Thermodynamic Analysis of the Two-Liquid Model for Anomalies of

Article pubs.acs.org/JPCB

Thermodynamic Analysis of the Two-Liquid Model for Anomalies of Water, HDL−LDL Fluctuations, and Liquid−Liquid Transition G. P. Johari* Department of Materials Science and Engineering, McMaster University, Hamilton, Ontario L8S 4L7, Canada

J. Teixeira Laboratoire Léon Brillouin (CEA/CNRS), CEA Saclay, 91191 Gif-sur-Yvette Cedex, France ABSTRACT: After reviewing the protocol-dependent properties of HDA, which thermally anneals to LDA, and the data gap over an unusually large T-range between HDA, LDA, and water, we investigate whether or not, despite HDA’s ill-defined state and distinction from a glass, the HDL−LDL fluctuations view of the two-liquid model can explain water’s anomalous behavior. An analysis of the density, ρ, compressibility, β, heat capacity, Cp, and thermal conductivity, κ, of water over a monotonic (continuous) path bridging this data gap shows the following: (i) Such a path between ρwater at 320 K and ρHDA yields an untenable thermal expansion coefficient of water. (ii) There is neither a continuous path between βwater at 353 K and βHDA, nor between Cp,water at 363 K and Cp,HDA. (iii) The same value of ρwater, of βwater, or of Cp,water at two temperatures separated by a maxima or a minima is incompatible with the HDL−LDL fluctuations view. (iv) κLDA at ∼140 K is about twice that of κ water at 253 K. (v) κHDA at 120 K is incompatible with κwater at T > 320 K. Thus, there is an internal inconsistency between the thermodynamics of HDA seen as a glass and that of water seen as an HDL−LDL mixture, which is incompatible with both the HDL−LDL fluctuations view and the liquid−liquid transition.

■

INTRODUCTION Temperature dependence of some properties of water is unusual among liquids.1,2 Briefly, on cooling from a temperature T of 320 K, water’s density, ρwater, increases sublinearly (less than anticipated), and then decreases at T < 277 K, showing a density maximum at 277 K. Its isothermal compressibility, βwater, shows a local minimum at ∼323 K, and the heat capacity, Cp,water, shows a local minimum at ∼315 K.1−5 When water is compressed at T below 306 K, its viscosity decreases,1−5 and when water is supercooled, its thermal expansion coefficient, α, becomes more negative and the temperature dependence of its βwater and Cp,water appear to show divergence.1−4 Since the 1960s, these properties have been explained in terms of two competing effects that become significant during the cooling of water from 298 K, namely,1,2,6 (i) strengthening or formation of the four-coordinated structure of H2O molecules which decreases the density, and (ii) decrease in the amplitudes of anharmonic (intermolecular) vibrations which increases the density. More recently, a two-liquid model has been used to explain water’s properties including those of its supercooled state. It postulates that there are two types of local structures in water that differ in the density by ∼20%. One structure is that of HDL, which would form when high-density amorphous ice, HDA, is heated, and the second is that of LDL which would form when the low-density amorphous ice, LDA, is heated. HDA is produced when hexagonal ice at 77 K collapses under an uniaxial pressure of 10−16 kbar (1.0−1.6 GPa). Its density, ρHDA, is 1.17 g mL−1 at 77 K and 1 bar. LDA forms when HDA © XXXX American Chemical Society

is thermally annealed by heating to T above 115 K at 1 bar pressure. Its density, ρLDA, is 0.94 g mL−1 at 77 K and 1 bar. (We use the material’s name as subscript to its property.) X-ray scattering data obtained on evaporation-cooled, micron-size droplets of superooled water7,8 were recently interpreted to conclude that water consists of fluctuating regions of HDL and LDL structures (clusters), whose relative amounts vary with T. Hence, it seemed to support the twoliquid model. On cooling, “patches” or clusters of LDL form and grow in the HDL matrix;8 i.e., the fluctuations change toward greater amount of LDL clusters. To quote from ref 8, “Measurements of the structure at deeply supercooled conditions show a continuous increase in tetrahedrality which becomes accelerated below the temperature of homogeneous ice nucleation. The two local structures are connected to the liquid−liquid critical point (LLCP) hypothesis in supercooled water and correspond to high-density liquid (HDL) and lowdensity liquid (LDL). We propose that both HDL and LDL behave as normal liquids and that the anomalous properties of water result from the transition between them, which occurs over a wide temperature range at ambient pressure.”8 Accordingly, properties of water are a superposition of the properties of its components, HDL and LDL, each of which, it was proposed, behaves as a normal liquid.8 On cooling at ambient pressure from ∼360 K to T < 314 K, the LDL fraction Received: July 6, 2015 Revised: September 30, 2015

A

DOI: 10.1021/acs.jpcb.5b06458 J. Phys. Chem. B XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry B

model may become acceptable, and one can justifiably proceed with theoretical and computational analyses based on these models. Here we search for such a bridging path from ρwater, αwater, βwater, Cp,water, and thermal conductivity, κwater, at T > 320 K to the corresponding properties of HDA and LDA at T < 150 K. Our method is basically the same as that used previously to examine the validity of the inference of a hidden λ-type transition in supercooled water,36 and to determine merits of the assumed reversibility of the HDA−LDA (solid−solid) transition, as well as to investigate whether LDA is the same as hyperquenched glassy water.30 We proceed by accepting Pettersson and Nilsson’s views8 that, (i) water is a mixture of HDL and LDL, (ii) water at T > 320 K is almost all HDL, (iii) the fraction of LDL increases on cooling, and, (iv) HDL and LDL behave as normal liquids. Simulation models for water were reviewed recently,8 so we do not discuss those here. We avoid the complications arising from the effects of solutes on water’s structure, and from the interfacial and other effects present in emulsified water droplets, in water imbibed in proteins, and in the water confined to nanometer-size pores of silica and other materials. Hence, we focus on bulk water’s properties. Without implying a structural hypothesis,8 or suggesting a discontinuous path between water and HDA, we search for a plausible (and reversible) path between the thermodynamic properties of bulk water, HDA, and LDA. In the subsequent section, we provide background to the two-liquid model, and discuss the ambiguous nature of HDA and its consequences for our interpretation. In the next section, we discuss the configurational thermodynamics and its relation to HDL and LDL. In the fourth section, we discuss the merits of the HDL−LDL fluctuations in the two-liquid model by using the ρ, α, β, Cp, and κ values. Discussion and conclusions are given in the last two sections.

increases at the expense of the HDL fraction. Water’s density anomaly could then be explained by invoking a change in the relative amounts of the HDL and LDL structures.7,8 Chandler9 questioned the likelihood of coexistence of two metastable liquid phases on fundamental grounds, and one of us argued that properties of HDA are protocol-dependent.10−12 There are difficulties in accepting the two-liquid model also because HDL does not exist at 1 bar pressure,12 and both HDA and LDA are unstable states. (An incomplete, 16-K-broad endotherm and the increase in its onset temperature with increase in the prior cooling rate observed for a specially treated HDA13 were mistaken12 as “typical signature of glass transition”.13) On the basis of X-ray emission spectroscopy data, Sellberg et al.14 reconsidered certain aspects of their previous view,7,8 but not the possibility of liquid−liquid transition. They used the following terms:7,8,14 “two liquid phases”, “two water structures”, “HDL and LDL patches”, “interconverting HDL−LDL clusters”, “bicontinuous two phases”, and “HDL−LDL fluctuations”. We use these terms interchangeably. Most recently, they suggested a fragile−strong transition, and vitrification of water at T near 227 K.15 The two-liquid model of ambient water3,7,8,14 was based on the assumptions that there is only one HDA, which is characterized by its unique density and structure, and that the HDA−LDA conversion (between two nonequilibrium solids) is reversible at 1 bar pressure and so is the HDL−LDL conversion. These assumptions overlooked the following findings: (i) There are several HDAs of varying behavior and structures,16−29 densities, 28 and other physical properties.22,24−29 (ii) Collapse of LDA produces a different HDA, then collapse of hexagonal ice.22,30 (iii) On heating at 1 bar, HDAs do not become HDL; instead, HDAs first anneal to a stress-free state and then anneal to LDA in the 115−125 K range,11,31 depending upon the heating rate. (iv) There is no latent heat of HDA−LDA conversion. (v) On heating at 1 bar, LDA first softens in the same manner as a glass softens to an ultraviscous liquid,32 and then on heating at 30 K/min rate crystallizes to cubic ice at T ≤ 148 K11,20,33 (Figure 7, curve 5, ref 33). (vi) On cooling at 1 bar pressure, LDA does not revert to HDA;34 i.e., HDA−LDA conversion is irreversible (see refs 11 and 12 for review). While thermal annealing of HDA at 1 bar pressure decreases the density and produces LDA,25,31 Gromnitskaya et al.26 originally found that thermal annealing of HDA under high pressure increases its density further. However, they did not perform an ex situ study of the still-higher density HDA and did not give it a new name. Eight variants of the amorphous ice have been listed.35 Out of these, LDA, HDA, and VHDA were argued to be different “states”.35 Thermodynamic properties of bulk water are known only at T > 248 K;5 those of the slowly heated HDA are known only at T < 115 K.31 HDL has not been found, and LDL exists at T < 148 K. Therefore, there is a data gap in water’s properties in the 248−148 K range. It is hoped that the HDL−LDL fluctuations view and the two-liquid model would be validated if only one could measure bulk water’s equilibrium properties in the 248− 148 K range. That has not been possible. We suggest that the model’s validity can also be assessed by connecting an equilibrium state’s property through a monotonic path that bridges the data gap in the 248−148 K range. If the estimated T-dependence of the property on that path is found to be internally consistent with the measured Tdependence of the property of bulk water, then the HDL−LDL fluctuation view, the liquid−liquid transition, and the two-liquid

■

AMBIGUOUS NATURE AND INSTABILITY OF HDAS For readers unfamiliar with the origin of the two-liquid model, we summarize its background. The model was an extension of Mishima et al’s 1984 finding23 that, under a load (uniaxial pressure) of ∼10−16 kbar, hexagonal ice kept inside a piston− cylinder apparatus at 77 K collapsed to an amorphous solid of density, ∼1.31 g mL−1, which they named high-density amorph, HDA.25 When HDA was heated at ambient pressure (1 bar), it converted to LDA,25 and when LDA at 130−140 K was pressurized through the ∼3 kbar range, it became HDA.34 (See details of curves a−c in Figure 2, ref 34.) At 1 bar, the density of HDA was found to vary between 1.105 and 1.25 g mL−1 (Figure 5, ref 28). Its value depends not only upon the time, temperature, and pressure protocols for collapsing ice crystals,21−24,26−28,30 but also upon the protocol of forced ejection of brittle HDA from the pressure vessel at 77 K. The forced ejection into liquid nitrogen causes strain-induced instability in solid HDA, whose consequences are observed in the HDA’s ex situ heating scan before it begins to anneal to its more stable state of LDA.11 Therefore, interpretation of the ex situ obtained data for HDAs at 1 bar pressure does not correlate with the in situ obtained data for HDAs at 10 kbar.24,37−40 At 1 bar pressure HDA is not formed by cooling HDL, nor HDL is formed by heating HDA. Therefore, HDA is not a glassy state. It behaves differently from a glass, and its thermodynamic state is neither well-defined nor unique, especially in view of the following: (1) The HDAs are regarded (i) as a poorly collapsed crystalline phase,21,37−42 (ii) as a B


Article


be heated in a time period of ∼24 s from 136 to 148 K at 30 K/ min in order to observe its “Tg-endotherm”.33 On heating, LDA begins to show thermal effects of crystallization to cubic ice at T ≤ 148 K.33 This T-range shifts to lower T when the heating rate of 10 K/h is used.31 To critically examine whether or not an unstable, timedependent state of LDA can be ergodic, we recall the concept of ergodenhypothese, which is attributed to Boltzmann, who described it in his papers on the kinetic theory of gases. (For citations and description of ergodenhypothese, see ref 46.) Boltzmann postulated that all microstates in phase space corresponding to the surface of constant energy can be, and are, accessed over a sufficiently long period of time. Accordingly, a system is defined as ergodic if the time average of its every measurable property gives the same result as the ensemble average. It is defined as nonergodic if it does not. No time scale was specified nor was any intermediate system state defined. In thermodynamic terms, a system is ergodic if its structure fluctuates, with equal probability, through all possible microstates consistent with a macrostate.46 However, when there is a broad distribution of relaxation times, not all modes of motion in an “ergodic” liquid come to equilibrium in a finite time. On the other hand, not all modes of motion kinetically freeze in a “nonergodic” glass, because localized motions of the Johari− Goldstein (JG) relaxation47 occur in its rigid structure, and homogeneous nucleation in some molecular glasses occurs in the regions of high molecular mobility of the JG relaxation.48 This seems inconsistent with ergodenhypothese which postulates that the state remains stable for an almost infinite period. In that sense, it seems that neither the ultraviscous LDL slowly nucleating and crystallizing to cubic ice in the 136−148 K range fulfills the requirements of ergodenhypothese, nor does the low-viscosity supercooled water instantaneously nucleating and crystallizing to hexagonal ice at T > 248 K. It is of some interest that cubic ice nucleates when ultraviscous LDL is heated beyond 148 K and hexagonal ice nucleates when water is supercooled through the 248 K range, as if the next lowenergy state is cubic ice for ultraviscous LDL and hexagonal ice for supercooled water at 248 K; the enthalpy of cubic ice is 35− 50 J/mol higher than that of hexagonal ice.

micro- and nanocrystalline mixture of high-pressure phases of ice,10,20,39 or (iii) as a high-density state that forms when pressure on a crystal exceeds the Born stability condition.29,38 Koza et al.19 performed very high-resolution X-ray scattering studies of the dynamic response of pressure-collapsed ice Ih, and concluded “Despite the apparent structural disorder of HDA, its dynamic response appears crystal-like with a close resemblance to ice XI pointing to an intriguing high degree of short range order.”19 (2) The HDAs crystallize on heating to at least six high-pressure phases of ice, namely, ices IV, V, VI, VII, VIII, and XII, and/or to crystallize to form their mixtures.43,44 Salzmann et al.33 concluded that, by controlling the pressure and the rate of heating during crystallization of HDA, one can obtain several of the high-pressure phases of ice that usually form on cooling water under a high pressure. These findings seem consistent with a previous suggestion10 that some HDAs may be a mixture of highly strained crystals of high-pressure phases of ice that are too small to show sharp Bragg lines in the X-ray diffraction, and when HDAs are heated from 77 K under different pressures, some of these small crystals grow competitively at the expense of others. (3) When hexagonal ice is pressurized at T higher than 77 K, up to 130 K, the pressure at which it begins to collapse is lower.37−40 This is the opposite of the liquid−glass transition pressure which is higher when T is higher. (4) Pressure needed to collapse ice is lower when the ice crystal size is small.10 (5) There is no evidence that HDA becomes HDL on isobaric heating or HDL reverts to HDA on isobaric cooling with a path hysteresis characteristic of glass−liquid−glass transition.12 Also, the cooling rate dependence of the endotherm’s onset temperature that was used as “a typical signature of a glass transition”13 is not a feature of the glass transition,12 and this alone ruled out the implication that HDA transforms to HDL on heating.12 (6) There is no latent heat of HDA to LDA conversion. At 1 bar pressure, HDA gradually becomes LDA on heating, but LDA does not transform back to HDA merely by slow cooling. However, the above given features of HDAs have been overlooked, and HDA and LDA are regarded as glassy states of HDL and LDL. In an intriguing manner, this has led to the hypothesis of a first order transition between HDL and LDL. In view of the ill-defined state of HDAs and no evidence of conversion on heating to HDL, we suggest that HDAs be regarded as collapsed states of hexagonal ice that appear noncrystalline in diffraction studies. In a detailed paper on thermodynamics of supercooled water, Holten et al.45 reviewed the experimental data and simulation studies, and analyzed the available data on ρwater, αwater, βwater, Cp,water, Cv,water, the speed of sound, and surface tension at T < 300 K.45 Despite the ill-defined state of HDAs, and lack of knowledge of variation of ρHDA and ρLDA (and ρHDL and ρLDL) with T at 1 bar and high pressures, they were able to theoretically model the thermodynamic data in terms of a virtual liquid−liquid critical point in supercooled water. They concluded that a phenomenological extension of the theoretical model can account for the data in the supercooled region, up to a pressure of 4 kbar (400 MPa). This seemed consistent with the two-state model, and now seems consistent also with the HDL−LDL fluctuation view.8 We mention one more aspect of LDL and its significance for our understanding of water as a mixture of two liquids. In the models fitted to thermodynamic data, ultraviscous LDL in the 136−148 K range is regarded as an ergodic state. This is despite the fact that it is unstable over periods of minutes and it had to

■

CONFIGURATIONAL AND VIBRATIONAL FEATURES AND THE STRUCTURE Properties of a solid are vibrational, and those of a liquid are configurational and vibrational. The configurational part is understood in terms of fluctuations in the local density at constant energy and constant volume, and interpreted in terms of a multiplicity of a liquid’s configurations. Each configuration corresponds to the state represented by one minimum among the numerous minima in a potential energy landscape used to describe thermodynamics of a liquid.49−51 The curvature of the minimum determines the vibrational features, and hence the vibrational part of the volume, enthalpy, entropy, elastic moduli, α, β, Cp, sound velocity c, and thermal conductivity κ, and the number of minima accessible at a given T determine the configurational part of the same properties. As a glass is heated toward the glass to liquid transition temperature, Tg, the phonon frequencies, vibrational amplitude, and anharmonic forces associated with the vibrations change, ρ slightly decreases, and α, Cp, and κ slightly increase. On continuous heating through T ≥ Tg, the configurational part begins to increase more rapidly than the vibrational part. At the heatingrate-dependent Tg, the enthalpy, entropy, and ρ do not change, C


Article

The Journal of Physical Chemistry B but α and Cp increase in a heating-rate-dependent, sigmoidshape manner. These features of a glass and liquid52−58 are illustrated for ρ, the bulk modulus, and refractive index in Figure 1A; for the volume, enthalpy, and entropy in Figure 1B; and for α and Cp in Figure 1C.

Figure 1. Illustration of the plots of the various properties in the glass−liquid transition range on heating the glass. (A) Plots of the density, refractive index, and bulk modulus. (B) Plots of the volume, enthalpy, and entropy. (C) Plots of the expansion coefficient and heat capacity. The plots are based upon the description and figures in the literature.52−58 The glass−liquid transition is indicated as Tg. Its value depends upon the rate of heating, and it is higher when the heating rate is higher.

Continuous increase in the LDL fraction decreases ρwater, which causes the ρwater − T plot to deviate downward, thus producing the ρwater maximum at 277 K.8 To express it in terms of structural change, when water is cooled from 314 K, the interstitial H2O molecule diffuses out of the first coordination shell of the HDL structure, which then rearranges to the coordination shell of LDL. An H2O molecule that had diffused out would add to the LDL cluster formed in its wake, or initiate formation of a new LDL cluster. The fraction of LDL at equilibrium with HDL at 277 K is not known, but there is more LDL than HDL, and there are “LDL patches that become more extended with decreasing temperature”.8 If ambient water is already a mixture of HDL−LDL clusters, then the properties of ambient LDL cannot be determined. That makes it difficult to estimate the equilibrium population of the HDL and LDL clusters from thermodynamic data. In contrast, mixture models1,2,6 attribute the water’s density maximum to two competing effects: (i) weakening or breaking down of the four coordinated structure of H2O molecules on heating, which increases ρwater, and (ii) increasing of the amplitude of anharmonic intermolecular vibrations that decrease ρwater. To express it as an effect of cooling from a high T, strengthening or formation of the four coordinated structure of H2O molecules decreases ρwater, and decrease in the amplitude of anharmonic intermolecular vibrations increases ρwater. Properties of LDA and HDAs are entirely vibrational, but those of water, LDL, and HDL are partly configurational and partly vibrational. So, although the two-liquid model appears similar to the mixture models of the 1960s,1,2,6 in reality it differs. The reason is that the two-liquid model would have four types of density fluctuations: (i) fluctuations within the HDL structure at fixed ρHDL, (ii) fluctuations within the LDL structure at fixed ρLDL, (iii) fluctuations that interconvert HDL and LDL at a fixed ρwater, as in a chemical process at equilibrium, and (iv) fluctuations of hydrogen-bonded structure at the HDL−LDL interface. The first two are intracluster fluctuations, the third is intercluster, and the fourth is interface fluctuations. Their time scales would differ at a fixed T, and all would contribute to water’s configurational properties, but their relative contribution would change differently with change in T.

Structural features of HDA obtained from X-ray scattering studies59 showed that its high density is due to the presence of an interstitial molecule, i.e., there is one “non-hydrogenbonded” H2O within the first coordination shell of the hydrogen-bonded structure, which was later confirmed by more detailed studies of HDA.60 (Note that, in such structure determination, one simplifies the analysis by expressing a site with a small probability of occupation by one H2O molecule as an interstitial.) The structure of HDL at ambient conditions is taken to be the same as the structure of one of the HDAs at 77 K, which has no configurational contribution to its thermodynamics, and the vibrational contribution is relatively small. The structure of LDL is taken to be the same as the structure of LDA at 77 K. At T > 314 K, water at 1 bar is HDL, or HDL with a negligibly small amount of LDL.8 (Thermodynamic equilibrium requires that both HDL and LDL be present, with decreasing fraction of LDL with increase in T.) According to Pettersson and Nilsson,8 when water is cooled from ∼314 K, H2O molecules begin to form increasingly more (or bigger and bulkier) patches arranged in the LDL structure (Figure 2, ref 8), thus leading to larger and more extensive fluctuations.8

THERMODYNAMIC PATHS IN THE DATA-GAP REGION Density and Thermal Expansion Coefficient. We first consider ρwater and αwater, the two anomalous properties of bulk water recognized centuries ago. As already noted here, ρwater is known only at T > 250 K, ρHDA at 1 bar is 1.17 g mL−1 at 77 K, and ρHDL is not known at any T or P.11,12 According to the HDL−LDL fluctuation view, water at 1 bar is HDL or mostly K HDL at T > 314 K, so ρwater at 320 K, ρ320 water , would be ρHDL at 61 320 K 320 K 320 K, ρHDL . From Kell’s data, ρwater is 0.989 36 g mL−1.61 K 77K Figure 2 shows the data points for ρ320 water and ρHDA. On the assumption that HDA is a glass, one expects ρHDA to 77K K decrease on heating from ρHDA of 1.17 g mL−1 to ρ320 HDL 320 K −1 (=ρwater ) of 0.9894 g mL in the manner illustrated in Figure 1A, i.e., to decrease slowly on heating up to a certain “presumed Tg” for a given heating rate, and then to decrease normally for a liquid. Accordingly, ρ77K HDA would first decrease to the density of HDA at Tg according to αHDA on the path labeled 1 in Figure 2 until the “presumed Tg” is reached. At this “Tg”, ρHDA = ρHDL. On further heating, ρHDL would decrease more rapidly along a 320 K continuous path to ρwater , like that of a normal liquid.

■

D


Article


Figure 3 and search for a plausible path between βHDA at 100 K and βwater (=βHDL) at T > 320 K. We determine βHDA at 100 K

Figure 2. Density of water at 320 K and of HDA at 77 K is plotted against the temperature. Two possible monotonic (continuous) paths, 1 and 2, are shown between water at 320 K and HDA at 77 K. Neither of these two paths yields a thermal expansion coefficient that would be consistent with the measured value61 for water at 320 K.

Figure 3. Compressibility of water plotted against the temperature. Data are taken from Kell.61

According to the data obtained by heating at the slower rate of 10 K/h in Figure 1 and Table 1 in ref 31, we use ∼114 K for this “Tg”. By taking αHDA as 9 × 10−5 K−1 for a typical −1 K [=1.17 − {9 amorphous solid, we obtain ρ114 HDA of 1.166 g mL −5 × 10 × 1.17 × (114 − 77)}], which would be the same as −1 K 114 K ρ114 to HDL . On heating, ρHDL would decrease from 1.166 g mL −1 K 320 K ρ320 (=ρ ) of 0.9894 g mL . From the slope of path 1 water HDL K 320 K between ρ114 and ρ in Figure 2 and the average ρ of HDL water HDL 1.078 g mL−1 (=(1.166 + 0.9894)/2) between 114 and 320 K, K −4 −1 we calculate ρ320 K [=(1.078) × {(1/0.9894) HDL as 8.0 × 10 K − (1/1.166)/(320 − 114)}]. We also estimate ρ320 HDL for an improbable case in which ρHDL follows path 2 from ρ77K HDA to −4 K 320 K ρ320 K−1, water in Figure 2. This path yields αHDL as 6.9 × 10 320 K which should be the same as αwater . The measured value of K −4 −1 61 K α320 K . Therefore, estimates of α320 water is 4.36 × 10 water of 6.9 −4 −1 −4 −1 × 10 K and 8.0 × 10 K based on the HDL−LDL K mixture model are 1.6−1.8-times the measured α320 water . The 61 uncertainty of ρwater is less than 0.1%, and that of ρHDA is at most 2%. These do not significantly affect our finding. We also consider other paths between HDA and water. Instead of being linear, if paths 1 and 2 in Figure 2 were to be convexly curved to the T-axis (lowest negative slope being at K 320 K), α320 water would be less than the measured value of 4.36 × −4 −1 K 77K 10 K , and α114 HDL and αHDL would be higher than 4.36 × −4 −1 10 K . Hence, αwater would seem to increase on cooling, which is the opposite of the decrease found for normal liquids. Alternatively, if paths 1 and 2 were concavely curved to the TK axis (highest negative slope being at 320 K), α320 water would be −4 −4 −1 more than the estimated 6.9 × 10 and 8.0 × 10 K , which are already more than the measured value of 4.36 × 10−4 K−1, −4 −1 K 77K and α320 K . So, HDL and αHDL would be less than 4.36 × 10 320 K neither the linear nor the curved paths give αwater close to the measured value. In summary, when water is taken to be HDL at K T > 320 K, the estimated α320 water is inconsistent with the K measured α320 . We could not search for a similar path water between LDA and water because only ρLDA at 77 K and 1 bar is known; ρwater at temperatures where water would be mostly LDL is not known. Isothermal Compressibility. For the usual analysis of the compressibility of water at a fixed T, we require the ρwater value at 1 bar, which is available, and the ρHDA or ρHDL values at high pressures, which are not known. So, we do an alternative analysis. We plot Kell’s data61 for βwater against T at 1 bar in

and 1 bar from Gromnitskaya et al.’s data.26 Their data show that (i) at 77 K and 1 bar (0.1 MPa) pressure, the bulk modulus of HDA, BHDA, is 10 GPa (Figure 4, ref 26); (ii) on heating at 0.5 kbar (0.05 GPa), BHDA decreases, from 10 GPa at 77 K to 9.8 GPa at 100 K (Figure 5, ref 26); and (iii) on depressurizing from 0.5 kbar to 1 bar, BHDA decreases by an insignificant amount (Figure 4, ref 26). By combining these, BHDA is 9.8 GPa at 100 K and 1 bar, or βHDA (= 1/BHDA) is 10 ± 0.1 Mbar−1. This value along with βwater from the literature61 are plotted against T in Figure 3. It is known that the ρwater maximum gradually vanishes on increase in pressure.1−6 Therefore, one would not expect βHDL and βLDL to be additive in an HDL−LDL mixture. However, if HDL were to behave as a normal liquid,8 βHDL would vary with T normally; i.e., βHDL would decrease with decrease in T progressively more slowly. (Details of such a decrease are given in the subsequent paragraph in parentheses.) Since water at T > 320 K is seen as HDL, βwater would be βHDL at T > 350 K, and the path from βHDL at T > 350 K to βHDA at 100 K would be curved, with its slope decreasing with decrease in T. However, the path from T > 350 to 100 K in Figure 3 is linear. Since βHDL would be higher than βHDA at T above the “presumed Tg”, we allow for argument here a 50% higher value for βHDL at 130 K and add this data point in Figure 3. The extrapolated path between βHDL at 130 K and βwater still shows no curvature. In conclusion, the path from βwater at T > 350 K to βHDL or βHDA is incompatible with the view that water is HDL at T > 320 K. (Two equations have been used for relating β with T. One is β = (1/c 2ρ) + (αTV/Cp),62 which contains the term T. The second is β = γ/ρc 2 given later in this paper, which does not contain the term T. Diaz Pena and Tardajos63 measured β of 12 liquids as a function of T. As it is easier to see the T dependence of β from their studies, we quote some of the data from their Table 2:63 For benzene, the β values are 966 TPa−1 at 298.15 K, 1044 TPa−1 at 308.15 K, 1128 TPa−1 at 318.15 K, and 1277 TPa−1 at 333.15 K. For n-hexane the β values are 1669 TPa−1 at 298.15 K, 1831 TPa−1 at 308.15 K, 2027 TPa−1 at 318.15 K, and 2385 TPa−1 at 333.15 K.63 The data show that β decreases on cooling, and the slope of the β against T plot decreases with decrease in T.) Heat Capacity. We now search for a continuous path between Cp,HDA at low T and Cp,water at T high enough that E


Article

The Journal of Physical Chemistry B water would be HDL. Figure 4A shows the plot of Cp,water against T in the 253−373 K range and the data for Cp,HDA ∼

generate a density maximum of the mixture. Such a change seems implausible. For further analysis, we plot ρwater from the data in Table 361 against T in Figure 4B to show a broad ρwater maximum. In this plot, there are numerous pairs of temperature at which ρwater is the same. One such pair is 293.16 and 262.8 K at which ρwater = 0.9982 g mL−1 (Table 3, ref 61). Obviously, two temperatures for the same ρwater are a consequence of the density maximum, and this maximum is attributed to the T-dependent variation in the relative amounts of normally behaving HDL and LDL.8 Therefore, we need to reconcile the occurrence of two such temperatures with the HDL and LDL fluctuations view.8 There are two ways of doing so: One is to assume that the relative amounts of fluctuating HDL and LDL at the two temperatures are the same on average, and the second is to assume that the relative amounts of HDA and LDA differ at the two temperatures, but the individual ρHDL and ρLDL are such that their combined amounts produce the same ρwater. The first requires that the relative population of LDL clusters be the same at both temperatures, which is inconsistent with the view that LDL clusters form and grow at the expense of HDL on cooling. The second, as argued above, is numerically implausible. Next we consider the minimum in βwater seen in the βwater−T plot in Figure 3, constructed also from the data listed by Kell.61 The minimum appears at βwater of 44.15 Mbar−1 at ∼320.7 K, and there are two temperatures, 363 and 284.6 K, at which βwater has the same value of ∼47.43 Mbar−1 (Table 3, ref 61). As mentioned earlier here, βHDL and βLDL cannot be linearly additive, and for normal liquid-like behavior of LDL, dβLDL/dT would not be negative. At T > 340 K, water is seen as HDL, and therefore, dβHDL/dT = dβwater/dT, and it is positive. One would not expect that increase in the LDL fraction on cooling from 363 K would cause the positive value of dβHDL/dT to become zero and then negative, overcompensating for the positive value of dβLDL/dT. One may argue that βLDL may not change monotonically with T. If so, LDL would not be a normal liquid. Lastly, we consider two temperatures across the minimum in Cp,water. The data in Figure 4A show a broad miminum at ∼315 K where Cp,water is 75.29 J mol−1 K−1. Here at 353 and 263 K, Cp,water is the same, 75.6 J mol−1 K−1. As water at T > 350 K is seen to be HDL, the known positive dCp,water/dT value at T > 350 K would be the positive dCp,HDL/dT value. Normal liquid behaviors of HDL and LDL require that dCp,LDL/dT have the same sign as dCp,HDL/dT, i.e., both be positive. So, as the LDL fraction in water increases on cooling from 353 to 263 K at the expense of HDL, dCp,water/dT as a mixture of HDL and LDL would remain positive. It may increase or decrease with change in T, but it would not become negative, and would not produce two temperatures with the same Cp,water. Therefore, the minimum in the Cp,water−T plot is also incompatible with the two-liquid model and/or the normal liquid behavior of HDL and LDL. Thermal Conductivity. As part of the background information, we recall that thermal conductivity, κ, of a crystalline solid is determined by the properties associated with the propagation of phonons.65,66 It is interpreted in terms of the equation κsolid = ρ Cντs, where ρ is the density, C the specific heat contribution from phonons, ν the phonon propagation velocity, and τs the time between two scattering events, which is limited by the T-, and P-invariant structure of a solid. Kittel65 theoretically discussed the magnitude of κ of glasses, and its T-dependence, and showed that the phonon

Figure 4. (A) Heat capacity of water plotted against the temperature. Data are taken from ref 4. (B) The plot of the density against the temperature showing the density maximum and identical values of the density at two temperatures. Data are taken from Kell.61

12.2 J mol−1 K−1 at 132 K, which is the maximum Cp,HDA reported in Figure 1, ref 64. (Whether or not the relatively rapid increase to Cp,HDA of ∼12.2 J mol−1 K−1 at 132 K64 is due to structure unfreezing, or due to some other endothermic effect prior to the onset of the deep exotherm, is not known.) The data in Figure 4A show that a linear path of Cp,water from T > 353 K, where water is seen as HDL,8 to 132 K yields Cp,HDA of ∼71 J mol−1 K−1 at 132 K. This value is much higher than the measured value of ∼12.2 J mol−1 K−1. If HDAs were glass, Cp,HDL would be higher than Cp,HDA (Figure 1C). So, we add an arbitrary value of 8 J mol−1 K−1 to 12.2 J mol−1 K−1, and include a data point at 20.2 J mol−1 K−1 at 132 K in Figure 4A. The extrapolated 71 J mol−1 K−1 value for HDA is still much higher. We conclude that there is no plausible path between Cp,HDA or Cp,HDL at 132 K and Cp,HDL (=Cp,water) at T > 353 K. Two Temperatures for the Same ρ, β, and Cp of Water. We now investigate whether the maximum in ρwater and the minimum in βwater and in Cp,water are consistent with the HDL− LDL fluctuation view or the two-liquid model. In general, a maximum or a minimum in the plot of a property, say x, against T appears only if the relative proportion of the components of the mixture changes very strongly with T, compensating for the normal T-dependence of the property x of each component. For example, assuming that dρ/dT is negative for LDL and HDL, i.e., both behave as normal liquids, the composition of the mixture should change dramatically with T in order to F


Article

The Journal of Physical Chemistry B concepts and the Debye equation are also applicable to glasses. On heating, κ of crystals decreases, and κ of the glasses increases. It is now generally accepted that κ of glass and other disordered solids is also determined by phonon propagation.66 In contrast, heat transport in liquids is affected by the changing local configurations, or density fluctuations. Bridgman67 used a simple physical picture of “the total energy transferred across a fixed point of any row of molecules per unit time as the product of the energy difference and the number of such energy steps contained in a row of molecules...”67 in a certain length. He thus deduced the first theoretical relation between κ, the sound velocity c, and molecular volume Vm (molecular mass divided by the density)67 ⎛ c ⎞ κliquid = 2.8kB⎜⎜ 2/3 ⎟⎟ ⎝ Vm ⎠

Figure 5. Thermal conductivity of water is plotted against the temperature with source of data from refs 75 and 76, as given. The continuous line is a guide to the eye. Also plotted is the thermal conductivity measured during the heating of HDA for sample A: (□) at 0.04 GPa at 7 K/h. For sample B: (∇) at 0.05 GPa at 12 K/h. The characteristic values of κ for HDA and LDA are indicated. These samples were prepared by freezing a new sample of water and forming HDA by isothermal pressure-collapse of hexagonal ice at T near 130 K. Data are taken from ref 24. The HDA and LDA samples had to be kept under slight pressure so that thermal contact between the sample and the hot wire used to determine κHDA and κLDA could be maintained. Therefore, one set of data is at 0.04 GPa (0.4 kbar) obtained on heating at 7 K/h, and the second set is at 0.05 GPa (0.5 kbar) obtained on heating at 12 K/h. Note that depressurizing at 130 K decreases κHDA slightly, and increases κLDA in Figure 1, ref 77. Therefore, κLDA at 1 bar and 130 K would be higher than the plotted κLDA.

(1)

where kB is the Boltzmann constant. According to Hirschfelder et al.,68 Eyring also gave a formula ⎛ ⎞ ⎛ 1 ⎞1/2 c κliquid = 2.8kB⎜⎜ 1/2 2/3 ⎟⎟ or κliquid = 2.8kBVm−2/3⎜ ⎟ ⎝ ρβ ⎠ ⎝ γ Vm ⎠ (2)

where γ = Cp/Cv, Cv is the heat capacity at a constant volume, β the isothermal compressibility, and ρ the density. According to eqs 1 and 2, κliquid varies with T as Cp, Cv, Vm, ρ, and β, or c = [γ/ρβ]1/2 vary with T. Biddle et al.69 used eqs 1 and 2 to relate κwater to interconversion of the HDL and LDL structures. They fitted an equation for κwater approaching a critical point, and suggested that κwater is strongly correlated with the anomalies of the thermodynamic properties associated with the existence of a liquid−liquid transition, noting that “The anomalous behavior of the speed of sound probably results from the existence of a region of the phase diagram in which it is thermodynamically rather inexpensive to convert water between the HDL to LDL structures, leading to a higher compressibility.”, and then concluding “On the other hand, the thermal conductivity and thermal diffusivity of supercooled water are strongly correlated with the anomalies of the thermodynamic properties associated with the existence of a liquid−liquid transition.”69 At high temperatures, molecular relaxation in liquids with or without hydrogen bonds occurs on the time scale of 10−11 to 10−9 s. It is known that κliquid decreases with increase in T at such temperatures,70,71 i.e., dκliquid/dT is negative, and that when a liquid vitrifies on supercooling,72 dκliquid /dT changes from negative to positive, or else when a glass is heated through Tg, dκliquid/dT changes from positive to negative.73 It is also worth noting that κliquid of binary mixtures, with or without intermolecular hydrogen bonds between the same type or different type of molecules, increases nonlinearly with increase in the amount of the component of higher κ.74 Figure 5 shows Benchikh et al.’s75 and Taschin et al.’s,76 data for κwater plotted against T. In the 250−350 K range, where density fluctuations occur on the time scale of 10−11 s (somewhat longer time scale at lower T) κwater increases with increase in T, i.e., dκwater/dT is positive. Since dκliquid/dT is usually negative, the increase in κwater with increase in T is unusual. A previous study had shown that κwater at 0.07 GPa (0.7 kbar) also increases on heating from 273 K (Figure 2, ref 77). Like the thermodynamic properties, κwater is determined by the structure as well as the dynamics. So, one expects that the

apparently anomalous dκwater/dT would be qualitatively related to the anomalous βwater, or to cwater = (γwater/ρwaterβwater)1/2; i.e., if Cp,water and βwater show an indication of divergence (Cv,water does not diverge) and the divergence is qualitatively explained by a two-state structural model, the data on κwater should also be explained by that model. However, it is difficult to see how, without knowing the values of γ, β, and ρ (or of c alone) of both HDL and LDL at ambient conditions, κwater could be related to the HDL−LDL fluctuations view. Figure 5 shows also the plots of κHDA and κLDA taken from Figure 1, ref 24. In that study, HDA was annealed to form LDA by heating from 80 K at the rate of 7 K/h and of 12 K/h. Thermal conductivity increased from κHDA of ∼0.6 W m−1 K−1 to κLDA of ∼1.2 W m−1 K−1. The plots also show that κHDA at 0.5 kbar and 100 K is slightly less than κwater at 1 bar and 298 K. As water at T > 320 K is mostly or all HDL behaving as normal liquid, a continuous path for κwater from T > 320 K to T near 120 K should yield κHDA at 120 K. In Figure 5, the linear path yields ∼0.5 W m−1K−1 at 120 K, which is significantly less than the measured value of ∼0.6 W m−1 K−1. Also, κLDA at 0.5 kbar is about twice the value of κwater at 1 bar and 298 K, and as κLDA increases on depressurizing,77 κLDA at 1 bar would have to be more than twice the value of κwater, as explained in Figure 5 caption. According to the HDL−LDL fluctuations view,8 water at T > 320 K would have mainly HDL patches, which would be large enough that phonon propagation can occur. Cooling to T below 314 K would produce increasingly more (and/or bigger) patches of H2O arranged in the LDL structure.8 Since HDA to LDA conversion increases κ, one expects that HDL to LDL G


Article

The Journal of Physical Chemistry B conversion would also increase κ, and κwater would increase on cooling. Figure 5 shows the opposite. In the context of fluctuations between two structures, it is worth discussing the types of fluctuations that occur in binary liquid mixtures and of their effect on κ. For example, in the water−methanol mixtures74 (see Figure 1, ref 74), concentration fluctuations would involve breaking and reforming of intermolecular hydrogen bonds mainly between the H2O molecules at low concentration of methanol, between CH3OH molecules at high concentrations, and between H2O and CH3OH molecules at intermediate concentrations. In this case, increasing the mole fraction of CH3OH, which is the component of lower κ, was found to decrease κ of the mixture.74 However, the study of another alcohol−water mixture, namely, C2H5OH−H2O, by Conde et al.78 showed a minimum in the adiabatic compressibility, βs, isotherms of the mixture (Figure 3, ref 78). The plots78 thus showed that βs is the same at two compositions of the mixture. In a previous study74 performed at a fixed T of 313 K, it was found that κ of C2H5OH−H2O mixture decreased monotonically with increase in C2H5OH over the entire mole fraction composition. It is possible that the change in the ratio, (1/Vm2/3ρ)1/2, dominates the change in βs in the C2H5OH−H2O mixtures, or else there is an unresolved interaction between C2H5OH and water that appears as a minimum in βs without an anomaly in κ, and it is absent in the CH3OH−H2O mixtures. If the HDL−LDL clusters were to behave like the hydrogenbonded clusters in the CH3OH−H2O mixtures, increase in the fraction of LDL on cooling ambient water would continuously decrease κwater from its value at 250 K to κLDA at 142 K. The plots in Figure 5 show that κLDA at 142 K is ∼1.2 W m−1 K−1 and κwater at 242 K is ∼0.49 W m−1 K,−1 i.e., instead of being lower, κLDA at 142 K is 2.5-times κwater at 250 K, where it would be mostly LDL.8 The magnitude of κLDA and the manner of variation of κwater with T, therefore, do not support the view that fraction of LDL increases on cooling. We find no plausible continuity between κwater and κHDA or between κwater and κLDA.

would then be comparable to the (conformational) equilibrium of two structural states of different densities of a large-size molecule with the relative population of the two states of the molecule itself (or of some of its segments) changing with T and the pressure, P. The equilibrium would be analogous to the intermolecular hydrogen-bond association equilibrium, such as that found between the bulkier (hydrogen-bonded) ring-dimers and (compact) monomers or the (compact) linear-dimers that form in, for example, acetic acid and in some alcohols.79 Sellberg et al.7 reported an X-ray diffraction study of micrometer-size water droplets. By making use of the natural phenomenon of evaporative cooling, they were able to supercool the droplets to significantly lower temperatures for their study. We point out that evaporative cooling concentrates impurities in a liquid. Increase in the impurity concentration changes the viscosity and the intermolecular hydrogen-bond equilibrium of water, extends the supercooling range by preventing crystallization, and changes the crystal phases formed on crystallization of the supercooled state, which may be cubic ice, hexagonal ice, their mixtures, or ice crystals with stacking faults. (A review of the freezing to such crystals was given recently.80) We also recall that, in an investigation of the effect of evaporation-enhanced impurities on the dielectric relaxation of hexagonal ice,81 it was found that when ∼100 mL of water at 298 K contained in a flask was “evacuated” by continuous pumping for several hours, it froze from the surface downward, and the hexagonal ice thus formed had enough impurities to decrease its dielectric relaxation or the orientation fluctuation time by several orders of magnitude. In a subsequent study by X-ray diffraction, Laksmono et al.15 reported crystallization of micrometer-size water droplets that had been evaporatively supercooled at an estimated rate of 103 to 104 K/s. The data appeared to show a slower increase in the nucleation rate on cooling the droplets below the estimated temperature of 232 K than anticipated on the basis of previous studies. They interpreted it as a result of rapid decrease in water’s diffusivity at T < 232 K, and regarded it as being consistent with a previous finding that micrometer-size droplets of water do not crystallize at T < 227 K when cooled at a rate of 106 to 107 K/s, but vitrify. Hence they hypothesized that the slower increase in the nucleation rate is connected with the proposed “fragile-to-strong” transition anomaly in water.15 Because of the known impurity enrichment on evaporative cooling and its consequences for water’s properties, we suggest caution in interpreting the nucleation and crystallization features of evaporatively supercooled droplets. For historical reasons, we should note that Gromnitskaya et al.26 had not only found that HDA further densifies on heating under pressure, producing a solid similar to that now known as VHDA35 (see also their earlier papers cited in ref 26), but also found that when hexagonal ice under 0.77 GPa pressure is heated from 77 K, its structure collapses on (isobaric) heating. Johari and Andersson24 later reported that when partially collapsed hexagonal ice is kept at 0.8 GPa and 130 K, it further collapses gradually with time isobarically and isothermally, indicating that a combination of P and T determines the rate of collapse of ice to HDAs. These findings were recently confirmed by Handle and Loerting,82 who reported that on heating from ∼77 to 155−170 K, hexagonal ice kept under a pressure of 9.0−9.5 kbar (0.9−0.95 GPa) isobarically converts to VHDA. The experiments neither show that the thus formed HDA is a glass, nor show it to be one out of the many known HDAs.

■

DISCUSSION An illustration of the thermally excited (pure) HDL liquid at high temperatures, fluctuations into tetrahedral patches in the HDL-dominated liquid, fluctuations into close-packed, HDLlike structures in the LDL-dominated liquid, and pure LDL at very low temperatures has appeared in Figure 2 in ref 8. We argued here that there would be a third structure, which would be at the interface of the LDL and HDL patches. Its contribution to thermodynamics and spectral features would be minimum or negligibly small when T is such that there are only a few nanometer-size patches of LDL or HDL and the total interfacial area or volume is small, but significant when T is such that the LDL patches at equilibrium are large and are as populated as the HDL patches. So, as the size of the patches changes with T, the ratio of the number of molecules at the interface to the number of molecules in the cluster’s bulk would change, adding to the change in both the thermodynamic and scattering features. Also, each cluster may have fluctuations within its own structure, which would contribute, depending upon its size, differently to the properties of water and their Tdependence. Although the (fluctuating) clusters of HDL and LDL structures interconvert at a fixed T, they remain at thermodynamic equilibrium without a change in the macroscopic volume and energy of liquid water. This equilibrium H


Article


fluctuations in water than the hypothesized fluctuations between HDL and LDL structures.8 (vii) Enhanced density fluctuations may be more plausible than the HDL−LDL fluctuations. It is well-known that properties of a glass depend upon the cooling and/or pressurizing rate54−58 during its formation and further on its annealing protocol. So, one may argue that, at least in that respect, HDAs resemble a glass. However, HDAs have not shown the characteristic thermodynamic features of a glass. We suggest that, in order to resolve the issue of the illdefined state of HDA, one may attempt to supercool bulk water at a fixed pressure of say 10 kbar through the phase boundary of ice VI and obtain ultraviscous liquid that in turn would vitrify on further cooling at that pressure. The glassy state formed at 10 kbar may be cooled to 77 K and depressurized to 1 bar, and the solid is extracted from the pressure vessel, and ex situ studied by calorimetric, spectral, and scattering methods. If it is found to show the same properties as HDA and also to anneal to LDA on heating, HDA may be similar to a glassy state of water. If that is not possible, then one more experiment with the HDAs and LDAs is needed. To elaborate, calorimetric measurements have been performed only by heating the samples to an apparent liquid state,33 but not on cooling the apparent liquid state to a glass in order to observe the thermal hysteresis of Cp, α, or other properties that characterize the glass−liquid−glass transition. In view of the (mistaken) second Tg of water deduced from the 16-K-wide, partial Cp-endotherm of a especially treated HDA, and its unjustifiable dependence on the prior cooling rate,12,13 it now seems necessary to ascertain the source of the observed Cp-endotherms by obtaining data on both the heating and the cooling paths between the apparent glassy and liquid states of the HDAs and of the LDAs, repeatedly, and perhaps also of hyperquenched glassy water and vapor-deposited amorphous solid water. Such measurements are now easily made by using commercial equipment. The available data show that HDA is not a glass, and the presumed HDL and LDL are not normal liquids. If that is accepted, the HDL−LDL fluctuations view, the two-liquid model, and the virtual liquid−liquid phase transition8 would all be consequences of a false premise, and reconsideration in the light of any of the above listed seven aspects would be unnecessary. Nevertheless, the methodology used to reach our conclusion would remain significant and useful. We suggest that our methodology may be used for constraining the number of inferences obtained when there is a data gap in the properties of a material over a wide T-range, and for resolving the debate on the inferred ergodic−nonergodic transitions of SiO2 and GeO2.84

It is possible that our findings merely show that the illdefined HDAs are not the glassy states of the presumed HDL. Structure factors obtained from detailed X-ray and neutron scattering experiments have already shown that there are five high-density amorphous solids and many more are possible.16 Also, neutron diffraction experiments on LDA have shown that there is not one but two LDAs, named LDAI and LDAII, which differ in their structure factor as well as the compressibility.83 Moreover, a calorimetric study of the HDA made by isobaric heating of hexagonal ice at P above 8.5 kbar (0.85 GPa) showed that when thus formed HDA was ex situ heated at 10 K min−1 rate at 1 bar, the onset temperature of its conversion to LDA was 129 ± 1 K and that of LDA’s crystallization to cubic ice was 166 ± 0.5 K.82 These onset temperatures differ from the corresponding ones for the two other HDAs and LDAs heated at the same rate of 10 K min−1, as reported in Figure 2, ref 11. This indicates that the HDA and LDA formed by isobaric heating82 are different from those in the previous studies (Figure 2, ref 11). If this finding is independently verified, it would demonstrate that LDA also is an ill-defined state. A thermodynamic analysis30 based on the free energy estimate had shown that LDA differs from the glassy water formed by hyperquenching its micrometer-size droplets. So, the protocoldependent LDAs would differ also from glassy water. Lastly, we recall that Holten et al.45 had used the same thermodynamic data of water that we did, but they reached a different conclusion. The difference is due to the fact that they45 made an effort to fit equations for approach of the data to a critical point, and we allowed the data to guide us. It is possible that by using nonsimple or abnormal behavior of HDL and LDL, the two-liquid model may fit the existing thermodynamic data at T below the homogeneous nucleation temperature. But that fitting would still require evidence that HDA is a glass that forms by cooling HDL at 1 bar or a higher pressure. Unfortunately, no such evidence has been found in the 31 years period since Mishima et al.’s study in 1984.23

■

CONCLUSION On the basis of the premise8 that HDL and LDL are formed by heating HDA and LDA and behave as normal liquids, we investigated whether the properties of water are consistent with the HDL−LDL fluctuations view and the related two-liquid model. The conclusions are as follows: (1) The ρwater, βwater, and Cp,water values at high temperatures, where water was suggested to be mostly or all HDL, are inconsistent with the measured values of HDA, and κwater is inconsistent with both κHDA and κLDA. These findings are incompatible with the HDL− LDL fluctuations view and the related model. (2) The same ρwater, βwater, and Cp,water values at two temperatures across their respective maximum or minimum are incompatible with the HDL−LDL fluctuations view. (3) The inference obtained from X-ray scattering studies7,8 needs to be reconsidered after taking into account the following features: (i) Properties of HDA are protocol-dependent, and HDA itself is an ill-defined state, not a glass. (ii) The density of HDAs is in the range 1.105−1.25 g mL−1 at 1 bar.28 (iii) HDA anneals to LDA, rapidly on heating at 1 bar; there is no latent heat of HDA−LDA conversion. (iv) HDA to LDA conversion on annealing at 1 bar occurs in several steps, and there are numerous HDA structures between the states of HDA and LDA.16−18 (v) There is no evidence to support the hypothesis that HDL−LDL transformation is reversible and HDL forms on heating HDA at 1 bar, or HDA forms on cooling HDL. (vi) There are more local density

■

AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Eisenberg, D.; Kauzmann, W. The Structure and Properties of Water; Oxford University Press: Oxford, U.K., 1969. (2) Davis, C. M., Jr.; Litovitz, T. A. Two-State Theory of the Structure of Water. J. Chem. Phys. 1965, 42, 2563−2576. I


Article

The Journal of Physical Chemistry B (3) Mallamace, F.; Corsaro, C.; Chen, S.-H.; Stanley, H. E. Transport and Dynamics in Supercooled Confined Water. In Liquid Polymorphism; Stanley, H. E., Ed.; John Wiley: New York, 2013. Adv. Chem. Phys. 2013, 152, 203−262. (4) Chaplin, M. F. Water Structure and Behavior; http://www.lsbu.ac. uk/water/index.html. Date accessed: 22 Sept 2015. (5) Angell, C. A. Supercooled water. Annu. Rev. Phys. Chem. 1983, 34, 593−630. (6) Davis, C. M., Jr.; Jarzinsky, J. Mixture Models of Water, in Water and Aqueous Solutions, Structure. Thermodynamics and Transport Properties; Horne, R. A., Ed.; Wiley Interscience: New York, 1972, Chapter 10, pp 377−423. (7) Sellberg, J. A.; Huang, C.; McQueen, T. A.; Loh, N. D.; Laksmono, H.; Sierra, R. G.; Hampton, C. Y.; Starodub, D.; DePonte, D. P.; Feldkamp, J.; et al. Ultrafast X-Ray Probing of Water Structure Below the Homogeneous Ice Nucleation Temperature. Nature (London, U. K.) 2014, 510, 381−384. (8) Pettersson, L. G. M.; Nilsson, A. The Structure of Water; From Ambient to Deeply Supercooled. J. Non-Cryst. Solids 2015, 407, 399− 417. (9) Chandler, D. Illusions of Phase Coexistence: Comments on “Metastable Liquid-Liquid transition... by J. C. Palmer et al., Nature 2014, 510, 385. arXiv preprint at arXiv:1407.6854v2 [cond-mat.statmech] 20 Aug 2014. Accessed: 23 Sept 2015. (10) Johari, G. P. On the Amorphization of Hexagonal Ice, the Nature of Water’s Low-Density Amorph, and the Continuity of Molecular Kinetics in Supercooled Water. Invited Lecture. Phys. Chem. Chem. Phys. 2000, 2, 1567−1577. (11) Johari, G. P. Comment on: “Relaxation Time of High-Density Amorphous Ice by P. H. Handle, M. Seidl, T. Loertin. Phys. Rev. Lett. 2012, 108, 225901.”; The α-Relaxation Time of Strained State of High-Density Amorphous Ice at T < Tg, its Tg, and its Transformations. Thermochim. Acta 2014, 589, 76−84. (12) Johari, G. P. Comment on “Water’s Second Glass Transition, K. Amann-Winkel, C. Gainaru, P. H. Handle, M. Seidl, H. Nelson, R. Böhmer, and T. Loerting. Proc. Natl. Acad. Sci. (U.S.A.) 2013, 110, 17720.”, and the Sub-Tg Features of Pressure-Densified Glasses. Thermochim. Acta 2015, 617, 208−218. (13) Amann-Winkel, K.; Gainaru, C.; Handle, P. H.; Seidl, M.; Nelson, H.; Böhmer, R.; Loerting, T. Water’s Second Glass Transition. Proc. Natl. Acad. Sci. U. S. A. 2013, 110, 17720−11725. (14) Sellberg, J. A.; McQueen, T. A.; Laksmono, H.; Schreck, S.; Beye, M.; DePonte, D. P.; Kennedy, B.; Nordlund, D.; Sierra, R. G.; Schlesinger, D.; et al. X-Ray Emission Spectroscopy of Bulk Liquid Water in “No-Man’s Land. J. Chem. Phys. 2015, 142, 044505. (15) Laksmono, H.; McQueen, T. A.; Jonas, A.; Sellberg, J. A.; Loh, N. D.; Huang, C.; Schlesinger, D.; Sierra, R. G.; Hampton, C. Y.; Nordlund, D.; Beye, M.; et al. Anomalous Behavior of the Homogeneous Ice Nucleation Rate in “No-Man’s Land”. J. Phys. Chem. Lett. 2015, 6, 2826. (16) Tulk, C. A.; Benmore, C. J.; Urquidi, J.; Klug, D. D.; Neuefeind, J.; Tomberli, B.; Egelstaff, P. A. Structural Studies of Several Distinct Metastable Forms of Amorphous Ice. Science 2002, 297, 1320−1323. (17) Guthrie, M.; Urquidi, J.; Tulk, C. A.; Benmore, C. J.; Klug, D. D.; Neuefeind, J. Direct Structural Measurements of Relaxation Processes during Transformations in Amorphous Ice. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 68, 184110. (18) Koza, M. M.; Schober, H.; Fischer, H. E.; Hansen, T.; Fujara, F. Kinetics of the High- to Low-Density Amorphous Water Transition. J. Phys.: Condens. Matter 2003, 15, 321−332. (19) Koza, M. M.; Schober, H.; Geil, B.; Lorenzen, M.; Requardt, H. Crystalline Inelastic Response of High-Density Amorphous Ice. Phys. Rev. B: Condens. Matter Mater. Phys. 2004, 69, 024204. (20) Seidl, M.; Amann-Winkel, K.; Handle, P. H.; Zifferer, G.; Loerting, T. From Parallel to Single Crystallization Kinetics in HighDensity Amorphous ice. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88, 174105.

(21) Koza, M. M.; May, R. P.; Schober, H. On the Heterogeneous Character of Water’s Amorphous Polymorphism. J. Appl. Crystallogr. 2007, 40, s517−s521. (22) Johari, G. P. Tests for Thermodynamic State of Water’s HighDensity Amorph. J. Chem. Phys. 2004, 121, 8428−8429. (23) Mishima, O.; Calvert, L. D.; Whalley, E. ‘Melting ice’ I at 77 K and 10 kbar: A New Method of Making Amorphous Solids. Nature (London, U. K.) 1984, 310, 393−395. (24) Johari, G. P.; Andersson, O. Water’s Polyamorphic Transitions and Amorphization of Ice under Pressure. J. Chem. Phys. 2004, 120, 6207−6213. (25) Mishima, O.; Calvert, L. D.; Whalley, E. An Apparently FirstOrder Transition Between Two Amorphous Phases of Ice Induced by Pressure. Nature (London, U. K.) 1985, 314, 76−78. (26) Gromnitskaya, E. L.; Stal’gorova, O. V.; Brazhkin, V. V.; Lyapin, A. G. Ultrasonic study of the nonequilibrium pressure-temperature diagram of H2O ice. Phys. Rev. B: Condens. Matter Mater. Phys. 2001, 64, 094205. (27) Nelmes, R. J.; Loveday, J. S.; Straessle, T.; Bull, C. L.; Guthrie, M.; Hamel, G.; Klotz, S. Annealed High-Density Amorphous Ice under Pressure. Nat. Phys. 2006, 2, 414−418. (28) Loerting, T.; Bauer, M.; Kohl, I.; Watschinger, K.; Winkel, K.; Mayer, E. Cryoflotation: Densities of Amorphous and Crystalline Ices. J. Phys. Chem. B 2011, 115, 14167−14175. (29) Johari, G. P.; Andersson, O. Mechanisms for Pressure- and Time-Dependent Amorphization of Ice under Pressure. Phys. Rev. B: Condens. Matter Mater. Phys. 2004, 70, 184108. (30) Johari, G. P. An Estimate of the Gibbs Energy of Amorphous Solid Waters and the Differences Between the Low-Density Amorph and Glassy Water. J. Chem. Phys. 2000, 112, 8573−8580. (31) Handa, Y. P.; Mishima, O.; Whalley, E. High-Density Amorphous Ice. III. Thermal Properties. J. Chem. Phys. 1986, 84, 2766−2270. (32) Johari, G. P. Liquid State of Low-Density Pressure-Amorphized Ice Above its Tg. J. Phys. Chem. B 1998, 102, 4711−4714. (33) Salzmann, C. G.; Radaelli, P. G.; Slater, B.; Finney, J. L. The Polymorphism of Ice: Five Unresolved Questions. Phys. Chem. Chem. Phys. 2011, 13, 18468−18480. (34) Mishima, O. Reversible First-Order Transition Between Two H2O Amorphs at ∼ 0 0.2 GPa and ∼ 135 K. J. Chem. Phys. 1994, 100, 5910−5912. (35) Loerting, T.; Winkel, K.; Seidl, M.; Bauer, M.; Mitterdorfer, C.; Handle, P. H.; Salzmann, C. G.; Mayer, E.; Finney, J. L.; Bowron, D. T. How Many Amorphous Ices are There? Phys. Chem. Chem. Phys. 2011, 13, 8783−8794. (36) Johari, G. P. On the Heat Capacity, Entropy and ‘Glass Transition’ of Vitreous Ice. Philos. Mag. 1977, 35, 1077−1090. (37) Andersson, O.; Johari, G. P. Time-Dependent Amorphization of Ice at 0.8−0.9 GPa. J. Chem. Phys. 2004, 121, 3936−3938. (38) Johari, G. P.; Andersson, O. Amorphization of Ice by Collapse Under Pressure, Vibrational Properties and Ultraviscous Water at 1 GPa. Proceedings of the NATO Advanced Research Workshop. Soft Matter under Exogenic Impacts, Fundamentals and Emerging Technologies; Rzoska, S. J., Mazur, V. A., Eds.;Springer: Berlin, 2007. (39) Johari, G. P.; Andersson, O. Vibrational and Relaxational Properties of Crystalline and Amorphous Ices. Thermochim. Acta 2007, 461, 14−43. (40) Andersson, O. Glass-Liquid Transition of Water at High Pressure. Proc. Natl. Acad. Sci. U. S. A. 2011, 108, 11013−1106. (41) Tse, J. S. Mechanical Instability in Ice I h. A Mechanism for Pressure-Induced Amorphization. J. Chem. Phys. 1992, 96, 5482−5487. (42) Tse, J. S.; Klug, D. D.; Tulk, C. A.; Swainson, I.; Svensson, E. C.; Loong, C.-K.; Shpakov, V.; Belosludov, V. R.; Belosludov, R. V.; Kawazoe, Y. The Mechanisms for Pressure-Induced Amorphization of Ice lh. Nature (London, U. K.) 1999, 400, 647−649. (43) Klotz, S.; Hamel, G.; Loveday, J. S.; Nelmes, R. J.; Guthrie, M. Recrystallisation of HDA Ice Under Pressure by in-situ Neutron Diffraction to 3.9 GPa. Z. Kristallogr. - Cryst. Mater. 2003, 218, 117− 122. J


Article

The Journal of Physical Chemistry B (44) Salzmann, C. G.; Kohl, I.; Loerting, T.; Mayer, E.; Hallbrucker, A. Pure Ices IV and XII from High-Density Amorphous Ice. Can. J. Phys. 2003, 81, 25−32. (45) Holten, V.; Bertrand, C. E.; Anisimov, M. A.; Sengers, J. V. Thermodynamics of Supercooled Water. J. Chem. Phys. 2012, 136, 094507. (46) Johari, G. P. Mechanical Relaxation and the Notion of TimeDependent Extent of Ergodicity During the Glass Transition. Phys. Rev. E 2011, 84, 021501. (47) Ngai, K. L. Relaxation and Diffusion in Complex Systems; Springer: New York, 2011, Chapter 2.3. (48) Okomato, N.; Oguni, M.; Sagawa, Y. Generation and Extinction of a Crystal Nucleus Below the Glass Transition Temperature. J. Phys.: Condens. Matter 1997, 9, 9187−9198. (49) Goldstein, M. Viscous Liquids and the Glass Transition: A Potential Energy Barrier Picture. J. Chem. Phys. 1969, 51, 3728−3739. (50) Stillinger, F. H.; Weber, T. A. Packing Structures and Transitions in Liquids and Solids. Science 1984, 225, 983−989. (51) Wales, D. J. Energy Landscapes: Applications to Clusters, Biomolecules and Glasses; Cambridge University Press: Cambridge, U.K., 2003. (52) Kauzmann, W. The Nature of the Glassy State and the Behavior of Liquids at Low Temperatures. Chem. Rev. 1948, 43, 219−256. (53) Davies, R. O.; Jones, G. O. Thermodynamics and Kinetic Properties of Glasses. Adv. Phys. 1953, 2, 370−410. (54) Moynihan, C. T.; Macedo, P. B.; Montrose, C. J.; Gupta, P. K.; De Bolt, M. A.; Dill, J. F.; Dom, B. E.; Drake, P. W.; Easteal, A. J.; Elterman, P. B.; et al. Thermodynamic and Transport Properties of Liquids near the Glass Transition Temperature. Structural Relaxation in Vitreous Materials. Ann. N. Y. Acad. Sci. 1976, 279, 15−35. (55) Wong, J.; Angell, C. A. Glass-Structure by Spectroscopy; Marcel Dekker, Inc.: New York, 1976. (56) Gutzow, I.; Schmelzer, J. The Vitreous State: Thermodynamics, Structure, Rheology and Crystallization; Springer: Berlin, 1995. (57) Donth, E.-J. The Glass Transition: Relaxation Dynamics in Liquids and Disordered Materials; Springer: Berlin, 2001. (58) Johari, G. P. Introduction to the Glassy State in the Undergraduate Curriculum. J. Chem. Educ. 1974, 51, 23−27. (59) Bosio, L.; Johari, G. P.; Teixeira, J. X-Ray Study of High-Density Amorphous Water. Phys. Rev. Lett. 1986, 56, 460−463. (60) Finney, J. L.; Hallbrucker, A.; Kohl, I.; Soper, A. K.; Bowron, D. T. Structures of High and Low Density Amorphous Ice by Neutron Diffraction. Phys. Rev. Lett. 2002, 88, 225503. (61) Kell, G. S. Density, Thermal Expansivity, and Compressibility of Liquid Water From 0° to 150 °C: Correlations and tables for Atmospheric Pressure and Saturation Reviewed and Expressed on 1968 Temperature Scale. J. Chem. Eng. Data 1975, 20, 97−105. (62) Antonenko, T. I.; Rykov, V. I. Temperature Dependence of Isothermal Compressibility of Liquids. Izvestiya Vuz. Fizika 1968, 11, 80−84; Soviet Physics Journal 1968, 11, 50−53. It contains the term T. (63) Diaz Pena, M.; Tardajos, G. Isothermal Compressibilities on nAlkanes and Benzene. J. Chem. Thermodyn. 1978, 10, 19−24. (64) Handa, Y. P.; Klug, D. D. Heat Capacity and Glass Transition Behavior of Amorphous Ice. J. Phys. Chem. 1988, 92, 3323−3325. (65) Kittel, C. Interpretation of the Thermal Conductivity of Glasses. Phys. Rev. 1949, 75, 972−974. (66) Berman, R. Thermal Conduction in Solids; Clarendon: Oxford, U.K., 1976. (67) Bridgman, P. W. The Thermal Conductivity of Liquids. Proc. Natl. Acad. Sci. U. S. A. 1923, 9, 341−345. (68) As quoted in Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley and Sons: New York, 1954. See also ref 69. (69) Biddle, J. W.; Holten, V.; Sengers, J. V.; Anisimov, M. A. Thermal Conductivity of Supercooled Water. Phys. Rev. E 2013, 87, 042302. (70) Horrocks, J. K.; McLaughlin, E. Temperature Dependence of the Thermal Conductivity of Liquids. Trans. Faraday Soc. 1963, 59, 1709−1716.

(71) Viswanath, D. S.; Rao, M. B. Thermal Conductivity of Liquids and Its Temperature Dependence. J. Phys. D: Appl. Phys. 1970, 3, 1444−1450. (72) Cahill, D. J.; Pohl, R. O. Thermal Conductivity of Amorphous Solids Above the Plateau. Phys. Rev. B: Condens. Matter Mater. Phys. 1987, 35, 4067−4073. See Figure 6. (73) Tonpheng, B.; Andersson, O. Crosslinking, Thermal Properties and Relaxation Behaviour of Polyisoprene Under High-Pressure. Eur. Polym. J. 2008, 44, 2865−2873. (74) Li, C. C. Thermal Conductivity of Liquid Mixtures. AIChE J. 1976, 22, 927−930. (75) Benchikh, O.; Fournier, D. A.; Boccara, C.; Teixeira, J. Photothermal Measurement of the Thermal Conductivity of Supercooled Water. J. Phys. (Paris) 1985, 46, 727−731. (76) Taschin, A.; Bartolini, P.; Eramo, R.; Righini, R.; Torre, R. Evidence of two distinct local structures of water from ambient to supercooled conditions. Nat. Commun. 2013, 4, 2401. (77) Andersson, O.; Suga, S. Thermal Conductivity of Amorphous Ices. Phys. Rev. B: Condens. Matter Mater. Phys. 2002, 65, 140201(R). (78) Conde, O.; Teixeira, J.; Papon, P. Analysis of Sound Velocity in Supercoled H2O, D2O, and Water−Ethanol Mixtures. J. Chem. Phys. 1982, 76, 3747−3753. (79) Johari, G. P. Molecular Architecture Dependent HydrogenBonded Motifs, Entropy Change, and Dielectric Permittivity of Alcohols. J. Chem. Phys. 2013, 139, 026101. (80) Johari, G. P.; Andersson, O. Effects of Stacking Disorder on Thermal Conductivity of Cubic Ice. J. Chem. Phys. 2015, 143, 054505. (81) Johari, G. P.; Whalley, E. The Dielectric Properties of Ice Ih in the Range 272−133 K. J. Chem. Phys. 1981, 75, 1333−1340. (82) Handle, P. H.; Loerting, T. Temperature-Induced Amorphisation of Hexagonal ice. Phys. Chem. Chem. Phys. 2015, 17, 5403−5412. (83) Winkel, K.; Bowron, D. T.; Loerting, T.; Mayer, E.; Finney, J. L. Relaxation Effects in Low Density Amorphous Ice: Two Distinct Structural States Observed by Neutron Diffraction. J. Chem. Phys. 2009, 130, 204502. (84) Pacaud, F.; Micoulaut, M. Thermodynamic Precursors, LiquidLiquid Transitions, Dynamic and Topological Anomalies in Densified Liquid Germania. J. Chem. Phys. 2015, 143, 064502.

K


Thermodynamic Analysis of the Two-Liquid Model for Anomalies of

Recommend Documents