Cavitation pressure in water

Eric Herbert, Sebastien Balibar and Frederic CaupinLaboratoire de Physique Statistique de l’Ecole Normale Superieure

associe aux Universites Paris 6 et Paris 7 et au CNRS24 rue Lhomond 75231 Paris Cedex 05, France

(Dated: June 30th, 2006)

We investigate the limiting mechanical tension (negative pressure) that liquid water can sustainbefore cavitation occurs. The temperature dependence of this quantity is of special interest forwater, where it can be used as a probe of a postulated anomaly of its equation of state. After a briefreview of previous experiments on cavitation, we describe our method which consists in focusing ahigh amplitude sound wave in the bulk liquid, away from any walls. We obtain highly reproducibleresults, allowing us to study in details the statistics of cavitation, and to give an accurate definitionof the cavitation threshold. Two independent pressure calibrations are performed. The cavitationpressure is found to increase monotonically from -26 MPa at 0oC to -17 MPa at 80oC. Whilethese values lie among the most negative pressures reported in water, they are still far away fromthe cavitation pressure expected theoretically and reached in the experiment by Angell and hisgroup [Zheng et al., Science 254, 829 (1991)] (around -120 MPa at 40oC). Possible reasons for thisdiscrepancy are considered.

PACS numbers: 64.60.Qb, 64.70.Fx, 64.30.+t

The liquid and vapor phases of a pure substance cancoexist at equilibrium only on a well defined line relat-ing pressure and temperature. Away from this coexis-tence line, one of the phase is more stable than the other.However, because of the existence of a liquid-vapor sur-face tension, if one phase is brought in the stability re-gion of the other, it can be observed for a finite time ina metastable state; the lifetime of this metastable statedecreases as one goes away from the coexistence line. Adetailed review about metastable states can be found inRef. [1].

We are more particularly interested in the case wherethe liquid is metastable compared to its vapor. Such astate may be prepared in two ways: by superheating theliquid above its boiling temperature, or by stretching itbelow its saturated vapor pressure. We use the secondmethod, and are able to reach negative pressures, that isto put the liquid under mechanical tension. This allowsus to study the nucleation of a bubble of the vapor phase,a phenomenon known as cavitation.

In this paper, we report our experimental resultson cavitation in water. Water is a fascinating sub-stance exhibiting many anomalies compared to other liq-uids. These anomalies arise mainly from the existenceof a coordinated hydrogen bond network between wa-ter molecules. For instance, in the temperature regionbelow the equilibrium freezing temperature (the liquid isthen in a metastable state compared to the solid phase, aphenomenon called supercooling), several thermodynam-ical properties of liquid water exhibit a large increase inamplitude when the temperature is decreased. Severalscenarios have been proposed to explain this behavior,however experiments which would help to decide betweenthem are hindered by the occurence of crystallization.Motivated by this debate, we have decided to study cav-itation in water, because it probes the cohesion between

water molecules and can give information about the liq-uid structure; indeed, it was recently emphasized [2] thatthe knowledge of the temperature dependence of cavi-tation could help to put more constraints on the phasediagram of water.

Cavitation is easily favored by impurities, known ascavitation nuclei. This explains why cavitation is oftenobserved close to equilibrium, and why results from dif-ferent experiments are widely scattered. The influence ofcavitation nuclei is more pronounced if large liquid vol-umes are studied over long time scales. The use of a fo-cused ultrasonic wave allows us to study a tiny volume ofbulk liquid without any wall, and during a short time. Wemeasure the statistics of cavitation with greater accuracythan in previous studies, and obtain clearly defined andreproducible cavitation thresholds. The pressure calibra-tion is checked by comparing two independent methods.

The paper is organized as follows. In the introduction,we give a brief account of the theory used to predict thecavitation pressure in a stretched liquid, and explain howthe different scenarios of water lead to different temper-ature behavior of the cavitation pressure. We briefly re-view in Sec. II the various experimental techniques thathave been developped to study this problem. In Sec. III,we give the details of our experimental methods, andemphasize the care taken to work with high purity wa-ter. Sec. IV details the methods of detection of cavita-tion events and of pressure calibration. This allows us topresent in Sec. V the results obtained for the cavitationpressure and statistics, as well as their good reproducibil-ity. The values found for the cavitation pressure disagreewith theory: we finally discuss in Sec. VI the possiblereasons for this discrepancy.

2

I. INTRODUCTION

A. Theoretical background

We begin with a summary of the basic theory of nucle-ation [3–5]. Here we deal only with homogeneous cavita-tion in a bulk liquid, not with heterogeneous cavitationtriggered on impurities or walls. In a liquid quenchedat a pressure P below its saturated vapor pressure Psat,the nucleation of the vapor phase represents a gain in en-ergy, proportional to the bubble volume. However, thisnucleation process is hindered by the energy cost of theliquid-vapor interface. For a spherical bubble of vapor ofradius R, the combination of these two terms writes:

E(R) =43πR3(P − Psat) + 4πR2σ (1)

where σ is the liquid-vapor surface tension. This resultsin an energy barrier Eb = (16πσ3)/[3 (Psat − P )2] (ata critical radius Rc = 2σ/(Psat − P )) which has to beovercome for the bubble to grow spontaneously. Thermalfluctuations of the system can trigger nucleation, at arate Γ = Γ0 exp [−Eb/(kBT )], where kB is Boltzmann’sconstant and T the absolute temperature. The prefactorΓ0 is the product of the number density of nucleationsites by an attempt frequency for nucleation; it is oftenapproximated by [6]:

Γ0 '(

43πRc

3

)−1kBT

h(2)

where h is Planck’s constant. For an experiment per-formed in a volume V and during a time τ , the cavitationprobability is Σ = 1− exp(−ΓV τ), and reaches 1

2 at thecavitation pressure

Pcav = Psat −(

16πσ3

3 kBT

1ln(Γ0V τ/ ln 2)

)1/2

(3)

where Psat is the saturated vapor pressure of the liquidat temperature T . One can see from Eq. 3 that the de-pendence of Pcav on Γ0V τ is weak, so that it can be con-sidered as an intrinsic property of the liquid. A moderateerror on Γ0 will not affect the estimate of Pcav; hence theapproximate Eq. 2 is sufficient. However, as a wide rangeof experimental parameters V and τ is available, we shallkeep in mind this dependence when comparing differentexperiments (see Sec. II F).

This basic theory assumes that the energy of the bub-ble can be separated into a volume and a surface term(Eq. 1), that is that the thickness of the bubble wall(liquid-vapor interface) can be neglected compared to thebubble radius. We call this approach the thin wall ap-proximation (TWA). The TWA is valid close to the coex-istence line, but becomes a crude approximation at largenegative pressures when the critical radius Rc becomesof the order ot the interfacial width. For water at 300 K,and V τ = 1 m3s, TWA predicts Pcav = −128 MPa and

Rc = 1.1 nm, close to the interface thickness (see Ref. [2]and references therein).

In addition to this oversimplification, TWA ignores afundamental feature of any first-order transition: it pos-sesses a spinodal limit. For a stretched liquid, this meansthat at a spinodal pressure Ps it ceases to be metastableand becomes macroscopically unstable. At Ps the liq-uid isothermal compressibility diverges and long wave-length density perturbations grow without limit; this cor-responds to a vanishing Eb. The existence of a spinodalline Ps(T ) where (∂P/∂V )T = 0 is easily understoodfrom the Van der Waals equation of state (EOS) for in-stance [1]; regardless of the EOS, it is a generic featureof all liquids.

A consistent theory of cavitation should thus improveTWA in two ways: (i) describe the nucleus of the newphase with a smooth profile between a low and a highdensity region; (ii) predict a vanishing nucleation barrierEb on the spinodal line. This can be achieved withinthe frame of density functional theory (DFT) [1]. Theprecise choice of the EOS will affect the prediction forthe cavitation pressure: we will now explain how this canhelp to constrain the theoretical description of water.

B. Motivation of the study of cavitation in water

To explain the singular properties of supercooled wa-ter, three scenarios have been proposed. In 1982,Speedy [7] extrapolated isotherms measured at positivepressure to estimate the spinodal pressure; he obtaineda spinodal line Ps(T ) with a minimum. Interestingly,Speedy showed that if the line of density maxima (LDM)of water reaches the spinodal line, thermodynamics re-quire that the slope of Ps(T ) changes sign at this partic-ular point. As many properties are singular on Ps(T ), inorder to explain water anomalies he proposed that Ps(T )would retrace up to positive pressure in the supercooledregion. On the other hand, Poole et al. [8] proposedthat the spinodal remains monotonic (in their picture theLDM avoids the spinodal), and explained water anoma-lies by the vicinity of a new critical point, terminatingthe coexistence line between two (low and high density)metastable liquids; this scenario has been substantiatedby all molecular dynamics simulations of water to date.A third scenario [9] proposed another consistent picture,where there is no second critical point, but where theincreases in the response functions of water are simplethermodynamical consequences of its density anomalies.

Recently, two different EOS illustrative of the two firstscenarios were used along with DFT to predict the cavita-tion pressure in water [2]. The first EOS was proposed bySpeedy [7] and shows a minimum in Ps(T ) (' −210MPaat 35 oC); the second EOS is calculated by molecular dy-namics simulations using the five-site transferable inter-action potential (TIP5P) [10] and leads to a monotonicPs(T ). The results for the cavitation pressure are shownin Fig. 1. We can see that DFT results for Pcav are less

3

-200

-150

-100

-50

260 280 300 320 340 360

Cavitation pressure (MPa)

Temperature (K)

FIG. 1: Cavitation pressure in water as a function of tem-perature, calculated within the TWA (dotted line), or us-ing DFT with Speedy’s EOS (solid line) or with the TIP5PEOS (dashed line). The parameters used are V = (10 µm)3,τ = 1 s, and Γ0 given by Eq. 2.

negative than TWA results: this is because for water,both the interfacial width and the critical radius for nu-cleation are around 1 nm. The main result is that thecavitation line Pcav(T ) exhibits the same qualitative be-havior as the spinodal line Ps(T ), that is with or withouta minimum. This finding motivated our experimental in-vestigation of the temperature dependence of Pcav. Tomeasure the shape of Pcav(T ) is of course not sufficientto settle the question of the existence of the postulatedsecond (liquid-liquid) critical point in water, but ratherwould provide a constraint that any overall picture of thephase diagram of water should be able to reproduce.

II. REVIEW OF THE PREVIOUSEXPERIMENTS

Observations of stretched water and other liquids weremade as early as in the seventeenth century, as docu-mented by Kell [11]. A detailed review of cavitation ex-periments is beyond the scope of this paper and can befound elsewhere [1, 12–15]. The values of the cavitationpressure are amazingly scattered, even between similarexperiments. In this section, we select for each differentexperimental method the reference giving the most nega-tive value. This will serve as a benchmark for evaluationof our own method.

A. Berthelot tube techniques

This method, first employed by Marcellin Berthelotin 1850 [16], consists in the following. A vessel is filledwith liquid water and sealed. If a gas bubble remains,the setup is warmed up until the bubble dissolves com-pletely; from the dissolution temperature, the liquid den-

sity is deduced. The vessel is then cooled down, watersticks to its walls and the pressure decreases, down tonegative pressure if the temperature is low enough. Thetemperature at which cavitation occurs is measured, andthe cavitation pressure is deduced using an extrapolatedEOS. Corrections may be made for the thermal expan-sion of the vessel. This method can be improved by theuse of an in situ pressure gauge: doing this, Hendersonand Speedy found cavitation at -16 MPa at 38oC [17] inglass capillaries (they also observed liquid water downto -20.3 MPa at 8.3oC [18]), and Ohde’s group reached aminimum value of -18.5 MPa at 53oC [19] in metal tubes.

B. Centrifugation

This method, first employed by Osborne Reynolds [20],consists in rotating at high speed a tube containing wa-ter. Because of the centrifugal force, a negative pressureis developped on the rotation axis: P = P0 − 1

2ρω2r2

where P0 is the pressure outside the tube, ρ is the waterdensity, and r is the distance between the center and theliquid-gas interface. The most negative value of Pcav wasobtained by Briggs [21] with boiled distilled water in aPyrex capillary tube. Briggs also investigated the tem-perature variation of Pcav: he found a minimum of -27.7MPa at 10oC, with Pcav = −2MPa at 0oC and −22MPaat 50oC.

C. Shock wave

Among cavitation experiments using shock waves, theone by Wurster et al. [22] is of particular interest. Aweakly focused shock wave is further focused by reflec-tion on a parabolic reflector. A fiber optic probe hy-drophone measures the reflectivity of a laser beam atthe fiber/water interface, from which the density of theliquid is deduced; the pressure is then estimated usingan extrapolation of Tait equation of state for water [23].With a rigid reflector, they find cavitation at -27 MPaon the hydrophone surface. On the other hand, with asoft reflector, they were able to reach ‘-59 MPa withoutcavitation at the fiber tip’. They claim that ‘the reasonfor this is that the adhesion of water to clean glass ishigher than the cohesion of water itself’ ; in fact, cavi-tation actually occurs away from the fiber tip [24]. Asthe study does not report any threshold for the onset ofcavitation away from the tip, we will use the value -27MPa for comparison (see Sec. II F), keeping in mind thatthis technique seems able to prepare liquid water at largenegative pressures, at least close to the fiber tip.

D. Acoustic cavitation

An acoustic wave can quench liquid water to negativepressure (during its negative swing). Standing and trav-

4

elling waves, focussed or not, were used by many dif-ferent groups. We will detail here the experiments byGalloway [25] and Greenspan and Tschiegg [26].

Galloway [25] used a standing wave produced by aspherical resonator. The sound amplitude at the centeris measured with a piezoelectric microphone. Gallowaydefines the threshold for cavitation as the point ‘at whichcavitation will occur at least once a minute, while a 10percent reduction in the peak sound pressure will not pro-duce any cavitation in a 15-minute interval’. He foundthat Pcav varies from−0.1MPa for distilled water satu-rated with air, to −20MPa for distilled water degassed at0.02 % saturation. The way to define the threshold is offundamental importance, because Galloway states that‘pressure 100 times greater than this threshold pressuremay be imposed on the sample for short lengths of time,of the order of seconds, without causing cavitation’; wealso learn from Finch [27] that Galloway [28] ‘generallyobtained much lower thresholds, of the order 1.5-2 MPa, with the higher values [−20MPa ] occurring only atcertain times, there being no obvious explanation for thechange’. Galloway also noticed a small increase of Pcav

with temperature (10 % betwen 5 and 45oC ). Greenspanand Tschiegg [26] used a standing wave focused in a cylin-der made of stainless steel to study carefully cleaned anddegassed water . They calibrated Pcav by the static pres-sure method (see IV D) and found Pcav = −16MPa (resp.−21MPa ) for an average waiting time for cavitation ofseveral minutes (resp. seconds).

E. Mineral inclusions

The principle of this method is similar to the Berthelottube method (see Sec. IIA), except that it uses micro-scopic vessels. It deserves a separate paragraph becausethe most negative pressures reported in water were ob-tained with this method.

Water trapped in small pockets (in the 10 − 100 µmrange) inside crystals can be found in nature. Angelland his group used synthetic inclusions [29, 30]. As theirfirst paper deals with inclusions of saline solutions [29],we will focus on the second one where Raman spectra ofthe inclusions indicated a low salt concentration [30].

Crystals (quartz, calcite and fluorite) are quench-fissured in pure water between 300 and 400oC. The frac-tured crystals are then sealed in Ag-Pd tubes with aknown amount of ultrapure water, and autoclaved. Dur-ing autoclaving, healing of the fissures traps water ininclusions at a desired density, depending on the auto-claving temperature and pressure. Angell and his groupthen followed Berthelot’s method to study these inclu-sions: the bubble remaining in the inclusion disappearsupon heating, at a temperature Td; when the sampleis cooled down, liquid water follows a nearly isochoricpath, until cavitation occurs at Tcav. To deduce Pcav,they have to rely on an EOS: they chose to extrapolatethe so-called HGK EOS to negative pressure. The HGK

EOS is a multiparameter EOS fitted ondata measuredat pressures where the liquid is stable; it is qualitativelysimilar to Speedy EOS, but quantitatively different, giv-ing for the coordinates of the minimum in the spinodalaround 60oC and -160 MPa.

For quartz inclusions, all inclusions in a given samplehave the same Td and hence the same density. Thereare two distinct cavitation behaviors. When Td > 250oC(autoclaving temperature higher than 400oC ), Tcav isthe same within ±2oC for all inclusions in a given sam-ple, whereas when Td < 250oC (high density inclusions),Tcav is scattered. For fluorite and calcite, Tcav is alwaysscattered, and the estimated Pcav is less negative thatin quartz. Angell and his group attribute the scatter toheterogeneous nucleation, and its source to ‘possibly sur-factant molecules cluster destroyed by annealing at thehigher temperatures’ .

For low density inclusions in quartz, Pcav is positive,and compares well with the maximum temperature atwhich liquid water can be superheated, as measured bySkripov [31]. The maximum tension is obtained in onesample with high density inclusions (0.91 g mL−1 andTd = 160oC); Angell and his group report that ‘some [in-clusions] could be cooled to -40oC without cavitation, andone was observed in repeated runs to nucleate randomlyin the range 40 to 47oC and occasionnaly not at all’ [30]:they estimate nucleation occured at Pcav ' −140MPa.The fact that ‘no inclusion that survived cooling to 40oCever nucleated bubbles during cooling to lower temper-atures’ was interpreted as an evidence that the isochorecrosses the metastable LDM, thus retracing to less neg-ative pressure at low temperature. This gives supportto Speedy’s scenario, at least in the sense that the LDMkeeps a negative slope, deep in the negative pressure re-gion in the P − T plane.

Further work on inclusions deals with the use ofBrillouin scattering to measure the sound velocity instretched liquid water [32]. This study reports tensionsbeyond -100 MPa at 20oC. Additionally, it was able toshow a volume change in a platelet-like inclusion, whichpoints out the difficulty with the isochoric assumptionmade to estimate Pcav; on the other hand, for roughlyspherical inclusions this assumption appears to be ap-propriate. It should be emphasized that in the work byAngell and his group , the inclusions in which they esti-mated Pcav ' −140MPa ‘were not of well-rounded form,like those on which the reliable and reproducible hightemperature data were obtained ’ [33].

To conclude with mineral inclusions, we shall mentiona recent work focusing on kinetic aspects, by measuringthe statistics of lifetimes of one inclusion at fixed tem-peratures [34]. The largest negative pressure achieved inthis work is −16.7 MPa at 258.3oC , and the lifetimesfollow a Poisson distribution.

5

TABLE I: Comparison between different cavitation experiments. Among the numerous and scattered values of the cavitationpressure in the litterature, only the most negative have been selected.

Method Ref. T V τ J = 1/(V τ) Wall PcavoC mm3 s mm−3 s−1 MPa

Berthelot [17] 40 1 20 5 10−2 Pyrex glass -16Berthelot [19] 53 47 5 4.3 10−3 stainless steel -18.5centrifugation [21] 10 0.38 10 2.6 10−1 Pyrex glass -27.7shock wave [22] 25 0.003 10−8 3.3 1010 silica fiber -27acoustic [26] 30 200 0.1 5 10−2 none -21inclusions [30] 40-47 4.2 10−6 1 2.4 105 quartz -140acoustic this work 20 2.1 10−4 4.5 10−8 1.1 1011 none -24

F. Comparison

Among the available measurements of the cavitationpressure in water, we select for each method those whichgive the most negative values. They are compared in Ta-ble I. We try to correlate the values to the parameterV τ which should affect the cavitation threshold as ex-pected from nucleation theory (see Sec. I A). We definethe experimental volume V (resp. time τ) as the volumein (resp. time during) which the pressure of the liquid iswithin 1% of its most negative value; when these quan-tities could not be inferred from the references, we usedan arbitrary estimate (shown by an italic font). We alsoindicate the type of walls in contact with stretched wa-ter (if any), because of their possible effect on Pcav. Wealso give the values corresponding to our results at 20oC;their measurement will be described in the following.

Figure 2 shows the cavitation pressure as a functionof the quantity J = 1/(V τ). For the sake of compar-ison, we have also plotted the prediction of TWA. Thelowest Pcav from Angell and his group [30] fall far awayfrom other experiments, but close to the theoretical esti-mate. The discrepancy with other experiments cannot beaccounted for by the difference in J ; furthermore, two ex-periments (shock wave, and present work) have a highervalue of J and give a less negative Pcav. One could thinkthat the nature of the wall plays a role: water adhesionmay be stronger on the quartz walls of the inclusions.However, we note that some acoustic experiments found' −20 MPa in the absence of walls, and that Strube andLauterborn [35], using the centrifugation method withquartz tubes, reached at best -17.5 MPa.

III. APPARATUS

To quench the liquid beyond the liquid-vapor equilib-rium line, we use an acoustic method. We tried to im-prove on previous acoustic experiments (see Sec. II D) inthe following ways. First, many of the acoustic experi-ments used rather long bursts, or even standing waves;this could enhance the sensitivity to minute quantitiesof dissolved gases because of a rectified diffusion pro-cess. Therefore we decided to use only short bursts. Fur-

-200

-150

-100

-50

0

10-3

10-1

101

103

105

107

109

1011

Cavitation pressure (MPa)

J (mm-3 s-1)

FIG. 2: Cavitation pressure as a function of the quantityJ = 1/(V τ). Data points are listed in Table I. The solid lineis the prediction of the TWA (Eq. 3).

thermore, we wanted to decrease the parameter V τ (seeSec. I A), by using high frequency ultrasound; with a1 MHz sound wave, we reach V τ ' 10−11 mm3s, evensmaller than the inclusion work. The use of a smallV τ reduces the effect of impurities and rules out theone of cosmic rays (their typical total flux at see levelis 240 m−2 s−1 [36]). We now present the experimentalsetup.

A. Generation and focusing of the acoustic pulses

Let us recall that some other acoustic experiments usedparameters similar to ours [37–39]; but they failed toreach pressures negative enough to produce cavitation inclean water (see Sec. II D), and they had to add impuri-ties on purpose. In order to reach more negative pressure,we chose a piezoelectric transducer with a hemispheri-cal shape. This ensures a very narrow focusing of thesound wave (in an ellipsoidal region 3.5mm3 in volume,see Sec. IVC). Another advantage is that the negativepressure is developed in the bulk liquid, far away fromany wall, which could trigger heterogeneous cavitation.These advantages were already used to study cavitation

6

oscilloscopedigital

oscilloscopedigital

RF outputhigh power

analog control

inputgate

inputRF

PC

NAND

high power RF amplifier

input

−40dB output

144.

5Ω

output

transformer 2:1step down

cellhigh pressure

waveform generator

RFgateoutput output

FIG. 3: Block diagram of the driving circuit of the transducer.The different units are designated in bold font. The solid linesrepresent the electrical connections, and the dashed lines theconnections for computer control. The shunt resistor and thestep-down transformer are used to improve the shape of theexcitation voltage.

in liquid helium [40–42].The transducer is a hemispherical shell, 16 mm inner

diameter, 20 mm outer diameter, made of material P762(Saint-Gobain Quartz), excited at resonance in its thick-ness mode at 1 MHz. Its impedance at resonance is real,equal to 26.5Ω.

B. Choice of the driving voltage characteristics

The transducers are driven with a radio-frequency am-plifier (Ritec Inc., GA 2500 RF). This amplifier is primar-ily designed to operate a 50Ω resistive load. To matchthe transducer impedance, we use a high power downsteptransformer and a resistor bridge. The best configurationfound is shown in Fig. 3. We continuously monitor thevoltage on the transducer side with a built-in -40 dBmonitor on the transformer. A typical excitation signalat the cavitation threshold is shown in Fig. 4. We seethat the voltage is nearly sinusoidal, although the enve-lope is not exactly rectangular; at the end of the pulse,there is also a small distortion followed by a slow relax-ation of the voltage. To characterize the excitation, wechose to measure the root mean square voltage on thelast undistorted cycle; we will refer to this quantity Vrms

as the excitation voltage in the following.Because the transducer is used at resonance, we need

to choose correctly the center frequency f of the electricburst. All other parameters being fixed, we have studiedthe variation of the cavitation threshold (see Sec. IVB)with f and found a rather shallow minimum at 1025 kHz.We used this value for the mechanical resonance fre-quency throughout the study. It is close to the electricalone, and it is constant over the whole pressure and tem-perature range.

-400

-200

0

200

400

0 2 4 6 8 10

Transducer voltage (V)

Time (µs)

FIG. 4: Excitation voltage of the transducer, driven with a4-cycles burst, at the cavitation threshold at T = 20oC andPstat = 1.7MPa. This corresponds to a peak power of 3.4 kW.

We need also to choose the burst length, on whichVcav depends because of the finite quality factor of thetransducer: the longer the burst, the lower Vcav. How-ever, there are two limitations: too short a burst makesVcav beyond the reach of the amplifier (especially at highstatic pressure), and too long a burst makes the nucle-ation time distributed over several cycles, thus complicat-ing the detection (see Sec. IV A). We found 4 to 6 cyclesbursts to be a good compromise with a low enough re-quired driving voltage, and a constant nucleation time(for given values of temperature and pressure). We havealso checked that Pcav does not change when the burstlength varies from 1 to 20 cycles.

C. Experimental cell

We have used two types of cells. Experiments requir-ing easy access to the focal region (e.g. calibration withan hydrophone, see Sec. IV C) were performed in sim-ple Pyrex or stainless steel (SS) containers, open to theatmosphere. The second type of cell used was designedfor high pressure operation (in particular for calibrationby the static pressure method, see Sec. IV D). The cor-responding setup is sketched in Fig. 5. The main bodyof the high pressure cell is a cylinder made of SS (5 mmthick). Its bottom is closed by a plate carrying the trans-ducer and its holder. The seal is made with an indiumwire. The tubing is made of SS (inner diameter 4 mm,outer diameter 6 mm). The connections are made by ar-gon welding or using SS high pressure seals (Sagana).Other seals (at the pressure gauge, at the bellow and be-tween Pneurop fittings) are in bulk Teflon, to avoid pollu-tion (see Sec. III E). A set of valves allows for pumping,filling, and pressure control. Before filling, the circuitis evacuated by pumping with an oil pump through anitrogen trap or a dry scroll pump; water can then betransferred in the cell under vacuum. Once the system

7

or

filling emptying

pressuregauge

thermostatedbath

to pump

ice

to amplifier

to computer

water

bellow

cell

FIG. 5: Sketch of the experimental setup. The high pressurepart contains the cell with the transducer, the pressure gaugeand the bellow for pressure control; it can be isolated fromthe rest by a valve. The use of two other valves allows evac-uation (with an oil pump through a nitrogen trap or a dryscroll pump), filling (the flask with degassed water being con-nected), and emptying (the collecting vessel being connectedand cooled with ice). The cell is immersed in a thermostatedbath (operated between -10 and 80oC). All the seals are madeof SS or bulk Teflon, except the one at the bottom of the cell,which is made with an indium wire.

is filled with liquid, the valve near the cell is closed, andthe pressure can be adjusted using a SS bellow, and mon-itored with a digital pressure gauge (Keller PAA-35S,range 0− 30 MPa, accuracy ±0.015MPa). The system isdesigned to sustain 24 MPa, but was operated below 10MPa. The rest of the circuit, operated at low pressure, isconnected with Pneurop fittings with Teflon O-rings. Allthe circuit was tested against leaks with a helium leakdetector (Alcatel ASM 110 Turbo CL).

D. Temperature control

When a temperature control was needed, the cells wereimmersed in a bath. The open cells were immersed in awater beaker on a heat plate (bath temperature regulatedwithin 0.1oC). The high pressure cell was immersed in acryothermostat (Neslab RTE 300), regulating the tem-perature between -10 and 80oC within 0.01oC.

We refer to the temperature of the liquid away fromthe acoustic focus. One may wonder about the actualtemperature inside the wave. Indeed, the liquid followsan adiabatic path, where the temperature is related tothe pressure by:

(∂T

∂P

)

S

=T Vmol αP

cP(4)

where Vmol and cP are the molar volume and heat ca-pacity at constant pressure, and αP is the thermal ex-pansion coefficient at constant pressure. When the liq-uid is stretched by the ultrasonic wave, it cools down orwarms up, depending on the sign of αP . To calculate thetemperature change, one should integrate Eq. 4 over theappropriate pressure and temperature range. We willlimit our discussion to an order of magnitude calcula-tion. If we start from the LMD (4oC at 0.1 MPa), whereαP is zero, there is no temperature change at low soundamplitude. To give numbers, let us use the tabulateddata [43] at 50oC and 0.1 MPa: Vmol = 1.82 10−5 m3,cP = 75.3 Jmol−1K−1, and αP = 4.4 10−4 K−1, we find(∂T/∂P )S = 3.4 10−8 KPa−1; with a negative swing ofthe wave of -20 MPa, we find a temperature change lessthan 0.7 K. We will neglect this effect and always referto the bath temperature.

E. Materials

In order to reach homogeneous cavitation, special caremust be taken in the preparation and handling of the wa-ter sample. For instance, dust particles or dissolved gasesare expected to trigger cavitation at less negative pres-sures. Materials of the handling system and the samplecell were chosen in order to avoid the so-called ‘containereffect’[44]. To do so, we checked the variation of the UVabsorption of ultrapure water after being heated at 80oCin contact with the material in an open Pyrex glass. Typ-ical checks are shown in Fig. 6. We kept the materialsshowing the smaller effect. The main materials involvedwere thus Pyrex glass and SS. Instead of the usual VitonO-rings, all seals (between Pneurop fittings, on the pres-sure gauge and on the bellow part) were made of bulkTeflon, except the bottom plate seal made of indium (seeSec. III C). The part of the pressure gauge in contactwith the liquid is made of a SS membrane.

Let us describe in details the materials used insidethe high pressure cell. We used ceramic-SS electricalfeedthroughs (CeramTec), argon welded to the cell bot-tom. A SS holder was designed to receive the trans-ducer. Its two electrodes were connected mechanically toSS wires, to avoid using tin solder. The SS wire for theouter electrode is crimped on a SS sheet, and the assem-bly is pressed on the electrode by a screwable part of theholder; for the inner electrode we shaped a thin SS sheetinto a spring to press onto the surface, and crimped iton the SS wire. We used 100 µm thick Teflon to insulatethis contact from the transducer holder. Finally the two

8

-0.2

0

0.2

0.4

0.6

0.8

150 200 250 300 350

Relative absorbance

Wavelength (nm)

Viton

Nylon

1%

ceramic

control

degassed water

steelstainless

ethanol

FIG. 6: UV absorption spectrum showing different ‘containereffects’. We used an automated spectrometer (Kontron In-struments, Uvikon 941) with quartz cuvettes (10mm thick-ness). The reference cuvette was filled with ultrapure wa-ter directly drawn from the water polisher and sealed witha Teflon cap. The solid curves show the results of our testfor several materials (designated by the labels), using the fol-lowing procedure: a piece of the material was immersed in50 mL ultrapure water in a Pyrex beaker, and the beakerwas heated at 80oC during 30 min, in contact with air. Af-ter cooling, water from this beaker was then used to rinseand fill the sample cuvette, and its UV absorption recorded.The control curve shows the results of this test for ultrapurewater without adding any material. Other materials used inthe cell (teflon tape, teflon tube and indium wire, not shown)give absorption spectra between the control and that of stain-less steel. For comparison, we also show the spectrum for a1% ethanol solution in water (dotted line), and for ultrapurewater degassed using the procedure describe in Sec. III F.

wires were fitted in two Teflon tubes, and clamped to thefeedthroughs conductors by a SS tube with a SS screwon its side.

F. Water sample

The results reported here were obtained with ultra-pure water drawn from a two stages water system (os-moser ELGA Purelab Prima, polisher ELGA PurelabUltra) which achieves a resistivity of 18.2MΩ cm and atotal oxidizable organic carbon less than 2 ppb. How-ever, it still contains dissolved gases, which are expectedto lower the cavitation threshold. To degas the water,we used the following method: a 250 mL Pyrex glass er-lenmeyer was modified to accept a Pyrex-SS fitting. Itwas cleaned with sulfochromic mixture, rinsed 3 timeswith ultrapure water and filled with 100mL of ultrapurewater. The erlenmeyer was connected to a diaphragmpump (BOC Edwards D-Lab 10-8, vacuum limit 8 mbar,pumping rate for air 10 Lmin−1) through a valve and aSS tubing, sealed with Teflon O-rings. After the end ofthe degassing, the valve is closed and water kept in con-

tact with its vapor only; as the high pressure cell can beevacuated to a fraction of millibar, we can transfer thewater sample into the cell without exposing water againto atmosphere.

The water was pumped continuously while beingshaken in an ultrasonic bath (Transsonic T425/H). Inprinciple, a better degassing is achieved at high temper-ature, because of the lower solubility of gases (arounda factor of 1.6 less at 80oC than at 20oC); however, athigh temperature, water evaporates with a high rate andcondenses in the tubing and in the pump, thus reducingthe pumping efficiency. Therefore we decided to keep thebath cold (20oC), by circulating tap water inside a coppertubing. We observe the following behavior: at the begin-ning, many bubbles appear on the erlenmeyer walls; weattribute this to air degassing on cracks or weak spotsin the glass. Then the bubbling slows down, and we candistinguish bubbles appearing in the bulk liquid, proba-bly at a pressure antinode of the ultrasonic bath. Finally,the bubbling decreases gradually, and after 30 min, onlylarge bubbles burst from time to time; we attribute thisto boiling in a degassed sample. Our observations aresimilar to previous ones [25, 27]. To check the waterquality after degassing, we measured its UV absorptionspectrum; Figure 6 shows that UV absorption is even lessthan before degassing: we attribute this to the removalof dissolved oxygen, which absorbs UV light below 250nm [45].

IV. OPERATION

We now turn to the measurements performed with ourexperimental setup. We first describe the methods ofdetection, then the statistics of cavitation, and finallyour two ways of converting the excitation voltage into anegative pressure.

A. Detection of the cavitation bubbles

When the pressure becomes sufficiently negative atthe focus, bubbles nucleate. In previous experimentson acoustic cavitation, bubbles were detected optically,by visual observation (directly or through a micro-scope) [25], high speed photography [46], light scatter-ing [27], or even chemiluminescence [47, 48]. They couldalso be detected acoustically, by the change in the pres-sure field used to produce cavitation [25] or by the soundemitted by cavitation (passive acoustic detection) [49].Greenspan and Tschiegg [26] used the change in the qual-ity factor of the resonator. Later on, Roy et al. [37] de-velopped an active acoustic detection scheme : a highfrequency sound wave is focused on the cavitation region,and backscattering is detected when bubbles are present;it is more sensitive than the passive detection method,leading to equal or less negative cavitation thresholds.

9

-8

-4

0

4

8

12 14 16 18 20

Transducer voltage (V)

Time (µs)

FIG. 7: Relaxation voltage of the transducer. The two traceswere recorded for two successive 4-cycles bursts with thesame experimental conditions (T = 20 oC, Pstat = 1.7MPa,Vcav = 163.3V). The solid line corresponds to the repro-ducible relaxation signal of the transducer coming back torest, without cavitation. The dashed line is an example ofthe random echo signal reflected on the nucleated bubble andreaching the transducer voltage with a delay tf after nucle-ation.

In the early stage of our experiment, we have investi-gated different detection methods in ethanol: light scat-tering, imaging on a CCD camera, passive acoustic de-tection and the ‘echo method’ [50]. All the methods werefound to be consistent with each other, that is to give si-multaneously the same diagnosis about the presence orabsence of a bubble. We will just describe here the ‘echomethod’ which we chose because of its simplicity to im-plement and its wide range of applicability. After thebubble is nucleated at the center of the hemisphere, therest of the ultrasonic wave reaches the focal region, andpart of it is reflected by the bubble surface, back to thehemispherical surface of the transducer. The reflectedwave is converted back into a voltage, superimposed onthe relaxation voltage of the transducer. Figure 7 showsa typical relaxation voltage with and without cavitation:the signals start to depart from each other at a timecorresponding to the time at which the pressure burstreaches its minimum, plus the time of flight tf of soundacross the transducer radius R = 8mm: tf = R/c, wherec is the sound velocity.

One of the main features of the cavitation phenomenonwe observe is its stochastic nature: if the acoustic burstsare simply repeated without changing any experimentalparameter, we observe randomly echoes with or withoutcavitation. As the relaxation voltage in the absence ofcavitation is very reproducible, it can be saved as a ref-erence and substracted from the following acquisitions.The cavitation events are then clearly detected from thelow remaining electrical noise, for example by readingfor each echo in a series the value of the peak-to-peakvoltage. Figure 8 shows a typical histogram of the cor-

0

10

20

30

40

50

0 5 10 15 20 25

Number of counts

Echo voltage (V)

FIG. 8: Histogram of the peak-to-peak value of successiveechoes. The left group of data shows the small noise on theexcitation voltage without cavitation, the right one shows thevarious amplitudes reached by echoes with cavitation. The1000 data points are distributed among 100 bins; the mainpeak (reaching 328 counts) is truncated for clarity.

responding values over 1000 bursts: they fall into twowell separated groups, which shows the reliability of thissimple method.

Our echo method is evocative of the active detectionmethod developped by Roy et al. [37]. To some extent,our method is simpler because it involves only one trans-ducer for producing and detecting the bubbles, avoidingthe need of a geometrical adjustment between the gener-ator and detector used by Roy et al..

B. Statistics of cavitation

The randomness of the cavitation phenomenon leads usto define the cavitation probability Σ for a given set ofparameters as the fraction of repeated bursts that exhibitcavitation, which is easily obtained from histograms suchas the one shown in Fig. 8.

When the excitation voltage Vrms is increased, all otherexperimental parameters being held constant, Σ increasesfrom 0 to 1 over a narrow range of Vrms values, as shownin Fig. 9. Because of their characteristic shape, we callthese curves ‘S-curves’. Their steepness allows us to de-fine accurately the cavitation threshold voltage Vcav, asthe value of Vrms at which Σ = 1/2.

We can investigate further the shape of the S-curves.The energy barrier for cavitation depends on the negativepressure reached, which in turn depends on the excitationvoltage Vrms. The cavitation rate (see Sec. IA) is relatedto these quantities by

Γ = Γ0 exp(−Eb [P (Vrms)]

kBT

)(5)

The cavitation probability writes Σ = 1 − exp(−ΓV τ).The threshold Vcav (or equivalently Pcav) are reached

10

0

0.2

0.4

0.6

0.8

1

130 140 150 160 170 180

Cavitation probability

Excitation voltage (V)

0

0.01

0.02

0.03

130 135 140 145 150

FIG. 9: Cavitation probability versus excitation voltage for4-cycles bursts at T = 20oC and Pstat = 1.7MPa. Each ofthe 25 data points was measured over 1000 repeated bursts.The standard deviation on the probability (calculated withthe binomial law) is shown as error bars. The data are wellfitted with Eq. 9 (solid line). The inset focuses on the lowprobability region, to show that zero probability is actuallyreached in the broad foot of the S-curve.

when

Eb [P (Vrms)] = kBT ln(

Γ0V τ

ln 2

)(6)

In the case of cavitation in a focused acoustic wave,the pressure varies in both space and time. By using anexpansion around the minimum Pmin of P (r, t), one cancalculate the effective V τ ; this was discussed in the caseof cavitation in liquid helium [6], and gives for one cycleof a spherical sinusoidal wave:

Vexp τexp =33/2 λ3 τ k2T 2

4π2 (∂ lnE/∂ ln|P |)2 Eb(Pmin)2(7)

Combining Eqs. 6 and 7, one obtains an implicit equationon Pcav. To solve it we need a theory for the energybarrier; for instance, if we use the prediction of the TWA,we find for water at 20 oC in a 1 MHz sound wave Pcav =−182.5MPa and:

Vexp τexp =(

λ

16.2

)3τ

16.2(8)

The values of V τ and Pcav do not depend much on themodel used, because of the logarithmic derivative in-volved in Eq. 7.

The probability as a function of voltage involves a dou-ble exponential, so that it varies fast around the thresh-old; using a linear expansion of Eb [P (Vrms)] around Vcav

in Eq. 5 will thus give a good approximation of Σ. Thisleads us to fit the experimental data shown in Fig. 9 withthe following function:

Σ = 1− exp− ln 2 exp

[ξ

(Vrms

Vcav− 1

)](9)

where ξ and Vcav are free parameters. ξ measures thesteepness of the probability curve, and is related to theenergy barrier through:

ξ = −Vcav

kBT

(∂Eb

∂V

)

Σ=1/2

(10)

Figure 9 shows that the fit with Eq. 9 reproduces wellthe data, including the typical asymetric shape (broadfoot and narrow head). The quality of the fit is discussedin Appendix A. Zero probability is actually reached inthe foot of the S-curve; in one case we checked that, forVrms in this region, no bubble was detected over 10000bursts. We measured the S-curves to a high level of accu-racy when we wanted to investigate in details the cavita-tion statistics. When we were only interested in the valueof the cavitation voltage, in order to gain time, we mea-sured the probability over 300 or 400 bursts at 4 voltagesaround the threshold.

We would like to emphasize that this analysis is an im-provement over the definition of the cavitation thresholdused in the experiments done by other groups. Indeed,when the variation of probability with pressure was suf-ficiently sharp, the threshold was often arbitrarily esti-mated by the experimenter. Sometimes, it seems thatonly the most negative value observed for Pcav was re-ported; for instance, Strube and Lauterborn [35] usedthe centrifugation method and observed a large scatterof Pcav not detailed in the previous work by Briggs [21].Only a few studies were concerned with statistics of thecavitation events [34, 51]. The good repeatability of theacoustic pulses and the use of automated data acquisitionallowed us to study these statistics with greater accuracyand more extensively.

One of the difficulties of our experiment lies in howto convert the excitation voltage of the transducer intoa value of the negative pressure reached at the focus.One way is to rely on a calculation to convert the mea-sured electrical power used by the transducer in acousticenergy [46]. To avoid the assumptions needed in thisprocedure, we prefer to use two independent methods ofcalibration that we shall now describe.

C. Pressure calibration with hydrophones

The first and most straightforward method uses cali-brated hydrophones. They are needle shaped piezoelec-tric hydrophones (Precision Acoustics): the sensor is adisc made of a 9 µm thick gold electroded Polyvinylidenedifluoride (PVdF) film. We have chosen disc diametersof 40 and 200 µm at the end of needles 300 and 460µm in diameter, respectively. The probe is very fragile,and cavitation on its surface causes irreversible damages.We have thus performed the calibration of the ceramictransducer with ultrapure degassed water; as the calibra-tion had to be done in an open tank, we worked only afew hours with the same water sample. To avoid cavita-tion, we had also to use excitation voltages significantly

11

-3

-2

-1

0

1

2

3

4 6 8 10 12 14

Pressure (MPa)

Time (µs)

FIG. 10: Response of the 40 µm needle hydrophone at thefocus. The hydrophone voltage has been converted into pres-sure using the manufacturer’s calibration data. The solid(resp. dotted) curve corresponds to an excitation voltageVrms = 7.21V (resp. 17.42V), that is 5.5% (resp. 13.4%)of the cavitation voltage. The time scale starts with the exci-tation voltage, and the acoustic wave reaches the focus afterthe time of flight over the transducer radius (tf = R/c, seeSec. IVA).

lower than the cavitation threshold. To determine thisthreshold, S-curves were measured immediately beforeand after the use of the hydrophone; to avoid damage,the hydrophone was removed from water while acquiringthe S-curves. We noticed a small drift in the cavitationvoltage, attributed to the change in water conductivitybecause of its exposure to air; indeed, the transducer elec-trodes are in contact with water and an increase in itsconductivity decreases the efficiency of the transducer fora given excitation voltage. However, this shift was lessthan 1.5% over the time needed for the calibration withone needle.

The needle hydrophone is inserted along the axis of thehemisphere. To find the position of the focus, we repeata given low amplitude acoustic burst and we look for thepostition which maximizes the peak-to-peak voltage ofthe hydrophone response. This can be done with an ac-curacy of around one quarter of the sensor diameter, us-ing micrometer screws. A typical signal thus obtained isshown in Fig. 10. The manufacturer provides calibrationdata for the complete hydrophone system, every 1 MHzfrom 1 to 20 MHz, with a stated uncertainty on the gainof 14 %. We have converted the voltage given by thehydrophone into pressure using both a direct conversionwith the gain tabulated at 1MHz, and a deconvolutiontechnique using the gain at all frequencies: because thewave has a small harmonic content (see Fig. 10), the twotechniques give no noticeable difference.

We have performed a detailed mapping of the acousticfield. The results are shown in Fig. 11. One can see thatthe focusing is sharp, with an ellipsoidal shape, 1.5mm(one wavelength λ) across and ' 3mm (' 2λ) along the

0

0.2

0.4

0.6

0.8

1

1.2

-2 -1 0 1 2 3

Normalized peak pressure

Position (mm)

FIG. 11: Map of the acoustic field. The data (normalized to1 at the focus) were taken with a 40 µm needle hydrophone inan open container at room temperature (' 25 oC), using 1-cycle bursts. Filled (resp. empty) symbols correspond to themaximum (resp. minimum) pressure in a burst. Diamonds(bottom) show a scan in the equatorial plane of the trans-ducer, whereas circles (shifted by 0.2 for clarity) show a scanalong its axis.

0

1

2

3

4

0 5 10 15 20 25

Peak pressure (MPa)

Excitation voltage Vrms (V)

FIG. 12: Peak pressures measured by the hydrophones as afunction of the transducer excitation voltage. The data weretaken in an open container at room temperature (' 25 oC,with 4-cycles bursts, the cavitation threshold voltage beingVcav = 130V. Filled (resp. empty) symbols correspond tothe maximum (resp. minimum) pressure in a burst. Circles(resp. triangles) were obtained with a 40 µm (resp. 200 µm)needle. The manufacturer gives a calibration uncertainty onthe gain of 14% , which means that the results from the twoneedles are consistent with each other.

transducer symmetry axis; for a complete sphere, onewould expect a spherical focus of diameter λ. From theknown variation of the sound velocity [43], we calculatethat the focal volume changes by less than 20% over therange of temperature and pressure used in the experi-ment.

12

We recorded the hydrophone signals for different val-ues of the excitation voltage of the transducer, typicallyup to 1/5 of the cavitation threshold, because distorsionssometimes appeared at this level. They could be eitherdue to heterogeneous cavitation on the hydrophone, orto nonlinearity of the hydrophone response at this largeamplitudes (although the manufacturer gives a pressurerange of use exceeding 20 MPa rms). In one case, wewent up to 14 MPa, but afterwards the hydrophone ap-peared to be broken. Figure 12 shows a set of resultsfor the same transducer and water sample: the differenthydrophones used give results consistent with each other.The relation between the peak pressure at the focus andthe excitation voltage is found to be linear, within theexperimental error. If we extrapolate up to the cavita-tion voltage, we obtain at 25oC: Pmin = −21MPa (resp.Pmin = −24.5MPa) for the 40 µm (resp. 200 µm) needle.The manufacturer gives a calibration uncertainty on thegain of 14% , which means that the results from the twoneedles are consistent with each other.

One may wonder if the presence of the hydrophone af-fects the pressure field. This seems unlikely because ofits needle shape which allows to handle it from a direc-tion that the acoustic wave reaches only after the focus.Building up of a stationary wave is not expected, becausethe size of the sensor tip is smaller than the sound wave-length (1.5mm), and because the acoustic impedance ofPVdF is close to the one of water. Furthermore, thetwo different sized hydrophones lead to the same results,given further support for our method.

The hydrophone signals have the shape of a modulatedsine wave and do not show any sign of shocks. But for thelarger pressure amplitudes involved to reach cavitation,one may wonder about the linearity of the focusing, whichwe assume when extrapolating the hydrophone measure-ments. In fact, because the sound velocity is an increas-ing function of pressure, one expects that nonlinearitieswill develop in the following way: the pressure at the fo-cus should deviate from a symmetric sine wave when theamplitude increases, exhibiting shallow and wide nega-tive swings and narrow and high positive peaks, withthe possibility of shock formation. This behavior is ob-served in numerical simulations of the spherical focusingof a wave in liquid helium, and confirmed experimen-tally [52]. However, these nonlinearities are not notice-able if the pressure amplitude is small compared to thespinodal pressure, which seems to be the case in our ex-periment (-26 MPa compared to -200 MPa, according toSpeedy EOS). Furthermore, we think that the nonlin-earities are much less pronounced in the hemisphericalgeometry, presumably because the velocity at the focusis not required to vanish, whereas it is for the sphericalgeometry [52, 53]. Anyhow, even if nonlinearities werepresent, they would lead to a less efficient build-up ofthe negative swing at large amplitudes, so that the linearextrapolation of the hydrophone measurements gives alower bound for Pcav.

0

2

4

6

8

1x105

1.5x105

2x105

2.5x105

Static pressure (M

Pa)

ρVcav

(kg V m-3)

0.1°C20°C40°C80°C 60°C

FIG. 13: Static pressure as a function of the product ρVcav

for 5 of the 15 temperatures investigated in run 0. The dot-ted lines are linear fits to the data; each set is labelled (onthe right) with the temperature. Experimental error bars areshown on each data point; most of the vertical errors are toosmall to be seen at this scale.

D. Calibration by the static pressure method

The second method of calibration we use is based onthe application of a static overpressure to the liquid. Itis similar to the method used by Briggs et al. [54] andGreenspan and Tschiegg [26], and also in our group tostudy cavitation in liquid helium [41, 42]. The range ofoverpressure explored in the present study is more thantenfold that of the previous ones. We produce this over-pressure with the bellow described in Sec. III C; we willrefer to it as the ‘static’ pressure Pstat to distinguish itfrom the acoustic pressure. When starting from a higherPstat, to reach the same value of Pcav at the focus, one hasto use a higher excitation voltage. It can be shown [52]that if the focusing is linear, the pressure swing in thewave ∆P = Pstat − Pmin is proportional to ρ(Pstat)Vrms,where ρ(Pstat) is the density of the liquid at rest; themarginal variation of ρ with pressure and temperaturewas taken into account in our analysis. Therefore, thedata Pstat versus ρ(Pstat)Vcav should fall on a line cross-ing the axis Vcav = 0 at the pressure Pcav. Taking thenonlinearities into account, the intercept thus obtainedshould give an upper bound for Pcav [52]. The results ofthe static pressure method will be presented in Sec. V A.As we shall see, its result at room temperature agrees wellwith the hydrophone calibration. At other temperatures,we use only the static pressure method.

V. RESULTS

A. Pressure dependence of the cavitation voltage

Let us now discuss the pressure dependence of the cav-itation voltage at a given temperature.

13

0

2

4

6

8

1.8x105

2x105

2.2x105

2.4x105

1st increase

1st decrease

2nd increase

2nd decrease

3rd increase

Static pressure (MPa)

ρ Vcav (kg V m

-3)

FIG. 14: Static pressure as a function of the product ρVcav

during the first pressure cycles of run 3 at 0.1 oC.

We begin with the results at high pressure. We givethe example of the first run (later referred to as run 0),where the number of temperatures investigated (15) waslarger than in subsequent runs. Figure 13 shows typicalresults for Pstat as a function of the product ρ(Pstat)Vcav.For each temperature, the data above 1 MPa fall on astraight line. In one of the following runs, we even pres-surized the sample cell up to 20 MPa: the correspondingdata point fell on the extrapolation of the straight lineobtained by fitting the data between 1 and 9 MPa. Thislinear behavior supports the validity of the static pres-sure method (see Sec. IV D), and shows that nonlineari-ties are weak. The pressure extrapolated to zero voltagewill be taken as the cavitation pressure; its variation withtemperature will be presented in Sec. VB.

The error bars in Fig. 13 come from the noise on Vrms

(at a given level, the standard deviation is less than 1%of the average), and from the less relevant fluctuationsin pressure (due to the part of the high pressure setupthat stands out of the thermostated bath, see Fig. 5).From the linear relation between Pstat and ρVcav , wehave estimated the uncertainty on Pcav [55]. For run 0,we find between ±0.3MPa and ±0.8MPa (see Sec. V B).

We can compare the results of both calibrations at25oC. The static pressure method gives −23.6±0.5MPa,whereas the 40 µm (resp. 200 µm) needle hydrophonegives Pmin = −21 ± 2.9MPa (resp. Pmin = −24.5 ±3.4MPa) (the uncertainty comes from the 14% uncer-tainty on each hydrophone gain). These values are re-markably close, which supports our whole calibrationprocedure. Because they shall give respectively a lowerand an upper bound of Pcav, their vicinity also confirmsthat nonlinearities are negligible.

To be exhaustive, we have to mention two problems en-countered in our study: a hysteretical behavior observedjust after filling the experimental cell, and an anomaly inthe variation of Vcav at low pressures.

Before filling, the cell is put under vacuum. It is cooled

to 4oC and then connected to the degassed water sampleat saturated vapor pressure and room temperature. Wa-ter flows inside the cell. After a few minutes, the isolationvalve is closed, and the pressure in the cell can be variedwith the bellow. During the first pressurization, we in-creased the pressure step by step beween 0 and 9 MPa,measuring at each step the cavitation voltage. We no-ticed that the curve obtained during the first pressuriza-tion always differed from the following curves. Further-more, the S-curves (see Sec. IV B) of this first pressuriza-tion were noisy or even hysteretical, and the histogramsof the echo signal did not exhibit a clear threshold; theseanomalies occured only for the first (low pressure) pointsof the first pressurization. On the other hand, if waterwas kept at 9 MPa for some time (typically half an hour),the following S-curves and histograms (at all pressures)were satisfactory, and the curves Pstat(ρVcav) obtainedby depressurizing or pressurizing the liquid fell on top ofeach other without hysteresis. These results are summa-rized in Fig. 14.

Another, more persistent anomaly, was detected: atlow pressure, an elbow appears in the curves Pstat(ρVcav),slightly more pronounced at lower temperature. We havechecked that the elbow was still present after pressurecycles (increasing Pstat from 0 to 9 MPa, keeping thesystem at 9 MPa for 2 hours, and then repeating themeasurements when decreasing Pstat back to 0); the cav-itation voltage in the elbow were reproducible and didnot show any hysteresis. This means that the source ofthe elbow (or more precisely its effect on cavitation) dis-appears above 1 MPa, and reappears below this value.

We see two possible reasons for this behavior. (i) Ei-ther cavitation nuclei are present, and give a cavitationthreshold which itself depends on the cavitation pressure,causing the failure of the static pressure method. Just af-ter filling, they would be of larger size than after the firstpressure cycle, thus explaining the observed hysteresis.These nuclei could be undissolved gas bubbles (possiblyentraped on solid particles). By Laplace’s law, for suchtrapped bubbles not to dissolve at 20 MPa (the high-est pressure reached in one run), the interface with theliquid should reach a radius of curvature of 7 nm. Itseems unlikely that impurities with the correct geometryand wetting properties could exist in sufficient concen-tration to explain our results. Morevover, we shall see inSec. V C that the statistics of cavitation are the same atall static pressures. (ii) Or the properties of the trans-ducer itself are pressure dependent, leading to an arti-fact in the pressure dependence of the cavitation voltage.As the mechanical resonance frequency was measured tobe pressure independent (see Sec. III B), we must lookfor another source of artifact. One can find appropriatesites for bubble trapping inside the transducer itself. Thepiezoelectric material is porous, made of ceramic grainsaround 10 µm, and, whereas the electrodes are not per-meable to water, the edge of the transducer is not, andis able to let the liquid enter the pores, thus changingthe efficiency of the transducer (e.g. by modifying its

14

-28

-24

-20

-16

260 280 300 320 340 360

Cavitation pressure (MPa)

Temperature (K)

FIG. 15: Cavitation pressure as a function of tempera-ture. Pcav was obtained with the static pressure method (seeSec. IV D). Run 0 (filled circles) is compared to our prelim-inary results [56] (empty circles). The uncertainties on Pcav

were calculated as described in Sec. VA.

dissipation and quality factor). Just before filling, thetransducer is completely dry. When Pstat increases from0 to 1 MPa (corresponding to a meniscus with a radiusof curvature of 140 nm), water would invade most of thepores volume, giving a measurable effect on Vcav. A fur-ther increase of Pstat would only affect the remaining freevolume (filled with vapor), without a noticeable changeof the transducer efficiency; but some sites with bottle-necks less than 7 nm would keep some vapor even at 20MPa, allowing the vapor phase to grow again inside thepores, thus explaining the absence of hysteresis duringthe following pressure cycles.

Despite the two anomalies just described, we are con-fident in the use of the static pressure method at highpressure, which is supported by the hydrophone cal-ibration, and the good reproducibility of results (seeSec. V D). In our data analysis, we decided to keep onlythe high pressure part of the curves Pstat(ρVcav) (that isfor 1 MPa ≤ Pstat ≤ 9MPa, as in Fig. 13).

B. Temperature dependence of the cavitationpressure

Figure 15 displays Pcav in run 0 (obtained as describedin Sec. V A) as a function of temperature. We can com-pare this work to our preliminary results [56]. At thattime, we used another hemispherical transducer, resonat-ing at 1.3 MHz. Pcav was also obtained with the staticpressure method, but Pstat varied only between 0 and3 MPa, and the problem of the low pressure elbow (seeSec. V A was not yet noticed. Keeping in mind thesedifferences, the agreement with the present work is sat-isfactory.

We find a monotonous temperature variation, withPcav becoming less negative as T is increased: it varies

from −26.4MPa at 0.1 oC to −16.5MPa at 80 oC. Thereis no obvious minimum, or if a minimum exists it is veryshallow. Anyhow, the experimental results disagree withboth theories as regards the magnitude of Pcav (' −24 in-stead of−120MPa). We will come back to this in Sec. VI.

Let us add a special comment concerning the low tem-perature part. Because of the negative slope of the melt-ing line of water in the P − T plane, stretched water atlow temperature is metastable against vapor and ice for-mation: this is called the doubly metastable region. Hen-derson and Speedy [18] have reported the largest pene-tration in this region: from −19.5 MPa at 0oC to −8MPaat -18oC. The present study exceeds these values, with−26MPa at 0.1oC. We have also observed cavitation at−0.6 oC, but as we kept Pstat = 8.5MPa to avoid bulkfreezing, we could not calibrate the pressure by the staticpressure method.

Before discussing the discrepancy between theory andthis experiment, we will report on how we have checkedits reproducibility.

C. Statistics of cavitation

The results reported in Secs. V A and V B involvedonly the measurement of the cavitation voltage. Rela-tively short acquisitions of S-curves (typically 4 values ofthe excitation voltage each corresponding to 400 repeatedbursts) are sufficient for this purpose. We have also in-vestigated the steepness ξ of the S-curves (see Sec. IVB,Eq. 9). To get enough accuracy on ξ, one needs muchlonger acquisitions: we used typically 25 values of theexcitation voltage each corresponding to 1000 bursts; de-tails about the accuracy of the S-curve parameters aregiven in Sec. A. At frep = 1.75Hz, this corresponds to4 hours, during which the experimental conditions mustremain stable. The temperature stability of the experi-mental region is excellent, controlled by the thermostatedbath. The pressure is more subject to fluctuations, be-cause of the temperature change of the emersed part ofthe handling sytem. We recorded the pressure and foundit to be always stable within a few percent.

The validity of the static pressure method shows thatPcav is independent of Pstat. We can thus convert theexcitation voltage Vrms used at any static pressure Pstat

into the minimum pressure Pmin reached in the wave:

Pmin = Pstat + (Pcav − Pstat)Vrms

Vcav(11)

where Vcav is determined by fitting the S-curve withEq. 9. The S-curves can now be plotted with the cav-itation probability versus Pmin, and fitted with:

Σ(Pmin) = 1− exp− ln 2 exp

[ξ

(Pmin

Pcav− 1

)](12)

Similarly to Eq. 10, ξ is related to the energy barrier for

15

0

0.2

0.4

0.6

0.8

1

20 22 24 26 28

Pstat = 0.48 MPa

Pstat = 1.04 MPa

Pstat = 1.60 MPa

Pstat = 5.57 MPa

Cavitation probability

-Pmin (MPa)

FIG. 16: Cavitation probability as a function of the minimumpressure reached in the wave for 4 different static pressures.Each point is an average over 1000 bursts. The data weretaken during run 0 at T = 4oC.

cavitation through:

ξ = −Pcav

kBT

(∂Eb

∂P

)

Pcav

(13)

The estimation of the uncertainty on the fitting parame-ters is discussed in Appendix A.

TABLE II: Results of the fit with Eq. 12 for the S-curves ofFig. 16. Pstat is measured for each burst, and then averagedfor the 1000 bursts with the same excitation voltage. To illus-trate the stability of Pstat during the acquisition of an S-curve,we give for each curve the average and the extreme values ofthis set of Pstat.

Pstat(MPa) ξ χ2

average min. max.0.48 0.45 0.51 45.2± 1.3 3.61.04 1.01 1.06 44.7± 1.1 2.61.60 1.58 1.62 44.0± 1.8 6.65.57 5.56 5.58 43.3± 1.2 3.1

In run 0, we have measured accurate S-curves at 4oCand several values of Pstat. They are compared in Fig. 16:the agreement is excellent. The values of ξ and the qual-ity of the fits are compared in Table II. The value ofPcav is the same by construction, but the fact that thesteepness of the curves is constant shows that the statis-tics of cavitation is not affected by the application of astatic pressure. Interestingly, this conclusion holds atPstat = 0.48MPa, in the elbow mentioned in Sec. V A:this rules out the possibility that cavitation nuclei areresponsible for this elbow. We have also studied the tem-perature dependence of ξ, measured at Pstat = 1.6MPa.The result is shown in Fig. 17. The value predicted byTWA is:

ξTWA = 2 ln(

Γ0V τ

ln 2

)(14)

35

40

45

50

260 280 300 320 340 360

ξ

Temperature (K)

FIG. 17: Steepness of the S-curves as a function of tempera-ture. Each data was measured in run 0, using 25 points withstatistics over 1000 bursts, at Pstat = 1.6MPa.

Using the values from Eqs. 2 and 8 (with Rc = 1 nm), onegets ξTWA ' 95, practically independent of temperaturein the range of interest. We have checked that this valuecan not be reduced to the measured ones by the exper-imental noise (see Appendix A). We will come back tothis discrepancy in Sec. VI.

D. Reproducibility of results

To check the reproducibility of the results, we have re-peated the measurements using the same procedure fora series of 8 runs, restricting the study to 4 values of thetemperature (0.1, 20, 40 and 80oC), and at each tem-perature 5 values of the pressure (around 1.1, 2.5, 4, 6and 8 MPa). Between each run, the cell was dried andevacuated in the following way. First, most of the waterwas transferred by evaporation/condensation from thecell heated at 80oC to a container cooled at 0oC, withoutcontact with the atmosphere (see Sec. III C, Fig. 5). Adry diaphragm pump was then used to evaporate the re-maining liquid, and finally the cell was pumped througha nitrogen trap by an oil pump to achieve a good vacuum.Alternatively, we could use a dry scroll pump for the lasttwo steps. The system was ready to be filled again withwater degassed as explained in Sec. III F in order to starta new run.

The curves Pcav(T ) obtained for the 8 runs are shownin Fig. 18. At each temperature, values of Pcav fall in a±1MPa pressure range, except at 0.1oC where the scat-ter reaches ±1.7MPa. This is only slightly larger thanthe uncertainty on Pcav that we expect from the staticpressure method, between ±0.3MPa and ±0.8MPa (seeSec. V A). The average value of Pcav varies from−26MPa at 0.1oC to −17 MPa at 80oC.

We have also considered the reproducibility of re-sults during the same run. As explained above, exceptfor the very first pressurization after filling, the curves

16

-28

-24

-20

-16

260 280 300 320 340 360

run 1

run 2

run 3

run 4

run 5

run 6

run 7

run 8

Cavitation Pressure (MPa)

Temperature (K)

FIG. 18: Cavitation pressure as a function of temperature,obtained with the static pressure method (see Sec. IV D), forruns 1 to 8. The error bars are omitted for clarity; they aresimilar to those in Fig. 15.

Pstat(ρVcav) showed no hysteresis. For some of the runsand temperatures, we have repeated the measurement ofPcav by the static pressure method after a time inter-val which could reach 9 days. For some of the 8 runs,we observed a small drift in the value of Vcav at a givenPstat (less than 1%). However, the value of Pcav wasalways found to be stable within ±0.2 MPa, except at0.1oC where the variation could reach ±0.75 MPa.

After each run, the UV absorption spectrum of the wa-ter collected by evaporation/condensation was recorded.As can be seen in Fig. 19, each spectrum shows a similarincrease of absorption (compared to ultrapure water) inthe range 190-240 nm. If one compares with the spectraobtained in ultrapure after contact with each material inthe cell (Sec. III E, Fig 6), we may attribute this increaseto the ceramic material of the transducer.

VI. DISCUSSION

The experimental method used in this study gives morenegative values for the cavitation pressure than others,except the one using mineral inclusions (see Sec. II). Inour case, the pressure was calibrated by two independentmethods in good agreement with each other. In addition,the acoustic technique allows a detailed investigation ofthe cavitation statistics and a clear definition of the cav-itation threshold. The results are highly reproducible, incontrast with other studies in which cavitation pressurescan be highly scattered and time-dependent, and some-times only their most negative values are reported. Thisis made possible because in our experiment, the negativepressure is developped during a short time, in a smallexperimental volume far from any wall: the influence ofimpurities, if any, is greatly suppressed.

However, despite these advantages, we have notreached the highly negative cavitation pressures observed

0

0.1

0.2

0.3

180 200 220 240 260 280 300

Relative absorbance

Wavelength (nm)

4

2

0

3

1

0

0.1

0.2

0.3

180 200 220 240 260 280 300

Relative absorbance

Wavelength (nm)

5

6

7

8

FIG. 19: UV absorption spectrum of the water collected fromthe experimental cell for 9 successive runs. We used an auto-mated spectrometer (Kontron Instruments, Uvikon 941) withquartz cuvettes (10mm thickness). The reference cuvette wasfilled with ultrapure water directly drawn from the water pol-isher and sealed with a Teflon cap. The label indicates therun number.

in the inclusion work. In fact, there is a large gap in cav-itation pressure data between the inclusion work (reach-ing Pcav < −100MPa at room temperature) and all otherexperiments (Pcav > −30MPa) (see Sec. II F). We canattribute the discrepancy between the present work andthe inclusion work to three possible reasons: (i) impuri-ties; (ii) error in the pressure estimate in the inclusionwork; and (iii) error in the pressure calibration of ourexperiment. We can rule out reason (iii), because of thegood agreement we find between two independent cali-bration methods (see Sec. IV C and IVD). We also em-phasize that our values of the cavitation pressure are closeto those reported in experiments were a direct measure-ment of the pressure was available (from the centrifugalforce [21] or with pressure gauges [17–19]). Let us exam-ine the two other hypotheses and their implications.

In Sec. VI A, we present some models of heteroge-neous cavitation, and compare their predictions for thestatistics of cavitation with our measurements. We have

17

also investigated different water samples, with differentpreparation or origin, in order to vary their possible con-tent in impurities; the results are presented in Sec. VI B.Finally, we discuss the pressure calibration in the inclu-sion work in Sec. VIC.

A. Impurities and cavitation statistics

In this paragraph we will discuss the possible effect ofimpurities on the statistics of cavitation. In particular,we are interested in the information that can be obtainedfrom the accurate measurements of the cavitation prob-ability versus excitation voltage (S-curve) presented inSec. IV B.

We can think of different kinds of impurities. Firstwe consider impurities which lower the energy barrierfor cavitation Eb(P ); let us call them type I impurities.These could be surfactant molecules, reducing the en-ergy cost associated with the creation of a liquid-vaporinterface, or dissolved molecules that change the localstructure of water (e.g. by disrupting the network ofhydrogen bonds). For example, the observed cavitationpressure could be explained within the TWA model if thesurface tension σ is changed to an effective value: for in-stance, σeff = 18.7mN m−1, would give Pcav = −24MPaat 20oC.

For simplicity we suppose that we have only one kindof type I impurities, that is they all lead to the samemodified energy barrier. If their concentration n0 is suf-ficiently high (typically n0(λ/16)3 À 1, see Eq. 8), thecavitation probability for a minimum pressure Pmin inthe wave will be given by Eq. 12 as for homogeneouscavitation, except that the cavitation pressure Pcav willbe lowered. We already know (see Fig. 9) that this modelprovides a good description of the data. The distributionof impurities can be more complicated: for instance, forsurfactant molecules at a concentration below the crit-ical micellar concentration, the size distribution of mi-cellar aggregates is a decreasing function of size [57]. IfEb reaches a minimum at an optimum aggregate size,the relevant concentration would then be the total con-centration of impurities with Eb within a small interval(' kBT ) around this minimum.

Now we consider a different kind of impurities, with adeterministic effect on cavitation; let us call them typeII impurities. We assume that to each impurity is associ-ated a threshold pressure Pt < Psat, and that cavitationwill occur if and only if the impurity is submitted toa pressure more negative than Pt. Type II impuritiescould be vapor bubbles, either stabilized by an organicshell [58], or pinned in a crevice on a hydrophobic par-ticle and stabilized by their radius of curvature [59, 60].The typical radius R of such bubbles can be estimatedwith Laplace’s law: R = 2σ/Pt; for Pt ' −25MPa andusing for σ the bulk surface tension of water, one findsR ' 5.8 nm.

The statistics of cavitation will depend on the dis-

tribution of type II impurities. We have investigatedseveral distributions (the calculations are given in Ap-pendix B). We have found that a 5th-power-law distri-bution between pressures P1 and P2 (see Appendix B,inset of Fig. 25) provides a good fit; it could be still im-proved by changing the distribution. The fitting pro-cedure leads to the total concentration of impuritiesn0 through n0λ

3/(P2 − P1)6 = 41.4MPa−6. P2 − P1

must be larger than the width of the probability curve,' 5MPa; this lead to a lower bound on the concentra-tion: n0 ≥ 1600mm−3.

Is it possible to distinguish between type I and type IIimpurities? The strongest difference between both kindof impurities is the dependence with the frequency ofthe sound wave. On one hand, for type I impurities, achange in the frequency will affect V and τ (see Eq. 7),but only slightly Pcav and ξ because of the logarithm inEq. 6. On the other hand, the S-curve for type II im-purities involves n0λ

3 (see Appendix B). In our prelim-inary study we used a transducer resonating at 1.3MHzinstead of the transducer at 1MHz; between these twofrequencies the value of λ3 changes by a factor of 2.2.We have already seen that the respective cavitation pres-sures were very close (see Fig. 15). If we compare ac-curate S-curves measured for each frequency under thesame experimental conditions, we find that the slopesare equal. However, one can argue that as each trans-ducer was used with a different water sample, this maybe a coincidence. To rule this out, we have repeatedthe measurement of accurate S-curves for a 1MHz and a1.3MHz transducers immersed in the same ultrapure wa-ter sample in an open Pyrex container. Figure 20 showsthat the slope is the same. We have normalized the ex-citation voltage by the cavitation voltage, assuming thecavitation pressure to be the same for both transduc-ers as found before. Using Eq. B8 with P2/Pcav = 0.8,the best fit is obtained with a 11/2-power law, andgives n0λ

3 [Pcav/(P2 − P1)]13/2 = (3.62± 0.10) 107 (resp.

(3.62 ± 0.12) 107) for the 1 MHz (resp. 1.3MHz) trans-ducer. In the case of cavitation on type II impurities,this parameter would change by a factor 2.2, which islarge enough to be detected by our accurate cavitationstatistics. We can therefore exclude the possibility thatcavitation is heterogeneous on type II impurities.

We have only considered type I impurities in large con-centration (n0(λ/16)3 À 1). If their concentration issmall, in addition to thermal activation, we have to takeinto account their probability of presence. In this case,we expect the slope of the S-curve to depend on the soundwave frequency in a similar way as for type II impurities,which is excluded.

We conclude that if cavitation is heterogeneous, it oc-curs on type I impurities. Their concentration shouldbe larger than 1/(λ/16)3 ' 2800mm−3 (based on the1.3MHz sound wave), and they should be present in allthe water samples we have tested. On the other hand, aswe have observed zero cavitation probability at moderatenegative pressures, impurities leading to a less negative

18

0

0.2

0.4

0.6

0.8

1

0.9 0.95 1 1.05

Cavitation probability

Vrms/V

cav

FIG. 20: Cavitation probability as a function of the exci-tation voltage normalized by the cavitation voltage. Eachfilled (resp. empty) circle shows the probability measuredwith a 1 MHz (resp. 1.3MHz) transducer, by countingthe cavitation events over 1000 bursts, at room temper-ature in an open Pyrex container. A fit with Eq. B8(solid line), using P2/Pcav = 0.8 and α = 11/2, give

n0λ3 [Pcav/(P2 − P1)]

13/2 = (3.62±0.10) 107 for both frequen-cies.

threshold should always be absent, or with a concentra-tion much less than 2800mm−3.

To give a consistent picture, the scenario of hetero-geneous cavitation should also explain the observed lowvalue of ξ (see Sec. V C). If we model the effect of typeI impurities on Eb by an effective surface tension σeff ,and use the predictions of TWA replacing σ by σeff , theexpression of ξ (Eq. 14) remains unaffected, because σcancels out in the calculation. But the factor Γ0 is differ-ent from the value estimated for homogeneous cavitation(Eq. 2): the density of nucleation sites is the density ofimpurities n0 instead of 1/(4πR3

c/3), and we write:

Γ0 = n0kBT

h(15)

This allows us to estimate n0 between 1.4 107 mm−3 at0.1oC and 9.1 104 mm−3 at 80oC. This is consistent withthe lower bound 2800 mm−3. The uncertainty on n0 islarge because of the exponential relation to ξ, but thestrong temperature dependence is a robust feature, forwhich we do not see any obvious explanation.

B. Different water samples

In this paragraph, we discuss the results obtained fordifferent water samples.

We have used high resistivity ultrapure water, degassedand transferred in the experimental cell under vacuum;inert materials were chosen to limit water contamination,and UV spectra of water before and after use show only

-28

-24

-20

-16

260 280 300 320 340 360

run 0

run 9

run 10

Cavitation Pressure (MPa)

Temperature (K)

(a)

0

0.2

0.4

0.6

0.8

1

19 20 21 22 23 24 25 26

run 0

run 9

run 10

Cavitation probability

-Pressure (MPa)

(b)

FIG. 21: Comparison between different water samples. Theinitial run (run 0, filled circles, as in Fig. 15) is shown forcomparison. In run 9 (empty circles), we used ultrapure waterfrom a different source (G Chromasolv for gradient elution,purchased from Sigma), and degassed with the method de-scribed in Sec. III F. In run 10 (empty squares), we usedmineral water (Speyside Glenlivet, dry residue at 180oC:58mg L−1)[61], distilled in a Pyrex glassware and degassedwith the same method. (a) Cavitation pressure as a functionof temperature; the error bars on Pcav are omitted for clarity:they all have a similar amplitude (see Fig. 15). (b) Cavitationprobability as a function of the minimum pressure reached inthe wave. The S-curves were obtained at T = 20 oC andPstat = 1.6MPa. The excitation voltages were converted intopressure using Eq. 11 and the value of Pcav shown in (a). Thesolid curves are fits with Eq. 12; they all give similar valuesof the steepness ξ: 43.3 ± 1 for run 0 (χ2 = 2.2), 44.6 ± 1.5for run 9 (χ2 = 3.9), and 48.1± 1.9 for run 10 (χ2 = 4.6).

a moderate increase in absorption below 240 nm. Succes-sive experimental runs with renewed water samples givereproducible values of Pcav. We have also tried to varythe water sample preparation or origin. Run 9 used ul-trapure water purchase from Sigma (G Chromasolv forgradient elution) and degassed with the same method asusual. Run 10 used a mineral water. We chose SpeysideGlenlivet for its low minerality (dry residue at 180oC:58mg L−1) [61]). It also has the advantage to be pack-

19

0

0.2

0.4

0.6

0.8

1

150 200 250 300

Relative absorbance

Wavelength (nm)

Sigma bottle

Sigma cell

Glenlivet cell

Glenlivet bottle

Glenlivet distilled

and degassed

FIG. 22: UV spectra of different water samples. The labelsindicate the water origin and preparation: Sigma refers to wa-ter used in run 9, and Glenlivet in run 10. Bottle is relativeto the UV spectrum of water directly coming out of the com-mercial bottle, whereas cell refers to the water coming out ofthe cell at the end of the run. We also show the spectrum ofGlenlivet water after distillation and degassing: these stepsdecrease the UV absorbance.

aged in glass bottles: mineral water in plastic bottlesshows a strong UV absorption. In addition, we distilledthe mineral water in Pyrex glassware before performingthe usual degassing. All the setup was rinsed with thecorresponding water before filling. We find the same Pcav

(Fig. 21 (a)) and the same cavitation statistics (Fig. 21(b)). The UV spectra are shown in Fig. 22: the watersamples coming out of the cell exhibit a larger absorbancethan usual. It is difficult to trace out the origin of thispollution, but we think it occured outside of the cell dur-ing the emptying. We have checked that it was not dueto a surfactant contamination by measuring the surfacetension with the Du Nouy method: we found for bothsamples before and after entering the cell the same valueas for our reference ultrapure water, σ = 71± 1mNm−1

at 25oC.In order to investigate the role of impurities, we also

performed experiments in dirty water. The cavitationvoltage in tap water, measured in an open glass container,was found to be slightly larger than the one of ultrapurewater in the same conditions. This effect is reduced ifwe cover the transducer with an insulating varnish: it ispresumably due to the higher conductivity of tap waterwhich reduces the transducer efficiency. Figure 23 showsthe comparison between S-curves obtained for tap andultrapure water and normalized by the respective cavita-tion voltages. This normalization assumes that the 50%probability threshold is the same in both samples; this issupported by the similarity of the centers of the S-curves.The low probability part of the curves are strikingly dif-ferent: ultrapure water goes smoothly to zero probabil-ity, whereas tap water gives a long tail with finite prob-ability. This shows that our experiment is sensitive to

0

0.2

0.4

0.6

0.8

1

0.8 0.9 1 1.1

Cavitation probability

Vrms/V

cav

FIG. 23: Cavitation probability as a function of the excita-tion voltage normalized to the cavitation voltage. Full (resp.empty) circles stand for ultrapure (resp. tap) water. The datawere recorded at room temperature in an open Pyrex glasscontainer. Each point corresponds to 1000 repeated bursts.The solid line is a fit of the data for ultrapure water withEq. 9. The center of these S-curves are similar, but the footfor tap water is much broader than for ultrapure water. Thisindicates the presence in tap water of a broad distribution ofdilute impurities triggering cavitation at pressures less nega-tive than the cavitation pressure of ultrapure water.

the numerous impurities present in tap water, some ofwhich trigger cavitation at moderate negative pressure.It shows also that the distribution of such impurities issufficiently broad and dilute to leave the narrow centerof the S-curves unaffected. This means that either ourexperiment reaches the homogeneous cavitation limit ofwater, or there are very different kind of impurities: di-lute, moderate cavitation pressure impurities that can beremoved by usual water purification, and a more abun-dant quantity of Type I impurities (see Sec. VI A). Thelatter have to be exceptionnaly calibrated and ubiqui-tous to explain the identical Pcav and statistics betweendifferent runs and samples. We have also investigatedthe cavitation threshold of water saturated with differ-ent gases (helium, nitrogen and acetylene, which have arelative solubility in water 1:1.7:106 at 20oC). This wasdone in sealed cells with a gas handling system. The cav-itation voltage varied only by a few percent compared todegassed ultrapure water.

The most important point is that, if impurities areresponsible for our results, all the water samples testedmust have the same impurities in sufficient concentra-tion. In the inclusion case, we shall recall that cav-itation pressures fall in a narrow range at high tem-perature, but exhibit a large scatter at low tempera-ture; only the largest negative pressures are reported(see Sec. II E). Angell and his group attribute the scat-ter to heterogeneous nucleation, and its source to ‘pos-sibly surfactant molecules cluster destroyed by anneal-ing at the higher temperatures’ (in excess of 400oC).

20

At low temperature, the impurities should have a lowenough concentration to be absent from some of the in-clusions, where homogeneous nucleation would thus takeplace; using the inclusion volume from Table I, this givesa typical concentration of one impurity per inclusion:n0 ' 1/(4.2 10−6 mm3) = 2.4 105 mm−3, larger than thethreshold at which our method becomes sensitive to im-purities (2.8 103 mm−3, see Sec. VI A). The same remarkapplies to all the experiments involving larger liquid vol-umes than the inclusions, and is able to explain why theyall obtain less negative cavitation pressures. It would beinteresting to check this issue by exposing water to hightemperatures before its use in our experiment. However,this presents some technical problems in the design of thecell; in addition, the Curie point of the piezoelectric ma-terial is 300oC: water should then be heated outside ofthe cell, or the transducer repoled after thermal cycling.

We now turn to the other reason proposed to under-stand the discrepancy between the present work and theinclusion work: (ii) error in the pressure estimate in theinclusion work.

C. Pressure calibration in the inclusion work andthe equation of state of water

How reliable is the pressure estimate in the inclusionwork? In this method, the quantity which is measuredis the cavitation temperature, from which the pressureis deduced from the known liquid density and the useof an EOS extrapolated in the negative pressure range.Alvarenga et al. [32] have shown that the assumptionof an inclusion of constant volume (required to know theliquid density) was not always reliable. To estimate Pcav,they rather used the measured change in sound velocityc before and after nucleation, and found values beyond−100MPa at room temperature. However, they still relyon an extrapolated EOS: they assume (dc/dP )T to keep aconstant value, equal to that at liquid-vapor equilibrium.

One possibility would then be that the extrapolatedEOS are wrong: thermodynamic properties of watershould then exhibit dramatic changes in a narrow rangebeyond this value to make all data compatible. Thiscould happen for example if the spinodal pressure Ps wasmuch less negative than expected from Speedy’s extrapo-lations or molecular dynamics simulations (see Sec. IA).Let us try to make this speculation more quantitative. Ifcavitation in our experiments is homogeneous, then wehave access to two quantities related to the EOS: theenergy barrier and its slope at Pcav. Indeed, we write:

Eb(Pcav) = kBT ln(

Γ0V τ

ln 2

)' 47.5 kBT (16)

taking for Γ0 the value from Eqs. 2 (with Rc = 1nm)and for V τ the value from Eq. 8. The slope is given by:

(∂Eb

∂P

)

Pcav

= −ξkBT

Pcav(17)

-60

-50

-40

-30

-20

-10

260 280 300 320 340 360

Pressure (MPa)

Temperature (K)

FIG. 24: Speculated spinodal pressure as a function of tem-perature. The values of the spinodal pressure (empty circles)were deduced from the cavitation pressures (filled circles) andsteepness of the S-curves (see Fig. 17) measured in run 0.

which ranges from 500 to 780 K MPa−1 from 0.1 to80oC (with the values of ξ from Fig. 17). By definition,Eb(Ps) = 0. To locate Ps, we extrapolate linearly Eb

to zero using the above values (Fig. 24). To get an ideaof the accuracy of such an extrapolation, we can look atthe curve Eb(P ) calculated with DFT [2]: it is convex,and nearly linear in the relevant range of Eb. We thusexpect the actual Ps to be slightly more negative thanthe extrapolated value.

The spinodal proposed is much less negative than pre-vious estimates (around −50 instead of −200MPa at300K). Although surprising, it is not impossible, be-cause of the possible uncertainties in the interparticle po-tential used in molecular dynamics simulations, or in thelengthy extrapolations of P (ρ) involved in Speedy’s EOS.In this alternative interpretation, the observed tempera-ture dependence of ξ is simply related to the temperaturedependence of the EOS.

VII. CONCLUSION

We have focused short bursts of high frequency ultra-sonic waves to study cavitation in water. A detailed ex-perimental procedure was followed to insure cleanlinessof the water sample. Statistics of cavitation could bemeasured with high accuracy, allowing a clear definitionof the cavitation threshold. We have used two indepen-dent methods to calibrate the negative pressure reachedin the wave, and found them to be in excellent agree-ment with each other. This allowed us to measure thetemperature dependence of the cavitation pressure. Ourresults were proven to be highly reproducible. We havefound a cavitation pressure that increases monotonicallywith increasing temperature, from −26 MPa at 0.1oC to−17MPa at 80oC.

Our values of the cavitation pressure are among the

21

most negative ones, with the only exception of thoseobtained in the work on mineral inclusions. In orderto understand this discrepancy, we propose two experi-mental checks. On one hand, further efforts should bemade to improve the purity of water. We have alreadytaken many precautions in this direction, and explainedthat the agreement of our results with previous stud-ies, their reproducibility and the statistics of cavitation,disfavor the assumption that nucleation occurs on impu-rities in our experiment. Nevertheless, a difference re-mains between the inclusion work and others, includingours. During the making of inclusions, water is broughtto high temperature and pressure: it would be interest-ing to check the effect of this procedure on cavitationpressures obtained by our method. On the other hand,the inclusion work lacks a direct measurement of the cav-itation pressure, because it has to rely on the use of anextrapolated EOS. All results could be made compati-ble if the EOS of water was much different from whathas been admitted up to now: in particular, the spin-odal pressure could be much less negative than previouslythought (around −50 instead of −200 MPa at 300 K).It is necessary to add experimental information on theEOS of water at large negative pressure: for this, weplan to combine our acoustic method to produce negativepressure with optical measurements to measure indepen-dently the density and speed of sound in the metastableliquid.

Acknowledgements

We thank A. Berthelot, V. Fourmond, B. Haas, andR. Melet for their participation at various stages of theexperiment. We thank E. Perez for providing us withultrapure water, A. Boudaoud, C. Guthmann, L. Mer-cury, R. Pecha, and E. Rolley, for helpful discussions,and C. Herrmann, G. Narcy, and J. Quintas for techni-cal assistance. We are grateful to M. McKenna and B. B.Chick from Ritec Inc. for their help and advices. Supportfrom ANR grants 05-BLAN-0084-01 and JC05-48942 isacknowledged.

APPENDIX A: ACCURACY OF THESTATISTICS OF CAVITATION

As explained in Sec. IVB, our method allows to re-peat many cavitation experiments under the same condi-tions. This determines accurately the probability of cavi-tation. We have measured the cavitation probability ver-sus excitation voltage curves (S-curves). Once the pres-sure has been calibrated with the static pressure method(Sec. IVD), the excitation voltage is converted into min-imum pressure reached in the wave. The S-curves arewell fitted by the relation expected for thermally acti-vated cavitation (Eq. 12). Here we estimate the errorbars on the fitting parameters Pcav and ξ. In the ab-

0

0.2

0.4

0.6

0.8

1

20 21 22 23 24 25 26

Cav

itat

ion p

robab

ilit

y

-Pressure (MPa)

-Pressure

dn/d

P

-P1

-P2

-P0

FIG. 25: Cavitation probability as a function of the minimumpressure reached in the wave. The data points are the same asin Fig. 9. The excitation voltage has been converted into pres-sure using Eq. 11 and Pcav = −23.845MPa. Only pressuresbetween −20 and −26 MPa are shown. The solid line is a fitwith Eq. 12. The best fit gives −Pcav = 23.847 ± 0.019MPaand ξ = 43.3 ± 1 with χ2 = 2.2. The inset shows the twodistributions of type II impurities tested (see Appendix B).Eq. B6 (corresponding to the monodisperse distribution) doesnot allow a good fit; the dotted line is drawn with P0 =−23MPa and n0λ

3 = 600 to pass through the high probabil-ity data. The dashed line is a fit with Eq. B8; P2 was set to−20MPa, and the best fit was obtained for α = 5: it givesn0λ

3/(P2 − P1)6 = (0.31± 0.01)MPa−6 and χ2 = 3.7.

sence of other noise, the uncertainty on the probabilityΣ comes from the finite number of repeated bursts usedfor its measurement, Nb; the standard deviation SD isgiven by the binomial law: σΣ =

√NbΣ(1− Σ). This

can be used in the fitting procedure to give a normal-ized mean-square deviation χ2 and error bars on the fit-ting parameters. The fit has to be limited to the regionwhere Σ is different from 0 and 1, in order to keep σΣ

non zero. For instance, applying this procedure to thedata in Fig. 25 gives χ2 = 2.2, Pcav = 23.85± 0.016MPaand ξ = 43.3± 1.

However, we must also account for the noise on theexcitation voltage Vrms applied to the transducer. Thevalue of Vrms is recorded for each burst; when the out-put level of the amplifier is set to a constant value, weobserve that Vrms follows a normal distribution: typicalvalues are a 0.6V SD for a 150V average. The corre-sponding average value and SD of Pmin are deduced fromthe affine relation Eq. 11 (typically a 0.1MPa SD fora −25MPa average). The noise on Pmin tends to en-large the S-curve and decrease the steepness ξ. To esti-mate this effect we have performed a numerical experi-ment in the following way. We assume that cavitationis a stochastic phenomenon occuring with the probabil-ity given by Eq. 12, with Pcav = Pcav,0 and ξ = ξ0. Wechoose Npts values of Pmin between Pm and PM. For eachPmin, we take Nb values Pi of a normal distribution with

22

TABLE III: Characterization of the fitting results of the simulated S-curves. The parameters used in the simulation were:−Pcav,0 = 24 MPa, ξ0 = 43, −Pm = 21.7MPa, −PM = 25.0MPa, σP = 0.1MPa, Nsimul = 1000, Npts = 15, and Nb = 1000.

average skewness standard deviation average asymptotic standard errors−Pcav(MPa) 24.003 0.111 0.0093 0.0093ξ 42.75 0.069 0.629 0.617χ2 1.064 1.51 0.448 –

an average Pmin and a SD σP ; for each Pi, a cavitationevent occurs with a probability Σ(Pi). The average ofthe Pi and of the number of cavitation events gives onedata point of an S-curve. Nsimul S-curves are simulatedthis way, and fitted with Eq. 12. This leads to Nsimul

pairs of fitting parameters, with their asymptotic stan-dard error bars (1-σ confidence interval), and values ofχ2. Table III give the statistical properties of the results;the simulation was performed using input values close tothe experimental ones (see caption of Table III). It canbe seen that Pcav is correctly obtained by the fit, whereasξ is slightly underestimated, as expected. Nevertheless,the noise on the excitation voltage is sufficiently smallcompared to the width of the S-curves to give reliablefitting parameters, with the actual values lying inside theerror bars. Therefore we can use the fitting parametersand asymptotic standard error bars obtained by fittingthe experimental data weighted by the SD on the prob-ability given by the binomial law. For example, the bestfit of the central region of the S-curve shown in Fig. 25gives −Pcav = 23.847± 0.019MPa and ξ = 43.3± 1 withχ2 = 2.2.

In Sec. V C, we have emphasized that the measured ξ(around 40) was significantly lower than the value pre-dicted by TWA (around 95). To check that this is notan artifact due to an enlargement of the width of theS-curve by the experimental noise, we have performedsimulations similar to the above ones, assuming ξ = 95:the noise does enlarge the S-curve, but only slighlty: wefind a fitting value of ξ = 90.3±1.3. The artifact is ruledout and the discrepancy confirmed.

APPENDIX B: CAVITATION PROBABILITY ONTYPE II IMPURITIES

We have discussed the role of type II impurities inSec. VIA. We give here the details of the calculationof the cavitation probability. Type II impurities have adeterministic effect on cavitation: it occurs if and onlyif the impurity is present at a position where the pres-sure exceeds a threshold defined by the impurity. Letus call dn/dP the distributions of these thresholds: theconcentration of impurities with a threshold between Pand P + dP is (dn/dP ) dP . We then calculate the cav-itation probability at a time where the pressure reachesits minimum Pmin at the focus; if the pressure at a point

r is P (r), we find:

Σ(Pmin) = 1− exp

(−

∫

V

dr∫ Psat

P (r)

dPdn

dP

)(B1)

where V is the total volume of liquid. In practice, becausethe sound wave is sharply focused, V can be replaced bythe volume of the focal region.

The simplest distribution is that of impurities all hav-ing the same threshold P0:

dn/dP = n0δ(P − P0) (B2)

where n0 is the concentration of impurities.This situationis experimentally realized in the case of electron bubblesin liquid helium [62]. Then

Σ(Pmin) = 1− exp [−n0V (P < P0|Pmin)] (B3)

with V (P < P0|Pmin) the volume in which the pressure isbelow the threshold P0, provided the minimum pressureis Pmin. At this stage we need an expression for P (r): wecould use the map of the acoustic field experimentallydetermined (see Sec. IV C), but the calculation becomessimpler if we take the approximation of a focused spher-ical wave:

P (r) = Pminsin(kr)

kr(B4)

where r = |r| and k = 2π/λ. If we introduce r0 such thatP (r0) = P0, we find:

Σ(Pmin) = 1− exp(−n0

43πr0

3

)(B5)

An expansion in power of ε = 1− (P0/Pmin) gives:

Σ(Pmin) = 1− exp

[−n0λ

3

√6

π2ε3/2

(1 +

920

ε + O(ε2))]

(B6)However, Eq. B6 fails to give a good fit to the ex-

perimental data, because of the pronounced inflexion ofthe S-curves. This leads us to consider more elaborateddistributions. We have tried power-laws, starting at apressure P2 and truncated at a pressure P1 (see inset ofFig 25):

dn/dP =αn0

(P2 − P1)α (P2 − P )α−1 (B7)

23

TABLE IV: Characterization of the fitting results of the simulated S-curves. The parameters used in the simulation were:α = 5, −P2 = 20 MPa, n0λ

3/(P2 − P1)6 = 0.31MPa−6, −Pm = 22.1MPa, −PM = 25.0MPa, σP = 0.1MPa, Nsimul = 1000,

Npts = 15, and Nb = 1000.

average skewness standard deviation average asymptotic standard errorsn0λ

3/(P2 − P1)6(MPa−6) 0.3090 0.118 0.0047 0.0046

χ2 1.061 0.791 0.397 –

This gives

Σ(Pmin) = 1− exp−n0λ

3(

−P2P2−P1

)α+1

K(α) ε3/2

× [1 + L(α)ε + O(ε2)

](B8)

with

K(α) =(

32π

)3/2 Γ(α + 2)Γ(α + 7

2 )(B9)

and

L(α) =74

+1310

α− 310

(α + 1)(α + 5)α + 7

2

(B10)

As can be seen from Eq. B8, P2 and n0λ3/(P2−P1)α+1

are dependent parameters. Therefore, to fit the data, wefix P2 at a value chosen as a pressure just above thelowest negative pressure at which a non zero probabil-ity is observed in the experiment, and find the best fitvalue for n0λ

3/(P2 − P1)α+1. To estimate the quality ofthe fit and the uncertainty, we have performed numer-ical experiments as explained in Appendix A, but nowwith Σ(Pmin) given by Eq. B8. The results of the simula-tion are reported in Table IV. As in Appendix A, we findthat the actual value of n0λ

3/(P2−P1)α+1 lies within theasymptotic error bars given by the fit. We will take thisvalue for the uncertainty in a fit of experimental data.An example is given in Fig. 25; the best fit, obtainedwith α = 5, is satisfactory (χ2 = 3.7).

[1] P. G. Debenedetti, Metastable liquids (Princeton Univer-sity Press, Princeton, 1996) and references therein.

[2] F. Caupin, Phys. Rev. E 71, 051605 (2005).[3] R. Becker and W. Doring, Ann. Phys. (Leipzig) 24, 719

(1935).[4] Y. B. Zel’dovich, Zh. Eksp. Teor. Fiz. 12, 525 (1942).[5] J. C. Fisher, J. App. Phys. 19, 1062 (1948).[6] M. S. Pettersen, S. Balibar, and H. J. Maris, Phys. Rev.

B 49, 12062 (1994).[7] R. J. Speedy, J. Phys. Chem. 86, 982 (1982).[8] P. H. Poole, F. Sciortino, U. Essmann, and H. E. Stanley,

Nature 360, 324 (1992).[9] S. Sastry, P. G. Debenedetti, F. Sciortino, and H. E.

Stanley, Phys. Rev. E 53 6144 (1996).[10] M. Yamada, S. Mossa, H. E. Stanley, and F. Sciortino,

Phys. Rev. Lett. 88, 195701 (2002).[11] G. S. Kell, Am. J. Phys. 51, 1038 (1983).[12] D. H. Trevena, J. Phys. D: Appl. Phys. 17, 2139 (1984).[13] C. T. Avedisian, J. Phys. Chem. Ref. Data 14 695 (1985).[14] T. G. Leighton, The acoustic bubble (Academic Press,

London, 1994).[15] F. Caupin and E. Herbert, submitted to C. R. Physique.[16] M. Berthelot, Ann. Chim. Phys. 30, 232 (1850).[17] S. J. Henderson and R. J. Speedy, J. Phys. E: Sci. Instr.

13 778 (1980).[18] S. J. Henderson and R. J. Speedy, J. Phys. Chem. 91

3062 (1987).[19] K. Hiro, Y. Ohde, and Y. Tanzawa, J. Phys. D: Appl.

Phys. 36 592 (2003).[20] O. Reynolds, Mem. Manch. Lit. Phil. Soc. 17, 159 (1878).[21] L. J. Briggs, J. Appl. Phys. 21 721 (1950).[22] C. Wurster, M. Kohler, R. Pecha, W. Eisenmenger, D.

Suhr, U. Irmer, F. Brummer, and D. Hulser, in Proceed-

ings of the 1st World Congress on Ultrasonics, Berlin,1995, edited by J. Herbertz (Universitat Duisburg-Essen,Duisburg, 1995), Part 1, p. 635.

[23] J. Staudenraus and W. Eisenmenger, Ultrasonics 31, 267(1993).

[24] R. Pecha, private communication[25] W. J. Galloway, J. Acoust. Soc. Am. 26 849 (1954).[26] M. Greenspan and C. E. Tschiegg, J. Res. Nat. Bur.

Stand. C 71 299 (1967).[27] R. D. Finch, J. Acoust. Soc. Am. 36 2287 (1964).[28] W. J. Galloway, private communication to R. D. Finch,

Ref. 6 of Ref. [27].[29] J. L. Green, D. J. Durben, G. H. Wolf, and C. A. Angell,

Science 249, 649 (1990).[30] Q. Zheng, D. J. Durben, G. H. Wolf, and C. A. Angell,

Science 254, 829 (1991).[31] V. P. Skripov, in Water and Steam, edited by J. Straub

and K. Scheffer (Pergamon, Elmsford, NY, 1980).[32] A. D. Alvarenga, M. Grimsditch, and R. J. Bodnar, J.

Chem. Phys. 98 8392 (1993).[33] Q. Zheng, J. Green, J. Kieffer, P. H. Poole, J. Shao, G.

H. Wolf, and C. A. Angell, in Proceedings of the NATOAdvanced Research Workshop on Liquids Under NegativePressure, Budapest, 2002, edited by A. R. Imre, H. J.Maris, and P. R. Williams (NATO Science Series, SeriesII: Mathematics, Physics and Chemistry, vol. 84, Kluwer,Dordrecht, 2002), p. 33.

[34] M. Takahashi, E. Izawa, J. Etou, and T. Ohtani, J. Phys.Soc. Jap. 71 2174 (2002).

[35] H. W. Strube and W. Lauterborn, Z. Angew. Phys. 29349 (1970).

[36] C. G. Wohl et al., Rev. Mod. Phys. 56, S1 (1984).[37] R. A. Roy, S. I. Madanshetty, and R. E. Apfel, J. Acoust.

24

Soc. Am. 87 2451 (1990).[38] C. K. Holland and R. A. Roy, J. Acoust. Soc. Am. 88

2059 (1990).[39] C. X. Deng, Q. Xu, R. E. Apfel, and C. K. Holland, J.

Acoust. Soc. Am. 100 1199 (1996).[40] J. A. Nissen, E. Bodegom, L. C. Brodie, and J. S. Semura,

Phys. Rev. B 40, 6617 (1989).[41] H. Lambare, P. Roche, S. Balibar, H. J. Maris, O. A.

Andreeva, C. Guthmann, K. O. Keshishev, and E. Rolley,Eur. Phys. J. B 2, 381(1998).

[42] F. Caupin and S. Balibar, Phys. Rev. B 64, 064507(2001).

[43] NIST Chemistry WebBook, Thermo-physical Properties of Fluid Systems,http://webbook.nist.gov/chemistry/fluid/

[44] A. D. Bangham and M. W. Hill, Nature 237 408 (1972).[45] L. J. Heidt, and L. Ekstrom, J. Am. Chem. Soc. C 79

1260 (1957).[46] G. W. Willard, J. Acoust. Soc. Am. 25 669 (1953).[47] L. A. Crum and J. B. Fowlkes, Nature 319 52 (1986).[48] J. B. Fowlkes and L. A. Crum, J. Acoust. Soc. Am. 83

2190 (1988).[49] A. A. Atchley, L. A. Frizzell, R. E. Apfel, C. Holland, S.

Madenshetty, and R. Roy, Ultrasonics 26, 280 (1988).[50] F. Caupin and V. Fourmond, in Proceedings of the NATO

Advanced Research Workshop on Liquids Under NegativePressure, Budapest, 2002, edited by A. R. Imre, H. J.

Maris, and P. R. Williams (NATO Science Series, SeriesII: Mathematics, Physics and Chemistry, vol. 84, Kluwer,Dordrecht, 2002), p. 307.

[51] D. Messino, D. Sette, and F. Wanderlingh, J. Acoust.Soc. Am. 35 1575 (1963).

[52] C. Appert, C. Tenaud, X. Chavanne, S. Balibar, F.Caupin, and D. d’Humieres, Eur. Phys. B 35 531 (2003).

[53] X. Chavanne, S. Balibar, F. Caupin, C. Appert, and D.d’Humieres, J. Low Temp. Phys. 126, 643 (2002).

[54] H. P. Briggs, J. B. Johnson, and W. P. Mason, J. Acoust.Soc. Am. 19 664 (1947).

[55] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W.T. Vetterling, Numerical Recipes (Cambridge UniversityPress, Cambridge, 1989), Chap. 14.2.

[56] E. Herbert E and F. Caupin, J. Phys. Condens. Mat. 17S3597 (2005).

[57] E. Ruckenstein and R. Nagarajan, J. Phys. Chem 792622 (1975).

[58] F. E. Fox and K. F. Herzfeld, J. Acoust. Soc. Am. 26984 (1954).

[59] M. Strasberg, J. Acoust. Soc. Am. 31 163 (1959).[60] A. A. Atchley and A. Prosperetti, J. Acoust. Soc. Am.

86 1065 (1989).[61] Speyside Glenlivet, http://www.speysideglenlivetwater.com/[62] J. Classen, C.-K. Su, M. Mohazzab, and H. J. Maris,

Phys. Rev. B 57, 3000 (1998).