We show that chiral symmetry can be broken spontaneously in one-component
systems with isotropic interactions, i.e. many-particle systems having maximal
a priori symmetry. This is achieved by designing isotropic potentials that lead
to self-assembly of chiral surfaces. We demonstrate the principle on a simple
chiral lattice and on a more complex lattice with chiral super-cells. In
addition we show that the complex lattice has interesting melting behavior with
multiple morphologically distinct phases that we argue can be qualitatively
predicted from the design of the interaction.Comment: 4 pages, 4 figure

E. Edlund

E. Eliel

E. N. Jacobsen

G. Blaschke

G. B. Folland

M. Nilsson Jacobi

O. Lindgren

English

arXiv

We show that chiral symmetry can be broken spontaneously in one-component systems with isotropic interactions, i.e., many-particle systems having maximal a priori symmetry. This is achieved by designing isotropic potentials that lead to self-assembly of chiral surfaces. We demonstrate the principle on a simple chiral lattice and on a more complex lattice with chiral supercells. In addition, we show that the complex lattice has interesting melting behavior with multiple morphologically distinct phases that we argue can be qualitatively predicted from the design of the interaction

Edlund, Erik

Lindgren, Oskar

Nilsson Jacobi, Martin

Chalmers Research

Chiral Surfaces Self-Assembling in One-Component Systems with Isotropic Interactions

Chalmers Publication Library

Chiral Surfaces Self-Assembling in One-Component Systems with Isotropic Interactions
E. Edlund, O. Lindgren, and M. Nilsson Jacobi*
Complex Systems Group, Department of Energy and Environment, Chalmers University of Technology,
SE-41296 Go¨teborg, Sweden
(Received 12 January 2012; published 20 April 2012)
We show that chiral symmetry can be broken spontaneously in one-component systems with isotropic
interactions, i.e., many-particle systems having maximal a priori symmetry. This is achieved by designing
isotropic potentials that lead to self-assembly of chiral surfaces. We demonstrate the principle on a simple
chiral lattice and on a more complex lattice with chiral supercells. In addition, we show that the complex
lattice has interesting melting behavior with multiple morphologically distinct phases that we argue can be
qualitatively predicted from the design of the interaction.
DOI: 10.1103/PhysRevLett.108.165502 PACS numbers: 81.16.Dn, 61.50.Ah, 81.10.h
Breaking of chiral symmetry plays a central role both in
fundamental physics, e.g., parity violation in the weak
interaction, and in biology where many types of biomole-
cules exist only as enantiomers. The commonly accepted
explanation for homochirality in chemical and biological
systems is spontaneous symmetry breaking. This phe-
nomenon involves two steps: first the formation of chiral
molecules and then a chiral specific catalysis that amplifies
a stochastic imbalance to a macroscopic scale. In hetero-
geneous systems, it is not hard to imagine that both steps
can be achieved, and there are indeed many chemical
systems that spontaneously deviate from racemic mixture
[1,2]. There are even several known molecules that show
chiral autocatalytic activity [3,4]. It is an interesting ques-
tion whether spontaneous breaking of chiral symmetry can
happen also in simpler systems, by which we here mean
systems with a higher degree of symmetry. The perhaps
simplest such class of systems would consist of a single
particle type with an isotropic pairwise interaction. For
several reasons, it appears to be much harder to form
homochiral states in such systems, with the exception of
chirality induced by finite size effects [5]. First, cluster
formation, which mimics the creation of molecules, typi-
cally results in achiral structures with dihedral group sym-
metries [6]. Second, (auto)catalysis is not easily achieved
in homogeneous isotropic systems. However, chiral sym-
metry can also be broken in another context, namely,
during crystallization. Again there are many examples of
heterogeneous systems where this is known to happen
[7,8], but no chiral crystals arising from isotropic interac-
tions have been reported, neither in explorative [9] nor
design [10] studies. In this Letter, we show that one-
component systems with carefully designed isotropic in-
teractions can self-assemble into chiral lattices. Our results
demonstrate that spontaneous breaking of chiral symmetry
can occur in many-particle systems with a maximal degree
of symmetry.
Chiral surfaces are of practical importance because of
their potential for use in chiral catalysis. The most impor-
tant applications are in the pharmaceutical industry,
where we learned the hard way that a therapeutically
well-behaved enantiomer may be toxic in the other enan-
tiomeric form. The classic example is the tragedy with
thalidomide, whose racemic form turned out to cause birth
defects after being extensively used as a sedative for preg-
nant women [11]. Today, the dominating method for syn-
thesizing large quantities of chiral products involves
various types of chiral catalysts in solution (homogeneous
catalysis) [12,13]. However, the development of chiral
surface catalysts (heterogeneous catalysis) receives
much interest due to their separability and reusability.
Several ways of producing chiral surfaces exist, for ex-
ample, through surface reconstruction induced by adsorbed
chiral molecules [14], cleaving achiral crystals along
planes of low symmetry such as the (643) surface of an
fcc structure [15], or self-assembly of molecules into chiral
structures on achiral surfaces [16,17]. One of the outstand-
ing challenges for all heterogeneous chiral catalysis is how
to make chiral surfaces with sufficiently large (macro-
scopic) active areas [18]. Simple models that exhibit
formation of chiral domains, such as the models presented
here, could prove useful in supporting progress in this area.
In this work, we focus on self-assembly at low tempera-
ture where structure formation is driven by minimization of
the potential energy. For a given configuration in a system
with isotropic interactions, the energy is defined by the
density distribution ðrÞ and is naturally described as a
quadratic form that sums all the contributions from the
pairwise potential. For our purposes, it is suitable to ex-
press the energy in reciprocal space, where the quadratic
form is diagonalized due to the translational invariance of
the isotropic interactions [19]:
Published by the American Physical Society under the terms of
the Creative Commons Attribution 3.0 License. Further distri-
bution of this work must maintain attribution to the author(s) and
the published article’s title, journal citation, and DOI.
PRL 108, 165502 (2012) P HY S I CA L R EV I EW LE T T E R S
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0031-9007=12=108(16)=165502(4) 165502-1 Published by the American Physical Society
E ¼
Z
dr1dr2ðr1ÞVðjr1  r2jÞðr2Þ
¼ 2
Z
dkj^ðkÞj2V^ðkÞ; (1)
where j^ðkÞj2dk ¼ Rjskj¼k dkj^ðkÞj2, ^ðkÞ is the standard
Fourier transform of ðrÞ, and V^ðkÞ is the radial Fourier
(Hankel) transform of VðrÞ [20],
V^ðkÞ ¼ 2
Z
rdrVðrÞJ0ðrkÞ:
The small jkj region describes the defining features of the
crystal structure, and, by designing the energy spectrum of
this region, particles can be made to self-assemble into
crystalline structures at low temperatures. We used this
observation in a recent study to show how to design the
interactions of a system so that it self-assembles into target
lattices [21]. Briefly, the method works as follows. The
symmetry of the lattice manifests in reciprocal space as a
restriction of the support of ^ðkÞ to a finite set of points
fGig. By choosing V^ðkÞ smooth and positive with zeros
coinciding with the reciprocal lattice Gi of the target
structure V^ðjGijÞ ¼ 0 [see Figs. 1(c) and 2(c)], we can
guarantee the target configuration to be a ground state.
This construction utilizes the fact that there are a finite
number of Bravais lattices, all with different structure
factors. The structure factors ^ðGiÞ will not affect the
energy, and hence all crystals with the same periodicity
will be ground states.
Chirality on a crystal surface is by definition equivalent
to the absence of axes of symmetry, and it can manifest
itself in fundamentally different ways. Every ideal crystal
structure consists of a mathematical lattice and a basis [22].
If either the basis or the lattice is chiral, the crystal will be
as well. But even if both lattice and basis are achiral, the
crystal can be chiral if the axes of symmetry of the lattice
do not coincide with those of the basis. In this case, the
reciprocal lattice representation is also achiral, and instead
the chirality is determined by the distribution of the struc-
ture factors (weights) over the reciprocal lattice points. We
exemplify these fundamentally different chiralities with
two lattices (crystal structures): a lattice of scalene tri-
angles and the snub hexagonal tiling. For each case, we
discuss how the chirality relates to the reciprocal dual of
the crystal and how this reflects on the energy spectrum of
the potential designed to self-assemble into the target
structure. We also demonstrate the crystallization with
snapshots of low energy states self-assembled when simu-
lated in the canonical ensemble.
The simplest chiral geometric form in two dimensions is
the scalene triangle, where the chirality depends on the
clockwise order of the side lengths. Hence the simplest
chiral crystal is one forming scalene (acute) triangles, an
oblique Bravais lattice. The oblique lattice is unique in the
sense that it is the only chiral Bravais lattice. In this case,
the chirality of the lattice is directly manifest in the recip-
rocal lattice and will form scalene triangles with the same
chirality in reciprocal space. Since the basis is trivial, there
is no freedom in the arrangement of its constituent parti-
cles, and the only requirement for self-assembly of such a
chiral structure is that the positive spectrum V^ðkÞ selects
for the reciprocal lattice of the target and that the system
has approximately the correct particle density. In Fig. 1(c),
a spectrum fulfilling these criteria is shown, together with
the corresponding potential and the result of a Monte Carlo
simulation with the potential. We see that the mirror sym-
metry of the system is indeed broken with a homochiral
lattice as the result.
As an example of a crystal where the chirality instead
emerges from the interplay between the basis and the
lattice, we use the snub hexagonal tiling and the crystal
formed by its vertices, shown in Fig. 2. The tiling and its
dual are the only uniform chiral tilings of the Euclidean
plane. This pattern closely resembles geometries observed
in experimental systems involving (anisotropic) molecules
with a triangular geometry [23]. An achiral superstructure
of vacancies determines the lattice periodicity, while the
orientation of the basis of six particles in relation to ad-
jacent vacancies determines the chirality. Unlike the pre-
vious example, here perturbations of the energy spectrum
at jGij are necessary to distinguish between crystals with
different bases, since they consists of more than one
FIG. 1 (color online). The simplest form of chiral lattice,
composed of scalene (acute) triangles with well separated side
lengths li. (a) A low energy state obtained from a MC simulation.
(b) The emerging chirality can be either left or right oriented. (c)
By selecting for the reciprocal lattice (the red peaks) of the target
structure, a potential (d) which causes self-assembly into one of
the two possible chiralities is obtained.
PRL 108, 165502 (2012) P HY S I CA L R EV I EW LE T T E R S
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165502-2
particle; see [21] formore details on the use of perturbations
to break degeneracies of the ground state. There are six pairs
of reciprocal lattice points representing wave vectors of
length jGij ¼
ﬃﬃﬃ
7
p jk0j, k0 being a primitive reciprocal
vector. The Fourier transform of the crystal ^ðkÞ at these
sites is equal to 1 or ^max ¼ 6. The chirality is deter-
mined by the order of the structure factors in the pairwise
sites. A negative perturbation of the energy spectrum at
this k will ensure that the ground state will have the correct
basis, as shown in Fig. 2(c). In the zero temperature limit, a
small deformation in the form of a small rotation of the
basis by 0.2 rad towards its chiral counterpart will be
visible: The spectrum will not be affected at its maxima
by small perturbations of the basis, while a rotation in-
creases the negative energy contribution from the other
peaks at that wavelength. The maxima correspond to the
hexagonal close-packed lattice which, together with the
vacancies introduced by the limited density, forms the snub
hexagonal tiling.
To test whether the target lattice can be assembled from
random initial configurations, we perform Monte Carlo
simulations of particles interacting with the obtained po-
tentials, allowing only local moves and with simulated
annealing at fixed density. During the temperature anneal-
ing, the particles organize into homochiral regions. The
boundaries where the chirality changes are energetically
disfavored, and the homochiral grains grow as the system
is annealed towards one of its ground states. The annealing
process is, however, very slow, as illustrated in Fig. 3,
where the time evolution is shown on an (approximately)
logarithmic time scale. Slow domain growth is typically
observed also in experimental systems, which is causing
practical difficulties in the synthesis of chiral surfaces [18].
Further simulations reveal that the snub hexagonal
model has a rich phase diagram, as demonstrated in
Fig. 4. It shows a large region of stability for the target
chiral configuration (e) as the temperature and magnitude
of the spectrum perturbation  (see Fig. 2) are varied. We
can understand the various neighboring phases in terms of
the target structure losing some of its properties while
retaining others. If the negative perturbation stabilizing
the basic hexagonal lattice is large, the preference for
particles to occupy sites on this lattice will remain even
at temperatures where the periodicity of the snub hexago-
nal lattice, induced by the locations of zeros in the
energy spectrum, disappears. The crystal melts through a
vacancy unbinding resulting in a hexagonal lattice with
randomly placed vacancies [Fig. 4(b)]. For even larger ,
FIG. 2 (color online). Snub hexagonal tiling, a chiral
Archimedean tiling. Note that the chirality does not originate
from the positions of the hexagons (they are organized in rhombs
and not parallelograms as they must when the tiles are regular
polygons) but from the orientations of the triangles in between.
(a)–(d) As in Fig. 1. In (a) two regions of opposite chirality are
highlighted. (c) illustrates the perturbation of magnitude  at the
maximum of ^.
FIG. 3. Time evolution of a self-assembling snub hexagonal
lattice, after (from the left) 250, 2500, 1:2 104, and 105 sweeps
of the MC algorithm (each sweep attempting to move each
particle on average once).
FIG. 4 (color online). A phase diagram for the snub hexagonal
model shows the equilibrium configuration as a function of
inverse temperature  and the magnitude  of the perturbation.
Without the perturbation, striped and stripelike morphologies are
observed (a), and a strong perturbation leads to vacancy unbind-
ing (b) before complete disorder (c) as the temperature increases.
A too strong perturbation results in aggregation of the vacancies
(d). In the central region (e), the chiral snub hexagonal lattice is
formed.
PRL 108, 165502 (2012) P HY S I CA L R EV I EW LE T T E R S
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the vacancies aggregate into clusters [Fig. 4(d)], not form-
ing the snub hexagonal lattice for any . If the negative
perturbation is small, the basis will dissolve at tempera-
tures below those where the periodicity of the lattice is
broken. A large variety of basis configurations are ob-
served in this region, although the dominant and most
stable morphology is striped [Fig. 4(a)]. We suspect that
the rich phase behavior observed here is typical for models
constructed through our design method, as the spectrum
fixes a set of features of the ground state with varying
strengths, each breaking down under different conditions.
This demonstrates that the energy spectrum approach to
designing potentials for targeted self-assembly can also be
useful when trying to understand and control complex
finite temperature phase behavior.
In summary, we demonstrate that isotropic pairwise
potentials can cause particle systems to self-assemble
into chiral surfaces with a varying degree of complexity.
In general, this work hints at some of the future possibil-
ities of nanoscale self-assembly, expanding on what we
know to be possible, and could be an inspiration for the
material science of tomorrow. Lately, there has been re-
markable progress on designing nano- and colloidal parti-
cles with exotic interaction potentials [24,25]. Still,
experimental realizations of systems with as complicated
interactions as we use here are not likely to appear in the
near future. For theoretical methods and experimental
techniques to converge and the field of self-assembly to
move from science to industry on a larger scale, further
progress from both directions is necessary.
O. L. and M.N. J. acknowledge support from the SuMo
Biomaterials center of excellence.
*mjacobi@chalmers.se
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Physical Review Letters

Chiral surfaces self-assembling in one-component systems with isotropic
  interactions

Chiral surfaces self-assembling in one-component systems with isotropic interactions

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