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Thermodynamics of stable and metastable equilibria in water in the T–P region
Physica B 265 (1999) 121—127
Thermodynamics of stable and metastable equilibria in water in the ¹—P region E.G. Ponyatovsky, V.V. Sinitsyn* Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscow Region, Russia
Abstract Water is considered as a regular solution of two structural components, their local configurations corresponding to the atomic short-range order of the lda and hda modifications of amorphous ice. The expression for the Gibbs potential, *G(¹,P,c), includes four model constants, i.e. the difference of the specific volumes, *», entropies, *S, and energies, *E, of the lda and hda components as well as the parameter º characterizing the mixing energy of the components in the binary solution. The concentration c of the hda component is determined from minimizing *G with respect to c. The ¹—P phase diagram calculated on the basis of this model agrees quantitatively with the experimental data and qualitatively with the computer calculations using the method of molecular dynamic simulation (MDS). 1999 Elsevier Science B.V. All rights reserved. Keywords: Phase diagram; Ices; Critical point; Amorphization
1. Introduction Ample experimental data on the thermodynamic properties of water in wide temperature and pressure ranges are available at present. There were many attempts to generalize these data and to draw an equation of state that would accurately represent the thermodynamic properties of water in as wide a temperature and pressure region as possible. Kell and Whally [1] suggested the equation of state that comprised 13 empirical constants and described the P—»—¹ relations of water in the temperature interval of 0°C to 150°C at pressures up to
* Corresponding author. Fax: #7-096-5764111; e-mail: [email protected].
1 kbar to an accuracy of about 20 ppm. Sato [2] managed to compose the equation of state for a yet wider temperature interval and approximated the density of water within 0.02% in the range between 240 and 423 K at pressures up to 0.1 kbar, but the number of fitting constants used in the P—»—¹ relations increased to 16. The earlier equations of state [1,2] well described the available experimental data and superseded tables that were cumbersome in practice. But they have some disadvantages due to the purely empiric way of their derivation. These equations include too many constants, do not explain the nature of the anomalous thermodynamic properties of water, particularly, the reason for a steep enhancement in the anomalous behavior of the heat expansion coefficient, the isothermal compressibility, the specific
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 3 4 1 - 6
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volume and the specific heat of supercooling of water as well as do not allow one to draw simple analytic expressions for these anomalies. There were also several attempts to give an analytic description of the anomalous behavior of supercooled water using expressions that are familiar from consideration of the phenomena near the critical point or near the spinodal line. This in principle correct model approach, however, does not have a detailed physical basis. To summarize, the nature of the anomalous behavior of supercooled water does not have even a qualitative interpretation, to say nothing of a quantitative one, up to the beginning of 1990s. This was closely connected to the lack of experimental data on the behavior of water in the amorphous state at high pressures and low temperatures. The work in this field was started in 1984 when Mishima, Whalley et al. [3—6] have found a new, more dense modification of amorphous ice. They have shown that two amorphous ice modifications could occur depending on the thermobaric conditions, i.e. low-density amorphous ice (lda) and high-density amorphous ice (hda). These two ice forms are very different in the specific volume, energy and which is most essential, in their atomic configuration. Further work by Mishima [7] has shown that a reversible first-order phase transition between these two disordered ice modifications took place at a temperature of 135 K and accompanied by a sharp volume change of 20% and a pressure hysteresis of &2 kbar. The occurrence of a reversible phase transition in one point implies the occurrence of a phase transition line. Its position in the ¹—P plane is essential, from the topological point of view, to understand the phenomena observed. The last experimental data [3—7] made a firm background for a thermodynamic model that presents an adequate view of the physics of the anomalies in supercooled water. Such model was developed in Refs. [8,9] where the temperature dependence of the thermodynamic parameters of supercooled water was calculated on the basis of the general model assumptions and quantitatively compared to all experimental data presently available. The purpose of this work is to put forth further arguments in order to detail the basic model state-
ments and to compare the data obtained within our model with other model calculations by computer simulations.
2. The basic model statements The basic model statements are: (i) Clusters of two kinds form the condensed disordered state of water, both liquid and amorphous. The clusters differ in their atomic configuration, specific volume, », and specific energy, E. The atomic configurations of the clusters correspond to the short-range order of either lda or hda amorphous ice modifications. (ii) The difference in the specific energies of the clusters is a linear function of the excitation of the system, i.e. of the concentration c of the high-energy clusters. (iii) The disordered state of water is considered as a regular solution of these clusters whereas the clusters are considered as two components of a binary system. (iv) The model thermodynamic potential is given by the same expression both in liquid and amorphous states. The set of the model statements will be discussed below, one by one: (i) The assumption of two main structure types in water is based on a good number of the experimental data. First, the occurrence of two amorphous ice modifications is firmly proved. Then, the earliest high-pressure neutron diffraction works on water [6,10] have already shown that the atomic radial distribution function (RDF) of water under a pressure of about 10 kbar reproduced, in fact, all the main features of the atomic RDF of hda ice under the same conditions. A more detailed structural study of water under high pressure in a wide temperature range was presented by BellisentFunel et al. [11,12]. They have found that the structure of water at atmospheric pressure approached the structure of lda ice as temperature was lowered whereas the structure of water at a pressure of 2.6 kbar resembled the structure of hda ice. (ii) The second assumption consists in a first approximation, to account for the energy change of
E.G. Ponyatovsky, V.V. Sinitsyn / Physica B 265 (1999) 121—127
the system both due to the cluster formation and due to the interaction between the clusters. It is possible, of course, to introduce quadratic and cubic terms and thus to account for triple and more complex kinds of interaction. It will be seen below that the set of the model constants we use is quite sufficient to describe the anomalies in the first approximation, and the constants themselves have a simple sense and can be determined immediately from the experimental data. (iii) The two-level statistical model with the linear dependence of the energy on excitation of the system proposed by Stra¨ssler and Kittel [13] has been shown [14,15] to correspond to the thermodynamic model of regular solutions with mixing energy º independent of temperature, pressure and the excitation degree. (iv) According to the available experimental data on the behavior of liquids below the melting point, the first derivatives of the thermodynamic potential in the glass transition point, ¹ , are continuous functions. The difference in the thermodynamic potentials of substances in liquid and glassy states is not large near ¹ , which is low for water, about 150 K [6,7,16]. The fourth assumption therefore does not markedly distort the shape of the ¹—P phase diagram in the temperature range under consideration, ¹*100 K, but considerably simplifies the analytic expressions of the model. Under the above assumptions, it is easy to write down the change in the thermodynamic potential, *G, due to formation of hda ice clusters within the lda phase in the molecular concentration equal to c: *G(¹,P,c)"G!G "(*E!¹*S#P*»)c#ºc(1!c) # R¹[c ln c#(1!c) ln(1!c)],
(1)
where *E"E !E , *S"S !S and *»" » !» . The ground state (component 1) is attributed to the structural state corresponding to the lda phase at atmospheric pressure and low temperatures. The excited state (component 2) is the hda structural state in its stability region at high pressures and low temperatures. Here ºc(1!c) is the mixing energy and !R[c ln c#(1!c) ln(1!c)] is the configuration entropy of mixing.
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We emphasize that Eq. (1) gives only that part of the Gibbs potential which is c-dependent. This part is responsible for the anomalies of the thermodynamic properties of supercooled water. The *G(c,¹,P) potential given by Eq. (1) as a function of c at fixed ¹ and P values can have either one or two minima. The line where condition *G (c )"*G (c ) holds the phase equilibrium
line for the lda and hda phases. The minimum *G (c) with lower c value cor responds to the lda phase, and the mimimum at the highest c value corresponds to the hda phase. Both phases may occur in the ¹—P region where *G has two minima. The deeper minimum corresponds to the more stable phase. The ¹—P region where *G has a single minimum is a single phase where only one phase occurs. The boundaries of the regions involved are two-phase instability lines (spinodals). One of the minima degenerates at each spinodal. The analytic expression for the equilibrium line is *E!¹*S#P*»"0
(2)
and those for the spinodals of either phase are of the form R¹ *G "!2º# . c(1!c) *c
(3)
Once the thermodynamic potential is known, all thermodynamic parameters, i.e. the specific volume, the entropy, the heat capacity, etc., can be analytically deduced as functions of three variables, ¹, P and c. But these three parameters are not independent. The equilibrium concentration is determined from minimization of *G at fixed ¹ and P. Taking this fact into account, the implicit form of the temperature and pressure dependence of c can be readily found for the phase in equilibrium under the given conditions:
*G *c
"(*E!¹*S#P*»)#(1!2c)º 2 . c # R¹ ln . (1!c)
(4)
The anomalous contributions to the thermodynamic properties and their dependencies on temperature and pressure can be derived on the basis of this relationship [8,9]. For example, the anomalous
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part of the specific heat is of the form *C "(*E#P*»#º!2cº) . *» . 2º!R¹/c(1!c)
(5)
Other expressions for the specific volume, thermal expansion and compressibility anomalies were also given in a previous work [9].
3. Results and discussion An important feature of the present model is that an analytic expression is given for the thermodynamic potential from the very outset. The expression includes four numeric constants, *E, *S, *» and º, and one temperature and pressure-dependent internal parameter, c. The *E, *S, *» values represent, respectively, the energy, entropy and volume difference between the lda and hda components, and º describes the interaction energy of these components in their binary solution. The best-fit values of the constants found earlier [9] are *»"!3.8 cm/mol, *S"4.225 J/ mol K, *E"1037 J/mol and º"3824 J/mol. The metastable ¹—P phase diagram of water calculated using these constants is presented in Fig. 1.The lda—hda phase equilibrium line is plotted as straight line E that terminates in a critical point, C . The lda and hda phase instability curves (spinodals) are shown as lines L and H, respectively. Fig. 1 also shows the experimental data on the lda & hda phase transition under pressure [7]. The parameters of the critical point were found to be ¹ "230 K and P "0.174 kbar. It follows from the above equations that the anomalies of the thermodynamic properties of water defined from the second derivatives of the thermodynamic potential (e.g., anomalies of the specific heat, *C , the thermal expansion coeffiN cient, *a, the isothermal compressibility, *c ) are 2 mainly related to the thermal and baric derivatives of parameter c rather than with the absolute values of parameter c at some ¹ and P values, that is, the anomalies are due to the parameter c sensitivity upon variation of the external conditions. Using Eq. (4), it is easy to calculate the functions c (¹) and .
Fig. 1. The metastable ¹—P phase diagrams of water calculated within the present thermodynamic model (solid lines) and using the MDS method (dashed lines) [20—22]. The experimental data on the lda & hda phase transition (open circles) are taken from Ref. [7]. Designations E, H, L and C are for the equilibrium line, the hda spinodal, the lda spinodal and the critical point calculated within the thermodynamic model. Similar designations with apostrophes are used for the MDS data.
c (P), thus, to visualize features of the thermo2 dynamic properties of water in the ¹—P plane. Moreover, these functions provide a quantitative description of the anomalies in the ¹—P region where the assumption of constant *E, *S, *» and º holds true, and a qualitative understanding of the main trends can be obtained in a more wide ¹—P region. Some general conclusions were extracted from the analysis of the parameter c variation with temperature and pressure [9]. The partial derivatives *c/*¹ and *c/*P as well as properties determined from the derivatives, *C , *a, *c tend to infinity N 2 upon intersection of the spinodals and in the close vicinity of the critical point. The anomalies rapidly subside upon recession from these specific elements of the ¹—P phase diagram. For example, parameter c is nearly constant, c+0.62, and anomalies are not the case in the temperature range 290—350 K at atmospheric pressure or over 250 K at P*P . To illustrate the parameter c behavior near the spinodals, Fig. 2a, b present several c(P) isotherms for the temperatures when the pressure axis
E.G. Ponyatovsky, V.V. Sinitsyn / Physica B 265 (1999) 121—127
intersects either one or two spinodals, as shown in the insets. In the temperature range between the critical point and the intersection point of the hda spinodal with the temperature axis, 170(¹(228 K, the pressure axis intersects two spinodals and the phase equilibrium line (inset in Fig. 2a). A representative c (P) curve for this range 2 is shown in Fig. 2a. At low pressures, P(P , there & is a single solution corresponding to lda. The second solution rises at the intersection point with the hda spinodal, P"P . This solution corresponds to & hda which is less stable up to the phase equilibrium pressure, P"P , but becomes more stable than lda # at P'P . Then the Gibbs potential minimum cor# responding to lda degenerates upon intersection of the lda spinodal at P"P . * Below 170 K, the pressure axis does not intersect the hda spinodal at positive P values therefore we have two c(P) solutions even at P"0 (Fig. 2b), in contrast to the range discussed above. This is the only feature in contrast with Fig. 2a, the others being almost the same. The other specific ¹—P region of the phase diagram where the thermodynamic properties exhibit an anomalous behavior is the supercritical region near the critical point. Contrary to the cases of intersection of the spinodals or the critical point,
125
the anomalies have no discontinuities here, which is due to the finite values of the partial derivatives *c/*¹ and *c/*P. Fig. 3 shows the isobars of the specific heat anomaly, *C (¹), calculated for pres. sures P"!0.1 kbar, 1 bar and 0.1 kbar. Fig. 3 also presents the anomaly of the specific heat at atmospheric pressure taken from the experimental data in Ref. [17], a good quantitative agreement between the experimental and calculated data quite distinct. One can see from the figure that a rather small pressure variation in the supercritical region results in a strong change of the *C (¹) curve. As . pressure increases up to P , the anomalies shift to lower temperatures, and the interval of the anomalous behavior contracts. The anomalies of the thermal expansion coefficient and the compressibility demonstrate a similar behavior [9]. Numerous theoretical works explaining anomalous properties of water can be nominally classified into three groups: (i) retracing spinodal scenario; (ii) second critical point scenario, and (iii) singularityfree scenario [18,19]. Our model corresponds to the second group of this classification. But the phase diagram with the second critical point involves elements of all three scenarios. If this second critical point were at a negative pressure, two spinodals would intersect
Fig. 2. The pressure dependencies of parameter c at several temperatures: a — ¹"200 K; b — ¹"170 K. Bold curves represent the more stable solutions whereas thin curves represent the less stable one. The corresponding cross-sections of ¹—P diagrams at these conditions are shown schematically on the insets.
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Fig. 3. The calculated anomalous part of the thermal dependence of the specific heat of water, *C (¹), at different pressures N (solid lines). Solid points are the experimental data [17] at atmospheric pressure.
the temperature axis at atmospheric pressure. If this were the case, the anomalous properties of water would be due to the onset of some instability point upon cooling. This point in the first scenario was accepted to be the point of the second intersection of the liquid—vapor spinodal with the temperature axis at atmospheric pressure. In the case when the second critical point is at a positive pressure, a supercritical region is always present in the phase diagram at positive pressures, as shown in Fig. 1. The anomalies in this region are finite and continuous (Fig. 3), as is the case in the third scenario. Thus, determination of the parameters of the second critical point is of utmost importance for the comprehension of the nature of the anomalies in supercooled water. We note that the occurrence of the second critical point of water has been first shown in the molecular-dynamic simulation (MDS) works [20—22]. The ¹—P phase diagram obtained in these works is plotted in Fig. 1 together with our calculated diagram. It is clearly seen that the two diagrams are in a good qualitative agreement, in spite of the quantitative difference. Both diagrams include the phase equilibrium line between two amorphous modifications of water which terminate in critical points. In both diagrams, the stability regions of each modification are limited by similarly shaped spinodals. But the parameters of the
critical point obtained using the MDS method were 200 K and 1.8 kbar [21], which is far from our calculated data. Moreover, the later MDS work [23] was carried out with a considerably higher calculation time, and the critical point in the proposed diagram was shifted to a negative pressure. Such ambiguity of the MDS calculation indicates that the data obtained with this method are only semiquantitative at present. However, the fact that two independent methods, one of them being a microscopic computer simulation and the other being a macroscopic thermodynamic approach, led to rather similar results is a strong argument for the occurrence of the second critical point in the phase diagram of water. Using the two-level model, we also analyzed consequences of the second critical point being at a negative pressure. It follows from the calculations carried out that model parameters *E, *S, *» and º become physically unreasonable even at a pressure as low as PK!50 bar. Therefore, Tanaka’s assumption that the critical point is in the field of negative pressures [23] seems to be unlikely and rather faulty. The present calculation data together with the available experimental data lead to a statement that the second critical pressure is situated at a positive pressure somewhat above P"1 atm, which gives rise to the observed anomalous behavior of various properties of supercooled water. Finally, we note that the two-level model used is only the first approximation and has its own limitations. One of them is the assumption that the model parameters *E, *S, *» and º are independent of temperature and pressure. In particular, temperature independent *S is inconsistent with the third thermodynamic law, which prevents from calculating the equilibrium line at low temperatures. To illustrate, Fig. 1 displays a good agreement between our calculated spinodal of low-density amorphous ice and the experimental data on the lda P hda phase transition whereas quantitative agreement is worse for the reverse transition. On the contrary, the MDS calculation data well fitted the hda P lda phase transition whereas the agreement was poor for the lda P hda transition. It can be assumed therefore that the actual lda—hda phase equilibrium line is somewhere between lines E and
E.G. Ponyatovsky, V.V. Sinitsyn / Physica B 265 (1999) 121—127
E in Fig. 1. So, a more accurate determination of the position of the critical point and the phase equilibrium line in the ¹—P phase diagram is a subject of further theoretical and experimental studies.
Acknowledgements The work was supported by the Russian Foundation for Basic Research (grant No. 98-02-16638). One of the authors (PEG) thanks the Conference Organizers for financial support.
References [1] [2] [3] [4]
G.S. Kell, E. Whalley, J. Chem. Phys. 62 (1975) 3496. H. Sato, Proc. 11th ICPS, Prague, 1989, 48. O. Mishima et al., Nature 310 (1984) 393. O. Mishima et al., Nature 314 (1985) 76.
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[5] Y.P. Handa, O. Mishima, E. Whalley, J. Chem. Phys. 84 (1986) 2766. [6] E. Whalley, J. Less-Common Metal 140 (1988) 361. [7] O. Mishima, J. Chem. Phys. 100 (1994) 5910. [8] E.G. Ponyatovsky et al., JETP Lett. 60 (1994) 360. [9] E.G. Ponyatovsky et al., J. Chem. Phys. 109 (1998) 2413. [10] A.Y. Wu et al., Mol. Phys. 47 (1982) 603. [11] M.-C. Bellissent-Funel et al., J. Chem. Phys. 87 (1987) 2231. [12] M.-C. Bellissent-Funel, L. Bosio, J. Chem. Phys. 102 (1995) 3727. [13] S. Stra¨ssler, C. Kittel, Phys. Rev. 139 (1965) A758. [14] I.L. Aptekar’, E.G. Ponyatovsky, Fiz. Met. Metalloved. 25 (1968) 777. [15] I.L. Aptekar’, E.G. Ponyatovsky, Fiz. Met. Metalloved. 25 (1968) 1049. [16] G.P. Johari, J. Chem. Phys. 102 (1995) 6224. [17] M. Oguni, C.A. Angell, J. Chem. Phys. 78 (1983) 7334. [18] S. Sastry et al., Phys. Rev. E 53 (1996) 6144. [19] L.P.N. Rebelo et al., J. Chem. Phys. 109 (1998) 626. [20] P.H. Poole et al., Nature 360 (1992) 324. [21] P.H. Poole et al., Phys. Rev. E 48 (1993) 4605. [22] H.E. Stanley et al., Physica A 205 (1994) 122. [23] H. Tanaka, Nature 380 (1996) 328.
Thermodynamics of stable and metastable equilibria in water in the ¹—P region E.G. Ponyatovsky, V.V. Sinitsyn* Institute of Solid State Physics, Russian Academy of Sciences, 142432 Chernogolovka, Moscow Region, Russia
Abstract Water is considered as a regular solution of two structural components, their local configurations corresponding to the atomic short-range order of the lda and hda modifications of amorphous ice. The expression for the Gibbs potential, *G(¹,P,c), includes four model constants, i.e. the difference of the specific volumes, *», entropies, *S, and energies, *E, of the lda and hda components as well as the parameter º characterizing the mixing energy of the components in the binary solution. The concentration c of the hda component is determined from minimizing *G with respect to c. The ¹—P phase diagram calculated on the basis of this model agrees quantitatively with the experimental data and qualitatively with the computer calculations using the method of molecular dynamic simulation (MDS). 1999 Elsevier Science B.V. All rights reserved. Keywords: Phase diagram; Ices; Critical point; Amorphization
1. Introduction Ample experimental data on the thermodynamic properties of water in wide temperature and pressure ranges are available at present. There were many attempts to generalize these data and to draw an equation of state that would accurately represent the thermodynamic properties of water in as wide a temperature and pressure region as possible. Kell and Whally [1] suggested the equation of state that comprised 13 empirical constants and described the P—»—¹ relations of water in the temperature interval of 0°C to 150°C at pressures up to
* Corresponding author. Fax: #7-096-5764111; e-mail: [email protected].
1 kbar to an accuracy of about 20 ppm. Sato [2] managed to compose the equation of state for a yet wider temperature interval and approximated the density of water within 0.02% in the range between 240 and 423 K at pressures up to 0.1 kbar, but the number of fitting constants used in the P—»—¹ relations increased to 16. The earlier equations of state [1,2] well described the available experimental data and superseded tables that were cumbersome in practice. But they have some disadvantages due to the purely empiric way of their derivation. These equations include too many constants, do not explain the nature of the anomalous thermodynamic properties of water, particularly, the reason for a steep enhancement in the anomalous behavior of the heat expansion coefficient, the isothermal compressibility, the specific
0921-4526/99/$ — see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 8 ) 0 1 3 4 1 - 6
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volume and the specific heat of supercooling of water as well as do not allow one to draw simple analytic expressions for these anomalies. There were also several attempts to give an analytic description of the anomalous behavior of supercooled water using expressions that are familiar from consideration of the phenomena near the critical point or near the spinodal line. This in principle correct model approach, however, does not have a detailed physical basis. To summarize, the nature of the anomalous behavior of supercooled water does not have even a qualitative interpretation, to say nothing of a quantitative one, up to the beginning of 1990s. This was closely connected to the lack of experimental data on the behavior of water in the amorphous state at high pressures and low temperatures. The work in this field was started in 1984 when Mishima, Whalley et al. [3—6] have found a new, more dense modification of amorphous ice. They have shown that two amorphous ice modifications could occur depending on the thermobaric conditions, i.e. low-density amorphous ice (lda) and high-density amorphous ice (hda). These two ice forms are very different in the specific volume, energy and which is most essential, in their atomic configuration. Further work by Mishima [7] has shown that a reversible first-order phase transition between these two disordered ice modifications took place at a temperature of 135 K and accompanied by a sharp volume change of 20% and a pressure hysteresis of &2 kbar. The occurrence of a reversible phase transition in one point implies the occurrence of a phase transition line. Its position in the ¹—P plane is essential, from the topological point of view, to understand the phenomena observed. The last experimental data [3—7] made a firm background for a thermodynamic model that presents an adequate view of the physics of the anomalies in supercooled water. Such model was developed in Refs. [8,9] where the temperature dependence of the thermodynamic parameters of supercooled water was calculated on the basis of the general model assumptions and quantitatively compared to all experimental data presently available. The purpose of this work is to put forth further arguments in order to detail the basic model state-
ments and to compare the data obtained within our model with other model calculations by computer simulations.
2. The basic model statements The basic model statements are: (i) Clusters of two kinds form the condensed disordered state of water, both liquid and amorphous. The clusters differ in their atomic configuration, specific volume, », and specific energy, E. The atomic configurations of the clusters correspond to the short-range order of either lda or hda amorphous ice modifications. (ii) The difference in the specific energies of the clusters is a linear function of the excitation of the system, i.e. of the concentration c of the high-energy clusters. (iii) The disordered state of water is considered as a regular solution of these clusters whereas the clusters are considered as two components of a binary system. (iv) The model thermodynamic potential is given by the same expression both in liquid and amorphous states. The set of the model statements will be discussed below, one by one: (i) The assumption of two main structure types in water is based on a good number of the experimental data. First, the occurrence of two amorphous ice modifications is firmly proved. Then, the earliest high-pressure neutron diffraction works on water [6,10] have already shown that the atomic radial distribution function (RDF) of water under a pressure of about 10 kbar reproduced, in fact, all the main features of the atomic RDF of hda ice under the same conditions. A more detailed structural study of water under high pressure in a wide temperature range was presented by BellisentFunel et al. [11,12]. They have found that the structure of water at atmospheric pressure approached the structure of lda ice as temperature was lowered whereas the structure of water at a pressure of 2.6 kbar resembled the structure of hda ice. (ii) The second assumption consists in a first approximation, to account for the energy change of
E.G. Ponyatovsky, V.V. Sinitsyn / Physica B 265 (1999) 121—127
the system both due to the cluster formation and due to the interaction between the clusters. It is possible, of course, to introduce quadratic and cubic terms and thus to account for triple and more complex kinds of interaction. It will be seen below that the set of the model constants we use is quite sufficient to describe the anomalies in the first approximation, and the constants themselves have a simple sense and can be determined immediately from the experimental data. (iii) The two-level statistical model with the linear dependence of the energy on excitation of the system proposed by Stra¨ssler and Kittel [13] has been shown [14,15] to correspond to the thermodynamic model of regular solutions with mixing energy º independent of temperature, pressure and the excitation degree. (iv) According to the available experimental data on the behavior of liquids below the melting point, the first derivatives of the thermodynamic potential in the glass transition point, ¹ , are continuous functions. The difference in the thermodynamic potentials of substances in liquid and glassy states is not large near ¹ , which is low for water, about 150 K [6,7,16]. The fourth assumption therefore does not markedly distort the shape of the ¹—P phase diagram in the temperature range under consideration, ¹*100 K, but considerably simplifies the analytic expressions of the model. Under the above assumptions, it is easy to write down the change in the thermodynamic potential, *G, due to formation of hda ice clusters within the lda phase in the molecular concentration equal to c: *G(¹,P,c)"G!G "(*E!¹*S#P*»)c#ºc(1!c) # R¹[c ln c#(1!c) ln(1!c)],
(1)
where *E"E !E , *S"S !S and *»" » !» . The ground state (component 1) is attributed to the structural state corresponding to the lda phase at atmospheric pressure and low temperatures. The excited state (component 2) is the hda structural state in its stability region at high pressures and low temperatures. Here ºc(1!c) is the mixing energy and !R[c ln c#(1!c) ln(1!c)] is the configuration entropy of mixing.
123
We emphasize that Eq. (1) gives only that part of the Gibbs potential which is c-dependent. This part is responsible for the anomalies of the thermodynamic properties of supercooled water. The *G(c,¹,P) potential given by Eq. (1) as a function of c at fixed ¹ and P values can have either one or two minima. The line where condition *G (c )"*G (c ) holds the phase equilibrium
line for the lda and hda phases. The minimum *G (c) with lower c value cor responds to the lda phase, and the mimimum at the highest c value corresponds to the hda phase. Both phases may occur in the ¹—P region where *G has two minima. The deeper minimum corresponds to the more stable phase. The ¹—P region where *G has a single minimum is a single phase where only one phase occurs. The boundaries of the regions involved are two-phase instability lines (spinodals). One of the minima degenerates at each spinodal. The analytic expression for the equilibrium line is *E!¹*S#P*»"0
(2)
and those for the spinodals of either phase are of the form R¹ *G "!2º# . c(1!c) *c
(3)
Once the thermodynamic potential is known, all thermodynamic parameters, i.e. the specific volume, the entropy, the heat capacity, etc., can be analytically deduced as functions of three variables, ¹, P and c. But these three parameters are not independent. The equilibrium concentration is determined from minimization of *G at fixed ¹ and P. Taking this fact into account, the implicit form of the temperature and pressure dependence of c can be readily found for the phase in equilibrium under the given conditions:
*G *c
"(*E!¹*S#P*»)#(1!2c)º 2 . c # R¹ ln . (1!c)
(4)
The anomalous contributions to the thermodynamic properties and their dependencies on temperature and pressure can be derived on the basis of this relationship [8,9]. For example, the anomalous
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part of the specific heat is of the form *C "(*E#P*»#º!2cº) . *» . 2º!R¹/c(1!c)
(5)
Other expressions for the specific volume, thermal expansion and compressibility anomalies were also given in a previous work [9].
3. Results and discussion An important feature of the present model is that an analytic expression is given for the thermodynamic potential from the very outset. The expression includes four numeric constants, *E, *S, *» and º, and one temperature and pressure-dependent internal parameter, c. The *E, *S, *» values represent, respectively, the energy, entropy and volume difference between the lda and hda components, and º describes the interaction energy of these components in their binary solution. The best-fit values of the constants found earlier [9] are *»"!3.8 cm/mol, *S"4.225 J/ mol K, *E"1037 J/mol and º"3824 J/mol. The metastable ¹—P phase diagram of water calculated using these constants is presented in Fig. 1.The lda—hda phase equilibrium line is plotted as straight line E that terminates in a critical point, C . The lda and hda phase instability curves (spinodals) are shown as lines L and H, respectively. Fig. 1 also shows the experimental data on the lda & hda phase transition under pressure [7]. The parameters of the critical point were found to be ¹ "230 K and P "0.174 kbar. It follows from the above equations that the anomalies of the thermodynamic properties of water defined from the second derivatives of the thermodynamic potential (e.g., anomalies of the specific heat, *C , the thermal expansion coeffiN cient, *a, the isothermal compressibility, *c ) are 2 mainly related to the thermal and baric derivatives of parameter c rather than with the absolute values of parameter c at some ¹ and P values, that is, the anomalies are due to the parameter c sensitivity upon variation of the external conditions. Using Eq. (4), it is easy to calculate the functions c (¹) and .
Fig. 1. The metastable ¹—P phase diagrams of water calculated within the present thermodynamic model (solid lines) and using the MDS method (dashed lines) [20—22]. The experimental data on the lda & hda phase transition (open circles) are taken from Ref. [7]. Designations E, H, L and C are for the equilibrium line, the hda spinodal, the lda spinodal and the critical point calculated within the thermodynamic model. Similar designations with apostrophes are used for the MDS data.
c (P), thus, to visualize features of the thermo2 dynamic properties of water in the ¹—P plane. Moreover, these functions provide a quantitative description of the anomalies in the ¹—P region where the assumption of constant *E, *S, *» and º holds true, and a qualitative understanding of the main trends can be obtained in a more wide ¹—P region. Some general conclusions were extracted from the analysis of the parameter c variation with temperature and pressure [9]. The partial derivatives *c/*¹ and *c/*P as well as properties determined from the derivatives, *C , *a, *c tend to infinity N 2 upon intersection of the spinodals and in the close vicinity of the critical point. The anomalies rapidly subside upon recession from these specific elements of the ¹—P phase diagram. For example, parameter c is nearly constant, c+0.62, and anomalies are not the case in the temperature range 290—350 K at atmospheric pressure or over 250 K at P*P . To illustrate the parameter c behavior near the spinodals, Fig. 2a, b present several c(P) isotherms for the temperatures when the pressure axis
E.G. Ponyatovsky, V.V. Sinitsyn / Physica B 265 (1999) 121—127
intersects either one or two spinodals, as shown in the insets. In the temperature range between the critical point and the intersection point of the hda spinodal with the temperature axis, 170(¹(228 K, the pressure axis intersects two spinodals and the phase equilibrium line (inset in Fig. 2a). A representative c (P) curve for this range 2 is shown in Fig. 2a. At low pressures, P(P , there & is a single solution corresponding to lda. The second solution rises at the intersection point with the hda spinodal, P"P . This solution corresponds to & hda which is less stable up to the phase equilibrium pressure, P"P , but becomes more stable than lda # at P'P . Then the Gibbs potential minimum cor# responding to lda degenerates upon intersection of the lda spinodal at P"P . * Below 170 K, the pressure axis does not intersect the hda spinodal at positive P values therefore we have two c(P) solutions even at P"0 (Fig. 2b), in contrast to the range discussed above. This is the only feature in contrast with Fig. 2a, the others being almost the same. The other specific ¹—P region of the phase diagram where the thermodynamic properties exhibit an anomalous behavior is the supercritical region near the critical point. Contrary to the cases of intersection of the spinodals or the critical point,
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the anomalies have no discontinuities here, which is due to the finite values of the partial derivatives *c/*¹ and *c/*P. Fig. 3 shows the isobars of the specific heat anomaly, *C (¹), calculated for pres. sures P"!0.1 kbar, 1 bar and 0.1 kbar. Fig. 3 also presents the anomaly of the specific heat at atmospheric pressure taken from the experimental data in Ref. [17], a good quantitative agreement between the experimental and calculated data quite distinct. One can see from the figure that a rather small pressure variation in the supercritical region results in a strong change of the *C (¹) curve. As . pressure increases up to P , the anomalies shift to lower temperatures, and the interval of the anomalous behavior contracts. The anomalies of the thermal expansion coefficient and the compressibility demonstrate a similar behavior [9]. Numerous theoretical works explaining anomalous properties of water can be nominally classified into three groups: (i) retracing spinodal scenario; (ii) second critical point scenario, and (iii) singularityfree scenario [18,19]. Our model corresponds to the second group of this classification. But the phase diagram with the second critical point involves elements of all three scenarios. If this second critical point were at a negative pressure, two spinodals would intersect
Fig. 2. The pressure dependencies of parameter c at several temperatures: a — ¹"200 K; b — ¹"170 K. Bold curves represent the more stable solutions whereas thin curves represent the less stable one. The corresponding cross-sections of ¹—P diagrams at these conditions are shown schematically on the insets.
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E.G. Ponyatovsky, V.V. Sinitsyn / Physica B 265 (1999) 121—127
Fig. 3. The calculated anomalous part of the thermal dependence of the specific heat of water, *C (¹), at different pressures N (solid lines). Solid points are the experimental data [17] at atmospheric pressure.
the temperature axis at atmospheric pressure. If this were the case, the anomalous properties of water would be due to the onset of some instability point upon cooling. This point in the first scenario was accepted to be the point of the second intersection of the liquid—vapor spinodal with the temperature axis at atmospheric pressure. In the case when the second critical point is at a positive pressure, a supercritical region is always present in the phase diagram at positive pressures, as shown in Fig. 1. The anomalies in this region are finite and continuous (Fig. 3), as is the case in the third scenario. Thus, determination of the parameters of the second critical point is of utmost importance for the comprehension of the nature of the anomalies in supercooled water. We note that the occurrence of the second critical point of water has been first shown in the molecular-dynamic simulation (MDS) works [20—22]. The ¹—P phase diagram obtained in these works is plotted in Fig. 1 together with our calculated diagram. It is clearly seen that the two diagrams are in a good qualitative agreement, in spite of the quantitative difference. Both diagrams include the phase equilibrium line between two amorphous modifications of water which terminate in critical points. In both diagrams, the stability regions of each modification are limited by similarly shaped spinodals. But the parameters of the
critical point obtained using the MDS method were 200 K and 1.8 kbar [21], which is far from our calculated data. Moreover, the later MDS work [23] was carried out with a considerably higher calculation time, and the critical point in the proposed diagram was shifted to a negative pressure. Such ambiguity of the MDS calculation indicates that the data obtained with this method are only semiquantitative at present. However, the fact that two independent methods, one of them being a microscopic computer simulation and the other being a macroscopic thermodynamic approach, led to rather similar results is a strong argument for the occurrence of the second critical point in the phase diagram of water. Using the two-level model, we also analyzed consequences of the second critical point being at a negative pressure. It follows from the calculations carried out that model parameters *E, *S, *» and º become physically unreasonable even at a pressure as low as PK!50 bar. Therefore, Tanaka’s assumption that the critical point is in the field of negative pressures [23] seems to be unlikely and rather faulty. The present calculation data together with the available experimental data lead to a statement that the second critical pressure is situated at a positive pressure somewhat above P"1 atm, which gives rise to the observed anomalous behavior of various properties of supercooled water. Finally, we note that the two-level model used is only the first approximation and has its own limitations. One of them is the assumption that the model parameters *E, *S, *» and º are independent of temperature and pressure. In particular, temperature independent *S is inconsistent with the third thermodynamic law, which prevents from calculating the equilibrium line at low temperatures. To illustrate, Fig. 1 displays a good agreement between our calculated spinodal of low-density amorphous ice and the experimental data on the lda P hda phase transition whereas quantitative agreement is worse for the reverse transition. On the contrary, the MDS calculation data well fitted the hda P lda phase transition whereas the agreement was poor for the lda P hda transition. It can be assumed therefore that the actual lda—hda phase equilibrium line is somewhere between lines E and
E.G. Ponyatovsky, V.V. Sinitsyn / Physica B 265 (1999) 121—127
E in Fig. 1. So, a more accurate determination of the position of the critical point and the phase equilibrium line in the ¹—P phase diagram is a subject of further theoretical and experimental studies.
Acknowledgements The work was supported by the Russian Foundation for Basic Research (grant No. 98-02-16638). One of the authors (PEG) thanks the Conference Organizers for financial support.
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