How far can one supercool a liquid before it crystallizes? How much can one stretch it before cavitation occurs ? In order to answer such questions, we have studied liquid helium, a model system. In this review, we show the limitations of the elementary ‘standard nucleation theory’. We then show that the existence of ‘spinodal’ limits needs to be considered in the framework of ‘density functional’ methods. We also briefly consider the possibility of nucleation by quantum tunnelling. The main emphasis is on cavitation and crystallization in liquid helium, but we also mention several connections with more classical systems, in particular water. Introduction: metastable liquids and negative pressures For a certain time, a liquid can stay in a metastable state, outside of the stability region in its phase diagram. For example, liquid water can be supercooled down to about −40 ◦C (233 K), [1] and overheated up to +280 ◦C at atmospheric pressure [2]. Water has also been stretched to −1400 bar, a remarkably large negative pressure [3]. Such a metastability is possible because the liquid–solid and the liquid–gas transitions are discontinuous, i.e. first order. As a result, interfaces between a liquid and its vapour or solid phase have a non-zero surface tension. For a more stable phase (solid or gas) to appear in a less stable one (the metastable liquid), an interface has to be created somehow, and there is an energy cost for that. As a consequence, there is an energy barrier against the nucleation of the stable phase, and metastability is possible. Nucleation is called ‘heterogeneous’ when it is influenced by the presence of defects, impurities, walls or radiation. This is the most common case in nature. For example, water droplets in clouds freeze around −20 ◦C, and this temperature depends on the pollution by dust particles and various chemicals. In the absence of defects, walls etc, nucleation is an intrinsic property of the system; it usually takes place very far from equilibrium conditions and it is called ‘homogeneous’. In this review, we mostly consider homogeneous nucleation, which is simpler to describe quantitatively. 1 Author to whom any correspondence should be addressed. 0953-8984/03/010075+08$30.00 © 2003 IOP Publishing Ltd Printed in the UK S75 S76 S Balibar and F Caupin Some people find it difficult to consider negative pressures, although they are present in everyday life, for example at the top of high trees [4, 5]. The pressure of a gas cannot be negative. Suppose that a gas is contained in a chamber closed by a piston. If one pulls the piston, the pressure P of the gas vanishes linearly with the density ρ inside according to the equation of state P(ρ). Condensed matter is different: at zero pressure, liquids and solids have a finite density due to attractive intermolecular interactions. Stretching a liquid or a solid means applying a positive stress to it, that is a negative pressure. At such a negative pressure, a liquid cannot be in equilibrium but it can stay metastable for a very long time. If our chamber had very clean, smooth and hydrophilic walls, and if it was filled with very pure water instead of a gas, we could pull the piston and reach a moderate negative pressure before vapour bubbles would nucleate. The pressure would follow an extension of the equation of state P(ρ) in a metastable region at negative pressure. In 1850, Berthelot held the world record for negative pressures, obtained by cooling down a very clean glass ampoule which he had first filled with water at high temperature and pressure and sealed. When cooled down, the water evolved along an isochore and the pressure decreased. Below a certain temperature depending on initial conditions, the water was under stress. Berthelot reached −50 bar. In 1991, Zheng et al [3] used the same method and reached −1400 bar with small water inclusions in quartz crystals. Finally, since an acoustic wave is an oscillation in density and pressure, it can induce negative pressures if its amplitude is larger than the static pressure in the medium where it propagates. A liquid can be taken far away from its stability region on the path of a largeamplitude acoustic wave. This is the method that we use for the study of homogeneous cavitation or crystallization in pure liquids. 1. Cavitation in helium 4, helium 3 and water