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# Molecular mechanism for cavitation in water under tension

Edited by Daan Frenkel, University of Cambridge, Cambridge, United Kingdom, and approved September 23, 2016 (received for review May 25, 2016)

## Significance

Cavitation, the formation of vapor-filled bubbles in a liquid at low pressures, is a powerful phenomenon with important consequences in nature and technology. For instance, cavitation bubbles may interrupt water flow in plants under dry conditions or severely damage the metal surfaces of machines such as pumps and propellers. Using molecular simulations, we have studied cavitation in water at strongly negative pressures and have revealed its molecular mechanism. We find that bubble growth is governed by the viscosity of the liquid. Although small bubbles are shaped irregularly, classical nucleation theory accurately describes the free energy barrier that impedes rapid bubble formation. Our simulations indicate that water can withstand negative pressures exceeding −120 MPa in agreement with recent experiments.

## Abstract

Despite its relevance in biology and engineering, the molecular mechanism driving cavitation in water remains unknown. Using computer simulations, we investigate the structure and dynamics of vapor bubbles emerging from metastable water at negative pressures. We find that in the early stages of cavitation, bubbles are irregularly shaped and become more spherical as they grow. Nevertheless, the free energy of bubble formation can be perfectly reproduced in the framework of classical nucleation theory (CNT) if the curvature dependence of the surface tension is taken into account. Comparison of the observed bubble dynamics to the predictions of the macroscopic Rayleigh–Plesset (RP) equation, augmented with thermal fluctuations, demonstrates that the growth of nanoscale bubbles is governed by viscous forces. Combining the dynamical prefactor determined from the RP equation with CNT based on the Kramers formalism yields an analytical expression for the cavitation rate that reproduces the simulation results very well over a wide range of pressures. Furthermore, our theoretical predictions are in excellent agreement with cavitation rates obtained from inclusion experiments. This suggests that homogeneous nucleation is observed in inclusions, whereas only heterogeneous nucleation on impurities or defects occurs in other experiments.

Due to its pronounced cohesion, water remains stable under tension for long times. Experimentally, strongly negative pressures exceeding

Due to the short time scale on which the transition takes place and the small volume of the critical bubble at experimentally feasible conditions, direct observation of cavitation at the microscopic level remains elusive. However, cavitation rates are directly accessible in experiment and some microscopic insight into the cavitation transition can be obtained from these data by means of the nucleation theorem (23), which relates the variation in the height of the free energy barrier separating the metastable liquid from the vapor phase upon change of external parameters to properties of the critical bubble (4, 21). The microscopic information that can be inferred is limited, and because not all quantities entering the nucleation theorem are known, ad hoc assumptions have to be introduced. For state points where cavitation is a rare event, classical nucleation theory (CNT) can be invoked to provide a qualitative understanding of the transition (24). However, although CNT provides a physically meaningful and appealingly simple picture of nucleation processes, the estimates for the nucleation rates obtained from CNT are known to differ substantially (up to many orders of magnitude) from those measured in experiments (22, 25, 26).

Computer simulations are a natural choice to investigate cavitation in water with molecular resolution on the time scales governing the emergence of microscopic bubbles in the liquid. Although cavitation in simple liquids has been studied extensively using computer simulations (27⇓⇓⇓⇓⇓–33), simulation studies of cavitation in water were focused on methodological aspects (34⇓–36) or performed at state points in vicinity of the vapor–liquid spinodal (37, 38). In this work, we apply a combination of several complementary computer simulation methods to identify the molecular mechanism of cavitation. A statistical committor analysis carried out on reactive trajectories reveals that the volume of the largest bubble in the system constitutes a good reaction coordinate for bubble nucleation. We compute the dynamics of nanoscale bubbles along this reaction coordinate and demonstrate that the pressure dependence of the bubble diffusivity can be reproduced by Rayleigh–Plesset (RP) theory generalized to include thermal fluctuations, thereby elucidating the crucial influence of viscous damping on bubble growth. Based on the Kramers formalism and the RP equation we obtain an analytical expression for the nucleation rate that yields excellent agreement with numerical results obtained for a wide range of pressures with a method akin to the Bennett–Chandler approach for the computation of reaction rate constants. The obtained rates are validated for selected points by comparison with estimates from transition interface sampling and support estimates obtained from inclusion experiments. To augment the microscopic picture of cavitation, we characterize the morphology of bubbles in water under tension and analyze the bubble surface in terms of its hydrogen bonding structure.

## Classical Nucleation Theory

Our investigations are guided by CNT, which posits that the decay of the metastable liquid under tension proceeds via the formation of a small vapor bubble, whose growth is initially opposed by a free energy barrier. According to Kramers theory (39, 40), the escape rate *k* from a well over a high barrier for a system moving diffusively in a potential *q* is given by *v* of the largest bubble in the system as the order parameter [committor calculations (41) indicate that *v* is indeed a good reaction coordinate; Fig. S1], and we replace the potential energy by the potential of mean force *v*, and *V* is the total volume of the system, and *ω* is related to the barrier curvature *κ* by *J* into a kinetic part

## Free Energy of Cavitation at Negative Pressures

Using umbrella sampling simulations, we have computed the equilibrium bubble density *Materials and Methods*). For large bubbles, **1** (42). The equilibrium bubble density is related to the Gibbs free energy *v* by *Materials and Methods*. Here we use information from molecular simulations to determine the value of *Supporting Information*).

We obtain a quantitative description of the cavitation free energy within CNT by examining the free energetic cost of the bubble interface, i.e., the free energy without the mechanical work *γ* is taken into account. In particular, the free energy of cavitation is reproduced by*v*. Here, the parameters **2**, which agree almost perfectly with the simulation data (dashed black lines), are shown in Fig. 1. Over the range of bubble volumes studied here, the value of *δ* obtained from the fit is positive, which indicates that the concave curvature of the interface decreases the surface tension *γ*, thereby favoring bubbles over droplets (a discussion of the curvature dependence of the surface tension is provided in *Supporting Information*).

## Bubble Morphology

At the conditions studied here, bubbles are essentially voids in the metastable liquid, which, for bubble volumes *A*). Larger bubbles are predominantly compact and may be viewed as resembling spheres with strongly undulating surfaces (34, 35). This observation is confirmed by computing the average asphericity of bubbles defined as *A*, the asphericity is only weakly dependent on pressure and decreases with increasing bubble volume.

The free energetic cost of forming bubbles in water is intimately connected to breaking and rearranging hydrogen bonds (HBs) at the interface. The hydrogen bonding structure at the liquid–vapor interface depends on the size of the bubble (43, 44). For small bubbles, HBs in the liquid are rearranged, and the fraction of broken HBs at the interface is similar to that of the bulk liquid, whereas in the case of large bubbles, the bubble surface becomes similar to the flat vapor–liquid interface. As shown in Fig. 2*B*, the number of broken HBs per molecule at the interface increases with bubble size, and the fraction of free OH groups at the interface decays roughly linearly with its mean curvature

## Bubble Dynamics

Because CNT with a curvature-dependent surface tension describes the free energy of cavitation very accurately, thus providing the volume **1** is the diffusivity

The RP equation is the equation of motion for the volume *v* of a spherical bubble evolving with internal pressure *m*, viscosity *η*, and surface tension *γ*:*Supporting Information*), and all results shown in Figs. 3 and 4 were obtained including this correction. Neglecting the inertial terms on the left-hand side of the RP equation, one finds*v* can be viewed as an overdamped motion on the CNT free energy

Because thermal fluctuations play an important role for microscopic bubbles, the RP equation is augmented with a random force *γ* yields a similar but slightly more complicated formula (*Supporting Information*).

A comparison between the diffusion constant *Materials and Methods*) is shown in Fig. 3. The viscosity at negative pressures needed in the formula for the diffusion constant was determined in molecular dynamics simulations using the Green–Kubo relation (Fig. S4). The analytical formula obtained from the RP–CNT approach underestimates the diffusivity only by about a factor of 2 compared with simulation results, which is remarkable considering that this estimate is obtained from a macroscopic approach based on hydrodynamics. Moreover, by virtue of the pressure dependence of

## Cavitation Rates

We are now in a position to predict cavitation rates according to Eq. **1** over a wide range of pressures, including the strongest tensions observed in experiment. As a point of comparison, we have computed cavitation rates numerically using a method akin to the divided-saddle method (48) based on the Bennett–Chandler (BC) (49, 50) approach and transition interface sampling (TIS) (51), respectively (*Materials and Methods*). The obtained cavitation rates, shown in Fig. 4, vary by more than 30 orders of magnitude over the studied range of pressures. The numerical results are accurately reproduced by CNT based on Eq. **1** with a curvature-dependent surface tension and the correct value of *Materials and Methods*), underestimates the cavitation rates by more than 15 orders of magnitude. This shortcoming illustrates the importance of including microscopic information, such as a curvature-dependent surface tension and the correct value of

By computing the cavitation pressure *Supporting Information*).

## Conclusions

At ambient temperature and strong tension, bubbles in metastable water are essentially voids in the liquid whose shape can deviate significantly from the assumption of a spherical nucleus made in CNT, depending on their size. Nonetheless, provided the dependence of the surface tension on the average curvature is included, the free energetics of bubble formation can be quantitatively described in the framework of CNT. We find that the curvature contribution favors the cavity over the droplet, i.e., *δ* in water, further study is required to elucidate the influence of the chosen water model and biasing toward certain cavity shapes on the obtained value of *δ*.

By including the effect of thermal fluctuations in the Rayleigh–Plesset equation, we obtain an estimate for the bubble diffusivity that accurately reproduces the pressure dependence found in simulation and scales inversely with the viscosity of the liquid. Combining the kinetic prefactor determined for this diffusivity with the equilibrium bubble density yields a CNT expression for the cavitation rate that reproduces the nucleation rates very well for negative pressures. However, the microscopic mechanism for cavitation is expected to change for higher pressures and temperatures, where the saturated vapor density is significantly higher than at the temperature studied here. At those conditions, similarly to droplet nucleation (52), the transport of molecules across the interface via evaporation and condensation will have a stronger influence on the kinetics of bubble growth, thereby diminishing the influence of viscous damping on the dynamics of the bubble.

The estimate for the cavitation pressure obtained from our rate calculations agrees well with the data from inclusion experiments, thus calling the conflicting results harvested by other techniques into question. Because the latter methods greatly underestimate the stability of water under tension, heterogeneous cavitation due to impurities is a likely explanation for this discrepancy.

## Materials and Methods

### Simulation Details.

We simulate

### Order Parameter.

We study homogeneous bubble nucleation from overstretched metastable water using the volume of the largest bubble as a local order parameter. Estimates for the volume *v* of each bubble present in the system are obtained by use of the V-method, which was developed to give thermodynamically consistent estimates for the bubble volume (35). (Note that the nomenclature was adapted to facilitate readability: *v**v* for the volume of a bubble corresponds to the average change in system volume due to the presence of such a bubble:

Here *ξ* is the preliminary bubble volume estimate from the grid-based method, i.e., the total volume of all vapor-like grid cubes belonging to the bubble, and *n* bubbles of size *ξ* are present. As such, *ξ* is added to or removed from the system. For large bubbles, i.e., for bubble volumes where **6** becomes*ξ* and

On average, because the vapor density in the interior of bubbles is negligible, volume estimates obtained by Eq. **7** are equal to those obtained by computing the equimolar dividing surface between liquid and the largest cavity for each configuration. As a result, the obtained estimates for the bubble volume fulfill the nucleation theorem (23), i.e.,

### Bubble Density.

To compute the equilibrium bubble density

### Detecting Hydrogen Bonds at the Liquid–Vapor Interface.

We identify molecules as belonging to the bubble surface when they are within

### Rate Calculation.

We use a method based on the Bennett–Chandler approach (49, 50) to obtain rates estimates without any assumptions about the dynamics of the bubble in the liquid. In addition to the states *A* (metastable liquid) and *B* (far enough to the right of the free energy barrier such that the system is committed to transitioning to the vapor phase), we introduce a state *S* around the dividing surface, akin to the approach taken in the divided-saddle method (48). An ensemble of trajectories, each *L* steps long, is generated by propagating checkpoints selected from the region *S* forward and backward in time. From these trajectories one then computes the time correlation function *B* at time *t* provided it is in *A* at time 0,

Here *S*. The ratio *S* relative to the equilibrium probability of state *A*, and it can be determined from the free energy

Nucleation rates calculated at

### Computation of the Diffusion Constant.

Because the volume of the largest bubble is a good reaction coordinate for the transition, its diffusivity can be computed via mean first passage times (65, 66). Assuming that the diffusion coefficient does not change significantly in the barrier region, i.e., *b* is the distance of the absorbing boundary from the top of the free energy barrier, approximated by an inverted parabola with curvature *b*. As a starting point at the top of the barrier we used equilibrium configurations created by umbrella sampling where the system contained a cluster of critical size and drew the particle velocities as well as the thermostat and barostat velocities at random from the appropriate Maxwell–Boltzmann distributions.

### Plain CNT.

As a point of comparison, we obtain an estimate for the cavitation rates from CNT with a constant surface tension

The equation above was obtained from Eqs. **1** and **5**, where

## Calibration of the Order Parameter

Below, we give a brief description on how the V-method, which is used in this work to obtain an estimate for the volume of the largest bubble, is parametrized to yield a thermodynamically consistent estimate for the bubble volume. For an in-depth description of the method used to detect bubbles in the metastable liquid, we refer the reader to ref. 35.

We use a grid-based procedure to detect bubbles in the system by clustering grid points that are not occupied by liquid-like water molecules. The preliminary size estimate *ξ* for the bubble is the total volume of all vapor-like cubes belonging to the same cluster. Here we use a grid of

As mentioned in *Materials and Methods*, the V-method is calibrated such that its estimate for the volume of the largest bubble *v* corresponds to the average change in system volume *V* due to the presence of a bubble. Because this change in system volume depends on the chosen thermodynamic state point, one needs to determine *v* as a function of the preliminary grid-based order parameter *ξ* to obtain the correct calibration at the state point of interest.

The data for *ξ* onto *v* we choose the fitting function, indicated by the gray line in Fig. S5, as

## Volume of the Largest Bubble as a Reaction Coordinate

The rate equation of CNT, Eq. **1**, is based on the assumption that the dynamics of bubble growth can be described as the diffusion of a bubble volume on the respective free energy surface. To quantify to which extent the volume of the largest bubble tracks the progress of the cavitation transition dynamically, i.e., whether the volume of the largest bubble is a reaction coordinate, we perform a statistical committor analysis, which correlates values of the chosen order parameter with the probability

We create reactive trajectories by propagating equilibrium configurations harvested by means of umbrella sampling close to the size of the critical bubble, which for a pressure of *N* is the number of shots and *σ* is the SE in the committor assuming Gaussian statistics.

The result of this analysis, shown in Fig. S1, reveals that the volume of the largest bubble in the system is a good reaction coordinate for cavitation in water. Higher values for the volume of the largest bubble correspond to higher committor probabilities, and the spread of the data is moderate. As such, the volume of the largest bubble is suitable for the computation of rates via Eq. **1** of the main text (65, 66). Further, the volume *v* parametrizes

In an effort to find correlations of

## Surface Free Energy and Curvature Dependence of the Surface Tension

In this section, we obtain a quantitative description of the cavitation free energy from CNT by examining the surface free energy, which allows us to compare the free energetic cost of forming a liquid–vapor interface for different pressures. We then discuss the obtained curvature dependence of the surface tension that favors bubbles over droplets.

The surface free energy *v*, *v* is formed (*Materials and Methods*); by subtracting this contribution, we can compare the cost of forming a bubble in the metastable liquid at different pressures directly. Furthermore, by fitting the surface free energy with a suitable functional form explained below, we elucidate the normalization constant

Remarkably, the resulting surface free energy *C* is related to **2**.

The functional form of the free energy in Eq. **2**, whose parameters are obtained from the fit described above, is identical to the variant of CNT incorporating a curvature-dependent surface tension proposed by Tolman (69), and as such it is tempting to identify the parameter *δ* with the Tolman length. However, a fundamental assumption required to obtain Eq. **2** in the framework of the theory is that the radius *r* of the bubble is large compared with the length *δ* (70, 71), and thus, the applicability of the Tolman formalism is questionable. However, when studying cavitation in water at ambient temperature, this shortcoming is only relevant for the theoretical exercise of extracting the Tolman length because Eq. **2** describes the free energy of cavitation accurately over the range of volumes

The value of *δ* obtained from the fit shown in Fig. S2 is positive, which indicates that the concave curvature of the interface decreases the surface tension *γ*. In the literature, there are conflicting reports on the dependence of the surface tension on curvature in water: refs. 4, 72, and 73 find that the bubble is free energetically favored over the droplet, whereas refs. 74⇓⇓⇓–78 arrive at the opposite conclusion. Notably, the value obtained from the fit for *δ* in water, further study is required to elucidate the influence of the chosen water model and biasing toward certain cavity shapes on the obtained value of *δ*.

## Pressure Dependence of the Cavitation Free Energy

The bubble surface free energy

For bubbles that do not occur spontaneously in the metastable liquid on the time scale of an unconstrained simulation, the bubble surface free energy is*ξ*, i.e., a largest bubble of size *ξ*, and the metastable liquid, on average. In this work, the chosen constraint is the preliminary grid-based bubble volume estimate described in ref. 35, but the following derivation is not limited to this specific bubble detection procedure. The pressure derivative is*v* is accurately reproduced by Eq. **S1** for all pressures (Fig. S5). The pressure derivatives for the respective terms in the equation above are**S9** vanishes because the change in *v*-dependent contribution remaining from Eq. **S6** is

In practice, the change in surface free energy given by Eq. **S10** is small due to the low compressibility of water. We compute the difference

## Curvature-Corrected Bubble Dynamics

Following the same procedure as in the main text, we obtain the cavitation rate estimate from CNT including a curvature-dependent surface tension *ω* of the free energy barrier.

Because the surface tension *r* explicitly, we cast the RP equation in terms of the bubble radius for simplicity. The RP equation with a curvature dependent surface tension **3**, describing the time evolution of the bubble radius *r* instead of its volume *v*, with an additional term *r*. Neglecting the inertial terms on the left hand side and the pressure *v*, we compute the diffusivity **5** only in the estimate for

The volume of the critical bubble **S15** predicts critical bubble volumes *δ* is small compared with the radius **S15** yields the estimate for the diffusivity *J* one requires the curvature

Inserting the quantities computed above into Eq. **1** gives the rate estimate when the curvature dependence of the surface tension is taken into account and the correct value of **S15** and **S16** were used.

## Viscosity of Water Under Tension

To estimate the diffusion coefficient *D* of a bubble in the framework of the RP equation,*η* of a fluid can be computed by use of the Green–Kubo relation (56),

We compute the pressure tensor *η* is obtained by fitting the emerging plateau for long times with a constant.

The resulting estimates for η are shown in Fig. S4. The shear viscosity increases with tension, consistent with the behavior at positive pressures reported in ref. 80. Due to the large scatter in the data and in absence of prior knowledge about the functional form of *η* is relatively large and, hence, the functional dependence of *p* cannot be reliably extracted from the data, we stress that the viscosity is not the only pressure-dependent factor entering Eq. **S19**. In particular, when using Eq. **S19** with the Kramers equation (Eq. **1**), the change in *η* with pressure is very small compared with the change in *η* will not significantly influence the estimates for the diffusion constant obtained from the RP equation in conjunction with CNT (let alone the estimate for the rates which are dominated by the change in free energy with pressure).

## Comparison of the Obtained Rates to Experimental Data

To put the cavitation pressure presented in Fig. 4 into context, we discuss its relation to experimental results obtained from different setups. Our estimate for the cavitation pressure, obtained from the cavitation rates calculated for typical experimental conditions,

Two different scenarios can explain the discrepancy between experiments (21): (*i*) either homogeneous cavitation occurs in water close to *ii*) homogeneous cavitation occurs close to

Finally, we note that heterogeneous nucleation also occurs in some inclusions. Indeed, in a given quartz sample containing many inclusions with similar liquid density, a wide range of

## Acknowledgments

We thank S. Garde, P. Geissler, V. Molinero, A. Patel, E. Sanz, A. Tröster, E. Vanden-Eijnden, C. Vega, and S. Venkatari for insightful comments. Calculations were carried out on the Vienna Scientific Cluster. The work of G.M., P.G., and C.D. was supported by the Austrian Science Foundation (FWF) under Grant P24681-N20 and within the Spezialforschungsbereich Vienna Computational Materials Laboratory (Grant F41). P.G. also acknowledges financial support from FWF Grant P22087-N16, and F.C. acknowledges support from the European Research Council under the European Union’s Seventh Framework Programme for Research and Technological Development Grant Agreement 240113. C.V. acknowledges financial support from a Marie Curie Integration Grant 322326-COSAAC-FP7-PEOPLE-CIG-2012 and a Ramon y Cajal tenure track. The team at Madrid acknowledges funding from the Ministerio de Economía y Competitividad Spanish Grants FIS2013-43209-P, FIS2016-78117-P, and FIS2016-78847-P.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: christoph.dellago{at}univie.ac.at.

Author contributions: G.M., F.C., C.V., and C.D. designed research; G.M., P.G., and C.D. performed research; G.M., M.A.G., F.C., J.L.F.A., C.V., and C.D. analyzed data; and G.M., M.A.G., P.G., F.C., J.L.F.A., C.V., and C.D. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

See Commentary on page 13549.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1608421113/-/DCSupplemental.

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- Article
- Abstract
- Classical Nucleation Theory
- Free Energy of Cavitation at Negative Pressures
- Bubble Morphology
- Bubble Dynamics
- Cavitation Rates
- Conclusions
- Materials and Methods
- Calibration of the Order Parameter
- Volume of the Largest Bubble as a Reaction Coordinate
- Surface Free Energy and Curvature Dependence of the Surface Tension
- Pressure Dependence of the Cavitation Free Energy
- Curvature-Corrected Bubble Dynamics
- Viscosity of Water Under Tension
- Comparison of the Obtained Rates to Experimental Data
- Acknowledgments
- Footnotes
- References

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