17 research outputs found
Free energy of the bcc-liquid interface and the Wulff shape as predicted by the Phase-Field Crystal model
The Euler-Lagrange equation of the phase-field crystal (PFC) model has been
solved under appropriate boundary conditions to obtain the equilibrium free
energy of the body centered cubic crystal-liquid interface for 18 orientations
at various reduced temperatures in the range .
While the maximum free energy corresponds to the
orientation for all values, the minimum is realized by the direction for small , and by the orientation for higher . The predicted dependence on
the reduced temperature is consistent with the respective mean field critical
exponent. The results are fitted with an eight-term Kubic harmonic series, and
are used to create stereographic plots displaying the anisotropy of the
interface free energy. We have also derived the corresponding Wulff shapes that
vary with increasing from sphere to a polyhedral form that differs
from the rhombo-dodecahedron obtained previously by growing a bcc seed until
reaching equilibrium with the remaining liquid
Hydrodynamic theory of freezing: Nucleation and polycrystalline growth
Structural aspects of crystal nucleation in undercooled liquids are explored
using a nonlinear hydrodynamic theory of crystallization proposed recently [G.
I. Toth et al., J. Phys.: Condens. Matter 26, 055001 (2014)], which is based on
combining fluctuating hydrodynamics with the phase-field crystal theory. We
show that in this hydrodynamic approach not only homogeneous and heterogeneous
nucleation processes are accessible, but also growth front nucleation, which
leads to the formation of new (differently oriented) grains at the solid-liquid
front in highly undercooled systems. Formation of dislocations at the
solid-liquid interface and interference of density waves ahead of the
crystallization front are responsible for the appearance of the new
orientations at the growth front that lead to spherulite-like nanostructures
Investigating nucleation using the phase-field method
The first order phase transitions, like freezing of liquids, melting of solids, phase separation in alloys, vapor condensation, etc., start with nucleation, a process in which internal fluctuations of the parent phase lead to formation of small seeds of the new phase. Owing to different size dependence of (negative) volumetric and (positive) interfacial contributions to work of formation of such seeds, there is a critical size, at which the work of formation shows a maximum. Seeds that are smaller than the critical one decay with a high probability, while the larger ones have a good chance to grow further and reach a macroscopic size. Putting it in another way, to form the bulk new phase, the system needs to pass a thermodynamic barrier via thermal fluctuations. When the fluctuations of the parent phase alone lead to transition, the process is called homogeneous nucleation. Such a homogeneous process is, however, scarcely seen and requires very specific conditions in nature or in the laboratory. Usually, the parent phase resides in a container and/or it incorporates floating heterogeneities (solid particles, droplets, etc.). The respective foreign surfaces lead to ordering of the adjacent liquid layers, which in turn may assist the formation of the seeds, a process termed heterogeneous nucleation. Herein, we review how the phase-field techniques contributed to the understanding of various aspects of crystal nucleation in undercooled melts, and its role in microstructure evolution. We recall results achieved using both conventional phase-field techniques that rely on spatially averaged (coarse grained) order parameters in capturing the phase transition, as well as molecular scale phase-field approaches that employ time averaged fields, as happens in the classical density functional theories, including the recently developed phase-field crystal models
Recent Developments in Modeling Heteroepitaxy/Heterogeneous Nucleation by Dynamical Density Functional Theory
Crystallization of supersaturated liquids usually starts by epitaxial growth or by heterogeneous
nucleation on foreign surfaces. Herein, we review recent advances made in modeling
heteroepitaxy and heterogeneous nucleation on flat/modulated surfaces and nanoparticles
within the framework of a simple dynamical density functional theory, known as the phase-field
crystal model. It will be shown that the contact angle and the nucleation barrier are nonmonotonous
functions of the lattice mismatch between the substrate and the crystalline phase.
In continuous cooling studies for substrates with lattice mismatch, we recover qualitatively the
Matthews–Blakeslee mechanism of stress release via the misfit dislocations. The simulations
performed for particle-induced freezing will be confronted with recent analytical results,
exploring thus the validity range of the latter. It will be demonstrated that time-dependent
studies are essential, as investigations based on equilibrium properties often cannot identify the
preferred nucleation pathways. Modeling of these phenomena is essential for designing materials
on the basis of controlled nucleation and/or nano-patterning