Polarization features of acoustic spectra in uniaxial and biaxial nematic liquid crystals
The results of investigation of uniaxial and biaxial nematic liquid crystals dynamics with molecules of the various forms are presented. These condensed matters possess internal spatial anisotropy and for their adequate description introduction of additional dynamic quantities is necessary. They are...
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| Cite this: | Polarization features of acoustic spectra in uniaxial and biaxial nematic liquid crystals / M.Y. Kovalevsky, L.V. Logvinova, V.T. Matskevych // Вопросы атомной науки и техники. — 2012. — № 1. — С. 221-224. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-1071142016-10-14T03:02:29Z Polarization features of acoustic spectra in uniaxial and biaxial nematic liquid crystals Kovalevsky, M.Y. Logvinova, L.V. Matskevych, V.T. Section D. Theory of Irreversible Processes The results of investigation of uniaxial and biaxial nematic liquid crystals dynamics with molecules of the various forms are presented. These condensed matters possess internal spatial anisotropy and for their adequate description introduction of additional dynamic quantities is necessary. They are vectors of spatial anisotropy and conformational degrees of freedom. Investigation of dynamics of the given condensed matters is based on Hamiltonian formalism in which framework the nonlinear dynamic equations for uniaxial and biaxial nematic liquid crystals are derived. Spectra of collective excitations are obtained and their polarization features are investigated. Представлены результаты исследования динамики одноосных и двухосных нематических жидких кристаллов с молекулами различной формы. Эти конденсированные среды обладают внутренней пространственной анизотропией, и для их адекватного описания необходимо введение дополнительных динамических переменных. Ими являются векторы пространственной анизотропии и конформационные степени свободы. Исследование динамики данных конденсированных сред базируется на гамильтоновом формализме, в рамках которого выведены нелинейные уравнения динамики для одноосных и двухосных нематических жидких кристаллов. Получены спектры коллективных возбуждений и исследованы их поляризационные особенности. Представлено результати досліджень динаміки одновісних та двовісних нематичних рідких кристалів з молекулами різної форми. Ці конденсовані середовища мають внутрішню просторову анізотропію, тож для їх адекватного опису необхідно введення додаткових динамічних змінних. Ними є вектори просторової анізотропії та конформаційні ступені свободи. Дослідження динаміки даних конденсованих середовищ базується на гамільтоновому формалізмі, у рамках якого виведені нелінійні рівняння динаміки для одновісних та двовісних нематичних рідких кристалів. Отримано спектри колективних збуджень та досліджено їх поляризаційні особливості. 2012 Article Polarization features of acoustic spectra in uniaxial and biaxial nematic liquid crystals / M.Y. Kovalevsky, L.V. Logvinova, V.T. Matskevych // Вопросы атомной науки и техники. — 2012. — № 1. — С. 221-224. — Бібліогр.: 6 назв. — англ. 1562-6016 PACS: 61.30Cz, 83.10.Bb http://dspace.nbuv.gov.ua/handle/123456789/107114 en Вопросы атомной науки и техники Belgorod State University |
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Section D. Theory of Irreversible Processes Section D. Theory of Irreversible Processes |
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Section D. Theory of Irreversible Processes Section D. Theory of Irreversible Processes Kovalevsky, M.Y. Logvinova, L.V. Matskevych, V.T. Polarization features of acoustic spectra in uniaxial and biaxial nematic liquid crystals Вопросы атомной науки и техники |
| description |
The results of investigation of uniaxial and biaxial nematic liquid crystals dynamics with molecules of the various forms are presented. These condensed matters possess internal spatial anisotropy and for their adequate description introduction of additional dynamic quantities is necessary. They are vectors of spatial anisotropy and conformational degrees of freedom. Investigation of dynamics of the given condensed matters is based on Hamiltonian formalism in which framework the nonlinear dynamic equations for uniaxial and biaxial nematic liquid crystals are derived. Spectra of collective excitations are obtained and their polarization features are investigated. |
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Kovalevsky, M.Y. Logvinova, L.V. Matskevych, V.T. |
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Kovalevsky, M.Y. Logvinova, L.V. Matskevych, V.T. |
| author_sort |
Kovalevsky, M.Y. |
| title |
Polarization features of acoustic spectra in uniaxial and biaxial nematic liquid crystals |
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Polarization features of acoustic spectra in uniaxial and biaxial nematic liquid crystals |
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Polarization features of acoustic spectra in uniaxial and biaxial nematic liquid crystals |
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Polarization features of acoustic spectra in uniaxial and biaxial nematic liquid crystals |
| title_full_unstemmed |
Polarization features of acoustic spectra in uniaxial and biaxial nematic liquid crystals |
| title_sort |
polarization features of acoustic spectra in uniaxial and biaxial nematic liquid crystals |
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Belgorod State University |
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2012 |
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Section D. Theory of Irreversible Processes |
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http://dspace.nbuv.gov.ua/handle/123456789/107114 |
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Polarization features of acoustic spectra in uniaxial and biaxial nematic liquid crystals / M.Y. Kovalevsky, L.V. Logvinova, V.T. Matskevych // Вопросы атомной науки и техники. — 2012. — № 1. — С. 221-224. — Бібліогр.: 6 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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2025-07-07T19:30:50Z |
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| fulltext |
POLARIZATION FEATURES OF ACOUSTIC SPECTRA IN
UNIAXIAL AND BIAXIAL NEMATIC LIQUID CRYSTALS
M.Y. Kovalevsky 1,2, L.V. Logvinova 1, V.T. Matskevych 2∗
1Belgorod State University, Belgorod, Russia
2National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine
(Received October 30, 2011)
The results of investigation of uniaxial and biaxial nematic liquid crystals dynamics with molecules of the various
forms are presented. These condensed matters possess internal spatial anisotropy and for their adequate description
introduction of additional dynamic quantities is necessary. They are vectors of spatial anisotropy and conformational
degrees of freedom. Investigation of dynamics of the given condensed matters is based on Hamiltonian formalism
in which framework the nonlinear dynamic equations for uniaxial and biaxial nematic liquid crystals are derived.
Spectra of collective excitations are obtained and their polarization features are investigated.
PACS: 61.30Cz, 83.10.Bb
1. INTRODUCTION
Nematic liquid crystals are orientationally or-
dered anisotropic liquids with spontaneously broken
symmetry to rotations in configuration space. Be-
sides, they possess the internal molecular structure
capable to be deformed in the process of evolution,
which also must be considered at the macroscopic
level. Hence, for the description of dynamics of
such condensed matters introduction of the addi-
tional variables connected both with broken symme-
try and with the form and the size of molecules is
necessary.
In the given work features of dynamics of uni-
axial and biaxial nematic liquid crystals taking into
account internal structure are considered, and possi-
bility of distribution of collective excitations and their
polarization structure is studied.
2. FEATURES OF DYNAMICS AND
POLARIZATION STRUCTURE OF
ACOUSTIC SPECTRA OF UNIAXIAL
NEMATIC LIQUID CRYSTALS
2.1. Nematics with rod-like molecules
For uniaxial nematics along with usual dynamic vari-
ables - densities of mass ρ, momentum πi and en-
tropy σ, the additional parameter - unit vector of
spatial anisotropy (director) n (x) is introduced [1].
Using Hamiltonian approach of [2, 3] we obtain dy-
namic equations of uniaxial nematic with rod-like
molecules [4]:
ρ̇ = −∇iπi, π̇i = −∇ktik, σ̇ = −∇k (σvk) ,
l̇ = −vs (x)∇sl(x) + l (x) ni (x)nj (x)∇jvi(x),
ṅj = −vs∇snj + δ⊥ij (n) nk∇kvi.
(1)
Momentum flux density looks like
tik = Pδik + ∂ε
∂πk
πi + ∂ε
∂∇knl
∇inl + ∂ε
∂l ninkl
+nkδ⊥il (n)
(
∂ε
∂nl
−∇j
∂ε
∂∇jnl
)
.
(2)
Here P ≡ −ε + δH
δζ
a
ζ
a
+ ∂ε
∂∇lni
∇lni is pressure, vi ≡
πk/ρ is macroscopic velocity, δ⊥ik (n) = δik − nink, ε
is energy density, l is length of a molecule.
Linearization of (1) near equilibrium state leads
to the system of linear and homogeneous equations
Dij (k, ω) δvj (k, ω) = 0, (3)
which have a nontrivial solution for the vanishing of
the determinant
detDij = ω6 + ω4I4 + ω2I2 = 0, (4)
where ω is frequency, k is wave vector and
I4 (k) = −k2c2 − c2λ (kn)2 ≤ 0,
I2 (k) = c4λ
(
k2 − (kn)2
)
(kn)2 ≥ 0.
(5)
Here λ ≡ l2
ρc2
∂2ε
∂l2 > 0 and c is sound velocity in
usual liquid. From (4) it is clear, that in unixial
nematic with rod-like molecules the propagation of
two acoustic oscillation modes ω2± (k) = c2± (k) k2 is
possible corresponding to the first and second sound.
The solution with a sigh (+) corresponds to the first
sound analogous to that in usual liquid. The so-
lution with a sigh (-) is new branch of excitations
caused by conformational degree of freedom - rod-like
molecule length. In spherical system of coordinates
∗Corresponding author E-mail address: matskevych@mail.ru
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1.
Series: Nuclear Physics Investigations (57), p. 221-224.
221
kn = k cos θ, where θ is polar angle, hence, velocities
c± look like [2]:
c± (θ) =
c√
2
[
1 + λ cos2 θ (6)
±
[(
1 + λ cos2 θ
)2 − λ sin2 2θ
]1/2
]1/2
.
Calculated angular values of extremum points (θ =
π/4) for sound velocity c− coincide with experimental
data [5, 6].
Let’s consider solutions of (3) corresponding to
modes ω = kc±. Expression for δv
(±)
j (k) we are
looking for in the form of decomposition on three or-
thogonal vectors:
δv
(±)
j (k) = kjδv
(±)
|| (k) + [k × n]jδv
(±)
1⊥ (k) +
+
[[k×n]×k]j
k δv
(±)
2⊥ (k) .
From (3) we find, that δv
(±)
1⊥ (k) = 0. Then
δv
(±)
j (k) = kjδv
(±)
|| (k) +
[[k × n] × k]j
k
δv
(±)
2⊥ (k) .
Hence, solutions corresponding to the first and sec-
ond sounds are superposition of longitudinal and
transversal components and the relation of these am-
plitudes has the form:
δv
(±)
2⊥
δv
(±)
||
=
−k5
(
c2
± − c2
)
+ λc2kk4
||
λc2k3
||k
2
⊥
≡ f± (θ) . (7)
Using (6) we rewrite (7) in terms of polar angle:
f± (θ) =
1 + λ cos2 θ cos 2θ
λ sin 2θ sin θ cos2 θ
∓
∓
√
(1 + λ cos2 θ)2 − λ sin2 2θ
λ sin 2θ sin θ cos2 θ
.
At λ << 1 the relation of amplitudes for the first
sound f+ (θ) becomes simpler:
f+ (θ) = −λ cos5 θ
4 sin2 θ
.
We can conclude, that at θ → 0 function f+ (θ) → ∞,
hence, at such values of polar angle sound is transver-
sal. At θ → π/2 function f+ (θ) → 0, hence, at such
values of polar angle sound is longitudinal. For the
second sound the relation of amplitudes f− (θ) looks
like
f− (θ) =
1 − λ sin2 θ cos 2θ
λ cos θ sin4 θ
. (8)
We can conclude, that at θ → 0 and θ → π/2 func-
tion f− (θ) → ∞, hence, at such values of polar angle
sound is transversal.
2.2. Nematics with disc-like molecules
The direction of orientation of such liquid crystals is
defined by unit vector of a normal to molecule plane.
Studying of dynamic behavior of uniaxial nematic
with disc-like molecules we will carry out similar to
earlier considered case of uniaxial nematic with rod-
like molecules. Dynamic equations of uniaxial ne-
matic with disc-like molecules are as follows [4]:
ρ̇ = −∇iπi, π̇i = −∇ktik, σ̇ = −∇k (σvk) ,
ḋ = −vs∇sd − dδ⊥lk (n)∇kvl, (9)
ṅj = −vs∇snj − niδ
⊥
jλ (n)∇λvi,
tik = Pδik + ∂ε
∂πk
πi + ∂ε
∂∇knl
∇inl + ∂ε
∂ddδ⊥lk (n)+
+nkδ⊥il (n)
(
∂ε
∂nl
−∇j
∂ε
∂∇jnl
)
.
Here d is molecule diameter. Dispersion equation has
the form (4), where coefficients Ia, a = 2.4 are as fol-
lows:
I4 (k) = −k2c2 − c2λ
(
k2 − (kn)2
)
< 0,
I2 (k) = c4λ
(
k2 − (kn)2
)
(kn)2 > 0,
(10)
where λ = d2
ρc2
∂2ε
∂d2 > 0. From here we come to
acoustic spectra ω2
± (k) = c2
± (k) k2. In this case also
two anisotropic velocities of acoustic waves exist [2]:
c± (θ) =
c√
2
[
1 + λ sin2 θ ±
±
[(
1 + λ sin2 θ
)2 − λ sin2 2θ
]1/2
]1/2
. (11)
Polarization structure of the received spectra of
collective excitations looks like
δv
(±)
j (k) = kjδv
(±)
|| +
[[k × n] × k]j
k
δv
(±)
2⊥ ,
and relation of amplitudes has the form
δv
(±)
2⊥ (k)
δv
(±)
|| (k)
=
−k5
(
c2
± − c2
)
+ λc2kk4
⊥
λc2k||k4
⊥
≡ g± (θ) .
(12)
Using (11) we rewrite (12) in terms of polar angle:
g± (θ) =
1
λ sin 2θ sin3 θ
(
1 − λ sin2 θ cos 2θ
)∓
∓
√(
1 + λ sin2 θ
)2 − λ sin2 2θ
λ sin 2θ sin3 θ
.
At λ << 1 the relation of amplitudes for the first
sound g+ (θ) becomes simpler:
g+ (θ) = −λ sin4 θ
4 cos θ
.
We can conclude, that at θ → 0 function g+ (θ) → 0,
hence, in this case sound is longitudinal. At θ → π/2
function g+ (θ) → ∞, hence, in this case sound is
transversal. At λ << 1 expression for c− does not
depend on molecule form, so the relation (8) is iden-
tical for uniaxial nematics with rod-like and disc-like
molecules.
222
3. FEATURES OF DYNAMICS AND
POLARIZATION STRUCTURE OF
ACOUSTIC SPECTRA OF BIAXIAL
NEMATIC LIQUID CRYSTAL
3.1. Nematics with ellipsoidal molecules
In the case of biaxial nematic with ellipsoidal mole-
cules the set of thermodynamic variables contains
additionally two unit and orthogonal vectors of spa-
tial anisotropy n (x),m (x) and three conformational
parameters u (x) , v (x) , p (x) describing sizes of long
and short molecule axes and an angle between them.
Acting further similarly to previously considered case
of uniaxial nematics, we obtain dynamic equations of
biaxial nematic with ellipsoidal molecules [4]:
ρ̇ = −∇iπi, π̇i = −∇ktik, σ̇ = −∇k (σvk) ,
ṅj (x) = −vs (x)∇snj (x) − Fiλj (x)∇λvi (x) ,
ṁj (x) = −vs (x)∇smj (x) − Giλj (x)∇λvi (x) ,
u̇ (x) = −vi (x)∇iu (x) − Fij (x)∇jvi (x) ,
v̇ (x) = −vi (x)∇iv (x) − Gij (x)∇jvi (x) ,
ṗ (x) = −vs (x)∇sp (x) − Hij (x)∇ivj (x) , (13)
tik = Pδik + ∂ε
∂πk
πi + ∂ε
∂∇knl
∇inl + ∂ε
∂∇kml
∇iml+
+ ∂ε
∂uFik + ∂ε
∂v Gik + ∂ε
∂pHik+
+Fikl
(
∂ε
∂nl
−∇j
∂ε
∂∇jnl
)
+ Gikl
(
∂ε
∂ml
−∇j
∂ε
∂∇jml
)
,
where P ≡ −ε + δH
δζ
a
ζ
a
+ ∂ε
∂∇lni
∇lni + ∂ε
∂∇lmi
∇lmi is
pressure, Fij , Gij , Hij and Fijk, Gijk , Hijk are some
functions of n (x),m (x) and u (x) , v (x) , p (x). Lin-
earization of (13) near equilibrium state leads to the
system of linear and homogeneous equations
δvj (k, ω) Dij (k, ω) = 0. (14)
Condition for the existence of a nontrivial solution
of (14) is the vanishing of the determinant det D̂ =
ω6 + ω4I4 + ω2I2 + I0 = 0, where coefficients Ia, a =
0, 2, 4 are some functions of k,F,G,H and F,G,H
are some functions of n (x),m (x),k and parameters
λα, α = 1, 2, 3:
λ1 ≡ u2
ρc2
∂2ε
∂u2
> 0, λ2 ≡ v2
ρc2
∂2ε
∂v2
> 0,
λ3 ≡ p2
ρc2
∂2ε
∂p2
> 0.
As a result we come to bicubic dispersion equation
ω6 + I4 (k, θ, ϕ)ω4 + I2 (k, θ, ϕ)ω2 + I0 (k, θ, ϕ) = 0.
(15)
From (15) it is clear, that in biaxial nematic
with ellipsoidal molecules in general case propaga-
tion of three acoustic oscillation modes ω2
1,2,3 (k) =
c2
1,2,3 (k) k2 is possible corresponding to the first, sec-
ond and third sounds. Detailed analysis of the ob-
tained spectra is given in [4].
Let’s consider solutions of (14) corresponding
to modes ω2
1,2,3 ≡ c2
1,2,3 (θ, ϕ) k2. Expression for
δv
(1,2,3)
j (k) we are looking for in the form of decom-
position on three orthogonal vectors:
δv
(1,2,3)
j (k)= kjδv
(1,2,3)
|| (k) + [k × l]jδv
(1,2,3)
1⊥ (k)+
+
[[k×l]×k]j
k δv
(1,2,3,)
2⊥ (k) .
From (14) we find that these solutions are su-
perposition of one longitudinal and two transver-
sal components. It can be shown, that at
θ → 0 sounds are cross-polarized with components
δv
(1,2,3)
1⊥ (k) , δv
(1,2,3,)
2⊥ (k); at θ → π/2 sounds are
cross-polarized with component δv
(1,2,3,)
2⊥ (k).
3.2. Nematics with discoidal molecules
Dynamic equations of biaxial nematic with discoidal
molecules are as follows [4]:
ρ̇ = −∇iπi, π̇i = −∇ktik, σ̇ = −∇k (σvk) ,
ṅj (x) = −vs (x)∇snj (x) − fiλj (x)∇λvi (x) ,
ṁj (x) = −vs (x)∇smj (x) − giλj (x)∇λvi (x) ,
q̇ (x) = −vi (x)∇iq (x) − fij (x)∇jvi (x) ,
ṫ (x) = −vi (x)∇it (x) − gij (x)∇jvi (x) ,
ṗ (x) = −vs (x)∇sp (x) − hij (x)∇ivj (x) , (16)
tik = Pδik + ∂ε
∂πk
πi + ∂ε
∂∇knλ
∇inλ + ∂ε
∂∇kmλ
∇imλ
+ ∂ε
∂qfik + ∂ε
∂t gik + ∂ε
∂phik + fikλ
(
∂ε
∂nλ
−∇j
∂ε
∂∇jnλ
)
+gikλ
(
∂ε
∂mλ
−∇j
∂ε
∂∇jmλ
)
.
Here n (x),m (x) are unit and orthogonal vectors of
spatial anisotropy, q (x) , t (x) , p (x) are conforma-
tional parameters describing sizes of long and short
molecule axes and an angle between them, fij , gij , hij
and fijk, gijk, hijk are some functions of n (x),m (x)
and q (x) , t (x) , p (x). Linearization of (16) near
equilibrium state leads to the system of linear and
homogeneous equations
δvj (k, ω)Dij (k, ω) = 0. (17)
Condition for the existence of a nontrivial solution
of (17) is the vanishing of the determinant det D̂ =
ω6 + ω4I4 + ω2I2 + I0 = 0, where coefficients Ia, a =
0, 2, 4 are some functions of k,f ,g,h and f ,g,h are
some functions of n (x),m (x),k and parameters λα,
α = 1, 2, 3:
λ1 ≡ q2
ρc2
∂2ε
∂q2
> 0, λ2 ≡ t2
ρc2
∂2ε
∂t2
> 0,
λ3 ≡ p2
ρc2
∂2ε
∂p2
> 0.
As a result we come to bicubic dispersion equation
ω6 + I4 (k, θ, ϕ)ω4 + I2 (k, θ, ϕ)ω2 + I0 (k, θ, ϕ) = 0.
(18)
From (18) it is clear, that in biaxial nematic with dis-
coidal molecules in general case propagation of three
acoustic oscillation modes ω2
1,2,3 (k) = c2
1,2,3 (k) k2
also is possible corresponding to the first, second and
third sounds. Detailed analysis of the obtained spec-
tra is given in [4].
223
Let’s consider solutions of (17) corresponding
to modesω2
1,2,3 ≡ c2
1,2,3 (θ, ϕ) k2. Expression for
δv
(1,2,3)
j (k) we are looking for in the form of decom-
position on three orthogonal vectors:
δv
(1,2,3)
j (k) = kjδv
(1,2,3)
|| (k) + [k × l]jδv
(1,2,3)
1⊥ (k)
+
[[k×l]×k]j
k δv
(1,2,3,)
2⊥ (k) .
From (17) we find that like in the previous case of
biaxial nematic with ellipsoidal molecules these solu-
tions are superposition of one longitudinal and two
transversal components. It can be shown, that at
θ → 0 sounds are cross-polarized with components
δv
(1,2,3)
1⊥ (k) , δv
(1,2,3,)
2⊥ (k); at θ → π/2 sounds are
cross-polarized with component δv
(1,2,3,)
2⊥ (k).
4. CONCLUSIONS
On the basis of Hamiltonian approach the dynamic
theory of uniaxial and biaxial nematic liquid crystals
with molecules of different geometry is constructed.
For all types of liquid crystals nonlinear dynamic
equations are derived, acoustic spectra of collective
excitations are received and their polarization struc-
ture is studied. It is shown that in the case of uni-
axial nematics the first sound is mostly longitudinal
and the second one is mostly transversal; for biaxial
nematics the first, second and third sounds mostly
possess transversal polarization.
References
1. P.G. De Gennes, J. Prost. The Physics of Liq-
uid Crystals. Oxford University Press, 1995,
597 p.
2. A.P. Ivashin, M.Y. Kovalevsky, L.V. Logvinova
// Int. J. Quantum Chemistry. 2004, v. 96,
p. 636-644.
3. A.P. Ivashin, M.Y. Kovalevsky, L.V. Logvinova
// Theor. Math. Phys. 2004, v. 140, N 3, p. 1316-
1327.
4. M.Y. Kovalevsky, L.V. Logvinova, V.T. Mats-
kevych // Physics of Elementary Particles and
Atomic Nuclei, 2009, v. 40, N 3, p. 704-753 (in
Russian).
5. J.V. Selinger, M.S. Spector, V.A. Greanya,
B.T. Weslowski, D.K. Shenoy, R. Shashidhar
// Phys. Rev. E. 2002, v. 66, 051708, 7 p.
6. V.A. Greanya, A.P. Malanovsky, B.T. Weslow-
ski, M.S. Spector, J.V. Selinger // Liquid Crys-
tals. 2005, v. 32, N 7, p. 933-941.
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