Recently, we adapted density-functional theory (DFT) to calculate the free energy surface, or reversible work W(n,v), of both homogeneous bubble formation within the pure-component superheated Lennard-Jones (LJ) liquid[2] and homogeneous droplet formation within the pure-component supercooled LJ vapor. (Here, n is the number of particles inside an embryo of a given spherical 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. For example, DFT reveals 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. DFT also suggests 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, which indicates that an embryo will more likely than not surmount the barrier along pathways that do not pass through the saddle point. Some of these conclusions also apply to vapor-to-liquid nucleation, though key differences between both free energy surfaces are noted. 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. Furthermore, unlike what was seen in previous theoretical approaches[3], a valley no longer appears beyond the saddle point. The surface beyond the ridge merely serves to channel the post-critical embryos towards the locus of instabilities. In the end, the DFT results 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.
For simplicity, the above DFT studies only considered the case of spherical embryos, a not unreasonable assumption for liquid droplets that may not be strictly applicable to bubble formation. Bubbles forming within superheated liquids are not constrained to maintain a spherical shape as previous molecular simulation studies of bubble nucleation[4,5] have demonstrated (although they do not deviate significantly from a sphere). We therefore explore the effects of the shape of the bubble on the physics of liquid-to-vapor nucleation. We present an extension of our DFT method to the study of the growth of bubbles of various non-spherical shapes. 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.
We also consider the simultaneous growth of two bubbles within the pure-component 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. We also speculate on what the appropriate signature is needed to identify these unstable growth stages and discuss how some current expressions for the rate of nucleation need to be modified to incorporate the different properties of the free energy surface that emerge from our studies.
Finally, we extend our constrained DFT method to the study of crystallization in the pure-component supercooled LJ liquid. The free energy surface for a given crystal structure is determined and compared to predictions of CNT. Our analysis of the work of formation of crystalline embryos also provides interesting insights into the ongoing debate concerning the appearance of spinodal (or barrierless) nucleation in deeply quenched LJ liquids[7].
[1] L. Gunther, Am. J. Phys., 71, 351 (2003).
[2] M. J. Uline and D. S. Corti, Phys. Rev. Letters, 99, 076102 (2007).
[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).
[7] L. S. Bartell and D. T. Wu, J. Chem. Phys., 127, 174507 (2007).