Nonlinear rotor dynamics of 2D molecular array: topology reconstruction
Molecular layers with rotational degrees of freedom and quadrupolar interaction between linear molecules are investigated theoretically. We found earlier that alternative orientation of the molecules along and perpendicular to an axis of the rectangular lattice is preferable. Here we find the integr...
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irk-123456789-1082272016-11-01T03:02:46Z Nonlinear rotor dynamics of 2D molecular array: topology reconstruction Lykah, V.A. Syrkin, E.S. Section F. Nonlinear Dynamics and Chaos Molecular layers with rotational degrees of freedom and quadrupolar interaction between linear molecules are investigated theoretically. We found earlier that alternative orientation of the molecules along and perpendicular to an axis of the rectangular lattice is preferable. Here we find the integral of motion and give the topology analysis of the possible dynamical phases and special points in the long-wave limit. We find the strong anisotropy in the angle space: directions of easy excitation ("valleys'') exist. We show the potential relief reconstruction in dependence on the adsorbed lattice anisotropy. Теоретически исследованы молекулярные слои с вращательными степенями свободы и квадрупольным взаимодействием между линейными молекулами. Мы нашли ранее, что предпочтительным является альтернативное упорядочение молекул с ориентацией вдоль и перпендикулярно оси прямоугольной решетки. Выведены уравнения движения, найден их интеграл и проведен топологический анализ возможных динамических фаз и особых точек в длинноволновом пределе. Показана сильная анизотропия в плоскости углов: существуют направления легкого возбуждения ("долины''). Мы показываем реконструкции потенциального рельефа в зависимости от анизотропии адсорбированной решетки. Теоретично досліджені молекулярні шари з обертальними ступенями свободи і квадрупольною взаємодією між лінійними молекулами. Ми знайшли раніше, що кращим є альтернативне впорядкування молекул з орієнтацією вздовж і перпендикулярно осі прямокутної гратки. Виведено рівняння руху, знайдено їх інтеграл і проведено топологічний аналіз можливих динамічних фаз і особливих точок для довгохвильової межі. Показано сильну анізотропію в площині кутів: існують напрями легкого збудження ("долини''). Ми показуємо реконструкцію потенційного рельєфу в залежності від анізотропії адсорбованої решітки. 2012 Article Nonlinear rotor dynamics of 2D molecular array: topology reconstruction / V.A. Lykah, E.S. Syrkin // Вопросы атомной науки и техники. — 2012. — № 1. — С. 346-350. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 63.22.+m; 65.40.Ba; 79.60.Dp http://dspace.nbuv.gov.ua/handle/123456789/108227 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| topic |
Section F. Nonlinear Dynamics and Chaos Section F. Nonlinear Dynamics and Chaos |
| spellingShingle |
Section F. Nonlinear Dynamics and Chaos Section F. Nonlinear Dynamics and Chaos Lykah, V.A. Syrkin, E.S. Nonlinear rotor dynamics of 2D molecular array: topology reconstruction Вопросы атомной науки и техники |
| description |
Molecular layers with rotational degrees of freedom and quadrupolar interaction between linear molecules are investigated theoretically. We found earlier that alternative orientation of the molecules along and perpendicular to an axis of the rectangular lattice is preferable. Here we find the integral of motion and give the topology analysis of the possible dynamical phases and special points in the long-wave limit. We find the strong anisotropy in the angle space: directions of easy excitation ("valleys'') exist. We show the potential relief reconstruction in dependence on the adsorbed lattice anisotropy. |
| format |
Article |
| author |
Lykah, V.A. Syrkin, E.S. |
| author_facet |
Lykah, V.A. Syrkin, E.S. |
| author_sort |
Lykah, V.A. |
| title |
Nonlinear rotor dynamics of 2D molecular array: topology reconstruction |
| title_short |
Nonlinear rotor dynamics of 2D molecular array: topology reconstruction |
| title_full |
Nonlinear rotor dynamics of 2D molecular array: topology reconstruction |
| title_fullStr |
Nonlinear rotor dynamics of 2D molecular array: topology reconstruction |
| title_full_unstemmed |
Nonlinear rotor dynamics of 2D molecular array: topology reconstruction |
| title_sort |
nonlinear rotor dynamics of 2d molecular array: topology reconstruction |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2012 |
| topic_facet |
Section F. Nonlinear Dynamics and Chaos |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/108227 |
| citation_txt |
Nonlinear rotor dynamics of 2D molecular array: topology reconstruction / V.A. Lykah, E.S. Syrkin // Вопросы атомной науки и техники. — 2012. — № 1. — С. 346-350. — Бібліогр.: 13 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT lykahva nonlinearrotordynamicsof2dmoleculararraytopologyreconstruction AT syrkines nonlinearrotordynamicsof2dmoleculararraytopologyreconstruction |
| first_indexed |
2025-07-07T21:10:18Z |
| last_indexed |
2025-07-07T21:10:18Z |
| _version_ |
1837024012563120128 |
| fulltext |
NONLINEAR ROTOR DYNAMICS OF 2D MOLECULAR
ARRAY: TOPOLOGY RECONSTRUCTION
V.A. Lykah 1 and E.S. Syrkin 2∗
1National Technic University “Kharkov Polytechnic Institute”, 61002, Kharkov, Ukraine
2B.Verkin Institute for Low Temperature Physics and Engeneering NAN, 61103, Kharkov, Ukraine
(Received November 4, 2011)
Molecular layers with rotational degrees of freedom and quadrupolar interaction between linear molecules are inves-
tigated theoretically. We found earlier that alternative orientation of the molecules along and perpendicular to an
axis of the rectangular lattice is preferable. Here we find the integral of motion and give the topology analysis of
the possible dynamical phases and special points in the long-wave limit. We find the strong anisotropy in the angle
space: directions of easy excitation (“valleys”) exist. We show the potential relief reconstruction in dependence on
the adsorbed lattice anisotropy.
PACS: 63.22.+m; 65.40.Ba; 79.60.Dp
1. INTRODUCTION
Low dimensional systems are objects of a great inter-
est either as models or as description of real objects.
1D and 2D models are a necessary stage of inves-
tigation of dynamics and thermodynamics of more
complex systems: crystals, atomic and molecular lat-
tices [1–3]. 1D chain models are applied to the de-
scription of linear lattice dynamics and thermody-
namics of molecular cryocrystals [4, 5] (that can be
generalized into 3D systems [4, 6]) in order to inves-
tigate nonlinear dynamics [7] and thermoconductiv-
ity [8]. Real objects where 1D and 2D models can be
applied to are adsorbed structures [9] or crystals with
low-dimensional motives. Some approximations are
necessary to simplify system’s description because of
the complexity of models even for 1D molecular chain.
For linear models such approximations are a force ma-
trix (harmonic expansion near equilibrium position)
instead of parameters of interaction between mole-
cules, and given parallel ordering of molecules [4–6].
In nonlinear consideration such approximations are
a model potential and 1D rotation [7–9]. In arti-
cles [10, 11] the equilibrium state, oscillation spec-
trum and density of states were found for 1D and 2D
molecular array with quadrupolar interaction.
In this paper we use approximations of one degree
of freedom for every molecule as in [7–9] and very stiff
translation potential as in [7,8], so translation vibra-
tions can be neglected. In this model for 1D [10] and
2D [11] adsorbed molecular systems, the structures
with alternative orientation ordering are stable and
are observed experimentally [9]. The article is con-
structed as follows. In the second section the poten-
tial energy is derived for the orientation interaction
of the molecules placed in a rectangular lattice on
an adsorbing plane and rotated in this plane. In the
third section the dynamic equations of the molecular
orientation oscillations are developed. In conclusion
we consider the whole picture of oscillations.
2. ENERGIES AND EQUATIONS OF
MOTION
In this paper we consider nonlinear rotational dynam-
ics of molecular 2D layer with realistic quadrupolar
interaction between two linear molecules [4].
There are another several contributions to the in-
teraction between the molecules. These contributions
have the same structure of terms [4] but their coeffi-
cients depend on the distance between molecules and
molecules’ parameters. Therefore, the model with
quadrupolar interaction between the molecules is the
simplest one but it reflects the most essential features
of real interactions and classical dynamics of the lin-
ear adsorbed molecules.
Let us consider the most realistic situation when
the adsorbed molecules rotate in the plane of the
substrate. Then the potential energy of the orien-
tational part of interaction between the molecules in
the 2D layer should be written [11]. Several sym-
metric orderings that can provide minimum of the
orientation potential energy in the adsorbed molecu-
lar layer with rectangular lattice were considered in
the work [11]. The absolute minimum of the poten-
tial energy is reached for the alternative ordering of
the molecules for both directions in the layer:
φ2n,2m = 0;φ2n,2m+1 = π/2;φ2n+1,2m = π/2;
or φ2n,2m = π/2;φ2n,2m+1 = 0;φ2n+1,2m = 0.
(1)
It means that each molecule has the nearest neighbors
that are oriented perpendicularly to it. Stability of
this structure was confirmed by investigation of small
∗Corresponding author E-mail address: syrkin@ilt.kharkov.ua
346 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 346-350.
oscillations [11]. It is important that these structures
are observed in the experiments [9].
Let us consider oscillations of the adsorbed mole-
cular layer. Lagrangian of the system is L = K − U
where U and K = 1
2
∑
Ji,j φ̇
2
i,j are the total potential
and kinetic energies that are obtained as the sum
of the energies for the {i, j}-th molecule with the
moment of inertia Ji,j = J0 and angle velocity φ̇i,j .
Then the system of Lagrange equations of the motion
is [11]:
J0φ̈i,j = 2Γ{a sin 2φi,j
+ b[sin 2(φi,j − φi+1,j) + sin 2(φi,j − φi−1,j)]
+ c[sin 2(φi,j + φi+1,j) + sin 2(φi,j + φi−1,j)]}
+ 2Γρ−5{−a sin 2φi,j
+ b[sin 2(φi,j − φi,j+1) + sin 2(φi,j − φi,j−1)]
+ c[sin 2(φi,j + φi,j+1) + sin 2(φi,j + φi,j−1)]}.
(2)
Here φi,j is the angle between the molecular axis in
the site i, j and axis Ox of rectangular lattice of the
molecules. The parameters of the interaction are:
Γ =
3Q2
4R5
0a
; a0 =
3
4
; a =
5
4
; b =
3
8
; c =
35
8
; ρ =
R0b
R0a
(3)
where Q is the quadrupolar moment of a molecule, ρ
is the parameter of anisotropy, the lattice constants
are R0a and R0b. Condition ρ ≥ 1 is applied only
to meet the requirement of crystallography where in-
equality R0b ≥ R0a is generally used.
The obtained system of equations (2) is nonlinear
and differential-difference. It is naturally to begin its
investigation starting with the linear approximation
for 1D [10] and for 2D rectangular [11] lattices. Long-
wave limit for arbitrary amplitudes and nonlinearity
for 1D case was investigated in [12]. We will consider
the long-wave limit for the 2D rectangular lattice.
Let us rewrite equations (2) for two sublattices.
They are caused by separating the molecules into two
subsystems in equilibrium state for even (φ2m,2n) and
odd (ψ2m±1,2n or ψ2m,2n±1) sites. Then the sites of
the same type are equivalent and the set of equations
can be transformed into a system of differential equa-
tions. Then we introduce more convenient variables
m = φ− ψ; p = φ+ ψ (4)
and dimensionless time t → τ = tω0; ω2
0 = Γ/J0.
Then the system of the equations can be written as
p̈− 8{(1 − 1
ρ5
)a cosm sin p+ (1 +
1
ρ5
)c sin 2p} = 0;
m̈− 8{(1 − 1
ρ5
)a cos p sinm+ (1 +
1
ρ5
)b sin 2m} = 0.
(5)
3. TOPOLOGY. SPECIAL POINTS
Very important information about dynamics of the
system can be obtained from the special (stationary
or fixed) points of the equations that are determined
by conditions ψ̇ = 0, ψ̈ = 0 and φ̇ = 0, φ̈ = 0. Then
the set of equations (5) can be rewritten as:
sin p [a(1 − 1
ρ5
) cosm+ 2c(1 +
1
ρ5
) cos p] = 0;
sinm [a(1 − 1
ρ5
) cos p+ 2b(1 +
1
ρ5
) cosm] = 0,
(6)
Solutions of equations (6) can be obtained by solving
several simple systems. The first simple system is
cosm = 0; cos p = 0. (7)
Its solution m1 = π/2 + πj; p1 = π/2 + πn coin-
cides with conditions (1) of energy minimum of the
molecular array. It coincides with the chain case.
The second simple system is
a(1 − 1
ρ5 ) cosm+ 2c(1 + 1
ρ5 ) cos p = 0;
sinm = 0.
(8)
It yields solutions
m2 = πj; p2 = arccos(± a(ρ5 − 1)
2c(ρ5 + 1)
); (9)
where sign + is for j = 2n + 1 and sign - is for
j = 2n. As we can see below these stationary points
correspond to low saddle points. As the system ap-
proaches a square lattice of the molecules (ρ → 1)
the deviation from the values p = π/2 + πj vanishes.
The third simple system of equations is
sin p = 0; sinm = 0. (10)
Its solutions are m3 = πj; p3 = πn. When j + n =
0,±2,±4... the solutions correspond to peaks of the
effective potential. The solutions of the type j + n =
2i + 1 correspond to the high saddle points. These
solutions coincide for 1D and 2D lattices.
The forth simple system of equations
sin p = 0;
a(1 − 1
ρ5 ) cos p+ 2b(1 + 1
ρ5 ) cosm = 0; (11)
has solutions
p4 = πj; m4 = arccos(± a(ρ5 − 1)
2b(ρ5 + 1)
). (12)
The system (11) is not satisfied for the chain (ρ→ ∞)
because of | cosm4| = a/2b > 1. The rectangular lat-
tice yields new solutions at condition | cosm| ≤ 1 or
ρbs ≤ ρ ≤ ρbg;
ρ5
bs =
a− 2b
a+ 2b
=
1
4
; ρ5
bg =
a+ 2b
a− 2b
= 4.
(13)
The rectangular array of the molecules has solutions
on condition | cosm| = 1, which coincide with (10). It
means that at conditions ρ = ρbg or ρ = ρbs the peaks
and the high saddle points of the effective potential
change their behavior.
Therefore all special points of the system of equa-
tions could be obtained from the solutions of the more
simple systems. This is the complete set of the special
points of the system.
347
4. INTEGRAL OF MOTION AND PHASES
We obtain integral of the rotational motion of the
molecular array: Wef = Wk + Wp. It includes “ki-
netic” Wk and “potential” Wp contributions:
Wk =
1
2
(ṁ2 + ṗ2);
Wp(m, p) = 4{(1 − 1
ρ5
)2a cosm cosp
+ (1 +
1
ρ5
)[b cos 2m+ c cos 2p]}.
(14)
Potential relief for Wp over the space of angles is
shown in Fig. 1. Analysis of linear oscillations with
arbitrary dispersion [10,11] has demonstrated strong
dependence of the relief on dispersion (wave number)
but for not large k the relief do not change strongly.
Existence of only one motion integral means that the
system is not integrable, nevertheless it is possible
to obtain some qualitative consequences about its
behavior. Motion of a molecular chain and rectan-
gular array is stochastic in space of angles because
of complex potential [13]. An excitation energy is
determined by temperature or by diffracting parti-
cles (electrons, neutrons etc.) in structure experi-
ments. We can point out several intervals of energy
with qualitatively different character of motion of the
molecules (dynamical phases) that can be related to
different structural phases.
1) The phase I (Wmin ≤Wef < WSL) is a region
of finite rotating oscillations of the molecules near
equilibrium positions m1, p1 (7):
Wmin = Wp(m1, p1) = −4(1 + 1
ρ5 )(b + c). (15)
They are ellipses’ centers at valley bottoms in
Fig. 1a − e. The molecular configuration is shown
in Fig. 1f . A structural data show the rotational
arrangement. High boundary of this region is energy
of lower separatrix WSL.
2) Wef �WSL is a narrow region of motion with
energy of the low separatrix that coincides with en-
ergy of the low saddle points m2, p2 in points (8):
WSL = 4{− a2(ρ5 − 1)2
2c(ρ5 + 1)ρ5
+ (1 +
1
ρ5
)(b − c)}; (16)
The low separatrix separates a finite and infinite mo-
tion (rotation) of the molecules in the chain or 2D
array. All thermodynamic characteristics have a pe-
culiarity because of high density of states [1] and the
order-disorder phase transition at this energy.
3) The phase II (WSL < Wef < WSH) is a region
of finite variation of p and infinite variation of m be-
tween the lower separatrixWSL and higher separatrix
WSH . A low-energy excitations along p � π/2 + πj
(’valleys’ on the potential) exist for the chain or
2D array. Excitations along ’valleys’ do not destroy
strong correlation between rotating molecules so that
p � π/2+πj along j-th ’valleys’. The molecular mo-
tion is limited in the angle space by the higher sep-
aratrix curves. Structural data for this phase show
rotational disorder (melting).
M M
M M
Level curves:
a b
c d
e
,f
V
V
V
V
V
V
V
V
V
Fig. 1. The effective potential Wp (14) relief
(normalized height) as a counter plot function of
coordinates m = φ − ψ (horizontal) and p = φ + ψ
(vertical). Variables change in range [0; +3π]. High
density of equidistant contours corresponds to high
gradient. The letter V points out ’valley’ of Wp.
The parameter of the lattice anisotropy takes values
ρ5 = 10; 4; 3; 2; 1 at a − e panels. Panel f shows
elementary cell of the lattice
4) Wef � WSH is a narrow region of motion
with energy of high separatrix that coincides with
energy of higher saddle point m3 = πj, p3 = πn at
j + n = 2i+ 1 (10):
WSH = 4{−(1 − 1
ρ5
)2a+ (1 +
1
ρ5
)[b + c]}. (17)
The high separatrix separates a finite and infinite mo-
tion (rotation) over p coordinate. All thermodynamic
characteristics have a peculiarity at this energy.
5) The phase III (WSH < Wef < WT ) is a region
of infinite variation of p and m coordinates but some
regions (islands) of angles near the peaks of the ef-
fective potential WT are forbidden. At these energies
the transitions between ’valleys’ are possible when
the system passes above the high separatrix.
6) Wef � WT is narrow region of motion with
mean energy near the tops m3 = πj, p3 = πn at
348
j + n = 2i of the effective potential (10):
WT = 4{(1 − 1
ρ5
)2a+ (1 +
1
ρ5
)[b+ c]}. (18)
At this energy all thermodynamic characteristics usu-
ally have a peculiarity because of high density of
states. The system undergoes the phase transition
into completely disordered phase.
7) The phase IV (WT ≤Wef ) corresponds to com-
pletely disordered phase.
All these stationary points and corresponding en-
ergy intervals give qualitatively the same behaviour
for the chain and 2D array of the adsorbed molecules.
In the chain case (ρ→ ∞), the fine structures of the
valleys, top and saddle points as well as correspond-
ing molecules’ orderings are shown in figures in [12].
5. THE TOPOLOGY RECONSTRUCTION
Significant changes in the topology appear due to the
extreme points connected with solutions p4,m4 (12)
of the forth simple system (11). These solutions exist
at ρbs ≤ ρ ≤ ρbg (13) and give two kinds of the new
saddle points with energy
WN = Wp(p4;m4)
= 4{− a2(ρ5 − 1)2
2b(ρ5 + 1)ρ5
+ (1 +
1
ρ5
)[−b+ c]}. (19)
When parameter ρ decreases, these solutions arise
at ρ = ρbg and the new saddle points coincide with
the old saddle points (17)
WN = WSH = 4
−6ab+ 2ac
a+ 2b
.
As ρ decreases, pairs of new saddle points split
around a position (m = −πj, p = π) (m = π, p = π),
(m = 0, p = π). The old saddle points turn out into
peaks meanwhile the old peaks keep their positions
and role. At ρ = ρbs either positions or energies
of the new saddle points coincide with the old top
points values. Further system evolution goes with
the old peaks and saddle points that keep their po-
sitions but change their functions. In the limit case
ρ = 1 the old and the new local peaks have equal
height WNp = WT . The new saddle points come
to the middle position between the old and the new
peaks.
To describe these changes of the system graph-
ically, we find the effective potential cross-sections
along the lines of the peaks and higher saddle points
p = 0;π. One of them is shown in top panel in Fig. 2.
The first equation in the second simple system (8)
is also the equation of the valley bottom. So one can
exclude the variable p. The cross-sections along the
valley bottom of the effective potential (14)
Wp(m) = (20)
= 4{− a2(ρ5 − 1)2
2c(ρ5 + 1)ρ5
cos2m+ (1 +
1
ρ5
)[b cos 2m− c]}
are shown in bottom panel in Fig. 2. We can see that
the bottom part of the effective potential changes
only quantitatively.
1
4
4
1
2
3
10
10
3
2
Fig. 2. Potential relief Wp (14) cross-sections at
different values of lattice anisotropy parameter ρ
(the same values ρ5 as in Fig. 1 are denoted at right
edge of each panel). Top panel shows cross-section
along top and high saddle points of the potential.
Bottom panel shows cross-section along bottom of
the valley
6. CONCLUSIONS
We considered the molecular adsorbed layer of linear
molecules with quadrupolar interaction. The mole-
cules have only one rotational degree of freedom in
the plane of adsorbing surface. In this model, the
alternative ordering of the molecules along and per-
pendicular to the axes of the rectangular unit cell is
most stable. This structure has two physically differ-
ent sites per unit cell (two different molecules’ orien-
tation) and doubled lattice constants. The theoret-
ically found equilibrium state coincides with exper-
imentally observed structures. Equations of motion
are derived in long-wave limit. Their motion integral
has been obtained. The topological analysis of the
system gives the following results.
In the bottom the topology of the effective poten-
tial completely coincides with the case of the molecu-
lar chain. The difference between 1D and 2D molecu-
lar structures consists in the dependence on the ratio
of the lattice constants in 2D case. The square molec-
ular lattice has the most stable equilibrium structure
(wide interval of the phase I in bottom at Fig. 2). As
the ratio of the lattice constants is growing the spec-
trum is softening and the chain limit comes quickly.
349
The top part of the effective potential consider-
ably changes its topology at ρ5 = 4, as ratio of the
lattice constants goes to 1 starting from the chain
case (ρ � 1). The square molecular lattice has i)
the most drastic changes of the topology, ii) the most
wide energy interval of the phase II and the most nar-
row interval of the phase III (see top panel in Fig. 2).
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