Unveiling enigmatic phase transitions of water in the supercooled region and no man’s land

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Using advanced sampling methods within the TIP4P/2005 water model, this study pinpoints the second critical point at temperature T c = 238±2 K and pressure P c = 288±30 bar, signifying an abrupt first-order to a gradual continuous phase transition. It also reveals a transition temperature T t =172±1 K at which a pivotal transformation unfolds, marking the stability switching of a two-step nucleation process, unveiling a previously unnoticed mid-density state bridging high- and low-density liquid states. These findings redefine our understanding of liquid-liquid phase transition, contributing to a comprehensive phase diagram for supercooled water including the elusive "no man’s land, unravelling its intricate complexity. Physical sciences/Chemistry/Theoretical chemistry/Molecular dynamics Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Phase transitions and critical phenomena Figures Figure 1 Figure 2 Figure 3 Figure 4 Introduction In the intricate landscape of water, a substance renowned for its perplexing nature, unexpected behaviors emerge, distinguishing it from conventional liquids. These peculiarities set the stage for a scientific enigma that has captivated researchers over several decades 1–4 . In 1992, Poole et al . proposed a revolutionary notion of liquid-liquid phase transition and the existence of a second critical point within the supercooled realm 5 . Since then, a prolific body of work 6–27 tirelessly validates Poole's groundbreaking concept, affirming the presence of a liquid-liquid transition in the supercooled region. Yet, within this expansive body of research, only a few select investigations have helped in disclosing the elusive second critical point. Despite these strides, the phase diagram of water in the supercooled region, particularly within the mysterious "no man's land," remains elusive. The intricate details of the real scenario behind the liquid-liquid phase transition persist as an ambiguous puzzle. As we embark on this scientific exploration, we confront the unknowns, seeking to unravel the complexities of supercooled water and the elusive liquid-liquid phase transition that defies complete elucidation. Mishima in his pivotal experimental work suggested a plausible location for the liquid- liquid critical point of water at temperature T = 230 ± 5K and pressure P = 50 ± 20 MPa, residing between the melting curves of ice III and ice V 7 . Kringle et al. 18 in a noteworthy contribution, reported that for temperatures below 245K, water’s structure can be expressed as a linear combination of two distinct forms. The data indicated the persistent presence of a barrier to nucleating crystalline ice, affording water the opportunity for thermal equilibration before undergoing crystallization. It should be noted that, for the low pressures examined in their experiments, no first-order phase transition was discerned, leading to the realization of high- and low-density liquid states (HDL and LDL) as localized domains within supercooled water. Debenedetti et al. ’s seminal investigations further corroborated the existence of a metastable LLPT under deeply supercooled conditions, as consistent with the 3D Ising universality class 19 , within TIP4P/2005 and TIP4P/Ice water models. Kim et al. achieved experimental verification of the LLPT in bulk supercooled water under pressure, unveiling structural alterations in liquid water at 205K indicative of a discontinuous LLPT transpiring between high- and low-density liquid phases within a pressure range spanning ambient conditions to 3.5 kbar 27 . In this study, we harnessed two distinct enhanced sampling methods to scrutinize the heterogeneous properties of the TIP4P/2005 water model 28 (known as one of the most reliable classical force fields for water phases) under NVT and NPT simulations, where N is the number of molecules and V is the volume. We posit that at T > T c the system heterogeneity arises from a continuous (second-order) transition, whereas at T < T c it stems from a first-order LLPT, thereby elucidating the second critical point of water in close concordance with known experimental observations. Moreover, we propose that the structural disparity between HDL and LDL is contingent on temperature, intensifying as temperature decreases. Additionally, we unveil a heretofore unreported mid-density liquid (MDL) state 29 and the resulting unique two-step nucleation process commencing at T ≃210K, rendering the inter-conversion between HDL and LDL states more plausible. Second critical point A distinctive feature of first-order phase transitions is the presence of a convex dip in the micro-canonical entropy, denoted as S ( E ) where E is total energy. This gives rise to a region with a negative slope in the statistical temperature 30–32 , often referred to as back bending or the S -loop denoted as \({T}_{S}\left(E\right)={\left(\partial S/\partial E\right)}_{V}^{-1}\) . To understand the supercooled region, we employed the generalized replica exchange (gREM) method, well-suited for first- order transitions associated with back bending in the statistical temperature, which is a metric characterizes the temperature of a system based on its energy distribution, allowing the study of complex thermodynamic behaviors and transitions through dynamical adaptation to render typically inaccessible regions in phase space reachable. A linear relationship between T s and E indicates a system with a relatively uniform energy distribution, while a transition from linearity to curvature signifies the occurrence of significant phenomena or transitions within the system. We conducted NVT gREM simulations with multiple replicas 33–35 and computed T s ( E ) from replica potential histograms using the statistical temperature weighted histogram analysis method (ST-WHAM) 36 . Figure 1a illustrates the T s ( E ) of the TIP4P/2005 water model through NVT simulations ( N = 1024, ρ = 1.0 g cm -3 ). As the temperature decreases from T = 238K, the back-bending graphs indicative of the first-order phase transition become evident. Each calculated T s ( E ) curve features two positive slopes corresponding to stable states and an unstable branch with a negative slope. The maxima and minima on each curve represent the spinodals of the two states. As the temperature is reduced, the energy differences (implying structural differences) between spinodals increase. Above T = 238K it indicates a continuous (second-order) transition associated with a heterogeneous state, while below T = 238K both the back bending and spinodals in the statistical temperature appear, indicating first-order transition between two different states. Thus, the inflection point T = 238K where two spinodals converge is a critical point associated with the supercold water under NVT simulations ( ρ = 1.0 g cm -3 ). The curvature of the T s ( E ) diminishes with rising temperature and disappears entirely at T ≃255K. It should be noted that this critical point exhibits similar phase transition features in simulations to the well-known critical point observed in the liquid-vapor phases of water 34 . The monotonic behavior of \(\beta \left(E\right)=1/{T}_{S}\left(E\right)\) is analyzed by examining its derivative with respect to energy, which provides a clear classification scheme of transitions in finite systems 37 . γ(E) = dβ(E)/dE = d 2 S/dE 2 ( 1 ) Figure 1b depicts the inverse temperature and its derivative as functions of energy per particle at temperatures higher, lower but proximate, and far lower than the critical temperature. In the top graph, at T > T c , the slope of the corresponding inflection point of β ( E ) at e = E/N = e tr is negative. Due to the monotonic temperature curve, a unique correspondence between β and E is possible, and there is no coexistence and latent heat between the phases. Such transitions can be categorized as second-order phase transitions 18,29 . This implies system heterogeneity due to the continuous transition and serves as strong evidence for the existence of the Widom line 38 in this region. In the middle graph, at temperatures lower but close to the critical point, γ tr = γ ( e tr ) > 0; the inverse temperature curve is non-monotonic and hence there is no unique correspondence between β and E . Physically, both phases coexist in the transition region 39,40 . Therefore, the system undergoes a first-order transition. In the bottom graph at T ≪ T c , both types of transitions occur unexpectedly. A continuous transition also appears within this temperature range. Later, using the umbrella sampling method, we will demonstrate that this second-order transition triggers a unique two-step nucleation in the system, influencing the LLPT under specific thermodynamic conditions. The “sub-phase” transitions in this graph with γ ( e tr ) > 0 are driven by surface effects, which depend on size and shape and vanish in the thermodynamic limit. To gain a more comprehensive understanding we employed the umbrella sampling method 41 under NPT simulations to compute the free energy of the system across various temperatures and pressures. In Fig. 2a, we plotted free energy against volume for a system comprising 512 water molecules, situated within the deeply supercooled region and ”no man’s land”. The umbrella sampling method, utilizing a harmonic restraint on the system’s volume as a collective variable, was instrumental in generating these results. A series of free-energy plots, exhibiting a negative slope in the pressure-temperature plane, aligning with the second critical point scenario 5 , are illustrated. The second critical point, occurring at T = 238 ± 2K and P = 288 ± 30 bar serves as the terminus for the first-order phase transition, giving rise to a continuous transition. At T < T c two structural forms of supercooled water (HDL and LDL) coexist in the transitional region. Their sizes gradually increase, resulting in augmented energy barriers, density disparities, and subsequently heightened structural distinctions between them. Figure 2b shows that the T/ρ profile of the coexistence basins under different pressures forms a distinctive bird’s beak shape, as suggested by Anisimov 42 . This graph illuminates that as temperature decreases, the structural disparity between HDL and LDL intensifies. It is noteworthy that the expansion of the high-density state comes to a halt at T ≃172K. The contrast in structure between HDL and LDL is visually represented in Fig. 2d via the radial distribution functions. Two-step nucleation via mid-density liquid state As the temperature drops below the critical point, both energy barrier and density disparity between coexisting states begin to escalate. Remarkably, at T ≃210K a precursor consistently emerges, instigating a unique two-step nucleation process 43–46 . A seminal computational study supporting this two-step mechanism was conducted by ten Wolde and Frenkel 43 , who investigated homogeneous nucleation in a Lennard-Jones system characterized by short-range attraction using Monte Carlo simulations. Their findings demonstrated that significant density fluctuations near the fluid-fluid critical point have a profound impact on the pathway to crystal nucleation. Notably, the nucleation barrier in proximity to the critical temperature experiences a substantial reduction. This distinctively anomalous nucleation process governs the LLPT from T ≃210K to lower temperatures (under varying pressures). Within this range and before reaching the coexistence state between HDL and LDL, the MDL state emerges, leading to heightened density fluctuations. This process is particularly distinctive since it can create a loop between LDL- and HDL-dominant regions, contingent on the temperature. To the best of our knowledge, this marks the first instance of two-step nucleation being observed between two liquid states (refer to Fig. 3a). Between T ≃210K and 172K (under different pressures), in addition to the increasing energy barrier, the two-step nucleation process arises when LDL dominates, especially when the energy difference between states is nearly equal. At T = 172 ± 1K, the system reaches a point where the energy barrier surpasses the energy difference between HDL and LDL significantly, compelling the system to undergo stability switching. Based on this phenomenon, from T < 172 K onward, two-step nucleation is witnessed when the HDL state prevails and LDL is metastable. The consistent energy difference between HDL and LDL during two-step nucleation im- plies the persistent and temperature-independent structural attributes of MDL. This newly emerged state, metastable in relation to LDL and HDL, plays a pivotal role in influencing inter- facial free energy, thereby reducing the barrier height. Recently, a prominent study experimentally confirmed the existence of mid-density ice 47 . Furthermore, our group has corroborated the presence of MDL in confined water 29 . A straightforward method to discern the distinction between one- and two-step nucleations is to examine the density / bond orientational order parameter ( Q 6 ) plane in the system 48 . The one-step process unfolds linearly, with both density and Q 6 increasing concurrently, ne- cessitating the overcoming of a single free energy barrier. Conversely, in two-step nucleation, the system undergoes a favorable density fluctuation initially, giving rise to a precursor that, in a subsequent step, based on thermodynamic conditions, leads to either a less ordered or more ordered state (see Fig. 3b). Figures 2c and 3c depict the free-energy barriers that separate HDL and LDL for the one-step nucleation process and the free energy of the MDL basins for the two-step nucleation at different system sizes. The barrier height increases with system size, following the expected N 2 / 3 scaling law for a first-order phase transition in the one-step process, and adheres to the same law for the two-step process as well. To compare the structural disparities between MDL and the other two states, we present the orientational tetrahedral order (OTO) q 49 of oxygen atoms in Fig. 3d. Neophytou et al. ’s 24 remarkable findings revealed that high-density networks were characterized by a profusion of highly entangled links and knots in the form of rings. In stark contrast, low-density networks primarily consisted of unknotted and unlinked rings. These stark differences in network structures led us to employ the OTO parameter for a more comprehensive analysis. The OTO focuses on the four nearest oxygen neighbors in water, and q varies from 0 for an ideal gas to 1 for a regular tetrahedron (see Supplementary Information for details). The high- density state exhibits a lower degree of order, while the low-density state displays a higher level of structural order. The mid-density state, situated between these two extremes, represents an intermediate level of order. Transition temperature In addition to the second critical point, an additional significant feature emerges along the coexistence line. The transition temperature Tt = 172 ± 1K marks the termination of the negative slope in the T/P graph, signifying the conclusion of HDL growth. At this critical temperature, owing to the substantial energy barrier relative to the energy difference between HDL and LDL, the two-step nucleation undergoes a first-order transition from being LDL-dominant to HDL-dominant. Above and below this transition temperature, the LDL-HDL transitions remain to be first-order while corresponding to LDL- and HDL-dominant, respectively, without merging into the continuous transition where LDL and HDL states can no longer be distinguished. In this regard, this temperature may not align with the traditional definition of a critical point, even though a previous simulation study suggested the second critical point around this transition temperature 19 , while it exhibits similar characteristics and even adheres to the Ising universality class. In Fig. 4a, we depict the OTO for two distinct states. While the LDL state gradually becomes more ordered with decreasing temperature, the structural evolution in the HDL state appears to stagnate. There is no observable difference in the HDL state between T = 175K and T = 165K, occurring before and after the transition temperature Tt = 172K. The phase diagram of the TIP4P/2005 water model within the supercooled region and the no man’s land is depicted in Fig. 4b. This phase diagram, closely aligned with experimental observations, substantiates the presence of a second critical point at 238±2K in water and introduces the transition temperature at 172±1K, where stability switching based on the two-step nucleation via MDL state takes place in the system. This newfound comprehension of supercooled water’s phase behavior holds the promise of reshaping the foundational underpinnings across diverse scientific domains. Declarations Data availability: The data that support the findings of this study are available from the corresponding author upon reasonable request. ACKNOWLEDGMENTS This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education(2021R1I1A1A01050280) and KISTI (KSC-2022-CRE-0268, KSC- 2022-CRE-0388, KSC-2022-CRE-0469, KSC-2023-CRE-0285). Author contributions S.P. performed all simulations and wrote the manuscript. K.S.K. revised the manuscript and supervised the project. Competing interests The authors declare no competing interests. Additional information Supplementary information The online version contains supplementary material available at Correspondence and request for data should be addressed to Kwang S. Kim. References Speedy, R. J. & Angell, C. A. Isothermal compressibility of supercooled water and evidence for a thermodynamic singularity at -45°c. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3611312","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":252504036,"identity":"1ab21992-1379-4f6a-a3a3-a10f8156ebf4","order_by":0,"name":"Kwang Kim","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyUlEQVRIiWNgGAWjYBACCWYGZgaGCgYGNqiAAZFazoC0MBOrhQGolLENxCRWi2Q7d7Lhz3mH8/j4zx9g+FHDYGzeQECLNDPv5mTebYeL2SSSGRh7jjGYyRwgoEUOqOUw47bDiW1AXzHwNjDYSBByGEjLwZ9zgFr4DzMw/iVGC8hhCbwNQC0MyQzMQFvMCGqRbObdbMxzLB3osGSDwzLHJIwJapE4f3az5I8a68T5/QcfPnxTY2M4g5AWFHAAHE+jYBSMglEwCigHAB2WNBom7lDEAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0002-6929-5359","institution":"Ulsan National Institute of Science and Technology","correspondingAuthor":true,"prefix":"","firstName":"Kwang","middleName":"","lastName":"Kim","suffix":""},{"id":252504037,"identity":"20064e76-cbe9-4937-9ab2-480ce20f1bfd","order_by":1,"name":"Saeed Pourasad","email":"","orcid":"","institution":"Ulsan National Institute of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Saeed","middleName":"","lastName":"Pourasad","suffix":""}],"badges":[],"createdAt":"2023-11-14 16:35:07","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3611312/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3611312/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":55330218,"identity":"3fdec6b8-d46a-456c-9224-b96901e854fb","added_by":"auto","created_at":"2024-04-25 19:11:32","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":546029,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003ePhase transitions in supercooled water at constant density. a \u003c/strong\u003eStatistical temperatures [\u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003ee\u003c/em\u003e)] plotted against energy per water molecule (\u003cem\u003ee\u003c/em\u003e=\u003cem\u003eE/N;\u003c/em\u003e \u003cem\u003eN \u003c/em\u003e=1024) at constant density (\u003cem\u003eρ\u003c/em\u003e= 1\u003cem\u003e.\u003c/em\u003e0 \u003cem\u003eg cm\u003c/em\u003e\u003csup\u003e\u003cem\u003e-3\u003c/em\u003e\u003c/sup\u003e), calculated using the generalized replica exchange method with the TIP4P/2005 water model. At and below \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e=238K, the appearance of back-bending lines signifies the onset of a 1\u003csup\u003est\u003c/sup\u003e-order transition, while above this temperature, the 1\u003csup\u003est\u003c/sup\u003e-order transition mechanism no longer applies. Around \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e=255K, the transition disappears completely. The graph displays multiple lines, each originating from 16 replicas extending in increments from \u003cem\u003eT\u003c/em\u003e to \u003cem\u003eT\u003c/em\u003e+30, with higher temperatures represented by the upper lines (see Supplementary Information for details). Error bars were computed using bootstrap analysis. \u003cstrong\u003eb\u003c/strong\u003e Inverse temperatures \u003cem\u003eβ\u003c/em\u003e(\u003cem\u003ee\u003c/em\u003e) and their derivatives \u003cem\u003eγ\u003c/em\u003e(\u003cem\u003ee\u003c/em\u003e) for three \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003ee\u003c/em\u003e) [the top third, sixth, and last dotted line in (a)]. The maxima of \u003cem\u003eγ\u003c/em\u003e(\u003cem\u003ee\u003c/em\u003e) at \u003cem\u003ee\u003c/em\u003e\u003csub\u003e\u003cem\u003etr\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e \u003c/em\u003eindicate transitions between structural phases. For \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e\u0026gt;\u003c/em\u003e238K (top-graph), a unique correspondence between \u003cem\u003eβ \u003c/em\u003eand \u003cem\u003ee \u003c/em\u003eis observed, signifying the absence of coexistence and characterizing a 2\u003csup\u003e\u003cem\u003end\u003c/em\u003e\u003c/sup\u003e-order transition. For \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e\u0026lt;\u003c/em\u003e238K (mid-graph), the presence of multiple intersections between \u003cem\u003eβ \u003c/em\u003eand \u003cem\u003ee \u003c/em\u003eindicates coexistence, a hall-mark of 1\u003csup\u003est\u003c/sup\u003e-order transition. For \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e≪238K (bottom-graph), both types of transitions occur unexpectedly.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-3611312/v1/69322131ed9825962d0e6d19.png"},{"id":55329765,"identity":"faf993f5-e79f-4d46-a1ff-37b920f5fa0d","added_by":"auto","created_at":"2024-04-25 19:03:30","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":544491,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSecond critical point scenario at various temperatures and pressures\u003c/strong\u003e. \u003cstrong\u003ea\u003c/strong\u003e Free energy versus volume graph for a system comprising 512 water molecules within the deeply supercooled region and ”no man’s land”. The coexistence between HDL and LDL at different \u003cem\u003eT\u003c/em\u003e and \u003cem\u003eP\u003c/em\u003e is highlighted. The symbol sizes are larger than the estimated uncertainties. \u003cstrong\u003eb\u003c/strong\u003e Presentation of the” bird’s beak” pattern depicting the presence of HDL and LDL under different temperatures. The red dotted line marks the location of the second critical point on the graph. The gray dot represents an average from two domains experiencing continuous transition. \u0026nbsp;\u003cstrong\u003ec\u003c/strong\u003e Demonstration of the increase in barrier height with system size following the \u003cem\u003eN\u003c/em\u003e\u003csup\u003e\u003cem\u003e2/3\u003c/em\u003e\u003c/sup\u003e (\u003cem\u003eT\u003c/em\u003e =230K) scaling law. \u003cstrong\u003ed\u003c/strong\u003e Radial distribution functions depicting two coexistence states.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-3611312/v1/fcc6b4bac3fc7a6ef9175894.png"},{"id":55329768,"identity":"ae574199-b715-4490-887a-12bbbb210350","added_by":"auto","created_at":"2024-04-25 19:03:30","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":380445,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTwo-step nucleation and mid-density state. a\u003c/strong\u003e Anomalous characteristic transitions that HDL/LDL states undergo prior to reaching the coexistence state, which capture the pre-coexistence state, emphasizing an anomalous process that occurs prior to the coexistence state depicted in Fig. 2a. The energy barrier (denoted as the red line) exhibits an inverse relationship with temperature. The light green line signifies the nearly constant energy difference between HDL/LDL during two-step nucleation. Stability switching occurs at the transition temperature \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e≃172K. \u0026nbsp;At \u003cem\u003eT \u003c/em\u003e=165K two-step nucleation emerges as the HDL state predominates. \u003cstrong\u003eb\u003c/strong\u003e Density-bond orientational order graphs showing characteristic anomalous nucleation mechanism. The two-step nucleation mechanism experiences a favorable density fluctuation along density first, culminating in a mid-density precursor that transitions to a stable state in a second step. \u003cstrong\u003ec\u003c/strong\u003e Demonstration of the increase in barrier height against system size, conforming to the \u003cem\u003eN\u003c/em\u003e\u003csup\u003e\u003cem\u003e2/3\u003c/em\u003e\u003c/sup\u003e scaling law (at \u003cem\u003eT \u003c/em\u003e=185K) for the energy of mid-density state. \u003cstrong\u003ed\u003c/strong\u003e Probability distribution of tetrahedral order \u003cem\u003eq\u003c/em\u003e, \u003cem\u003eP\u003c/em\u003e(\u003cem\u003eq\u003c/em\u003e), for three distinct states at constant temperature.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-3611312/v1/7606fb5138c37fc03514fced.png"},{"id":55330219,"identity":"07c78f8c-4dc6-49d3-b122-56bda64b1c50","added_by":"auto","created_at":"2024-04-25 19:11:33","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":347732,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eTransition temperature and phase diagram. a\u003c/strong\u003e Probability distribution of tetrahedral order q for HDL/LDL states at different temperatures. Before and after the transition temperature \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e∼172K, the HDL state maintains the same order without displaying any discernible difference. \u003cstrong\u003eb\u003c/strong\u003e Phase diagram within the supercooled region and the no man’s land for the TIP4P/2005 water model. The red dot signifies the location of the second critical point, while the orange dot denotes the transition temperature. The dark blue line presents the coexistence state between HDL and LDL (extracted from Fig. 2a) and the regions characterized by two-step nucleation (extracted from Fig. 3a) are denoted by the green points.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-3611312/v1/99079c4f26ac2022a0b29aea.png"},{"id":55330347,"identity":"76165523-3a53-49ec-af4c-e87dff82fcdd","added_by":"auto","created_at":"2024-04-25 19:13:37","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1770596,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3611312/v1/aaaae9c2-1c40-43f0-8a2e-5746e949962c.pdf"},{"id":55329767,"identity":"d160de87-acec-4e09-95da-38c86f1177f1","added_by":"auto","created_at":"2024-04-25 19:03:30","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":704983,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cbr\u003e\u003c/p\u003e","description":"","filename":"SuppInfo.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3611312/v1/32f84f119d4f37de4ee4abe4.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Unveiling enigmatic phase transitions of water in the supercooled region and no man’s land","fulltext":[{"header":"Introduction","content":"\u003cp\u003eIn the intricate landscape of water, a substance renowned for its perplexing nature, unexpected behaviors emerge, distinguishing it from conventional liquids. These peculiarities set the stage for a scientific enigma that has captivated researchers over several decades\u003csup\u003e1\u0026ndash;4\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eIn 1992, Poole \u003cem\u003eet al\u003c/em\u003e. proposed a revolutionary notion of liquid-liquid phase transition and the existence of a second critical point within the supercooled realm\u003csup\u003e5\u003c/sup\u003e. Since then, a prolific body of work\u003csup\u003e6\u0026ndash;27\u003c/sup\u003e tirelessly validates Poole's groundbreaking concept, affirming the presence of a liquid-liquid transition in the supercooled region. Yet, within this expansive body of research, only a few select investigations have helped in disclosing the elusive second critical point. Despite these strides, the phase diagram of water in the supercooled region, particularly within the mysterious \"no man's land,\" remains elusive. The intricate details of the real scenario behind the liquid-liquid phase transition persist as an ambiguous puzzle. As we embark on this scientific exploration, we confront the unknowns, seeking to unravel the complexities of supercooled water and the elusive liquid-liquid phase transition that defies complete elucidation.\u003c/p\u003e\u003cp\u003eMishima in his pivotal experimental work suggested a plausible location for the liquid- liquid critical point of water at temperature \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;230\u0026thinsp;\u0026plusmn;\u0026thinsp;5K and pressure \u003cem\u003eP\u003c/em\u003e\u0026thinsp;=\u0026thinsp;50\u0026thinsp;\u0026plusmn;\u0026thinsp;20 MPa, residing between the melting curves of ice III and ice V\u003csup\u003e7\u003c/sup\u003e. Kringle \u003cem\u003eet al.\u003c/em\u003e\u003csup\u003e18\u003c/sup\u003e in a noteworthy contribution, reported that for temperatures below 245K, water\u0026rsquo;s structure can be expressed as a linear combination of two distinct forms. The data indicated the persistent presence of a barrier to nucleating crystalline ice, affording water the opportunity for thermal equilibration before undergoing crystallization. It should be noted that, for the low pressures examined in their experiments, no first-order phase transition was discerned, leading to the realization of high- and low-density liquid states (HDL and LDL) as localized domains within supercooled water.\u003c/p\u003e\u003cp\u003eDebenedetti \u003cem\u003eet al.\u003c/em\u003e\u0026rsquo;s seminal investigations further corroborated the existence of a metastable LLPT under deeply supercooled conditions, as consistent with the 3D Ising universality class \u003csup\u003e19\u003c/sup\u003e, within TIP4P/2005 and TIP4P/Ice water models. Kim \u003cem\u003eet al.\u003c/em\u003e achieved experimental verification of the LLPT in bulk supercooled water under pressure, unveiling structural alterations in liquid water at 205K indicative of a discontinuous LLPT transpiring between high- and low-density liquid phases within a pressure range spanning ambient conditions to 3.5 kbar\u003csup\u003e27\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eIn this study, we harnessed two distinct enhanced sampling methods to scrutinize the heterogeneous properties of the TIP4P/2005 water model\u003csup\u003e28\u003c/sup\u003e (known as one of the most reliable classical force fields for water phases) under \u003cem\u003eNVT\u003c/em\u003e and \u003cem\u003eNPT\u003c/em\u003e simulations, where \u003cem\u003eN\u003c/em\u003e is the number of molecules and \u003cem\u003eV\u003c/em\u003e is the volume. We posit that at \u003cem\u003eT\u0026thinsp;\u0026gt;\u0026thinsp;T\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e the system heterogeneity arises from a continuous (second-order) transition, whereas at \u003cem\u003eT\u0026thinsp;\u0026lt;\u0026thinsp;T\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e it stems from a first-order LLPT, thereby elucidating the \u003cem\u003esecond critical point\u003c/em\u003e of water in close concordance with known experimental observations. Moreover, we propose that the structural disparity between HDL and LDL is contingent on temperature, intensifying as temperature decreases. Additionally, we unveil a heretofore unreported mid-density liquid (MDL) state\u003csup\u003e29\u003c/sup\u003e and the resulting unique two-step nucleation process commencing at \u003cem\u003eT\u003c/em\u003e ≃210K, rendering the inter-conversion between HDL and LDL states more plausible.\u003c/p\u003e"},{"header":"Second critical point","content":"\u003cp\u003eA distinctive feature of first-order phase transitions is the presence of a convex dip in the micro-canonical entropy, denoted as \u003cem\u003eS\u003c/em\u003e(\u003cem\u003eE\u003c/em\u003e) where \u003cem\u003eE\u003c/em\u003e is total energy. This gives rise to a region with a negative slope in the statistical temperature\u003csup\u003e30\u0026ndash;32\u003c/sup\u003e, often referred to as back bending or the \u003cem\u003eS\u003c/em\u003e-loop denoted as \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({T}_{S}\\left(E\\right)={\\left(\\partial S/\\partial E\\right)}_{V}^{-1}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\u003cp\u003eTo understand the supercooled region, we employed the generalized replica exchange (gREM) method, well-suited for first- order transitions associated with back bending in the statistical temperature, which is a metric characterizes the temperature of a system based on its energy distribution, allowing the study of complex thermodynamic behaviors and transitions through dynamical adaptation to render typically inaccessible regions in phase space reachable.\u003c/p\u003e\u003cp\u003eA linear relationship between \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eE\u003c/em\u003e indicates a system with a relatively uniform energy distribution, while a transition from linearity to curvature signifies the occurrence of significant phenomena or transitions within the system. We conducted \u003cem\u003eNVT\u003c/em\u003e gREM simulations with multiple replicas\u003csup\u003e33\u0026ndash;35\u003c/sup\u003e and computed \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003eE\u003c/em\u003e) from replica potential histograms using the statistical temperature weighted histogram analysis method (ST-WHAM) \u003csup\u003e36\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eFigure 1a illustrates the \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003eE\u003c/em\u003e) of the TIP4P/2005 water model through \u003cem\u003eNVT\u003c/em\u003e simulations (\u003cem\u003eN\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1024, \u003cem\u003eρ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1.0 \u003cem\u003eg cm\u003c/em\u003e\u003csup\u003e\u003cem\u003e-3\u003c/em\u003e\u003c/sup\u003e). As the temperature decreases from \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;238K, the back-bending graphs indicative of the first-order phase transition become evident. Each calculated \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003eE\u003c/em\u003e) curve features two positive slopes corresponding to stable states and an unstable branch with a negative slope. The maxima and minima on each curve represent the spinodals of the two states. As the temperature is reduced, the energy differences (implying structural differences) between spinodals increase. Above \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;238K it indicates a continuous (second-order) transition associated with a heterogeneous state, while below \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;238K both the back bending and spinodals in the statistical temperature appear, indicating first-order transition between two different states. Thus, the inflection point \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;238K where two spinodals converge is a critical point associated with the supercold water under \u003cem\u003eNVT\u003c/em\u003e simulations (\u003cem\u003eρ\u003c/em\u003e\u0026thinsp;=\u0026thinsp;1.0 \u003cem\u003eg cm\u003c/em\u003e\u003csup\u003e\u003cem\u003e-3\u003c/em\u003e\u003c/sup\u003e). The curvature of the \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e(\u003cem\u003eE\u003c/em\u003e) diminishes with rising temperature and disappears entirely at \u003cem\u003eT\u003c/em\u003e ≃255K. It should be noted that this critical point exhibits similar phase transition features in simulations to the well-known critical point observed in the liquid-vapor phases of water\u003csup\u003e34\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eThe monotonic behavior of \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\beta \\left(E\\right)=1/{T}_{S}\\left(E\\right)\\)\u003c/span\u003e\u003c/span\u003e is analyzed by examining its derivative with respect to energy, which provides a clear classification scheme of transitions in finite systems\u003csup\u003e37\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003e\u003cem\u003eγ(E)\u0026thinsp;=\u0026thinsp;dβ(E)/dE\u0026thinsp;=\u0026thinsp;d\u003c/em\u003e \u003csup\u003e \u003cem\u003e2\u003c/em\u003e \u003c/sup\u003e \u003cem\u003eS/dE\u003c/em\u003e \u003csup\u003e \u003cem\u003e2\u003c/em\u003e \u003c/sup\u003e (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e)\u003c/p\u003e\u003cp\u003eFigure 1b depicts the inverse temperature and its derivative as functions of energy per particle at temperatures higher, lower but proximate, and far lower than the critical temperature. In the top graph, at \u003cem\u003eT\u0026thinsp;\u0026gt;\u0026thinsp;T\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e, the slope of the corresponding inflection point of \u003cem\u003eβ\u003c/em\u003e(\u003cem\u003eE\u003c/em\u003e) at \u003cem\u003ee\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eE/N\u003c/em\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003ee\u003c/em\u003e\u003csub\u003e\u003cem\u003etr\u003c/em\u003e\u003c/sub\u003e is negative. Due to the monotonic temperature curve, a unique correspondence between \u003cem\u003eβ\u003c/em\u003e and \u003cem\u003eE\u003c/em\u003e is possible, and there is no coexistence and latent heat between the phases. Such transitions can be categorized as second-order phase transitions\u003csup\u003e18,29\u003c/sup\u003e. This implies system heterogeneity due to the continuous transition and serves as strong evidence for the existence of the Widom line\u003csup\u003e38\u003c/sup\u003e in this region.\u003c/p\u003e\u003cp\u003eIn the middle graph, at temperatures lower but close to the critical point, \u003cem\u003eγ\u003c/em\u003e\u003csub\u003e\u003cem\u003etr\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eγ\u003c/em\u003e(\u003cem\u003ee\u003c/em\u003e\u003csub\u003e\u003cem\u003etr\u003c/em\u003e\u003c/sub\u003e )\u0026thinsp;\u003cem\u003e\u0026gt;\u003c/em\u003e\u0026thinsp;0; the inverse temperature curve is non-monotonic and hence there is no unique correspondence between \u003cem\u003eβ\u003c/em\u003e and \u003cem\u003eE\u003c/em\u003e. Physically, both phases coexist in the transition region\u003csup\u003e39,40\u003c/sup\u003e. Therefore, the system undergoes a first-order transition. In the bottom graph at \u003cem\u003eT\u003c/em\u003e ≪\u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e, both types of transitions occur unexpectedly. A continuous transition also appears within this temperature range. Later, using the umbrella sampling method, we will demonstrate that this second-order transition triggers a unique two-step nucleation in the system, influencing the LLPT under specific thermodynamic conditions. The \u0026ldquo;sub-phase\u0026rdquo; transitions in this graph with \u003cem\u003eγ\u003c/em\u003e(\u003cem\u003ee\u003c/em\u003e\u003csub\u003e\u003cem\u003etr\u003c/em\u003e\u003c/sub\u003e )\u0026thinsp;\u003cem\u003e\u0026gt;\u003c/em\u003e\u0026thinsp;0 are driven by surface effects, which depend on size and shape and vanish in the thermodynamic limit.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eTo gain a more comprehensive understanding we employed the umbrella sampling method\u003csup\u003e41\u003c/sup\u003e under \u003cem\u003eNPT\u003c/em\u003e simulations to compute the free energy of the system across various temperatures and pressures. In Fig.\u0026nbsp;2a, we plotted free energy against volume for a system comprising 512 water molecules, situated within the deeply supercooled region and \u0026rdquo;no man\u0026rsquo;s land\u0026rdquo;. The umbrella sampling method, utilizing a harmonic restraint on the system\u0026rsquo;s volume as a collective variable, was instrumental in generating these results.\u003c/p\u003e\u003cp\u003eA series of free-energy plots, exhibiting a negative slope in the pressure-temperature plane, aligning with the second critical point scenario\u003csup\u003e5\u003c/sup\u003e, are illustrated. The second critical point, occurring at \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;238\u0026thinsp;\u0026plusmn;\u0026thinsp;2K and \u003cem\u003eP\u003c/em\u003e\u0026thinsp;=\u0026thinsp;288\u0026thinsp;\u0026plusmn;\u0026thinsp;30 bar serves as the terminus for the first-order phase transition, giving rise to a continuous transition. At \u003cem\u003eT\u0026thinsp;\u0026lt;\u0026thinsp;T\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e two structural forms of supercooled water (HDL and LDL) coexist in the transitional region. Their sizes gradually increase, resulting in augmented energy barriers, density disparities, and subsequently heightened structural distinctions between them.\u003c/p\u003e\u003cp\u003eFigure 2b shows that the \u003cem\u003eT/ρ\u003c/em\u003e profile of the coexistence basins under different pressures forms a distinctive bird\u0026rsquo;s beak shape, as suggested by Anisimov\u003csup\u003e42\u003c/sup\u003e. This graph illuminates that as temperature decreases, the structural disparity between HDL and LDL intensifies. It is noteworthy that the expansion of the high-density state comes to a halt at \u003cem\u003eT\u003c/em\u003e ≃172K. The contrast in structure between HDL and LDL is visually represented in Fig.\u0026nbsp;2d via the radial distribution functions.\u003c/p\u003e"},{"header":"Two-step nucleation via mid-density liquid state","content":"\u003cp\u003eAs the temperature drops below the critical point, both energy barrier and density disparity between coexisting states begin to escalate. Remarkably, at \u003cem\u003eT\u003c/em\u003e ≃210K a precursor consistently emerges, instigating a unique two-step nucleation process\u003csup\u003e43\u0026ndash;46\u003c/sup\u003e. A seminal computational study supporting this two-step mechanism was conducted by ten Wolde and Frenkel\u003csup\u003e43\u003c/sup\u003e, who investigated homogeneous nucleation in a Lennard-Jones system characterized by short-range attraction using Monte Carlo simulations. Their findings demonstrated that significant density fluctuations near the fluid-fluid critical point have a profound impact on the pathway to crystal nucleation. Notably, the nucleation barrier in proximity to the critical temperature experiences a substantial reduction.\u003c/p\u003e\u003cp\u003eThis distinctively anomalous nucleation process governs the LLPT from \u003cem\u003eT\u003c/em\u003e ≃210K to lower temperatures (under varying pressures). Within this range and before reaching the coexistence state between HDL and LDL, the MDL state emerges, leading to heightened density fluctuations. This process is particularly distinctive since it can create a loop between LDL- and HDL-dominant regions, contingent on the temperature. To the best of our knowledge, this marks the first instance of two-step nucleation being observed between two liquid states (refer to Fig.\u0026nbsp;3a).\u003c/p\u003e\u003cp\u003eBetween \u003cem\u003eT\u003c/em\u003e ≃210K and 172K (under different pressures), in addition to the increasing energy\u003c/p\u003e\u003cp\u003ebarrier, the two-step nucleation process arises when LDL dominates, especially when the energy difference between states is nearly equal. At \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;172\u0026thinsp;\u0026plusmn;\u0026thinsp;1K, the system reaches a point where the energy barrier surpasses the energy difference between HDL and LDL significantly, compelling the system to undergo stability switching. Based on this phenomenon, from \u003cem\u003eT\u0026thinsp;\u0026lt;\u003c/em\u003e\u0026thinsp;172 K onward, two-step nucleation is witnessed when the HDL state prevails and LDL is metastable.\u003c/p\u003e\u003cp\u003eThe consistent energy difference between HDL and LDL during two-step nucleation im- plies the persistent and temperature-independent structural attributes of MDL. This newly emerged state, metastable in relation to LDL and HDL, plays a pivotal role in influencing inter- facial free energy, thereby reducing the barrier height. Recently, a prominent study experimentally confirmed the existence of mid-density ice\u003csup\u003e47\u003c/sup\u003e. Furthermore, our group has corroborated the presence of MDL in confined water\u003csup\u003e29\u003c/sup\u003e.\u003c/p\u003e\u003cp\u003eA straightforward method to discern the distinction between one- and two-step nucleations is to examine the density\u003cem\u003e/\u003c/em\u003ebond orientational order parameter (\u003cem\u003eQ\u003c/em\u003e\u003csub\u003e\u003cem\u003e6\u003c/em\u003e\u003c/sub\u003e) plane in the system\u003csup\u003e48\u003c/sup\u003e. The one-step process unfolds linearly, with both density and \u003cem\u003eQ\u003c/em\u003e\u003csub\u003e\u003cem\u003e6\u003c/em\u003e\u003c/sub\u003e increasing concurrently, ne- cessitating the overcoming of a single free energy barrier. Conversely, in two-step nucleation, the system undergoes a favorable density fluctuation initially, giving rise to a precursor that, in a subsequent step, based on thermodynamic conditions, leads to either a less ordered or more ordered state (see Fig.\u0026nbsp;3b).\u003c/p\u003e\u003cp\u003eFigures 2c and 3c depict the free-energy barriers that separate HDL and LDL for the one-step nucleation process and the free energy of the MDL basins for the two-step nucleation at different system sizes. The barrier height increases with system size, following the expected \u003cem\u003eN\u003c/em\u003e2\u003cem\u003e/\u003c/em\u003e3 scaling law for a first-order phase transition in the one-step process, and adheres to the same law for the two-step process as well.\u003c/p\u003e\u003cp\u003eTo compare the structural disparities between MDL and the other two states, we present the orientational tetrahedral order (OTO) \u003cem\u003eq\u003c/em\u003e\u003csup\u003e49\u003c/sup\u003e of oxygen atoms in Fig.\u0026nbsp;3d. Neophytou \u003cem\u003eet al.\u003c/em\u003e\u0026rsquo;s\u003csup\u003e24\u003c/sup\u003e remarkable findings revealed that high-density networks were characterized by a profusion of highly entangled links and knots in the form of rings. In stark contrast, low-density networks primarily consisted of unknotted and unlinked rings. These stark differences in network structures led us to employ the OTO parameter for a more comprehensive analysis.\u003c/p\u003e\u003cp\u003eThe OTO focuses on the four nearest oxygen neighbors in water, and \u003cem\u003eq\u003c/em\u003e varies from 0 for an ideal gas to 1 for a regular tetrahedron (see Supplementary Information for details). The high- density state exhibits a lower degree of order, while the low-density state displays a higher level of structural order. The mid-density state, situated between these two extremes, represents an intermediate level of order.\u003c/p\u003e"},{"header":"Transition temperature","content":"\u003cp\u003eIn addition to the second critical point, an additional significant feature emerges along the coexistence line. The transition temperature \u003cem\u003eTt\u003c/em\u003e\u0026thinsp;=\u0026thinsp;172\u0026thinsp;\u0026plusmn;\u0026thinsp;1K marks the termination of the negative slope in the \u003cem\u003eT/P\u003c/em\u003e graph, signifying the conclusion of HDL growth. At this critical temperature, owing to the substantial energy barrier relative to the energy difference between HDL and LDL, the two-step nucleation undergoes a first-order transition from being LDL-dominant to HDL-dominant. Above and below this transition temperature, the LDL-HDL transitions remain to be first-order while corresponding to LDL- and HDL-dominant, respectively, without merging into the continuous transition where LDL and HDL states can no longer be distinguished. In this regard, this temperature may not align with the traditional definition of a critical point, even though a previous simulation study suggested the second critical point around this transition temperature\u003csup\u003e19\u003c/sup\u003e, while it exhibits similar characteristics and even adheres to the Ising universality class.\u003c/p\u003e\u003cp\u003eIn Fig.\u0026nbsp;4a, we depict the OTO for two distinct states. While the LDL state gradually becomes more ordered with decreasing temperature, the structural evolution in the HDL state appears to stagnate. There is no observable difference in the HDL state between \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;175K and \u003cem\u003eT\u003c/em\u003e\u0026thinsp;=\u0026thinsp;165K, occurring before and after the transition temperature \u003cem\u003eTt\u003c/em\u003e\u0026thinsp;=\u0026thinsp;172K.\u003c/p\u003e\u003cp\u003eThe phase diagram of the TIP4P/2005 water model within the supercooled region and the\u003c/p\u003e\u003cp\u003eno man\u0026rsquo;s land is depicted in Fig.\u0026nbsp;4b. This phase diagram, closely aligned with experimental observations, substantiates the presence of a second critical point at 238\u0026plusmn;2K in water and introduces the transition temperature at 172\u0026plusmn;1K, where stability switching based on the two-step nucleation via MDL state takes place in the system. This newfound comprehension of supercooled water\u0026rsquo;s phase behavior holds the promise of reshaping the foundational underpinnings across diverse scientific domains.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eData availability:\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe data that support the findings of this study are available from the corresponding author upon reasonable request.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eACKNOWLEDGMENTS\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis\u0026nbsp;research\u0026nbsp;was\u0026nbsp;supported\u0026nbsp;by\u0026nbsp;the\u0026nbsp;Basic\u0026nbsp;Science\u0026nbsp;Research\u0026nbsp;Program\u0026nbsp;through\u0026nbsp;the\u0026nbsp;National\u0026nbsp;Research\u0026nbsp;Foundation\u0026nbsp;of\u0026nbsp;Korea (NRF)\u0026nbsp;funded\u0026nbsp;by\u0026nbsp;the\u0026nbsp;Ministry\u0026nbsp;of\u0026nbsp;Education(2021R1I1A1A01050280)\u0026nbsp;and\u0026nbsp;KISTI (KSC-2022-CRE-0268,\u0026nbsp;KSC-\u0026nbsp;2022-CRE-0388,\u0026nbsp;KSC-2022-CRE-0469,\u0026nbsp;KSC-2023-CRE-0285).\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eS.P. performed all simulations and wrote the manuscript. K.S.K. revised the manuscript and supervised the project.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAdditional information\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eSupplementary information The online version contains supplementary material available at\u003c/p\u003e\n\u003cp\u003eCorrespondence and request for data should be addressed to Kwang S. Kim.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eSpeedy, R. J. \u0026amp; Angell, C. A. Isothermal compressibility of supercooled water and evidence for a thermodynamic singularity at -45\u0026deg;c. \u003cem\u003eThe Journal of Chemical Physics \u003c/em\u003e\u003cstrong\u003e65\u003c/strong\u003e, 851\u0026ndash;858 (1976).\u003c/li\u003e\n\u003cli\u003eAngell, C. A., Sichina, W. J. \u0026amp; Oguni, M. 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Using advanced sampling methods within the TIP4P/2005 water model, this study pinpoints the second critical point at temperature \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e= 238±2 K and pressure \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e\u003c/sub\u003e= 288±30 bar, signifying an abrupt first-order to a gradual continuous phase transition. It also reveals a transition temperature \u003cem\u003eT\u003c/em\u003e\u003csub\u003e\u003cem\u003et\u003c/em\u003e\u003c/sub\u003e=172±1 K at which a pivotal transformation unfolds, marking the stability switching of a two-step nucleation process, unveiling a previously unnoticed mid-density state bridging high- and low-density liquid states. These findings redefine our understanding of liquid-liquid phase transition, contributing to a comprehensive phase diagram for supercooled water including the elusive \"no man’s land, unravelling its intricate complexity.\u003c/p\u003e","manuscriptTitle":"Unveiling enigmatic phase transitions of water in the supercooled region and no man’s land","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-04-25 19:03:25","doi":"10.21203/rs.3.rs-3611312/v1","editorialEvents":[],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nature-communications","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"NCOMMS","sideBox":"Learn more about [Nature Communications](http://www.nature.com/ncomms/)","snPcode":"","submissionUrl":"https://mts-ncomms.nature.com/","title":"Nature Communications","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"ejp","reportingPortfolio":"Nature Communications","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"770c8db6-f4e3-4d05-8dd1-7f8dbf6d5c4d","owner":[],"postedDate":"April 25th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[{"id":26667826,"name":"Physical sciences/Chemistry/Theoretical chemistry/Molecular dynamics"},{"id":26667827,"name":"Physical sciences/Physics/Statistical physics, thermodynamics and nonlinear dynamics/Phase transitions and critical phenomena"}],"tags":[],"updatedAt":"2024-04-25T19:03:25+00:00","versionOfRecord":[],"versionCreatedAt":"2024-04-25 19:03:25","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3611312","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3611312","identity":"rs-3611312","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}