Liquid-Phase Transition in Water (NIMS Monographs) 4431569146, 9784431569145

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NIMS Monographs

Osamu Mishima

Liquid-Phase Transition in Water

NIMS Monographs Series Editor Naoki OHASHI, National Institute for Materials Science, Tsukuba, Ibaraki, Japan Editorial Board Mikiko TANIFUJI, National Institute for Materials Science, Tsukuba, Japan Takahito OHMURA, National Institute for Materials Science, Tsukuba, Ibaraki, Japan Yoshitaka TATEYAMA, National Institute for Materials Science, Tsukuba, Ibaraki, Japan Takashi TANIGUCHI, National Institute for Materials Science, Tsukuba, Ibaraki, Japan Kazuya TERABE, National Institute for Materials Science, Tsukuba, Ibaraki, Japan Masanobu NAITO, National Institute for Materials Science, Tsukuba, Ibaraki, Japan Nobutaka HANAGATA, National Institute for Materials Science, Tsukuba, Ibaraki, Japan Kenjiro MIYANO, National Institute for Materials Science, Tsukuba, Ibaraki, Japan

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Osamu Mishima

Liquid-Phase Transition in Water

Osamu Mishima National Institute for Materials Science Tsukuba, Ibaraki, Japan

ISSN 2197-8891 ISSN 2197-9502 (electronic) NIMS Monographs ISBN 978-4-431-56914-5 ISBN 978-4-431-56915-2 (eBook) https://doi.org/10.1007/978-4-431-56915-2 © National Institute for Materials Science, Japan 2021 This work is subject to copyright. All rights are reserved by the National Institute for Materials Science, Japan (NIMS), whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms, or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of applicable copyright laws and applicable treaties, and permission for use must always be obtained from NIMS. Violations are liable to prosecution under the respective copyright laws and treaties. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. NIMS and the publisher make no warranty, express or implied, with respect to the material contained herein. This Springer imprint is published by the registered company Springer Japan KK part of Springer Nature. The registered company address is: Shiroyama Trust Tower, 4-3-1 Toranomon, Minato-ku, Tokyo 1056005, Japan

Preface

In the Galileo era, four elements of “earth, water, air, and fire” were thought to explain Nature. The understanding of these elements expanded significantly after the Scientific Revolution of the seventeenth century. However, only the understanding of water has been extremely limited. To stretch a point, our understanding of water is as if it were in the Galileo era. Of course, water is incredibly special to us, and the knowledge of water has become so rich owing to many advanced measurements. However, compared to gases and crystals, liquids, especially water, exhibit complex behavior and are difficult to understand. Now, the situation of water is beginning to change with the development of pressure generation technology and improvement in computer performance. From experiments and simulations supported by these technologies, a concept of “polyamorphism (or amorphous polymorphism)” has newly emerged. Specifically, two distinctly different liquid phases are considered to exist for water. They are supercooled liquids, and a sharp first-order transition occurs between them. When heated, they merge at the liquid-liquid critical point to form a supercritical liquid state. The fluctuation between the two waters in the supercritical state explains the complex properties of water. This introductory book on the liquid-phase transition in water focuses on experimental thermodynamic data of liquid water, supercooled water, and amorphous solid water at various pressures and temperatures. I will describe how the two-water scenario initially evolved experimentally and show that this scenario is likely correct. Acknowledgement I thank Naoto Kawai, Shoichi Endo, Osamu Fukunaga, Edward Whalley, Katsutoshi Aoki, H Eugene Stanley, and Yoshiharu Suzuki; without them, my research on water would be difficult to continue. Tsukuba, Japan

Osamu Mishima

v

Contents

1 Liquid–Liquid Critical Point Hypothesis of Water . . . . . . . . . . . . . . . . . 1.1 Cold Water: A Strange Liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Low-Density and High-Density Amorphous Ices . . . . . . . . . . . . . . . . 1.3 Transition Between Two Amorphous Ices . . . . . . . . . . . . . . . . . . . . . . 1.4 Polyamorphism in Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Liquid–Liquid Critical Point Hypothesis of Water . . . . . . . . . . . . . . . 1.5.1 Cause of the Liquid–Liquid Critical Point . . . . . . . . . . . . . . . 1.5.2 Two Amorphous Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Notes on the Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Indirect Experimental Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 6 12 13 15 17 18 19 22 23

2 Volume of Liquid Water and Amorphous Ices . . . . . . . . . . . . . . . . . . . . . 2.1 Volume of Liquid Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Volume of Accessible Liquid Water . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Vitrification of Water Under Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Softening of Low-Density and High-Density Amorphous Ices . . . . . 2.5 Apparently Discontinuous Transition Between Two Amorphous Ices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Liquid–Liquid Transition in Dilute Aqueous Solution . . . . . . . . . . . . 2.6.1 Decompression of Diluted Salt Water (LiCl–H2 O) and Its Volume Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Volume of Diluted Glycerol Aqueous Solution . . . . . . . . . . . 2.7 Liquid–Liquid Transition in Pure Water . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Speculation of the Volume of Water in the No-Man’s Land . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 32 36 42

3 Metastable Melting Lines of Crystalline Ices . . . . . . . . . . . . . . . . . . . . . . 3.1 Pressure–Temperature Phase Diagram of H2 O . . . . . . . . . . . . . . . . . . 3.2 Metastable Melting Line of Ice Ih . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59 59 60

43 47 48 50 51 52 54

vii

viii

Contents

3.3 Metastable Melting Line of Ice III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Metastable Melting Lines of Ice IV and Ice V . . . . . . . . . . . . . . . . . . 3.5 Metastable Melting Line of Ice PNP-XIV . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 67 72 72

4 Gibbs Energy of Liquid Water and the Liquid–Liquid Critical Point Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Construction of Gibbs Energy Surface of Water . . . . . . . . . . . . . . . . . 4.2 Gibbs Energy Surface of Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Projection in Energy-Axis Direction . . . . . . . . . . . . . . . . . . . . 4.2.2 Projection in Temperature-Axis Direction . . . . . . . . . . . . . . . 4.2.3 Projection in Pressure-Axis Direction . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 78 80 82 82 84

5 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

87 88

6 Appendix: Melting Points of Emulsified H2 O Ices . . . . . . . . . . . . . . . . . 6.1 Melting Points of Emulsified H2 O Ice Ih . . . . . . . . . . . . . . . . . . . . . . . 6.2 Melting Points of Emulsified H2 O Ice III . . . . . . . . . . . . . . . . . . . . . . 6.3 Melting Points of Emulsified H2 O Ice IV . . . . . . . . . . . . . . . . . . . . . . 6.4 Melting Points of Emulsified H2 O Ice V . . . . . . . . . . . . . . . . . . . . . . . 6.5 Melting Points of Emulsified H2 O Ice VI . . . . . . . . . . . . . . . . . . . . . . 6.6 Melting Points of Emulsified H2 O Ice PNP-XIII (Ice XII?) . . . . . . . 6.7 Melting Points of Emulsified H2 O Ice PNP-XIV . . . . . . . . . . . . . . . .

91 91 96 97 100 102 103 106

Chapter 1

Liquid–Liquid Critical Point Hypothesis of Water

1.1 Cold Water: A Strange Liquid Water is everywhere and is important to us. Therefore, you might think that water is well understood by modern science. Indeed, our knowledge of water has become extraordinarily rich [1]. But what about understanding water? Water is a complex substance with many peculiar properties that are difficult to understand. Many people are not aware of the strangeness of water. “Do you know the temperature of the bottom of quiet ice water? (Fig. 1.1)” I asked many scientists. There is no ice at the bottom of the glass. Their answers ranged from −3 °C to room temperature for various reasons, and the correct answer rate was roughly 10%. If you measure the water temperature at the bottom, it is always 4∼5 °C. In general, as temperature drops, the material shrinks. However, strangely, when temperature drops below 4 °C, the volume of water begins to expand, and its density decreases. As a result, light and cold 0 °C water floats on heavy and warm 4 °C water. The same thing happens in a winter pond. Even though the surface of the pond freezes, the water temperature at the bottom is kept at 4 °C, and the fish live in the warmest water at the bottom of the pond. Why does the volume of water expand at low temperatures? This strange phenomenon was found in the seventeenth century, and there has been no clear answer to this simple question. According to the explanation proposed in the late nineteenth century [2], a part of water forms a bulky crystalline ice-like structure, and its amount increases as temperature decreases. Later, this so-called mixture model became less popular because no such structure was found in subsequent experiments and simulations. In the explanation of the twentieth century [3, 4], each water molecule (H2 O) has a bent shape based on quantum mechanics, and due to its electrical properties, one water molecule combines with four surrounding water molecules to form a network structure (Fig. 1.2). The bond between two water molecules is called the hydrogen bonding. The volume of the network structure expands homogeneously at low temperature. However, the reason for the expansion is complex, and this socalled continuum model is difficult to understand immediately. Various sophisticated © National Institute for Materials Science, Japan 2021 O. Mishima, Liquid-Phase Transition in Water, NIMS Monographs, https://doi.org/10.1007/978-4-431-56915-2_1

1

2 Fig. 1.1 Ice water in a glass. The bottom water temperature indicates the strangeness of water. Colored photo: courtesy of Keiko Mishima. Adapted from [108], with permission from Elsevier

Fig. 1.2 The network of H2 O molecules. Red ball: oxygen. White ball: hydrogen. Dotted line: hydrogen bonding

1 Liquid–Liquid Critical Point Hypothesis of Water

3

228 K 235 K

1.1 Cold Water: A Strange Liquid

1.10

Volume (cm 3 / g)

ice

1.05

? supercooled (ice)

stable water

water 4C

1.00 200

250 -50

350 (K)

300 0

50 ( C)

100

Temperature Fig. 1.3 Strange volume change of water at low temperature

water models were derived from the mixture model and the continuum model. Many models are confusing and suggest that it is far from understanding water. Liquid water can be cooled below 0 °C (or 273 K in absolute temperature scale) without freezing. The behavior of this metastable supercooled water is even more mysterious. The volume of supercooled water increases more significantly at lower temperature, and the water always changes to crystalline ice (ice Ih) at about 235 K (−38 °C) (Fig. 1.3). Experiments by Angell and coworkers in the 1970s implied that, if the supercooled water does not crystallize at 235 K, it will behave extremely strangely at 228 K (−45 °C) [5, 6]. For example, the water temperature does not drop below 228 K even if water is strongly cooled (meaning the specific heat capacity becomes infinitely large), and the volume of water changes without any force applied (that is, the compressibility becomes infinitely large). Experiments down to 235 K showed that the volume of water tends to become infinite at 228 K. Does the supercooled liquid water vaporize at 228 K? Speedy thought that liquid water does not exist at temperatures below 228 K [7]. Is that true? You may be surprised to find that, despite many sophisticated measurements of this important and ubiquitous liquid, there is still debate about this very primitive water mystery. Because we do not know what water is, the understanding of aqueous solutions is not complete. Life activities can only be carried out in aqueous solutions, but their understanding is still inadequate.

4

1 Liquid–Liquid Critical Point Hypothesis of Water

1.2 Low-Density and High-Density Amorphous Ices It became experimentally clear in the early 1980s that liquid water at room temperature becomes a glass when cooled extremely rapidly [8–10]. That is, it did not crystallize below 235 K, and each water molecule stopped moving, preserving the disordered molecular arrangement of the liquid. Since 1935, it has been known that the slow deposition of water vapor on a very cold substrate in a vacuum produces deposits with disordered molecular arrangements [11]. When heated, the deposits softened and crystallized at about 150 K. Unlike this, Mayer experimented with rapid cooling of liquid water [10]. He used a nebulizer to create micrometer-sized water droplets which were then collided at high speed against a wall cooled to liquid nitrogen temperature (77 K, or −196 °C) in a vacuum. Heat was rapidly transferred from water to the substrate by conduction. This was the method to make the first amorphous metal from a liquid in 1960 [12, 13]. Each fine water droplet was instantaneously cooled, and the volume-expanded supercooled liquid water was frozen (Fig. 1.4). The structure of the molecular arrangement in each droplet was like the annealed structure of vapor deposits and somewhat similar to the bulky crystal structure of ice Ih. Today, this water material with the structure of frozen glass is called low-density amorphous ice (LDA). The density is about 0.94 g/cm3 , which is close to the density of ice Ih. Fig. 1.4 Deposits of extremely rapidly cooled water droplets. Color photo: courtesy of Prof. Thomas Loerting. Adapted from [109] with permission from the PCCP Owner Societies

1.2 Low-Density and High-Density Amorphous Ices

273 K (0 °C)

water

251 K

temperature

Fig. 1.5 Melting of ice Ih by compression. The black dashed line is the metastable melting line [20]. The red dashed line is the extrapolated metastable melting line. Adapted figure from [14, 16]

5

ice Ih

high-pressure ice

?

77 K 0

~1 GPa pressure

At about the same time in 1984, two laboratories independently reported the existence of high-density amorphous ice (HDA), which was different from lowdensity amorphous ice [14, 15]. Around that time, I was working as a post-doctoral fellow at National Research Council of Canada (NRC) in Ottawa and discovered the high-density amorphous ice [14]. I wrote in detail how the experiment started [16]. My boss at the time, Whalley, also briefly wrote the beginning of the experiment in his conference papers [17–19]. At NRC, I tried “pressure melting” of ice at liquid nitrogen temperature. When ice (ice Ih) is compressed at a temperature slightly lower than 273 K (0 °C), the ice melts into liquid water (Fig. 1.5). When ice is compressed at temperatures below 251 K (−22 °C), it undergoes a phase transition to high-pressure crystalline ice. However, if this phase transition does not occur, the ice will be further compressed and will eventually melt into supercooled liquid water on the metastable melting line of ice Ih. This metastable melting line is shown by the black bold dashed line in Fig. 1.5 [20]. At lower temperatures, the movement of molecules slows down and conversion to high-pressure crystalline ice becomes difficult to occur. At liquidnitrogen temperature (77 K), ice Ih may be compressed to the extrapolated melting line (red dashed line in Fig. 1.5). I compressed the ice at 77 K to see if it would melt under pressure. The volume of ice decreased rapidly by about 20% at about 1 gigapascal (1 GPa, or about 10,000 atmospheres), and the ice became a high-density state [14] (Fig. 1.6). When pressure was reduced to 1 atmospheric pressure (1 atm) at 77 K, the high-density state was recovered, and its density was about 1.17 g/cm3 . Experiments such as X-ray diffraction and Raman scattering at low temperatures showed that it was an amorphous solid with disordered and densely arranged water molecules; it was named as high-density amorphous ice (HDA). Its structure was similar to that of liquid water. By compressing ice Ih at low temperature, a large amount of high-density amorphous ice could be produced.

Fig. 1.6 Change in volume of ice Ih during compression at 77 K. Adapted figure from [14]

1 Liquid–Liquid Critical Point Hypothesis of Water

0 77 K Piston displacement (mm)

6

2

4

0

1 Pressure (GPa)

On the other hand, Heide cooled low-density amorphous ice (and ice Ih) to 20 K (−253 °C) or less in a vacuum and irradiated it with electron beams. He found a large structural change to another amorphous state in the electron-diffraction images [15]. He identified the resulting amorphous state as the high-density amorphous ice reported in 1974 [21, 22]. The amorphous ice reported in 1974 showed a somewhat sharp peak in the so-called oxygen–oxygen correlation function, and this indicated the existence of water molecules that entered the gap in the hydrogen-bonding network. However, subsequent experiments were unable to reproduce this ice with the sharp peak, and its existence was questioned. The ice made by Heide was reproducible [23, 24], and its structure resembled that of high-density amorphous ice made by compressing ice Ih at low temperature.

1.3 Transition Between Two Amorphous Ices When high-density amorphous ice produced under high pressure was decompressed and heated at 1 atm, its volume gradually increased slightly, and then the ice rapidly expanded at about 120 K and turned into low-density amorphous ice [14] (Fig. 1.7). This structural change was observed in our X-ray-diffraction experiment. My first surprise was that the structure of low-density amorphous ice did not change much with increasing temperature. This suggested that the low-density structure was unique and stable. Initially, I thought that the amorphous structure gradually relaxed from high density to low density around 120 K as temperature increased. If so, as shown by the thick gray line in Fig. 1.7, the relaxation temperature of 120 K would be nearly constant even if pressure is increased. This is because the change in temperature (heating) is often more effective in activating molecules than the change in

1.3 Transition Between Two Amorphous Ices

7

temperature

LDA

ex p.

~150 K

?

xpected)

~120 K

relaxation (e

HDA ?

? LDA

negative pressure

HDA

77 K

1 atm

LDA

HDA

high pressure

Fig. 1.7 The stability of HDA in the pressure–temperature diagram. Initially, HDA was expected to show a gradual volume change on the gray line when heated under pressure. However, HDA rapidly changed to LDA on the blue line. Adapted figure from [16]

pressure (compression and decompression). However, in heating experiments under pressure, the supposed relaxation temperature changed significantly with pressure, as shown by the blue experimental line in Fig. 1.7. This was my second surprise. These unexpected observations suggested a discontinuous “phase transition” between two different amorphous structures [16]. If the change at 120 K and 1 atm is a true phase transition, then a reverse phase transition from the low-density phase to the high-density phase should occur. On July 3, 1984, I compressed the low-density amorphous ice at 77 K using a piston-cylinder high-pressure apparatus to confirm the presence or absence of this amorphous-amorphous phase transition. At about 0.6 GPa, the piston suddenly advanced rapidly without sound, and despite supplying high-pressure gas (air) to the press machine to advance the piston, the increase in gas pressure, namely the increase in sample pressure, stopped. I was surprised and thought the pressure apparatus had broken. I immediately stopped the experiment. However, the apparatus was not broken. The next day, when I did the same experiment again, the piston suddenly advanced at the same pressure, and the volume of low-density amorphous ice decreased rapidly by about 20% (Fig. 1.8). The low-density amorphous ice changed to high-density amorphous ice. This change was apparently a discontinuous first-order phase transition. There were various discussions about water, such as the mixture model and the continuum model, but no one imagined that there is a discrete transition between two structures in which water molecules are randomly arranged. “We (water researchers) have been stupid!”; this was what I felt in this amorphous-amorphous transition.

Fig. 1.8 Change in volume of low-density amorphous ice during compression at 77 K. Adapted figure from [25]

1 Liquid–Liquid Critical Point Hypothesis of Water

Piston displacement (mm)

8

0

2

4

0

1 Pressure (GPa)

2

Whalley immediately realized the importance of the transition, and this discovery was published in 1985 [25]. Subsequent experiments showed that the low-density amorphous ice and the highdensity amorphous ice were clearly different [26–28] (Fig. 1.9). For example, when the phase boundary between low-pressure low-density amorphous ice and highpressure high-density amorphous ice was visually observed at temperatures above 120 K through a transparent window of a high-pressure device called the diamondanvil cell, the initially zigzagging phase-boundary line became sharper and straighter as temperature increased [26]. The movement of this boundary line showed that, as the temperature increased, water molecules moved, and that these amorphous

Fig. 1.9 High-density amorphous ice (HDA) and low-density amorphous ice amorphous ices (LDA) at 1 atm. Transparent HDA was obtained by annealing HDA at about 140 K under pressure; see [110]. LDA was made by heating HDA at 1 atm. These video-frame photos were taken just before and just after the rapid volume change at about 130 K. Photos were taken at National Institute for Research in Inorganic Materials at Tsukuba (now National Institute for Materials Science) and used in [111] and then [108]. Adapted from [108] with permission from Elsevier

1.3 Transition Between Two Amorphous Ices 200

Temperature (K)

Fig. 1.10 Transitions between low density and high-density amorphous ices and their crystallization in the pressure–temperature diagram. Adapted from [28], with the permission of AIP Publishing. Data also from [30, 112]

9

150

HDA HDA

100

50 0

0.2

0.4

0.6

0.8

Pressure (GPa)

ices were deformed by stress. The more distinct boundary line at higher temperature clearly indicated the existence of two different types of amorphous ice. At temperatures above 150 K, these amorphous ices crystallized, and phase boundaries between various crystalline ices appeared. These amorphous and crystalline ices were identified by Raman-scattering measurements and phase relationships [26]. High-pressure experiments supported discontinuity of the transition between the two amorphous ices. For example, the difference in transition pressure between compression and decompression suggested the pressure hysteresis in the first-order transition (Fig. 1.10). The equilibrium phase-boundary pressure between the two amorphous ices was estimated to be 0.15–0.2 GPa by thermodynamic consideration [29] and was about 0.2 GPa by experiment [28]. In addition, the physical properties of the two amorphous ices were distinctly different. As a result of ultrasonic experiments [30] and thermal measurements [31] it was found that low-density amorphous ice tends to deform more easily and is more difficult to transfer heat at higher pressures. On the other hand, high-density amorphous ice showed the opposite properties. Figure 1.11 shows the results of the analysis of neutron-diffraction experiments and Raman-scattering measurements of these two amorphous ices. Low-density amorphous ice resembles ice Ih (and ice Ic), while the high-density amorphous ice resembles liquid water. Are these amorphous ices irregular aggregates of extremely small ice crystals? This was the first question and was controversial. These amorphous ices softened and deformed upon heating, suggesting a possibility of structural changes. However, the X-ray-diffraction image of the low-density amorphous ice did not change, and the phase boundary between the two amorphous ices was clearly observed in the diamond-anvil-cell experiment. In other words, amorphous structures were stable even after softening and deformation, and no signs of crystal growth were detected at temperatures below about 150 K. For low-density amorphous ice, it was transformed into crystalline ice Ic when heated to about 150 K at 1 atm. The state during the transformation was cooled

10

1 Liquid–Liquid Critical Point Hypothesis of Water

Radial distribution function

Raman spectra

O-O

O-O

O-H

80 K

77 K

12 K

Ih 220 K

Ic 77 K

Ic 12 K

80 K

77 K

12 K

298 K

263 K

263 K

4

2

0

2

Intensity

g OO (r)

4

0

2

0 2

liquid water 0 0

5

r (angstrom)

10

0

200

3000

3400

Raman shift (cm -1)

Fig. 1.11 Neutron diffraction results and Raman spectra of low-density and high-density amorphous ices, liquid water, and ice Ih (Ic). The oxygen–oxygen partial radial distribution functions, goo (r), were obtained by structure-refinement simulations using neutron-diffraction data. r: oxygen– oxygen distance. Courtesy of Prof. John L. Finney. Adapted from [33]: Copyright (2002) by the American Physical Society. Similar profiles are shown in [34, 113, 114]. The O–O and O–H Raman spectra were adapted from [110] and [115], respectively, with permissions from Elsevier

to 77 K and compressed. Two transitions occurred: the transition from low-density amorphous ice to high-density amorphous ice at about 0.6 GPa and the amorphization from crystalline ice to high-density amorphous ice at about 1 GPa [25] (Fig. 1.12). This suggested that the two phases are mixed during the transformation from lowdensity amorphous ice to ice Ic upon heating at 1 atm, and that they are distinctly different. This was later reconfirmed by Raman-scattering experiments [32]. In other words, if low-density amorphous ice is a collection of microcrystals, crystal growth (not nucleation) will progress gradually at about 150 K, creating a continuous crystalline state. If this were compressed at 77 K, there would be only the transition from the crystal to high-density amorphous ice, and the transition pressure would shift gradually from 0.6 to 1 GPa as the crystal growth progresses. However, this was not

Ic+LDA

Ic→HDA

0

11

→HDA

Fig. 1.12 Change in volume of the ice during compression at 77 K. The ice was made by heating low-density amorphous ice to 148 K at 1 atm before compression. Adapted figure from [25]

Piston displacement (mm)

1.3 Transition Between Two Amorphous Ices

2

4

0

1 Pressure (GPa)

2

the case. This implied that the low-density amorphous ice was not a collection of small crystals. Finney et al. analyzed their neutron-diffraction data of amorphous ices and concluded that if low-density amorphous ice made from high-density amorphous ice is a collection of microcrystals, the number of water molecules in each microcrystal should be no more than a few dozen [33]. This size corresponds to one or two crystal unit cells. With this small size, low-density amorphous ice may be said to have a disordered structure. The same view for high-density amorphous ice was given by Klotz et al. from their neutron-diffraction data [34]. High-density amorphous ice appeared to have a more disordered structure than the low-density amorphous ice because it showed more blurred diffraction images and broader peaks in the Raman spectrum than the low-density amorphous ice (Fig. 1.11). By the way, Kamb analyzed X-ray-diffraction data of liquid water and estimated that, if liquid water is an irregular assembly of microcrystals of various kinds of ices, the number of water molecules in each crystal is less than a few dozen [35]. This means that the structures of lowdensity and high-density amorphous ices are just as disordered as the structure of liquid water. In general, there are innumerable ways to randomly arrange atoms and molecules. Therefore, amorphous ice with randomly arranged water molecules will have an infinite number of different types of disordered structures, due to differences in the preparation method. However, experimental data showed that, when the internal stresses were relieved by heat treatment, the disordered structures were classified into two groups, both of which had innumerable structures. One group had the same “low-density” characteristics, and the other group had the same “high-density” characteristics. Intermediate disordered structures might exist between the two groups, but they were thermodynamically unstable. Therefore, there was a gap in amorphous

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1 Liquid–Liquid Critical Point Hypothesis of Water

structure between the two groups. The existence of such different amorphous structures was called polyamorphism (amorphous polymorphism); polyamorphism of the amorphous structure corresponded to polymorphism of the crystal structure.

1.4 Polyamorphism in Water Since the 1927 report on the transition of liquid helium, it has been known that a pure substance can have plural liquid phases, causing a sudden change in liquid [36]. The transition in liquid helium occurred between the fluid and superfluid phases [37] and was caused by the quantum effect known as Bose–Einstein condensation [38]. However, discontinuous transitions in general liquids and glasses, especially discontinuous volume changes, have been questioned. With the development of high-pressure generation technology in the twentieth century, it became clear that the crystal discontinuously changes into another crystal of higher density as it is compressed. It was then speculated that the structure of the liquid would gradually change from a low-density crystal-like structure to a high-density crystal-like structure depending on the pressure. In 1967, Rapoport hypothesized that liquid has two local states (or structures, or species): a low-density state and a high-density state. He then calculated the two-state model in which the ratio of these two species changes continuously as the pressure changes. This theory predicted existence of a discontinuous phase transition of the liquid at low temperature, but he did not consciously discuss this discontinuity. This was because it was generally believed that the change in liquid should be smooth, and because there was no experiment that showed liquid discontinuity [39]. In 1970, Hemmer and Stell published an interaction potential between two particles that showed a discontinuous fluid–fluid transition at low temperatures. However, they did not consider the transition between two liquid phases [40]. On the other hand, Aptheker, like Rapoport, calculated the two-state model and applied it to liquid germanium and liquid silicon. He expected the existence of discontinuous structural changes (phase transitions) in these liquids and the existence of liquid–liquid critical points [41]. Although the idea that liquid changes continuously between different local structures (species) has been considered by some researchers, the first-order liquid–liquid transition, encountered in theoretical calculation at low temperature, was not taken seriously. One reason for this was the criticism of the liquid mixture model. For example, criticism of the mixture model for room-temperature water existed then [42] and now [43]. High-pressure experiments were conducted to study changes in liquids and amorphous materials. In 1953, Bridgman and Šimon reported that when glasses were compressed, continuous densification eventually began to occur [44]. In one experiment, compression of amorphous silicon (and amorphous germanium) resulted in a sudden decrease in electrical resistance. This suggested that a pressure-induced discontinuous structural transition occurred [45]. However, another experiment implied that the change in the resistance might be due to pressure-induced crystallization of the amorphous silicon [46]. Some researchers considered discontinuous

1.4 Polyamorphism in Water

13

liquid–liquid or amorphous-amorphous transitions from their experiments, but none of the experiments clearly showed discontinuous transitions. The volumetric discontinuity of amorphous ice was important in establishing a new concept, polyamorphism, that there is discontinuity in the isotropic disordered structure of a single-component condensed matter. Although the discontinuous structural changes exhibited by multi-component liquids and liquid crystals were well known, the clear discontinuous change in the volume of an isotropic pure material was first discovered in amorphous ice. The gas is not a condensate, but both the gas and the liquid are different disordered phases of the same pure substance, and the gas–liquid phase transition is first-order. Therefore, the relationship between the two amorphous ices was like the discontinuous relationship between gas and liquid. Polyamorphism of amorphous ice raised many questions and at the same time led to strong criticism, confusion, doubt, and neglect. In general, the glassy state softens with increasing temperature, becoming crystalline or liquid. Observations using a diamond-anvil cell showed that these low-density and high-density amorphous ices deformed under stress as temperature increased. In other words, water molecules began to move, and structural relaxation occurred. When the temperature was further increased, the low-density amorphous ice turned into low-density crystalline ice at low pressure and the high-density amorphous ice turned into highdensity crystalline ice at high pressure. If these amorphous ices did not crystallize, the low-density amorphous ice might become low-density liquid water at low pressure and the high-density amorphous ice might become high-density liquid water at high pressure. And it was naturally imagined that the two liquid waters would change discontinuously with each other. This is because both structure and properties were distinctly different between the low-density and high-density amorphous phases. Or do low-density amorphous ice and high-density amorphous ice become the same liquid phase at high temperature? Or is the high-density amorphous ice produced by compressing crystalline ice basically different in structure from liquid water? Or can we say that the transition between the two amorphous ices is indeed discontinuous? In addition, there may be a third amorphous ice besides the low-density and highdensity amorphous ices. Despite these many questions, progress in experimental study was slow. The reason was that there were few high-pressure experimenters in this new field of research and that many researchers did not believe in polyamorphism at all. They believed that the isotropic disordered structure of pure materials had to change continuously through many intermediate amorphous structures.

1.5 Liquid–Liquid Critical Point Hypothesis of Water In the early 1990s, Poole, a graduate student in the Stanley group at Boston University, and Sciortino and Essmann, postdoctoral researchers in the same group, simulated supercooled water at temperatures around 228 K. As a result of the calculation, it was found that a discontinuous pressure-induced transition occurs between two amorphous phases. One has a low-density amorphous ice-like structure, and the other

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1 Liquid–Liquid Critical Point Hypothesis of Water

has a high-density amorphous ice-like structure. Therefore, the Stanley group thought that, as one of the liquid water scenarios, the liquid water might separate into highdensity amorphous ice and low-density amorphous ice at a pressure and temperature called a critical point. Discontinuous amorphous-amorphous transition occurs at temperatures below this critical point. This simulation result was published in 1992 [47]. The two amorphous solid phases were then considered to be two liquid phases, which were thought to vitrify into respective amorphous ices at low temperature. This view of liquid water at low temperature is called the liquid–liquid critical point hypothesis or the liquid–liquid phase transition hypothesis [48]. The low-density liquid water is stable at low temperature and low pressure, whereas the high-density liquid water is stable at high temperature or low temperature and high pressure. In other words, the discontinuous amorphous-amorphous transition found in the low-temperature and high-pressure experiment becomes a discontinuous liquid– liquid transition at high temperature [49] (Fig. 1.13). At even higher temperature, the two liquid phases mix at a critical point to form a single liquid phase. There, low-density and high-density states fluctuate, forming a seemingly homogeneous supercritical state. This supercritical state is the cause of the strangeness of water.

(K)

400

GLCP

600

( C)

Temperature

G 200

L 400

hypothesized

0

LLCP 200 Tg

Tg

-200 0

0

0.2

0.4

Pressure (GPa) Fig. 1.13 Hypothesized liquid–liquid transition (LLT) between low-density liquid water (LDL) and high-density liquid water (HDL) in the pressure–temperature diagram. Adapted figure from [49]. LDL and HDL vitrify at low temperature and become low density amorphous ice (LDA) and high-density amorphous ice (HDA), respectively. The metastable equilibrium LLT line ends at the hypothesized liquid–liquid critical point (LLCP) at high temperature. The LLT becomes a nonequilibrium LDA-HDA transition line at low temperature (the dashed line). GLCP is the critical point of the gas phase (G) and the liquid phase (L). Tg: glass-transition temperature

1.5 Liquid–Liquid Critical Point Hypothesis of Water

15

Broadly speaking, both the water mixture model and the water continuum model may be consistently explained by the supercritical state. Initially, this liquid–liquid critical point hypothesis was regarded as a ridiculous idea and was laughed at by many researchers. However, experimental results showing the apparently discontinuous transition between two amorphous ices and simulation results showing the discontinuous transition between two liquid phases led water research in a new direction [49–52].

1.5.1 Cause of the Liquid–Liquid Critical Point Why are there two types of liquid water? As shown in Fig. 1.14, if there are two different types of local arrangements of three water molecules, a first molecule, a second hydrogen-bonded molecule, a third non-bonded molecule (1) or a hydrogenbonded molecule (2), each water molecule belongs to one of the two arrangements, or two states. This classification is just a rough idea, but according to Rapaport, who assumed two different states in a liquid, the discrete transition between the two phases and the existence of a critical point are theoretically possible at low temperature [39, 53–55]. On the other hand, if two types of attractive forces act between two adjacent particles that make up a liquid, and each force type tries to place the two particles at different distances, a discontinuous liquid–liquid transition occurs at low temperature and a critical point exists [40, 56–59] (Fig. 1.15). For example, experiments on the structure of water showed that one water molecule is always bound to the four surrounding water molecules (Fig. 1.2). This group of five water molecules in total was called the Walrafen pentamer. The Stanley group considered this pentamer to be a single particle with four binding hands and thought that this model particle would bind to the adjacent pentamer particles. In this case, roughly speaking, there may be two low-energy states: (1) a high-density state where the two model particles are close together and (2) a low-density state where they are slightly apart [60] (Fig. 1.15). The two states correspond to two forces: (1) van der Waals force and (2) hydrogenbonding force. When two types of forces act between two model particles, liquid water separates into two phases at low temperature.

Fig. 1.14 Two possible arrangements of three water molecules

1 Liquid–Liquid Critical Point Hypothesis of Water

pair interaction potential

16

distance

Fig. 1.15 Generalized pair potential with two cohesive energies and the corresponding configurations of the Walrafen pentamers [56, 60]. Red ball: oxygen atom. Black bar: hydrogen bonding. Hydrogen atoms are omitted. Van der Waals force (1) and hydrogen bonding (2) between two pentamers. Adapted figure from [49]

sta

llin

LDL sity

Cry

Den

ity

HD L

0

Fig. 1.16 Energy landscape of low-density and high-density liquids (LDL basin and HDL basin). Adapted by permission from Prof. Pablo G. Debenedetti: Nature [61], COPYRIGHT (2014)

Free Energy

Years later, the research group led by Debenedetti simulated supercooled water and rigorously confirmed that the energy of the low-density water structure group and that of the high-density water structure group are relatively stable [61] (Fig. 1.16). There are continuous intermediate disordered structures between these two groups,

1.5 Liquid–Liquid Critical Point Hypothesis of Water

17

but the intermediate structures are high-energy and unstable; they would appear under the supercritical condition, along with low-density and high-density structures. As temperature drops, the structure of either the low-density group or the high-density group appears, and the unstable intermediate structure does not appear. When pressure changes at low temperature, a discontinuous phase transition occurs between the two groups. By the way, the structures of each group are not completely disordered, but slightly ordered. However, this order has nothing to do with the crystal structure. There are potential barriers between the structures of each amorphous group and that of crystalline ice. Therefore, these low-density and high-density amorphous structures are metastable. That is, the high-density and low-density liquid states can exist for some time before crystallization begins. For the basin of the low-density liquid water (LDL) in Fig. 1.16, the slope of its high-crystallinity side is the potential barrier between low-density water and ice Ic [61].

1.5.2 Two Amorphous Structures Experimentally, high-density amorphous ice was created by the low-temperature compression of ice Ih. It was debated whether this mechanically collapsed ice Ih could be regarded as a glass of liquid water [62, 63]. Note that the structure of amorphous ice is not necessarily the same as that of liquid. It was argued that high-density amorphous ice and liquid water are unrelated due to slight differences between their structures, but in general, it is difficult to discuss the relationship between hard amorphous ice and fluid liquid water [64]. However, the structure of amorphous ice gradually relaxes and softens as temperature increases. The structure of high-density amorphous ice, if sufficiently relaxed at high pressure, may belong to the structural group of highdensity liquid water. In other words, the structure may be located somewhere within the HDL basin in Fig. 1.16. Similarly, the low-density amorphous structure created by decompressing high-density amorphous ice at low temperature may be related to liquid water at low pressure. Experiments seem to indirectly support this view, as described in the following chapters. That is, amorphous ice and liquid water are closely related. Simulation suggested such an ambiguous relationship between the liquid state and the amorphous solid [65]. Simulations can reveal low-density and high-density structures of amorphous ice and liquid water. Differences in their molecular network structure can be easily recognized from the statistics [66] and illustrations [67, 68] of the network ring (Fig. 1.17). The high-density structure is characterized by a large ring network of about eight water molecules connected in a row (HDA in Fig. 1.17). Each ring is collapsed, and some of the water molecules of the ring are close to each other. These rings are connected to each other, and each connection is also distorted to form a ring network consisting of about eight water molecules. These deformed and complex rings form a dense structure throughout. The bonds between water molecules are bent and easily broken, creating defects [69]. It can be said that high-density liquid water having

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1 Liquid–Liquid Critical Point Hypothesis of Water

Fig. 1.17 Illustration of network structures of low-density amorphous ice (LDA) and high-density amorphous ice (HDA). These structures almost reproduce the experimental diffraction data of the two amorphous ices (Fig. 1.11). Colors of the original figures in [68] are changed: courtesy of Prof. David T. Limmer. Conceptually similar structures are shown in [67]

such a structure is fluid and unstructured. Hot liquid water at 1 atm can be thought of as hot high-density water (Fig. 1.13). On the other hand, the low-density structure is an aggregate of small annular polyhedra in which about 6 water molecules are connected (LDA in Fig. 1.17). There is a large space in this hard-to-collapse polyhedron. This polyhedron creates a viscous three-dimensional network structure. The low-density structure has two types of local network structures: right-handed and left-handed [70]. The network structure is easily distorted, but bonds between water molecules are difficult to break. Low-density water is a structured liquid that is orders of magnitude more viscous than high-density water. It is difficult to compare the viscosities of low-density and highdensity liquids at the same temperature, but supposing and roughly speaking their difference in viscosity at room temperature, if high-density water is our familiar liquid water, the low-density water may correspond to oil or syrup. Water below 220 K at 1 atm can be considered low-density water. A liquid lump of low-density water tends to form crystalline ice, and it is difficult to observe the lump experimentally unless it is in a low-temperature glassy state. When the low-density structure is compressed to high pressure or heated to high temperatures, the polyhedron collapses, and the low-density structure changes to the high-density structure.

1.5.3 Notes on the Hypothesis It had been thought that the structure of simulated liquid water is uniform, and this was a reason to criticize the water mixture model. However, a new method of analyzing the structure of water was discovered, which revealed that simulated water

1.5 Liquid–Liquid Critical Point Hypothesis of Water

19

is a mixture of two different local structures (or two states): low-density and highdensity structures [71, 72]. It is apparent that the room-temperature water at 1 atm looks homogeneous due to thermal movement of water molecules. However, the two local structures (states) were hidden in the room-temperature water [73]. The ratio of these two states changed with temperature and pressure, and it was determined by simulation. The ratio of the high-density state was high under high temperature, while the ratio of the low-density state was high under low temperature and low pressure. When this ratio was 5:5 in the supercritical water, the high-density state and the low-density state transformed into each other and strongly fluctuated. On the liquid– liquid transition line of the pressure–temperature phase diagram, low-density water and high-density water were discontinuously separated and coexisted (see Fig. 1.13). Upon leaving the phase boundary, water in one of the two states became dominant. It was pointed out that even if a critical point exists in supercooled water, it is impossible to identify the critical point by experiments [74]. At the critical point, the two states mix and fluctuate significantly, and it takes long time to attain the equilibrium condition. During that time, the supercooled water will crystallize, so the location of the critical point cannot be determined directly in experiments. However, this does not deny the liquid–liquid critical point hypothesis, nor does it deny the existence of the discontinuous phase transition between two waters. In fact, simulations showed that there is a liquid–liquid critical point in the region of supercooled water [75]. Simulations of gradually varying the forces between two adjacent water molecules showed that the position of the liquid–liquid critical point moves smoothly between the stable water and metastable supercooled water regions. If the force between water molecules is changed, the simulation result will be different. Several water models have been proposed and used in simulations. For example, in the models called ST2, TIP4P-ice, and TIP4P/2005, there is certainly a critical point in the supercooled water [61, 76]. It has been suggested that the critical point also exists in many other models (TIP4P, TIP4P-EW, TIP5P, TIP5P-E, iAmoeba, WAIL, E3B3 etc.) [77, 78]. On the other hand, there are some models, such as the so-called mW model and the SPC/E model, for which there is no critical point in the supercooled liquid state. You can find out which model can reproduce real water better by comparing the results of each model with the experimental data. For many realistic simulations, the liquid–liquid critical point hypothesis is preferred. It should be noted that the first-principles calculation of liquid water using no empirical parameter also implies the validity of the hypothesis [79].

1.6 Indirect Experimental Evidence Many experimental results have been accumulated over the last 30 years. Next, let us take a quick look at these results. Supercooled water tends to crystallize into ice at temperatures below the so-called homogeneous nucleation temperature, and two lowtemperature amorphous ices also crystallize when heated. The region of pressure and temperature where crystallization is extremely easy to occur was called “no-man’s land” [49]. Because of this crystallization, no one has ever seen the phase separation

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1 Liquid–Liquid Critical Point Hypothesis of Water

of the two supercooled liquid waters visually. Controversy has been repeated since the early 1990s because it was not possible to experimentally confirm directly that liquid water had the liquid–liquid critical point and the liquid–liquid phase transition. Initially, due to the small number of experiments and the fact that the concept and definition of amorphous ice, as well as those of polyamorphism, differed among researchers, the interpretation and discussion of the experimental results were chaotic. The term (true) polyamorphism was proposed by Wolf [50, 80] and became gradually popular after 1995 [49, 50, 52], but research on it was just beginning. The liquid–liquid critical point hypothesis was criticized in several respects around 2000. Opposite, alternative, or cautious opinions to this hypothesis were (1) the possibility that there is no relationship between amorphous ice and liquid water [62] (which is consistent with the so-called stability-limit hypothesis [7]), (2) the possibility that the transition between the two amorphous structures is continuous [81] (which is consistent with the so-called singularity-free scenario [82]), and (3) the possibility of a third amorphous phase [83–85]. Experimental data on amorphous ice and supercooled liquid water were accumulated by several research groups, and after controversy about these data, the liquid–liquid critical point hypothesis became a powerful scenario for understanding water around 2010. However, the existence of the liquid–liquid phase transition in simulated water had been the subject of intense debate in the 2010s [86, 87]. It appears to me that much of the experimental and simulation evidence now supports the hypothesis, and that the alternative scenarios (1, 2, and 3) are not significant. The followings are examples of experiments. To avoid crystallization of supercooled water, many experiments were conducted using micrometer-sized emulsions, amorphous ice, confined water, thin-film water, and dilute aqueous solutions. The results of these indirect experiments, which did not use bulk pure water, were consistent with the liquid–liquid critical point hypothesis. Some of them implied the existence of the discontinuous transition between the two liquid waters and suggested the existence of the critical point [88–90]. Bellissent-Funel and Bosio showed that the neutron-diffraction data of emulsified supercooled liquid water could be represented by a linear combination of the structures of low-density and high-density amorphous ices [91]. On the other hand, Soper and Ricci showed that high-pressure low-temperature bulk liquid water could be represented by a mixture of two liquid structures and that these structures were considerably similar to those of low-density and high-density amorphous ices [92]. An international collaborative research team led by Nilsson conducted experiments in which micrometer-sized liquid water was cooled to temperatures below 235 K in a vacuum and studied structural fluctuation in the supercooled water by X-ray measurements during a short period of about a millisecond until the supercooled water crystallized [93]. From this study, the research team claimed that the fluctuation of the liquid structure related to the liquid–liquid critical point occurred at about 228 K. Apart from this, infrared spectroscopy of a nanometer-sized water film recently reported that water was a mixture of two structural motifs [94]. Bellessent-Funel suggested that when emulsified liquid water was cooled, its structure approached the structure of low-density amorphous ice at low pressure

1.6 Indirect Experimental Evidence

21

and the structure of high-density amorphous ice at high pressure [95]. In fact, when micrometer-sized liquid water was cooled extremely rapidly in a vacuum, it became low-density amorphous ice [10], and when cooled rapidly at high pressure, it became high-density amorphous ice [96]. When the temperature of amorphous ices was raised or lowered at 1 atm, the amorphous ices were reversibly softened or frozen. This meant the presence of glass transitions, and implied that each amorphous ice became viscous liquid when heated. Amann-Winkel et al. found that low-density amorphous ice and high-density amorphous ice had different glass-transition temperatures at 1 atm [97]. In other words, they claimed that there were two liquid states. Note that the reason high-density water can exist at 1 atm is because it is a metastable state. Using emulsions of dilute aqueous solutions in high-pressure experiments, Suzuki and I suggested that the discontinuous transition between two amorphous ices became a discontinuous transition between two liquids of water as temperature increased [98]. Klotz et al. showed that low-density amorphous ice and high-density amorphous ice coexisted during the amorphous-amorphous transition at a constant pressure (about 0.3 GPa) and a constant temperature (130 K) [99]. Furthermore, Winkel et al. showed that these amorphous ices coexisted during the transition at 140 K under pressure. Because this temperature was near the glass-transition temperatures of these ices, the coexistence of two different structures supported the first-order liquid–liquid transition [100]. What we want to know is the plausibility of the existence of the liquid–liquid critical point in actual water. To know this, thermodynamics will be a basic and robust tool. As described in the following chapters, I studied the thermodynamic data obtained from experiments at various pressures and temperatures. As a result, I believed in the existence of the discontinuous liquid–liquid transition. That is, I believed in the discontinuous phase transition between low-density amorphous ice and high-density amorphous ice, and I believed in the thermodynamic connection between each amorphous ice and liquid water. These amorphous ices were hard at low temperature and were not in the thermodynamic equilibrium liquid state, but they softened as temperature increased. This softening allowed us to discuss two amorphous ices and two liquid waters. I also instantly melted various ice crystals into liquid water in the no-man’s land for a moment. The melted water crystallized immediately in the no-man’s land, but it was possible to know the liquid state before crystallization. That is, the melting lines of various crystalline ices were used to infer the liquid state of the no-man’s land, suggesting the existence of a liquid–liquid critical point for water. Its location was very roughly estimated to be ≈0.05 GPa (500 atm) and ≈223 K (−50 °C) [88, 89]. Finally, I made the Gibbs energy surface of low-temperature liquid water. This energy surface was consistent with the liquid– liquid critical point hypothesis [88], indicating that the hypothesis is promising. This book does not deal with the structural and dynamic properties of water. The pressure–temperature region where anomalous changes in the thermodynamics of water were observed overlapped with the region where anomalous changes in dynamics were observed [101, 102]. The heterogeneous movement of the network structure of water molecules was suggested by simulation [103]. The structure and

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1 Liquid–Liquid Critical Point Hypothesis of Water

dynamics of water appear to be consistent with the liquid–liquid critical point hypothesis. Please refer to other review articles for more information on these points [77].

1.7 Philosophy Even if theory and simulation show the existence of the liquid–liquid critical point of water, it cannot prove that actual water really has this critical point. Experimental proof of the hypothesis is essential. The critical point is related to supercooled water, and supercooled water tends to crystallize. Therefore, the experiments are not easy. In addition, the accuracy of these experiments is limited, and detailed analysis is difficult. For this reason, in many cases, the result of each experiment and its interpretation have been controversial, with repeated criticisms of the experiments that may support the hypothesis. Many researchers also believe that the liquid–liquid critical point hypothesis should be proven using bulk pure water. In other words, they believe that “even if experiments without bulk pure water provide evidence to support the hypothesis, it is indirect evidence; it may not deny the hypothesis but does not prove it either; a clear liquid–liquid phase separation should be observed visually in a glass container”. They are skeptical of the hypothesis because there is no rigorous experiment to prove it reliably. Obtaining clear evidence by experiment can be difficult. In such a case, the hypothesis will be evaluated by accumulating a lot of circumstantial evidence and looking at the whole. Most experimental results of indirect experiments were consistent with the hypothesis. And, over time, evidence that more strongly suggests the validity of the hypothesis is emerging. However, these results were often forgotten because they could not provide the conclusive proof for the hypothesis. If the liquid–liquid critical point hypothesis of water is simpler and better than the other scenarios, and if all direct and indirect experimental results, realistic simulations, and the theoretical conditions for generating the critical point are consistent with the existence of a liquid–liquid critical point, then this hypothesis can be argued to be correct. Although the pressure and temperature at the liquid–liquid critical point have not yet been determined, I believe that various experimental and theoretical results consistent with this hypothesis have been sufficiently accumulated. Whether or not to accept this hypothesis should be decided by the consensus of many researchers, but I accept it on the condition that “until one well-founded experimental evidence emerges that is completely inconsistent with the hypothesis”. If the liquid–liquid critical point hypothesis of water is accepted as correct, the liquid water we usually see becomes a supercritical state of low-density and highdensity waters. The equations of the critical phenomena would allow us to describe and predict the complex properties of water in a concise and unified way. The properties of water would be essentially explained by the ratio of the low-density state to the high-density state, which is determined by the pressure and temperature at the critical point. Anisimov’s group made an empirical equation for water based on the

1.7 Philosophy

23

two-state model and explained the complex thermodynamic properties of water in the ratio of the low-density state to the high-density state. This equation had a critical point, and over a wide range of pressures and temperatures around the critical point, the equation calculated very well the strange thermodynamic properties of water that the experiment showed [104]. In addition, the so-called Ising model was recently applied to two liquid phases of water. The model showed that a liquid–liquid critical point exists when the lowpressure low-temperature liquid is low-density and low-entropy [105]. In fact, experiments suggested that this condition was met. This experimental suggestion will be discussed in Chap. 4. Until now, it has been difficult to understand water, but this situation is changing. In the future, it may be possible to theoretically calculate the entire complex thermodynamic behavior of H2 O over a wide range of pressure–temperature regions. Soper concisely summarized “Hence the second (liquid–liquid) critical point model of water appears to fit into the category of a model that may be useful for understanding, but formally incorrect for, the real material. With this point of view one might say therefore that water behaves as if there were a second critical point below the homogeneous nucleation temperature” [Courtesy of Alan K. Soper [106] Copyright (2017), with permission from Elsevier]. Therefore, it is considered that the volume of water increases below 4 °C at 1 atm because the ratio of the high-density state to the low-density state is gradually changing in water. Research has also been done to explain the dynamic properties of water, such as viscosity at low temperatures and high pressures, using the ratio [102, 107].

References 1. Ball P (1999) H2 O: a biography of water. Weidenfeld and Nicolson, London. ISBN-13:9780297643142 2. Röntgen WC (1892) Ueber die constitution des flüssigen wassers. Ann Phys Chem 281:91–97. https://doi.org/10.1002/andp.18922810108 3. Bernal JD, Fowler RH (1933) A theory of water and ionic solution, with particular reference to hydrogen and hydroxyl ions. J Chem Phys 1:515–548. https://doi.org/10.1063/1.1749327 4. Pople JA (1951) Molecular association in liquids II. A theory of the structure of water. Proc Royal Soc Lond A 205:163–178. https://doi.org/10.1098/rspa.1951.0024 5. Speedy RJ, Angell CA (1976) Isothermal compressibility of supercooled water and evidence for a thermodynamic singularity at −45°C. J Chem Phys 65:851–858. https://doi.org/10.1063/ 1.433153 6. Angell CA (1982) Supercooled water. In: Franks F (eds) Water and aqueous solutions at subzero temperatures. Water (A comprehensive treatise) 7:1–81 Springer, Boston. https://doi. org/10.1007/978-1-4757-6952-4_1 7. Speedy RJ (1982) Stability-limit conjecture. An interpretation of the properties of water. J Phys Chem 86:982–991. https://doi.org/10.1021/j100395a030 8. Brüggeller P, Mayer E (1980) Complete vitrification in pure liquid water and dilute aqueous solutions. Nature 288:569–571. https://doi.org/10.1038/288569a0 9. Dubochet J, McDowall AW (1981) Vitrification of pure water for electron microscopy. J Microsc 124:3–4. https://doi.org/10.1111/j.1365-2818.1981.tb02483.x

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Chapter 2

Volume of Liquid Water and Amorphous Ices

2.1 Volume of Liquid Water First, let us think the relationship between liquid water and two amorphous ices in terms of volume. Volume depends on pressure and temperature. Figure 2.1 shows the volume of H2 O water obtained by experiments. The unit of the temperature axis is absolute temperature (K). The unit of pressure axis is gigapascal (GPa). 0.1 GPa corresponds to the pressure when 1 ton of force is applied to a fingertip, and it is close to the pressure at the bottom of the Pacific Ocean Trench. 1 atmospheric pressure (1 atm) is almost 0 GPa (0.0001 GPa or 0.1 MPa) in Fig. 2.1. The unit of volume axis is cubic centimeter per gram (cm3 /g). The minimum volume of water at 1 atm is 1 cm3 /g at 277 K (4 °C). Historically, the weight of 1 cm3 of water at 277 K was defined as a unit of weight (1 g) and determined the weight (or mass) of all matter from quarks to neutron stars. Figure 2.1 of liquid water and amorphous ices may give a vague idea of the relationship between these phases. That is, (1) connection between liquid water and low-density amorphous ice (LDA) at low pressure [1], (2) connection between liquid water and high-density amorphous ice (HDA) at high pressure [2], and (3) the discontinuous relationship between low-density and high-density amorphous ices at low temperatures [3, 4]. These volume relationships may give an idea of the liquid–liquid critical point hypothesis. For a long time, the mixture model and the continuum model of water have been disputed using only the experimental results of high-temperature water. In the stalemate between these models, the existence of two amorphous ices played an important role in the emergence of a new scenario, the liquid–liquid critical point hypothesis. Although the volumetric data were incomplete in the no-man’s land in Fig. 2.1, the figure had been a mental anchor for some researchers who believed in the liquid–liquid critical point hypothesis. Figure 2.2 shows the high-temperature part of Fig. 2.1. The Tm line (thick black line) corresponds to the melting temperature of ice crystals. Liquid water is stable at temperatures above the Tm line, and various ice crystals are stable at temperatures below this line. The volume of stable water has been measured with high accuracy. © National Institute for Materials Science, Japan 2021 O. Mishima, Liquid-Phase Transition in Water, NIMS Monographs, https://doi.org/10.1007/978-4-431-56915-2_2

29

30

1.2

no man’ s land (1)

1.1

liquid water

1.0

(3) 0.9

A

HD

Volume (cm3 /g)

LDA

0.8

(2) 0.7

0 100

200

0.4 300

Temperature (K)

0.8

re ssu

a) (GP

Pre

Fig. 2.2 Experimental volume data of liquid water. Data from [10, 18, 20, 21, 71–79]. The pink [10] and blue data (Mishima unpublished preliminary data) at temperatures lower than the Tm line were obtained using emulsified water

1.1

1.0

0.9

TH

Volume (cm /g)

Fig. 2.1 Experimental volume data of liquid water and amorphous ices. High-density amorphous ice (HDA) was made by compression of ice Ih below 150 K and low-density amorphous ice (LDA) was made by heating HDA at 1 atm. Crystallization occurred easily in the no-man’s land where it was difficult to measure the properties of liquid water. Note that the volume of LDA is approximately equal to the volume of crystalline ice Ih

2 Volume of Liquid Water and Amorphous Ices

0.8

Tm 200

300

0.7

400

Temperature (K)

0 Pa)

0.5 (G 0 . 1 ure s s Pre

To calculate the volume of water at any pressure and temperature, several empirical formulas were developed to accurately match these experimental values. In one formula [5], the volume of water was calculated by adding 40 terms, and in another formula [6], 56 terms were added. An international organization determined which formula was appropriate and recommended the use of that formula. The data at temperatures below the Tm line in Fig. 2.2 are the volume of supercooled water. When an ice-crystal nucleus appears in the supercooled water, ice grows from this nucleus, and the entire supercooled water becomes crystalline, making it

2.1 Volume of Liquid Water

31

impossible to measure the volume of the supercooled liquid. By using pure water at 1 atm, the supercooled liquid state could be maintained for a while, and accurate volume measurement was possible before crystallization started. However, even with ultrapure water, the supercooled water crystallized when the temperature was extremely low. As a method of measuring the properties of low-temperature supercooled water, it is necessary to measure it in a short time before crystallization starts, or to keep the supercooled water stable for a relatively long period of time. In the latter case, empirically, micrometer-sized emulsified droplets [7, 8] or water confined in micrometer-sized tubes [9] were used. The volume of supercooled water (pink and light blue data) in Fig. 2.2 was measured using water emulsion [10]. Emulsion is many micrometer-sized water droplets dispersed in oil (Fig. 2.3). A small amount of surfactant was used to make the emulsion, and the surfactant located at the interface between oil and water and was insoluble inside the water droplet. Even if some water droplets crystallized, their crystal growth did not extend to other water droplets because they were separated from other water droplets. The volume of water was calculated by subtracting the volume of oil from the volume of the entire emulsion. Strictly speaking, even if the emulsion was properly prepared using suitable oil and surfactant as noted in [11], the properties of emulsified water might be different from those of bulk pure water. I compared in the pressure–temperature region of stable water the volume of water in freshly prepared emulsions with those of bulk pure water. They were the same within 2–3% experimental error. The volume of supercooled water in Fig. 2.2 smoothly connects to the volume of high-temperature water. This indicates that the volume of the emulsified water can be regarded as that of bulk pure water within an experimental error of about 2–3%. The same conclusion Fig. 2.3 Water emulsion: liquid water droplets (1–10 µm in diameter) in oil matrix

OIL

H2 O

(1–10 μm)

Fig. 2.4 Comparison of heat capacity at 1 atm between bulk pure water (black point) and emulsified water (open circle). Data from [8, 80–83]

2 Volume of Liquid Water and Amorphous Ices

Heat capacity (J/mol K)

32

100

emulsified water bulk water

90

80

70 240

260

280

300

Temperature (K)

may be drawn for the heat capacity of emulsified water (Fig. 2.4). The point is that the water of the emulsion is nearly the same as bulk pure water. Even with the use of the emulsion, at sufficiently low temperatures, many water droplets rapidly became ice crystals simultaneously, making the experiment of supercooled water difficult. This crystallization temperature was called the homogeneous nucleation temperature (TH ) (the dashed line in Fig. 2.2). Because of this crystallization, the volume data of the supercooled water was interrupted at low temperature. The pressure–temperature region below this TH line is the no-man’s land: the region of easy crystallization (Fig. 2.1). However, it should be noted that supercooled water in the no-man’s land can be experimentally studied. Liquid water can exist for a very short time before the crystallization begins [12, 13]. Besides, when water is confined in a nanometersized shape, it does not crystallize in the no-man’s land and can be studied [14–17]. Further, when the emulsion of dilute aqueous solution is pressurized and cooled, the homogeneous nucleation does not occur, and the solution smoothly becomes a highdensity glassy solution (or a high-density amorphous solid of the aqueous solution). The experiments of the aqueous solutions will be described in Sect. 2.6. The green data in Fig. 2.2 is the volume of water at negative pressures. The water volume was estimated from measurements and analysis of the micrometer-sized water inclusions in quartz [18]. The volume at negative pressures connects smoothly to the volume at positive pressures.

2.2 Volume of Accessible Liquid Water The volume of liquid water showed that water tends to phase separate at low temperature.

2.2 Volume of Accessible Liquid Water

33

The overall volume surface was estimated from the data in Fig. 2.2 and is shown in Fig. 2.5. In the low-pressure low-temperature region, as both temperature and pressure decrease, the surface becomes upwards, as indicated by the blue arrow. This “turn up” surface (rapid change in slope of volume surface) represents the strangeness of water. The change in volume when water is compressed is called compressibility, and the change in volume when water is heated is called thermal expansion coefficient, or expansivity. Therefore, the slope of the volume surface corresponds to compressibility and expansivity. The “turn up” of the volume surface indicates that both the water compressibility and expansivity change more significantly at lower temperatures and lower pressures. We can recognize this “turn up” clearly by rotating the three-dimensional volume surface of Fig. 2.5 properly. That is, when we change our viewpoint on the surface of Fig. 2.5 properly, many volume values at high pressures and high temperatures overlap on one monotonous line rather precisely (the green line in Fig. 2.6). The overlap may be related to the liquid–gas phase transition. The water with the volume above this green line is anomalous compared with the “normal” water at high pressures and high temperatures. We can recognize that the volume of the room temperature water at 1 atm (the red point in Fig. 2.6) is slightly above this green “volume” line. In other words, the ambient water is slightly low-density and slightly strange. It is known that, when the temperature of water decreases at 1 atm, its compressibility starts to increase at 319 K (46 °C) [19]. Because compressibility is the slope of the volume surface in the pressure direction, the increase in compressibility corresponds to the appearance of the thin black line with a slope larger than that of the green line at temperatures lower than about 319 K (Fig. 2.6). If you look carefully at the water volume surface, water at high pressures and low temperatures also has a strange feature. As indicated by the red arrow in Fig. 2.5, the temperature change of the volume is slightly “turn down”, or “convex upward”, Fig. 2.5 Turn up and turn down of the volume of liquid water at low temperatures. The thin red line is the boundary between the convex upward region at low temperature and the convex downward region at high temperature and corresponds to the line of the temperature of the minimum expansivity (Tme) [22, 24]

1.0 0.9 0.8

TH

Tm

Tme 0.7

0

200

300

400

Temperature (K)

0.5

1.0

ss

Pre

Pa)

(G ure

Volume (cm /g)

1.1

34

2 Volume of Liquid Water and Amorphous Ices

Fig. 2.6 Rotated Fig. 2.5. The red point is the volume of our familiar 300-K water, and it locates slightly above the green base line to which the volume of high-pressure high-temperature water collapsed

1.0 0.9 0.8 200

400

0.4 0.8 eratu re (K re Pressu (GPa) )

Temp

0

Volume (cm /g)

1.1

0.7

at low temperature. That is, the volume of low-temperature water decreases larger as temperature decreases. Using bulk pure water, researchers of several different institutes performed precise measurements of volume, heat of compression, and speed of sound at high pressures [20–23]. Analyzing these data, expansivity was calculated, and these independent experimental results showed that the “turn down” volume change at high pressure was small but certainly occurred. This “turn down” was also suggested by the experiment of the emulsified supercooled water [10]. On the other hand, the volume change was “convex downward” at high temperature. The boundary between the high temperature “convex downward” region and the low temperature “convex upward” region was estimated using the experimental data [22, 24] and is shown by the thin red line in Fig. 2.5. The boundary corresponds to the line of temperature of the minimum expansivity (Tme). The volume of water decreased more rapidly at lower temperatures below this thin red line. The two experimental facts, namely “the volume increase of low-pressure and low-temperature water” and “the volume decrease of high-pressure low-temperature water”, indicated that the volume of water changed a lot by pressure at low temperatures. With Fig. 2.1 in mind, the question was whether a discontinuous change in volume would occur in liquid water at much lower temperatures. To speculate it, the pressure dependence of expansivity at low temperature was examined. As illustrated in the left column of Fig. 2.7, at low pressure, the expansivity is positive at high temperature and negative at low temperature. In the case of high-pressure water, it is large positive at high temperature, it becomes smaller as temperature decreases, and becomes large again at low temperature. The experimental values of expansivity of liquid water between 246 and 403 K and between 0 GPa and 0.6 GPa are shown by the black dots in Fig. 2.8. Ter Minassian et al. made an empirical equation of the expansivity based on their experimental values [22], and the green lines in the Fig. 2.8 are the calculated expansivity of the equation. Note that the volume of water, derived from the expansivity equation,

2.2 Volume of Accessible Liquid Water

35

low pressure

high pressure

slope: expansivity

slope: expansivity

very

negative

positive

volume

volume

positive

positive

zero

temperature

high

low

0

low

temperature

high

temperature

Tme

expansivity

expansivity

TMD

low

very positive

high

zero 0

low

temperature

high

Fig. 2.7 Relation between volume and expansivity. Expansivity relates to the slope of volume surface along the temperature axis. TMD: temperature of maximum density (or minimum volume). Tme: temperature of minimum expansivity

agrees with the experimental volume in Fig. 2.2, including the volume near 200 K [10] and the volume at negative pressure [18]. When the green lines in Fig. 2.8 are extrapolated to low temperatures by calculating the equation, the lines tear the expansivity surface up and down at about 0.11 GPa and 190 K, indicating occurrence of a phase separation of water into low-density and high-density states. The lowtemperature extrapolation of another empirical equation of water volume [5] also indicated the existence of the phase separation at about 0.06 GPa and 220 K. That is, precise experimental data of liquid water under pressure indicated that liquid water tends to separate into two phases at a low temperature. Note the suggested separation point is not the liquid–liquid critical point but probably relates to the low-temperature stability limit of the liquid state (Fig. 2.9); the red separation point of the expansivity surface in Fig. 2.9 might locate on the spinodal line of high-density liquid water as will be illustrated later in Fig. 2.23. Simulation of the ST2 model of water showed both “turn up” and “turn down” behaviors and a phase separation at a low temperature [25, 26]. The so-called IAPWS-95 equation [6], which was recommended for volume calculation of high-temperature liquid water by the international organization, could not be extrapolated to low temperatures under pressure because it neglected the “turn down” tendency of water volume observed by Ter Minassian et al. [22].

2 Volume of Liquid Water and Amorphous Ices

Expansivity (10

-3

-1

K )

36

0.5

0 0.

-0.5 400

0. 350

Tempe

300

rature

(K)

250 22

0

K

2 0. ure

4

(G

6

) Pa

0 ess r

P

Fig. 2.8 Expansivity surface of liquid water. Experimental data (black points) from [20, 22]. Holten et al. [24] extracted numerical values of expansivity from the graph of [22], and I calculated expansivity using the volume data of [20]. TMD: the temperature of maximum density. The slope of the expansivity surface in the temperature direction is zero at the thin red Tme line. Pressure difference between adjacent green lines is 0.04 GPa. The lowest temperature is 220 K. Note that the directions of the pressure and temperature axes are different from those in Fig. 2.5

As illustrated in Fig. 2.7, expansivity is zero at the temperature where volume is minimum, or density is maximum (TMD). The pressure dependence of TMD and that of the temperature of the minimum expansivity (Tme) were calculated using the empirical equation [22] and shown by the red dashed and red solid lines, respectively, in Figs. 2.8 and 2.9.

2.3 Vitrification of Water Under Pressure As noted in Sect. 1.2, cooling micrometer-sized liquid–water droplets quickly enough in vacuum caused liquid water to vitrify into low-density amorphous ice (the blue arrow in Fig. 2.10). This vitrification was well confirmed in experiments of plural research institutes [1, 27–29]. It was difficult to measure the properties of the water during the rapid cooling. This was because water crystallized when cooled slowly for measurement. However, some experiments suggested that liquid water at about 1 atm gradually changed from high density to low density as temperature decreased [17, 30–32], which will be discussed in Chap. 3. When the emulsified liquid water was cooled rapidly under high pressure, the water vitrified into high-density glassy water (the red arrow in Fig. 2.10) [33]. Raman spectra of the glassy water resembled that of the high-density amorphous ice made by cold compression of crystalline ice Ih. The high-density glassy water made from

2.3 Vitrification of Water Under Pressure

37

0.45

300

0.4

Temperature (K)

e

TM D

Tm

0.3 0.2

0.5

0 -1

-∞

200

TH +∞ -0.1

0

0.1

0.2

0.3

0.4

0.5

Pressure (GPa) Fig. 2.9 Pressure dependence of the temperature of maximum density (TMD) and the temperature of minimum expansivity (Tme). Holten et al. [24] derived TMD values from experimental volume data of [75] (black open circles) and [10] (green filled points). Green open points from [18]. Black point at 1 atm locates at 277 K (4 °C). Light blue line: Caldwell experimental data [88]. Thick red dashed line is the TMD line in Fig. 2.8. The TMD line continuously changes into a line of the temperature of minimum density at low temperatures after the maximum pressure (thin red dashed line). The red dashed line and gray lines are obtained using equation of [22], and the numbers indicate constant expansivity values (10–3 K−1 ). Grey experimental points near the line have the same expansivity; see Fig. 2.8. Black dashed TH line: homogeneous nucleation line [89]. Red point: separation point of the calculated expansivity surface. The red Tme line is obtained using the equation of [22], and the red open points are determined using experimental data of [20, 22] Fig. 2.10 Vitrification of liquid water

1.2

1.0

0.9

A

HD

0.8 0.7

0 100

200

0.4 300

Temperature (K)

0.8

r ssu Pre

Pa) e (G

Volume (cm3 /g)

1.1

liquid water

LDA

38

2 Volume of Liquid Water and Amorphous Ices

liquid water transformed rapidly to low-density amorphous ice at low pressure as the high-density amorphous ice made by cold compression of crystalline ice did. Therefore, the high-density amorphous states made by the two different methods well resemble. To rapidly cool the high-pressure liquid water compressed inside a high-pressure vessel, some contrivance was required. This was because the high-pressure vessel also needed to be cooled together. We put emulsified liquid water in a 0.3 mmsize hole of a small pressure gasket and compressed it to about 0.3 GPa using two elongated conical steel rods [33] (Fig. 2.11). We cooled a soft metal (indium) to liquid-nitrogen temperature (77 K) and quickly pressed the soft metal to the thin rods with the gasket. The tip temperature of the rods decreased from room temperature to 130 K or less within 0.1 s by this “hammer quenching” method. Immediately thereafter, the entire high-pressure device was immersed in liquid nitrogen. In this way, high-pressure emulsified water (high-density liquid water) could be vitrified to high-density amorphous ice reproducibly. This high-density amorphous ice (or highdensity glassy water) was decompressed to 1 atm at 77 K and heated. It transformed into low-density amorphous ice, like the high-density amorphous ice made by cold compression of ice Ih. The phase change at 1 atm was confirmed by Raman-scattering measurements as well as crude X-ray-diffraction experiment. The same method was applied to dilute salt water [33, 34]. The solution could be vitrified without emulsification. That is, when 2-mol% lithium-chloride (LiCl)

high-pressure water (emulsion)

Fig. 2.11 A devise for rapid cooling of high-pressure water [33]. Water emulsion was squeezed between truncated surfaces of two rods, and then cooled rapidly with cold soft indium metal. The tip diameter of the truncated surface is 1 mm. Reproduced from [33], with the permission of AIP publishing

2.3 Vitrification of Water Under Pressure

39

aqueous solution was rapidly cooled at a pressure higher than 0.2 GPa, it became high-density glass with a water structure like that of high-density amorphous ice [34]. Solute molecules seemed to be dispersed uniformly in it. The high-density glass of solvent water made by vitrification under pressure was assumed to be the same with the high-density amorphous ice made by cold compression. These glass and amorphous ice are henceforth referred as high-density amorphous ice in this book. Gruner’s group has devised another way to vitrify high-pressure water. Initially, they tried cryopreservation of a small protein crystal under pressure. The protein crystal was made from a “mother” aqueous solution, and water was included in the crystal. The crystal was covered by a small amount of oil and compressed to 0.2 GPa using high-pressure helium gas that did not liquefy even at low temperature [35, 36]. In the high-pressure helium gas, the protein crystal was brought into contact with a wall of high-pressure vessel that was cooled to liquid-nitrogen temperature. The protein crystal was cooled from room temperature together with the water in the crystal. Thus, high-density amorphous ice was produced in the crystal. The protein crystal was decompressed to 1 atm, and its structure was examined by X-ray-diffraction measurement at low temperatures. As shown in the left panel of Fig. 2.12a, a halo pattern of high-density amorphous ice was observed together Fig. 2.12 Low-temperature X-ray-diffraction images of a a protein crystal infiltered with water and b bulk “mother solution. The crystal and solution were cooled under pressure, decompressed at low temperature, and heated at 1 atm. X-ray measurements were performed during the heating [36]. Composite figure: courtesy of Prof. Sol M. Gruner. Adapted with permission of the International Union of Crystallography. https://doi. org/10.1107/S00218898070 48820

(a) water in a protein crystal

(b) water of bulk solution 80 K

150 K

40

2 Volume of Liquid Water and Amorphous Ices

with the diffraction points of the protein crystal and a broad diffraction ring of the coating oil. When the crystal was heated, the high-density amorphous ice changed to low-density amorphous ice in the protein crystal (the right panel of Fig. 2.12a). In the field of cryopreservation of protein crystals and bio-tissues, various “mother” solutions were cooled rapidly at pressures higher than about 0.2 GPa, and high-density amorphous ice was often made [36–38]. The X-ray-diffraction image of the vitrified solution (that is high-density amorphous ice) is shown in Fig. 2.12b. The image of the low-density amorphous ice made by heating this highdensity amorphous ice at 1 atm is also shown. The X-ray-diffraction profiles of these low- and high-density amorphous ices are shown in Fig. 2.13. The profiles

in protein solution

Intensity

bulk

in protein solution bulk 0

1

2

3

4

-1

Q (Å ) Fig. 2.13 X-ray scattered intensity of amorphous ices at 1 atm. High density amorphous ice (HDA) vitrified under pressure (in protein and solution) and HDA made by cold compression of ice Ih (bulk) were heated at 1 atm to make low density amorphous ice (LDA). Q: momentum transfer. Two vertical lines: common peak positions of HDA and LDA. Thin dashed line: diffraction of coating oil. Adapted profiles of amorphous ices in protein and solution [36]: courtesy of Prof. Sol M. Gruner. Adapted with permission of the International Union of Crystallography. https://doi.org/ 10.1107/S0021889807048820. Adapted profiles of bulk amorphous ices: courtesy of Prof. Gyan P. Johari. Adapted figure with permission from [84]. Copyright (1986) by the American Physical Society

2.3 Vitrification of Water Under Pressure

HDL with LiCl

Tg

0.1 mm

Piston displacement

Fig. 2.14 Piston displacement of 4.8 mol% LiCl–H2 O emulsion during cooling under pressure, showing change in volume at glass transition. The solution becomes glassy below Tg where the rate of volume decrease becomes small. Reproduced from [40], with the permission of AIP Publishing

41

HDA with LiCl

at 0.45 GPa

140

150

160

170

Temperature (K)

of high-density amorphous ice made by cold compression of ice Ih and the lowdensity amorphous ice made by heating it are also shown for comparison. Although the X-ray-diffraction peak positions in protein and solution are slightly shifted from those of bulk pure amorphous ices, these high-pressure-cooling experiments clearly showed high-density liquid water (or solution) became high-density amorphous ice when cooled under pressure. When emulsion of dilute aqueous solution was used, a large amount of highdensity amorphous ice could be made by cooling the emulsion relatively slowly under pressure [39, 40]. When the volume change of the emulsion was measured during the cooling, only the glass transition from high-density liquid to high-density amorphous ice was observed (Fig. 2.14). The high-density amorphous ice (glass) containing a small amount of solute changed to low-density amorphous ice at low pressures. In the case of the dilute LiCl aqueous solution, it is phase separated into low-density amorphous ice of pure water and high-density amorphous ice of high solute concentration, as observed when the solution was rapidly cooled at 1 atm [29]. In the case of the dilute aqueous solution of glycerol, high-density amorphous ice containing the solute turned into low-density amorphous ice showing no noticeable phase separation. When this low-density amorphous ice of glycerol solution was compressed, it completely converted again to high-density amorphous ice [41] (see Sect. 2.6.2).

42

2 Volume of Liquid Water and Amorphous Ices

2.4 Softening of Low-Density and High-Density Amorphous Ices The amorphous ices are stiff at low temperature, and it is easily imagined that water molecules of the amorphous ices do not move around easily. The network structure of water molecules will not change immediately when pressure and temperature are changed. Therefore, when its volume was measured under a certain pressure– temperature condition, the volume was different depending on the procedure how the amorphous ice had been brought to the pressure–temperature condition. Even so, the amorphous ice could be divided into two types, and their volumes were clearly different. The volume of low-density amorphous ice and that of high-density amorphous ice are shown by the LDA and HDA portions in Fig. 2.1, respectively. The volumes of these unrelaxed amorphous ices were scattered and ambiguous. The unrelaxed high-density amorphous ice was called uHDA. When uHDA was strongly compressed to higher pressures, its volume initially decreased slowly, then largely, and slowly again, like a sigmoid curve [42, 43]. The gradual densification was observed during compression of various glasses since 1953 [44] and was different from the distinct and sharp transition between the low-density and high-density amorphous ices [43]. Water molecules in low-density and high-density amorphous ices started to move when temperature was increased. Along with this, low-density amorphous ice and high-density amorphous ice slowly relaxed toward their equilibrium state at the given pressure and temperature. When the unrelaxed high-density amorphous ice made by compressing ice Ih at 77 K was heated under high pressure, its structure and properties changed slightly by the relaxation, or annealing [13, 45]. For example, as shown by Loerting et al., when heated at about 1.1 GPa, its volume started to decrease slowly from about 100 K, and the volume reduction almost finished around 160 K [46]. The high-density amorphous ice relaxed (annealed) under high pressure was called very high-density amorphous ice, or vHDA. On the other hand, high-density amorphous ice relaxed by heating at low pressure, like 0.2 GPa, was called expanded highdensity amorphous ice [47], or eHDA, because its volume gradually increased by the relaxation. There were numerous and consecutive intermediate states between eHDA, vHDA, and uHDA, and these states changed continuously as the pressure and temperature were changed [48]. Because of the continuity, every state between eHDA, vHDA, and uHDA belonged to the same high-density amorphous phase. A lot of experiments indicated that these relaxed (annealed) amorphous ices became soft when heated, as reviewed in [49]. Water molecules in low- and highdensity amorphous ices started moving at about 130–140 K around 0–0.3 GPa in the laboratory timescale. In one experiment, low-density amorphous ice was pushed with a blunted conical indenter [50], and in another experiment, low- and high-density amorphous ices were sandwiched between two flat surfaces of the diamond anvils [51] (Fig. 2.15). It was observed in both experiments that when the temperature of these amorphous ices increased above 130–140 K, the amorphous ices deformed slowly, meaning that the water molecules had moved. The movement of water molecules,

2.4 Softening of Low-Density and High-Density Amorphous Ices

(a)

43

(b) squeeze

[side view] [side view]

[top view]

~1 cm 1 mm

Fig. 2.15 Illustration of the mechanical deformation test of amorphous ices to infer large movement of water molecules on heating. a Indenter loading. Adapted figure from [50]: courtesy of Prof. Gyan P. Johari. b Hard pushing between flat surfaces [51]. Adapted figure from [86]

or relaxation of amorphous ices, was suggested in thermal measurements, dielectric measurements, and volume measurements, too [49]. In addition, it was shown in a small-angle X-ray-diffraction experiment of high-density amorphous ices at 130 K and 1 atm that water molecules in the amorphous ice diffused as in the high-density liquid water [52]. Looking at these experimental results, both low- and high-density amorphous ices at about 140 K may be roughly regarded as glasses that have begun to mechanically soften and as very viscous liquids simultaneously. The word of “liquid” has a fluid image, but these low-temperature liquids might look like a very viscous syrup. At about 140 K at 0.1 GPa and at about 150 K at 0.2 GPa, these viscous liquids seemed to be in nearly equilibrium states [53, 54].

2.5 Apparently Discontinuous Transition Between Two Amorphous Ices The transition between low-density amorphous ice and high-density amorphous ice occurred discontinuously and reversibly with a volume jump of about 20% (Fig. 2.16). The transition pressure during compression was higher than the pressure of the reverse transition during decompression. The higher the temperature, the smaller the difference in the transition pressure between the compression and decompression processes (see Fig. 1.10). The transition was always sharp, and the volume change of about 20% was nearly constant, even if the temperature changed from 77 to 150 K.

44

2 Volume of Liquid Water and Amorphous Ices

LDA

Volume (cm /g)

at 135 K 1.0

HDA 0.8

0

0.5

1.0

Pressure (GPa) Fig. 2.16 Compression-induced and decompression-induced LDA-HDA transition of H2 O at about 135 K. Results of two experiments are shown. Dashed line: adapted figure from [4], with the permission of AIP Publishing. Solid line: adapted figure from the supporting information of [85]: courtesy of Dr. Yoshiharu Suzuki

Figure 2.17a shows two photos when low-density and high-density amorphous ices were strongly squeezed between two diamond anvils. The left photo was taken at 120 K, and the right at 135 K. In each photo, the pressure of the sample increases gradually from 0 GPa at the left side of the sample area to a few GPa in the right side; this is because the sample pressure in the center of the diamond anvil is generally higher than the pressure of the edge of the sample as shown in Fig. 2.17b. Therefore, the ice at the edge was low-density amorphous ice, and the ice at the center is the high-density amorphous ice. These amorphous ices were identified by the Raman-scattering measurement and the known low-temperature phase relations of crystalline and amorphous ices [51]. The boundary between low-pressure low-density amorphous ice and high-pressure high-density amorphous ice became sharper and straighter at higher temperatures (for example, the 135 K photo compared to the 120 K photo). This indicated that, during heating from 120 to 135 K, water molecules moved, the ice deformed, and the stress in the amorphous ices became smooth. The clear single boundary line at high temperature was consistent with the sharp transition between the low- and high-density amorphous ices. Crystallization appeared to start at about 145–150 K. When amorphous ices were strongly squeezed to much higher pressures, another sharp transition line was observed at a few GPa, and it corresponded to the pressureinduced crystallization of high-density amorphous ice [55]. There was no other detectable transition line, which indicated that there was no more amorphous phase. There were only two kinds of amorphous ices: low-density and high-density amorphous ices. When the structural relaxation (or annealing) of each amorphous ice was disregarded, nearly all experiments performed at different laboratories, such as neutron

2.5 Apparently Discontinuous Transition Between Two Amorphous Ices

(a)

LDA

HDA

HDA

135 K

120 K

LDA

Fig. 2.17 Visual observation of the transition between low- and high-density amorphous ices (LDA and HDA) under pressure. a Photos at 120 K (left) and at 135 K (right). b Illustration of the photos: adapted figure from [86]. Powder of LDA, made by the vapor-deposition method, was squeezed between two transparent diamond anvils, which created high pressure in the center. LDA transformed to HDA in the center, and the LDA-HDA boundary was observed. Boundary line became more distinct as temperature increased [51]. Blue and red lines: pressure distribution in LDA and in HDA

45

boundary

(b) [side view]

LDA

squeeze

transparent anvil

HDA

LDA

HDA

pressure

[top view] HDA LDA

experiments, x-ray-diffraction measurements, and Raman-scattering measurements, indicated that there were only two structures during the transition between lowdensity and high-density amorphous ices [52, 54, 56–58]. Low-density and highdensity structures of these amorphous ices coexisted during the amorphous-toamorphous transition occurring at a fixed pressure–temperature condition, and the ratio of the two structures changed continuously. We visually observed temporal transition from well-relaxed high-density amorphous ice to low-density amorphous ice at a fixed temperature near 120 K and at a fixed pressure of 1 atm (Fig. 2.18). One place of high-density amorphous ice suddenly expanded, and the transition started from this point and propagated, making cracks due to its volume expansion [57]. That is, the transition occurred heterogeneously

46

2 Volume of Liquid Water and Amorphous Ices

X'

HDA

0.5 0.0

X

LDA X

X’

110 min

1.0

X'

0.5

LDA X

X’

305 min

1.0

X'

115.4 K, 1 atm

HDA

0.0

X

LDA

Raman intensity

45 min

1.0

HDA

0.73 HDA + 0.27 LDA

HDA

HDA

0.5

LDA 0.0

X

1.8 mm

LDA X

X’

3000

3200

3400

3600

Raman frequency (cm -1)

Fig. 2.18 Transition of high-density amorphous ice (HDA) to low-density amorphous ice (LDA) during annealing at 115.4 K at 1 atm. The sample (about 1.8 mm in size) locates inside the dashed lines of the left photos. After the time indicated in the left, the sample was cooled to 25 K and microscopic Raman measurements were done at each point of the sample. The transition started at X and propagated over time (top to bottom) to X’. Thin black line in the photos: cracks : HDA : LDA : colors indicate that the Raman spectrum at each point was expressed by a linear combination of HDA and LDA spectra (the right panel). The center panel shows the change in the HDA/LDA ratio along X-X’ line of the sample. Adapted figures from [57]

even though both pressure and temperature were the same all over the amorphous ice. We could vaguely recognize the boundary region between the low-density area and the high-density area during the transition. In this boundary region, the ratio of the two amorphous structures changed gradually. This was quite different from the homogeneous relaxation phenomena in the high-density amorphous ice, which was observed just before the transition to low-density amorphous ice. During the relaxation, volume of the high-density amorphous ice expanded gradually and homogeneously all over the amorphous ice even if there might be local inhomogeneity of density. At a low temperature, for example at 115.4 K and 1 atm, the transition from highdensity amorphous ice to low-density amorphous ice took several hours (Fig. 2.18). At a high temperature, say 131 K and 1 atm, the transition completed in 0.03 seconds. When temperature was increased rapidly, the amorphous ice shattered because of the sudden volume expansion at the transition (Fig. 2.19). Many experiments showed that the transition between the two amorphous ices was first-order. Strictly speaking, this expression is incorrect. Although we can apply thermodynamics to glassy states because thermodynamics is the experimental relation

2.5 Apparently Discontinuous Transition Between Two Amorphous Ices

47

Fig. 2.19 Quick transition from high-density amorphous ice to low density amorphous ice at about 130 K. The high-density amorphous ice was relaxed by heating under pressure, recovered at 1 atm, and heated. The ice, located in the center of the left panel, suddenly changed to low-density amorphous ice with a rapid increase in volume (right panel). Photo from [87]

of the conservation of energy, we cannot apply the concept of equilibrium phase transition to the amorphous ices. This is because the water molecules hardly move at temperatures lower than the glass-transition temperature, and the amorphous ices are not in the equilibrium condition. This was the reason why we used the term “first-order like”, or “apparently first-order”, and why the boundary between lowand high-density amorphous ices was drawn by a dashed line in Fig. 1.13. However, in experiments near or above the glass-transition temperature, the transition between the low-density and high-density amorphous structures might be regarded as the firstorder transition. Indeed, the large pressure hysteresis of the transition was observed at 135–140 K at which amorphous ices started to soften. This strongly suggested the existence of a distinct first-order transition between low-density liquid water and high-density liquid water near the glass-transition temperature [4, 43, 52, 54, 57–59].

2.6 Liquid–Liquid Transition in Dilute Aqueous Solution When bulk pure water was used, it was extremely difficult to observe the first-order transition between low-density liquid and high-density liquid because of crystallization in the no-man’s land. However, when emulsion of dilute aqueous solution (such as LiCl aqueous solution and glycerol aqueous solution) was cooled at high pressure, the crystallization was suppressed; the no-man’s land disappeared. We could study supercooled liquid solution at low temperatures, and the existence of the first-order liquid–liquid transition was suggested.

48

2 Volume of Liquid Water and Amorphous Ices

2.6.1 Decompression of Diluted Salt Water (LiCl–H2 O) and Its Volume Change Emulsion of dilute lithium-chloride (LiCl) aqueous solution was compressed at room temperature and cooled under pressure. Then, the produced cold high-density liquid solution (HDL with LiCl), or high-density glassy solution (HDA with LiCl), was decompressed from high pressure at a constant rate at various low temperatures. The temperature of the emulsion, as well as the change in volume, was measured during the decompression [39, 40] (Figs. 2.20 and 2.21). ← (LDL) ← HDL

h n atio cle nu us eo en og om

180

ssion

decompre

HDL with salt equi -HDL LDL

no-man’ s land

160

A LDL

y l. boundar

Temperature (K)

200

Tg

LDL

140

HDA with salt LDL(LDA) ← HDL(HDA)

0

0.1

0.2

0.3

Pressure (GPa) Fig. 2.20 Transition of emulsified 4.8 mol% LiCl–H2 O solution on decompression. Thin lines are the temperature of the emulsion during decompression. Light blue line is the grass-transition line between liquid (HDL) and glass (HDA). The labeled phases were identified, or suggested, by X-ray and Raman measurements at 1 atm. Crystallization of LDL occurred on the green line. The transition above 150 K and homogeneous nucleation above about 170 K were assumed to be the subsequent HDL-LDL-ice I transition: HDL-to-LDL transition (the narrow region between the red line and the blue dashed line) and LDL-to-ice I crystallization (green line). Black dashed line: the supposed metastable equilibrium LDL-HDL boundary. A is the region between the red and black dashed lines. Adapted from [40], with the permission of AIP Publishing

2.6 Liquid–Liquid Transition in Dilute Aqueous Solution

a

4.8 mol% LiCl - H 2 O

49

b Piston displacement (a.u.: background subtracted)

Piston displacement (a.u.: background subtracted)

LDL LDL LDL

LDL HDL

LDL -HDL LDL

HDL

Tg

co

mp

re

decompression

ss

ion

HDL with salt Tg 140

0

0.1

ary

ound

quil. b

DL e

LDL-H

de

equil. boundary

A

A HDL with salt

0.2

Pressure (GPa)

0.3

150

160

170

Temperature (K)

180

0.1

0.3

0.2

re

ssu

Pre

0

a) (GP

Fig. 2.21 Background-subtracted piston displacement during decompression of emulsified 4.8 mol% LiCl–H2 O solution. a Volume change during decompression. b Three-dimensional illustration of the volume change. Thin lines are the volume changes of the emulsion during decompression. Blue Tg line is the grass transition between liquid (HDL) and glass (HDA). During decompression, the volume of the emulsion started to increase at the red line due to the appearance of LDL in the solution and finished at the dark blue line. Crystallization of LDL occurs on the green line. The same two-step transition was supposed to occur above 150 K. Black dashed line: the supposed location of the metastable equilibrium LDL-HDL boundary. A is the region between the red and black dashed lines. Adapted from [40], with the permission of AIP Publishing

When high-density glassy solution was decompressed at temperatures lower than 140 K, low-density amorphous ice (LDA) appeared in the high-density glassy solution (HDA with salt) as observed during decompression of bulk pure high-density amorphous ice (see Fig. 1.10). On decompression at temperatures slightly higher than 140 K, the high-density glassy solution changed to high-density liquid solution on the glass-transition line (the light blue Tg line in Fig. 2.20). After that, volume of the high-density liquid solution started to increase on the red line and stopped increasing on the dark blue line (Fig. 2.21). The rapid increase in volume in the purple zone in Fig. 2.21 was due to the appearance of low-density structure in high-density liquid solution [40]. During decompression of high-density liquid solution at a temperature slightly higher than 145 K, two distinct transitions were detected. First, low-density structure appeared in high-density liquid in the purple zone as mentioned above. And then, sudden crystallization of this low-density structure occurred on the green line, which was detected by the sharp increase in temperature (Figs. 2.20 and 2.21b). At temperatures higher than 150 K, the high-density liquid solution suddenly started to crystallize on the red line, and this red line became the known homogeneous nucleation line of the solution at temperatures higher than about 170 K. What is important is that the crystallization at high temperatures started on the line which continued smoothly from the low-temperature red line. This fact suggested

50

2 Volume of Liquid Water and Amorphous Ices

that the crystallization was triggered by the transition from high-density amorphous structure to low-density amorphous structure. That is, the homogeneous nucleation at high temperatures would be a two-step process: appearance of low-density amorphous structure and immediate crystallization of the low-density structure [39, 40]. Simulation supported this two-step scenario of the homogeneous crystallization [60]. Conversely, the homogeneous nucleation line (TH line) might be regarded as the transition line from high-density liquid to low-density liquid during decompression. Furthermore, because transition generally occurs easily at high temperatures, and because the pressure hysteresis of the transition is small at high temperature, the homogeneous nucleation line might be considered as the equilibrium liquid–liquid transition line at high temperatures. On the other hand, the pressure of the equilibrium liquid–liquid transition of pure water was estimated to be about 0.23 GPa at 150 K [61], which will be mentioned in the next section. By considering that the liquid– liquid transition pressure would be slightly lower than 0.23 GPa for solution as indicated in the next chapter, the equilibrium phase-boundary line between low- and high-density solvent waters was supposed to be something like the black dashed line in Figs. 2.20 and 2.21. Existence of the first-order liquid–liquid transition in the solution was hinted by the pressure hysteresis of the transition from high-density liquid solution to low-density liquid [39, 40]. The high-density liquid was decompressed, and the change in volume was monitored by measuring the displacement of the piston of the high-pressure apparatus. The displacement involved the volume change of the solvent water, that of the emulsion matrix, and the deformation of the high-pressure apparatus. The volume change of the solvent water was qualitatively estimated by subtracting a smooth arbitrary background and is shown in Fig. 2.21. As shown in Fig. 2.21b, when the aqueous solution was decompressed at temperatures between 150 and 170 K, its volume changed smoothly until the pressure of the red line. That is, the highdensity structure of water continued smoothly to the pressure lower than the supposed equilibrium phase boundary between low- and high-density liquid waters (the black dashed line). Therefore, high-density liquid existed in the region of low-density liquid (the region A in Figs. 2.20 and 2.21), meaning existence of a pressure hysteresis in the transition from high-density liquid water to low-density liquid water. This pressure hysteresis was observed at temperatures sufficiently higher than the glass-transition temperature of the solution. This implied the liquid-to-liquid transition in the solution was first-order.

2.6.2 Volume of Diluted Glycerol Aqueous Solution In the high-pressure experiments of emulsions of dilute glycerol solution, the transition from high-density water to low-density water during decompression and the reverse transition from low-density water to high-density water during compression were clearly observed [41] (Fig. 2.22). It was confirmed by Raman-scattering measurement that the structure of solvent water in the low-density aqueous solution

2.6 Liquid–Liquid Transition in Dilute Aqueous Solution

1.1 0.03 mf glycerol aqueous solution com

pres

sion

Volume (cm /g)

Fig. 2.22 The volume changes of the emulsified glycerol-H2 O solution of 0.03 mol fraction during compression and decompression at different temperatures. The volume change of the solution is not as sharp as that of pure water. Reproduced from [41], with the permission of AIP publishing

51

LDL

1.0 152 K 138 K 138 K

152 K

HDL

0.9 decom

pressio

0

0.2

n

0.4

Pressure (GPa) corresponded to the structure of low-density amorphous ice, and the structure in the high-density aqueous solution corresponded to that of high-density amorphous ice. Transition with pressure hysteresis also occurred at temperatures near or just above the glass-transition temperature of about 150 K, indicating presence of the first-order liquid–liquid transition. Suzuki changed the solute concentration of various kinds of aqueous solutions and studied the transition pressure and the pressure hysteresis of these solutions [61]. Extrapolating the concentration to zero (pure water), he estimated the equilibrium phase-boundary pressure between low-density liquid phase and high-density liquid phase of pure H2 O water. He concluded it is about 0.23 GPa at 150 K. It should be noted that these polyamorphic transition did not depend on the preparation history of the sample, meaning that the well-relaxed equilibrium amorphous states, or liquid states, were studied (See Supporting Information of [62]).

2.7 Liquid–Liquid Transition in Pure Water Strictly speaking, we cannot apply the result of aqueous solution to pure liquid water. That is, even though the aqueous solution showed the first-order liquid–liquid transition, the change in pure water might be a higher-order transition in which the change would occur continuously [63]. Then, how about the pressure hysteresis of the liquid–liquid transition of pure liquid water? The hysteresis at 135–140 K was described in the last part of Sect. 2.5, and the first-order transition between the soften amorphous ices was suggested. It should be noted that the hysteresis of transition was also suggested by the experiments of liquid water confined inside a nanometer-sized tubes at much higher temperatures [64]. Furthermore, importantly, bulk pure high-density liquid water was observed at pressures lower than 0.23 GPa

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2 Volume of Liquid Water and Amorphous Ices

and at about 150 K [53]. The pressure was lower than the equilibrium liquid–liquid transition pressure (0.23 GPa), and the temperature was higher by a few degrees than the glass-transition temperature. In other words, the equilibrium pure high-density liquid water existed in the region of low-density liquid water. This experimental observation showed the hysteresis of the transition from high-density liquid water to low-density liquid water, and strongly indicated the existence of the first-order liquid–liquid transition in pure water. Are the low-density and high-density liquid phases, existing at a low-temperature side of the no-man’s land, the same materials as the low-density and high-density liquid phases at the high-temperatures side of the no-man’s land? It was experimentally shown that the structure of liquid water at high temperatures could be represented by a mixture of the structures of the two amorphous ices [65, 66]. This suggested that the low-temperature liquids near 150 K and the high-temperature liquids above about 200 K are connected in the no-man’s land. It also suggested the existence of the low-density liquid water whose structure resembled that of the low-density amorphous ice.

2.8 Speculation of the Volume of Water in the No-Man’s Land Assuming the existence of the liquid–liquid critical point and refereeing the theoretical suggestion of the volume of supercooled water [67, 68], the volume of water in the no-man’s land was speculated and drawn in Fig. 2.23. The volume surface is twisted at low temperature, indicating the discontinuity between low-density water and high-density water. Both low-density and high-density liquid phases below the critical temperature has each stability limit that is called the spinodal line; the liquid water is unstable in the shadow region in Fig. 2.23, and the boundary of the shadow region is the spinodal line. Note that the experimental values are scattered below the glass-transition temperature (Tg) because the water molecules of amorphous ices are difficult to move in the laboratory timescale. The glass transition of liquid near its stability limit may be an interesting research topic in the future [69]. The question is whether low-density and high-density amorphous ices become liquid when compressed and decompressed to each spinodal line at low temperatures, respectively. The slope of the volume surface at the liquid–liquid critical point is vertical. On the other hand, the slope of the surface in the temperature direction is horizontal at the temperature of maximum density (TMD), as shown in Fig. 2.7. Therefore, the slope changes from horizontal to vertical in the pressure–temperature range between the red dashed TMD line and the critical point in Fig. 2.23. In other words, the critical point must be on the low-pressure low-temperature side of the TMD line. This restricts the location of the critical point. If the pressure dependence of TMD in Fig. 2.9 is correct, and if the liquid–liquid transition line located near the homogeneous nucleation line

2.8 Speculation of the Volume of Water in the No-Man’s Land

53 1.2

LLCP Tg

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HD

HD

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Tg

0 100

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0.4 300

Temperature (K)

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re ssu

a)

(GP

Pre

Fig. 2.23 The volume of liquid and glassy water speculated from experimental data. The glass transition near the high-pressure limit of low-density liquid water (the blue Tg line between LDA and LDL at high pressure) is speculated. Empty circle: liquid–liquid critical point (LLCP). Red dashed TMD line: the locus of the temperature of maximum density at different pressures. Red solid Tme line: the locus of the temperature of minimum expansivity. The equilibrium LDL-HDL transition occurs on the black dashed line whose pressure and temperature are assumed to be those of the homogeneous nucleation line. Note the point where TMD and Tme lines meet is not the LLCP

as discussed in Sect. 2.6.1, the critical point should locate below about 0.13 GPa and above about 210 K. In Fig. 2.23, when liquid water is cooled at pressures lower that of the red point located on the spinodal line of high-density liquid water, its volume increases significantly near the red point. On the other hand, its volume decreases when cooled at pressures slightly higher than that of the red point. This red point corresponds to the separation point of the expansivity surface where the line of maximum density (TMD) and the line of minimum expansivity (Tme) meets (see Fig. 2.9). By applying the Ising model, a simple theory with two critical points (a gas–liquid critical point and a liquid–liquid critical point) was made, and the volume change of water at low temperature was calculated [70] (Fig. 2.24). This theoretical volume qualitatively reproduced not only the well-known strange volume change of water but also the approximate overlap of the high-pressure high-temperature volume values on the “volume” line (the green line in Fig. 2.6) and the “turn down” volume change of the high-pressure low-temperature water in Fig. 2.5. It should be noted that the model did not consider the degree of freedom of molecules and had some limitation regarding estimation of entropy and related properties of real water.

54

2 Volume of Liquid Water and Amorphous Ices

(b)

GLCP 1.5

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0.6

0.2

0.4

r

ssu

Pre

0

a)

P e (G

Fig. 2.24 Volume of Ising model of water. The volume was calculated using three equations reported in [70]. Red dashed and red solid lines on the surface: temperature of maximum density (TMD) and temperature of minimum expansivity (Tme), respectively. Thin black dashed lines: equilibrium transition lines. a: The liquid–liquid critical point (LLCP) and the gas–liquid critical point (GLCP) are shown. b: Enlarged view near LLCP

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52. Perakis F, Amann-Winkel K, Lehmkühler F, Sprung M, Mariedahl D, Sellberg JA, Pathak H, Späh A, Cavalca F, Schlesinger D, Ricci A, Jain A, Massani B, Aubree F, Benmore CJ, Loerting T, Grübel G, Pettersson LGM, Nilsson A (2017) Diffusive dynamics during the high-to-low density transition in amorphous ice. Proc Natl Acad Sci USA 114:8193–8198. https://doi.org/ 10.1073/pnas.1705303114 53. Stern JN, Seidl-Nigsch M, Loerting T (2019) Evidence for high-density liquid water between 0.1 and 0.3 GPa near 150 K. Proc Natl Acad Sci USA 116:9191–9196. https://doi.org/10.1073/ pnas.1819832116 54. Handle PH, Loerting T (2018) Experimental study of the polyamorphism of water. I. The isobaric transitions from amorphous ices to LDA at 4 MPa. J Chem Phys 148:124508. https:// doi.org/10.1063/1.5019413 55. Hemley RJ, Chen LC, Mao HK (1989) New transformations between crystalline and amorphous ice. Nature 338:638–640. https://doi.org/10.1038/338638a0 56. Yoshimura Y, Stewart ST, Mao HK, Hemley RJ (2007) In situ Raman spectroscopy of lowtemperature/high-pressure transformations of H2 O. J Chem Phys 126:174505. https://doi.org/ 10.1063/1.2720830 57. Mishima O, Suzuki Y (2002) Propagation of the polyamorphic transition of ice and the liquidliquid critical point. Nature 419:599–603. https://doi.org/10.1038/nature01106 58. Klotz S, Strässle Th, Nelmes RJ, Loveday JS, Hamel G, Rousse G, Canny B, Chervin LC, Saitta AM (2005) Nature of the polyamorphic transition in ice under pressure. Phys Rev Lett 94:025506. https://doi.org/10.1103/PhysRevLett.94.025506 59. Winkel K, Mayer E, Loerting T (2011) Equilibrated high-density amorphous ice and its firstorder transition to the low-density form. J Phys Chem B 115:14141–14148. https://doi.org/10. 1021/jp203985w 60. Bullock G, Molinero V (2013) Low-density liquid water is the mother of ice: on the relation between mesostructure, thermodynamics and ice crystallization in solutions. Faraday Discuss 167:371–388. https://doi.org/10.1039/c3fd00085k 61. Suzuki Y (2017) Effect of solute nature on the polyamorphic transition in glassy polyol aqueous solutions. J Chem Phys 147:064511. https://doi.org/10.1063/1.4998201 62. Suzuki Y (2019) Effect of OH groups on the polyamorphic transition of polyol aqueous solutions. J Chem Phys 150:224508. https://doi.org/10.1063/1.5095649 63. Chatterjee S, Debenedetti PG (2006) Fluid-phase behavior of binary mixtures in which one component can have two critical points. J Chem Phys 124:154503. https://doi.org/10.1063/1. 2188402 64. Zhang Y, Faraone A, Kamitakahara WA, Liu KH, Mou CY, Leão JB, Chang S, Chen SH (2011) Density hysteresis of heavy water confined in a nanoporous silica matrix. Proc Natl Acad Sci USA 108:12206–12211. https://doi.org/10.1073/pnas.1100238108 65. Bellissent-Funel MC, Bosio L (1995) A neutron scattering study of liquid D2 O under pressure and at various temperatures. J Chem Phys 102:3727–3735. https://doi.org/10.1063/1.468555 66. Soper AK, Ricci MA (2000) Structures of high-density and low-density water. Phys Rev Lett 84:2881–2884. https://doi.org/10.1103/PhysRevLett.84.2881 67. Poole PH, Sciortino F, Grande T, Stanley HE, Angell CA (1994) Effect of hydrogen bonds on the thermodynamic behavior of liquid water. Phys Rev Lett 73:1632–1635. https://doi.org/10. 1103/PhysRevLett.73.1632 68. Jeffery CA, Austin PH (1999) A new analytic equation of state for liquid water. J Chem Phys 110:484–496. https://doi.org/10.1063/1.477977 69. Giovambattista N, Loerting T, Lukanov BR, Starr FW (2012) Interplay of the glass transition and the liquid-liquid phase transition in water. Sci Rep 2:390. https://doi.org/10.1038/srep00390 70. Cerdeiriña CA, Troncoso J, González-Salgado D, Debenedetti PG, Stanley HE (2019) Water’s two-critical-point scenario in the Ising paradigm. J Chem Phys 150:244509. https://doi.org/10. 1063/1.5096890 71. Adams LH (1931) Equilibrium in binary systems under pressure. I. An experi-mental and thermodynamic investigation of the system, NaCl-H2 O, at 25°. J Am Chem Soc 53:3769–3813. https://doi.org/10.1021/ja01361a020

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72. Kell GS, Whalley E (1975) Reanalysis of density of liquid water in range 0–150 °C and 0–1 kbar. J Chem Phys 62:3496–3503. https://doi.org/10.1063/1.430986 73. Bradshaw A, Schleicher K (1976) Compressibility of distilled water and seawater. Deep-Sea Res 23:583–593. https://doi.org/10.1016/0011-7471(76)90002-4 74. Hilbert R, Tödheide K, Franck EU (1981) PVT data for water in the ranges 20 to 600 °C and 100 to 4000 bar. Ber Bunsenges Phys Chem 85:636–643. https://doi.org/10.1002/bbpc.198108 50906 75. Hare DE, Sorensen CM (1987) The density of supercooled water. II. Bulk samples cooled to the homogeneous nucleation limit. J Chem Phys 87:4840–4845. https://doi.org/10.1063/1.453710 76. Sotani T, Arabas J, Kubota H, Kijima M (2000) Volumetric behaviour of water under high pressure at subzero temperature. High Temp-High Press 32:433–440. https://doi.org/10.1068/ htwu318 77. Asada S, Sotani T, Arabas J, Kubota H, Matsuo S, Tanaka Y (2002) Density of water at subzero temperature under high pressure: measurements and correlation. J Phys Condens Matter 14:11447–11452. https://doi.org/10.1088/0953-8984/14/44/498 78. Guignon B, Aparicio C, Sanz PD (2010) Specific volume of liquid water from (253 to 323) K and pressures up to 350 MPa by volumetric measurements. J Chem Eng Data 55:3338–3345. https://doi.org/10.1021/je100083w 79. Romeo R, Albo PAG, Lorefice S, Lago S (2016) Density measurements of subcooled water in the temperature range of (243 and 283) K and for pressures up to 400 MPa. J Chem Phys 144:074501. https://doi.org/10.1063/1.4941580 80. Angell CA, Oguni M, Sichina WJ (1982) Heat capacity of water at extremes of supercooling and superheating. J Phys Chem 86:998–1002. https://doi.org/10.1021/j100395a032 81. Tombari E, Ferrari C, Salvetti G (1999) Heat capacity anomaly in a large sample of supercooled water. Chem Phys Lett 300:749–751. https://doi.org/10.1016/S0009-2614(98)01392-X 82. Archer DG, Carter RW (2000) Thermodynamic properties of the NaCl + H2 O system. 4. Heat capacities of H2 O and NaCl(aq) in cold-stable and supercooled states. J Phys Chem B 104:8563–8584. https://doi.org/10.1021/jp0003914 83. Voronov VP, Podnek VE, Anisimov MA (2019) High-resolution adiabatic calorimetry of supercooled water. J Phys Conf Ser 1385:012008. https://doi.org/10.1088/1742-6596/1385/ 1/012008 84. Bosio L, Johari GP, Teixeira J (1986) X-ray study of high-density amorphous water. Phys Rev Lett 56:460–463. https://doi.org/10.1103/PhysRevLett.56.460 85. Suzuki Y (2018) Experimental estimation of the location of liquid-liquid critical point for polyol aqueous solutions. J Chem Phys 149:204501. https://doi.org/10.1063/1.5050832 86. Mishima O (2010) Polyamorphism in water. Proc Jpn Acad B 86:165–175. https://doi.org/10. 2183/pjab.86.165 87. Mishima O (2004) Polyamorphism in water and the second critical point hypothesis. Netsu Sokutei 31:23–28. https://doi.org/10.11311/jscta1974.31.23 88. Caldwell DR (1978) The maximum density points of pure and saline water. Deep-Sea Res 25:175–181. https://doi.org/10.1016/0146-6291(78)90005-X 89. Kanno H, Speedy RJ, Angell CA (1975) Supercooling of water to −92°C under pressure. Science 189:880–881. https://doi.org/10.1126/science.189.4206.880

Chapter 3

Metastable Melting Lines of Crystalline Ices

3.1 Pressure–Temperature Phase Diagram of H2 O The pressure–temperature phase diagram of H2 O is shown in Fig. 3.1 where liquid water and crystalline ices are in thermally equilibrium conditions. The diagram is closely related to the so-called Gibbs energy of each phase. The Gibbs energy depends on pressure and temperature. Each phase, indicated by the label in Fig. 3.1, has the lowest Gibbs energy in its pressure–temperature domain compared with other phases. The Gibbs energies of two adjacent phases become the same on their phase-boundary line. Three phases have the same Gibbs energy at a triple point where three phaseboundary lines meet. It should be noted that, at the triple point, the angle between any two boundary lines must be less than 180°. Although each phase tends to be transformed into a more stable phase when brought beyond its phase boundary, it still can exist as a metastable state before its transformation to the more stable phase. Therefore, the phase-boundary line does not stop at the triple point. For example, the melting line of any ice phase continues beyond the triple point as a metastable melting line from the ice to supercooled liquid water (the loci of the thin dots in Fig. 3.1). If the melting line continues smoothly into the no-man’s land at low temperature, this suggests that metastable equilibrium liquid water exists along the melting line in the no-man’s land. It should be also noted that the phase-boundary line between two crystalline phases passes through the point where their melting lines intersect. Thermodynamic properties of any phase can be defined by using the Gibbs energy. Once we know the Gibbs energy of a phase in a wide pressure–temperature region, that is once the Gibbs energy surface of the phase is known as functions of pressure and temperature, we can theoretically derive all the thermodynamic properties of the phase from the slope and curvature of the surface. For example, volume is the slope in the direction of the pressure axis, and entropy is related to the slope in the direction of the temperature axis. We can also obtain information about the volume and entropy of adjacent phases from the slope of the equilibrium phase-boundary line in Fig. 3.1 in accordance with the so-called Clausius–Clapeyron relation. This is because the phase boundary is related to the Gibbs energies of these phases, and the © National Institute for Materials Science, Japan 2021 O. Mishima, Liquid-Phase Transition in Water, NIMS Monographs, https://doi.org/10.1007/978-4-431-56915-2_3

59

Fig. 3.1 Phase diagram of H2 O. Our familiar ice is ice Ih. The stable regions of ice Ih, II, III, V, and VI are indicated by the bold lines. Thin dots are metastable melting points obtained by experiments of emulsified ices

3 Metastable Melting Lines of Crystalline Ices 300

liquid water VI

Temperature (K)

60

250

Ih

V

III

II

200

150 0

0.4

0.8

Pressure (GPa)

volume and entropy are related to these Gibbs energies. The phase-boundary lines between ice Ih and ice III, between ice III and ice V, and between ice V and ice VI are almost parallel to the temperature axis, as shown in Fig. 3.1. This indicates that these ices have different volume and nearly the same entropy according to the Clausius–Clapeyron relation. On the other hand, the phase boundaries between ice II and the other ices (ice Ih, ice III, and ice V) are not parallel to the temperature axis. This indicates that both volume and entropy of ice II are different from those of the other ices. The difference in entropy is related to the orientation of water molecule in ice, or location of hydrogen atoms. The orientation of water molecules in ice II is ordered, and the ice II is a proton-ordered ice. Therefore, its entropy is small. On the other hand, the other ices have large entropy because the orientation of their water molecules is random; they are proton-disordered ices. This chapter only discusses with proton-disordered ices with large entropy.

3.2 Metastable Melting Line of Ice Ih In order to experimentally detect the metastable melting line of ice Ih, the emulsion of ice Ih was compressed at various low temperatures [1, 2]. The ice Ih emulsion is micrometer-sized ice-Ih particles dispersed in oil. Generally, when the bulk ice Ih was cooled below 251 K and compressed beyond the pressure of the equilibrium phaseboundary line between ice Ih and high-pressure ice, the ice became the high-pressure ice as shown in Fig. 1.5 of Sect. 1.2. But that did not really happen when the ice

3.2 Metastable Melting Line of Ice Ih

force liquid

temperature

Fig. 3.2 Illustration of the experimental set-up (left) and the change in the emulsion temperature, Temulsion , during compression (right). Tcylinder : temperature of cylinder. The temperature of the emulsion started to change at the red point. Green TH point: homogeneous nucleation temperature on the melting line. Adapted figure from [2]

61

Tcylinder Temulsion

ice Ih 200 K

Tcylinder Temulsion

TH

Temulsion Tcylinder

pressure

emulsion was compressed. Many particles of ice Ih suppressed this crystal-to-crystal transition, and ice could be further compressed beyond its phase boundary. The ice finally melted on its metastable melting line as illustrated in Sect. 1.2. Importantly, the melting line continued smoothly into the no-man’s land below the homogeneous nucleation temperature, TH [1, 2]. Specifically, the ice-Ih emulsion was first made as follows. The emulsion of liquid water was made at room temperature at 1 bar, put in a piston-cylinder high-pressure apparatus, and cooled below the homogeneous nucleation temperature (about 235 K) at 1 atm; the liquid was transformed into the ice Ih in the no-man’s land. Then, the ice emulsion was heated, or cooled, to a given cylinder temperature and compressed (Fig. 3.2). The cylinder temperature was held nearly constant during the compression, and compression rate was fixed constant even during the melting transition [2]. Therefore, the ice Ih was forced to melt by pressure. As illustrated by the black line of the upper part of the right graph in Fig. 3.2, when the ice emulsion was compressed at a cylinder temperature higher than the homogeneous nucleation temperature on the ice-Ih melting line (the green point in Fig. 3.2), the temperature of the emulsion started to change at the red point and decreased along the melting line. This is because water molecules of the ice took away energy from its surroundings to move around as liquid water. After the entire ice melted, the temperature of the emulsion returned to the cylinder temperature. When the ice Ih emulsion was compressed at a cylinder temperature lower than the homogeneous nucleation temperature, a sudden increase in the emulsion temperature and a sudden decrease in volume were simultaneously observed at the red points as shown in the lower part of the right graph in Fig. 3.2. The temperatures of the emulsions of H2 O and D2 O ices during compression are shown by the thin black lines in Fig. 3.3a and b, respectively. Importantly, the increase in the emulsion temperature and the decrease in volume below the homogeneous nucleation temperature began exactly on the low-temperature extension of the melting line (Red points below TH in Fig. 3.3). Therefore, the low-temperature

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3 Metastable Melting Lines of Crystalline Ices

(a)

H2 O

water

Temperature (K)

250

supercooled water

200

0

0.5

1.0

1.5

Pressure (GPa)

(b) m

D2 O

water

Temperature (K)

250

supercooled water

200

0

0.5

1.0

1.5

Pressure (GPa) Fig. 3.3 Temperature of emulsified ice Ih during compression. a H2 O. b D2 O. Red point: compression induced melting point of ice Ih. Melting data of ice Ih and high-pressure ices are available in [12, 16]. See also the appendix of this book. Various thermodynamic data is available in the supplementary material of [16]. Tm : melting temperature. TH : homogeneous nucleation temperature. Green point: homogeneous nucleation temperature on the melting line. Adapted figure from [2]

changes were most likely related to the melting of the ice. It would be reasonable to think that ice Ih melted on the melting line, and immediately after the melting, the produced liquid was transformed into a high-pressure ice (Fig. 3.4). That is, the metastable equilibrium liquid water existed for a moment in the no-man’s land [1].

3.2 Metastable Melting Line of Ice Ih Fig. 3.4 Virtual melting. Ice Ih transforms to high pressure ice (HP ice) via high density liquid phase (HDL)

63

Ice Ih

HDL

HP Ice

endothermic melting exothermic crystallization

Generally, the crystal-crystal transition via a liquid phase, that is melting and crystallization, is called “virtual melting” [3] (Fig. 3.4). This kind of transition was visually observed in a colloidal crystal [4]. The transition at temperatures below the homogeneous nucleation temperature was considered melting and crystallization. In total, the transition looked like a crystal-crystal transition from ice Ih to a high-pressure ice. It should be noted that the observed transition was not an equilibrium crystal-crystal transition. The equilibrium phase-boundary line between ice Ih and the high-pressure ice would be almost parallel to the temperature axis and should pass through the intersection of their melting lines. Therefore, if the observed transition was the equilibrium crystal-crystal transition, the transition pressure should be lower than the melting pressure of ice Ih. When ice Ih is compressed at a constant rate and is transformed into a protonordered high-pressure ice, such as ice II, on the equilibrium crystal-crystal phase boundary, the heat would be evolved during the transition, and the temperature of these ices would increase. However, when the high-pressure ice is the protondisordered ice, the temperature would not change much during the equilibrium transition. This is because the molecular orientation of ice Ih and that of the high-pressure ice are random, and their entropy difference is small. The small entropy difference means almost no heat evolution nor heat absorption during the transition. Therefore, the change in temperature during the transition would be small. When the crystal–liquid–crystal transition occurred at the melting pressure of ice Ih (Fig. 3.3), heat should be generated. This is because ice Ih has a larger volume than the high-pressure ice. While ice Ih is compressed from the pressure of the equilibrium

Fig. 3.5 Expected melting lines (Tm ) near Kauzmann temperature (TK ). The TK is assumed to be a liquid property

3 Metastable Melting Lines of Crystalline Ices

liquid

temperature

64

quid

ooled li

superc

TK (P)

pressure

crystal-crystal transition to the melting pressure of ice Ih, more mechanical energy is stored in ice Ih than the high-pressure ice. This mechanical energy is released at the melting pressure of ice Ih, and the emulsion temperature is increased by the energy. In the pressure–temperature phase diagram, the slope of the ice-Ih melting line approached parallel to the temperature axis as temperature decreased. It means that heat absorbed during melting becomes small according to the Clausius–Clapeyron relation. That is, melting at low temperatures hardly reduce the emulsion temperature. If the line becomes parallel at a certain temperature, then the entropy of ice and that of water becomes the same at that temperature. That is, ice melts only by mechanical force without any heat absorption. The temperature at which the liquid has the same entropy as the crystal is called Kauzmann temperature, TK . It is unknown whether the Kauzmann temperature is a liquid property, or it depends on the crystalline structure. When the Kauzmann temperature is assumed to be a liquid property and depends on pressure, TK (P), the melting line of any crystalline phase should become parallel to the temperature axis at a temperature on the TK (P) line (Fig. 3.5). As shown in Fig. 3.6, the compression-induced melting line of H2 O ice Ih started to shift to the high-pressure side around 160 K and 0.5 GPa just before the slope of the melting line became parallel to the temperature axis. The shift indicated that nucleation of liquid phase in ice Ih was difficult to occur and/or that the thermal equilibrium condition between ice Ih and supercooled water was broken. The movement of water molecules became slow at low temperatures and the liquid would start to vitrify. Indeed, when ice Ih was compressed at a temperature lower than 160 K, water in a glassy state (high-density amorphous ice) started to emerge [1]. Anyway, the metastable equilibrium high-density liquid state existed down to about 160 K at 0.5 GPa along the smooth melting line of ice Ih.

3.3 Metastable Melting Line of Ice III

65

300 water

lti me

supercooled water

ng

Temperature (K)

250

200

Tg

150 HDA orp

am n

atio hiz

100

50 0

0.5

1.0

1.5

Pressure (GPa)

Fig. 3.6 Compression-induced melting and amorphization of ice Ih [1]. Tm : melting temperature of high-pressure ice. Tg: glass-transition temperature of dilute LiCl–H2 O solution. TH : homogeneous nucleation temperature of supercooled liquid water at high pressure. Adapted figure from [1]

3.3 Metastable Melting Line of Ice III In the above-mentioned compression experiment of the ice-Ih emulsion, ice Ih was transformed into various high-pressure ices. When I decompressed the emulsions of these high-pressure ices, I observed metastable melting of the high-pressure ices along their extrapolated melting lines. This arose the following question. If the liquid– liquid critical point of water exists as shown in Fig. 1.13, how do the ice melting lines behave in its vicinity? In an evening of the 1996 Gordon Conference, I asked this question to Poole and Sciortino in a chat at a pub of the conference. We discussed and drew various melting lines on the water phase diagram. Then, after I returned to my laboratory at Tsukuba, I did experiments. I decompressed emulsions of highpressure H2 O ices and detected their melting lines at low pressures (Fig. 3.7). The existence of the low-density liquid water and the transition between the low-density and high-density liquids were examined. The experimental results and its significance were discussed, evaluated, summarized, and published together with Stanley [2]. The validity of the results was later reconfirmed using the emulsion of D2 O ices [5]. Meanwhile, the use of the melting line to study the liquid–liquid transition was also employed independently by Togaya. In 1997, he reported the high-pressure high-temperature experiment of the melting line of graphite and indicated a possible

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3 Metastable Melting Lines of Crystalline Ices

(a)

(b)

300

300

H2O 280

VI

Ih

V

260

260

Temperature (K)

III IV

220

200

IV III PNP-XIV

240

220

200 TH

PNP-XIII

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TH

PNP-XIV

180

160

280

PNP-XIII

Ih

Temperature (K)

D2O

VI

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0.6

Pressure (GPa)

0.8

1.0

160

0

0.2

0.4

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Pressure (GPa)

Fig. 3.7 Temperature of emulsified high-pressure ices during decompression. a: H2 O. b: D2 O. TH : homogeneous nucleation temperature. Adapted figures from [2]

existence of a liquid–liquid transition in carbon [6]. The data might be less accurate due to the severe experimental condition of about 4700 K and 5 GPa, and further experimental support was desired. Returning to the water experiment, importantly, a smooth melting line of D2 O ice III was observed at pressures lower than about 0.3 GPa as shown by the purple line in Fig. 3.8 [5]. The smoothness indicated that high-density liquid water became low-density liquid water continuously along the melting line. Specifically, the temperature of the emulsified D2 O ices during decompression is shown by thin black lines in Fig. 3.8. When the emulsion was decompressed, its temperature started to change at the purple melting line of ice III, and decreased along the melting line, as in the case of ice Ih during compression. At temperatures lower than 237 K, or the homogeneous nucleation temperature on the ice-III melting line, both sample temperature and sample volume suddenly increased simultaneously on the smoothly extrapolated melting line of ice III to low temperature and low pressure. As in the case of ice Ih, this implied that the ice III melted into liquid water, and then the liquid crystalized immediately. The smooth continuation of the purple melting line of ice III from about 0.3 GPa to nearly 1 atm indicated that metastable equilibrium liquid water continued to low temperature along the melting line. The especially important observation was the steeply curved melting line around 237 K and 0.02 GPa. Note the melting line already steeply curved above the homogeneous nucleation temperature of 237 K. According to the Clausius–Clapeyron relation, the slope of the melting line indicated that liquid water at high pressure had a highdensity and/or high-entropy property, and liquid water at temperatures lower than

3.3 Metastable Melting Line of Ice III

250

D2O III 240

Temperature (K)

Fig. 3.8 Enlarged view of Fig. 3.7b. Melting lines of D2 O high-pressure ices observed during decompression. Thin black lines are temperature of the emulsified water during decompression. Although the melting line of the emulsified ice III was detected near 1 atm, the ice III was not obtained around 200–230 K at 1 atm. Adapted figure with permission from [5]: Copyright (2000) by American Physical Society

67

230

V

IV

220

210 0

0.05

0.1

0.15

Pressure (GPa)

237 K near 1 atm had a low-density and/or low-entropy property. Therefore, the melting line of D2 O ice III indicated that, as the temperature decreased, liquid water continuously changed from high-density liquid to low-density liquid at about 237 K near 1 atm along the melting line. The existence of low-density liquid water was also suggested by the infrared spectroscopy [7], the density estimation [8], the sound-velocity measurement [9], the short-duration X-ray measurement [10], and the pulse-heating IR measurement [11]. Many of these and other cooling experiments at 1 atm indicated the change from high-density liquid water to low-density liquid water occurs continuously near 230 K. Regarding the D2 O water in Fig. 3.8 [5], the pressure and temperature at which the curvature of the melting line of D2 O ice III was largest was close to the pressure and temperature where the maximum of compressibility of D2 O water was observed by the X-ray measurement [10]. All these observations are consistent with the smooth change between low-density water and high-density water in the supercritical region of the liquid–liquid critical point as discussed in the next section.

3.4 Metastable Melting Lines of Ice IV and Ice V Melting of ice IV and that of ice V were detected in the same manner as for ice III, as shown in Fig. 3.8. The slope of the orange melting line of D2 O ice IV changed abruptly unlike the continuous change of the slope of the purple melting line of ice III. In addition, the green melting line of ice V also appeared to bend abruptly. The

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3 Metastable Melting Lines of Crystalline Ices

transition after the bending was thought to be the melting of ice IV (ice V) to lowdensity liquid water and the immediate crystallization of the liquid to ice Ih, as in the case of ice III. The sudden slope change implied the existence of abrupt change in the liquid state. In other words, the existence of the first-order phase transition between low-density liquid water and high-density liquid water was implied. In Fig. 3.8, the orange line of ice IV after the bending (the orange line below 220 K) was nearly parallel to the temperature axis or slightly inclined to the high-pressure side at low temperature (the orange dotted line), suggesting that the entropy of lowdensity liquid water might be smaller than that of ice IV. That is, crystal was more disordered than liquid. This strange result was probably caused by an experimental error; the temperature range, where the presumed melting to low-density liquid was observed, was between 210 and 220 K, and this temperature range was too narrow to determine the correct slope of this line. Moreover, there was a possibility that the melting of other high-pressure ices, such as ice PNP-XIII and ice VI, could make it difficult to detect melting of ice IV. This strange slope was corrected by the experiments of solutions. When emulsion of dilute saltwater (2.0 mol % LiCl–H2 O solution) was studied in the same decompression experiment, the slope of the transition line below 210 K slightly inclined toward low pressure and low temperature (Fig. 3.9). 280

Temperature (K)

decompression

260

IV

HDL with LiCl l-H

240 %

O

2

C Li

ol

220

2

m

200

LDL + HDL with LiCl

ice IV + HDL with LiCl

180

ice Ih (or Ic) 160

140 0.0

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0.8

Pressure (GPa) Fig. 3.9 Decompression-induced melting (liquidus) line of ice IV in 2.0 mol % LiCl–H2 O solution. The solution was emulsified, and ice IV melted in each micrometer-sized particle. Adapted with permission from [14]: Copyright (2011) American Chemical Society

3.4 Metastable Melting Lines of Ice IV and Ice V 280 0.0

LiCl-H 2O

mo

l%

l%

2.0

0.0

mo

mo

l%

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HDL

4.8

l%

mo

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l%

mo

l%

Temperature (K)

Fig. 3.10 Concentration dependence of melting lines of ice Ih (black) and ice IV (orange) in LiCl–H2 O solution. The homogeneous nucleation lines (blue) are assumed to be the LDL-HDL boundary lines. Green line: the phase boundary between ice Ih and ice IV. Data from [12, 14, 17]. Adapted figure with permission from [14]: Copyright (2011) American Chemical Society

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ice IV LDL

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0.6

Pressure (GPa)

The difference between the LiCl–H2 O experiment of Fig. 3.9 and the pure-D2 O experiment of Fig. 3.8 is the difference between melting ice IV in an aqueous solution and melting ice IV in pure liquid water. In the case of the salt solution, melting point depression occurred at high pressures as shown by the orange lines above 0.1 GPa in Fig. 3.10; the orange melting temperature decreased as the concentration of the solute increased, and the melting line shifted to the lower temperature side. The melting point depression of ice Ih was also observed as shown by the black lines in Fig. 3.10. The melting point depression is illustrated in the three-dimensional pressure–temperature-concentration graphs in Fig. 3.11. These graphs correspond to Fig. 3.10. The gray surface in Fig. 3.11a represents the liquidus temperature of ice Ih above which ice dissolves completely into high-density liquid solution. Black dots are experimental data of 0.0, 2.0, 3.2, 4.8, and 7.7 mol% LiCl solutions. The gray surface was made by interpolating, and extrapolating, these black experimental data. When concentration increases, and when pressure increases, liquidus temperature lowers. The surface of the liquidus temperature of ice IV where the ice IV melts completely into high-density liquid solution is shown by the orange surface above 0.1 GPa in Fig. 3.11c. Orange dots are data of 0.0, 2.0 and 4.8 mol% solutions. The orange dots are almost vertical at about 0.1 GPa. This orange vertical surface was assumed to be the melting surface between ice IV and low-density liquid solution, and the melting point depression was not observed (see also Fig. 3.10). The green surface at about 0.3 GPa in Fig. 3.11e is the equilibrium transition surface between ice Ih and ice IV in the solution. The blue surface in Fig. 3.11b is the homogeneous nucleation temperature of ice Ih (Ic) in the solutions. Blue dots are experimental data of the 0.0, 2.0, 3.2, 4.8,

70

3 Metastable Melting Lines of Crystalline Ices

(c) 280

240 220 200

260 240 220

0

0

sure

0.4

(GPa

)

0.6 10

2

0

n

atio 8 ntr nce %) Co (mol 6

4

0.2

Pres

220 200

sure

0.4

(GPa

)

0.6 10

0

2

Co 4 nce 6 (m ntra 8 1 ol % tio 0 0 ) n

n

atio 8 ntr nce %) Co (mol 6

(d) 0.0 2.0

240 220 200

Temperature (K)

260 4.8

mo

mo

mo

mol% 0.0 260 ol% .0 m

2 l%

m 4.8

240

ol%

220 200

180

0.2

0.4

0.2

ure Press

) (GPa

280 l%

l%

180

0

Pressure (GPa)

180 0.6 0.4

(e) 280

0

240

180

2 4

0.2

Pres

260

200

180

0

280

10

(

l mo

%)

Temperature (K)

260

Temperature (K)

280

Temperature (K)

(b) Temperature (K)

(a)

0 0

0.2

0.4

Pressure (GPa)

10

% ol

)

(m

Fig. 3.11 Three-dimensional (temperature–pressure-concentration) illustration of both melting surfaces and transition surfaces of ices in LiCl–H2 O solutions. a: Gray melting surface of ice Ih to high-density liquid water. b: Blue surface of the homogeneous nucleation of ice Ih (Ic) in the solutions. The blue surface was assumed to correspond to the phase-boundary surface between low-density liquid water and high-density liquid water. c: Melting surface of ice IV to high-density liquid water (the top orange surface). The vertical orange surface at about 0.1 GPa in c and d was assumed to be the melting surface between ice IV and low-density liquid water. e: The green surface at about 0.3 GPa is the transition surface between ice Ih and ice IV. Dots in a and b are experimental data of the 0.0, 2.0, 3.2, 4.8, and 7.7 mol% LiCl solutions. Dots in c are data of 0.0, 2.0 and 4.8 mol% solutions. The surfaces were made by interpolating, and extrapolating, these experimental data. Data from [12, 14, 17]

and 7.7 mol% solutions. The blue surface was assumed to correspond to the phaseboundary surface between low-density liquid water and high-density liquid water as discussed in Sect. 2.6.1. As shown by the orange almost vertical line at 0.1 GPa in the Figs. 3.10 and 3.11, the concentration of solute hardly affected the presumed melting of ice IV to low-density liquid. If ice IV in the solution contained no salt internally, and if this pure ice IV melted into the pure low-density liquid water in the solution, the vertical orange line would be the melting line from pure ice IV to pure low-density liquid water; the solute concentration would not influence the location of the melting line of ice IV to the low-density liquid water. There are some experimental supports for the viewpoint that salt does not dissolve into ice IV nor low-density liquid water. In Figs. 3.10 and 3.11e, three crossing points

3.4 Metastable Melting Lines of Ice IV and Ice V

71

between the black ice-Ih melting line and the orange ice-IV melting line are shown by the green filled squares. When the solute concentration was changed, the locus of these green crossing points (the green line at 220–240 K) was nearly parallel to the temperature. The phase boundaries between pure ice Ih and pure ice IV are shown by the green open squares of 0.0 mol% [12]. The locus of these squares (the green dashed line of pure water) is also nearly parallel to the temperature axis. The green solid line and the green dashed line make the green vertical surface at 0.3 GPa as shown in Fig. 3.11e. This implies that salt hardly affect the pressure of the phase boundary between ice Ih and ice IV. Because ice Ih does not dissolve salt, ice IV in the solution experiment was likely pure ice IV, containing no salt inside the ice. Conversely, if salt dissolved in ice IV, the energy of ice IV would be difference from that of pure ice IV. In this case, the phase-boundary line between salty ice IV and pure ice Ih (or the green solid line) would incline as the salt concentration was increased; but this was not observed. Another experiment has suggested that the salt (LiCl) dissolves in high-density amorphous ice but almost insoluble in low-density amorphous ice [13]. Assuming that ice IV in the solution is pure ice IV, the decompressed pure ice IV did not melt until the almost vertical orange line at about 0.1 GPa in Fig. 3.10. This suggests that the almost vertical orange line was the melting line from pure ice IV to pure low-density liquid water. In the pure H2 O (D2 O) experiments, the pressures and temperatures at which the melting lines of ice IV and ice V bend, and the pressure and temperature at which the slope of the melting line of ice III changes most strongly locate on one smooth line: the blue solid/dashed line of Fig. 3.12. This was easily explained as the liquid–liquid phase transition line and its extension. The bending of the melting curves of ice V and ice IV indicate sudden change between high-density and low-density liquids, and the 280

Suggested melting curves of H 2O ices 260

Temperature (K)

III V

240 LLCP?

IV

tr uid -liq uid liq

Fig. 3.12 Supposed melting lines of high-pressure ices and the hypothesized liquid–liquid critical point and the liquid–liquid transition. The liquid–liquid transition is assumed to relate to the homogeneous nucleation as discussed in the Sect. 2.6.1. Adapted figure with permission from [5]: Copyright (2000) by American Physical Society

220

LDL (Ih or Ic)

180 0

? ition ans

200

HDL

0.2

0.4

Pressure (GPa)

0.6

72

3 Metastable Melting Lines of Crystalline Ices

smooth change of the melting curve of ice III indicates a continuous change between the two liquids. Assuming both the experimental melting data of the high-pressure ices and the analysis of the data were correct, the critical point would be located between the melting curve of ice III and the melting curve of ice V. The critical point was considered to locate somewhere around 0.05 GPa and 230 K for D2 O. For H2 O, its location was roughly suggested to be 0.05 GPa and 223 K from analogy of D2 O and from the comparison between mathematical equation of the critical phenomenon and experimental thermodynamic data [14, 15]. However, although the slope of the ice-V melting line appeared to change discontinuously, it might change continuously. Then, the critical point would be located between the melting curve of ice V and the melting curve of ice IV. Therefore, further study of the LLCP location was necessary.

3.5 Metastable Melting Line of Ice PNP-XIV The melting line of an unidentified crystalline ice, tentatively named as ice PNP-XIV (possible new phase-XIV) in [2], is shown in Fig. 3.7. When the ice PNP-XIV was decompressed, its temperature decreased along the brown line and then increased along the exothermic homogeneous nucleation line (the TH line) of high-density liquid water. Its melting line continued smoothly into the no-man’s land at low temperatures, which implied existence of metastable equilibrium liquid water (highdensity liquid water) along the brown melting line down to 160–170 K at about 0.2 GPa.

References 1. Mishima O (1996) Relationship between melting and amorphization of ice. Nature 314:546– 549. https://doi.org/10.1038/384546a0 2. Mishima O, Stanley HE (1998) Decompression-induced melting of ice IV and the liquid-liquid transition in water. Nature 392:164–168. https://doi.org/10.1038/32386 3. Levitas VI, Henson BF, Smilowitz LB, Asay BW (2004) Solid-solid phase transformation via virtual melting significantly below the melting temperature. Phys Rev Lett 92:235702. https:// doi.org/10.1103/PhysRevLett.92.235702 4. Peng Y, Wang F, Wang Z, Alsayed AM, Zhang Z, Yodh AG, Han Y (2015) Two-step nucleation mechanism in solid-solid phase transitions. Nat Mater 14:101–108. https://doi.org/10.1038/ nmat4083 5. Mishima O (2000) Liquid-liquid critical point in heavy water. Phys Rev Lett 85:334–336. https://doi.org/10.1103/PhysRevLett.85.334 6. Togaya M (1997) Pressure dependences of the melting temperature of graphite and the electrical resistivity of liquid carbon. Phys Rev Lett 79:2474–2477. https://doi.org/10.1103/PhysRevLett. 79.2474 7. Mallamace F, Broccio M, Corsaro C, Faraone A, Majolino D, Venuti V, Liu L, Mou CY, Chen SH (2007) Evidence of the existence of the low-density liquid phase in supercooled, confined water. Proc Natl Acad Sci USA 104:424–428. https://doi.org/10.1073/pnas.0607138104

References

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8. Liu D, Zhang Y, Chen CC, Mou CY, Poole PH, Chen SH (2007) Observation of the density minimum in deeply supercooled confined water. Proc Natl Acad Sci USA 104:9570–9574. https://doi.org/10.1073/pnas.0701352104 9. Pallares G, Azouz MEM, González MA, Aragones JL, Abascal JLF, Valeriani C, Caupin F (2014) Anomalies in bulk supercooled water at negative pressure. Proc Natl Acad Sci USA 111:7936–7941. https://doi.org/10.1073/pnas.1323366111 10. Kim KH, Späh A, Pathak H, Perakis F, Mariedahl D, Amann-Winkel K, Sellberg JA, Lee JH, Kim S, Park J, Nam KH, Katayama T, Nilsson A (2017) Maxima in the thermodynamic response and correlation functions of deeply supercooled water. Science 358:1589–1593. https://doi.org/ 10.1126/science.aap8269 11. Kringle L, Thornley WA, Kay BD, Kimmel GA (2020) Reversible structural transformations in supercooled liquid water from 135 to 245 K. Science 369:1490–1492. https://doi.org/10. 1126/science.abb7542 12. Krüger Y, Mercury L, Canizarès A, Marti D, Simon P (2019) Metastable phase equilibria in the ice II stability field. A Raman study of synthetic high-density water inclusions in quartz. Phys Chem Chem Phys 21:19554–19566. https://doi.org/10.1039/C9CP03647D 13. Suzuki Y, Mishima O (2000) Two distinct Raman profiles of glassy dilute LiCl solution. Phys Rev Lett 85:1322–1325. https://doi.org/10.1103/PhysRevLett.85.1322 14. Mishima O (2011) Melting of the precipitated ice IV in LiCl aqueous solution and polyamorphism of water. J Phys Chem B 115:14064–14067. https://doi.org/10.1021/jp203669p 15. Holten V, Anisimov MA (2012) Entropy-driven liquid-liquid separation in supercooled water. Sci Rep 2:713. https://doi.org/10.1038/srep00713 16. Holten V, Sengers JV, Anisimov MA (2014) Equation of state for supercooled water at pressures up to 400 MPa. J Phys Chem Ref Data 43:043101. https://doi.org/10.1063/1.4895593 17. Kanno H, Speedy RJ, Angell CA (1975) Supercooling of water to −92°C under pressure. Science 189:880–881. https://doi.org/10.1126/science.189.4206.880

Chapter 4

Gibbs Energy of Liquid Water and the Liquid–Liquid Critical Point Hypothesis

4.1 Construction of Gibbs Energy Surface of Water The pressure and temperature dependence of the Gibbs energy of H2 O liquid water at temperatures higher than 238 K is shown in Fig. 4.1. The important point is that the slope of this Gibbs energy surface in the direction of the pressure axis corresponds to the volume of the liquid water. Generally, the slope of the surface is different at different pressure and different temperature, meaning that volume differs when pressure and temperature is different. That is, if the slope along the pressure axis of this Gibbs energy surface is plotted at each pressure and temperature, it becomes the volume surface shown in Fig. 2.2. Similarly, the slope of the surface in the direction of the temperature axis relates to the entropy surface. Also, the curvature of the Gibbs energy surface is related to the slope of the volume surface and the slope of the entropy surface, that is, the compressibility and the specific heat capacity. Furthermore, combining the slope and curvature of the Gibbs energy surface, it is possible to know all thermodynamic properties of liquid water and their fluctuations at any pressure and temperature in accordance with the definition of the properties. Conversely, when we experimentally know the thermodynamic properties such as volume and specific heat capacity at various pressures and temperatures, we can construct the Gibbs energy surface which reproduce those experimental properties. The Gibbs energy surface of ice Ih and that of liquid water are shown by the white and red surfaces in Fig. 4.2a and b, respectively. These surfaces can be constructed using experimental values of the volume at various pressures and temperatures and the specific heat capacity at various temperatures at 1 atm. In the case of ice Ih, there is no experimental data at high pressures and high temperatures where ice melts. Therefore, the Gibbs energy of ice Ih in the high-pressure high-temperature region was estimated by extrapolating the empirical equation that was made to accurately calculate the Gibbs energy of ice Ih in its stable pressure–temperature region [1]. The slope of the white ice Ih surface in the direction of the pressure axis (Fig. 4.2a) is larger than that of the red surface of liquid water (Fig. 4.2b). This indicates that the volume of ice is larger than the that of water. The slope of the red water surface © National Institute for Materials Science, Japan 2021 O. Mishima, Liquid-Phase Transition in Water, NIMS Monographs, https://doi.org/10.1007/978-4-431-56915-2_4

75

76

4 Gibbs Energy of Liquid Water and the Liquid–Liquid Critical Point …

Fig. 4.1 The Gibbs energy surface, G (P, T), of H2 O liquid water. P: pressure. T: temperature. V: volume. KT : compressibility. S: entropy. Cp: heat capacity. Temperature range: 238–298 K

V KT G

S Cp

5 kJ/mol liquid water

240 260

T (K) 280 1

0.8

0.6

0.4

0.2

0

P (GPa)

(Fig. 4.2b) in the direction of the temperature axis is negative and steeper than that of white ice (Fig. 4.2a). This indicates that the entropy of water is larger than that of ice; that is, the liquid water is more disordered than ice Ih. When these two Gibbs energy surfaces are superimposed, the lower-value surface becomes energetically more stable (Fig. 4.2c). That is, ice (the white surface) is stable at low pressure and low temperature, and liquid water (the red surface) is stable at high pressure and high temperature. The line of intersection between the two surfaces is the pressure and temperature where water and ice have the same Gibbs energy, which is the melting line of the ice. Since the pressure and temperature of the melting line of ice is known experimentally, the relative positional relationship between the white and red surfaces can be determined from the melting line. The result of subtracting the Gibbs energy of ice from that of water is shown in Fig. 4.2d. The white ice-Ih surface becomes the reference plane in this figure. The red surface corresponds to the difference in Gibbs energy between water and ice, and its slope and curvature corresponds to the difference of their thermodynamic properties. For example, the slope of the red surface of Fig. 4.2d in the pressure-axis direction is the volume difference between water and ice which can be experimentally obtained by subtracting the volume of ice from that of water. Given the numerical volume data of high-pressure crystalline ices, we can also construct the Gibbs energy surfaces of high-pressure ices. The volumes of highpressure ices were estimated by extrapolation and interpolation of the available experimental volume data (Fig. 4.3). The intersection of the ice surface and the liquid– water surface corresponds to the melting line of the high-pressure ice (Fig. 4.4a). The intersection where two ice surfaces meet corresponds to the equilibrium phaseboundary line between these ices (Fig. 4.4b). The phase-boundary line was regarded to be parallel to the temperature axis because the molecular orientation of these crystalline ices is random. That is, the ices would have similar entropy, and their

4.1 Construction of Gibbs Energy Surface of Water

77

(a)

(b)

G

G

5 kJ/mol

5 kJ/mol

ice Ih

liquid

240

240

260

260

T (K)

280 1

0.8

0.6

0.4

0.2

0

T (K) 280 1

P (GPa)

(c)

0.8

0.6

0.4

0.2

0

P (GPa)

(d) melting line of ice Ih

melting line of ice Ih

1

G

5 kJ/mol

ΔG (kJ/mol)

ice Ih

0

-1 -2

liquid

liquid

240

240

260

260

T (K)

ice Ih

280 1

0.8

0.6

0.4

P (GPa)

0.2

0

T (K)

280 0.5

0.4

0.3

0.2

0.1

0

P (GPa)

Fig. 4.2 Gibbs energy surfaces of liquid water and crystalline ice Ih. a: Gibbs energy of ice Ih [1]. b: Gibbs energy of liquid water. c: Superimpose of the surface of ice Ih and the surface of liquid water. The line of intersection between the two surfaces is the melting line of ice Ih. d: Difference in Gibbs energy between liquid water and ice Ih. The basal plane corresponds to the Gibbs energy of ice Ih

phase-boundary lines would be parallel to the temperature axis according to the Clausius—Clapeyron relation. By using the experimentally obtained melting lines and crystal-crystal boundary lines, we can determine the relative positional relationship among these Gibbs energy surfaces (Fig. 4.4c).

4 Gibbs Energy of Liquid Water and the Liquid–Liquid Critical Point …

Fig. 4.3 Volume of high-pressure ices. The volume, shown by each line, was estimated by extrapolating and interpolating available experimental volume data. Black data points: adapted figure from [23], with the permission of AIP Publishing. Data from [24, 25]

235 ~ 240 K

1.1 volum

e of ic

e Ih a

t 240

K

1.0

Volume (cm 3 /g)

78

0.9 III V

0.8

IV

XII PNP-XIII PNP-XIV

0.7

0

0.2

VI

0.4

0.6

0.8

1.0

Pressure (GPa)

4.2 Gibbs Energy Surface of Water The Gibbs energies of ices and liquid water have been calculated in 1998 using the experimental values available at that time [2], and the surface of Gibbs-energy difference between liquid water (and amorphous ices) and ice Ih was constructed. The surface was consistent with the liquid–liquid critical point hypothesis. Now, the previous surface was slightly revised by adding new experimental data. Especially, the newly estimated liquid–liquid transition pressure of 0.23 GPa at 150 K [3] provided better consistency than the previously-used transition pressure of 0.2 GPa. The revised surface is shown in Fig. 4.5a. There are a light-blue flat summit region and a pink mountain-slope region in the surface. The locus of the black points is the melting line of ice Ih. The slope of the Gibbs energy of low-density amorphous ice (the light blue LDA region) in the direction of pressure is nearly zero, and this means that the volume difference between the low-density amorphous ice and ice Ih is almost zero. That is, the volume of low-density amorphous ice is nearly the same with that of ice Ih. The slope of the Gibbs energy of high-density amorphous ice (the pink HDA region) in the direction of pressure is negative, and this indicates that the volume of the high-density amorphous ice is smaller than that of ice Ih. On the light blue dotted line in Fig. 4.5a, the Gibbs energy of the pink high-pressure water (the mountain slope of HDA/HDL) becomes the same with the energy of the blue low-pressure water (the flat LDA/LDL summit). This line corresponds to the phase boundary between the two liquid waters: the liquid–liquid transition line. As discussed in Sect. 2.6.1, this line is assumed to be the homogeneous nucleation line of ice Ih (the TH line). Discontinuity in slope between the summit surface and the mountain slope surface means the first-order phase transition; both volume and

4.2 Gibbs Energy Surface of Water

79

(b)

(a)

Ih-IV phase boudary

melting line of ice IV

G

G

ice IV

5 kJ/mol

5 kJ/mol

ice Ih

ice IV 240

240

liquid

260

260

T (K) 280 0.5

0.4

0.1

0.2

0.3

T (K) 280

0

P (GPa)

1

0.8

0.6

0.4

0.2

0

P (GPa)

(c) melting line of ice IV

melting line of ice Ih

1

ΔG (kJ/mol)

0 ice IV

-1

ice Ih

-2 liquid 240 260

T (K)

280 0.5

0.4

0.3

0.2

0.1

0

P (GPa)

Fig. 4.4 Gibbs energy surfaces of liquid water (red), ice IV (orange), and ice Ih (white). a: Superimpose of the surface of ice IV and the surface of liquid water. b: Superimpose of the surface of ice IV and the surface of ice Ih. c: Difference in Gibbs energy between liquid water and ice Ih and between ice IV and ice Ih. The ice Ih surface is a reference plane. The phase-boundary line between ice Ih and ice IV is nearly parallel to the temperature axis [26]

entropy change discontinuously. This phase-boundary line is suggested to disappear at the liquid–liquid critical point (LLCP) around 0.05 GPa and 223 K as discussed in Sect. 3.4. When temperature is increased at 1 atm, the low-density liquid water smoothly becomes the high-density liquid water.

80

4 Gibbs Energy of Liquid Water and the Liquid–Liquid Critical Point …

(a)

(b) (HDA)

2

2

LDA Gw-GIh (kJ/mol)

1

(LDA)

LDL

HDA

Gw-GIh (kJ/mol)

HDL

150 200 250 ) (K T 300

Tg

ice Ih

0.6

0.2

LLCP

ice Ih

150 200 250 T (K) 300

0

HDL

liquid

-1

0.4

Tg

LDL

HDA

0

0

-1

1

LDA

0.6

P (GPa)

0

0.2

0.4

P (GPa)

Fig. 4.5 Difference between the Gibbs energy of H2 O liquid water, Gw , and that of ice Ih, GIh at 140–340 K and 0–0.8 GPa. The hypothetical liquid–liquid transition line (light blue dotted line) ends at the liquid–liquid critical point. a: Gibbs energy constructed by using experimental data. b: Illustration of the Gibbs energy surface. Dashed lines indicate metastable regions of low-density and high-density structures. The grey pressure–temperature region is the no-man’s land. Adapted figures from [2, 27]

4.2.1 Projection in Energy-Axis Direction Figures 4.6, 4.7, and 4.8 show projections of Fig. 4.5a. In the pressure and temperature diagram (Fig. 4.6), we can see the melting lines of ices. The lines of the homogeneous nucleation temperature of liquid water (TH ), crystallization temperature of 300 VI Ih PNP-XIII

Temperature (K)

Fig. 4.6 Projection along the energy axis of the H2 O three-dimensional diagram of Fig. 4.5. Grey region: no-man’s land. Open circle: solution data. See Sect. 3.4 about the melting of ice IV in the no-man’s land. Adapted figure from [2] and with permission from [18]: Copyright (2000) by American Physical Society

250

III V

PNP-XIV

IV

TH

200

TH No man’ s land Tg (HDL)

Tx Tx

150 0

0.2

0.4

0.6

Pressure (GPa)

0.8

4.2 Gibbs Energy Surface of Water

81

LDA (140-150 K) 20 0

K

A HD

III

K)

II

V

50 -1 40 (1

XI PPN

IV

0

Ih

PN XI

PVI 30

25

0K

-1 0

V

Gibbs energy (kJ/mol)

1

0.2

0

K

0.4

0.6

0.8

Pressure (GPa) Fig. 4.7 Projection along the temperature axis of the three-dimensional diagram of Fig. 4.5. Light blue dot: Gibbs energy at the homogeneous nucleation line, or at the hypothetical liquid–liquid transition line. Adapted figure from [2]

LDL (0.1

MPa)

1

III IV V Ih

0

HD L

PNP

.1

(0 a) a

GP

GP

a

a

a

GP

GP

4

Pa

0.1

0.

G

3 0.

5

0.2

I

0.

MP

-XII

IV P-X

PN

Gibbs energy difference (kJ/mol)

LDA (0.1 MPa)

-1 150

200

250

300

Temperature (K) Fig. 4.8 Projection along the pressure axis of the three-dimensional diagram of Fig. 4.5. Light blue dot: Gibbs energy at the homogeneous nucleation line, or at the hypothetical liquid–liquid transition line. Adapted figure with permission from [14]: Copyright (1992) American Chemical Society. LDA data from [10]

82

4 Gibbs Energy of Liquid Water and the Liquid–Liquid Critical Point …

amorphous ices (TX ), and the glass-transition temperatures (Tg) of pure water (pink filled circle) and dilute aqueous solutions (pink open circle) are also drawn. The grey region between TH and TX lines is the no-man’s land. It should be noted that the crystallization temperature of the green dashed Tx line is slightly higher than, or nearly the same as, the glass-transition temperature (the pink Tg line) of high-density liquid water at about 0.2 GPa [4]. This means that we can study the liquid state in the narrow temperature domain between Tx and Tg. As discussed in Sects. 2.6 and 2.7, the first-order liquid–liquid transition was suggested to occur at about 150 K and 0.23 GPa. The orange open circles have been obtained by using dilute salt (LiCl-H2 O) solutions, and it was regarded as the melting line of pure ice IV to pure low-density liquid water as discussed in Sect. 3.4. The almost vertical slope of the orange melting line below 215 K indicated small difference in entropy between ice IV (and ice Ih) and the low-density liquid water.

4.2.2 Projection in Temperature-Axis Direction In the Gibbs energy difference and pressure diagram (Fig. 4.7), the slopes of the colored lines with ice names correspond to the volume difference between the labelled phases and ice Ih. The crossing point of the line of each crystalline ice and the horizontal line of ice Ih is the crystal-crystal transition pressure, and the pressure is assumed to be independent on temperature. The equilibrium transition pressure of two liquid phases, or two relaxed amorphous ices (LDA and HDA), is about 0.23 GPa at about 150 K. Possible new phases, ice PNP-XIII and ice PNP-XIV, were made by the crystallization of emulsified supercooled liquid water under pressure. Their densities were estimated from Figs. 4.5a and 4.7 to be about 1.29 g/cm3 (0.775 cm3 /g in specific volume) at 1 atm, allowing some experimental errors. It is about the density of ice XII at 1 atm (Fig. 4.3). The ice XII was made by cooling liquid water to 250 K at about 0.5 GPa [5], as well as by heating high-density amorphous ice under pressure [4]. The location of the melting line of ice XII was proposed [6–8], and the melting line of PNP-XIII in Fig. 4.6 is close to the proposed melting line of ice XII. Therefore, ice PNP-XIII may be ice XII. On the other hand, ice PNP-XIV is likely a new ice phase.

4.2.3 Projection in Pressure-Axis Direction The Gibbs energy difference between the low-density amorphous ice (LDA) and ice Ih is shown in the energy and temperature diagram (Fig. 4.8). The Gibbs energy of the thin film of low-density amorphous ice was experimentally obtained at temperatures lower than 150 K at 1 atm [9, 10] and shown by the solid blue LDA line. On the other hand, there is no experimental data of bulk pure low-density liquid water because it crystallizes easily. Therefore, its Gibbs energy at 1 atm (0.1 MPa) is speculated and

4.2 Gibbs Energy Surface of Water

83

drawn by the dashed blue LDL line in Fig. 4.8. The Gibbs energy of water above 235 K at 1 atm was calculated using the heat capacity of Fig. 2.4 [11] (the red HDL line at 0.1 MPa). There were arguments on the thermodynamic connection between the red HDL line and the blue LDA line [9, 12–14]. If the room-temperature liquid water becomes low-density amorphous ice continuously when cooled at 1 atm, the Gibbs energy surface needs to be bent at around 220–240 K [12]. Regarding the Gibbs energy at 1 atm (0.1 MPa) in Fig. 4.8, the slope of the Gibbs energy in the direction of temperature axis is slightly negative at low temperatures and very negative at high temperatures. This indicates the existence of low-entropy liquid at low temperatures and the existence of high-entropy liquid at high temperatures. The curvature is large around 220–240 K, and this indicate the heat capacity of water is large around 220–240 K, which is consistent with the experimental large heat capacity near 228 K [12]. In the three-dimensional Fig. 4.5, both the Gibbs energy surface of low-density amorphous ice (LDA) and that of low-density liquid water (LDL) are drawn to be approximately parallel to the reference plane of ice Ih. That is, both low-density amorphous ice and low-density liquid water have volume and entropy like those of ice Ih. If so, as shown in Fig. 4.5, the overall Gibbs energy surface of liquid water and amorphous ices can be made satisfactory, indicating the existence of liquid–liquid transition and the existence of liquid–liquid critical point. Conversely, if the density or entropy of low-pressure low-temperature liquid water is large, the Gibbs energy surface must be different from that of Fig. 4.5, and the liquid–liquid critical point hypothesis would be wrong. The heating experiments of the low-density amorphous ice at 1 atm showed small change in the heat capacity at the glass transition to low-density liquid water [15]. Additionally, the structure of low-density liquid water was like that of low-density amorphous ice [16], indicating the density of the low-density liquid is low. This is consistent with the flat surface over the LDA-LDL glass transition in the Fig. 4.5. Furthermore, the short-time X-ray measurement at 1 atm showed the compressibility maximum of liquid water at about 230 K [17], and the measurement of the ice III melting line showed the large change in its slope at about the same pressure and temperature [18]. Measurements of confined water, as well as the thin film of water, also indicated the change of water state around 230 K [19–22]. Indeed, there have been so many indications of the change in the properties of liquid water around 220–240 K at 1 atm. These observations imply that the slope of the Gibbs energy surface changes around this temperature. In addition, the aqueous solution experiments hinted that the melting curve of ice IV into low-density liquid water has a positive slope in the pressure–temperature diagram, as discussed in the Sect. 3.4. This implied that the Gibbs energy of low-density liquid water near 1 atm has the values of the open orange circle in Fig. 4.8, which is consistent with the flat LDL surface with low entropy. These results are not inconsistent with the flat LDA/LDL surface. Within experimental uncertainty, there is no contradiction in the experimental thermodynamic data with the liquid–liquid critical point hypothesis.

84

4 Gibbs Energy of Liquid Water and the Liquid–Liquid Critical Point …

References 1. Feistel R, Wagner W (2006) A new equation of state for H2 O ice Ih. J Phys Chem Ref Data 35:1021–1047. https://doi.org/10.1063/1.2183324 2. Mishima O, Stanley HE (1998) Decompression-induced melting of ice IV and the liquid-liquid transition in water. Nature 392:164–168. https://doi.org/10.1038/32386 3. Suzuki Y (2017) Effect of solute nature on the polyamorphic transition in glassy polyol aqueous. J Chem Phys 147:064511. https://doi.org/10.1063/1.4998201 4. Stern JN, Seidl-Nigsch M, Loerting T (2019) Evidence for high-density liquid water between 0.1 and 0.3 GPa near 150 K. Proc Natl Acad Sci USA 116:9191–9196. https://doi.org/10.1073/ pnas.1819832116 5. Lobban C, Finney JL, Kuhs WF (1998) The structure of a new phase of ice. Nature 391:268–270. https://doi.org/10.1038/34622 6. Chou IM, Blank JG, Goncharov AF, Mao HK, Hemley RJ (1998) In situ observations of a high-pressure phase of H2 O ice. Science 281:809–812. https://doi.org/10.1126/science.281. 5378.809 7. Salzmann C, Kohl I, Loerting T, Mayer E, Hallbrucker A (2002) The Raman spectrum of ice XII and its relation to that of a new “high-pressure phase of H2 O ice.” J Phys Chem B 106:1–6. https://doi.org/10.1021/jp012755d 8. Salzmann CG (2019) Advances in the experimental exploration of water’s phase diagram. J Chem Phys 150:060901. https://doi.org/10.1063/1.5085163 9. Speedy RJ, Debenedetti PG, Smith RS, Huang C, Kay BD (1996) The evaporation rate, free energy, and entropy of amorphous water at 150 K. J Chem Phys 105:240–244. https://doi.org/ 10.1063/1.471869 10. Smith RS, Matthiesen J, Knox J, Kay BD (2011) Crystallization kinetics and excess free energy of H2 O and D2 O nanoscale films of amorphous solid water. J Phys Chem A 115:5908–5917. https://doi.org/10.1021/jp110297q 11. Speedy RJ (1987) Thermodynamic properties of supercooled water at 1 atm. J Phys Chem 91:3354–3358. https://doi.org/10.1021/j100296a049 12. Angell CA, Shuppert J, Tucker JC (1973) Anomalous properties of supercooled water. Heat capacity, expansivity, and proton magnetic resonance chemical shift from 0 to −38°. J Phys Chem 77:3092–3099. https://doi.org/10.1021/j100644a014 13. Johari GP, Fleissner G, Hallbrucker A, Mayer E (1994) Thermodynamic continuity between glassy and normal water. J Phys Chem 98:4719–4725. https://doi.org/10.1021/j100068a038 14. Speedy RJ (1992) Evidence for a new phase of water: water II. J Phys Chem 96:2322–2325. https://doi.org/10.1021/j100184a056 15. Johari GP, Hallbrucker A, Mayer E (1987) The glass-liquid transition of hyperquenched water. Nature 330:552–553. https://doi.org/10.1038/330552a0 16. Soper AK, Ricci MA (2000) Structures of high-density and low-density water. Phys Rev Lett 84:2811–2884. https://doi.org/10.1103/PhysRevLett.84.2881 17. Kim KH, Späh A, Pathak H, Perakis F, Mariedahl D, Amann-Winkel K, Sell-berg JA, Lee JH, Kim S, Park J, Nam KH, Katayama T, Nilsson A (2017) Maxima in the thermodynamic response and correlation functions of deeply supercooled water. Science 358:1589–1593. https://doi.org/ 10.1126/science.aap8269 18. Mishima O (2000) Liquid-liquid critical point in heavy water. Phys Rev Lett 85:334–336. https://doi.org/10.1103/PhysRevLett.85.334 19. Faraone A, Liu L, Mou CY, Yen CW, Chen SH (2004) Fragile-to-strong liquid transition in deeply supercooled confined water. J Chem Phys 121:10843. https://doi.org/10.1063/1.183 2595 20. Mallamace F, Broccio M, Corsaro C, Faraone A, Majolino D, Venuti V, Liu L, Mou CY, Chen SH (2007) Evidence of the existence of the low-density liquid phase in supercooled, confined water. Proc Natl Acad Sci USA 104:424–428. https://doi.org/10.1073/pnas.0607138104

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21. Mallamace F, Branca C, Broccio M, Corsaro C, Mou CY, Chen SH (2007) The anomalous behavior of the density of water in the range 30 K < T < 373 K. Proc Natl Acad Sci USA 104:18387–18391. https://doi.org/10.1073/pnas.0706504104 22. Kringle L, Thornley WA, Kay BD, Kimmel GA (2020) Reversible structural transformations in supercooled liquid water from 135 to 245 K. Science 369:1490–1492. https://doi.org/10. 1126/science.abb7542 23. Mishima O (1994) Reversible first-order transition between two H2 O amorphs at ~0.2 GPa and ~135 K. J Chem Phys 100:5910–5912. https://doi.org/10.1063/1.467103 24. Gagnon RE, Kiefte H, Clouter MJ, Whalley E (1990) Acoustic velocities and densities of polycrystalline ice Ih, II, III, V, and VI by Brillouin spectroscopy. J Chem Phys 92:1909–1914. https://doi.org/10.1063/1.458021 25. Noya EG, Menduiña C, Aragones JL, Vega C (2007) Equation of state, thermal expansion coefficient, and isothermal compressibility for ices Ih, II, III, V, and VI, as obtained from computer simulation. J Phys Chem C 111:15877–15888. https://doi.org/10.1021/jp0743121 26. Krüger Y, Mercury L, Canizarès A, Marti D, Simon P (2019) Metastable phase equilibria in the ice II stability field. A Raman study of synthetic high-density water inclusions in quartz. Phys Chem Chem Phys 21:19554–19566. https://doi.org/10.1039/C9CP03647D 27. Mishima O (2010) Polyamorphism in water. Proc Jpn Acad B 86:165–175. https://doi.org/10. 2183/pjab.86.165

Chapter 5

Outlook

Is the liquid–liquid critical point hypothesis of water correct? My answer is “it is probably correct, but not absolutely sure, because there is no definitive experimental proof”. In my opinion, many indirect experiments have been done to test the hypothesis, and there is, so far, no fatal contradiction. Assuming that the interpretation of the experimental results in this book is correct, I accept the hypothesis under the condition that “until the appearance of the experimental evidence that clearly contradicts the hypothesis”. The hypothesis will be examined by repeating precise experiments and by looking for a definitive counterexample. Also, experiments on water polyamorphism should be conducted to study the effect of electric field, magnetic field, solution composition, confinement, and so on. Water is certainly peculiar showing the minimum volume at 4 °C at 1 atm. And water is related to wide research fields [1, 2]. The understanding of the two kinds of water will facilitate the explanation of the complex water behavior. For example, the aqueous solution may be discussed from the polyamorphic viewpoint that the solute melts into either liquid state. Our familiar home cooking may be related to changing the ratio of low-density water to high-density water. The supercritical state of the liquid–liquid critical point at low temperature may activate subtle structural changes of proteins and their chemical reactions without damaging the proteins by heat. To know the effect of drugs, structures of proteins are examined by low temperature X-ray diffraction. When water-containing protein crystals are cooled at 1 atm, water inside becomes the low-density crystalline ice or the low-density amorphous ice, and this may break the protein crystals. Based on the polyamorphism of water, a method was already developed to vitrify water to high density amorphous ice under high pressure so as not to break the crystals [3]. Perhaps this was the first application of the water polyamorphism although the high-pressure cooling of a protein crystal itself was first conducted in 1973 [4]. A cause of distinct polyamorphism in a disordered system is that there are multiple energetically stable distances between two adjacent elements [5]. Whatever the element, if the condition of the energetically stable multiple interaction is satisfied, polyamorphism and the critical point may appear. Like water, it is expected that © National Institute for Materials Science, Japan 2021 O. Mishima, Liquid-Phase Transition in Water, NIMS Monographs, https://doi.org/10.1007/978-4-431-56915-2_5

87

88

5 Outlook

liquid of silicon atoms with four bonding hands would have two low- and high-density liquid states. Experiments indeed showed that amorphous silicon exhibits polyamorphism [6]. Moreover, it has been clearly shown by high-pressure high-temperature experiments that the first-order liquid–liquid transition occurs in phosphorus [7]. Oxide of aluminum and yttrium [8] and triphenyl phosphite [9, 10] also showed the discontinuous polyamorphic transition. A book on polyamorphism in various materials has been already published [11]. Water has two liquid phases; it is simply interesting. But, so what? Note that we can feel, think, and act. And remember that more than half of our body is made of water. How can our activities be done in the water? This question stimulates our curiosity. If the structure and properties of liquid water can be controlled by changing solute composition, electromagnetic field, confinement, etc., it will be possible for us to utilize water at will in future. And, we probably have a mathematical tool; the thermodynamic properties of water can be described by the equation of critical phenomenon [12, 13]. The Ising model on water polyamorphism was developed [13] and may be applied to broad research fields. Hopefully, in the twenty-first century, our understanding of water, liquids, and other disordered systems may advance dramatically. Although we must not make any catastrophic material like the “ice nine” of the novel “Cat’s Cradle” [14], the understanding of polyamorphism will be useful to create sophisticated liquid functions.

References 1. Ball P (1999) H2 O: a biography of water. Weidenfeld and Nicolson, London. ISBN-13:9780297643142 2. Debenedetti PG, Klein ML (2017) Chemical physics of water. Proc Natl Acad Sci USA 114:13325–13326. https://doi.org/10.1073/pnas.1719350115 3. Kim CU, Kapfer R, Gruner SM (2005) High-pressure cooling of protein crystals without cryoprotectants. Acta Cryst D61:881–890. https://doi.org/10.1107/S090744490500836X 4. Thomanek UF, Parak F, Mössbauer RL, Formanek H, Schwager P, Hoppe W (1973) Freezing of myoglobin crystals at high pressure. Acta Cryst A29:263–265. https://doi.org/10.1107/S05 67739473000677 5. Buldyrev SV, Malescio G, Angell CA, Giovambattista N, Prestipino S, Saija F, Stanley HE, Xu L (2009) Unusual phase behavior of one-component systems with two-scale isotropic interactions. J Phys Condens Matter 21:504106. https://doi.org/10.1088/0953-8984/21/50/ 504106 6. Deb SK, Wilding M, Somayazulu M, McMillan PF (2001) Pressure-induced amorphization and an amorphous-amorphous transition in densified porous silicon. Nature 414:528–530. https:// doi.org/10.1038/35107036 7. Katayama Y, Mizutani T, Utsumi W, Shimomura O, Yamakata M, Funakoshi K (2000) A first-order liquid-liquid phase transition in phosphorus. Nature 403:170–173. https://doi.org/ 10.1038/35003143 8. Aasland S, McMillan PF (1994) Density-driven liquid-liquid phase separation in the system Al2 O3 -Y2 O3 . Nature 369:633–636. https://doi.org/10.1038/369633a0 9. Ha A, Cohen I, Zhao X, Lee M, Kivelson D (1996) Supercooled liquids and polyamorphism. J Phys Chem 100:1–4. https://doi.org/10.1021/jp9530820

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10. Kurita R, Tanaka H (2004) Critical-like phenomena associated with liquid-liquid transition in a molecular liquid. Science 306:845–848. https://doi.org/10.1126/science.1103073 11. Stanley HE (ed) (2013) Liquid polymorphism. Adv Chem Phys 152. Wiley, New York. ISBN: 9781118453445. https://doi.org/10.1002/9781118540350 12. Anisimov MA, Duška M, Caupin F, Amrhein LE, Rosenbaum A, Sadus RJ (2018) Thermodynamics of fluid polyamorphism. Phys Rev X 8:011004. https://doi.org/10.1103/PhysRevX.8. 011004 13. Cerdeiriña CA, Troncoso J, González-Salgado D, Debenedetti PG, Stanley HE (2019) Water’s two-critical-point scenario in the Ising paradigm. J Chem Phys 150:244509. https://doi.org/10. 1063/1.5096890 14. Vonnegut K, Cat’s Cradle. ISBN-13: 978-0385333481

Chapter 6

Appendix: Melting Points of Emulsified H2 O Ices

6.1 Melting Points of Emulsified H2 O Ice Ih

P (MPa)

T (K)

25.80

270.38

46.94

268.36

60.90

265.92

74.88

265.06

83.89

264.47

105.93

264.09

100.37

263.18

107.52

262.64

134.58

261.62

118.20

261.52

124.81

260.51

125.46

260.25

126.94

259.87

133.51

259.86

135.51

259.81

133.63

259.39

149.33

258.77

145.92

257.15

162.43

256.33

169.99

255.40

193.04

254.37

175.67

254.36

181.13

254.00

© National Institute for Materials Science, Japan 2021 O. Mishima, Liquid-Phase Transition in Water, NIMS Monographs, https://doi.org/10.1007/978-4-431-56915-2_6

(continued) 91

92

6 Appendix: Melting Points of Emulsified H2 O Ices

(continued) P (MPa)

T (K)

183.00

253.72

184.90

253.15

208.93

252.23

210.72

250.99

230.62

250.64

228.83

248.20

244.52

246.77

242.62

246.15

252.45

245.75

251.26

244.58

263.91

242.66

279.49

242.60

272.70

241.82

293.99

240.92

282.05

240.53

280.37

240.41

287.88

239.72

290.96

239.24

292.02

238.39

305.84

236.40

312.88

236.03

304.03

235.63

327.16

235.30

309.95

234.93

316.54

234.22

325.44

233.39

324.66

233.35

319.84

233.30

332.75

233.17

328.67

231.63

330.53

231.47

337.23

230.97

352.90

230.37

336.33

229.29

346.93

228.82

359.08

228.23

349.33

227.58 (continued)

6.1 Melting Points of Emulsified H2 O Ice Ih

93

(continued) P (MPa)

T (K)

358.47

226.85

354.95

226.32

359.69

225.42

359.99

225.27

361.71

225.15

371.72

224.76

367.74

223.74

387.93

223.23

381.79

222.91

379.17

219.56

388.77

219.40

383.30

219.20

390.15

218.99

398.83

218.89

388.92

218.64

392.24

218.49

406.45

218.01

393.23

217.35

391.44

216.86

395.39

216.79

395.23

216.53

399.79

216.06

401.62

215.59

410.47

214.23

404.71

214.06

406.94

213.69

429.56

212.51

406.50

210.64

428.33

210.46

417.65

210.27

423.62

210.19

438.19

210.09

432.38

209.45

428.48

209.38

425.34

209.22

424.06

208.98

440.94

208.74 (continued)

94

6 Appendix: Melting Points of Emulsified H2 O Ices

(continued) P (MPa)

T (K)

429.21

208.11

428.56

207.20

450.27

205.29

430.55

205.22

449.44

204.72

451.82

203.87

421.33

203.78

442.42

203.63

445.52

203.47

444.53

203.21

448.94

202.57

454.02

202.23

447.19

202.11

469.89

201.27

462.09

200.24

443.63

200.12

443.31

199.76

465.76

199.52

468.21

199.33

468.93

199.17

444.72

199.11

468.79

199.00

459.23

198.95

456.61

198.52

452.73

198.42

454.77

198.23

449.96

198.11

447.82

197.62

455.10

197.42

447.58

196.85

446.62

196.76

473.06

196.45

455.32

196.28

444.86

195.45

458.82

194.67

458.10

194.23

454.87

194.15 (continued)

6.1 Melting Points of Emulsified H2 O Ice Ih

95

(continued) P (MPa)

T (K)

464.29

194.11

457.70

194.00

464.37

193.78

449.66

193.59

464.74

192.71

473.22

192.71

460.69

192.53

474.02

191.40

459.83

191.39

472.59

191.26

477.64

190.99

472.38

190.81

476.18

189.87

473.88

189.42

458.72

189.09

482.86

188.90

480.26

188.76

481.11

186.88

460.26

185.41

476.69

184.50

462.77

184.39

490.79

184.25

474.35

183.17

493.83

182.82

492.91

182.16

497.36

181.78

491.40

181.65

492.87

178.82

482.02

178.66

493.03

178.42

492.20

174.41

500.32

173.36

491.87

173.02

509.96

172.97 (continued)

96

6 Appendix: Melting Points of Emulsified H2 O Ices

(continued) P (MPa)

T (K)

500.05

172.07

507.41

169.75

507.05

168.90

6.2 Melting Points of Emulsified H2 O Ice III

P (MPa)

T (K)

359.00

256.00

288.00

254.00

233.00

252.00

192.00

250.00

158.50

248.00

130.50

246.00

104.50

244.00

81.50

242.00

63.50

240.00

46.50

238.00

33.50

236.00

24.00

234.00

29.17

235.71

37.14

236.20

34.29

236.78

39.41

236.82

38.84

237.40

49.08

237.93

51.93

238.18

54.77

238.92

54.20

239.46

63.30

239.83

67.86

240.24

74.11

241.48

103.69

242.84

93.45

242.84 (continued)

6.2 Melting Points of Emulsified H2 O Ice III

97

(continued) P (MPa)

T (K)

91.18

242.96

92.32

243.04

111.09

244.61

121.90

245.02

148.64

247.16

186.18

250.01

206.09

250.87

233.40

251.74

252.17

252.40

247.62

252.32

285.73

253.39

296.54

253.96

298.25

253.43

295.41

253.35

309.06

254.42

336.93

255.28

371.07

256.39

361.96

255.90

394.39

256.60

446.73

257.51

534.90

258.58

6.3 Melting Points of Emulsified H2 O Ice IV

P (MPa)

T (K)

444.75

254.72

421.16

254.52

407.12

252.77

407.27

252.50

375.58

249.79

391.97

248.99

357.73

248.04

337.85

245.99

334.70

245.87 (continued)

98

6 Appendix: Melting Points of Emulsified H2 O Ices

(continued) P (MPa)

T (K)

340.07

245.81

314.18

244.22

312.43

244.20

307.36

243.04

304.49

242.00

298.40

241.13

292.91

241.06

284.60

240.78

281.27

240.51

281.63

240.49

281.05

240.26

280.90

240.04

274.22

239.40

267.20

238.76

252.25

237.24

245.87

235.77

243.39

235.75

245.10

235.05

231.41

234.47

234.96

234.32

226.74

233.79

223.53

233.15

219.42

232.78

220.92

232.69

220.57

232.37

220.71

232.37

223.31

232.37

209.75

230.33

193.41

229.20

193.32

228.87

187.39

228.16

179.09

228.06

188.18

227.84

176.41

227.05

178.50

227.03

178.82

227.00

178.22

226.21 (continued)

6.3 Melting Points of Emulsified H2 O Ice IV

99

(continued) P (MPa)

T (K)

167.27

225.53

162.20

224.38

161.09

224.32

164.48

223.92

155.22

222.81

145.37

221.67

140.67

221.02

139.06

220.54

136.08

220.06

141.17

219.78

129.87

219.54

127.04

218.81

130.47

218.60

128.89

218.51

130.14

218.46

125.22

218.35

132.84

218.35

120.58

217.98

122.88

217.82

125.06

217.72

119.23

217.43

123.22

217.34

120.69

217.07

117.58

216.94

120.29

216.89

116.89

216.83

118.77

216.73

121.06

216.55

109.52

216.05

109.62

215.92

118.03

215.90

116.36

215.88

113.45

215.67

112.06

215.38

109.82

215.36

107.50

215.32

111.22

215.11 (continued)

100

6 Appendix: Melting Points of Emulsified H2 O Ices

(continued) P (MPa)

T (K)

115.44

214.99

110.39

214.81

106.51

214.30

107.51

214.26

110.12

214.18

109.68

213.55

108.92

213.42

110.86

213.08

110.11

212.96

110.81

212.67

104.43

212.24

110.75

211.78

105.37

211.60

112.75

211.47

104.19

211.27

104.81

211.22

112.04

211.18

109.23

211.17

114.06

210.26

97.09

210.07

98.00

210.07

113.22

208.99

111.34

208.89

103.94

207.65

107.04

207.63

6.4 Melting Points of Emulsified H2 O Ice V

P (MPa)

T (K)

669.00

275.00

578.50

270.00

492.00

265.00

413.50

260.00

346.00

255.00 (continued)

6.4 Melting Points of Emulsified H2 O Ice V

101

(continued) P (MPa)

T (K)

286.50

250.00

233.50

245.00

187.00

240.00

148.00

235.00

117.00

230.00

86.00

225.00

75.50

223.00

75.00

220.00

71.50

215.00

627.75

273.15

560.26

268.67

593.59

271.40

524.18

266.49

501.31

265.06

490.85

264.45

496.36

264.43

494.98

265.04

468.54

263.61

456.14

262.30

442.92

262.39

408.21

259.80

405.18

259.69

372.12

256.60

375.43

257.39

369.09

256.63

365.51

256.60

342.37

254.54

331.63

253.64

321.44

252.53

314.28

251.89

295.00

250.62

274.89

250.21

279.02

249.57

286.46

249.66

284.53

249.45

259.19

247.82

242.11

245.64 (continued)

102

6 Appendix: Melting Points of Emulsified H2 O Ices

(continued) P (MPa)

T (K)

234.95

245.53

222.00

244.10

228.89

243.61

234.40

242.59

221.72

242.59

193.35

240.18

182.06

238.87

172.42

236.46

150.38

235.03

120.63

231.51

113.19

229.57

114.30

229.22

105.76

228.03

106.86

227.12

78.76

223.96

89.23

226.43

74.63

221.45

74.63

219.36

75.73

219.25

75.46

215.70

6.5 Melting Points of Emulsified H2 O Ice VI

P (MPa)

T (K)

984.00

298.00

937.50

295.00

862.50

290.00

792.50

285.00

724.50

280.00

663.00

275.00

603.50

270.00

547.00

265.00

496.00

260.00

446.00

255.00 (continued)

6.5 Melting Points of Emulsified H2 O Ice VI

103

(continued) P (MPa)

T (K)

400.00

250.00

356.50

245.00

316.00

240.00

283.00

235.00

997.77

299.11

979.40

297.69

990.88

298.17

912.83

293.81

856.21

289.77

843.21

288.46

781.99

284.63

743.74

281.96

718.49

279.65

663.39

275.45

636.61

273.30

599.12

269.89

569.28

267.27

542.50

264.75

524.90

263.28

493.53

260.24

466.75

257.77

457.57

256.51

437.68

254.68

415.49

252.16

393.30

249.06

361.93

245.91

356.57

244.76

6.6 Melting Points of Emulsified H2 O Ice PNP-XIII (Ice XII?)

P (MPa)

T (K)

925.88

282.34

832.54

278.19 (continued)

104

6 Appendix: Melting Points of Emulsified H2 O Ices

(continued) P (MPa)

T (K)

749.41

272.48

720.25

270.69

685.98

268.90

658.27

267.14

641.50

266.02

613.06

264.79

621.08

264.45

594.83

263.33

565.01

262.71

585.75

262.13

569.95

260.85

559.58

260.30

555.63

259.31

557.60

259.06

530.94

258.12

517.61

257.21

507.24

256.74

503.85

255.80

494.96

255.15

481.63

254.26

473.24

253.15

459.41

252.44

454.47

251.79

451.51

251.35

443.61

251.02

448.05

250.80

429.78

249.61

418.92

248.45

407.56

248.43

404.11

247.95

402.13

247.54

393.74

246.93

411.02

247.00

420.40

246.64

402.62

245.84

382.82

245.24

360.91

243.70 (continued)

6.6 Melting Points of Emulsified H2 O Ice PNP-XIII (Ice XII?) (continued) P (MPa)

T (K)

355.07

242.51

346.04

241.72

336.16

240.58

320.85

238.84

308.51

237.90

303.08

237.36

291.72

236.38

296.16

235.85

286.29

234.88

274.93

234.01

280.36

233.59

262.59

232.32

260.61

231.25

252.71

230.51

249.25

229.82

248.76

229.70

245.30

229.35

233.95

227.85

229.01

227.17

228.52

226.90

224.57

226.22

214.69

224.66

209.75

223.21

211.23

223.26

199.38

221.80

187.53

221.42

188.03

221.01

188.52

220.60

188.52

220.19

177.16

219.19

181.61

218.68

157.90

216.66

154.94

215.88

149.02

214.96

145.56

214.00

138.65

212.41

105

106

6 Appendix: Melting Points of Emulsified H2 O Ices

6.7 Melting Points of Emulsified H2 O Ice PNP-XIV

P (MPa)

T (K)

1345.56

282.21

1186.29

273.67

1120.67

269.65

1047.12

264.36

1020.13

264.66

1022.78

264.70

965.10

260.51

935.46

257.14

919.59

256.75

914.83

256.75

898.42

255.99

797.88

248.38

795.77

247.06

784.65

246.50

784.65

246.55

783.07

246.33

740.20

242.40

696.28

238.43

664.54

235.36

640.72

232.84

632.26

231.47

622.73

231.43

602.62

229.33

608.97

229.16

578.28

225.23

575.11

225.02

549.71

221.30

528.01

220.53

527.48

219.51

516.37

218.18

505.79

216.48

479.86

214.13

477.74

212.59

462.40

211.14

468.75

210.92 (continued)

6.7 Melting Points of Emulsified H2 O Ice PNP-XIV (continued) P (MPa)

T (K)

450.23

209.04

425.35

205.46

408.42

203.15

395.72

201.18

383.02

199.05

381.43

198.58

366.62

196.36

344.39

192.60

338.04

191.87

335.40

190.89

303.65

186.02

301.00

184.74

292.54

181.49

283.01

180.08

269.25

175.09

262.37

174.23

257.61

173.08

254.44

171.03

237.50

167.20

107