We present results from particle simulations with isotropic medium range
interactions in two dimensions. At low temperature novel types of aggregated
structures appear. We show that these structures can be explained by
spontaneous symmetry breaking in analytic solutions to an adaptation of the
spherical spin model. We predict the critical particle number where the
symmetry breaking occurs and show that the resulting phase diagram agrees well
with results from particle simulations.Comment: 4 pages, 4 figure

A. Belavin

E. Edlund

K. H. Fischer

M. Nilsson Jacobi

O. Lindgren

R. Hinton

R. Hoyle

R. J. Baxter

English

arXiv

Crossref

Physical Review Letters

Novel Self-Assembled Morphologies from Isotropic Interactions

We present results from particle simulations with isotropic medium range interactions in two dimensions. At low temperature novel types of aggregated structures appear. We show that these structures can be explained by spontaneous symmetry breaking in analytic solutions to an adaptation of the spherical spin model. We predict the critical particle number where the symmetry breaking occurs and show that the resulting phase diagram agrees well with results from particle simulations

Edlund, Erik

Lindgren, Oskar

Nilsson Jacobi, Martin

Chalmers Research

Chalmers Publication Library

Novel Self-Assembled Morphologies from 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 20 April 2011; revised manuscript received 22 June 2011; published 15 August 2011)
We present results from particle simulations with isotropic medium range interactions in two
dimensions. At low temperature novel types of aggregated structures appear. We show that these structures
can be explained by spontaneous symmetry breaking in analytic solutions to an adaptation of the spherical
spin model. We predict the critical particle number where the symmetry breaking occurs and show that the
resulting phase diagram agrees well with results from particle simulations.
DOI: 10.1103/PhysRevLett.107.085501 PACS numbers: 81.16.Dn, 05.65.+b, 75.10.Hk, 89.75.Kd
Understanding the principles behind the spontaneous
formation of structured morphologies is of interest both
as a fundamental scientific question and in engineering
applications where the possibility of using self-assembly
to produce novel materials provides a compelling co-
mplement to traditional blueprinted fabrication [1,2].
Consequently there is growing interest in exploring inter-
actions that can facilitate self-fabrication of materials with
novel properties [3–5]. The more general question of what
structures are possible to self-assemble from a given class
of interactions has however not been addressed, with some
notable exceptions, e.g., simulation-based analysis of
polyhedral packing [6].
To examine the possibilities of self-assembly from a
theoretical angle, in this Letter we consider systems with
pairwise isotropic interactions, frequently used as coarse-
grained models of more complex mesoscale systems such
as colloidal systems [7], particles in an ambient fluid [8],
and spin glasses [9]. We show that typical aggregates
appearing as low-temperature configurations in particle
simulations with randomly generated medium range iso-
tropic interactions can be predicted analytically using an
adaptation of the spherical spin model [10]. The morphol-
ogies, many of them novel and surprisingly complex, can
be systematically classified by their spontaneous breaking
of the rotational symmetry.
To formulate a solvable model of a particle system we
start by considering a lattice spin system with Hamiltonian
of the form
H ¼X
ij
Uijsisj; (1)
where si 2 f0; 1g, and si ¼ 1 represents a particle at lattice
site i and si ¼ 0 represents vacuum. The total number of
particles is set by the normalization
P
isi ¼ N. The inter-
action Uij is an effective isotropic potential with a hard
core repulsion corresponding to the lattice spacing. In a
spin glass metal the interaction is mediated by a polariza-
tion of the Fermi sea [11] while in a colloidal system it
could involve a surface polymer induced steric hindrance
competing with a depletion attraction [12].
The discrete model described by Eq. (1) can equiva-
lently be formulated as a continuous one (si 2 R) with
auxiliary constraints,
P
is
m
i ¼ N 8 m. To make the model
analytically tractable we relax the constraints to include
only the first two moments. The result is the spherical spin
model with an external field, with Hamiltonian Hs ¼P
ijUijsisj þ h
P
isi. In general there is no guarantee that
this approximation will produce relevant results, but as we
will see, typically there is a strong correspondence between
the discrete and continuous models. The main idea in this
study is that the ground states in the continuous model can
be derived analytically and that these can be used to predict
the low-temperature morphologies of the discrete particle
systems.
A necessary condition for an energy minimum in the
spherical model is
P
jðUij  ijÞsj ¼ h=2, where  is the
identity matrix, and  comes from the constraint on the
second moment. Translational invariance implies that
si ¼ c is a solution. Nonconstant solutions si ¼ vi þ c
exist if  is an eigenvalue of the matrix U corresponding
to the eigenvector v. From the constraints it follows that
c ¼ , where  is the density, and V1Piv2i ¼ ð1 Þ,
where V is the number of lattice sites or the volume. The
ground states are defined by the eigenvector(s) correspond-
ing to the lowest eigenvalue of U.
It can be shown, e.g., using translational invariance, that
all isotropic interaction matrices have the Fourier modes as
eigenvectors. This also follows from the observation that
any such matrix Uij can be expressed as a linear combina-
tion of matrices of the form lþm, l; m ¼ 0; 1; . . . , where
the two components are the discrete Laplace operator
defined as usual þ or along the diagonals  (note that
þ and  commute). The Fourier modes are eigenfunc-
tions of the Laplace operator so the two arguments lead to
the same conclusion. However, the latter argument points
to a subtlety: the eigenfunctions of the Laplacian depend
on the boundary conditions, where the Fourier harmonics
result from periodic boundaries. Requiring that the func-
tions converge to zero at infinity instead results in cylin-
drical harmonic eigenfunctions with angular modulations
localized around a nucleation point. These localized
PRL 107, 085501 (2011) P HY S I CA L R EV I EW LE T T E R S
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configurations bear similarity to topological defects, such
as the Belavin-Polyakov monopole, that are important for
the low-temperature behavior of, e.g., Heisenberg ferro-
magnets [13].
Our central result is that, when there are too few parti-
cles to occupy a translational invariant ground state, the
eigenfunctions of Bessel type J!ð2krÞ cosð!Þ deter-
mine the low-temperature behavior of self-assembling
particle systems in two dimensions. The eigenvalues, i.e.,
the energies, are given by the radial Fourier transform of
the potential as [14,15]
EðkÞ ¼ 2
Z 1
0
rdrUðrÞJ0ð2krÞ; (2)
so that the wave number  ¼ argminkEðkÞ defines an
eigenfunction with minimal energy. The energy spectrum
is degenerate since EðkÞ is independent of !, reflecting the
rotational invariance around the nucleation point.
However, this degeneracy is broken by the nonlinearity
in the mapping from the spherical model Hs to the discrete
lattice system (1), where a single base frequency ! to-
gether with its overtones (similar to the harmonic over-
tones of a square wave) are energetically favored. The
spontaneous symmetry breaking of the ground state, from
O(2) to one of its isotropy subgroups D!, is equivalent to
the behavior in a bifurcation problem [16,17]. The ground
states of the spherical model of relevance to the self-
assembly problem in (1) are therefore of the form
X1
n¼0
anJn!ð2rÞ cosðn!Þ þ cJ0ð2kmÞ (3)
in the limit km ! 0, which turns the last term (which we
refer to as the mass builder) into the translational invariant
constant c in the minimization ofHs. However, the km ! 0
limit is only relevant for infinite structures (with nonzero
global density). For an aggregated structure with finite
mass a small but nonzero km is needed to localize the
solution (resulting in a nonzero local density but a global
density of zero).
As we will see later, the sum in Eq. (3) is dominated by
the first few terms and in practice most structures are
adequately described by the zeroth and first term.
Notable exceptions are finite lattices, see, e.g., Fig. 1(e),
which grow by successively including higher terms.
To connect the above results to the particle model (1),
we map the continuous ground states (si 2 R) to particle
configurations (si 2 f0; 1g) by applying a step function
threshold. The energy of the resulting discrete configura-
tions depends on how the mapping distorts the power
spectrum relative to the energy spectrum (2). While the
main weight of the discrete configurations typically re-
mains localized to , contributions from overtones that
appear in the discrete system can have a large effect on
the energy. These effects are difficult to quantify analyti-
cally. Instead we generate possible candidates for ground
states using Eq. (3), i.e., linear combinations of Bessel
functions with angular frequencies 0, !, 2!, and 3!
with  ¼ 1, as well as a mass-building Bessel function
with ! ¼ 0 and km chosen such that the first zero of the
mass builder is slightly larger than the th zero of J! for
 ¼ 1; 2; 3. The configurations were mapped to binary
valued configurations on a lattice with a threshold that
defines different masses. The energy of the resulting con-
figurations was calculated for 1000 random first-order
spline potentials. We used aggregating potentials, i.e.,
potentials with a global minimum in the energy spectrum
at k ¼ 0. In addition, their arguments where scaled so that
the interior minimum in the energy spectrum resided at
 ¼ 1 as demonstrated in Fig. 2(a) and were attractive at
the lattice distance to ensure correct resolution of the
lattice. For each potential and mass we recorded the
configuration with the lowest energy, which resulted in a
limited number of favored morphologies, shown in Fig. 1.
FIG. 1 (color). Predicted morphological alphabet for aggregating potentials generated by Eq. (3) (with !  8). The morphologies
shown are energetically preferable for a large fraction of random potentials. Red background indicates a strong signal while blue
background signifies weaker signal.  denotes the number of active radial oscillations of the ground frequency after the threshold
mapping.
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The method predicts many novel structures [e.g.,
Figs. 1(a), 1(c), 1(d), 1(g), 1(h), and 1(m)], as well as
simpler ones like disks and localized lattices [e.g.,
Figs. 1(b), 1(e), 1(l), and 1(q)]. The latter are observed
on the atomic scale [18], while the former morphologies
are only expected to appear at higher particle numbers.
To summarize the predictions of the adapted spherical
spin model, we expect a particle system with an aggregat-
ing potential to show (I) morphologies from a limited
alphabet (Fig. 1) with (II) a single dominant (base) wave-
length  determined by the minimum of the spectrum and
(III) an ! degeneracy where a single potential can self-
assemble into many different morphologies, depending on
external parameters.
To test these predictions we performed off-lattice simu-
lations of particle systems. We constructed 1200 random
interaction potentials (piecewise constant and 3rd order
splines with the same restrictions on the spectrum as
above) and did Monte Carlo annealing at different particle
numbers. Examples of the resulting particle configurations
are shown in Figs. 2 and 3 and are in good agreement with
(I). Of the simulated particle systems, approximately 86%
[Fig. 3(a)] annealed to predicted morphologies shown in
Fig. 1 and 9% [Fig. 3(c)] to morphologies described by
Eq. (3) but absent in Fig. 1 due to the limited parameter
range considered (e.g., !  8). The latter were in general
similar to those in Fig. 1. The strengths of the signals of the
predicted morphologies in Fig. 1 were also consistent with
how frequent the corresponding configurations were ob-
served in simulations. The remaining 5% of the morphol-
ogies exhibited hierarchies of Bessel-like structures; for an
example, see Fig. 3(b).
By expressing the observed particle configurations in the
Bessel basis we confirm that (II) the dominant wave num-
ber is accurately predicted by the minimum in the energy
spectrum of the potential. Examples of this are shown in
Figs. 2(b)–2(g), where we note that while the point sym-
metry of the preferred configuration for a single potential
changes with varying particle number, the dominant (non-
mass-building) active wavelength is approximately con-
stant and equal to the predicted value . This switching
of symmetry group is observed frequently in the simula-
tions and shows that (III) the ! degeneracy is not just a
mathematical curiosity in our model. For a given potential
the particle number sets the morphology, an important
fact that could facilitate the use of standard techniques
such as density gradient centrifugation [19] for differential
selection of morphologies [20].
The spectral analysis also allows us to analytically pre-
dict a phase diagram for aggregating potentials. The spec-
trum shown in Fig. 2(a) is typical for the potentials we
consider in that it is oscillatory and has a global minimum
at k ¼ 0. In a spin system a global minimum at k ¼ 0
FIG. 2 (color online). Self-assembled morphologies for a single potential. (a) A typical energy spectrum of a random potential. The
wavelength of the morphologies is determined by the interior minimum  of the spectrum when the number of particles N is less than
Nc, the number required to build a disk with radius corresponding to k0; see Eq. (4). (b)–(g) Transitions between different symmetry
groups for the potential of (a) as the particle number is decreased. (Top) Configurations from annealed particle simulations,
(middle) corresponding states of the spherical model with contours added at a level giving correct number of particles, and
(bottom) Bessel spectra (arb. units) of the particle configurations expanded on the form of (3), summed over !. The first maximum
corresponds to the mass builder km, the second should be compared to  ¼ 1 predicted from the spectrum in (a), and the rest are
overtones.
a b c
FIG. 3. Examples of configurations from particle simulations
with randomly generated potentials. All configurations except
(b) are well represented by functions on the form (3) with the
first four terms included, most requiring only one or two.
(a) Morphologies predicted as common by our method; see
Fig. 1. (b) An example of a hierarchical structure, not directly
describable in our theory, where each part of the usual Bessel
structure serves as the nucleation point for a separate Bessel
structure. (c) Morphologies that, while being simply represent-
able by Eq. (3), are not predicted in Fig. 1.
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implies a ferromagnetic ground state. For a particle system
it implies that in the high particle limit we can use large
scale interface minimization to argue that the ground state
is dominated by the mass builder, typically a closely
packed disk with a surface that is smooth or has wavelike
indentations [similar to Fig. 2(c)]. For a given number of
particles there is a maximal size of such a disk and for low
particle numbers this effectively excludes the small k part
of the energy spectrum in the minimization that determines
. This causes a transition as the particle number is de-
creased, from a disk to the more complex Bessel-based
morphologies we observe. This transition happens when
the energy of the disk rises above the energy of the lowest
interior minimum EðÞ. We denote this point k0, as illus-
trated in Fig. 2(a). The critical particle number Nc can be
predicted through
Nc ¼ f
a2
b2
k20
; (4)
where f is the packing fraction, a is the radius of the
particles, and b  0:293 is a numerical constant corre-
sponding to the optimal k to describe a disk of radius 1,
found through argmaxk
R
1
0 J0ð2krÞrdr=kJ0ðkÞk. The
phase diagram predicted by Eq. (4) is shown in Fig. 4
together with simulation results.
Note that though the results presented in this Letter are
limited to two dimensions, the analytic solution of the
spherical model is the same in higher dimensions with
the Bessel harmonics being replaced with spherical
harmonics. Whether the connection to the particle systems
remains equally clear is a question for future work, though
our preliminary investigations suggest this being the case.
We conclude that isotropic pairwise additive potentials
can give rise to complex morphologies appearing between
the atomic and macroscopic scale and that the frequently
occurring structures form a limited morphological alpha-
bet. The patterns are generically constructed as discretized
linear combinations of a few Bessel functions, which can
be understood and analytically predicted from an adapta-
tion of the spherical spin model. Further, we analytically
calculate the phase diagram showing where these patterns
occur for different potentials. The accuracy of the predic-
tions is surprising considering the complexity involved in
determining the ground states in a many particle system.
The methodology we present applies to the entire class of
isotropic interaction potentials and provides new theoreti-
cal understanding of self-assembly processes.
O. L. and M.N. J. acknowledge support from the SuMo
Biomaterials center. We thank Kolbjørn Tunstrøm for valu-
able comments and discussions.
*mjacobi@chalmers.se
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FIG. 4 (color online). Decreasing the number of particles
causes a transition from disklike structures at the macroscale
to more complex morphologies at the mesoscale. The figure
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for packing fraction f versus k0 calculated [see Fig. 2(a)] from
respective spectra for 33 random potentials. Inset: Four examples
of potentials used.
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085501-4


Edlund, Erik,

Lindgren, Oskar,

Nilsson Jacobi, Martin,

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