One of the main challenges of thermodynamics is to predict and measure accurately the properties of metastable fluids. Investigation of these fluids is hindered by their spontaneous transformation by nucleation into a more stable phase. We show how small closed containers can be used to completely prevent nucleation, achieving infinitely long-lived metastable states. Using a general thermodynamic framework, we derive simple formulas to predict accurately the conditions (container sizes) at which this superstabilization takes place and it becomes impossible to form a new stable phase. This phenomenon opens the door to control nucleation of deeply metastable fluids at experimentally feasible conditions, having important implications in a wide variety of fields.

Metastable fluids are found everywhere: in clouds, plants, as magma erupting from volcanoes, in stars or comets. Proper understanding and accurate description of these fluids remains one of the main challenges in thermodynamics and is crucial in many phenomena of scientific and industrial relevance.^{1} One example is cavitation of water at large negative pressures,^{2–6} which causes damage to tissues, pumps, and propellers. Despite major efforts, it is still a phenomenon which is not completely understood, as evidenced by the deviations with respect to predictions by Classical Nucleation Theory (CNT)^{1,2,7,8} and even between different experimental techniques.^{3,6} In industry, much attention has been given to explosive boiling and rapid phase transitions, where metastable liquids such as liquefied nitrogen or natural gas vaporize violently.^{9} In a broader context, proper predictions of nucleation rates rely on a precise knowledge of the properties of the metastable phase. A striking example occurs even for the simple case of condensation of Argon, where the predictions of CNT deviate more than 20 orders of magnitude from experiments.^{10} An accurate description of metastable fluids can help understand all these phenomena and will have important implications in a wide range of areas.

While *equilibrium* properties of fluids are experimentally available, it is a challenge to measure metastable fluid properties. By their own labile nature, they tend to transform to a stable new phase via a nucleation process triggered by fluctuations. The nucleation rate according to CNT is^{1}

where *K* is a kinetic prefactor and Δ*W** is the height of the nucleation barrier to overcome, corresponding to the reversible work of formation of a critical size nucleus beyond which the growth of the new phase becomes energetically favorable. The nucleation barrier dictates the life-time of a metastable state and depends on the distance from equilibrium. Metastable states are possible only until the spinodal is reached. At and beyond this limit, the barrier to nucleation disappears and the phase change occurs spontaneously and often vigorously. Accurate experimental investigations of metastable fluids in the vicinity of the spinodal have until now been almost impossible, since they require the metastable phase to be preserved at least long enough to measure its properties.

It is known that bubbles and droplets can be stabilized in closed systems^{11} or by controlling the fluctuations in the number of particles, as in the Gauge Cell Method.^{12–14} It is also well-known that phase transitions can be deferred in small and strongly confined systems. Examples include the melting point depletion in molecular clusters,^{15–17} capillary condensation and evaporation in nanoporous materials,^{18,19} and delay of crystallization in small confinements^{20,21} or of vaporization around nanoparticles.^{22} Here we show that it is not only possible to delay these phase transitions, but to go one step further and completely *impede* nucleation in small closed systems. We shall see that this superstabilization of fluids takes place when the new phase becomes unavailable, typically due to limitations in volume and number of particles. This can be achieved using nanocontainers of particular sizes, and we shall predict the conditions under which we can obtain infinitely long-lived metastable phases. At these conditions, the properties of the metastable fluid can be investigated experimentally, facilitating the development of accurate equations of state for metastable regions and even a precise location of the spinodal.

To demonstrate that the phenomenon is not linked to any special interactions or equations of state, we resort to a general thermodynamic description. Moreover, we discuss this effect in both homogeneous condensation (droplet formation) and cavitation (bubble formation). For illustrative purposes, and for its practical interest, we use water as example. But our analysis holds for any substance, and the same ideas are applicable to other phase transitions, e.g., crystallization and liquid-liquid phase transitions.

The key to understand the phenomenon is a careful analysis of how phase transitions take place in closed containers, i.e., in the canonical ensemble, rather than at the usual open conditions, where pressure or chemical potential are controlled. Let us consider a single-component fluid with *N* total number of molecules confined inside a fixed volume *V* at a constant temperature *T*. The discussion does not rely on any exact shape of the container, but for illustrative purposes we represent the total volume as a sphere with radius R (see inset in Fig. 1). Moreover, we restrict our analysis to homogeneous nucleation and assume that the surface energy of the container wall is negligible in the case of condensation (i.e., that no condensation can take place on the walls). To analyze the peculiarities of nucleation in small systems, we use the modified bubble/droplet model^{23} that has been shown to give very similar predictions for the properties of the critical cluster/bubble as more advanced theories, e.g., density functional theory.^{24} This model considers the potential formation of a bubble or droplet of *n* molecules at the center of the container. The bubble/droplet and the exterior phase are both assumed to have the thermodynamic properties of a homogeneous phase separated by an interface at *r* with constant surface tension, σ. In the *NVT* (canonical) ensemble, the appropriate thermodynamic potential is the Helmholtz energy. By assuming that the bubble/droplet is perfectly spherical, the variation in the total Helmholtz energy of the system at constant *T* becomes^{24}

where we have used the subscript “*n*” to denote either a droplet or a bubble at the center of the container, and “e” denotes the exterior phase. Furthermore, *P* is the pressure, μ is the chemical potential, and *V*_{n} is the volume of the droplet/bubble. The stationary states of the Helmholtz energy obey *dF* = 0. This leads to equality of the chemical potentials of both phases and the Laplace relation:

Since both *N* and *V* are fixed, the system must also satisfy the constraint:

Regardless of the equations of state used to represent the phases, the energy landscape to form a perfectly spherical bubble or droplet of *n* molecules and radius *r* is

where

In an open system, the stationary conditions, Eqs. (3) and (4), have only one solution, corresponding to the critical cluster/bubble in CNT. However, in the canonical ensemble, Eqs. (3)–(5) can have two solutions for a given initial density. One corresponds again to a critical cluster or bubble (i.e., a saddle point in the energy landscape of Eq. (6)), whereas the other one represents a stable bubble or droplet (i.e., a minimum). For illustrative purposes, Fig. 1 shows the radii corresponding to these two solutions for water-cavitation at 300 K, as a function of the density and for spherical containers of different radii. The results are obtained using the reference (IAPWS) equation of state^{25} and the surface tension values tabulated in Ref. 26. A similar model was recently used by Marti *et al.* in a very insightful study on the effect of surface tension on the temperature of liquid-vapor homogenization and the determination of density in fluid inclusions.^{27}

For each container radius, *R*, the lower branch (solid line) represents the size of the critical bubble, and the upper branch (dashed line) is the stable bubble. Remarkably, both solutions merge at a particular density indicating the existence of a minimum stable bubble or droplet size, below which there are no longer any stationary solutions to the conditions dictated by Eqs. (3)–(5).

Under those circumstances, the energy landscape, Eq. (6), is a monotonically increasing function of cluster/bubble-size, and no bubbles or droplets form. This special state has been referred to as “bubble spinodal”,^{27} or “superspinodal”,^{12,13,28–30} although it is not an intrinsic property of the fluid, since it depends fully on the confining volume. In simulations in the canonical ensemble or in experiments performed in small closed systems, the dramatic consequence of this effect is that nucleation is impossible even at conditions where nucleation rates are considerable in large or open systems. Hence, the fluid will remain indefinitely in its metastable state due to the impossibility of transforming to the new stable phase, giving rise to this superstability effect.

In the case of cavitation, the effect is associated with the compressibility of the liquid. For small bubbles, the negative pressure in the Laplace equation becomes so large, that the liquid phase will stretch to fill the entire container, thereby collapsing the bubble.^{24,31} For droplet condensation, this situation happens because droplet formation occurs at the expenses of the surrounding vapor. For sufficiently small systems, the formation of the liquid drop can deplete the supersaturation so much that sufficiently large droplets cannot form.^{23}

By comparing the stationary solutions for *R* = 14 nm, 8 nm, and 4 nm in Fig. 1, we see that the density beyond which there are no longer any stationary solution, corresponding to a situation where cavitation is impossible, shifts to lower values and hence higher metastabilities as the container becomes smaller. For instance, at a density of 967 kg/m^{3} (Isochore 2), no bubbles can form in containers smaller than 14 nm. At an initial density of 950 kg/m^{3} (Isochor 1), a bubble can form in a 14 nm container, but no bubbles form in containers smaller than 8 nm. The minimum stable bubble size changes with density according to the solid line in Fig. 1. In practical terms, this effect will always give a lower limit to how few particles and how small the total volume can be before nucleation becomes impossible. Knowing the equation of state, this limit can be calculated numerically by looking for the conditions at which the two solutions of Eqs. (3)–(5) merge. Alternatively, to get a better idea of what the main parameters controlling this superstability phenomenon are, one can derive approximate formulas that predict the minimum *volume* of the container below which no droplet or bubble can form:

Subscript *l* and *g* denote liquid and gas, respectively, κ is the compressibility, ρ is the density, and the superscript * indicates that the properties are evaluated at the stationary solution. These approximate formulas have been obtained by expanding the densities of both phases to first order around the densities at the stationary solution, using Eqs. (3) and (4), and then locating the extrema

*n*

^{*}/

*N*(for droplets) and

We have tested Eqs. (8) and (9) for several single-component fluids, and they are found to be valid within 0.1%. This means that they capture the underlying physics of the minimum container-size. The equations show that the surface tension and the compressibility of the *external phase* are always the important parameters. They also show that, in the case of cavitation, as the compressibility of the liquid goes to zero, the minimum container-size disappears. In addition, the previous expressions demonstrate that this superstabilization occurs for container volumes significantly larger than the volume of the critical cluster/bubble (see also Fig. 1). Thus, it is not a consequence of being unable to accommodate the critical embryo. The first order curvature correction to surface tension did not affect the validity of Eqs. (8) and (9). Higher order curvature corrections may play a more significant role for small drops/bubbles and near the spinodals.^{32,33}

From a practical point of view, one is interested in the container-size below which no nucleation can happen for a given initial density or supersaturation,

These approximated equations have been obtained from Eqs. (8) and (9), by using Eq. (4) for the critical cluster size, and expanding the pressures and densities of both phases to first order around the density corresponding to the CNT solution, retaining only the leading corrections.

Figure 2 plots the minimum radius of the container below which cavitation and condensation become impossible for water at 300 K. In Figs. 2(a) and 2(b), the solid lines show the exact results for the minimum container-sizes, obtained numerically from the modified bubble/droplet model, Eqs. (3)–(5), using the IAPWS equation of state. The dashed lines show the minimum container-sizes predicted with the approximate formulas Eqs. (10) and (11). The average accuracy of the formulas when applied to water is 8% for both bubble and droplet-formation. It is evident that in very small closed containers, one can arrest homogeneous nucleation at very large supersaturations and overstretchings, which will immediately lead to nucleation in open systems. This can be used to control the transformation and to explore the properties of fluids at strongly metastable conditions. The presented formulas have several possible applications in simulations and theoretical considerations of nucleation phenomena. As we will now discuss, the idea can also be used to design nanoscale experiments for investigation of metastable properties.

There has been a lot of recent interest in exploring water cavitation at large negative pressures in quartz inclusions.^{5,6} The motivation is to understand the anomalies of water, specially locating the line of maximum density (LMD) at negative pressures and a possible re-entrant behavior of the spinodal. In these experiments, water was confined in sealed inclusions in quartz of typical sizes ∼*V* = 500 μm^{3}, and the onset of cavitation was monitored by first heating the system until all vapor condensed filling the inclusion (the saturation line was exceeded), and subsequently cooling it along an isochor to generate large negative pressures. Cavitation of water at very large negative pressures, down to −140 MPa,^{5,6} which is close to the limit of spontaneous cavitation predicted by CNT, were reported.

From our results, we predict that very small inclusion volumes can be used to control and preclude completely cavitation, potentially reaching even larger negative pressures than mentioned above. This may allow the equation of state of metastable water to be accurately determined, or to locate precisely the spinodal, where this superstabilization effect is expected to end violently since the fluid becomes mechanically unstable. For instance, Fig. 3 plots the size of the inclusions required to completely suppress cavitation down to the LMD for three illustrative isochores. Containers of this size could be implemented with excellent control using sealed nanoporous materials, for instance MCM or SBA crystals.^{19,34}

The key idea of these experiments would be to use very small containers to frustrate and control cavitation. The same idea could be utilized to study strongly supersaturated water vapor. Using small hydrophobic nanoinclusions (to prevent heterogeneous condensation on the walls), or nanopores,^{19} it could be in principle possible to keep steam indefinitely in a strongly metastable state. At those conditions one could measure or simulate accurately the properties of metastable steam, of interest for many industrial applications.

In summary, we have shown that condensation and cavitation can be completely prevented in small closed containers. It is thus possible to keep fluids indefinitely at strongly metastable conditions that would lead to immediate nucleation in large or open volumes. It is worth emphasizing that this superstabilization is a general thermodynamic phenomenon linked to the impossibility of forming a new phase in a small closed volume. It holds irrespective of any capillary approximation or assumption of a constant surface tension, as evidenced by the fact that it also appears in density functional calculations in closed volumes.^{24,31,35} At these conditions, that are experimentally feasible, it would be possible to control and prevent the explosive boiling of liquids, the cavitation of overstretched fluids, and to measure accurate equations of state for fluids in deeply metastable regions. This would thus have striking consequences in a wide variety of scientific and industrial phenomena.

The work was partially supported by the Spanish MINECO through Grant No. FIS2011-22603. ETH Zurich is thanked for guest professorships to S.K. and D.B.