Origin of the emergent fragile-to-strong transition in supercooled water

Significance

Upon cooling, the liquid dynamics generally slow down, with a rate that either keeps constant for a strong liquid or monotonically increases for a fragile one. However, water, silica, and some metallic liquids do not obey this general rule and the rate exhibits a maximum. This unusual phenomenon is known as the fragile-to-strong transition and its origin remains very controversial. Here we show that the fragility of water is only apparent and that it originates from a crossover between a high-temperature liquid without locally favored structures and a low-temperature one full of them, both of which have Arrhenius dynamics. Given its generality, this explanation of dynamic anomalies may be common to any liquids having a tendency to form locally favored structures.

Abstract

Liquids can be broadly classified into two categories, fragile and strong ones, depending on how their dynamical properties change with temperature. The dynamics of a strong liquid obey the Arrhenius law, whereas the fragile one displays a super-Arrhenius law, with a much steeper slowing down upon cooling. Recently, however, it was discovered that many materials such as water, oxides, and metals do not obey this simple classification, apparently exhibiting a fragile-to-strong transition far above $T_{\mathrm{g}}$. Such a transition is particularly well known for water, and it is now regarded as one of water’s most important anomalies. This phenomenon has been attributed to either an unusual glass transition behavior or the crossing of a Widom line emanating from a liquid–liquid critical point. Here by computer simulations of two popular water models and through analyses of experimental data, we show that the emergent fragile-to-strong transition is actually a crossover between two Arrhenius regimes with different activation energies, which can be naturally explained by a two-state description of the dynamics. Our finding provides insight into the fragile-to-strong transition observed in a wide class of materials.

A fragile-to-strong transition, or a dynamic crossover from fragile to strong behavior, has attracted considerable attention in the water and glass science community. Historically, the first example of such a transition was discovered in a supercooled state of water (1, 2). Although direct observation of such a transition in bulk water was preempted by ice crystallization, many pieces of evidence have accumulated in confined (3, 4), low-density amorphous (5, 6), high-density amorphous (6), and vapor-deposited amorphous water (5), which consistently show Arrhenius behavior near $T_{\mathrm{g}}$. These observations, albeit indirectly, strongly support the existence of a fragile-to-strong transition in deeply supercooled water.

Several scenarios have been proposed to account for this transition. We mention here the three most relevant ones for our discussion. The first one was proposed by Angell and coworkers (1, 2, 7), on the basis of experimental observations showing that water is fragile above the homogeneous nucleation temperature $T_{\mathrm{H}}$, while strong near the glass transition temperature $T_{\mathrm{g}}$. It ascribes water’s dynamic crossover to the glass transition, based either on mode-coupling theory (MCT) (8) or on Adam–Gibbs theory (2, 7). However, this scenario does not explain why the dynamical slowing down appears at a temperature much higher than the glass transition temperature $T_{\mathrm{g}}\sim 136$ K (9). This limitation was circumvented by the introduction of the Widom line ($T_{\mathrm{W}}$) by Stanley and coworkers (10). The Widom line is the name given to a family of lines emanating from the liquid–liquid critical point (11–13) where thermodynamic susceptibilities are maximized (10). This scenario explains the dynamic anomalies with a crossover from the power-law behavior of high-density liquid water to the Arrhenius behavior of low-density liquid water at the Widom line. Finally, a two-state scenario was proposed by Tanaka (14, 15) on the basis of the existence of two types of local structures (two states) in liquid water, which interprets the fragile-to-strong transition as a dynamic crossover from the high-temperature disordered state to the low-temperature ordered state, via a mixed state of the two. The advantages of the two-state description are that (i) it can explain both thermodynamic and dynamic anomalies in the same framework and (ii) it provides an account of water anomalies that does not hinge on the power-law divergences of the glass transition or the second critical point, while still being able to accommodate them.

The fragile-to-strong transition provides a comprehensive description of water’s dynamic anomalies, but its physical origin remains elusive. This is because all three scenarios provide a reasonably good description above $T_{\mathrm{W}}$ and predict different behavior only below $T_{\mathrm{W}}$. New measurements and simulations in this deeply supercooled regime are thus needed to understand the nature of the fragile-to-strong transition in liquid water.

Great progress on the detection of liquid water in the deeply supercooled regime has been made recently. For example, the maximum of isothermal compressibility, known as the Widom line, has been observed at 229.2 K in water droplets by femtosecond X-ray scattering (16). Moreover, the diffusion coefficient of water has been determined inside the “no man’s land,” through a measurement of the ice growth rate of an $\sim 7$-nm-thick water film on a polycrystalline ice substrate (17). In this article, via extensive computer simulations of two water models and analyses of experimental data below $T_{\mathrm{W}}$, we provide evidence strongly supporting a microscopic two-state scenario, while running against the scenarios based on either the glass transition or the Widom line for the fragile-to-strong transition in supercooled water.

In the two-state scenario, water can be regarded as a dynamical mixture of two types of local structures (or two states), whose fraction changes with temperature and pressure, accounting for the thermodynamic anomalies of water (14, 15, 18–26). Evidence supporting the existence of two states in liquid water has been found by Raman spectroscopy (27–29), femtosecond mid-IR pump-probe spectroscopy (30), time-resolved optical Kerr effect spectroscopy (31), X-ray absorption (32), and emission spectroscopies (33). Despite the great success of the two-state model for water’s thermodynamic anomalies, its relevance for dynamical properties is far from clear. Recently, Singh et al. (34) reported a good fitting of a two-state model to water’s dynamic properties. However, this work and previous two-state models for thermodynamics were based on a phenomenological two-state description of physical quantities and lack of microscopic support: In other words, the basic two states were unidentified at a molecular level and their presence was presupposed.

With the help of a microscopic structural descriptor $\zeta$ (19), two types of local structures (high-temperature disordered state denoted by $\rho$ and low-temperature ordered state denoted by $S$ hereafter) were successfully detected from a bimodal distribution of $\zeta$, providing a solid microscopic basis for a two-state description of water (35) (SI Appendix, Fig. S1). Typical configurations of $\rho$ and S states are given in Fig. 1 A and B, respectively. A good correlation can be found between $\zeta$ (Fig. 1C) and local density (Fig. 1D), which underlies a two-state description of water’s density anomaly.

Fig. 1.

Water’s local structures and their correlations to local density and mobility. ($A$ and $B$) Typical snapshots of water’s local structures for $\rho$ and $S$ states. The dotted red lines represent H bonds and the solid blue lines show the tetrahedral structure. ($C$–$F$) Different fields for a sample configuration for the TIP5P water at 0.1 MPa and 250 K. All molecules (only oxygen atoms are shown) are colored in red if the field is low or blue if it is high. ($C$) $\zeta$ field. ($D$) Inverse local density field ($6-N_{\mathrm{f}\mathrm{s}}$, where $N_{\mathrm{f}\mathrm{s}}$ is the number of first-shell neighbors). ($E$) ${\zeta }^{\mathrm{C}\mathrm{G}}$ field. ($F$) Inverse mobility field ($1-\mathrm{\Delta }{r}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left({\tau }_4\right)$), where $\mathrm{\Delta }{r}_{\mathrm{m}\mathrm{a}\mathrm{x}}\left({\tau }_4\right)$ is the maximum distance one molecule travels during a time period of ${\tau }_4$ (${\tau }_4$: dynamic heterogeneity time scale) (SI Appendix, sections VI and VII). In $C$–$E$ a simple low-pass filter by performing a rolling time average of each field over ${\tau }_4$ is used to remove thermal fluctuations. In $C$, $D$ and $E$, $F$, $58\%$ and $26\%$ of molecules with a higher field are colored in blue, respectively.

Here we also find that the molecular mobility (Fig. 1F) is correlated “not” to $\zeta$, but instead to its coarse-grained version ${\zeta }^{\mathrm{C}\mathrm{G}}$ (Fig. 1E) (SI Appendix, section II). This result is confirmed by a good correlation between ${\zeta }^{\mathrm{C}\mathrm{G}}$ and mobility (SI Appendix, Fig. S2), suggestive of a dynamic bimodality: The slow water has larger ${\zeta }^{\mathrm{C}\mathrm{G}}$ than the fast one. This finding can be understood from the fact that, while the structure in a fluid is a local property, the dynamics of a water molecule are instead intrinsically coupled to those of its neighbors (nonlocal). The structural and dynamical bimodalities, linked via spatial coarse graining, together provide a microscopic structural basis for a unified description of water’s thermodynamic and dynamical anomalies.

Under the following two assumptions, (i) any pure state (either fast or slow water state) follows an Arrhenius law and (ii) the lifetime of a state is shorter than the typical dynamical timescale (e.g., rotational time), a dynamical quantity $X$ can be expressed by a generalized Arrhenius law as (14, 15)

$$X=X_0\exp \left(\frac{E_{\mathrm{a}}^{\rho ,X}+\mathrm{\Delta }{E}_{\mathrm{a}}^X\cdot s^{\mathrm{D}}}{k_{\mathrm{B}}T}\right),$$ [1]

where $E_{\mathrm{a}}^{S,X}$ and $E_{\mathrm{a}}^{\rho ,X}$ are the activation energies for the slow and fast water, respectively, $\mathrm{\Delta }{E}_{\mathrm{a}}^X=E_{\mathrm{a}}^{S,X}-E_{\mathrm{a}}^{\rho ,X}$ is the activation energy difference, $X_0$ is the prefactor, $s^{\mathrm{D}}$ is the fraction of slow water, and $k_{\mathrm{B}}$ is the Boltzmann constant. The fraction $s^{\mathrm{D}}$, following two-state behavior (SI Appendix, Fig. S3), can be empirically described by a two-state equation (14, 15, 18, 35, 36) as

$$s^{\mathrm{D}}=\frac{1}{1+\exp \left(\frac{\mathrm{\Delta }{E}^{\mathrm{D}}-T\mathrm{\Delta }{\sigma }^{\mathrm{D}}+P\mathrm{\Delta }{V}^{\mathrm{D}}}{k_{\mathrm{B}}T}\right)},$$ [2]

where $\mathrm{\Delta }{E}^{\mathrm{D}}$, $\mathrm{\Delta }{\sigma }^{\mathrm{D}}$, and $\mathrm{\Delta }{V}^{\mathrm{D}}$ are fitting parameters. At ambient pressure, the term $P\mathrm{\Delta }{V}^{\mathrm{D}}$ in Eq. 2 is negligible.

We note that Singh et al. (34) have recently proposed a different two-state model by assuming that fast water follows Vogel–Fulcher–Tammann (VFT) instead of Arrhenius behavior. This analysis is based on the presence of a critical point at low pressure ($T_c=228.2$ K and $P_c=0$ MPa) (37) and a divergence of the dynamics at a finite temperature ($T=147.75$ K). This model seems inconsistent with experimental measurements that found neither a critical point (16, 38) nor divergent dynamics (5, 6, 17) at ambient pressure.

Fig. 2 A and B (Upper) shows water’s reorientational time ${\tau }_2$ (squares) and inverse diffusion coefficient $1/D$ (circles) down to $\sim 30$ K below the Widom line in TIP5P and ST2 water at 0.1 MPa. The dramatic slowing down of dynamics by $\sim 7$ and $\sim 4$ orders of magnitude for TIP5P and ST2 water, respectively, can be nicely described by Eqs. 1 and 2 (solid curves). We note that the same data can also be well described by the Widom line scenario for the dynamics (SI Appendix, Figs. S4 and S5). By the Widom line scenario, we refer to the description of the apparent fragile-to-strong transition as a power-law–Arrhenius crossover at $T_{\mathrm{W}}$. The two-state scenario is consistent with the presence of the Widom line itself. However, the two-state scenario and the Widom line scenario make very different predictions for the dynamics.

Fig. 2.

(A–C) Two-state scenario for the fragile-to-strong transition in TIP5P ($A$), ST2 ($B$), and real water ($C$). (A–C, Upper) The reorientational time ${\tau }_2$ (in picoseconds) and inverse diffusion coefficient $1/D$ (in $10^6\mathrm{s}{\mathrm{m}}^{-\mathrm{2}}$) in log scale. The solid curves are fits to the two-state model. (A–C, Lower) The curvature $K$ [in units of 160 (kcal/mol)² for models and 40 (kcal/mol)² for real water], the rate $R$ (in units of 40 kcal/mol for models and 25 kcal/mol for real water) of $\ln \left(D/D_0\right)$, and the fraction $s^{\mathrm{D}}$ of slow water as a function of inverse temperature. In $A$ and $B$, ${\kappa }_T$ and $C_P$ maximum lines were taken from ref. 10. In $C$, the ${\kappa }_T$ maximum line was taken from ref. 16, black circles are from ref. 17, and red squares are from ref. 39. We can see by comparing $A$–$C$ that the width of the dynamic transition is much narrower for water models than for real water. This originates from the fact that these water models, although they capture the essential features of real water, have a much stronger tendency to form locally favored structures than real water, as can be seen in the two-state model parameters listed in SI Appendix, Tables S1–S3.

In the Widom line scenario, water’s dynamics follow a power law above $T_{\mathrm{W}}$ (in agreement with MCT) and an Arrhenius law below $T_{\mathrm{W}}$ (8, 10, 40). Although experimental measurements of diffusion (41, 42), viscosity (41, 42), and relaxation time (41, 43, 44) support a power-law behavior of liquid water above $T_{\mathrm{W}}$ at low pressure, there are at least three difficulties with this interpretation: (i) The relation between the experimental power-law divergence temperature ($T_{\mathrm{M}\mathrm{C}\mathrm{T}}$) and $T_{\mathrm{g}}$, $T_{\mathrm{M}\mathrm{C}\mathrm{T}}\simeq 1.6T_{\mathrm{g}}$ (42), violates the empirical rule $T_{\mathrm{M}\mathrm{C}\mathrm{T}}\simeq 1.2T_{\mathrm{g}}$ found in other glass formers; (ii) the large difference between $T_{\mathrm{g}}$ and $T_{\mathrm{W}}$ causes unrealistic prediction of either a too long relaxation time at $T_{\mathrm{g}}=136$ K or a too high glass transition temperature ($T_{\mathrm{g}}/T_{\mathrm{m}}\simeq 0.65\sim 0.7$) for water models (SI Appendix, section V); and (iii) the Widom line scenario cannot explain the new experimental diffusion data below $T_{\mathrm{W}}$ (SI Appendix, Fig. S6). On the other hand, the two-state scenario not only gives reasonable reorientational timescales ($10^3$ s for TIP5P and 10 s for ST2) and diffusion coefficients at $T_{\mathrm{g}}$, but also provides a quantitative description of experimental diffusion data over a wide temperature range (126 K $<T<298$ K) (Fig. 2C).

From Eqs. 1 and 2, we can define a rate $R$ (first derivative) and a curvature $K$ (second derivative) of $\ln \frac{X}{X_0}$ with respect to $\beta \equiv \frac{1}{k_{\mathrm{B}}T}$ by

$$\begin{array}{ccc}\hfill R=\frac{\partial \ln \frac{X}{X_0}}{\partial \beta }& =\hfill & E_{\mathrm{a}}^{\rho ,X}+\beta \mathrm{\Delta }{E}_{\mathrm{a}}^X\mathrm{\Delta }{E}^{\mathrm{D}}\left[{\left(s^{\mathrm{D}}-\frac{1}{2}+\frac{1}{2\beta \mathrm{\Delta }{E}^{\mathrm{D}}}\right)}^2\right.\hfill \\ \hfill & \hfill & -\left.{\left(\frac{1}{2}-\frac{1}{2\beta \mathrm{\Delta }{E}^{\mathrm{D}}}\right)}^2\right],\hfill \end{array}$$ [3]

$$\begin{array}{ccc}\hfill K=\frac{{\partial }^2\ln \frac{X}{X_0}}{\partial {\beta }^2}& =\hfill & 2\beta \mathrm{\Delta }{E}_{\mathrm{a}}^X{\left(\mathrm{\Delta }{E}^{\mathrm{D}}\right)}^2{s}^{\mathrm{D}}\left(s^{\mathrm{D}}-1\right)\hfill \\ \hfill & \hfill & \times \left(s^{\mathrm{D}}-\frac{1}{2}+\frac{1}{\beta \mathrm{\Delta }{E}^{\mathrm{D}}}\right).\hfill \end{array}$$ [4]

The curves of $s^{\mathrm{D}}$, $R$, and $K$ are plotted in Fig. 2 A–C, Lower. The curvature $K$, as a measure of deviation from Arrhenius behavior, goes to zero when $s^{\mathrm{D}}\to 0$ or $s^{\mathrm{D}}\to 1$, suggesting an Arrhenius behavior of water in the one-state regimes. At $0<s^{\mathrm{D}}<1$ (two-state regime), $K$ exhibits two peaks (positive and negative), indicative of two types of non-Arrhenius behaviors (convex and concave), as shown in Fig. 2 A–C, Upper. Here we define the temperatures at half height of the two peaks of $K$ by $T_{\mathrm{m}\mathrm{i}\mathrm{x}}^{+}$, $T_{\mathrm{d}\mathrm{s}}^{+}$, $T_{\mathrm{d}\mathrm{s}}^{-}$, and $T_{\mathrm{m}\mathrm{i}\mathrm{x}}^{-}$ from high to low temperature.

With these characteristic temperatures we define three regimes (Fig. 2): the one-state (fast water) regime (red region) at $T>T_{\mathrm{m}\mathrm{i}\mathrm{x}}^{+}$, the two-state regime (light green region) at $T_{\mathrm{m}\mathrm{i}\mathrm{x}}^{-}<T<T_{\mathrm{m}\mathrm{i}\mathrm{x}}^{+}$, and the one-state (slow water) regime (blue region) at $T<T_{\mathrm{m}\mathrm{i}\mathrm{x}}^{-}$. Within the two-state regime we can also identify a specific band, i.e., the dynamic Schottky (DS) band (dark green region) at $T_{\mathrm{d}\mathrm{s}}^{-}<T<T_{\mathrm{d}\mathrm{s}}^{+}$. Eq. 4 tells us that the dynamics of water should show a crossover from an Arrhenius behavior ($K\sim 0$ and constant $R$) in the fast-water dominant regime ($s^{\mathrm{D}}\sim 0$) to another Arrhenius behavior ($K\sim 0$ and constant $R$) in the slow-water dominant regime ($s^{\mathrm{D}}\sim 1$). As shown in Fig. 2C, the diffusion coefficient measured in bulk water indeed shows such crossover behavior, in agreement with measurements in confined (3, 4) and amorphous water (5, 6). The second Arrhenius behavior is usually called strong behavior, according to Angell’s scheme (45).

In the narrow DS band, $s^{\mathrm{D}}\sim \frac{1}{2}$, $R$ maximizes (Eq. 3), and $K\to 0$ (Eq. 4), indicative of an Arrhenius behavior inside the two-state regime (Fig. 2). Importantly, the DS band is located at $\sim 20$ K below $T_{\mathrm{W}}$. The Arrhenius behavior and the maximum of $R$ explain why the Widom line scenario predicts an apparently “strong” behavior with a too large activation energy below $T_{\mathrm{W}}$ (SI Appendix, Figs. S4–S6). However, from Eqs. 3 and 4, we know that the apparent strong behavior in the DS band originates from the maximal rate of dynamic slowing down upon cooling and is fundamentally different from the inherent strong behavior in the slow-water dominant regime.

We now show that the presence of dynamic bimodality can be directly inferred from the study of dynamic heterogeneities. The four-point susceptibility ${\chi }_4\left(t\right)$ is a measure of the fluctuations of dynamics, i.e., dynamic heterogeneities (SI Appendix, section VI). ${\chi }_4\left(t\right)$ has a maximum at a dynamical timescale ${\tau }_4$ (SI Appendix, Figs. S7–S10). For normal glass-forming liquids, the maximum ${\chi }_4\left({\tau }_4\right)$ increases monotonically when approaching the glass transition temperature $T_{\mathrm{g}}$ upon cooling. Contrary to this glass phenomenology, for TIP5P and ST2 water both translational and rotational susceptibilities ${\chi }_4^{\mathrm{T},\mathrm{R}}\left({\tau }_4\right)$ maximize near $T_{s^{\mathrm{D}}=\frac{1}{2}}$ in the DS band where the system is half fast and half slow water (Fig. 3), similar to the maximization of the thermodynamic response functions near the Widom line (10). The maximization of dynamic heterogeneity can also be seen from the behavior of the stretching parameter $\beta$ (SI Appendix, section IV), which describes the deviation of molecular dipole reorientation from the Debye process ($\beta =1$). Non-Debye behavior ($\beta <1$) is usually attributed to heterogeneous dynamics. Fig. 3 shows that $\beta$ minimizes at $T_{s^{\mathrm{D}}=\frac{1}{2}}$, again confirming the maximization of dynamic heterogeneity in the DS band. We argue that this is a unique feature of the two-state scenario, which is known as the Schottky anomaly (19, 46). We stress that this feature cannot be explained by the scenarios based on the glass transition, where ${\chi }_4\left({\tau }_4\right)$ usually increases monotonically when approaching $T_{\mathrm{g}}$, and thus in principle no ${\chi }_4\left({\tau }_4\right)$ maximization is expected to occur above $T_{\mathrm{g}}$. Moreover, the observed maximization occurs at $\sim 20$ K below the Widom line, which indicates a significant difference between thermodynamic and dynamic fluctuations, in agreement with the crucial role of coarse graining that we found in Fig. 1 and SI Appendix, Figs. S2 and S3.

Fig. 3.

(A and B) Dynamic heterogeneity in TIP5P ($A$) and ST2 ($B$) water. Both translational (black circles) and rotational (red squares) four-point susceptibilities ${\chi }_4^{\mathrm{T},\mathrm{R}}\left({\tau }_4\right)$ maximize around $T_{s^{\mathrm{D}}=\frac{1}{2}}$ in the DS band (dark green region), providing strong evidence of the two-state behavior. The stretching parameter $\beta$ (blue triangle), which is determined by fitting the stretched exponential function to the second Legendre polynomial of the time correlation function of the molecular dipole moment (SI Appendix, section IV), also shows a minimum around $T_{s^{\mathrm{D}}=\frac{1}{2}}$ in the DS band. The locations of heat capacity maxima of the two water models are indicated by arrows.

It was shown that ${\chi }_4$ may suffer from finite size effects in a simulation of 1,000 particles, if the correlation length exceeds 6-particle size (47). Here we calculated the dynamic correlation length ${\xi }_4$ from the spatial correlation function of dynamic heterogeneity (48) (SI Appendix, section VIII) in TIP5P water (SI Appendix, Fig. S12). A maximum correlation length of $\sim 6$ Å or $\sim 2$ molecular size was found at $T_{s^{\mathrm{D}}=\frac{1}{2}}$ (SI Appendix, Fig. S13). This agrees with a maximal structural correlation length of $\sim 4$ Å at 229.2 K that was reported recently by femtosecond X-ray scattering (16). This short correlation length confirms that the finite size effects, if any, should be negligible in our systems. We also performed several independent microsecond simulations to check the sample-to-sample fluctuations at low temperatures (SI Appendix, Table S4), which provide the error bars in Fig. 3. Clearly the maximization of ${\chi }_4$ that we found in Fig. 3 is not a consequence of statistical errors.

Despite the very long simulation times (SI Appendix, Table S4), due to the significant increase of the structural relaxation time, the lowest-temperature data still suffer from large statistical fluctuations. Nevertheless, we can see the clear increase of ${\chi }_4\left({\tau }_4\right)$ at the lowest temperature. We speculate that this may reflect a general tendency of the dynamic susceptibility to monotonically increase upon cooling. Such behavior is known even for a system with only local dynamics, obeying the Arrhenius behavior, which may be the case for water (Eq. 1) (49). We note that the temperature is still too far away from the glass transition temperature $T_{\mathrm{g}}$ to have glassy dynamical heterogeneity. In relation to this, it should be noted that the two-state model itself cannot provide information on glassy behavior.

In normal glass-forming liquids, rotational motion decouples from translational diffusion below $\sim 1.2T_{\mathrm{g}}$ (50), which is known as the breakdown of the Stokes–Einstein and Stokes–Einstein–Debye relations ($D{\tau }_2=2a^2/9$, where $a$ is an effective hydrodynamic radius) and which is believed to be a consequence of glassy dynamic heterogeneity. However, as shown in Fig. 4, the breakdown happens much earlier ($\sim 2T_{\mathrm{g}}$) than $1.2T_{\mathrm{g}}$ for TIP5P and ST2 water, which along with the abnormally high MCT temperature $T_{\mathrm{M}\mathrm{C}\mathrm{T}}\simeq 1.6T_{\mathrm{g}}$ (42) brings into question the glass transition scenario. Here we show that this anomalous behavior can also be naturally explained by the two-state scenario. Applied to diffusive and rotational motions, Eq. 1 gives another explanation for the Stokes–Einstein–Debye relation by

$$\begin{array}{ccc}\hfill D{\tau }_2& =\hfill & D_0{\tau }_0\exp \left[\frac{E_{\mathrm{a}}^{\rho ,\tau }-E_{\mathrm{a}}^{\rho ,D}}{k_{\mathrm{B}}T}\right]\cdot \exp \left[\frac{\mathrm{\Delta }{E}_{\mathrm{a}}^{\tau }-\mathrm{\Delta }{E}_{\mathrm{a}}^D}{k_{\mathrm{B}}T}\cdot s^{\mathrm{D}}\right]\hfill \\ \hfill & \approx \hfill & D_0{\tau }_0\exp \left[\frac{\mathrm{\Delta }{E}_{\mathrm{a}}^{\tau }-\mathrm{\Delta }{E}_{\mathrm{a}}^D}{k_{\mathrm{B}}T}\cdot s^{\mathrm{D}}\right].\hfill \end{array}$$ [5]

The second equation in Eq. 5 is valid only if the activation energies for rotation and diffusion are equal in the fast-water dominant state ($E_{\mathrm{a}}^{\rho ,\tau }\approx E_{\mathrm{a}}^{\rho ,D}$); i.e., rotation is coupled to diffusion. This is true, since the Stokes–Einstein–Debye relation holds at high temperature, where $s^{\mathrm{D}}\sim 0$. This is also confirmed by our fitting result (SI Appendix, Tables S1 and S2). In the fast-water dominant state ($T>T_{\mathrm{m}\mathrm{i}\mathrm{x}}^{+}$, red in Fig. 4), translational motion couples to reorientation. However, the activation energy for reorientation becomes considerably higher than translation in the slow-water dominant state (SI Appendix, Tables S1 and S2), so the reorientation will slow down much faster than translation upon cooling, which leads to the breakdown of the Stokes–Einstein–Debye relation (Eq. 5). It can be seen clearly in Fig. 4 that the fast-water dominant state follows the Stokes–Einstein–Debye relation quite well, and the decoupling behavior can be perfectly described by the prediction of the two-state model (Eq. 5) without any adjustable parameter, indicating that the anomalous breakdown mainly comes from the growth of the slow-water dominant state upon cooling and not from glassiness.

Fig. 4.

(A and B) Breakdown of the Stokes–Einstein–Debye relation in TIP5P ($A$) and ST2 ($B$) water. Solid and dotted lines represent the two-state and individual fast-water contributions, respectively, indicating that the growth of slow water upon cooling results in the breakdown of the Stokes–Einstein–Debye relation in supercooled water. The effective hydrodynamic radius $a=1.3$ Å and 1.2 Å was estimated from the high-temperature data for TIP5P and ST2 water, respectively.

Finally, we compare the temperature dependence of water’s diffusion coefficient with a typical fragile liquid ($o$-Terphenyl) and a strong liquid ($\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{2}}$) in Fig. 5. The two-state model quantitatively describes water’s fragile-to-strong transition as a crossover from one Arrhenius to another Arrhenius behavior. Water’s dynamical slowing down starts much farther above $T_{\mathrm{g}}$ than the case of a normal fragile liquid, $o$-Terphenyl. This feature, along with the abnormally high MCT temperature $T_{\mathrm{M}\mathrm{C}\mathrm{T}}\simeq 1.6T_{\mathrm{g}}$ and the early breakdown of the Stokes–Einstein–Debye relation at $\sim 2T_{\mathrm{g}}$, strongly suggests an apparently fragile behavior of water essentially different from that of normal fragile glass formers. The former comes from the formation of two states (fast and slow water) below $T_{\mathrm{m}\mathrm{i}\mathrm{x}}^{+}\simeq T_{\mathrm{M}}\simeq 2T_{\mathrm{g}}$, while the latter originates from the glass transition at $T_{\mathrm{g}}$. These features, together with the failure of the power law at high pressure (43) and the maximization of dynamic heterogeneity, provide strong evidence for the two-state scenario.

Fig. 5.

Angell plot of experimental liquid diffusivities. Water’s diffusion slows down rapidly when entering the two-state mixture regime ($T_{\mathrm{m}\mathrm{i}\mathrm{x}}^{+}>T>T_{\mathrm{m}\mathrm{i}\mathrm{x}}^{-}$) from high temperature, showing its apparent fragile nature far above $T_{\mathrm{g}}$ compared to a normal fragile liquid, $o$-Terphenyl. Near the Widom line (${\kappa }_T$ max), water’s diffusion apparently follows the Arrhenius law (curvature $k\simeq 0$), but deviates from the typical strong behavior of $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{2}}$. Water shows its inherent strong nature like $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{2}}$ when leaving the two-state mixture regime and entering the slow-water dominant regime ($T<T_{\mathrm{m}\mathrm{i}\mathrm{x}}^{-}$). ${\kappa }_T$ maximum line of water was taken from ref. 16, oxygen diffusivity in $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{2}}$ from ref. 51, $o$-Terphenyl diffusivity from refs. 52 and 53, and water diffusivity from refs. 17 and 39.

The two-state scenario predicts a DS band, where dynamic heterogeneity maximizes and the dynamics apparently obey the Arrhenius law. This feature naturally explains the observations of a strong behavior of water just below the Widom line. However, here we have shown that this apparent strong behavior, as a two-state feature, is fundamentally different from water’s inherent strong nature near $T_{\mathrm{g}}$, as can be seen from its large deviation from the typical strong behavior of $\mathrm{S}\mathrm{i}{\mathrm{O}}_{\mathrm{2}}$ at the Widom line (${\kappa }_T$ maximum line) in Fig. 5. We also note that the calculated maximal dynamic correlation length of $\sim 6$ Å together with a maximal structural correlation length of $\sim 4$ Å reported recently by femtosecond X-ray scattering (16) supports the two-state scenario based on local structural ordering, but runs against scenarios relying on an extended length scale from either a glass transition or a critical point, at least at ambient pressure.

Water-like dynamic anomalies in the form of fragile-to-strong transitions, which were pioneered by Angell and coworkers (1, 2, 54), have been seen in many glass formers such as tetrahedral liquids (e.g., refs. 55 and 56) and metallic liquids (e.g., ref. 57), and they are also located far above the glass transition temperature, as in the case of water. On noting that many of these liquids have a tendency to form locally favored structures in the form of tetrahedral or icosahedral structures, we argue that these behaviors may also be caused by a similar two-state feature originating from local structural ordering (36), instead of glassiness. We hope that our work will initiate further research along this direction.