Viscosity of deeply supercooled water and its couplingto molecular diffusionAmine Dehaoui, Bruno Issenmann, and Frédéric Caupin1

Institut Lumière Matière, UMR5306 Université Claude Bernard Lyon 1-CNRS, Université de Lyon, Institut Universitaire de France, 69622 Villeurbanne, France

Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved August 12, 2015 (received for review May 7, 2015)

The viscosity of a liquid measures its resistance to flow, withconsequences for hydraulic machinery, locomotion of microorgan-isms, and flow of blood in vessels and sap in trees. Viscosity increasesdramatically upon cooling, until dynamical arrest when a glassy stateis reached. Water is a notoriously poor glassformer, and the super-cooled liquid crystallizes easily, making the measurement of itsviscosity a challenging task. Here we report viscosity of watersupercooled close to the limit of homogeneous crystallization. Ourvalues contradict earlier data. A single power law reproduces the50-fold variation of viscosity up to the boiling point. Our resultsallow us to test the Stokes–Einstein and Stokes–Einstein–Debyerelations that link viscosity, a macroscopic property, to the molecu-lar translational and rotational diffusion, respectively. In molecularglassformers or liquid metals, the violation of the Stokes–Einsteinrelation signals the onset of spatially heterogeneous dynamics andcollective motions. Although the viscosity of water strongly decoup-les from translational motion, a scaling with rotational motion re-mains, similar to canonical glassformers.

supercooled water | viscosity | Stokes–Einstein relations

Water, considered as a potential glassformer, has been a long-lasting topic of intense activity. Its possible liquid–glass

transition was reported 50 years ago to be in the vicinity of 140 K(1, 2). However, ice nucleation hinders the access to this transitionfrom the liquid side. Bypassing crystallization requires hyper-quenching the liquid at tremendous cooling rates, ca. 107  K · s−1(3). As a consequence, many questions about supercooled andglassy water and its glass–liquid transition remain open (4–7).As an example, crystallization of water is accompanied by one of

the largest known relative changes in sound velocity, which has beenattributed to the relaxation effects of the hydrogen bond network(8, 9). Indeed, whereas the sound velocity is around 1,400 m · s−1 inliquid water at 273 K, it reaches around 3,300 m · s−1 in ice at 273 Kand a similar value in the known amorphous phases of ice at 80 K(10). Such a large jump is usually the signature of a strong glass, i.e.,one in which relaxation times or viscosity follow an Arrhenius lawupon cooling. However, pioneering measurements on bulk super-cooled water by NMR (11) and quasi-elastic neutron scattering(12), as well as recent ones by optical Kerr effect (8, 9), reveal alarge super-Arrhenius behavior between 340 and 240 K, similar towhat is observed in fragile glassformers (13, 14). The temperaturedependence of the relaxation time is well described by a power law(8, 9), as expected from mode-coupling theory (15, 16), whichusually applies well to liquids with a small change of sound velocityupon vitrification. Based on these and other observations, it hasbeen hypothesized that supercooled water experiences a fragile-to-strong transition (17). This idea has motivated experimental effortsto measure dynamic properties of supercooled water and has re-ceived some indirect support from experiments on nanoconfinedwater (18–20) and from simulations (21, 22).In usual glassformers, many studies have focused on the coupling

or decoupling between the following dynamic quantities: viscosity(η) and self or tracer diffusion coefficients for translation (Dt) androtation (Dr). If objects as small as molecules were to followmacroscopic hydrodynamics, one would expect that the precedingquantities would be related through the Stokes–Einstein (SE),

Dt ∝T=η, and Stokes–Einstein–Debye (SED), Dr ∝T=η, relations,where T is the temperature. These relations are indeed obeyed bymany liquids at sufficiently high temperature. However, they mightbreak down at low temperature. Pioneering experiments wereperformed by the groups of Sillescu (23–25) and Ediger (26–28)where a series of molecular glassformers were investigated. SErelation is obeyed at sufficiently high temperature but violatedaround 1.3Tg, where Tg is the glass transition temperature, thusindicating decoupling between translational diffusion and viscosity.In contrast, it was observed for ortho-terphenyl (23, 24, 26)that rotational diffusion and viscosity remain strongly coupled (i.e.,obey the SED relation) even very close to Tg. A corollary is thattranslational and rotational diffusion decouple from each other atlow temperature. These observations imply that deeply supercooledliquids exhibit spatially heterogeneous dynamics (29–31). Dynamicheterogeneities have been confirmed by direct observations of sev-eral single fluorescent molecules immersed in ortho-terphenyl (32)or nanorods immersed in glycerol (33). Physically different systemsalso show analogous behavior. Colloids near the colloidal glasstransition violate SE but obey SED (34). In the metallic alloyZr64Ni36, SE relation is even violated without supercooling, morethan 35% above the liquidus temperature (35). This has also beenrelated to the emergence of dynamic heterogeneities (36).For water, SE already breaks down at ambient temperature, which

corresponds to around 2.1  Tg (Tg ’ 136 K). Molecular dynamicssimulations (37–39) have proposed that this occurs concurrently todynamic heterogeneities caused by a putative liquid–liquid criticalpoint. However, SE and SED also fail by application of high pressureat 400 K (40) where no liquid–liquid transition is expected. To gainmore insight, the test of SE and SED in supercooled water deservesfurther investigation. Translational self-diffusion coefficient Dt (41)and rotational correlation time τr (assumed to scale as 1=Dr) (42)

Significance

Water is the most ubiquitous liquid but also the most anomalous.In usual fluids far from their glass transition, viscosity and mo-lecular diffusion are coupled through the Stokes–Einstein re-lations. For water, viscosity already starts decoupling fromtranslational diffusion below 20°C. Simulations have suggested aconnection with the putative separation of supercooled waterinto two distinct liquid phases. Whereas experimental diffusiondata extend far in the supercooled region, accurate viscosity datawere lacking due to the readiness of supercooled water to crys-tallize under the slightest perturbation. Using Brownianmotion ofspheres suspended inwater, we havemeasured its viscosity downto −34°C without freezing. We find that whereas viscosity de-couples increasingly from molecular translation upon cooling, itremains coupled to rotation.

Author contributions: F.C. designed research; A.D. performed research; B.I. and F.C. ana-lyzed data; and B.I. and F.C. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: frederic.caupin@univ-lyon1.fr.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1508996112/-/DCSupplemental.

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have thus been measured down to the homogeneous crystallizationtemperature (238 K) at ambient pressure. Their comparison revealsa decoupling between rotation and translation that increases withsupercooling (42), similar to glassformers. However, viscosity dataare needed for a direct test of SE and SED relations. Quite sur-prisingly, there are only two sets of data for the viscosity η at sig-nificant supercooling. Using Poiseuille flow in capillaries, Hallett(43) reached 249.35 K, and Osipov et al. (44) reached 238.15 K.However, the two sets disagree below 251 K, with an 8% differenceat 249 K, beyond the reported uncertainties. The measurements inref. 44 are suspected of errors (45) because of the small capillarydiameter used. Here we report η at ambient pressure down to239.27 K. Our study completes the knowledge of the main dynamicparameters of water down to the homogeneous crystallization limitand allows us to check the coupling of viscosity to molecular trans-lation or rotation, as has been done for usual glassformers.

Results and DiscussionWe studied the Brownian motion of polystyrene spheres withphysical radius a= 175± 3  nm using dynamic differential mi-croscopy (46) (see Materials and Methods for details). In brief, aseries of microscope images of a capillary containing the sus-pension of spheres in water is recorded. The image differencesare analyzed to give the decorrelation time τðqÞ of the lightscattered by the suspension at a given wavenumber q. τðqÞ decaysas 1=q2, as expected for Brownian motion, and the coefficientgives the inverse of the translational diffusion coefficient D ofthe polystyrene spheres (Fig. 1, Inset). For these mesoscopicobjects, the SE relation holds

D=kBT6πηah

 , [1]

where kB is the Boltzmann constant, T is the temperature, and ah isthe hydrodynamic radius of the sphere. Thus, our measurement ofD allows us to deduce η. The uncertainty on η ranges from 2.3% atthe highest temperature to 2.9% at the lowest temperature (Mate-rials and Methods). We measured four different samples whichwere able to reach 239.43, 244.45, 241.86, and 239.27 K withoutcrystallization. The results are shown in Fig. 1, Fig. S1, and Tables

S1 and S2 and compared with literature values in Fig. S2. Goodagreement is found for the stable liquid above 273 K. We confirmHallett’s measurements on supercooled water (43) and extendthem by 10 K. We find however systematically lower values thanOsipov et al. (44) at low temperature.Referring to previous work (47) showing an increase of the

density of water inside small μm-sized capillaries, Cho et al. (45)argued that this effect might have changed the viscosity of water inthe experiment of Osipov et al., who used quartz capillaries withradius R= 1  μm. We propose another possible bias: electroosmo-sis. The quartz surface being charged, it induces a layer of counterions in the liquid close to the surface. Imposing mass flow throughthe capillary generates a current, which in turn creates a voltageopposing the flow (48). This leads to an increased viscosity com-pared with the bulk one. Detailed calculations are given in Mate-rials and Methods. The main ingredient is the surface potential ϕ onthe capillary walls. We found that a temperature-independentvalue ϕ= 4 mV is sufficient to increase our viscosity values to thoseof ref. 44. The effect scales as 1=R2, which explains why it did notaffect the data of Hallett, who used R= 100  μm.To analyze the data over a broad temperature range, we have

combined our data with literature values, selecting in eachtemperature interval among the most accurate data available(see Supporting Information and Tables S3 and S4 for details).Fig. 2 compares different usual parameterizations of the tem-perature variation of viscosity from the boiling point to themaximum supercooling temperature. We start with two non-diverging functional forms of the viscosity. The first is thevenerable Arrhenius law [ηðTÞ= η0 exp½Ea=ðkBTÞ�], obeyed bystrong glassformers. It reflects thermally activated transport overthe energy barrier Ea. For water (Fig. 2), ln η varies more rapidlythan inverse temperature, indicative of a fragile behavior. Thesecond is the parabolic law [Arrhenius law above T0 switching toηðTÞ= η0 exp½J2ð1=T − 1=T0Þ2 +Ea=ðkBTÞ� below T0] introducedby Elmatad et al. (49). They were able to collapse with this lawtransport properties of 58 fragile glassformers and also simula-tion data, including several models of water (50). However, whenapplied to real water (see Supporting Information for detailsabout the fitting procedure), the parabolic law is not able toreproduce the data correctly, showing deviations much largerthan the experimental uncertainties at high and low tempera-tures. In addition, in contrast to the analysis with the paraboliclaw of simulated structural relaxation time for five water models(50), T0 = 305.15 K is significantly above the experimental tem-perature of maximum density (277.14 K), and the parameterJ=T0 is 3.6 instead of 7.4± 0.4 for simulations.Next we test two models with an apparently diverging viscosity.

The first is the Vogel–Fulcher–Tammann (VFT) equation,ηðTÞ= η0 exp½BT0=ðT −T0Þ�. This originally empirical fit can beinterpreted within the Adam–Gibbs theory (51), which identifiesT0 as the Kauzmann temperature. At T0, the supercooled liquidwould have the same entropy as the crystalline phase. T0 wouldhowever never be reached because it lies below Tg. The VFT fitperforms slightly better than the parabolic law but falls beyondthe experimental uncertainties at high and low temperatures. Inaddition, the fit value for T0 = 168.9 K is above Tg ’ 136 K forwater, which is not in line with the Adam–Gibbs picture. Finally,we consider a power law, ηðTÞ= η0ðT=Ts − 1Þ−γ. This form issuggested by two physical pictures. One is mode-coupling theory(15, 16), with Ts identified as the mode-coupling temperatureTs =Tc >Tg. Obviously, viscosity cannot diverge above Tg; thisdivergence would be avoided thanks to the thermally activatedprocess of hopping of molecules, which would lead to a de-parture from the power law slightly above Ts (16). The otherpicture was proposed by Speedy and Angell (52), who noticedthat many thermodynamic and dynamic properties of water seemto have a singular behavior, extrapolating via power laws to adivergence at a common temperature Ts. They proposed two

0

5

10

15

20

240 250 260 270 280 290

(mPa

s)

Temperature (K)

Vis

cosi

ty

1

10

51

τ (s

)

q (μm-1)

Fig. 1. Viscosity of water: data from this work (purple circles), Hallett (43)(blue squares), Collings and Bajenov (70) (green diamonds), and Osipov et al.(44) (black crosses). The error bars are only displayed below 249  K for clarity.The curve shows the calculated effective viscosity (based on our data) thatwould be measured by Poiseuille flow in a capillary with radius 1  μm (44) andsurface potential ϕ= 4 mV (see Materials and Methods for details). (Inset)Image decorrelation time τðqÞ (pluses) showing a 1=ðDq2Þ dependence (redline). The data shown are for run 1 at 239.43  K.

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interpretations: the existence of “a λ transition associated withthe cooperative formation of an open hydrogen-bond net-work” or a “limit of mechanical stability for the supercooledliquid phase.” The power law gives an excellent fit over the whole134-K interval. It would extrapolate to a singular temperatureTs = 225.66± 0.18 K. This is above Tg but slightly below the ho-mogeneous crystallization temperature Th, so that a true di-vergence cannot be directly observed by experiment. The quality ofthe power law fit was noted earlier but using data down to 249 Konly (52). A power law also gives the best fit to Dt up to 500 K andto τr up to 450 K but with lower values of Ts (see Figs. S3 and S4,Table S5, and Supporting Information for details about the errorbars on the fit parameters). This is at odds with mode-couplingprediction of a common Ts values for all dynamic properties orwith the assumption of a common singularity temperature.We now turn to the discussion of the coupling between viscosity

and molecular translation and rotation. In Fig. 3, we test the SE andSED relations by plotting as a function of T the quantities Dtη=Tand η=ðτrTÞ, which should be constant if SE and SED relationshold. This is the case at high temperature, but the relations startbeing violated below around 350 K. The violation of SE is stronger,reaching 70% at 239 K, with an increasing trend as the temperaturedecreases. The maximum violation of SED is only around 18% at239 K. This behavior is reminiscent of what is observed for ortho-terphenyl (23, 24, 26), although the viscosity range investigated forwater is more modest. We also emphasize once more that SE vi-olation in water occurs far above Tg. Our findings contrast withmolecular dynamic simulations (MDS) of the ST2 model for water(37, 38), which find that SED is violated even more strongly thanSE in the same temperature range. The reason for the discrepancymay lie in the different quantities used to check SED. Indeed,NMR experiments access τr only, whereas MDS can access both τrand the rotational diffusion coefficient Dr of the water moleculesand find that the usual assumption Dr ∝ 1=τr is not valid (37, 38).UsingDr, ST2MDS find that SED holds at high temperature and isviolated upon cooling. If 1=τr is used instead, SED is never valid.However, we note that MDS also use a proxy for the viscosity: theyreplace it by the alpha-relaxation time τα, assuming η∝ τα. In acareful study of model atomic and molecular systems (53), it wasfound that τα=η is temperature-dependent, which causes biases inthe evaluation of SE and SED using τα. Therefore, it appears highlydesirable to determine both η and τr in MDS of a water model, tocompare with the available experimental data.

Several models have attempted to improve on SE. A simplehydrodynamic model (54) treating water molecules as spheres withradius a moving into a surrounding continuous medium introducesa slip boundary condition, which modifies the proportionality co-efficients between Dt, Dr, and T=η. This model describes well thedata above 360 K, with a molecular diameter 2a= 0.25  nm and aslip length of 0.23a. However, the low-temperature data analyzedwith this model yield an unphysical 60% decrease of a. Anotherexplanation of the departure from SE and SED is given bymode coupling theory (16), which predicts a universal behavior:η∝ 1=Dt ∝ τr. Although this should only occur close to the modecoupling temperature Tc (typically 1.3Tg), experiments on Zr64Ni36(35) have found a constant Dtη up to 2Tc, 500 K above the liquidustemperature (1283 K). Dtη is also found to be nearly constant forwater between 259 and 300 K. However, when our data are in-cluded, Dtη increases sharply between 259 and 249 K, whereas η=τrlevels off (Fig. S5). The observed behavior of Dtη thus disagreeswith the mode coupling predictions.Other modifications of SE and SED are the fractional SE (FSE)

and fractional SED (FSED) relations, with Dt and τr proportionalto ðη=TÞζ. Such relations or close variants were originally in-troduced for tracer diffusion in various stable liquids (55, 56) andfor diffusion in supercooled ortho-terphenyl (23, 24). In the lattercase, the FSE exponent exhibits a crossover from ζ=−1 at hightemperature to ζ=−0.79 at low temperature. A similar crossover ofthe FSE exponent was observed for Tris-naphthylbenzene fromζ=−1 to ζ=−0.77 (27). The FSE relation has since been tested fora variety of liquids and found to give a good description far from theglass transition (57). A putative universal crossover from FSE withζ=−1 to ζ ’ −0.85 upon cooling was suggested by the analysis ofdata for nine liquids (58). Simulations of models of supercooledfragile liquids (East models) find ζ ’ −0.7 to −0.8 (59, 60).Fig. 4 investigates the validity of FSE and FSED relations for

water. In the case of translation, a transition between two FSErelations with exponents 0.94 and 0.67 has been proposed earlier(57, 61) but was based on the flawed data from ref. 44. Using ournew data, we see that a single exponent ζ is still not sufficient. Thelimited experimental range does not allow deducing reliable ex-ponents at high and low temperatures. However, the data areconsistent with the ST2 MDS (37, 38), which find a crossover be-tween ζ=−1 at high temperature and ζ=−0.8 at low temperature.MDS of another water model (mW) find a crossover from ζ=−1to ζ=−0.75 (62). The overall picture is that a similar crossover in

1 2T

0/(T-T

0)

VFT

1 10T

s/(T-T

s)

power-law

0 1J (1/T-1/T

0)

parabolic

1

10

3 4

this workHallett 1963Collings 1983Kestin 1985

Vis

cosi

ty (m

Pa s)

1000/T (K-1)

Arrhenius

0

10

-3

0

3

Fig. 2. Parameterizations of viscosity. The four panelsinclude four datasets: this work (purple circles) and refs.43 (blue squares), 70, (green diamonds), and 71 (redtriangles). (Left) Arrhenius plot, showing an apparentactivation energy increasing from 1,560 to 6,410 Kupon cooling (solid lines). (Left Center) Paraboliclaw, with best fit (χ2 = 14.2) parameters T0 = 305.15  K,η0 = 2.323  10−6   Pa  s, J= 1,112  K, Ea = 1,769  K. (CenterRight) VFT representation, with best fit (χ2 = 10.5)parameters T0 = 168.9  K, η0 = 4.44210−5   Pa  s and B=2.288. (Right) Power law representation, with bestfit (χ2 = 0.91) parameters Ts = 225.66± 0.18  K, η0 =ð1.3788± 0.0026Þ10−4   Pa  s, and γ = 1.6438± 0.0052.(Top) The normalized residuals ðηexp − ηfitÞ=σexp, whereηexp and ηfit are the experimental and fitted viscos-ity, respectively, and σexp is the experimental un-certainty (1 SD). Note that the vertical scale of TopRight is different.

12022 | www.pnas.org/cgi/doi/10.1073/pnas.1508996112 Dehaoui et al.

the FSE exponent is observed for real water, simulations of watermodels (37, 38, 62), and experiments on ortho-terphenyl (23, 24),Tris-naphthylbenzene (27), and seven other liquids (58).In the case of rotation, we find that FSED experimentally

holds with a single exponent ζ= 0.97 over more than 2 decades.We emphasize that this would not be the case if the low-temper-ature data from Osipov et al. (44) were used instead of ours (Fig.4). The success of FSED for experiments contrasts with the ST2MDS results, which find a crossover between two FSED relationswith different exponents. However, as noted above, more simu-lations, calculating η rather than τα, are needed to allow a directcomparison. As for FSE, the experimental result for FSED inwater resembles that for ortho-terphenyl where ζ= 1 (23, 24). Wealso note that a crossover from FSED with ζ= 1 to ζ ’ 0.85 uponcooling was suggested by the analysis of data for six liquids (58);however, the structural relaxation time was used instead of τr,preventing a direct comparison.When the coupling between dynamic quantities is considered,

water behaves similarly to canonical glassformers. The close anal-ogy with ortho-terphenyl is striking. However, a major difference inthe case of water is that SE violation already occurs far above Tg.By analogy with what is known for molecular glassformers, thissuggests the existence of dynamic heterogeneities in water. Theyhave been observed in MDS (37–39) which were discussed in thecontext of a liquid–liquid transition, which exists in at least one ofthe interaction potential studied, ST2 (63). However, we note that

MDS of another water-like model (mW) for which there is noliquid–liquid transition (64) find a violation of SE relation andcrossover to a FSE relation with ζ=−0.75 (62), in the temperaturerange where a growing correlation length is detected (65). Variouswater models thus predict correlated regions whose size increasesupon cooling. Experiments have found that SE and SED in waterare also violated at high temperature by application of high pres-sure (40). In that case it was attributed to the rigidity of the firstneighbors shell in water and to the invariant number of hydrogenbonds at high pressure. What would be the effect of pressure on SEand SED at the lowest temperatures? MDS of a water model witha liquid–liquid transition have investigated SE breakdown in thesupercooled liquid under pressure (39). They find that the tem-perature at which SE starts being violated correlates with the locusof heat capacity maxima emanating from the liquid–liquid criticalpoint seen in the simulations. Experiments are underway to extendour viscosity measurements to higher pressure.

Materials and MethodsSamples. We use a dispersion of polystyrene spheres (Duke Scientific; meanphysical diameter 350± 6  nm) in water, placed in a rectangular borosilicate cap-illary (Vitrotubes; 20  μm thick and 200  μmwide). The capillary is glued with epoxyglue (Araldite; Huntsman Advanced Materials) at both ends, to prevent any ad-vective motion of the spheres (e.g., induced by Marangoni effects). The spheresare small enough to exhibit Brownian motion, and their density close to 1 pre-vents settling on the experimental timescale. The volume fraction of the

1

1.2

1.4

1.6

1.8

1

1.1

1.2

250 300 350 400 450 500

norm

aliz

edη/

(τr T

)

Temperature (K)

norm

aliz

ed D

tη/T

Fig. 3. Test of the Stokes–Einstein and Stokes–Einstein–Debye relations. Dtη=T(Upper) and η=ðτrTÞ (Lower) are plotted as a function of temperature. Theviscosity data used are the same as in Fig. 2 (same colors), and Dt and τr werecalculated at the temperatures of the viscosity data using the power law fitsgiven in Table S5. Only the combined uncertainty (1 SD) without the datasymbol is displayed for clarity. The data were further normalized by their valueat 362.25  K. SE and SED relations would thus correspond to the horizontaldotted lines. SE and SED hold at high temperature, but they are violated byaround 70% and 18% at low temperature, respectively.

10-10

10-9

10-8

ζ=-1

ζ=-0.8

0.1

1

10

10-6 10-5

τ r (ps)

η/T (Pa s K-1)

Dt (

m2 s

-1)

Fig. 4. Test of the fractional Stokes–Einstein and Stokes–Einstein–Debyerelations. Dt (Upper) and τr (Lower) are plotted as a function of η=T. Theviscosity data used are the same as in Fig. 2 (same symbols and colors), and Dt

and τr were calculated at the temperatures of the viscosity data using thepower law fits given in Table S5. FSE (Upper) does not hold with a singleexponent ζ, but the data suggest a crossover from ζ =−1 to −0.8 (solid lines)upon cooling, as observed in ST2 MDS (37). FSED relation (Lower) holds withζ= 0.97 (solid line). The low temperature data from Osipov et al. (44) (blackcrosses) deviate from the FSED fit.

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commercial dispersion is 1%.We checked that the dilution of the particles had noinfluence on the measurements. However, the more diluted they were, the largersupercooling that could be reached. Thus, for a good balance between signal andsupercooling, we used 100 times dilutions (0.01% volume fraction), which allowedreaching around 239  K in two capillaries. Below that temperature the samplesystematically crystallized before the measurements could be performed. Theeffect of the purity of the solution in which the spheres are suspended was alsochecked, by centrifugating a sample and replacing the supernatant with ultrapurewater (Direct-Q3 UV; Millipore) several times. This did not change the results.

Experimental Setup. The capillary filled with the suspension of spheres is gluedon a microscope slide placed on the Peltier temperature stage (Linkam LTS120) of an upright microscope (Zeiss AxioScope) equipped with a ×  100 long-working distance objective (Mitutoyo Plan Apo, NA 0.7). The numerical ap-erture of the light source was chosen as small as possible (Ns = 0.1), and thediaphragm aperture was maximum. The intensity of illumination had noinfluence on the results. To calibrate the actual temperature inside thecapillary down to 239  K, we used the same capillaries as for the viscositymeasurement but filled with pure chemicals whose known melting points(66) served as reference. The temperature calibration was repeated just aftereach of the runs reported in this work. The temperature uncertainty is0.15  K. The spheres’ motion was recorded with a CCD camera (ProsilicaGX1050, 1,024 × 1,024 pixels2; Allied Vision Technologies) able to reach up to112 fps. The frequency of the camera was checked by filming a LED flashingwith a tunable frequency. No significant deviation was observed betweenthe imposed flashing frequency and the one deduced from the movies (lessthan 0.01% at 100 fps).

Dynamic Differential Microscopy. Typical data consist of a sequence of 500images, acquired at 10–100 fps depending on temperature, with 8 ms ex-posure time. Images are processed using a home-written code (MATLAB;Mathworks) based on Fourier analysis as explained in ref. 46. This yields thedecorrelation time τ as a function of the wave vector q. The diffusion co-efficient D for a given movie was obtained by least-squares fitting y = ln τ asa function of x = lnq with the function y =−lnD− 2x. A typical curve isshown in Fig. 1, Inset. All movies were analyzed with the same q interval.Changing the boundaries by 10% does not affect the results.

Viscosity Values. To convert the values of Dmeasured by dynamic differentialmicroscopy into the viscosity η of water, the starting point is the Stokes–Einstein relation for a Brownian sphere (Eq. 1). We have converted our Dvalues into viscosity η using

ηðTÞ= ηðT0Þ TT0DðT0ÞDðTÞ , [2]

where T0 = 293.15  K is a reference temperature at which the viscosity is knownwith high accuracy (67): ηðT0Þ= 1.0016± 0.0017 mPa  s. The experimental valuefor DðT0Þ was set by repeating the measurement at T0 in 12 independentcapillaries, which gave D0 = 1.086± 0.027  μm2 · s−1. The final viscosity mea-surements were performed in four independent runs in four different capil-laries, with 1–11 movies per capillary (typically 3–5) at each temperature. Thedetailed viscosity dataset and smoothed values are given in Tables S1 and S2,respectively.

Hydrodynamic Radius. As a further check of the procedure, we can also calculatewith Eq. 1 the hydrodynamic radius ah. At T0 we find ah = 189.1± 4.0  nm basedon the 12 independent measurements. This is slightly higher than the valuegiven by the manufacturer: 179± 5  nm. This small difference can be explainedby the effect of confinement on Brownian motion. Several corrections havebeen proposed for a sphere of radius ah confined between two parallel wallsseparated by a distance d (68). All corrections are similar for the large value ofd=ð2ahÞ ’ 50 (weak confinement) corresponding to our experiment. To esti-mate the effect, we use the Oseen formula (equation 1a in ref. 68). For a spherewhose center is at a distance z from one of the confining walls,

DkDSE

= 1−9ah16

�1z+

1d − z

�, [3]

where Dk is the parallel diffusion coefficient in confinement and DSE is thebulk diffusion coefficient given by Eq. 1. The experiment averages betweenz= ah and z=d − ah, which yields

DkDSE

= 1−98

ahd − 2ah

ln�

ahd − ah

�. [4]

This relation combined with the experimental values of D0 yields ah =179.9± 3.7  nm in excellent agreement with the manufacturer value. Thisvalidates the procedure. It is important to note that the correction factor toDSE in Eq. 4 is independent of viscosity and therefore of temperature. Thethermal expansion of polystyrene is also negligible in the temperature rangeinvestigated. This justifies the use of Eq. 2 to convert D into η at all tem-peratures. The validity of our approach is further corroborated by the ex-cellent agreement with Hallett’s data which cover the range 239.15–273.15 K(Fig. 1 and Fig. S2).

Measurement Uncertainty. We take for the intrinsic relative uncertainty(1 SD) on η the 2.3% SD of the measurements on 12 independent capillariesat T0 = 293.15  K. Because the viscosity is well described by a power law[ηðTÞ= η0ðT=Ts − 1Þ−γ], the effect of the 0.15  K temperature uncertainty wastaken into account to give the total relative uncertainty (1 SD) at tempera-ture T using fð0.023Þ2 + ½0.15γ=ðT − TsÞ�2g1=2. The resulting uncertainty rangesfrom 2.3% at the highest temperature to 2.9% at the lowest temperature.Analysis of the data scatter around the smoothed values (Fig. S1) is fully con-sistent with this calculated uncertainty.

Electroosmotic Effect. To explain the discrepancy of Hallet’s and our datawith the data of Osipov et al. (44), we propose that the latter measure-ments were affected by an electroosmotic effect. Viscosity was deducedfrom a Poiseuille flow in a 1-μm-radius quartz capillary. The surface chargespresent at the quartz surface are advected by water, creating an electriccurrent in the capillary. As this capillary is an open electric circuit, a voltageappears between the ends of the capillary that cancels this electric current.The resulting fluid flow is thus a combination of the pressure gradient andthe electric field (48). It can be shown that this leads to the measurement ofan effective viscosity

ηeffðTÞ=ηðTÞ

1 −aðTÞηðTÞ

, [5]

with

aðTÞ= 8e20R2

e2ðTÞϕ2ðTÞσðTÞ > 0, [6]

where R is the radius of the capillary, « is the relative dielectric permittivity ofwater, ϕ is the electric potential of the capillary surface, and σ is the electricalconductivity of the fluid. The viscosity deduced with usual Poiseuille flowformulas is thus overestimated. The effect decreases with the radius of thecapillary as 1=R2 and becomes negligible for wide capillaries, such as used byHallett (R= 100  μm).

To model the electroosmotic effect in supercooled water, we extrapolateformulas for the conductivity (69) σðTÞ and the permittivity (66) eðTÞ of stablewater:

σðTÞ= σ0 exp�−A−

BT−

CT2 −

DT3

�, [7]

with σ0 = 100  μSm−1, A=−1.7756, B=−903.75  K, C = 0.41340106   K2, and D=0.81046108   K3, and

eðTÞ= e1 + e2T + e3T2, [8]

with e1 =249.21, e2 =−0.79069  K−1, and e3 = 7.299710−4   K−2.Finally, we combine Eqs. 5–8 and compute the effective viscosity from our

measured viscosities, looking for the value of the surface potential ϕ thatreproduces Osipov’s data. Assuming a temperature-independent ϕ for sim-plicity, we find ϕ= 4 mV (Fig. 1), a reasonable value.

ACKNOWLEDGMENTS. We thank Marine Borocco for her help at the earlystage of the experiments, Luca Cipelletti and Cécile Cottin-Bizonne for discus-sion about dynamic differential microscopy, Abraham D. Stroock and Anne-Laure Biance for discussion about electrokinetics, and Livia E. Bove for a criticalreading of the manuscript. We acknowledge funding by the European Re-search Council under the European Community’s FP7 Grant Agreement 240113and by the Agence Nationale de la Recherche Grant 09-BLAN-0404-01.

12024 | www.pnas.org/cgi/doi/10.1073/pnas.1508996112 Dehaoui et al.

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Supporting InformationDehaoui et al. 10.1073/pnas.1508996112Viscosity ValuesA total of 369 viscosity measurements were collected from four runswith four independent capillaries. The individual data points aregiven in Table S1. For convenience, we also provide smoothed datain Table S2 which were used in Fig. 1. We have tried several simplesmoothing functions (combining positive and negative powers ofT and exponentials of these), but the best fit with the lowestnumber of parameters was found to be given by a power law:ηðTÞ= η0ðT=Ts − 1Þ−γ, with η0 = 1.306910−4   Pa  s, Ts = 224.80 K,and γ = 1.7044, reduced χ2 = 0.79. The normalized residuals be-tween the data and the smoothed values are shown in Fig. S1A.The corresponding histogram (Fig. S1B) is close to a Gaussian withunit variance. This fully confirms the measurement uncertaintycalculated as explained in Materials and Methods.

Comparison with Literature Values of the ViscosityFig. S2 shows the relative deviation of a set of viscosity data (43, 44,70) from our smoothed values. We can see that our data are inagreement with those of refs. 43 and 70 in the region of overlap.There might exist a small systematic bias between our data and ref.43 at the lowest temperatures. This might originate from a tem-perature error. Indeed, Hallett (43) states, “The temperature wasmaintained uniform, and could be measured to ±0.1°C for an in-dividual measurement. At the lowest temperature, the uncertaintywas somewhat larger, ±0.3°C, as equilibrium had not been at-tained.”At 249 K, 0.3°C translates into a 2% uncertainty on η. Thisis to be compared with the combination of the different errorslisted by Hallett, which yields a total of 0.5%, and to his quoted“estimated errors of a single observation of about 1%” (43). Notethat we have used a 1% SD for all of Hallet’s data. Hallett alsowrites, “A possible defect in this technique is that the warmer waterwhich enters the measuring system is not cooled sufficiently quicklyto the bath temperature, leading to apparently smaller values ofviscosity,” and then provides three pieces of evidence suggestingthis effect is negligible. However, we note that he carried out thecorresponding tests at 273.15 K, and the effect might be larger atlower temperatures. We conclude that the small negative deviationof ref. 43 data from ours at low temperature is not significant. Incontrast, the large positive deviation of ref. 44 data from ours is farabove the reported error bars. We attribute this discrepancy to abias of the Poiseuille flow experiment of ref. 44, due to electro-osmotic effects as proposed in Materials and Methods.

Selection of Data for Viscosity, Self-Diffusion Coefficient,and Rotational Correlation TimeFor the calculations reported in this work, we need experimentalvalues of η, Dt, and τr with reliable error bars. An internationalformulation for the viscosity of water is available (67). However,its “combined expanded uncertainty with a coverage factor k = 2”at ambient pressure is 1%, which corresponds to a SD of 0.5%,larger than several of the datasets included in the formulation.Therefore, we have chosen to select a series of datasets with asmaller uncertainty, to cover the full temperature range between239.15 and 362.25 K. In case of overlap between the temperatureranges of the data, only the data with the lowest uncertainty werekept. The same protocol was used for Dt up to 498.2 K and τr upto 451.63 K. The details of the datasets are given in Table S3. Theviscosity data below 250 K are the smoothed values from thepresent work (Table S2).In preparing this selection, we discovered several issues with

the reports on self-diffusion of water by Price et al. (41). First, wecould not reproduce the power law fit of the data given in ref. 41.

We found that the original fit had been performed with a least-squares method, that is, without weighting the data by the ex-perimental uncertainty. For a variation of Dt of more than 1decade, this results in giving excessive weight to the high-tempera-ture data. Then, we were surprised by the low stated uncertainty,1%, actually given as follows: “The accuracy of each diffusionmeasurement should be within 1%,” with a reference to a work byanother group (72), where special care was given to error analysis. Inparticular, it was stressed in ref. 72 that the main source of absoluteerror is the temperature inaccuracy in the experiment. Price et al.estimate their temperature uncertainty to 0.2 K. At low tempera-ture, this corresponds to a 2.8% uncertainty on Dt. Moreover, inanother study on D2O, Price et al. (73) reported their error bars onDt in the form of “80% confidence limit from Monte Carlo simu-lations.” They vary from an optimistic 0.2% at room temperature to4.3% at low temperature. In ref. 41, the apparatus was calibrated bysetting Dt at 298.15 K to its value determined accurately by tracerdiffusion by Mills (74). We have therefore compared the dataof Price et al. and Mills in the overlapping temperature range,from 275.8 to 298.15 K. Mills’ data being more accurate (0.2%uncertainty), we interpolate them with the usual power law fit[DtðTÞ=D0ðT=Ts − 1Þγ] to calculate a reference Dt at the tem-peratures of the data of Price et al. Table S4 gives the relativedeviations between the data sets at six temperatures. The some-times large values confirm that the stated 1% uncertainty is toooptimistic. We have thus decided to ascribe to the data of Priceet al. a more realistic 3% uncertainty.

Fit to the Viscosity Data with the Parabolic LawIn their analysis of the data for 58 experimental and 6 simulatedfragile glassformers, Elmatad et al. (49) used the followingequation:

ηðTÞ= η0 exp

"J2�1T−

1T0

�2#, [S1]

or an equivalent formula when the structural relaxation time τwas used instead of the viscosity η. They obtained a collapse ofthe data on a universal parabola when plotted in reduced units;yet we note that they did not check the compatibility of the fitwith the experimental uncertainties. We tried the same approachfor the viscosity of real water but obtained very poor fits. Thenwe noticed in a footnote of ref. 50 that η0 “may itself multiply anArrhenius temperature dependent factor.” Also, in figure 2C inref. 50, the simulated τ data for several water models were col-lapsed using an Arrhenius law above T0 switching to the para-bolic law below T0. Therefore, and to ensure continuity at T0, weused the following fitting function:

ηðTÞ= η0 exp

"J2�1T−

1T0

�2

+Ea

kBT

# for  T <T0 [S2]

= η0 exp�Ea

kBT

� for  T >T0. [S3]

The temperature T0 was varied to find the best fit with the lowestreduced χ2. The result is shown in Fig. 2, and the parameters aregiven in its legend. The fit quality is poor (χ2 = 14.2) with de-viations much larger than the experimental uncertainties. Even ifwe do not constrain the parameters η0 and Ea to the same values

Dehaoui et al. www.pnas.org/cgi/content/short/1508996112 1 of 8

on each side of T0 (which causes a small discontinuity at T0), thefit quality remains poor (χ2 = 10.8).

Power Law Fits to the Data on Self-Diffusion Coefficient andRotational Correlation TimeSince the pioneering work of Speedy and Angell (52), power lawfits of the properties of water have become common. However,the quality of the fits and the associated error bars are not alwaysdiscussed. Recently, based on their high-accuracy rotational re-laxation time data, Qvist et al. (42) provide detailed informationabout its power law fit. They quote errors on the fit parameters,obtained “with the Monte-Carlo method over 10000 syntheticdata sets.” We have checked that standard nonlinear fittingroutines that take into account the experimental error bars giveparameters and parameter errors identical to that of Qvist et al.To confirm this for the present work on viscosity, we generated1,024 synthetic data sets as follows. We use the same list oftemperatures as in the experimental data selection. To generateone synthetic set, we choose at each temperature a viscosityvalue as a random variable generated from a Gaussian distri-bution with mean equal to the value predicted by the best fit ofthe experimental data and variance equal to the experimentalSD. We then fit each synthetic data set to obtain 1,024 values forTs, γ, and η0. Their average and SDs are found to be exactly thesame as given in Table S5. The average and SD of the reduced χ2

are 1 and 0.2, respectively. This is consistent with χ2 = 0.91 fromreal data. To show how sensitive χ2 is to the choice of Ts, we tryforcing the value of Ts and fit γ and η0. If we shift Ts by plus orminus two error bars from the best fit value, χ2 becomes 1.22or 1.26, respectively; shifting by plus or minus 2 K gives χ2 = 4.1or 3.2, respectively.Qvist et al. (42) also provide a figure with power law fits to η

and Dt in their supplementary information. However, the tem-perature ranges of the fits are limited to 236.2–309.8 K for τr and273–373 K for η and Dt. We have extended the power law fits to

larger temperature intervals. For each quantity A (A= η, Dt, orτr), the data from the above selection were fitted with a powerlaw function: A0ðT=Ts − 1Þ−γ. The quality of the fit was assessedby the value of its reduced χ2 and the plot of the normalizedresiduals, ðAexp −AfitÞ=σexp, where Aexp and Afit are the experi-mental and fitted quantity, respectively, and σexp is the experi-mental uncertainty (1 SD). The minimum of the temperatureinterval was fixed at the lowest available temperature, and itsmaximum was adjusted until the fit quality was satisfactory. Figs.S3 and S4 display the fits and the normalized residuals, andTable S5 gives the results for the temperature interval and the fitparameters. Based on the error bars on the fit parameters, the Tsvalues for the three quantities are not consistent with each other.We note that it might be possible to reconcile them by designingappropriate background functions, but this was not attempted.The various tests presented in this work (usual and fractional

Stokes–Einstein and Stokes–Einstein–Debye relations, modecoupling predictions) involve combinations between η and Dtand between η and τr. Because they reproduce accurately thedata over broader temperature intervals, the power law fits to Dtand τr were used to calculate these quantities, whereas the ex-perimental data were used for η.

Test of the Mode Coupling PredictionsMode coupling theory (16) predicts a universal behavior:η∝ 1=Dt ∝ τr. Although this should only occur close to the modecoupling temperature Tc (typically 1.3  Tg), experiments on Zr64Ni36(35) have found a constant Dtη up to 2Tc, 500 K above the liq-uidus temperature (1,283 K). Fig. S5 shows Dtη and η=τr as afunction of temperature. Dtη is also found to be nearly constantfor water between 259 and 300 K. However, when our data areincluded, Dtη increases sharply between 259 and 249 K, whereasη=τr levels off. The observed behavior of Dtη thus disagrees withthe mode coupling predictions.

-4

-2

0

2

4

240 250 260 270 280 290

run 1run 2run 3run 4

Nor

mal

ized

resi

dual

s

T emperature (K)

A

0

20

40

-4 -2 0 2 4

Cou

nt

R ange

B

Fig. S1. Deviation between the raw viscosity data (Table S1) and their power law fit. (A) Normalized residuals, ðηexp − ηfitÞ=σexp, where ηexp and ηfit are theexperimental and fitted values, respectively, and σexp is the experimental uncertainty (1 SD). (B) Histogram of the normalized residuals, compared with aGaussian distribution with variance 1.

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-5

0

5

10

15

20

240 250 260 270 280 290

Hallett 1963Osipov 1977Collings 1983

Rel

ativ

e de

viat

ion

(%)

Temperature (K)

+1 SD

-1 SD

Fig. S2. Relative deviation of literature data from our smoothed viscosity data (Table S2). Data from three references (43, 44, 70) are displayed with their errorbars (Table S3); the symbols are identified in the legend. The two red curves show the uncertainty (1 SD) of our measurements.

10-10

10-9

10-8

011

Price 1999Mills 1973Easteal 1989Krynicki 1978

Diff

usio

n co

effic

ient

Dt (

m2 s-1

)

Ts/(T-T

s)

-4

-2

0

2

4

Nor

mal

ized

resi

dual

s

Fig. S3. Power law fit to the self-diffusion coefficient data. (Lower) Dt for the selected data sets (Table S3) is plotted as a function of Ts=ðT − TsÞ, where Ts is thebest fit parameter of the power law fit (Table S5). (Upper) Normalized residuals, ðDt,exp −Dt,fitÞ=σexp, where Dt,exp and Dt,fit are the experimental and fittedvalues, respectively, and σexp is the experimental uncertainty (1 SD).

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-2

0

2

norm

aliz

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10

1 10

Qvist 2012Hindman 1974

τ r (ps)

Ts/(T-T

s)

Fig. S4. Power law fit to the rotational relaxation time data. (Lower) τr for the selected data sets (Table S3) is plotted as a function of Ts=ðT − TsÞ, where Ts isthe best fit parameter of the power law fit (Table S5). (Upper) Normalized residuals, ðτr,exp − τr,fitÞ=σexp, where τr,exp and τr,fit are the experimental and fittedvalues, respectively, and σexp is the experimental uncertainty (1 SD).

0.8

1

1.2

1.4

norm

aliz

ed D

tη

0.8

1

1.2

250 300 350 400 450 500

norm

aliz

edη/τ r

T emper ature (K)

Fig. S5. Test of the mode coupling predictions. Dtη (Upper) and η=τr (Lower) are plotted as a function of temperature. The selected viscosity data (Table S3) usethe same color code as in Fig. 2. Dt and τr were calculated at the temperatures of the viscosity data using the power law fits of Table S5. Only the combineduncertainty (1 SD) without the data symbol is displayed for clarity. The data were further normalized by their value at 362.25  K. There is no sign of reaching aconstant value at low temperature.

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Table S1. Raw values of the viscosity

Temperature, K η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s

Run 1239.43 13.75 13.89 13.68 14.04 13.99240.47 12.77 12.64 12.27 12.64 13.08 12.80 12.69241.42 11.30 11.23 11.43 11.47 11.38 11.98242.46 10.21 10.09 10.49 10.23 10.21 10.37243.41 9.389 9.522 9.567 9.462 9.628 9.345 9.345 9.404244.45 8.527 8.661 8.686 8.479 8.420 8.420 8.588 0.000245.40 7.863 7.884 7.977 8.008 8.008 7.854 7.956 8.360 8.281 8.029 8.247249.38 5.542 5.562 5.802 5.513 5.770253.36 4.400 4.413 4.316 4.376 4.413273.16 1.825 1.841 1.841 1.845 1.849283.15 1.293 1.280 1.295 1.278 1.287293.15 1.025 1.001 1.045 1.033 1.032

Run 2244.4 8.468245.4 7.763 7.714 7.763246.4 7.326 7.146 7.222 6.943247.4 6.655 6.569 6.527 6.640 6.418248.3 6.259 6.259 6.159 6.032249.4 5.791253.4 4.407 4.463 4.438 4.413 4.425263.2 2.675 2.693 2.684 2.667 2.711283.1 1.326 1.295 1.308 1.342 1.322293.1 1.017

Run 3241.86 10.41 10.56 10.34 9.675 9.706 9.582242.82 9.322 9.366 9.605 9.279 9.381 9.590243.78 8.588 8.775 8.737 8.905 8.612 8.445244.74 8.292 8.214 8.192 8.008 8.384 8.147245.80 7.461 7.535 7.526 7.215 7.292 7.327246.76 6.955 6.931 6.725 6.638 6.667247.72 6.334 6.481 6.400 6.282 6.282 6.347248.68 6.035 6.000 6.131 6.230249.64 5.588 5.649 5.649 5.479 5.441 5.337250.69 5.251 5.287 5.198251.65 4.891 5.033 4.899252.61 4.566 4.580 4.718253.57 4.404 4.404 4.386254.63 4.122 4.090 4.090255.59 3.879 3.884 3.894256.55 3.624 3.665 3.665257.51 3.501 3.475 3.572258.47 3.332 3.298 3.281259.52 3.199 3.155 3.205260.48 3.019 3.059261.44 2.958 2.921 2.926262.40 2.795 2.786 2.744263.46 2.643 2.618 2.614264.42 2.530 2.515 2.538265.38 2.406 2.427 2.374266.34 2.268 2.306 2.306267.30 2.228 2.234 2.186268.35 2.122 2.144 2.080269.31 2.046 2.067 2.051270.27 1.926 1.976 1.958271.23 1.894 1.846 1.883272.29 1.827 1.831 1.823273.25 1.722 1.776 1.772 1.834 1.810 1.810274.21 1.693 1.728 1.735275.17 1.683 1.634 1.679276.13 1.585 1.606 1.612277.18 1.546 1.554 1.557278.14 1.494 1.505 1.487279.15 1.469 1.459 1.452

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Table S1. Cont.

Temperature, K η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s η, mPa s

280.15 1.408 1.401 1.415281.15 1.404 1.388 1.410282.15 1.321 1.342 1.340283.15 1.314 1.299 1.281 1.316 1.322 1.324284.15 1.273 1.266 1.269285.15 1.209 1.223 1.228286.15 1.191 1.180 1.196287.15 1.149 1.165 1.162288.15 1.114 1.145 1.118289.15 1.104 1.099 1.107290.15 1.066 1.056 1.076291.15 1.050 1.040 1.047292.15 1.015 1.006 1.011293.15 0.9922 1.001 0.9922 0.9987 1.019 0.9836

Run 4239.27 13.74 13.66 13.74 13.61240.33 12.11 12.02 11.83 12.11 12.11241.38 10.69 10.88 10.81 10.79 10.88242.44 9.922 9.809 9.972 9.793 9.890243.50 9.001 8.829 9.151 8.987 9.041244.55 8.285 8.152 7.960 8.130 8.163245.61 7.577 7.375 7.502 7.410 7.576246.66 6.983 6.782 6.897 6.843 6.976247.72 6.347 6.347 6.321 6.400 6.295248.68 5.830 5.577 5.818 5.970 5.919249.73 5.395 5.433 5.367 5.491 5.452273.15 1.764 1.794 1.802 1.772 1.790283.15 1.328 1.316 1.350 1.334 1.312293.15 0.9878 0.9878 0.9836 0.9922 0.9922

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Table S2. Smoothed values of the viscosity

Temperature, K η, mPa s

239.15 14.22240.15 12.68241.15 11.39242.15 10.29243.15 9.354244.15 8.545245.15 7.842246.15 7.226247.15 6.684248.15 6.203249.15 5.775250.15 5.392251.15 5.048252.15 4.738253.15 4.456254.15 4.201255.15 3.967256.15 3.754257.15 3.558258.15 3.379259.15 3.213260.15 3.059261.15 2.917262.15 2.785263.15 2.663264.15 2.548265.15 2.442266.15 2.342267.15 2.248268.15 2.161269.15 2.078270.15 2.001271.15 1.928272.15 1.859273.15 1.794274.15 1.732275.15 1.674276.15 1.619277.15 1.567278.15 1.517279.15 1.470280.15 1.425281.15 1.382282.15 1.341283.15 1.302284.15 1.265285.15 1.229286.15 1.195287.15 1.163288.15 1.132289.15 1.102290.15 1.073291.15 1.046292.15 1.020293.15 0.994

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Table S3. Selected datasets for viscosity, self-diffusion coefficient, and rotational correlation time

Primary data Selected data

First author andreference Year Accuracy, % Temperature range, K Number of data Temperature range, K Number of data

ViscosityHallett (43) 1963 1 249.15–273.15 25 250.15–273.15 24Collings (70) 1983 0.2* 274.15–343.15 12 274.15–343.15 12Kestin (71) 1985 0.5* 297.85–491.95 12 343.35–491.95 7Dehaoui 2015 2.9–2.3 239.15–298.15 55 239.15–249.15 11

Diffusion coefficientMills (74) 1973 0.2 274.15–318.15 7 274.15–318.15 7Krynicki (75) 1978 5 275.2–498.2 12 383.2–498.2 6Easteal (76) 1989 0.2 323.15–363.15 4 323.15–363.15 4Price (41) 1999 3 237.8–298.2 26 237.8–268.6 19

Rotational correlation timeHindman (77) 1974 2.5 242.2–451.6 46 312.5–451.6 30Qvist (42) 2012 1.5–0.5 236.2–309.8 21 236.2–309.8 21

The given accuracy value corresponds to 1 SD.*From table 1 in ref. 67.

Table S4. Comparison of the data of Price et al. (41) and the reference data from Mills (74),interpolated using a power law

Temperature, K

Diffusion coefficient, 10−9  m2 · s−1

Deviation, %Price et al. (41) Interpolated from Mills (74)

275.80 1.21 1.217 −0.55281.20 1.44 1.448 −0.55286.80 1.67 1.709 −2.3287.10 1.73 1.724 0.35292.60 1.89 2.003 −5.7297.30 2.16 2.259 −4.4

Table S5. Best fits obtained with a power law A0(T=Ts −1)−γ

Quantity Temperature range, K Number of data Reduced χ2 Ts, K γ A0

Viscosity 239.15–373.15 49 0.91 225.66± 0.18 1.6438±0.0052 137.88±0.26  μPasSelf-diffusion coefficient 237.8–498.2 36 1.62 213.96± 0.35 −2.0801± 0.0086 16  , 077± 78  μm2   s−1

Rotational relaxation time 236.18–451.63 51 0.61 223.05± 0.14 1.8760±0.0065 217.89±0.90  fs

The reported parameter errors correspond to a 68.3% confidence interval.

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