11 research outputs found
Crystallization: colloidal suspense
According to classical nucleation theory, a crystal grows from a small nucleus that already bears the symmetry of its end phase—but experiments with colloids now reveal that, from an amorphous precursor, crystallites with different structures can develop
Phase-field modeling of eutectic structures on the nanoscale: the effect of anisotropy
A simple phase-field model is used to address anisotropic eutectic freezing on the nanoscale in two (2D) and three dimensions (3D). Comparing parameter-free simulations with experiments, it is demonstrated that the employed model can be made quantitative for Ag-Cu. Next, we explore the effect of material properties, and the conditions of freezing on the eutectic pattern. We find that the anisotropies of kinetic coefficient and the interfacial free energies (solid-liquid and solid-solid), the crystal misorientation relative to pulling, the lateral temperature gradient, play essential roles in determining the eutectic pattern. Finally, we explore eutectic morphologies, which form when one of the solid phases are faceted, and investigate cases, in which the kinetic anisotropy for the two solid phases are drastically different
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
Orientation-field models for polycrystalline solidification: grain coarsening and complex growth forms
We compare two versions of the phase-field theory for polycrystalline solidification, both relying on the concept of orientation
fields: one by Kobayashi et al. [Physica D 140 (2000) 141] and the other by Henry et al. [Phys. Rev. B 86 (2012) 054117]. Setting
the model parameters so that the grain boundary energies and the time scale of grain growth are comparable in the two models, we
first study the grain coarsening process including the limiting grain size distribution, and compare the results to those from experiments
on thin films, to the models of Hillert, and Mullins, and to predictions by multiphase-field theories. Next, following earlier
work by Gránásy et al. [Phys. Rev. Lett. 88 (2002) 206105; Phys. Rev. E 72 (2005) 011605], we extend the orientation field to the
liquid state, where the orientation field is made to fluctuate in time and space, and employ the model for describing of multi-dendritic
solidification, and polycrystalline growth, including the formation of “dizzy” dendrites disordered via the interaction with foreign
particles
Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview
Here we review the basic concepts and applications of the phase-field-crystal (PFC) method,
which is one of the latest simulation methodologies in materials science for problems, where
atomic- and microscales are tightly coupled. The PFC method operates on atomic length and
diffusive time scales, and thus constitutes a computationally efficient alternative to molecular
simulation methods. Its intense development in materials science started fairly recently following
the work by Elder et al. [Phys. Rev. Lett. 88 (2002), p. 245701]. Since these initial
studies, dynamical density functional theory and thermodynamic concepts have been linked to
the PFC approach to serve as further theoretical fundamentals for the latter. In this review, we
summarize these methodological development steps as well as the most important applications
of the PFC method with a special focus on the interaction of development steps taken in hard
and soft matter physics, respectively. Doing so, we hope to present today’s state of the art in
PFC modelling as well as the potential, which might still arise from this method in physics and
materials science in the nearby future
Nonlinear hydrodynamic theory of crystallization
We present an isothermal fluctuating nonlinear hydrodynamic theory of crystallization in
molecular liquids. A dynamic coarse-graining technique is used to derive the velocity field, a
phenomenology which allows a direct coupling between the free energy functional of the
classical density functional theory and the Navier–Stokes equation. In contrast to the
Ginzburg–Landau type amplitude theories, the dynamic response to elastic deformations is
described by parameter-free kinetic equations. Employing our approach to the free energy
functional of the phase-field crystal model, we recover the classical spectrum for the phonons
and the steady-state growth fronts. The capillary wave spectrum of the equilibrium
crystal–liquid interface is in good qualitative agreement with the molecular dynamics
simulations
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
Thermodynamics, formation dynamics and structural correlations in the bulk amorphous phase of the phase-field crystal model
We investigate bulk thermodynamic and microscopic structural properties of amorphous solids in the framework of the phase-field crystal (PFC) model. These are metastable states with a non-uniform density distribution having no long-range order. From extensive numerical simulations we determine the distribution of free energy density values in varying size amorphous systems and also the point-to-set correlation length, which is the radius of the largest volume of amorphous one can take while still having the particle arrangements within the volume determined by the particle ordering at the surface of the chosen volume. We find that in the thermodynamic limit, the free energy density of the amorphous tends to a value that has a slight dependence on the initial state from which it was formed – i.e. it has a formation history dependence. The amorphous phase is observed to form on both sides of the liquid linear-stability limit, showing that the liquid to amorphous transition is first order, with an associated finite free energy barrier when the liquid is metastable. In our simulations this is demonstrated when noise in the initial density distribution is used to induce nucleation events from the metastable liquid. Depending on the strength of the initial noise, we observe a variety of nucleation pathways, in agreement with previous results for the PFC model, and which show that amorphous precursor mediated multi-step crystal nucleation can occur in colloidal systems.</p
Heterogeneous nucleation of/on nanoparticles: a density functional study using the phase-field crystal model
Crystallization of supersaturated liquids usually starts by heterogeneous nucleation. Mounting evidence shows
that even homogeneous nucleation in simple liquids takes place in two steps; first a dense amorphous
precursor forms, and the crystalline phase appears via heterogeneous nucleation in/on the precursor cluster.
Herein, we review recent results by a simple dynamical density functional theory, the phase-field crystal
model, for (precursor-mediated) homogeneous and heterogeneous nucleation of nanocrystals. It will be
shown that the mismatch between the lattice constants of the nucleating crystal and the substrate plays a
decisive role in determining the contact angle and nucleation barrier, which were found to be non-monotonic
functions of the lattice mismatch. 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
Phase-field modeling of polycrystalline solidification, from needle crystals to spherulites: a review
Advances in the orientation-field-based phase-field (PF) models made in the past are reviewed.
The models applied incorporate homogeneous and heterogeneous nucleation of growth centers
and several mechanisms to form new grains at the perimeter of growing crystals, a phenomenon
termed growth front nucleation. Examples for PF modeling of such complex polycrystalline
structures are shown as impinging symmetric dendrites, polycrystalline growth forms (ranging
from disordered dendrites to spherulitic patterns), and various eutectic structures, including
spiraling two-phase dendrites. Simulations exploring possible control of solidification patterns
in thin films via external fields, confined geometry, particle additives, scratching/piercing the
films, etc. are also displayed. Advantages, problems, and possible solutions associated with
quantitative PF simulations are discussed briefly