## 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 κ* _{T}* (compressibility at constant temperature

*T*) increasing upon cooling, because

*S*(0) = (ρ/

*m*)

*k*

_{B}

*T*κ

*, where ρ is the liquid density,*

_{T}*m*is the molecular mass, and

*k*

_{B}is the Boltzmann constant. The analysis in (

*1*) involves Ornstein-Zernike formalism, which is reliable near a critical point when

*S*(0) and κ

*diverge, density fluctuations are enhanced, and their correlation length ξ becomes much larger than the molecular spacing. However, in (*

_{T}*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 κ* _{T}* 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, ρ

_{0}+ ρ

_{1}

*T*+ ρ

_{γ}(

*T*/

*T*

_{s}– 1)

^{γ}, suggested by Speedy and Angell (

*5*), where

*T*

_{s}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 κ

_{T}by ρ/ρ′. Using the power law with

*T*

_{s}between 225 and 227 K yields an extremely flat trend for κ

*at low temperature (Fig. 1), which, taking into account the error bars, casts doubt on the existence of a maximum.*

_{T}Any choice of extrapolation for the density is arbitrary. Still, if a choice yields a maximum in κ* _{T}* at a temperature

*T*

_{m}, one can discuss its statistical significance. In their supplementary materials, Kim

*et al*. tested the null hypothesis “Measured value at

*T*

_{m}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 κ

*, the procedure gives*

_{T}*p*= 11.8% for H

_{2}O. However, the premise of the test is questionable. It starts by assuming that the actual κ

*(*

_{T}*T*) curve exhibits a maximum, and calculates the probability for an experiment to miss it. Here, instead, we first assume that the actual κ

*(*

_{T}*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 10

^{6}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 κ

*reported in (*

_{T}*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 κ* _{T}*. In our opinion, from the

*S*(

*q*) data in (

*1*), there is no definitive evidence for or against a maximum in ξ or in κ

*. No conclusion can be drawn about which scenario should be preferred to explain the strange behavior of water.*

_{T}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 κ* _{T}* 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 κ

*at negative pressure either monotonic or with maxima (*

_{T}*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 κ

*along isobars around –100 MPa and 265 K (*

_{T}*8*). We emphasize that although a κ

*maximum is a necessary condition for the validity of both the second critical point scenario (*

_{T}*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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