Domain-walls formation in binary nanoscopic finite systems
Using a simple one-dimensional Frenkel-Kontorowa type model, we have demonstrated that finite commensurate chains may undergo the commensurate-incommensurate (C-IC) transition when the chain is contaminated by isolated impurities attached to the chain ends. Monte Carlo (MC) simulation has shown that...
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irk-123456789-1527812019-06-13T01:26:33Z Domain-walls formation in binary nanoscopic finite systems Patrykiejew, A. Sokołowski, S. Using a simple one-dimensional Frenkel-Kontorowa type model, we have demonstrated that finite commensurate chains may undergo the commensurate-incommensurate (C-IC) transition when the chain is contaminated by isolated impurities attached to the chain ends. Monte Carlo (MC) simulation has shown that the same phenomenon appears in two-dimensional systems with impurities located at the peripheries of finite commensurate clusters. 2014 Article Domain-walls formation in binary nanoscopic finite systems / A. Patrykiejew, S.Sokołowski // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23605:1-9 — Бібліогр.: 32 назв. — англ. 1607-324X arXiv:1407.2408 DOI:10.5488/CMP.17.23605 PACS: 64.70.Rh, 68.55.Ln, 64.60.an http://dspace.nbuv.gov.ua/handle/123456789/152781 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Using a simple one-dimensional Frenkel-Kontorowa type model, we have demonstrated that finite commensurate chains may undergo the commensurate-incommensurate (C-IC) transition when the chain is contaminated by isolated impurities attached to the chain ends. Monte Carlo (MC) simulation has shown that the same phenomenon appears in two-dimensional systems with impurities located at the peripheries of finite commensurate clusters. |
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Patrykiejew, A. Sokołowski, S. |
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Patrykiejew, A. Sokołowski, S. Domain-walls formation in binary nanoscopic finite systems Condensed Matter Physics |
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Patrykiejew, A. Sokołowski, S. |
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Patrykiejew, A. |
| title |
Domain-walls formation in binary nanoscopic finite systems |
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Domain-walls formation in binary nanoscopic finite systems |
| title_full |
Domain-walls formation in binary nanoscopic finite systems |
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Domain-walls formation in binary nanoscopic finite systems |
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Domain-walls formation in binary nanoscopic finite systems |
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domain-walls formation in binary nanoscopic finite systems |
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Інститут фізики конденсованих систем НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/152781 |
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Domain-walls formation in binary nanoscopic finite systems / A. Patrykiejew, S.Sokołowski // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23605:1-9 — Бібліогр.: 32 назв. — англ. |
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Condensed Matter Physics |
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AT patrykiejewa domainwallsformationinbinarynanoscopicfinitesystems AT sokołowskis domainwallsformationinbinarynanoscopicfinitesystems |
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2025-07-14T04:16:36Z |
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1837594414442086400 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 2, 23605: 1–9
DOI: 10.5488/CMP.17.23605
http://www.icmp.lviv.ua/journal
Domain-walls formation in binary nanoscopic finite
systems
A. Patrykiejew, S. Sokołowski
Department for the Modelling of Physico-Chemical Processes, Faculty of Chemistry,
Maria Curie-Sklodowska University, 20031 Lublin, Poland
Received February 24, 2014, in final form May 6, 2014
Using a simple one-dimensional Frenkel-Kontorowa type model, we have demonstrated that finite commensu-
rate chains may undergo the commensurate-incommensurate (C-IC) transition when the chain is contaminated
by isolated impurities attached to the chain ends. Monte Carlo (MC) simulation has shown that the same phe-
nomenon appears in two-dimensional systems with impurities located at the peripheries of finite commensu-
rate clusters.
Key words: binary mixture, commensurate-incommensurate transitions, Monte Carlo simulation, finite
systems, Frenkel-Kontorova model
PACS: 64.70.Rh, 68.55.Ln, 64.60.an
1. Introduction
In modern nanotechnologies one often deals with very small systems of countable numbers of atoms
or molecules. In such cases, the finite size and boundary effects are large and bound to significantly
affect the properties of the system with respect to its bulk counterpart [1]. Another important problem
is the purity of small systems. While tiny amounts of impurities may be unimportant in macro-scale, the
behavior of nanoscopic systems is considerably influenced even by a small number of impurity atoms
[2]. Among the systems in which the presence of impurities may be of importance are those exhibiting
the C-IC transition.
The C-IC transitions have been experimentally observed in a variety of systems including adsorbed
films, [3–6] intercalated compounds [7, 8] composite crystals [9] and magnetically ordered structures
of rare-earth compounds [10]. Theoretical studies of C-IC transitions have focused on the domain wall
description of incommensurate phases [11–14]. According to the domain wall formalism, the IC phase
is a collection of C domains separated by domain walls. The density within the domain walls may be
lower or higher than the density of commensurate domains. In the former case, the walls are light and
superlight, while in the latter the walls are heavy and superheavy [13].
The simplest theoretical approach which predicts the formation of domain walls is the one-dimen-
sional Frenkel-Kontorova model [15]. The original FK model assumes that an infinite chain of atoms
interacting via harmonic potential at zero temperature is subjected to a periodic (sinusoidal) external
field. Depending on the misfit between the equilibrium distance of the harmonic potential, the period
and amplitude of the external field, the FK model is capable of describing the C-IC transition. The FK
model has been extended to two-dimensional systems [16, 17] to mixtures [18], systems with disorder
[19] and has also been used to study finite chains [20–22].
The NPT Monte Carlo simulation has demonstrated [23, 24] that finite one-dimensional chains, ei-
ther uniform or subjected to periodic field, exhibit structures that cannot appear in infinite chains. In
particular, it has been shown that the chain experiences very large density fluctuations. In the case of
chains on a periodic substrate, a number of different structures (registered, free floating, domain-wall
© A. Patrykiejew, S. Sokołowski, 2014 23605-1
http://dx.doi.org/10.5488/CMP.17.23605
http://www.icmp.lviv.ua/journal
A. Patrykiejew, S. Sokołowski
incommensurate and resulting from the chain fragmentation) have been found to appear during a single
run.
In one of our recent papers [25], we have shown that finite two-dimensional clusters of Kr adsorbed
on graphite undergo the C-IC phase transition when contaminated by small amounts of Ar atoms. Com-
puter simulation has demonstrated that the transition occurs already when the boundaries of a finite
krypton island are covered with a single layer of argon atoms.
In this paper we address the issue of the influence of impurities on the behavior of finite one- and two-
dimensional (1D and 2D) systems. We are interested in the effects of impurities located at the peripheries
of finite 1D chains and 2D clusters of atoms subjected to the periodic external field. The field is assumed
to be strong enough to enforce the formation of commensurate structures in pure systems and we con-
sider the possibility of the C-IC transition driven by the presence of impurities. The paper is organized
as follows. In the next section we discuss the behavior of one-dimensional systems in the framework
of a modified Frenkel-Kontorova model. Then, in the third section we consider two-dimensional finite
systems studied by Monte Carlo simulation. In the final section we summarize our findings.
2. One-dimensional Frenkel-Kontorova model
At first, we have considered a simple 1D finite chain of atoms at zero temperature and used the
Frenkel-Kontorova (FK) model [15]. The energy of a finite chain consisting of N atoms subjected to a
periodic external potential and containing impurities can be written as follows:
E = 1
2
{
N−1∑
i=1
Ki [xi+1 −xi −bi ]2 +
N∑
i=1
vi [1−cos(2πxi /a)]
}
, (2.1)
where Ki and bi are the elastic constant and the equilibrium distance for the pair (i , i +1) and vi is the
amplitude of the external field for the i -th particle. Having introduced the displacements ui = xi /a −pi
(i = 1,2, . . . N ) with p being a positive integer (in this work we set p = 2), the energy given by eqn.(2.1)
can be rewritten in units of K0a2/2 (K0 being the elastic constant for a pure chain) as follows:
E =
N−1∑
i=1
K̂i [ui+1 −ui −mi ]2 +
N∑
i=1
v̂i [1−cos(2πui )]. (2.2)
In general, the elastic constants K̂i (the misfits mi ) can assume one of three possible values k0,0 ≡ 1,
k0,im or kim,im (m0,0,m0,im ormim,im) depending on the composition of the pair (i , i+1), and the amplitude
v̂i = vi /a2K0 is equal either to v0 or to vim.
In order to find the equilibrium configuration of the chain, the energy should attain its minimum
value, specified by the condition stating that the forces fi =−∂E/∂ui = 0 for all i . We consider the systems
in which a single impurity atom is located at one end of the chain (class I) and the systems with two
impurity atoms located at both ends of the chain (class II) and put k0,im = kim and m0,im = mim. The
behavior of pure chains depends on m0 and v0. For the assumed value of m0 = −0.1, pure chains are
commensurate when v0 exceeds the critical value v0,c ≈ 0.004950. The calculations have been done for
N between 21 and 401 and v0 = 0.006.
The first series of calculations have been carried out for kim = 1.0, while vim and mim have been
varied. Figure 1 shows the example of the results obtained for the chains with N = 41,mim =−0.25 and
different values of vim. The lower and upper panels of figure 1 show the results for the systems of class I
and II, respectively. It is evident that the systems belonging to both classes exhibit a qualitatively similar
behavior. Of course, the systems of the class I lack the symmetry of atomic displacements with respect
to the central atom. For vim lower (higher) than vim,l (vim,u), the chain remains commensurate, while
for intermediate values of vim, there appear incommensurate structures with domain walls. When vim
is lower than vim,l, the energy cost to put the impurity out of registry position is low and the impurity
can exhibit large displacements from the commensurate position, while the rest of the chain assumes a
commensurate structure due to the domination of the surface over the elastic interaction. On the other
hand, when vim > vim,u, the impurity is strongly pinned by the surface potential and the chain retains
23605-2
Domain-walls in finite systems
Figure 1. (Color online) Atomic displacements vs. atomic positions for the systems of class I (part a) and
II (part b) and different values of vim (given in the figure). The calculations have been done N = 41,
mim = −0.25 and kim = 1.0. The insets show the changes of the average nearest-neighbor distance vs.
vim The regions marked by C and ks (k = 1,2,3,4) correspond to the commensurate structure to the
incommensurate structures with k domain walls, respectively.
the commensurate structure. Figure 1 shows that when vim > vim,u, the displacements of impurities are
considerably lower than when vim < vim,l.
For intermediate values of vim, the gain in elastic energy due to transition into the incommensurate
structure is larger than the loss of surface energy. The insets to figure 1 show the changes of the average
nearest-neighbor distance with vim. In both classes of systems, we have found a series of transitions
characterized by a different number of domain walls. The calculations for chains with N up to 401 have
shown that in longer chains a larger number of structures appear, and it seems that in the limit of N →∞
the transitions form a harmless staircase [11].
The values of vim,l and vim,u change with mim and there is a critical value of mim,c for which the
difference ∆vim = vim,u−vim,l goes to zero (figure 2). In the particular examples considered here (N = 41
and kim = 1.0), mim,c ≈ −0.1856 for the systems of class I and II. The inset to figure 2 shows that ∆vim
scales withmim,c−mim as follows:
∆vim∝ (mim,c−mim)1/2, k = 1,2. (2.3)
The same scaling appears for the transitions between different incommensurate structures, but the value
ofmim,c for each transition is different (figure 2).
We have then investigated the effects due to changes in the magnitude of kim. Figure 3 gives the
example of results for the systems of class I. We see that the values of vim at the transitionsC −1s, 1s−2s
and 2s −3s are nearly independent of kim, while those corresponding to transitions 3s −2s, 2s −1s and
1s −C (leading to the recovery of commensurability), exhibit a logarithmic dependence on kim.
The logarithmic dependence of vim on kim at transitions 3s−2s, 2s−1s and 1s−C has the same origin
as the C-IC transition in a pure FK model [11]. On the other hand, the mechanism of transitions leading to
commensurability when vim becomes very low is different. For sufficiently low vim, the energy cost to put
23605-3
A. Patrykiejew, S. Sokołowski
Figure 2. The phase diagrams showing the dependence of the difference between the upper and lower
values of vim at the transition points (∆vim(α,β)) between different structures (α,β) and the impurity
misfit mim(α,β). The calculations have been done for N = 41 and kim = 1.0 and for the classes I and II.
The inset shows the scaling plot for the C-IC transition.
the impurity out of registry position and to restore the commensurate positions in the rest of the chain is
low. Consequently, the impurity can exhibit a large displacement from the commensurate position, while
the rest of the chain assumes a commensurate structure due to the domination of surface energy over
the elastic energy. This is illustrated in the inset to figure 3, which shows atomic displacements for the
Figure 3. (Color online) The phase diagrams showing the dependence of vim on kim at the transition
points between different structures. The calculations have been done for N = 41 and mim = −0.24. The
inset shows the atomic displacements versus actual atomic positions in the commensurate phase for
vim = 0.003 and vim = 0.017 when kim = 1.0. Filled symbols mark the impurity atom.
23605-4
Domain-walls in finite systems
systems with kim = 1.0 and vim = 0.003 and vim = 0.017, while the rest of the parameters have been kept
the same as in the main figure. Of course, there is an asymmetry of displacements in the chain but it is
very small. In the commensurate phase at high values of vim = 0.017, the displacements at both ends are
nearly the same.
Another question is whether a single impurity can drive the incommensurate system into commen-
surability? The answer is no. On the other hand, two impurities located at both ends of the chains do lead
to the recovery of commensurability when vim is sufficiently high. This is just the same as with the rope
pinned either to one or two walls. In the first case, the rope hangs down freely. In the latter, the rope
pinned to the opposite walls can be expanded to some extent. In the harmonic approximation, the chain
always retains integrity, although when the interaction potential allows for dissociation, the chain may
rupture [23, 24, 26] rather than restore the C structure.
3. Two-dimensional finite clusters
The phenomenon of reentrant commensurability is not restricted to the above discussed simple 1D
model. It also appears in more realistic 2D systems in the ground state and at finite temperatures. To
demonstrate this, we have performed MC simulation in the canonical ensemble for finite clusters dec-
orated with impurities at the boundaries. The particles of A and B (impurity) have been placed in the
substrate field
vi (x, y) =−Vi {cos(q1r)+cos(q2r)+cos[(q1 −q2)r]} (3.1)
corresponding to the graphite basal plane, where Vi is the amplitude of the potential for the i th compo-
nent and q1 and q2 are the reciprocal lattice vectors of the graphite basal plane. The initial configurations
have been a hexagon of a perfectly ordered (
p
3×p
3)R30◦ structure consisting of NA z atoms of A and
NB atoms of B placed along the one edge (case I), the adjacent three edges (case II) or along all six edges
(case III) of the cluster. The interaction between the particles has been represented by the LJ(12,6) poten-
tial with the fixed σA A = 1.46a (a = 2.46 Å is the graphite lattice constant taken as the unit of length) and
εA A/kB = 170 K (assumed to be a unit of energy) andV ∗
A =VA/εA A = 0.12. In order to reduce the tendency
of B atoms towards aggregation, we have put εBB = 0.5εA A , while εAB has been assumed to be equal to
εA A . The parameters σBB =σAB and VB have been varied.
The simulations have been carried out for the systems with NA = 271 and NB = 11 (case I), NB = 33
(case II) and NB = 66 (case III). Besides, we have also performed some runs for a larger system corre-
sponding to case III, with NA = 811 and NB = 114 as well as with NA = 1189 and NB = 138. The simula-
tion has been performed at reduced temperatures T ∗ = kBT /εA A between 0.005 and 0.3. The formation
of domain walls in the system has been monitored using the order parameter [25, 27]
φ(r) = cos(q1r)+cos(q2r)+cos[(q1 −q2)r], (3.2)
and assuming that the atom is commensurate (incommensurate) when φ > 0 (φ É 0). We have calcu-
lated the average numbers of incommensurate atoms A and B neglecting the atoms located at the patch
boundaries, i.e., those with less than 5 nearest neighbors.
It has been found that for any value of σ∗
BB = σBB /a between 1.34 and 1.37 and for sufficiently low
and sufficiently high values of V ∗
B , the systems remain commensurate, while for the intermediate values
of V ∗
B , there appear networks of heavy walls separating commensurate domains.
In general, the results at low temperatures are qualitatively very similar to those obtained for 1D FK
model. In particular, we have observed the formation of IC structures in the cases I, II and III, though
the structure of the domain-wall networks is different in all cases. In general, the walls tend to assume
the orientations perpendicular to the edges decorated with the impurity atoms. Therefore, in case III, we
have found regular networks like that given in the leftmost part offigure 4. On the other hand, the systems
with only one or three adjacent edges decorated by impurities form irregular networks of domain walls
(see the middle and rightmost panels to figure 4).
The region of V ∗
B over which the IC phase is stable at low temperatures gradually decreases when
the size of impurity atoms increases, so that for sufficiently large impurity atoms, only the C phase occurs
independently of the magnitude ofV ∗
B (see figure 5). The calculations for the larger systemwith NA = 811
23605-5
A. Patrykiejew, S. Sokołowski
Figure 4. (Color online) The examples of snapshots for the system with σ∗
BB = 1.36 and V ∗
B = 0.12 at
T∗ = 0.02 corresponding to the three cases considered. Shaded circles represent B atoms. Open (filled)
circles correspond to the commensurate (incommensurate) A atoms.
and NB = 114 (case III) have shown that maximum value of σ∗
AB above which the C-IC transition does not
appear slightly increases with the system size. Moreover, the simulations performed for larger systems
have also demonstrated that at low temperatures the size of C domains does not change and only their
number increases. This result is quite similar to that obtained for 1D FK model for the chains of different
length. This is demonstrated by the snapshots given in the leftmost part of figure 4 and in figure 6.
The impurity induced changes of the surface stress are responsible for the C-IC transition, which
leads to the compression of the finite island. When the amplitude of the impurity surface potential, V ∗
B ,
is very low, the presence of impurities at the cluster boundaries does not appreciably affect the strain of
the atoms in the cluster. Therefore, the surface stress of the impurity decorated island does not match
differently from that of the pure system. For sufficiently large values of V ∗
B , the impurities are strongly
pinned over the surface potential minima as well as do not enlarge the strain in the island. Therefore,
for sufficiently low and high V ∗
B , the system retains the C structure. For intermediate values of V ∗
B , the
competing atom-atom and atom-external field interactions lead to a sufficiently large increase of a surface
stress in order to trigger the C-IC transition. One should note a similarity of the herein reported impurity
driven C-IC transition to the behavior of metal nanowires [28] and to the formation of IC phases in the
subjected to uniaxial compression self-assembledmonolayer gold nanoparticles supported on a fluid [29].
The results presented here describe only the low temperature behavior of the systems. The simulation
has demonstrated that upon an increase of temperature, the IC structure gradually changes and the C
structure is restored at temperatures still below the melting point of the cluster. The effects of thermal
excitations on the domain-wall networks will be discussed elsewhere.
Figure 5. (Color online) The phase diagram showing the locations (V ∗
B versus σ
∗
AB ) of C-IC transition in
the limit of T∗ → 0 in the systems with NA = 271 and different concentration of impurity atoms.
23605-6
Domain-walls in finite systems
Figure 6. (Color online) The example of snapshot for the system with NA = 1189 and NB = 138, σAB =
1.365 and V ∗
B = 0.08 recorded at T∗ = 0.01 K. Shaded circles represent B atoms. Open and filled circles
correspond to the commensurate and incommensurate A atoms.
4. Summary
In this work we have studied the impurity driven commensurate-incommensurate transitions in one-
and two-dimensional finite systems at zero temperature. In the case of one-dimensional finite chains,
we have considered the situations in which the impurity is located at one and two ends of the chain. It
has been shown that in both situations, the C-IC transition occurs when the amplitude of the external
field experienced by the impurity atoms falls into the region between the lower and upper threshold
values. These limiting values of vim depend upon the parameters characterizing the interaction between
the atoms in the main chain, the amplitude of external field acting on the main chain atoms and the
interaction between the main chain and the impurities. The number of solitons (domain walls) in the IC
structure is different for the chains with one and two ends decorated with the impurity atoms and also
depends upon the vim. This behavior is a consequence of different symmetry properties of the system.
In the case of two-dimensional finite clusters, a very similar C-IC transition has been found. In par-
ticular, we have observed that the transition occurs when only a part of the cluster boundary is covered
with a single row of impurity atoms. Also, by analogy to the results of one-dimensional FK model, the
transition occurs only between the lower and upper threshold values of vim.
It should be noted that the C-IC transition observed in two-dimensional systems occurs only when the
parameters entering the potential describing the interactions between A A, AB , and BB atoms are suit-
ably chosen. First of all, the components A and B should not exhibit the tendency towards mixing. Oth-
erwise, the impurity atoms B are likely to penetrate the patch and different scenarios are possible. One,
is the formation of a mixed commensurate phase. This has been found in the case of Ar-Kr finite patches
at sufficiently high temperatures [25], when the AB interaction potential parameters were obtained us-
ing the standard Lorentz-Berthelot mixing rules. The same mixture exhibits a different behavior at low
temperatures, and does undergo the C-IC transition, although with the domain walls preferentially made
of Ar atoms. Another requirement is that atoms A should tend to order into the C phase, while atoms
B should order into the IC phase. This imposes certain restrictions on the choice of σ∗
A A , σBB and the
values of V ∗
A and V ∗
B . In particular, σ
∗
A A and V ∗
A should assume the values ensuring that a pure A patch
is commensurate, but are likely to undergo the C-IC transition when the density exceeds the monolayer
capacity or when subjected to an external force due to the presence of impurity atoms along the patch
boundaries. Thus, V ∗
A cannot be too low or to high and the misfit between the sizes of surface lattice and
adsorbate atoms is rather small. A good example of such a system is krypton adsorbed on graphite [5]. In
fact, the parameters σ∗
A A and V ∗
A used here are rather close to those describing the krypton adsorbed on
the graphite basal plane [25]. On the other hand, the values of σBB and V ∗
B should favor the formation
of IC phase by pure component B . Thus, the misfit between the size of B atoms and the surface lattice
23605-7
A. Patrykiejew, S. Sokołowski
should be sufficiently large (it can be positive as well as negative) and V ∗
B should be sufficiently small.
However, there is still another requirement for the appearance of C-IC transition in finite patches. The AB
interaction should be sufficiently strong so that the layer of B atoms along the patch boundary is stable.
Otherwise, the atoms B would prefer to form clusters rather than stay at the patch boundary. Besides,
this condition is also necessary to exert a sufficiently large force upon the atoms A inside the finite patch
to trigger the C-IC transition. Usually, the C-IC transition occurs only when the film density exceeds the
density of a fully filled commensurate phase. For the C-IC transition to occur in finite patches, the stability
of the C phase should be low enough so that a rather small force exerted by a thin layer of impurity atoms
along the patch boundary is able to drive the C-IC transition. From our earlier studies of adsorption of
Ar-Kr, Ar-Xe and Kr-Xe mixtures [25, 30–32] on graphite it follows that only in the case of Ar-Kr mixture
the C-IC transition occurs at submonolayer densities in finite patches of adsorbed phase.
One expects that at higher temperatures, thermal excitations may considerably influence the effects
observed. This problem is currently under study and the results will be published elsewhere.
Acknowledgements
This work was supported by ERA under the Grant PIRSES 268498.
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Формування домен-стiнки в бiнарних наноскопiчних
скiнченних системах
А. Патрикєєв, С. Соколовскi
Вiддiл моделювання фiзико-хiмiчних процесiв, Унiверситет Марiї Кюрi-Склодовської, Люблiн, Польща
Використовуючи просту одновимiрну типову модель Френкеля-Конторової, ми показали, що скiнченнi
спiввимiрнi ланцюжки можуть зазнавати спiввимiрний-неспiввимiрний (C-IC) перехiд, якщо ланцюжок
забруднений iзольованими домiшками, дочепленими до кiнцiв ланцюжка. За допомогою методу Монте
Карло (MC) показано, що таке ж явище виникає в двовомiрних системах з домiшками, розмiщеними на
периферiї скiнчених спiввимiрних кластерiв.
Ключовi слова: бiнарна сумiш, спiввимiрнi-неспiввимiрнi переходи, симуляцiї Монте Карло, скiнченнi
системи, модель Френкеля-Конторової
23605-9
http://dx.doi.org/10.1073/pnas.1101630108
http://dx.doi.org/10.5488/CMP.15.23601
http://dx.doi.org/10.1063/1.3699330
http://dx.doi.org/10.1088/0953-8984/25/1/015001
Introduction
One-dimensional Frenkel-Kontorova model
Two-dimensional finite clusters
Summary
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