Abstract
The Stokes–Einstein relation has long been regarded as one of the hallmarks of transport in liquids. It predicts that the selfdiffusion 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 OHstretch 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 lowdensity amorphous solid H_{2}O. 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.
Main
We first present our experimental results on water confined in MCM41S nanotubes. We measure the selfdiffusion D by nuclear magnetic resonance, and we measure the translational relaxation time τ by using incoherent, quasielastic neutron scattering^{1,2} (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},
with ζ≈0.62.
As a first step to obtain a structural interpretation of this fractional Stokes–Einstein behaviour, we turn to measurements of the infrared spectrum^{1,2,7,8,9}. For water, this spectrum can be split into two contributions, one resembling the spectrum of highdensity amorphous (HDA) solid H_{2}O and the other resembling the spectrum of lowdensity amorphous (LDA) solid H_{2}O. We interpret these two contributions as corresponding to water molecules with more HDAlike local structure, or more LDAlike local structure, respectively^{10}. Figure 1b shows the relative populations of molecules with locally LDAlike structure and molecules with locally HDAlike structure calculated by decomposition of the infrared spectra. With decreasing T, the LDAlike population increases, whereas the HDAlike 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 LDAlike local structure (and a corresponding decrease of the population of the molecules with HDAlike 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 LDAlike local structure (and, correspondingly, with HDAlike local structure), we calculate the derivatives of the relative populations with respect to temperature (Fig. 1c). The derivatives of the relative populations become noticeably nonzero 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, T_{max}≈245 K, approaching the Widom temperature 225 K for bulk water^{11}.
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 surface^{12} show that hydrophilic silicaconfined 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 MCM41S confined system, only negligible hysteresis was observed by means of Xray scattering and calorimetric experiments^{13,14}. Thus, it is plausible that the MCM41S confined water provides information regarding bulk water.
As experiments on bulk water at T<250 K are impractical owing to crystallization, we carry out constantT and constantdensity molecular dynamics simulations of N=512 water molecules interacting with the TIP5P potential^{15} 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 rootmeansquare 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 relation^{16} 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 LDAlike 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 Q_{i} (ref. 17), defined as
where φ_{j 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 Q_{i} vary between Q_{i}=0 for the limit of uncorrelated angles and Q_{i}=1 for the perfect tetrahedral network.
For each molecule, we calculate its local orientational order Q_{i}. We assign a locally ‘LDAlike’ molecule if Q_{i}>0.8, and an ‘HDAlike’ molecule if Q_{i}≤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 lowdensityliquidlike structure (locally tetrahedral structure in the first shell with Q≈0.9) and (2) the highdensityliquidlike structure (nontetrahedral 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 n_{L} and n_{H} the number of molecules with ‘LDAlike’ and ‘HDAlike’ local structures, respectively. The relative populations are defined as n_{L}/N and n_{H}/N, as shown in Fig. 2b.
We observe in Fig. 2b a gradual increase in LDAlike local structures, and a decrease in HDAlike local structure. The derivatives of the relative populations of the LDAlike and the HDAlike molecules with respect to temperature (Fig. 2c) show that the change does not have a sharp onset at T_{×}≈320 K. For each species (LDAlike and HDAlike), the maximum change defines a temperature T_{max}≈255 K for the structural evolution ^{19,20}. Like our experimental results, T_{×}>T_{max} 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 point^{21,22}, we also study the twoscale spherically symmetric Jagla ramp model^{23,24} 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 constanttemperature and constantvolume ensemble.
We examine the Stokes–Einstein relation and relative populations along a constantpressure path P=0.30 that remains in the onephase 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 HDAlike and LDAlike 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 ‘LDAlike’ molecule if n<2, and an ‘HDAlike’ molecule if n≥2. Thus, we can calculate the relative populations n_{L}/N and n_{H}/N. Figure 4b shows that the relative population of the more ordered phase (HDAlike for the Jagla model and LDAlike for water) starts to increase sharply at T_{×}≈0.6, whereas the relative population of the lessordered 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 T_{max}≈0.4 below the onset temperature of the sharp changes of the relative populations, that is, T_{max}<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 constantpressure specific heat is the most commonly examined of these response functions, and the locus of the specificheat maxima is often referred to as the Widom line^{19}. 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 pressure^{25}.
For the Jagla model, T_{max} coincides closely with the Widom temperature T_{W}(P). The Jagla model has a liquid–liquid critical point at T_{c}=0.375, P_{c}=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 T_{W} and T_{max} 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 point^{26}. For TIP5P, we find that T_{W}≈255 K, identical to T_{max}. 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 T_{W}≈225 K(ref. 11) and T_{max}≈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 LDAlike and HDAlike 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 expected^{16,27,28}; molecules with a locally tetrahedral geometry are more ‘sluggish’ than less wellnetworked 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 structure^{29}. For these reasons, our structurebased interpretation for the failure of the Stokes–Einstein relation is particular to water, and the breakdown occurs at almost twice the glasstransition temperature T_{g}. An explanation for the failure of the Stokes–Einstein relation for liquids in general, where the breakdown typically occurs within 20–40% of T_{g}, 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.
Methods
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 signaltonoise 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 MM41S annotates) have hydration levels of H∼0.5 (0.5 g of H_{2}O per gram of MM415).
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 hydrogenbonded or nonhydrogenbonded OHS oscillators by considering different hydrogenbonded geometries^{30,31,32}. We work in an intermediate picture between the continuous and discrete models, assuming the existence of a percolating hydrogenbond network with a characteristic hydrogenbond 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 lowdensity amorphous LDA ice^{30}. 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 hydrogenbonded 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 T_{c}=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 HDAlike 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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Acknowledgements
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 DMR0427239. 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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DOI: https://doi.org/10.1038/nphys1328
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