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Relation between the Widom line and the breakdown of the Stokes–Einstein relation in supercooled water

Contributed by H. E. Stanley, March 24, 2007 (received for review October 29, 2006)
Abstract
Supercooled water exhibits a breakdown of the Stokes–Einstein relation between the diffusion constant D and the alpha relaxation time τ_{α}. For water simulated with two different potentials, TIP5P and ST2, we find that the temperature of the decoupling of diffusion and alpha relaxation correlates with the temperature of the maximum in specific heat that corresponds to crossing the Widom line T _{W}(P). Specifically, we find that our results for Dτ_{α}/T collapse onto a single “master curve” if temperature is replaced by T − T _{W}(P). We further find that the size of the mobile molecule clusters (dynamical heterogeneities) increases sharply near T _{W}(P). Moreover, our calculations of mobile particle cluster size <n(t*)>_{w} for different pressures, where t* is the time for which the mobile particle cluster size is largest, also collapse onto a single master curve if T is replaced by T − T _{W}(P). The crossover to a more locally structured low density liquid (LDL) as T → T _{W}(P) appears to be well correlated both with the breakdown of the Stokes–Einstein relation and with the growth of dynamic heterogeneities. Our results are consistent with the possibility that the breakdown of the SE relation in supercooled water at low pressures is associated with the hypothesized liquid–liquid phase transition.
A 17th century study of the density maximum at 4°C (1) demonstrates the long history of water science. Since that time, dozens of additional anomalies of water have been discovered (2–5), including the sharp increase upon cooling of both the constantpressurespecific heat C _{P} and the isothermal compressibility K _{T}. These anomalies of water become more pronounced as water is supercooled. To explain these properties, a liquid–liquid (LL) critical point has been proposed (6). Emanating from any critical point there must be loci of extrema of thermodynamic response functions such as C _{P} and K _{T}. These loci must coincide as the critical point is approached, because response functions are proportional to powers of the correlation length, and the locus of the correlation length maxima asymptotically close to the critical point defines the Widom line T _{W}(P) (7).
A number of other phenomena have been correlated with T _{W}(P). Some of these phenomena are found only in experiments, such as the sharp drop in the temperature derivative of the zerofrequency structure factor or the appearance of a boson peak, both observed by quasielastic neutron scattering (QENS) (8). Others are found only in simulations, such as the crossover in the relaxation time of the fluctuations in orientational order parameter Q (P.K., S.V.B., and H.E.S., unpublished calculations) or the maximum in the temperature derivative of the number of hydrogen bonds per molecule (9). Finally, some anomalies that correlate with the Widom line are found in both experiments and simulations, such as the dynamic “fragiletostrong” crossover in the diffusion constant (7, 10–12), or the sharp drop in the temperature derivative of the mean squared displacement (7, 10, 13).
In the supercooled region of the pressure–temperature phase diagram, the dynamic properties of water show dramatic changes (14, 15). One basic relation among dynamic properties is the Stokes–Einstein (SE) relation where D is the diffusion constant, T is the temperature, k _{B} is the Boltzmann constant, η is the viscosity, and a is the effective hydrodynamic radius of a molecule. This expression provides a relation between mass and momentum transport of a spherical object in a viscous medium. The SE relation describes nearly all fluids at T ≳ (1.2–1.6) T _{g}, where T _{g} is the glass transition temperature. Because the hydrodynamic radius a is roughly constant, Dη/T is approximately independent of T (16–18). However, in most liquids, for T ≲ (1.2–1.6) T _{g}, Dη/T is no longer a constant (19–28). For the case of water, the breakdown of the SE relation occurs at ≈1.8 T _{g}, in the same temperature region in which many of the unusual thermodynamic features of water occur (5, 11, 14, 29).
Our aim is to evaluate to what degree the SE breakdown can be correlated with the presence of thermodynamic anomalies and the onset of spatially heterogeneous dynamics, and how these features relate to the location of the Widom line (22, 30–33). From prior studies of water, we can already form an expectation for the correlation between the SE breakdown and the Widom line by combining three elements: (i) the Widom line is approximately known from the extrapolated powerlaw divergence of K _{T} (34); (ii) the locus of points T _{D}(P) where D extrapolates to zero is also known, and nearly coincides with T _{W}(P) at low pressures (see Fig. 1 of ref. 12); (iii) the SE relation has been found to fail in liquids generally at the temperature T _{D}(P) (35). Combining these three results, one might not be surprised if the breakdown of the SE relation should occur near to the Widom line for P < P _{C}, and it should continue to follow T _{D}(P) for P > P _{C}. We will see that our results are consistent with this expectation, but reveal some unexpected insights.
Our results are based on molecular dynamics (MD) simulations of N = 512 waterlike molecules interacting via the TIP5P potential (36, 37), which exhibits a LL critical point at approximately T _{C} ≈ 217 K and P _{C} ≈ 340 MPa (37, 38). We carry out simulations in the isothermal isobaric ensemble for three different pressures P = 0, 100, and 200 MPa, and for temperatures T ranging from 320 K down to 230 K for P = 0 and 100 MPa, and down to 220 K for 200 MPa. We also analyze MD simulations of N = 1,728 waterlike molecules interacting via the ST2 potential (39, 40), which displays a LL critical point at T _{C} ≈ 245 K and P = 180 MPa (41). The simulations for the ST2 model are performed at fixed volume and temperature (40).
Results
We explore the possible relation between the Widom line and the breakdown of the SE relation (Eq. 1 ). To locate the Widom line, we first analyze the isobaric heat capacity C _{P} for the TIP5P (Fig. 1 a) and the ST2 (Fig. 1 b) models.
We next calculate the diffusion constant via its asymptotic relation to the meansquared displacement, where r _{j}(t) is the position of oxygen j at time t. It is difficult to accurately calculate the viscosity η in simulations, so we instead calculate the alpha relaxation time τ_{α} (which is expected to have nearly the same T dependence as η) as the time at which the coherent intermediate scattering function decays by a factor of e. Here ρ(q, t) ≡ ∑_{j} ^{N}exp[−i q·r _{j}(t)] is the Fourier transform of the density at time t, r _{j}(t) is the position of oxygen j at time t, q is a wave vector and the brackets denote an average over all q = q and many equilibrium starting configurations. We calculate F(q, t) at the value of q corresponding to the first maximum in the static structure factor F(q, 0). It is important that we use the coherent scattering function (as opposed to the incoherent, or selfscattering function), because we want to capture collective relaxation that contributes to η. We expect that τ_{α} calculated this way would behave similar to η and hence the SE relation (Eq. 1 ) can be written as We see that both τ_{α} and D for the TIP5P (Fig. 2) display rapid changes at low T.
Fig. 3 a shows Dτ_{α}/T as a function of T for the TIP5P model. At high T, Dτ_{α}/T remains approximately constant (42). At low T, Dτ_{α}/T increases, indicating a breakdown of the SE relation (1), which occurs in the same T region near the Widom line T _{W}(P). To test whether there is a correlation between the SE breakdown and T _{W}(P), we plot Dτ_{α}/T against T − T _{W}(P) (Fig. 3 b) and find the unexpected result that all of the curves for different pressures overlap within the limits of our accuracy for the TIP5P model. Hence, Dτ_{α}/T is a function only of T − T _{W}(P), from which it follows that the locus of the temperature of the breakdown of the SE relation is correlated with T _{W}(P).
Because we find a correlation between T _{W}(P) and the breakdown of the SE relation, the hypothesized connection between the SE breakdown and the onset of dynamical heterogeneities (DH) suggests a connection between T _{W}(P) and the onset of DH. To investigate the behavior of the dynamic heterogeneities, we study the clusters formed by the 7% most mobile molecules (43), defined as molecules with the largest displacements during a certain interval of time t. The clusters of the highly mobile molecules are defined as follows. If in a pair of mobile molecules determined in the interval [t _{0}, t _{0} + t], two oxygens at time t _{0} are separated by less than the distance corresponding to position of the first minimum in the pair correlation function (0.315 nm in TIP5P and 0.350 nm in ST2), this pair belongs to the same cluster.
The weightaveraged mean cluster size measures the average cluster size to which a randomly chosen molecule belongs, where <n(t)> is the numberaveraged mean cluster size. We show <n(t)>_{w} in Fig. 4 for TIP5P as a function of observation time interval t for different T at P = 0 MPa (Fig. 4 a). The behavior at higher P is qualitatively the same (Fig. 4 b). At low T, <n(t)>_{w} has a maximum at the time t* associated with the breaking of the cage formed by the neighboring molecules (see (44) and the references therein). Both the magnitude and the time scale t* of the peak grow as T decreases. At high T, this peak merges and becomes indistinguishable from a second peak with fixed characteristic time ≈0.5 ps. By evaluating the vibrational density of states, we associate this feature with a low frequency vibrational motion of the system, probably the O–O–O bending mode (45).
To probe the temperature dependence of <n(t)>_{w}, we plot the peak value <n(t*)>_{w} in Fig. 4 c as a function of T for P = 0, 100, and 200 MPa for the TIP5P model. At high T, <n(t*)>_{w} is nearly constant, because at high T, clusters result simply from random motion of the molecules. Upon cooling near the Widom line, <n(t*)>_{w} increases sharply. When <n(t*)>_{w} is plotted as a function of T − T _{W}(P) (see Fig. 4 d), we find that (similar to the behavior of Dτ_{α}/T) the three curves for P = 0, 100, and 200 MPa overlap, and it is apparent that the pronounced increase in <n(t*)>_{w} occurs for T ≈ T _{W}(P).
Finally, we consider the correspondence between DH and static structural heterogeneity in the supercritical region; this region is characterized by large fluctuations spanning a wide range of structures, from HDLlike to LDLlike. To quantify these structural fluctuations, we calculate for the TIP5P model the temperature derivative of a local tetrahedral order parameter Q (46). In Fig. 5, we show (∂Q/∂T)_{P} at different T for P = 0, 100, and 200 MPa, and find maxima (10) at T _{W}(P) (7, 47). The maxima in (∂Q/∂T)_{P} indicates that the change in local tetrahedrality is maximal at T _{W}(P), which should occur when the structural fluctuations of LDLlike and HDLlike components is largest. We see that the growth of the dynamic heterogeneity coincides with the maximum in fluctuation of the local environment. Also, because Q quantifies the orientational order, the fact that we find that (∂Q/∂T)_{P} has a maximum at approximately the same temperature where C _{P} = T(∂S/∂T)_{P} has a maximum, supports the idea that water anomalies are closely related to the orientational order present in water.
To further test whether the breakdown of the SE relation in water is associated with the Widom line, we study another model of water, ST2, which displays a LL phase transition at low temperatures (6). Fig. 6 a shows Dτ_{α}/T as a function of T for the ST2 model. At high T, Dτ_{α}/T remains approximately constant. At low T, Dτ_{α}/T increases, indicating a breakdown of the SE relation (2), which, similar to the case for the TIP5P model, occurs near the Widom line T _{W}(P). For P < P _{C}, we plot Dτ_{α}/T against T − T _{W}(P) (Fig. 6 b) and we again find, similar to the TIP5P model, all of the curves for different pressures overlap within the limits of accuracy, suggesting that Dτ_{α}/T is a function only of T − T _{W}(P).
To alternatively quantify the temperature where the SE relation breaks down, we use the observation that by plotting parametrically logD as a function of log(τ_{α}/T), one can identify the crossover temperature T _{×}(P) between the two regions by the intersection of the high T SE behavior and the low T “fractional SE behavior” D(τ_{α}/T) ^{ξ} = const (19, 40). Fig. 7 shows the locus of T _{×} in the P–T plane of the ST2 model. We confirm that the same collapse of Dτ_{α}/T can be found by replacing T with T − T _{×}(P), demonstrating (Fig. 7) that the locus of SE breakdown defined by T _{×}(P) approximately tracks T _{W}(P) for P < P _{C}. For P > P _{C}, there is no Widom line and we can see the drastic behavior change of T _{×}(P), which supports the hypothesis that the SE relation breakdown is correlated to the LL phase transition. There is also some difference between the ST2 and TIP5P models in the relative location of the breakdown of the SE relation and T _{W}(P), evidenced by the fact that the magnitude of the SE breakdown is different for ST2 at T = T _{W}(P). As a result, the results for the two models will not collapse when plotted together. Moreover, we find that Dη/T from the experimental data plotted against T − T _{W}(P) (Fig. 6 b), coincide with the ST2 results, suggesting that the data collapse likely exists for water when T is replaced by T − T _{W}(P). The experimental data for D were taken from ref. 15, and the experimental data for η were taken from refs. 48–50. The T _{W} for water at P = 1 atm is taken to be the temperature of the C _{p} ^{max} which is ≈225 K (11, 47). For P > P _{C} (when there is no Widom line), water behaves similar to simple glass forming liquids, and so we expect the breakdown of the SE relation for water to be similar to other simple liquids. Specifically, the SE relation is believed to break down at T ≈ (1.2–1.6)T _{g} (19), roughly coinciding with the temperature T _{MCT}(P), where the mode coupling description of the dynamics fails (19, 35). This has been verified for the SPC/E model of water (51), and we expect the same is true for the ST2 and TIP5P models.
To test whether there is an increase in dynamic heterogeneities at T _{W}(P) as found for the TIP5P, we show <n(t)>_{w} for different T along an isochore of density ρ = 0.83 g/cm ^{3} for the ST2 model in Fig. 8 a. We show an isochore because only isochoric data are available from ref. 41. As in the case of TIP5P, we find the emergence of a second time scale larger than 0.5 ps in <n(t)>_{w} near the crossing of the Widom line. Similarly, <n(t*)>_{w} increases near the Widom line temperature (see Fig. 8 b). Hence, the sharp growth of DH and the appearance of a second time scale in <n(t)>_{w} both occur near the Widom line. We also find that the magnitude of <n(t*)>_{w} is larger for the ST2 model than for the TIP5P model at T _{W}(P), consistent with the above observation that the breakdown of the SE relation is more pronounced for the ST2 model than for the TIP5P model.
Discussion and Summary
We have shown that the breakdown of the SE relation for P < P _{C} can be correlated with the Widom line emanating from the LL critical point. In particular, rescaling T by T − T _{W}(P) yields data collapse of Dτ_{α}/T for different pressures. Rapid structural changes occur for T near T _{W}(P), where larger LDL “patches” emerge upon cooling (52–54). The size of the dynamic heterogeneities also increases sharply as the Widom line is crossed. The breakdown of the SE relation can be understood by the fact that diffusion at low T is dominated by regions of fastest moving molecules (DH), whereas the relaxation of the system as a whole is dominated by the slowest moving molecules. Consistent with this, we find that the growth of mobile particle clusters occurs near the Widom line and also near the breakdown of the SE relation for P < P _{C}. Thus, the SE breakdown and sharp growth in dynamic heterogeneities in water are consistent with the LL critical point hypothesis (2–6). Our results are also consistent with recent experimental findings in confined water (8, 13, 14).
Acknowledgments
We thank C. A. Angell, S.H. Chen, G. Franzese, J. M. H. Levelt Sengers, S. Han, L. Liu, M. G. Mazza, F. Sciortino, M. Sperl, K. Stokely, B. Widom, L. Xu, Z. Yan, E. Zaccarelli, and especially S. Sastry for helpful discussions and the National Science Foundation Chemistry Program for support. We also thank the Boston University Computation Center, Yeshiva University, and ACEnet for allocation of computational time.
Footnotes
 ^{‡}To whom correspondence may be addressed. Email: pradeep{at}physics.bu.edu or hes{at}bu.edu

Author contributions: P.K., S.V.B., S.R.B., P.H.P., F.W.S., and H.E.S. designed research; P.K., S.V.B., S.R.B., P.H.P., F.W.S., and H.E.S. performed research; P.K., S.V.B., S.R.B., P.H.P., F.W.S., and H.E.S. contributed new reagents/analytic tools; P.K., S.V.B., S.R.B., P.H.P., F.W.S., and H.E.S. analyzed data; and P.K., S.V.B., P.H.P., F.W.S., and H.E.S. wrote the paper.

The authors declare no conflict of interest.
 Abbreviations:
 LL,
 liquid–liquid;
 SE,
 Stokes–Einstein.
 © 2007 by The National Academy of Sciences of the USA
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