# A numerical study on the enhancement and suppression of crystal nucleation

Promotion on 20 October 2009

Summary of thesis

First-order phase transitions often start with nucleation, which refers to the spontaneous formation of a microscopic amount of the new phase due to thermal fluctuations. If the size of such a fluctuation exceeds a certain threshold, the so-called critical nucleus size, the new phase has a high probability to grow to macroscopic dimensions. Although this process is well understood from a phenomenological perspective, the design of nucleation agents to facilitate the growth of high-quality crystals requires insight into the physical mechanism on the microscopic scale.

The research presented in this Thesis aims to assess the physical mechanism of nucleation on the microscopic scale using many-particle simulations. The main emphasis of this work is on homogeneous and heterogeneous crystal nucleation of nano-colloids and proteins from dilute solution, but also other aspects of this field of research are addressed, such as alternative compute hardware, nearest neighbor algorithms, and the origin of geometrical frustration in liquids.

In Chapter 3 we discuss a topic common to all numerical simulations: the compute hardware. Simulations are typically performed on a computer's Central Processing Unit (CPU), such as a Pentium® or an Athlon® processor. The CPU's architecture is highly optimized for the execution of complex applications, dedicating significantly more transistors to program flow control and data caches than to so-called arithmetic logic units (ALUs), which perform integer and floating point operations. However, numerical simulations are often limited by the performance of these ALUs. Recent fully programmable Graphic Processing Units (GPUs) offer an alternative to CPUs as compute hardware. Driven by the demands of 3D video games they are designed to process large amounts of image data in parallel. In Chapter 3 we demonstrate that a conventional Molecular Dynamics simulation can be rewritten to run entirely on a GPU reducing its run-time by up to a factor 80 for an n-squared algorithm. Further, we show how cell lists domain decomposition can be parallelized efficiently resulting in a maximum 35-fold speed-up. We conclude this chapter with a discussion on the advantages and limitations of graphics hardware for the use in scientific simulations.

The next chapter concerns homogeneous crystal nucleation of nano-colloids from dilute solution. The model system consists of spherical particles interacting via a Lennard-Jones pair potential under conditions below the triple point. Here, the vapor is the meta-stable parent phase, the crystal the stable final phase, and the liquid is an intermediate meta-stable phase with a free energy inbetween the other two phases. Using forward-flux sampling paired with a local bond-order parameter we find no evidence for a direct vapor-crystal phase transition. Instead, using a density-based order parameter, we find the transition to proceed via a two-step process: first a meta-stable liquid droplet forms and subsequently a crystal emerges within this droplet. Both nucleation events are independent and can be treated separately. Our findings confirm earlier simulations based on quasi-equilibrium umbrella sampling. Our simulations reveal that a minimum droplet size is required to host a stable crystal cluster, and that the crystal is always covered by a liquid mono-layer. The overall nucleation rate is limited by the vapor-liquid nucleation step, and hence depends sensitively on the vapor pressure. We compare our simulation results to predictions from classical nucleation theory (CNT) and literature results using umbrella sampling. We find good qualitative agreement, which suggests that this phase transition can be treated by an equilibrium theory.

In Chapter 5 we extend the study on vapor-crystal nucleation of Lennard-Jones particles to the case of heterogeneous nucleation on smooth weakly-adsorbing surfaces. We investigate the microscopic mechanism of nucleation in the presence of an attractive planar wall, planar circular patch, and hemi-spherical pore. Nucleation in a pore is of particular interest due to experimental evidence that a porous medium may function as universal'' nucleation agent for protein crystallization. As in the case of homogeneous nucleation we observe the vapor-crystal nucleation pathway to proceed via a intermediate liquid phase, and that both nucleation events can be treated independently. For liquid nucleation, we find good qualitative agreement with CNT. Only for the circular patch there exists a narrow range of parameters for which CNT predicts a weak double-peak in the free-energy barrier that is not reproduced by our simulations. We find crystal nucleation to be spontaneous, that is without any noticeable barrier, for both the planar wall and the circular patch, provided the liquid droplet exceeds a minimum size. For nucleation in a hemi-spherical pore CNT predicts a double-peaked free-energy barrier for a wider range of parameters, and here the behavior is nicely reproduced by our simulation. We find that the first peak corresponds to filling the (microscopic) pore, and the second barrier separates the liquid from growing into the bulk. On increasing the pore radius the computed overall nucleation rate goes through a minimum before it approaches, in the limit of an infinitely large pore, the rate for nucleation on a planar wall. In contrast to the planar wall and circular patch, crystal nucleation is not induced by the surface of the pore, but proceeds homogeneously in the bulk. Although it is possible for a crystal to form within the pore, our simulations suggest that it is much more likely that crystal nucleation occurs after the liquid has grown out of the pore. In conclusion, our simulations provide further evidence that porous media with a broad distribution of pore sizes can significantly enhance crystal nucleation from dilute solution, and that the process is dominated by a few pores with optimal size.

The final chapter of this Thesis discusses the origin of geometrical frustration in liquids. Geometrical frustration is conjectured to prevent crystallization and therefore help glass formation. It arises from a competition between the local and global packing. Locally, particles can arrange to achieve a density higher than that of a crystal. But such a local order cannot be used to fill space. Once the system density exceeds a certain threshold a global packing becomes favorable. During the crystallization process particles have to leave their local arrangement in order to adapt to the global lattice. For hard-sphere systems this leads to the entropic free-energy barrier to crystal nucleation. This contrasts the behavior of two-dimensional hard disks, for which hexagonal packing is both locally and globally preferred, and crystallization is particularly easy. In three-dimensional (3D) Euclidean space, there exist two local structures that are considered to cause geometrical frustration. The first is the icosahedron as the smallest maximum kissing-number Voronoi polyhedron, and the second is the tetrahedron as the smallest volume that can show up in a Delaunay tessellation. Because the icosahedron can be constructed from almost-perfect tetrahedra, both are often used interchangeably, which leaves the true origin of frustration unclear. In this Chapter we leave the familiarity of our 3D world and investigate crystallization of four-dimensional (4D) hard (hyper-)spheres, because in 4D the smallest maximum kissing-number polyhedron \emph{is} commensurable with the densest crystal lattice. This allows us to assess whether it is the icosahedron or the tetrahedron that causes the frustration. Our simulations reveal a free-energy barrier that is significantly larger in 4D than in 3D at comparable conditions, clearly identifying tetrahedral order as the origin of frustration. Moreover, the high free-energy barrier to crystallization makes 4D hard spheres a suitable model system to test theories of glass transition and jamming. In the appendix of this chapter we extend the discussion to 5D and 6D hard spheres presenting equations of state and quantifying the growing dissimilarity between higher-dimensional fluid and crystal phases using second- and third-order bond-order invariants.

PhD promotion of
Jacobus (Koos) van Meel
promotor
Prof. Dr. D. Frenkel
when
10:00, Tuesday, 20 October 2009
where
Agnietenkapel, UvA, Amsterdam