a more complex model involving several relaxation channels with different energy gaps. However, such a description is not required to account for the experimental observations.

Despite being rather phenomenological, our model is also in broad accordance with the results of previous 2D infrared experiments: from the modelled time-dependent excited state populations we can readily extract 2D infrared spectra (see Supplementary Discussion 1 and Supplementary Fig. 5 for details). The time-dependent centre line slope, which is commonly used as a measure for the excitation frequency memory in 2D infrared experiments, extracted from our model decays with a time constant of ∼140 fs, which agrees well with the previously reported decays of the centre line slope of bulk H 2 O (50–180 fs)12,13,14 (see Supplementary Fig. 6 for details). In turn, this means that the commonly used 2D infrared plots are not very sensitive to dispersive vibrational dynamics at the edges (that is, at the blue edge for the OH stretching band of H 2 O) of the vibrational bands. Our model is further in broad agreement with state-of-the-art ab-initio molecular dynamics simulations that predict the first spectral moment of the transient spectra to equilibrate with a ∼140-fs time constant after excitation at 3,600 cm−1 (ref. 5), whereas our model predicts a ∼180-fs equilibration (see Supplementary Discussion 2 and Supplementary Fig. 7 for details).

We now proceed with modelling the results at the interface. For interfacial water molecules, the bending vibration has been reported to be red-shifted, with respect to bulk water42. Hence, in modelling the infrared pump/HD-SFG probe experiments, we assume the overtone of the bending vibration to be located at 3,190 cm−1. It has further been observed that spectral diffusion of interfacial water is slower than in bulk water28,29. Several factors influence the spectral diffusion rate at the interface: At the interface, the density of water is reduced by a factor of two, the number of water molecules in the gas phase being negligible. Hence, the reduced number of oscillators that can accept the vibrational energy is expected to reduce the spectral diffusion rate by a factor of 2 (refs 28, 29). Moreover, molecular dynamics simulations have indicated that the spectral dynamics at the interface is slowed-down by even more than a factor of 2 in comparison to bulk water, as the H-bond switching, which also contributes to the spectral equilibration, is three times slower at the interface compared with the bulk43. In Fig. 3, we show the effect of a twofold and the effect of a threefold reduction of the spectral diffusion rate (reduced k inter ) on the value of τ 1 as obtained from our model with all other parameters (except the position of the bending overtone) kept the same as for bulk water. As can be seen from the dashed red line in Fig. 3, a twofold reduction of the spectral diffusion rate leads to an enhanced dispersion of τ 1 . Reducing k inter by a factor of 3 quantitatively describes the interfacial effects on the vibrational population relaxation, as measured in our infrared pump/HD-SFG probe experiments for excitation frequencies ranging from 3,100 to 3,500 cm−1. We note that comparison between the model and data is inappropriate for ω pump =3,700 cm−1, because the experiment probes, at this frequency, the outermost non-H-bonded OH oscillators, pointing towards the air phase38. These OH groups are rather isolated because of the lack of H-bonding and have been demonstrated to have very different coupling to other H 2 O molecules compared with H-bonded OH groups (ref. 38), which is not accounted for in our model.

In conclusion, we report a strong frequency dependence of the vibrational relaxation of the OH stretch vibrations of water, both in the bulk and at its surface. For bulk water, the vibrational relaxation time shows a smooth increase from ∼250 fs at the red side of the OH stretching band to ∼550 fs at the blue edge of the infrared absorption band. A similar increase by a factor of two from the red side to the blue side has been reported for the OH vibration of HOD dissolved in D 2 O (ref. 24), showing that also for HDO, spectral diffusion from the red and blue edges of the OH stretch band is sufficiently slow for the heterogeneity in H-bond strengths to be expressed in variations of the vibrational lifetime. Despite the very fast spectral diffusion and accelerated vibrational relaxation of the OH stretch vibration in pure H 2 O, we show that this intrinsic heterogeneity persists in H 2 O. For interfacial H 2 O, the variation of the vibrational decay time is even more pronounced than for bulk water, with τ 1 increasing from ∼150 fs at 3,100 cm−1 to ∼750 fs at 3,500 cm−1.