Recently, we adapted density-functional theory (DFT) to calculate the free energy surface, or reversible work W(n,v), of bubble formation for the pure-component superheated Lennard-Jones (LJ) liquid[2], where n is the number of particles inside a bubble of volume v. The DFT calculations, which constrain the number of particles located inside the bubble for a fixed radius, indicate that W(n,v) is quite different from what is predicted from CNT. In particular, DFT suggests that liquid-to-vapor liquid nucleation is more appropriately described by an “activated instability”. As the free energy barrier is surmounted, W(n,v) abruptly ends along a locus of instabilities. Further growth of the post-critical bubbles must necessarily proceed via a mechanism appropriate for an unstable system. Also in contrast to the classical picture of bubble nucleation, DFT reveals that the saddle point, which still corresponds to the critical bubble, is not the only pathway an embryo may take in order to cross the activation barrier. The ridge corresponding to the maximum free energy for each n that leads to the critical bubble is not steep, suggesting that an embryo will more likely than not surmount the barrier along pathways that do not pass through the saddle point.
Given the very different view of bubble nucleation that emerges from our DFT method, we next turn our attention to homogeneous droplet formation with a bulk supercooled vapor. Specifically, using our modified DFT method, we investigate whether the molecular mechanism of vapor-to-liquid nucleation is likewise described by a set of activated instabilities. We find that the free energy surface, W(n,v), of droplet formation maintains most of the characteristic features obtained in earlier theoretical approaches (e.g., [3]). While still exhibiting a ridge of maximum volumes (similar to the ridge found in the bubble surface), a valley now appears beyond the ridge that funnels the post-critical droplets towards the bulk liquid phase. In contrast to the bubble surface, W(n,v) for droplet formation does not abruptly end after the ridge is surmounted. Nevertheless, limits of stability do appear when the liquid clusters become too dense, such that the surrounding supercooled vapor can no longer maintain its vapor-like density. In this case, one of the faces of the valley abruptly ends at a locus of instability, though the valley itself continues indefinitely towards the liquid phase. Despite the appearance of these instabilities, the overall picture, or the net effect of droplet nucleation, has not really changed. In the end, the DFT analysis of vapor-to-liquid nucleation serves to highlight the important differences between droplet and bubble nucleation, indicating that any future descriptions of bubble formation cannot solely rely on ideas that have emerged from the study of droplet formation.
In addition, we also present an extension of our DFT method to the study of the growth of bubbles of various (non-spherical) shapes. Bubbles forming within superheated liquids are not constrained to maintain a spherical shape. In fact, molecular simulation studies of bubble nucleation[4,5] demonstrate that pre-critical bubbles are not perfectly spherical (although they do not deviate significantly from a sphere). Since our previous analysis of bubble formation was developed for spherical embryos, we now explore the effects of shape on the physics of bubble nucleation. Specifically, we present results for the work of prolate and oblate spheroid bubbles. The density profiles surrounding these embryos are now a function of two spatial dimensions as opposed to the single one needed when spherical symmetry is imposed. As a result, the aspect ratio and not just the volume of these spheroids is another important parameter in constructing the free energy surface, i.e., the free energy surface is properly described by W(n,v,c/a), where c is the length of the major axis and a is the length of the minor axis. We find that a locus of instabilities still exists for these non-spherical shapes, so that the previous conclusion regarding an activated instability as the appropriate mechanism describing bubble nucleation and growth remains unchanged.
Finally, we consider the simultaneous growth of two bubbles within the superheated LJ liquid. By varying both the sizes of and center-to-center separations between the two bubbles, we discuss how the resulting density profiles and free energy surfaces provide insights into the importance of coalescence during the unstable growth phase of bubbles. A recent simulation study of Zahn[6] on the boiling of a water-like fluid indicates that phase separation is initiated by the formation of several tiny cavities, which in turn coalesce to form a larger cavity that appears to then grow further into a macroscopic bubble.
[1] L. Gunther, Am. J. Phys., 71, 351 (2003).
[2] M. J. Uline and D. S. Corti, Phys. Rev. Letters, submitted.
[3] P. Schaaf, B. Senger, and H. Reiss, J. Phys. Chem. B, 101, 8740 (1997).
[4] T. Kinjo and M. Matsumoto, Fluid Phase Equilibria, 144, 343 (1998).
[5] B. R. Novak, E. J. Maginn and M. J. McCready, Phys. Rev. B, 75 085413 (2007).
[6] D. Zahn, Phys. Rev. Letters, 93, 227801 (2004).