Technical Comments

Comment on “Maxima in the thermodynamic response and correlation functions of deeply supercooled water”

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Science  18 May 2018:
Vol. 360, Issue 6390, eaat1634
DOI: 10.1126/science.aat1634

Abstract

Kim et al. recently measured the structure factor of deeply supercooled water droplets (Reports, 22 December 2017, p. 1589). We raise several concerns about their data analysis and interpretation. In our opinion, the reported data do not lead to clear conclusions about the origins of water’s anomalies.

The structure factor of a fluid, S(q), where q is the wave vector change, is related by Fourier transform to the pair correlation function of molecules in the fluid. In water, S(q) exhibits a rise at low q, whose amplitude increases as temperature decreases and water becomes supercooled, as observed by Kim et al. (1). This anomalous behavior is related to κT (compressibility at constant temperature T) increasing upon cooling, because S(0) = (ρ/m)kBTκT, where ρ is the liquid density, m is the molecular mass, and kB is the Boltzmann constant. The analysis in (1) involves Ornstein-Zernike formalism, which is reliable near a critical point when S(0) and κT diverge, density fluctuations are enhanced, and their correlation length ξ becomes much larger than the molecular spacing. However, in (1), S(0) and ξ remain small [S(0) < 0.1 and ξ < 0.42 nm], making ξ very sensitive to the choice of splitting S(q) between normal and anomalous components (2). Note also that ξ is not a cluster size, as figure 4B of (1) could suggest.

Kim et al. also deduced κT from S(0). Knowledge of ρ is required but is only available above 239.74 K (3). To analyze data down to ~227.7 K, Kim et al. used an extrapolation intermediate between two formulas (3, 4) [figure S8A of (1)]. Wölk and Strey (4) overestimated available ρ (3) by 2.6 kg m−3 at low temperature and their formula should not be used. The intermediate choice in (1) has a similar issue. The remaining formula (3) is a higher-quality, polynomial fit (maximum deviation 0.24 kg m−3). We tried an empirical power law, ρ0 + ρ1T + ργ(T/Ts – 1)γ, suggested by Speedy and Angell (5), where Ts and γ are adjustable parameters. The fit has the same quality as the polynomial, but it extrapolates to lower densities. For a given value of S(0), changing the extrapolation from ρ to ρ′ multiplies κT by ρ/ρ′. Using the power law with Ts between 225 and 227 K yields an extremely flat trend for κT at low temperature (Fig. 1), which, taking into account the error bars, casts doubt on the existence of a maximum.

Fig. 1 A choice of extrapolation for the density ρ different from that in Kim et al. yields a very flat isothermal compressibility κT at low temperature.

The solid lines show monotonic hyperbolic tangent functions that fit the data sets within their standard error (reduced χ2 = 0.7).

Any choice of extrapolation for the density is arbitrary. Still, if a choice yields a maximum in κT at a temperature Tm, one can discuss its statistical significance. In their supplementary materials, Kim et al. tested the null hypothesis “Measured value at Tm is lower than the values at lower temperatures” with a p-value analysis: “Given the probability densities corresponding to the measured data points, [the authors] draw a set of values and evaluate whether it fulfills the null hypothesis or not.” For the probability p to get no maximum in κT, the procedure gives p = 11.8% for H2O. However, the premise of the test is questionable. It starts by assuming that the actual κT(T) curve exhibits a maximum, and calculates the probability for an experiment to miss it. Here, instead, we first assume that the actual κT(T) curve is monotonic, and we calculate the probability that an experiment gives an artificial maximum. We have tried two ansätze for the assumed monotonic curve, which both fit the data in (1) within their standard error: (i) a parabola; (ii) a flatter function, a tanh[b(x/c – 1)] + d (Fig. 1). For each temperature reported in the experiment, we drew synthetic data from a Gaussian distribution around the ansatz function, with a standard deviation equal to the experimental standard error. We repeated the procedure for 106 data sets and measured the probability p* to find a maximum value at a temperature strictly above the lowest one. The result is p* = 35.3% for the parabolic ansatz and 61.2% for the flatter ansatz. The maximum in κT reported in (1) is not statistically significant.

We have not considered other effects, such as a possible revision of the temperature estimate (6), nor have we considered how changing the extrapolation for ρ would change the normal component [calculated from hard spheres at density ρ (1)] used to decompose S(q), which would in turn alter the results for ξ and κT. In our opinion, from the S(q) data in (1), there is no definitive evidence for or against a maximum in ξ or in κT. No conclusion can be drawn about which scenario should be preferred to explain the strange behavior of water.

An argument is also put forward in (1) to rule out the critical point–free model (7): If the evaporating droplets cross a liquid-liquid transition at ambient pressure, the authors would “expect a discontinuous change with the coexistence of two peaks in the structure factor.” We agree that a first-order phase change would be directly observable in S(q) if a phase change occurred. However, we note that when the droplet temperature crosses 273.15 K, although S(q) changes continuously, this is not sufficient to conclude that ice does not exist. What happens is that water remains liquid in a metastable state with respect to ice, beyond the line of liquid-ice equilibrium. Metastability is a characteristic feature of first-order transitions. Typical conditions to avoid nucleation of the more stable phase and promote metastability are cleanliness, small sample size, and short experimental time scales. These conditions are met by micrometer-sized droplets evaporating in vacuum. Even if they cross the location of a first-order equilibrium line (either the well-known ice-liquid line or the putative liquid-liquid line), they might therefore still remain metastable with respect to the other phase, without any abrupt change in S(q).

We conclude by mentioning the first experimental report (8) of the long-sought line of maxima in κT along isobars, overlooked by Kim et al. It was proposed (9) that such a line, possibly hidden by the line of homogeneous ice nucleation at positive pressure, might emerge in the experimentally accessible region at negative pressure. Early experiments in this region (9) found a minimum in sound velocity along an isochore, and a first equation of state for stretched water was obtained from interpolation of sound velocity (10). However, because of the limited amount of data, different interpolations of similar quality yielded curves for κT at negative pressure either monotonic or with maxima (10). Recently, more sound velocity minima were measured in samples at other densities, which further constrained the possible interpolations: They all consistently lead to a line of maxima in κT along isobars around –100 MPa and 265 K (8). We emphasize that although a κT maximum is a necessary condition for the validity of both the second critical point scenario (11) and the singularity-free interpretation (12), it is not sufficient to allow a decision between them. The exciting possibility of observing more anomalies of water at positive pressure by reaching extreme supercooling, as in (1) and (6), calls for more experiments.

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