On sound wave spectra in biaxial nematic crystals
On the basis of the Hamiltonian approach dynamics of biaxial nematics surveyed and the deduction of the nonlinear equations of ideal hydrodynamics, taking into account the shape, (disk-shaped and rod-shaped of a molecule) and conformation degree of freedom are given. The densities of additive integr...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Cite this: | On sound wave spectra in biaxial nematic crystals / М.Y. Kovalevsky, А.L. Shishkin // Вопросы атомной науки и техники. — 2001. — № 6. — С. 309-314. — Бібліогр.: 17 назв. — англ. |
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| author | Kovalevsky, М.Y. Shishkin, А.L. |
| author_facet | Kovalevsky, М.Y. Shishkin, А.L. |
| citation_txt | On sound wave spectra in biaxial nematic crystals / М.Y. Kovalevsky, А.L. Shishkin // Вопросы атомной науки и техники. — 2001. — № 6. — С. 309-314. — Бібліогр.: 17 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | On the basis of the Hamiltonian approach dynamics of biaxial nematics surveyed and the deduction of the nonlinear equations of ideal hydrodynamics, taking into account the shape, (disk-shaped and rod-shaped of a molecule) and conformation degree of freedom are given. The densities of additive integrals of motion and relevant fluxes are represented in the terms of a thermodynamic potential. The spectra of linear oscillations in these liquid crystals surveyed. Two branches of ultrasonic oscillations are found and the character of an anisotropy of both velocities of sounds is found out.
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ON SOUND WAVE SPECTRA IN BIAXIAL NEMATIC CRYSTALS
М.Yu. Kovalevsky, А.L. Shishkin
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: mik@kipt.kharkov.ua
On the basis of the Hamiltonian approach dynamics of biaxial nematics surveyed and the deduction of the
nonlinear equations of ideal hydrodynamics, taking into account the shape, (disk-shaped and rod-shaped of a
molecule) and conformation degree of freedom are given. The densities of additive integrals of motion and relevant
fluxes are represented in the terms of a thermodynamic potential. The spectra of linear oscillations in these liquid
crystals surveyed. Two branches of ultrasonic oscillations are found and the character of an anisotropy of both
velocities of sounds is found out.
PACS: 61.30 Cz, 62.60+v
I. INTRODUCTION
The liquid crystal (LC) dynamic equations in the
traditional phenomenological approach are obtained
proceeding from the symmetry considerations as
accounted for by the conservation laws. In this case, the
accurate accounting for non-linear items presents a
problem. The Hamiltonian approach is exactly more
consistent [1]. This approach provides for an effective
method for construction of the non-linear dynamic
equations describing the transfer phenomena in various
condensed media. In the Hamiltonian approach, the
Poisson bracket (PB) structure for reduced description
parameters that settles completely the medium state at
the macroscopic level is of primary importance.
Selection of the reduced description parameters in
the case of LC depends on a number of factors. Some
hydrodynamic parameters are associated with the
Hamiltonian symmetry properties, manifested as the
presence of dynamic equations conditioned by the
differential conservation laws. The molecular shape is
another factor influencing the set of hydrodynamic
parameters. In LC, there is a correlation between the
molecular shape and the structure of hydrodynamic
equations. The Poisson bracket structure of
hydrodynamic parameters has been shown to be of
different form for disc- and rod-shaped molecules [2-6].
This molecular shape effect is manifested physically as
different signs of the reactive coefficient in the
hydrodynamic equations, different possibilities of
ferroelectric state realization and spectral peculiarities
of the polarized light absorption. In [5,6], a specific
feature of the mentioned correlation between the
molecular shape and hydrodynamics has been clarified,
manifested, firstly, as distinctions in the Poisson
brackets structure for the reduced description
parameters, and, secondly, as an increased number of
the mentioned parameters at the hydrodynamic
evolution stage. From the mathematical standpoint, the
appearance of an additional quantity is due to the
requirement of the closed Poisson bracket algebra for
the entire set of hydrodynamic variables. Physically, this
is associated with the fact that several characteristic
lengths and relaxation time values are present in such a
medium.
Near the phase transition temperature in sufficiently
strong electric or magnetic external fields, additional
components of the order parameter are to be taken into
account in low-dimensional cases (n<3), the parameter
being a symmetric and hole-free tensor [7−10]. In this
connection, analogy with the super-fluid Bose-liquid
should be noted; for the latter, near the phase transition
region, or in the super-fluid aerogel state, the set of the
reduced description parameters should also be extended
in order to take into account not only the order
parameter phase but also its module [11,12]. At last, the
parameter set is associated with the character of
spontaneous symmetry disturbance in the system. The
elasticity theory is formulated as part of the continuous
medium mechanics, based on the concept of
spontaneously disturbed translation symmetry. It is just
the deformation tensor that represents the dynamic
quantity in the set of reduced description parameters
associated with such symmetry disturbance. That tensor
can be presented in a certain manner in terms of the
distortion one [13]. The latter quantity reflects
completely the strain character of the continuous
medium, but its introduction as an additional dynamic
quantity is redundant as a rule. The LC hydrodynamic
theory can also be viewed as continuous medium
mechanics with spontaneously disturbed symmetry. In
this case, there is the symmetry disturbance with respect
to rotations in the configuration space. It has been
shown [4-6] that additional hydrodynamic parameters
associated with this symmetry disturbance can also be
presented in terms of the distortion tensor for many
uniaxial LC.
The hydrodynamics of biaxial LC has been
considered in [14-17]. The liquid crystalline ordering is
described by a field of three unit vectors. For this LC
class, the typical case is the full spontaneous symmetry
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 309-314. 53
disturbance with respect to rotations in the configuration
space O(3). The mentioned works, however, do not
present in explicit form the expressions for all reactive
flux densities of the motion integrals in terms of the
energy functional. The molecular shape effect on the
dynamic equations for this LC class is also not
elucidated.
In this work, some additional hydrodynamic
quantities associated with spontaneous symmetry
disturbance with respect to rotations in the configuration
space are introduced in terms of the distortion tensor.
The densities and fluxes of the additive motion integrals
are presented in terms of the thermodynamic potential.
The ideal hydrodynamic equations are derived. The
spectra of collective excitations for biaxial LC are
studied taking into account their molecular shape.
Firstly, let the Hamiltonian formalism be considered
briefly for the case of LC. Let the system Lagrangian be
presented as
∫ −′≡−= HxxxxFdHkLL )())(,(3)
.
,( αϕϕαϕϕ
,where )
.
,( ϕϕkL is the kinematic part of the
Lagrangian; ∫= )(3)( ϕεϕ xdн - the Hamiltonian;
))(,( xxF ′ϕα , a certain functional of dynamic variables
ϕα(x). From the stationary action principle, the motion
equations for the ϕα(x) field components are presented
as follows:
( )
( ) ( )
)()(
,',J
x
xF
x
xF
xx
′
−
′
≡
βδ ϕ
αδ
αδ ϕ
βδ
ϕα β . (1.1)
Let the Poisson brackets for the arbitrary functional
values A(ϕ), B(ϕ) be defined by the equality
{ } ∫ ∫
′
′−′≡
)(
),(1
)(
33,
x
B
xxJ
x
A
xdxdBA
βδ ϕ
δ
α β
αδ ϕ
δ
.
Then, the equations (1.1) take on the Hamiltonian form:
{ }Hxx ),()(
.
αϕαϕ = . (1.2)
The Hamiltonian approach allows one to construct
differential conservation laws without specifying
explicitly the Hamiltonian form, but accounting only for
its symmetry properties. To this purpose, note that the
motion equation for the density a(x) of the arbitrary
physical quantity ∫= )(3 xxadA can be presented [4]
as:
( ) { } { }
{ }∫ ∫ ′−−′+′′=
∇−≡
1
0
))1((),(3)(
),()(,),(=xa
xxxxadkxxdxka
xkakxAHxa
λελλ
ε
(1.3)
These relationships permit to express the flux
densities in terms of the Poisson brackets for the
corresponding densities of the additive motion integrals.
Assuming in (1.3) ( ) ( )xxa ρ= , where ( )xρ is the
substance density, and regarding
( ){ } 0, , )(3 =∫= xМxxdM ερ ,
we obtain the differential law of mass conservation:
{ }∫ ∫ ′−−′+′′=
− ∇=
1
0
))1((),(3)(
,)()(
.
xxxxdkxxdxkj
xkjkx
λελρλ
ρ
(1.4)
Here, ( )xkj is the mass flux density. At
( ) ( )xxa kπ= , (where ( )xkπ is the momentum
density) and under condition of the translation
invariance of energy density,
{ }∫ ∇== )()(, ,)(3 xixiPxixdiP εεπ
we obtain the differential law of momentum
conservation
{ }∫ ∫ ′−−′+′′+
+−=
− ∇=
1
0
))1((),(3
)()(
),()(
.
xxxxidkxxd
ikxxikt
xiktkxi
λελπλ
δε
π
(1.5)
Here, ( )xikt is the momentum flux density. At
( ) ( )xxa ε= , we obtain the differential law of energy
conservation from (1.3):
{ }∫ ∫ ′−−′+′′=
− ∇=
1
0
))1((),(3
2
1
)(
,)()(
.
xxxxdkxxdxkq
xkqkx
λελελ
ε
(1.6)
where ( )xkq is the energy flux density.
2. THE POISSON BRACKET STRUCTURE
FOR CLASSIC CONTINUOUS MEDIA
In terms of the Lagrangian variables iζ , the
position of a particle in a medium is characterized by the
functions ),( tkx ζ . The Lagrangian coordinates have
the physical sense of particle coordinates in the initial
position complying with the strain-free medium state.
Let the system Lagrangian be written as:
∫== ),(3 -
∂ ζ
∂
πζ ε
x
dHH , kLL ,
( ) ( )ζζζ π ixidkL
.3∫= ,
where )),(()(
∂ ζ
∂
ζπεζε
x
= is the energy density and
)(ζπ k is the momentum density in the Lagrangian
variables; kL - the kinematic Lagrangian part. Let the
latter expression be transformed using the Eulerian
54
)(
),,(13)(
.
x
H
xxJxdx
′
′−∫ ′=
βδ ϕ
δ
ϕα βαϕ
variables ),( tix ζ and introducing the displacement
vector of continuous medium particles ( )txui , :
),()( txiuitix += ζ . (2.1)
Bearing in mind that
( ) ( ) )()( , x
.1.
xiujjixijbiuxjibjx ∇−≡−= δ (2.2)
let the kinematic Lagrangian part be presented in the
form of ( ) )(
.
)()(
.
xxxkuxkpkL ψσ−= , where
)(1))()()(()( xikbxixxixkp −∇−= ψσπ is the
generalized momentum density in the Eulerian
coordinates. The variable )(xψ that is canonically
conjugate with the entropy density )(xσ is believed to
be cyclic. Using this kinematic part, one can obtain the
PB for the variables )(xσ , )(xψ , )(),( xiuxiπ ,
proceeding from such transformations that retain the
invariant kinematic part of the Lagrangian
( ) ( )
0)()(
, )( , )(
==
==
xxip
xbxxiaxiu
δ σδ
δ ψδ
where the functions ( ) ( )xbxia , are independent of
)(),(),(),(р xxiuxxi ψσ . As a result, we have the PB
set:
{ }
{ } ( ) ( ),)(i(x),
, )()()(),(
xixxx
xxxkibxixku
ψδπψ
δπ
∇′−−=′
′−′=′
{ }
( ) ( ){ } ( ),,
,)()()(),(
xxxx
xxixxxi
′−=′
′−∇−=′
δψσ
δσσπ
{ }
)()(
)()()(),(
xxjxi
xxixjxjxi
′−∇′−
−′−∇ ′=′
δπ
δπππ
(2.3)
Now, let us take into account the fact that if ρ~ is the
substance density referred to the strain-free unit volume,
then the true density ρ is defined as:
( ) )(det~ xijbx ρρ = (2.4)
Let’s find PB for the variables ( ) ( )', xxk ρπ taking into
account (2.2), (2.3), (2.4):
{ } )()()(),( xxixxxi ′−∇ ′=′ δρρπ (2.5)
In the follow-up, the energy density ( )xε is assumed
to be translationally invariant, thus, not being dependent
on the displacement vector ( )xku , but rather on its
derivatives ikwkui ≡∇ ( ikw is the distortion vector)
or, the same, on the strain matrix )(xstb . Therefore, it
is convenient to select directly the bst(x) as a dynamic
variable along with others. For bst(x), the only non-zero
PB has the form
{ } )()()(),( xxjxkibxkjbxi ′−∇−=′ δπ . (2.6)
The PB set (2.3),(2.5),(2.6) is the construction base
for hydrodynamic type equations for normal liquids,
crystals and LC.
3. DYNAMICS OF BIAXIAL NEMATIC
CRYSTALS
In the LC biaxial nematic phase, the complete
rotational symmetry spontaneous disturbance takes
place. Along with the dynamic variables describing the
isotropic liquid state, namely, the mass density ( )xρ ,
the momentum density )(xiπ , the entropy density ( )xσ
, additional parameters associated with the rotational
symmetry disturbance in the configuration space should
be determined. Two anisotropy axes have been included
in the reduced description parameters in [14–17]; these
works have a phenomenological character that is not
connected with the Hamiltonian formalism. In the case
of disc-shaped LC, let the unit and orthogonal symmetry
axes, characterizing the rotational invariance
disturbance ( ) ( )xnxm , and the shape factor ( )xp , be
defined as
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ,
,
xaxbxbxa
xiaxbxibxa
xim
xaxbxbxa
xiaxbxibxa
xin
−
−
≡
+
+
≡
(3.1)
( ) ( )
( ) ( )
−≡
xbxa
xbxa
xp
1
2
1
)( .
The vectors ( )xa , ( )xb
in terms of the strain matrix
( )xijb have the form )(1)( xkjbkexja ≡ ,
)(2)( xkjbkexjb ≡ . Here, keke 2,1 are the
orthogonal constant vectors defining the anisotropy axes
in the strain-free state ( ) ( )xaxa = , ( ) ( )xbxb
= . The
orientation degrees of freedom ( ) ( ) ( )xpxnxm ,, are
defined not as local functions of the strain field u , but
as those of their derivatives ( )xkui∇ . Therefore, the
local relations of four quantities ( ) ( ) ( )xpxnxm ,, with
nine ( )xkui∇ ones will not result in overflow of the
dynamic equation system, and the quantities
( ) ( ) ( )xpxnxm ,, can be considered as independent
variables. According to (3.1) and taking into account
(2.6), we obtain the PB for the variables
( ) ( ) ( )xkmxknxk ,,π :
{ }
( ) , )('
)()()(),(
xxxjif
xjnixxxjnxi
′−∇ ′+
+∇′−=′
δλλ
δπ
(3.2)
55
{ }
( ) , )('
)()()(),(
xxxjig
xjmixxxjmxi
′−∇ ′+
+∇′−=′
δλλ
δπ
where the quantities ( )xjif λ , ( )xjig λ and ))(( xfkj
⊥δ
are defined by the equalities:
( )
( ) ( ))()()()()(
))((
ximxnxmxinxjmxp
xnjinxjif
λλ
λδλ
+−
−⊥≡
,
( )
( )( ) ( ))()()()()(1
))((
ximxnxmxinxjnxp
xmjimxjig
λλ
λδλ
+−−
−⊥≡
,
)()())(( xjfxkfkjxfkj −≡⊥ δδ .
The scalar parameter ( )xp is defined, according to
(3.1), by the angle between the strained LC axes. It
follows from (3.1) accounting for (2.6),(2.3) that the
nonzero PB for the variable ( )xp with other biaxial
nematic crystal hydrodynamic parameters has the form
{ }
( ) ( ) )()()()()()(1
)(2)()()(),(
xxlxlmximxlnxinxp
xpxpixxxpxi
′−∇ ′′′−′′′−×
×′−∇′−=′
δ
δπ
(3.3)
The Poisson brackets (3.2),(3.3) along with the
(2.3),(2.5),(2.6) ones form closed algebra of
hydrodynamic quantities for the biaxial nematic crystals
with disc-shaped molecules. It is just this instance that
makes it necessary to include the p(x) quantity in this set
of variables.
Now let us consider the motion equation in the
Hamiltonian form (1.2), taking into account the
(3.2),(3.3),(2.3) PB form and explicit expressions for the
flux density of additive integrals (1.4),(1.5). Assuming
the Hamiltonian has the Galilean invariant form
∫ ∇∇Φ+=
),,,,,,(
)(2
)(2
3 pmmnn
x
xixdH σρ
ρ
π
.
(3.4)
we obtain the following dynamic equations for this
biaxial phase:
( )( ) )()(
.
, )()(
.
xiixxivxix πρσσ − ∇=− ∇= ,
)()(
.
xiktkxi − ∇=π , (3.5)
( ) ,)()()()(
.
xivxjifxjnsxsvxjn λλ ∇−∇−=
( ) ( ) )()()( xivxjigxjmsxsvxjm λλ ∇−∇−= ,
( ) ( )
( ) )()()()()(
)(1)(2)()(
.
xlvkxlmxkmxlnxkn
xpxpxpsxsvxp
∇−×
×−+∇−=
.
Here,
( )
( )x
xi
xi
x
xiv
ρ
π
∂ π
∂ ε
=≡
)(
)(
)( is the medium unit
mass velocity. The momentum flux density )(xikt can
be found using the PB set (2.3),(2.5),(3.2),(3.3) and the
formula (1.5). As a result, we obtain the following
expression for this flux in terms of energy density:
=)(xikt ( ) ( )xiktxikt ′+0 , (3.6)
ik
kiikt δ
∂ ρ
∂
ρ
∂ σ
∂
σ
ρ
ππ
Φ−
Φ
+
Φ
+=0 ,
+∇
∇
+∇
∇
=′
jmi
jmk
jni
jnk
ikt
∂
∂ ε
∂
∂ ε
( ) ( ) . 12 kmimknin
p
pp
lmj
j
lmiklg
lnj
j
lniklf
−−−
−
∇
∇−+
∇
∇−+
∂
∂ ε
∂
∂ ε
∂
∂ ε
∂
∂ ε
∂
∂ ε
According to (3.4), the second law of thermodynamics
has the form
( ) ( )
( ) ( )
⊥
∇
∇+⊥+
+⊥
∇
∇+⊥+
++++=
kdmmik
imj
jkdmmik
im
kdnnik
inj
jkdnnik
in
dp
pidivTddd
δ
∂
∂ ε
δ
δ
δ ε
δ
∂
∂ ε
δ
δ
δ ε
∂
∂ ε
πσρρµε
(3.7)
where T - is the temperature; µ - the chemical
potential.
The formulae (3.5,6) represent a complete set of
equations for the ideal hydrodynamics of biaxial nematic
crystals consisting of disc-shaped molecules.
In the case of rod-shaped molecules, let the unit and
orthogonal anisotropy axes characterizing the rotational
invariance disturbance, ( ) ( )xnxm , and the shape factor
( )xp be defined as
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ,
,
xAxBxBxA
xiAxBxiBxA
xim
xAxBxBxA
xiAxBxiBxA
xin
−
−
=
+
+
=
(3.8)
( ) ( )
( ) ( )
−=
xBxA
xBxA
xp
1
2
1
)( . (3.9)
Here, the vectors )(1
1)( xkjbkexjA −≡ and
)(1
2)( xkjbkexjB −≡ are set in terms of the reverse
matrix ( )xkjb 1− .Taking into account this definition (3.9)
and the relation (2.6), we obtain the Poisson brackets
56
{ }
( ) ,)('
)()()(),(
xxxjif
xjnixxxjnxi
′−∇ ′+
+∇′−=′
δλλ
δπ
(3.10)
{ }
( ) ,)('
)()()(),(
xxxjig
xjmixxxjmxi
′−∇ ′+
+∇′−=′
δλλ
δπ
{ }
( ) ( ) )()()()()()(1
)(2)()()(),(
xxlxlmximxlnxinxp
xpxpixxxpxi
′−∇ ′′′−′′′−×
×′+∇′−=′
δ
δπ
where the following notations are used:
( ) ( )
( ),)()()()(
)())((
ximxnxmxin
xjmxpxnijnxjif
λλ
δλλ
+×
×+⊥−≡
( ) ( )( )
( ).)()()()(
)(1))((
ximxnxmxin
xjnxpxmijmxjig
λλ
δλλ
+×
×−+⊥−≡
Making further treatment in a manner similar to the
above case of disc-shaped molecules, it is not difficult to
derive equations for ideal hydrodynamics of rod-shaped
molecules:
( )( ) )()(
.
, )()(
.
xiixxivxix πρσσ − ∇=− ∇= ,
)()(
.
xiktkxi − ∇=π , (3.11)
( ) ,)()()()(
.
xivxjifxjnsxsvxjn λλ ∇−∇−=
( ) ( ) )()()( xivxjigxjmsxsvxjm λλ ∇−∇−= ,
( ) ( )
( ) )()()()()(
)(1)(2)()(
.
xlvkxlmxkmxlnxkn
xpxpxpsxsvxp
∇−×
×−−∇−=
In this case, the momentum flux density has the form
=)(xikt ( ) ( )xiktxikt ′+0 ,
ik
ki
ikt δΦ−
∂ ρ
Φ∂
ρ+
∂ σ
Φ∂
σ+
ρ
ππ
=
0 , (3.12)
( ) ( ) .12
δ
δ ε
++−
∂
∂ ε
−+
+
δ
δ ε
+∇
∇∂
∂ ε
+∇
∇∂
∂ ε
=′
lmiklgkmimknin
p
pp
lniklfjmi
jmk
jni
jnk
ikt
4. SPECTRA OF COLLECTIVE
EXITATIONS
Let’s consider the spectra of collective excitations
based on the equations for biaxial nematic crystals. The
equilibrium state of such a medium is believed to be
homogeneous, strain-free ( 21/=p ), while the medium
as a whole is at rest ( 0=kv ). From equations (4.2) we
obtain the dispersion equation for determination of the
collective excitation spectra:
( ) ( ) 02
2
4
12det =
∂
∂
−
∂
∂
− kjRkiR
p
P
jkikij
ε
ρρ
δω
where ( ) ( ) ( )( )kmimkninkiR
−≡ . Expanding the
determinant, we obtain
( ) ( )( ) 024
242 =+− kLkL
ωωω (4.1)
Here, the following notations are used:
( ) ( )( ) 02
||
22222
4 >+≡×+= AkckAnmkckkL
,
( ) ( ) ( ) ( ) ( )( )[ ]222
||
22
42
2 mknkkmknkAckL ++=
where
02 >
∂
∂
=
ρ
P
c , 02
2
4
1
>
∂
∂
=
p
A
ε
ρ
.
Thus, we see that in the biaxial nematic crystals under
consideration, propagation of two acoustic vibration
branches is quite possible:
( ) ( ) ( ) ( ) 22
242
442
12 k
k
k
ckLkLkLk
±≡−±=±
ω
(4.2)
corresponding to the 1st and 2nd sound, respectively. The
solution of (4.2) with the positive sign complies with a
branch similar to the 1st sound occurring in the normal
liquid. The solution with the negative sign corresponds
to a new excitation branch due to the presence of LC
biaxiality; in general, the shape factor p appears for
such LC states. For both solutions, the sound speed
anisotropy is of significant importance. Introducing the
unit vector kke /
≡ , in the spherical coordinates
ϕθ cossin=me
, ϕθ sinsin=ne , θcos=le
,
where θ,ϕ are the polar and asimuthal angles,
respectively, defining the direction of the wave vector
e . In terms of these variables, the speeds c± (4.2) have
the following form
( ) ( )
( ) ] .2/12cos22sin2sin2sin4
22sin12sin12
,22
θ+ϕθθλ−
−θλ+±θλ+=
ϕθ±
c
c
(4.3)
In the Figures, the character of anisotropy, as defined
by (4.3), is presented (three vectors lnm
,, form the
Cartesian orthogonal coordinate system).
Comparing formulae (4.3) to the results presented in
[14–16], it should be noted that in the latter, the
additional modes related to the disturbed symmetry with
respect to rotations in the configuration space are of
dissipative character where the reactive component is
absent. The accounting for the shape factor in the
hydrodynamic equations for biaxial nematic crystals
results in the reactive component being in the second
sound spectrum that is already under adiabatic
approximation.
The linearization of the hydrodynamic equations
(3.11),(3.12) corresponding to the case of rod-shaped
57
molecules results in the same dispersion equation and,
respectively, in the spectra defined by (4.3). This result
is similar to the case of uniaxial nematic crystals
considered before [8] where the distinction in
hydrodynamic equations due to the molecular shape
does not manifest itself in the spectra under the main
approximation and appears only while accounting for
the next approximation, namely, the dissipative one.
a)
b)
Fig. 1. The angular dependence of speeds c+ (a) and
c– (b) at the parameter value λ≡A/c2=0.1
a)
b)
Fig. 2. The angular dependence of speeds c+ (a) and
c– (b) at the parameter value λ≡A/c2=1
58
REFERENCES
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2. G.E. Volovik. About connection between
shape of molecule and hydrodynamic in nematics //
Pis'ma Zh. Eksper. Teor. Fiz. 1980, v.31, p. 297-
300.
3. V.V. Lebedev, E.M. Kats. Dynamics of
Liquid Crystals. Nauka: “Moscow”, 1988, 144 p.
4. A.A. Isayev, M.Yu. Kovalevsky,
S.V. Peletminsky. On construction of Poisson
brackets and dynamics of liquid crystals // Mod.
Phys. Lett. B. 1994, v. 8, №11, p. 677-686.
5. M.Yu. Kovalevsky, V.V. Kuznetsov.
Hamiltonian dynamics of biaxial nematics // Dokl.
Akad. Nauk Ukrainy. 1999, №12, p. 90-95.
6. M.Yu. Kovalevsky, A.L. Shishkin. To hydro-
dynamic of uniaxial nematics with conformation
degree of freedom // The Journal of Kharkov
National University. Physical series ''Nuclei,
Particles, Fields''. 2001, №510, Issue 1/13, p. 31-
36.
7. G.E. Volovik, E.M. Kats. On nonlinear
hydrodynamics of liquid crystals // Zh. Eksper.
Teor. Fiz. 1981, v. 81, p. 240-248.
8. X. Zhuang, L. Marrucci, Y.R. Shen. Surface-
Monolayer Induced Bulk Alignment of Liquid
Crystals. // Phys. Rev. Lett. 1994, v. 73, p. 1513-
1516.
9. T. Qian, P. Sheng. Generalized
hydrodynamical equations for nematic liquid
crystals // Phys. Rev. E. 1998, v. 58, p. 7475-7485.
10. S. Blenk, H. Ehrentraut, W. Muschik.
Statistical foundation of macroscopic balances for
liquid crystals in alignment tensor formulation //
Physica A. 1991, v. 175, p. 119-138.
11. L.P. Pitaevsky. Bose-Einstein condensation
in magnetic traps. Introduction in a theory // Usp.
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12. E. Zaremba, T. Nikuni, A. Griffin, Dynamics
of Trapped Bose Gases at Finite Temperatures // J.
Low Temp. Phys. 1999, v. 116, p. 277-345.
13. L.D. Landay, E.M. Lifshits. Elasticity
Theory. Nauka: “Moskow”, 1965, 203 p.
14. W.M. Saslow. Hydrodynamics of biaxial
nematics with arbitrary nonsingular textures //
Phys. Rev. A. 1982, v. 25, p. 3350-3359.
15. M. Liu. Hydrodynamic theory of biaxial
nematics // Phys. Rev. A. 1981, v. 24, №5, p. 2720-
2726.
16. H. Brand, H. Pleiner. Hydrodynamics of
biaxial discotics // Phys. Rev. A. 1981, v. 24, №5,
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17. C. Papenfub, J. Verhas, W. Muschik. A
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804.
59
ON SOUND WAVE SPECTRA IN BIAXIAL NEMATIC CRYSTALS
I. INTRODUCTION
REFERENCES
|
| id | nasplib_isofts_kiev_ua-123456789-80039 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:24:26Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kovalevsky, М.Y. Shishkin, А.L. 2015-04-09T16:21:18Z 2015-04-09T16:21:18Z 2001 On sound wave spectra in biaxial nematic crystals / М.Y. Kovalevsky, А.L. Shishkin // Вопросы атомной науки и техники. — 2001. — № 6. — С. 309-314. — Бібліогр.: 17 назв. — англ. 1562-6016 PACS: 61.30 Cz, 62.60+v https://nasplib.isofts.kiev.ua/handle/123456789/80039 On the basis of the Hamiltonian approach dynamics of biaxial nematics surveyed and the deduction of the nonlinear equations of ideal hydrodynamics, taking into account the shape, (disk-shaped and rod-shaped of a molecule) and conformation degree of freedom are given. The densities of additive integrals of motion and relevant fluxes are represented in the terms of a thermodynamic potential. The spectra of linear oscillations in these liquid crystals surveyed. Two branches of ultrasonic oscillations are found and the character of an anisotropy of both velocities of sounds is found out. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Kinetic theory On sound wave spectra in biaxial nematic crystals О звуковых спектрах в биаксиальных нематических кристаллах Article published earlier |
| spellingShingle | On sound wave spectra in biaxial nematic crystals Kovalevsky, М.Y. Shishkin, А.L. Kinetic theory |
| title | On sound wave spectra in biaxial nematic crystals |
| title_alt | О звуковых спектрах в биаксиальных нематических кристаллах |
| title_full | On sound wave spectra in biaxial nematic crystals |
| title_fullStr | On sound wave spectra in biaxial nematic crystals |
| title_full_unstemmed | On sound wave spectra in biaxial nematic crystals |
| title_short | On sound wave spectra in biaxial nematic crystals |
| title_sort | on sound wave spectra in biaxial nematic crystals |
| topic | Kinetic theory |
| topic_facet | Kinetic theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80039 |
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