Equations of state of ice VI and ice VII at high pressure and high temperature Lucile Bezacier, Baptiste Journaux, Jean-Philippe Perrillat, Hervé Cardon, Michael Hanfland, and Isabelle Daniel Citation: The Journal of Chemical Physics 141, 104505 (2014); doi: 10.1063/1.4894421 View online: http://dx.doi.org/10.1063/1.4894421 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On equation of state, elastic, and lattice dynamic stability of bcc bismuth under high pressure: Ab-initio calculations J. Appl. Phys. 115, 053702 (2014); 10.1063/1.4863785 Equation of state of rhenium and application for ultra high pressure calibration J. Appl. Phys. 115, 043511 (2014); 10.1063/1.4863300 Equation of state for gadolinium gallium garnet crystals: Experimental and computational study Appl. Phys. Lett. 95, 141902 (2009); 10.1063/1.3237163 The temperature-pressure-volume equation of state of platinum J. Appl. Phys. 105, 013505 (2009); 10.1063/1.3054331 Semiempirical pressure-volume-temperature equation of state: Mg Si O 3 perovskite is an example J. Appl. Phys. 102, 123506 (2007); 10.1063/1.2822458
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THE JOURNAL OF CHEMICAL PHYSICS 141, 104505 (2014)
Equations of state of ice VI and ice VII at high pressure and high temperature Lucile Bezacier,1 Baptiste Journaux,2 Jean-Philippe Perrillat,2 Hervé Cardon,2 Michael Hanfland,1 and Isabelle Daniel2 1
European Synchrotron Radiation Facility, 6 rue Jules Horowitz, 38000 Grenoble, France Laboratoire de Géologie de Lyon, UMR 5276 CNRS, Ecole Normale Supérieure de Lyon – Université Claude Bernard Lyon 1, 2 rue Raphael Dubois, 69622 Villeurbanne Cedex, France 2
(Received 16 May 2014; accepted 20 August 2014; published online 10 September 2014) High-pressure H2 O polymorphs among which ice VI and ice VII are abundant in the interiors of large icy satellites and exo-planets. Knowledge of the elastic properties of these pure H2 O ices at high-temperature and high-pressure is thus crucial to decipher the internal structure of icy bodies. In this study we assess for the first time the pressure-volume-temperature (PVT) relations of both polycrystalline pure ice VI and ice VII at high pressures and temperatures from 1 to 9 GPa and 300 to 450 K, respectively, by using in situ synchrotron X-ray diffraction. The PVT data are adjusted to a second-order Birch-Murnaghan equation of state and give V0 = 14.17(2) cm3 mol−1 , K0 = 14.05(23) GPa, and α 0 = 14.6(14) × 10−5 K−1 for ice VI and V0 = 12.49(1) cm3 mol−1 , K0 = 20.15(16) GPa, and α 0 = 11.6(5) × 10−5 K−1 for ice VII. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4894421] I. INTRODUCTION
The accurate pressure-volume-temperature (PVT) properties of ice phases at high pressure and high temperature are important for assessing a number of geophysical issues. It includes the physical properties of the inner layers of the internal structure of icy satellites such as Europa, Ganymede, Titan,1 and water-rich exo-planets2, 3 or the possible chemical exchanges between those deep layers. Practically, this requires comparing the properties of theoretical pure H2 O ices with those of ices with more realistic and complex compositions, including salts for instance, e.g., Refs. 4–10. Inside those large icy satellites, a layer of ice VI should be present down to 500 km deep beneath shells of ices Ih, III, and V, i.e., until pressure reaches ca. 2 GPa.1, 11 Ice VII is also expected to be present during the early history of those satellites when mixtures of ices and rocks could be found up to 5 GPa, e.g., in the infancy of Titan.12 Ice VII likely comprises the interior of hundreds of extra-solar planetary bodies that are largely composed of H2 O. Accurate PVT Equations of State (EoS) of ices VI and VII are mandatory to understand the structure and petrology of their icy mantles. Numerous experiments have been carried out under dynamic pressure using shock-waves,13–15 under static highpressure using either diamond anvil cells16–20 or large-volume presses.21, 22 Nonetheless, only a few experimental data are available for ice VII and virtually none for ice VI in the pressure range relevant to the interior of icy satellites.16, 21 Ice VI has a tetragonal unit cell (P42 /nmc) and crystallizes at 1.1 GPa when lowering temperature down to 270 K. The structures were first determined by Kamb23, 24 and Block.25 The PVT equation of state of D2 O ice VI has recently been determined by neutron powder diffraction in the range 0–2.6 GPa and 120–330 K by Fortes et al.21 Volume measurements at 263 K,26 248 K,27 237 K,28 and 230– 0021-9606/2014/141(10)/104505/6/$30.00
350 K29 have been reported previously but to our best knowledge no PVT data of pure H2 O ice VI are available as yet. Ice VII has a body-centered-cubic unit cell (Pn-3m) and is the stable high-pressure phase of H2 O at room temperature above 2.3 GPa.13, 30 After high-pressure phases of ice were first recognized in 1900 by Tammann, Bridgman31 first determined the compressibility of ice VII and measured the melting curve of both ices VI and VII. Numerous melting curves of ice VII were established up to 108 GPa16, 31–38 and the stability of ice VII was extensively studied to 128 GPa.39–43 The studies of Fei et al.16 and Frank et al.44 improved our knowledge of the thermal expansion of ice VII. In this study, we present a new set of X-ray diffraction data on polycrystalline H2 O-ice VI at high pressure and high temperature and we confirm the ice VII PVT EoS with extensive data in the 2.2–10.1 GPa range. Measurements were performed between 300–340 K and 300–450 K for ice VI and VII, respectively. Then PVT EoS of ice VI and ice VII are determined from a 2nd order Birch-Murhaghan fit to PVT data. II. EXPERIMENTAL METHODS
The experiments were performed in an externally heated diamond-anvil cell45 equipped with diamond anvils of 500-μm culet. MilliQ water (18 m) and a small ruby sphere as a pressure gauge were loaded in a 200-μm diameter sample chamber drilled in a pre-indented (70 μm) T-301 stainless steel gasket lined with platinum to avoid chemical reactions between water and the gasket. High temperatures were achieved with an external resistive coil around the DAC, which was thermally insulated with ceramic fiber tape for a high thermal stability. Temperature was measured with a commercially calibrated K-type thermocouple. Neither dissolution nor alteration of the ruby spheres was observed up to the highest temperature investigated in the
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J. Chem. Phys. 141, 104505 (2014) TABLE I. Unit cell parameters of ice VI and ice VII as a function of pressure and temperature. Temperature error is ±0.4 K.
FIG. 1. X-ray diffraction patterns of ice VI (top) and ice VII (bottom) at 1.34 GPa, 300.7 K and 4.46 GPa, 380.3 K, respectively. Data are displayed as circles, the best Le Bail fit as a black curve, and the residuals below the patterns as a grey curve. The most intense peaks are labeled.
present study. Ruby spheres were doped with 3600 ppm of Cr3+ and annealed to avoid internal strain.46 Pressure was determined from the shift of the R1 fluorescence line.47–49 The temperature dependence of the R1 shift was calibrated for the present batch of ruby spheres and the data were best fitted with a second order polynomial.50 The pressure uncertainty is always smaller than 50 MPa. The emission line from a Nelamp was collected simultaneously to the fluorescence of the ruby to provide an internal absolute calibration. Temperature changes during the measurements were within ±0.4 K. Experiments were conducted at the ID09A beamline at the European Synchrotron Radiation Facility (Grenoble, France). Synchrotron X-ray radiation with a monochromatic wavelength of 0.4139 Å and a beamsize of 10 μm × 15 μm was used for in situ powder diffraction analysis. The diffraction data were collected with a mar555 flatpanel detector (marresearchTM ) located at a distance of 400 mm from the sample. The geometry of the set-up was calibrated using standard NBS silicon powder with data processed using the FIT2D software.51 Each diffraction pattern was collected with an exposure time of 1 s. Lattice parameters were refined using the Le Bail procedure,52 with the EXPGUI-GSAS package (see Fig. 1).53, 54 EOSFit 5.2 was used to determine the coefficients of PV and PVT equations of state.55 Data are presented in Table I and displayed in Figure 2.
P (GPa)
T (K)
1.26(2) 1.29(2) 1.34(2) 1.38(2) 1.38(2) 1.47(4) 1.52(2) 1.54(3) 1.58(3) 1.63(3) 1.64(2) 1.65(3) 1.69(3) 1.72(4) 1.76(2) 1.83(2) 1.93(2) 2.08(2) 2.17(2) 2.26(3) 2.39(3) 2.55(3) 2.81(4) 1.26(2) 1.31(2) 1.36(2) 1.40(2) 1.44(2) 1.58(2) 1.64(3) 1.71(3) 1.78(3) 1.82(4) 1.86(5) 1.92(5) 1.98(5) 2.03(5) 2.06(5) 2.19(6) 2.23(4) 2.27(3) 2.28(3) 2.38(4) 2.48(4) 2.56(3)
300.7 300.6 300.7 300.5 300.7 300.5 300.7 298.7 298.8 299.0 300.7 299.0 299.1 299.3 300.6 300.6 300.5 300.6 300.7 300.7 300.7 300.7 300.7 310.5 310.4 310.2 316.2 316.5 316.6 321.9 322.4 322.2 327.5 328.2 328.4 333.7 334.4 334.4 340.0 340.5 340.7 340.7 340.7 340.6 340.7
P (GPa) 2.71(2) 3.17(2) 3.29(2) 3.56(2) 4.20(3) 4.81(2) 5.79(5) 6.29(4) 6.85(2) 7.31(2) 7.64(2)
T (K) 300.6 300.5 300.5 300.5 300.6 300.8 300.5 300.5 300.5 300.5 300.5
Ice VI V (Å3 ) 217.944(7) 217.542(6) 216.988(4) 216.536(5) 216.559(6) 215.820(3) 214.996(3) 214.814(4) 214.406(3) 213.955(3) 213.573(3) 213.669(4) 213.118(5) 212.780(4) 212.457(3) 211.897(2) 210.673(3) 209.385(3) 208.386(5) 207.620(3) 206.347(3) 205.207(2) 202.779(3) 216.673(4) 216.279(4) 215.771(4) 215.529(5) 215.060(4) 213.545(4) 213.284(3) 212.652(4) 212.016(3) 211.811(3) 211.567(4) 210.931(4) 210.648(3) 210.307(5) 210.013(5) 209.485(4) 209.093(5) 208.648(4) 208.618(5) 207.960(5) 206.936(3) 206.233(3) Ice VII V (Å3 ) 37.156(1) 36.542(1) 36.430(1) 36.123(1) 35.586(1) 35.009(1) 34.152(1) 33.786(1) 33.284(1) 33.033(1) 32.701(2)
a (Å)
c (Å)
6.1845(1) 6.1815(1) 6.1757(1) 6.1729(1) 6.1725(1) 6.1658(1) 6.15858(5) 6.1569(1) 6.15347(5) 6.1492(1) 6.14588(5) 6.1471(1) 6.1419(1) 6.1391(1) 6.13609(5) 6.13141(4) 6.11950(5) 6.1085(1) 6.0993(1) 6.09364(5) 6.08197(5) 6.07183(4) 6.04892(5) 6.1746(1) 6.1708(1) 6.1665(1) 6.1648(1) 6.1604(1) 6.1470(1) 6.1450(1) 6.1392(1) 6.1342(1) 6.1321(1) 6.1298(1) 6.1244(1) 6.12213(5) 6.1192(1) 6.1168(1) 6.1124(1) 6.1090(1) 6.1042(1) 6.1048(1) 6.0988(1) 6.0894(1) 6.0832(1)
5.6981(1) 5.6931(1) 5.6893(1) 5.6825(1) 5.6840(1) 5.6769(1) 5.66851(5) 5.66676(4) 5.66237(4) 5.65814(4) 5.65427(4) 5.65451(4) 5.6494(1) 5.64562(5) 5.64271(4) 5.63642(4) 5.62570(5) 5.61135(5) 5.6015(1) 5.59133(5) 5.57840(4) 5.56614(4) 5.54201(6) 5.68306(5) 5.67972(5) 5.67424(5) 5.67105(5) 5.66684(4) 5.65147(5) 5.64825(4) 5.64207(5) 5.63438(4) 5.63280(4) 5.63055(4) 5.62363(4) 5.62021(4) 5.6165(1) 5.6130(1) 5.60699(5) 5.6027(1) 5.5996(1) 5.59776(5) 5.5910(1) 5.58063(4) 5.57310(4)
a (Å) 3.33689(1) 3.31841(2) 3.31501(1) 3.30568(2) 3.28922(2) 3.27136(2) 3.24425(3) 3.23281(3) 3.21671(4) 3.20861(5) 3.1978(1)
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J. Chem. Phys. 141, 104505 (2014) TABLE I. (Continued.)
TABLE I. (Continued.)
P (GPa)
T (K)
8.12(2) 8.54(2) 9.03(1) 9.22(2) 5.49(4) 9.78(2) 2.88(2) 2.88(2) 2.81(2) 2.78(3) 2.76(3) 2.67(2) 2.55(3) 2.48(3) 2.33(3) 2.28(3) 2.19(5) 2.18(5) 2.15(2) 2.14(2) 2.22(3) 2.20(2) 4.87(3) 5.59(4) 6.26(2) 6.93(2) 7.64(2) 8.49(2) 8.93(3) 9.25(3) 9.66(3) 9.96(3) 2.73(3) 2.75(4) 2.79(3) 2.86(4) 2.90(5) 2.93(3) 3.04(3) 3.15(3) 3.24(3) 3.24(3) 3.34(3) 3.47(3) 3.52(5) 3.62(2) 3.75(4) 3.80(2) 3.88(5) 4.00(3) 4.28(4) 4.28(3) 4.38(4) 4.46(3) 4.60(3) 4.94(3) 5.28(3) 6.04(3) 6.88(2)
300.6 300.6 300.6 300.6 300.5 300.5 340.4 340.3 340.5 340.5 340.6 340.6 340.7 340.7 340.6 340.6 340.6 340.6 340.7 340.7 340.6 340.6 340.7 340.7 340.7 340.7 340.7 340.7 340.7 340.7 340.7 340.7 380.1 380.1 380.1 380.1 380.1 380.1 380.1 380.1 380.2 380.2 380.2 380.2 380.2 380.3 380.2 380.3 380.2 380.2 380.3 380.3 380.4 380.3 380.3 380.3 380.3 380.2 380.3
Ice VII V (Å3 ) 32.406(2) 32.102(2) 31.872(2) 31.730(2) 34.502(1) 31.577(2) 37.155(1) 37.170(1) 37.196(1) 37.238(1) 37.317(1) 37.394(1) 37.622(1) 37.758(1) 38.001(1) 38.070(1) 38.220(1) 38.221(1) 38.218(1) 38.223(1) 38.202(1) 38.212(1) 35.082(1) 34.561(1) 33.967(1) 33.481(1) 32.871(1) 32.391(1) 32.095(1) 31.962(1) 31.693(1) 31.569(1) 37.726(1) 37.684(1) 37.588(1) 37.486(1) 37.403(1) 37.374(1) 37.214(1) 37.040(1) 36.896(1) 36.941(1) 36.766(1) 36.565(1) 36.498(1) 36.385(1) 36.249(1) 36.167(1) 36.065(1) 35.980(1) 35.670(1) 35.699(1) 35.543(1) 35.473(1) 35.423(1) 35.142(1) 34.816(1) 34.118(1) 33.363(1)
a (Å)
P (GPa)
T (K)
Ice VII V (Å3 )
3.1882(1) 3.1782(1) 3.1706(1) 3.1659(1) 3.25549(2) 3.1607(1) 3.33685(2) 3.33732(1) 3.33811(2) 3.33935(2) 3.34172(4) 3.34402(2) 3.35078(3) 3.35482(2) 3.36200(3) 3.36404(2) 3.36846(3) 3.36849(4) 3.36839(4) 3.36854(4) 3.36792(2) 3.36823(3) 3.27363(3) 3.25733(3) 3.23856(2) 3.22304(3) 3.20334(3) 3.18767(4) 3.17795(4) 3.17354(5) 3.16461(5) 3.16049(5) 3.35388(1) 3.35262(1) 3.34977(2) 3.34676(2) 3.34429(1) 3.34343(1) 3.33864(2) 3.33342(1) 3.32910(1) 3.33045(2) 3.32518(1) 3.31912(1) 3.31709(1) 3.31365(2) 3.30952(1) 3.30704(1) 3.30392(1) 3.30133(1) 3.29181(1) 3.29270(2) 3.28789(1) 3.28573(1) 3.28418(2) 3.27549(2) 3.26533(2) 3.24337(2) 3.21924(2)
7.98(3) 8.89(3) 9.06(3) 9.43(3) 9.69(2) 5.71(5) 5.64(5) 5.71(6) 6.14(7) 6.37(3) 6.55(2) 6.94(2) 7.27(3) 7.82(3) 8.58(4) 10.11(3) 9.67(3) 9.37(4) 4.25(5) 4.39(5) 4.83(5) 5.12(5)
380.2 380.2 380.2 380.2 380.2 380.2 450.6 450.7 451.1 451.1 451.2 449.9 449.9 449.9 449.9 449.9 450 450.1 450.1 449.7 449.8 449.9
32.613(1) 32.118(1) 31.959(1) 31.760(1) 31.538(1) 34.498(1) 34.725(1) 34.734(1) 34.492(1) 34.342(1) 34.201(1) 33.829(1) 33.512(1) 33.118(1) 32.570(1) 31.702(1) 31.833(1) 31.947(1) 36.093(1) 35.885(1) 35.351(1) 34.965(2)
a (Å) 3.19493(2) 3.17870(3) 3.17343(2) 3.16684(2) 3.15944(3) 3.25535(2) 3.26248(2) 3.26275(2) 3.25518(2) 3.25044(2) 3.24598(2) 3.23418(3) 3.22402(2) 3.21135(3) 3.19354(3) 3.16493(2) 3.16928(3) 3.17304(2) 3.30478(3) 3.29840(3) 3.28198(2) 3.2700(1)
FIG. 2. Pressure-volume data for ice VI (a) and ice VII (b). Lines represent the equation of state at the different temperatures. All data sets are well adjusted by a second-order Birch-Murnaghan equation of state. (a) Diamonds and circles represent data for ice VI at 300 and 340 K, respectively. (b) Diamonds, squares, and circles represent data at 300, 340, and 450 K, respectively. Data from Fei et al.16 and from Loubeyre et al.17 in the same pressure range are displayed for comparison, as triangles (black 300 K, grey 350 K and white 450 K) and stars (black 300 K), respectively.
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III. RESULTS AND DISCUSSION
An example of refined ice VI and ice VII diffraction patterns is given in Figure 1. Peaks remain sharp over the whole (P,T) range and all patterns could be fitted with space groups P42 /nmc and Pn-3m for ice VI and VII, respectively. Table I gives the lattice parameters and volume for each data point. In the following, subscript 0 on the volume V and the bulk modulus K stands for ambient conditions, here taken as P0 = 105 Pa and T0 = 300 K. The evolution of the volume of ice VI and ice VII as a function of pressure at room temperature was fitted to a second-order Birch-Murnaghan equation of state hereafter designated as BM2:56 7 5 V0 3 3 V0 3 P = K0 − (1) 2 V V and at high-temperature to the Birch-Murnaghan EoS with V (P , T ) = V0 (P , T0 ) exp(α0 (T − T0 ))
(2)
K(T ) = K0 + (T − T0 )(∂K/∂T )P .
(3)
and
This formulation includes all second derivatives of the volume with respect to the intensive variables P and T, and is usually suitable to fit most experimental PVT data sets collected from room temperature up to 1000 K.55 A f-F plot confirms a BM2 EoS for both ice VI and ice VII.55 The f-F plot is another visual quality assessment tool for the finite strain EoS, where Eulerian strain f is plotted versus normalized stress F. Based on the f-F plot one may characterize the compressional behavior of a material. The value of K0 estimated from the f-F plot is consistent with the results from the BM2 EoS fit (Figure 3). (∂K/∂T)P is set to 0 since no dependence on temperature was noticeable.
A. Ice VI
There were until now very scarce PVT data on ice VI. D2 O-ice VI has been investigated with neutron diffraction;21 H2 O ice VI has been submitted to shock-wave up to 2.1 GPa,57 and thermodynamical models have been developed by Tchijov et al.29 A least square fit of the data to a secondorder Birch-Murnaghan EoS yields V0 = 14.17(2) cm3 mol−1 , K0 = 14.01(27) GPa at 300 K. These results are in good agreement with the parameters proposed by Fortes et al.21 (Table II). The value of V0 obtained in the present study is only 1.8% higher than the V0 = 13.91(4) cm3 mol−1 of D2 O in Fortes et al.21 The K0 value determined for H2 O in the present experiment is 13% lower than the value K0 = 16.11(69) of D2 O reported by Fortes et al.21 and TABLE II. PV parameters at 300 K and PVT parameters; zero-pressure volumes V0 , bulk moduli K0 , pressure derivative K0 , thermal expansion coefficient α for pure H2 O ice VI and VII in the present study and in the literature. Values refer to the Birch-Murnaghan formalism unless specified. V and BM2 stand for Vinet and second order Birch Murnaghan equation of state formalism, respectively. V0 (cm3 mol−1 ) K0 (GPa) Ice VI—PV H2 O, this study D2 O (Fortes et al.21 )
FIG. 3. f-F plots based on the Birch-Murnaghan EoS for ice VI (top) and ice VII (bottom). These plots confirm that data are well fitted by a second-order Birch-Murnaghan equation of state.
K0
14.17(2) 13.91(4)
14.01(27) 4 16.1(7) 4
α 0 × 105 (K−1 )
Ice VII—PV H2 O, this study Liu69 Munro et al.15,a D2 Oa (Munro et al.15 ) Hemley et al.40 Fei et al.16 Wolanin et al.19 Loubeyre et al.17,a V BM2 Frank et al.44
12.42(2) 12.71(9) 12.2(2) 12.0(3) 12.3(3) 12.32(5) 12.37(9)
20.77(26) 24.6(9) 22.5(24) 24.9(29) 23.7(9) 23.7(6) 14.9(8)
4 4.4 4 4 4.15(7) 4 5.4 (1)
14.51 12.7(3) 12.4(3)
4.26 20.2(20) 21.3(13)
7.75 4 4.4(1)
Ice VI—PVT H2 O, this study D2 O (Fortes et al.21 )
14.17(2) 13.98(2)
14.05(23) 4 16.7(3) 4
14.6(14) 17.9(10)
Ice VII—PVT H2 O, this study Fei et al.16
12.49(1) 12.29(5)
20.15(17) 4 24.1(6) 4
11.58(54) 11.39(68)
a
EoS parameters re-calculated from reported unit-cell volumes under 20 GPa.
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contradicts the common thought that the EoS of H2 O and D2 O should not be very different, as concluded by Munro et al.15 on H2 O and D2 O ice VII, and by Loubeyre et al.58 after experiments on H2 and D2 that showed negligible difference between their EoS. The latter statement is also supported by experiments on LiH and LiD showing only slightly different V0 but same bulk modulus.59 However, a new analysis of the H2 O and D2 O data by Munro et al.15 within the BirchMurnaghan EoS (Table II) shows that D2 O ice VII appears ca. 10% stiffer than H2 O ice VII. As already highlighted by Fortes et al.,21 stronger interactions between interpenetrating framework in D2 O ice VI than in H2 O ice VI may well explain the observed difference in bulk modulus between H2 O and D2 O high-pressure ices. Any detailed interpretation would require either single crystal diffraction data or theoretical calculations that are beyond the scope of the present contribution. In Table II, PVT equations of state are presented for second-order Birch-Murnaghanan equation of state and give V0 = 14.17(2) cm3 /mol, K0 = 14.05(23) GPa, α 0 = 14.6(14) 10−5 K−1 , and (∂K/∂T)P = 0 for ice VI (Fig. 2). This fit adjusts well to the data and can only be compared with the experimental results by Fortes et al.21 on D2 O giving V0 = 13.98(2) cm3 mol−1 , K0 = 16.71(33) GPa and α 0 = 17.9(10) 10−5 K−1 . The present values of K0 and α 0 are 15.9% and 18.1% lower than those obtained for D2 O by Fortes et al.21 The present results may also be compared to the bulk moduli determined by Brillouin spectroscopy28 and ultrasonic methods.27 The present isothermal bulk modulus KT is converted into adiabatic bulk modulus KS using the relation,60 KS =
C P KT , (CP − α 2 KT V T )
(4)
where V is the molar volume (cm3 mol−1 ), α is the volume coefficient of thermal expansion (K−1 ), and CP is the isobaric specific heat capacity (J K−1 mol−1 ) calculated with the data from Tchijov et al.61 At 0.77 GPa and 237.6 K, our result leads to KS = 17.05 GPa, which is in excellent agreement with the values of KS = 18.1(9) GPa at 0.77 GPa, 237.6 K28 and KS = 16.0 GPa at 0.80 GPa, 248.1 K.27 B. Ice VII
The unit cell parameters of ice VII calculated from the three observable diffraction peaks are consistent with a cubic structure, and are in agreement with neutron diffraction measurements.62, 63 A least square fit to the data yields V0 = 12.42(2) cm3 mol−1 , K0 = 20.8(3) GPa for a BM2 EoS. These results are in very good agreement with those of Fei et al.,16 i.e., V0 = 12.3(5) cm3 mol−1 , K0 = 23.7(6) GPa. The f-F plot obtained for ice VII (Figure 3) is difficult to fully reconcile with either a second-order or a third-order BirchMurnaghan EoS, we therefore decided to use for ice VII a BM2 EoS that has less variables and facilitates comparison with ice VI. The present PV data are compared to several compression studies of ice VII in Table II. The value of V0 = 12.42(2) cm3 mol−1 determined in this study is consistent with all previous studies on ice VII except apparently with
the one by Loubeyre et al.17 who reported a value of 14.51 cm3 mol−1 using a Vinet equation of state (K0 = 7.75). We re-calculated a BM2 fit to the latter data set (taking into account only unit-cell volumes under 20 GPa) and we obtained V0 = 12.73(8) cm3 mol−1 and K0 = 16.7(6) GPa, which is in good agreement with the present results (Figure 2(b)). The observed deviation between published PVT models of ice VII is largely due to the lack of constraint on the value of V0 since ice VII is not quenchable. In this study, we proposed new values for the PVT EoS of ice VII supported by numerous data points just above 2 GPa, close to its lower limit of stability. In Table II, PVT equations of state are presented for a BM2 EoS and give V0 = 12.49(1) cm3 mol−1 , K0 = 20.15(17) GPa, and α 0 = 11.58(54) 10−5 K−1 . These new results agree again very well with those obtained by Fei et al.16 on ice VII, i.e., V0 = 12.29(5) cm3 mol−1 , K0 = 24.1(6) GPa, and α 0 = 11.4(7) × 10−5 K−1 . The change in volume between ice VI to ice VII across the transition has been estimated to −1.936(8) Å3 per molecule for D2 O21 at 2.15 GPa, 300 K, and −1.71 Å3 per molecule for H2 O.64 The present measurements show a decrease of the volume of −1.699 Å3 per molecule at 2.75 GPa, 300.6 K, which is in good agreement with Fortes et al.21 As expected from its lower PT stability field ice VI is more compressible than ice VII, K0 of ice VI is approximately ∼40% lower than the one of ice VII. Finally, the dependence of K0 and α on temperature was not detectable in the low-pressure range 0–10 GPa. Thermal properties of ices VI and VII may not change significantly in the low-pressure range relevant to icy satellites but this may not apply at the higher pressures characteristics of the interiors of planets or exo-planets. It is now widely accepted that the ice phases present in these H2 O-rich bodies are not pure and may incorporate various impurities such as salts NaCl, KCl, MgSO4 ,3, 7, 50, 65 or gases CO2 , CH4 .66–68 Experimentally ice VII can incorporate colossal amount of salt with molar concentration as high as 1:6, like for LiCl · 6D2 O.4 The high-pressure range investigated in this study aimed at filling the lack of PVT data on ices VI and VII. Their elastic properties obtained in the 0–10 GPa and 300–450 K pressure and temperature range relevant to the interiors of icy bodies will definitely help interpreting the recent results on the experimental petrology of ices65, 67 and ultimately will contribute to deciphering the structure of the icy satellites.
ACKNOWLEDGMENTS
We acknowledge the European Synchrotron Radiation Facility (Grenoble, France) for provision of beamtime at beamline ID09A. The authors are grateful to W. Crichton for fruitful discussions. 1 A.
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THE JOURNAL OF CHEMICAL PHYSICS 141, 104505 (2014)
Equations of state of ice VI and ice VII at high pressure and high temperature Lucile Bezacier,1 Baptiste Journaux,2 Jean-Philippe Perrillat,2 Hervé Cardon,2 Michael Hanfland,1 and Isabelle Daniel2 1
European Synchrotron Radiation Facility, 6 rue Jules Horowitz, 38000 Grenoble, France Laboratoire de Géologie de Lyon, UMR 5276 CNRS, Ecole Normale Supérieure de Lyon – Université Claude Bernard Lyon 1, 2 rue Raphael Dubois, 69622 Villeurbanne Cedex, France 2
(Received 16 May 2014; accepted 20 August 2014; published online 10 September 2014) High-pressure H2 O polymorphs among which ice VI and ice VII are abundant in the interiors of large icy satellites and exo-planets. Knowledge of the elastic properties of these pure H2 O ices at high-temperature and high-pressure is thus crucial to decipher the internal structure of icy bodies. In this study we assess for the first time the pressure-volume-temperature (PVT) relations of both polycrystalline pure ice VI and ice VII at high pressures and temperatures from 1 to 9 GPa and 300 to 450 K, respectively, by using in situ synchrotron X-ray diffraction. The PVT data are adjusted to a second-order Birch-Murnaghan equation of state and give V0 = 14.17(2) cm3 mol−1 , K0 = 14.05(23) GPa, and α 0 = 14.6(14) × 10−5 K−1 for ice VI and V0 = 12.49(1) cm3 mol−1 , K0 = 20.15(16) GPa, and α 0 = 11.6(5) × 10−5 K−1 for ice VII. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4894421] I. INTRODUCTION
The accurate pressure-volume-temperature (PVT) properties of ice phases at high pressure and high temperature are important for assessing a number of geophysical issues. It includes the physical properties of the inner layers of the internal structure of icy satellites such as Europa, Ganymede, Titan,1 and water-rich exo-planets2, 3 or the possible chemical exchanges between those deep layers. Practically, this requires comparing the properties of theoretical pure H2 O ices with those of ices with more realistic and complex compositions, including salts for instance, e.g., Refs. 4–10. Inside those large icy satellites, a layer of ice VI should be present down to 500 km deep beneath shells of ices Ih, III, and V, i.e., until pressure reaches ca. 2 GPa.1, 11 Ice VII is also expected to be present during the early history of those satellites when mixtures of ices and rocks could be found up to 5 GPa, e.g., in the infancy of Titan.12 Ice VII likely comprises the interior of hundreds of extra-solar planetary bodies that are largely composed of H2 O. Accurate PVT Equations of State (EoS) of ices VI and VII are mandatory to understand the structure and petrology of their icy mantles. Numerous experiments have been carried out under dynamic pressure using shock-waves,13–15 under static highpressure using either diamond anvil cells16–20 or large-volume presses.21, 22 Nonetheless, only a few experimental data are available for ice VII and virtually none for ice VI in the pressure range relevant to the interior of icy satellites.16, 21 Ice VI has a tetragonal unit cell (P42 /nmc) and crystallizes at 1.1 GPa when lowering temperature down to 270 K. The structures were first determined by Kamb23, 24 and Block.25 The PVT equation of state of D2 O ice VI has recently been determined by neutron powder diffraction in the range 0–2.6 GPa and 120–330 K by Fortes et al.21 Volume measurements at 263 K,26 248 K,27 237 K,28 and 230– 0021-9606/2014/141(10)/104505/6/$30.00
350 K29 have been reported previously but to our best knowledge no PVT data of pure H2 O ice VI are available as yet. Ice VII has a body-centered-cubic unit cell (Pn-3m) and is the stable high-pressure phase of H2 O at room temperature above 2.3 GPa.13, 30 After high-pressure phases of ice were first recognized in 1900 by Tammann, Bridgman31 first determined the compressibility of ice VII and measured the melting curve of both ices VI and VII. Numerous melting curves of ice VII were established up to 108 GPa16, 31–38 and the stability of ice VII was extensively studied to 128 GPa.39–43 The studies of Fei et al.16 and Frank et al.44 improved our knowledge of the thermal expansion of ice VII. In this study, we present a new set of X-ray diffraction data on polycrystalline H2 O-ice VI at high pressure and high temperature and we confirm the ice VII PVT EoS with extensive data in the 2.2–10.1 GPa range. Measurements were performed between 300–340 K and 300–450 K for ice VI and VII, respectively. Then PVT EoS of ice VI and ice VII are determined from a 2nd order Birch-Murhaghan fit to PVT data. II. EXPERIMENTAL METHODS
The experiments were performed in an externally heated diamond-anvil cell45 equipped with diamond anvils of 500-μm culet. MilliQ water (18 m) and a small ruby sphere as a pressure gauge were loaded in a 200-μm diameter sample chamber drilled in a pre-indented (70 μm) T-301 stainless steel gasket lined with platinum to avoid chemical reactions between water and the gasket. High temperatures were achieved with an external resistive coil around the DAC, which was thermally insulated with ceramic fiber tape for a high thermal stability. Temperature was measured with a commercially calibrated K-type thermocouple. Neither dissolution nor alteration of the ruby spheres was observed up to the highest temperature investigated in the
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J. Chem. Phys. 141, 104505 (2014) TABLE I. Unit cell parameters of ice VI and ice VII as a function of pressure and temperature. Temperature error is ±0.4 K.
FIG. 1. X-ray diffraction patterns of ice VI (top) and ice VII (bottom) at 1.34 GPa, 300.7 K and 4.46 GPa, 380.3 K, respectively. Data are displayed as circles, the best Le Bail fit as a black curve, and the residuals below the patterns as a grey curve. The most intense peaks are labeled.
present study. Ruby spheres were doped with 3600 ppm of Cr3+ and annealed to avoid internal strain.46 Pressure was determined from the shift of the R1 fluorescence line.47–49 The temperature dependence of the R1 shift was calibrated for the present batch of ruby spheres and the data were best fitted with a second order polynomial.50 The pressure uncertainty is always smaller than 50 MPa. The emission line from a Nelamp was collected simultaneously to the fluorescence of the ruby to provide an internal absolute calibration. Temperature changes during the measurements were within ±0.4 K. Experiments were conducted at the ID09A beamline at the European Synchrotron Radiation Facility (Grenoble, France). Synchrotron X-ray radiation with a monochromatic wavelength of 0.4139 Å and a beamsize of 10 μm × 15 μm was used for in situ powder diffraction analysis. The diffraction data were collected with a mar555 flatpanel detector (marresearchTM ) located at a distance of 400 mm from the sample. The geometry of the set-up was calibrated using standard NBS silicon powder with data processed using the FIT2D software.51 Each diffraction pattern was collected with an exposure time of 1 s. Lattice parameters were refined using the Le Bail procedure,52 with the EXPGUI-GSAS package (see Fig. 1).53, 54 EOSFit 5.2 was used to determine the coefficients of PV and PVT equations of state.55 Data are presented in Table I and displayed in Figure 2.
P (GPa)
T (K)
1.26(2) 1.29(2) 1.34(2) 1.38(2) 1.38(2) 1.47(4) 1.52(2) 1.54(3) 1.58(3) 1.63(3) 1.64(2) 1.65(3) 1.69(3) 1.72(4) 1.76(2) 1.83(2) 1.93(2) 2.08(2) 2.17(2) 2.26(3) 2.39(3) 2.55(3) 2.81(4) 1.26(2) 1.31(2) 1.36(2) 1.40(2) 1.44(2) 1.58(2) 1.64(3) 1.71(3) 1.78(3) 1.82(4) 1.86(5) 1.92(5) 1.98(5) 2.03(5) 2.06(5) 2.19(6) 2.23(4) 2.27(3) 2.28(3) 2.38(4) 2.48(4) 2.56(3)
300.7 300.6 300.7 300.5 300.7 300.5 300.7 298.7 298.8 299.0 300.7 299.0 299.1 299.3 300.6 300.6 300.5 300.6 300.7 300.7 300.7 300.7 300.7 310.5 310.4 310.2 316.2 316.5 316.6 321.9 322.4 322.2 327.5 328.2 328.4 333.7 334.4 334.4 340.0 340.5 340.7 340.7 340.7 340.6 340.7
P (GPa) 2.71(2) 3.17(2) 3.29(2) 3.56(2) 4.20(3) 4.81(2) 5.79(5) 6.29(4) 6.85(2) 7.31(2) 7.64(2)
T (K) 300.6 300.5 300.5 300.5 300.6 300.8 300.5 300.5 300.5 300.5 300.5
Ice VI V (Å3 ) 217.944(7) 217.542(6) 216.988(4) 216.536(5) 216.559(6) 215.820(3) 214.996(3) 214.814(4) 214.406(3) 213.955(3) 213.573(3) 213.669(4) 213.118(5) 212.780(4) 212.457(3) 211.897(2) 210.673(3) 209.385(3) 208.386(5) 207.620(3) 206.347(3) 205.207(2) 202.779(3) 216.673(4) 216.279(4) 215.771(4) 215.529(5) 215.060(4) 213.545(4) 213.284(3) 212.652(4) 212.016(3) 211.811(3) 211.567(4) 210.931(4) 210.648(3) 210.307(5) 210.013(5) 209.485(4) 209.093(5) 208.648(4) 208.618(5) 207.960(5) 206.936(3) 206.233(3) Ice VII V (Å3 ) 37.156(1) 36.542(1) 36.430(1) 36.123(1) 35.586(1) 35.009(1) 34.152(1) 33.786(1) 33.284(1) 33.033(1) 32.701(2)
a (Å)
c (Å)
6.1845(1) 6.1815(1) 6.1757(1) 6.1729(1) 6.1725(1) 6.1658(1) 6.15858(5) 6.1569(1) 6.15347(5) 6.1492(1) 6.14588(5) 6.1471(1) 6.1419(1) 6.1391(1) 6.13609(5) 6.13141(4) 6.11950(5) 6.1085(1) 6.0993(1) 6.09364(5) 6.08197(5) 6.07183(4) 6.04892(5) 6.1746(1) 6.1708(1) 6.1665(1) 6.1648(1) 6.1604(1) 6.1470(1) 6.1450(1) 6.1392(1) 6.1342(1) 6.1321(1) 6.1298(1) 6.1244(1) 6.12213(5) 6.1192(1) 6.1168(1) 6.1124(1) 6.1090(1) 6.1042(1) 6.1048(1) 6.0988(1) 6.0894(1) 6.0832(1)
5.6981(1) 5.6931(1) 5.6893(1) 5.6825(1) 5.6840(1) 5.6769(1) 5.66851(5) 5.66676(4) 5.66237(4) 5.65814(4) 5.65427(4) 5.65451(4) 5.6494(1) 5.64562(5) 5.64271(4) 5.63642(4) 5.62570(5) 5.61135(5) 5.6015(1) 5.59133(5) 5.57840(4) 5.56614(4) 5.54201(6) 5.68306(5) 5.67972(5) 5.67424(5) 5.67105(5) 5.66684(4) 5.65147(5) 5.64825(4) 5.64207(5) 5.63438(4) 5.63280(4) 5.63055(4) 5.62363(4) 5.62021(4) 5.6165(1) 5.6130(1) 5.60699(5) 5.6027(1) 5.5996(1) 5.59776(5) 5.5910(1) 5.58063(4) 5.57310(4)
a (Å) 3.33689(1) 3.31841(2) 3.31501(1) 3.30568(2) 3.28922(2) 3.27136(2) 3.24425(3) 3.23281(3) 3.21671(4) 3.20861(5) 3.1978(1)
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J. Chem. Phys. 141, 104505 (2014) TABLE I. (Continued.)
TABLE I. (Continued.)
P (GPa)
T (K)
8.12(2) 8.54(2) 9.03(1) 9.22(2) 5.49(4) 9.78(2) 2.88(2) 2.88(2) 2.81(2) 2.78(3) 2.76(3) 2.67(2) 2.55(3) 2.48(3) 2.33(3) 2.28(3) 2.19(5) 2.18(5) 2.15(2) 2.14(2) 2.22(3) 2.20(2) 4.87(3) 5.59(4) 6.26(2) 6.93(2) 7.64(2) 8.49(2) 8.93(3) 9.25(3) 9.66(3) 9.96(3) 2.73(3) 2.75(4) 2.79(3) 2.86(4) 2.90(5) 2.93(3) 3.04(3) 3.15(3) 3.24(3) 3.24(3) 3.34(3) 3.47(3) 3.52(5) 3.62(2) 3.75(4) 3.80(2) 3.88(5) 4.00(3) 4.28(4) 4.28(3) 4.38(4) 4.46(3) 4.60(3) 4.94(3) 5.28(3) 6.04(3) 6.88(2)
300.6 300.6 300.6 300.6 300.5 300.5 340.4 340.3 340.5 340.5 340.6 340.6 340.7 340.7 340.6 340.6 340.6 340.6 340.7 340.7 340.6 340.6 340.7 340.7 340.7 340.7 340.7 340.7 340.7 340.7 340.7 340.7 380.1 380.1 380.1 380.1 380.1 380.1 380.1 380.1 380.2 380.2 380.2 380.2 380.2 380.3 380.2 380.3 380.2 380.2 380.3 380.3 380.4 380.3 380.3 380.3 380.3 380.2 380.3
Ice VII V (Å3 ) 32.406(2) 32.102(2) 31.872(2) 31.730(2) 34.502(1) 31.577(2) 37.155(1) 37.170(1) 37.196(1) 37.238(1) 37.317(1) 37.394(1) 37.622(1) 37.758(1) 38.001(1) 38.070(1) 38.220(1) 38.221(1) 38.218(1) 38.223(1) 38.202(1) 38.212(1) 35.082(1) 34.561(1) 33.967(1) 33.481(1) 32.871(1) 32.391(1) 32.095(1) 31.962(1) 31.693(1) 31.569(1) 37.726(1) 37.684(1) 37.588(1) 37.486(1) 37.403(1) 37.374(1) 37.214(1) 37.040(1) 36.896(1) 36.941(1) 36.766(1) 36.565(1) 36.498(1) 36.385(1) 36.249(1) 36.167(1) 36.065(1) 35.980(1) 35.670(1) 35.699(1) 35.543(1) 35.473(1) 35.423(1) 35.142(1) 34.816(1) 34.118(1) 33.363(1)
a (Å)
P (GPa)
T (K)
Ice VII V (Å3 )
3.1882(1) 3.1782(1) 3.1706(1) 3.1659(1) 3.25549(2) 3.1607(1) 3.33685(2) 3.33732(1) 3.33811(2) 3.33935(2) 3.34172(4) 3.34402(2) 3.35078(3) 3.35482(2) 3.36200(3) 3.36404(2) 3.36846(3) 3.36849(4) 3.36839(4) 3.36854(4) 3.36792(2) 3.36823(3) 3.27363(3) 3.25733(3) 3.23856(2) 3.22304(3) 3.20334(3) 3.18767(4) 3.17795(4) 3.17354(5) 3.16461(5) 3.16049(5) 3.35388(1) 3.35262(1) 3.34977(2) 3.34676(2) 3.34429(1) 3.34343(1) 3.33864(2) 3.33342(1) 3.32910(1) 3.33045(2) 3.32518(1) 3.31912(1) 3.31709(1) 3.31365(2) 3.30952(1) 3.30704(1) 3.30392(1) 3.30133(1) 3.29181(1) 3.29270(2) 3.28789(1) 3.28573(1) 3.28418(2) 3.27549(2) 3.26533(2) 3.24337(2) 3.21924(2)
7.98(3) 8.89(3) 9.06(3) 9.43(3) 9.69(2) 5.71(5) 5.64(5) 5.71(6) 6.14(7) 6.37(3) 6.55(2) 6.94(2) 7.27(3) 7.82(3) 8.58(4) 10.11(3) 9.67(3) 9.37(4) 4.25(5) 4.39(5) 4.83(5) 5.12(5)
380.2 380.2 380.2 380.2 380.2 380.2 450.6 450.7 451.1 451.1 451.2 449.9 449.9 449.9 449.9 449.9 450 450.1 450.1 449.7 449.8 449.9
32.613(1) 32.118(1) 31.959(1) 31.760(1) 31.538(1) 34.498(1) 34.725(1) 34.734(1) 34.492(1) 34.342(1) 34.201(1) 33.829(1) 33.512(1) 33.118(1) 32.570(1) 31.702(1) 31.833(1) 31.947(1) 36.093(1) 35.885(1) 35.351(1) 34.965(2)
a (Å) 3.19493(2) 3.17870(3) 3.17343(2) 3.16684(2) 3.15944(3) 3.25535(2) 3.26248(2) 3.26275(2) 3.25518(2) 3.25044(2) 3.24598(2) 3.23418(3) 3.22402(2) 3.21135(3) 3.19354(3) 3.16493(2) 3.16928(3) 3.17304(2) 3.30478(3) 3.29840(3) 3.28198(2) 3.2700(1)
FIG. 2. Pressure-volume data for ice VI (a) and ice VII (b). Lines represent the equation of state at the different temperatures. All data sets are well adjusted by a second-order Birch-Murnaghan equation of state. (a) Diamonds and circles represent data for ice VI at 300 and 340 K, respectively. (b) Diamonds, squares, and circles represent data at 300, 340, and 450 K, respectively. Data from Fei et al.16 and from Loubeyre et al.17 in the same pressure range are displayed for comparison, as triangles (black 300 K, grey 350 K and white 450 K) and stars (black 300 K), respectively.
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III. RESULTS AND DISCUSSION
An example of refined ice VI and ice VII diffraction patterns is given in Figure 1. Peaks remain sharp over the whole (P,T) range and all patterns could be fitted with space groups P42 /nmc and Pn-3m for ice VI and VII, respectively. Table I gives the lattice parameters and volume for each data point. In the following, subscript 0 on the volume V and the bulk modulus K stands for ambient conditions, here taken as P0 = 105 Pa and T0 = 300 K. The evolution of the volume of ice VI and ice VII as a function of pressure at room temperature was fitted to a second-order Birch-Murnaghan equation of state hereafter designated as BM2:56 7 5 V0 3 3 V0 3 P = K0 − (1) 2 V V and at high-temperature to the Birch-Murnaghan EoS with V (P , T ) = V0 (P , T0 ) exp(α0 (T − T0 ))
(2)
K(T ) = K0 + (T − T0 )(∂K/∂T )P .
(3)
and
This formulation includes all second derivatives of the volume with respect to the intensive variables P and T, and is usually suitable to fit most experimental PVT data sets collected from room temperature up to 1000 K.55 A f-F plot confirms a BM2 EoS for both ice VI and ice VII.55 The f-F plot is another visual quality assessment tool for the finite strain EoS, where Eulerian strain f is plotted versus normalized stress F. Based on the f-F plot one may characterize the compressional behavior of a material. The value of K0 estimated from the f-F plot is consistent with the results from the BM2 EoS fit (Figure 3). (∂K/∂T)P is set to 0 since no dependence on temperature was noticeable.
A. Ice VI
There were until now very scarce PVT data on ice VI. D2 O-ice VI has been investigated with neutron diffraction;21 H2 O ice VI has been submitted to shock-wave up to 2.1 GPa,57 and thermodynamical models have been developed by Tchijov et al.29 A least square fit of the data to a secondorder Birch-Murnaghan EoS yields V0 = 14.17(2) cm3 mol−1 , K0 = 14.01(27) GPa at 300 K. These results are in good agreement with the parameters proposed by Fortes et al.21 (Table II). The value of V0 obtained in the present study is only 1.8% higher than the V0 = 13.91(4) cm3 mol−1 of D2 O in Fortes et al.21 The K0 value determined for H2 O in the present experiment is 13% lower than the value K0 = 16.11(69) of D2 O reported by Fortes et al.21 and TABLE II. PV parameters at 300 K and PVT parameters; zero-pressure volumes V0 , bulk moduli K0 , pressure derivative K0 , thermal expansion coefficient α for pure H2 O ice VI and VII in the present study and in the literature. Values refer to the Birch-Murnaghan formalism unless specified. V and BM2 stand for Vinet and second order Birch Murnaghan equation of state formalism, respectively. V0 (cm3 mol−1 ) K0 (GPa) Ice VI—PV H2 O, this study D2 O (Fortes et al.21 )
FIG. 3. f-F plots based on the Birch-Murnaghan EoS for ice VI (top) and ice VII (bottom). These plots confirm that data are well fitted by a second-order Birch-Murnaghan equation of state.
K0
14.17(2) 13.91(4)
14.01(27) 4 16.1(7) 4
α 0 × 105 (K−1 )
Ice VII—PV H2 O, this study Liu69 Munro et al.15,a D2 Oa (Munro et al.15 ) Hemley et al.40 Fei et al.16 Wolanin et al.19 Loubeyre et al.17,a V BM2 Frank et al.44
12.42(2) 12.71(9) 12.2(2) 12.0(3) 12.3(3) 12.32(5) 12.37(9)
20.77(26) 24.6(9) 22.5(24) 24.9(29) 23.7(9) 23.7(6) 14.9(8)
4 4.4 4 4 4.15(7) 4 5.4 (1)
14.51 12.7(3) 12.4(3)
4.26 20.2(20) 21.3(13)
7.75 4 4.4(1)
Ice VI—PVT H2 O, this study D2 O (Fortes et al.21 )
14.17(2) 13.98(2)
14.05(23) 4 16.7(3) 4
14.6(14) 17.9(10)
Ice VII—PVT H2 O, this study Fei et al.16
12.49(1) 12.29(5)
20.15(17) 4 24.1(6) 4
11.58(54) 11.39(68)
a
EoS parameters re-calculated from reported unit-cell volumes under 20 GPa.
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contradicts the common thought that the EoS of H2 O and D2 O should not be very different, as concluded by Munro et al.15 on H2 O and D2 O ice VII, and by Loubeyre et al.58 after experiments on H2 and D2 that showed negligible difference between their EoS. The latter statement is also supported by experiments on LiH and LiD showing only slightly different V0 but same bulk modulus.59 However, a new analysis of the H2 O and D2 O data by Munro et al.15 within the BirchMurnaghan EoS (Table II) shows that D2 O ice VII appears ca. 10% stiffer than H2 O ice VII. As already highlighted by Fortes et al.,21 stronger interactions between interpenetrating framework in D2 O ice VI than in H2 O ice VI may well explain the observed difference in bulk modulus between H2 O and D2 O high-pressure ices. Any detailed interpretation would require either single crystal diffraction data or theoretical calculations that are beyond the scope of the present contribution. In Table II, PVT equations of state are presented for second-order Birch-Murnaghanan equation of state and give V0 = 14.17(2) cm3 /mol, K0 = 14.05(23) GPa, α 0 = 14.6(14) 10−5 K−1 , and (∂K/∂T)P = 0 for ice VI (Fig. 2). This fit adjusts well to the data and can only be compared with the experimental results by Fortes et al.21 on D2 O giving V0 = 13.98(2) cm3 mol−1 , K0 = 16.71(33) GPa and α 0 = 17.9(10) 10−5 K−1 . The present values of K0 and α 0 are 15.9% and 18.1% lower than those obtained for D2 O by Fortes et al.21 The present results may also be compared to the bulk moduli determined by Brillouin spectroscopy28 and ultrasonic methods.27 The present isothermal bulk modulus KT is converted into adiabatic bulk modulus KS using the relation,60 KS =
C P KT , (CP − α 2 KT V T )
(4)
where V is the molar volume (cm3 mol−1 ), α is the volume coefficient of thermal expansion (K−1 ), and CP is the isobaric specific heat capacity (J K−1 mol−1 ) calculated with the data from Tchijov et al.61 At 0.77 GPa and 237.6 K, our result leads to KS = 17.05 GPa, which is in excellent agreement with the values of KS = 18.1(9) GPa at 0.77 GPa, 237.6 K28 and KS = 16.0 GPa at 0.80 GPa, 248.1 K.27 B. Ice VII
The unit cell parameters of ice VII calculated from the three observable diffraction peaks are consistent with a cubic structure, and are in agreement with neutron diffraction measurements.62, 63 A least square fit to the data yields V0 = 12.42(2) cm3 mol−1 , K0 = 20.8(3) GPa for a BM2 EoS. These results are in very good agreement with those of Fei et al.,16 i.e., V0 = 12.3(5) cm3 mol−1 , K0 = 23.7(6) GPa. The f-F plot obtained for ice VII (Figure 3) is difficult to fully reconcile with either a second-order or a third-order BirchMurnaghan EoS, we therefore decided to use for ice VII a BM2 EoS that has less variables and facilitates comparison with ice VI. The present PV data are compared to several compression studies of ice VII in Table II. The value of V0 = 12.42(2) cm3 mol−1 determined in this study is consistent with all previous studies on ice VII except apparently with
the one by Loubeyre et al.17 who reported a value of 14.51 cm3 mol−1 using a Vinet equation of state (K0 = 7.75). We re-calculated a BM2 fit to the latter data set (taking into account only unit-cell volumes under 20 GPa) and we obtained V0 = 12.73(8) cm3 mol−1 and K0 = 16.7(6) GPa, which is in good agreement with the present results (Figure 2(b)). The observed deviation between published PVT models of ice VII is largely due to the lack of constraint on the value of V0 since ice VII is not quenchable. In this study, we proposed new values for the PVT EoS of ice VII supported by numerous data points just above 2 GPa, close to its lower limit of stability. In Table II, PVT equations of state are presented for a BM2 EoS and give V0 = 12.49(1) cm3 mol−1 , K0 = 20.15(17) GPa, and α 0 = 11.58(54) 10−5 K−1 . These new results agree again very well with those obtained by Fei et al.16 on ice VII, i.e., V0 = 12.29(5) cm3 mol−1 , K0 = 24.1(6) GPa, and α 0 = 11.4(7) × 10−5 K−1 . The change in volume between ice VI to ice VII across the transition has been estimated to −1.936(8) Å3 per molecule for D2 O21 at 2.15 GPa, 300 K, and −1.71 Å3 per molecule for H2 O.64 The present measurements show a decrease of the volume of −1.699 Å3 per molecule at 2.75 GPa, 300.6 K, which is in good agreement with Fortes et al.21 As expected from its lower PT stability field ice VI is more compressible than ice VII, K0 of ice VI is approximately ∼40% lower than the one of ice VII. Finally, the dependence of K0 and α on temperature was not detectable in the low-pressure range 0–10 GPa. Thermal properties of ices VI and VII may not change significantly in the low-pressure range relevant to icy satellites but this may not apply at the higher pressures characteristics of the interiors of planets or exo-planets. It is now widely accepted that the ice phases present in these H2 O-rich bodies are not pure and may incorporate various impurities such as salts NaCl, KCl, MgSO4 ,3, 7, 50, 65 or gases CO2 , CH4 .66–68 Experimentally ice VII can incorporate colossal amount of salt with molar concentration as high as 1:6, like for LiCl · 6D2 O.4 The high-pressure range investigated in this study aimed at filling the lack of PVT data on ices VI and VII. Their elastic properties obtained in the 0–10 GPa and 300–450 K pressure and temperature range relevant to the interiors of icy bodies will definitely help interpreting the recent results on the experimental petrology of ices65, 67 and ultimately will contribute to deciphering the structure of the icy satellites.
ACKNOWLEDGMENTS
We acknowledge the European Synchrotron Radiation Facility (Grenoble, France) for provision of beamtime at beamline ID09A. The authors are grateful to W. Crichton for fruitful discussions. 1 A.
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