Equilibrium configurations of director in a planar nematic cell with one spatially modulated surface
We study two-dimensional equilibrium configurations of nematic liquid crystal (NLC) director in a cell confined between two parallel surfaces: a planar surface and a spatially modulated one. The relief of the modulated surface is described by a smooth periodic sine-like function. The director easy...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Equilibrium configurations of director in a planar nematic cell with one spatially modulated surface / M.F. Ledney, O.S. Tarnavskyy, A.I. Lesiuk, V.Yu. Reshetnyak // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33604: 1–11. — Бібліогр.: 27 назв. — англ. |
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irk-123456789-1562102019-06-19T01:25:13Z Equilibrium configurations of director in a planar nematic cell with one spatially modulated surface Ledney, M.F. Tarnavskyy, O.S. Lesiuk, A.I. Reshetnyak, V.Yu. We study two-dimensional equilibrium configurations of nematic liquid crystal (NLC) director in a cell confined between two parallel surfaces: a planar surface and a spatially modulated one. The relief of the modulated surface is described by a smooth periodic sine-like function. The director easy axis orientation is homeotropic at one of the bounding surfaces and is planar at the other one. Strong NLC anchoring with both surfaces is assumed. We consider the case where disclination lines occur in the bulk of NLC strictly above local extrema of the modulated surface. These disclination lines run along the crests and troughs of the relief waves. In the approximation of planar director deformations we obtain analytical expressions describing a director distribution in the bulk of the cell. Equilibrium distances from disclination lines to the modulated surface are calculated and their dependences on the cell thickness and the period and depth of the surface relief are studied. It is shown that the distances from disclination lines to the modulated surface decrease as the depth of the relief increases. Дослiджено двовимiрнi рiвноважнi конфiгурацiї директора нематичного рiдкого кристалу (НРК) в комiрцi обмеженiй двома паралельними поверхнями: планарною i просторово модульованою. Форма рельєфу модульованої поверхнi описується гладкою перiодичною синусоподiбною функцiєю. Розглянуто випадок наявностi дисклiнацiйних лiнiй в об’ємi НРК строго над локальними екстремумами модульованої поверхнi, що проходять вздовж гребенiв i западин хвиль рельєфу. Виникнення дисклiнацiйних лiнiй зумовлене рiзним типом орiєнтацiї легкої осi директора на обмежуючих поверхнях комiрки (гомеотропна орiєнтацiя на однiй з обмежуючих поверхонь i планарна на iншiй) та жорстким зчепленням НРК з поверхнями. В наближеннi плоских деформацiй директора отримано аналiтичнi вирази для розподiлу директора в об’ємi комiрки. Розраховано рiвноважнi вiдстанi вiд дисклiнацiйних лiнiй до модульованої поверхнi та знайдено їх залежнiсть вiд параметрiв комiрки (перiоду та глибини хвиль рельєфу), а також вiд товщини комiрки. Показано, що вiдстанi вiд дисклiнацiйних лiнiй до модульованої поверхнi зменшуються зi зростанням глибини рельєфу. 2016 Article Equilibrium configurations of director in a planar nematic cell with one spatially modulated surface / M.F. Ledney, O.S. Tarnavskyy, A.I. Lesiuk, V.Yu. Reshetnyak // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33604: 1–11. — Бібліогр.: 27 назв. — англ. 1607-324X PACS: 61.30.-v, 61.30.Hn, 61.30.Jf DOI:10.5488/CMP.19.33604 arXiv:1609.04980 http://dspace.nbuv.gov.ua/handle/123456789/156210 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We study two-dimensional equilibrium configurations of nematic liquid crystal (NLC) director in a cell confined
between two parallel surfaces: a planar surface and a spatially modulated one. The relief of the modulated
surface is described by a smooth periodic sine-like function. The director easy axis orientation is homeotropic
at one of the bounding surfaces and is planar at the other one. Strong NLC anchoring with both surfaces is
assumed. We consider the case where disclination lines occur in the bulk of NLC strictly above local extrema of
the modulated surface. These disclination lines run along the crests and troughs of the relief waves. In the approximation of planar director deformations we obtain analytical expressions describing a director distribution
in the bulk of the cell. Equilibrium distances from disclination lines to the modulated surface are calculated and
their dependences on the cell thickness and the period and depth of the surface relief are studied. It is shown
that the distances from disclination lines to the modulated surface decrease as the depth of the relief increases. |
| format |
Article |
| author |
Ledney, M.F. Tarnavskyy, O.S. Lesiuk, A.I. Reshetnyak, V.Yu. |
| spellingShingle |
Ledney, M.F. Tarnavskyy, O.S. Lesiuk, A.I. Reshetnyak, V.Yu. Equilibrium configurations of director in a planar nematic cell with one spatially modulated surface Condensed Matter Physics |
| author_facet |
Ledney, M.F. Tarnavskyy, O.S. Lesiuk, A.I. Reshetnyak, V.Yu. |
| author_sort |
Ledney, M.F. |
| title |
Equilibrium configurations of director in a planar nematic cell with one spatially modulated surface |
| title_short |
Equilibrium configurations of director in a planar nematic cell with one spatially modulated surface |
| title_full |
Equilibrium configurations of director in a planar nematic cell with one spatially modulated surface |
| title_fullStr |
Equilibrium configurations of director in a planar nematic cell with one spatially modulated surface |
| title_full_unstemmed |
Equilibrium configurations of director in a planar nematic cell with one spatially modulated surface |
| title_sort |
equilibrium configurations of director in a planar nematic cell with one spatially modulated surface |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/156210 |
| citation_txt |
Equilibrium configurations of director in a planar nematic cell with one spatially modulated surface / M.F. Ledney, O.S. Tarnavskyy, A.I. Lesiuk, V.Yu. Reshetnyak // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33604: 1–11. — Бібліогр.: 27 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT ledneymf equilibriumconfigurationsofdirectorinaplanarnematiccellwithonespatiallymodulatedsurface AT tarnavskyyos equilibriumconfigurationsofdirectorinaplanarnematiccellwithonespatiallymodulatedsurface AT lesiukai equilibriumconfigurationsofdirectorinaplanarnematiccellwithonespatiallymodulatedsurface AT reshetnyakvyu equilibriumconfigurationsofdirectorinaplanarnematiccellwithonespatiallymodulatedsurface |
| first_indexed |
2025-07-14T08:35:42Z |
| last_indexed |
2025-07-14T08:35:42Z |
| _version_ |
1837610715722022912 |
| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 3, 33604: 1–11
DOI: 10.5488/CMP.19.33604
http://www.icmp.lviv.ua/journal
Equilibrium configurations of director in a planar
nematic cell with one spatially modulated surface
M.F. Ledney, O.S. Tarnavskyy, A.I. Lesiuk, V.Yu. Reshetnyak
Physics Department, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine
Received February 18, 2016, in final form May 31, 2016
We study two-dimensional equilibrium configurations of nematic liquid crystal (NLC) director in a cell confined
between two parallel surfaces: a planar surface and a spatially modulated one. The relief of the modulated
surface is described by a smooth periodic sine-like function. The director easy axis orientation is homeotropic
at one of the bounding surfaces and is planar at the other one. Strong NLC anchoring with both surfaces is
assumed. We consider the case where disclination lines occur in the bulk of NLC strictly above local extrema of
the modulated surface. These disclination lines run along the crests and troughs of the relief waves. In the ap-
proximation of planar director deformations we obtain analytical expressions describing a director distribution
in the bulk of the cell. Equilibrium distances from disclination lines to the modulated surface are calculated and
their dependences on the cell thickness and the period and depth of the surface relief are studied. It is shown
that the distances from disclination lines to the modulated surface decrease as the depth of the relief increases.
Key words: nematic liquid crystal, spatially modulated surface, topological defects, conformal mapping
PACS: 61.30.-v, 61.30.Hn, 61.30.Jf
1. Introduction
Over the recent years the number of researches in liquid crystals (LCs) with the aim to design low
power LC displays with high resolution has been increasing very fast. In this field, bistable LC systems
have proved to be very promising. A planar nematic liquid crystal (NLC) cell with at least one spatially
modulated boundary surface that has the shape of a smooth or broken periodic line is one of the well-
known bistable LC systems [1–4]. Such systems are used in ZBD technology [5–8], which allows one to
create ergonomic display devices. While the cell of standard LC display requires a constant power supply,
bistable LC systems only need the power to switch between different optic states caused by the existence
of two or more stable director configurations. In the operation modes that do not require a frequent
change of images, the employment of bistable LC systems as basic components of display devices is espe-
cially profitable since this reduces the total power consumption by such devices.
Themodelling of display devices based on bistable LC systems aiming at finding the optimal operating
mode requires the knowledge of the director distribution in each stable state. The director stable states
between which the switching is performed can have structure defects of disclination type caused by the
relief surface, flexopolarization, etc.
In most theoretical studies of NLC director configurations, only semi-infinite NLC cells bounded by
one spatially modulated surface were considered due to the complexity of calculations [9–15]. Many
works studied the LC director distribution only in a thin NLC layer adjacent to the boundary surface.
In particular, the peculiarities of NLC elastic interaction with spatially modulated surface of semi-infinite
NLC cell were studied in [9, 10, 14]. The interaction of NLC with the relief surface was studied theoreti-
cally in the case of two-dimensional spatial modulation of the surface [11] and in the case of twisted LC
director structure in the cell [12, 13]. The technique of the director structure calculation in semi-infinite
NLC above the surface with the profile of periodically broken line was proposed in [15]. In [3], possible
stable LC director configurations between a planar and a smooth spatially modulated substrates were
© M.F. Ledney, O.S. Tarnavskyy, A.I. Lesiuk, V.Yu. Reshetnyak, 2016 33604-1
http://dx.doi.org/10.5488/CMP.19.33604
http://www.icmp.lviv.ua/journal
M.F. Ledney, O.S. Tarnavskyy, A.I. Lesiuk, V.Yu. Reshetnyak
experimentally studied. Stable director states and switching between them were studied in an NLC cell
with one grooved substrate [16]. The authors of [17, 18] carried out numerical calculations of stable di-
rector configurations between two sawtooth substrates. Two-dimensional director configurations in NLC
cell with one sawtooth substrate were obtained [19, 20], while taking into account the finiteness of the
cell thickness.
In general, there are many studies on the director configuration states. However, the finiteness of
the cell thickness was considered only in some particular cases of sawtooth substrate relief. Besides that,
there is no analytical description of LC director configuration which would take into account the finite-
ness of cell thickness and the presence of defects in a cell with smooth periodic surface relief. In the
present paper, we propose a technique which allows one to calculate two-dimensional equilibrium con-
figurations of the NLC director in a planar nematic cell of finite thickness. The cell is bounded by two
surfaces one of which is planar and the other one is smooth and spatially modulated. The anchoring of
NLC with bounding surfaces is assumed to be strong. If the orientation of director easy axis at one bound-
ing surface is homeotropic and at the other one it is planar, then disclination lines can occur in the bulk
of NLC [3]. In this paper we calculate equilibrium LC director configurations with the structure defects
in the bulk. The relief of the modulated surface is described by a smooth periodic sine-like function. This
function is not arbitrarily taken, but is chosen in a special way for the given values of the cell thickness,
the period and depth of the relief. We find equilibrium distances from disclination lines to the relief sur-
face and study their dependence on the cell thickness and on the period and depth of the surface relief.
It is worth noting that the knowledge of the defects location is necessary in nanoparticles trapping and
arrangement by topological defect systems in LC cells [21–25].
2. The problem geometry and basic equations
Let us consider a planar cell of NLC which is bounded in the direction of the coordinate axis OY by
two surfaces: a planar surface and spatially modulated one (see figure 1). We assume that the shape of
the modulated surface relief is described by a smooth sine-like function periodic in the coordinate X with
the period λ. The NLC anchoring energy is supposed to be infinitely large. The director easy axis at the
planar surface is assumed to be directed along the axis OX , while at the spatially modulated surface it is
homeotropic, i.e., perpendicular to the surface.
We consider the case where under the stated conditions for the director at bounding surfaces, the
director field in the bulk of the cell is not continuous. Therefore, the disclination lines with the strength
“±1/2” occur strictly above the local maxima andminima of the spatially modulated surface. These discli-
NLC
director
X
Y
O
defects
L
A D
B
λ
G NLC
b
b
b
b
b
Figure 1. The problem geometry: L is the cell thickness, λ, D are the spatial period and the depth of the
modulated surface relief, A, B are the distances from the disclination lines to the cell relief surface, G is
the region in which, for symmetry reasons, the angle θ(X ,Y ) between the director and OX -axis should
be found.
33604-2
Equilibrium configurations of director
nation lines are perpendicular to XOY plane and, correspondingly, parallel to the relief waves (see fig-
ure 1). In a general case, disclination lines can adopt a zigzag form. Nevertheless, as was shown in [26],
if elastic constants are equal K1 = K2 = K3 = K , then the disclination lines are straight. Here and further
on, we assume that the one elastic constant approximation holds.
The deformations of the director field in the cell are assumed to be planar, that is, lying in the
plane XOY . Since the system is homogeneous in the direction of OZ -axis, we write the NLC director
in the form
n = i ·cosθ(X ,Y )+ j ·sinθ(X ,Y ), (1)
where i, j are unit vectors of the axes of Cartesian coordinate system. The twisting deformations (n·rotn =
0) are not involved in this case and the free energy of the NLC cell is equal to F =
K
2
∫
V
(∇θ)2 dV . Then,
the director distribution in the bulk of NLC is described by Lapalce’s equation ∆X ,Y θ = 0. For symmetry
reasons, we seek a solution θ(X ,Y ) of Laplace’s equation in the region G marked by a dashed line in
figure 1. It is evident that at the boundary of region G, the unknown function θ(x, y) should satisfy the
following conditions (see figure 2):
θS =
−π/2, if x =−π/2, y1 É y < ã,
0, if x =−π/2, ã < y É h,
0, if −π/2É x Éπ/2, y = h,
0, if x =π/2, b̃ < y É h,
−π/2, if x =π/2, y2 É y < b̃,
α−π/2, if (x, y) ∈Γ,
(2)
where x = kX , y = kY are dimensionless coordinates, k = 2π/λ, ã = y1 + a, b̃ = y2 +b are dimensionless
distances from the disclination lines to the coordinate plane xOz, α is the angle between a tangent line
to the curve Γ at a given point and the positive direction of Ox-axis, a = k A, b = kB .
The function
w = sn
(
2
π
K (m)z; m
)
(3)
α
b
b
ã
b̃
x
y
−π/2 π/20
h
l y2
y1
d
G
Γ
z
Figure 2. The geometry of region G in which the director angle θ(x, y) is sought as a solution of Laplace’s
equation: • — disclination lines, y1, y2 are distances from Ox-axis to a crest and a trough of the relief
surface, respectively, ã, b̃, h are distances from the defects and from the upper cell surface to Ox-axis, Γ
is the curve which describes the form of the relief surface, α is the angle between the tangent to Γ and the
direction of Ox-axis, l = kL, d = kD are dimensionless thickness of the cell and the dimensionless depth
of the modulated surface relief, k = 2π/λ.
33604-3
M.F. Ledney, O.S. Tarnavskyy, A.I. Lesiuk, V.Yu. Reshetnyak
x
y
−π/2 π/2
0
h
z
Figure 3. The rectangle region of complex plane z , which is conformally mapped on the upper half
plane Imw Ê 0 by function (3).
is a conformal mapping which maps the rectangle with the height h and the width π in the complex
plane z = x + iy (see figure 3) onto the upper half plane of the complex plane w . Here, sn(z; m) is the
Jacobian elliptic function, K (m) is the complete elliptic integral of the first type. The rectangle vertexes
are mapped onto the points −1/m, −1, 1, 1/m of the real axis of complex plane w , where parameter m is
a root of the equation (see [27])
K (m)
K
(p
1−m2
) =
π
2h
. (4)
Let us cut out a semi-circle of radius R centred at point w = 1 of the real axis from the upper half
plane Im w Ê 0 (see figure 4), where
2É R É
1
m
−1. (5)
This obviously implies that
0É m É
1
3
. (6)
The upper half plane with the cut out semi-circle can be conformally mapped onto the upper half
plane of complex plane η using the Zhukovsky transformation [27]
η(w) =
1
2
(
R
w −1
+
w −1
R
)
. (7)
p
s
0-1 1
R
ϕ
−1/m 1/m
w
Figure 4. The hatched region of complex plane w is conformally mapped on the upper half plane ImηÊ 0
by function (7), while on the other hand it is mapped on the region G in figure 2 by the function which is
inverse to (3).
33604-4
Equilibrium configurations of director
On the other hand, the transformation z = πF (w ; m)/[2K (m)], which is inverse to (3), maps the half
plane Im w Ê 0 with the cut out semi-circle of radius R onto the rectangle of the complex plane Im z Ê 0
(region G in figure 2), which is clipped from below by a certain smooth curve Γ. Here, F (w ; m) is the
elliptic integral of the first type. The curve Γ is the image of semi-circle w = 1+Reiϕ , where ϕ ∈ [π,0]. The
tangent lines to curve Γ at points with x = ±π/2 prove to be parallel to Ox-axis due to the principle of
angle-preservation in conformal mapping [27]. Combining functions (3) and (7) we obtain the following
transformation
η(z) =
1
2
[
R
sn
(
2K (m)z/π;m
)
−1
+
sn
(
2K (m)z/π;m
)
−1
R
]
, (8)
which maps the region G belonging to the complex plane z onto the upper half plane of the complex
plane η. Here, the lower boundary of G which is the smooth curve Γ is given in the form
z =
π
2K (m)
F
(
1+Reiϕ ;m
)
, where ϕ ∈ [π,0]. (9)
3. The function of the surface relief
Let us find a real function describing the relief of the cell modulated surface (curve Γ in figure 2).
Since function (7) maps the semi-circle w = 1+Reiϕ, where ϕ ∈ [π,0], onto the real axis interval [−1,1] of
the complex plane η= p + is, formula (9) can be rewritten as follows:
x(p)+ iy(p)= f (p)=
π
2K (m)
F
(
1+R
(
p + i
√
1−p2
)
; m
)
, (10)
where −1É p É 1.
Differentiating (10) with respect to the variable p and substituting p = cosϕ (since −1É p É 1 for the
points of curve Γ), we obtain
x′(ϕ) =−eβ(ϕ) cosα(ϕ) sinϕ,
y ′(ϕ)=−eβ(ϕ) sinα(ϕ) sinϕ,
(11)
where α = Imln f ′(p), β = Reln f ′(p). A prime is used to mark the derivatives with respect to the argu-
ment.
Substituting p = cosϕ into f ′(p) and expanding the functions of ln(1+x) type at |x| < 1 into Maclaurin
series, we find:
β(ϕ) = ln
(
π
2K (m)
1
p
1−m2
1
sinϕ
)
+γ(ϕ), (12)
α(ϕ) =
1
2
∞
∑
k=1
(−1)k+1
k
[
(
2
R
)k
−
(
mR
m −1
)k
−
(
mR
1+m
)k
]
sin kϕ, (13)
where
γ(ϕ)=−
1
2
∞
∑
k=1
(−1)k+1
k
[
(
2
R
)k
+
(
mR
m −1
)k
+
(
mR
1+m
)k
]
cos kϕ.
From equations (11), taking into account the explicit form of β(ϕ) (12), we obtain the expression for
the curve Γ in parametric form
x(ϕ) =−
π
2
+
π
2K (m)
1
p
1−m2
π
∫
ϕ
eγ(ϕ) cosα(ϕ)dϕ,
y(ϕ) =y1 +
π
2K (m)
1
p
1−m2
π
∫
ϕ
eγ(ϕ) sinα(ϕ)dϕ,
(14)
33604-5
M.F. Ledney, O.S. Tarnavskyy, A.I. Lesiuk, V.Yu. Reshetnyak
where ϕ ∈ [π,0]. Formulae (14) take into account that ϕ = π corresponds to the beginning of curve Γ at
the point (−π/2, y1) (see figure 2). y -coordinates of the ends of curve Γ can be obtained by taking the
imaginary part of formula (9) at ϕ= π, 0:
y1 =
π
2K (m)
F
(
1
p
1−m2
√
1−
1
(R −1)2
;
√
1−m2
)
,
y2 =
π
2K (m)
F
(
1
p
1−m2
√
1−
1
(R +1)2
;
√
1−m2
)
.
(15)
Therefore, the relief of the modulated surface is described by the curve Γ and, as follows from (14)
and (15), is determined by the values of two dimensionless parameters m and R. For the given values
of geometric parameters of the NLC cell, namely, the cell thickness L, the depth D and period λ of the
modulated surface relief, the values of parameters m and R are found from the following system of
equations
h− y1 =
2π
λ
L,
y2 − y1 =
2π
λ
D,
(16)
where (4) and (15) must be taken into account. It is necessary that the obtained parameters m and R
should meet physically justifiable conditions (5) and (6). Thus, R = 1/m−1 [see condition (5)] corresponds
to the case L = D where the upper planar surface lies on the lower relief surface of the cell. If R < 2, then
the curve Γ, which describes the surface relief (see figure 2), becomes a broken line rather than a smooth
curve. It is worth noting that for m > 1/3, inequality (5) cannot be satisfied at all. Conditions (5), (6) define
the region of admissible values of L, D , λ. This region which is found using (16) is presented in figure 5.
It is obvious that physically admissible values of the cell thickness L and relief depth D should meet the
condition 0 É D É L. Our conformal mapping technique is capable of describing the director field not
for all L, D , λ, but only for those values of the parameters that correspond to the points of the region S.
Further on we consider only such values of parameters L, D , λ from region S in figure 5 for which there
exists a unique solution of system (16).
Let us consider a particular case m = 0 which corresponds to a semi-infinite cell (L → ∞). Then,
from (15) we have y1 = arcosh(R−1), y2 = arcosh(R+1). Dimensionless relief depth d = y2−y1 reaches the
maximum possible value dmax = arcosh3 at R = 2 (see figure 5). At a fixed but still large parameter R ≫ 1
(h →∞, and the value of y1 is finite), taking into account (12) and (13), from (14) we have x(ϕ) ∼π/2−ϕ
and y(ϕ) ∼ y1+1/R+cosϕ/R, where y1 ∼ ln(2R)−1/R. Hence, the function describing the curve Γ can be
easily obtained in an explicit form y(x) = ln(2R)+sin x/R. So, 2/R is the depth of the surface relief. Thus,
in a semi-infinite cell with a small surface relief depth, the form of the relief is described by function
sin(x).
L/λ
D/λ
O
D = L
K (2
p
2/3)
4K (1/3)
≈ 0.39
arcosh3
2π
≈ 0.28
S
Figure 5. The hatched region S corresponds to admissible values of the cell thickness L, the surface relief
depth D and period λ for which a director distribution and equilibrium locations of the defects can be
calculated.
33604-6
Equilibrium configurations of director
4. The director field
Now we find the director field θ(x, y) in the region G (see figure 2). By conformal mapping η(z) (8),
the boundary points of region G in the complex plane z are mapped onto the points of the real axis of the
complex plane η= p + is. Points p1 and p2 of the real axis p are the images of the defect points in the re-
gion G, where −1/m É p1 É−1, 1 É p2 É 1/m. The conformal mapping of the region G boundaries taking
into account boundary conditions (2) for function θ(x, y) yields the following values of function θ(p, s) on
the real axis p:
θ(p,0) =
0, if −∞< p < p1 , p2 < p <+∞,
−
π
2
, if p1 < p <−1, 1 < p < p2 ,
−
π
2
+α, if −1 É p É 1,
(17)
where α is the angle between the tangent line to the curve Γ at a point z = x + iy and the positive di-
rection of Ox-axis. As follows from the properties of conformal mapping [27], the angle α = arg f ′(p) =
Imln f ′(p), which is given by (13) [a point z = f (p) belongs to the curve Γ at −1 É p É 1]. Therefore,
in order to find the harmonic function θ(η) = θ(p, s) we have to solve the Dirichlet problem for the up-
per half plane Imη Ê 0. This can be done by using the Poisson integral [27]: θ(η) =
1
π
∫∞
−∞
sθ(t ,0)dt
(p − t)2 + s2
.
Substituting the value of θ(p,0) (17) at the real axis Reη into the integral, we obtain
θ(η) =−
1
2
p2
∫
p1
s dt
(p − t)2 + s2
+
1
π
1
∫
−1
sα(t)dt
(p − t)2 + s2
. (18)
After the substitution t = cosϕ, we use the value of α(ϕ) taken from (13) in the second integral. As a
result, the integration yields
θ(η) = Im
{
ln
√
η−p1
η−p2
+
1
2
∞
∑
k=1
(−1)k
k
[
(
2
R
)k
−
(
mR
m −1
)k
−
(
mR
1+m
)k
]
(
η−
√
η2 −1
)k
}
. (19)
Substituting η from formula (8) into (19) and reducing power series, after simple transformations, we
obtain the following expression
θ(z) = Imln
√
√
√
√
1−2ξ(ã)g (z)+ g 2(z)
1+2ξ(b̃)g (z)+ g 2(z)
[
1− mR
1−m
g (z)
][
1+ mR
1+m
g (z)
]
1+ 2
R g (z)
, (20)
where
ξ(t) =−
1
2
R
dn−1
(
2
πK (m)t ;
p
1−m2
)
−1
+
dn−1
(
2
π
K (m)t ;
p
1−m2
)
−1
R
,
g (z)= η−
√
η2 −1 = R
[
sn
(
2
π
K (m)z; m
)
−1
]−1
.
Here, the function dn(z; k) is the delta amplitude.
5. The equilibrium positions of defects
The equilibrium distances ã and b̃ from disclination lines to the coordinate plane xOz should provide
the minimum value of the NLC free energy. By applying Green’s first identity, the NLC free energy of
region G per unit length along Oz-axis (see figure 2) can be expressed in the form of a closed contour
integral
F =
K
2
∫
G
∫
(∇θ)2 dxdy =
K
2
∮
∂G
θ
∂θ
∂n
dl , (21)
33604-7
M.F. Ledney, O.S. Tarnavskyy, A.I. Lesiuk, V.Yu. Reshetnyak
0.00 0.05 0.10 0.15 0.20
0.3
0.6
0.9
1.2
1.5
1
2
3
4
5
6
a /π, b /π
d /π
Figure 6. Equilibrium distances a (solid curves) and b (dashed curves) from disclination lines to the cell
modulated surface versus the surface relief depth d . l /π=0.58 (1); 1.0 (2); 1.4 (3); 1.8 (4); 2.2 (5); 2.6 (6);
3.0 (7).
where ∂/∂n is the normal derivative taken at the points of curve ∂G , which restricts the region G, dl is
an arc element of the curve ∂G. It is obvious that points (−π/2, ã) and (π/2, b̃), which are points corre-
sponding to defects, lie at the curve ∂G and are the singular points of the free energy integral (21). For
this reason, the integral (21) is calculated along the boundary of the region G without two small areas
around the defect points. After parametrization of the curve ∂G taking into account the bypass of the
defect points, the free energy functional is further numerically minimized with respect to the positions
of the defects.
Equilibrium distances from disclination lines to the cell modulated surface, a = k A and b = kB , as
functions of dimensionless depth of the surface relief d = kD are calculated for several values of the
cell dimensionless thickness l = kL. The corresponding curves are presented in figure 6. The distances
are measured in the units of the region G width which is equal to π (see figure 2). As the surface relief
depth d increases, the values of equilibrium distances a and b decrease monotonously, but the difference
between them |a −b| grows. Thus, the distance from the modulated surface to a disclination line located
above a relief crest decreases faster than the distance to a disclination line located above a relief trough
as the relief depth grows.
Equilibrium distances a and b from disclination lines to the cell modulated surface, calculated as
functions of the cell thickness l for several values of the relief depth d are given in figure 7. As the cell
1.0 1.5 2.0 2.5 3.0
0.2
0.4
0.6
0.8
l/π
a/π, b/π
1
2
3
Figure 7. Equilibrium distances a (solid curves) and b (dashed curves) from disclination lines to the cell
modulated surface versus the cell thickness l . d/π= 0.05 (1); 0.1 (2); 0.2 (3).
33604-8
Equilibrium configurations of director
(a) (b)
Figure 8. Director field lines (a) and the distribution of the director deviation angle θ (b) in the region G.
thickness l grows, the distances a and b increase monotonously until they reach certain constant values.
Thus, for a given surface relief depth d , the positions of defects become practically independent of the cell
thickness when latter is sufficiently large. The greater the relief depth d of the modulated surface is, the
faster the equilibrium distances a and b reach saturation. It is worth noting that, for a given relief depth
d , the form of the cell surface relief is not invariant with respect to the changes in the cell thickness l . This
is just a peculiarity of the technique used by us. However, this variation of the relief proves to be small,
and we do not take it into account. In particular, as calculations show, for the relief depth d/π= 0.2, the
change in l with the range as in figure 7 corresponds to a relief variation that does not exceed 6 percent
of the relief depth d . Since for a given relief depth d , the defect locations are independent of the cell
thickness l starting from its certain value lc . π (see figure 7), it follows that the approximation m = 0
can be justifiably used in the case of a thick cell where l = kL ≫ 1. If the relief depth of a thick cell is
small d = kD . 1 (while L ≫ D), then the relief of the modulated surface can be described sufficiently
well by the function y(x) ∼ sin x/R. Here, R is found from system (16). Therefore, in the case of cells
with L ≫ λ, the proposed technique allows one to calculate the director field and equilibrium locations
of the defects, if the relief depth D <λ arcosh3/(2π) ≈ 0.28λ.
In figure 8 (a), director field lines in the region G are presented. The following NLC cell parameters
were used for calculations: L = 2.8 µm, λ= 10 µm and D = 1.4 µm, which are close to the typical ones [3]
and correspond to m = 0.1789 and R = 2.9894. The distribution of the director deviation angle θ(x, y) in
the region G for the given cell parameters is presented in figure 8 (b).
Nowwe assume that the NLC anchoring at the cell planar surface is homeotropic and strong. The NLC
anchoring at the spatially modulated surface is assumed to be planar: the director easy axis is tangent
to the surface and lies in the plane XOY . Similar to the above considered case, disclination lines parallel
to the relief wave crests and troughs occur in the bulk of NLC above the local extrema of the modulated
surface. It is easy to see that the value of angle θ in the region G is obtained by adding π/2 to the value
given by formula (20). It is evident that in this case for a given cell thickness L, as well as depth D and
period λ of the cell surface relief, the distances from disclination lines to the surface prove to be the
same as those in the case of planar orientation of the director at the cell planar surface and homeotropic
orientation of the director at the modulated surface.
6. Conclusions
We have studied two-dimensional equilibrium configurations of the NLC director in the cell bounded
by two parallel surfaces: a planar surface and a spatially modulated one. TheNLC director anchoringwith
cell surfaces is assumed to be strong. We have considered the case where the disclination lines occur in
33604-9
M.F. Ledney, O.S. Tarnavskyy, A.I. Lesiuk, V.Yu. Reshetnyak
the bulk of NLC above local extrema of the modulated surface relief. These disclination lines, which are
caused by different orientation of the director easy axis at two bounding surfaces, namely, homeotropic
at one surface and planar at the other one, run along the crests and troughs of the relief. The relief of
the modulated surface is described by a smooth sine-like periodic function. This function is taken in a
special way for particular measurable parameters of the cell, namely, the cell thickness L, period λ and
depth D of the surface relief. Thus, for the given values of L, λ, D system (16) together with (4) and (15)
yields the values of m and R. For the obtained values of m and R, formulae (14) define a sine-like relief
profile. In particular, for a thick cell L ≫ λ with a small relief depth D . λ, the relief profile is described
by the function sin x/R. The region of the admissible values of cell parameters L, λ, D has been found, for
which the proposed technique allows one to calculate a director distribution and equilibrium locations
of the defects (see figure 5).
In the approximation of planar director deformations, an analytically calculated director field is given
by (20). Minimizing the free energy (21) of the system we have found equilibrium distances a = 2πA/λ
and b = 2πB/λ from disclination lines to the cell relief surface. We have studied the dependence of
these distances on the cell thickness L and the period λ and depth D of the surface relief. The distances
from disclination lines to the cell modulated surface decrease as the relief depth increases (see figure 6),
whereas the distance between the modulated surface and the disclination line located above a relief
wave crest decreases faster than the distance corresponding to the disclination line located above a relief
trough. The increase in the cell thickness results in greater equilibrium distances from disclination lines
to the modulated surface. However, they become practically independent of the thickness for sufficiently
large thickness value (see figure 7). The presented in figures 6, 7 equilibrium distances a and b from
disclination lines to the relief surface can be used in constructing NLC cells, one surface of which has a
sine-like relief profile.
The knowledge of the defects location and its dependence on the cell parameters can be useful for
investigations in trapping and self-assembling of nanoparticles by topological defects in NLC cells.
Acknowledgements
The authors express sincere gratitude to I.P. Pinkevich for helpful advice during the result discussions.
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Рiвноважнi конфiгурацiї директора в планарнiй нематичнiй
комiрцi з однiєю просторово модульованою поверхнею
М.Ф. Ледней, О.С. Тарнавський, А.I. Лесюк, В.Ю. Решетняк
Київський нацiональний унiверситет iменi Тараса Шевченка, Київ, Україна
Дослiджено двовимiрнi рiвноважнi конфiгурацiї директора нематичного рiдкого кристалу (НРК) в комiрцi
обмеженiй двома паралельними поверхнями: планарною i просторово модульованою. Форма рельєфу
модульованої поверхнi описується гладкою перiодичною синусоподiбною функцiєю. Розглянуто випадок
наявностi дисклiнацiйних лiнiй в об’ємi НРК строго над локальними екстремумами модульованої поверх-
нi, що проходять вздовж гребенiв i западин хвиль рельєфу. Виникнення дисклiнацiйних лiнiй зумовлене
рiзним типом орiєнтацiї легкої осi директора на обмежуючих поверхнях комiрки (гомеотропна орiєнтацiя
на однiй з обмежуючих поверхонь i планарна на iншiй) та жорстким зчепленням НРК з поверхнями. В на-
ближеннi плоских деформацiй директора отримано аналiтичнi вирази для розподiлу директора в об’ємi
комiрки. Розраховано рiвноважнi вiдстанi вiд дисклiнацiйних лiнiй до модульованої поверхнi та знайдено
їх залежнiсть вiд параметрiв комiрки (перiоду та глибини хвиль рельєфу), а також вiд товщини комiрки.
Показано, що вiдстанi вiд дисклiнацiйних лiнiй до модульованої поверхнi зменшуються зi зростанням
глибини рельєфу.
Ключовi слова: нематичний рiдкий кристал, просторово модульована поверхня, топологiчнi дефекти,
конформне вiдображення
33604-11
http://dx.doi.org/10.1088/0022-3727/42/8/085114
http://dx.doi.org/10.1088/0022-3727/43/49/495105
http://dx.doi.org/10.1103/PhysRevE.82.011707
http://dx.doi.org/10.1103/PhysRevE.86.041706
http://dx.doi.org/10.1038/ncomms8180
http://dx.doi.org/10.1002/adma.201103791
http://dx.doi.org/10.1039/C0SM00437E
http://dx.doi.org/10.1021/nl204030t
http://dx.doi.org/10.1103/PhysRevE.90.032503
http://dx.doi.org/10.1038/ncomms1709
Introduction
The problem geometry and basic equations
The function of the surface relief
The director field
The equilibrium positions of defects
Conclusions
|