(296d) Homogeneous Nucleation and Growth: From Equilibrium-Based Analysis of the Embryos to the Dynamics of the Phase Transition

Homogeneous nucleation is the activated process by which the embryo of a new phase is formed due to the spontaneous fluctuations in local density within a bulk metastable fluid and in the absence of impurities, solid surfaces or external perturbations. According to classical nucleation theory (CNT) [1], if an embryo of the new phase is smaller than some critical size, the embryo collapses back into the metastable fluid; if the embryo exceeds this critical size, it grows to a macroscopic new phase. Previously, Punnathanam and Corti [2] and Uline and Corti [3] adapted a constrained density-functional theory (DFT) to generate the free energy surface for equilibrium embryos of volume v that contain n particles of the Lennard-Jones fluid. In order to generate the density profile, ρ(r), of an equilibrium (n,v) embryo, analogous to the one within CNT, they solved for a spherically symmetric density profile that yields the minimum free energy of the system subject to the following constraint: ∫dv ρ = n. In the DFT approach, as opposed to CNT, the non-uniformity of the density throughout the embryo and the surrounding mother-phase, as well as the interaction between the embryo particles and mother-phase, is taken into account. The resulting free energy surface revealed some interesting behavior that was not predicted by the less rigorous treatment of CNT. For instance, in the DFT free energy landscape, appears a flat ridge of local maxima in (n,v) space, which begins with a pseudo-saddle point . Another new and interesting feature of the generated free energy surface is the emergence of a locus of instabilities. This locus of instabilities, which is the result of including the interaction between the embryo and the mother-phase, sets a limit on the region in the (n,v) space for which the surrounding system can maintain a stable density profile that represents the metastable bulk phase. These features of the (n,v) free energy surface have also been validated via umbrella-sampling Monte-Carlo (MC) simulation. Based on its current definition, this (n,v) equilibrium embryo analysis fails, however, to eliminate redundancies when assigning different configurations to their corresponding order parameter. This means that a single configuration can be equivalently assigned to more than one point in the (n,v) space. This in turn leads to points with similar or even identical density profiles. In order to remove the redundancies inherent in the equilibrium (n,v) embryo definition, we apply the idea of a bond constraint or cluster connectivity [4] to our analysis. This additional constraint is applied to the DFT analysis by excluding those embryos that contain a region with a density resembling that of the surrounding mother-phase. We also apply the bond constraint to the generation of the free energy via the use of umbrella sampling MC simulations. For both the Lennard-Jones fluid and a water-like system, we ensure that all the particles assigned to a given embryo are surrounded by or connected to the cluster of the new phase formed within the bulk mother-phase. The application of the bond constraint to the equilibrium analysis of (n,v) embryos results in a new free energy landscape that is more instructive in explaining the dynamics of the nucleation and growth. The growth of an embryo of the new phase now becomes a more properly defined trajectory on the free energy surface, which is crucial in proposing a tractable dynamical theory for the kinetics of homogeneous nucleation and growth. Given a much improved definition of an equilibrium embryo, we then discuss an initial attempt to bridge the gap between our knowledge of equilibrium (n,v) embryos and the dynamics of homogeneous nucleation and growth. We perform a series of isothermal-isobaric molecular dynamics (MD) simulations in order to follow the time evolution of different (n,v) embryos formed within the metastable Lennard-Jones fluid. The initial configurations for these MD simulations are snapshots taken form equilibrated (n,v) bubbles generated via an isothermal-isochoric MC simulation. Here, an impermeable spherical boundary of volume v is introduced at the center of the simulation box, which serves to define the boundary of the bubble. After that the system is allowed to sample only those configurations in which exactly n particles are contained inside volume v. After sufficient equilibration steps, a random configuration is saved, providing the starting point for the subsequent MD simulation, but now the boundary used to generate the bubble is removed. With the explained procedure, we demonstrate that for cavities as well as small particle bubbles there exists a limiting radius below which all the embryos collapse and the system traverses a pathway that leads back to the uniform liquid phase (as was predicted by the DFT analysis of [2] and [3]). Beyond that critical radius, the MD simulations reveal an instability that results in a phase transition, whereby the metastable liquid evolves toward the vapor phase. The overall density of the simulation cell is followed throughout the MD simulations and provides a clear indication of the phase transition. If the system remains as a metastable liquid, the density slowly approaches that of the bulk superheated liquid at the imposed temperature and pressure. On the other hand, if the bubble starts to grow and the phase transition occurs, the overall density rapidly drops. We also compare the locus of instabilities and its proximity to the flat ridge of the free energy surface as predicted by the DFT analysis and the region of phase transitions revealed by our MD simulations.

[1] L. Gunther, Am. J. Phys. 71, 351 (2003). [2] S. Punnathanam and D. S. Corti, J. Chem. Phys. 123, 164101(2005). [3] M. J. Uline and D. S. Corti, Phys. Rev. Letters 99, 076102 (2007). [4] F. H. Stillinger, J. Chem. Phys. 38, 1486 (1963).