536 research outputs found
Stability of Soft Quasicrystals in a Coupled-Mode Swift-Hohenberg Model for Three-Component Systems
In this article, we discuss the stability of soft quasicrystalline phases in
a coupled-mode Swift-Hohenberg model for three-component systems, where the
characteristic length scales are governed by the positive-definite gradient
terms. Classic two-mode approximation method and direct numerical minimization
are applied to the model. In the latter approach, we apply the projection
method to deal with the potentially quasiperiodic ground states. A variable
cell method of optimizing the shape and size of higher-dimensional periodic
cell is developed to minimize the free energy with respect to the order
parameters. Based on the developed numerical methods, we rediscover decagonal
and dodecagonal quasicrystalline phases, and find diverse periodic phases and
complex modulated phases. Furthermore, phase diagrams are obtained in various
phase spaces by comparing the free energies of different candidate structures.
It does show not only the important roles of system parameters, but also the
effect of optimizing computational domain. In particular, the optimization of
computational cell allows us to capture the ground states and phase behavior
with higher fidelity. We also make some discussions on our results and show the
potential of applying our numerical methods to a larger class of mean-field
free energy functionals.Comment: 26 pages, 13 figures; accepted by Communications in Computational
Physic
Stability of Two-Dimensional Soft Quasicrystals
The relative stability of two-dimensional soft quasicrystals is examined
using a recently developed projection method which provides a unified numerical
framework to compute the free energy of periodic crystal and quasicrystals.
Accurate free energies of numerous ordered phases, including dodecagonal,
decagonal and octagonal quasicrystals, are obtained for a simple model, i.e.
the Lifshitz-Petrich free energy functional, of soft quasicrystals with two
length-scales. The availability of the free energy allows us to construct phase
diagrams of the system, demonstrating that, for the Lifshitz-Petrich model, the
dodecagonal and decagonal quasicrystals can become stable phases, whereas the
octagonal quasicrystal stays as a metastable phase.Comment: 11 pages, 7 figure
Multiple-scale structures: from Faraday waves to soft-matter quasicrystals
For many years, quasicrystals were observed only as solid-state metallic
alloys, yet current research is now actively exploring their formation in a
variety of soft materials, including systems of macromolecules, nanoparticles
and colloids. Much effort is being invested in understanding the thermodynamic
properties of these soft-matter quasicrystals in order to predict and possibly
control the structures that form, and hopefully to shed light on the broader
yet unresolved general questions of quasicrystal formation and stability.
Moreover, the ability to control the self-assembly of soft quasicrystals may
contribute to the development of novel photonic or other applications based on
self-assembled metamaterials. Here a path is followed, leading to quantitative
stability predictions, that starts with a model developed two decades ago to
treat the formation of multiple-scale quasiperiodic Faraday waves (standing
wave patterns in vibrated fluid surfaces) and which was later mapped onto
systems of soft particles, interacting via multiple-scale pair potentials. The
article reviews, and substantially expands, the quantitative predictions of
these models, while correcting a few discrepancies in earlier calculations, and
presents new analytical methods for treating the models. In so doing, a number
of new stable quasicrystalline structures with octagonal, octadecagonal and
higher-order symmetries, some of which may, it is hoped, be observed in future
experiments.Comment: 22 pages, 22 figures, 1 table. Comments welcom
Stability of Quasicrystals Composed of Soft Isotropic Particles
Quasicrystals whose building blocks are of mesoscopic rather than atomic
scale have recently been discovered in several soft-matter systems. Contrary to
metallurgic quasicrystals whose source of stability remains a question of great
debate to this day, we argue that the stability of certain soft-matter
quasicrystals can be directly explained by examining a coarse-grained free
energy for a system of soft isotropic particles. We show, both theoretically
and numerically, that the stability can be attributed to the existence of two
natural length scales in the pair potential, combined with effective three-body
interactions arising from entropy. Our newly gained understanding of the
stability of soft quasicrystals allows us to point at their region of stability
in the phase diagram, and thereby may help control the self-assembly of
quasicrystals and a variety of other desired structures in future experimental
realizations.Comment: Revised abstract, more detailed explanations, and better images of
the numerical minimization of the free energ
Self-assembly of two-dimensional binary quasicrystals: A possible route to a DNA quasicrystal
We use Monte Carlo simulations and free-energy techniques to show that binary
solutions of penta- and hexavalent two-dimensional patchy particles can form
thermodynamically stable quasicrystals even at very narrow patch widths,
provided their patch interactions are chosen in an appropriate way. Such patchy
particles can be thought of as a coarse-grained representation of DNA multi-arm
`star' motifs, which can be chosen to bond with one another very specifically
by tuning the DNA sequences of the protruding arms. We explore several possible
design strategies and conclude that DNA star tiles that are designed to
interact with one another in a specific but not overly constrained way could
potentially be used to construct soft quasicrystals in experiment. We verify
that such star tiles can form stable dodecagonal motifs using oxDNA, a
realistic coarse-grained model of DNA
Numerical Methods for Quasicrystals
Quasicrystals are one kind of space-filling structures. The traditional
crystalline approximant method utilizes periodic structures to approximate
quasicrystals. The errors of this approach come from two parts: the numerical
discretization, and the approximate error of Simultaneous Diophantine
Approximation which also determines the size of the domain necessary for
accurate solution. As the approximate error decreases, the computational
complexity grows rapidly, and moreover, the approximate error always exits
unless the computational region is the full space. In this work we focus on the
development of numerical method to compute quasicrystals with high accuracy.
With the help of higher-dimensional reciprocal space, a new projection method
is developed to compute quasicrystals. The approach enables us to calculate
quasicrystals rather than crystalline approximants. Compared with the
crystalline approximant method, the projection method overcomes the
restrictions of the Simultaneous Diophantine Approximation, and can also use
periodic boundary conditions conveniently. Meanwhile, the proposed method
efficiently reduces the computational complexity through implementing in a unit
cell and using pseudospectral method. For illustrative purpose we work with the
Lifshitz-Petrich model, though our present algorithm will apply to more general
systems including quasicrystals. We find that the projection method can
maintain the rotational symmetry accurately. More significantly, the algorithm
can calculate the free energy density to high precision.Comment: 27 pages, 8 figures, 6 table
Universal self-assembly of one-component three-dimensional dodecagonal quasicrystals
Using molecular dynamics simulations, we study computational self-assembly of
one-component three-dimensional dodecagonal (12-fold) quasicrystals in systems
with two-length-scale potentials. Existing criteria for three-dimensional
quasicrystal formation are quite complicated and rather inconvenient for
particle simulations. So to localize numerically the quasicrystal phase, one
should usually simulate over a wide range of system parameters. We show how to
universally localize the parameters values at which dodecagonal quasicrystal
order may appear for a given particle system. For that purpose, we use a
criterion recently proposed for predicting decagonal quasicrystal formation in
one-component two-length-scale systems. The criterion is based on two
dimensionless effective parameters describing the fluid structure which are
extracted from radial distribution function. The proposed method allows
reducing the time spent for searching the parameters favoring certain solid
structure for a given system. We show that the method works well for
dodecagonal quasicrystals; this results is verified on four systems with
different potentials: Dzugutov potential, oscillating potential which mimics
metal interactions, repulsive shoulder potential describing effective
interaction for core/shell model of colloids and embedded-atom model potential
for aluminum. Our results suggest that mechanism of dodecagonal quasicrystal
formation is universal for both metallic and soft-matter systems and it is
based on competition between interparticle scales.Comment: 8 pages, 6 figure
Controlled self-assembly of periodic and aperiodic cluster crystals
Soft particles are known to overlap and form stable clusters that
self-assemble into periodic crystalline phases with density-independent lattice
constants. We use molecular dynamics simulations in two dimensions to
demonstrate that, through a judicious design of an isotropic pair potential,
one can control the ordering of the clusters and generate a variety of phases,
including decagonal and dodecagonal quasicrystals. Our results confirm
analytical predictions based on a mean-field approximation, providing insight
into the stabilization of quasicrystals in soft macromolecular systems, and
suggesting a practical approach for their controlled self-assembly in
laboratory realizations using synthesized soft-matter particles.Comment: Supplemental Material can be obtained through the author's website
at: http://www.tau.ac.il/~ronlif/pubs/ClusterCrystals-Supp.pd
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