3062
J. Phys. Chem. 1987, 91, 3062-3068
The change in s* vs. pR behavior is interesting since available data from solubility and chromatographic retention studies do not indicate any similar density dependence.20.21 Thus, while s* measurements appear consistent with relative fluid-phase solubilities, and effectively probe certain specific solute-solvent interactions, a quantitative relationship between s* values and supercritical fluid solubilities is not evident. The origin of these differences may be the fact that the s*value most strongly reflects the immediate solute environment rather than the effects of a larger solvation sphere. Data for the subcritical liquids of NH3,C o 2 , CzH6, and SF6 obtained at room temperature are also given in Table I. The x* values determined from the liquefied gas spectrum of C2H6and SF6for 2-nitroanisole correspond to the s* shown in Figure 3 for the supercritical fluid at similar values of reduced density. Therefore, the solvent power of a subcritical liquefied gas and a supercritical fluid are approximately equal at like densities. The present data also show that both supercritical SF6and Xe are solvents with properties comparable to other nonpolar solvents (e.g., ethane) with a very low effective polarity. Xe, which has
somewhat larger s* values than SF6, should prove to be a useful supercritical solvent for three reasoris. First, it has convenient critical properties (T,= 289.8 K, P, = 58.0 atm). Second, since it is a monatomic gas, it does not have specific solvent-solute interactions. Finally, for practical applications requiring detection or analysis, it provides an optimum spectroscopic window. The ability of liquid Xe to solubilize polymers22 and large hydroc a r b o n ~has ~ ~been attributed to its polarizability; however, the present s* measurements lead to the prediction that its solvent powers should be somewhat inferior to other supercritical fluids of common interest (e.g., C02). The relative s* values of Xe and sF6are surprising, however, and suggest the importance of specific interactions in these systems. Future studies will be aimed at understanding the specific solventsolute interactions that occur in supercritical solvents and the dependence of these interactions on the temperature and density.
Acknowledgment. The authors acknowledge the support of the
US.Department of Energy, Office of Basic Energy Science, under Contract DE-AC06-76RLO- 1830. (22) Everett, D. H.; Stagemen, J. F. J. Chem. SOC.,Faraday Trans. 2
(20) Yonker, C. R.; Gale, R. W.; Smith, R. D. J. Chromatogr. 1986,371, 83. (21) Gitterman, M.; Procaccio, I. J . Chem. Phys. 1983, 78, 2648.
1978, 230.
(23) Rentzepis, P. M.; Douglas, D. C. Nature (London) 1981, 293, 165.
Temperature of Maximum Density in Water at Negative Pressure Stephen J. Hendersont and Robin J. Speedy* Chemistry Department, Victoria University of Wellington, Wellington, New Zealand (Received: September 17, 1986)
The locus of the temperature of maximum density in stretched water is reported to pressures below -200 bar for H20, D,O, and an HDO mixture. The water samples were stretched in a fine helical capillary by the Berthelot tube principle. Pressure in the sample was measured by monitoring the unwinding of the helix, using the Bourdon tube principle.
Introduction A liquid which is under tension, or stretched, is metastable with respect to the liquid plus vapor system. Nucleation of the vapor phase results in the sudden loss of tension. Nucleation and growth of a vapor bubble is called cavitation and is accompanied by an audible click. The maximum negative pressure that a liquid can sustain without cavitating is its tensile strength. Experimental studies of water over the past three centuries' have yielded tensile strengths which vary by orders of magnitude, from about 1 to 277 bar. This range reflects both the difficulty of attaining tensions in liquids and of measuring them. Theoretical estimates of the tensile strength of water are considerably higher, varying from 2702to 6000 bar3 at 10 O C . Several reviews have been written on stretched Nucleation can occur homogeneously from spontaneous fluctuations in the local liquid structure which produces a viable bubble. However, the lack of reproducibility of nucleating tensions between different workers indicates that cavitation probably occurs at heterogeneous nuclei such as the solid surfaces in contact with the sample. The critical nucleating factors(s), though, have not been isolated. Indications are that cleanliness of both the water and solid surfaces in contact with it are very important. The importance of working with small samples and scrupulously clean conditions, which is well recognized in studies of superheated and Current address: Research School of Chemistry, Australian National University, Canberra, A.C.T., Australia.
0022-3654/87/2091-3062$01.50/0
supercooled liquids, is often overlooked in work on stretched liquids and probably accounts for the wide range of reported nucleating tensions. Most previous work on stretched liquids has focussed on the problem of measuring the maximum tension that the sample will sustain before cavitation. Because of the difficulty of maintaining samples under tension, there have been few reports of the physical properties of stretched liquids. Water is the most important and most intensively studied liquid. The recent progress* which has been made toward mapping out its properties in the superheated and supercooled regions and the questions raised by its profoundly anomalous character highlight the need for information about its behavior at negative pressure. For example, water is anomalous in that it expands when cooled below 4 OC at atmospheric pressure. The temperature of maximum density (TMD) is the locus where the thermal expansivity of water is zero, Le., a = V ' ( I ~ V / I ~=T0.) , The TMD is a unique (1) Kell, G. S. Am. J . Phys. 1983, 51(11), 1038. (2) Kwak, H.; Panton, R. J . Phys. D . 1985, 18, 647. (3) Benson, S.; Gerjoy, E. J . Chem. Phys. 1949, 17, 914. (4) Hayward, A. T. J. Am. Sci. 1971, 59, 434. (5) Apfel, R.E. Sci. Am. 1972, 227(6), 58. (6) Trevena, D. H. Contemp. Phys. 1976, 17(2), 109. (7) Henderson, S. J.; Speedy, R. J., following paper in this issue.
(8) Angell, C. A. "Superheated and Supercooled Water" in Wuter and Steam: Their Properties and Current Industrial Applications: Pergamon: New York. 1980.
0 1987 American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 11, 1987 3063
Water at Negative Pressure 2500
500
2000 1500 1000
250
& n
500
1
g
o
(0
il
&
-500 0
-1000
-1500
-2000 -2500 -150-100 -50
-250
0
50 100 150 200 250 300 350
-25
0
Figure 1. The conjectured limits of stability'O for liquid water (H,O). T,,, = melting line; Tb = boiling line; pa= limit of stability; a = thermal expansivity; CP = critical point.
temperature on an isobar (for pressures below 2 kbar) where water exhibits a maximum density. It is depressed and the density maximum becomes shallower with increasing p r e s s ~ r e so , ~ it is to be expected that the T M D shifts to higher temperature and becomes sharper at negative pressures. Likewise, the viscosity of water is expected to increase as the liquid is stretched, in contrast to other liquids in which viscosity increases with increasing pressure. Any analytic equation of state which describes both the liquid and the gas states of matter must, like van der Waals equation, display minima and maxima on the p-V isotherms below the critical temperature. The locus of the minima, where (dp/dV), = 0 and the isothermal compressibility KT = Vl(aV/dp), diverges, is called the spinodal line (p,) or limit of stability for the liquid. Beyond this line, K , would be negative. In the positive pressure region, ps defines a limit of superheating, and at negative pressure ps defines the ultimate tensile strength. Figure 1 shows a previously estimates of the locus of ps for water. The initial motivation for the present work was to test some of the implications of figure 1. For example, thermodynamic consistency requires that the sign of the expansivity is the same as the sign of the slope dp,/dT of the line p,, so that the T M D locus must intercept p s where dp,/dT changes sign.1° Thermodynamic consistency also requires that the expansivity diverge at p,, from which it follows that a changes from +m to --co where the lines a = 0 and ps intersect. In practice, cavitation will always intervene to prevent the stability limit from being reached so these extreme predictions are not directly observable. Nonetheless, we aimed to follow the T M D line as far as possible into the negative pressure region if only to give some experimental respectability to the previously speculative discussion of the properties of stretched water. Relevant areas of the equilibrium phase diagram in Figure 1 emphasize the extended and largely unresearched regions of p-T space where p < 0. The maximum tension observed12 in pure water is only 277 bar at 10 OC. Experimental p V T datal3 exist (9) Angell, C. A.; Kanno, H. Science (Washington, D.C.) 1976, 193, 1121. (IO) Speedy, R. J. J . Phys. Chem. 1982, 86, 982. ( 1 1) Speedy, R. J. J . Phys. Chem. 1982, 86, 3002. (12) Briggs, L.J. J . Appl. Phys. 1950, 21, 721. (13) Meyer, J. Abh. Dtsch. Bunsen. Ges. 1911, 6, 1.
25
75
50
Temperature /
Temperature / OC
100
OC
Figure 2. p T behavior of a Berthelot tube containing water.
to only -34 bar and no other properties have been measured. Our work extends the range of p V T measurement below -200 bar, and while we have not exceeded Briggs world record tension of 277 bar, our samples sustained much greater tensions than his at lower temperatures. In a subsequent paper: we report the locus of the melting point of ice to -200 bar for both H 2 0 and D20.
Experimental Section 1. Introduction. BerthelotI4 devised a method for generating negative pressures in liquids, which has been widely used and is now described. A capillary tube that was sealed at one end and drawn to a fine point at the other was filled with water. The tube was cooled, drawing a small volume of air into the drawn out point, which was then sealed in a flame. Consider Figure 2 for the explanation of the subsequent p-T behavior of the capillary tube. The tube was heated, causing the water to expand to fill the whole B). A further temtube and to dissolve any trapped air (A perature increase made the pressure in the water rise sharply (B C). Upon cooling the tube (C B D), the water continued to occupy the whole volume of the tube a t a temperature lower than that at which the water had first filled the tube (B), so that the water was then under tension or negative pressure a t D. Continued cooling (D E) caused the tension to increase until F) as the bubble the water column ruptured or cavitated (E reappeared somewhere in the tube. The negative pressure at E is the experimental tensile strength of the sample at the temperature a t E. If cavitation does not occur at E, the tension increases as the temperature is reduced to a maximum at G after which the thermal expansivity becomes negative, so the tension decreases to some point H. At H , water is metastable with respect to both boiling (cavitation) and freezing; i.e., the sample is simultaneously superheated and supercooled. Penetration into the double metastable region for H 2 0 was first reported by Hunt and Jackson15 and later by Hayward4 who applied a pressure of -0.2 bar to H 2 0 at a temperature of -5 "C in a manometer. A much more extensive penetration is reported for both H20and D 2 0in this paper and ref 16. No other researchers have reported data in this area. Ice is always nucleated before the vapor phase in these experi-
-
--
-
-
-
Berthelot, M . Ann. Chim. (Phys.) 1850, 30, 232. Hunt, J. D.; Jackson, K. A. J . Appl. Phys. 1966, 37, 254. (16) Henderson, S . J.; Speedy, R. J. J. Phys. E. 1980, 13, 778. (14) (15)
3064 The Journal of Physical Chemistry, Vol. 91, No. 11, 1987
Q-
mirror
+helix
Figure 3. Apparatus used by MeyerI3to stretch water and other liquids.
ments. If the water contains any dissolved gas, then freezing will force the gas out of solution. The appearance of the first bubble will nucleate boiling so the tension will be lost a moment from when freezing begins. The magnitude of the tension a t G is determined mainly by the temperature at B which is set by the size of vapor bubble trapped in the tube at the time of sealing. The other factor affecting the tension at G is the capillary wall thickness. The tension will increase with the wall thickness because thinner tubes deform more under tension. Berthelot estimated that a tension of about 50 bar for water was achieved in his experiments. The main disadvantage of Berthelot’s original method is the problem of measuring pressure, which involved an estimation from the thermal properties of glass and the enclosed liquid.” The Berthelot technique has since undergone several modifications in the way that negative pressure is estimated.16 The method used in this work will now be described. The straight tube of Berthelot was replaced by a helical capillary supporting a small mirror underneath (see Figure 4). When the pressure inside the capillary helix changed, the helix responded by making the mirror rotate slightly. This is an example of a Bourdon tqbe which is widely used for pressure measurement. The angular position of the mirror was measured as a displacement by observing a laser beam reflected from the mirror onto a screen, curved along the circumference of an arc with the mirror at its centre. The calibration response was measured by connecting the capillary to a high-pressure line incoporating a pressure gauge. After disconnecting the capillary from the line, the helix was sealed off at the upper end, trapping a vapor bubble with the water. Cycling of temperature generated tension in the water as in the Berthelot tube method. The first use of a similar BerthelotBourdon technique was made by Meyer” (seeFigure 3), although the current work16 originated independently. Meyer used a much larger sample volume which limited his penetration into the metastable regions of water to -34 bar for water. 2. Preparation ofrhe Water Samples. Double-distilled water was redistilled from a Pyrex distillation apparatus. Demineralized water normally contains traces of organic impurities18 which are either partly volatile or may be cracked into volatile products by (17) Lewis, G. M. Proc. Phys. Soc. 1961, 78, 133. (18) Petrick, H., et al. Mikrochim. Acra 1981, ZZ, 277.
Henderson and Speedy the combined action of water and heat during distillation. The redistilled water was then filtered through a 0.45-pm Sartorious membrane filter (cellulose nitrate) and further distilled from an alkaline permanganate solution (0.8 g of KMnO, and 5 g of NaOH in 1 L of water). The alkaline permanganate was present to oxidize organic impurities in the water,I9 which are probably ubiquitous. Pyrocatalytic distillationz0is more effective than permanganate for the removal of organics, but the cost of such a method was difficult to justify for this work as subsequent improvement in water tensions achievable was uncertain. The water was distilled again into a collecting flask. Finally, the water passed through a 0.1-pm Sartorious membrane filter before reaching the external environment. These filters are not effective in removing nonrigid organic impurities, which may shear through them.2’ Distillation systems are usually vented to the external laboratory air. A direct vent was undesirable for this work due to the resultant contamination from airborne particles, so a clean air supply was devised for the still to create an internal atmosphere compatible with the water purity. Industrial compressed dry nitrogen was passsed over a column of activated charcoal (18 mesh) and then passed through a 0.3-pm Matheson gas filter before entering and passing through the still. The still was constructed entirely of Pyrex glass and Teflon (polytetrafluoroethylene),except for the filters. The use of plastics other than Teflon is usually a v ~ i d e d ’ to ~ ~prevent ’~ contamination from organics. Quartz is preferred as a construction material to Pyrex, since Pyrex is a source of ionic contamination, but the tensile strength of water is probably unaffected by trace ionic content so the constructional difficulty and cost of quartz was not justified. Storage of ultrapure water is generally recognized as difficult,21 so it was used soon after preparation. Charcoal was not used directly in contact with the water for the removal of organic impurities. If charcoal is used for this purpose, an elaborate procedure is required for the purification of the charcoal itself, due to the presence of contaminants in most samples of active charcoal.z0 Samples of heavy water (DzO) used were of 99.75% purity manufactured by Merck and were used without further purification. Samples H D O (6.69 mol % D 2 0 in HzO) were made by combination of H20and DzO prepared as stated above. The H D O samples were prepared for Raman spectroscopy studies.z2 3. Constructionand Filling of the Helices. The apparatus used for construction of the Pyrex helices has been reported by Y ~ r k e . ~ ~ Pyrex tubing of either 6 mm 0.d. (wall thickness 1 mm) or 4 mm 0.d. (wall thickness 0.8 mm) was drawn out to form a capillary of 0.05-0.20 mm 0.d. The capillary dimensions were not constant over the length used. About 11 turns were wound on to a cylindrical ceramic (pyrophyllite) former which had a slight taper to facilitate the removal of the glass helix. A small electric furnace surrounded the capillary to soften the glass as it was wound onto the former. Earlier formers used were 8 and 19 mm in diameter, but the majority of helices were made on a 4-mm former. The smaller former enabled a smaller volume of water to be used as both the overall length of the helix was reduced and a smaller 0.d. capillary could be used. The cross section of the capillary remained circular as it was wound around the former. The capillary was etched by total immersion in dilute HF (5-12%) berore filling with water. The HF was simultaneously drawn through the capillary. The etching time was altered to give the correct calibration response for the helix. Etching in H F produced the following results: (a) it altered the physical dimensions of the capillary in order that an appropriate torsional movement occurred when the internal pressure changed; (b) it (19) Bangham, A. D.; Hill, M. W. Nature (London) 1972, 237, 408. (20) Conway, B. E., et al. Anal. Chem. 1973, 45(8), 1331. (21) Kaye, W. J . Colloid Interface Sci. 1974, 46(3), 543. (22) Green, J. L.; Lacey, A. R.; Sceats, M. G.; Henderson, S. J.; Speedy, R. J., to be published. (23) Yorke, S . G . J. Sei. Imtrum. 1945, 22, 196. Yorke, S. G . J . Sei. Instrum. 1948, 25, 16.
The Journal of Physical Chemistry, Vol. 91,No. 11, I987 3065
Water a t Negative Pressure
I
c
SS huh press= tubing
citing epoxy resin
Mite capillary
epoxy resin
litRT
innerwkdow mirrur
e p x y resin
Figure 4. Scale diagram of capillary helix mounted in cell for calibration.
thoroughly cleaned the glass surface; (c) and it enhanced the strength of the capillary. It is well-known that etched glass capillaries can withstand high internal pressure. Capillaries of 0.1-0.5 mm 0.d. drawn from tubing of 9 mm 0.d. and wall thickness 3 mm could withstand 2.9 kbar internal pressure24 after etching in 5% HF. A further increase in pressure resistance of the glass is made2s with the use of 52% H F rather than 5% HF. The capillaries in this work were used at pressures up to 2 kbar. Abrasion cracks and surface inclusions (particles of foreign matter which become bonded to the surface while hot) provide the two serious strength-impairing flaws on the surface of glass, but both types of flaws are removed with HF. After etching, care must be taken to avoid contacting the capillary with other hard solid surfaces otherwise fracture may result when the capillary is stressed. After etching, the water sample to be used was sucked through the capillary for several minutes to flush out the HF. Suction was removed and the lower end of the capillary was sealed with a microtorch. A short length of solid Pyrex rod with the mirror attached was then fused on to the lower seal. The capillary was mounted in a metal holder as shown in Figure 4. A 2-cm-long Pyrex collar (5 mm o.d., mm i.d.) was fastened over the capillary above the helix to allow the capillary to be firmly clamped in a block in the metal frame. The tail passed through a guide hole in a thin Teflon disk. High sensitivity of the system is of prime importance and critically depends on this pivot. Some “sticking” in the mirror’s position was noted for the helices of smaller capillary diameter where tapping the system achieved consistent mirror positions, with some loss of precision. A precision pressure gauge, consisting of a quartz Bourdon tube (a helix of 16 turns) with an optical readout, has been developed by Texas Instruments Inc. as a commercial unit,26 having an (24) Yamada, H. Rev. Sci. Instrum. 1974, 45(5), 640. (25) Williams, R. K. Reu. Sci. Instrum. 1978, 49(5), 591. (26) Damrel, J. B. Instrum. Control Syst. 1963, 36, 87.
accuracy comparable to that of a manometer or dead weight tester but faster to operate. 4. Pressure Calibration. The capillary helix was placed in the cell shown in Figure 4. The upper end of the helix was glued with epoxy resin into pressure tubing for calibration against a 500-bar Heise gauge. The calibration response was independent of temperature, so only one calibration was needed per helix. The repeatability of the calibration response of the capillary helix arises because it is made of glass. At temperatures well below their . softening points (820 O C for Pyrex) and for low strain, glasses obey Hooke’s law. There is no permanent (plastic) deformation of a macroscopic sample upto the stress at which fracture occurs.27 Calibration data were represented by an empirical equation between applied pressure P (i.e., the pressure difference between the inside and outside of the capillary helix) and the laser beam position on the screen, x x =
aP + bP2
(1)
where a and b are constants. The applied pressure P is related to the absolute pressure by p = (P 1 atm). The helices usually unwound with increase in pressure, but some displayed the reverse or nil behavior. When coefficient a was small, b was usually zero. The nonlinear term in eq 2 typically contributed 5% of the x value for the maximum P. The diversity of the calibration responses was due to the nonuniformity of the capillary: the inner bore was not always coaxial with the outer surface of the capillary and the 0.d. of the capillary varied along its length. An important characteristic of the calibration response equation is its accuracy when it is extrapolated to negative pressure. This was difficult to establish, though was investigated in two ways: (1) Comparison of predicted pressures a t the origin from reduced calibration data sets (e.g., 200-500 bar values only) were made with the known value of P = 0 using
+
x =k
+ aP + bP2
(2)
The constant term k is introduced into eq 2 so that the equation is not constrained to go through the origin by its functional form. The deviation of k from zero evaluated by using data from +200 to +500 bar is considered comparable to the error from an extrapolation to -200 bar using data from positive pressure via eq 1; the error at -200 bar is thus estimated at less than 5%. (2) Comparison of TMD’s evaluated from this work was made with those extrapolated from equations of state. The agreement is reasonable as shown in figures 8, 9, and 10. After the calibration response was recorded, the capillary was broken beneath the epoxy resin connection to the pressure line. The open end of the capillary was sealed with a microtorch. Much care was needed to get the vapor bubble size required at this seal. For several helices, two different successful seals were achieved, thus obtaining in each case two distinct sets of experimental data from the same helix. 5. Temperature Effects. The temperature was increased until the water pressure was about 400 bar. This pressure was maintained for about an hour before the temperature was lowered and p-T data were recorded at intervals of 10 OC. Temperature was measured with a platinum resistance thermometer and pressure was measured from the laser beam position, so points in p-T space were produced in the positive and negative pressure regions. Temperature accuracy was limited to +0.1 OC by thermal gradients in the cell. Temperature measurements of gaseous systems are among the most difficult because gases have low thermal conductivities and low heat capacities. Typically, the problems of sufficient heat capacity, thermometer immersion depth, and thermal lag are 20 times worse in air than in water.28 There were two temperature-dependent effects superposed on the variation of water pressure with temperature that caused the mirror under the helix to rotate: (27) Hollaway, D. G. The Physical Properties of Glass; London, 1973. (28). Nicholas, J. V.; White, D. R. Traceable Temperatures; Science Information Division, D.S.I.R.: Wellington, 1982.
Henderson and Speedy
3066 The Journal of Physical Chemistry, Vol. 91, No. 11, 1987 500
* L
200 13
'
150
$!
3
ln
??
n
-250 -25
0
25
50
75
(a) The cell sometimes moved slightly relative to the screen. The resultant correction to x was small and was measured by recording the laser beam position on the screen that was reflected from the outer window of the cell. (b) The mirror under the helix also rotated as a function of temperature at constant pressure. This movement, called background data, was sometimes a major correction, though its magnitude varied considerably between helices. It was measured by recording the mirror position as a function of temperature while a vapor bubble was present in the capillary. A vapor bubble could always be forced into a capillary containing stretched water by chilling the top of the capillary with liquid nitrogen to induce cavitation. This method gave data only for temperatures below the temperature at point B in Figure 2, so background data at higher temperatures were obtained by linear extrapolation. The background data represented a reference line of x values where the pressure is equal to the vapor pressure of the water, Le., p = 0. The source of this background movement was unknown. Background data and normal p-T data were recorded alternately until two successive sets of background data were either equivalent or sufficiently close to establish the reliability of the p T data in between. 6. Volume Calibration. The volume of the water sample in the helical capillary, along with the helix dimensions, varies with T, p , and the angle of unwinding, 6. If the functional form of that variation was known, then the volume of the helix could be calibrated by using the positive pressure measurements where the volume of water is known. This calibration could be extrapolated to yield the volume of the sample in the negative pressure region of the p T curves of Figures 5 , 6 , and 7. However, the variation of the helix volume as a function of T, p , and 6 is a complex problem in mechanics which is unsolved even for an idealized helix. We attempted to represent the helix volume by the first few terms of a Taylor's series about the p = 1 bar intercept (e.g., eq 3) and to determine the constants of the equation by fitting the experimental points at positive pressure. The negative pressure volumes which resulted from that procedure showed variations of 0.05% to 0.4% (due in part to a very high correlation between the terms which were retained in the Taylor series) from those estimated by extrapolation of two equations of (29) Chen, C-T.; Fine, R. A,; Millero, F. J. J . Chem. Phys. 1977, 66(5), 2142. (30) Fine, R. A.; Millero,
F. J. J . Chem. Phys. 1975, 63(1), 89.
50
-250 ' ' -25
100
Temperature / OC Figure 5. Experimental p-T curves for H20.
100
'
'
" 0
"
'
'
'
25
"
'
'
I
'
'
'
50
'
I
75
"
"
'
'
100
Temperature / OC
Figure 6. Experimental p - T curves for D20.
1
I
-250 -25
0
500 450 400 350 300 250
3
200
p!
150
' 3
z
100 50 0 -50
-100 -150 -200 25
50
75
100
Temperature I OC
Figure 7. Experimentalp - T curves for HDO.
When the relatively small pressure range involved (0 to -250 bar) is considered, volumes calculated by extrapolation of the equations of ~ t a t e ~are ~ , likely ' ~ to be reliable to better than 0.1% and therefore to be more accurate than estimates of negative pressure volumes based on our own results. For that reason, we do not report estimates of the volumes. 7 . Locating the TMD from the Experimental Curve. H 2 0 , D20, or an H 2 0 / D 2 0 mixture each have a unique line in p-T space where (dV/dT), = 0. The intersection of these lines with the experimental p T curves of corresponding samples can be found. The T M D on an experimental p T curve is located at a lower temperature than that at the p-T curve minimum due to the small thermal expansivity of the glass capillary, so it is necessary to estimate correction terms from the turning point coordinates. Twelve experimental p T curves for H20, D20, and H D O which traversed the minimum are shown in Figures 5, 6,
The Journal ofphysical Chemistry, Vol. 91, No. 11, 1987 3067
Water at Negative Pressure TABLE I: Estimation of the Values of K; at the Turning Points of the p - T Curves in Eq 4 for H20(Hn),D,O (Do),and HDO (HDo) data set
T, “C
H1
69.81 59.80 45.02
H2
79.76 69.88 59.87 50.58
H3
49.83 41.88
H4
69.83 59.87 49.40
D1
69.74 59.83 47.53
D2
79.76 69.75 55.89
D3
49.84 24.27
D4
79.79 69.84 55.38
D5
79.83 69.83 59.80 52.70
HDl
80.31 69.88 59.88 44.9 1
HD2
69.86 59.83 56.28
HD3
79.83 59.75 53.1 1
p,
dp/dT, bar/K
bar
106~,, 106a,, bar-l K-’
268.2 151.4 1.o (-18 1.6) 299.2 189.2 86.5 1.o (-203.3) 65.3 1 .o (-138.8) 201.5 114.6 1.o (-129.8)
12.10 11.17 9.03
41.91 42.71 44.15
563.41 515.1 422.6
11.50 10.74 9.75 8.61
42.30 42.82 43.46 44.18
610.3 568.9 518.2 461.6
8.66 7.50
43.42 44.20
455.7 399.5
9.22 8.22 6.41
42.68 43.13 44.16
567.8 517.0 418.1
209.4 108.9 1.o (-140.9) 215.0 119.4 1.o (-184.7) 143.5 1.o (-17.7) 236.4 132.9 1.o (-1 86.7) 304.5 180.6 69.8 1.o (-1 87.9)
10.60 9.62 7.83
43.07 43.77 44.87
554.8 497.9 409.0
9.87 9.17 7.84
43.76 44.16 44.95
610.7 560.7 473.6
7.75 3.03
43.21 46.59
427.7 183.1
10.83 9.93 8.24
43.50 44.00 44.93
609.0 560.3 469.8
13.02 11.76 10.28 9.09
42.68 43.42 44.23 44.88
603.4 557.1 499.1 449.7
10.06 9.45 8.66 7.03
42.23 42.62 43.12 44.21
611.4 566.7 515.6 419.5
9.78 8.91 8.54
43.61 44.15 44.38
572.5 519.2 497.8
11.47 9.52 8.67
42.64 43.79 44.28
612.4 517.3 476.9
311.7 209.8 119.1 1.o (-144.5) 125.9 32.0 1.o (-228.9) 273.7 61.4 1.o (-210.0)
1 0 6 ’, ~
T P
bar’l
data set
. H2
3.83 2.54 1.57 (-0.0) 9.90 9.25 8.68 8.25 (7.10) 8.05 7.79 (7.23) 17.84 18.60 19.50 (20.59)
data set
p, bar
bar-’
bail
dp/dT, bar/K T, OC p, bar
HI H2 H3 H4
T, ‘C 7.99 8.29 7.58 7.09
-181.6 -203.3 -138.8 -129.8
50.77 50.98 50.30 50.32
0.00 7.10 7.23 20.59
-0.195 -0.170 -0.172 -0.140
7.36 7.69 6.95 6.47
-181.5 -203.2 -138.7 -129.7
D1 D2 D3 D4 D5
14.35 14.50 12.26 14.77 14.72
-140.9 -184.7 -17.7 -186.7 -187.9
50.38 50.92 49.37 50.87 50.90
4.47 11.70 10.68 10.14 3.84
-0.180 -0.158 -0.165 -0.162 -0.181
13.75 13.94 11.67 14.19 14.14
-140.8 -184.6 -17.6 -186.6 -187.8
7.69 9.33 9.12
-144.5 -228.9 -210.0
50.49 51.19 50.98
12.42 10.54 9.25
-0.157 -0.160 -0.164
HD1 HD2 HD3
106~,, lo%,’,
7.08 -144.4 8.73 -228.8 8.50 -209.9
’TP = turning point. b T M D = temperature of maximum density.
17.54 16.32 15.30 14.09 (12.42) 13.92 13.01 12.74 (10.54) 9.91 9.51 9.60 (9.25)
expansivity of glass (Pyrex) = 9.9 X 10” K-’(25-400 OC, manufacturer); K, is the isothermal compressibility of ~ a t e r ; * ~ , ~ O and ~ gisl the “effective” isothermal compressibility of the internal helix volume. The compressibility of Pyrex is 2.75 X bar-’. The value of K; is expected to be larger because it incorporates the additional volume change from the AO, which is observed (Table I) except for data set H1. This apparent exception probably arises from cumulative error in the evaluation of K;. The point (po,T0)is chosen as the intersection of the p-T curve with the T M D curve so eq 4 becomes
vg= v, awAT - K,AP = agAT + ~glAp
+ Kgl)
P, bar -203.3 -203.4 -203.3 -203.5 -186.7 -186.6 -186.7 -186.7 -228.9 -228.9 -229.0 -229.1
TABLE III: Estimation of the Values of TMD near the Turning Points of the p - T Curves from Eq 5 TP TMD~
8.35 6.99 6.10 (4.47) 17.11 15.88 14.21 (11.70) 10.70 10.68 (1 0.68) 11.81 11.43 10.88 (IO.14) 2.90 3.14 3.34 3.48 (3.84)
(where AT = ( T - To), Ap = (p - po)) and because the internal volume of the glass capillary (V,) is the same as the volume of the water (V,)
(dP/dT) = (a, - a g ) / ( K w
T, OC 8.29 8.39 8.40 8.17 14.77 14.72 14.69 14.69 9.33 9.34 9.31 9.28
‘TP = turning point.
(derivatives evaluated at (po,To)) ( 3 )
+ KB’)
4 5 6 7 4 5 6 7 4 5 6 7
HD2
V(p,T) = V(po,To)+ (aV/aT)pAT + ( ~ V / ~ P ) T+A P
( & / A T ) = (a, - a g ) / ( K ,
polynom order
D4
and 7. Error bars for the points are smaller than the point size shown. Despite problems in estimating the volumes, the TMD’s can be located precisely and provide a good test of any equation of state. Consider a Taylor’s expansion about the point (po,To)on a p - T curve characteristic of some water sample. The volume of the capillary (or water) is given by higher terms
TABLE 11: Variation in Some p - T Curve Minima due to the Choice of the Interpolating Function
(4)
where a, is the thermal expansivity of ~ a t e r ; * ~ag , ~ is O the thermal
(dp/dT) = - ( Y ~ / ( K ,
+ K:)
(as a, =O)
(5)
Both K, and ~ gmust l be estimated at the associated T M D coordinate for each p - T curve. Equations of state29v30are used to estimate K, values; this involves the use of the equations outside their valid pressure ranges. The extrapolation error is probably small as the equations of state are well behaved in this region and K, is only a very slowly varying function along the p-T curves (see Table I). The values of K, for H D O are calculated from both the equations of state by adding contributions from each weighted according their mole fraction in the sample. The values of K; are calculated by using eq 4 at several coordinates at positive pressure where (dp/dT), a,, ag,and K, are known, and these are linearly extrapolated against pressure to the negative pressure at the turning point. The calculated and extrapolated values of K; are shown in Table I. Their variation with pressure is usually small. Substitution of K, and IC; into eq 6 gives the value of (dp/dT) at the coordinates of the TMD for the particular p-T curve. The T M D is then found from the first derivative of the fourth-order polynomial equation representative of the experimental curve (see Table 111). The TMD results are plotted in Figures 8, 9, and 10. The minimum of the p - T curve is shallow and consequently its position is sensitive to error in the data points. The main error
3068 The Journal of Physical Chemistry, Vol. 91, No. 11, 1987
b
n
200
200
160
160
100
100
60
60
o
b
D
\
f
E
-60
??
-100
n -100
-160
-160
-200
-200
-260
-250
-3OOt' 0 1
'
'
2
'
3
I
4
'
'
6
'
8
'
7
-300
I
8
9
Temperature I OC Figure 8. TMD of H 2 0 at negative pressure: -, Chen et work +, Meyer." 200
0
0 , this
1
100
60
0
\
$
-60
Lo ln
a
2
-100
-160
-200
E
-300 -260 9
1
2
3
4
5
6
7
6
9
Temperature / O C Figure 10. TMD of HDO at negative pressure: -, from Chen, et a129 and Fine and miller^:^^ 0 , this work. produced by changing the order of the fitting polynomial for three curves. Data with a larger mean square residual about the polynomial show greater turning point variation when the polynomial order is varied.
160
6
-60
ul ln
a
n
o
\
ln
2
Henderson and Speedy
9
10
11
12
13
14
15
Discussion The original object of this work was to map out the pVT properties of stretched water as far as possible into the negative pressure domain and, thereby, to test the unusual predictions of the stability limit conjecture. Two problems prevented the achievement of that aim. Firstly, we were not able to stretch water much beyond -200 bar, which corresponded to only a 1% expansion over its ambient pressure volume. While this improvement is almost an order of magnitude better than the range of Meyer's work,13 it still leaves room for another order of magnitude extension to reach the conjectured stability limits.1° Secondly, because the volume variation of the helices is a complex and poorly understood problem, estimates of the sample volumes at negative pressure are probably less reliable than extrapolations of volumes from the positive pressure region. Despite these problems, the location of the T M D is quite insensitive to the helix mechanics and the T M D measurements provide a stringent test of any equation of state for water. The maximum difference between our results and the two equations of state d i s c u ~ s e is d ~0.3 ~ ~OC ~ ~(average = 0.15 "C). Similar agreement is also observed with the more comprehensive equation of Haar, Gallagher, and KelL31 The first two equations do not contain a spinodal line, while the third predicts a spinodal line similar to that illustrated in Figure 1. The T M D data presented here are not extensive enough to provide a test of the stability limit conjecture.I0 Registry No. H20, 7732-18-5. (31) Haar, L.: Gallagher, J. S.: Kell, G. S. Water and Steam: Straub, J., Scheffler, K., Eds.: Pergamon: New York, 1980.
