The Stokes–Einstein relation has long been regarded as one of the hallmarks of transport in liquids. It predicts that the self-diffusion constant D is proportional to (τ/T)−1, where τ is the structural relaxation time and T is the temperature. Here, we present experimental data on water confirming that, below a crossover temperature T×≈ 290 K, the Stokes–Einstein relation is replaced by a ‘fractional’ Stokes–Einstein relation D∼(τ/T)−ζ with ζ≈3/5 (refs 1, 2, 3 4, 5, 6). We interpret the microscopic origin of this crossover by analysing the OH-stretch region of the Fourier transform infrared spectrum over a temperature range from 350 down to 200 K. Simultaneous with the onset of fractional Stokes–Einstein behaviour, we find that water begins to develop a local structure similar to that of low-density amorphous solid H2O. These data lead to an interpretation that the fractional Stokes–Einstein relation in water arises from a specific change in the local water structure. Computer simulations of two molecular models further support this interpretation.
We first present our experimental results on water confined in MCM-41-S nanotubes. We measure the self-diffusion D by nuclear magnetic resonance, and we measure the translational relaxation time τ by using incoherent, quasi-elastic neutron scattering1,2 (QENS). Thus, the Stokes–Einstein relation,
can be tested. Our data (Fig. 1a) confirm equation (1) at high temperatures, but show that, on cooling below a crossover temperature T×≈290 K, the Stokes–Einstein relation (1) gives way to a ‘fractional Stokes–Einstein relation’ 2,3,4,5,6,
As a first step to obtain a structural interpretation of this fractional Stokes–Einstein behaviour, we turn to measurements of the infrared spectrum1,2,7,8,9. For water, this spectrum can be split into two contributions, one resembling the spectrum of high-density amorphous (HDA) solid H2O and the other resembling the spectrum of low-density amorphous (LDA) solid H2O. We interpret these two contributions as corresponding to water molecules with more HDA-like local structure, or more LDA-like local structure, respectively10. Figure 1b shows the relative populations of molecules with locally LDA-like structure and molecules with locally HDA-like structure calculated by decomposition of the infrared spectra. With decreasing T, the LDA-like population increases, whereas the HDA-like population decreases. The fractional Stokes–Einstein crossover temperature T× seems to roughly coincide with the onset of the increase of the population of molecules with LDA-like local structure (and a corresponding decrease of the population of the molecules with HDA-like local structure), consistent with the possibility that the changes in intramolecular vibrational properties may be connected to the onset of fractional Stokes–Einstein behaviour.
To more clearly see the change in the relative populations of molecules with LDA-like local structure (and, correspondingly, with HDA-like local structure), we calculate the derivatives of the relative populations with respect to temperature (Fig. 1c). The derivatives of the relative populations become noticeably non-zero at the same value of the crossover temperature, T×≈290 K. In contrast, we find that the maximal rate of change of the vibrational spectrum occurs at a much lower temperature, Tmax≈245 K, approaching the Widom temperature 225 K for bulk water11.
As these experiments examine water confined to cylindrical pores of ≈2 nm diameter, it is natural to question whether the findings might be instructive for understanding bulk water at low T. There are two reasons to believe that the answer is yes: (1) computer simulations of confined water on a hydrophilic surface12 show that hydrophilic silica-confined water has similar behaviour to bulk water, indicating that the hydrophilic surfaces do not have serious effects on the properties of water, except for significantly lowering the freezing temperature and stabilizing the liquid phase, which enables the study of the supercooled region made impossible in bulk water owing to crystallization; (2) the presence of hysteresis in a temperature cycle (on cooling/heating) is a signature of an interaction between water and silica. However, for the MCM-41-S confined system, only negligible hysteresis was observed by means of X-ray scattering and calorimetric experiments13,14. Thus, it is plausible that the MCM-41-S confined water provides information regarding bulk water.
As experiments on bulk water at T<250 K are impractical owing to crystallization, we carry out constant-T and constant-density molecular dynamics simulations of N=512 water molecules interacting with the TIP5P potential15 at a fixed density ρ=1 g cm−3. In addition, direct access to the molecular coordinates makes it possible to connect the changes in D to changes in the local molecular structure.
The relaxation time τ is defined as the time when the coherent intermediate scattering function decays by a factor of e for the wave vector q of the first peak of the static structure factor. The diffusion coefficient is computed from the root-mean-square displacement of the oxygens as a function of temperature. Analogous to the experimental results in Fig. 1a, we show the simulation results of TIP5P water for D as a function of τ/T (Fig. 2a). We see that below T×≈320 K the Stokes–Einstein relation crosses over to a fractional Stokes–Einstein relation16 of equation (2) with ζ=0.77.
We next use our simulations to make a connection to local structure, rather than intramolecular vibration. This provides a more intuitive connection to the real space structure of the fluid. Similar to the experimental approach, we wish to relate the onset of the fractional Stokes–Einstein relation to the emergence of LDA-like local structure in the liquid. We identify different local structures by carrying out a direct calculation for each molecule i=1,2,…,N of the local tetrahedral structural order parameter Qi (ref. 17), defined as
where φj k is the angle formed by the lines joining the oxygen atom of molecule i with pairs of its four nearest neighbours j and k. The possible values of Qi vary between Qi=0 for the limit of uncorrelated angles and Qi=1 for the perfect tetrahedral network.
For each molecule, we calculate its local orientational order Qi. We assign a locally ‘LDA-like’ molecule if Qi>0.8, and an ‘HDA-like’ molecule if Qi≤0.8. Our decomposition to different structural groups is based on the probability density function P(Q). As shown in Fig. 2d of ref. 18, for density ρ= 1 g cm−3, P(Q) changes with temperature. At very low temperature (T=240 K), P(Q)has one dominant peak near Q≈0.9, indicating a more tetrahedral local structure similar to ice. At intermediate temperatures, P(Q)starts to develop a shoulder at Q≈0.5, indicating a change in the population of the local structure; and at higher temperatures (T=340 K), P(Q) has a broader distribution with two peaks at Q≈0.5 and Q≈0.9, respectively. Thus, the decomposition of the local structure to (1) the low-density-liquid-like structure (locally tetrahedral structure in the first shell with Q≈0.9) and (2) the high-density-liquid-like structure (non-tetrahedral structure with Q<0.8) is a reasonable and valid decomposition based on the number of hydrogen bonds and the distribution of the local tetrahedral order parameter Q. We can count nL and nH the number of molecules with ‘LDA-like’ and ‘HDA-like’ local structures, respectively. The relative populations are defined as nL/N and nH/N, as shown in Fig. 2b.
We observe in Fig. 2b a gradual increase in LDA-like local structures, and a decrease in HDA-like local structure. The derivatives of the relative populations of the LDA-like and the HDA-like molecules with respect to temperature (Fig. 2c) show that the change does not have a sharp onset at T×≈320 K. For each species (LDA-like and HDA-like), the maximum change defines a temperature Tmax≈255 K for the structural evolution 19,20. Like our experimental results, T×>Tmax for TIP5P, indicating that the change to fractional Stokes–Einstein behaviour can be connected with the emergence of more highly structured regions of the liquid, rather than the maximal rate of change.
To explore whether this behaviour is specific to water, or if similar behaviour occurs for other pure substances that may possibly possess a liquid–liquid critical point21,22, we also study the two-scale spherically symmetric Jagla ramp model23,24 of a liquid (Fig. 3). This model reproduces the thermodynamic and dynamical anomalies of liquid water, as well as exhibiting a liquid–liquid critical point. For the Jagla potential (Fig. 3)23, we implement a discrete molecular dynamics simulation for N=1,728 particles interacting through step potentials in a constant-temperature and constant-volume ensemble.
We examine the Stokes–Einstein relation and relative populations along a constant-pressure path P=0.30 that remains in the one-phase region (path α of Fig. 3)23. Figure 4 shows for the Jagla model the analogue of Figs 1 and 2. We find for T>T× the normal Stokes–Einstein relation with ζ=1, and for T<T×≈0.6 a fractional Stokes–Einstein relation with ζ≈0.87.
Analogous to the structural analysis done for TIP5P, we next consider the relative population of HDA-like and LDA-like molecules in the Jagla liquid. In the Jagla liquid, the changes in the structure can be deduced from the number of particles in the first and second coordination shell. In our study, for each molecule, we calculate its number of nearest neighbours, n, within a distance r/a≤1.3 (first minimum in pair correlation function). We define a locally ‘LDA-like’ molecule if n<2, and an ‘HDA-like’ molecule if n≥2. Thus, we can calculate the relative populations nL/N and nH/N. Figure 4b shows that the relative population of the more ordered phase (HDA-like for the Jagla model and LDA-like for water) starts to increase sharply at T×≈0.6, whereas the relative population of the less-ordered phase (LDA for the Jagla model) starts to decrease, just as observed experimentally and in the TIP5P model.
Similar to the experimental and TIP5P results, the temperature derivative of relative populations (Fig. 4c) shows that the maximum change occurs at a temperature Tmax≈0.4 below the onset temperature of the sharp changes of the relative populations, that is, Tmax<T×. The maxima we observe in the rates of change of the different species of molecules is reminiscent of the observation that thermodynamic response functions exhibit maxima in the supercooled region of water. The constant-pressure specific heat is the most commonly examined of these response functions, and the locus of the specific-heat maxima is often referred to as the Widom line19. Hence, it is natural to expect that there may be some connection between the location of the maximal rates of change in the molecular structure we measure and the Widom temperature for that pressure25.
For the Jagla model, Tmax coincides closely with the Widom temperature TW(P). The Jagla model has a liquid–liquid critical point at Tc=0.375, Pc=0.245 (ref. 24), and we consider cooling along the P=0.3 isobar, near to the critical pressure. As response functions must diverge at the critical point, the locus of any maxima must become asymptotically close as we approach the critical point. Therefore, the coincidence between TW and Tmax is expected. As a result, the breakdown of the Stokes–Einstein relation is influenced by the liquid–liquid critical point.
For the experimental results and TIP5P water, we work at a pressure that is much lower than that of the expected critical pressure for each system. As we work far away from the critical point, the positions of the temperature maxima of the various response functions may differ by a significant amount, as they must coincide only near the critical point26. For TIP5P, we find that TW≈255 K, identical to Tmax. This suggests that the loci for the maximum in specific heat and the maximal rate of change of the local structure coincide even at relatively low pressure. However, for water, we found a larger difference, with TW≈225 K(ref. 11) and Tmax≈245 K.
Both our experimental findings and our simulation results are consistent with the possibility that, in water, the fractional Stokes–Einstein relation (2) sets in near the temperature where the relative populations of molecules with LDA-like and HDA-like local structures start to rapidly change. A structural origin for the failure of the Stokes–Einstein relation can be understood by recognizing that the Stokes–Einstein relation defines an effective hydrodynamic radius. The different species have different hydrodynamic radii, so when their relative populations change, the classical Stokes–Einstein relation (based on the assumption of the fixed hydrodynamic radius) breaks down. Moreover, a connection between the local structure of water and its dynamics is expected16,27,28; molecules with a locally tetrahedral geometry are more ‘sluggish’ than less well-networked molecules. This effect also occurs in solutions, where a failure of the scaling between diffusion and relaxation has been interpreted in terms of changes in the local network structure29. For these reasons, our structure-based interpretation for the failure of the Stokes–Einstein relation is particular to water, and the breakdown occurs at almost twice the glass-transition temperature Tg. An explanation for the failure of the Stokes–Einstein relation for liquids in general, where the breakdown typically occurs within 20–40% of Tg, must involve understanding how the intermittency of the molecular motion couples to diffusion and relaxation mechanisms approaching the glass transition. Such a general explanation should also be applicable to water, where the emergence of intermittency of the dynamics occurs at unusually high T owing to water’s unusual thermodynamics and corresponding changes in the fluid structure.
We carried out infrared absorption measurements at ambient pressure in the oxygen–hydrogen stretching (OHS) vibrational spectral regions. We used a Bomem DA8 Fourier transform spectrometer, operating with a Globar source, in combination with a KBr beam splitter and a DTGS/KBr detector. To avoid saturation effects, we used the attenuated total reflection geometry, which is generally insensitive to sample thickness. The spectra of interest were recorded at a resolution of 4 cm−1, automatically adding 200 repetitive scans to obtain a good signal-to-noise ratio and highly reproducible spectra. They were then appropriately normalized by taking into account the effective number of absorbers. Our spectra for confined water were not smoothed, and the only manipulation used was the baseline adjustment. Investigated samples (water confined in MM-41-S annotates) have hydration levels of H∼0.5 (0.5 g of H2O per gram of MM-41-5).
Different approaches, such as mixture and continuous models developed in terms of scattering theory, have been applied to analyse the spectra of water by means of Gaussian components unambiguously classified as hydrogen-bonded or non-hydrogen-bonded OHS oscillators by considering different hydrogen-bonded geometries30,31,32. We work in an intermediate picture between the continuous and discrete models, assuming the existence of a percolating hydrogen-bond network with a characteristic hydrogen-bond lifetime of the order of a picosecond. The Gaussian components’ peaks are located at: (I) 3,120 cm−1, (II) 3,220 cm−1, (III) 3,500 cm−1 and (IV) 3,620 cm−1.
The network contributes to the OHS vibrational spectra with two main collective modes. The first, at 3,120 cm−1, is assigned to fully bonded water molecules, owing to its similarity to the OHS band observed in low-density amorphous LDA ice30. The second, at 3,220 cm−1, is associated with water molecules having an average degree of connectivity larger than that of dimers, but lower than those in a hydrogen-bonded tetrahedral structure. The highest wavenumber components are the only ones present in bulk water in the temperature region 600<T<640 Knear the liquid–gas critical point Tc=647 K, and so they are ascribed to dimeric and monomeric water.
In summary, the OHS spectra of water can be decomposed into three main groups: (I), (II and III) and (IV), where (II–III and IV) represent the HDA-like water. Components I and II make the largest contribution to the OHS spectra in the supercooled region. Using this classification, the relative population of local structures in water can be calculated on the basis of the decomposition, which is defined as the relative area (the ratio of each component’s integrated area to the total OHS integrated area)1. The overall continuity we found between supercooled and amorphous water (see, for example, Fig. 3 of ref. 1) in the infrared spectral parameter supports the experimental validity of our data interpretation.
For the microscopic average correlation time, our experiment used incoherent QENS and the spectral data were analysed by using the relaxing cage model. The typical wave vector is in the range 0.2 Å−1<q<2 Å−1. A proper choice of the energy resolution and of the dynamical range is also important for this type of QENS experiment. We use two spectrometers, one with an energy resolution of 0.8 μV and a dynamical range of ±11 μV and the other with an energy resolution of 20 μV and a dynamical range of ±0.5 mV. Further experimental details are given in refs 1, 2.
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We thank C. A. Angell, P. G. Debenedetti and V. Molinero for helpful discussions, and the NSF Chemistry Program for support. X.L.M. acknowledges the support by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. F.W.S. acknowledges NSF grant DMR-0427239. S.V.B. acknowledges partial support through the Bernard W. Gamson Computational Science Center at Yeshiva College.
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Xu, L., Mallamace, F., Yan, Z. et al. Appearance of a fractional Stokes–Einstein relation in water and a structural interpretation of its onset. Nature Phys 5, 565–569 (2009). https://doi.org/10.1038/nphys1328
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