Activated Instability of Homogeneous Bubble Nucleation and Growth Mark J. Uline and David S. Corti * School of Chemcial Engineering, Purdue University, West Lafayette, Indiana 47907, USA (Received 9 February 2007; published 16 August 2007) For the superheated Lennard-Jones liquid, the free energy of forming a bubble with a given particle number and volume is calculated using density-functional theory. As conjectured, a consequence of known properties of the critical cavity [S. N. Punnathanam and D. S. Corti, J. Chem. Phys. 119, 10 224 (2003)], the free energy surface terminates at a locus of instability. These stability limits reside, however, unexpectedly close to the saddle point. A new picture of homogeneous bubble nucleation and growth emerges from our study, being more appropriately described as an ‘‘activated instability.’’ DOI: 10.1103/PhysRevLett.99.076102 PACS numbers: 82.60.Nh, 64.70.Fx Homogeneous bubble nucleation is the activated process by which the vapor phase is formed from a bulk super- heated liquid in the absence of impurities or solid surfaces [1]. According to classical nucleation theory (CNT) [1], if an embryo of the vapor phase is less than some critically sized bubble, the embryo collapses back into the super- heated liquid; if the embryo exceeds this critical size, the bubble grows to macroscopic size. Within CNT, the homo- geneous nucleation rate is expressed in terms of the free energy, or reversible work, of forming embryos of various sizes. Since the vapor embryo is compressible, the free energy of formation is usually expressed as a function of the radius and internal pressure (or equivalently the num- ber of particles n) of the (spherical) bubble [1,2]. The critical bubble then corresponds to the saddle point in free energy space: a maximum in the radius and a mini- mum in the internal pressure (or n). In addition, the free energy surface continues on indefinitely beyond the saddle point, serving to channel the embryo toward the lower lying minimum corresponding to the bulk vapor phase (e.g., Fig. 4 of Ref. [2]). Though this subsequent growth of nuclei may be rapid, the post-critical embryos never- theless follow well-defined pathways that describe the reversible change of n and radii. In obtaining an expression for the nucleation rate, CNT also invokes another feature of the free energy surface, namely, that the region about the saddle point is sharply peaked [1]. Hence, only a small area centered about the saddle point describes the most likely transition paths between a precritical embryo and the vapor phase. Recently, Punnathanam and Corti [3,4] focused on the relevance of cavities to bubble nucleation (a cavity is a spherical region devoid of particle centers), where density- functional theory (DFT) was employed to study cavity formation within the superheated Lennard-Jones (LJ) liq- uid [4]. For cavity radii less than some critical size, the superheated liquid was found to be stable (DFT yielded convergent liquidlike density profiles around the cavity). Beyond this critical cavity size, the superheated liquid became unstable (no convergent liquidlike density profile was obtained). Molecular simulation verified the existence of the critical cavity [3]. A stability analysis revealed that the lowest eigenvalue of the matrix generated from the second-functional derivative of the grand potential went to zero at the critical cavity size [4], indicating that the critical cavity represents a true thermodynamic limit of stability. Also, the radius of the critical cavity  c was found to be a lower bound to the radius of the critical bubble, and the work of forming the critical cavity W c was found to be a tight upper bound to the work of forming the critical bubble W b [4]. These results suggest that cavities play an impor- tant role in the process of bubble nucleation, a conclusion that is not inconsistent with the apparent dominant role that cavities play in the initial stages of phase transitions in liquids as seen in previous molecular simulation studies [5]. Specifically, the existence of a critical cavity necessarily implies that the free energy surface Wn; v of bubble formation, where n is the number of particles inside a bubble of volume v, is very different from what follows from CNT [2]. The critical cavity, or terminus of the n  0 profile, should be in a sense ‘‘felt’’ throughout Wn; v. In other words, we suspect that a limit of stability will be reached for each n, with the radius of the bubble at this stability limit increasing with an increase in n. For small n, W at the limit of stability should decrease with an increase in n (because W c >W b ). Along n  n  , where n  is the number of particles inside the critical bubble, a maximum of W with respect to v will appear before the stability limit is reached, since the critical bubble corresponds to a saddle point. For some n in between n  0 (where no maximum appears [3]) and n  n  , other maxima should also develop before limits of stability are reached. Since the stability limits near the saddle point are located after maxima (beyond the barrier), they may be irrelevant to the early stages of bubble nucleation. This conjectured view of the free energy surface is different from previous constructions of the bubble surface and offers a new and intriguing picture of the molecular mechanism of bubble nucleation. For one, a locus of in- stability appears, describing the values of n and v at which PRL 99, 076102 (2007) PHYSICAL REVIEW LETTERS week ending 17 AUGUST 2007 0031-9007= 07=99(7)=076102(4) 076102-1  2007 The American Physical Society