The Anomalies and Criticality of Liquid Water

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Downloaded by guest on September 25, 2021 a water liquid Shi of Rui criticality and anomalies The itr fdsic ye flclsrcue ic h ieof time the since www.pnas.org/cgi/doi/10.1073/pnas.2008426117 structures local of types distinct of mixture a successful been not nucleation. ice has inevitable experi- water the the liquid to in However, due LLCP 12–16). for (7, search scenario another mental LLCP as regarded the been to low- has support ices, ices, amorphous amorphous two high-density TIP4P/Ice of and most presence and two the TIP4P/2005 in Experimentally, water, LLCP (11). for the (10). models of pro- classical existence water been further the accurate real of has have evidence Zerze to model of strong and vided compared ST2 surface Sciortino, overstructured the Debenedetti, free-energy Recently, be However, the to 9). in cor- regarded basins (8, HDL two water and of metastable presence LDL in the to respectively) identifying HDL, responding by and model (LDL same liquids the between high-density transition liquid–liquid be and associated low- existence the can 2014, and very In LLCP it the the 7). confirmed of that (6, firmly coworkers anomalies proposed and water’s Debenedetti and of origin simulations thermodynamic water, the numerical model in by one gas–liquid ST2, the besides (LLCP) point critical anoma- the and dynamic compression hand, cooling. under upon other maximum transition fragile-to-strong mobility the the On include lies cooling. heat and upon compressibility ther- the as capacity of such increase functions rapid response the modynamic and maximum density the include W fluctuations dynamical long- the anomalies water’s settle to from clue far a too provide is debate. would it standing findings because experimen- Our water the water, LLCP. liquid in real the of negligible for region almost is accessible MPa Nevertheless, criticality tally 173 there. the around and that LLCP find that K the we reveal 184 of presence we around the land, cross suggesting man’s lines and no two compressibility the the the in of measurements results waters. diffusivity we experimental which model recent exists, for on it Based simulations if LLCP hundred-microsecond the by at confirm cross should lines fluctu- maxima dynamical and ation thermodynamic This two model. the two-state that hierarchical predicts our model comprehensive on thermodynamic, based a structural, data make experimental dynamic of and we set First, large water. a of liquid to analysis impact of strategy LLCP unique its a the of liq- and propose locate origin we of the LLCP Here leaving unclear. factor the elusive, anomalies water’s remained structure of has experimental existence properties water’s the the on However, in water. DNLS uid and forma- coexistence a the LFTS cooperative for evidence of a of direct found is existence recently We there the LFTS. if of tion rationalize (LLCP) point can thermo- local critical water’s also liquid–liquid Phe- explains but of water. only anomalies types not dynamic liquid model two (DNLS)—in this dis- nomenologically, structure of and long- normal-liquid (LFTS) coexistence R of structure ordered to dynamical matter tetrahedral back favored a the dating structures—locally been model, on has two-state 2020) relies 29, anomalies A April review water’s debate. for (received standing of 2020 9, September origin approved and The NJ, Princeton, University, Princeton Debenedetti, G. Pablo by Edited Japan 153-8505, Tokyo Tokyo, of University eateto hsc,Zein nvriy aghu302,Cia and China; 310027, Hangzhou University, Zhejiang Physics, of Department rmasrcua on fve,wtrhsbe considered been has water view, of point structural a From liquid–liquid the discovered coworkers and Stanley 1992, In mat norpae 15.Temdnmcanomalies Thermodynamic (1–5). planet significant our have on that properties impacts anomalous many has ater a,b n aieTanaka Hajime and | iudlqi transition liquid–liquid | w-tt model two-state b,1 | rtclpoint critical | ontgen, ¨ b eateto udmna niern,Isiueo nutilScience, Industrial of Institute Engineering, Fundamental of Department doi:10.1073/pnas.2008426117/-/DCSupplemental at online information supporting contains article This 1 the under Published Submission. y Direct PNAS a is article This interest.y competing no declare authors The static the to maximum, fluctu- fluctuation order-parameter thermodynamic the of i.e., line ations, symmetry the of is line which the plane, refer so-called we the Thus, criticality), anomaly. without Schottky (even function response At namic 40). 21, (20, parameter is ulse coe 5 2020. 15, October published First H.T. and R.S. paper. research; the performed wrote H.T. and R.S. R.S. research; and data; designed analyzed H.T. contributions: Author long-standing the water. its of of identifying criti- resolution mystery final and the the LLCP toward to The crucial the distance is for water. location the searching real on Therefore, for point. depends anomalies cal largely with the criticality associated for of criticality responsible effect the is or on LLCP feature consensus no the two-state still the is There of (28). which feature two-state anomalies, its water-like to coarse-grained shows owing but (42)—a (43) LLCP model no mW model—has singularity-free water example, the For does i.e., shown singularity, (41). anomalies any scenario been water-like on hinge has of nec- necessarily a emergence It not not the anomalies. is that water-like its LLCP theoretically for the in condition of criticality 40). existence essary liquid–liquid 21, the and (20, However, feature framework. LLCP two-state incor- an naturally the can with cooperativity porate with transition model two-state liquid–liquid the Thus, a to lead such accessing has directly water in real difficulty experimentally. to structures the microscopic model to two-state due the elusive remained of water. validity liquid the in (LFTS) (DNLS) However, structure structure identifying tetrahedral normal-liquid by disordered favored model and locally two-state of the coexistence supports the further of 31–39) structure (24, descriptors 25, the structural microscopic of various at with Analysis waters anomalies model (18–30). water’s level explains phenomenological successfully a has model two-state R owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To ae eas ti o a rmteLLCP. the from far too anoma- is of the it state because on liquid water accessible influence experimentally little the of has belief, properties criticality spread K lous the widely 184 that the around find to located we contrary is However, water MPa. real 173 of and approach LLCP unique the This that diagram. the phase suggests the as in LLCP lines fluctua- maxima the dynamical tion LLCP detect and thermodynamic to where exception- between propose the intersection search we region However, experimental Here the the challenging. water. ally makes in liquid located ice be our in might of on exists crystallization liquid (LLCP) inevitable liquid–liquid mysterious a point whether yet is critical question essential fundamental most One planet. the is Water Significance ngn(7.I h at2dcds h oenvrino the of version modern the decades, 2 last the In (17). ontgen ¨ ntetosaemdl h rcino LFTS, of fraction the model, two-state the In can cooperative, is it if LFTS, of formation the Theoretically, PNAS NSlicense.y PNAS | coe 7 2020 27, October s /,teei eko thermody- of peak a is there 1/2, = | . y o.117 vol. y https://www.pnas.org/lookup/suppl/ | s o 43 no. 1 = s /2 steorder the is , | nthe in 26591–26599 T -P

APPLIED PHYSICAL SCIENCES Schottky line (24). In principle, if the LLCP exists, this line s = 1 − g(rH−bond) ∝ AFSDP. [1] should coincide with the symmetry line of critical fluctuations, In simulations, we can also estimate s by the analysis of micro- where the correlation length has a local maximum, which is called scopic structures of water. Focusing on water’s second shell of the Widom line (44). nearest neighbors (59), we introduced the ζ parameter to mea- Recently, significant progress has been made to measure the sure local translational order (24). It is defined for molecule i structural, thermodynamic, and dynamic properties of liquid as the difference between the distance dj 0i of the closest neigh- water via advanced experimental tools (45–48) in the deeply bor molecule j 0 not H-bonded to molecule i and the distance supercooled state. Even covering the no man’s land, these recent 00 d 00 of the furthest neighbor molecule j H-bonded to molecule measurements provide valuable information on the anomalous j i behaviors of water. i: ζ(i) = dj 0i − dj 00 i . Here we emphasize that considering H- This article presents a structure-based two-state model that bonding is a crucial feature of this method (24, 38). As was successfully describes the experimentally measured structural, shown in refs. 24, 38, 49, and 50, ζ has a bimodal distribu- thermodynamic, and dynamic properties of liquid water in a uni- tion. We can estimate the fraction of LFTS, s, by decomposing fied manner. This model predicts a special line of dynamical fluc- the ζ-distribution into two Gaussian functions (the high-ζ LFTS tuation maximum—we call it the dynamic Schottky line—whose and low-ζ DNLS distributions). We confirm the above rela- existence is supported by hundred-microsecond simulations of tion 1 for both simulated waters and real water (see Fig. 1A). model waters and recent experimental measurements in the no For simulations, we confirm the relation of s estimated from man’s land. The crucial point is that this line is located at a sub- the ζ-distribution with g(rH−bond) for TIP4P/2005 and TIP5P. stantially lower temperature than the Widom line at ambient For real water, we confirm the relation between AFSDP and pressure (49, 50), suggesting its unique role in characterizing the g(rH−bond), which are estimated from k- and real-space analysis dynamics of water. If there is the LLCP, this line should inter- of the experimentally measured O–O correlations, respectively. sect with the line of thermodynamic fluctuation maximum, i.e., The free energy of a nonideal mixture of the two types of local the Widom line, exclusively at the LLCP. Using this strategy, we structures (two states) can be written as (19, 20, 24, 40) predict the location of the LLCP around 184 K and 173 MPa for real water. This location, which is within the range of previous G = Gρ + s∆G + kBT [s ln s + (1 − s) ln(1 − s)]+ Js(1 − s), estimations, suggests that water’s anomalies mainly come from [2] the two-state feature and not from the liquid–liquid criticality. It is because the experimentally accessible temperature-pressure where ∆G is the free-energy difference between LFTS and region of liquid water is too far from the LLCP. We also provide DNLS, s is the fraction of LFTS, J is the strength of cooperativ- practical guidance for the future experimental detection of the ity, and kB is the Boltzmann constant. ∆G can be further written criticality in liquid water. by ∆G = ∆E − T ∆σ + P∆V , where ∆E, ∆σ, and ∆V char- acterize the energy, entropy, and volume differences between Results LFTS and DNLS, respectively. The coexistence of the two states reaches equilibrium if ∂G/∂s = 0, which leads to the following Water’s Structure. The most direct way to experimentally access relation: the liquid structure is to measure the structure factor by scattering experiments. For ordinary liquids such as Lennard–Jones ∂G s = ∆G + k T ln + J (1 − 2s)= 0. [3] liquids, the first diffraction peak of the structure factor is gener- ∂s B 1 − s ally located at the wavenumber of kp = 2π/r1, corresponding to the interparticle distance r1. However, liquids with local tetrahe- Then the fraction s of LFTS can be obtained analytically from dral order, such as H2O, C, Si, SiO2, and GeO2, do not follow this Eq. 3 if the cooperativity is negligible (J ∼ 0; see below) or s 1 general rule but commonly have a peak at a wavenumber lower (19, 20), than kp . This peak is widely known as the first sharp diffraction 1 peak (FSDP), whose origin was a long-standing mystery (51, 52). s = ∆E−T∆σ+P∆V . [4] 1 + exp k T We recently found for liquid water, similarly to the case of SiO2 B (52), that the FSDP is located around k = 3π/2r1, correspond- We show in Fig. 1B the T , P dependence of s (symbols) esti- ing to the most extended period of density waves associated mated from accurate X-ray scattering experiments (45, 60), using with a tetrahedral structure made of five water molecules (10). Thus, the LFTS and DNLS with and without tetrahedral symmetry should have the lowest-wavenumber peaks at k = 3π/2r1 and k = 2π/r1, respectively. We indeed found that the apparent A B / X-ray (2014) first diffraction peak of the oxygen–oxygen (O–O) partial struc- X-ray (2016) 0.6 ture factor is composed of such two peaks (10). Although the 0.5 two-state feature was suggested by spectroscopic measurements, 0.4 such as Raman (53, 54), femtosecond mid-IR pump-probe (55), 0.3 time-resolved optical Kerr (56), and X-ray absorption (57) spec- s troscopies, the presence of the FSDP provides experimental 0.2 0 0.1 evidence for the two-state model from microscopic structures. P (MPa) 100

200 225 0 Moreover, the integrated intensity of the FSDP, AFSDP, propor- 250 275 300 300 tional to the number of LFTS, provides a direct measure of the 325 400 (K) fraction of LFTS, s (Eq. 1). Furthermore, since the LFTS, i.e., T the inside of its hydrogen-bonded (H-bonded) ﬁrst shell (r ≤ Fig. 1. (A) The fraction of LFTS, s, in water models obtained by the struc- rH−bond = 3.5 A),˚ is protected from the penetration of surround- tural descriptor ζ versus 1 − g(rH−bond) for TIP4P/2005 (circles) (24) and TIP5P ing water molecules (i.e., g(rH−bond) = 0) (58), only the DNLS (squares) water. The integrated intensity (AFSDP) of the FSDP in SOO(k) ver- can contribute to the O–O radial distribution function (RDF), sus 1 − g(rH−bond) obtained from X-ray scattering data of liquid water (45) ˚ is shown by diamonds (see the right vertical axis). (B) The fraction of LFTS, g(r), at r = 3.5 A. Therefore, we can estimate the fraction of s, of liquid water directly estimated from the experimental X-ray scattering LFTS, s, approximately from RDF (24), using the following data (45, 60) as a function of temperature T and pressure P. The solid lines relation: are the two-state ﬁts by Eq. 4.

26592 | www.pnas.org/cgi/doi/10.1073/pnas.2008426117 Shi and Tanaka Downloaded by guest on September 25, 2021 Downloaded by guest on September 25, 2021 eoetebcgon otiuin rmDL.Tebu n e col- of side red each right of and the pressure on blue The shown respectively. is The pressures, isobar lower DNLS. and from higher curves represent contributions dashed ors The background predictions. model the two-state the denote are curves solid the and capacity heat Isobaric Density (A) water. ibility liquid of data namic 2. Fig. density of dependence (see (G background the control two-state the taking thermo- form, reorganize experimental P dimensionless of we the set in convenience, large equations For a data. to model dynamics simultaneously two-state now our can we fit data, scattering the from determined Anomalies. Thermodynamic Water’s anomalies. control water’s directly of data strength scattering the We experimental the 50). by (∆E 49, solely parameters 38, mined key (24, three basis these that structural note microscopic than the water simulated entropy on for and estimations models the density, with energy, agreement in lower DNLS, has the LFTS the that seen in listed are parameters (Eq. model two-state the by described Eq. h n Tanaka and Shi be contributions. might background unacceptable fittings physically constraints, with such done without that note We (see details). contributions DNLS background free the of the on constraints physical energy 12 introduce decreasing we with characteristics, decrease increase should should DNLS of DNLS ity of density the decreasing with example, sim- For for requirements liquids. physical ple basic are share represent that properly curves features which monotonic dashed DNLS, from The contributions common range 1B. background the wide Fig. the a in with determined over describes pressure pendently symbols) well and (solid curves) temperature data (solid of experimental model the two-state of our all that see can coefficient expansion mal C A r 200 = efidta h re parameter order the that find We 1. IAppendix SI κ T G w-tt-oe nlsso xeietlymaue thermody- measured experimentally of analysis Two-state-model 6,6,6) (C 66). 65, (62, ρ P sarfrnesaepit ota ecneasily can we that so point, state reference a as MPa n t eiaie,esrn h ipelqi features simple-liquid the ensuring derivatives, its and T o h eal) nFg ,w ltthe plot we 2, Fig. In details). the for hra h opesblt n etcapac- heat and compressibility the whereas , C P hra xaso coefficient expansion Thermal ) 6) h ybl ersn xeietldata, experimental represent symbols The (68). stemlcompressibility isothermal ρ, tcnide be indeed can It S2. Table Appendix, SI α ρ P eair erterfrnepoint reference the near behaviors ) n sbrcha capacity heat isobaric and , A–C. D B ρ sn h re parameter order the Using stemlcompress- Isothermal (B ) (61–64). .Tevle ftefitting the of values The 4). , s ∆σ (T IAppendix SI T T r and , , ae nthese on Based . 308.15 = P s ) α (T P a ewell be can ∆V 6,6) (D) 67). (62, , κ P T C deter- ) ) o the for ther- , and K P T inde- We . , P s eetyt h oa ciaineeg,adtu,the thus, and time coefficient energy, diffusion quantity, of activation inverse dynamic a total of the dependence to ferently water. clusters, of LFTS properties of fraction the whereas fraction the model: LFTS, two-state nonlocal LFTS of hierarchical are this a the incorporate neighbors developed To in we nearest 50). effect, those whose (49, LFTS LFTS are predominantly an molecules also precisely, water more slow but cluster, locally words, 50). determined other (49, not environment In nearest-neighbor is parameter its of molecule order influence a the coarse-grained under of the dynamics (i.e., that the shell since (i.e., found first structures the recently local to to we not parameter water, linked order micro- directly the ST2 LFTS is identifying and dynamics By water’s TIP5P anomalies. in dynamic scopically the to apply Anomalies. Dynamic Water’s structural experimental 1B). (Fig. the data by from described determined independently commonly we is the parts fittings, anomalous our in the below). fittings. that (see emphasize thermody- criticality the also any the and We to describes resorting model well without model mea- our anomalies two-state namic of 70) our that validity to (69, note the data We data up indicating the density pressures S2), by the Fig. negative parameterized predicts at is nicely which sured MPa, pressures. model, 0 negative our to above that model our find extrapolate validity We we the fittings, check our fit- further of all To the describes the range. for well over temperature-pressure model quantitatively MPa wide water two-state 0.1 of our anomalies 2, at thermodynamic Fig. these K only in used 255 shown have As data above we tings. capacity analysis, data the heat thermodynamic on formation the the ice in of effects observed Appendix rious (SI been K also 255 due below nucleation have experimen- the ice errors beyond by deviations tal influenced Significant size. less smaller 4 be the to of to S1 results expected Fig. are the Appendix, small which with of (SI agrees nucleation measurements the model the to Our due during probably crystals 2A), (Fig. ice MPa 0.1 at K 300 and where 71): 50, 49, lses(lwwtr htvre rm0t pncooling upon 1 to 50): (49, 0 from that found varies to feature that similar a water) shares (slow clusters data Experimental 72). suggest (49, temperatures high at holds constant, relations, Stokes–Einstein the temperature. that reference a as K pressure-dependent following the energy: predomi- by activation is expressed is pressur- structure upon which down liquid ization, slows weakly the dynamics water’s where DNLS, nantly temperatures, high and At respectively, molecules, water fast and slow nti oe,so n atwtrmlclscnrbt dif- contribute molecules water fast and slow model, this In 10, 4, of capillaries using measurements density of results The u oteheacia aue h fraction the nature, hierarchical the to Due τ R E ,i ie ytefloiggnrlzdAreislw(19, law Arrhenius generalized following the by given is ), λ µm a S 1 = s and ietydtrie h hroyai properties, thermodynamic the determines directly , s imtr(1 iarewt ahohrbelow other each with disagree (61) diameter D X for a eapoiae ytetosaeequation two-state the by approximated be can E = PNAS a ρ η X E n for 0 and r h ciaineeg otiuin from contributions energy activation the are a 0 ρ = s | u otelclodrn nldn up including ordering local the to but ) T T E coe 7 2020 27, October r a .T vi h osbespu- possible the avoid To S1B). Fig. , 0 s + h aetosaepcuecnalso can picture two-state same The λ o hroyais epreviously We thermodynamics. for 1/D exp P 1/D λ ∆ sapyia xoetensuring exponent physical a is V and D n h oainlrelaxation rotational the and , E a X τ eew take we Here . a ρ IAppendix, (SI MPa −110 R τ + | s R eg,teviscosity the (e.g., constant = k D B . s µm o.117 vol. otostedynamic the controls , T D T ∆ imtrcapillary, diameter , E P ∆E a | s eedneof dependence , (T a o 43 no. s and T D = , r P E fLFTS of 273.15 = D which ), a S T η/ | − ∼ η T 26593 the , , s E 255 [5] A). D a = P ρ ) .

APPLIED PHYSICAL SCIENCES D 1 s = , [6] clusters by pressurization since LFTS has a larger specific volume ∆E D−T∆σD+P∆V D+P2b than DNLS. Moreover, Eq. 5 predicts a dynamic crossover from 1 + exp k T B ρ S one Arrhenius (with activation energy Ea ) to another (with Ea ) D where ∆E D, ∆σD, and ∆V D denote the effective energy, as s increases from 0 to 1 upon cooling. Such an Arrhenius- entropy, and volume difference between the slow and fast water, to-Arrhenius crossover was indeed observed experimentally in respectively. We introduce the second-order term, bP 2, to better the T -dependent diffusion coefficient D (Fig. 3A) (48), which describe the P dependence of the dynamic properties of liquid accounts for the apparent fragile-to-strong transition behavior water. This parameter b determines the curvature of the dynamic in liquid water (83), as discussed in refs. 49 and 50. The diffusion data (48) also deny any singular behavior of liquid water Schottky line, TsD= 1 (P), at which the number of fast and slow 2 in a wide range of temperature from the melting point to the water equals glass-transition temperature Tg = 136 K at ambient pressure, consistent with the dielectric measurement of ultraviscous water ∆E D + P∆V D + P 2b (81). Our hierarchical two-state model provides a simple yet TsD= 1 = . [7] 2 ∆σD physically relevant explanation to water’s dynamic anomalies, without resorting to the glass transition or the critical singularity. Eqs. 5 and 6 have been used to fit the available experimental viscosity (η), diffusion (D), and rotational relaxation time Dynamic Schottky Line. In the two-state model, thermodynamic (τR) simultaneously (Fig. 3). The parameters for the dynamic fluctuations maximize at the line of Ts= 1 (P), where LFTS anomalies of liquid water are provided in SI Appendix, Table S3. 2 By fitting Eqs. 5 and 6 to the experimental data of D, η, and DNLS have the equal fractions, i.e., s = 1/2. This line and τR simultaneously, our hierarchical two-state model (solid is essentially equivalent to the so-called Widom line (44) if curves) remarkably well describes the water’s dynamic behaviors there is the LLCP. However, this line is not necessarily linked (symbols) in the wide temperature-pressure range. The effective to criticality, unlike the Widom line defined in connection to activation energy for water rotation is determined to be 13.7 and the LLCP. On the Widom line, the thermodynamic response 35.2 kJ/mol for the fast and slow water, respectively, which agree functions maximize, reflecting the maximal critical fluctuations. well with the experimental values of 14.2 and 34 kJ/mol, mea- However, in the two-state model without LLCP, these functions maximize at T 1 (P), reflecting the maximal (noncritical) struc- sured by ultrafast infrared spectroscopy at high temperatures s= 2 (80) and dielectric spectroscopy at low temperatures (81, 82), tural fluctuations between the two states. It is also the case respectively. In the model, the mobility maximum (or the vis- for silica and mW water, in which thermodynamic response cosity minimum) as a function of pressure at low temperatures functions maximize even without the LLCP (43, 84). For con- (Fig. 3 B–D) is a direct consequence of the reduction of LFTS venience, we called the line of s = 1/2 the static Schottky line in the two-state terminology to distinguish it from the Widom line (49, 50). One striking prediction from our hierarchical two-state model AB is that dynamical fluctuations maximize on the line of T D 1 (P) s = 2 (dynamic Schottky line), where sD = 1/2, but not on the static Schottky line. In this work, we directly confirm the presence of the dynamic Schottky line by showing the maximization of dynamic heterogeneity at T D 1 (P) in deeply supercooled s = 2 TIP4P/2005 water via extensive (hundred microseconds; SI Appendix, Table S4) molecular dynamics simulations (Fig. 4A and SI Appendix, Figs. S3–S5 and Table S5). The existence of the dynamic Schottky line is also evident from the rate, R, of dynamic slowing down upon cooling. Usually, R monotonically increases for a fragile liquid or keeps almost con- C D stant for a strong one. In contrast, liquid water does not obey this general rule, and the rate shows a distinct maximum, which is known as the so-called “fragile-to-strong transition” (83, 90–92). Our hierarchical two-state model predicts that R should maximize on the dynamic Schottky line upon cooling (49, 50), obeying the following equation (at P = 0):

∂ log D0 ∂ ln D0 R = D = D ∂β ln (10)∂β E ρ β∆E ∆E D 1 1 = a + a sD − + δ 2 − − δ 2 , ln (10) ln (10) 2 2

Fig. 3. Two-state-model analysis of the experimentally measured dynamic [8] data of liquid water. (A) The T dependence of the diffusion coefficient D (48, 73). (B) The P dependence of the diffusion coefficient D for various T where β ≡ T −1. Here δ = 2β∆E D−1, whose absolute value is (48, 74, 75). (C) The P dependence of the viscosity η for various T (27, 76, one order of magnitude smaller than 1/2, and thus, we neglect 77). (D) The P dependence of the rotational relaxation time τR for various it. We have confirmed that R indeed maximizes on the dynamic T (78, 79). Here τR is the time scale characterizing the rotational motion of water molecules obtained from the spin-lattice relaxation time measured Schottky line for TIP4P/2005 and TIP5P models, as predicted by by nuclear magnetic resonance. The symbols represent experimental data, our hierarchical two-state model (SI Appendix, Fig. S5). Thanks and the curves are the fits of our hierarchical two-state model. The blue to the recent experimental measurement of the diffusion coef- and red colors represent lower and higher temperatures, respectively. The ficient (48), we are now able to calculate R of liquid water temperature of each isotherm is shown on the right sides of B–D. down to the no man’s land. As shown in Fig. 5A, R maximizes

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(49, ters) neighbors parameter nearest order the to dynamic up (the coarse-grained not one is parameter the by behavior order deter- but dynamic LFTS) static be the the cannot that by molecule found controlled a have of We dynamics locally. mined the and since dynamics the simple between parameter link that the order contrast, In (the LFTS). controlled structures of directly local is the dynamic tetra- behavior by and of thermodynamic impact thermodynamic The the the in anomalies. on difference ordering the structural on hedral focusing LLCP, the of of LLCP the it of make location the accurately. problems the water of these real estimate accuracy All to the simulations. challenging by for extremely con- limited used two-state is model and latter water critical the the whereas mixes tributions, the that from suffers arbitrariness former model fittings’ the water However, particular results. a features experimental of studies with two-state numerical the of and comparison point a or critical the adjustable both many esti- for involving these parameters model because data theoretical is thermodynamic a It of experimental to fittings 99). to 0 the and either from from 98, range made 97, pressure were refs. mations the e.g., and (see, temper- K the MPa 228 in 195 to scatter 168 largely from results range the ature but water, real of LLCP the understanding for iden- crucial anomalies. and is water’s existence LLCP of the origin the of for location searching properties the (96), gen- tifying physical vicinity liquid’s point its the critical near on a only impact Since significant the water. a exclude liquid has in necessarily erally LLCP not an does of it presence criticality, invoking anomalies without dynamic and thermodynamic water’s explains cessfully Water. in LLCP for line Schottky dynamic (43). report the water. previous of real a existence with agrees the which supports model, mW strongly in LLCP the of absence at the suggests lines Schottky dynamic and static the of departure In the ). S4 cases, both Fig. In Appendix , S5). (SI Fig. water Appendix, TIP4P/2005 and 50) (49, In TIP5P the (star). In are water water. circles TIP4P/2005 TIP5P The water. for for reported MPa LLCP 258 of K, locations 216 and water TIP4P/2005 and curve) h n Tanaka and Shi 4. Fig. A eew rps nqesrtg oetmt h location the estimate to strategy unique a propose we Here the locate to used been have approaches various Previously, ∼ 200 C ( and TIP5P, (B) TIP4P/2005, (A) of diagram phase The T ,floigorheacia w-tt oe,which model, two-state hierarchical our following K, s D = 1 2 lhuhorheacia w-tt oe suc- model two-state hierarchical our Although tikbu uv) epciey rs taseilpit hc sntigbtteLC.Islcto setmtdt e12K 1 P for MPa 216 K, 172 be to estimated is location Its LLCP. the but nothing is which point, special a at cross respectively, curve), blue (thick eso w yai cotylnsotie rmtefitn odfuindt ihfree with data diffusion to fitting the from obtained lines Schottky dynamic two show we C, κ T aiarpre o I5 4)adTPP20 2)wtr h qae r h yai eeoeet aiadtrie for determined maxima heterogeneity dynamic the are squares The water. (26) TIP4P/2005 and (44) TIP5P for reported maxima s D rtefato fLT clus- LFTS of fraction the or , T -P s h rne(8,mgna(9,adgentinls(4 r h oain fLC eotdfrTIP5P for reported LLCP of locations the are (44) triangles green and (89), magenta (88), orange the B, ie,tefato of fraction the (i.e., eedne,which dependences, B s rtefraction the or , h imn stemxmmo h aeo yai lwn down, slowing dynamic of rate the of maximum the is diamond the B, Wwtrmdl.In models. water mW ) h lc 8) rne(6,mgna(5,gen(7,advoe rage 1)aethe are (11) triangles violet and (87), green (25), magenta (86), orange (85), black the A, s snot is pt t ers egbr ed otesgicn reduction significant the (T to line parameter, Schottky leads order dynamic neighbors the nearest of its LFTS of to coarse-graining spatial up the Thus, and DNLS. isolated rather by be surrounded may LFTSs and random, spatiotemporally rmtebhvoso h ttcaddnmcShtk lines Schottky dynamic experimental recent and Although static water. supercooled the deeply of the in behaviors the from models classical accurate most the and of K water. that one of model (188.6 as TIP4P/Ice LLCP regarded the the widely for is (11) to Notably, reported close 1). recently very (Table MPa) 174.6 is estimations location location previous this estimated of unique that note the range a We water the 5C). at real (Fig. within cross MPa of is 173 they LLCP and that the K 184 locates find around successfully and two- analysis hierarchical not, This our or point. by cross predicted model lines state Schottky dynamic and (43) report S6). previous Table the a and for with S6–S8 agreement Figs. lines, Appendix, in inter- Schottky (SI 4C), no dynamic (Fig. i.e., our and model LLCP, static mW Moreover, of the (11). absence between methods the section the predicts standard to correctly more same close method the using model, in recently TIP4P/2005 by determined the MPa) model for 185.0 and MPa K 216 (173.1 LLCP LLCP and the pre- K locates TIP5P the method 172 and our near at TIP4P/2005 example, located For for is respectively. LLCP which water, of point, locations state reported unique viously shown the As at 85–89). 44, meet 25, 4 (11, Fig. determined methods in firmly standard been more already other have by LLCPs the where models, and infinite, becomes length to equal the correlation be at should cross the eventually, where and, other LLCP each approach order lines the Schottky of value more the be on parameter spa- influence may the little Thus, LFTS has LFTSs. correlation coarse-graining an other tial by structural and surrounded increases, frequently the more steeply and LLCP, LFTS the of length approach (T pressure line and Schottky static the than eew aepooe nqesrtg olct h LLCP the locate to strategy unique a proposed have we Here static the whether check water, real to method this apply We water for works indeed strategy this that confirmed have We A A and s and D h ttcaddnmcShtk lines, Schottky dynamic and static the B, erteLC.A eut h ttcaddynamic and static the result, a As LLCP. the near h ttcaddnmcShtk ie indeed lines Schottky dynamic and static the B, s PNAS D = s . 1 2 C isa oe eprtrsadpressures and temperatures lower at lies ) | coe 7 2020 27, October b s = and s 1 2 D 5) stetemperature the As (50). ) codnl,tedynamic the Accordingly, . b | = (Eq. 0 o.117 vol. o I5 ae (SI water TIP5P for R, o Wmdl In model. mW for 7) | T s= o 43 no. 2 1 ti purple (thin | 26595 s D

APPLIED PHYSICAL SCIENCES A C

Exp. Exp.

B Supercritical Exp.

Fig. 5. The criticality of the liquid–liquid transition in liquid water. (A) The rate of dynamic slowing down, R, determined from the experimental diffusion data [circles (73) and squares (48); Fig. 3A]. The blue curve is the prediction of our hierarchical two-state model. (B) The experimentally measured correlation length ξ (black ball) (93) in the vicinity of the GLCP [located at 647.3 K and 22.12 MPa (93)] obeys a theoretical scaling function (colored surface; Eq. 9) (94). The Widom line (black curve) at ρ = ρc follows the power law. (C) The phase diagram of liquid water determined from the experimental data. The static and dynamic Schottky lines, T (thin purple curve) and T (thick blue curve), respectively, meet at the LLCP (star). Its location is estimated to s= 1 sD= 1 2 2 be around 184 K, 173 MPa. The circle is the κT maximum obtained by X-ray scattering (47), and the diamond is the R maximum determined in this work (A). In C, the black dashed curves represent the phase boundaries (95), and the squares are homogeneous nucleation temperatures (95). The contour lines of the correlation length of 3 to 7 A˚ estimated by Eq. 9 (94) are displayed by red dot-dashed curves. The magenta region represents the T-P range, where the experimental thermodynamic data are analyzed in this work. The correlation length exceeds twice of the molecule size only in the vicinity of the LLCP (red region) and rapidly decays to the molecule scale at T/Tc − 1 > 0.3, and thus, it becomes negligible in the experimentally accessible T-P region (magenta region).

measurements (47, 48) in the no man’s land support the existence Criticality in Water. Since the retracting spinodal scenario pro- of these two lines, new experiments at higher pressures and lower posed by Speedy and Angell (65), the power law ﬁttings to temperatures are highly desirable for determining the static and water anomalies have become extremely popular because such dynamic Schottky lines accurately, proving the very existence of behaviors are anticipated from the second-critical point (or the the LLCP, and determining its location for real water. Widom line) scenario (7, 44, 101). Indeed, the thermodynamic

Table 1. Estimated locations of LLCP and the key parameters of the two-state model for real water and three water models

H2O TIP4P/2005 TIP5P mW

∆E (K) −1,952 −1,815 −3,356 −2,261 ∆σ −8.32 −7.66 −13.1 −11.5 Ts=1/2 (K) at 0.1 MPa 235 237 256 197 † ‡ § ¶ # Tc (K) 184 (168*–228 ) 172 (173 –193 ) 216 (210 –217 ) − † ‡ § ¶ # Pc (MPa) 173 (195*–0 ) 216 (185 –135 ) 258 (310 –340 ) −

The ranges of previous estimations are shown in the parentheses. Here the values of ∆E and ∆σ are expressed in the unit of temperature (K) and 1, respectively, by dividing them by kB. *Ref. 100. †Ref. 97. ‡Ref. 11. §Ref. 85. ¶Ref. 89. #Ref. 88.

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Fig. length, in relation lines contour length the correlation by the shown calculate theoretically can we LP yiptigteeprmnal eemndodrparam- order eter determined experimentally the inputting By LLCP. nvraiycas(0,wihhsrcnl encnre firmly length, confirmed correlation been the recently has (11), which (40), class universality 5B). (Fig. water of length GLCP correlation (Eq. measured function experimentally scaling the The (96). universality and exponents, cal T iey o h LP h est steodrprmtr The parameter. order the respec- is point, density critical exponents the the critical GLCP, at the value For its tively. and parameter order the the Near criti- fluids. length simple the correlation any although to criticality, the universal Ising GLCP, be water the should real of behavior of Here example cal (GLCP) typical (11). point a simulations critical as gas–liquid numerical the by which consider confirmed (40), we been class also universality to Ising has belong (3D) should three-dimensional LLCP with the associated phenomena the critical as The weaker becomes when generally pears point length, critical a correlation accessible of experimentally effect the The in water liquid of lies properties anomalous the the of on location of influence water. the origin of its determine physical elucidate to the and critical elucidating LLCP is for can- it fittings So, anomalies, empirical issue. water’s power Thus, the and (71). two-state resolve well the not work that equally fact fittings the law criticality. from mode-coupling arises the difficulty anomalies or The critical line the Widom from the with anticipated the associated are Similarly, which power 103). by 104), analyzed 102, (44, often and laws 47, quite been laws (43, also criticality power have anomalies of by dynamic sign described a often as quite regarded been have anomalies ihtetemdnmcsaigvariable scaling thermodynamic the with (94), function .O ihm,H .Saly h eainhpbtenlqi,sproldadglassy and supercooled liquid, between metastable relationship of The behaviour Stanley, Phase E. H. Stanley, Mishima, E. H. O. Essmann, 7. U. Sciortino, F. Poole, H. P. 6. liquid Metastable water: glassy and Supercooled Sciortino, F. Loerting, T. Handle, H. P. 5. .P Gallo P. 4. water. glassy and Supercooled Debenedetti, G. P. 3. Kauzmann, W. Eisenberg, S. D. water. Supercooled 2. Angell, A. C. 1. ∼ c ic h iudlqi rniinsol eogt h Ising the to belong should transition liquid–liquid the Since hrfr,w osdrteipc fteLC nteanoma- the on LLCP the of impact the consider we Therefore, water. water. land. no-man’s (2017). a 13344 and (s), solid amorphous (s), 76(2003). 1726 2005). Oxford, Press, University − 4 = Θ 1| t28t 4 esrdb -a cteiga ambi- at scattering X-ray by measured K 347 to 228 at A ˚ − Nature Nature ae:Atl ftoliquids. two of tale A Water: al., et β ˆ s where , .A result, a As S9). Fig. , ξ ξ Θ 2–3 (1998). 329–335 396, (1992). 324–328 360, (T c eoe oprbet h oeua ie(96). size molecular the to comparable becomes 1 = ξ , xed wc h oeuesz (∼ size molecule the twice exceeds , ρ) β ˆ b ˆ n h above-estimated the and /2, T 0 = ≈ ξ and c eoe hre n vnulydisap- eventually and shorter becomes , b ˆ steciia temperature. critical the is .326 (z c ˆ 2 nu e.Py.Chem. Phys. Rev. Annu. h tutr n rpriso Water of Properties and Structure The r constants, are 1) + and ξ hudas byEq. obey also should , −1/2 ξ ξ ν ˆ erae n prahsthe approaches and decreases hm Rev. Chem. 0 = olw h nvra scaling universal the follows rc al cd c.U.S.A. Sci. Acad. Natl. Proc. (T olwn h DIsing 3D the following .63, .Py.Cnes Matter Condens. Phys. J. /T β c 4370 (2016). 7463–7500 116, ˆ z − 9–3 (1983). 593–630 34, and ˆ = ieydescribes nicely 9) 1) c − −1 ξ ν ˆ ν ˆ Θ T 9)na the near (93) (Θ , r h criti- the are T c and ξ 9 -P − oEq. to 6 hc is which , 13336– 114, erthe near Θ )only A) ˚ region. 1669– 15, Θ c (Oxford 9 c )|T are [9] (SI 9, / ξ 3 .Msia eesbefis-re rniinbtentoH two between transition first-order two Reversible between Mishima, transition O. first-order apparently 13. An Whalley, E. Calvert, L. Mishima, models realistic O. two in 12. point critical Second Zerze, H. G. Sciortino, F. two Debenedetti, of G. coexistence P. the for 11. function scattering the in evidence Direct Tanaka, H. Shi, R. 10. u hsclpcue hscnlso sfrhrspotdb the Si, by supported liquid further that is fact conclusion of This validity the picture. supports physical range our pressure and quantities temperature dynamic wide three a the in experimental and thermodynamic the four and the of model data two-state agreement hierarchical excellent The our assumption. between any without scatter- alone experimental data the ing from determined directly is parameter anomalies, the order that the phenomenological of emphasize previous dence We to analyses. model contrast two-state in experiments, scat- X-ray tering from obtained our information on by structural based microscopic data is the which experimental criticality, without of existing model analysis the two-state the hierarchical of from set conclusion with comprehensive this associated derived the criticality the have by of We not formation LLCP. and the DNLS) the of (i.e., sea feature a in two-state are LFTS the water by liquid primarily of caused region state temperature-pressure liquid accessible the tally of influence properties little physical the of indicates water. on of finding measurements criticality This the experimental of length. water, previous molecu- of correlation the region with of the liquid order accessible agreement the experimentally in in the be (Eq. in correla- to class size is critical universality lar length Ising experimental the correlation 3D the the of the that on of function for solely estimation length scaling based this tion theoretical is that LLCP The stress the We data. water of MPa. real 173 location of and the LLCP the K locate 184 to struc- around us local allows of nature which cooperativity hierarchical ordering, the tural model’s on two-state information our crucial provides that shown have We Conclusion in anomalies water-like LLCP. clear the shows of experi- model absence the mW the by the in supported that also least is fact result at the This water, region. liquid liquid accessible of mentally properties the influence eeu c ulainln,weetecreainlnt may homo- length the correlation above the just where exceed MPa line, nucleation 200 ice near geneous detectable become any may without them describe nicely 58). can (28, model anomalies, criticality water-like two-state exhibit also the 107), (84, and LLCP an have to not aa oit o h rmto fSineadteMtuih Foundation. Mitsubishi the the and from Science of JP20H05619) Pro- Promotion and the Specially supported for JP25000002 and Society partly Grants JP18H03675) Japan was (KAKENHI Grant study Research (KAKENHI This (A) moted water. Research ST2 Scientific of by data length correlation the ACKNOWLEDGMENTS. information. Availability. Data of behaviors. convergence anomalous the water’s to of understanding lead physical and will the transition findings associated liquid–liquid criticality our the of that with detection hope experimental We the to desirable. contribute highly are region this .J .Palmer C. J. 8. .J .Ple,P .Poe .Sirio .G eeeet,Avne ncomputational in Advances Debenedetti, G. P. Sciortino, F. Poole, H. P. Palmer, C. J. 9. hs ecnld htalwtrsaoaisi h experimen- the in anomalies water’s all that conclude we Thus, h rtclflcutosaengiil tabetpesr but pressure ambient at negligible are fluctuations critical The Nature and pressure. by induced ice of phases amorphous water. of water. liquid in structures local of types models. water-like and (2018). water 9129–9151 in transition liquid-liquid the of studies 3 K. ∼135 ∼ 8–8 (2014). 385–388 510, Science 7 eatbelqi-iudtasto namlclrmdlo water. of model molecular a in transition liquid-liquid Metastable al., et .Ce.Phys. Chem. J. .Ftr xeietlivsiain in investigations experimental Future 5C). (Fig. A ˚ l td aaaeicue nteatceadsupporting and article the in included are data study All 8–9 (2020). 289–292 369, PNAS SiO eaegaeu oJ .Ple o rvdn swith us providing for Palmer C. J. to grateful are We | 9051 (1994). 5910–5912 100, 2 coe 7 2020 27, October n Wwtr hc r considered are which water, mW and , s .A.Ce.Soc. Chem. Am. J. hc steoii falo the of all of origin the is which , Nature | o.117 vol. 67 (1985). 76–78 314, 8827 (2020). 2868–2875 142, 2 mrh at amorphs O | o 43 no. T hm Rev. Chem. , predicts 9) P . GPa ∼0.2 depen- | 26597 118,

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