Wigner crystals for a planar, equimolar binary mixture of classical, charged particles
We have investigated the ground state configurations of an equimolar, binary mixture of classical charged particles (with nominal charges Q₁ and Q₂) that condensate on a neutralizing plane. Using efficient Ewald summation techniques for the calculation of the ground state energies, we have identifie...
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| Cite this: | Wigner crystals for a planar, equimolar binary mixture of classical, charged particles / M. Antlanger, G. Kahl // Condensed Matter Physics. — 2013. — Т. 16, № 4. — С. 43501:1-9. — Бібліогр.: 21 назв. — англ. |
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irk-123456789-1208512017-06-14T03:04:17Z Wigner crystals for a planar, equimolar binary mixture of classical, charged particles Antlanger, M. Kahl, G. We have investigated the ground state configurations of an equimolar, binary mixture of classical charged particles (with nominal charges Q₁ and Q₂) that condensate on a neutralizing plane. Using efficient Ewald summation techniques for the calculation of the ground state energies, we have identified the energetically most favorable ordered particle arrangements with the help of a highly reliable optimization tool based on ideas of evolutionary algorithms. Over a large range of charge ratios, q = Q₂ / Q₁, we identify six non-trivial ground states, some of which show a remarkable and unexpected structural complexity. For 0.59 \lesssim q < 1 the system undergoes a phase separation where the two charge species populate in a hexagonal arrangement spatially separated areas. Ми дослiдили конфiгурацiї основного стану еквiмолярної бiнарної сумiшi класичних заряджених части-нок (з номiнальними зарядами Q₁ i Q₂), якi конденсуються на нейтралiзуючiй площинi. Використовуючи ефективний метод пiдсумовування Евальда для обчислення основних енергетичних станiв, ми iденти-фiкували найбiльш енергетично вигiднi впорядкованi розташування частинок за допомогою високо на-дiйного оптимiзацiйного методу, який ґрунтується на iдеях еволюцiйних алгоритмiв. Для великої областi змiни коефiцiєнта зарядiв, q Æ Q₂/Q₁, ми iдентифiкували шiсть нетривiальних основних станiв, деякi з яких демонструють значну i неочiкувану структурну складнiсть. Для 0.59 . q Ç 1, система зазнає фазо-вого розшарування, де зарядженi сорти заселяють просторово вiдокремленi областi з гексагональним розташуванням. 2013 Article Wigner crystals for a planar, equimolar binary mixture of classical, charged particles / M. Antlanger, G. Kahl // Condensed Matter Physics. — 2013. — Т. 16, № 4. — С. 43501:1-9. — Бібліогр.: 21 назв. — англ. 1607-324X PACS: 52.27.Lw, 64.70.K-, 64.75.St, 73.20.Qt DOI:10.5488/CMP.16.43501 arXiv:1308.5791 http://dspace.nbuv.gov.ua/handle/123456789/120851 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We have investigated the ground state configurations of an equimolar, binary mixture of classical charged particles (with nominal charges Q₁ and Q₂) that condensate on a neutralizing plane. Using efficient Ewald summation techniques for the calculation of the ground state energies, we have identified the energetically most favorable ordered particle arrangements with the help of a highly reliable optimization tool based on ideas of evolutionary algorithms. Over a large range of charge ratios, q = Q₂ / Q₁, we identify six non-trivial ground states, some of which show a remarkable and unexpected structural complexity. For 0.59 \lesssim q < 1 the system undergoes a phase separation where the two charge species populate in a hexagonal arrangement spatially separated areas. |
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Antlanger, M. Kahl, G. Wigner crystals for a planar, equimolar binary mixture of classical, charged particles Condensed Matter Physics |
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Antlanger, M. Kahl, G. |
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Antlanger, M. |
| title |
Wigner crystals for a planar, equimolar binary mixture of classical, charged particles |
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Wigner crystals for a planar, equimolar binary mixture of classical, charged particles |
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Wigner crystals for a planar, equimolar binary mixture of classical, charged particles |
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Wigner crystals for a planar, equimolar binary mixture of classical, charged particles |
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Wigner crystals for a planar, equimolar binary mixture of classical, charged particles |
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wigner crystals for a planar, equimolar binary mixture of classical, charged particles |
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Інститут фізики конденсованих систем НАН України |
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Wigner crystals for a planar, equimolar binary mixture of classical, charged particles / M. Antlanger, G. Kahl // Condensed Matter Physics. — 2013. — Т. 16, № 4. — С. 43501:1-9. — Бібліогр.: 21 назв. — англ. |
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Condensed Matter Physics |
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AT antlangerm wignercrystalsforaplanarequimolarbinarymixtureofclassicalchargedparticles AT kahlg wignercrystalsforaplanarequimolarbinarymixtureofclassicalchargedparticles |
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1837105402543603712 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 4, 43501: 1–9
DOI: 10.5488/CMP.16.43501
http://www.icmp.lviv.ua/journal
Wigner crystals for a planar, equimolar binary
mixture of classical, charged particles
M. Antlanger1,2, G. Kahl1
1 Institut für Theoretische Physik and Center for Computational Materials Science (CMS), Vienna University of
Technology, Wiedner Hauptstraße 8–10, A–1040 Wien, Austria
2 Laboratoire de Physique Théorique (UMR 8627), Université Paris-Sud and CNRS, Bât. 210, F–91405 Orsay
Cedex, France
Received August 14, 2013, in final form September 24, 2013
We have investigated the ground state configurations of an equimolar, binary mixture of classical charged parti-
cles (with nominal chargesQ
1
andQ
2
) that condensate on a neutralizing plane. Using efficient Ewald summation
techniques for the calculation of the ground state energies, we have identified the energetically most favorable
ordered particle arrangements with the help of a highly reliable optimization tool based on ideas of evolution-
ary algorithms. Over a large range of charge ratios, q ÆQ
2
/Q
1
, we identify six non-trivial ground states, some
of which show a remarkable and unexpected structural complexity. For 0.59 . q Ç 1, the system undergoes a
phase separation where the two charge species populate in a hexagonal arrangement spatially separated areas.
Key words: Wigner crystals, binary mixture of charged systems, ground states, Ewald summation, evolutionary
algorithm
PACS: 52.27.Lw, 64.70.K-, 64.75.St, 73.20.Qt
1. Introduction
The identification of the ordered ground state configurations of classical charged particles is known in
literature as theWigner problem [1]. In two dimensions and at vanishing temperature these charges form
a hexagonal lattice [2–4]. In rather recent investigations, this problem has been extended: one example
is the bilayer problem where the charges are confined between two planes separated by a finite distance
[5, 6]. Due to the availability of closed, analytic expressions for the potential energy of this particular
system, the complete set of its ground state configurations could be identified.
In the present contribution we return to the single layer problem and consider an equimolar, binary
mixture of charged particles (with nominal valuesQ
1
andQ
2
), that condensate at vanishing temperature
on a neutralizing plane. For given values Q
1
and Q
2
, the ordered equilibrium configurations of the par-
ticles at vanishing temperature are imposed by the requirement that the potential energy is minimized.
With the help of efficient and highly accurate Ewald summation techniques [7], the lattice sum of this
system can be evaluated for any particle arrangement on an arbitrary, two-dimensional lattice. Employ-
ing suitable optimization techniques, the parameters of these lattices are then optimized in such a way
as to minimize the lattice sum. In this contribution we have used an optimization tool that is based on
ideas of evolutionary algorithms (EAs) [8, 9]. Within this concept, any possible two-dimensional lattice
is considered as an individual, to which a fitness value is assigned. These individuals are then exposed
on the computer to an artificial evolution: via creation and mutation operations a large number of indi-
viduals is produced; in the former procedure a pair of new individuals is created from a pair of parent
individuals that are selected according to their fitness values. Along this evolution, only the best, i.e., the
fittest, individuals are expected to survive and are thus retained. Bearing in mind that we are looking for
the individual (= structure) with the lowest lattice sum, we assign a high fitness value to an energetically
favorable ordered structure. EA-based optimization algorithms have turned out to be highly efficient and
© M. Antlanger, G. Kahl, 2013 43501-1
http://dx.doi.org/10.5488/CMP.16.43501
http://www.icmp.lviv.ua/journal
M. Antlanger, G. Kahl
reliable tools for identifying ordered equilibrium structures in a broad variety of condensed matter sys-
tems, in general [9–12], and for quite a few two-dimensional systems, in particular [13–17].
We point out that our setup ignores features of a possible experimental verification of the prob-
lem: “real” particles are of finite size which sediment from a (three-dimensional) solution onto the (two-
dimensional) substrate where they eventually self-organize. By contrast, in the present contribution we
consider an idealized version of the problem where point particles interact exclusively via Coulomb po-
tentials. Further, we neglect the process that leads to the self-assembly of the particles and simply look
at the different, final results of self-organization. These simplifications are justified in view of the fact
that the original Wigner problem has been formulated to an idealized situation similar to the one that we
consider.
In total we have identified six non-trivial ground states. They are characterized by a broad variety
of structural complexity, which is the result of an intricate competition between the interactions of the
two charges. Introducing the charge ratio q as the only relevant parameter of the system (defined as
q Æ Q
2
/Q
1
with 0 É q É 1 due to simple consideration and due to symmetry arguments) we identified
two structures that show a remarkable stability over relatively large q-ranges: (i) one of them can be de-
scribed via two intertwining, commensurate square sublattices, one of them populated with charges Q
1
,
the other with charges Q
2
; this structure shows, in addition, among all ground states the highest energy
gain with respect to a suitably defined reference state; (ii) in the other ordered equilibrium configuration,
strongly distorted, but symmetric hexagonal tiles cover the entire two-dimensional space hosting in their
interior pairs of the weaker charges. For 0.59 . q Ç 1, the system undergoes a phase separation where
the two spatially separated phases are represented by hexagonal lattices populated with either species of
charges. The remaining four non-trivial ground states are dominated by distorted, asymmetric hexago-
nal arrangements of chargesQ
1
, hosting in their interior pairs and triplets of chargesQ
2
. For the limiting
values, i.e., q Æ 0 and q Æ 1, we obtain the expected hexagonal particle arrangements.
The paper is organized as follows. In the subsequent section we briefly introduce our model system.
Section 3 is devoted to the methods we have used: both the Ewald summation technique and our opti-
mization tool (based on ideas of evolutionary algorithms) are briefly summarized; further we introduce a
suitable state of reference for our energetic considerations. In section 4we thoroughly discuss the results.
The contribution is closed with concluding remarks.
2. Model
We consider an equimolar mixture of classical charges with the particles being confined to a pla-
nar (i.e., two-dimensional) geometry. The point charges (with nominal values Q
1
and Q
2
) are located at
positions r
i
and r
j
and interact via an unscreened Coulomb interaction
©(r
i j
)Æ
Q
i
Q
j
r
i j
(2.1)
with r
i j
Æ jr
i
¡r
j
|.
Since the total number density (i.e., number of particles per unit area), ½, can be scaled out via the
distances, its actual quantity is irrelevant for further considerations. In the equimolar case, for the partial
number densities we obtain ½
1
Æ ½
2
Æ ½/2.
For convenience we introduce the parameter q Æ Q
2
/Q
1
, i.e., the ratio between the two types of
charges. Since negative values of q lead to a divergent potential energy and taking into account the sym-
metry q$ 1/q we can restrict ourselves to the range 0É q É 1; thus, we assume that chargeQ
1
is stronger
than chargeQ
2
. Note that we recover the classical Wigner problem for q Æ 1.
To compensate for the charges, we introduce a uniform, neutralizing background on the plane, spec-
ified by a charge density ¾, which is given by
¾Æ¡½
1
Q
1
¡½
2
Q
2
Æ¡Q
1
1Åq
2
½. (2.2)
43501-2
Wigner crystals for a planar, equimolar binary mixture
3. Method
The present contribution is devoted to a complete identification of the ground state configurations of
an equimolar mixture of point charges, i.e., the ordered equilibrium structures at vanishing temperature.
Following the basic laws of thermodynamics, the particleswill arrange under these conditions in an effort
to minimize the corresponding thermodynamic potential. For our system (i.e., fixed particle number N
and density ½), we have to minimize the potential energy, which reduces at vanishing temperature to the
lattice sum of the ordered particle configuration.
Among the numerous optimization schemes available in literature, we have opted for an optimization
algorithm that is based on ideas of EAs [8, 9]. Our choice is motivated by the fact that this strategy has
turned out to be highly successful in related problems for a wide variety of soft matter systems, including
in particular problems in two-dimensional geometries [13–17]. For a comprehensive presentation of this
optimization algorithm and of the related computational and numerical details, we refer the reader to
[9, 18]. For completeness, we note that our optimization scheme is capable of identifying only an ordered,
i.e., periodic, ground state configuration of systems.
The quantity that has to be minimized is the lattice sum of an ordered particle configuration. Taking
into account the long-range character of the interactions (2.1), this quantity can most conveniently be
calculated via Ewald sums [7]. For the separation of r - and k-space contributions, we have used the cutoff
values r
Æ 15/
p
½ and k
Æ 10
p
½, respectively; for the Ewald summation parameter we use®Æ 0.3. This
set of numerical parameters ensures a relative accuracy of 10¡5 for the evaluation of the internal energy
[19, 20]. This upper limit for the accuracy was verified by comparing our Ewald summation-based results
to the data obtained for the symmetric Wigner bilayer [5, 6] by Šamaj and Trizac, who used an essentially
exact route for their computations, based on a rapidly convergent series expansion of the energy.
In an effort to specify the ordered structures, we introduce for convenience a q-dependent reference
energy. For the one component system (i.e., q Æ 1) the ground state energy (per particle) of point charges,
arranged on a neutralizing plate in a hexagonal lattice, is given by (Q ÆQ
1
ÆQ
2
)
E
0
(q Æ 1)Æ¡C
M
p
½Q
2
, (3.1)
C
M
Æ 1.960515789 being the Madelung constant of this particular particle arrangement.
We extend this expression continuously to q É 1 along the following lines: we imagine the system to
be split up into two infinitely large regions, labeled °Æ 1 and °Æ 2, each of them hosting exclusively the
respective charges, Q
1
and Q
2
, and each of them being locally charge neutral. The two regions share a
common border. Introducing local number densities, ½
(°)
i
, for species i in region ° (i Æ 1,2 and °Æ 1,2),
we arrive at the following relations:
½
(1)
1
Q
1
Å¾Æ 0, ½
(1)
2
Æ 0 in region 1,
½
(2)
1
Æ 0, ½
(2)
2
Q
2
Å¾Æ 0 in region 2.
Together with (2.2), we obtain
½
(1)
1
Æ
1Åq
2
½ in region 1, ½
(2)
2
Æ
1Åq
2q
½ in region 2. (3.2)
With these values for the local number densities and assuming that the charges will form hexagonal
lattices in the respective regions, we obtain — with the help of equation (3.1) — the total energy per
particle for this system
E
0
¡
q
¢
Æ ¡C
M
µ
1
2
q
½
(1)
1
Q
2
1
Å
1
2
q
½
(2)
2
Q
2
2
¶
Æ ¡C
M
p
½Q
2
r
1Åq
2
1Åq
3/2
2
. (3.3)
In passing we note the following: for a finite system (as studied, for instance, in computer simula-
tions), the contribution originating from the border between the two coexisting regions to the total en-
ergy of the systemmust not be neglected. However, since in our idealized approach we deal with systems
43501-3
M. Antlanger, G. Kahl
of infinite extent, the contribution of this border vanishes as the size and the number of particles of the
system tend to infinity.
In an effort to characterize the emerging ground state configurations, we have evaluated the two-
dimensional orientational bond order parameters,ª
4
andª
6
[21], defined via
ª
n
Æ
¯
¯
¯
¯
¯
1
N
i
N
i
X
jÆ1
exp[in£
j
℄
¯
¯
¯
¯
¯
, n Æ 4,6. (3.4)
N
i
is the number of nearest neighbors of a tagged particle with index i and £
j
is the angle of the vector
connecting this particle with particle j with respect to an arbitrary, but fixed orientation.
4. Results
In an effort to identify the complete set of ordered ground state configurations of our system speci-
fied in section 2, we have performed extensive EA-runs, taking into account up to 20 particles per species
and per unit cell. Up to 5000 individuals were created for a given state point. Calculations have been per-
formed on a discrete q-grid with a spacing of¢q Æ 0.01; thus, in this contribution¢q defines the accuracy
in the location of the boundaries between ground states. The respective minimum energy configurations
were retained as the ground state particle arrangements.
In figure 1 we display the energy (per particle), E(q), of these ground state configurations and the
energy difference, ¢E(q)Æ [E(q)¡E
0
(q)℄, with respect to the reference energy, E
0
(q), as defined in equa-
tion (3.3). Note that over the entire q-range ¢E(q) is very small, i.e., less than 5 ¢ 10
¡3. The fact that the
differences between the competing structures are that small is a fingerprint of the long-range nature of
the Coulomb interaction.
E(q) decays with increasing q: it connects the limiting value at q Æ 0 [E(q Æ 0) Æ E
0
(q Æ 0) Æ
¡C
M
p
½Q
2
/2
p
2], obtained for a pure system of charges Q
1
and a number density ½
1
Æ ½/2 with the
other reference state at q Æ 1, where the charges are indistinguishable (i.e., Q
1
Æ Q
2
Æ Q) and thus
E(q Æ 1)Æ E
0
(q Æ 1)Æ¡C
M
p
½Q
2. In the intermediate q-range, the curve seems — at first glance — to be
a smooth, monotonous function. The subtle details, which reflect the structural changes of the system as q
varies, become visible only if we subtract from E(q) the reference energy, E
0
(q), i.e.,¢E(q)Æ E(q)¡E
0
(q).
This function is now non-monotonous and shows kinks for particular q-values which can be associated
with the structural changes. For convenience, the vertical broken lines in figure 1 indicate the limits of
0 0.2 0.4 0.6 0.8 1
q
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
E
(q
)
[ρ
1
/2
Q
2
]
0 0.2 0.4 0.6 0.8 1
-0.005
-0.004
-0.003
-0.002
-0.001
0
0.001
∆
E
(q
)
[ρ
1
/2
Q
2
]
phase separation1 2 3 4 5 6
Figure 1. (Color online) Energy per particle of the ground state configuration, E(q) (blue line), and energy
difference (per particle), ¢E(q) Æ [E(q)¡E
0
(q)℄ (black line), with respect to the reference energy, E
0
(q),
as defined in equation (3.3) as functions of q . Vertical broken lines indicate the limits of stability of the
six identified non-trivial ground states. The horizontal broken line marks the q -values where the phase
separated system is the energetically most favorable one.
43501-4
Wigner crystals for a planar, equimolar binary mixture
stability of the six non-trivial identified ground states. The fact that these kinks are sometimes more or
less pronounced is related to three issues: (i) the limited accuracy of our energy evaluation (see discussion
above), (ii) the finite grid-size ¢q underlying our investigations, and (iii) the intersection angle between
the E(q)-curves of two neighboring ground state structures. For 0.59 . q Ç 1, the energy of the (phase
separated) reference state, E
0
(q), attains values that are smaller than the energy of the respective ground
states identified in our EA search, indicating that the system undergoes a phase separation. This phe-
nomenon, i.e., the formation of two infinitely large regions populated with only one species of charges
representing the coexisting phases, cannot be grasped with our EA-based optimization tool, since it relies
on a finite number of charges per unit cell.
The table provides an overview of the ground state configurations that we have identified for 0É q É
1; the structures themselves are depicted in figures 2–5.
Table 1. Overview of the identified ground state configurations and the respective q -ranges. The struc-
tures themselves are depicted in figures 2–5.
q-range ground state structure
0.00 hexagonal lattice formed by chargesQ
1
0.00Ç q . 0.04 structure 1
0.05 . q . 0.09 structure 2
0.10 . q . 0.25 structure 3
0.26' q structure 4
0.27 . q . 0.28 structure 5
0.29 . q . 0.59 structure 6
0.60 . q Ç 1 phase separation
1.00 hexagonal lattice formed by the
(indistinguishable) chargesQ
1
andQ
2
For q Æ 0, charges with non-vanishing nominal values, Q
1
, arrange — as expected — in a hexagonal
lattice (not depicted) with number density ½
1
Æ ½/2; the other, chargeless particles do not interact with
any particle species and thus occupy arbitrary positions.
For 0Ç q . 0.04, chargesQ
1
form a hexagonal lattice which is to a high degree regular (structure 1, de-
picted in the left hand panel of figure 2): the order parameterª
6
varies betweenª
6
(q Æ 0.01)Æ 0.99972
and ª
6
(q Æ 0.04) Æ 0.99577. The deviation from its ideal value, ª
6
Æ 1, stems from a slight distortion of
these hexagons (highlighted in the corresponding panel): a central “axis” (dotted line in the correspond-
Figure 2. (Color online) Left hand panel: structure 1 (identified for q Æ 0.02), right hand panel: struc-
ture 2 (identified for q Æ 0.05). Blue: charges Q
1
, green: charges Q
2
. Lines highlight the hexagonal units
discussed in the text. The dotted lines mark the central axes of the hexagonal units (cf. text).
43501-5
M. Antlanger, G. Kahl
ing panel), connecting the “upper” and “lower” vertices of the hexagon, is formed by two line-segments
of equal length; the distances of the vertices left (right) to this central axis from the center of the hexagon
differ in their length by less than Å4% (¡3%). Thus, the left hand half of the hexagon is slightly larger
than its counterpart located to the right of the central axis. This enlarged space is imposed by the fact that
this area hosts charges Q
2
which form a zig-zag pattern within the ground state configuration, oriented
parallel to the central axis of the hexagon; within this line, charges Q
2
are equidistant. The other part of
the hexagon located right to its central axis remains empty.
In the adjacent q-range, 0.05. q . 0.09, charges Q
1
maintain their hexagonal arrangements (struc-
ture 2, displayed in the right hand panel of figure 2). As compared to structure 1, the distortion of the
hexagons (highlighted in the corresponding panel) is now considerablymore pronounced:ª
6
(q Æ 0.05)Æ
0.90598 while ª
6
(q Æ 0.09) Æ 0.83643. The “upper” and the “lower” vertices of the hexagon are still con-
nected via a central axis (dotted line in the corresponding panel), consisting of two line segments of equal
length; however, the distances of the vertices to left of this central axis from the center of the hexagon
are now by up to 21% longer, while the corresponding distances of the vertices on the opposite side of
the axis are less than 5% shorter. This conceivable asymmetry (see the highlighted hexagon in the cor-
responding panel) is imposed by the increased nominal value of the charge Q
2
: the zig-zag arrangement
of particles, observed for this species of charges in structure 1 has been replaced by a parallel arrange-
ment of pairs of particles which are aligned in the direction of the central axis of the hexagon; while the
intra-pair distance is very short, the inter-pair distance is quite large.
In the relatively wide range of 0.10. q . 0.25 the two species of charges arrange in two intertwining,
commensurate square lattices (structure 3, cf. figure 3). It has to be emphasized that both sublattices
remain perfect over the entire q-range of stability, i.e.,ª
4
Æ 1 for 0.10 . q . 0.25. The structural stability
of this particular ground state is also reflected by the fact that structure 3 is characterized by the highest
energy gain compared to the energy value of the reference structure, E
0
(q) (see figure 1).
Figure 3. (Color online) Structure 3 (identified for q Æ 0.10). Blue: chargesQ
1
, green: chargesQ
2
.
Around the value q ' 0.26, charges Q
1
form hexagonal structural units which are in their shape
reminiscent of gems or diamonds. These six-particle rings form in a head-to-tail arrangement parallel
lanes: adjacent six-particle rings of neighboring lanes share vertices, while the remaining edges form
equilateral triangles. This ground state is referred to as structure 4; it is depicted in the left hand panel of
figure 4 and the six-particle rings are highlighted. Each of the six-particle units hosts in its center an es-
sentially equilateral triangle of chargesQ
2
. The positions of the six surrounding charges Q
1
are imposed
by the condition that the smallest distance of these charges from any of the three inner charges (Q
2
) has
the same value; this requirement induces the particular shape of the six-particle rings. The triangular
arrangements that fill up the interstitial space are not populated by charges Q
2
.
The ground state identified for 0.27 . q . 0.28 (denoted as structure 5 and depicted in the right
hand panel of figure 4) differs only in one feature from structure 4: the lanes, formed by the head-to-tail
arrangements of the six-particle rings (highlighted in the corresponding panel) are now antiparallel. In
43501-6
Wigner crystals for a planar, equimolar binary mixture
Figure 4. (Color online) Left hand panel: structure 4 (identified for q Æ 0.26), right hand panel: struc-
ture 5 (identified for q Æ 0.27). Blue: charges Q
1
, green: charges Q
2
. Lines highlight the hexagonal units
discussed in the text.
Figure 5. (Color online) Left hand panel: structure 6 (identified for q Æ 0.30), right hand panel: structure 6’
(identified for q Æ 0.35. Blue: chargesQ
1
, green: chargesQ
2
. Lines highlight the hexagonal units discussed
in the text. The dotted lines mark the central axes of the hexagonal units (cf. text).
this configuration, the neighboring rings of adjacent lanes share edges, which leads now to the formation
of rhombic four-particle arrangements which are again void ofQ
2
charges.
Finally, at q ' 0.29 structure 6 emerges and remains the ground state over the relatively large interval
0.29. q . 0.59 (see left hand panel of figure 5). Its basic unit is an elongated hexagon (highlighted in the
corresponding panel): aligned in parallel and sharing the edges with neighboring tiles, they completely
cover the two-dimensional space. The direction perpendicular to the longest elongation of this hexagon
is considered for the following discussion as the central axis (dotted in the corresponding panel). For
structure 6 this axis is also the symmetry axis of the hexagon. The four edges originating from the central
axis have the same lengths, say l
1
; similarly, the remaining two edges of the hexagon (oriented parallel
to the central axis) assume another, equal value, say l
2
. Each of these hexagons hosts a pair of charges
Q
2
, located on a line perpendicular to the central axis and separated by a distance, which decreases as q
is increased. By increasing the charge ratio q , l
1
decreases from 1.297/
p
½ (at q Æ 0.29) to 1.269/
p
½ (at
q Æ 0.59), while l
2
increases from 1.29/
p
½ (at q Æ 0.29) to 1.324/
p
½ (at q Æ 0.59). At the cross-over, i.e.,
l
1
' l
2
(observed for q ' 0.36, our optimization tool identifies a closely related, energetically degenerate
structure, denoted as structure 6’ and depicted in the right hand panel of figure 5: now that all edges
of the basic hexagon are equal, these units are no longer forced to align in parallel, but are capable of
choosing an alternative, non-parallel arrangement: imposed by the internal angle between the edges of
the basic hexagon, these units arrange in a grain-like super-structure.
43501-7
M. Antlanger, G. Kahl
For 0.59 . q Ç 1.00, we find that E
0
(q) Ç E(q) (see figure 1), indicating that the phase separated
particle configuration is energetically more stable than any other ordered structure identified by our
optimization tool. Due to the limitation in the number of particles per unit cell, the EA-based search for
ground state configurations proposes — depending on the number of particles per cell — configurations
with increasing complexity, all of them being characterized by an energy value E(q) that is larger than the
corresponding value E
0
(q). Thus, the demixed state, formed by two separate hexagonal, ordered regions
and each of them being populated by one species of charge, is the ground state in this q-range.
Finally, for q Æ 1, we recover the one-component hexagonal monolayer (not displayed).
5. Conclusions
In this contribution we have investigated the ground state configurations of an equimolar, binary
mixture of classical charged particles (with nominal charges Q
1
and Q
2
), that self-assemble on a neu-
tralizing plane. Our investigations are based on reliable Ewald summation techniques which permit an
efficient evaluation of the ground state energies (= lattice sums). With the help of reliable optimization
tools, which are based on ideas of evolutionary algorithms, we are able to identify the ordered ground
states of the system: by searching essentially among all possible two-dimensional lattices, this algorithm
identifies the energetically most favorable particle arrangement for a given charge ratio q ÆQ
2
/Q
1
. Apart
from the expected, trivial hexagonal lattices for q Æ 0 and q Æ 1, we could in total identify six ground state
configurations for 0Ç q . 0.59.
Quite unexpectedly, these particle arrangements show a remarkable structural complexity which
is the result of the energetic competition between the charge-charge interactions. Throughout, a pro-
nounced impact of the weaker charges, Q
2
, on the sublattices formed by the stronger charges, Q
1
, could
be observed: this holds even if the corresponding q-values are rather small (i.e., q ' 0.05). Except for a
purely square particle arrangement (which is stable over a remarkably large q-range and which shows
the highest energy gain among all ground states with respect to a suitably defined reference state — see
also the discussion about energies below), the ground states can be described on the basis of asymmetric,
sometimes strongly distorted six-particle arrangements formed by charges Q
1
, which host in their inte-
rior simple two- or three-particle configurations of chargesQ
2
. A deeper insight into the mechanisms that
govern the formation of ground states is gained by introducing a phase separated reference state, where
the two species of charges populate in hexagonal arrangements spatially separated areas. Comparing the
energies of our ground state configurations, E(q), with the energy of this reference state, E
0
(q), we find
that these two functions differ by less than 5 ¢ 10
¡3, a fact that represents a characteristic fingerprint of
the long-range Coulomb interactions. An analysis of this energy difference as a function of q reveals that
transitions from one ground state to an adjacent one become visible as (more or less pronounced) kinks
in this function. Based on these considerations we could show that for 0.59 . q Ç 1, the energetically most
favorable particle arrangement is the (ideal) phase separated state.
Acknowledgements
This work was financially supported by the Austrian Research Fund (FWF) under Proj. No. P23910-
N16 and Projekt Amadée under Proj. No. FR 10/2012. The authors gratefully acknowledge discussions
with Martial Mazars (Paris-Orsay), Ladislav Šamaj (Bratislava), and Emmanuel Trizac (Paris-Orsay).
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Вiгнерiвськi кристали для планарної еквiмолярної сумiшi
класичних заряджених частинок
M. Антлангер1,2, Г. Каль1
1 Iнститут теоретичної фiзики i центр обчислювального матерiалознавства (CMS), Вiденський
технологiчний унiверситет, A–1040 Вiдень, Австрiя
2 Лабораторiя теоретичної фiзики (UMR 8627), Унiверситет Парi-Сюд, F–91405 Орсе, Францiя
Ми дослiдили конфiгурацiї основного стану еквiмолярної бiнарної сумiшi класичних заряджених части-
нок (з номiнальними зарядамиQ
1
iQ
2
), якi конденсуються на нейтралiзуючiй площинi. Використовуючи
ефективний метод пiдсумовування Евальда для обчислення основних енергетичних станiв, ми iденти-
фiкували найбiльш енергетично вигiднi впорядкованi розташування частинок за допомогою високо на-
дiйного оптимiзацiйного методу, який ґрунтується на iдеях еволюцiйних алгоритмiв. Для великої областi
змiни коефiцiєнта зарядiв, q ÆQ
2
/Q
1
, ми iдентифiкували шiсть нетривiальних основних станiв, деякi з
яких демонструють значну i неочiкувану структурну складнiсть. Для 0.59 . q Ç 1, система зазнає фазо-
вого розшарування, де зарядженi сорти заселяють просторово вiдокремленi областi з гексагональним
розташуванням.
Ключовi слова: кристали Вiгнера, бiнарна сумiш заряджених систем, основнi стани, пiдсумовування
Евальда, еволюцiйний алгоритм
43501-9
http://dx.doi.org/10.1016/j.physrep.2010.11.004
http://dx.doi.org/10.1063/1.1901585
http://dx.doi.org/10.1063/1.3006065
http://dx.doi.org/10.1039/b906832e
http://dx.doi.org/10.1063/1.3509380
http://dx.doi.org/10.1209/0295-5075/82/18001
http://dx.doi.org/10.1039/b717205b
http://dx.doi.org/10.1021/la900421v
http://dx.doi.org/10.1088/0953-8984/22/10/104105
http://dx.doi.org/10.1088/0953-8984/23/40/404206
http://dx.doi.org/10.1039/c2sm26043c
http://dx.doi.org/10.1080/08927029208049126
http://dx.doi.org/10.1063/1.2816570
http://dx.doi.org/10.1103/RevModPhys.60.161
Introduction
Model
Method
Results
Conclusions
|