Cavitation pressure in water

Eric Herbert, Sébastien Balibar, and Frédéric Caupin

Phys. Rev. E 74, 041603 – Published 16 October 2006

Abstract

We investigate the limiting mechanical tension (negative pressure) that liquid water can sustain before cavitation occurs. The temperature dependence of this quantity is of special interest for water, where it can be used as a probe of a postulated anomaly of its equation of state. After a brief review of previous experiments on cavitation, we describe our method which consists in focusing a high amplitude sound wave in the bulk liquid, away from any walls. We obtain highly reproducible results, allowing us to study in detail the statistics of cavitation, and to give an accurate definition of the cavitation threshold. Two independent pressure calibrations are performed. The cavitation pressure is found to increase monotonically from $- 26 MPa$ at $0 ° C$ to $- 17 MPa$ at $80 ° C$ . While these values lie among the most negative pressures reported in water, they are still far away from the cavitation pressure expected theoretically and reached in the experiment by Angell and his group [Zheng et al., Science 254, 829 (1991)] (around $- 120 MPa$ at $40 ° C$ ). Possible reasons for this discrepancy are considered.

18 More

Received 30 June 2006

DOI:https://doi.org/10.1103/PhysRevE.74.041603

Authors & Affiliations

Eric Herbert, Sébastien Balibar, and Frédéric Caupin

Laboratoire de Physique Statistique de l’Ecole Normale Supérieure associé aux Universités Paris 6 et Paris 7 et au CNRS, 24 rue Lhomond 75231 Paris Cedex 05, France

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 74, Iss. 4 — October 2006

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Cavitation pressure in water as a function of temperature, calculated within the TWA (dotted line), or using DFT with Speedy’s EOS (solid line) or with the TIP5P EOS (dashed line). The parameters used are $V = {(10 μ m)}^{3}$ , $τ = 1 s$ , and $Γ_{0}$ given by Eq. (2).Reuse & Permissions
Figure 2
Cavitation pressure as a function of the quantity $J = 1 ∕ (V τ)$ . Data points are listed in Table . The solid line is the prediction of the TWA [Eq. (3)].Reuse & Permissions
Figure 3
Block diagram of the driving circuit of the transducer. The different units are designated in bold font. The solid lines represent the electrical connections, and the dashed lines the connections for computer control. The shunt resistor and the step-down transformer are used to improve the shape of the excitation voltage.Reuse & Permissions
Figure 4
Excitation voltage of the transducer, driven with a four-cycle burst, at the cavitation threshold at $T = 20 ° C$ and $P_{stat} = 1.7 MPa$ . This corresponds to a peak power of $3.4 kW$ .Reuse & Permissions
Figure 5
Sketch of the experimental setup. The high pressure part contains the cell with the transducer, the pressure gauge, and the bellow for pressure control; it can be isolated from the rest by a valve. The use of two other valves allows evacuation (with an oil pump through a nitrogen trap or a dry scroll pump), filling (the flask with degassed water being connected), and emptying (the collecting vessel being connected and cooled with ice). The cell is immersed in a thermostated bath (operated between $- 10$ and $80 ° C$ ). All the seals are made of SS or bulk Teflon, except the one at the bottom of the cell, which is made with an indium wire.Reuse & Permissions
Figure 6
UV absorption spectrum showing different “container effects.” We used an automated spectrometer (Kontron Instruments, Uvikon 941) with quartz cuvettes ( $10 mm$ thickness). The reference cuvette was filled with ultrapure water directly drawn from the water polisher and sealed with a Teflon cap. The solid curves show the results of our test for several materials (designated by the labels), using the following procedure: a piece of the material was immersed in $50 − mL$ ultrapure water in a Pyrex beaker, and the beaker was heated at $80 ° C$ during $30 \min$ , in contact with air. After cooling, water from this beaker was then used to rinse and fill the sample cuvette, and its UV absorption recorded. The control curve shows the results of this test for ultrapure water without adding any material. Other materials used in the cell (teflon tape, teflon tube, and indium wire, not shown) give absorption spectra between the control and that of stainless steel. For comparison, we also show the spectrum for a 1% ethanol solution in water (dotted line), and for ultrapure water degassed using the procedure described in Sec. 3F (dashed line).Reuse & Permissions
Figure 7
Relaxation voltage of the transducer. The two traces were recorded for two successive four-cycle bursts with the same experimental conditions ( $T = 20 ° C$ , $P_{stat} = 1.7 MPa$ , $V_{cav} = 163.3 V$ ). The solid line corresponds to the reproducible relaxation signal of the transducer coming back to rest, without cavitation. The dashed line is an example of the random echo signal reflected on the nucleated bubble and reaching the transducer voltage with a delay $t_{f}$ after nucleation.Reuse & Permissions
Figure 8
Histogram of the peak-to-peak value of successive echoes. The left group of data shows the small noise on the excitation voltage without cavitation, the right one shows the various amplitudes reached by echoes with cavitation. The 1000 data points are distributed among 100 bins; the main peak (reaching 328 counts) is truncated for clarity.Reuse & Permissions
Figure 9
Cavitation probability vs excitation voltage for four-cycle bursts at $T = 20 ° C$ and $P_{stat} = 1.7 MPa$ . Each of the 25 data points was measured over 1000 repeated bursts. The standard deviation on the probability (calculated with the binomial law) is shown as error bars. The data are well fitted with Eq. (9) (solid line). The inset focuses on the low probability region, to show that zero probability is actually reached in the broad foot of the S curve.Reuse & Permissions
Figure 10
Response of the $40 − μ m$ needle hydrophone at the focus. The hydrophone voltage has been converted into pressure using the manufacturer’s calibration data. The solid (respectively, dotted) curve corresponds to an excitation voltage $V_{rms} = 7.21 V$ (respectively, $17.42 V$ ), that is 5.5% (respectively, 13.4%) of the cavitation voltage. The time scale starts with the excitation voltage, and the acoustic wave reaches the focus after the time of flight over the transducer radius ( $t_{f} = R ∕ c$ , see Sec. 4A).Reuse & Permissions
Figure 11
Map of the acoustic field. The data (normalized to 1 at the focus) were taken with a $40 − μ m$ needle hydrophone in an open container at room temperature $(≃ 25 ° C)$ , using one-cycle bursts. Filled (respectively, empty) symbols correspond to the maximum (respectively, minimum) pressure in a burst. Diamonds (bottom) show a scan in the equatorial plane of the transducer, whereas circles (shifted by 0.2 for clarity) show a scan along its axis.Reuse & Permissions
Figure 12
Peak pressures measured by the hydrophones as a function of the transducer excitation voltage. The data were taken in an open container at room temperature ( $≃ 25 ° C$ , with four-cycle bursts, the cavitation threshold voltage being $V_{cav} = 130 V$ ). Filled (respectively, empty) symbols correspond to the maximum (respectively, minimum) pressure in a burst. Circles (respectively, triangles) were obtained with a $40 − μ m$ (respectively, $200 μ m$ ) needle. The manufacturer gives a calibration uncertainty on the gain of 14%, which means that the results from the two needles are consistent with each other.Reuse & Permissions
Figure 13
Static pressure as a function of the product $ρ V_{cav}$ for 5 of the 15 temperatures investigated in run 0. The dotted lines are linear fits to the data; each set is labeled (on the right) with the temperature. Experimental error bars are shown on each data point; most of the vertical errors are too small to be seen at this scale.Reuse & Permissions
Figure 14
Static pressure as a function of the product $ρ V_{cav}$ during the first pressure cycles of run 3 at $0.1 ° C$ .Reuse & Permissions
Figure 15
Cavitation pressure as a function of temperature. $P_{cav}$ was obtained with the static pressure method (see Sec. 4D). Run 0 (filled circles) is compared to our preliminary results (Ref. [56]) (empty circles). The uncertainties on $P_{cav}$ were calculated as described in Sec. 5A.Reuse & Permissions
Figure 16
Cavitation probability as a function of the minimum pressure reached in the wave for four different static pressures. Each point is an average over 1000 bursts. The data were taken during run 0 at $T = 4 ° C$ .Reuse & Permissions
Figure 17
Steepness of the S curves as a function of temperature. Data were measured in run 0, using 25 points with statistics over 1000 bursts, at $P_{stat} = 1.6 MPa$ .Reuse & Permissions
Figure 18
Cavitation pressure as a function of temperature, obtained with the static pressure method (see Sec. 4D), for runs 1–8. The error bars are omitted for clarity; they are similar to those in Fig. 15.Reuse & Permissions
Figure 19
UV absorption spectrum of the water collected from the experimental cell for nine successive runs. We used an automated spectrometer (Kontron Instruments, Uvikon 941) with quartz cuvettes ( $10 mm$ thickness). The reference cuvette was filled with ultrapure water directly drawn from the water polisher and sealed with a Teflon cap. The label indicates the run number.Reuse & Permissions
Figure 20
Cavitation probability as a function of the excitation voltage normalized by the cavitation voltage. Each filled (respectively, empty) circle shows the probability measured with a $1 − MHz$ (respectively, $1.3 − MHz$ ) transducer, by counting the cavitation events over 1000 bursts, at room temperature in an open Pyrex container. A fit with Eq. (B8) (solid line), using $P_{2} ∕ P_{cav} = 0.8$ and $α = 11 ∕ 2$ , gives $n_{0} λ^{3} {[P_{cav} ∕ (P_{2} - P_{1})]}^{13 ∕ 2} = (3.62 \pm 0.10) 10^{7}$ for both frequencies.Reuse & Permissions
Figure 21
Comparison between different water samples. The initial run (run 0, filled circles, as in Fig. 15) is shown for comparison. In run 9 (empty circles), we used ultrapure water from a different source (G Chromasolv for gradient elution, purchased from Sigma), and degassed with the method described in Sec. 3F. In run 10 (empty squares), we used mineral water (Speyside Glenlivet, dry residue at $180 ° C$ : $58 mg L^{- 1}$ ) (Ref. [61]), distilled in a Pyrex glassware and degassed with the same method. (a) Cavitation pressure as a function of temperature; the error bars on $P_{cav}$ are omitted for clarity: they all have a similar amplitude (see Fig. 15). (b) Cavitation probability as a function of the minimum pressure reached in the wave. The S curves were obtained at $T = 20 ° C$ and $P_{stat} = 1.6 MPa$ . The excitation voltages were converted into pressure using Eq. (11) and the value of $P_{cav}$ shown in (a). The solid curves are fits with Eq. (12); they all give similar values of the steepness $ξ$ : $43.3 \pm 1$ for run 0 $(χ^{2} = 2.2)$ , $44.6 \pm 1.5$ for run 9 $(χ^{2} = 3.9)$ , and $48.1 \pm 1.9$ for run 10 $(χ^{2} = 4.6)$ .Reuse & Permissions
Figure 22
UV spectra of different water samples. The labels indicate the water origin and preparation: Sigma refers to water used in run 9, and Glenlivet in run 10. Bottle is relative to the UV spectrum of water directly coming out of the commercial bottle, whereas cell refers to the water coming out of the cell at the end of the run. We also show the spectrum of Glenlivet water after distillation and degassing: these steps decrease the UV absorbance.Reuse & Permissions
Figure 23
Cavitation probability as a function of the excitation voltage normalized to the cavitation voltage. Full (respectively, empty) circles stand for ultrapure (respectively, tap) water. The data were recorded at room temperature in an open Pyrex glass container. Each point corresponds to 1000 repeated bursts. The solid line is a fit of the data for ultrapure water with Eq. (9). The center of these S curves are similar, but the foot for tap water is much broader than for ultrapure water. This indicates the presence in tap water of a broad distribution of dilute impurities triggering cavitation at pressures less negative than the cavitation pressure of ultrapure water.Reuse & Permissions
Figure 24
Speculated spinodal pressure as a function of temperature. The values of the spinodal pressure (empty circles) were deduced from the cavitation pressures (filled circles) and steepness of the S curves (see Fig. 17) measured in run 0.Reuse & Permissions
Figure 25
Cavitation probability as a function of the minimum pressure reached in the wave. The data points are the same as in Fig. 9. The excitation voltage has been converted into pressure using Eq. (11) and $P_{cav} = - 23.845 MPa$ . Only pressures between $- 20$ and $- 26 MPa$ are shown. The solid line is a fit with Eq. (12). The best fit gives $- P_{cav} = 23.847 \pm 0.019 MPa$ and $ξ = 43.3 \pm 1$ with $χ^{2} = 2.2$ . The inset shows the two distributions of type-II impurities tested (see Appendix B). Equation (B6) (corresponding to the monodisperse distribution) does not allow a good fit; the dotted line is drawn with $P_{0} = - 23 MPa$ and $n_{0} λ^{3} = 600$ to pass through the high probability data. The dashed line is a fit with Eq. (B8); $P_{2}$ was set to $- 20 MPa$ , and the best fit was obtained for $α = 5$ : it gives $n_{0} λ^{3} ∕ {(P_{2} - P_{1})}^{6} = (0.31 \pm 0.01) {MPa}^{- 6}$ and $χ^{2} = 3.7$ .Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics