Isotope effects in supercooled H$_2$O and D$_2$O and a corresponding-states-like rescaling of the temperature and pressure

Water shows anomalous properties that are enhanced upon supercooling. The unusual behavior is observed in both H$_2$O and D$_2$O, however with different temperature dependences for the two isotopes. It is often noted that comparing the properties of the isotopes at two different temperatures (i.e., a temperature shift) approximately accounts for many of the observations with a temperature shift of 7.2 K in the temperature of maximum density being the most well-known example. However, the physical justification for such a shift is unclear. Motivated by recent work demonstrating a corresponding-states-like rescaling for water properties in three classical water models that all exhibit a liquid-liquid transition and critical point (B. Uralcan, et al., J. Chem. Phys. 150, 064503 (2019)), the applicability of this approach for reconciling the differences in temperature- and pressure-dependent thermodynamic properties of H$_2$O and D$_2$O is investigated here. Utilizing previously published data and equations-of-state for H$_2$O and D$_2$O, we show that the available data and models for these isotopes are consistent with such a low temperature correspondence. These observations provide support for the hypothesis that a liquid-liquid critical point, which is predicted to occur at low temperatures and high pressures, is the origin of many of water's anomalies.


Introduction
Water is an unusual liquid that has been extensively investigated for over a century. 1,2 17][18][19][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 or 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.
4][25] Several classical water models have been rigorously shown to have an LLCP. 14,26,27 F[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). 25cause the universal scaling associated with a critical point is in addition to a "normal" (nondiverging) component, experiments typically need to be done very close to the critical point to unambiguously observe the universal scaling behavior. 252][33][34] More generally, Pitzer 31 and Guggenheim 34 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 n Pitzer's formulation, the Helmholtz free energy is a universal function, F = F(T/A, V/R0), where A and R0 are a characteristic energy and length scales 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, Tc, and volumes, Vc, as the scale parameters.
For water, the experimental data is 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 challenging -a point that was made even in the initial work of Angell and Speedy. 6As 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 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. 35Subsequent work suggested that shifting the temperature scale for D2O by the difference in temperature of maximum density, δTMD ≅ 7.2 K, between D2O and H2O (at atmospheric pressure) resulted in corresponding states for densities of the two isotopes. 36,37 owever, because a 7.2 K temperature shift was less successful for other properties, in practice δTMD came to be used as an adjustable parameter without any specific physical significance associated with it.Instead of shifting the temperatures to match the TMD'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 TMD at atmospheric pressure and a reference pressure related to the enthalpy and volume changes for water upon melting (also at atmospheric pressure). 38ecently, Uralcan, et al., investigated possible scaling relationships between 3 classical water models that are known to have an LLCP (ST2, TIP4P/2005, and TIP5P). 39By analyzing the patterns of extrema (density maximum and minimum, compressibility, etc.) in the P-T plane they found a "correspondingstates-like rescaling" for the pressure and temperature.Specifically, they found that for reduced temperatures,  � , and pressures,  � , the patterns of extrema for the models approximately collapsed onto universal curves when: In Eqn. 1, Tc and Pc are the critical temperature and pressure, respectively, for the LLCP of a given water model, while Tmax, P(Tmax) and Pmin are related to characteristics of the TMD line in the P-T plane for that model.Specifically, Pmin is the minimum pressure along the TMD line, and Tmax is maximum temperature on the TMD line, which occurs at P = P(Tmax).Uralcan, et al., also included a small rotation in the P-T plane, which we will assume is small for the water isotopes and can be ignored.It is important to note that Eqn. 1 is different from the reduced temperatures and pressures associated with the liquid-vapor critical point:  �  =     and  �  =     , where we have added the superscript LV to distinguish the LVCP from the LLCP.Because of these differences Uralcan et al., referred to Eqn. 1 as a "correspondingstates-like rescaling."However, we will simply refer to the low temperature "corresponding states" for H2O and D2O while keeping in mind this importance distinction.
Following the approach of Uralcan, et al., 39 here we investigate whether a scaling relationship similar to Eqn. 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 is also available for D2O, with considerably less data on other isotopes such as, H2 18 O, H2 17 O and D2 17 O.Therefore, we will consider the relationship between H2O and D2O.To facilitate the analysis, we use published EoS's for supercooled H2O 25 and supercooled D2O. 40We find that a simple scaling relationship for pressures and temperatures, which is analogous to Eqn. 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.

Methods
In Eqn. 1, there are 4 unknowns for each isotope,   ,   , (  ), and   .Because these values are uncertain for H2O and D2O (assuming for now that the LLCP hypothesis is correct), the specific form of the reduced temperatures and pressures in Eqn. 1 was not convenient to use in the search for a correspondence between H2O and D2O.Instead, it was convenient to work with the actual temperatures and pressures that were used as inputs to the EoS's for both H2O and D2O.If Eqn. 1 describes low temperature corresponding states for H2O and D2O, then there must be a linear relationship between the temperatures and pressures for the isotopes that produces the correspondence such that: where Ti (Pi) for i = H or D refer to the temperatures (pressures) for H2O and D2O, respectively.A second benefit of using Eqn. 2 to express the corresponding temperatures and pressures for D2O and H2O is that it is "agnostic" with respect to the possible existence and location of an LLCP.
As mentioned above, Eqn. 1 is different than the usual equations for  �  and  �  .In the form of Eqn.
2, the corresponding temperatures and pressures for D2O and H2O relative to the LVCP, are where Hruby and co-workers that relied upon their high-quality measurements of the density and is valid from 254 K to 298 K and from atmospheric pressure to 100 MPa. 40For 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. 41We will refer to these as the NIST H2O and D2O EoS.
Because the densities are more commonly encountered than the molar volumes, below we compare the D2O densities -multiplied by the ratio of the molar masses -to the H2O densities.For a given D2O density,    , the mass scaled density is   ′ = ( 2 / 2 ) •    , where  2 and  2 are the molar masses of H2O and D2O.To optimize the parameters from Eqn. 2 (i.e., , ∆, , and ∆), we calculated the H2O density at 1 K intervals from 249 to 293 K at 0.101325, 20, 40, 60, 80, and 100 MPa using the supercooled H2O EoS. 25 Those temperatures and pressures were then converted into their corresponding D2O values using Eqn. 2 for a trial set of parameters, and the corresponding D2O densities were calculated with the supercooled D2O EoS. 40The parameters were then adjusted to minimize the average absolute deviation, ∆  , between the H2O and D2O densities.For properties    (  ,   ) and    (  ,   ) calculated (or measured) at a series of points, i, ∆  was calculated as where   is the number of data points and ∆  is the relative deviation at each data point.Once the best fit values for the parameters were determined by comparing the densities, they were subsequently used without further adjustment to compare the isothermal compressibility, expansivity, speed of sound, and isobaric heat capacity of H2O and D2O.We also extended the comparison outside the range of validity of the supercooled D2O EoS to investigate range of temperatures and pressures over which the low temperature correspondence provides reasonable estimates of the various properties.
In addition to comparing properties computed with the H2O and D2O EoS's, it was also useful to compare experimentally measured H2O properties at various temperatures and pressures, to the values calculated with the D2O EoS, at the corresponding TD and PD.In some cases, we calculated the reduced residuals between the experimental data (   ( , ,  , )) and corresponding values calculated with the D2O EoS (  () ( , ,  , )) using published estimates of the absolute experimental uncertainty for the data. 30,42 he reduced residual for a given data point i,  , , is given by where   is the associated absolute experimental uncertainty. 30These values could then be compared to the reduced residuals calculated for the H2O data and H2O EoS.

Results
Figure 1 shows the correspondence between the H2O and (mass-scaled) D2O densities -  =   (  ,   ) and   ′ =   ′ (  ,   ), respectively -for the set of parameters that minimizes the average absolute deviation, ∆  (see Eqn. 4).The optimized parameters are:  = 1.00576, ∆ = 4.00 K,  = 1.0187, and ∆ = 10.362MPa.Fig. 1a shows   calculated using (i) the supercooled H2O EoS (solid red line) along with the NIST H2O EoS (dashed red line). 25,41 imilarly,   ′ was calculated with EoS's for supercooled (open blue circles and diamonds) and normal D2O (solid blue circles) states. 40,41 lthough the supercooled D2O EoS is nominally valid for   ≤ 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.Fig. S1 shows the relative deviation, ∆  , (see Eqn. 4b) between the densities calculate with the supercooled D2O and H2O EoS's.The differences are of the order of 10 -4 , and they show some systematic trends.For example, the differences between the densities for PD ≤ 100 MPa are generally the smallest for TH ~ 267 K (TD ~ 273 K).At lower temperatures,   ′ is less than   at low pressures, but larger at higher pressures, while the opposite trend is found at temperatures > 270 K.
Given that both supercooled EoS's use polynomials in various ways, such systematic differences are not too surprising., 25 and the NIST H2O EoS, 41 respectively.The open blue circles (diamonds) correspond to D2O densities calculated using the supercooled D2O EoS within (outside) its range of validity. 40The filled blue circles show D2O densities calculate with NIST D2O EoS.The H2O and D2O densities along the H2O melting line, Tm, are also shown.
While the results in Fig. 1 compare densities calculated using the chosen H2O and D2O equations-ofstate 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 experimental data for H2O densities along with estimates of the absolute experimental uncertainty, that they used to develop their EoS. 30We used their results as input for the D2O EoS to calculate the corresponding D2O densities and the reduced residuals (see Eqn. 4). Figure 2 shows the results for the data of Hare and Sorensen, 43 and Sotani, et al. 44 For the range where the supercooled D2O EoS is valid, -1 <  , < 1 for most of the data, with  , (min) = −2.1, and  , (max) = 1.5.The average absolute value of the reduced residuals, (| , |), is 0.42.For comparison, using the supercooled H2O EoS on the same data gives (| , |) = 0.39.As seen in the figure, including data with pressures up to 200 MPa (i.e., outside the valid range for the supercooled D2O EoS), the correspondence is still quite good.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. 2b).
Fig. 2. a) Comparison of measured H2O densities (red circles), 43,44   , to the corresponding D2O densities,   ′ , calculated with the supercooled D2O EoS (blue circles). 25,40 ) The reduced residuals (Eqn.As discussed in the introduction, Uralcan, et al. used the lines of extrema in the P-T plane, particularly focusing on the density maxima, to find approximate corresponding states for three classical water models. 39Figure 3 shows the loci of the density maxima for H2O,    (  ,   ), (red circles) and the corresponding values for D2O,    (  ,   ), (blue diamonds). 40,45,46 Te good overlap for the density maxima seen in Fig. 3 is unsurprising given the results shown in Fig. 1.However, based upon the results presented by Uralcan, et al., it also suggests that the other properties will show a similar correspondence.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 4a   , calculated with the supercooled and NIST EoS's for both isotopes.For D2O,    is consistently less than the corresponding values for H2O, but the trends versus temperature and pressure are nicely reproduced.As seen in Fig. 5a, an overall scale factor,  ≈ 1.015, significantly improves the overlap (i.e.,    ≈  •    ).Fig. S3 compares the H2O compressibility data to the corresponding D2O values calculated with the EoS's, and Fig. 5b shows the deviations of the H2O and D2O EoS's relative to the H2O compressibility data (where the scale factor for    , , is included in the calculation). 6,7,50 A observed above for the density and the expansivity, the compressibility calculated using the D2O EoS's using the low temperature correspondence produces similar deviations relative to the H2O data compared to the H2O EoS's, except in this case    is consistently about 1.5% smaller than    (see discussion below).
Accurate measurements of the speed of sound are available for H2O,   =   (  ,   ), over a wide range of temperatures and pressures (Fig. 6, red squares). 25,42,54 Te speed of sound is inversely proportional to the square root of the density, so to compare between D2O and H2O, we use the massscaled D2O speeds,  ′ = � 2  2 ⁄   (  ,   ).However, after correcting for the mass differences, the corresponding states still show systematic differences in between the H2O and D2O.To illustrate this Fig. 6 shows  •  ′ calculated with the supercooled D2O EoS (blue circles).The value of the overall scale factor,  = 0.992, which was determined by minimizing ∆  (see Eqn. 4) between   and  •  ′ over the range of validity for the supercooled D2O EoS (shown by the black dotted lines in Fig. 6).For that range and with  = 0.992, ∆  = 0.00093.The figure also shows   (red lines) and  •  ′ (blue dashed lines) calculated with the NIST EoS for each isotope.At pressures above 100 MPa, where the supercooled D2O EoS begins to deviate more noticeably, the correspondence for H2O and D2O calculated with the NIST EoS is still quite good.Figure 7 shows the isobaric heat capacities for H2O (   ) and D2O (   ) versus temperature for 0.1 MPa ≤ PH ≤ 400 MPa and the corresponding range of D2O pressures.Red symbols show experimental results for H2O, 30,[57][58][59] along with the results from the NIST H2O EoS (solid red lines).(Other data, which extends to lower temperatures than the range of validity for the supercooled D2O EoS, are not shown.) For D2O,    was calculated with the supercooled EoS (solid blue line) for PD = 10.465MPa (which corresponds to PH = 0.101325 MPa), while at higher pressures the NIST D2O EoS was used (dashed blue lines).The agreement between the H2O data and the corresponding values obtained with the D2O EoS's is acceptable given that there is limited data and considerable uncertainty in the measurements at high pressures. 59Furthermore, apparently the only data available for supercooled water is at atmospheric pressure. 40,45 t is interesting to note that while the H2O EoS's predict that    decreases at low temperatures for PH > 100 MPa (see Fig. S4), the H2O data and the D2O EoS suggest that    stops decreasing and perhaps goes through a minimum. 59Because the heat capacity is likely to be sensitive to quantum effects, 60 experiments comparing supercooled H2O and D2O would be useful for developing a better understanding of how such effects influence the low temperature correspondence described here.EoS.9]61 The solid and dashed blue lines show the corresponding values for    from the supercooled and NIST D2O EoS's, respectively.
The low temperature correspondence between H2O and D2O shows systematic deviations for some properties, such as the compressibility (Fig. 5) and the speed of sound (Fig. 6).However, it is noteworthy that using the standard corresponding states scaling associated with the LVCP (see Eqn. 3) also results in systematic deviations between H2O and D2O for these properties.For example, for temperatures near the LVCP and H2O pressures ≤ 100 MPa, the speed of sound for D2O is systematically less than H2O such that an overall scale factor of ~1.015 significantly reduces the differences (Fig. S5).This is compared to the results for low temperatures, where, as discussed above, a scale factor of 0.992 produces a better correspondence (Fig. 6).Similarly, multiplying the D2O compressibility,    , by 0.982 reduces ∆  in the vicinity of the LVCP (see Fig. S6), compared to a scale factor of ~1.015 for the low temperature correspondence (see Fig. 5).Because the isothermal compressibility is proportional to the square of the volume fluctuations, 12 the experimental results show that the fluctuations for D2O are smaller (larger) than the corresponding fluctuations for H2O near the LLCP (LVCP).On the other hand, the expansivity is proportional to the product of the volume and entropy fluctuations, 12 so the apparent lack of systematic differences between    and    (Fig. 4) indicates that the reduced volume fluctuations in D2O are compensated by entropy fluctuations.While the heat capacity is proportional to the square of the entropy fluctuations, the large uncertainties in both the data and the EoS predictions for    and    make it difficult to assess their relative magnitudes in supercooled water.
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 temperature.Therefore, it is instructive to consider the temperatures at which the low and high temperature correspondences produce comparable results.Figure 8 shows the differences in densities between D2O and H2O -calculated with the NIST EoS's -using the low temperature correspondence (Eqn.2), () =   ′ −   (dark blue symbols), and the liquid-vapor correspondence (Eqn.3), () =   ′ −   (light blue symbols).For the range of pressures shown, the low temperature correspondence is more accurate for TH < 347 K, while the liquid-vapor correspondence is more accurate for TH > 378 K.The red circles in Fig. 8 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 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.)64] For example, Uralcan, et al., found a correlation between the liquid-vapor spinodal and the LLCP in three classical water models. 39In a twostate 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 Te low temperature correspondence between H2O and D2O also suggests a connection between the two critical points.Figure 9 shows several lines of extrema for H2O and D2O versus reduced temperature,  � , and pressure,  � .For this figure, Eqn. 1 has been used to calculate  � and  � , and the values of   ,   ,   , (  ), and   for H2O were taken from Table III and Fig. 13 in ref. 30 The values for D2O in Eqn. 1 were then calculated from the H2O values using the low temperature correspondence (Eqn.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 H2O liquidvapor spinodal and suggests 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 liquidvapor spinodal for three water models found by Uralcan, et al. 39 for H2O and D2O, respectively. 40,46,47,50 Th red stars show the location of H2O liquid-vapor spinodal 9 near the H2O liquid-vapor critical point (red square).In lieu of reliable data for the liquid-vapor spinodal at low temperatures, the dotted red and green lines shows the spinodal calculated for the TIP4P/2005 model 30 and derived from a two-state model, 64 respectively.The NIST EoS's were used to find the compressibility minima and speed of sound maxima at positive pressures for D2O (blue diamonds and triangles) and H2O (red diamonds and triangles).For negative pressures, the data of Pallares, et al. are   shown as open red diamonds and triangles for the κmin and wmax, respectively. 46

Discussion
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 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.
For the analysis presented here, the parameters in Eqn. 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 luding 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, ∆T, and ∆P.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 Eqn. 2).
6][67][68] For example, classical simulations cannot predict the changes in TMD for the different isotopes of water. 65Recent 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. 68In addition, those calculations follow the low temperature correspondence described here reasonably well.In particular, the TMD and liquid-vapor spinodal lines, which are determined from the simulations, essentially overlap (see Fig. S7).
(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 found for H2O and D2O might break down at even lower temperatures.
For example, previous results found that, while NQE's 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 were less important at higher temperatures. 69cause the potential energy surface (PES) for a collection of water molecules does not depend on the isotope, the differences between the isotopes comes from their behavior on the PES. 70In this context, it is useful to consider supercooled water's properties in the potential energy landscape (PEL) framework. 71The 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, and their behavior is dominated the properties of these minima and the infrequent transitions it makes between minima.These properties (such as the number of minima versus 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 n the PEL framework, isotopes of water will show corresponding behavior if they inhabit portions of the PEL that are similar (statistically).The low temperature correspondence describe 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 he 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 temperaturedependent changes in structure of the (essentially single-component) high temperature liquid can account for the isotopic differences. 66Of 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 Tresults presented here indicate that there is an approximate, low temperature correspondence for the thermodynamic properties of H2O and D2O.][78] The amount of highly accurate, pressure-dependent dynamic data that is 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. S8).More work is needed to assess the extent to which the low temperature correspondence applies to other dynamical properties.

Conclusions
When comparing thermodynamic properties of supercooled H2O and D2O, a simple linear relationship between the temperatures and pressures of the isotopes (Eqn.2) produces a correspondence such that   (  ,   ) ≈   (  ,   ), where   and   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 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.scaling factor, here taken to be 1.015, significantly improves the correspondence between the isotopes for the range of temperature and pressures shown here.Note that for the low temperature correspondence (Fig. 6), the scaling factor, λ, is less than 1.

Fig. 1 .
Fig. 1.Comparison of the H2O density,   (  ,   ), to the mass-scaled D2O density,   ′ (  ,   ).The bottom and top axes show the temperatures for H2O and D2O, respectively.The H2O pressures, PH, are shown in the figure, and the corresponding pressures for D2O, PD, are obtained from Eqn. 2b.The solid and dotted red lines show   calculated with the supercooled H2O EoS of Holten, et al., 25 and the NIST

5 )
between the measured H2O and calculated H2O densities (red circles) are not appreciably different that residuals for the measured H2O densities and the calculated D2O values (blue circles).The open (filled) blue circles show D2O points that are within (outside) the range of validity of the supercooled D2O EoS, while the filled (open) red circles show points within (outside) the range of validity.

Fig. 4 .
Fig. 4. a) Comparison of the expansivity calculated with H2O (red lines) and D2O (blue lines) EoS's.The supercooled EoS's were used for TH, TD < 300 K and PH, PD ≤ 100 MPa, otherwise the NIST EoS's were used.The open red circles show the expansivity for H2O derived from speed of sound measurements.53,54b) Differences between H2O experimental expansivity data43,48,52 and values calculated with the supercooled D2O (blue diamonds) and H2O (red circles) EoS's.
Fig. 5. a) Comparison of the isothermal compressibility calculated with the H2O (red lines) and D2O (blue lines) EoS's.The supercooled EoS's were used for TH, TD < 300 K and PH, PD ≤ 100 MPa, otherwise the NIST EoS's were used.The open red circles show the compressibility for H2O derived from speed of sound measurements. 53, 54b) Deviations between H2O experimental compressibility data 6, 7, 50 and values calculated with the supercooled and NIST D2O EoS's (dark and light blue circles, respectively) and supercooled and NIST H2O EoS's (red circles and triangles, respectively) EoS's.The D2O compressibility has been multiplied by µ = 1.015 in a) and for the calculation of the deviations in b).

Fig. 6 .
Fig.6.Speed of sound comparison.H2O data for Taschin, et al.,55 Belogol'skii, et al.,56 and Lin andTrusler (open squares circles)53 and the corresponding values calculated with the supercooled D2O EoS (blue circles) and the NIST D2O EoS (blue crosses).The black dotted line shows the range where the supercooled D2O EoS is valid.In addition to the expected correction by the square root of the masses, all the D2O sound speeds are multiplied by λ = 0.992.This value minimizes the average absolute deviation, ∆  , for the H2O data compared to the supercooled D2O EoS values in the range where it is valid.The red and dashed blue lines show the sound speeds calculated with the NIST EoS for H2O and D2O, respectively.For the NIST results, the H2O pressures, PH, are 0.101325, 10, 20, 40, 60, 75, 100, 125, 150, 175, 200, 225, and 250 MPa from bottom to top, and the corresponding D2O pressures are calculated using Eqn. 2.

Fig. 8 .
Fig. 8. Density differences between H2O and D2O calculated using the low temperature correspondence (Eqn.2, dark blue symbols and lines) and the liquid-vapor correspondence (Eqn.3, light blue symbols and lines).The results shown are for PH = 0.101325 MPa (solid lines), 20 MPa (circles), 40 MPa (dashed lines), 60 MPa (dotted lines), 80 MPa (+'s), and 100 MPa (diamonds).The low temperature correspondence is more accurate for TH < ~350 K.The red circles show where the low and high temperature correspondence cross at each pressure.

Fig. 9 .
Fig. 9. Extrema lines for thermodynamic properties of H2O and D2O versus reduced temperature,  � and pressure,  � (see Eqn. 1).The open red and blue circles show the loci of density maxima,    and    ,

Fig. S2 .
Fig. S2.H2O expansivity data (red circles) and the corresponding values calculated with the supercooled D2O EoS (blue squares).Fig. 4b show the differences between these values for the range of validity of the supercooled D2O EoS.The red and blue dashed lines show the expansivity at PH = 0.101325, 50 and 100 MPa (and the corresponding D2O pressures) calculated with supercooled H2O and D2O EoS, respectively.

Fig. S4 .
Fig. S4.Isobaric heat capacity for H2O comparison of experimental values to NIST H2O EoS at 200, 300, and 400 MPa.At these higher pressures, the NIST EoS decreases as the temperature decreases whereas the H2O data is approximately independent of temperature.

Fig. S5 .
Fig. S5.Speed of sound, w, for H2O (red lines) and D2O (blue lines) versus the reduced temperature,  �  =     , for reduced pressures,  �  =     , of 0.0046, 1.0, 1.81, 2.72, 3.63 and 4.53, where the critical pressures for H2O and D2O are 22.065 and 21.671 MPa, respectively.a) To account for the expected mass effects, the speed of sound for D2O has been multiplied by square root of the masses.However, the massscaled D2O speeds are consistently less than the corresponding H2O values.b) Including an additional

Fig. S6 .
Fig. S6.Compressibility for H2O (red lines) and D2O (blue lines) versus the reduced temperature,  �  =     , for reduced pressures,  �  =     , of 0.0046, 1.0, 1.81, 2.72, 3.63 and 4.53, where the critical pressures for H2O and D2O are 22.065 and 21.671 MPa, respectively.a) The D2O compressibilities are consistently larger than the corresponding H2O values.b) Including an additional scaling factor, here taken to be 0.982, significantly improves the correspondence between the isotopes for the range of
=   , exhibits corresponding states if   (  ,   ) =   (  ,   ),   ) =    (  ,   ).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