A coherent picture for water at extreme negative pressure

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1 A coherent picture for water at extreme negative pressure Mouna El Mekki Azouzi 1, Claire Ramboz 2, Jean-François Lenain 3 and Frédéric Caupin 1 1 Laboratoire de Physique de la Matière Condensée et Nanostructures, Université Claude Bernard Lyon 1 et CNRS, Institut Universitaire de France, 43 boulevard du 11 novembre 1918, Villeurbanne, France 2 Institut des Sciences de la Terre d Orléans, UMR 6113 CNRS/Universités Orléans-Tours, 1A rue de la Férollerie, Orléans Cedex, France 3 GRESE (Groupe de Recherche Eau-Sol-Environnement), Université de Limoges FST, 123 Avenue A. Thomas, Limoges 87060, France Materials and methods The quartz sample studied was synthesized in a previous study 1 following the thermal crack sealing method 2 using a internally-heated apparatus at 750 MPa and 530 C during 13 days. This fragment of quartz is around 450 µm-thick, 2 mm-wide and 1 mm 3 - volume, contains several thousand inclusions in the µm 3 volume range. For this study, we chose in the sample the water fluid inclusion that has the highest value of negative pressure around 120 MPa. It is located 80 µm from one crystal surface, and it is 20 µm-long and 4 µm-wide tube, with a volume of 570 µm 3 (Fig. 1). The quartz fragment was placed on a heating-cooling stage (Linkam THMS 600) mounted on a microscope (Olympus BHS). Phase changes in the inclusions were observed with a x50 longworking distance objective (Olympus). The temperature cycles of the stage are controlled using 1

2 a computer program. Phase changes are recorded using a black and white camera with a CMOS 2/3 sensor, 1280 x 1024 pixels 2 (Marlin) at a rate of 13 fps. Generation of negative pressure To explain the procedure, we refer to Fig. 1 of the main text, starting with a bi-phasic fluid inclusion, first containing liquid and vapour (a). Upon warming along the liquid vapour-equilibrium (a-b), the bubble disappears at the homogenization temperature T h (b), from which the liquid density is obtained using the known equation of state 3. For the inclusion studied, T h = C and = kg m 3. Then, upon isochoric cooling, the bubble does not reappear and the liquid is stretched (b-c), until cavitation occurs at T cav (c), bringing back the system to equilibrium (a). Cavitation statistics In a preliminary study, an inclusion nucleating around 84 MPa was selected, and statistics were measured by repeating step-like cooling cycles down to several predefined temperatures, as in Ref. 4. An issue with step-like cooling is the finite slope of the step: for large metastability, nucleation may occur during pre-cooling, before the desired temperature is reached. Therefore, in the present study, we have chosen to repeat ramp-like cooling cycles: the temperature decreases linearly with time, at a constant cooling rate r. We have studied a single inclusion at three cooling rates, r = 2, 5 and 10 K min 1, measuring 61, 33, and 60 values of T cav, respectively. For a given r, the N r values of T cav were sorted in ascending order to give a list (T i ) 1 i Nr. The survival probability was deduced as Σ(T i ) = (i 1/2)/N r. 2

3 Nucleation rate The nucleation rate is Γ = Γ 0 exp [ E b /(k B T )], where Γ 0 is a prefactor, and E b is the free energy barrier that has to be overcome. We take Γ 0 = N A 2σ/(πm) 5, 6, where N A is Avogadro s constant, m the mass of one molecule, and σ is the surface tension of water. There is some uncertainty in the value of the prefactor Γ 0, but, because of the exponential in the rate, changes by several orders of magnitude affect the results only marginally. In the present experiment, the rate Γ ranges from to m 3 s 1. Calculation of the free energy barrier to nucleation The survival probability Σ is related to the nucleation rate by a reasoning analogous to that used to describe the collisions and mean free path in a gas. For an inclusion of volume V to survive until time t + dt, it must have survived until time t, and no nucleation must have occurred during the time interval dt: Σ(t + dt) = Σ(t) (1 ΓV dt). (1) This gives a differential equation for Σ. We integrate it from the time t h (or temperature T 0 ) at which the cooling starts, with Σ(t h ) = 1. Using the above expression for Γ, and noting that T = T h r(t t h ), we have: t ln Σ = = 1 r [ E ] b(t ) dt (2) k B T (t ) [ Γ 0 (T )V exp E ] b(t ) dt. (3) k B T t h Γ 0 (t )V exp T0 T Because E b /(k B T ) appears in an exponential, we can safely replace t h and T h by infinity, and Γ 0 (T ) by its value at T = 328 K, a reference temperature in the middle of the experimental range of T cav. The choice of T does not affect the conclusions of the study. 3

4 We then use a second order temperature expansion: which leads to: E b (T ) k B T = E ( ) b(t ) T + ξ k B T T ( ) 2 T 2 κ T 1, (4) ( r ln Σ = Γ 0 (T ) V exp E ) { [ ( )]} b(t ) π T k B T 2κ e ξ2 ξ κ T 2κ 1 erf + 2κ 2 T 1 (5) where erf stands for the error function. A first order expansion (κ = 0) would give a straight line in Fig. 2, inset. As the data exhibit a curvature, the experiment is able to capture a finite κ. The parameters E b (T )/(k B T ), ξ and κ are obtained by taking the collapsed data from the three data sets (Fig. 2, inset), and fitting simultaneously ln( r ln Σ) as a function of T using Eq. 5. The main plot in Fig 2 also shows that these parameters gives Σ in good agreement with each data set taken separately. However, because the experiment involves a finite number of ramps, the parameters obtained from the fit are subject to statistical errors. Statistical errors To evaluate the errors on the parameters, we have simulated the experimental procedure, generating a list of random values of T cav that follows the survival law given by Eq. 5. The 1024 simulated data sets thus obtained were fitted as above. The results are given in Table 1. Classical nucleation theory CNT assumes the cavitation nucleus to be a sphere of radius R filled with vapour and separated from the liquid by an infinitely thin boundary 6. The competition between volume and surface energy results in a free energy barrier: E CNT b = 16π 3 σ 3 (P sat P liq ) 2, (6) 4

5 Table 1: Best fit parameters of the data using Eq. 5. To estimate the statistical errors, 1024 simulations of the experiment have been performed, with the best fit parameters to the experiment as input. They were in turn fitted using Eq. 5 which yield a distributions of parameter values. As these distributions are not always gaussian, in addition to their mean and std. dev., we also give the median and inter-quartile range (IQR); half of the distribution lies within 1 IQR of the median. There is a bias in κ, but it is smaller than the large error bar and will be ignored. The bias in T min, the temperature of the minimum of E b /(k B T ), due to extreme negative fluctuations, is reduced by taking the median instead of the mean, and will also be ignored. The IQRs are taken for error bars in the analysis. Experiment Simulations Parameter Mean Std. dev. Median IQR E b (T )/(k B T ) ξ κ std dev. of fit residuals T min

6 where σ is the liquid-vapour bulk surface tension, P sat the saturated vapour pressure, and P liq the pressure of the metastable liquid. The free energy barrier is reached for a critical radius R c = 2σ/(P sat P liq ). Along the isochore followed in the experiment, using the tabulated σ 7, we find E CNT b > 74 k B T, which yields a rate Γ < m 3 s 1, negligible when compared to the experimental Γ: nucleation should never occur! Tolman length correction Using Eq. 2 of the main text and the equations of CNT, one finds that by including a Tolman length δ, the CNT results are changed as follows: R c = 2σ P sat P liq + 2δ, (7) σ(r c ) = σ + δ (P sat P liq ), (8) E b = 16π ( σ δ (P ) 3 sat P liq ). 3 (P sat P liq ) 2 σ (9) We have used the last equation to fit the experimental data for E b with a constant δ, which gives δ = nm. Shift between T min and T LDM E b /(k B T ) reaches a minimum at T min not only because the isochore reaches a maximum negative pressure at T LDM, but also because the temperature variation of σ 3 /T (Eq. 6). T min is thus the solution of: T 2 P P sat ( ) P = 3 T dσ T σ dt 1 (10) We use the IAPWS EoS to compute the left hand side, and the tabulated surface tension data 7 to compute the right hand side; solving for T min gives T min = T LDM K = K. Interestingly, the shift T min T LDM remains identical if one multiplies sigma by a constant factor, because dσ/σ 6

7 is involved in Eq. 10. In the experiment, the nucleation statistics yield T min = ± 4.3 K. This value, which does not involve the use of any EoS, is consistent with that derived from CNT using the isochore from the IAPWS EoS (with or without the Tolman length correction). Nucleation theorem The results for E b /(k B T ) can also be analysed using the nucleation theorem 8. Without assuming any specific microscopic model for nucleation, it relates the properties of the critical nucleus to the variation of the free energy barrier. For an isothermal experiment, this theorem yields directly the excess number n c of molecules in the critical nucleus: ( ) Eb µ liq T = n c, (11) where µ liq is the chemical potential of the metastable liquid. For cavitation, the critical bubble involves a region where less molecules are present, and n c is negative. The version of the nucleation theorem for an experiment following an isochoric path is more complex, involving also the excess entropy S c of the critical nucleus: ( ) Eb µ liq ( ) T = n c S c µ liq. (12) Introducing v liq and s liq the volume and entropy per particle in the liquid, respectively, we write n c = V c /v liq and S c = n c s liq + S prof, the sum of the entropy loss due to the missing molecules, and the entropy gain due to the inhomogeneous profile of the nucleus. The temperature derivative of E b is the quantity directly accessible to the experiment (Eq. 4). Using the Gibbs- Duhem relation, Eq. 12 can be transformed into: ( ) ( ) Eb P = V c S prof. (13) T T 7

8 This gives a lower bound on V c : V c ( ) T P ( ) Eb T. (14) CNT can be used as a guide to get a reasonable estimate for V c. Within CNT, as vap liq, we have the following relations: V c = 4 3 πr c 3 < 0 and S prof = 4πR c 2 dσ dt > 0, (15) and the ratio of the first to second terms in Eq. 13 using the extrapolation of the EoS measured at positive pressure 3 is roughly 2. Assuming the same ratio for the experiment, V c is around twice the value of the lower bound. 1. Shmulovich, K. I., Mercury, L., Thiéry, R., Ramboz, C. & El Mekki, M. Experimental superheating of water and aqueous solutions. Geochimica et Cosmochimica Acta 73, (2009). 2. Bodnar, R. & Sterner, S. Synthetic fluid inclusions in natural quartz. II. Application to PVT studies. Geochimica et Cosmochimica Acta 49, (1985). 3. IAPWS. Revised release on the IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use (International Association for the Properties of Water and Steam, 2009); available via Rev.pdf. 8

9 4. Takahashi, M., Izawa, E., Etou, J. & Ohtani, T. Kinetic characteristic of bubble nucleation in superheated water using fluid inclusions. Journal of the Physical Society of Japan 71, (2002). 5. Blander, M. & Katz, J. Bubble nucleation in liquids. AIChE Journal 21, (1975). 6. Debenedetti, P. G. Metastable liquids (Princeton University Press, 1996). 7. IAPWS. Release on the surface tension of ordinary water substance (International Association for the Properties of Water and Steam, 1994); available via 8. Oxtoby, D. W. & Kashchiev, D. A general relation between the nucleation work and the size of the nucleus in multicomponent nucleation. The Journal of Chemical Physics 100, (1994). 9

Supplemental Material for Curvature-dependence of the liquid-vapor surface tension beyond the Tolman approximation

Supplemental Material for Curvature-dependence of the liquid-vapor surface tension beyond the Tolman approximation Nicolas Bruot and Frédéric Caupin Institut Lumière Matière, UMR536 Université Claude Bernard