Maier-Saupe nematogenic fluid: field theoretical approach
We adopt a field theoretical approach to study the structure and thermodynamics of a homogeneous Maier-Saupe nematogenic fluid interacting with anisotropic Yukawa potential. In the mean field approximation we retrieve the standard Maier-Saupe theory for liquid crystals. In this theory the density is...
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Holovko, M. di Caprio, D. Kravtsiv, I. 2017-06-10T18:27:56Z 2017-06-10T18:27:56Z 2011 Maier-Saupe nematogenic fluid: field theoretical approach/ M. Holovko, D. di Caprio, I. Kravtsiv // Condensed Matter Physics. — 2011. — Т. 14, № 3. — С. 33605: 1-12. — Бібліогр.: 24 назв. — англ. 1607-324X PACS: 64.70.M-, 64.10.+h, 05.70.Fh DOI:10.5488/CMP.14.33605 arXiv:1202.4548 https://nasplib.isofts.kiev.ua/handle/123456789/120019 We adopt a field theoretical approach to study the structure and thermodynamics of a homogeneous Maier-Saupe nematogenic fluid interacting with anisotropic Yukawa potential. In the mean field approximation we retrieve the standard Maier-Saupe theory for liquid crystals. In this theory the density is expressed via the second order Legendre polynomial of molecule orientations. In the Gaussian approximation we obtain analytical expressions for the correlation functions, the elasticity constant, the free energy, the pressure, and the chemical potential. We also use Ward symmetry identities to set a simple condition for the correlation functions. Subsequently we find corrections due to fluctuations and show that density now contains Legendre polynomials of higher orders. Ми застосовуємо теоретико-польовий пiдхiд для вивчення структурних i термодинамiчних властивостей однорiдного нематичного плину Майєра-Заупе з анiзотропною взаємодiєю типу Юкави. У наближеннi середнього поля нами отримано стандартну теорiю Майєра-Заупе для рiдких кристалiв. У цiй теорiї одночастинкова функцiя розподiлу виражається через полiном Лєжандра другого порядку взаємної орiєнтацiї частинок. У гаусiвському наближеннi нами отримано аналiтичнi вирази для кореляцiйних функцiй, константи еластичностi, вiльної енергiї, тиску i хiмiчного потенцiалу. За допомогою тотожностi Ворда нами встановлено просту умову для кореляцiйних функцiй. Нами також знайдено поправки внаслiдок флуктуацiй i показано що вираз для одночастинкової функцiї розподiлу вже мiстить полiноми Лєжандра вищих порядкiв M. Holovko and D. di Caprio are grateful for the support to the National Academy of Sciences of Ukraine (NASU) and to the Centre National de la Recherche Scientifique (CNRS) (project no. 21303), I. Kravtsiv is grateful for the support to the French embassy in Ukraine (the grant of the French government for PhD programs in partnership). The authors also thank Dr. O.V. Patsahan for the useful comments. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Maier-Saupe nematogenic fluid: field theoretical approach Нематичний плин Майєра-Заупе: теоретико-польовий пiдхiд Article published earlier |
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Maier-Saupe nematogenic fluid: field theoretical approach |
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Maier-Saupe nematogenic fluid: field theoretical approach Holovko, M. di Caprio, D. Kravtsiv, I. |
| title_short |
Maier-Saupe nematogenic fluid: field theoretical approach |
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Maier-Saupe nematogenic fluid: field theoretical approach |
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Maier-Saupe nematogenic fluid: field theoretical approach |
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Maier-Saupe nematogenic fluid: field theoretical approach |
| title_sort |
maier-saupe nematogenic fluid: field theoretical approach |
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Holovko, M. di Caprio, D. Kravtsiv, I. |
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Holovko, M. di Caprio, D. Kravtsiv, I. |
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2011 |
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Нематичний плин Майєра-Заупе: теоретико-польовий пiдхiд |
| description |
We adopt a field theoretical approach to study the structure and thermodynamics of a homogeneous Maier-Saupe nematogenic fluid interacting with anisotropic Yukawa potential. In the mean field approximation we retrieve the standard Maier-Saupe theory for liquid crystals. In this theory the density is expressed via the second order Legendre polynomial of molecule orientations. In the Gaussian approximation we obtain analytical expressions for the correlation functions, the elasticity constant, the free energy, the pressure, and the chemical potential. We also use Ward symmetry identities to set a simple condition for the correlation functions. Subsequently we find corrections due to fluctuations and show that density now contains Legendre polynomials of higher orders.
Ми застосовуємо теоретико-польовий пiдхiд для вивчення структурних i термодинамiчних властивостей однорiдного нематичного плину Майєра-Заупе з анiзотропною взаємодiєю типу Юкави. У наближеннi середнього поля нами отримано стандартну теорiю Майєра-Заупе для рiдких кристалiв. У цiй теорiї одночастинкова функцiя розподiлу виражається через полiном Лєжандра другого порядку взаємної орiєнтацiї частинок. У гаусiвському наближеннi нами отримано аналiтичнi вирази для кореляцiйних функцiй, константи еластичностi, вiльної енергiї, тиску i хiмiчного потенцiалу.
За допомогою тотожностi Ворда нами встановлено просту умову для кореляцiйних функцiй. Нами також знайдено поправки внаслiдок флуктуацiй i показано що вираз для одночастинкової функцiї розподiлу вже мiстить полiноми Лєжандра вищих порядкiв
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120019 |
| citation_txt |
Maier-Saupe nematogenic fluid: field theoretical approach/ M. Holovko, D. di Caprio, I. Kravtsiv // Condensed Matter Physics. — 2011. — Т. 14, № 3. — С. 33605: 1-12. — Бібліогр.: 24 назв. — англ. |
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2025-11-25T23:28:33Z |
| last_indexed |
2025-11-25T23:28:33Z |
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| fulltext |
Condensed Matter Physics, 2011, Vol. 14, No 3, 33605: 1–12
DOI: 10.5488/CMP.14.33605
http://www.icmp.lviv.ua/journal
Maier-Saupe nematogenic fluid: field theoretical
approach
M. Holovko1, D. di Caprio2, I. Kravtsiv1∗
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
2 Laboratoire d’Electrochimie, Chimie des Interfaces et Modélisation pour l’Energie (LECIME) ENSCP,
Chimie ParisTech, Case 39, 4 Pl. Jussieu, 75005 Paris, France
Received June 29, 2011, in final form August 4, 2011
We adopt a field theoretical approach to the study of the structure and thermodynamics of a homogeneous
Maier-Saupe nematogenic fluid interacting with anisotropic Yukawa potential. In the mean field approximation
we retrieve a standard Maier-Saupe theory for liquid crystals. In this theory, the single-particle distribution
function is expressed via the second order Legendre polynomial of molecule orientations. In the Gaussian
approximation we obtain analytical expressions for correlation functions, free energy, pressure, chemical po-
tential, and elasticity constant. Subsequently we find corrections due to fluctuations and show that the single-
particle distribution function now contains Legendre polynomials of higher orders. We also use Ward symmetry
identities to set a simple condition for correlation functions.
Key words: Maier-Saupe nematogenic fluid, field theoretical approach, correlation function, thermodynamics
PACS: 64.70.M-, 64.10.+h, 05.70.Fh
1. Introduction
Maier-Saupe nematogenic fluid [1] is one of the simplest models that account for the isotropic-
nematic phase transition in the liquid crystal phase. The properties of this model have been inten-
sively studied by the liquid theory methods such as integral equations for correlation functions [2–7].
In the integral equation theory there is a problem of the correctness of taking the fluctuation effects
into account, the treatment of which depends on closure relations used in integral equations. In
order to treat the fluctuations more properly and to control the level of this treatment, in this paper
we will apply the field theoretical approach. This is the first time the field theoretical approach is
applied to the description of anisotropic molecular fluids.
The method we are proposing focuses on fluctuations of the field at a given point and implements
a perturbative scheme by expanding the Hamiltonian on density fluctuations. In the past, the
statistical field theory proved to be successful in the description of a variety of systems with
Coulomb [8–12] and Yukawa-type interactions [13, 14]. In this work we show that this approach
also reproduces the familiar results for anisotropic systems, notably the mean field Maier-Saupe
theory. Subsequently we go beyond this approximation and obtain an analytical expression for
the pair correlation function. In the Gaussian approximation we also obtain new results for the
main structural and thermodynamic properties of the system. The expressions we derive contain
the orientational order parameter allowing us to compare the results for the isotropic and nematic
phases. Finally, we calculate the correction to the mean field single-particle distribution function
due to fluctuations which is expressed in terms of the fourth order Legendre polynomials of molecule
orientations. Our results for the pair correlation functions predict the appearance of Goldstone
modes in the system which is in full agreement with the theory of de Gennes [15].
∗E-mail: ivankr@icmp.lviv.ua
c© M. Holovko, D. di Caprio, I. Kravtsiv, 2011 33605-1
http://dx.doi.org/10.5488/CMP.14.33605
http://www.icmp.lviv.ua/journal
M. Holovko, D. di Caprio, I. Kravtsiv
For the purpose of simplification, in this paper we consider a fluid of point particles. However, in
the future we hope to modify the obtained results for non-point particles using the mean spherical
results [2, 3, 7] as it was done for a non-point ionic system [16].
2. The model and field theory formalism
We consider a molecular fluid of particles interacting via an anisotropic Yukawa-type potential
ν(r12 ,Ω1Ω2):
ν(r12 ,Ω1Ω2) =
A
r12
e−αr12P2(cos θ12)
=
A
r12
e−αr12
1
5
∑
m
Y ∗
2m(Ω1)Y2m(Ω2), (2.1)
where r12 denotes the distance between particles 1 and 2, Ω = (θ, φ) are orientations of particles,
P2(cos θ12) = (3 cos2 θ12 − 1)/2 is the second order Legendre polynomial of relative molecule ori-
entations, Ylm(Ω) are standard spherical harmonics [17] without the normalization factor 1/
√
4π,
A is the amplitude of the interaction, and α is the inverse range.
In a series of papers on ionic and Yukawa fluids [8, 13, 14] it was shown that it is possible to
describe these fluids using the field theoretical approach. In this paper we will develop this approach
for the description of an anisotropic molecular fluid with the interaction of the form (2.1).
Within the field-theoretical formalism, the Hamiltonian is a functional of density field and can
be written as
βH [ρ(r,Ω)] = βHentr[ρ(r,Ω)] + βH int[ρ(r,Ω)]
=
∫
ρ(r,Ω)
[
ln(ρ(r,Ω)ΛRΛ
3
T)− 1
]
drdΩ
+
β
2
∫
ν(r12 ,Ω1Ω2)ρ(r1 ,Ω1)ρ(r2 ,Ω2)dr1dr2dΩ1dΩ2 , (2.2)
where β = 1/kBT is the inverse temperature, dΩ = (1/4π) sin θdθdφ is the normalized angle
element, ρ(r,Ω) is particle density per angle such that
∫
ρ(r,Ω)dΩ = ρ(r), ΛT is the thermal de
Broglie wavelength of the molecules, and the quantity Λ−1
R is the rotational partition function for
a single molecule [17].
As in previous papers [8, 13, 14], we adopt the canonical ensemble approach. We fix the number
of particles by the condition
∫
ρ(r)dr = N or 1
V
∫
ρ(r)dr = ρ, where V is the volume and ρ is
the average density of the system. To verify this condition in a formally unconstrained calculus we
introduce a Lagrange multiplier λ such that
δβH [ρ(r,Ω)]
δρ(r,Ω)
= λ. (2.3)
The partition function ZN [ρ(r,Ω)] can be expressed as
ZN [ρ(r,Ω)] =
∫
Dρ(r,Ω) exp{−βH [ρ(r,Ω)]},
where Dρ(r,Ω) denotes functional integration over all possible density distributions such that the
total number of particles is N . The logarithm of the partition function gives the Helmholtz free
energy
βF = − lnZN . (2.4)
Due to the character of the interparticle interaction, the considered system is characterized by
two non-dimensional parameters: non-dimensional density ρ∗ = ρ/α3 and non-dimensional inverse
33605-2
Maier-Saupe nematogenic fluid: field theoretical approach
temperature β∗ = −βAα = 1/T ∗. As we will see in our calculations, the third non-dimensional
parameter appears M∗ = −4πρβA/α2 = 4πρ∗β∗.
In order to calculate the functional integral, we expand the Hamiltonian around the real angle-
dependent density ρ(Ω) which in the homogeneous case does not depend on r:
βH [ρ(Ω) + δρ(r,Ω)] =
=
∫
(
ρ(Ω) + δρ(r,Ω)
)
[
ln
(
ρ(Ω)
ρ
)
+ ln
(
1 +
δρ(r,Ω)
ρ(Ω)
)
− 1
]
drdΩ
+
β
2
∫
ν(r12 ,Ω1Ω2)
(
ρ(Ω1) + δρ(r1 ,Ω1)
)(
ρ(Ω2) + δρ(r2 ,Ω2)
)
dr1dr2dΩ1dΩ2 . (2.5)
3. Mean field approximation
In order to obtain thermodynamic properties of the considered fluid we need to calculate the
partition function. The lowest order approximation for the partition function is the saddle point
for the functional integral which is the mean field approximation (MFA) from the physical point
of view. In the canonical formalism it corresponds to fixing the Lagrange parameter λ such that
the relation (2.3) is true for the average density.
Expanding the logarithm in (2.5), we obtain
δβH [ρ(r,Ω)]
δρ(r1 ,Ω)
= ln
ρ(Ω)
ρ
+ β
∫
ν(r12 ,Ω1Ω2)ρ(Ω2)dr2dΩ2 . (3.1)
The second term on the right-hand side of equation (3.1) equals
β
∫
ν(r12 ,Ω1Ω2)ρ(Ω2)dr2dΩ2 =
= β
∫
A
r12
e−αr12dr2
∫
1
5
∑
m′
Y ∗
2m′(Ω2)Y2m′(Ω1)
∑
lm′′
ρlm′′Ylm′′ (Ω2)dΩ2
= β
1
5
ν
∑
m
ρ2mY2m(Ω1), (3.2)
where we have used
ν =
∫
A
r12
e−αr12dr12 =
4πA
α2
. (3.3)
If we choose the value of parameter λ to be
eλ ≡
(
∫
dΩ exp
[
−β
1
5
ν
∑
m
ρ2mY2m(Ω)
])−1
≡ 1
Z
(3.4)
then from (3.1) we get the following equation for density within MFA:
ρ(Ω) =
ρ
Z
exp
[
−β
1
5
ν
∑
m
ρ2mY2m(Ω)
]
≡ ρf(Ω), (3.5)
where f(Ω) is the single-particle distribution function and the averages can be calculated according
to 〈. . .〉Ω =
∫
f(Ω)(. . .)dΩ.
If we multiply both sides of equation (3.5) by Y2m(Ω) and integrate by dΩ we will obtain
∑
l′m′
ρl′m′
∫
dΩYl′m′(Ω)Y2m(Ω) = ρ〈Y2m(Ω)〉Ω ,
ρ2m = ρ〈Y2m(Ω)〉Ω . (3.6)
33605-3
M. Holovko, D. di Caprio, I. Kravtsiv
0,10 0,15 0,20 0,25
0,0
0,2
0,4
0,6
0,8
1,0
1/M*
S = S (1/M*)
S = 0.324
S = 0.435
S
3,0 3,5 4,0 4,5 5,0
8
9
10
11
12
13
14
Isotropic
S = 0.324, M* = 4.484
S = 0.435, M* = 4.587
T*
*
Nematic
Figure 1. Dependence of orientational order pa-
rameter S on parameter 1/M∗.
Figure 2. Density-temperature phase diagram.
In normal nematics, the orientational distribution function f(Ω) is axially symmetric with respect
to a preferred direction n and depends only on the angle θ between the molecular orientation Ω
and n [2]. This means that only quantities independent of angle φ, in the plane perpendicular to n,
yield non-zero averages and therefore for any m 6= 0 the averages 〈Y2m(Ω)〉Ω equal 0. As a result,
we obtain a well-known Maier-Saupe equation [15]
〈Y20(Ω)〉Ω =
1
Z
∫
Y20(Ω) exp
[
−β
1
5
νρ〈Y20(Ω)〉ΩY20(Ω)
]
dΩ. (3.7)
In terms of the orientational order parameter S and reduced unit M∗, equations (3.5) and (3.7)
can be rewritten as follows:
ρ(Ω)
ρ
=
exp
[
3
2M
∗S cos2 θ
]
1
∫
0
exp
[
3
2M
∗Sx2
]
dx
, (3.8)
S = 〈P2(cos θ)〉Ω =
1√
5
〈Y20(Ω)〉Ω = −1
2
+
3
2
1
∫
0
x2 exp
[
3
2M
∗Sx2
]
dx
1
∫
0
exp
[
3
2M
∗Sx2
]
dx
. (3.9)
The order parameter S can take on values from 0 to 1 with values S > 0 corresponding to
the nematic phase. Equation (3.9) is self-consistent and must be solved numerically. The resulting
relationship between S and M∗ is presented in figure 1. The theory predicts a weak first-order
phase transition from the isotropic phase with S = 0 to the nematic phase with S > 0. The
smallest value of the order parameter S = 0.324 corresponding to M∗ = 4.484 defines the stability
of the isotropic phase. A stable nematic phase is given by the solution that minimizes the free
energy which in the MFA can be presented in the form
βF
N
= ln
(
ρΛ3
TΛR
)
− 1 +
1
2
M∗S2. (3.10)
As a result, the stable nematic phase appears at M∗ = 4.587 and the value of the order parameter
at the transition is S = 0.435. The region between M∗ = 4.484 and M∗ = 4.587 corresponds to the
two-phase region which separates the isotropic and the nematic phases. The corresponding phase
diagram in “density-temperature” coordinates is presented in figure 2.
33605-4
Maier-Saupe nematogenic fluid: field theoretical approach
4. Fluctuation and correlation effects: Gaussian approxim ation
In the MFA, fluctuations are neglected. In this section we take them into account. To this end
we should expand the Hamiltonian. From (2.5), the quadratic term in the Hamiltonian equals
βH2 [ρ(r,Ω)] =
1
2
∫
1
ρ(Ω)
δρ2(r,Ω)drdΩ
+
β
2
∫
ν(r12 ,Ω1Ω2)δρ(r1 ,Ω1)δρ(r2 ,Ω2)dr1dr2dΩ1dΩ2 . (4.1)
Expanding on the Fourier components
δρ(r,Ω) =
∑
k
δρ(k,Ω)eikr, (4.2)
we obtain the expression for the quadratic term in the k-space
βH2 [ρ(k,Ω)] =
V
2
∑
k
∫
dΩ1dΩ2δρ(k,Ω1)δρ(−k,Ω2)
×
(
δΩ1Ω2
ρ(Ω1)
+
4πβA
k2 + α2
1
5
∑
m
Y ∗
2m(Ω1)Y2m(Ω2)
)
. (4.3)
4.1. Correlation functions
The expression for a pair correlation function h is
h(r12 ,Ω1Ω2)〈ρ(r1 ,Ω1)〉 〈ρ(r2 ,Ω2)〉 = 〈δρ(r1 ,Ω1)δρ(r2 ,Ω2)〉
−〈δρ(r1 ,Ω1)〉 〈δρ(r2 ,Ω2)〉 − δ(r1 − r2)δΩ1Ω2
〈ρ(r1 ,Ω1)〉. (4.4)
The second term on the right-hand side of equation (4.4) disappears like in the homogeneous case
〈δρ(r,Ω)〉 = 0. The first term equals
〈δρ(r1 ,Ω1)δρ(r2 ,Ω2)〉 =
1
ZN
∫
D (δρ(r,Ω)) e−βH2[ρ(r,Ω)]δρ(r1 ,Ω1)δρ(r2 ,Ω2)
=
∑
k
eik(r1−r2)〈δρ(k,Ω1)δρ(−k,Ω2)〉
=
∑
k
eik(r1−r2)
∫
D (δρ(k,Ω)) e−βH2[ρ(k,Ω)]δρ(k,Ω1)δρ(−k,Ω2)
∫
D (δρ(k,Ω)) e−βH2[ρ(k,Ω)]
. (4.5)
As in the basis (4.2) the Hamiltonian is of diagonal form, and the Gaussian integral (4.5) yields
〈δρ(k,Ω1)δρ(−k,Ω2)〉 =
1
V
(
δΩ1Ω2
ρ(Ω1)
+
4πβA
k2 + α2
1
5
∑
m
Y ∗
2m(Ω1)Y2m(Ω2)
)−1
. (4.6)
The inverse of the matrix in brackets is
(
δΩ1Ω2
ρ(Ω1)
+
4πβA
k2 + α2
1
5
∑
m
Y ∗
2m(Ω1)Y2m(Ω2)
)−1
= h(k,Ω1Ω2)ρ(Ω1)ρ(Ω2) + δΩ1Ω2
ρ(Ω1). (4.7)
Identity (4.7) is in essence the Ornstein-Zernike equation in the random phase approximation
(RPA) for point particles [18, 19]
h(k,Ω1Ω2) = C(k,Ω1Ω2) +
∫
dΩ3C(k,Ω1Ω3)h(k,Ω2Ω3)ρ(Ω3) (4.8)
33605-5
M. Holovko, D. di Caprio, I. Kravtsiv
with the closure
C(r12 ,Ω1Ω2) = −βν(r12 ,Ω1Ω2), (4.9)
where C(k,Ω1Ω2) and h(k,Ω1Ω2) are the Fourier transforms of the direct and pair correlation
functions, respectively.
In (4.8) we can expand f(k,Ω1Ω2) on spherical harmonics
f(k,Ω1Ω2) =
∑
lmnn′
flmnn′(k)Y ∗
lm(Ω1)Ynn′(Ω2). (4.10)
Due to the closure (4.9) and symmetry properties of the nematic we can write f(r12 ,Ω1Ω2) in the
form
f(r12 ,Ω1Ω2) =
∑
m
f22m(r12)Y
∗
2m(Ω1)Y2m(Ω2), (4.11)
where
C22m(r) = −1
5
βA
r
e−αr. (4.12)
This reduces to the following equation for harmonics
h22m(k) = C22m(k) + 〈Y 2
2m(Ω)〉Ω ρh22m(k)C22m(k), (4.13)
resulting in the following expression for the harmonics of a pair correlation function
h22m(k) = −1
5
4πβA
k2 + α2 + 〈Y 2
2m(Ω)〉Ω 1
54πρβA
, (4.14)
which is a renormalized, “effective” Yukawa potential in the k-space. In the r-space
h22m(r) = −1
5
βA
r
exp
[
−r
√
α2 + 〈Y 2
2m(Ω)〉Ω
1
5
4πρβA
]
, (4.15)
where 〈Y 2
2m(Ω)〉Ω = (1/ρ)
∫
dΩρ(Ω)|Y 2
2m(Ω)|.
Dependencies of quantities 〈Y 2
2m(Ω)〉Ω on the product SM∗ are presented in figure 3. Due to
the normalization condition of functions Y2m(Ω), for SM∗=0, the averages 〈Y 2
2m(Ω)〉Ω = 1.
4.2. Correction to the single-particle distribution funct ion
Correction to the single-particle distribution function due to Gaussian fluctuations can be found
according to
ρCR(Ω)
ρ
=
fMF(Ω)
Z ′
(
1 + [h(r,Ω1Ω2)− C(r,Ω1Ω2)]
)
∣
∣
∣
∣
Ω1→Ω2
r1→r2
=
1
Z ′
e
3
2
M∗S cos2 θ
(
1 +
∑
m
Y ∗
2m(Ω)Y2m(Ω)[h22m(r) + βν(r)]
)
∣
∣
∣
∣
r→0
, (4.16)
where the normalization constant Z ′ can be found from condition
∫
ρCR(Ω)dΩ = ρ. Since
lim
r→0
[h22m(r) + βν(r)] = −1
5
β∗
[
√
1− 1
5
〈Y 2
2m(Ω)〉ΩM∗ − 1
]
, (4.17)
33605-6
Maier-Saupe nematogenic fluid: field theoretical approach
0 2 4 6 8 10
0,0
0,5
1,0
1,5
2,0
2,5
3,0
3,5
4,0 <Y2
2m
(
SM*
(3)
(2)
(1)
<Y2
22
( (1)
<Y2
21
( (2)
<Y2
20
( (3)
Figure 3. Dependence of quantities 〈Y 2
2m(Ω)〉Ω on parameter SM∗ for m = 0, 1, 2.
then the corrected single-particle distribution function has the form
ρCR(Ω)
ρ
=
e
3
2
SM∗ cos2 θ
(
1 + β∗ − 1
5β
∗
∑
m
|Y2m(Ω)|2
√
1− 1
5 〈Y 2
2m(Ω)〉ΩM∗
)
∫
dΩ e
3
2
M∗S cos2 θ
(
1 + β∗ − 1
5β
∗
∑
m
|Y2m(Ω)|2
√
1− 1
5 〈Y 2
2m(Ω)〉ΩM∗
) . (4.18)
We can also approximate the corrected single-particle distribution function in an exponential form
as
ρEXP(Ω)
ρ
=
1
Z ′′
exp
[
3
2
M∗S cos2 θ − 1
5
β∗
∑
m
|Y2m(Ω)|2
√
1− 1
5
〈Y 2
2m(Ω)〉ΩM∗
]
, (4.19)
where Z ′′ is the normalization constant such that
∫
ρEXP(Ω)dΩ = ρ.
Note that
|Y2m(Ω)|2 =
∑
l
5
(2l + 1)
1
2
(
2 2 l
m −m 0
)(
2 2 l
0 0 0
)
Yl0(Ω), (4.20)
where l = 0, 2, 4;
(
2 2 l
m −m 0
)
and ( 2 2 l
0 0 0 ) are the corresponding Clebsch-Gordon coefficients [17].
We can see that in the Gaussian approximation the dependence of the single-particle distribu-
tion function on β∗ and ρ∗ is more complicated than in the MFA: ρ(Ω) now depends not only on
M∗ but there is also a direct β∗-dependence and a 〈Y 2
2m(Ω)〉Ω-dependence. We also see that in the
Gaussian approximation the single-particle distribution function contains Legendre polynomials of
the second and fourth orders of molecule orientations whereas in the linear approximation only
Legendre polynomials of the second order are present. From expression (4.19) it is readily seen
that the role of the fluctuation term increases with an increase of inverse temperature β∗.
4.3. Free energy, pressure, and chemical potential
For a homogeneous system, the part of the Helmholtz free energy responsible for field interaction
can be calculated by integrating with respect to the coupling parameter λ:
F − Fid =
V
2
∫
drdΩ1dΩ2ρ(Ω1)ρ(Ω2)ν(r,Ω1Ω2)
1
∫
0
dλ[1 + h(λ, r,Ω1Ω2)], (4.21)
33605-7
M. Holovko, D. di Caprio, I. Kravtsiv
where
h(λ, r,Ω1Ω2) =
∑
m
h2m(λ, r)Y ∗
2m(Ω1)Y2m(Ω2)
=
∑
m
(
−λβA
r
1
5
exp
[
−r
√
α2 + 〈Y 2
2m(Ω)〉Ω
1
5
4πρβAλ
])
Y ∗
2m(Ω1)Y2m(Ω2). (4.22)
Expression (4.21) in terms of parameter B = 1
54πρβA yields
β (F − Fid) = V
Bρ
2α2
〈Y20(Ω)〉2Ω
+ V
∑
m
[
−
(
α2 + 〈Y 2
2m(Ω)〉ΩB
)3/2
12π
+
α3
12π
+
α〈Y 2
2m(Ω)〉ΩB
8π
]
. (4.23)
Having an explicit expression for the free energy, we can find the pressure:
βP =− β
[
∂
∂V
F
]
T,N
= ρ+
Bρ
2α2
(
〈Y20(Ω)〉2Ω + ρ
∂
∂ρ
〈Y20(Ω)〉2Ω
)
−
∑
m
(
−
(
α2 + 〈Y 2
2m(Ω)〉ΩB
)3/2
12π
+
α3
12π
+
α〈Y 2
2m(Ω)〉ΩB
8π
)
+
ρB
8π
∑
m
(
1
ρ
〈Y 2
2m(Ω)〉Ω +
∂
∂ρ
〈Y 2
2m(Ω)〉Ω
)
(
α−
(
α2 + 〈Y 2
2m(Ω)〉ΩB
)1/2
)
, (4.24)
where the derivatives of the averages 〈. . .〉Ω are equal to
∂
∂ρ
〈
Y 2
2m(Ω)
〉
Ω
=
B
ρα2
(
〈Y20(Ω)〉Ω + ρ
∂
∂ρ
〈Y20(Ω)〉Ω
)
×
(
〈
Y 2
2m(Ω)
〉
Ω
〈Y20(Ω)〉Ω −
〈
Y 2
2m(Ω)Y20(Ω)
〉
Ω
)
, (4.25)
∂
∂ρ
〈Y20(Ω)〉Ω =
B
[
〈Y20(Ω)〉3Ω −
〈
Y 2
20(Ω)
〉
Ω
〈Y20(Ω)〉Ω
]
ρα2 − ρB
[
〈Y20(Ω)〉2Ω − 〈Y 2
20(Ω)〉Ω
] . (4.26)
In the isotropic phase 〈Y20(Ω)〉Ω = 0 and expression (4.24) considerably simplifies:
βP = −β
[
∂
∂V
F
]
T,N
= ρ+
∑
m
(
α2
(
α2 +B2m
)1/2
12π
− α3
12π
− B2m
(
α2 +B2m
)1/2
24π
)
, (4.27)
where B2m ≡ 〈Y 2
2m(Ω)〉ΩB. This expression is similar to the one obtained in [13, 20], where there
is a supplementary summation over m and the quantity B is replaced by B2m . The chemical
potential µ of the fluid can be found from expressions (4.23) and (4.24) as µ = (F + PV ) /N and
equals
βµ = ln
(
ρΛ3
TΛR
)
+
B
α2
(
〈Y20(Ω)〉2Ω +
1
2
ρ
∂
∂ρ
〈Y20(Ω)〉2Ω
)
+
B
8π
∑
m
(
1
ρ
〈Y 2
2m(Ω)〉Ω +
∂
∂ρ
〈Y 2
2m(Ω)〉Ω
)
(
α−
(
α2 + 〈Y 2
2m(Ω)〉ΩB
)1/2
)
. (4.28)
4.4. Broken symmetry problem and the elasticity constant
A specific feature of the considered molecular fluid is a broken symmetry which appears in the
absence of an orienting external field. In [2, 3, 7] using the Lovett-Mou-Buff-Wertheim equation [21,
33605-8
Maier-Saupe nematogenic fluid: field theoretical approach
22] an exact relation for orientationally non-uniform fluids was obtained:
∇∇∇Ω1
ln ρ(Ω1) =
∫
dr12dΩ2C(r12 ,Ω1Ω2)∇∇∇Ω2
ρ(Ω2), (4.29)
where C(r12 ,Ω1Ω2) is the direct correlation function which in the RPA is given by equation (4.9).
Equation (4.29) is also known as the integro-differential form of the Ward identity [2, 7, 23].
The angular gradient operator ∇∇∇Ω decomposes into 3 spherical components ∇0 , ∇+ , ∇− . As ∇0
is oriented in the direction of liquid crystal, it vanishes due to rotational invariance. For other
components, the following relations hold [17]:
∇±Ylm(Ω) = [l(l + 1)−m(m± 1)]
1/2
Yl,m±1 . (4.30)
The direct correlation function can be expanded as [2]
C(r12 ,Ω1Ω2) =
∑
lml′m′
Clml′m′(r12)Y
∗
lm(Ω1)Yl′m′(Ω2). (4.31)
Due to the axial symmetry of a nematic, m = m′. From (4.29) we derive the following relation for
the average 〈Y 2
21(Ω)〉Ω :
C221ρ〈Y 2
21(Ω)〉Ω = 1, (4.32)
where
C221 =
∫
dr12C221(r12) = −1
5
4πβA
α2
. (4.33)
Due to condition (4.32), harmonic h221(k) diverges in the limit k → 0 signalling the appearance of
Goldstone modes in the system. This phenomenon is responsible for a number of unique properties
of nematics such as elastic behavior and critical light scattering. The other harmonics of the pair
correlation function should be finite according to the phenomenological theory of de Gennes [15].
In the considered approximation (4.9) we have
C220 = C221 = C222 (4.34)
and due to (4.32) and (4.33)
(
α2 +
1
5
4πρβA
〈
Y 2
2m(Ω)
〉
Ω
)1/2
= α
(
1−
〈
Y 2
2m(Ω)
〉
Ω
〈Y 2
21(Ω)〉Ω
)1/2
. (4.35)
As we can see from figure 3, for m = 2, the ratio
〈
Y 2
22(Ω)
〉
Ω
/
〈
Y 2
21(Ω)
〉
Ω
< 1 and h222(r) is well
defined. However, for m = 0, the ratio
〈
Y 2
20(Ω)
〉
Ω
/
〈
Y 2
21(Ω)
〉
Ω
> 1 and h220(r) is not defined. For
this reason, a problem arises also for thermodynamic properties and for the density correction.
This problem is connected with approximation (4.9) which leads to (4.34). We should mention
that in previous works [2, 3, 7] carried out in the mean spherical approximation equation (4.34)
is not true and the problem being discussed does not appear. We think that for point particles
we can also solve this problem by introducing an additional isotropic interaction. We will consider
this aspect of the problem in a separate paper.
Broken symmetry is connected with elasticity properties of the considered fluid which are
described by the elasticity constant K. Formal expression for K as proposed by Poniewiersky and
Stecki [24] within our model reads
βK =
1
6
∫
drdΩ1dΩ2r
2 ∂ρ(Ω1)
∂ cos θ1
∂ρ(Ω2)
∂ cos θ2
nx(Ω1)nx(Ω2)C(r,Ω1Ω2)
= 10πρ2S2
∫
r4C221(r)dr = −12πρ2S2βA
α4
. (4.36)
33605-9
M. Holovko, D. di Caprio, I. Kravtsiv
0 5 10 15 20 25 30 35
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
T*
K*
3 4 5 6 7
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
3200
3400
3600 K*
T*= 3.5
T*= 5
Figure 4. Temperature dependence of the elas-
ticity constant.
Figure 5. Density dependence of the elasticity
constant.
Another way to calculate elastic constants comes from the theory of hydrodynamic fluctuations [24]
and can be found as
βK = 3
[
lim
k→0
k2h221(k)
〈Y 2
21(Ω)〉2Ω
〈Y20(Ω)〉2Ω
]−1
= −12πρ2S2βA
α4
. (4.37)
As we see, both expressions for K are identical.
Dependencies of the reduced elasticity constant K∗ = −K/Aα2 = 12πS2ρ∗2 on temperature T ∗
and density ρ∗ are presented in figures 4 and 5. One can see that with increasing temperature the
elasticity constant decreases and with increasing density it increases. The effect is more pronounced
respectively for larger density and for larger temperature. This result is in agreement with the
previous result [7] obtained in the framework of the mean spherical approximation for a non-point
model of the nematogenic Maier-Saupe fluid.
5. Conclusions
In this paper, for the first time the field theoretical approach is applied to the description of
correlation functions and thermodynamic properties of molecular anisotropic fluids. As an example
we consider a Maier-Saupe nematogenic fluid with the Yukawa potential of interparticle interaction.
By expanding the Hamiltonian in powers of density fluctuations we examine the system in the mean
field and Gaussian approximations.
In the mean field approximation, we obtain analytical expressions for the single-particle distri-
bution function and the orientational order parameter which are in agreement with the classical
Maier-Saupe theory for nematic ordered fluids.
In the Gaussian approximation we find analytical expressions for the pair correlation function,
the free energy, the pressure, the chemical potential, and the elasticity constant. We calculate the
correction to the mean field single-particle distribution function due to fluctuations and show that
the corrected single-particle distribution function has a more complex dependence on density and
temperature compared to the MFA. In contrast to the MFA single-particle distribution function, it
includes the fourth order Legendre polynomials of molecule orientations in addition to the second
order ones. We also use Ward symmetry identity to derive a simple expression for the average of
spherical harmonic
〈
Y 2
21(Ω)
〉
. One consequence of this condition is that for the system to be stable,
the distance-dependent part of the interaction should be attractive. We show that in the Gaussian
approximation the harmonic h221(k) diverges in the limit k → 0. Such a situation occurs at the
phase transition from an isotropic to a nematic phase. This change in symmetry causes collective
fluctuations known as the Goldstone modes. However, in the RPA for the considered system, the
harmonic h220(k) is not defined. We hope to solve this problem in our next paper by introducing
additional isotropic interparticle interactions.
33605-10
Maier-Saupe nematogenic fluid: field theoretical approach
Acknowledgements
M. Holovko and D. di Caprio are grateful for the support to the National Academy of Sciences
of Ukraine (NASU) and to the Centre National de la Recherche Scientifique (CNRS) (project
no. 21303), I. Kravtsiv is grateful for the support to the French embassy in Ukraine (the grant of the
French government for PhD programs in partnership). The authors also thank Dr. O.V. Patsahan
for the useful comments.
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33605-11
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http://dx.doi.org/10.1103/PhysRevE.60.2912
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M. Holovko, D. di Caprio, I. Kravtsiv
Нематичний плин Майєра-Заупе: теоретико-польовий пiдхiд
М. Головко1, Д. дi Капрiо2, I. Кравцiв1
1 Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
2 Лабораторiя електрохiмiї, хiмiї поверхонь i енергетичного моделювання,
Вiддiлення хiмiї вищої нацiональної школи ПарiТех, пл. Жуссю, 4, 75005 Париж, Францiя
Ми застосовуємо теоретико-польовий пiдхiд для вивчення структурних i термодинамiчних власти-
востей однорiдного нематичного плину Майєра-Заупе з анiзотропною взаємодiєю типу Юкави. У
наближеннi середнього поля нами отримано стандартну теорiю Майєра-Заупе для рiдких криста-
лiв. У цiй теорiї одночастинкова функцiя розподiлу виражається через полiном Лєжандра другого
порядку взаємної орiєнтацiї частинок. У гаусiвському наближеннi нами отримано аналiтичнi вира-
зи для кореляцiйних функцiй, константи еластичностi, вiльної енергiї, тиску i хiмiчного потенцiалу.
За допомогою тотожностi Ворда нами встановлено просту умову для кореляцiйних функцiй. Нами
також знайдено поправки внаслiдок флуктуацiй i показано що вираз для одночастинкової функцiї
розподiлу вже мiстить полiноми Лєжандра вищих порядкiв.
Ключовi слова: нематичний плин Майєра-Заупе, теоретико-польовий пiдхiд, кореляцiйна функцiя,
термодинамiка
33605-12
Introduction
The model and field theory formalism
Mean field approximation
Fluctuation and correlation effects: Gaussian approximation
Correlation functions
Correction to the single-particle distribution function
Free energy, pressure, and chemical potential
Broken symmetry problem and the elasticity constant
Conclusions
|