Thermodynamic properties of ice, water and their mixture under high pressure

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1 Glaciers-Ocam-Almosplterc Interactions (Proceedings of the International Symposium held at St Petersburg, September 1990). IAHS Publ. no. 08,1991. Thermodynamic properties of ice, water and their mixture under high pressure.e. CHIZHO & O.. NAGORNO Moscow Institute of Engineering and Physics, Kashirskoe Shosse 31, Moscow , USSR ABSTRACT The equilibrium melting of ice under high pressure is analyzed. Experimental data from various authors is used to build a thermal equation of the state (TES) of ice. Observations of thermal properties of water are also used to construct the TES of water at temperatures below 0 C. This equation describes all anomalous properties of water and is in good agreement with Bridgman's data. Ice melting under high pressure is studied numerically by means of integrating the system of ordinary differential equations. Phase boundaries of pure ice and water on P- diagram, isoentropes of ice-water mixture, and melting curve of ice are investigated. INTRODUCTION The models of deformation and flow of ice within the Greenland and Antarctic Ice Sheets, especially with respect to its compressibility, require information on thermodynamic properties of ice and water at temperature T < 0 C and under high pressure. Static (Bridgman, 1911) and dynamic (Larson, 1984; Fomin, 1985) loading experiments helped distinguish a phase transition (melting) of ice in a certain domain of pressure and temperature variations. The resulting ice-water mixture may considerably change mechanical properties of material. Using published data, we have constructed thermal equations of the state (TES) of ice I (in hydrostatic approximation) and water at T < 0 C and have determined their specific heat capacities. We have also numerically investigated the equation of the state of ice-water mixture in the domain of their equilibrium coexistence and have determined the bounds of phase equilibrium on the P- diagram. THERMODYNAMIC PROPERTIES OF ICE TES of simple substances connects three thermodynamic parameters: pressure P, temperature T, and specific volume 463

2 . E. Chizhov &O.. Nagornov 464. When the equilibrium melting of ice is discussed, it is convenient to choose P and T as independent variables, for they are equal in both phases. Then the TES of ice has the following form: i = x (P,T) (1) Indices 1 and in equation (1) and later in the text refer to ice and water, respectively. Equation (1) is commonly used to describe the deformable solid body in case the loading stress is much greater than the strength of this solid body (hydrostatic approximation). The strength of ice with respect to compression, tension, and shear stress is not greater than several dozens of bars (Landolt-Bornstein, 1980) whilst the ice melting effects take place under the pressure of the order of bar. Such a pressure exists, for example, in Antarctic Ice Sheet at the depth of about 1 km. Below we shall neglect the possible effect of shear stresses on ice compression and will use hydrostatic approximation. Equation (1) can be built using experimental data on the isothermal compressibility /3 = -1/ (3 /3P) and coefficient of thermal expansion a = 1/ (S /3T) by means of integrating the differential expression presented below d -TÏ-!- = -/3 dp + a dt ii T l l The path of integration on P-T diagram of ice is shown on Fig. 1 (line ABC). Line 1 is the melting curve of ice I. The path starts at A, where P = 0 bar, T = 0 C and = " 3 m 3 kg 1. The desired TES of ice should have the following form: l = ioexp - J /3 T1 (P',T) dp' + J a xi (P o,t') dt' Here T is temperature in C. We shall also denote absolute temperature as T. The processing of experimental published data (Butkovich, 1959; Gammon e_t al., 1983; Brockamp & on Ruter, 1969; Gagnon e_t al ) made it possible to determine a (P,T) and /3 (P,T). The expressions are as follows: a (P,T) = A + A T + A T + A T 3 Tl 1 o ' /3 (P,T) = Tl ' ' ' Tl 1 + m /3 P 1 T 1

3 465 Thermodynamic properties of ice and water under high pressure T 1 /3 T 1 where A = A = A A = ~ 10, = s /3 = The units of a and (3 T 1 from above are K~ and bar, respectively. Coefficient is close to 5. According to Brockamp & on Ruter (1969) m. = 4.4, while m = 5.37 according to Gagnon e_t al. (1986) P bar) -000 C, B, c 1-800, -400 B A T («C) FIG. 1 Phase P-T diagrams of ice and water. 1 - melting curve of ice I; ABC, A B C - paths of integration. Definitely the TES of ice I can be expressed as follows : = l " (T) 1 + m 13 P) 1/m a 1 T 1 ' r T (T) = a (P,T') dt' The specific heat capacity of ice I is given by

4 . E. Chizhov &.O.. Nagornov 466 -p r da C (P,T) = C (P,T) - T f + a dp PI K / / 1 ' ' PI 1 v o' ' ' K J p 1 [ a T Tl where C (P,T) = T (J kg~ a K 1 ); and a are obtained from equation (). THERMODYNAMIC PROPERTIES OF WATER A thermal equation of the state of water at T < 0 C was constructed, basing on works of Kell (1967), Minassian et al. (1981), Grindley & Lind (1971), Angel et aj^ (1973), Zemansky (1968), by means of integrating the expression d -^-5- = -13 dp + a T T dt along the path ABC (see Pig. 1). The reference point is P = 0 bar and T 1 = K. The result of integration is TES of water: o e *P { - J/T < P '< T K 1> dp ' + O + f K «T < P ' T ;> T dt ;} where 4 10 /3 (P,T ) = S b P T ' Kl' i i = a (P,T ) = A + T v ' K ; C + r A = a + a T +at 1 K 3 K B = a + a T + a T + a T T + a r 4 5K 6K 7K 8 C = a +a T +a T + a T K 11K 1K T = P + a P + a P The expressions for a and /3 were achieved by Minassian et al. (1981), Grindley & T Lind (1971) by precise measurements. One may find the coefficients a. (i = 1,,...,14) and b. (i = 0, 1,...,4) in the work of Minassian et al. (1981)! The specific heat capacity of water is given by:

5 467 Thermodynamic properties of ice and water under high pressure C (P,T) p \ / / C (P,T) U>4 da 3T + a dp' The observations of supercooled water's properties have indicated that specific heat of water at constant volume is very close to J kg -1 K * when T varies from -6 to 5 C and P = 0 bar. After that it may be determined from the following thermodynamic equation (Zemansky, 1968) T a K T P 0 T The received TES's of ice and water together with C (i = 1,) is a sufficient basis for constructing all PI thermodynamic functions needed for mathematical simulations. CALCULATION OF ICE-MELTING UNDER HIGH PRESSURE Line 1 in Fig. 1 is the melting curve of ice I. It corresponds to such values of P and T when coexistence of ice I and water in the form of a two-phase mixture is permissible. Let Z be the mass share of water in the mixture. The specific volume and specific entropy S are presented as follows: = (1 Z) + Z ' 1 S = (1 Z) S + Z S ' 1 Below we give the equation connecting the parameters of pure phases and the mixture: av av ap = (1 - Z) i T av i 1ST" C T PI K ( - v 1, av 3P (. T av 3T v - (. " C T P K where q is the specific heat of melting of ice I. We have investigated the process of adiabatic loading of ice-water mixture, when S = const. In such case the partial derivative in equation (4) may be replaced by d/dp. With attention fixed on the parameters of mixture during adiabatic loading we invoke two additional

6 . E. Chizhov & O.. Nagornov 468 conditions of equilibrium: Clausius-Clapeyron equation dt dp T ( K v (5) and the equation for the specific heat of melting dq dp c - c P PI T T ( K q q ( a v ^ T - 1 a 1 Tl The system of ordinary differential equations (4)-(6) has been solved numerically by Runge-Kutta method. The values of P changed from P = 0 to P = 100 bar by step AP = 100 bar. The initial conditions at P = 0 were T = 0 C 3-1 and q = J kg. The initial value of specific volume varied_ from = " m kg -1 (ice; Z = 0) to = m kg 1 (mixture; Z = 0.8). The bounds of mixture on P- diagrams - left bound = (P,T(P)) and right bound = (P,T(P)) - were determined together with calculations of (P), T(P), q(p) and Z(P). The relations = (P) and T = T(P) are the adiabatic equations of state of mixture and the melting curve, respectively. (s; P(bar) (10 3 m 3 kg-') FIG. Phase P- diagrams of ice-water mixture. Solid lines - phase boundaries of ice water; dashed lines isoentropes of ice-water mixture under various initial conditions.

7 469 Thermodynamic properties of ice and water under high pressure RESULTS We have compared the calculated values of specific volumes and with experimental observations of Bridgman and other researchers. The results agree quite well: the relative error of is about 1-% and that of - a share of percent. Fig. displays the P- diagram of ice-water mixture. Left and right curves are the bounds of ice-water mixture area. Dashed lines are isoentropes at P = 0 bar, T = 0 C and initial values of Z: 1 - Z = 0.0, - Z = 0., 3 - Z = = 0.4, 4 - Z = 0.6, 5 - Z = 0.8. The final values of Z on lines 1-4 at P = 100 bar are 0.19, 0.49, 0.65 and 0.88 respectively. Hence, if the sample of ice, initially at P = 0 bar, T = 0 C, is subjected to quasistatic adiabatic compression along the melting curve, only about 0% of ice will transfer into water. The obtained isoentropes satisfy the inequality (d /SP ) < 0. It points out the transformation of shock wave into continuous wave of compression while it travels through the ice sample. Larson's (1984) experiments confirm such an inference. The repeated results may form the basis for processing experimental data on the properties of ice within ice sheet and for mathematical simulations of ice behavior under high pressure with respect to its compressibility and phase transitions. REFERENCES Angell, C.A., Shuppert, J. & Tucker, J.C. (1973) Anomalous properties of supercooled water. Heat capacity, expansivity and proton magnetic resonance chemical shift from 0 to -38. ÇL. Phys. Chem. 77 (6), Bridgman, P.W. (1911) Water, in the liquid and five solid forms, under pressure. Proc. Am. Acad. Arts Sci. 47 (13), Brockamp, B. & on Ruter, H. (1969) Die Abhângigkeit der elastischen Parameter des eises vom hydrostatischen Druck bis zu 400 bars. Zeit. fur Geophysik B. 35 (3), Butkovich, T.R. (1959) Thermal expansion of ice. J_,_ Appl. Phys. 30 (3), Fomin,.A. (1985) Action of Explosion in Ice with Melting in Pressure Wave (In Russian). Masters Thesis, Moscow. Gagnon, R.E., Kiefte, H., Clouter, M.J. & Whalley, E. (1986) Acoustic contsants of ice lh, up to.8 kbar, by Brillouin spectroscopy. II Symp. on the Physics and Chemistry of Ice. C1-3-C1-8. Gammon, P.H., Kiefte, H., Clouter, M.J. & Denner, W.W. (1983) Elastic constants of artificial and natural ice samples by Brillouin spectroscopy. J^_ Glaciol. 9 (103),

8 . E. Chizhov &0.. Nagomov 470 Grindley, T. & Lind, J.E. (1971) PT properties of water and mercury. i_ Chem. Phys. 54 (9), Kell, G.S. (1967) Precise representation of volume properties of water at one atmosphere, i. Chem. and Enq. Data 1 (1), Landolt-Bornstein (1980) Numerical Data and Functional Relationships in Science and Technology. New Series. Group I. vol. 4.. Larson, D.B. (1984) Shock-wave studies of ice under uniaxial strain conditions. >L_ Glaciol. 30 (105), Minassian, L.T., Prusan, P. & Saulard, A. (1981) Thermodynamic properties of water under pressure of 5 kbar and between 8 and 10 C. Estimations in supercooled region down to -40 C. i_ Chem. Phys. 75 (6), Zemansky, M.W. (1968) Heat and Thermodynamics. N.Y., McGraw Hill.

Data Provided: A formula sheet and table of physical constants are attached to this paper.

Data Provided: A formula sheet and table of physical constants are attached to this paper. DEPARTMENT OF PHYSICS AND ASTRONOMY Spring Semester (2016-2017) From Thermodynamics to Atomic and Nuclear Physics