Dielectric response of light, heavy and heavy-oxygen water: isotope eﬀects on the hydrogen-bonding network’s collective relaxation dynamics †

Isotopic substitutions largely aﬀect the dielectric relaxation dynamics of hydrogen-bonded liquid water; yet, the role of the altered molecular masses and nuclear quantum eﬀects has not been fully established. To disentangle these two eﬀects we study the dielectric relaxation of light (H 216 O), heavy (D 216 O) and heavy-oxygen (H 2 18 O) water at temperatures ranging from 278 to 338 K. Upon 16 O/ 18 O exchange, we find that the relaxation time of the collective orientational relaxation mode of water increases by 4–5%, in quantitative agreement with the enhancement of viscosity. Despite the rotational character of dielectric relaxation, the increase is consistent with a translational mass factor. For H/D substitution, the slow-down of the relaxation time is more pronounced and also shows a strong temperature dependence. In addition to the classical mass factor, the enhancement of the relaxation time for D 216 O can be described by an apparent temperature shift of 7.2 K relative to H 216 O, which is higher than the 6.5 K shift reported for viscosity. As this shift accounts for altered zero-point energies, the comparison suggests that the underlying thermally populated states relevant to the activation of viscous flow and dielectric relaxation differ.


Introduction
The substitution of hydrogen for deuterium in water has strong impact on its dynamic properties, such as viscosity, 1-5 translational and rotational self-diffusion, 5,6 resonance frequency of molecular vibrations, 5,7 echo decay time for hydrogen-bond vibrations obtained from Raman-terahertz (THz) spectroscopy, 8 as well as dipole-dipole correlation times measured by dielectric relaxation [9][10][11][12][13][14][15] and THz time-domain spectroscopies. [16][17][18] Relative to light water (H 2 16 O), the different dynamics of heavy water (D 2 16 O) can be largely explained by the altered strength of its hydrogen-bonding network, which arises from nuclear quantum effects, 5,8,[19][20][21][22][23][24][25] e.g. different zero-point energy and reduced delocalization of deuterium due to its higher nuclear mass. 5 Despite these differences, numerous dynamic properties of H 2 16 O and D 2 16 O collapse onto a single curve when nuclear quantum effects are accounted for by an apparent temperature shift, DT. [2][3][4]8,16 That is, the properties of D 2 16 O at T + DT equal those of H 2 16 O at T, reflecting that increased thermal fluctuations of D 2 16 O mimic enhanced quantum fluctuations of H 2 16 O. 8,19,20,23,26,27 Besides these nuclear quantum effects, classical mechanics already predict a change in the properties of water for different isotopes due to their different nuclear masses. 5 To assess these classical mass effects, it is instructive to compare the properties of H 2 16 O to those of D 2 16 O and heavy-oxygen water, H 2 18 O. 1,5,8,28,29 Both, 16 O/ 18 O and H/D substitutions, result in a nearly identical increase in the molecular mass of water. For instance, these increased masses give rise to a trivial increase in the volumetric densities. 28 For molecular vibrations, for which resonance frequencies scale with the inverse square root of the oscillators' reduced mass, isotope shifts due to classical mass effects are most pronounced for H/D exchange: the center frequency of the OD stretching band is $ ffiffi ffi 2 p times lower than the OH stretching frequency, whereas heavy oxygen -which increases the reduced mass only by B0.6% -hardly affects these local dynamics. 5,7 In case of collective dynamics involving more than one water molecule, classical mass effects are arguably less straightforward to predict. The macroscopic viscosity is one example for properties that reflect such non-local dynamics. [30][31][32][33] For non-associated liquids, Eyring proposed as early as in 1936 that the thermally activated translation of a single molecule provides a good description for viscous flow, which in turn predicts viscosity to scale with the square root of the molecular mass. 34 For hydrogen-bonded liquids, however, correlated molecular motions, i.e. the restructuring of the hydrogen-bonding network alters viscosity. [30][31][32][33] This structural rearrangement also involves rotational motions of water molecules, 33,35,36 for which the rate scales with the inverse square root of the moments of inertia. 6 Viscometric experiments using isotopic 16 O/ 18 O substitution, which alters the moments of inertia of a water molecule by B0.6% but increases its molecular mass by B13%, have however shown that the B5% increase in viscosity is consistent with a translational mass factor, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 1 Conversely, the substantial increase in viscosity (15-30%) upon H/D exchange is largely attributed to nuclear quantum effects. 2,8 Empirically, this enhancement has been modelled using the same translational mass factor (5%) as found for 16 O/ 18 O substitution and an apparent temperature shift of 6.5 K to account for (temperature-dependent) nuclear quantum effects. [2][3][4]8 Yet, for translational diffusion -which is intimately related to viscosity -the ratio of diffusivities of H 2 16 O and D 2 16 O seems to be even lower than the translational mass factor at elevated temperatures. 5 Based on MD simulations, 36 such deviations have been attributed to coupling to rotational motions. Furthermore, Raman-THz echoes have been shown to be rather insensitive to classical mass effects, 8 despite their correlation with structural aspects related to viscosity of aqueous solutions. 37 As such, the role of isotope effects in the collective dynamics of water has remained elusive.
Such collective dynamics are also reflected in the dynamics probed with dielectric relaxation spectroscopy (DRS), which is sensitive to dipolar reorientation of an ensemble of water molecules. At 298 K, the dielectric spectrum of water is dominated by an intense relaxation mode centered at B20 GHz, which is attributed to dipole fluctuations upon rearrangement of the hydrogen-bonding network. [38][39][40][41][42] To quantitatively reproduce the static permittivity of water using ab initio molecular dynamics simulations, correlated motion of water molecules across a few hydration shells has to be taken into account, which has led to the notion that dielectric relaxation reports on the collective relaxation dynamics of water. 43 Despite the collectivity of the detected dynamics, the corresponding relaxation time of 8.3 ps at 298 K is consistent with what one would expect for uncorrelated, diffusive motion of dipolar water molecules in the liquid phase and the variation of the relaxation time with temperature is correctly predicted by the Stokes-Einstein-Debye equation over a remarkably wide temperature range. [44][45][46] Hence, viscous friction appears to dominate the dielectric relaxation dynamics. Consistent with this notion, the relaxation time of water increases upon H/D exchange. [9][10][11][12][13][14][15][16] Yet, the increase in viscosity and the increase in relaxation time upon H/D exchange do not agree quantitatively 6,9 and are certainly different for H 2 16 O/D 2 16 O mixtures. 12 Therefore, in addition to altered viscosity, librational and vibrational motions have been invoked to explain the slow-down of the dipolar reorientation dynamics. 12 Overall, besides water in the gas-phase, where the variation of rotational transitions are accurately predicted by the altered moments of inertia, 47 the origin of isotope effects in dielectric relaxation dynamics is not yet fully understood. In particular, the role of classical mass and nuclear quantum effects has not been disentangled.
In this work, we separate such classical mass effects from nuclear quantum effects by studying the broadband dielectric response of H 2 16 O, H 2 18 O, and D 2 16 O at temperatures ranging from 278 to 338 K to provide a unifying view on isotope effects on the dielectric response of water. In line with previous findings, we observe that, relative to H 2 16

Dielectric relaxation spectroscopic measurements (DRS)
DRS probes the frequency-dependent macroscopic polarization of a sample induced by a low-amplitude, oscillating electric field with field frequency n, 48 commonly expressed in terms of the complex permittivity, e 0 (n): where e 0 (n) and e 00 (n) are the frequency-dependent dielectric permittivity and loss representing the real and imaginary components of polarization, respectively. For dipolar liquids, polarization at microwave frequencies predominantly stems from rotational motion of species with a permanent dipole moment. Upon applying an external field with low frequency, a molecular ensemble will rearrange according to the field resulting in a polarization as measured by e 0 . With increasing field frequency, molecules cannot instantaneously follow the oscillating field, giving rise to a decrease in e 0 and a peak in e 00 .
In the present study,ê n ð Þ spectra were recorded using an Anritsu Vector Network Analyzer (model MS4647A). The range 0.05 r n/GHz r 50 was covered using a frequency-domain reflectometer, equipped with a coaxial open-ended probe based on 1.85 mm connectors. Measurements at 50 r n/GHz r 125 were performed using an open-ended probe, connected with 1 mm connectors to an external frequency converter module (Anritsu 3744A mmW). 15 To calibrate the setup, air, conductive silver paint, and H 2 16 O were used as calibration standards using the dielectric spectra of H 2 16 O reported in literature. 15 The temperature was varied from 278 to 338 K at increments of 10 K. To maintain constant temperature, the samples were placed into a double-walled sample holder connected to a Julabo F12-ED thermostat. The temperature stability in the center of the sample was estimated to be AE0.5 K. The spectra were recorded 6-8 times for each water isotope and this set of experiments was repeated at least once.

Results and discussion
Spectral analysis and qualitative trends In Fig. 1a we show the dielectric spectrum of H 2 18 O at 298 K, which exhibits a dominant relaxation at B20 GHz. This relaxation is characterized by a dispersion in e 0 and a peak in e 00 . For H 2 16 O, this relaxation mode has been attributed to the collective rearrangement of its three-dimensional hydrogen-bonding network 38-42 and can be excellently described by a Debye-type relaxation with its center position characterized by the collective relaxation time, t c . At higher frequencies (450 GHz), the spectra of water deviate from a single Debye relaxation, 13,[15][16][17][39][40][41]45,[49][50][51][52][53] which is often accounted for by an additional Debye relaxation -the so-called fast relaxation -with relaxation time t f . 13,[15][16][17]39,45,49,50,53 In line with these earlier studies, we model the experimental spectra of all three isotopic species of water with a combination of two Debye-type relaxations: where S c and S f are the relaxation strengths (amplitudes) of the collective and the fast Debye relaxation, respectively. e N is the infinite-frequency permittivity, which comprises polarization contributions at frequencies above our experimentally accessible range. The corresponding static dielectric constant, e s , equals to S c + S f + e N . The last term of eqn (2) accounts for (minor) contributions due to the dc conductivity, k, of the samples. e 0 is the permittivity of free space. We note that the thus obtained conductivities (k o 0.01 S m À1 ) reflect the experimental uncertainty (given by the accuracy of the calibration with conductive silver paint), rather than the sample conductivity.
To reduce the parameter space when fitting eqn (2)  O to the values reported in the same reference. We perform the fits for each spectrum individually by minimizing the sum of the squared deviations between simulated and experimental data on a logarithmic scale. For all three water isotopes, the averages of the thus obtained parameters and the corresponding error bars (estimated as three times the standard deviation) are listed in Tables S1-S3 in the ESI. † Similar to previous findings for H 2 16  the detected frequency range, in particular at elevated temperatures. Thus, we restrict the quantitative analysis to the collective relaxation. In Fig. 1b, the fitted dielectric losses for all studied isotopic species at four temperatures are displayed. For each species, we observe a shift of the dominant relaxation to higher frequencies upon increasing temperature, which demonstrates that the relaxation is thermally activated. Simultaneously, the relaxation strength (i.e. S c + S f , the area of the loss peaks; see Fig. S2, ESI †) markedly decreases, as enhanced thermal fluctuations counter dipolar alignment. 54,55 Upon isotopic substitution, we find the loss peak for D 2 16 O to shift to lower frequencies at all temperatures as compared to H 2 16 O, consistent with previous studies. [9][10][11][12][13][14][15] This red-shift is also reflected by the longer t c relaxation times of D 2 16 O (Fig. 2).
For H 2 18 O, we detect a weaker, yet significant, shift to lower frequencies, which is again accompanied by the higher values of t c (Fig. 2). As such, we find that not only H/D exchange, but also 16  presumably reflect the shift of the fast relaxation to frequencies significantly higher than those covered by our experiments.

The effect of temperature on the relaxation time of the cooperative mode
To quantify the thermal activation of t c , we use the Eyring theory, 39,56 which assumes that the relaxation pathway passes through a thermally activated transition state: where DH # is the enthalpy, and DS # is the entropy of activation. k B , h, and R are the Boltzmann, Planck, and gas constants, respectively, and T is the thermodynamic temperature. Using this approach, we find that assuming DH # and DS # to be independent of temperature does not fully reproduce the observed variation of t c as a function of temperature: the experimental ln t c k B T h values versus 1/T deviate from linearity (see eqn (3), dashed lines in Fig. 3), in agreement with previous findings. 39,42,45,57 This deviation indicates that DH # and DS # are temperature-dependent, which can be accounted for by the isobaric heat capacity of activation, Dc p # : 39,57 O as a function of temperature, as obtained by fitting eqn (2) to the experimental complex permittivity spectra. Inset: Relaxation time of the fast relaxation (t f ). The corresponding error bars represent the triple standard deviation.  The effect of viscosity on the relaxation dynamics As the Stokes-Einstein-Debye equation 45 predicts t c to be proportional to the macroscopic viscosity Z, we first investigate the relationship between t c and Z. For both H 2 16 O and D 2 16 O, our data for t c indeed scale linearly with Z (viscosity data were taken from ref. 2), but this variation is not identical for the two isotopic species (see Fig. S3, ESI †). For H 2 18 O, although literature data for its viscosity are scarce and limited to 288-308 K, 1 we find that the plots of t c vs. Z for H 2 16 O and H 2 18 O collapse onto a single line (Fig. S3, ESI †), indicating that the relaxation times are affected by viscosity in the same manner.
To gain further information about the isotope effects, it is instructive to compare the enhancements of the viscosities and relaxation times. In Fig. 4a 1,5,8 Rotational motions, which would scale with the square root of the ratio of the moments of inertia (% ffiffiffiffiffiffiffiffiffiffiffi 1:006 p % 1:003), seem to play a minor role. Fig. 4a shows that the relative increase in the relaxation time and viscosity upon H/D exchange is more pronounced (B15-35%) than for 16 O/ 18 O substitution (B4-5%), which therefore cannot be explained solely by translational mass effects. Rather, the temperature dependence of both ratios is a strongly indicative of nuclear quantum effects. 8 For viscosity, such differences between D 2 16 O and H 2 16 O have been accounted for by an apparent temperature shift (DT = 6.5 K) in addition to the translational mass factor. 2 The data in Fig. 4b (7)) as an adjustable parameter, we obtain this factor to be 1.054, in quantitative agreement with ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi S5, ESI †). As such, our data strongly suggest that classical mass effects on the dielectric relaxation time scale with the square root of water's molecular mass.
The determined value for DT = 7.16 K is in close agreement with DT = 7.2 K obtained earlier from THz spectroscopic measurements (performed at n 4 100 GHz), 16 although in ref. 16 the mass factor has not been included. Interestingly, the enhancement for t c upon H/D exchange is consistently higher than for Z (Fig. 4a), in line with previous findings. 6,9,12 Given the mass factor is the same for both quantities, this difference is reflected by the higher DT for t c , as compared to the temperature shift reported for Z (DT = 6.5 K). [2][3][4] Hence, the slightly different isotope effect for viscosity and relaxation time upon H/D exchange suggests that the role of the relevant molecular motions involved in the thermal activation of viscous flow and dipolar reorientation are somewhat different.

Conclusions
We study the effect of isotopic substitution on the dielectric relaxation of water at temperatures ranging from 278 to 338 K. In line with earlier reports on H 2 16 O, the spectra of D 2 16 O and H 2 18 O at frequencies ranging from 0.05 to 125 GHz can be excellently described with two Debye relaxations. The dominant relaxation is centered at 8-40 GHz -depending on temperature and isotopic composition -and has been ascribed to the cooperative dipolar rearrangement of water's three-dimensional hydrogen-bonded network. Upon H/D exchange, we find a 20-35% increase in the corresponding relaxation time, whereas 16 O/ 18 O substitution results in a markedly weaker (B4-5%) slow-down of the dynamics. Analysis of the thermal activation of the relaxation using the extended Eyring theory shows that the activation parameters for  Table 1.