Melting curve and fluid equation of state of carbon dioxide at high pressure and high temperature Valentina Giordano, Frédéric Datchi, Agnès Dewaele
To cite this version:
Valentina Giordano, Frédéric Datchi, Agnès Dewaele. Melting curve and fluid equation of state of car- bon dioxide at high pressure and high temperature. Journal of Chemical Physics, American Institute of Physics, 2006, 125, pp.054504. 10.1063/1.2215609. hal-00088068
HAL Id: hal-00088068 https://hal.archives-ouvertes.fr/hal-00088068 Submitted on 28 Jul 2006
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ccsd-00088068, version 1 - 28 Jul 2006 † ∗ I for III fteepae a eqece oro temperature room to quenched be can phases these of nesvl tde nteps 5yas i.1 Fig. I for phase years. determined presently 15 CO as < past diagram P the phase the in shows studied intensively accuracy. high melt- state, with of the measured equation in are fluid these the kinks provided in by discontinuities or detected curve be ing can transitions Such ietefis-re hs rniinfudi phosporus pres- past, in recent found high the transition at in phase discoveries first-order fluids the important like to simple led of has other sure investigation the On the conditions. hand, and supercritical ma- scientific in or of sciences synthesis high planetary number terial as at large such dioxide a domains, technological in carbon important and is water density like fluids molecular eauetoohrmlclrpae xs:paeI for II phase exist: > phases T molecular other two perature etn uv n udeuto fsaeo abndoieat dioxide carbon of state of equation fluid and curve Melting lcrncades [email protected] address: Electronic lcrncades [email protected] address: Electronic h oi hs iga fcro ixd a been has dioxide carbon of diagram phase solid The nweg ftetemdnmcpoete fsimple of properties thermodynamic the of Knowledge 2 rsalzsit yia oeua oi,tecubic the solid, molecular typical a into crystallizes 7 n hs Vfor IV phase and K 470 0GPa 20 > P 4 hc rnfrst h rhrobcphase orthorhombic the to transforms which , 0GPa 10 rnfre otemdnmcvle nodrt aclt t calculate to T order process. in values relaxation thermodynamic thermal to a transformed of presence frequenci the B Brillouin postulating and and veloc ultrasonic Interferometric sound between and velocity range. index sound refractive probed the determine the sho to melt, conducted in the curve with equilibrium melting in tempe the phase and solid pressure the identify of to measurements in-situ and equilibrium ihpesr narssieyhae imn ni el T cell. 11 anvil to diamond up resistively-heated temperature a room in pressure high nltcfruaino h est ihrsett pressur to respect with the density in the of formulation analytic udaoe50Ki escmrsil hnpeitdfo var from predicted than compressible less is K 500 above fluid 2,3 h etn uv n udeuto fsaeo abndioxid carbon of state of equation fluid and curve melting The tro eprtr n . GPa, 0.5 and temperature room At . .INTRODUCTION 1. P 5,6 − nvri´ iree ai ui,10reLuml 51 Pa 75015 Lourmel, rue 140 Curie, Marie et Universit´e Pierre For . T nvri´ iree ai ui,10reLuml 51 Pa 75015 Lourmel, rue 140 Curie, Marie et Universit´e Pierre ag .- P n 0-0 n cuaewti % u resu Our 2%. within accurate and K 300-700 and GPa 0.1-8 range > P > T MM,Pyiu e iiu ess NSUR7590, UMR CNRS Denses, Milieux des Physique IMPMC, MM,Pyiu e iiu ess NSUR7590, UMR CNRS Denses, Milieux des Physique IMPMC, E,Bo CEA, I/´ preetd hsqeT´oiu tAppliqu´ee, Th´eorique et Physique DIF/D´epartement de ES ooSinic,SsoFoetn F) Italy (FI), Fiorentino Sesto Scientifico, Polo LENS, 2Gaadhg tem- high and GPa 12 0 K 500 . ˆ 1 t otl 2 18 Bruy`eres-le-Chˆatel, 91680 France 12, Postale ite 2,3,7 ± 0 ic both Since . . P n 800 and GPa 1 aetn .Giordano M. Valentina Dtd uy2,2006) 29, July (Dated: Fr´ed´eric Datchi temperature g` sDewaele Agn`es 1 . oanweeeprmna aaaeasn sunknown. is absent the are in data accuracy experimental their where but domain have formulations, Attempts state various of on bridged. equations based phenomenological be derive to to made remains been that data thou- of few a sets of temperatures K curve estimated Hugoniot sand and the GPa 70 along to points up few a K given lim- 980 have been and iments far GPa so 0.8 have to measurements experiments ited PVT pressure high pressures. static high in at extended state of and equation new that at then K clear 130 is needed. by are It lower measurements temperature Kennedy pressure. melting and this Grace a by gives extrapolation reported which curve the melting with the disagreement of strong in actually sbsdo aau o36Kad11 P published GPa 1.17 Yoo and and K Iota 1 366 Fig. by to in up reported Bridgman data one by on The based known. is barely melting the still the i.e. is with, domains, curve, solid begin To and fluid fluid. the between pressure limit high the on studies few oyei tutr paeV ihfu-odcoordinated a atoms four-fold carbon to with V) transforms (phase pres- dioxide structure higher carbon polymeric at temperatures, Finally, that and metastable. proposed sures actually been is has it III phase III, phase to reverting without ± hr sas ako xeietldt ntefluid the on data experimental of lack a also is There ncnrs ihtesldpae,teehv envery been have there phases, solid the with contrast In yvsa bevto ftesolid-fluid the of observation visual by K 5 si vdne n ol erpoue by reproduced be could and evidenced is es † igta oi stesal hs along phase stable the is I solid that wing n eprtr spooe,suitable proposed, is temperature and e t ftefli hs.Adseso fthe of dispersion A phase. fluid the of ity 16,17,18 eBilunsudvlcte eethen were velocities sound Brillouin he eeuto fsaeo udCO fluid of state of equation he aue aa pcrsoywsused was spectroscopy Raman rature. emligln a eemndfrom determined was line melting he ∗ ospeoeooia models. phenomenological ious ilunsatrn xeiet were experiments scattering rillouin 12 8,9,10,11 hr stu a ewe hs two these between gap a thus is There . aebe eemndunder determined been have e n11 n n igepitmeasured point single one and 1914 in 2 t4Gaad60K h atris latter The K. 640 and GPa 4 at i,Fac,and France, ris, . i,France ris, ihpesr n high and pressure high 14,15 t hwta the that show lts hl hc-aeexper- shock-wave while , 2 An . 13 , 2
The most sophisticated of these formulations has been ing the gas directly into the cell. Because the cell is proposed by Span and Wagner (SWEOS)19, whose valid- globally heated, this setup provides minimal tempera- ity range was estimated to extend from the triple point ture gradients across the cell, as found from numerous (P = 0.518 MPa, T = 216.59K) to 0.8GPa and 1100K. previous experiments performed in the same temperature This is based on a phenomenological form of the Helmoltz range21,23,24: the maximum difference observed between free energy with 42 free parameters fitted to a large bank the sample and other parts of the cell is 5 K. A K-type of thermodynamic experimental data available in the low- thermocouple was placed in contact with the sides of one pressure range. Although very accurate within its valid- of the diamond anvils and fixed in place by high tem- ity range, its adequacy in the extrapolated range is ques- perature ceramic cement. An independent and in-situ tionable, as shows the comparison between the experi- measure of temperature was provided by using the ruby mental Hugoniot curve and the predicted one19. Among as a thermometer, as described in Ref. [21]. This tem- the other available models, the Pitzer-Sterner equation perature agreed with that of the thermocouple within ±5 of state (PSEOS)20 has been considered a good one at K up to 600 K, as typically observed for this setup. At high pressure because it was constrained by the shock higher temperatures the ruby R-doublet is very broad wave measurements. and no more resolved, so that this temperature measure- We present here the first study of fluid CO2 at static ment becomes increasingly inaccurate. We thus relied on pressures above 1 GPa. First, we measured the melting the temperature given by the thermocouple. The reli- curve up to 800 K in order to determine the fluid stability ability of the temperature reading is supported by the domain. Using Brillouin scattering and interferometric very good reproducibility of the present results between measurements, we then determined the sound velocity experiments performed on different samples, and by the and refractive index of fluid CO2 up to 700 K and 8 GPa. continuous aspect of the melting line as shown below. From the measured sound velocity, we derived the fluid On this basis, we can confidently claim that the error on equation of state in the same P −T range, which extends temperature is within ±5 K. the previously covered pressure range by a factor of 10. The melting curve was determined by visually monitor- This paper is organized as follows. In a first section, ing the solid–fluid equilibrium. The sample was brought we describe the experimental details of our experiments. to its room temperature melting pressure (0.55 GPa) Then in section III we present the results obtained for where a single crystal was grown by finely tuning the the melting curve, the refractive index and sound veloc- load and kept in equilibrium with its melt. The sample ity and compare them with available literature data. In was then slowly heated and compressed in order to main- section IV, the method used to derive the equation of tain the coexistence of the crystal and the fluid, as shown state is given along with the results. Finally, the conclu- in the inset of Fig. 2. The difference in refractive index sion summarizes our main findings. between the fluid and solid phases remained large enough to observe a good contrast between the two phases up to our highest pressure. A melting point was defined by the 2. EXPERIMENTAL DETAILS measurement of pressure and temperature when equilib- rium was observed and stable in P and T . This method The present experiments were conducted in a mem- allows a fine sampling of the melting curve and precludes brane diamond-anvil cell (mDAC) made of high- any error that could originate from metastabilities such temperature resistant alloy (PER72 from Aubert et Du- as under-cooling or over-pressurization. val). CO2 (99.99 % purity) samples were loaded by con- The coexistent solid phase was identified by Raman densing the gas at 273K and 35bars in a high-pressure spectroscopy all along the melting curve. Raman spec- vessel. A golden ring about 10 µm large separated the tra were collected in a back-scattering geometry using + sample from the gasket material (rhenium) in order to the 514.53 nm line from an Ar laser and a T64000 Ra- avoid any possible chemical reaction at high tempera- man spectrometer (Jobin-Yvon-Horiba, f =0.64 m, 1800 2+ ture. A ruby ball, a small amount of SrB4O7:Sm grooves/mm grating, 100 µm entrance slits) equipped and a 15 µm-size cubic BN crystal were loaded with with a ℓN2-cooled CCD array detector. the sample. Pressure was primarily determined from the The refractive index of the fluid was measured by an 5 7 2+ D0 − F0 luminescence line of SrB4O7:Sm and cross- interferometric technique described in detail in Refs. [25] checked with the one obtained from the Raman shift of and [26]. Briefly, this method involves measurements of the TO mode of c-BN, using the pressure scales reported the interference patterns obtained by illuminating the in Refs. [21,22,23]. The estimated uncertainty on pres- sample with parallel white light (500 <λ< 700 nm) sure was about 0.02 GPa at 300 K, 0.05 GPa at 500 K on one hand and monochromatic light (λ = 632.8 nm, and 0.1GPa at 700 K. Fabry-Perot rings) on the other hand. By combining the The whole mDAC was heated by means of a ring- two interference patterns, we were able to determine both shaped resistive heater enveloping the cell, whose tem- thickness and refractive index n of the sample. The rela- perature can be regulated within 1 K using an electronic tive uncertainty on the refractive index measured by this module. Heating was done in air or in a reducing at- method is approximately 5 × 10−3. mosphere (mixture of Ar and 2%H2) obtained by flush- Brillouin scattering experiments were performed using 3 a 6-pass tandem Fabry-Perot interferometer from JRS are two fit parameters. Fitting this expression to our data Scientific27. The scattered light from a Ar+ laser excita- gives a = 0.403(5) GPa and b = 2.58(1). Figure 2 shows tion was collected in a back-scattering geometry. In this that this form reproduces very well our data, with a rms case, the Brillouin frequency shift is given by: deviation of 3.7 K. The Kechin equation was proposed as an improvement over the Simon-Glatzel one and is ∆σ =2nv /λ c (1) s 0 0 able to represent melting curves with negative slopes or where vs, λ0 = 514.53 mn and c0 are respectively the going through a maximum melting temperature; it can sound velocity, the excitation wavelength and the speed be expressed as: of light in vacuum. The frequency shift was determined 1 with an accuracy of 2 × 10−3 cm−1. P − P b T = T 1+ 0 exp[c(P − P )] (3) 0 a 0 3. RESULTS AND DISCUSSION In the case of CO2 , the melting temperature is a regular increasing function of pressure and a fit to the Kechin 3.1. The melting curve equation only gives a small improvement over the S-G law. The obtained parameters are: a = 0.443(9) GPa, × −3 −1 The melting curve of CO2 was determined in three sep- b = 2.41(3) and c = 4.7(7) 10 GPa , with a rms arate experiments covering the temperature range 300- deviation of 3.5 K. 800 K and pressure range 0.55-11.1 GPa. The measured In addition to the visual observation of melting, we melting points are shown in Fig. 2 along with the avail- used Raman spectroscopy in order to identify the solid able literature data. The different runs showed very good phase in equilibrium with its melt. Fig. ?? reports spec- reproducibility, within the uncertainty of our pressure tra collected along the melting curve at 430, 733 and 800 and temperature readings. The large number of collected K after subtraction of the liquid diffusion background. data constrain very well the melting curve in the covered Spectra of solid phase I at 300 K, 2.1 GPa and of solid range. phase IV at 580 K, 15 GPa are added for comparison. De- Our data are in excellent agreement with those re- spite the line broadening at high temperatures, the three ported by Bridgman12 where they overlap (300
(numbers in parentheses give the standard error on the When expressed on a logarithmic scale for ρ and P , an last digit). analytical relation was found that well reproduces our densities in the covered P − T range:
2 3 4. CALCULATION OF THE EQUATION OF i j STATE ln(ρ)= aij T (ln(P )) (8) Xi=0 Xj=0
From the sound velocity data, corrected for the ther- with the density expressed in g/cm3, P in GPa and T in mal relaxation, the equation of state of fluid CO can be 2 K. The aij parameters, obtained by a least-squares fit, obtained by recursively integrating the following equa- are listed in Table I. tions: The evolution of n with density, according to our new equation of state, is well described by the polynomial 2 ∂ρ 1 T α form: = 0 2 + (6) ∂P (vs ) Cp T 2 − 3 n(ρ)=1+0.21(1)ρ +0.04(1)ρ 0.017(5)ρ (9)
Fitting Eq. (5) to the thermodynamic values of the 2 ∂cp − ∂ V sound velocity deduced from our calculations leads = T 2 (7) ∂P ∂T to the following coefficients: a0 = 0.9249(5), a1 = T P −0.000392(1), b0 =0.2683(6) and b1 =0.000197(1). where ρ, V, α, and cp are the density, specific volume, As mentioned in the Introduction, there are a number volume thermal expansion and isobaric heat capacity re- of phenomenological equations of state for fluid CO2 . spectively. The method consists in integrating Eq. (6) in These equations have usually been parameterized in or- a small temperature range around the desired isotherm, der to accurately reproduce the properties at low pressure by first neglecting the second term giving the correction and around the critical point; their predictions at high between isothermal and adiabatic compressions. From pressure need to be tested against experimental data. ∞ 0 the value of vs given by Eq. (5), vs is calculated using In Fig. 10 (a) and (b) we compare our calculated EoS Eqs. (A3)-(A4). The second term of Eq. (6) is then eval- with the SWEOS and the PSEOS, respectively. As it uated using the deduced value for α =1/V (∂V/∂T ) and can be seen, our results show that the fluid becomes the one of cp obtained by integrating Eq. (7). Eq. (6) is less compressible than predicted by the SWEOS above integrated again and the procedure iterated until conver- 500 K, with a difference in density which grows continu- gence. This procedure was first tested using the sound ously larger with temperature, up to -4.2% at 8 GPa and velocity derived from the SWEOS (which directly gives 700 K. The PSEOS densities remain within about 2.2% 0 vs ); the calculated ρ, α and cp were in excellent agree- of our data and the general agreement is better than for ment with those directly derived from the SWEOS. the SWEOS, although this EoS also tends to overesti- The integration starts at a point P0 for which density, mate the density at HP-HT. α, cp and cv need to be known. We chose P0 =0.25 GPa, Comparison with the classical molecular dynamics cal- value above which Eq. (5) provides a reliable representa- culations of Belonoshko and Saxena39 is reported in tion of our data at all temperatures. The starting values Fig. 10 (c). For these simulations, an effective inter- for the thermodynamic quantities were calculated using molecular pair potential of the exp-6 form was used. A the SWEOS, which provides a consistent frame for the fit to the densities obtained at various P -T points was evaluation of any thermodynamic property. The uncer- provided by the authors for 0.5
heats include several contributions, one of 12 very good agreement with the work of Bridgman up to which, cvib, is due to the vibrational degrees of freedom 366 K, but significantly deviate at higher T from data of of the molecules. In the condition ωτv ≃ 1 (τv being the other authors2,13. Furthermore, we found that the solid thermal relaxation time), the contribution to the specific phase in equilibrium with the fluid remains phase I up heat from the vibrational modes is modulated by their to 800 K, indicating that the I-IV-Fluid triple point is degree of relaxation. For ωτv ≫ 1, these modes will be located at higher temperature than previously thought. frozen-like and their contribution to the specific heats Secondly, we have reported the first measurements of relaxes out33, so that the effective specific heats will be ∞ 0 the refractive index and Brillouin frequency shift in the ci = ci − cvib, with i = p, v. The unrelaxed sound temperature and pressure range [300–700]K and [0.1– velocity may then be calculated as: 8] GPa. From the deduced sound velocities, corrected for the effect of thermal relaxation, we have calculated the 2 ∞ ∞ dP (v∞) = (cp /cv ) (A2) equation of state of fluid CO2 with an estimated uncer- dρ T tainty of ∆ρ = 2% (Eq. 8, table I), and compared it with various EoS proposed in the literature. The model and thus: of Pitzer and Sterner provides the best agreement at the 2 0 0 ∞ ∞ highest P-T conditions, but all tested models were found (v0/v∞) = (cp/cv)(cv /cp ) (A3) to overestimate the compressibility of the fluid at high temperature. This work should thus be useful in order The value of cvib may be deduced from the knowledge to better constrain these EoS in the high pressure regime. of the vibrational frequencies νi and their degeneracies gi, using the Planck-Einstein formula (R is the ideal gas constant): APPENDIX A: THERMAL RELAXATION AND DISPERSION OF THE SOUND VELOCITY. g (hν )2/k T 2 exp(−hν /k T ) c = R i i B i B (A4) vib (1 − exp(−hν /k T ))2 The thermodynamic definition of the adiabatic sound Xi i B 0 velocity vs is: To our knowledge, there are no data on the vibrational dP frequencies of the fluid in our P − T range; however, (v0)2 = (c0/c0) (A1) s p v dρ the measurements made on solid I showed that these T modes are very weakly pressure and temperature depen- 0 0 40,41,42 where cp and cv are the thermodynamic (or“static”) val- dent , which should also be valid for the fluid. The ues of the specific heats at constant pressure and constant estimation of cvib was thus based on the gas values for volume respectively. As long as the frequency ω at which the vibrational frequencies34, and assumed to be pressure the sound waves are probed remain below any relaxation independent.
1 Y. Katayama, T. Mizutani, W. Utsumi, O. Shimomura, 11 M. Santoro, J. F. Lin, H. K. Mao, and R. J. Hemley, J. M. Yamakata, and K. ichi Funakoshi, Nature 403, 170 Chem. Phys. 121, 2780 (2004). (2000). 12 P. W. Bridgman, Phys. Rev. 3, 126 (1914). 2 V. Iota and C. S. Yoo, Phys. Rev. Lett. 86, 5922 (2001). 13 J. D. Grace and G. C. Kennedy, J. Phys. Chem. Solids 28, 3 F. A. Gorelli, V. M. Giordano, P. R. Salvi, and R. Bini, 977 (1967). Phys. Rev. Lett. 93, 205503 (2004). 14 D. Tsiklis, L. Linshits, and S. Tsimmerman, Teplofizich- 4 B. Olinger, The Journal of Chemical Physics 77, 6255 eskie Svoistva Veshchestv i Materialov 3, 130 (1971), (in (1982). Russian). 5 R. C. Hanson, J. Phys. Chem. 89, 4499 (1989). 15 V. Shmonov and K. Shmulovich, Doklady Akademiia Nauk 6 K. Aoki, H. Yamawaki, and M. Sakashita, Phys. Rev. B SSSR 217, 935 (1974), (in Russian). 48, 9231 (1993-I). 16 W. J. Nellis, A. C. Mitchell, F. H. Ree, M. Ross, N. C. 7 C. S. Yoo, V. Iota, and H. Cynn, Phys. Rev. Lett. 86, 444 Holmes, R. J. Trainor, and D. J. Erskine, J. Chem. Phys. (2001). 95, 5268 (1991). 8 V. iota, C. S. Yoo, and H. Cynn, Science 283, 1510 (1999). 17 G. L. Schott, High Press. Res. 6, 187 (1991). 9 C. S. Yoo, H. Cynn, F. Gygi, G. Galli, V. Iota, M. Nicol, 18 V. N. Zubarev and G. S. Telegin, Sov. Phys. Dokl. 7, 34 S. Carlson, D. Hausermann, and C. Mailhiot, Phys. Rev. (1962). Lett. 83, 5527 (1999). 19 R. Span and W. Wagner, J. Phys. Chem. Ref. Data 25, 10 V. Iota and C. S. Yoo, Physica Status Solidi (b) 223, 427 1509 (1996). (2001). 20 K. S. Pitzer and S. M. Sterner, J. Chem. Phys. 101, 3111 7
(1994). Figures 21 F. Datchi, R. LeToullec, and P. Loubeyre, J. Appl. Phys. 81, 3333 (1997). 22 H. K. Mao, J. Xu, and P. M. Bell, J. Geophys. Res. 91, • Fig. 1: (Color online) Phase diagram of CO2 as 4673 (1986). presently known up to 20 GPa from the work of 23 F. Datchi and B. Canny, Phys. Rev. B 69, 144106 (2004). 24 Ref. [2], and modified to take into account the F. Datchi, P. Loubeyre, and R. LeToullec, Phys. Rev. B experimental results of Ref. [3]. The dashed line 61, 6535 (2000-II). 25 R. LeToullec, P. Loubeyre, and J. P. Pinceaux, Phys. Rev. is the kinetic barrier between phases III and II. B 40, 2368 (1989). The symbols show the experimental melting points 12 26 A. Dewaele, J. H. Eggert, P. Loubeyre, and R. L. Toullec, available in the literature. ◦: Bridgman ; ×: 13 2 Phys. Rev. B 67, 094112 (2003). Grace and Kennedy ; : Iota and Yoo . 27 J. R. Sandercock, U. S. Patent 4, 225 (1980). 28 F. Simon and G. Glatzel, A. Anorg. Allgem. Chem. 178, 309 (1929). • Fig. 2: (Color online) Experimental melting 29 V. V. Kechin, Phys. Rev. B 65, 052102 (2001). points determined from three separate exper- 30 T. K. H. Shimizu and S. Sasaki, Phys. Rev. B (Rapid iments (•, H, N) along with literature data (◦ Comm.) 47, 11567 (1993-I). 12 13 31 from Bridgman , from Grace and Kennedy J. M. H. L. Sengers, J. Straub, and M. Vicentini-Missoni, and from Iota and Yoo2). The solid line is J. Chem. Phys. 54, 5034 (1971). 32 the fit to our data using the the Simon-Glatzel L. L. Pitaevskaya and A. V. Bilevich, Russian Journal of melting law [Eq. (2) with a=(0.403 ± 0.005) GPa, Physical Chemistry 47, 126 (1973). 33 b=(2.58 ± 0.01)]; the dotted lines are guides to the J. Lamb, Physical Acoustics (Academic Press, W. P. Ma- son, 1965), chap. 4, p. 203. eyes. The inset shows a photograph of the single 34 W. H. Pielemeier, H. Saxton, and D. Telfair, J. Chem. crystal (a) in equilibrium with its melt (b) taken Phys 8, 106 (1939). at 560 K. The pressure gauges are visible: ruby 35 2+ M. C. Henderson and L. Peselnick, The Journal of the (c), cBN (d) and SrB4O7:Sm (e). acoustical society of America 29, 1074 (1957). 36 W. M. Madigosky and T. A. Litovitz, J. Chem. Phys. 34, 489 (1961). • Fig. ??: Raman spectra collected on the solid 37 R. W. Gammon, H. L. Swinney, and H. Z. Cummins, Phys. phase in equilibrium with the melt along the Rev. Lett. 19, 1467 (1967). melting curve at 430 K, 733 K and 800 K. A 38 Q. H. Lao, P. E. Schoen, and B. Chu, J. Chem. Phys. 64, spectrum of phase I at 300 K and 2.1 GPa and one 3547 (1976). 39 of phase IV at 580 K and 15 GPa (dashed line) are A. Belonoshko and S. K. Saxena, Geochimica et Cos- shown for comparison. mochimica Acta 55, 3191 (1991). 40 R. C. Hanson and L. H. Jones, Journal of Chemical Physics 75, 1102 (1981). 41 • Fig. 4: (Color online) Refractive index n of R. Lu and A. M. Hofmeister, Phys. Rev. B 52, 3985 (1995- fluid CO as a function of pressure. Diamonds: II). 2 42 this work; crosses: Ref. [30]; squares: Ref. [31]. V. M. Giordano, F. A. Gorelli, and R. Bini, submitted (2006). Inset: same data plotted as a function of the SWEOS predicted density ρSW . The dashed line is the fit to our data and those of Ref. [31] [Eq. (4)].
• Fig. 5: Brillouin spectrum from the CO2 sample collected at 700 K and 8 GPa. The intensity was scaled to emphasize the two Brillouin peaks B1 and B2 (Stokes and anti-Stokes components). The weak features at ∼ ±0.8 cm−1 are the ”ghosts” of the neighboring interference orders resulting from the tandem arrangement of the Sandercock interferometer27.
• Fig. 6: (Color online) Evolution of the sound velocity as determined by the present Brillouin scattering and refractive index measurements, along isotherms at 300(+), 400(), 500(•), 600(◦) and 700K(H). The error bars on sound velocities include the ones for pressure and are more impor- tant at low pressure due to the larger sensitivity of 8
the sound velocity to variations of pressure. The FIGURES inset shows a zoom of the 0-2 GPa region.
• Fig. 7: Comparison between present sound velocity measurements (circles) at 300 K (a) and 400 K (b) with the available literature data. Squares: Pitaevskaya and Bilevich32; triangles: Shimizu et al30. The sound velocities predicted from the SWEOS are reported as dashed lines.
• Fig. 8: Comparison between the thermodynamic sound velocity obtained by ”correcting” the Bril- louin velocities for the effect of thermal relaxation (◦), with the ultrasonic () and SWEOS (dashed line) velocities at 300 K (a) and 400 K (b).
• Fig. 9: Calculated densities along several isotherms in the range 300-700 K.
• Fig. 10 (Color online) The relative deviation (ρexp − ρeos)/ρexp of our calculated densities with respect to the predictions from SWEOS19 (a), PSEOS,20 (b) and the simulations of Belonoshko and Saxena39 (c) is reported for different isotherms between 300 and 700 K.
Tables
• Tab. I: Values of the parameters aij obtained by fitting Eq. (8) to the calculated densities. 9
1000 800 800 Fluid IV 600 700 T (K) II
400 I III 600 0 5 10 15 20 P (GPa)
T (K) 500 FIG. 1:
400 b a 300 d c e
0 2 4 6 8 10 12 P (GPa)
FIG. 2: 10
1.45 580 K 15 GPa 600 500 400 300 T(K) 800 K 11.1 GPa 1.40 n 733 K 8.8 GPa
1.35
Intensity (a.u.) 430 K 1.9 GPa 1.4 300 K 2.1 GPa 1.3 n 100 200 300 400 500 600 1.30 Refractive index 1.2 ν -1 (cm ) 0.5 1.0 1.5 3 2.0 FIG. 3: ρ (g/cm ) 1.25 SW
0 1 2 3 4 5 P(GPa)
FIG. 4: 11
3 12x10 5
11 B1 B2 4
Intensity (a.u.) 10 3
(Km/s) 3
-0.5 0.0 0.5 S
-1 v σ (cm )
2 (Km/s)
FIG. 5: 2 S v
1
0 1 2 1 P (GPa)
0 1 2 3 4 5 6 7 8 P (GPa)
FIG. 6: 12
2.0 2.0
1.5 1.5 (Km/s) (Km/s)
s (a) s v v 1.0 1.0 (a) 0.5
0.1 0.2 0.3 0.4 0.5 0.6 0.0 0.2 0.4 0.6 0.8 3.0 3
2.5 (b)
2.0 2 (Km/s)
s 1.5 v (Km/s) s v 1.0 (b) 1
0.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 2.5 P (GPa) P (GPa)
FIG. 7: FIG. 8: 13
0 (a) 2.0 -1 300 K 400 K (%) 500 K ) -2
3 600 K 700 K
δρ/ρ -3 -4 (g/cm 1.5 (b) 300 K 1 400 K
density 500 K
(%) 0 600 K 1.0 700 K -1 δρ/ρ -2 1 2 3 4 5 6 7 8 P (GPa) 2 (c) 0 -2 FIG. 9: (%) -4
δρ/ρ -6 -8
1 2 3 4 5 6 7 8 P (GPa)
FIG. 10: 14
TABLES TABLE I:
j i 0 1 2 3 0 0.6521(5) 0.0301(7) −0.0139(8) −0.0150(6) −5 −5 1 −0.000700(2) 0.000520(3) 8.1(3) · 10 4.0(2) · 10 −7 −7 −7 −8 2 2.14(2) · 10 −2.75(2) · 10 −1.28(3) · 10 −2.1(2) · 10