22,252 research outputs found
An alternative to the Allen-Cahn phase field model for interfaces in solids - numerical efficiency
The derivation of the Allen-Cahn and Cahn-Hilliard equations is based on the
Clausius-Duhem inequality. This is not a derivation in the strict sense of the
word, since other phase field equations can be fomulated satisfying this
inequality. Motivated by the form of sharp interface problems, we formulate
such an alternative equation and compare the properties of the models for the
evolution of phase interfaces in solids, which consist of the elasticity
equations and the Allen-Cahn equation or the alternative equation. We find that
numerical simulations of phase interfaces with small interface energy based on
the alternative model are more effective then simulations based on the
Allen-Cahn model.Comment: arXiv admin note: text overlap with arXiv:1505.0544
Sharp-Interface Limit of a Fluctuating Phase-Field Model
We present a derivation of the sharp-interface limit of a generic fluctuating
phase-field model for solidification. As a main result, we obtain a
sharp-interface projection which presents noise terms in both the diffusion
equation and in the moving boundary conditions. The presented procedure does
not rely on the fluctuation-dissipation theorem, and can therefore be applied
to account for both internal and external fluctuations in either variational or
non-variational phase-field formulations. In particular, it can be used to
introduce thermodynamical fluctuations in non-variational formulations of the
phase-field model, which permit to reach better computational efficiency and
provide more flexibility for describing some features of specific physical
situations. This opens the possibility of performing quantitative phase-field
simulations in crystal growth while accounting for the proper fluctuations of
the system.Comment: 21 pages, 1 figure, submitted to Phys. Rev.
Sharp Interface Limits of the Cahn-Hilliard Equation with Degenerate Mobility
In this work, the sharp interface limit of the degenerate Cahn-Hilliard
equation (in two space dimensions) with a polynomial double well free energy
and a quadratic mobility is derived via a matched asymptotic analysis involving
exponentially large and small terms and multiple inner layers. In contrast to
some results found in the literature, our analysis reveals that the interface
motion is driven by a combination of surface diffusion flux proportional to the
surface Laplacian of the interface curvature and an additional contribution
from nonlinear, porous-medium type bulk diffusion, For higher degenerate
mobilities, bulk diffusion is subdominant. The sharp interface models are
corroborated by comparing relaxation rates of perturbations to a radially
symmetric stationary state with those obtained by the phase field model.Comment: 27 pages, 2 figure
Phase-Field Model of Cell Motility: Traveling Waves and Sharp Interface Limit
This letter is concerned with asymptotic analysis of a PDE model for motility
of a eukaryotic cell on a substrate. This model was introduced in [1], where it
was shown numerically that it successfully reproduces experimentally observed
phenomena of cell-motility such as a discontinuous onset of motion and shape
oscillations. The model consists of a parabolic PDE for a scalar phase-field
function coupled with a vectorial parabolic PDE for the actin filament network
(cytoskeleton). We formally derive the sharp interface limit (SIL), which
describes the motion of the cell membrane and show that it is a volume
preserving curvature driven motion with an additional nonlinear term due to
adhesion to the substrate and protrusion by the cytoskeleton. In a 1D model
problem we rigorously justify the SIL, and, using numerical simulations,
observe some surprising features such as discontinuity of interface velocities
and hysteresis. We show that nontrivial traveling wave solutions appear when
the key physical parameter exceeds a certain critical value and the potential
in the equation for phase field function possesses certain asymmetry.Comment: 7 pages, 3 figure
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
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