66 research outputs found
Defects and multistability in eutectic solidification patterns
We use three-dimensional phase-field simulations to investigate the dynamics
of the two-phase composite patterns formed upon during solidification of
eutectic alloys. Besides the spatially periodic lamellar and rod patterns that
have been widely studied, we find that there is a large number of additional
steady-state patterns which exhibit stable defects. The defect density can be
so high that the pattern is completely disordered, and that the distinction
between lamellar and rod patterns is blurred. As a consequence, the transition
from lamellae to rods is not sharp, but extends over a finite range of
compositions and exhibits strong hysteresis. Our findings are in good agreement
with experiments.Comment: 6 pages, 8 figure
Remarks on some open problems in phase-field modelling of solidification
International audienceThree different topics in phase-field modelling of solidification are discussed, with particular emphasis on the limitations of the currently available modelling approaches. First, thin-interface limits of two-sided phase-field models are examined, and it is shown that the antitrapping current is in general not sufficient to remove all thin-interface effects. Second, orientation-field models for polycrystalline solidification are analyzed, and it is shown that the standard relaxational equation of motion for the orientation field is incorrect in coherent polycrystalline matter. Third, it is pointed out that the standard procedure of incorporating fluctuations into the phase-field approach cannot be used in a straightforward way for a quantitative description of nucleation
Lattice Boltzmann simulations of 3D crystal growth: Numerical schemes for a phase-field model with anti-trapping current
A lattice-Boltzmann (LB) scheme, based on the Bhatnagar-Gross-Krook (BGK)
collision rules is developed for a phase-field model of alloy solidification in
order to simulate the growth of dendrites. The solidification of a binary alloy
is considered, taking into account diffusive transport of heat and solute, as
well as the anisotropy of the solid-liquid interfacial free energy. The
anisotropic terms in the phase-field evolution equation, the phenomenological
anti-trapping current (introduced in the solute evolution equation to avoid
spurious solute trapping), and the variation of the solute diffusion
coefficient between phases, make it necessary to modify the equilibrium
distribution functions of the LB scheme with respect to the one used in the
standard method for the solution of advection-diffusion equations. The effects
of grid anisotropy are removed by using the lattices D3Q15 and D3Q19 instead of
D3Q7. The method is validated by direct comparison of the simulation results
with a numerical code that uses the finite-difference method. Simulations are
also carried out for two different anisotropy functions in order to demonstrate
the capability of the method to generate various crystal shapes
Grain coarsening in two-dimensional phase-field models with an orientation field
In the literature, contradictory results have been published regarding the
form of the limiting (long-time) grain size distribution (LGSD) that
characterizes the late stage grain coarsening in two-dimensional and
quasi-two-dimensional polycrystalline systems. While experiments and the
phase-field crystal (PFC) model (a simple dynamical density functional theory)
indicate a lognormal distribution, other works including theoretical studies
based on conventional phase-field simulations that rely on coarse grained
fields, like the multi-phase-field (MPF) and orientation field (OF) models,
yield significantly different distributions. In a recent work, we have shown
that the coarse grained phase-field models (whether MPF or OF) yield very
similar limiting size distributions that seem to differ from the theoretical
predictions. Herein, we revisit this problem, and demonstrate in the case of OF
models [by R. Kobayashi et al., Physica D 140, 141 (2000) and H. Henry et al.
Phys. Rev. B 86, 054117 (2012)] that an insufficient resolution of the small
angle grain boundaries leads to a lognormal distribution close to those seen in
the experiments and the molecular scale PFC simulations. Our work indicates,
furthermore, that the LGSD is critically sensitive to the details of the
evaluation process, and raises the possibility that the differences among the
LGSD results from different sources may originate from differences in the
detection of small angle grain boundaries
Phase-field crystal study of grain-boundary premelting
We study the phenomenon of grain-boundary premelting for temperatures below
the melting point in the phase-field crystal model of a pure material with
hexagonal ordering in two dimensions. We investigate the structures of
symmetric tilt boundaries as a function of misorientation for two different
inclinations and compute in the grand canonical ensemble the disjoining
potential V(w) that governs the fundamental interaction between crystal-melt
interfaces as a function of the premelted layer width w. The results reveal
qualitatively different behaviors for high-angle grain boundaries that are
uniformly wetted, with w diverging logarithmically as the melting point is
approached from below, and low-angle boundaries that are punctuated by liquid
pools surrounding dislocations, separated by solid bridges. This qualitative
difference between high and low angle boundaries is reflected in the
w-dependence of the disjoining potential that is purely repulsive (V'(w)<0 for
all w) above a critical misorientation, but switches from repulsive at small w
to attractive at large w for low angles. In the latter case, V(w) has a minimum
that corresponds to a premelted boundary of finite width at the melting point.
Furthermore, we find that the standard wetting condition (the grain boundary
energy is equal to twice the solid-liquid free energy) gives a much too low
estimate of the critical misorientation when a low-temperature value of the
grain boundary energy is used. In contrast, a reasonable estimate is obtained
if the grain boundary energy is extrapolated to the melting point, taking into
account both the elastic softening of the material at high temperature and
local melting around dislocations.Comment: 24 pages, 13 figures, some figure files with reduced resolution
because of submission size limitations. In the 2nd version, some parts (and
figures) have been modified, especially in Sec. V (discussion
Unified derivation of phase-field models for alloy solidification from a grand-potential functional
In the literature, two quite different phase-field formulations for the
problem of alloy solidification can be found. In the first, the material in the
diffuse interfaces is assumed to be in an intermediate state between solid and
liquid, with a unique local composition. In the second, the interface is seen
as a mixture of two phases that each retain their macroscopic properties, and a
separate concentration field for each phase is introduced. It is shown here
that both types of models can be obtained by the standard variational procedure
if a grand-potential functional is used as a starting point instead of a
free-energy functional. The dynamical variable is then the chemical potential
instead of the composition. In this framework, a complete analogy with
phase-field models for the solidification of a pure substance can be
established. This analogy is then exploited to formulate quantitative
phase-field models for alloys with arbitrary phase diagrams. The precision of
the method is illustrated by numerical simulations with varying interface
thickness.Comment: 36 pages, 1 figur
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