6,895 research outputs found
Morphological instability, evolution, and scaling in strained epitaxial films: An amplitude equation analysis of the phase field crystal model
Morphological properties of strained epitaxial films are examined through a
mesoscopic approach developed to incorporate both the film crystalline
structure and standard continuum theory. Film surface profiles and properties,
such as surface energy, liquid-solid miscibility gap and interface thickness,
are determined as a function of misfit strains and film elastic modulus. We
analyze the stress-driven instability of film surface morphology that leads to
the formation of strained islands. We find a universal scaling relationship
between the island size and misfit strain which shows a crossover from the
well-known continuum elasticity result at the weak strain to a behavior
governed by a "perfect" lattice relaxation condition. The strain at which the
crossover occurs is shown to be a function of liquid-solid interfacial
thickness, and an asymmetry between tensile and compressive strains is
observed. The film instability is found to be accompanied by mode coupling of
the complex amplitudes of the surface morphological profile, a factor
associated with the crystalline nature of the strained film but absent in
conventional continuum theory.Comment: 16 pages, 10 figures; to be published in Phys. Rev.
On trivial words in finitely presented groups
We propose a numerical method for studying the cogrowth of finitely presented
groups. To validate our numerical results we compare them against the
corresponding data from groups whose cogrowth series are known exactly.
Further, we add to the set of such groups by finding the cogrowth series for
Baumslag-Solitar groups and prove
that their cogrowth rates are algebraic numbers.Comment: This article has been rewritten as two separate papers, with improved
exposition. The new papers are arXiv:1309.4184 and arXiv:1312.572
Permutations generated by a depth 2 stack and an infinite stack in series are algebraic
© 2015, Australian National University. All rights reserved. We prove that the class of permutations generated by passing an ordered sequence 12... n through a stack of depth 2 and an in nite stack in series is in bi-jection with an unambiguous context-free language, where a permutation of length n is encoded by a string of length 3n. It follows that the sequence counting the number of permutations of each length has an algebraic generating function. We use the explicit context-free grammar to compute the generating function:(formula presented) where cn is the number of permutations of length n that can be generated, and (formula presented) is a simple variant of the Catalan generating function. This in turn implies that (formula presented
Ordering kinetics of stripe patterns
We study domain coarsening of two dimensional stripe patterns by numerically
solving the Swift-Hohenberg model of Rayleigh-Benard convection. Near the
bifurcation threshold, the evolution of disordered configurations is dominated
by grain boundary motion through a background of largely immobile curved
stripes. A numerical study of the distribution of local stripe curvatures, of
the structure factor of the order parameter, and a finite size scaling analysis
of the grain boundary perimeter, suggest that the linear scale of the structure
grows as a power law of time with a craracteristic exponent z=3. We interpret
theoretically the exponent z=3 from the law of grain boundary motion.Comment: 4 pages, 4 figure
Amplitude expansion of the binary phase field crystal model
Amplitude representations of a binary phase field crystal model are developed
for a two dimensional triangular lattice and three dimensional BCC and FCC
crystal structures. The relationship between these amplitude equations and the
standard phase field models for binary alloy solidification with elasticity are
derived, providing an explicit connection between phase field crystal and phase
field models. Sample simulations of solute migration at grain boundaries,
eutectic solidification and quantum dot formation on nano-membranes are also
presented.Comment: 11 pages, 8 figure
Noise and dynamical pattern selection
In pattern forming systems such as Rayleigh-Benard convection or directional
solidification, a large number of linearly stable, patterned steady states
exist when the basic, simple steady state is unstable. Which of these steady
states will be realized in a given experiment appears to depend on unobservable
details of the system's initial conditions. We show, however, that weak,
Gaussian white noise drives such a system toward a preferred wave number which
depends only on the system parameters and is independent of initial conditions.
We give a prescription for calculating this wave number, analytically near the
onset of instability and numerically otherwise.Comment: 12 pages, REVTEX, no figures. Submitted to Phys. Rev. Let
Phase Diagram and Commensurate-Incommensurate Transitions in the Phase Field Crystal Model with an External Pinning Potential
We study the phase diagram and the commensurate-incommensurate transitions in
a phase field model of a two-dimensional crystal lattice in the presence of an
external pinning potential. The model allows for both elastic and plastic
deformations and provides a continuum description of lattice systems, such as
for adsorbed atomic layers or two-dimensional vortex lattices. Analytically, a
mode expansion analysis is used to determine the ground states and the
commensurate-incommensurate transitions in the model as a function of the
strength of the pinning potential and the lattice mismatch parameter. Numerical
minimization of the corresponding free energy shows good agreement with the
analytical predictions and provides details on the topological defects in the
transition region. We find that for small mismatch the transition is of
first-order, and it remains so for the largest values of mismatch studied here.
Our results are consistent with results of simulations for atomistic models of
adsorbed overlayers
Grain boundary motion in layered phases
We study the motion of a grain boundary that separates two sets of mutually
perpendicular rolls in Rayleigh-B\'enard convection above onset. The problem is
treated either analytically from the corresponding amplitude equations, or
numerically by solving the Swift-Hohenberg equation. We find that if the rolls
are curved by a slow transversal modulation, a net translation of the boundary
follows. We show analytically that although this motion is a nonlinear effect,
it occurs in a time scale much shorter than that of the linear relaxation of
the curved rolls. The total distance traveled by the boundary scales as
, where is the reduced Rayleigh number. We obtain
analytical expressions for the relaxation rate of the modulation and for the
time dependent traveling velocity of the boundary, and especially their
dependence on wavenumber. The results agree well with direct numerical
solutions of the Swift-Hohenberg equation. We finally discuss the implications
of our results on the coarsening rate of an ensemble of differently oriented
domains in which grain boundary motion through curved rolls is the dominant
coarsening mechanism.Comment: 16 pages, 5 figure
Phase-field approach to heterogeneous nucleation
We consider the problem of heterogeneous nucleation and growth. The system is
described by a phase field model in which the temperature is included through
thermal noise. We show that this phase field approach is suitable to describe
homogeneous as well as heterogeneous nucleation starting from several general
hypotheses. Thus we can investigate the influence of grain boundaries,
localized impurities, or any general kind of imperfections in a systematic way.
We also put forward the applicability of our model to study other physical
situations such as island formation, amorphous crystallization, or
recrystallization.Comment: 8 pages including 7 figures. Accepted for publication in Physical
Review
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