850 research outputs found
Quantitative Phase Field Model of Alloy Solidification
We present a detailed derivation and thin interface analysis of a phase-field
model that can accurately simulate microstructural pattern formation for
low-speed directional solidification of a dilute binary alloy. This advance
with respect to previous phase-field models is achieved by the addition of a
phenomenological "antitrapping" solute current in the mass conservation
relation [A. Karma, Phys. Rev. Lett 87, 115701 (2001)]. This antitrapping
current counterbalances the physical, albeit artificially large, solute
trapping effect generated when a mesoscopic interface thickness is used to
simulate the interface evolution on experimental length and time scales.
Furthermore, it provides additional freedom in the model to suppress other
spurious effects that scale with this thickness when the diffusivity is unequal
in solid and liquid [R. F. Almgren, SIAM J. Appl. Math 59, 2086 (1999)], which
include surface diffusion and a curvature correction to the Stefan condition.
This freedom can also be exploited to make the kinetic undercooling of the
interface arbitrarily small even for mesoscopic values of both the interface
thickness and the phase-field relaxation time, as for the solidification of
pure melts [A. Karma and W.-J. Rappel, Phys. Rev. E 53, R3017 (1996)]. The
performance of the model is demonstrated by calculating accurately for the
first time within a phase-field approach the Mullins-Sekerka stability spectrum
of a planar interface and nonlinear cellular shapes for realistic alloy
parameters and growth conditions.Comment: 51 pages RevTeX, 5 figures; expanded introduction and discussion; one
table and one reference added; various small correction
Towards a quantitative phase-field model of two-phase solidification
We construct a diffuse-interface model of two-phase solidification that
quantitatively reproduces the classic free boundary problem on solid-liquid
interfaces in the thin-interface limit. Convergence tests and comparisons with
boundary integral simulations of eutectic growth show good accuracy for
steady-state lamellae, but the results for limit cycles depend on the interface
thickness through the trijunction behavior. This raises the fundamental issue
of diffuse multiple-junction dynamics.Comment: 4 pages, 2 figures. Better final discussion. 1 reference adde
Phase-Field Formulation for Quantitative Modeling of Alloy Solidification
A phase-field formulation is introduced to simulate quantitatively
microstructural pattern formation in alloys. The thin-interface limit of this
formulation yields a much less stringent restriction on the choice of interface
thickness than previous formulations and permits to eliminate non-equilibrium
effects at the interface. Dendrite growth simulations with vanishing solid
diffusivity show that both the interface evolution and the solute profile in
the solid are well resolved
Phase-field modeling of microstructural pattern formation during directional solidification of peritectic alloys without morphological instability
During the directional solidification of peritectic alloys, two stable solid
phases (parent and peritectic) grow competitively into a metastable liquid
phase of larger impurity content than either solid phase. When the parent or
both solid phases are morphologically unstable, i.e., for a small temperature
gradient/growth rate ratio (), one solid phase usually outgrows and
covers the other phase, leading to a cellular-dendritic array structure closely
analogous to the one formed during monophase solidification of a dilute binary
alloy. In contrast, when is large enough for both phases to be
morphologically stable, the formation of the microstructurebecomes controlled
by a subtle interplay between the nucleation and growth of the two solid
phases. The structures that have been observed in this regime (in small samples
where convection effect are suppressed) include alternate layers (bands) of the
parent and peritectic phases perpendicular to the growth direction, which are
formed by alternate nucleation and lateral spreading of one phase onto the
other as proposed in a recent model [R. Trivedi, Metall. Mater. Trans. A 26, 1
(1995)], as well as partially filled bands (islands), where the peritectic
phase does not fully cover the parent phase which grows continuously. We
develop a phase-field model of peritectic solidification that incorporates
nucleation processes in order to explore the formation of these structures.
Simulations of this model shed light on the morphology transition from islands
to bands, the dynamics of spreading of the peritectic phase on the parent phase
following nucleation, which turns out to be characterized by a remarkably
constant acceleration, and the types of growth morphology that one might expect
to observe in large samples under purely diffusive growth conditions.Comment: Final version, minor revisions, 16 pages, 14 EPS figures, RevTe
Multi-phase-field analysis of short-range forces between diffuse interfaces
We characterize both analytically and numerically short-range forces between
spatially diffuse interfaces in multi-phase-field models of polycrystalline
materials. During late-stage solidification, crystal-melt interfaces may
attract or repel each other depending on the degree of misorientation between
impinging grains, temperature, composition, and stress. To characterize this
interaction, we map the multi-phase-field equations for stationary interfaces
to a multi-dimensional classical mechanical scattering problem. From the
solution of this problem, we derive asymptotic forms for short-range forces
between interfaces for distances larger than the interface thickness. The
results show that forces are always attractive for traditional models where
each phase-field represents the phase fraction of a given grain. Those
predictions are validated by numerical computations of forces for all
distances. Based on insights from the scattering problem, we propose a new
multi-phase-field formulation that can describe both attractive and repulsive
forces in real systems. This model is then used to investigate the influence of
solute addition and a uniaxial stress perpendicular to the interface. Solute
addition leads to bistability of different interfacial equilibrium states, with
the temperature range of bistability increasing with strength of partitioning.
Stress in turn, is shown to be equivalent to a temperature change through a
standard Clausius-Clapeyron relation. The implications of those results for
understanding grain boundary premelting are discussed.Comment: 24 pages, 28 figure
Instability and Spatiotemporal Dynamics of Alternans in Paced Cardiac Tissue
We derive an equation that governs the spatiotemporal dynamics of small
amplitude alternans in paced cardiac tissue. We show that a pattern-forming
linear instability leads to the spontaneous formation of stationary or
traveling waves whose nodes divide the tissue into regions with opposite phase
of oscillation of action potential duration. This instability is important
because it creates dynamically an heterogeneous electrical substrate for
inducing fibrillation if the tissue size exceeds a fraction of the pattern
wavelength. We compute this wavelength analytically as a function of three
basic length scales characterizing dispersion and inter-cellular electrical
coupling.Comment: 4 pages, 3 figures, submitted to PR
Eutectic Colony Formation: A Stability Analysis
Experiments have widely shown that a steady-state lamellar eutectic
solidification front is destabilized on a scale much larger than the lamellar
spacing by the rejection of a dilute ternary impurity and forms two-phase cells
commonly referred to as `eutectic colonies'. We extend the stability analysis
of Datye and Langer for a binary eutectic to include the effect of a ternary
impurity. We find that the expressions for the critical onset velocity and
morphological instability wavelength are analogous to those for the classic
Mullins-Sekerka instability of a monophase planar interface, albeit with an
effective surface tension that depends on the geometry of the lamellar
interface and, non-trivially, on interlamellar diffusion. A qualitatively new
aspect of this instability is the occurence of oscillatory modes due to the
interplay between the destabilizing effect of the ternary impurity and the
dynamical feedback of the local change in lamellar spacing on the front motion.
In a transient regime, these modes lead to the formation of large scale
oscillatory microstructures for which there is recent experimental evidence in
a transparent organic system. Moreover, it is shown that the eutectic front
dynamics on a scale larger than the lamellar spacing can be formulated as an
effective monophase interface free boundary problem with a modified
Gibbs-Thomson condition that is coupled to a slow evolution equation for the
lamellar spacing. This formulation provides additional physical insights into
the nature of the instability and a simple means to calculate an approximate
stability spectrum. Finally, we investigate the influence of the ternary
impurity on a short wavelength oscillatory instability that is already present
at off-eutectic compositions in binary eutectics.Comment: 26 pages RevTex, 14 figures (28 EPS files); some minor changes;
references adde
Phase-Field Approach for Faceted Solidification
We extend the phase-field approach to model the solidification of faceted
materials. Our approach consists of using an approximate gamma-plot with
rounded cusps that can approach arbitrarily closely the true gamma-plot with
sharp cusps that correspond to faceted orientations. The phase-field equations
are solved in the thin-interface limit with local equilibrium at the
solid-liquid interface [A. Karma and W.-J. Rappel, Phys. Rev. E53, R3017
(1996)]. The convergence of our approach is first demonstrated for equilibrium
shapes. The growth of faceted needle crystals in an undercooled melt is then
studied as a function of undercooling and the cusp amplitude delta for a
gamma-plot of the form 1+delta(|sin(theta)|+|cos(theta)|). The phase-field
results are consistent with the scaling law "Lambda inversely proportional to
the square root of V" observed experimentally, where Lambda is the facet length
and V is the growth rate. In addition, the variation of V and Lambda with delta
is found to be reasonably well predicted by an approximate sharp-interface
analytical theory that includes capillary effects and assumes circular and
parabolic forms for the front and trailing rough parts of the needle crystal,
respectively.Comment: 1O pages, 2 tables, 17 figure
Universal Dynamics of Phase-Field Models for Dendritic Growth
We compare time-dependent solutions of different phase-field models for
dendritic solidification in two dimensions, including a thermodynamically
consistent model and several ad hoc models. The results are identical when the
phase-field equations are operating in their appropriate sharp interface limit.
The long time steady state results are all in agreement with solvability
theory. No computational advantage accrues from using a thermodynamically
consistent phase-field model.Comment: 4 pages, 3 postscript figures, in latex, (revtex
Phase field modelling of grain boundary premelting using obstacle potentials
We investigate the multi-order parameter phase field model of Steinbach and
Pezzolla [I. Steinbach, F. Pezzolla, A generalized field method for multiphase
transformations using interface fields, Physica D 134 (1999) 385-393]
concerning its ability to describe grain boundary premelting. For a single
order parameter situation solid-melt interfaces are always attractive, which
allows to have (unstable) equilibrium solid-melt-solid coexistence above the
bulk melting point. The temperature dependent melt layer thickness and the
disjoining potential, which describe the interface interaction, are affected by
the choice of the thermal coupling function and the measure to define the
amount of the liquid phase. Due to the strictly finite interface thickness also
the interaction range is finite. For a multi-order parameter model we find
either purely attractive or purely repulsive finite-ranged interactions. The
premelting transition is then directly linked to the ratio of the grain
boundary and solid-melt interfacial energy.Comment: 12 page
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