Isotope effects in supercooled H₂O and D₂O and a corresponding-states-like rescaling of the temperature and pressure

Water is an unusual liquid that has been extensively investigated for over a century.^1,2 Early work by Angell, Speedy and co-workers, which showed that many of water’s anomalous properties are enhanced upon supercooling, perhaps signaling a singularity, has generated tremendous ongoing interest in this area.^3–11 The results of numerous experiments, theories, models, and simulations on supercooled water have been the subject of excellent reviews.^1,10,12–14 Currently, two related theories, the liquid–liquid critical point (LLCP) hypothesis and the singularity-free scenario, have the most experimental support.^1,15–20 The LLCP hypothesis proposes that at low temperatures and high pressures, water has two thermodynamically distinct (metastable) liquid phases, typically called the high- and low-density liquid or HDL and LDL, respectively, that are separated by a first-order phase transition. The HDL–LDL coexistence line ends in a critical point—the LLCP. In that case, water beyond the critical point is an inhomogeneous mixture of two locally favored structures.^21,22 For the singularity-free scenario, there are still two locally favored structures that have different dependences on temperature and pressure, but these never lead to phase separation.

If water has an LLCP, then it belongs to the 3D Ising model universality class.^23–25 Several classical water models have been rigorously shown to have an LLCP.^14,26,27 Furthermore, various two-state models based on the physics associated with an LLCP can reproduce most of the available experimental data over a wide range of temperatures and pressures.^{24,25,28–30} One of those models is also the basis for the recommended equation-of-state (EoS) for supercooled water by the International Association for the Properties of Water and Steam (IAPWS).²⁵ 

Because the universal scaling associated with a critical point is in addition to a “normal” (non-diverging) component, experiments typically need to be done very close to the critical point to unambiguously observe the universal scaling behavior.²⁵ However, even far away from a critical point, a corresponding behavior is observed for many fluids (to a greater or lesser extent, depending on the fluid).^31–34 More generally, Pitzer³¹ and Guggenheim³⁴ demonstrated the conditions necessary for “perfect” liquids to exhibit corresponding states and began the (ongoing) discussion of the deviations from this behavior expected for real liquids.^32,33 In Pitzer’s formulation, the Helmholtz free energy is a universal function, F = F(T/A, V/$R 0 3$⁠), where A and R₀ are a characteristic energy and length scale associated with the molecular interaction potential, respectively. He noted that it is convenient, but not essential, to choose the liquid–vapor critical point (LVCP) temperatures, T_c, and volumes, V_c, as the scale parameters.

For water, the experimental data are relatively far from a putative LLCP, so assessing if it follows the scaling behavior for the 3D Ising model is challenging. Conversely, using the experimental observations to predict the location of any possible singularity is also challenging—a point that was made even in the initial work of Speedy and Angell.⁶ As discussed below, we are interested in the isotopes of water and how a possible second critical point influences their properties. It is important to note that we are not concerned with the universal power law scaling expected in the immediate vicinity of an LLCP. Instead, we are interested in investigating the corresponding states (in Pitzer’s sense) for the isotopes over a wider range of temperatures and pressures. It was noted early on that while various properties of H₂O suggested a singularity at ∼228 K, the corresponding results for D₂O indicated a singularity at ∼233 K.¹⁰ Analysis of the melting curves of H₂O and D₂O led to similar conclusions.³⁵ Subsequent work suggested that shifting the temperature scale for D₂O by the difference in temperature of maximum density, δT_MD ≅ 7.2 K, between D₂O and H₂O (at atmospheric pressure) resulted in corresponding states for densities of the two isotopes.^36,37 However, because a 7.2 K temperature shift was less successful for other properties, in practice, δT_MD came to be used as an adjustable parameter without any specific physical significance associated with it. Instead of shifting the temperatures to match the T_MD’s, Limmer and Chandler suggested that the appropriate temperature and pressure scales for producing corresponding states in water and various classical water models were the T_MD at atmospheric pressure and a reference pressure related to the enthalpy and volume changes for water upon melting (also at atmospheric pressure).³⁸ 

Following the approach of Uralcan et al.,³⁹ here, we investigate whether a scaling relationship similar to Eq. (1) produces low temperature corresponding states for the isotopes of water. If it does, the range of temperatures and pressures over which the correspondence holds between the isotopes will provide some evidence of the range over which a possible critical point exerts its influence on water’s properties. Besides extensive data available on H₂O, considerable data are also available for D₂O, with considerably less data on other isotopes, such as H₂¹⁸O, H₂¹⁷O, and D₂¹⁷O. Therefore, we will consider the relationship between H₂O and D₂O. To facilitate the analysis, we use published EoS’s for supercooled H₂O²⁵ and supercooled D₂O.⁴⁰ We find that a simple scaling relationship for pressures and temperatures, which is analogous to Eq. (1), produces corresponding states for H₂O and D₂O for pressures up to ∼200 MPa and temperatures below ∼300 K for various properties, including the density, isothermal compressibility, and speed of sound. Furthermore, the resulting deviations from strict corresponding states follow patterns that are similar to the deviations observed for the corresponding states of H₂O and D₂O when they are referenced to the LVCP.

A given thermodynamic property, Xⁱ, exhibits corresponding states if $X H T H , P H = X D T D , P D$⁠, where i = H or D refers to H₂O or D₂O, respectively. Because the thermodynamic response functions can be determined from the molar volume as a function of temperature and pressure, V_m(T, P), we searched for suitable values for the parameters in Eq. (2)—β, ΔT, γ, and ΔP—that provided best match for $V m H T H , P H = V m D T D , P D$⁠. To facilitate the search, it was important to use EoS’s for H₂O and D₂O that included as much of the supercooled region as possible. For D₂O, we used the recent EoS developed by Hruby and co-workers that relied upon their high-quality measurements of the density and is valid from 254 to 298 K and from atmospheric pressure to 100 MPa.⁴⁰ For the corresponding range of temperatures and pressures for H₂O, there are several choices for the EoS that give essentially identical molar volumes. We chose to use the EoS described in Holten et al.,²⁵ which is the EoS for supercooled H₂O recommended by the International Association for the Properties of Water and Steam (IAPWS). Below, we will refer to these as the supercooled H₂O or D₂O EoS. For temperatures and pressures above the melting line of H₂O and D₂O (i.e., “normal” water), we used the REFPROP software package from the National Institute of Standards and Technology, which is based on the IAPWS EoS for H₂O and D₂O, to calculate and compare the properties of interest.⁴¹ We will refer to these as the NIST H₂O and D₂O EoS.

Figure 1 shows the correspondence between the H₂O and (mass-scaled) D₂O densities—ρ^H = ρ^H(T_H, P_H) and $ρ D ′ = ρ D ′ T D , P D$⁠, respectively—for the set of parameters that minimizes the average absolute deviation, Δ_abs [see Eq. (4)]. The optimized parameters are as follows: β = 1.005 76, ΔT = 4.00 K, γ = 1.0187, and ΔP = 10.362 MPa. This figure shows ρ^H calculated using (i) the supercooled H₂O EoS (red solid line) along with the NIST H₂O EoS (red dashed line).^25,41 Similarly, ρ^D′ was calculated with EoS’s for supercooled (blue open circles and diamonds) and normal D₂O (blue solid circles) states.^40,41 Although the supercooled D₂O EoS is nominally valid for P_D ≤ 100 MPa, the correspondence with the H₂O densities is also reasonably accurate up to 200 MPa and 300 K. Furthermore, the correspondence between normal H₂O and D₂O (i.e., above their melting points) calculated using the NIST EoS’s is also generally good for T < ∼300 K and P ≤ 200 MPa. Figure S1 shows the relative deviations [see Eq. (4b)], which are of the order of 10⁻⁴, between the densities calculated with the supercooled D₂O and H₂O EoS’s.

While the results in Fig. 1 compare densities calculated using the chosen H₂O and D₂O equations-of-state for supercooled water, it is also useful to compare the measured H₂O densities to the corresponding D₂O densities calculated using both the supercooled D₂O EoS and the NIST D₂O EoS. Caupin and Anisimov compiled the experimental data for H₂O densities along with the estimates of the absolute experimental uncertainty that they used to develop their EoS.³⁰ We used their results as an input for the D₂O EoS to calculate the corresponding D₂O densities and the reduced residuals [see Eq. (4)]. Figure 2 shows the results for the data of Hare and Sorensen⁴³ and Sotani et al.⁴⁴ For the range where the supercooled D₂O EoS is valid, −1 < r_X,i < 1 for most of the data, with $r X , i min =−2.1$ and $r X , i max =1.5$⁠. The average absolute value of the reduced residuals, ave(|r_X,i|), is 0.42. For comparison, using the supercooled H₂O EoS on the same data gives ave(|r_X,i|) = 0.39. It is interesting to note that, in contrast to the relative deviations between the supercooled H₂O and D₂O EoS’s (Fig. S1), the reduced residuals calculated for supercooled D₂O EoS relative to the H₂O data do not show any obvious systematic trends [Fig. 2(b)]. The low temperature correspondence also correctly accounts for the differences in the experimental T_MD values for H₂O^44–49 and D₂O^40,47,50 (see Fig. S2).

Generally, the various derivatives of the molar volumes with respect to temperature and pressure will be more sensitive to the deviations from the corresponding states picture and thus could reveal more about the isotopic differences beyond what might be expected in a classical picture. Figure 3(a) compares the thermal expansivity, $α p =− 1 V ∂ V ∂ T p$⁠, for H₂O (⁠$α p H$⁠) and D₂O (⁠$α p D$⁠), calculated with their respective equations-of-state. Figure S3 shows a comparison of H₂O expansivity data with the supercooled D₂O EoS results, and Fig. 3(b) shows the differences between the H₂O data^43,46,51 and the values calculated with the supercooled D₂O and H₂O EoS’s (blue diamonds and red circles, respectively). The results in Fig. 3 indicate that the D₂O EoS’s are largely able to reproduce the H₂O expansivity data at low temperatures. Furthermore, Fig. 3(b) suggests that the deviations of supercooled D₂O and H₂O EoS’s with respect to the H₂O data are comparable.

Figure 4(a) compares the isothermal compressibility, $κ T =− 1 V ∂ V ∂ P T$⁠, calculated with the supercooled and NIST EoS’s for both isotopes. For D₂O, $κ T D$ is consistently less than the corresponding values for H₂O, but the trends vs temperature and pressure are nicely reproduced. As seen in Fig. 4(a), an overall scale factor, μ ≈ 1.015, significantly improves the overlap (i.e., $κ T H ≈μ⋅ κ T D$⁠). Figure S4 compares the H₂O compressibility data to the corresponding D₂O values calculated with the EoS’s, and Fig. 4(b) shows the deviations of the H₂O and D₂O EoS’s relative to the H₂O compressibility data.^6,7,48 As observed above for the density and the expansivity, the compressibility calculated using the D₂O EoS’s and the low temperature correspondence produces similar deviations relative to the H₂O data compared to the H₂O EoS’s, except in this case $κ T D$ is consistently about 1.5% smaller than $κ T H$ (see discussion below). The low temperature correspondence also adequately accounts for the isotopic differences in the speed of sound^52,54,55 (see Sec. A and Fig. S5 of the supplementary material) and the isobaric heat capacity^56–59 (see Sec. B and Figs. S6 and S7 of the supplementary material).

The isotope effects near the liquid vapor critical point for water have been investigated previously, and both H₂O and D₂O exhibit the expected universal scaling.^60,61 It is noteworthy that applying the standard corresponding states analysis to the data near the LVCP [see Eq. (3)] results in systematic deviations between H₂O and D₂O for properties, such as the compressibility and the speed of sound (see Figs. S8 and S9), that are similar to the deviations observed in the low temperature correspondence for these properties (Figs. 4 and S5). Because the isothermal compressibility is proportional to the square of the volume fluctuations,¹² the experimental results in Fig. 4 (Fig. S8) show that the fluctuations for D₂O are smaller (larger) than the corresponding fluctuations for H₂O near the LLCP (LVCP). However, the expansivity is proportional to the product of the volume and entropy fluctuations,¹² so the apparent lack of systematic differences between $α p H$ and $α p D$ (Fig. 3) indicates that the reduced volume fluctuations in D₂O are compensated by increased entropy fluctuations.

Above ∼300 K, the low temperature correspondence gets progressively worse (as expected). Conversely, the correspondence predicted between H₂O and D₂O near the LVCP gets worse at lower temperatures. Therefore, it is instructive to consider the temperatures at which the low and high temperature correspondences produce comparable results. Figure 5 shows the differences in densities between D₂O and H₂O—calculated with the NIST EoS’s—using the low temperature correspondence [Eq. (2)], δρ(LL) = ρ^D′ − ρ^H (dark blue symbols), and the liquid–vapor correspondence [Eq. (3)], δρ(LV) = ρ^D′ − ρ^H (light blue symbols). For the range of pressures shown, the low temperature correspondence is more accurate for T_H < 347 K, while the liquid–vapor correspondence is more accurate for T_H > 378 K. The red circles in Fig. 5 show the points at which deviations calculated using the low and high temperature correspondences cross. It is interesting to note that at ambient pressure, this temperature is ∼350 K, which is near the isothermal compressibility minimum for H₂O. The isothermal compressibility minimum has been suggested to be an indicator of the point at which the two-state character of water begins to have an appreciable influence on the properties of water. (However, see the discussion below regarding the transition between “two-state” and “one-state” descriptions of liquid water.)

Previous investigations of the properties of stretched water have noted the influence of the liquid–vapor spinodal on water’s thermodynamic properties.^{9,29,30,62–64} For example, Uralcan et al. found a correlation between the liquid–vapor spinodal and the LLCP in three classical water models.³⁹ In a two-state model, the liquid–vapor spinodal of the high-temperature state contributes a term to its Gibb’s free energy, which then influences the equilibrium fraction of each state as a function of temperature and pressure.^29,30,64 The low temperature correspondence between H₂O and D₂O also suggests a connection between the two critical points. Figure 6 shows several lines of extrema for H₂O and D₂O vs reduced temperature, $T ̂$⁠, and pressure, $P ̂$⁠. For this figure, Eq. (1) has been used to calculate $T ̂$ and $P ̂$⁠, and the values of T_c, T_max, P_c, P(T_max), and P_min for H₂O were taken from Table III and Fig. 13 of Ref. 30. The values for D₂O in Eq. (1) were then calculated from the H₂O values using the low temperature correspondence [Eq. (2)]. The red/blue diamond shows the location of the LLCP for H₂O and D₂O (which are the same, by construction), while the red and blue squares show the LVCP or H₂O and D₂O, respectively. It is noteworthy that using the low temperature correspondence places the D₂O LVCP nearly on the H₂O liquid–vapor spinodal and suggests that the liquid–vapor spinodal for D₂O will closely follow the H₂O spinodal. This observation is similar to the correlation between distances from the LLCP to various points on the liquid–vapor spinodal for three water models found by Uralcan et al.³⁹ 

If water has an LLCP, then both H₂O and D₂O should exhibit the universal scaling expected of the 3D Ising model in the immediate vicinity of the critical point. For example, Holten et al.²⁵ showed that a two-state model based on a LLCP could account for the experimental data for both H₂O and D₂O. They also noted that “(w)hile the critical part of the thermodynamic properties of H₂O and D₂O follow the law of corresponding states (the critical amplitudes a and k are the same) the regular parts do not follow this law.” The results presented here should be consistent with those observations. In particular, a consistent treatment of the “regular parts” could presumably be developed that also accounts for the low temperature correspondence discussed here and thus account for the behavior in the immediate vicinity of the critical point and over the larger range of temperatures and pressures. Further research is needed to explore this possibility in detail.

The parameters in Eq. (2) for the corresponding states were determined by minimizing the difference in the molar volumes using the supercooled H₂O and D₂O EoS’s.^25,40 Including other properties and data in the optimization, using a different choice for weighting the contribution of various data, and/or changing the range of temperatures and pressures would undoubtedly change the specific values obtained for β, γ, ΔT, and ΔP. One possible outcome of such changes could be a reduction in some of the systematic differences observed between the isotopes for properties such as the speed of sound and the isothermal compressibility that were described above. However, while further refinements of the correspondence described here will be valuable, they seem unlikely to change the main observation, which is that the properties of H₂O and D₂O at supercooled temperatures are brought into correspondence with a linear scaling of temperatures and pressures that includes a non-zero offset term for each [see Eq. (2)].

The differences between H₂O and D₂O are ultimately derived from nuclear quantum effects (NQEs).^65–68 For example, classical simulations cannot predict the changes in T_MD for the different isotopes of water.⁶⁵ Recent simulations that include NQEs on the thermodynamic properties of H₂O and D₂O for a wide range of pressures and temperatures (including the supercooled states) largely reproduce the experimental results for the density and isothermal compressibility.⁶⁸ In addition, those calculations follow the low temperature correspondence described here reasonably well. In particular, the T_MD and liquid–vapor spinodal lines, which are determined from the simulations, essentially overlap (see Fig. S10). (The corresponding locations of the LLCP for H₂O and D₂O, which are determined by fitting the simulation results to a two-state model, are also similar, but the agreement is not as good.) Because the results presented here are based on data at T > 235 K (at 0.1 MPa), it leaves open the possibility that the low temperature correspondence might break down at even lower temperatures. For example, previous results found that, while NQEs are important in the description of low-density amorphous ice (LDA) and hexagonal ice at very low temperatures, the difference between quantum and classical MD simulations was less important at higher temperatures.⁶⁹ 

Because the potential energy surface (PES) for a collection of water molecules does not depend on the isotope, the differences between the isotopes come from their behavior on the PES.⁷⁰ In this context, it is useful to consider supercooled water’s properties in the potential energy landscape (PEL) framework.⁷¹ The PEL is a hypersurface that represents the potential energy for a system as a function of the coordinates of all the atoms in the system. At sufficiently low temperatures, liquids primarily reside in local minima on the PEL, the system's behavior is dominated by the properties of these minima and the infrequent transitions it makes between minima. These properties (such as the number of minima vs energy and their curvature) can be used to determine the partition function for the liquid. For supercooled water, a simple model for the PEL (the Gaussian PEL) can account for water’s anomalous properties and is consistent with the results of classical MD simulations and two-state models of the LLCP.^72,73 In the PEL framework, the isotopes of water will show the corresponding behavior if they inhabit portions of the PEL that are similar (statistically). The low temperature correspondence described here indicates that this occurs when D₂O is at slightly higher pressures and temperatures relative to H₂O. These differences can presumably be modeled by differences in the zero-point energies associated with the local minima in the PEL and also anharmonic effects on the vibrational component of the free energy.^71,73 The low temperature correspondence between the isotopes is similar to the widely noted idea that the structure of liquid D₂O at a given temperature (above the melting point) is similar to that of H₂O at a somewhat higher temperature. The primary difference between the low and high temperature cases is that as the temperature increases, the influence of the local minima in the PEL on the thermodynamics (and dynamics) is reduced, the fraction of the low-temperature structural motif decreases, and temperature-dependent changes in the structure of the (essentially single-component) high temperature liquid can account for the isotopic differences.⁶⁶ Of course, defining the transition when water is best described as an inhomogeneously broadened, single-component liquid and one that is best described by a two-state model depends on one’s definitions and is subject to considerable debate.^2,74,75

The results presented here indicate that there is an approximate, low temperature correspondence for the thermodynamic properties of H₂O and D₂O. It is also well known that many of water’s dynamic properties are potentially consistent with the LLCP hypothesis,^1,6 with the D₂O results typically showing a shift to higher temperatures (for the same pressure) that is similar to those observed for the thermodynamic properties.^10,76–78 The amount of highly accurate, pressure-dependent dynamic data that are available for both H₂O and D₂O limits the ability to perform a detailed comparison of the low temperature correspondence in most cases. However, for the self-diffusion in supercooled H₂O and D₂O,^76,79 the low temperature correspondence appears to provide a reasonable description of the results (see Fig. S11). More work is needed to assess the extent to which the low temperature correspondence applies to other dynamical properties.

When comparing the thermodynamic properties of supercooled H₂O and D₂O, a simple linear relationship between the temperatures and pressures of the isotopes [Eq. (2)] produces a correspondence such that $X H T H , P H ≈ X D T D , P D$⁠, where X^H and X^D are properties, such as the molar volume, expansivity, isothermal compressibility, and speed of sound, for H₂O and D₂O. This approximate, low temperature correspondence for the isotopes, which is distinct from the usual corresponding states associated with the liquid–vapor critical point, is generally good for temperatures below ∼300 K and pressures below ∼200 MPa. The most plausible physical origin for the low temperature correspondence is a liquid–liquid critical point for supercooled water. Based on the range of temperatures and pressures that produce a correspondence between the properties of H₂O and D₂O, these results support the idea that some of water’s most notable anomalies, such as the existence of the density maximum at near ambient temperatures, are related to the LLCP in the deeply supercooled region.

The supplementary material includes Secs. A and B, which describe the isotope effects for the speed of sound and isobaric heat capacity, respectively, and Figs. S1–S11.

The author would like to thank Bruce D. Kay, Nicole R. Kimmel, and Frédéric Caupin for helpful discussions and Jan Hruby for help implementing his D₂O equation-of-state. This work was supported by the U.S. Department of Energy (DOE), Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, Condensed Phase and Interfacial Molecular Science program, Grant No. FWP 16248.

The author has no conflicts to disclose.

Greg A. Kimmel: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.