# (625c) Activated Instability of Homogeneous Nucleation in Metastable Fluids

#### AIChE Annual Meeting

#### 2009

#### 2009 Annual Meeting

#### Engineering Sciences and Fundamentals

#### Nucleation and Growth

#### Thursday, November 12, 2009 - 4:05pm to 4:30pm

Homogeneous nucleation is the activated process by which the new phase (vapor, liquid or solid) is formed from a bulk metastable fluid (superheated liquid, supercooled vapor or liquid) in the absence of impurities or solid surfaces. According to classical nucleation theory (CNT)[1], if an embryo of the new phase is less than some critical size, the embryo collapses back into the metastable fluid; if the embryo exceeds this critical size, the nucleus ?reversibly? grows to macroscopic size. The critical nucleus corresponds to the saddle point in free energy space (if the embryo is not treated as incompressible): a maximum in the radius and a minimum in the number of particles, n, inside the embryo. The free energy surface continues on indefinitely beyond the saddle point, serving to channel the embryo toward the much lower lying minimum corresponding to the new, bulk phase. Though this subsequent growth of nuclei may be rapid, the post-critical embryos nevertheless follow well-defined pathways that describe the reversible change of n and radii. CNT also invokes another feature of the free energy surface, namely that the region about the saddle point is sharply peaked. Hence, only a small area centered about the saddle point describes the most likely transition paths between a pre-critical embryo and the new phase.

Recently, we adapted density-functional theory (DFT) to describe the phenomena of homogeneous nucleation in a metastable fluid comprised of Lennard-Jones particles[2]. In the DFT approach, as opposed to CNT, the non-uniformity of the density throughout the embryo and the surrounding mother-phase is taken into account. In order to generate a free energy surface analogous to the one within CNT, we solve for the density profile that yields the minimum free energy of a system comprised of a spherical embryo of a given radius λ, which contains a given number of particles n surrounded by the metastable mother-phase. The difference between the calculated free energy of the (n, λ) embryo system and the one for the corresponding uniform system is the minimum work of formation of the embryo, W(n,λ). Focusing on bubble nucleation within a superheated liquid, the obtained free energy surface reveals some interesting properties not predicted by CNT. For a large enough number of particles inside the bubble, W(n,λ) goes through a maximum with respect to the radius, analogous to what is seen in CNT. Increasing the radius further beyond this maximum, however, one reaches a critical radius beyond which no liquid-like density profile can be found that corresponds to the local minimum of the free energy functional. This in turn means there is a limit to the radius of the bubble for a given number of particles, in which we can have a locally stable superheated liquid. In this way our DFT approach, as opposed to CNT, predicts that the free energy surface terminates at a locus of instabilities. Similar results are found for the free energy surface that describes droplet nucleation within a supercooled vapor.

Characterizing the spherical embryo via the radius and number of particles requires the optimization of the free energy functional subject to the n, λ constraint. In addition to the traditional constrained functional optimization method using undetermined Lagrange multipliers, we also employ the number-conserving functional derivative method introduced by Gal [3]. This new approach expedites our numerical calculations considerably and yields equivalent accurate solutions using the method of Lagrange multipliers. Furthermore, we investigate in greater detail the thermodynamic stability of the (n,λ) embryo, particularly at the predicted limits of stability, by studying the proper eigenvalue integral equation that follows from the second functional derivative of the free energy. The developed methodology reveals that the smallest eigenvalue of the second functional derivative operator approaches zero exactly at the same locus of instabilities predicted by our DFT calculations.

We also extend our DFT method to fluids more complex than the relatively simple system of Lennard-Jones particles. We adapt the association model developed by Chapman et al. [4], based on an extension of Wertheim's theory [5], which is able to capture several important features of water-like systems. We demonstrate that homogeneous bubble and droplet nucleation within our water-like system is also described by a free energy surface that is qualitatively similar to the one for the Lennard-Jones fluid. In this way, we demonstrate that the interesting and new properties of the free energy surface, in particular the existence of a locus of instabilities and a relatively flat ridge of transition states all corresponding to the same value of the activation barrier, are not just limited to the Lennard-Jones fluid.

In addition, we perform a series of molecular dynamics (MD) simulations to further validate the predicted instabilities generated by bubbles and droplets beyond or below some critical size. For example, we initialize the superheated liquid in which is contained a spherical bubble of a given n and λ. The (n,λ) embryo-constraint is then relaxed and we allow the liquid to evolve dynamically on its own via additional MD steps. With this procedure, we test if a given configuration described by a given n and λ does in fact lead to a phase change, or whether the bubble collapses rapidly enough to maintain stability, as would be predicted in either case by our DFT stability analysis.

Finally, we also apply the shell particle formulation [6] to rigorously define the volume of an embryo with a given number of particles. Interestingly, we have shown that the thermodynamic properties of a system containing a shell particle can be straightforwardly related to the system that that is not described by a shell particle. In this way, we can straightforwardly calculate the free energy of a shell particle system from our previous DFT results. We discuss how the use of the rigorous shell particle formulation alters some but not all of the properties of the free energy surface of embryo formation.

[1] L. Gunther, Am. J. Phys., 71, 351 (2003).

[2] M. J. Uline and D. S. Corti, Phys. Rev. Letters, 99, 076102 (2007).

[3] T. Gal, Phys. Rev. A, 63, 022506 (2001).

[4] W. G. Chapman, K. E. Gubbins, G. Jackson and M. Radosz, Ind. Eng. Chem. Res., 29, 1709 (1990).

[5] M. S. Wetheim, J. of Stat. Phys., 35, 19 (1983).

[6] D. S. Corti and G. Soto-Campos , J. Chem. Phys., 108, 7959 (1998).