Instability of the Nematic-like Phases Filled with Impurities Against the Formation of Modulated Structures
We examine spatial distribution of impurity rigid-sphere-like macroparticles in the mesomorphic liquid crystal host. Using continuum statistical mechanical theories, we analyze the thermodynamic conditions necessary for a modulated lamellar-structure to appear. There is a long-range effective intera...
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| Zitieren: | Instability of the Nematic-like Phases Filled with Impurities Against the Formation of Modulated Structures / A.V. Kleshchonok, V.Yu. Reshetnyak, V.A. Tatarenko // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 524-532. — Бібліогр.: 22 назв. — англ. |
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| author | Kleshchonok, A.V. Reshetnyak, V.Yu. Tatarenko, V.A. |
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| citation_txt | Instability of the Nematic-like Phases Filled with Impurities Against the Formation of Modulated Structures / A.V. Kleshchonok, V.Yu. Reshetnyak, V.A. Tatarenko // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 524-532. — Бібліогр.: 22 назв. — англ. |
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| description | We examine spatial distribution of impurity rigid-sphere-like macroparticles in the mesomorphic liquid crystal host. Using continuum statistical mechanical theories, we analyze the thermodynamic conditions necessary for a modulated lamellar-structure to appear. There is a long-range effective interaction between the impurity particles. This interaction is considered as being responsible for the formation of superstructures. In the general case, this interaction includes two components: a van der Waals-type direct interaction and an indirect interaction (through the director-field distortions). The last one depends on both the temperature of a sample and the concentration of particles. This effective interaction controls the structure and properties of the system. Analytical solutions for a director-field distortion, density inhomogeneity of the host medium, temperature of the formation of a modulated structure, and its spatial period are obtained. The proposed theoretical approach can be applied to other anisotropic and inhomogeneous systems.
Розглянуто просторовий розподiл домiшкових жорстких сфероподiбних макрочастинок у мезоморфному рiдкокристалiчному носiї. Використовуючи континуальнi статично-механiчнi теорiї, проаналiзовано термодинамiчнi умови, необхiднi для появи модульованих шаруватих надструктур. Мiж домiшковими частинками присутня ефективна далекодiйна взаємодiя. Вважається, що вона вiдповiдає за формування структур. У загальному випадку така взаємодiя мiстить два внески: пряму взаємодiю типу Ван-дер-Ваальса та непряму (через поле викривлення директора). Остання залежить вiд температури зразка та концентрацiї частинок. Ця ефективна взаємодiя контролює структуру та властивостi системи. Було одержано аналiтичнi розв’язки для розподiлу поля директора, неоднорiдностi густини носiя, температури формування модульованих структур та їх просторового перiоду. Запропонований теоретичний пiдхiд може бути застосований до iнших анiзотропних та неоднорiдних систем.
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SOFT MATTER
524 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
INSTABILITY OF THE NEMATIC-LIKE PHASES FILLED
WITH IMPURITIES AGAINST THE FORMATION
OF MODULATED STRUCTURES
A.V. KLESHCHONOK,1 V.YU. RESHETNYAK,1 V.A. TATARENKO2
1Taras Shevchenko National University of Kyiv, Physics Faculty
(2/1, Academician Glushkov Prosp., Kyiv 03022, Ukraine; e-mail: avklesh@ gmail. com )
2G.V. Kurdyumov Institute for Metal Physics, Nat. Acad. of Sci. of Ukraine
(36, Academician Vernadsky Blvd., Kyiv 03680, Ukraine)
PACS 61.30.Dk, 61.30.Jf,
82.70.Dd, 82.70.Kj, 83.80.Xz
c©2010
We examine spatial distribution of impurity rigid-sphere-like ma-
croparticles in the mesomorphic liquid crystal host. Using con-
tinuum statistical mechanical theories, we analyze the thermody-
namic conditions necessary for a modulated lamellar-structure to
appear. There is a long-range effective interaction between the
impurity particles. This interaction is considered as being respon-
sible for the formation of superstructures. In the general case, this
interaction includes two components: a van der Waals-type direct
interaction and an indirect interaction (through the director-field
distortions). The last one depends on both the temperature of a
sample and the concentration of particles. This effective interac-
tion controls the structure and properties of the system. Analytical
solutions for a director-field distortion, density inhomogeneity of
the host medium, temperature of the formation of a modulated
structure, and its spatial period are obtained. The proposed theo-
retical approach can be applied to other anisotropic and inhomo-
geneous systems.
1. Introduction
Nematic liquid crystal (LC) colloids have attracted a lot
of attention in recent years [1–5]. These materials have
all properties of the LC phase but a non-trivial behav-
ior of the impurity subsystem is possible as well. For
instance, the modulated structures, which would have
great influence on the physical properties of entire sys-
tem, can be formed. Macromolecules or other particles
with relatively large molecular mass and diameter of
about 10-1000 nm (that is much larger than these pa-
rameters for typical nematic molecules) can act as im-
purities. When these impurities are introduced into the
nematic host crystal, a distortion of its structure occurs,
because of the presence of ’tight’ anchoring between LC
molecules and the impurity surface. The director field
distortions can occupy a much bigger volume than that
per one impurity particle. If the areas of deviations in-
duced by different impurities are overlapping, each of
those particles will react upon the director deviation. In
other words, this means that the effective interaction be-
tween impurity particles through the nematic host crys-
tal will take place. The indirect interaction energy can
be much larger than the energy of the direct van der
Waals interaction. All of these effects result in the key
relevant property of such a system, namely, in the forma-
tion of modulated structures with certain spatial period.
Zumer with collaborators carried out the experiments
and simulations of the formation of superstructures in
LC (see, e.g., [5]). It has been known long ago that an
interaction induced by a director deformation plays the
crucial role in systems based on a filled liquid crystal
[6–9]. Moreover, the main problem of a filled LC focuses
certainly on the system stability which will be considered
in the present paper.
There are plenty of applications of the systems based
on filled liquid crystals. Such applications as diffraction
gratings, different magnetooptical switches, as well as
information processors and even medical devices [1, 10,
11] are only few of examples.
In this paper, a theoretical description of the instabil-
ity of a homogeneous system associated with the forma-
tion of modulated structures is presented in the manner
of Ref. [12].
INSTABILITY OF THE NEMATIC-LIKE PHASES FILLED WITH IMPURITIES
2. Statement of the Problem
Let us consider a highly dispersed system that occupies
a volume V and consists of two interacting subsystems
– nematic host and impurity sphere-like particles.
The behavior of the system depends of the phase
state of the liquid crystal host. Let us assume that
the sample is isotropic above a certain temperature
T1. The corresponding global space-symmetry group is
S1 ≡ R3 ∧O(3).
It may be presumed that, below T1, the nematic phase
of the host medium takes place and is characterized by
the existence of the axis of a preferred orientation of
molecules and a uniform impurity distribution. In con-
trast to isotropic liquids in the mesophase, a long-range
correlation between orientations of elongated molecules
can exist. The symmetry group SN ≡ R3 ∧ D∞h re-
sponds to this state [13] (D∞h – cylinder group with the
axis of rotation C∞ which is directed along the axis of a
preferred orientation of liquid crystal molecules).
On the further cooling of the sample below a certain
temperature T0c (T1 ≥ T0c ≥ TM , where TM is the melt-
ing point temperature of the liquid crystal), the uniform
distribution of interacting impurities can disappear, and
the so-called focal conic texture can be formed due to
the appearance of impurity layers of the same thickness
which can quasifreely glide over one another. The host
medium molecules have kept the direction perpendicu-
lar to the impurity layers, on the average. The fact that
layers glide over one another indicates the absence of
a long-range translation order in their plane. Such a
phase is somewhat similar to the phase called smectic
A. The smectic A phase differs from the nematic one
by breaking the translation symmetry along the direc-
tion that is perpendicular to the layer plane. Thus,
the space symmetry group corresponding to this state
is SA ≡
(
R2 × Z
)
∧ D∞h, where Z is the group of dis-
crete translations along the axis of the preferential ori-
entation of host medium molecules, the Oz direction for
definiteness, and R2 is a group of two-dimensional con-
tinuous translations. Thus, the phase transition that
is accompanied by the appearance of (modulated) peri-
odic structures is similar, in respect to symmetry, to the
phase transition between nematic and smectic A.
We will characterize the bulk density (the number
of molecules per unit volume) of the system with the
function of spatial distribution ρ(r) through “lattice
sites” {r} of the quasilattice. Moreover, ρ(r) � 1/Vm,
where Vm is the natural volume of one host medium
molecule. The mean position-independent density of
Fig. 1. Geometry of the problem
the host medium can be determined with the expression
ρ0 ≡ V −1
∫
V
ρ(r)dr; ρ0 � 1/Vm.
To describe a spatial distribution of impurities em-
bedded into a host medium, the quasi-lattice-gas rep-
resentation can be used. Impurity particle centers are
located in “interstices” which are distributed randomly
throughout the lattice {R} which is embedded in the
host medium lattice {r}. The fraction c0 of occupied
sites {R} is determined by the relative concentration of
impurity particles: c0 ≡ N−1
∑
R∈V
C(R) c0 � 1. Here,
C(R) is a stochastic function being equal to 1, if the “lat-
tice site” with radius vector R is occupied by an impu-
rity particle (from the system {R1, R2, R3, ...} ) and is
equal to 0 otherwise: C(R) = { 1R=(R1,R2,R3, ...),
0R 6=(R1,R2,R3, ...);
c(R)
is the one-particle impurity distribution function which
is determined by the formula c(R) = 〈C(R)〉 [14], where
the averaging is realized over the ensemble of impurity
particles under additional constrained condition of the
impurities number conservation; N = V/Vm is the total
number of “lattice sites”, where liquid crystal molecules
can be arranged. There is same number of “interstice”
accessible for the penetration of an impurity, but only
some of them will be occupied. To be exact, we have(
π
√
2/6
)
V/Vim ≈ 0.7405V/Vim � N , where Vim is the
natural volume of one sphere-like impurity particle [14].
Therefore, it is supposed that the largest rest of “sites”
of the embedded lattice remains vacant. In other words,
c(R)� 1.
The nematic host is described with a director vector
field n(r) which is directed along the primary orientation
of nematic molecules in a vicinity of the point {r}. The
initial director orientation n0 is chosen along the axis
n0 = (0, 0, 1). The geometry of the problem is illustrated
in Fig. 1.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 525
A.V. KLESHCHONOK, V.YU. RESHETNYAK, V.A. TATARENKO
The elastic part of the free energy of a host-medium
is determined by the well-known expression [15]:
ΔFel =
1
2
∫
V
{
K1(div n)2 +K2 (n · rotn)2+
+ K3 [n× rotn]2
}
dr. (1)
Here, K1, K2, and K3 are Frank–Oseen coefficients. In
expression (1), we have neglected the saddle-splay elas-
ticity characterized by the divergence coefficient, K24
[16, 17], since we do not consider the effect of external
boundaries of the system at issue. Nevertheless, when
the nematic cell is sufficiently small, but is considerably
thicker than the anchoring extrapolation length [17], a
modulated phase in the planar nematic may arise, as a
result of saddle-splay distortions. The positive modules
Kj (j = 1, 2, 3) are implicit functions of the absolute
temperature T , through their dependence on the den-
sity distribution ρ(r) and mainly on the scalar measure
Q (T, ρ(r), c0) that describes a long-range orientation or-
dering of the system. We suppose that Kj don’t depend
on r and will express them through the mean order pa-
rameter S (T, ρ(r), c0) ≡ V −1
∫
V
Q (T, ρ(r), c0) dr:
Kj (ρ(r); Q (T, ρ(r), c0)) ∼= Kj (ρ0; S (T, ρ0, c0)) + ....
It is quite natural to consider that the energy of a
wetting-like bond between an every surface section of
an individual impurity sphere-like macroparticle and the
nematic host depends on the orientation of a director
and the surrounding environment density. Then the to-
tal energy of such an interaction of all impurities with
the medium can be written as
ΔFrel =
∑
R∈V
c(R)
∫
V
g (r−R,n, ρ;Q (T, ρ, c0)) dr , (2)
where g (r−R,n, ρ;Q (T, ρ, c0)) is the phenomenologi-
cal density of such an interaction.
In the self-consistent field (correlationless) approxima-
tion, let us consider explicitly the director orientation
independent term of the free energy [18]:
Fpos = ΔFunfloc + kBT
∑
R∈V
c(R) ln c(R)+
+kBT
∑
R∈V
(1− c(R)) ln (1− c(R)) +
+
1
2
∫ ∫
V
ρ(r)ρ(r′)E (r− r′;S) +
+kBT
∫
V
ρ(r) ln
(
ρ(r)
ρ0
)
dr. (3)
Here, ΔFunfloc = 1
2
∑
R,R′∈V
c(R)c(R′)U(R − R′) is a
contribution of the direct impurity–impurity pairwise in-
teraction (van der Waals-type) with the potential energy
U(R −R′). Two next terms describe a contribution of
the impurity subsystem mixing entropy to the free en-
ergy. The quantity E(r− r′;S) characterizes the nonlo-
cal interaction of a pair of physically small host volume
elements. The last term is host medium subsystem con-
figurational entropies in the self-consistent field approx-
imation.
Thus, the full free energy of the system is
F = F0 + ΔFel + ΔFrel + ΔFpos, (4)
where F0 is independent of the impurity distribution con-
figuration, and depends on three variables: n(r) is the
director vector field, ρ(r) is the distribution function of
the nematic liquid crystal density, and c(R) is the one-
particle distribution function of impurities.
3. State Stability Loss
To obtain the statistical-thermodynamic state equations
for the “equilibrium liquid crystal – impurities” system
and to find the conditions of stability loss, we will con-
sider small deviations from the initial uniform state:
c(R) ≡ c0 +δc(R), n(r) ≡ n0 +δn(r), ρ(r) ≡ ρ0 +δρ(r).
Let us expand the energy density of the interaction be-
tween nematic molecules and impurity particles in a se-
ries in the variables δρ(r), δc(R), and δn(r) to within the
second order (see details in Ref. [12]). Further, we will
use the representation of this series in terms of the pa-
rameters g0 ≡ g(r−R,n0, ρ0;Q0) (Q0 ≡ Q(T, ρ0, c0))
and the derivatives of g(r−R,n0, ρ;Q), which are taken
at values of their arguments for the initial pseudohomo-
geneous conditions, n = n0 and ρ = ρ0. For example,
the derivatives ∂2g(r−R,n0,ρ;Q)
∂δnj∂ρ
|n0,ρ0
can be written as
g′′njρ(r−R).
For the further consideration of stability-loss condi-
tions, we will apply the static fluctuation-wave method
526 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
INSTABILITY OF THE NEMATIC-LIKE PHASES FILLED WITH IMPURITIES
[19] following the rules
f̃(k) =
1
v
∫
V
f(r)e−ikrdr,
f(r) =
1
N
∑
k∈BZ
f̃(k)eikr ∼=
v
(2π)3
∫
k∈BZ
f̃(k)eikrdk.
With the wave vector k = 0 – the parameter of Fourier
transform, the integration and the summation are car-
ried out over all wave vectors of the first Brillouin zone
(BZ), and v = V/N is the volume of a primitive unit
cell of the notionally chosen quasilattice. The reality
of the function under transformation, f(r), provides
an additional constraint on the Fourier components:
f̃(k) = f̃∗(−k). As mentioned above, the initial director
direction is chosen as n0 = (0, 0, 1); thus, the director
normalization leads to the relation∑
k
(
δñ(k) · δñ∗(k)
)
= −2 (n0 · δñ(0)) . (5)
While minimizing functional (4), we take constraint
(5) into account by means of the indefinite Lagrange
multiplier, λ (see also Appendix A). The constraints,∑
R∈V δc(R) = 0 and
∫
V
δρ(r)dr = 0, which correspond
to the conservation of the numbers of impurity parti-
cles and nematic-LC-molecules, respectively, lead to the
following properties of the ’zeroth’ Fourier components:
δρ̃(0) = 0 and δc̃(0) = 0.
With the second-order accuracy in δc̃(k), δρ̃(k), and
δñ(k), let us find functional F (4) by taking into account
the latest constraints, as well as the director normaliza-
tion (5).
In the case of k 6= 0 after applying the Lagrange–
Euler equations (for instance, in the form of ∂F
∂(Reδρ̃(k)) +
i ∂F
∂(Imδρ̃(k)) = 0 and ∂F
∂(Reδñj(k)) + i ∂F
∂(Imδñj(k)) = 0 (j =
x, y, z)), one can obtain a system of linearized equa-
tions of state in a vicinity of the stability-loss point,
Tbif(ρ0, c0) (see Ref. [12] and Appendix A).
To simplify the system of equations, we will specify the
density of interaction energy between the host molecules
and surfaces of the impurities. The following form of
the interaction corresponds to the homeotropic bound-
ary condition [15] on the impurity surface, which is re-
alized in most experiments:
g(r−R,n0, ρ;Q) ∼=
∼= −
(n(r) · (r−R))2
|r−R|2
β(r−R,n0, ρ;Q), (6)
g′ni(r−R) = g′ni |n0,ρ0
∼=
∼= −2
(r−R)i(r−R)z
|r−R|2
β(r−R,n0, ρ;Q), (7)
for |r−R| ≥ dim. Here, β is a function which charac-
terizes the radial dependence of the density of interac-
tion energy between the impurities and the liquid-crystal
molecules. In the case of |r−R| < dim (where ρ(r) ≡ 0),
we will suppose that g ≡ 0. The Fourier components of
various derivatives of g(r−R,n0, ρ;Q) (6) are presented
in Appendix B.
To obtain a non-trivial solution of the linearized
Lagrange–Euler equations (which is true near the point
of uniform-state stability loss), we require that the deter-
minant of the above-mentioned system of equations be
equal to zero (see Appendix A). This condition allows us
to find the solution bifurcation temperature Tbif(ρ0, c0),
where a loss of the homogeneous-state stability against
an inhomogeneous one can occur.
In such a way, we can obtain the director-distortion
profile of a host nematic LC and the spatial density dis-
tribution of its molecules:
δñj(k) ≈ − 1
K1K3k4
{[
(K3 −K1)(k · g̃′n(k))kj+
+K1k
2g̃′nj (k)
]
δc̃(k) +
[
c0(K3 −K1)kz g̃′′nzρ(0)kj+
+c0K1k
2g̃′′nzρ(0)δjz
]
δρ̃(k)
}
, (8)
δρ̃(k) ≈
≈
{
−g̃′ρ(k) +
c0
K1K3k4
[(K3 −K1)(k · g̃′n(k))kz+
+ K1k
2g̃′nz (k)
]
g̃′′nzρ(0)
}
δc̃(k)×
×
{
kBTbif
ρ0
+ vẼ(k) + c0g̃
′′
ρρ(0) −
− c20
K1K3k4
[
(K3 −K1)k2
z +K1k
2
] [
g̃′′nzρ(0)
]2}−1
, (9)
(δjz is the Kronecker symbol). The existence condition
of nontrivial solutions of the system of respective lin-
earized equations (see Appendix A), in the case at issue,
(6) and (7), is as follows:
1
v
kbTbif
c0(1− c0)
+
1
v
Ũ(k)−
− 1
K1K3k4
[
(K3 −K1)|(k · g̃′n(k))|2 +K1k
2|g̃′n(k)|2
]
−
−
∣∣∣∣g̃′ρ(k)− c0
K1K3k4
[(K3 −K1)(k · g̃′n(k))kz+
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 527
A.V. KLESHCHONOK, V.YU. RESHETNYAK, V.A. TATARENKO
+K1k
2g̃′nz (k)
]
g̃′′nzρ(0)
∣∣∣∣2{kBTbif
ρ0
+ vẼ(k) + c0g̃
′′
ρρ(0) −
− c20
K1K3k4
[
(K3 −K1)k2
z +K1k
2
] [
g̃′′nzρ(0)
]2}−1
∼= 0. (10)
As seen even from this particular relation and from the
non-linearized impure-LC equilibrium equations based
on a general expression for the free energy, one can
define the Fourier components of the effective pair-
interaction energy (both direct and indirect ones) be-
tween the sphere-like impurity particles (in particular,
by means of a nematic-LC medium). This interaction,
which acts as a collective mechanism of formation of
modulated structures, can be presented in the following
way:
Ṽeff(k) ∼= Ũ(k)−
− v
K1K3k4
[
(K3 −K1)|(k · g̃′n(k))|2 +K1k
2|g̃′n(k)|2
]
−
−v
∣∣∣∣g̃′ρ(k)− c0
K1K3k4
[(K3 −K1)(k · g̃′n(k))kz+
+K1k
2g̃′nz (k)
]
g̃′′nzρ(0)
∣∣∣∣2{kBTbif
ρ0
+ vẼ(k) + c0g̃
′′
ρρ(0) −
− c20
K1K3k4
[
(K3 −K1)k2
z +K1k
2
] [
g̃′′nzρ(0)
]2}−1
, (11)
where k 6= 0. It is easy to understand that
limk→0 Ṽeff(k) 6= Ṽeff(0). The non-analytic features of
Ṽeff(k) in a vicinity of k = 0, including Ṽeff(0), which
are concerned with the self-interaction exclusion, will be
published elsewhere. Nevertheless, we will consider this
dependence on the distance from the point k = 0 in a
subsequent section of this work.
We note that the elastic constant, K2, which is re-
sponsible for the resistance to a twist distortion of a
nematic LC, does not enter into expressions (8)–(11) for
sphere-like impurity particles. Thus, in the general case,
the effective interaction energy between the impurities
depends on the temperature, nematic-medium density,
and concentration of impurities. This feature allows con-
trolling the structure and properties of a system under
study by means of changing the external thermodynamic
conditions.
4. Numeric Calculations
Let us estimate some numerical parameters. With a goal
of numerical calculations by formulas (10) and (11), we
will neglect the energy parameters of the direct interac-
tion, Ũ(k), for the pairwise van der Waals-type interac-
tion between impurities in further discussions. It can be
done because their values seem to be lower by several
orders than the respective indirect-interaction parame-
ters (related to distortions of the director-vector field)
at typical interimpurity distances. In addition, as ex-
pected, Ẽ(k) describing the ’interaction’ between small
host-volume elements may be compensated by kBT
ρ0
(see
the denominator of the third term on a right-hand side
of (11)).
Let us specify the radial factor of the density of inter-
action energy between the host molecules and the impu-
rity surface (6) as follows [3]:
β(r−R,n0, ρ;Q) ∼= Wρ exp (−κ(|r−R| − dim)/σm)
(12a)
or
β(r−R,n0, ρ;Q) ∼= Wρδ+ (−κ(|r−R| − dim)/σm)
(12b)
for |r−R| > dim, where ρ(r) > 0 is the host-LC vol-
ume density, and δ+(ξ) is the asymmetric Dirac delta
function δ+(+0) = 1 [20]. W and κ are (fitting) pa-
rameters which depend on the surfactant kind and its
concentration on the impurity surface and characterize
the interaction energy between an impurity particle (at
R) and a nematic-LC element (at r) with a characteristic
linear size of σm
∼= 3
√
Vm. (Further, we use values of the
κ parameter in (12a) to make the ’screened’ impurity-
nematic interaction range of the order of the ’length’
of several nematic molecules.) The parametrization of
expressions (12a) and (12b) results from both the ex-
plicit form of functional dependences (2) and (3) with
regard for (4) and the fact that, without a nematic-LC
medium, the contribution concerned with director dis-
tortions must also disappear in (4) and (11).
We considered different types of the nematic host
medium (PAA, MBBA, 5CB, 8CB), but the results
turned out to be similar. Therefore, we will illustrate
them only for impurities in 8CB. Values of the elastic-
ity parameters, K1 and K3, are considered to have de-
pendence on the temperature according to experimental
data [21]. Calculations were carried out for the impu-
rity size dim = 5 × 10−4 cm that is the most common
experimental result.
528 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
INSTABILITY OF THE NEMATIC-LIKE PHASES FILLED WITH IMPURITIES
In the case of typical nematics, our numerical estima-
tions demonstrate that (11) can be reduced to
Ṽeff(k) ∼= Ũ(k)− W 2ρ2
0
vK1K3k4
×
×
[
(K3 −K1)|(k · G̃′n(k))|2 +K1k
2|G̃′n(k)|2
]
+
+
∣∣∣[(K3 −K1)(k · G̃′n(k))kz +K1k
2G̃′nz (k)
]∣∣∣2
[(K3 −K1)k2
z +K1k2]
, (13)
g̃′n(k) ≡ Wρ0
v G̃′n(k). We note that the third term on
the right-hand side of (13) gives a considerable contri-
bution to the total effective interaction energy. How-
ever, this contribution is neglected in the literature con-
cerned with phenomenological theories, but it was of-
ten investigated numerically [22]. The origin of this
term in (13) is related to a spatial inhomogeneity of
the nematic-host density, δρ̃(k), even in spite of the
explicit absence of derivatives with respect to the den-
sity. In addition, one can notice that the second and
third terms in (13) have competitive character, which
results in the formation of a local minimum of the effec-
tive interparticle interaction energy (13) corresponding
to certain nonzero values of the modulation wave vec-
tor.
Let us examine the simplified formula (13) for Fourier
components (with allowable wave vectors, k 6= 0) of
the renormalized impurity-impurity interaction energy,
ϒ̃(k) = vṼeff (k)
W 2ρ20
, which does not depend on the intrinsic
structure of the notionally chosen quasilattice describ-
ing the LC medium. For the chosen parametrization of
the interaction energy between the LC medium and the
impurity particles (6), (12a), and (12b), expression (13)
turns out to be dependent on both the magnitude, |k|,
and the kz component of the wave vector k. Surfaces
for ϒ̃(k) (corresponding to (13) and interaction energies
between the impurities with the radius dim = 5 × 10−4
cm) are plotted in Figs. 2 and 3.
Let us determine the wave vector k0 (k0 < 2π/(2dim))
which minimizes ϒ̃(k) and generates a modulated struc-
ture, i.e. it corresponds to its period that cannot be
smaller than the impurity radius. For impurity par-
ticles of the radius dim = 5 × 10−4 cm within the
host LC with K1 6= K3(6= K2), k0z = 2853 cm−1,
k0x = k0y = 1554 cm−1, which corresponds to impurity-
impurity periods az = 4.4dim ax = ay = 8.1dim
for the case of (12a) and k0z = 4010 cm−1, k0x =
a
b
Fig. 2. Dependence of Fourier components of the effective pairwise
interimpurity interaction on the wave vector for particles with the
radius dim = 5×10−4 cm in an LC medium with K1 6= K3(6= K2)
for (12a) (a); and (12b) (b)
k0y = 1777 cm−1, which corresponds to impurity-
impurity periods az = 3.1dim ax = ay = 7.1dim for
case of (12b). For the impurity particles of the ra-
dius dim = 5 × 10−4 cm within the host LC with
K1 = K3(6= K2), k0z = 2354 cm−1, k0x = k0y = 1636
cm−1, which corresponds to impurity-impurity periods
az = 5.3dim ax = ay = 7.7dim for the case of (12a)
and k0z = 5110 cm−1, k0x = k0y = 2529 cm−1, which
corresponds to impurity-impurity periods az = 3.4dim
ax = ay = 5dim for case of (12b). These collections
of periods weakly change along the entire temperature
interval of the nematic phase of an LC (306-313 K for
8CB). One can see that there are the long-range and
quasioscillatory characters of this interaction in its en-
ergy dependence on the interimpurity distance. Our cal-
culations demonstrate that a larger screening parame-
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5 529
A.V. KLESHCHONOK, V.YU. RESHETNYAK, V.A. TATARENKO
a
b
Fig. 3. Dependence of Fourier components of the effective pairwise
interimpurity interaction on the wave vector for particles with the
radius dim = 5×10−4 cm in the LC medium with K1 = K3(6= K2)
for (12a) (a); and (12b) (b)
ter corresponds to a smaller value of the wave vector
k0.
5. Conclusions
We have considered the stability loss against the forma-
tion of modulated structures, basing on the phenomeno-
logical theory of filled liquid crystals.
We have obtained the system of linearized equa-
tions which describe a distortion of the director field
and deviations of the nematic-density distribution and
the concentration of impurity particles from their ini-
tial homogeneous-state values. These equations take
into account the structure and the anisotropy of prop-
erties of the ’nematic LC + impurity’ system under
study.
We have obtained the condition of homogeneous-
distribution stability loss against the formation of mod-
ulated structures. This condition allows one to calculate
the temperature of stability loss and to estimate the pe-
riod of a formed structure, for instance, along a direction
of the undistorted director.
An expression for the Fourier components of the ef-
fective pairwise interaction energy between the impurity
particles is derived. It takes into account the direct and
indirect (by means of the nematic-LC medium) contri-
butions to this interaction.
We find that, along with an indirect contribu-
tion to the effective impurity-impurity interaction by
means of induced director-field distortions, a signifi-
cant contribution to its energy is given by the interac-
tion of impurities by means of spatial inhomogeneities
of the nematic-LC density associated with nematic-
director distortions and caused by impurities. Ex-
actly such indirect contributions act as a part of a
collective mechanism of formation of modulated struc-
tures.
There are the long-range and quasioscillatory char-
acters of this interaction in its energy dependence on
the interimpurity distance. In the realistic case of
K1 6= K3( 6= K2), the k-dependence of Fourier compo-
nents for such an interaction is non-analytic in a vicinity
of k = 0.
Generally, the interaction energy depends on the tem-
perature, density of a nematic-LC medium, and concen-
tration of impurities. This property allows controlling
the structure and properties of the studied system by
changing the external thermodynamic conditions.
APPENDIX A
In the case of k 6= 0, after applying the Lagrange–Euler equations,
we obtain the following system of linearized equations of state in
a vicinity of the stability-loss point, Tbif(ρ0, c0), for a uniform
distribution of the system characteristics:[
K1k
2
x +K2k
2
y +K3k
2
z + c0(g̃′′nxnx (0)− λ̃)
]
δñx(k) +
+
[
(K1 −K2)kxky + c0g̃
′′
nxny
(0)
]
δñy(k) +
+
[
(K1 −K3)kxkz + c0g̃
′′
nxnz
(0)
]
δñz(k) +
+
[
g̃′nx (k)
]
δc̃(k) +
[
c0g̃
′′
nxρ
(0)
]
δρ̃(k) = 0, (14)
[
(K1 −K2)kxky + c0g̃
′′
nxny
(0)
]
δñx(k) +
+
[
K1k
2
y +K2k
2
x +K3k
2
z + c0(g̃′′nyny (0)− λ̃)
]
δñy(k) +
+
[
(K1 −K3)kykz + c0g̃
′′
nynz (0)
]
δñz(k) +
+
[
g̃′ny (k)
]
δc̃(k) +
[
c0g̃
′′
nxρ
(0)
]
δρ̃(k) = 0, (15)
530 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
INSTABILITY OF THE NEMATIC-LIKE PHASES FILLED WITH IMPURITIES
[
(K1 −K3)kxkz + c0g̃
′′
nxnz (0)
]
δñx(k) +
+
[
(K1 −K3)kykz + c0g̃
′′
nynz
(0)
]
δñy(k) +
+
[
K1k
2
z +K3(k2
x + k2
y) + c0(g̃′′nznz (0)− λ̃)
]
δñz(k) +
+
[
g̃′nz (k)
]
δc̃(k) +
[
c0g̃
′′
nzρ
(0)
]
δρ̃(k) = 0, (16)
[
c0g̃
′′
nxρ
(0)
]
δñx(k) +
[
c0g̃
′′
nyρ
(0)
]
δñy(k) +
+
[
c0g̃
′′
nzρ
(0)
]
δñz(k) +
[
g̃′ρ(k)
]
δc̃(k) +
+
[
c0g̃
′′
ρρ(0) + vẼ(k) + (kBT )/ρ0
]
δρ̃(k) = 0, (17)
[
g̃′∗nx (k)
]
δñx(k) +
[
g̃′∗ny (k)
]
δñy(k) +
[
g̃′∗nz (k)
]
δñz(k) +
+
1
v
[
Ũ(k) +
kBT
c0(1− c0)
]
δc̃(k) +
[
g̃′∗ρ (k)
]
δρ̃(k) = 0, (18)
where λ̃ = 2λ/c0 is a renormalized Lagrange multiplier. The sys-
tem of equations (14)-(18) is obtained in the general case, i.e. with-
out any assumptions about the interaction between LC molecules
and impurities, which allows us to use it under various anchoring
conditions on the impurity surface.
For k = 0 separately, applying the Lagrange–Euler equations,
∂F
∂δñx(0)
= ∂F
∂δñy(0)
= ∂F
∂δñz(0)
= 0, one can find a system of
equations for the Lagrange-multiplier with the use of Eq.(5):[
g̃′′nxnx (0)− λ̃
]
δñx(0) + g̃′′nxny (0)δñy(0) +
+g̃′′nxnz (0)δñz(0) = −g̃′nx (0), (19)
g̃′′nxny (0)δñx(0) +
[
g̃′′nyny (0)− λ̃
]
δñy(0) +
+g̃′′nynz (0)δñz(0) = −g̃′ny (0), (20)
g̃′′nxnz (0)δñx(0) + g̃′′nynz (0)δñy(0) +
+
[
g̃′′nznz (0)− λ̃
]
δñz(0) = λ̃− g̃′nz (0). (21)
APPENDIX B
Let us determine the Fourier components which are the parts of
(10) and (11):
g̃′nx (k) =
8π
v
kxkz
k2
×
×
∞∫
dim
dsβ0(s)s2
[
3 sin(ks)
(ks)3
−
3 cos(ks)
(ks)2
−
sin(ks)
ks
]
, (22)
g̃′ny (k) =
8π
v
kykz
k2
×
×
∞∫
dim
dsβ0(s)s2
[
3 sin(ks)
(ks)3
−
3 cos(ks)
(ks)2
−
sin(ks)
ks
]
, (23)
g̃′nz (k) = −
8π
3v
∞∫
dim
dsβ0(s)s2
sin(ks)
ks
+
8π
3v
3k2
z − k2
k2
×
×
∞∫
dim
dsβ0(s)s2
[
3 sin(ks)
(ks)3
−
3 cos(ks)
(ks)2
−
sin(ks)
ks
]
, (24)
g̃′ρ(k) = −
4π
3v
∞∫
dim
dsβ′ρ0(s)s2
sin(ks)
ks
+
4π
3v
3k2
z − k2
k2
×
×
∞∫
dim
dsβ′ρ0(s)s2
[
3 sin(ks)
(ks)3
−
3 cos(ks)
(ks)2
−
sin(ks)
ks
]
, (25)
where dim is the impurity particle radius, β0(s) = β(|r−R|, ρ0),
β′ρ0(s) = β′ρ(|r−R|, ρ0)|ρ0 , s = |r−R|, and, k2 = |k| =
k2
x + k2
y + k2
z .
In the case of k = 0, we obtain
g̃0(0) = −
4π
3v
∞∫
dim
dsβ0(s)s2, (26)
g̃′nx (0) = g̃′ny (0) = 0, g̃′nz (0) = 2g̃0(0), (27)
g̃′ρ(0) = g̃′0ρ (0) = 0, g̃′′ρρ(0) = g̃′′0ρρ (0),
g̃′′nzρ(0) = 2g̃′0ρ (0), (28)
g̃′′nxnx (0) = g̃′′nyny (0) = g̃′′nznz (0) = 2g̃′0(0), (29)
g̃′′nxny (0) = g̃′′nynz (0) = g̃′′nynz (0) =
= g̃′′nxρ(0) = g̃′′nyρ(0) = 0. (30)
Therefore, considering the case of parametrization (6) and Eqs.
(19)–(21), one can obtain the Lagrange multiplayer: λ = c0g̃0(0).
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Received 13.10.09
НЕСТАБIЛЬНIСТЬ НЕМАТИЧНО-ПОДIБНИХ
ФАЗ, НАПОВНЕНИХ ДОМIШКАМИ, ЩОДО
ФОРМУВАННЯ МОДУЛЬОВАНИХ
СТРУКТУР
А.В. Клещонок, В.Ю. Решетняк, В.А. Татаренко
Р е з ю м е
Розглянуто просторовий розподiл домiшкових жорстких сфе-
роподiбних макрочастинок у мезоморфному рiдкокристалiчно-
му носiї. Використовуючи континуальнi статично-механiчнi те-
орiї, проаналiзовано термодинамiчнi умови, необхiднi для по-
яви модульованих шаруватих надструктур. Мiж домiшкови-
ми частинками присутня ефективна далекодiйна взаємодiя.
Вважається, що вона вiдповiдає за формування структур. У
загальному випадку така взаємодiя мiстить два внески: пря-
му взаємодiю типу Ван-дер-Ваальса та непряму (через поле
викривлення директора). Остання залежить вiд температу-
ри зразка та концентрацiї частинок. Ця ефективна взаємодiя
контролює структуру та властивостi системи. Було одержано
аналiтичнi розв’язки для розподiлу поля директора, неоднорi-
дностi густини носiя, температури формування модульованих
структур та їх просторового перiоду. Запропонований теорети-
чний пiдхiд може бути застосований до iнших анiзотропних та
неоднорiдних систем.
532 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 5
|
| id | nasplib_isofts_kiev_ua-123456789-56193 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 2071-0194 |
| language | English |
| last_indexed | 2025-11-28T03:09:15Z |
| publishDate | 2010 |
| publisher | Відділення фізики і астрономії НАН України |
| record_format | dspace |
| spelling | Kleshchonok, A.V. Reshetnyak, V.Yu. Tatarenko, V.A. 2014-02-13T19:58:28Z 2014-02-13T19:58:28Z 2010 Instability of the Nematic-like Phases Filled with Impurities Against the Formation of Modulated Structures / A.V. Kleshchonok, V.Yu. Reshetnyak, V.A. Tatarenko // Український фізичний журнал. — 2010. — Т. 55, № 5. — С. 524-532. — Бібліогр.: 22 назв. — англ. 2071-0194 PACS 61.30.Dk, 61.30.Jf, 82.70.Dd, 82.70.Kj, 83.80.Xz https://nasplib.isofts.kiev.ua/handle/123456789/56193 We examine spatial distribution of impurity rigid-sphere-like macroparticles in the mesomorphic liquid crystal host. Using continuum statistical mechanical theories, we analyze the thermodynamic conditions necessary for a modulated lamellar-structure to appear. There is a long-range effective interaction between the impurity particles. This interaction is considered as being responsible for the formation of superstructures. In the general case, this interaction includes two components: a van der Waals-type direct interaction and an indirect interaction (through the director-field distortions). The last one depends on both the temperature of a sample and the concentration of particles. This effective interaction controls the structure and properties of the system. Analytical solutions for a director-field distortion, density inhomogeneity of the host medium, temperature of the formation of a modulated structure, and its spatial period are obtained. The proposed theoretical approach can be applied to other anisotropic and inhomogeneous systems. Розглянуто просторовий розподiл домiшкових жорстких сфероподiбних макрочастинок у мезоморфному рiдкокристалiчному носiї. Використовуючи континуальнi статично-механiчнi теорiї, проаналiзовано термодинамiчнi умови, необхiднi для появи модульованих шаруватих надструктур. Мiж домiшковими частинками присутня ефективна далекодiйна взаємодiя. Вважається, що вона вiдповiдає за формування структур. У загальному випадку така взаємодiя мiстить два внески: пряму взаємодiю типу Ван-дер-Ваальса та непряму (через поле викривлення директора). Остання залежить вiд температури зразка та концентрацiї частинок. Ця ефективна взаємодiя контролює структуру та властивостi системи. Було одержано аналiтичнi розв’язки для розподiлу поля директора, неоднорiдностi густини носiя, температури формування модульованих структур та їх просторового перiоду. Запропонований теоретичний пiдхiд може бути застосований до iнших анiзотропних та неоднорiдних систем. en Відділення фізики і астрономії НАН України Український фізичний журнал М'яка речовина Instability of the Nematic-like Phases Filled with Impurities Against the Formation of Modulated Structures Нестабільність нематично-подібних фаз, наповнених домішками, щодо формування модульованих структур Article published earlier |
| spellingShingle | Instability of the Nematic-like Phases Filled with Impurities Against the Formation of Modulated Structures Kleshchonok, A.V. Reshetnyak, V.Yu. Tatarenko, V.A. М'яка речовина |
| title | Instability of the Nematic-like Phases Filled with Impurities Against the Formation of Modulated Structures |
| title_alt | Нестабільність нематично-подібних фаз, наповнених домішками, щодо формування модульованих структур |
| title_full | Instability of the Nematic-like Phases Filled with Impurities Against the Formation of Modulated Structures |
| title_fullStr | Instability of the Nematic-like Phases Filled with Impurities Against the Formation of Modulated Structures |
| title_full_unstemmed | Instability of the Nematic-like Phases Filled with Impurities Against the Formation of Modulated Structures |
| title_short | Instability of the Nematic-like Phases Filled with Impurities Against the Formation of Modulated Structures |
| title_sort | instability of the nematic-like phases filled with impurities against the formation of modulated structures |
| topic | М'яка речовина |
| topic_facet | М'яка речовина |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/56193 |
| work_keys_str_mv | AT kleshchonokav instabilityofthenematiclikephasesfilledwithimpuritiesagainsttheformationofmodulatedstructures AT reshetnyakvyu instabilityofthenematiclikephasesfilledwithimpuritiesagainsttheformationofmodulatedstructures AT tatarenkova instabilityofthenematiclikephasesfilledwithimpuritiesagainsttheformationofmodulatedstructures AT kleshchonokav nestabílʹnístʹnematičnopodíbnihfaznapovnenihdomíškamiŝodoformuvannâmodulʹovanihstruktur AT reshetnyakvyu nestabílʹnístʹnematičnopodíbnihfaznapovnenihdomíškamiŝodoformuvannâmodulʹovanihstruktur AT tatarenkova nestabílʹnístʹnematičnopodíbnihfaznapovnenihdomíškamiŝodoformuvannâmodulʹovanihstruktur |