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